Pore-Scale Geochemical Processes 9780939950966

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REVIEWS in MINERALOGY & GEOCHEMISTRY Volume 80

geochemical society

PORESCALE GEOCHEMICAL PROCESSES EDITORS: Carl I. Steefel, Simon Emmanuel, Lawrence M. Anovitz

MINERALOGICAL SOCIETY OF AMERICA GEOCHEMICAL SOCIETY Series Editor: Ian P. Swainson

2015

80_Cover_degruter.indd 1

ISSN 1529-6466

2015-07-20 7:07:51 PM

REVIEWS in MINERALOGY and GEOCHEMISTRY Volume 80

2015

Pore-Scale Geochemical Processes EDITORS Carl I. Steefel Lawrence Berkeley National Laboratory, U.S.A.

Simon Emmanuel The Hebrew University of Jerusalem, Israel

Lawrence M. Anovitz Oak Ridge National Laboratory, U.S.A.

Front-cover: Colored pH values that correlate with reaction rates on calcite grain surfaces from a high performance reactive transport simulation of CO2 injection in a capillary tube experiment . See Molins S, Trebotich D, Yang L, Ajo-Franklin JB, Ligocki TJ, Shen C, Steefel CI (2014) Pore-scale controls on calcite dissolution rates from flow-through laboratory and numerical experiments. Environ Sci Technol 48:7453–7460 for description of experiment. Back-cover: SEM micrograph of a pore within an artificially weathered dolomite sample, Duperow Formation, Montana, USA. Figure courtesy of Marco Voltolini and Jonathan AjoFranklin, Lawrence Berkeley National Laboratory.

Series Editor: Ian Swainson MINERALOGICAL SOCIETY of AMERICA GEOCHEMICAL SOCIETY

Reviews in Mineralogy and Geochemistry, Volume 80 Pore-Scale Geochemical Processes ISSN 1529-6466 ISBN 978-0-939950-96-6

Copyright 2015

The MINERALOGICAL SOCIETY of AMERICA 3635 Concorde Parkway, Suite 500 Chantilly, Virginia, 20151-1125, U.S.A. www.minsocam.org The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner’s consent that copies of the article can be made for personal use or internal use or for the personal use or internal use of specific clients, provided the original publication is cited. The consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other types of copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. For permission to reprint entire articles in these cases and the like, consult the Administrator of the Mineralogical Society of America as to the royalty due to the Society.

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FROM THE SERIES EDITOR As the 80 volume of Reviews in Mineralogy and Geochemistry, this edition marks some historical changes in faces. It is the first volume since Jodi Rosso became the Executive Editor of Elements after many years in the position of Series Editor with this journal. I am very grateful to Jodi for her continuing support. It also marks the return of two other editors: it is now eighteen years since a volume on a similar topic was issued, Volume 34: Reactive Transport in Porous Media, for which Carl Steefel was also a Volume Editor; Lawrence Anovitz was also a Volume Editor for Volume 33: Boron Mineralogy, Petrology, and Geochemistry. th

All supplemental materials associated with this volume can be found at the MSA website. Errata will be posted there as well. Ian P. Swainson, Series Editor Vienna, Austria June 2015

PREFACE The pore scale is readily recognizable to geochemists, and yet in the past it has not received a great deal of attention as a distinct scale or environment that is associated with its own set of questions and challenges. Is the pore scale merely an environment in which smaller scale (molecular) processes aggregate, or are there emergent phenomena unique to this scale? Is it simply a finer-grained version of the “continuum” scale that is addressed in larger-scale models and interpretations? We would argue that the scale is important because it accounts for the pore architecture within which such diverse processes as multi-mineral reaction networks, microbial community interaction, and transport play out, giving rise to new geochemical behavior that might not be understood or predicted by considering smaller or larger scales alone. Fortunately, the last few years have seen a marked increase in the interest in pore-scale geochemical and mineralogical topics, making a Reviews in Mineralogy and Geochemistry volume on the subject timely. The volume had its origins in a special theme session at the 2012 Goldschmidt meeting in Montreal where at least some of the contributors to this volume gave presentations. From the diversity of pore-scale topics in the session that spanned the range from multi-scale characterization to modeling, it became clear that the time was right for a volume that would summarize the state of the science. Based in part on the evidence in the chapters included here, we would argue that the convergence of state of the art microscopic characterization and high performance pore scale reactive transport modeling has made it possible to address a number of long-standing questions and enigmas in the Earth 1529-6466/15/0080-0000$00.00

http://dx.doi.org/10.2138/rmg.2015.80.00

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FROM THE SERIES EDITOR As the 80 volume of Reviews in Mineralogy and Geochemistry, this edition marks some historical changes in faces. It is the first volume since Jodi Rosso became the Executive Editor of Elements after many years in the position of Series Editor with this journal. I am very grateful to Jodi for her continuing support. It also marks the return of two other editors: it is now eighteen years since a volume on a similar topic was issued, Volume 34: Reactive Transport in Porous Media, for which Carl Steefel was also a Volume Editor; Lawrence Anovitz was also a Volume Editor for Volume 33: Boron Mineralogy, Petrology, and Geochemistry. th

All supplemental materials associated with this volume can be found at the MSA website. Errata will be posted there as well. Ian P. Swainson, Series Editor Vienna, Austria June 2015

PREFACE The pore scale is readily recognizable to geochemists, and yet in the past it has not received a great deal of attention as a distinct scale or environment that is associated with its own set of questions and challenges. Is the pore scale merely an environment in which smaller scale (molecular) processes aggregate, or are there emergent phenomena unique to this scale? Is it simply a finer-grained version of the “continuum” scale that is addressed in larger-scale models and interpretations? We would argue that the scale is important because it accounts for the pore architecture within which such diverse processes as multi-mineral reaction networks, microbial community interaction, and transport play out, giving rise to new geochemical behavior that might not be understood or predicted by considering smaller or larger scales alone. Fortunately, the last few years have seen a marked increase in the interest in pore-scale geochemical and mineralogical topics, making a Reviews in Mineralogy and Geochemistry volume on the subject timely. The volume had its origins in a special theme session at the 2012 Goldschmidt meeting in Montreal where at least some of the contributors to this volume gave presentations. From the diversity of pore-scale topics in the session that spanned the range from multi-scale characterization to modeling, it became clear that the time was right for a volume that would summarize the state of the science. Based in part on the evidence in the chapters included here, we would argue that the convergence of state of the art microscopic characterization and high performance pore scale reactive transport modeling has made it possible to address a number of long-standing questions and enigmas in the Earth 1529-6466/15/0080-0000$00.00

http://dx.doi.org/10.2138/rmg.2015.80.00

Pore-scale Geochemical Processes ‒ Preface and Environmental Sciences. Among these is the so-called “laboratory-field discrepancy” in geochemical reaction rates, which may be traceable in part to the failure to consider porescale geochemical issues that include chemical and physical heterogeneity, suppression of precipitation in nanopores, and transport limitations to and from reactive mineral surfaces. This RiMG volume includes contributions that review experimental, characterization, and modeling advances in our understanding of pore-scale geochemical processes. The volume begins with chapters authored or co-authored by two of the éminences grises in the field of pore-scale geochemistry and mineralogy, two who have made what is perhaps the strongest case that the pore-scale is distinct and requires special consideration in geochemistry. The chapter by Andrew Putnis gives a high level overview of how the pore-scale architecture of natural porous media impacts geochemical processes, and how porosity evolves as a result of these. The chapter makes the first mention of what is an important theme in this volume, namely the modification of thermodynamics and kinetics in small pores. In a chapter authored by Røyne and Jamtveit, the authors investigate the effects of mineral precipitation on porosity and permeability modification of rock. Their principal focus is on the case where porosity reduction results in fracturing of the rock, in the absence of which the reactions will be suppressed due to the lack of pore space. The next chapter by Emmanuel, Anovitz, and Day-Stirrat addresses chemo-mechanical processes and how they affect porosity evolution in geological media. The next chapter by Anovitz and Cole provides a comprehensive review of the approaches for characterizing and analyzing porosity in porous media. Small angle neutron scattering (SANS) plays prominently as a technique in this chapter. Stack presents a review of what is known about mineral precipitation in pores and how this may differ from precipitation in bulk solution. Liu, Liu, Kerisit, and Zachara focus on porescale process coupling and the determination of effective (or upscaled) surface reaction rates in heterogeneous subsurface materials. Micro-continuum modeling approaches are investigated by Steefel, Beckingham, and Landrot, where the case is made that these may provide a useful tool where the computationally more expensive pore and pore network models are not feasible. The next chapter by Noiriel pursues the focus on characterization techniques with a review of X-ray microtomography (especially synchrotron-based) and how it can be used to investigate dynamic geochemical and physical processes in porous media. Tournassat and Steefel focus on a special class of micro-continuum models that include an explicit treatment of electrostatic effects, which are particularly important in the case of clays or clay-rich rock. Navarre-Sitchler, Brantley, and Rother present an overview of our current understanding of how porosity increases as a result of chemical weathering in silicate rocks, bringing to bear a range of characterization and modeling approaches that build toward a more quantitative description of the process. In the next chapter, Druhan, Brown, and Huber demonstrate how isotopic gradients across fluid–mineral boundaries can develop and how they provide insight into pore-scale processes. Yoon, Kang, and Valocchi provide a comprehensive review of lattice Boltzmann modeling techniques for pore-scale processes. Mehmani and Balhoff summarize mesoscale and hybrid models for flow and transport at the pore scale, including a discussion of the important class of models referred to as “pore network” that typically can operate at a larger scale than is possible with the true pore-scale models. Molins addresses the problem of how to represent interfaces (solid–fluid) at the pore scale using direct numerical simulation. In addition to thanking the scientists who have contributed their time and effort to preparation of this volume and presentations at the short course in Prague, we would like to thank Ian Swainson for his patience and hard work in preparing the volume for publication. We iv

Pore-scale Geochemical Processes ‒ Preface would also like to thank those who provided reviews of the chapters in the volume, including Sergi Molins, Bhavna Arora, Qingyun Li, Susan Brantley, Alexis Navarre-Sitchler, Lauren Beckingham, Alejandro Fernandez-Martinez, Kate Maher, Marco Voltolini, Masa Prodanovic, Uli Mayer, Anja Røyne, Li Li, Francois Renard, Chris Huber, Jennifer Druhan, and Chongxuan Liu. Alex Speer at the Mineralogical Society of America provided critical advice during the development stage of the volume and the planning of the short course. We would also like to thank Sara Hefty for her help in organizing the Short Course held in Prague before the 2015 Goldschmidt Meeting. Thanks also go to the Geochemical Society for providing funds for student travel grants to attend the Short Course. Carl I. Steefel, Lawrence Berkeley National Laboratory Simon Emmanuel, The Hebrew University of Jerusalem Lawrence M. Anovitz, Oak Ridge National Laboratory

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Transient Porosity Resulting from Fluid–Mineral Interaction and its Consequences Andrew Putnis

INTRODUCTION ....................................................................................................................1 FLUID–MINERAL INTERACTIONS: PRESSURE SOLUTION ..........................................2 FLUID–MINERAL INTERACTIONS: MINERAL REPLACEMENT .................................3 POROSITY AND FELDSPAR–FELDSPAR REPLACEMENT ............................................5 SECONDARY POROSITY ASSOCIATED WITH MINERAL REPLACEMENT— A UNIVERSAL PHENOMENON .....................................................................................7 IMPLICATIONS OF MICROPOROSITY— SUPERSATURATION AND CRYSTAL GROWTH .........................................................8 Critical and threshold supersaturation ...........................................................................9 Crystallization experiments in confined media ...........................................................11 DISCUSSION .........................................................................................................................16 CONCLUSIONS.....................................................................................................................19 ACKNOWLEDGMENTS.......................................................................................................19 REFERENCES .......................................................................................................................19

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Pore-Scale Controls on Reaction-Driven Fracturing Anja Røyne, Bjørn Jamtveit

INTRODUCTION ..................................................................................................................25 FIELD-SCALE OBSERVATIONS .........................................................................................26 Reaction-induced fracturing: The effect of porosity ...................................................26 Reaction-induced clogging and closure of fluid pathways ..........................................29 PORE-SCALE MECHANISMS ............................................................................................30 Fracturing around expanding grains ............................................................................30 Intragrain fracturing ....................................................................................................31 Growth in pores ...........................................................................................................33 FUNDAMENTAL PROPERTIES OF CONFINED FLUID FILMS......................................35 The disjoining pressure of confined fluid films ...........................................................36 Transport in confined fluid films .................................................................................39 vii

Pore-Scale Geochemical Processes ‒ Table of Contents INTERFACE-DRIVEN TRANSPORT ON THE PORE SCALE ..........................................39 CONCLUDING REMARKS ..................................................................................................41 ACKNOWLEDGMENTS.......................................................................................................42 REFERENCES .......................................................................................................................42

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Effects of Coupled Chemo-Mechanical Processes on the Evolution of Pore-Size Distributions in Geological Media Simon Emmanuel, Lawrence M. Anovitz, Ruarri J. Day-Stirrat

INTRODUCTION ..................................................................................................................45 PORE-SIZE EVOLUTION DURING MINERAL PRECIPITATION AND DISSOLUTION .......................................................................................................47 PORE-SIZE EVOLUTION DURING MECHANICAL COMPACTION .............................51 CHEMO-MECHANICAL COUPLING AND PORE-SIZE EVOLUTION ..........................54 CONCLUDING REMARKS ..................................................................................................56 ACKNOWLEDGMENTS.......................................................................................................58 REFERENCES .......................................................................................................................58

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Characterization and Analysis of Porosity and Pore Structures Lawrence M. Anovitz, David R. Cole

INTRODUCTION ..................................................................................................................61 PETROPHYSICAL APPROACHES ......................................................................................63 Direct methods.............................................................................................................63 Imaging methods .........................................................................................................72 Downhole porosity logs ...............................................................................................83 SCATTERING METHODS ....................................................................................................87 Theoretical basis of scattering experiments ................................................................91 The two-phase approximation and its limitations .......................................................93 Sample preparation ......................................................................................................94 Geometrical principles of small-angle scattering experiments ...................................99 Contrast matching......................................................................................................109 Reduction and analysis of SAS data..........................................................................111 IMAGE ANALYSIS..............................................................................................................120 Sample preparation and image acquisition ................................................................120 Combining imaging and scattering data ....................................................................123 Three-point correlations ............................................................................................125 Monofractals and multifractals .................................................................................129 Lacunarity, succolarity, and other correlations ..........................................................132 Comparisons of multiple techniques .........................................................................138 CONCLUSIONS...................................................................................................................140 viii

Pore-Scale Geochemical Processes ‒ Table of Contents ACKNOWLEDGMENTS.....................................................................................................142 REFERENCES .....................................................................................................................143

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Precipitation in Pores: A Geochemical Frontier Andrew G. Stack

INTRODUCTION ................................................................................................................165 RATIONALE ........................................................................................................................166 PORE-SIZE-DEPENDENT PRECIPITATION ....................................................................170 Effects of precipitation on porosity and permeability ...............................................170 Observations of precipitation in pores .......................................................................172 ATOMIC-SCALE ORIGINS OF A PORE SIZE DEPENDENCE ......................................175 Substrate and precipitate effects ................................................................................175 EFFECTS IN SOLUTION....................................................................................................180 TRANSPORT .......................................................................................................................184 CONCLUSIONS AND OUTLOOK .....................................................................................186 ACKNOWLEDGMENTS.....................................................................................................187 REFERENCES .....................................................................................................................187

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Pore-Scale Process Coupling and Effective Surface Reaction Rates in Heterogeneous Subsurface Materials Chongxuan Liu, Yuanyuan Liu, Sebastien Kerisit, John Zachara

INTRODUCTION ................................................................................................................191 THEORETICAL CONSIDERATION OF EFFECTIVE REACTION RATES ...................193 Well-mixed conditions...............................................................................................195 Mass transport limited conditions .............................................................................195 INTRINSIC RATES AND RATE CONSTANTS .................................................................197 Approaches to calculate molecular-scale reaction rates ............................................197 Molecular-scale rates of uranyl sorption reactions at mineral surface sites ..............198 Molecular-scale rates of elementary mechanisms of mineral growth and dissolution ..........................................................................................199 GRAIN-SCALE REACTIONS, SUB-GRAIN PROCESS COUPLING, AND EFFECTIVE RATES .............................................................................................199 Pore-scale variability in reactant concentrations at the sub-grain scale ....................201 Effective reaction rates and rate constants.................................................................202 SUBGRID VARIATIONS IN REACTANT CONCENTRATIONS AND EFFECTIVE RATE CONSTANTS UNDER FLOW CONDITIONS ............................205 Pore-scale concentration variations under flow conditions .......................................205 Effective rate constants .............................................................................................207 CONCLUSIONS...................................................................................................................208 ACKNOWLEDGMENT .......................................................................................................209 REFERENCES .....................................................................................................................209 ix

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Micro-Continuum Approaches for Modeling Pore-Scale Geochemical Processes Carl I. Steefel, Lauren E. Beckingham, Gautier Landrot

INTRODUCTION ................................................................................................................217 MAPPING OF MODEL PARAMETERS FROM IMAGE ANALYSIS ..............................218 Porosity ......................................................................................................................219 Mineral volumes ........................................................................................................220 Mineral surface area ..................................................................................................221 Diffusivity ..................................................................................................................222 Permeability...............................................................................................................226 Imaging issues impacting parameter estimation........................................................228 Image resolution ........................................................................................................229 MICRO-CONTINUUM MODELING APPROACHES .......................................................229 Volume averaging of porosity and mineral volume fractions ....................................230 Multi-continuum approaches .....................................................................................234 Resolution of nanoscale reaction fronts ....................................................................237 SUMMARY AND PATH FORWARD ..................................................................................242 ACKNOWLEDGMENTS.....................................................................................................243 REFERENCES .....................................................................................................................243

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Resolving Time-dependent Evolution of Pore-Scale Structure, Permeability and Reactivity using X-ray Microtomography Catherine Noiriel

INTRODUCTION ................................................................................................................247 Resolving pores using X-ray microtomography........................................................248 Absorption contrast X-ray microtomography ...........................................................249 Phase contrast X-ray microtomography ....................................................................252 X-ray fluorescence microtomography (XFMT) ........................................................252 Dual-energy X-ray microtomography .......................................................................253 PORE-SCALE CHARACTERIZATION .............................................................................253 Image segmentation ...................................................................................................253 Porosity determination, pore geometry and pore-space distribution .........................254 Solid-phase distribution and quantification ...............................................................256 Specific and reactive surface area measurements ......................................................257 Fracture characterization ...........................................................................................258 Multi-resolution imaging ...........................................................................................260 COMBINING EXPERIMENTS, 3-D IMAGING AND NUMERICAL MODELING........261 X-ray microtomography for monitoring reactive transport .......................................262 X-ray microtomography for monitoring geomechanical evolution ...........................264 EMERGING APPLICATIONS.............................................................................................264 Effect of mineral reaction kinetics on evolution of the physical pore space .............264 x

Pore-Scale Geochemical Processes ‒ Table of Contents Rates of dissolution/precipitation reactions in porous rocks .....................................265 Rates of dissolution/precipitation reactions in fractures ...........................................267 Porosity and permeability development in porous media and fractures ...................269 Effects of texture and mineralogy on complex porosity–permeability relationships and transport ....................................................................................271 Effects of pore-scale heterogeneity on permeability reduction .................................274 CONCLUDING THOUGHTS..............................................................................................274 ACKNOWLEDGMENTS.....................................................................................................276 REFERENCES .....................................................................................................................276

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Ionic Transport in Nano-Porous Clays with Consideration of Electrostatic Effects Christophe Tournassat, Carl I. Steefel

INTRODUCTION ................................................................................................................287 CLASSICAL FICKIAN DIFFUSION THEORY .................................................................288 Diffusion basics .........................................................................................................288 Anion, cation and water diffusion in clay materials ..................................................290 Diffusion under a salinity gradient ............................................................................293 KD values obtained from static and diffusion experiments ........................................293 From classic diffusion theory to process understanding ...........................................295 CLAY MINERAL SURFACES AND RELATED PROPERTIES ........................................295 Electrostatic properties, high surface area, and anion exclusion ...............................295 Adsorption processes in clays ...................................................................................298 CONSTITUTIVE EQUATIONS FOR DIFFUSION IN BULK, DIFFUSE LAYER, AND INTERLAYER WATER .......................................................................................303 From real porosity distributions to reactive transport model representation .............303 Diffusive flux in bulk water .......................................................................................303 Diffusive flux in the diffuse layer ..............................................................................304 Interlayer diffusion ....................................................................................................307 Approximations for Nernst-Planck equation for bulk and EDL water......................308 RELATIVE CONTRIBUTIONS OF CONCENTRATION, ACTIVITY COEFFICIENT AND DIFFUSION POTENTIAL GRADIENTS TO TOTAL FLUX...........................................................................................................310 Model system ............................................................................................................310 Example 1: Constant ionic strength...........................................................................310 Example 2: Gradient in ionic strength and tracer concentration ...............................310 Example 3: Gradient in ionic strength and no tracer gradient ...................................313 Links to experimental diffusion results .....................................................................313 Diffusive transport equation for porous medium with interlayer and EDL water .....314 Summation of bulk and diffuse layer diffusive fluxes over an interface ...................315 Differentiation of the flux at interface between two numerical grid cells .................318 APPLICATIONS...................................................................................................................319 Code limitations ........................................................................................................319 Simultaneous diffusion calculations of anions, cations, and neutral species ............320 Diffusion under a salinity gradient ............................................................................321 Interlayer diffusion ....................................................................................................322 xi

Pore-Scale Geochemical Processes ‒ Table of Contents SUMMARY AND PERSPECTIVES ...................................................................................323 ACKNOWLEDGMENTS.....................................................................................................325 REFERENCES .....................................................................................................................325

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How Porosity Increases During Incipient Weathering of Crystalline Silicate Rocks Alexis Navarre-Sitchler, Susan L. Brantley, Gernot Rother

INTRODUCTION ................................................................................................................331 METHODS FOR POROSITY AND PORE-SIZE DISTRIBUTION QUANTIFICATION .......................................................................................................333 Sorption and intrusion techniques .............................................................................333 Electron and optical microscopy ...............................................................................334 Neutron scattering .....................................................................................................335 Fractal nature of rocks ...............................................................................................337 CASE STUDIES ...................................................................................................................339 Weathering of felsic to intermediate composition rocks ...........................................339 Weathering of mafic rocks .........................................................................................344 LINKING FRACTAL SCALING AND PORE-SCALE OBSERVATIONS TO WEATHERING MECHANISMS .............................................................................346 SUMMARY ..........................................................................................................................349 ACKNOWLEDGMENTS.....................................................................................................350 REFERENCES .....................................................................................................................351

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Isotopic Gradients Across Fluid–Mineral Boundaries Jennifer L. Druhan, Shaun T. Brown, Christian Huber

INTRODUCTION ................................................................................................................355 A conceptual model of isotope partitioning at the pore scale ...................................356 Organization of article ...............................................................................................358 NOTATION ...........................................................................................................................358 A note on fractionation ..............................................................................................361 EXAMPLES OF ISOTOPIC ZONING ACROSS FLUID–SOLID BOUNDARIES ...........362 Alpha recoil ...............................................................................................................362 Diffusive fractionation ...............................................................................................366 Dissolution.................................................................................................................370 Precipitation...............................................................................................................376 TREATMENT OF TRANSIENT ISOTOPIC PARTITIONING AND ZONING ................381 ACKNOWLEDGMENTS.....................................................................................................384 REFERENCES .....................................................................................................................384

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Lattice Boltzmann-Based Approaches for Pore-Scale Reactive Transport Hongkyu Yoon, Qinjun Kang, Albert J. Valocchi

INTRODUCTION ................................................................................................................393 REACTIVE TRANSPORT MODELING OF BIOGEOCHEMICAL PROCESSES AT THE PORE SCALE ............................................................................396 Fluid Flow ................................................................................................................396 Multicomponent reactive transport ............................................................................397 Mineral precipitation and dissolution .......................................................................398 Biofilm dynamics ......................................................................................................399 LATTICE BOLTZMANN METHODS FOR FLOW AND REACTIVE TRANSPORT .....401 LBM for fluid dynamics ............................................................................................401 LBM for multi-component reactive transport ...........................................................403 Update of solid–pore geometry .................................................................................407 LBM for biofilm dynamics ........................................................................................409 LB-BASED ALGORITHMS FOR OTHER APPLICATIONS ............................................412 Multiphase reactive transport ....................................................................................412 Electrokinetic transport .............................................................................................412 Coupled LBM-DNS for multicomponent reactive transport processes.....................413 THREE-DIMENSIONAL CHARACTERIZATION OF PORE TOPOLOGY ...................414 LB-BASED APPLICATIONS FOR PRECIPITATION, DISSOLUTION, AND BIOFILM GROWTH AND THEIR IMPACT ON FLOW ALTERATION ..........416 Pore cementation/dissolution and flow feedback ......................................................416 Microfluidic experiments for pore cementation and flow blocking ..........................418 Example results illustrating feedback between flow and biofilms ............................420 FUTURE RESEARCH DIRECTIONS ...............................................................................423 ACKNOWLEDGMENTS ....................................................................................................424 RERERENCES .....................................................................................................................424

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Mesoscale and Hybrid Models of Fluid Flow and Solute Transport Yashar Mehmani, Matthew T. Balhoff

INTRODUCTION ................................................................................................................433 PORE-SCALE MODELING ................................................................................................434 Direct pore-scale modeling .......................................................................................434 Pore-network models .................................................................................................437 Network modeling of solute transport .......................................................................439 HYBRID MODELING .........................................................................................................449 CONCLUSIONS...................................................................................................................453 ACKNOWLEDGMENTS.....................................................................................................454 REFERENCES .....................................................................................................................454 xiii

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Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes Sergi Molins

INTRODUCTION ................................................................................................................461 PORE-SCALE PROCESSES ...............................................................................................463 Flow ...........................................................................................................................463 Multicomponent reactive transport ............................................................................463 Surface reactions .......................................................................................................464 DIRECT NUMERICAL SIMULATION ..............................................................................465 Interface representation .............................................................................................465 Micro-continuum and multiscale approaches............................................................466 SURFACE AREA ACCESSIBILITY AND EVOLUTION IN MINERAL REACTIONS ..469 Transport control on rates ..........................................................................................469 Surface area evolution ...............................................................................................470 PORE-SPACE EVOLUTION ...............................................................................................472 Level set method........................................................................................................472 Phase-field method ....................................................................................................474 Continuum and multiscale approaches ......................................................................474 UPSCALING OF SURFACE REACTIONS BY VOLUME AVERAGING ........................476 SUMMARY AND OUTLOOK ............................................................................................477 ACKNOWLEDGMENTS.....................................................................................................479 REFERENCES .....................................................................................................................479

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 1-23, 2015 Copyright © Mineralogical Society of America

Transient Porosity Resulting from Fluid–Mineral Interaction and its Consequences Andrew Putnis The Institute for Geoscience Research (TIGeR) Curtin University, Perth, Australia and Institut für Mineralogie, University of Münster Münster, Germany [email protected]; [email protected]

INTRODUCTION The term porosity is very widely used in geosciences and normally refers to the spaces between the mineral grains or organic material in a rock, measured as a fraction of the total volume. These spaces may be filled with gas or fluids, and so the most common context for a discussion of porosity is in hydrogeology and petroleum geology of sedimentary rocks. While porosity is a measure of the ability of a rock to include a fluid phase, permeability is a measure of the ability for fluids to flow through the rock, and so depends on the extent to which the pore spaces are interconnected, the distribution of pores and pore neck size, as well as on the pressure driving the flow. This chapter will be primarily concerned with how reactive fluids can move through ‘tight rocks’ which have a very low intrinsic permeability and how secondary porosity is generated by fluid–mineral reactions. A few words about the meaning of the title will help to explain the scope of the chapter: (i) “Fluid–mineral interaction”: When a mineral is out of equilibrium with a fluid, it will tend to dissolve until the fluid is saturated with respect to the solid mineral. We will consider fluids to be aqueous solutions, although many of the principles described here also apply to melts. The generation of porosity by simply dissolving some minerals in a rock is one obvious way to enhance fluid flow. Dissolution of carbonates by low pH solutions to produce vugs and even caves would be one example. However, when considering the role of fluid–mineral reaction during metamorphism the fluid provides mechanisms that enable re-equilibration of the rock, i.e., by replacing one assemblage of minerals by a more stable assemblage. This not only involves the dissolution of the parent mineral phases, but the reprecipitation of more stable product phases while the rock remains essentially solid through the whole process, even though the reactions require permeability for fluid transport. This latter aspect of fluid–mineral interaction will be one focus of this chapter. The interpretation of mechanisms of reactions in rocks is based on studying the microstructural development associated with the reaction. The microstructure of a rock describes the relationships between the mineral grains and organic material in the rock, their size, shape, and orientation. When reactions involve fluids we look for microstructural and chemical evidence for the presence of fluid and fluid pathways. In that sense the porosity can be considered as an integral part of the microstructure, in that it is the space occupied by the fluid phase, and the distribution of the fluid phase is just as important to the rock properties as the distribution of the minerals. 1529-6466/15/0080-0001$05.00

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Putnis

The individual mineral grains in a rock may also have a microstructure that reflects the processes taking place within the crystalline structure as a result of the geological history of the rock. Examples would include exsolution from an initial solid solution, or transformation twinning, or the formation of dislocation arrays during deformation. The rock and mineral microstructures can provide important information on how the rock formed and its subsequent thermal and deformational history. The microstructure is thus a direct result of the mechanisms of the processes that take place when a rock or mineral reacts to lower the overall free energy of the system under the imposed physical and chemical conditions. The microstructure formed during any process is always a balance between the thermodynamics (reducing the free energy) and the kinetics (the time available and the rates of the processes involved). As such, a microstructure may change over time, for example by coarsening or recrystallization to reduce the surface energy. (ii) “Transient porosity”: If we consider that the porosity is that part of the microstructure occupied by the fluid phase, then it is also subject to re-equilibration over a period of time. Just as porosity in poorly consolidated sediments is modified during diagenesis, compaction and pressure solution, porosity generated by fluid–mineral reactions will also be modified with time. The rate of fluid–mineral reactions is orders of magnitude faster than solid state reactions in which mass transfer takes place by slow diffusional processes (Dohmen and Milke 2010; Milke et al. 2013) and so some porosity microstructures generated by fluid–mineral reactions may be considerably “more transient” than microstructures generated by solid state mechanisms, e.g., exsolution. The “closure temperature” at which a specific microstructure or chemical distribution is frozen in time and available for study is much higher when the reequilibration mechanism depends on solid state diffusion than if reactant or product transport through fluids is involved. (iii) The “Consequences” discussed in this chapter will be mainly concerned with the supersaturation of fluids in pores and hence the role of porosity in controlling nucleation and growth of secondary minerals in the pore spaces.

FLUID–MINERAL INTERACTIONS: PRESSURE SOLUTION When an aqueous fluid interacts with a mineral with which it is not in equilibrium (i.e., is either undersaturated or supersaturated with respect to that mineral) the mineral will either tend to dissolve or grow until equilibrium is reached, i.e., until the fluid is saturated with respect to that mineral. The solubility of a mineral can also be increased by applied stress and this leads to the phenomenon of pressure solution whereby grains in contact with one another and under compression dissolve at these pressure points. The resulting fluid is then supersaturated with respect to a free mineral surface and can reprecipitate in the pore spaces (Fig. 1). The newly precipitated phase may have the same major chemical composition as the dissolved phase, but can be recognized as an overgrowth both texturally and from trace element and isotope geochemistry (Rutter 1983; Gratier et al. 2013). This mechanism of compaction reduces the porosity of a rock and leads to well-known relationships between depth of burial, porosity and rock density (e.g., Bjørlykke 2014 and refs therein). The reprecipitated phase can also act as a cement to bind the particles during the lithification process. Pressure solution is usually invoked to describe compaction during burial of mineralogically simple sedimentary rocks (such as sandstone and limestone) at relatively low temperatures. Theoretical models of pressure solution usually assume mono-mineralic rocks. Even when two minerals with different dissolution rates or hardnesses are in contact, chemical reactions between them and the solution are usually not considered, although the enhancement of the pressure solution of quartz by the presence of clays has been recognized for many years (Renard et al. 2007). The effects of pressure solution on rock microstructure can be

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3

Precipitate

Figure 1. Schematic drawing of pressure solution. A material under applied stress dissolves at the stress points, where the spherical grains are in contact, and is transported through the solution to sites of lower stress where it reprecipitates. Compaction by pressure solution reduces the porosity and permeability of a rock.

readily identified, as can the reduction in pore space. Such effects are principally of concern to evaluating fluid flow in sedimentary rocks and have been summarized in many books and review papers (e.g., Jamtveit and Yardley 1997; Parnell 1998). However, the general principles in pressure solution, i.e., dissolution–transport– precipitation, also apply to deeper crustal rocks, where the transport of material by pressure solution creep can result in large scale rock deformation (Gratier et al. 2013).

FLUID–MINERAL INTERACTIONS: MINERAL REPLACEMENT A more general scenario than that applied to simple pressure solution is the situation that when a mineral or rock reacts with a fluid with which it is out of equilibrium, it will start to dissolve and result in a new fluid composition at the fluid–mineral interface which is supersaturated with respect to some other mineral phase, or phases. Thus the dissolution of the parent phase may result in the precipitation of a new, more stable phase. From basic thermodynamic considerations, the more stable phase will be less soluble in the specific fluid composition than the parent dissolving phase. The spatial relationships between the dissolution and precipitation will depend on the rate-controlling step in the sequence of processes of dissolution–transport–precipitation. If dissolution and transport are fast relative to precipitation then the components in the supersaturated fluid may migrate some distance through a rock before precipitation takes place, and the parent and product phases may be spatially separated. This would be the case if overall the system was rate limited by precipitation. However, from a study of natural rock textures as well as experimental reactions, a more common situation is that dissolution is the rate-controlling step and that precipitation is fast relative to dissolution (Wood and Walther 1983; Walther and Wood 1984; Putnis 2009). In such a case precipitation is closely coupled to the dissolution, and may actually take place on the surface of the dissolving parent phase.

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Close spatial and temporal coupling of dissolution and precipitation leads to pseudomorphic mineral replacement in which the product phase has the same external dimensions and shape as the parent. The mechanism of such interface-coupled dissolution–precipitation has been described in a number of review papers (Putnis 2009; Ruiz-Agudo et al. 2014) and so only the briefest account will be given here. In the rest of this chapter the term “parent phase” refers to the mineral which is dissolving in the fluid, and the “product phase” is the reprecipitated phase from the resultant solution. The important point to note is that porosity generation in the product phase is a necessary prerequisite for pseudomorphic replacement to take place. The general principles can be illustrated with the aid of a schematic diagram (Fig. 2). The dissolution of even a few monolayers of the parent mineral may result in a localized supersaturation at the fluid–mineral interface relative to a new phase. This phase could then nucleate on the dissolving mineral surface (Fig. 2a,b). Nucleation would be enhanced if there is some crystallographic matching (epitaxy) between the parent and the product phase (e.g., one feldspar replaced by another, Niedermeier et al. (2009); Norberg et al. (2011)). However the lack of epitaxial relations between parent and product phases does not preclude pseudomorphic replacement as shown by the replacement of calcite by apatite, which have no obvious structural features in common (Kasioptas et al. 2011). In the former case where there is good crystallographic matching across the parent product interface, a single crystal of the parent can be replaced by a single crystal of the product. In the latter case the product phase is polycrystalline. The coupling between the dissolution and precipitation ensures that the new phase preserves the external shape of the parent. The next step in the process is crucial. Unless there are transport pathways maintained between the external fluid reservoir and the interface between the parent and the product phases, an impermeable layer of the new phase could isolate the parent from the fluid and the reaction would stop with equilibrium only established between the new rim and the fluid. Thus for the reaction to proceed instead of armoring the reacting crystal (e.g., Velbel 1993), the new phase must have interconnected porosity (Fig. 2c). The generation of porosity depends on two factors: (i) the molar volume difference between the parent and product phases, and (ii) the relative solubility of the parent and product phases in the specific reactive fluid. If the molar volume associated with a replacement process increases, as is the case of aragonite being replaced by calcite (Perdikouri et al. 2011) this will tend to militate against porosity generation. If under the conditions of any experiment or natural system calcite is

a

b

c

d

e

Figure 2. Schematic drawing of a pseudomorphic replacement reaction. When a solid (a) is out of equilibrium with an aqueous solution, it will begin to dissolve. The resulting fluid composition at the interface with the solid may become supersaturated with respect to another phase and (b) precipitate on the surface of the dissolving solid. This sets up a feedback between dissolution and precipitation. For the dissolution– precipitation reaction to continue (c–e), the precipitated phase must have interconnected porosity, allowing fluid access to the reaction interface and diffusion of components through the fluid

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more stable than aragonite, then calcite will be less soluble than aragonite. This means that more of the aragonite will be dissolved than calcite precipitated and some Ca2+ and CO32ions will go into the solution. This second factor will tend to outweigh the first, as for any reaction to proceed, the thermodynamics will dictate that the product must be less soluble, in the specific solution from which it precipitates, than the parent. If interconnected porosity is generated then the reaction will proceed and result in a porous, and hence turbid, pseudomorph of the original phase (Fig. 2d,e). Turbidity or cloudiness in a crystal is usually associated with porosity or the presence of many fine inclusions and can be the first indication that a mineral is the result of a secondary replacement. The model salt system in which such a mechanism has most thoroughly been examined is KBr–KCl–H2O (Putnis and Mezger 2004; Putnis et al. 2005; Pollok et al. 2011; Raufaste et al. 2011). In particular, Pollok et al. (2011) have analyzed the progressive replacement of one salt by another, taking into account the solid:fluid ratio, the changing chemistry of the fluid, and the evolution of the chemistry of the solid. As there is complete solid solution between KBr and KCl, the replacement proceeds progressively in composition, by continuously reequilibrating the solid composition as the fluid composition evolves (Putnis and Mezger 2004). In this system the porosity has also been shown to evolve with time while the replaced and porous crystal remains in contact with the fluid with which it has chemically equilibrated (Putnis et al. 2005; Raufaste et al. 2011). Thus textural equilibration, also by dissolution and reprecipitation, follows chemical equilibration and porosity may ultimately be completely removed. The dynamic nature of porosity and its eventual annihilation by coarsening has also been observed in experiments on feldspar replacement by Norberg et al. (2011). Although in the examples above the porosity is a dynamic and transient feature of fluid– mineral interaction, it is possible that a fluid-filled pore may also be stable especially on grain boundaries, depending on the balance of surface tensions between two solid surfaces and the solid–fluid interfaces (von Bargen and Waff 1986; Watson and Brennan 1987; LaPorte and Watson 1991; Lee et al. 1991).

POROSITY AND FELDSPAR–FELDSPAR REPLACEMENT The presence of turbidity and related porosity in feldspars from many different geological environments has been noted in the literature for many years. After the initial studies which demonstrated that turbidity in feldspars was due to porosity and the association with crystallization in a fluid-rich environment (Folk 1955; Montgomery and Brace 1975), detailed petrography in alkali feldspars (Parsons 1978; Walker 1990; Worden et al. 1990; Guthrie and Veblen 1991; Waldron et al. 1993; Walker et al. 1995; Parsons and Lee 2009) demonstrated that porosity was often associated with fluid-induced sub-solidus alteration and coarsening, e.g., cryptoperthite coarsening to patch perthite, as well as orthoclase coarsening to microcline. More recently it has been demonstrated experimentally that the development of patch perthite from cryptoperthite involves porosity generation (Norberg et al. 2013). In the examples above the replacement of one feldspar by another was essentially isochemical and driven by the reduction in strain energy. Strained feldspar (as is the case with the coherently exsolved alkali feldspars and the coherent transformation twinning in orthoclase) is more soluble than patch perthite and microcline in which the internal interfaces are incoherent, and hence the generation of porosity is a result of this difference in solubility rather than any molar volume effects. Porosity is also generated when a feldspar is replaced by another with different composition. The most common example is albitization which generally refers to the replacement of any feldspar by almost pure albite. It is one of the most common alumino-silicate reactions in

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the shallow crust of the earth (Perez and Boles 2005) and takes place whenever feldspar is in contact with saline aqueous solutions. Albitization is common during the diagenesis of arkosic sediments as well as taking place at higher hydrothermal temperatures. The porosity associated with the pseudomorphic albitization of natural plagioclase has been noted by many authors (e.g., Boles 1982; Lee and Parsons 1997; Lee and Lee 1998; Engvik et al. 2008). For example, albitization is widespread in the crystalline rocks of the Bamble Sector, south Norway where the replacement of plagioclase by albite is also associated with reddening of the rock. Figure 3a is a BSE SEM image of a polished cross-section of a specimen from a Norwegian outcrop showing a partially replaced plagioclase grain. On the right of this figure the plagioclase phase is oligoclase and on the left the darker contrast phase is albite. The black spots are pores which have remained unchanged since the albitization event over a billion years ago (Nijland et al. 2014). The largest pores are up to several microns and the smallest are below the resolution of the image. TEM observations of the same sample (Engvik et al. 2008) show that the pore size varies down to a few nanometers. The pale laths in Figure 3a are hematite, which is associated with the albitization and causes reddening. The pseudomorphic albitization reaction for an intermediate plagioclase (Merino 1975; Perez and Boles 2005) could be represented by: NaAlSi3O8·CaAl2Si2O8 + H4SiO4 + Na+ = 2 NaAlSi3O8 + Ca2+ + Al3+ + 4 OHwhere the silica required for the reaction is in solution together with the Na+ ions. The equation makes the approximation that intermediate plagioclase has the same molar volume as albite. To account for the porosity some of the silicate must also stay in solution, and this is not taken into account in the equation above. This requires that both Ca and Al are released into solution. Note: A pseudomorphic replacement reaction implies preservation of the external volume and shape of the original parent crystal (i.e., the total volume of the product phase plus generated pore volume equals the volume of the parent crystal.). The albitization equation above is written to preserve volume and this inevitably implies mobility of Al. However whether a reaction is pseudomorphic or not depends entirely on the composition and pH of the fluid phase. In determining mass balance during mineral replacement (e.g., using Gresens’ analysis) an assumption has to be made as to whether the volume is preserved, as above, or whether a specific element (typically Al) is immobile. This problem discussed in detail in Putnis (2009) and in Putnis and Austrheim (2012).

a

10 μm

b

100 nm

Figure 3. (a) A back-scatter electron (BSE) image of an albitization reaction front in a plagioclase single crystal from partly albitized rock in the Bamble Sector, south Norway. The smooth textured phase on the right is the parent oligoclase and the darker phase on the left is the product albite. The black spots in the albite are pores. The lighter laths are sericite. (b) During replacement of feldspars hematite may crystallize within the pore-spaces, giving the rock a reddish color, such as commonly seen in pink granites.

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Although the pseudomorphic replacement of plagioclase by albite generates porosity and hence may increase the permeability of a rock, complications arise if the ions released to solution also precipitate another phase. For example, the release of Al creates the possibility of reaction with excess silica in solution to form kaolinite, which would reduce the permeability. In experiments on the albitization of plagioclase, Hövelmann et al. (2010) found that needles of pectolite (NaCa2Si3O8OH) formed within the pores generated within the albite. These experiments were carried out at 600 ºC and 2 kbars, and at those temperatures reaction rims up to 50 μm wide formed within 14 days. Norberg et al. (2011) studied the microstructural evolution during experimental albitization of K-rich alkali feldspar, a reaction that only involved the ion-exchange of K by Na. They found that the ion-exchange mechanism was not a simple diffusional exchange, but involved interface-coupled dissolution–precipitation and the generation of porosity. Furthermore they made the important observation that the porosity and connectivity of the pores is dynamic and continuously evolves during the replacement process. A similar observation was made by Putnis and Mezger (2004) and Raufaste et al. (2011) for the replacement of KBr by KCl. Albitization is a ubiquitous process in the Earth’s crust because of the wide availability of Na-rich saline aqueous solutions. However it is also common to find plagioclase replaced by K-feldspar, again with the generation of porosity (e.g., Schermerhorn 1956; Harlov et al. 1998; Putnis et al. 2007). The replacement of one feldspar by another in nature, whether it be plagioclase replaced by K-feldspar or by albite, is frequently associated with the precipitation of hematite, which gives these feldspars a pink color. A study of pink feldspars in granites from a number of localities (Putnis et al. 2007; Plümper and Putnis 2009) has shown that the hematite forms nano-rosettes within the pore spaces generated during the feldspar replacement (Fig. 3b). The conclusion from these studies was that pink feldspars indicated that the original rock has been infiltrated by an Fe-bearing solution and that the feldspar replacement also initiated hematite precipitation, possibly due to a local change in pH at the reaction interface. In general, the presence of porosity and hence turbidity in natural feldspars is characteristic of replaced feldspars (Parsons 1978; Worden et al. 1990) although the absence of porosity does not necessarily indicate that the phase is primary. Dalby et al. (2010) report on albite whose trace element composition indicates that it is a product of secondary albitization, i.e., the replacement of a pre-existing feldspar by albite. However, the albite is not turbid and they suggest that this could be due to textural equilibration. Martin and Bowden (1981) have also reported non-turbid secondary albite in granite. As noted above, porosity generation and its interconnectivity has been shown in experimental systems to be a dynamically evolving process.

SECONDARY POROSITY ASSOCIATED WITH MINERAL REPLACEMENT—A UNIVERSAL PHENOMENON Although the section above has emphasized porosity generation during interface-coupled dissolution–precipitation in feldspars, Putnis and Putnis (2007) and Putnis (2009) have argued that mineral replacement by this mechanism is the universal mechanism of re-equilibration of solids in the presence of a fluid phase and that porosity generation is an integral part of this mechanism. Many examples of other systems have been described in previously published reviews and will not be given again here (see Putnis 2002, 2009; Putnis and Austrheim 2010, 2012; Putnis and John 2010; and Ruiz Agudo et al. 2014). The porosity generated may take a number of forms. In the examples described above, the porosity appears as apparently unconnected pore space at the spatial scale of the observation

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(e.g., Fig. 3b). However, during the replacement process there must be connectivity between the fluid reservoir and the reaction front suggesting that the porosity and permeability is a dynamically evolving feature (Norberg et al. 2011; Raufaste et al. 2011). However, when there is a large volume difference between the parent and product, the porosity takes on a form which resembles fluid-induced fracturing (Røyne et al. 2008; Jamtveit et al. 2009; Navarre-Sitchler et al. 2015, this volume; Røyne and Jamtveit 2015, this volume). Fluid-induced fracturing is usually associated with replacement involving a positive volume change, such as in hydration reactions during weathering (Fletcher et al. 2006; Buss et al. 2008) and the serpentinization of olivine (Iyer et al. 2008). However, Janssen et al. (2010) found that during the reaction of ilmenite (FeTiO3) with acid to form rutile (TiO2), a pseudomorphic replacement which involves a 40% reduction in solid volume, the reaction interface moves through the crystal by the generation of microfractures (Fig. 4). Rutile nucleates as nanoparticles from the solution formed in the microfractures. At present there is no clear understanding of the factors that control the morphology of the porosity.

IMPLICATIONS OF MICROPOROSITY— SUPERSATURATION AND CRYSTAL GROWTH Fluid–mineral interaction is a ubiquitous process throughout the geological history of a rock, from its first formation through to weathering and destruction. The porosity and its

Figure 4. A back-scatter electron (BSE) image of a cross-section through an ilmenite crystal (FeTiO3) partly replaced by an Fe-poor phase, which eventually results in the whole crystal being replaced by rutile. The “leaching” of the Fe from the ilmenite takes places by the generation and propagation of fractures (dark) though the unaltered ilmenite and precipitation of rutile from the fluid in the fractures. (See Janssen et al. 2010).

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distribution play a major role in controlling the processes involved at every stage. The reaction products of fluid–mineral interaction nucleate from the fluid phase, and the free energy drive for precipitation can be expressed in terms of the supersaturation of the fluid with respect to the precipitating phase or phases. In the rest of this chapter we will review the concept of supersaturation, the degree of supersaturation required to nucleate a new phase, and how these concepts need to be modified when discussing nucleation in a porous rock. Because crystallization in porous rocks has such wide fields of interest and many diverse applications, it has been discussed for many years from different viewpoints. The porosity and permeability of a reservoir rock is a key parameter which determines the efficiency of fluid flow, whether this be oil, gas, or hot water, and therefore precipitation within the pore space and hence permeability reduction is a major concern (Jamtveit and Yardley 1997). In a different field, salt crystallization in porous stonework and concrete is known to cause sufficient damage to break up the stone due to the “force of crystallization” associated with growing crystals pushing on pore walls (Rodriguez-Navarro and Doehne 1999a; Scherer 1999; EspinozaMarzal and Scherer 2010). The force of crystallization is related to the supersaturation (Steiger 2005a,b). The mechanism of frost heaving is another closely related area of research where undercooled (i.e., supersaturated) water within the pore spaces in fine-grained silt migrates to the surface and crystallizes forming an ice hill (a pingo) whose weight is supported by the force of crystallization of the ice below (Ozawa 1997). In all of these studies the fundamental premise is that a high fluid supersaturation (or undercooling in the case of ice crystallization) can be maintained in finely porous materials relative to the supersaturation at which nucleation takes place in an open system. Furthermore, the finer the porosity the higher the supersaturation in the fluid that can be maintained before crystallization can take place. To understand why this may be the case we will first examine the various concepts used to describe supersaturation and its relationship to nucleation.

Critical and threshold supersaturation Supersaturation in an open system is defined as the ratio between the activity product in solution and the solubility constant of the mineral. For a sparingly soluble salt such as barite, the degree of saturation is defined as :  

Ba  SO  / K 2

2

4

sp

.

(1)

For nucleation to take place a certain critical supersaturation where  > 1 must be reached to provide a driving free energy. This is usually derived from classical nucleation theory (CNT), which is simply based on thermodynamic arguments of balancing the increase in surface energy in forming a nucleus with the decrease in free energy associated with the nucleation (Fig. 5a). As the supersaturation increases, the value of the activation energy for nucleation decreases and at some critical value of the supersaturation the nucleation rate very rapidly increases (Fig. 5b). This value is taken as the critical supersaturation. Although CNT has obvious shortcomings, such as the definition of a surface energy when applied to a sub-critical nucleus, equations derived from this theory have been applied to experimental nucleation data for decades. From CNT (Nielsen 1964) the general form of the equation which describes the rate of heterogeneous nucleation in a solution, J, (i.e., the number of nuclei formed in a fixed volume) is:  3Y 2  J   exp  3 3 , 2   k T (ln S ) 

(2)

where  is a shape related factor,  is the fluid–mineral interfacial energy of the nucleus, Y is the volume of a growth unit in the nucleus, k is Boltzmann’s constant, T is temperature and

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S is the supersaturation (in this case more simply defined as the ratio of concentration in the solution to the concentration at equilibrium). The pre-exponential factor  is related to the rate at which the nucleus can grow to a supercritical size, and hence it involves diffusion of growth units to the surface of the nucleus. This equation shows that the value of the nucleation rate is a very sharp function of the supersaturation (Fig. 5b) and also that salts with a large surface energy () (and hence low solubility) and large molar volume (Y ) require a higher value of supersaturation for nucleation to become measurable. The shape of the nucleation rate function (Fig. 5b) leads to the general concept of a critical supersaturation, at which nucleation becomes spontaneous. This equation is based on the assumption that the supersaturation is a fixed value, reached instantaneously and takes no account of how the supersaturation is reached, in other words it does not take into account gradients in supersaturation either in space or in time. However, the actual value of the supersaturation that is achieved in any given system depends on the rate of change of supersaturation. This is a well-known phenomenon in cooling melts for example: the cooling rate will determine the degree of undercooling at which crystallization takes place. In an aqueous solution, supersaturation gradients will also exist due to cooling and evaporation rates as well as due to the diffusion-controlled compositional gradients in space around a nucleus. In any real system, supersaturation gradients are unavoidable and so must be included in any treatment of nucleation from solution. The actual value of the supersaturation at which nucleation becomes measurable, as a function of supersaturation gradients, however achieved, is termed the threshold supersaturation. We shall also extend this term to mean the actual value of supersaturation at which nucleation can take place when the solution is confined within a porous rock. Some general principles of the relationship between supersaturation gradients and the threshold supersaturation at nucleation in many salt crystallization systems were determined as a function of cooling rate and have been investigated both experimentally and theoretically (Nyvlt 1968; Nyvlt et al. 1985; Kubota et al. 1978, 1986, 1988). The general relationship between the cooling rate (i.e., supersaturation rate) and the degree of undercooling (threshold supersaturation) is: Supersaturation rate  (Threshold supersaturation)m or R  K ( th )m

(3)

Nucleation rate J

where K and m are empirical coefficients, and Ωth is the threshold supersaturation.

a

b

Ωcrit Supersaturation Ω

Figure 5. (a) Schematic drawing of the change in free energy as a function of nucleus size according to classical nucleation theory (CNT). The curve “eq” represents the free energy of a particle formed at equilibrium where only a surface energy term would be relevant. Curves (1) to (4) show the situation with increasing supersaturation where the reduction in free energy is dominated by the volume free energy of the nucleus. (b) A consequence of CNT is that the nucleation rate is a very sharp function of the supersaturation and defines an approximate value of the critical supersaturation for nucleation.

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Crystallization experiments in confined media Crystallization of ice. When aqueous solutions are confined to occupy a specific volume, we need to consider additional factors which could contribute to the threshold supersaturation for crystallization. A confined solution could be a droplet in a pore space or a thin film in a grain boundary. There has been an active interest in crystallization in small droplets for centuries, ever since the observation that liquid fogs and clouds could persist well below the frost point of water. The degree of undercooling possible in small droplets continues to be a subject of intensive research, especially since the radiative properties of ice clouds and the climatic consequences depend on the size of cloud particles. It is well known that small water droplets in clouds can be supercooled to at least -30 ºC and in some cases as low as -40 ºC—supercooling, meaning that the water is lowered to that temperature without crystallization as ice (e.g., McDonald 1953; Prupacher 1995). Numerous experiments on the freezing points of ultrapure water droplets have demonstrated an approximately linear dependence of the degree of undercooling on droplet size (Fig. 6) (for a summary see Prupacher 1995). These experiments were designed to measure homogeneous nucleation temperatures. The addition of any particles of impurities or dust would immediately provide seeds and promote heterogeneous nucleation of ice. However, even for carefully designed experiments to avoid heterogeneous nucleation, the interpretation of the results is made complicated by the presence of the water–air interface, and it has been argued that the nucleation rate at this interface is orders of magnitude faster than within the bulk droplet (Tabazadeh et al. 2002; Shaw et al. 2005). If the interface between the water droplet and its surroundings is an important factor then two opposing effects have to be taken into account: first, according to CNT the nucleation rate at a given supercooling should decrease as the droplet size decreases, but second, a smaller droplet will have a larger surface to volume ratio and hence the surface nucleation rate will increase. For pure water the experimental results suggest that the surface nucleation rate could become comparable to the volume nucleation rate at droplet radii below ~4 μm (Duft and Leisner 2004; Earle et al. 2010). Ice crystallization at large values of undercooling is made more complex because of the changing viscosity and hydrogen bonding at such low temperatures and further discussion of ice is beyond the scope of this chapter. Nevertheless the ice example illustrates two important points: (i) that the volume of the solution droplet is important, and (ii) that the interface between the solution and its surroundings will also play a role.

Supercooling (0C)

30

35

40

45 -6 10

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Drop Diameter (cm)

Figure 6. The relationship between the supercooling required to crystallize ice from ultra-pure water, as a function of the drop diameter. The full line is for a cooling rate of 1 ºC s-1 and the dashed line for a cooling rate of 1 ºC.min-1. The data for the lines are from a large number of experiments analyzed by Prupacher (1995).

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Crystallization of salts in small pores. Various experimental and theoretical approaches have been applied to the question of threshold supersaturation when an aqueous solution is confined in a porous medium. As well as the need to consider pore size, salt crystallization introduces the need to consider diffusion of ions through a solution to the crystallizing nucleus, and therefore the inevitability of compositional gradients forming around a crystal. As we have noted above, compositional gradients suggest supersaturation gradients in space as well as time and this will have an impact on the threshold supersaturation. Experiments designed to investigate these effects on the crystallization of sparingly soluble salts have been used by Prieto et al. (1988, 1989, 1990, 1994) and Putnis et al. (1995). The porous medium used is silica hydrogel and crystallization is induced by the counterdiffusion of ions from opposite ends of a diffusion column (Fig. 7). The silica hydrogel contains ~95 vol% of solution within an interconnecting porous network where the pore diameters are typically from 100 to 500 nm although secondary pores of up to 10 μm are also common (Henisch 1988). The gel inhibits advection and convection and transport of ions is purely by diffusion through the pore spaces. It is known that under these circumstances nucleation is inhibited and the threshold supersaturation is high. It is possible to determine the rate of change of supersaturation as well as the actual threshold supersaturation at the point of nucleation within the column (Prieto et al. 1994; Putnis et al. 1995). The results for a number of different salts (Fig. 8) follow the same empirical law that was found in cooling experiments, i.e., that Supersaturation rate  (Threshold supersaturation)m. Note the wide range of threshold supersaturation depending on the solubility (and hence the surface energy) of the salt, consistent with the expectations from classical nucleation theory. However, the meaning of the slopes of the straight lines in the ln–ln plots in Figure 8 is still not well understood (Prieto 2014). Although these experiments are a good demonstration of the effect of supersaturation rate on the threshold supersaturation, the reason for the suppression of nucleation in the silica hydrogel is not clear. The implication is that it is due to the suppression of advection and convection relative to crystal growth in an open system and that diffusion of the ions through the gel is also reduced due to the small pore size and the tortuosity of the diffusion paths. This hypothesis has been tested by Nindiyasari et al. (2014) in experiments on the growth of CaCO3 by counter-diffusion in gelatin hydrogels with different pore sizes and porosity distribution.

Solution reservoirs

Ba2+ solution

Gel column

SO42solution

Gel column

CrystallizaƟon zone

Figure 7. Schematic of the experimental set-up for double diffusion crystal growth experiments where a gel column separates the reacting components, in this case Ba2+ and SO42-. Counter-diffusion of the components through the gel column eventually supersaturates the fluid and results in precipitation (of BaSO4 in this case).

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Figure 8. ln–ln plot showing the relationship between the supersaturation rate and the threshold supersaturation for 4 different phases determined from crystallization experiments such as shown in Figure 7 (data from Prieto et al. 1994).

Although the waiting times for nucleation increased significantly and systematically with increasing gelatin content of the gel (and hence reduced total pore volume and pore size), separate diffusion experiments showed that reduced diffusion was not the reason for the inhibition of nucleation in the smaller pore sized gels. What inhibits nucleation in small pores? A number of possible explanations have been advanced for the observation that nucleation is inhibited in small pores: (i) Nucleation is a stochastic process that involves random collisions between ions to form clusters. In classical nucleation theory these clusters need to grow to a critical size where the gain in surface energy of the cluster is counterbalanced by the volume reduction in free energy due to forming the solid from a supersaturated solution (Fig. 5a). In the case of sparingly soluble salts, the probability of these random collisions will be small because the number of ions in a given volume is small, even when the supersaturation is high. The induction time is an important feature of classical nucleation theory and is the time between the creation of the supersaturation and the detection of nucleation. The induction time ti is given by: ti  1 / JV

(4)

where J is the nucleation rate and V is the volume of the solution. At the detection of nucleation, the nucleation rate is taken to be one nucleus in the given volume at the induction time ti. Using the standard equations from classical nucleation theory (Kashchiev 2000; Kashchiev and Rosmalen 2003), Prieto (2014) has calculated the supersaturation and induction times for barite precipitation as a function of pore size, assuming that in the nucleation equations the relevant volume V is the volume of a single pore. Figure 9 from Prieto (2014) shows the calculated critical supersaturation and induction times calculated for pore sizes of 0.1 μm and 1 μm; these were compared to the experimental values from the gel experiments of Prieto et al. (1994) and Putnis et al. (1995). Although there is always some uncertainty about the relevant values of the parameters used in the calculations, the results show a clear dependence of

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14000

Supersaturation

12000 10000 0.1 μm

8000 1 μm

6000

4000 2000 0

5 10 Log induction time (s)

15

Figure 9. Calculated curves from CNT for the relationship between the supersaturation and the induction time for nucleation in pores of size 0.1 μm and 1 μm (after Prieto 2014).

critical supersaturation on the pore size. These results are also consistent with the experimental observations by Nindiyasari et al. (2014) that higher supersaturation values are needed for crystallization from solutions confined to smaller pores. (ii) For crystal growth in small pores the crystal size will be limited by the pore size and crystals whose geometry is characterized by a large surface to volume ratio will have a higher solubility than the bulk solubility. Pore-size-controlled solubility (PCS) has been advanced by Emmanuel and Berkowitz (2007), Emmanuel and Ague (2009) and Emmanuel et al. (2010) as an explanation for the preferential precipitation of salts in larger pore volumes. When the effects of surface energy are taken into account, it is possible that a solution could be undersaturated in small pores while being supersaturated in large pores. Determining the solubility of crystals in small pores is experimentally difficult. One method which has successfully been used to measure the composition of solutions at the point of nucleation or dissolution inside a pore, as well as determining the pore size, is nuclear magnetic resonance spectroscopy (NMR). For Na in solution, the intensity of the NMR peaks can be correlated with solution composition and the relaxation rate to the pore size (Rijniers et al. 2004, 2005). Therefore for Na salts it is possible to determine the solubility as a function of pore size. For Na2CO3, Rijniers et al. (2005) found that the solubility inside 30-nm-sized pores hardly differs from bulk solubility whereas for 10-nm pores the solubility was more than twice the bulk solubility. According to Espinoza-Marzal and Scherer (2010), PCS needs to be taken into account in pores smaller than 0.1 μm, although the effect of pore size on solubility will be greater for sparingly soluble salts which have a higher surface energy. PCS is likely to be an important factor when considering precipitation in pores in the submicron range (Emmanuel and Ague

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2010). Therefore, PCS is another possible explanation for the high threshold supersaturation required to nucleate salts in the gel experiments described above. (iii) The role of the pore surface. Neither of the previous two possible explanations for the effect of pore size on threshold supersaturation consider the role of the pore wall itself. To test whether the surface chemistry of the pores controlled CaCO3 precipitation, Stack et al. (2014) introduced supersaturated solutions into a manufactured controlled-pore glass (CPG) which contained both macropores (~32 nm) and nanopores (~8 nm). In some experiments the pore walls were functionalized with a monolayer of anhydride with a polar functional group. Precipitation was studied in situ using small angle X-ray scattering, a non-destructive method which allows the researcher to discriminate between different densities in a sample, as well as ex situ by scanning transmission electron microscopy. The results showed that in the native CPG, precipitation only took place in the macropores, whereas in the functionalized CPG, precipitation also took place in nanopores, suggesting that a favorable surface chemistry could promote nucleation, even in nanopores. As mentioned above, the Emmanuel and Berkowitz (2007) model for pore-size-controlled solubility may be relevant for nanometer-sized pores, but not for pores with sizes of several microns. Mürmann et al. (2013) modified this model by considering that the surface charge on the pore walls will interact with the ions in the fluid and change their activity and hence the supersaturation of the fluid. Using a numerical simulation they calculated the relationship between pore size and the crystallization as a function of supersaturation and showed that by including the surface charge the pore radius at which pore-size-controlled solubility is relevant is shifted from the nanometre to the micrometre range. The model could also be relevant to the observations in Stack et al. (2014), where a polar functional group modifies the surface charges on the pore walls and enhances nucleation, although the interpretation of the enhancement in that case involves matching of the crystal structure with the substrate (Lee et al. 2013), rather than the surface charge modifying ion activities. (iv) The nucleation mechanism. Most interpretations of crystal growth in pores have assumed a classical model of nucleation, i.e., that crystallization first involves the formation of a critical nucleus and then grows by adding further growth units to the cluster. However, such an equilibrium model of balancing positive surface energy terms with negative free energy reduction for nucleation is not likely to be relevant to a situation where nucleation takes place at very high values of supersaturation, as is the case in small pores. Furthermore, there is increasing evidence that even at moderate supersaturation a “non-classical” model of crystal growth involving the oriented attachment and assembly of sub-critical clusters can better explain experimental observations (Niederberger and Cölfen 2006; Gebauer and Cölfen 2011; De Yoreo 2013). The morphology of crystals grown at high supersaturations in hydrogels also suggests a non-classical nucleation model in which larger crystals are made up by the oriented attachment of smaller crystallites (e.g., Grassmann et al. 2003; Nindiyasari et al. 2014). Nindiyasari et al. (2014) also noted that the mosaic spread of calcite aggregates grown in gels increases when the pore size is smaller. In the non-classical model, pre-nucleation nanoclusters exist in supersaturated solutions effectively reducing the activity of the ions in solution. Thus the concepts we use to define supersaturation at the point of crystal growth would need to be redefined.

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In a geological context, the evolution of porosity, its generation and destruction through fluid–mineral interaction is of utmost importance. In a sedimentary basin undergoing diagenesis, for example, compaction processes reduce the primary porosity in an unconsolidated sediment through pressure solution and reprecipitation of material into the pore spaces. At the same time, increasing temperature and pressure will induce porosity-generating reactions between saline solutions and the minerals in the sediment. We have seen how porosity generation is an integral aspect of the re-equilibration of a rock by interface-coupled dissolution–precipitation, but that this porosity is also a transient feature which may be annihilated by recrystallisation. Even when porosity is preserved as is the case in the albitized feldspar in Figure 3a, permeability can be drastically reduced if the pores lose connectivity. This effectively closes the mineral to further fluid flow and the pores are preserved as fluid inclusions. As long as fluid flow through a porous rock is still possible, the pore spaces serve as sites for crystal growth, further reducing the porosity and potentially clogging the rock. The reduction of permeability has obvious consequences for the extraction of fluids from a rock as in oil and gas recovery and in geothermal reservoirs. It is therefore important to understand the factors which control such secondary mineralization and pore size appears to be a major factor controlling nucleation and growth. In this final section we give some further examples of the role of pore size on crystallization and its consequences and interpret these in terms of the experimental results quoted above. (i) The porous sandstones of the Lower Triassic Bunter Formation in north-west Germany have been studied as potential reservoir rocks for gas storage. However, due to the proximity of salt domes the pore fluids are highly saline and salt cementation in the pore space significantly deteriorates the reservoir quality, reducing the porosity from ~30% to 5% and the permeability by up to 4 orders of magnitude (Putnis and Mauthe 2001). In some parts of this Formation the sandstone has small scale (~1 mm) periodic variations in grain size, allowing a study of the effect of pore size on halite precipitation (Putnis and Mauthe 2001). Figure 10 shows an

Figure 10. Micrograph of a thin section of a sandstone, with a periodic variation of quartz grain size, which has been partly cemented by halite (NaCl). The sandstone has subsequently been injected with a dark resin which reveals the residual porosity. The unfilled pores are all in the finer grain size sandstone while the larger pores between the coarser grains have been fully cemented by halite. The width of the micrograph is 4.2 mm.

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optical micrograph of such a sandstone sample in which the residual porosity is shown by the dark resin which is injected into the sample before preparing the thin section. The resin only penetrates the finer sandstone layers, as the coarser layers are fully cemented with halite. Quantitative measurements of the pore size distributions in these rocks showed that the size of the pores left uncemented by salt is decreased as the porosity and permeability is reduced, so that the first pores to be salt-cemented were the largest with pore diameters in the tens of micron range. Progressively smaller pores are filled until the residual pore size in the most cemented samples is submicron. If we assume that the NaCl concentration in solution was uniform throughout (but see below), our conclusion would be that the solution in the small pores was able to maintain a higher supersaturation before crystallization, compared to the larger pores. The persistence of open pores in the submicron range could be the result of pore-size controlled solubility, but this does not explain the differences when the pore sizes are much larger. Some other systematic effect of surface to volume ratio of the pores, such as the surface-charge model suggested by Mürmann et al. (2013) or effects related to solute transport into pores of different sizes may be important. A similar permeability reduction problem has afflicted the development of a geothermal plant within the Upper Triassic Rhaetian sandstone succession in northern Germany, where anhydrite precipitation has effectively reduced the flow of hot water below that needed to be economic (Wagner et al. 2005). Analysis of core samples showed that anhydrite precipitation was restricted to regions of relatively high primary porosity, while regions of low porosity remained uncemented. In the discussion so far the assumption has been that the solute concentration in pore fluid throughout the pore-space of a rock is homogeneous and that the same concentration of solute could result in solution in large pores being supersaturated while that in small pores was undersaturated. However, this is likely to be an oversimplification particularly as the layer of a reactive fluid in contact with a mineral surface is different from the bulk composition (Putnis et al. 2005; Ruiz-Agudo et al. 2012). Therefore, the definition of ‘bulk’ in this context would depend on pore size. Sequential centrifugation has been widely used to extract pore water from a rock and it has been demonstrated that with increasing centrifugal speed water is progressively extracted first from larger pores and then from smaller pores (Edmunds and Bath 1976). Therefore, changing solution concentrations expelled from a rock with increasing centrifugal speed have been interpreted as an indication that larger pores have different solute concentrations than small pores and that concentration gradients exist in reacting rocks (Yokoyama et al. 2011 and references therein). The fact that reactive fluid infiltration into a rock is a heterogeneous phenomenon depending on local flow paths is a commonplace observation with implications for local rock strength, deformation, and differential stress (Mukai et al. 2014; Wheeler 2014). (ii) Salt crystallization in porous rocks—weathering and disintegration. The relationship between pore size and threshold supersaturation for crystal growth has dramatic consequences when salts grow in the pores of rock and concrete. Confined crystals growing from a supersaturated solution exert a stress (crystallization pressure) on the pore walls through a thin film of supersaturated fluid between the crystal and the pore wall (Steiger 2005a,b). Supersaturation may be generated through evaporation so the point at which a solution reaches the threshold supersaturation will determine the size of the crystallization pressure. Rapid evaporation increases the supersaturation rate and hence the threshold supersaturation. The presence of small pores, in which the threshold supersaturation can be very high, results in greater damage during fluid evaporation. Repeated wetting and drying of salt solutions generates sufficient damage to eventually disintegrate sandstone, limestone, and concrete.

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Figure 11. Photograph of honeycomb weathering in sandstone, Nobby’s Beach, Newcastle, Australia.

Salts with high surface energy such as the sodium sulfate hydrates require high values of supersaturation to nucleate and so are capable of causing the greatest damage, whereas halite has always been considered as having a very high nucleation rate even at low supersaturation. However, the very common phenomenon of honeycomb weathering of rocks in coastal environments (Fig. 11) and deserts with saline groundwater suggests that even halite crystallization in porous rocks enhances weathering and erosion (Wellman and Wilson 1965; Rodriguez-Navarro et al. 1999b; Doehne 2002). Recent work on NaCl crystallization in solutions confined in microcapillaries (Desarnaud et al. 2014) has shown that nucleation takes place at considerably higher values of supersaturation than previously assumed, at S ~1.6 where S is defined as the ratio of concentration relative to the bulk solubility value. At such a high value the crystallization pressure would be more than sufficient to exceed the tensile strength of sedimentary rocks (Derluyn et al. 2014). (iii) Mineral vein formation involves the supersaturation of an aqueous solution and subsequent precipitation in a narrow dilatational feature in a rock. Because the veins form after the formation of the host rock, the vein characteristics and the microstructures of the minerals within the vein have been used to provide information about paleo-stress fields, deformation mechanisms which may have caused the dilatation as well as fluid pathways (Bons et al. 2012). Vein formation is ultimately related to the formation of fractures in a rock, and fracture mechanics is a topic beyond the scope of this chapter (but see Røyne and Jamtveit 2015, this volume). However the generation of supersaturation does relate to some of the issues discussed here. There are a number of ways that supersaturation can be generated, and the most commonly discussed are changes in temperature and pressure, which reduce the solubility of a mineral, changes in the chemical environment, such as pH and Eh (redox conditions), and fluid mixing and changes in composition of a fluid due to interaction with the host rock. These factors have been reviewed by Bons et al. (2012). However, another rarely mentioned possibility is that diffusional and advective transport of material and fluid from the pores in the bulk rock into a dilational feature will initiate supersaturation due to the fact that an open space cannot

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sustain the same level of supersaturation as in the small pores in a rock (Putnis et al. 1995). In such a case the fluid transport may be normal to the vein rather than fluid flow along an open fracture (Fisher and Brantley 1992; Fisher et al. 1995) and equilibrium solubility values would not be relevant in determining the amount of fluid required to precipitate a given amount of vein material. In such veins, the mineral growth originates from a small fracture which then forms a median line within the vein as the fracture widens. The mineral growth is then from this median line towards the vein wall (antiaxial) compared with mineral growth in an open fracture which starts from the walls and grows inwards (syntaxial). The various styles of vein formation and the mineral microstructures associated with different types of veins have been discussed by Elburg et al. (2002) and Bons et al. (2012).

CONCLUSIONS Although the fundamental origin of the relationship between pore size and fluid supersaturation is still not totally understood, and may be the result of a number of interacting parameters, there is no doubt that the pore structure exerts a very significant influence on every aspect of crystal growth. In sedimentary rocks undergoing diagenesis, porosity generation through fluid–mineral re-equilibration with increasing T, P is contemporaneous with porosity destruction though compaction and recrystallization. The interplay between these processes will form a lively topic of research for years to come. Similarly, in low permeability crystalline rocks the feedback is between reaction-driven porosity generation and which allows further fluid infiltration providing the mechanism for large-scale metamorphic and metasomatic reactions.

ACKNOWLEDGMENTS The early experiments on the generation of porosity during salt replacement by Christine V. Putnis, University of Münster and her colleagues led to the development of our understanding of mineral replacement mechanisms, and then together with Håkon Austrheim and Bjørn Jamtveit, University of Oslo to the wider applications to fluid-rock interaction. Many of the concepts relating to supersaturation in porous media have been developed through many years of discussions with Manuel Prieto, University of Oviedo. I thank Sue Brantley and Carl Steefel for comments that have improved this chapter. Financial support from the Marie Curie-Sklodowska Action of the European Union (FlowTrans: Flow and Transformation in Porous Media PITN-GA-2012-316889) is gratefully acknowledged.

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Prieto M (2014) Nucleation and supersaturation in porous media (revisited). Mineral Mag 78:1437–1447 doi: 10.1180/minmag.2014.078.6.11 Prieto M, Viedma C, Lopez-Aceueodo V, Martin-Vivaldi JL, Lopez-Andres S (1988) Mass transfer and supersaturation in crystal growth in gels. Application to CaSO4.2H2O. J Cryst Growth 92:61–68 Prieto M, Fernández-Díaz L, Lopez-Andres S (1989) Supersaturation evolution and first precipitate location in crystal growth in gels; application to barium and strontium carbonates. J Cryst Growth 98:447–460 Prieto M, Putnis A, Fernández-Díaz L (1990) Factors controlling the kinetics of crystallization: supersaturation evolution in a porous medium. Application to barite crystallization. Geol Mag 127:485–495 Prieto M, Putnis A, Fernández-Díaz L, Lopez-Andres S (1994) Metastability in diffusing-reacting systems. J Cryst Growth 142:225–235 Pruppacher HR (1995) A new look at homogeneous ice nucleation in supercooled water drops. J Atmos Sci 52:1924–1933 Putnis A (2002) Mineral replacement reactions: from macroscopic observations to microscopic mechanisms. Mineral Mag 66:689–708 Putnis A (2009) Mineral replacement reactions. Rev Mineral Geochem 30:87–124 Putnis A, Austrheim H (2010) Fluid induced processes: Metasomatism and Metamorphism. Geofluids 10:254– 269 Putnis A, Austrheim H (2012) Mechanisms of metasomatism and metamorphism on the local mineral scale: The role of dissolution–reprecipitation during mineral reequilibration. In: Metasomatism and the Chemical Transformation of Rock. Harlov DE and Austrheim H (eds). Springer-Verlag, Berlin, p 139–167 Putnis A, John T (2010) Replacement processes in the Earth’s Crust. Elements 6:159–164 Putnis A, Mauthe G (2001) The effect of pore size on cementation in porous rocks. Geofluids 1:37–41 Putnis CV, Mezger K (2004) A mechanism of mineral replacement: isotope tracing in the model system KCl– KBr–H2O. Geochim Cosmochim Acta 68:2839–2848 Putnis A, Putnis CV (2007) The mechanism of reequilibration of solids in the presence of a fluid phase. J Solid State Chem 180:1783–1786 Putnis A, Prieto M, Fernández-Díaz L (1995) Fluid supersaturation and crystallization in porous media. Geol Mag 132:1–13 Putnis CV, Tsukamoto K, Nashimura Y (2005) Direct observation of pseudomorphism: Compositional and textural evolution at a solid–fluid interface. Am Mineral 90:1902–1912 Putnis A, Hinrichs R, Putnis CV, Golla-Schindler U, Collins L (2007) Hematite in porous red-clouded feldspars: evidence of large-scale crustal fluid–rock interaction. Lithos 95:10–18 Raufaste C, Jamtveit B, John T, Meakin P, Dysthe D (2011) The mechanism of porosity formation during solvent-mediated phase transformations. Proc R Soc Math Phys Eng Sci 467:1408–1426 Renard F, Ortoleva P, Gratier JP (2007) Pressure solution in sandstones: influence of clays and dependence on temperature and stress. Tectonophys 280:257–266 Rijniers LA, Magusin PCMM, Huinink HP, Pel L, Kopinga K (2004) Sodium NMR relaxation in porous materials. J Mag Res 167:25–30 Rijniers LA, Huinink HP, Pel L, Kopinga K (2005) Experimental evidence of crystallization pressure inside porous media. Phys Rev Lett 94:075503 Rodriguez-Navarro C, Doehne E (1999a) Salt weathering: Influence of evaporation rate, supersaturation and crystallization pattern. Earth Surf Proc and Landf 24:191–209 Rodriguez-Navarro C, Doehne E, Sebastian E (1999b) Origins of honeycomb weathering: The role of salts and wind. Geol Soc Am Bull 111:1250–1255 Røyne A, Jamtveit B (2015) Pore-scale controls on reaction. Rev Mineral Geochem 80:25-44 Røyne A, Jamtveit B, Malthe-Sørenssen A (2008) Controls on weathering rates by reaction-induced hierarchical fracturing. Earth Planet Sci Lett 275:364–369 Ruiz-Agudo E, Putnis CV, Rodriguez-Navarro C, Putnis, A (2012) The mechanism of leached layer formation during chemical weathering of silicate minerals. Geol 40:47–950 Ruiz-Agudo E, Putnis CV, Putnis A (2014) Coupled dissolution and precipitation at mineral–fluid interfaces. Chem Geol 383:132–146 Rutter EH (1983) Pressure solution in nature, theory and experiment. J Geol Soc 140:725–740 Scherer GW (1999) Crystallization in pores. Cem Concr Res 34:1613–1624 Schermerhorn LJG (1956) The granites of Trancoso (Portugal): a study in microclinization. Am J Sci 254:329– 348 Shaw RA, Durant AJ, Mi Y (2005) Heterogeneous surface crystallization observed in undercooled water. J Phys Chem B 109:9865–9868 Stack AG, Fernandez-Martiniez A, Allard LF, Banuelos JL, Rother G, Anovitz LM, Cole DR, Waychunas GA (2014) Pore-size dependent calcium carbonate precipitation controlled by surface chemistry. Environ Sci Technol 48:6177–6183 Steiger M (2005a) Crystal growth in porous materials I: The crystallization pressure of large crystals. J Cryst Growth 282:455–469

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Steiger M (2005b) Crystal growth in porous materials II: Influence of crystal size on crystallization pressure. J Cryst Growth 282:470–481 Tabazadeh A, Djikaev YS, Reiss H (2002) Surface crystallization of supercooled water in clouds. PNAS 99:15873–15878 Velbel MA (1993) Formation of protective surface layers during silicate-mineral weathering under wellleached, oxidizing conditions. Am Mineral 78:405–414 von Bargen N, Waff HS (1986) Permeabilities, interfacial areas and curvatures of partially molten systems: Results of numerical computations of equilibrium microstructures. J Geophys Res 91:9261–9276 Wagner R, Kühn M, Meyn V, Pape H, Vath U, Clauser C (2005) Numerical simulation of pore space clogging in geothermal reservoirs by precipitation of anhydrite. Int J Rock Mech Min Sci 42:1070–1081 Waldron KA, Parsons I, Brown WL (1993) Solution–redeposition and the orthoclase–microcline transformation: evidence from granulites and relevance to 18O exchange. Mineral Mag 57:687–695 Walker FDL (1990) Ion microprobe study of intragrain micropermeability in alkali feldspars. Contrib Mineral Petrol 106:124–128 Walker FDL, Lee MR, Parsons I (1995) Micropores and micropermeable texture in alkali feldspars: geochemical and geophysical implications. Mineral Mag 59:505–534 Walther JV, Wood BJ (1984) Rate and mechanism in prograde metamorphism. Contrib Mineral Petrol 88:246– 259 Watson EB, Brennan JM (1987) Fluids in the lithosphere, 1. Experimentally-determined wetting characteristics of CO2–H2O fluids and their implications for fluid transport, host-rock physical properties, and fluid inclusion formation. Earth Planet Sci Lett 85:497–515 Wellman HW, Wilson AT (1965) Salt weathering: neglected geological erosive agent in coastal and arid environments. Nature 205:1097–1098 Wheeler J (2014) Dramatic effects of stress on metamorphic reactions. Geology 42:647–650 Wood BJ, Walther JV (1983) Rates of hydrothermal reactions. Science 222:413–415 Worden RH, Walker FDL, Parsons I, Brown WL (1990) Development of microporosity, diffusion channels and deuteric coarsening in perthitic alkali feldspars. Contrib Mineral Petrol 104:507–515 Yokoyama T, Nakashima S, Murakami T, Mercury L, Kirino Y (2011) Solute concentration in porous rhyolite as evaluated by sequential centrifugation. Appl Geochem 26:1524–1534

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 25-44, 2015 Copyright © Mineralogical Society of America

Pore-Scale Controls on Reaction-Driven Fracturing Anja Røyne and Bjørn Jamtveit Physics of Geological Processes (PGP) Departments of Physics and Geoscience University of Oslo P.O. Box 1048 Blindern N-0316 Oslo Norway [email protected] [email protected]

INTRODUCTION Fluid migration through reactive rocks invariably leads to modifications of the rock porosity and pore structure. This, in turn, provides feedback on the fluid migration process itself. Reactions may lead to increases or decreases of the rock permeability. When the volume of solids decreases, either by an increase in the rock density or by transport of mass out of the system, the corresponding increase in porosity will enhance fluid transport and the continued propagation of the reaction front (Putnis 2015, this volume). In contrast, reactions that increase the solid volume will fill the pore space and may reduce permeability (Hövelmann et al. 2012). In this case, the reaction will only proceed if the stress generated by the volumeincreasing process is large enough to create a fracture network that will enable continued fluid flow. Reaction-induced fracturing is particularly relevant during fluid migration into high-grade metamorphic and slowly cooled magmatic rocks with very low initial porosity, but may also be important during reactive transport in more porous rocks where growth processes within the pore space exerts forces on the pore walls (Jamtveit and Hammer 2012). In this article we attempt to shed some light on the factors that determine whether volume-increasing reactions and growth in pores will reduce or increase permeability. We will start by describing field-scale examples of reaction-driven fracturing, and use a Discrete Element Model (DEM) to analyze how the resulting pattern and the rate and progress of reaction depend on the initial porosity of the rock. Ultimately, however, stress generation is related to growth processes taking place at the pore scale. We will therefore zoom in and describe pore-scale growth processes and how these are associated with fracturing and the production of new reactive surface area and new transport channelways for migrating fluids. Stress generation by growth in pores requires that crystals continue to grow even after having ‘hit’ the pore wall. This implies that the fluid from which the crystals precipitate is not squeezed out from the reactive interface by the normal stress generated by the growth, but can be kept in place as a thin film by opposing forces that operate at very small scales. To understand the dynamics of crystal growth against confining pore walls, we need to zoom in even further and examine interface processes taking place at the nanometer scale. Hence, the last part of this chapter focuses on the nanometer-scale morphology of the reacting interface and the mechanical and transport properties of the fluids confined along reactive grain boundaries. 1529-6466/15/0080-0002$05.00

http://dx.doi.org/10.2138/rmg.2015.80.02

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Reaction-induced fracturing: The effect of porosity The potential importance of reaction-driven fracturing is highest when the confining pressure is low or the rock is under high differential stress. In such a situation, even modest reaction-driven stresses may cause failure and fracture propagation. Spheroidal weathering is an example of a process where reactions produce pronounced fracturing under the low confining pressures that prevail near the Earth’s surface (Fletcher et al. 2006; Røyne et al. 2008). This kind of weathering has been described for most rock types in a wide range of climate zones (Chapman and Greenfield 1949). Although spheroidal weathering, per definition, involves surface-parallel fracturing and spalling of layers at the margin of rock blocks, which become progressively rounded (core stones), the progress of this mode of weathering and the fracture patterns produced are sensitive to the initial porosity of the rock type, as outlined below. A representative example of spheroidal weathering of a rock with very low initial porosity is shown in Figure 1. Interactions between the dolerite and oxidizing groundwaters generate a characteristic sequence of mm-to-cm-thick spalls that separates the weathered product from almost completely fresh dolerite. During progressive weathering, the central core stone will often split into two or more daughter core stones by a process called twinning. This process is also driven by the stresses generated through the volume-increasing reactions at the outer margin of the fresh core stone (Røyne et al. 2008). Because rocks are elastically extremely stiff, only a very small volume change «1% may generate stresses high enough to crack the rock. Figure 2 shows spheroidal weathering of an andesitic intrusion with an original porosity of ~8% (Jamtveit et al. 2011). During spheroidal weathering, reaction-driven expansion and spalling occur along with domain-dividing fractures to form smaller “twins” and “triplets”. In contrast to the situation described for the dolerite above, there are no sharp reaction fronts or interfaces, and individual core stones show pronounced progress of weathering reactions tens of centimeters inside the innermost onion-skin fracture. In fact, most core stones show significant production of weathering products in the pores throughout the entire rock volume, with no remaining completely unaltered andesite.

Figure 1. Spheroidal weathering of doleritic sill intrusion from the Karoo Basin, South Africa. Reactiondriven fracturing produces a number of spalls (‘onion-skin’-like fractures) that result in a rounded ‘core stone’ from an initially angular dolerite block, cut out by pre-existing joints (left). Continued weathering eventually produces tensile stresses inside the core stone that are high enough to make the original core stone (outlined by solid lines) divide into two or more daughters (dashed lines) (right). [Modified from Røyne et al. 2008].

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Figure 2. Block of an andesitic intrusion subdivided into smaller domains (I–III) by internal fracturing. Domain III was further subdivided into IIIA and IIIB, before IIIA split into twins IIIAa and IIIAb. The brown-colored striation is a combination of Liesegang bands and onion-skin fractures. [Modified from Jamtveit et al. 2011].

Recent numerical simulations have provided us with a more complete understanding of the differences described above. Until recently, most models focused on rocks with very low initial porosity where transport in unreacted rock occurred mainly by slow grain boundary diffusion. In such systems, a sharp reaction front separates completely unreacted rocks from rocks with a high extent of reaction. Moreover, the (1-D) propagation rate of this front is controlled mainly by the transport properties of the unfractured fresh rock (Rudge et al. 2010). However, in more porous rocks, the transport rates in unreacted rocks will be fast compared to the chemical reaction kinetics, and the reaction fronts may become broader with a more gradual transition from extensively reacted rock to fresh rock. A 2-D model describing reaction-driven fracturing of rocks with variable porosity was recently presented by Ulven et al. (2014a). This model is a discrete element model (DEM) with a reaction-diffusion solver developed by Ulven et al. (2014b). It simulates the deformation and fracturing of a solid with constant intergranular porosity undergoing a local volume-increasing chemical reaction. Fluid flow in fractures is assumed to be effectively instantaneous compared to the rate of other relevant processes. In a natural system, the progress of the fluid-driven, volume-increasing reactions, as described by the model above, is controlled by two main parameters: rock porosity (), and the shape of the initial domain undergoing volatilization. For the circular domains shown in Figure 3, porosity variation will control both the relative rates of reaction kinetics and transport, often expressed by the dimensionless Damkohler number ( 1/), and the amount of the mobile reactant that the rock can contain ( ). High porosity implies low  and high  (see Ulven et al. 2014a). Figure 3 shows plots of reaction progress versus time for four different porosities, as well as fracture patterns developed at 50% reaction progress. As demonstrated by Ulven et al. (2014a), for porosities less than about 2.5%, the ‘overall’ reaction rate (rate averaged over the entire domain) is proportional to N, where N is in the range 0.45–2 and high N-values

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(high sensitivity to initial porosity) corresponds to reactions with small volume changes. For initial porosities greater than 2.5%, the overall rate becomes increasingly less sensitive to the porosity, and is controlled mainly by the kinetics of the reaction. In a natural system, the shape of the domains undergoing reaction will always deviate from a circle or a sphere. This has some important consequences for the reaction progress, as sharp edges or other regions with high curvature tend to act as foci for differential stress and thus control the timing of the onset of fracturing. Figure 4 shows examples of fracture patterns and reaction progress at 50% reaction for an initial porosity of 0.05%. This figure indicates that the total reaction rates are fairly similar after fracturing has commenced. However, the onset of fracturing (the steep part of the curves) is significantly different for different geometries. One implication of this is that spheroidal weathering of jointed rocks may progress very differently in blocks of different shape. Reaction-driven fracturing is thus expected to produce a characteristic size distribution of the resulting core stones as described by Fletcher and Brantley (2010) and Ulven et al. (2014a).

Figure 3. Total reaction progress versus time (central graph) for four different porosities: (A)  = 2.5%, (B)  = 0.9%, (C)  = 0.3%, and (D)  = 0.05%. Shade shows extent of reaction, from dark for unreacted material to light for completely reacted material Panels show fracture patterns and local reaction progress at 50% total reaction for systems with fluid flow in the fractures. The volume change of the reaction was 0.8% in all cases. [Modified from Ulven et al. 2014a].

Figure 4. Reaction progress versus time for four different initial shapes (central graph). The panels show the four different domains (1-4) at 50% reaction progress for an initial porosity of 0.05%. [Modified from Ulven et al. 2014a].

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Reaction-induced clogging and closure of fluid pathways Although reactions may produce fractures and porosity, mineral growth in pores may also lead to clogging. This may significantly decrease the permeability and therefore fluid migration through initially porous rocks, and through rocks that are initially fractured due to tectonic and thermal processes. Tectonic deformation may, according to the Gutenberg– Richter law, riddle the crust with fractures on all scales (Molnar et al. 2007), and provide pervasive permeability and reactive surface area in brittly deformed rocks. When fluidconsuming reactions occur synchronously with tectonic deformation, the externally imposed stress may completely control the reaction rate and progress if permeability is generated at a faster rate than the infilling of pores by reaction products. On the other hand, if the precipitation rate is fast compared to the rate of fracturing and permeability generation, the progress of reaction may be modest even in extensively fractured rocks. Figure 5 shows a possible example of such a system: an extensively faulted, fractured, but only partly serpentinized, dunite.

Figure 5. Extensively fractured, faulted, and partly serpentinized dunite from the Leka ophiolite complex, Central Norwegian Caledonides. Despite the extensive fracturing and faulting observed, the average extent of serpentinization varies from 40–50 to 70–80% in the various domains of this outcrop.

Extrapolation of the kinetic data of the serpentinization reaction based on powder experiments (Malvoisin et al. 2012) predicts that hydration of the original rock should be complete within tens of years, a nearly instantaneous process on the time scales of plate tectonics. Yet, the extent of reaction is only about 50%. Clearly, the supply of water to the olivine surfaces must have been limited, in spite of the extensive fracturing process. Continuous clogging of thermally and tectonically induced fractures related to the strong increase in solid volume associated with the rapid olivine hydration reaction (> 30%) is a possible explanation for this observation (Malvoisin and Brunet 2014).

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We will now take a closer look at the pore-scale mechanisms that lead to the stresses and fracture patterns discussed in the preceding section. The key requirement for reactions to take place is that water has access to the reactive surface. In the examples given below, we will show how fracturing can allow the reaction to proceed, and how the closure of nanoscale fluid pathways will cause the reaction to stop. In all of these examples, two conditions are absolutely necessary for fracturing to take place. The first is some thermodynamic overstepping of the reaction. In geological settings, reactions are often assumed to take place very close to equilibrium. This cannot, however, lead to reaction-driven fracturing. The minimal (Griffith) requirement for fracture propagation to take place is that the elastic strain energy released during fracture propagation is at least equal to the amount of energy required to form two new fracture surfaces (Fletcher et al. 2006; Røyne et al. 2011b). When growing minerals perform work on their surroundings by displacing their confining walls, elastic strain energy builds up in the system. This energy comes from the chemical energy available in the reaction, and it requires that minerals precipitate from a supersaturated solution or that the Gibbs free energy, G, of the reaction is sufficiently negative. As a crystal grows against the confinement of neighboring minerals, the stress on the growing crystal surface will increase. When the energy penalty for growing a crystal under normal stress becomes larger than the energy gained by precipitating material, growth will cease (Gibbs 1876; Steiger 2005a). The thermodynamic overstepping of the reaction therefore gives an upper bound for the mechanical stresses that can be generated from it. The second prerequisite is the continued supply of reactants to the reaction site. In order to form a fracture, the material must be mechanically strained, meaning that the volume inside a pore must continuously increase until the strain is large enough for the fracture threshold to be reached. This requires that reactants must be transported to the reacting site while the reaction is taking place, even after the precipitated material has filled the original pore space. Since transport through solid phases is usually much too slow to be relevant, this transport must take place through a liquid film that is confined at the reactive interface, despite the normal stress that builds up across this interface (Taber 1916; Espinosa Marzal and Scherer 2008). Nanometer-thick confined fluid films can in many cases sustain significant normal stresses across them without being squeezed out. The normal stress is sustained by the disjoining pressure of the confined fluid film, which is a function of the film thickness (de Gennes et al. 2003; Israelachvili 2011). The maximum disjoining pressure depends on the details of the fluid and its confining surfaces, as will be further described in the last part of this chapter.

Fracturing around expanding grains As demonstrated by Jamtveit et al. (2008), the swelling associated with olivine hydration may cause fracturing and a concomitant permeability increase (Fig. 6). In the upper 10 km of the Earth’s crust, olivine hydration may produce stresses exceeding 300 MPa, greater than that required to fracture rocks, overcome the compressive stress, and cause frictional failure (Kelemen and Hirth 2012). However, in a ductile matrix, stresses may be released through creep processes that are not necessarily associated with significant permeability generation. Thus, serpentinization of olivine may be a more effective mechanism in generating fluid pathways in a rock when the olivine grains are surrounded by a brittle matrix than in a rock containing an increasing amount of mechanically weak serpentine minerals that may accommodate volume changes in a ductile manner. Without this fracturing, grains that were not connected to a pre-existing fracture network would be virtually inaccessible to fluids. The reaction would then not be able to proceed, even though, thermodynamically, the olivine grains are out of equilibrium with the fluids that percolate through the rock.

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Figure 6. Micrograph of reaction-driven fracturing around partly serpentinized olivine crystals in a plagioclase matrix from a troctolite from the Duluth Igneous Complex. A dense network of microfractures connects individual olivine crystals and provides permeable pathways for fluid migration. Small olivine grains in unfractured regions are virtually unaltered [Reprinted from Jamtveit B, Malthe-Sørenssen A, Kostenko O, Reaction enhanced permeability during retrogressive metamorphism, Earth and Planetary Science Letters, Vol. 267, p. 620–627, Copyright (2008), with permission from Elsevier].

Intragrain fracturing In nature, some volume-expanding reactions are very efficiently shut down by the formation of an impenetrable layer of precipitated mineral on the surface of the reactive grain (Prieto et al. 2003). The newly formed layer will be in equilibrium with the surrounding fluid, and the reactive grain is inaccessible to the fluids. However, it is common for volume-expanding reactions to proceed to near or full completion. This may happen if the reaction causes fracturing of the reactive grain. Fracturing exposes new reactive surface and allows access to unreacted parts of the grain, thereby greatly accelerating the process. The fracturing of the reacting mineral grain is essential for the progress of any volume-increasing reaction, including almost all volatilization reactions. Without the continuous generation of new reactive surface area by fracturing, such reactions will normally result in the formation of a passivating layer of product phase and very limited reaction progress. Replacement reactions mediated by a fluid phase take place through a coupled dissolution–precipitation reaction (Putnis 2002). When fluid comes into contact with the reactive mineral, dissolution takes place at the mineral surface. This immediately creates a supersaturated solution with respect to the replacing mineral, which will then precipitate in the immediate vicinity of the dissolving surface. A nanometer-scale confined fluid film between the parent and daughter phases will allow the reaction to continue. If the supersaturation with respect to the precipitating phase is high, there will be enough chemical energy available for the precipitating material to exert a mechanical stress on the reacting grain, and cause fracturing, when precipitation is confined within dissolution pits and wedges as illustrated below. Experimental works that demonstrate this include the replacement reactions: aragonite  calcite (Perdikouri et al. 2011), leucite  analcime (Putnis et al. 2007; Jamtveit et al. 2009),

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scolecite  tobermorite (Dunkel and Putnis 2014), and olivine  serpentine (Malvoisin et al. 2012). All of these reactions are associated with a volume increase, and all of them produce fresh surface area by reaction-driven fracturing in unconfined hydrothermal experiments. Figure 7 shows the interface morphology formed during the replacement of the zeolite mineral scolecite (CaAl2Si3O10·3H2O) by tobermorite (Ca5Si6O17(OH)2·5H2O) after 3 days in a 2M-NaOH solution at 200 ºC (Dunkel and Putnis 2014). During the replacement, tobermorite precipitates in dissolution pits formed at the scolecite surface. These pits develop into wedgeshaped cavities. When fibrous tobermorite grows from the supersaturated solution towards the scolecite ‘walls’, it exerts a stress on them, and the tips of the wedge-shaped pits act as stress concentrators and drive the growth of fractures into the parent phase. These fractures expose fresh reactive surfaces, which allow the process to repeat itself.

Figure 7. Scolecite reacting to form tobermorite. a) Fractures into scolecite emanating from dissolutioncontrolled, wedge-shaped depressions on the scolecite surface. b) Radial growth of fibrous tobermorite precipitating in etch pits formed on the reacting scolecite surface. [Modified from Dunkel and Putnis 2014.]

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A similar process has been shown to operate during serpentinization of olivine. The interface between olivine and the hydrated product is always sharp, even at the nanometer scale. However, the interface is always rough with extensive pitting during incipient olivine dissolution. An amorphous proto-serpentine phase precipitates in these pits and causes local stress concentration and fracturing (Plümper et al. 2012). Pitting thus prepares fracturing during serpentinization, in a way similar to the case for the scolecite–tobermorite replacement reaction. When fracturing starts, the permeability of the system increases and when water gains access to fresh fracture surfaces, more pits are formed by dissolution, allowing new fractures to nucleate and grow producing the observed hierarchical fracture pattern associated with the commonly observed mesh texture that form during partial serpentinization of olivine crystals. The examples above describe growth in cavities produced by dissolution processes that are directly coupled to local precipitation. In the following, we will examine reaction effects on the porosity and permeability of a medium with a significant initial porosity.

Growth in pores If a fluid that flows through a network of pores becomes supersaturated with respect to a solid phase due to dissolution of reactive minerals, changes in temperature or pressure, or for any other reason, precipitation may take place in the open pore space. If pores become filled with solid material, this will reduce the permeability of the rock. However, when the driving force for crystallization, i.e., the supersaturation or undercooling of the fluid phase, is sufficiently high, it will be energetically favorable for the system to continue precipitation of solid material even after the crystal has grown to fill its available pore space. By exerting a mechanical stress on the pore wall transmitted through the disjoining pressure of the confined-fluid film, the growing crystal can make the pore expand elastically and thus make room for continued growth. If stresses become high enough, this will cause fractures to form, and thus new fluid pathways are opened (Fig. 8). Fracturing caused by the pressure exerted by mineral growth in porous rocks is a serious issue in a broad range of Earth and Environmental sciences, including conservation science, geomorphology, geotechnical engineering, and concrete materials science (Scherer 1999; Flatt et al. 2014). The ability of growing crystals to lift imposed loads has been demonstrated in classical experiments (Becker and Day 1905; Correns 1949); see also Flatt et al. (2007) and Taber (1916, 1929). Crystallization pressures that exceed local failure thresholds are thought to be the key process responsible for the evolution of damage during salt weathering (Scherer 1999; Espinosa Marzal and Scherer 2008), frost heave in soils (Dash et al. 2006), and frost cracking of rocks (Walder and Hallet 1985; Murton et al. 2006). It may also lead to vein formation (Fletcher and Merino 2001; Røyne et al. 2011b) and displacive fabrics in the neighboring minerals (Watts 1978). A crystal growing in a pore will stop growing when the stress on the crystal surface approaches the maximum crystallization pressure. However, if the stress is sufficient to open a fracture, the stress on the crystal surface will decrease, thus enabling further growth. When strain rates are slow, as can be the case during precipitation in pores and cracks, fracture propagation takes place through a kinetic process known as subcritical crack growth (Atkinson 1987). Røyne et al. (2011b) showed how to couple the rates of crystal growth and fracture propagation when a crystal grows from a supersaturated solution inside the aperture of a fracture (see Fig. 9). As long as there is an unlimited supply of supersaturated solution, fracture propagation causes the stress on the crystal surface to decrease and the rate of fracture propagation will accelerate until complete failure takes place.

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Figure 8. Secondary electron image of the surface of fresh (A) and weathered (B) andesite. Note the large subspherical pore in A. Dark arrows in B indicate inferred pre-existing pores that are now filled with a fine grained mixture of ferrihydrite and calcite. White arrows indicate microfractures at grain boundaries, inferred to have formed during growth in the pre-existing pores [Used with permission from John Wiley and Sons, from Jamtveit B, Kobchenko M, Austrheim H, Malthe-Sorenssen A, Røyne A, Svensen H (2011) Porosity evolution and crystallization-driven fragmentation during weathering of andesite, Journal of Geophysical Research-Solid Earth, Vol. 116, B12201, Fig. 5.]

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1

0.8

V*, p*

II III

0.6 I 0.4 V

0.2

2w c a

0

0

5

10

15

20

t*

Figure 9. Insert: Conceptual model of a crystal (of radius a) growing in the aperture of a penny-shaped fracture (radius c, maximum opening w). A confined-fluid film between the crystal surface and the fracture walls sustains the continued crystal growth as long as the normal stress, , is smaller than the maximum disjoining pressure of the fluid film. Graph: Dimensionless normal stress on the crystal surface (normalized to the maximum crystallization pressure: dashed line) and equivalent pressure (the fluid pressure inside the penny-shaped fracture that would create the same driving force for fracture propagation: solid line) as a function of dimensionless time for a given set of crystal growth and fracture propagation parameters. The dash-dot line shows the stress on the crystal face that would develop if fracture propagation had not initiated at point II. At I, the crystal has grown to fill the entire fracture; at III, lateral crystal growth can no longer keep up with the fracture propagation rate, causing the stress on the crystal face to be larger than the equivalent fluid pressure. [Used with permission from Røyne A, Meakin P, Malthe-Sørenssen A, Jamtveit B, Dysthe DK (2011) Crack propagation driven by crystal growth. EPL, Vol. 96, 24003, doi:10.1209/02955075/96/24003].

In nature, the stress generated during crystal growth in the pores of a rock depends on the properties of the fluid film confined between the crystal surface and the pore wall as well as on the continued supply of supersaturated solution through transport in the fluid phase. We will discuss these issues in more details in the following two sections.

FUNDAMENTAL PROPERTIES OF CONFINED FLUID FILMS As we have shown in the preceding sections, the processes that modify the porosity and permeability of rocks on the pore scale depend critically on the nature and presence of confined fluid films present in microfractures and along grain boundaries. When fluids are confined at reactive grain boundaries, they play a critical role in determining the force that is exerted by a growing crystal on its surroundings (Taber 1916; Espinosa Marzal and Scherer 2008) and whether a stressed grain boundary will heal or remain open (Renard et al. 2012; Houben et al. 2013). Confined fluids form transport pathways through low-permeability rocks (Alcantar et al. 2003), and fluids at grain boundaries also control macroscopic elastic (Tutuncu and Sharma 1992; Schult and Shi 1997) and yield properties (Risnes and Flaageng 1999; Megawati et al. 2013). For rocks that are stressed near failure, fluids confined at fracture tips control the fracture propagation threshold (Clarke et al. 1986; Røyne et al. 2011a).

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It is extremely difficult to make direct measurements of the properties of nanometer-thick fluid films between mineral grains, but there are elegant ways in which the effects of such confined films can be observed. One of the most simple, yet highly illustrative, examples is the experiment by Bouzid et al. (2011). A solution of NaCl was introduced in a micrometric glass capillary, open on both ends (Fig. 10). As water evaporated to the dry external atmosphere, the solution gradually became supersaturated with respect to halite. After some time, solid halite crystals nucleated on both air–liquid interfaces. The crystals continued to grow until they completely filled the capillary diameter, apparently clogging the tube. The system was then left undisturbed for three months. At this point, the authors observed that a small bubble had formed inside the fluid that was trapped between the halite cylinders. Such vapor cavities, which had nucleated inside the bulk liquid, can only form when the pressure in the liquid has decreased substantially below zero. Instead of the halite crystals completely shutting off the transport pathway between the fluid in the capillary and the outside atmosphere, a fluid film must have persisted between the crystal and the glass wall. Water could continue to evaporate from the surface of this film, continuously pulling water out from the reservoir inside. Despite the presence of the water film, the crystals were not mechanically free to move inwards; instead, the depletion of water caused the pressure of the trapped solution to decrease. The wetting of the halite–glass interface was strong enough to prevent the gas–liquid interface from receding towards the middle of the capillary. In due course, the bulk fluid inside the capillary became thermodynamically unstable, and nucleation of vapor-filled cavities occurred. In summary, evaporation of water caused this, initially extremely simple, system to follow a complex pathway: 1) increased concentration of sodium chloride; 2) the first phase transition, with nucleation and growth of salt crystals in the regions of highest salt concentration, the air– water interfaces; 3) decrease in fluid pressure; and 4) the second phase transition, nucleation of a vapor bubble. Given the complexity that resulted from this very simple setup, it is no surprise that the coupling between transport and reactions on different scales may lead to a variety of patterns in geological systems.

NaCl

NaCl

Figure 10. Halite crystals (dark grey), trapping a saturated solution of NaCl (light grey) in which a vapor bubble has formed. Note that the space between the halite crystals and the walls of the capillary is highly exaggerated in order to illustrate the negative curvature of the air–liquid surface. [Modified from Bouzid et al. 2011.]

The disjoining pressure of confined fluid films The example described above illustrates how fluid films can persist and allow slow fluid transport, even in systems that seem to be completely clogged. Importantly, fluid films can persist even when their confining surfaces are squeezed together with a significant pressure, due to externally imposed stress or due to the stress generated during growth of a mineral. We will now address the conditions that allow fluid films to persist under compressive stress, starting with the fundamental thermodynamics. Because atoms that form part of the surface of a material have fewer neighbors than those in the bulk, there is an excess free energy associated with all surfaces, called the surface energy, , of the material. This is also the energy that needs to be added to the system in order to create one unit area of new surface.

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The intrinsic surface energy of a material, , may be defined as the excess free energy per unit area of a surface under vacuum conditions. However, in most applications, a surface of one material will be interacting with another gas, fluid, or solid phase. The measurable quantity is then not the intrinsic surface energy of the material, but rather the interfacial energy of phases 1 and 2 in contact, 12. The concept of interfacial energies allows us to analyze the thermodynamics of a fluid sandwiched between two solid surfaces. Consider a system consisting of two semi-infinite, parallel solid surfaces of materials 1 and 2, separated by a distance h in a fluid medium 3 (Fig. 11), with total interfacial energy U(h). If the surface separation is large enough to prevent any interaction between the solid surfaces, the energy of the system is U(∞) = 13 + 23. Upon bringing the solids into dry contact, the energy becomes U(0) = 12.

J23 h J13

J23 J13

P(h)

h

J12

Figure 11. Surface energies and surface forces. Left: solid 1 and solid 2 separated by a thick film of liquid (3), U     13   23 . Right: solid–solid contact, with energy U  0   12 . Middle: solids separated by a thin film, U  h   13   23  P  h  .

As the film thickness decreases continually towards zero, the energy does not jump discontinuously from 13 + 23 to 12. Instead, at sufficiently small separations, the interaction between the solid surfaces across the confined liquid film gives rise to an additional energy contribution, P(h). We may then write the energy of the system as U(h) = 13 + 23 + P(h), where P(∞) = 0 and P(0) = 12 - (13 + 23). When the surface separation changes, the change in interfacial energy gives rise to a measurable force per unit area: F d U (h) d P (h)   . ¨ d d

(1)

This force, which may be attractive or repulsive, is referred to as a surface force. The corresponding pressure is called the disjoining pressure of the thin film (de Gennes et al. 2003; Israelachvili 2011). In the foregoing sections, where we have discussed the stability of fluid films, we have implicitly referred to repulsive disjoining pressures. Forces between solids surfaces in a fluid medium arise from a number of processes, many of which are not yet properly understood. The most well-known theory, named DLVO theory after Derjaguin and Landau (1941) and Verwey and Overbeek (1948), contains two contributions to the surface forces. The first is the van der Waals force, which is a function of the polarizabilities of the materials involved, and is characterized by the Hamaker constant, AH , of the interfacial system. Values for the Hamaker constant of a range of surfaces in air and water are available in the literature (Bergström 1997; Israelachvili 2011). For symmetric systems, where an interfacial layer separates two surfaces of the same material, the van der Waals force is always attractive, but for asymmetric systems, such as the ice–water–air

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interface, it may be repulsive (Dash et al. 2006). For two flat surfaces, the van der Waals energy U vdW is given by (Israelachvili 2011) U vdW  

AH . 12 h2

(2)

The corresponding force is given by the derivative of this function. Relations for other surface geometries are given in Israelachvili (2011). The second contribution arises due to the overlap of the electric double layers associated with charged surfaces. Most solid surfaces become charged in liquid environments. The electric double layer interaction energy, U EDL , between planar surfaces depends exponentially on the separation distance. For symmetric interfaces, monovalent electrolytes and surface potentials below about 25 mV, U EDL can be found in terms of the surface potential  or surface charge  as (Israelachvili 2011): U EDL 

2 0  2  h /  D 2 D 2  h /  D  0e  e . D 0

(3)

Here, D is the Debye screening length, which is a function of the ionic strength of the electrolyte. In pure water, D is close to 1 m, while in concentrated solutions it is on the order of a few tenths of a nanometer. The DLVO theory has been experimentally validated using the Surface Forces Apparatus (SFA) (Israelachvili and Adams 1978; Israelachvili 2011), and more recently also with colloidal probe Atomic Force Microscopy (AFM) (Butt et al. 2005), in a range of systems. However, despite its advantages, the DLVO theory is not sufficient to predict the full interaction of a given pair of surfaces. One reason is that in high electrolyte concentrations, specific ion effects that are not accounted for in the DLVO framework become important (Boström et al. 2001). Also, even for moderate electrolyte concentration, one will need to know the surface charge or surface potentials; these are parameters that depend on the pH and ionic concentration of the pore fluid for any given mineral surface. There is a clear need for more data on this for geologically relevant systems, and progress is being made. Very recently, the atomic force microscope (AFM) has evolved to such an extent that it is possible to image the adsorption and dynamics of ions on a mineral surface (Ricci et al. 2013; Siretanu et al. 2014). This is giving us new insight into the complex processes of hydration and the electric double layer formation, and through this an exciting possibility to obtain a better understanding of wetting and surface interactions in geological systems. At small separations, approaching a few molecular diameters, the continuum DLVO theory breaks down and other forces, that might be orders of magnitude larger than those described by DLVO, come into play. These forces depend on the molecular structure of the surfaces and intervening fluid, and include the hydrophobic attraction, hydration or hydrophilic repulsion, oscillatory solvation forces, ion correlation forces, and others (Israelachvili 2011). In geological systems, where large pressures can be expected, these forces may be the most important ones (Alcantar et al. 2003; Anzalone et al. 2006). Unfortunately, the theoretical framework for predicting these forces accurately is still lacking (although semi-empirical relations exist for specific cases such as hydrophobic attraction and hydration repulsion (Donaldson et al. 2014)). Only more recently has attention been turned to high electrolyte concentrations. These systems are more complex, but their behavior is still consistent with the molecular picture that has been obtained for the lower solution concentrations (Baimpos et al. 2014). Interestingly, at these concentrations adhesive interaction forces are found to be largely due to solute

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and solvent correlation forces (Lesko et al. 2001; Espinosa-Marzal et al. 2012; Baimpos et al. 2014). Ion correlation forces have been suggested to play a key role in controlling the cohesion of cement (Lesko et al. 2001), and we may speculate that they are a critical factor in determining the cohesive properties of natural rocks as well.

Transport in confined fluid films When confined films become very thin, fluid and molecular transport becomes highly surface specific. The fluid viscosity may approach that of a solid-like material (Ruths and Israelachvili 2010), while on the other hand, diffusive ion transport may be significantly enhanced (Duan and Majumdar 2010). For thicker films, it is possible to make some generalizations about their properties. For instance, experiments typically show that the viscosity of a confined aqueous fluid film does deviate significantly from the bulk value until the film thickness is only a few molecular diameters (Horn et al. 1989). The diffusive properties of the film are also close to bulk values, and films as thin as 3–5 molecular diameters have been found to have a diffusivity that is less than one order of magnitude lower than that in bulk water (Alcantar et al. 2003). This implies that, in most cases, confined fluid films can be treated essentially as bulk fluids in terms of transport properties. However, the large surface-to-fluid ratio of confined fluid films can give rise to surfaceor fluid-specific properties that should not be ignored. For instance, the charge and wetting properties of the pore walls can significantly affect both advective and diffusive transport properties (Wang 2014). Since molecular species are affected in different ways by the properties of the pore walls, diffusion may cause individual species to become either depleted or enriched relative to that the bulk solution (Roach et al. 1988; Heidug 1995; Bresme and Cámara 2006). Natural mineral interfaces typically display some degree of roughness on the nanoscale. If a normal force is applied across such a boundary it will lead to gradients in the disjoining pressure in the confined fluid. In this case, the process known as pressure diffusion can cause solutes of smaller molecular volume to flow in the direction of the pressure gradient, and therefore towards regions of decreasing grain boundary width where the disjoining pressure is large (Heidug 1995). Because of the complexity of reaction-driven fracturing in geosystems, much remains to be understood and discovered. Nanoscale experiments and modeling will play an important role in the development of a comprehensive understanding of the coupling between dissolution, precipitation, and transport, as well as how these processes are coupled with deformation and fracturing.

INTERFACE-DRIVEN TRANSPORT ON THE PORE SCALE The transport of material through a rock is not governed by its pore structure and permeability alone, but also by the driving forces for fluid migration. In the systems that we are discussing here, flows driven by differences in interfacial energies form an important class of transport phenomena. We will first discuss flow driven by the contact between a wetting fluid and a non-wetting fluid or gas phase. This is important during weathering of rocks near the Earth’s surface, but also during water flooding of oil reservoirs and CO2 injection into saturated rocks. We limit the discussion here to a few cases that are particularly relevant for the coupling between transport and precipitation; a more comprehensive review on the physics of pore-scale multiphase and multicomponent transport has been given by Steefel et al. (2013). The following can be applied to any system containing two immiscible fluids where one is more wetting than the other. The difference in wetting properties is a result of the difference in

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Figure 12. Creeping of a saturated NaCl solution, first as a thin layer of crystals on the side of the beaker (left) followed by secondary creeping on the first structure that formed, leading to a porous mass of crystals (right). [Reprinted with permission from (Bouzid M, Mercury L, Lassin A, Matray JM (2011) Salt precipitation and trapped liquid cavitation in micrometric capillary tubes. Journal of Colloid and Interface Science, Vol. 360, p. 768–776, doi:10.1016/j.jcis.2011.04.095). Copyright (2011) American Chemical Society.]

solid–liquid interfacial energies, which are much smaller for wetting than for the non-wetting fluids. Where the two fluid phases and the solid phase meet, the angle between the surface of the solid and that of the wetting fluid will be close to zero degrees. When confined inside a narrow pore or slit, the geometry imposed by the walls will cause the surface of the wetting fluid to curve inwards with a radius given by the pore opening. The drive in the system to minimize surface area will then manifest itself as a capillary pressure, pulling the wetting liquid towards the interface. In a vertical capillary tube, the balance between the capillary pressure and gravity determines the height to which the fluid will rise inside the tube. In a water-wetting, oil-saturated reservoir rock, capillary pressure will pull the water into the pores of the rock and the oil will be pushed out. In the experiment of Bouzid et al. (2011), the substantially negative capillary pressure at the air-water interface at the exit of the halite–glass channel caused fluid to be pulled out of the fluid reservoir between the halite crystals. While mineral growth in pores can severely restrict pressure-driven fluid flow, it can also, in some cases, accelerate interface-driven fluid transport. A good example of this is the phenomenon of creeping salts (van Enckevort and Los 2013). If a salt solution is left in an open beaker in the lab, then, for some salts, one can return days later and find a crust of salt crystals covering the walls of the beaker all the way to the top, sometimes even down on the other side of the beaker and onto the benchtop—with most of the liquid solution gone (see Fig. 12). What has happened is that salt crystals have precipitated at the location where supersaturation is reached first, which is at the contact line between the salt solution and the beaker wall. Because water readily wets the salt crystals, the salt solution will climb up to the top of the newly precipitated material, where it again becomes supersaturated due to evaporation, which leads to more precipitation. With time, a porous structure builds upwards, allowing the salt solution to climb out of the beaker. The upward fluid flow and enhanced evaporation that is created by this crystallization leads to accelerated drying of the salt solution. A similar effect has been shown for salt crystallization due to evaporation in a hydrophobic porous medium (Sghaier et al. 2014). In the absence of free boundaries (fluid–gas or fluid–fluid interfaces), there is yet another interfacial driving force that may drive fluid transport in porous systems, and that is the thermomolecular flow, where a temperature gradient generates a gradient in the disjoining pressure. This is now understood to be the main driving mechanism for frost heave, which is

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the displacive growth of ice lenses in porous soils (Wettlaufer and Worster 2006), and at least some instances of frost cracking of intact rocks (Murton et al. 2006). Although water expands when it freezes to ice, this is in most cases not sufficient to form fractures and the extents of frost heave that are observed in nature. Instead, water is supplied to the freezing front from unfrozen parts of the material by the thermomolecular flow, which causes fluid to flow from warmer to colder temperatures. The fluid flow and build-up of ice will cease when the pressure on the ice lens from the displaced overburden is enough to balance the thermomolecular pressure. It is this continued supply of water from neighboring parts of the material that creates sufficient volume expansion to create fractures and damage. In the case of the ice lens, the freezing of ice acts as a sink, helping to sustain the fluid flow. A mineral that grows in a pore will also deplete the solute concentration around it and cause diffusion solute from the surrounding reservoir. If the supersaturation of the fluid was initially uniform, and crystal nucleation was to take place in all pores simultaneously, then the supersaturation would soon be consumed without any significant build-up of solid material. However, in sufficiently small pores, crystals may be inhibited from precipitating even at high levels of supersaturation. This is due to the energy penalty associated with the large surfaceto-volume of crystals confined inside a small volume of a different material. As a rule of thumb, the solubility of a salt crystal in a pore increases significantly in pores that are below 1 μm in size (Steiger 2005b). In a rock that contains a distribution of small and large pores, the small pores may act as reservoirs for supersaturated or subcooled fluid that feed the growth of crystals in larger pores. The pore-size distribution and connectivity can therefore have an important effect on the spatial distribution of precipitated material (Emmanuel and Berkowitz 2007), as well as on the damage of the material due to crystallization pressures (Scherer 1999; Steiger 2005b).

CONCLUDING REMARKS By zooming in from the field scale to the pore and interface scales, we have shown that whenever fluid-driven reactions involve positive volume changes, the reaction will be shut down unless some mechanism ensures continued supply of fluid to the reactive surfaces. This requires a percolating network of fluid channels. Fluid supply is normally maintained through fractures or pore networks with apertures exceeding micrometer size. However, the transport to the reacting surfaces often takes place through nanometer-scale fluid films. These films can often sustain a significant normal stress without being squeezed out. For reactions to generate new fractures, which is often necessary to get access to the interior of reacting grains or to grains that are embedded inside a tight matrix, a significant overstepping of the relevant reaction is required. When crystals precipitate from a supersaturated solution, the growth process may elastically displace the confining surfaces. When the elastic strain reaches some critical value, this can result in fracture growth and the opening of new fluid pathways. The energy needed for the creation of new surfaces is thus taken from the chemical energy available in the reaction. Even without fracturing, the coupling between reaction and transport in porous reactive rocks is highly complex. In order to better understand what determines the rates of advance of reaction fronts, whether reactions will come to a halt or not, and the evolution of the permeability of the rock, we need a better understanding of forces, transport and reaction kinetics under nanoscale confinement.

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Røyne & Jamtveit ACKNOWLEDGMENTS

This project was supported by a Center of Excellence grant to PGP, and a FRINATEK postdoc grant 222300 to AR, both from the Norwegian Research Council. BJ was supported by an Alexander von Humboldt Research Award from the German Alexander von Humboldt Foundation, and part of this work was carried out at the Department of Mineralogy at the University of Münster. We benefitted from comments and discussions with Andrew Putnis, Francois Renard, Paul Meakin, and Carl Steefel. Figures 3 and 4 were prepared by Ole Ivar Ulven at PGP.

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Risnes R, Flaageng O (1999) Mechanical properties of chalk with emphasis on chalk–fluid interactions and micromechanical aspects. Oil Gas Science Tech 54:751–758 Roach DH, Lathabai S, Lawn BR (1988) Interfacial layers in brittle cracks. J Am Ceram Soc 71:97–105 Røyne A, Jamtveit B, Mathiesen J, Malthe-Sørenssen A (2008) Controls on rock weathering rates by reactioninduced hierarchical fracturing. Earth Planet Sci Lett 275:364–369, doi:10.1016/j.epsl.2008.08.035 Røyne A, Bisschop J, Dysthe DK (2011a) Experimental investigation of surface energy and subcritical crack growth in calcite. J Geophys Res 116:B04204, doi:10.1029/2010jb008033 Røyne A, Meakin P, Malthe-Sørenssen A, Jamtveit B, Dysthe DK (2011b) Crack propagation driven by crystal growth. EPL (Europhys Lett) 96:24003, doi:10.1209/0295-5075/96/24003 Rudge JF, Kelemen PB, Spiegelman M (2010) A simple model of reaction-induced cracking applied to serpentinization and carbonation of peridotite. Earth Planet Sci Lett 291:215–227 Ruths M, Israelachvili JN (2010) Surface Forces and Nanorheology of Molecularly Thin Films. In: Springer Handbook of Nanotechnology. Bhushan B (ed) Springer Berlin Heidelberg, pp. 857–922 Scherer GW (1999) Crystallization in pores. Cem Concr Res 29:1347–1358 Schult A, Shi G (1997) Hydration swelling of crystalline rocks. Geophys J Int 131:179–186 Sghaier N, Geoffroy S, Prat M, Eloukabi H, Ben Nasrallah S (2014) Evaporation-driven growth of large crystallized salt structures in a porous medium. Phys Rev E 90, doi:10.1103/PhysRevE.90.042402 Siretanu I, Ebeling D, Andersson MP, Stipp SL, Philipse A, Stuart MC, van den Ende D, Mugele F (2014) Direct observation of ionic structure at solid-liquid interfaces: a deep look into the Stern Layer. Sci Rep 4:4956, doi:10.1038/srep04956 Steefel CI, Molins S, Trebotich D (2013) Pore scale processes associated with subsurface CO2 injection and sequestration. Rev Mineral Geochem 77:259–303 Steiger M (2005a) Crystal growth in porous materials—I: The crystallization pressure of large crystals. J Cryst Growth 282:455–469 Steiger M (2005b) Crystal growth in porous materials—II: Influence of crystal size on the crystallization pressure. J Cryst Growth 282:470–481 Taber S (1916) The growth of crystals under external pressure. Am J Sci 41:532–556 Taber S (1929) Frost heaving. J Geol 37:428–461 Tutuncu AN, Sharma MM (1992) The influence of fluids on grain contact stiffness and frame moduli in sedimentary rocks. Geophysics 57:1571–1582 Ulven OI, Jamtveit B, Malthe-Sørenssen A (2014a) Reaction driven fracturing of porous rocks. J Geophys Res:doi:10.1002/2014JB011102 Ulven OI, Storheim H, Austrheim H, Malthe-Sørenssen A (2014b) Fracture initiation during volume increasing reactions in rocks and applications for CO2 sequestration. Earth Planet Sci Lett 389:132–142, doi:10.1016/j.epsl.2013.12.039 van Enckevort WJP, Los JH (2013) On the creeping of saturated salt solutions. Cryst Growth Des 13:1838– 1848, doi:10.1021/cg301429g Verwey EJW, Overbeek JTG (1948) Theory of The Stability of Lyophobic Colloids. Elsevier, Amsterdam Walder J, Hallet B (1985) A theoretical-model of the fracture of rock during freezing. Geol Soc Am Bull 96:336–346 Wang Y (2014) Nanogeochemistry: Nanostructures, emergent properties and their control on geochemical reactions and mass transfers. Chem Geol 378-379:1-23, doi:10.1016/j.chemgeo.2014.04.007 Watts NL (1978) Displacive calcite: Evidence from recent and ancient calcretes. Geology 6:699–703 Wettlaufer JS, Worster MG (2006) Premelting dynamics. Annu Rev Fluid Mech 38:427–452

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 45-60, 2015 Copyright © Mineralogical Society of America

Effects of Coupled Chemo-Mechanical Processes on the Evolution of Pore-Size Distributions in Geological Media Simon Emmanuel Institute of Earth Sciences The Hebrew University of Jerusalem Edmond J. Safra Campus Givat Ram, Jerusalem, 91904 Israel [email protected]

Lawrence M. Anovitz MS 6110 PO Box 2008 Geochemistry and Interfacial Sciences Group Oak Ridge National Laboratory Oak Ridge, Tennessee 37831-6110, USA

Ruarri J. Day-Stirrat Shell International Exploration and Production Inc. Shell Technology Center Houston R1004B 3333 Highway 6, South Houston, Texas 77082, USA INTRODUCTION The pore space in rocks, sediments, and soils can change significantly as a result of weathering (see Navarre-Sitchler et al. 2015, this volume), diagenetic, metamorphic, tectonic, and even anthropogenic processes. As sediments undergo compaction during burial, grains are rearranged leading to an overall reduction in porosity and pore size (Athy 1930; Hedberg 1936; Neuzil 1994; Dewhurst et al. 1999; Anovitz et al. 2013). In addition, geochemical reactions can induce the precipitation and dissolution of minerals, which can either enhance or reduce pore space (e.g., Navarre-Sitchler et al. 2009; Emmanuel et al. 2010; Stack et al. 2014; Anovitz et al. 2015). During metamorphism too, mineral assemblages can change, altering rock fabrics and porosity (Manning and Bird 1995; Manning and Ingebritsen 1999; Neuhoff et al. 1999; Anovitz et al. 2009; Wang et al. 2013). As the pore space in geological media strongly affects permeability, evolving textures can influence the migration of water, contaminants, gases, and hydrocarbons in the subsurface. Although models—including the Kozeny–Carman equation (Kozeny 1927; Bear 1988)— exist to predict the relationship between porosity and permeability, they are often severely limited, in part because little is known about how pore size, pore geometry, and pore networks evolve in response to chemical and physical processes (Lukasiewicz and Reed 1988; Costa 2006; Xu and Yu 2008). In the case of geochemical reactions, calculating the change in total porosity due to the precipitation of a given mass of mineral is straightforward. However, predicting the way in which the precipitated minerals are distributed throughout the pores remains a non-trivial challenge (Fig. 1; Emmanuel and Ague 2009; Emmanuel et al. 2010, Hedges and Whitlam. 2013; Wang et al. 2013; Stack et al. 2014; Anovitz et al. 2015). 1529-6466/15/0080-0003$05.00

http://dx.doi.org/10.2138/rmg.2015.03

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t1 t0

Figure 1. Schematic representation of the effect on pore-size distribution of mineral growth. As pores shrink, the distribution shifts to the left but the peaks also shrink, resulting in the reduction in overall porosity.

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Similarly, the impact of mechanical processes on the development of pore-size distributions and the anisotropy of pore networks has yet to be resolved to a satisfactory level (Day-Stirrat et al. 2008a,b, 2010). An important step towards understanding how pores evolve in different geological scenarios is to develop mathematical frameworks that can describe the effects of geochemical reactions and mechanical compaction on porous rock matrices. Here, we present three different phenomenological models that describe the effect of geochemical reactions and mechanical processes on the evolution of pore-size distributions in geological media. As both of these processes can alter pore size, we also develop a framework for coupling the geochemical and mechanical modes of evolution. In addition, we review some of the attempts to use these models to simulate actual field and laboratory data, and explore some of the open questions and directions for future research.

PORE-SIZE EVOLUTION DURING MINERAL PRECIPITATION AND DISSOLUTION Mineral precipitation typically reduces the size of voids while dissolution increases them, and the use of powerful computational methods has enabled such processes to be simulated directly at the pore scale. Lattice-Boltzmann models have been used to map the evolution of the solid–fluid interface (e.g., Kang et al. 2002, 2005, 2007; Huber et al. 2014), while pore network models—which replace the complex geometry of the pore space with a network of interconnected pores and channels—have provided insight into the scale dependence of geochemical rate laws (Li et al. 2006; Meakin and Tartakovsky 2009). However, although porescale models can be powerful tools, they are unable to resolve processes that operate at scales smaller than the grid size. In many rocks, pore sizes are highly multiscalar, ranging from the nanometer scale up to hundreds of microns or larger; modeling processes that simultaneously occur at these different scales represents a major challenge. In addition, there is significant evidence (Rother et al. 2007; Anovitz et al. 2013b; Hedges and Whitelam 2013; Kolesnikov et al. 2014) that as pores become smaller the physical properties of the fluids contained within them are strongly altered, further complicating matters. In any given representative volume there may be many millions of individual voids of varying size and shape. Thus, if geochemical processes are size-dependent, it may be more practical to adopt an approach that attempts to simulate the evolution of pores across a wide range of spatial scales. One approach that couples a statistical description of pore size with a mechanistic view of their evolution involves using a partial differential form of a continuity equation to track temporal changes to pores and pore-size distributions (Or et al. 2000; Leij et al. 2002). In its simplest geochemical formulation, the equation effectively tracks changes to the pore size frequency as a result of precipitation or dissolution:   vfr  fr  , t r

(1)

where fr is a function describing the number of pores, characterized by a radius r, per unit volume of rock, sediment, or soil; the variable can be thought of as akin to a probability density function for pore size. In this framework the parameter v is the rate of change of pore size in units of distance per unit time and represents the rate of contraction or growth of the pores. This rate parameter is dependent on the mineral being precipitated or dissolved, as well as environmental conditions such as solution chemistry, temperature, and pressure, and the parameter can be dependent on pore size. This framework represents a much simplified version of other models describing pore size evolution (Or et al. 2000; Leij et al. 2002) in that it assumes that the total number of pores is conserved and that pores can only change as a result of geochemical processes. As a result, additional diffusional and sink terms are not included.

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In the simplest scenario, v is a constant, and it can be related to a standard expression for the geochemical reaction rate by (Lasaga 1998)





b

v    m k a  1 ,

(2)

where vm is the molar volume of the precipitating or dissolving mineral, k is a rate constant,  is the bulk degree of saturation (i.e., the ratio of the ion activity product to the solubility product), and a and b are positive exponents. In this convention if the sign of v is positive, dissolution occurs and pore size increases. Importantly, a constant value for vimplies that the pores of all sizes respond to geochemical conditions in the same way so that all pores contract or expand at the same rate. Moreover, such behavior implies that precipitation will cause small pores to close first, although during diagenesis some studies have reported that this is not necessarily the case (Putnis and Mauthe 2001; Emmanuel et al. 2010; Anovitz et al. 2013a, 2015) Deviation from a uniform pore size reduction could be due to preferred flow paths and higher reactant fluxes though large pores. Alternatively, the effects of interfacial energy in very small pores could make v size-dependent (Emmanuel and Berkowitz 2007; Emmanuel and Ague 2009; Emmanuel et al. 2010; Hedges and Whitelam 2013; Rother et al. 2007; Anovitz et al. 2013b; Kolesnikov et al. 2014). Such effects will depend both on pore size and pore geometry. For pores with convex walls (see Fig. 2) as opposed to pores with concave walls (e.g., spherical pores) the expression could have the form (Emmanuel and Ague 2009): b

a    2 m  cos     ,  1 v    m k   exp     RTr     

(3)

where  is the interfacial energy,  is the dihedral angle between adjacent pore walls, R is the gas constant, and T is temperature. Crucially, this size-dependent modification effectively reflects the increased solubility of tiny crystals with high curvature. This means that grains with low curvature (large grains) can grow in big pores at the same time that smaller, high-curvature grains are prevented from growing in small pores. However, this effect is only expected to be important in nano-scale pores, or at very low levels of supersaturation. Elevated temperatures, too, may reduce size-dependent effects. For a much more detailed discussion of the solid–fluid

2r

Figure 2. Schematic diagram showing the geometry for simulated crystal growth. Crystal growth occurs on the convex walls of a square cuboid-shaped pore; as the pore is filled during mineralization, the pore size decreases and crystal curvature increases. The dihedral angle is marked as , and the characteristic size of the pore is indicated by the dimension r. Adapted from Emmanuel and Ague (2009).

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interface at the nanometer-scale and its influence on fluid structure and fluid–solid interactions the reader is referred to other papers (e.g., Zhang et al. 2004, 2006; Vlcek et al. 2007; Mamontov et al. 2009; Wesolowski et al. 2012). While Equation (1) possesses the relatively straightforward structure of an advection equation, the variable fr (i.e., the density of pores of a given size) is not a commonly reported parameter, in part due to the difficulty of assigning a representative size to the irregularly shaped pores typically found in natural porous media (Anovitz et al. 2009, 2013a, 2015). Descriptions of the various techniques used to measure pore-size distributions are given by Anovitz and Cole (2015, this volume) and Navarre-Sitchler et al, (2015, this volume). Such methods often report size distributions in terms of the volume of pores of a given size (or, more correctly, in a certain interval of sizes) per volume of sample. Using such data a cumulative porosity curve can be obtained so that the total porosity  in the interval r1 to r2 is given by the following integral: r2

  r dr ,

(4)

r1

where r is the probability density function for pore size multiplied by the total porosity. Practically speaking, r can be calculated from pore size data by numerically differentiating the cumulative porosity curve with respect to pore size (e.g. Anovitz et al. 2015). If the pores of a given size possess a characteristic volume, Vr, then fr = r / Vr, and Equation (1) becomes   fr / Vr    vr / Vr    . t r

(5)

For the simplest case in which the pores are spherical it follows that

  4 r 3  4 r 3    r /   v /   r  3  3      , t r and using the quotient rule it can be shown that the expression simplifies to    vr  3vr  r    . t  r r  

(6)

(7)

The extra term on the right hand side of the equation is a geometric term that effectively serves to increase overall porosity when pores expand, and reduce overall porosity when pores shrink. Similar expressions can be developed for different pore geometries; for example, in the case of cylindrical pores undergoing radial shrinkage it can be shown that the geometric term is 2 vr / r, while for cuboid pores with uniaxial shrinkage the term is vr / r. These equations can be readily solved for a range of initial distributions by numerically solving 1D partial differential equations. In its basic form the model assumes a constant supply of reactants, although in principle it could be coupled to a reactive transport equation that conserves mass. While a fully coupled model would be more realistic, the main way in which pore-size distributions evolve can be shown by considering Equation (7) on its own. When v is constant, the model, as expected, predicts that as pores are filled, smaller pores are the first to be filled. To demonstrate the way pores of different sizes are predicted to change, Figure 3 shows a simulation of a bimodal distribution: the peak initially located at 30 μm is much higher than the peak at 60 μm; however, by the end of the simulation both peaks are approximately equal in height. While

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the v parameter determines the rapidity with which the curves evolve, the model predicts that under constant conditions pore-size distributions will evolve along fixed paths. Although such models contain a number of simplifying assumptions, they can nevertheless provide important insights into the processes governing the evolution of pore-size distributions during mineralization. In a study examining varying levels of quartz cementation in sandstone from the Stø formation in the Barents Sea (Fig. 4; Emmanuel et al. 2010), reduction in pore size was found to be non-uniform across different sizes, so that pores with radii smaller than approximately 10 μm appeared to undergo very little change (Fig. 5a). Fluid flow in the system was thought to be minimal, and the study used a complex diffusion-reaction equation with moving boundary conditions to simulate the evolution of pore sizes. However, Equation (7) on its own can be used to model the evolution of the porous matrix from high porosity to low porosity in a much more straightforward way. Using the pore-size distribution in the least-cemented sample as an initial condition, the constant-v model performs adequately at simulating the shift of the peak initially located at 13.5 μm to ~12.3 μm (Fig. 5b). However, the model also predicts that the peak originally appearing at 8.5 μm should shift to around 7.4 μm and that the peak at 2 μm should disappear altogether. In contrast, in the low-porosity sample, both measured peaks remain unchanged. Significantly, the preferential filling of larger pores has been reported in other studies of sandstones (Putnis and Mauthe 2001; Anovitz et al. 2013a, 2015) and this type of behavior may be relatively common. There are a number of mechanisms that could lead to the preferential filling of larger pores. Clay coatings in small pores may inhibit quartz cementation, and the timing of clay mineral formation, as well as the extent of coverage, may be critical factors. Alternatively, limited connectivity and subsequent limited solute flux in small pores may restrict the supply of dissolved silica available for quartz precipitation (Emmanuel et al. 2010). However, both simulations and measurements indicate that interfacial energy effects could also play an important role (Emmanuel and Berkowitz 2007; Emmanuel and Ague 2009; Emmanuel et al. 2010): when using the size-dependent expression for v in Equation (3), the model produces an excellent match with the measured data (Fig. 5c). Critically, in this simulation only two variables were changed to obtain a good fit: k and  All other parameters were taken from the 0.01

t = 0 ky t = 500 y t = 1000 y

0.009 0.008

−1

Ir [Pm ]

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

10

20

30

40

50

60

70

80

90

100

pore radius [ Pm] Figure 3. Simulation of the temporal evolution of pore-size distributions based on Equation (7). The initial bimodal distribution is based on the sum of two normal distributions. In the simulation v is arbitrarily set to 10 μm ky-1.

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b

500 mm

500 mm

Figure 4. Scanning electron microscope images of sandstone samples from the Stø Formation (Barents Sea). (a) A high porosity (~21%) sample, and (b) a low porosity sample (~12%) with a high degree of quartz cementation. Note that the pores typically have convex walls.

literature and are given in the caption for Figure 5. Importantly, this model suggests that the level of supersaturation during cementation was extremely low with = 1.0004, a value that is similar to that calculated by Emmanuel et al. (2010) using a more complex reactive transport model. At higher levels of supersaturation, pores smaller than 10 μm are also filled, so that the model behavior evolves towards that of the constant-v model. One of the main limitations of the pore-size distribution approach presented here is that although the sizes of the pores can be predicted, their spatial ordering and connectivity are not expressed. As pores get smaller, flow through throats can be restricted, and when throats 10 μm, but predicts that smaller pores will close off. Time in the simulations is arbitrary so that the duration of the simulation is equal to 1 unit of time; the v parameter is 1.2 mm. (c) Simulated distribution using the interfacial energy model of Equation 3 and the measured distribution. The initial condition is the same as that in (b). Note that a much better fit is obtained for pores in the 90º the term is intrinsically negative. An externally applied pressure produces a force that acts over the area of the circle contact and is expressed as D2P/4 where P is applied pressure. At equilibrium, where applied force is equal to the resistance, we have D 2 P D cos   . 4

(10)

Simplifying this equation yields: D

4  cos  . P

(11)

This is known as the Washburn Equation. Assuming a contact angle of 130º and a surface tension of 485 dyne cm-1, it takes a pressure of only 0.5 psi for mercury to enter pores approximately 360 μm in diameter. For smaller pores such as those encountered in tight sands or shale, 60,000 psi pressure can result in mercury accessing pores as small as 3 nm in diameter.

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These examples assume a spherical pore shape that is, of course, an oversimplification. For a case where the rock is dominated by phyllosilicates where slit-like pore openings dominate the Washburn expression can be modified to W

2  cos  , P

(12)

where W is the width between plates. In a typical mercury intrusion porosimetry experiment a dry sample is placed into a container, which is then evacuated to remove contaminant gases and vapors (usually water). While the container is still evacuated mercury is allowed to fill the container. This creates a system that consists of a solid, a non-wetting liquid (mercury), and mercury vapor. In the next step pressure is increased toward ambient. This causes mercury to enter the larger openings in the sample, and the amount that does so is reflected in a volume change (Jennings 1987; Pittman 1992). At this point, at ambient pressure pores of diameters large than about 12 mm have been filled. The sample container is then placed in a pressure vessel and attached to a pressurization system that allows the pressure on the system to be increased up to approximately 60,000 psi (414 MPa) (a typical maximum value for commercial instruments. As per Equation (12) this will force mercury into pores as small as approximately 0.003 m in diameter. Regardless of the pore geometry and the model employed to quantify it, the volume of mercury forced into the pore (and other void spaces) increases as pressure increases. Therefore, increasing the applied pressure on the mercury surrounding the porous sample produces unique pressurevolume curves such as those shown in Figure 5 for the Dolgeville formation, a Utica shale (Eigmati et al. 2011). These curves are dependent on (a) pore size distribution and tightness, (b) rock type and (c) saturation history (intrusion versus the extrusion process).

Figure 5. Mercury intrusion and extrusion data and pore size distribution of intrusion data for Utica shale, Dolgeville Formation from a depth of 5,197 ft. [Reproduced from Eigmati MM, Zhang H, Bai B, Flori R (2011) Submicron-pore characterization of shale gas plays. Society of Petroleum Engineers SPE-144050MS, with permission from the Society of Petroleum Engineers].

According to Ramakrishanan et al. (1998) self-consistent pore-body/pore-throat ratios can be obtained from the measured hysteresis between the intrusion and extrusions curves. The intrusion curve is controlled by pore-throats whereas the extrusion curve is controlled by pore radii, and pore connectivity. This hysteresis behavior can be attributed to variations in the saturation process, alterations due to advancing and receding contact angles and mercury trapped in pores. These curves are typically normalized to the total amount of mercury in the sample at the end of the intrusion cycle (Venkataramanan et al. 2014). An initial estimate of

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the pore-body/pore-throat ratio is the ratio of radii for a given mercury-saturation volume. This initial ratio can further be averaged over all mercury saturations. The initial estimate can be refined by correcting the measured extrusion curve for mercury trapped in the sample. At any point along the extrusion curve, the mercury saturation in the sample (Simb) is the sum of the disconnected (Sdc) and connected (Sc) network of mercury saturations Simb  Sdc  Sc .

(13)

At the end of the intrusion cycle, the amount of mercury saturation in the sample, Si, is unity. At the end of the extrusion cycle, there is a residual disconnected network of mercury in the sample, Sres. Using the initial and final saturations with Land’s Equation, which relates the residual to the initial saturation (Ramakrishnan and Wasan 1986), the amount of connected and disconnected saturations at any point along the extrusion curve can be calculated:

Sres  Si / 1  CSi  .

(14)

The total disconnected mercury saturation at the end of the extrusion cycle is the sum of the disconnected mercury saturation at any point and the disconnected saturation that could arise from the connected phase from further extrusion:

Sres  Sdc  Sc / 1  CSc  .

(15)

From Equation (14), with Si = 1, the variable C is calculated. Equations (13) and (15) lead to a quadratic equation of the form, aSdc2 + bSdc + k = 0, where a = C, b = -C(Sres + Simb) and k = Simb[CSres – 1] + Sres. Typically there is a fairly constant horizontal shift between the intrusion and modified extrusion curves when plotting cumulative saturation against capillary pressure. Procedurally, a single pore-body/pore-throat ratio is estimated by computing a scaling factor that matches the measured intrusion and corrected extrusion curves. It is possible to transpose the mercury injection data to represent water–oil (w-o) or waterair (w-a) capillary pressure curves using the Leverett J-function (Tiab and Donaldson 2004, 2012). Pc ( k / )0.5 J ,  cos 

(16)

where Pc is the pressure difference between the non-wetting and wetting phase, k is the permeability in darcies (measured prior to mercury intrusion),  is the porosity (also measured prior to mercury intrusion),  is the interfacial surface tension and  is the contact angle. From this expression we can generate equalities that lead to estimates of other capillary behavior based on mercury intrusion data. PcHg Pcw-o Pcw-a 0.5   k /  .     ( w-o )cos0 ( w-a )cos0 (Hg )cos130

(17)

It is important to keep in mind that mercury porosimetry suffers as a measure of pore size for several reasons (Clarkson et al. 2012a). First as noted above it is limited to pore sizes generally greater than about 2 nm which excludes the micropores observed in organic-rich shales that contribute to significant matrix porosity. Second, the method can distort, compress and damage the pore structure of highly compressible clay-rich samples and possibly others because of the high intrusion pressures required for measurement (up to 60,000 psi). Finally, the method requires a reasonable estimate of the interfacial surface tension and contact angle for pore size calculations from the Washburn Equation which are not well constrained for shale.

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Imaging methods A broad range of direct imaging methods are available to describe the nature of porosity and its association with minerals in rock materials. These include optical light microscopy, standard scanning electron microscopy (SEM) with energy dispersive X-ray spectroscopy (EDX), focused ion beam SEM (FIB SEM), transmission electron microscopy (TEM), nuclear magnetic resonance imaging (NMRI) and X-ray tomography. In this section we provide background and examples of all but TEM and tomography, the latter of which is covered in detail by (Noiriel 2015, this volume). Optical petrology. Optical petrology is the most straight forward, accurate and repeatable means of evaluating the pore system and associated mineralogy of seals and reservoir rock samples (core, sidewall core, drill cuttings and outcrop). In order to better identify pores and fractures, one of two types of epoxy, normal blue or rhodamine-B for fluorescence under ultra-violet light, are impregnated into the rock to highlight the pore system. The finished thin section is viewed under plane-polarized, cross-polarized and/or ultra-violet light to examine a two-dimensional cross section through a rock, estimate the bulk mineral composition, and make important observations regarding grain fabric and texture by point counting or image analysis of the mineralogy, texture, diagenesis, pore system and reservoir quality of the sample. However, despite the large amount of information available, the actual three-dimensional grain relationships and details of the intergranular pore structure were always beyond our reach. While digital image analysis is not new, important recent advances in petrography include pattern recognition and pattern classification software for description and quantification of rock-ore geometric characteristics. These approaches built on the early work of Ehrlich and coworkers (Ehrlich et al. 1984; Ehrlich and Etris 1990) who pioneered the arena of Pore Image Analysis (PIA) to determine the size, shape and relative proportions of different pore types through computer-based thin-section porosity analysis. It is possible to define several hundred variables for each field of view using this technique. PIA is used in conjunction with MICP data to develop physical models for the determination of capillary pressure characteristics related to pore-type and pore-throat size (Fig. 6). Currently the two more popular free-ware platforms for implementing PIA are Fiji/Image J™ and Image-Pro™.

Figure 6. Differential intrusion vs pore size of the four samples from the Dolomicrite Facies of the Copper Ridge Dolomite, Ohio. This graph reflects the amount of pore space for a given pore size diameter. The higher spikes indicate the pore sizes that contribute the most to the overall porosity. Thin section photos at 4× for each sample show the type of porosity reflected by the graph.

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To facilitate mineral assessments in concert with pore identification, carbonate stains (alizarin Red-S for calcite and potassium ferricyanade for ferroan carbonate) and/or feldspar stain are applied to the 30-m thin section (Warne 1981). Applying such stains to the rock chip before gluing and sectioning leaves the polished surface pristine for other analytical approaches as needed. Also any sedimentary structures, morphology, bioclasts, crystals habit, textures and fabric of the thin section are noted. From the calculated mineralogy and pore system estimation, the reservoir or seal quality can be estimated with references to potential problems (acid, fines migration and fresh water sensitivity). Scanning electron microscopy (SEM). Through the application of SEM and EDX systems, earth scientists are now able to go one step beyond thin section analysis—to look down into the pores, identify the smallest minerals, and examine the distribution of these minerals within the pores (cf. Welton 2003). Other advantages of the SEM over optical petrography are ease of sample preparation (for certain types of applications), greater depth of field and resolution, and a significantly higher magnification range (most SEM analysis of rocks involves magnifications between 10x to 20,000x). When examining an SEM micrograph for the first time, the major problem is one of scale. However, with minimal training and experience, the user can soon identify minerals and textures previously observed only in thin section. This is not to say that the SEM replaces thin section analysis; instead, the SEM complements thin section analysis by providing a different type of information which—when used in combination with other techniques—provides important new information to help characterize pore features in rocks. Huang et al. (2013) provide a very nice summary of SEM operational aspects and selected imaging applications with emphasis on shale. As they note, the mechanics of the modern scanning electron microscope (SEM) system allow for various imaging and detecting techniques that can be used to study different aspects of the composition of shale and other rock materials at very high resolution. Scanning electron microscopy, unlike conventional light microscopy, produces images by recording various signals resulting from interactions of an electron beam with the sample as it is scanned in a raster pattern across the sample surface. A fine electron probe, with a spot size from a few angstroms to several hundred nanometers, is generated by focusing electrons emanating from an electron source (conventionally called the electron gun) onto the surface of the specimen using a series of electro-magnetic lenses. The primary electron beam interaction with the sample generates a number of different types of signals: (a) secondary electrons useful for 3-D textural assessment, (b) backscattered electrons used for characterizing composition and crystalline structure, (c) characteristic X-rays used for element-specific mapping and mineral identification and (d) photons resulting in cathodoluminescence (CL) indicative of certain trace impurities in minerals (Goldstein et al. 2003). Figure 7 illustrates the value of combining different SEM approaches to describe a geologic sample, in this case, a gas shale from the Utica formation of the Appalachian Basin, USA. In this analysis FEI QEMSCAN™ software was used to generate a very detailed mineral map of an organic-rich shale demonstrating the power of combining X-ray emission assessment, which yields individual elemental signal intensity, with backscattered-electron signals based on average atomic number contrast. The resolution for this type of analysis is typically on the order of 1–2 μm per pixel (Swift et al. 2014). It cannot be over stated, however, that when imaging clay-rich samples or other fine-grained or chemically complex materials it is paramount to have confirmatory data from X-ray diffraction for accurate mineral identification. Another important consideration is the quality of the polish. Even with the utmost care during grinding, there is nearly always the chance for creation of artificial pores and fractures that once imaged yield an inaccurate accounting of true porosity. In many cases the softer clays can be smeared and stretched producing unnatural textures (Fig. 8). Additionally,

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Figure 7. Mineralogical and textural images of a Utica gas shale with a high total organic carbon (TOC) content, Wood Co., West Virginia, 9,503 ft depth. (A) X-ray mineral map with a pixel size of about 2 μm2. (B) Backscattered electron image (BSE) of a small portion of (A). (C) Higher-resolution BSE image showing textural relationships among carbonate, sulfide, and pores. [Arthur M, Cole DR (2014) Unconventional hydrocarbon resources: prospects and problems. Elements, Vol. 10, p. 257–264 with permission from the Mineralogical Society of America]

Figure 8. SEM image of a Utica shale sample where surface grinding has led to smearing and degradation of the surface especially those areas rich in clay.

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artificial pores are commonly created at the contact between soft phases such as clays and more resilient phases such as quartz or feldspar. To circumvent this problem many labs now use argon ion milling to complete the polishing process thus preserving true mineral textures and pore structures (Erdman and Drenzek 2013). High-resolution SEM of ion-milled samples has yielded remarkable new insights such as the occurrence of pores and pore networks contained within the organic matter of gas shale (e.g., Loucks et al. 2009, 2012; Ambrose et al. 2010; Klimentidis et al. 2010; Sondergeld et al. 2010; Curtis et al. 2012). Interestingly, in some over-mature organic-rich rocks, it appears that as much as 50% of the volume of the original organic matter may consist of pores smaller than 100 nm (e.g., Passey et al. 2010; Heath et al. 2011). An example of this is given in Figure 9, which shows submicron pores in organic matter intercalated with clay and carbonate in a very mature shale from the Utica formation, Wood County, West Virginia. What is interesting is that wherever organic matter is “protected” by resilient grains of carbonate or quartz, we see little or no development of submicron pores. Conversely were organic matter is wrapped by clay we see abundant pore formation. This suggests that the clay can deform and accommodate the volume expansion due to gas generation whereas the stiffer grains prohibit the volume expansion, hence they lack pores. The point here is that SEM coupled with ion milling is very valuable in revealing details that reflect the evolution of the pore system in complex lithologies.

Figure 9. A FIB/SEM slice of the Utica shale revealing the nature of nanopores hosted primarily in the organic-dominated regions associated with clays, in this case, illite. The bright grain in the upper left is pyrite.

Focused ion beam (FIB) SEM. Focused ion beam (FIB) SEM is finding a growing number of applications in the earth sciences (Goldstein et al. 2003). This method uses serial sectioning and imaging in order to produce sets of sequential SEM images (generally several hundred) that permit a three-dimensional (3-D) visualization of minerals, organics and pores. From these 3-D images one can calculate porosity, pore-size distribution, kerogen volume percentage, and permeability (e.g., Heath et al. 2011; Zhang and Klimentidis 2011; Curtis et al. 2012; Landrot et al. 2012; Huang et al. 2013). In a typical FIB-SEM system, an extraction field is applied to a gallium (Ga) liquid metal ion source to field emit Ga ions and form a Ga beam. Due to its relatively high atomic mass,

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the Ga beam not only can be used to generate electron and ion images, but also may be used to mill samples to remove material. A cross-section of the sample is milled by a Ga FIB beam and is imaged simultaneously by the SEM (Fig. 10, Curtis et al. 2012). Thicknesses of the milled slices depend on the milling ion current, with high currents (on the order of nA) milling thicker and less precise slices greater than about 25 nm. In most FIB/SEM systems, the electron beam and ion beam are positioned between 50 and 54° from each other (Fig. 10). Because of this configuration, any reasonably flat location on a sample can be milled and imaged (Silin and Kneafsey 2012). To avoid repositioning the sample so that the surface is orthogonal to the SEM following each FIB slice, the SEM images are adjusted to account for this angle.

Figure 10. (A) Schematic diagram of the sectioning and imaging procedure in a focused ion beam–scanning electron microscopy (FIB-SEM) system. (B) Backscattered electron (BSE) image of a site on a shale surface prepared in cross section. Pt strip = platinum strip. HV = high voltage; TLD = through lens detector; WD = working distance; HFW = horizontal field width. [From Curtis ME, Sondergeld CH, Ambrose RJ, Rai CS (2012) Microstructural investigation of gas shales in two and three dimensions using nanometer-scale resolution imaging. American Association of Petroleum Geologists Bulletin, Vol. 96, p. 665–677; AAPG© 2012, reprinted by permission of the AAPG whose permission is required for further use.]

For most advanced studies, measurements take advantage of multiple detector types for detecting secondary (SE) and backscattered electrons (BSE) and X-rays. A BSE detector is preferable for imaging because it minimizes surface electron charging. However, an SE detector can also produce satisfactory images. Because most geological materials are nonconductive, super electron charges can build up on the sample surface. To mitigate this issue, one must use a low voltage electron beam (2–5 kV) on the SEM side when a high electron beam current (0.17–1.4 nA) is applied. Silin and Kneafsey (2012) provide a good discussion of some of the issues encountered with FIB/SEM applications that we summarize here. One of the major limitations of FIB/SEM is the extremely small size of the sample area. Therefore, when performing nanometer-scale interrogations of fine-grained, low porosity materials like shale, it is important to consider the scale of the observation in the context of the scale of interest. Volumes of 20 μm x 10 μm x 5 μm are typically imaged, whereas the rock unit the sample came from may be 10 s of m thick with a lateral extent of 10s or 100s of km. Hence the sample may be 20 to nearly 30 orders of magnitude smaller than the lithologic unit. A case in point is illustrated in images shown in Figure 11A–C. This FIB/SEM sample comes from the Utica formation, Wood County, West Virginia at a depth of 9,502.7 feet (Arthur and Cole 2014). In this deep part of the Appalachian Basin the Utica is roughly 300 feet thick; and the formation extends north, northwest and northeast for several hundred kms. The pores (Fig. 11C) within this sample occur primarily within the kerogen (Fig. 11B), and exhibit a fair degree of connectivity, but the image is only ~ 20 m across.

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Figure 11. (A) Dual beam FIB/SEM reconstructed volume of a Utica shale sample described in Figure 7. (B) 3-D reconstructed images of kerogen (red) and (C) pore (blue) distributions. The 3-D reconstructed volumes have dimensions of 20 × 17 × 6.5 mm. [B and C reproduced from Arthur M, Cole DR (2014) Unconventional hydrocarbon resources: prospects and problems. Elements, Vol. 10, p. 257–264 with permission from the Mineralogical Society of America.]

A second issue impacting pore assessment is that sampling bias must be taken into account. Most geologic materials exhibit some form of heterogeneity that may cross a variety of length scales. For example, in the case of shales they are usually anisotropic in the form of thin laminae and contain pores ranging in scale from 10s of μm down to below 10 nm (see Fig. 9). The porosity may vary within a given layer and between layers, as might permeability both of which are also typically anisotropic. The rheologic integrity of the sample may also be affected by the nature of the sampling, sample handling, desiccations, especially for clay rich materials, and machining, which all place stress on the sample. Further, even the FIB milling process, particularly at high beam currents, can adversely impact the sample leading to varying forms of artificial porosity. Small cracks in the sample may result from machining, so insights into their nature such as the presence of clay particles within the fracture need to be pursued and their origins evaluated. Lower currents are recommended for polishing and prior to image collection. Determination of the size of the representative elementary volume is required to effectively see the 3-D pore structure for flow simulations and to scale up pore-scale results to answer reservoir-scale questions. Clearly this is a difficult task given the extent of heterogeneity observed in most geological materials. Silin and Kneafsey (2012) point out that for shale samples this can be done independently of the fracture network, which imposes another scale of interest. Despite these various issues, nanoscale imaging via FIB/SEM has a number of advantages. As noted above, from the images one can obtain a fundamental understanding of the 3-D nature of pore space (Fig. 11C), pore connectivity, and the location and distribution of mineral and organic phases (Fig. 11B). The images provide a foundation for conceptual model building

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that leads to quantification of permeability and fluid flow. Recently this approach has been used to estimate the accessible surface area in the Lower Tuscaloosa sandstone (Landrot et al. 2012). Mineral distributions mapped in 2-D by SEM/EDX were coupled with dual-beam FIB/ SEM and X-ray-based micro tomographs of seclect regions within the samples to quantify the connected pore network. Core-scale NMR imaging. A wide range of NMR spectroscopic methods are available for non-destructive characterization of porous materials. According to Mitchell and Fordham (2014) the majority of NMR spectrometers are high field instruments (magnetic field strength B0 > 1.5 T) used for chemical analysis and biomedical studies. In these applications, the use of a high magnetic field offers several advantages. The inherent signal-to-noise ratio (SNR) of the measurement is improved by increasing the field strength, enabling better spectral (chemical) and spatial (image) resolution. Furthermore, nuclei with lower gyromagnetic ratios than 1H are more readily accessed (e.g., 23Na, 19F, 31P, 13C, 2H) for studies of molecular structure and chemical reaction monitoring (Mitchell and Fordham 2014). High-field NMR also offers the advantage of shorter radio frequency (rf) probe recovery times, allowing the detection of short relaxation time components in solids. Unfortunately, high field strengths can bring complications, especially in studies of heterogeneous materials (e.g., liquid-saturated porous media). The solid/fluid magnetic susceptibility contrast in such samples results in pore-scale magnetic field distortions (so-called “internal gradients”). Molecular diffusion through these internal gradients leads to an enhanced decay of transverse magnetization. Additionally, the field dependence of relaxation times prevents high field measurements from being compared directly to low field studies. In the majority of chemical and medical applications, the advantages of high field significantly outweigh the disadvantages. However, this is not the case for laboratory studies of fluids in rock. It is the magnetic field dependence of relaxation times that necessitates laboratory instruments and logging tools to operate at similar frequencies; if laboratory data are to be used for log calibration, the measurements must be based on consistent spin physics. Consequently, the industry standard for laboratory NMR core analysis has been set at an 1H resonance frequency of 0 = 2 MHz, corresponding to a magnetic field strength of B0 = 0.05 T. The use of low field also limits the influence of internal gradients, enabling quantitative analysis (Mitchell et al. 2010). More recently researchers have been exploring the use of other field strengths to assess core such as 0.3 T as reviewed by Mitchell and Fordham (2014). Regarding pore assessment in rock cores, low-field NMR measures the response of hydrogen protons inside an external magnetic field. Therefore the signal response comes from the water or oil saturated in the rock and not the rock itself (Bryan et al. 2013). The protons in the oil or water are polarized in the direction of this static magnetic field, called the longitudinal direction. Another magnetic field is then applied as a radio frequency pulse to “tip” the protons onto the perpendicular transverse plane, where they rotate in phase with one another. As the protons give off energy to one another and to their surroundings, the magnetic signal in the transverse plane decays. This is known as transverse relaxation, or T2 (Coates et al. 1999). In homogeneous magnetic fields such as those generated in NMR laboratory instruments, two types of relaxation exist in fluids: bulk relaxation and surface relaxation (Straley et al. 1997). When a bulk fluid is placed in the NMR the measured transverse relaxation is bulk relaxation, or energy transferred between protons in the fluid. Bulk relaxation is a property of the fluid, related to local motions such as molecular tumbling and diffusion (Kleinberg and Vinegar 1996; Straley et al. 1997). If solids are present, surface relaxation occurs at the fluid– solid interface where the hydrogen protons are constricted by the grain surfaces and therefore transfer energy to these surfaces. When samples of saturated porous media are measured, the amplitude of the T2 measurement is directly proportional to porosity, and the decay rate

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is related to the pore sizes and the fluid type and its viscosity in the pore space. Short T2 times generally indicate small pores with large surface-to-volume ratios and low permeability. Conversely, longer T2 times indicate larger pores with higher permeability. Hydrogen nuclei in thin interlayers of clay water experience high NMR relaxation rates because the water protons are close to grain surfaces and interact with surfaces frequently. Additionally, if the pore volumes are small enough that water is able to diffuse easily back and forth across the water-filled pores, then the relaxation will reflect the surface-to-volume ratio of the pores. Thus, water in small clay pores with larger surface-to-volume ratios will exhibit fast relaxation rates and therefore short T2 porosity components. Because porosities are not equal in a given lithofacies, especially one with a significant mix of clays and clastic grains, capillary-bound or clay-bound waters are not very mobile, but free water can be. This can set up a scenario of two approximately equal porosities, but with entirely different mobility regimes that can be distinguished by their T2 time distributions. Figure 12A shows an example of T2 behavior of water in different sizes of pores. One observes a much faster relaxation time for water contained in the smaller pores, in this case, clay. Water alone is a low viscosity fluid, thus its bulk relaxation is slow, on the order of approximately 2000 ms. However, when water is imbibed into pores of varying size, the water T2 distribution is essentially analogous to a pore size distribution because surface relaxation is so much faster than bulk relaxation of water. Heavy oil and bitumen are highly viscous and therefore exhibit very fast relaxation times. Figure 12B compares the spectrum of a bulk sample of bitumen to its signal inside sand. Due to the high viscosity, the relaxation times for heavy oil and bitumen occur at approximately the same T2 locations whether the fluid exists in bulk form or in a porous matrix. The presence of gas in pores poses an interesting problem for NMR that has been discussed for example by Sigal and Odusina (2001). Gas has much less hydrogen per unit volume than liquids such as water or oil and as such will yield a neutronderived porosity that is too low. By comparing the density logs described below against NMR derived measurements one can distinguish liquid-filled versus gas-filled pore systems. A pore-size distribution can be obtained from a fully water-saturated rock core using the NMR T2 distribution as follows: 1 A 1 ,   T2 V T2,B

(18)

where  is the relaxivity, A/V is the ratio of the area to volume of a pore (an uncertain value for fractal pore surfaces), and T2,B is the relaxation of the bulk fluid. For spherical pores with radius r, A/V  3/r. Therefore, the NMR T2 distribution can be converted to pore-size r distribution by suitable selection of the relaxivity . Typically for natural porous matrices the magnitude of  ranges between 1 and 10 μm s-1. NMR Cryoporometry (NMRC) is a recent technique developed at the University of Kent in the UK for measuring total porosity and pore size distributions (Strange et al. 1993). It makes use of the Gibbs-Thomson effect wherein small crystals of a liquid in the pores melt at a lower temperature than the bulk liquid. The melting point depression is inversely proportional to the pore size. The technique is closely related to that of the use of gas adsorption to measure pore sizes (Kelvin Equation). Both techniques are particular cases of the Gibbs Equations; the Kelvin Equation is the constant temperature case, and the Gibbs–Thomson Equation is the constant pressure case (Mitchell et al. 2008). To make a cryoporometry measurement, a liquid is imbibed into the porous sample, the sample cooled until all the liquid is frozen, and then warmed slowly while measuring the quantity of the liquid that has melted. According to Mitchell et al. (2008), it is similar to DSC thermoporosimetry, but has higher resolution, as the signal detection does not rely on transient heat flows, and the measurement can be

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Figure 12. (A) Low-field NMR spectra of water in sand and clay. (B) Low-field NMR spectra of oil sand compared to bulk bitumen. [Reproduced from Bryan et al. 2013, Heavy oil reservoir characterization using low field NMR. AAPG Search and Discovery Article #90170; AAPG©2013, reprinted by permission of the AAPG whose permission is required for further use.]

made arbitrarily slowly (Fig. 13). It is suitable for measuring pore diameters in the range 2 nm–2 μm. NMR may be used as a convenient method of measuring the quantity of liquid that has melted, as a function of temperature, making use of the fact that the T2 relaxation time in a frozen material is usually much shorter than that in a mobile liquid. It is also possible to adapt the basic NMR experiment to provide structural resolution in spatially dependent pore size distributions (Strange and Webber 1997) or to provide behavioral information about the confined liquid (Alnaimi et al. 2004). Atomic Force Microscopy. Atomic force microscopy (AFM) is a relatively new tool being used to characterize pores and pore features in complex rock matrices down to the atomic scale. This method cannot only be used to obtain topographic images of surfaces, but it also can simultaneously identify different materials on surfaces at high resolution (e.g., Javadpour 2009; Javadpour et al. 2012). This recent interest in AFM application for rock characterization can be traced to the emergence of unconventional shale-gas reservoirs and the interest

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Figure 13. Comparison between pore size distributions generated by (A) NMR cryoporometry and gas adsorption and (B) NMR cryoporometry and DSC thermoporometry, using naphthalene as the probe. [Reproduced from Strange JH, Mitchell J, Webber JBW (2003) Pore surface exploration by NMR. Magnetic Resonance Imaging, Vol. 21, p. 221–226 with permission from Magnetic Resonance Imaging.]

reservoir engineers have in quantifying the wetting and flow behavior of hydrocarbon gases and fluids at the nanoscale. The unique capabilities of the AFM make it ideal for interrogating nanopores, organic materials (e.g., kerogen), minerals and diagenetic microfractures in shale. Additionally, it can be used to measure the localized bulk modulus of elasticity on a surface, which has implications for geophysical modeling and even designs for hydraulic fracturing (e.g., Kopycinska-Müller et al. 2007).

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In AFM, a flexible cantilever acts as a spring to determine the net force between a coating at the tip of the cantilever and a sample, or substrate. Local attractive and repulsive forces between the tip and the sample bend the cantilever arm. Deflection of the cantilever is optically detected and converted into an electrical signal to determine force curves versus distance using Hook’s law. The detection system that has become the standard employs a focused laser beam that is reflected from the rear of the cantilever onto a detector. By the optical lever principle, a small displacement of the cantilever is converted to a measurably large deflection in the position of the reflected spot in a photodetector. Vertical deflection can be quantified by comparing the spot location from the top and bottom halves of the detector (e.g., Javadpour 2009). An example of nondestructive AFM imaging of a nanoporous system is shown in Figure 14. Nanopores and nanogrooves can be seen in this mudrock sample where the imaging dimensions were 4 × 4 × 0.6 μm3. This rock sample was cut using a cryogenic ultramicrotome. The flat surface was scanned using a triangular AFM tip (7–10 nm tip radius, 50 N m-1 spring constant) to reveal the topography and pore network. Pores on the order of 30 nm in diameter and nanogrooves with widths of approximately 60 nm were observed in a larger depression that may have been occupied by a single grain that was plucked from the sample during preparation. This is, in fact, one key issue that can hamper the study of nanoporous samples is the creation of artificial pores when preparing surfaces from grinding and polishing or using microtome techniques. The use of ion milling can generally help mitigate this problem. Images of nanopore features and grain boundaries derived from AFM can form the basis of models designed to assess the nature of fluid and gas flow in nanoscale networks (e.g., Javadpour et al. 2007; Javadpour 2009). Furthermore, AFM can be used to characterize the surfaces of discrete phases such as micas that have been reacted with fluids not in equilibrium with the phase to quantify the extent of dissolution and generation of surface pores (e.g., Hu et al. 2011a; Shao et al. 2011). Coverage of the voluminous AFM literature on mineral dissolution leading to pore evolution is beyond the scope of this article.

Figure 14. AFM image of nanopores and nanogroves in a fine-grained mudrock [Reproduced from Javadpour F (2009) Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstones). Journal of Canadian Petroleum Technology, Vol. 48, p. 16–21 with permission from the Society of Petroleum Engineers.]

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Downhole porosity logs There are a number of downhole logging methods used to determine porosity: resistivity, density, neutron, NMR and sonic logs. Below we briefly describe the first four of these in rather general terms. Further details on these methods can be found in Tiab and Donaldson (2004, 2012), Ellis and Singer (2008) and Crain’s Petrophysical Handbook (https://spec2000. net/01-index.htm). Resistivity logs. Resistivity logging works by characterizing the rock or sediment in a borehole by measuring its electrical resistivity, which is the ability to impede the flow of electrical current. Resistivity is a fundamental material property that represents how strongly a material opposes the flow of electric current. This helps to differentiate between formations filled with salty waters (good conductors of electricity) and those filled with hydrocarbons (poor conductors of electricity). In these logs, resistivity is measured using four electrical probes to eliminate the resistance of the contact leads. The log must run in holes containing electrically conductive mud or water. Resistivity and porosity measurements are used to calculate water saturation. Resistivity is expressed in ohms or ohms/m, and is frequently charted on a logarithm scale versus depth because of the large range of resistivity. The distance from the borehole penetrated by the current varies with the tool, from a few centimeters to one meter. The foundation for using resistivity to assess formation properties is derived from the empirical Archie’s law (Archie 1942, 1947; Tiab and Donaldson 2004; Ellis and Singer 2008) that relates the in situ electrical conductivity of a sedimentary rock to its porosity and brine saturation: 1 Ct  Cwm Swn , a

(19)

where  denotes the porosity, Ct the electrical conductivity of the fluid saturated rock, Cw represents the electrical conductivity of the brine, Sw is the brine saturation, m is the cementation exponent of the rock, n is the saturation exponent and a the tortuosity factor. The cementation exponent models how much the pore network increases the resistivity, as the rock itself is assumed to be non-conductive. For unconsolidated sand the m value is roughly 1.3; for sandstones between 1.8 and 2 and for limestones, 1.7 to 4. The saturation exponent n models the dependence of the conductivity on the presence of non-conductive fluid (hydrocarbons) in the pore-space, and is related to the wettability of the rock. Its value is usually very close to 2. These exponents are typically determined from measurements on core plugs whereas the brine conductivity can be measured on produced water. The tortuosity factor a is meant to correct for variation in compaction, porestructure and grain size, and lies between 0.5 and 1.5. Recast in terms of electrical resistivity, the equation reads Rt  a m S w n Rw ,

(20)

where Rt refers to the fluid saturated rock resistivity, and Rw represents brine resistivity. The factor

F

a R  0 m  Rw

(21)

is also called the formation factor, where R0 is the resistivity of the rock filled with only water (Sw = 1). The factor

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Rt  Sw n R0

(22)

is also called the resistivity index. As noted above Archie’s law is a purely empirical relation attempting to describe ion flow (mostly sodium and chloride) in clean, consolidated sands, with varying intergranular porosity. Electrical conduction is assumed not to be present within the rock grains or in fluids other than water. For brine-saturated intervals (Sw = 1) Archie’s law can be written: log Ct  log Cw  m log .

(23)

Therefore, plotting the logarithm of the measured in situ electrical conductivity against the logarithm of the measured in situ porosity (a so-called Pickett plot), should yield a straightline relationship according to Archie’s law. The slope is equal to the cementation exponent m and the intercept is equal to the logarithm of the in situ brine conductivity. This relationship, however, breaks down when dealing with rocks containing appreciable amounts of clay. In this case one can apply the Waxman–Smits Equation that tries to correct for this (Waxman and Smits 1968). Density and neutron logs. Density logs take advantage of the well-known linear relationship between density and porosity (e.g., Ellis et al. 2003). In the case of a binary system of a framework of rock with a density ma and a portion of the volume filled with a fluid of density f it is given by b  f   ma (1  ),

(24)

where b is the bulk density of the formation and , the porosity, or volume fraction that is not rock, or “matrix.” It is assumed to be saturated with a fluid of known density. This relationship can be recast to determine porosity 

 b  ma   a  b, b  f  ma 

(25)

where the scaling constants a and b are not constants but depend on the formation parameters specific to the zone being investigated: a

1 ma and b  .  f  ma   f  ma 

(26)

Thus, to estimate porosity properly, two important parameters must be known: the rock matrix (or grain) density (ma) and the density of the saturating fluid (f) since they determine the slope and intercept of this simple relationship. Density logs measure the electron density of the formation, which is related to the formation density. The logging tools contain a cesium-137 -ray source that irradiates the formation with 662 keV -rays. These -rays interact with electrons in the formation through Compton scattering (inelastic scattering of a photon by a quasi-free charged particle) and lose energy. Once the energy of the -rays has fallen below 100 keV, photoelectric absorption dominates: -rays are eventually absorbed by the formation. The amount of energy loss by Compton scattering is related to the number of electrons per unit volume of formation. Since, for most elements of interest (below Z = 20), the ratio of atomic weight, A, to atomic number, Z, is close to 2, -ray energy loss is related to the amount of matter per unit volume, i.e., formation

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density. A -ray detector located some distance from the source, detects surviving -rays and sorts them into several energy windows. The number of high-energy -rays is controlled by Compton scattering, hence by formation density. The number of low-energy -rays is controlled by photoelectric absorption, which is directly related to the average atomic number, Z, of the formation, hence to lithology. Modern density logging tools include two or three detectors, which allow compensation for some borehole effects, in particular for the presence of mud cake between the tool and the formation. Since there is a large contrast between the density of the minerals in the formation and the density of pore fluids, porosity can easily be derived from measured formation bulk density if both mineral and fluid densities are known. Neutron porosity measurements employ a neutron source to measure the Hydrogen Index in a reservoir, which is directly related to porosity (e.g., Ellis et al. 2003, 2004). The Hydrogen Index (HI) of a material is defined as the ratio of the concentration of hydrogen atoms per cm3 in the material, to that of pure water at 75 °F. As hydrogen atoms are present in both water and oil filled reservoirs, measurement of this ratio allows estimation of the amount of liquid-filled porosity. Neutron porosity logging tools contain an americium–beryllium neutron source, which irradiates the formation with neutrons. These neutrons lose energy through elastic collisions with nuclei in the formation. Once their energy has decreased to thermal level (see below), they diffuse randomly away from the source and are ultimately absorbed by a nucleus. Hydrogen atoms have essentially the same mass as the neutron; therefore hydrogen is the main contributor to the slowing down of neutrons. A detector at some distance from the source records the number of neutron reaching this point. Neutrons that have been slowed down to thermal level have a high probability of being absorbed by the formation before reaching the detector. The neutron count rate is therefore inversely related to the amount of hydrogen in the formation. Since hydrogen is mostly present in pore fluids (water, hydrocarbons) the count rate can be converted into apparent porosity. Modern neutron logging tools usually include two detectors to compensate for some borehole effects. Porosity is derived from the ratio of count rates at these two detectors rather than from count rates at a single detector. For a discussion of neutron logs beyond this simple over view consult Gilchrist (2008) for details on interpretation of compensated neutron log responses and Fricke et al. (2008) for a summary of downhole pulse neutron techniques. The combination of neutron and density logs takes advantage of the fact that lithology has opposite effects on these two porosity measurements. The average of neutron and density porosity values, therefore, is usually close to the true porosity, regardless of lithology. Another advantage of this combination is the “gas effect.” Gas, being less dense than liquids, translates into a density-derived porosity that is too high. Gas, on the other hand, has much less hydrogen per unit volume than liquids: neutron-derived porosity, which is based on the amount of hydrogen is, therefore, too low. If both logs are displayed on compatible scales, they overlay each other in liquid-filled clean formations and are widely separated in gas-filled formations. NMR logs. NMR logging, a subcategory of electromagnetic logging, measures the induced magnetic moment of hydrogen nuclei (protons) contained within the fluid-filled pore space of porous media (reservoir rocks). Unlike conventional logging measurements (e.g., acoustic, density, neutron, and resistivity), which respond to both the rock matrix and fluid properties and are strongly dependent on mineralogy, NMR-logging measurements only respond to the presence of hydrogen protons. Because these protons primarily occur in pore fluids, borehole NMR measurements can provide different types of formation porosity-related information (e.g., Allen et al. 1997). First, they tell how much fluid is in the formation. Second, they provide details about the formation pore size and structure that are not usually available from conventional porosity logging tools. This can lead to a better description of fluid viscosity and mobility—whether the fluid is bound by the formation or free to flow. Finally, in some cases, NMR logs can be used to determine the type of fluid—water, oil or gas.

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As noted above the NMR behavior of a fluid in the pore space of a reservoir rock is different from the NMR behavior of the fluid in bulk form. For example, as the size of pores containing water decreases down to the micron scale, the differences between the apparent NMR properties of the water in the pores and the water in bulk form increases (e.g., Cole et al. 2013). Micro-porosity associated with clays and with some other minerals typically contains water that, from an NMR perspective, appears almost like a solid. Water in such micro-pores has a very rapid “relaxation time.” Because of this rapid relaxation, this water is more difficult to see with NMR logging tools than, for example, producible water associated with larger pores. Fortunately, modern NMR logging methods can see essentially all the fluids in the pore space, and the porosity measurement made by these tools is thus characterized as being a “total-porosity” measurement (Coates et al. 1999). An added feature of NMR is the fact that measurements of the formation made when the magnetic resonance imaging logging (MRIL) tool is in the wellbore can be duplicated in the laboratory by NMR measurements made on rock cores recovered from the formation. This ability to make repeatable measurements under very different conditions is what makes it possible for researchers to calibrate the NMR measurements to the petrophysical properties of interest (such as pore size) to the end user of MRIL data (Murphy 1995; Cherry 1997). Although conventional porosity tools, such as neutron, density, and sonic, exhibit a bulk response to all components of the volumetric model (i.e., matrix plus pore-fluid), they are more sensitive to matrix materials than to pore fluids (e.g., Coates et al. 1999). Furthermore, the responses of these tools are highly affected by the borehole and mudcake, and the sensitive volumes of these tools are not as well defined as that of the NMR imaging tool. NMR porosity is essentially matrix-independent—that is, the tools are sensitive only to pore fluids. The difference in various NMR properties—such as relaxation times (T1 and T2) and diffusivity (D)—among various fluids makes it possible to distinguish (in the zone of investigation) among bound water, mobile water, gas, light oil, medium-viscosity oil, and heavy oil (Fig. 15). Clay-bound water, capillary-bound water, and movable water occupy different pore sizes and locations. Hydrocarbon fluids differ from brine in their locations in the pore space, usually occupying the larger pores. They also differ from each other and brine in viscosity and diffusivity. NMR logging uses these differences to characterize the fluids in the pore space. In terms of the measurement process, before a formation is interrogated with an NMR tool, the protons in the formation fluids are randomly oriented. When the tool passes through the formation, the tool generates magnetic fields that activate those protons. First, the tool’s permanent magnetic field aligns, or polarizes, the spin axes of the protons in a particular direction. Then the tool’s oscillating field is applied to tip these protons away from their new

Figure 15. The typical qualitative values of D, T1 and T2, for different fluid types and rock pore sizes demonstrate the variability and complexity of the T1 and T2 relaxation measurements using NMR.

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equilibrium position. When the oscillating field is subsequently removed, the protons begin tipping back, or relaxing, toward the original direction in which the static magnetic field aligned them (Fukushima and Roeder 1981). Specified pulse sequences are used to generate a series of so-called spin echoes, which are measured by the NMR logging tool and are displayed on logs as spin-echo trains (e.g., Coates et al. 1999; Kleinberg 1999; Kleinberg and Jackson 2001). These spin-echo trains constitute the raw NMR data. To generate a spin-echo train an NMR tool measures the amplitude of the spin echoes as a function of time. Because the spin echoes are measured over a short time, an NMR tool travels no more than a few inches in the well while recording the spin-echo train. The recorded spin-echo trains can be displayed on a log as a function of depth. The initial amplitude of the spin-echo train is proportional to the number of hydrogen nuclei associated with the fluids in the pores within the sensitive volume. Thus, this amplitude can be calibrated to give a porosity. The amplitude of the spin-echo-train decay can be fit very well by a sum of decaying exponentials, each with a different decay constant. The set of all the decay constants forms the decay spectrum or transverse-relaxation-time (T2) distribution (see discussion above for corebased NMR measurements). In water-saturated rocks, it is a single exponential with a decay constant proportional to pore size; that is, small pores have small T2 values and large pores have large T2 values (Kenyon 1997). At any depth in the wellbore, the rock samples probed by the NMR tool will have a distribution of pore sizes. In essence, a key function of the NMR tool and its associated data-acquisition software is to provide an accurate description of the T2 distribution at every depth in the wellbore. Hence, the multi-exponential decay represents the distribution of pore sizes at that depth, with each T2 value corresponding to a different pore size. Properly defined, the area under the T2-distribution curve is equal to the initial amplitude of the spin-echo train. Hence, the T2 distribution can be directly calibrated in terms of porosity.

SCATTERING METHODS Small and ultrasmall angle scattering (U)SAS techniques provide powerful, relatively new, uniquely useful tools for characterizing rock porosity and the properties of confined fluids. In wide-angle X-ray diffraction experiments, familiar to many geochemists in the context of laboratory-scale or synchrotron radiation sources, one probes the structure of materials on an atomic-length scale. Each crystallographic phase of a mineral produces a distinctive wide-angle X-ray diffraction pattern, which serves as a characteristic “fingerprint” of that mineral phase. A typical laboratory-scale X-ray diffraction instrument might measure diffraction at angles above 5° 2 and, for CuK radiation, d-spacings below about 17.67 Å. On the other hand, using one of several different geometries, small-angle scattering experiments analyze nano-scale structures (typically greater than ~10 Å) by measuring the intensity of the diffracted beam at significantly smaller-angles and/or longer wavelengths. Although natural materials are typically disordered and heterogeneous at these scales, the contrast in scattering length density between the mineral phases and the pore space of a rock produces a scattering signal in the small-angle regime that reflects the pore structure of the rock as a whole, in part because the contrast between the minerals and the pores is significantly larger than that between the mineral phases themselves. In this section, we explain the principles of smallangle diffraction experiments, how the data may be analyzed to obtain information about the pore space, and what opportunities and obstacles are presented to the experimenter. While most SAS techniques, with the exception of spin-echo approaches, provide data in inverse space (as do other, more familiar diffraction experiments), all interrogate and average relatively large volumes when compared to standard microscopy. While transmission electron microscopy (TEM) provides an obvious and very useful method of characterizing nanoscale porosity it is difficult to use it to statistically quantify the structures of porous materials given

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the wide variation in length scales involved. This is because, while electron microscopy can provide detailed images of pores at high magnifications, the total volume of the rock imaged is, of necessity, very small. In fact, Howard and Reed (2005) calculated that if all the material that has ever been in focus in all of the transmission electron microscopes in the world were gathered together it would total less than 1 cm3. Thus tools that can provide a more statistically representative quantification of the pore structure are highly useful complements to highmagnification imaging techniques. This is provided by small angle scattering experiments, which allow characterization of porosity over a very wide range of scales, from nanometer to 10’s of mm (extendable to cm ranges with the integration of low magnification, high resolution imaging techniques, see below), and interrogate all types of void space in the rock, from nanopores to fractures for sample sized from < ~ 1 mm to ~ 2.5 cm, integrating volumes up to ~ 30 mm3 (Anovitz et al. 2009). Scattering techniques can be broadly classified either in terms of the instrument geometry and data acquistion scheme used (e.g., pinhole, Bonse–Hart, spin-echo, time-of-flight etc.) or the type of radiation employed (neutron, X-ray or light), but all interrogate all types of void space in the rock. X-ray sources range from laboratory-sized instruments to those at synchrotron sources and typically have relatively small beam sizes. Synchrotron sources typically have high flux rates as well, which has the advantages of short counting times, good statistics, and the ability to map variations within a sample. Neutron sources can be continuous, typically reactor-generated (e.g., HFIR/ORNL, NCNR/NIST, ILL), or pulsed, including spallation sources (e.g., LANSCE/LANL, ISIS, SNS/ORNL) or pulsed reactors (IBR-2, Dubna, Russia). The latter, or the addition of choppers at the former allows measurements to be made in timeof-flight mode, and potentially provide the opportunity for analysis of fast reaction kinetics. Neutron sources typically have much lower intensities than X-ray sources, however, and thus neutron scattering experiments require longer counting times, but typically have larger beam sizes and thus interrogate and average a larger sample volume (~ 30 mm3, Anovitz et al. 2009). This latter can be advantageous, as the beam is typically much larger than the grain sizes of typical (although not all) rock materials and thus neutron scattering may provide a more statistically meaningful, quantified understanding of pore structures. There are also significant and useful differences in how X-rays and neutrons interact with the sample. X-rays interact electromagnetically with the electron clouds around atoms. In contrast, neutrons interact either with atomic nuclei via the short-range strong force, or with unpaired orbital electrons via a magnetic dipole interaction. Thus, while X-ray scattering intensity is a function of atomic number, neutron scattering is not, allowing the two techniques to provide complementary information. Because the neutron-nucleus potential is approximately a -function, only spherically symmetrical scattering occurs from a single-fixed nucleus. That is, neutrons have a constant form factor with scattering angle. In X-ray scattering studies of condensed matter, however, the distribution of electrical charge within the atom produces a form factor so that the amplitude of a scattered photon depends upon scattering angle. In addition, there is no obvious or simple pattern connecting the neutron scattering length b to the atomic number of the atoms Z, as there is for X-rays. Because neutron scattering is a nuclear, rather than an electronic effect, it is sensitive to isotopic variations, and two nuclear isotopes, such as 1H and 2H, may have dramatically different neutron scattering cross-sections, despite having the same chemical identity. Therefore, the scattering length density of the sample, or a fluid in it can be experimentally controlled. For instance, if the sample is soaked in an H2O/D2O mixture with a scattering length density matched to the rock matrix, connected (effective) porosity and total porosity can be separated. In addition, because of the nature of the neutron/sample interaction neutrons are much more penetrating than X-rays, and can be used to study magnetic (Shull and Smart 1949) as well as structural effects. However, unlike X-rays, neutrons can activate elements

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in a sample, and samples must be scanned for radioactivity after having been in a neutron beam. In rare cases activation may be large enough to prevent release of the sample after analysis. Despite this potential difficulty, these differences make X-ray and neutron scattering complementary approaches, presenting an opportunity for an experimenter investigating a complicated material such as natural porous media. The initial studies of small angle scattering were those of Guinier (1937). He and others pioneered the theoretical and practical basis of the approach. (cf. Kratky 1938; Debye and Bueche 1949; Guinier and Fournet 1955; Debye et al. 1957; see also the comprehensive review of Hammouda 2008). In these techniques collimated radiation is deflected by small angles from areas in a sample across which there are variations in scattering length density (see below), and is measured as a differential cross section d/d or the number of particles (neutrons or photons) scattered per unit time per unit solid angle, divided by the incident flux. Small angle scattering experiments measure the angular dependence of scattering intensity, either without considering changes in the energy of the scattered particle or rejecting particles that are not scattered elastically. In neutron scattering studies of porous media, the incident neutrons interact with the sample by scattering from the short-range potential of the atomic nuclei making up the sample. The wavelengths of neutrons used in scattering studies of condensed matter are much longer than the range of these nuclear forces. As a result, the neutron-nucleus potential may be described by a simplified phenomenological model that takes the potential to be a -function located at the position of each nucleus and assigned a strength (b). The scattering length b is an empirically determined property of the neutronnucleus interaction that depends upon the nuclear isotope and spin state. This makes neutron scattering sensitive to nuclear isotopes, which is the physical basis for the contrast matching techniques described below. In a neutron diffraction experiment, the measured differential cross section d/d may be decomposed into two distinct contributions, known as coherent and incoherent scattering as: 2





d 2 2  b eiQ  rl  N b2  b d   l     incoherent

(27)

coherent

The terms ‘coherent’ and ‘incoherent’ refer to interference effects in the wavefunction of the scattered neutron. The size of each component depends upon the strength and variation of the scattering lengths b throughout the sample. The first component, coherent scattering, provides information about the structure of the material. The relative distances between the atoms making up the sample determine the phase factors eiQ.rl, the superposition of the scattering wavelets making the amplitude of the coherent scattering dependent upon scattering angle. The strength of coherent scattering is given by the average of the scattering lengths in the sample b . The second component, incoherent scattering, provides no information about the relative distances between atoms in the sample. Unlike coherently scattered neutrons, the intensity of the incoherently scattered neutrons is independent of angle. In this case, one may think of the incident neutrons as interacting with each nucleus of the sample separately. The 2 intensity of incoherent scattering is given by the variation in scattering lengths b2  b , whether because of nuclear spin state or isotopic identity. In small-angle diffraction studies, incoherent scattering produces a flat, ‘background-like’ signal which may limit the analysis of the scattering data at larger angles (smaller scales). One important consequence of this effect is that the presence of hydrogen in a sample, for which the incoherent scattering cross section is very large, tends to generate a large background. In some experiments this can be overcome by substituting D2O for H2O in experimentally altered or synthetic materials. This is not, of course, possible for analysis of natural rock materials. Similarly, strong incoherent scattering

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from vanadium is often used for calibration of detectors in various kinds of experiments. Coherent and incoherent scattering also occur during the scattering of electrons by X-rays, the incoherent scattering in this case corresponding to free recoil scattering of individual electrons under the impact of a photon (Compton scattering). Because the photons in Compton scattering may be considered to be interacting with each electron in the sample separately, there are no coherent interference effects in the wavefunction of the scattered photon that reveal the relative positions of electrons in the sample. The coherent and incoherent scattering cross sections, which may be found in standard tables, are defined in terms of the scattering lengths. The coherent cross section is given by: 2

c  4 bc2  4  b ,

(28)

and the incoherent cross section is given by:



 i  4 bi2  4  b 2  b

2

.

(29)

These tables typically provide the bound scattering lengths, which assume that the scattering nucleus is fixed in sample. In a neutron scattering study, the incident neutrons are entering a potential field produced by the sample given by a sum of -function scatters in space. For wide-angle diffraction studies one is probing length scales comparable to interatomic distances. However, in smallangle study, one does not resolve interatomic distances and a coarse-grained picture of the experimental system may be adopted. The scattering length density ((r), see below for a definition) is a continuous function of position that assigns a scattering length to every point in space. As discussed further below, the measured scattering intensity, given as a function of angle, is related to local differences in scattering length density (r) by means of a Fourier transform. In particular, for a porous rock, the differences in scattering length density (r) between the pore space and different mineral phases produce its distinctive small-angle diffraction pattern. In this way, one may test or parameterize a model of the pore space and distribution of minerals in a rock by performing small-angle X-ray and neutron diffraction experiments. Neutron and X-ray techniques commonly examine size ranges from approximately 1 nm to 10 m, which can be adjusted somewhat by selecting appropriate wavelengths. X-ray sources typically use energies from 10–30 keV (0.124–0.041 nm wavelengths). Neutron energy ranges are are typically described as hot, which are in thermal equilibrium with the temperature of a hot graphite block in the reactor core ( ~ 0.04–0.1 nm), thermal, which are in thermal equilibrium with the temperature of the cooling water around the reactor core ( ~ 0.1–0.3 nm) and cold, which are in thermal equilibrium with the temperature of a cold source such as liquid H2 ( ~ 0.3–2.0 nm). It is typically longer wavelength thermal and cold neutrons that are used for small angle scattering experiments. The kinetic energies of neutrons are in the meV range, with: En  meV   81.805 /  2 ,

(30)

for  in Å, much lower than those of photons of similar wavelengths. Small angle light scattering techniques (e.g., Jung et al. 1995; Alexander and Hallett 1999; Bittelli et al. 1999; Cipelletti and Weitz 1999; Holoubek et al. 1999; Stone et al. 2002; Chou and Hong 2004, 2008; Liao et al. 2005) have been less commonly employed for analysis of porous samples, but have the potential ability to extend the size range investigated because the wavelengths of the visible laser light used are significantly longer than typical X-ray and neutron values. As

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in standard petrographic analysis, however, samples for SALS analysis will have to be thin enough to permit transparency and reduce multiple scattering, given the long wavelengths used. In addition, a combination of scattering and imaging techniques allows characterization of porosity over a yet wider range of scales, from nanometer to centimeter. Scattering data yield information about the shape and size of scatterers, including the scattering contrast and volume, overall and cumulative porosities, pore distribution geometry (mass fractal behavior), the nature of the pore/rock interface (surface fractal behavior), characteristic lengths, and the surface area to volume ratio. A number of early studies (Debye and Bueche 1949; Riseman 1952; Vineyard 1954, Guinier et al. 1955; Debye et al. 1957; Blech and Averbach 1965; Tchoubar and Méring 1969; Vonk 1976) defined the mathematics of the scattering experiment, and began examining particle size distributions and porous materials. To our knowledge, however, the earliest study to report SAS data on rock material is that of Hall et al. (1983) who used neutron scattering to examine the pore size distribution of shales. They noted the asymmetry of the scattering, and that it was related to the bedding plane of the samples and calculated pore size distributions and cumulative pore volumes, and found that the pore volume distribution appeared to be bimodal, a result confirmed in later work (e.g., Swift et al. 2014). While a number of rock materials have been studied (cf. Schmidt 1989; Radlinski 2006), including: coals and hydrocarbon source rocks (Bale and Schmidt 1984; Reich et al. 1990; Haenel 1992; Radlinski et al. 1996, 1999, 2000a,b, 2004; Radlinski and Radlinska 1999; Sastry et al. 2000; Sen et al. 2001, 2002a; McMahon et al. 2002; Prinz et al. 2004; Avdeev et al. 2006; Connolly et al. 2006; Radlinski 2006; Mares et al. 2009; Melnichenko et al. 2009, 2012; Sakurovs et al. 2009; Mastalerz et al. 2012, 2013; Cai et al. 2014; Thomas et al. 2014), sandstones, shales, and carbonates (Hall et al. 1983, 1986; Mildner et al. 1986; Wong and Howard 1986; Triolo et al. 2000, 2006; Sen et al. 2002b; Lebedev et al. 2004, 2006, 2007; Giordano et al. 2007; Anovitz et al. 2009, 2011, 2013a,b, 2014, 2015a,b; Favvas et al. 2009; Jin et al. 2011, 2013; Clarkson et al. 2012a,b,c, 2013; Clarkson and Haghshenas 2013; Ruppert et al. 2013; Wang et al. 2013; Bahadur et al. 2014; Barbera et al. 2014 Swift et al. 2014), clays and soils (Borkovec et al. 1993; Knudsen et al. 2004), sulfides (Xia et al. 2014), and igneous rocks (Lucido et al. 1985, 1988, 1991; Floriano et al. 1994; Kahle et al. 2004, 2006; Anovitz et al. 2011; Bazilevskaya et al. 2013; Navarre-Sitchler et al. 2013). Only the more recent of these studies, however, have transitioned from examining small angle scattering of rocks as materials, often noting the apparent fractal character of pore surfaces, to using these data to understand geological processes in broader contexts.

Theoretical basis of scattering experiments As noted above, the primary variable in the analysis of scattering data is the scattering length density. The average scattering length density  for a particle is simply the sum of the scattering lengths (b)/unit volume, the local average of scattering lengths (strength) over a local subvolume. Thus:

 

n

xbc  i 

i 1

vm

,

(31)

where bc(i) is the bound coherent scattering length for the ith of n atoms in the formula, and Vm is the molecular volume such that: vm 

m N Av d

(32)

where m is the molecular mass of the phase, NAv is Avogadro’s number, and d is the phase

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density. For X-rays bc(i) is replaced by Zire, where Zi is the atomic number of the ith atom, and re is the classical radius of the electron (2.81 × 10-13 cm). For a group of identical, randomly oriented particles the intensity of the coherent, elastic scattering is dependent only on the magnitude of Q, the scattering vector, and is given as: I  q   N ( V )2 P  Q  S  Q  ,

(33)

where N is the number of particles per unit volume, V is the volume of the particles, P(Q) is a form factor that depends on the shape of the particles, S(Q) is a structure factor that describes the inter-particle correlation structure, and   (r )  c

(34)

the scattering density difference between the scattering particles and the surrounding solvent. Q is the scattering momentum transfer. For neutron or photon scattering, Q is defined as the difference between the incident and scattered wavevectors ki – kf. Since elastic interactions are characterized by zero energy transfer |ki| = |kf|, Q is thus an inverse-space distance, defined as: Q  ( 4  /  ) sin()  2  / d ,

(35)

where d is the real-space correlation length and 2 is the angle between the incoming and scattered beams, the scattering angle and  is the wavelength of the neutrons or photons. (Note that in some cases this angle is defined as , in which case sin  above becomes sin /2.) Scattering data is commonly presented as intensity as a function of Q, or as some transform of that plot (see below). Scattering intensity is derived from the correlation function (r) of the material. In standard diffraction analysis diffraction intensity is derived from the lattice structure of the material under study, and the static structure factor S(Q) is, therefore, compose of a sum of -function peaks. However, for a system without long-range order, there are no -function peaks. Thus, as noted by Radlinski (2006, see there for additional review of scattering concepts discussed below), the correlation function (r) is analogous for a disordered material to the lattice structure in crystals. Following Debye and Bueche (1949) and Debye et al. (1957) the correlation function is defined as: (r ) 

Av

  A B

Av

(36)

,

where  refers to value of the property (e.g., scattering length density for neutrons, electron density for X-rays, or the dielectric constant for light) whose variation provides scattering contrast, referred to as scattering length density for both X-rays and neutrons. From this basic relationship it can be derived (Guinier and Fournet 1955) that: I  Q   2

   r  exp  i Q  r  d.

(37)

For isotropic media this becomes: I  Q   4  2



 r  0

sin  Qr  Qr

dr.

(38)

The key point here is that the correlation function and the scattering function are Fourier pairs. This concept should be familiar to most geochemists, as diffraction patterns and crystal structures are simply another example of the same relationship. In wide-angle diffraction, one

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is typically studying arrangements of point scatterers while in small-angle diffraction one examines a coarse-grained picture. In both regimes, one may find either disorder or long-range order.

The two-phase approximation and its limitations In his discussion of small angle neuton scattering Radlinski (2006) noted that “for a wide range of substances, the SAS data for geological materials and porous media can generally be interpreted using a two-phase approximation.” This is because the scattering length density of most minerals is both roughly the same (3–7 × 1010 cm-2) and significantly greater than that for an empty pore (~0 cm-2), and the scattering intensity is a function of the square of the difference. This greatly simplifies analysis of scattering data. As will be discussed below this also facilitates combination of scattering and imaging data, as the images need only be considered in binary (mineral/pore) form. While the two-phase approach based on a combination of scattering and imaging data is computationally sound, Anovitz et al. (2009, 2013a) pointed out a caveat in its application that must be considered when analyzing scattering data from mineralogically complex rocks. Table 1 shows the scattering length densities, and contrasts of those minerals relative to vacuum (empty pores), and dolomite. If we consider a calcite-dolomite-pore system with all three phases of the same size, then 1/3 of the scattering surfaces are calcite-pore, 1/3 is dolomite-pore, and 1/3 is calcite-dolomite. From Table 1 it is clear that the contrast due to the calcite-dolomite scattering is much smaller than that from the mineral-pore interfaces for both X-ray and neutron scattering. However, for neutron scattering mineral-pore scattering can Table 1. Scattering length densities and contrasts for selected minerals. All scattering length density values ×1010 cm-2, and contrast values are ×1020 cm-2.

Mineral

Neutron

X-ray

Neutron

ratio

X-ray

Neutron

Contrast with Vacuum

Contrast with Dolomite

X-ray (10 keV)

Scattering Length Densities

Almandine

35.84

6.316

1284.51

39.89

32.20

139.24

0.81

Brucite

20.69

2.325

428.08

5.41

79.19

11.22

9.55

Calcite

22.98

4.723

528.08

22.31

23.67

1.12

0.48

Diopside

27.74

4.867

769.51

23.69

32.49

13.69

0.30

Dolomite

24.04

5.416

577.92

29.33

19.70

0.00

0.00

Enstatite

26.95

5.152

726.30

26.54

27.36

8.47

0.07

Graphite

19.16

7.534

367.11

56.76

6.47

23.81

4.49

Hematite

42.80

7.293

1831.84

53.19

34.44

351.94

3.52

Magnesite

25.44

6.328

647.19

40.04

16.16

1.96

0.83

Magnetite

41.92

7.010

1757.29

49.14

35.76

319.69

2.54

Muscovite

23.87

3.793

569.78

14.39

39.60

0.03

2.63

Pyrite

41.10

3.831

1689.21

14.68

115.10

291.04

2.51

Quartz

22.45

4.185

504.00

17.51

28.78

2.53

1.52

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become much more important where hydrous minerals (e.g. brucite in Table 1) are involved. Similarly, as the pore fraction becomes small the influence of otherwise relatively weak, but abundant, mineral-mineral boundaries becomes more significant. It is also clear from Table 1 that the ratio between X-ray and neutron scattering is quite variable, depending on the minerals involved. Thus for rocks with complex mineralogy the possibility exists to combine SAXS and SANS to better understand the structure of the sample.

Sample preparation A key factor in obtaining high-quality SAS results is the method used for sample preparation. This is because several factors compete to improve or adversely affect the data. The first is counting statistics. For a given sample, the larger the area, and the thicker the sample, the greater the scattering, but also, for increased thickness, the greater the absorption and probability of multiple scattering. For X-ray scattering this is typically not a problem, as the large flux, especially on synchrotron-based instruments, often makes counting times quite short. For neutrons, however, increasing the count rate can be a significant advantage, as neutron experiments are often “flux limited”. This is because neutrons are relatively weakly interacting with most materials, and the flux of even the most intense neutron sources (currently the Spallation Neutron Source at the Oak Ridge National Laboratory, Tennessee, USA) is relatively low, especially when compared with those from modern synchrotron X-ray sources. While sample area is only limited by beam size and available sample size, however, the thickness of the sample cannot be increased infinitely. This is both because of beam attenuation and because of multiple scattering effects that distort the scattering pattern in ways that, while generally predictable (scattering intensity is shifted from low to higher Q, especially at lower Q), are difficult to model and correct for. The macroscopic bound atom scattering cross section of a material may be calculated as:  B   k k ,

(39)

k

k  N nk ,

(40)

N AV , M

(41)

M  nk mk ,

(42)

N 

k

where mk is the atomic mass of element k, nk is the number of atoms of element k per scattering unit, NAV is Avogadro’s number,  is the mass density, n is the number density of the scattering units, k is the number density of atoms of element k, and k is the total (coherent plus incoherent) bound atom scattering cross section for element k. The standard unit of absorption cross section is the barn. 1 barn = 10-28 m2 or 10-24 cm2. The macroscopic bound atom scattering cross section is not, however, equivalent to the macroscopic total scattering cross section S, which depends on a number of other factors such as temperature, neutron energy, and the structure and dynamics of the sample. This is given as: 

S  dEf  d 0

4

d 2 , ddEf

(43)

where (d2/ddEf) is he macroscopic differential cross section for scattering into a solid angle

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 and energy Ef. S may be significantly different from b, especially for hydrous materials. The macroscopic total cross section T is then defined as the sum of the macroscopic total scattering cross section S and the macroscopic absorption cross section A. As with all transmission techniques absorption from this source may be described using the well-known Beer–Lambert law as: I  I 0 exp   A t  ,

(44)

where A is the macroscopic absorption cross section in units of inverse distance, and t is the distance through or into the material. The value of the absorption cross section, however, depends on the wavelength and type of the radiation employed. As noted above, neutrons primarily interact with the nucleus or unpaired orbital electrons. For neutrons then, there are several types of absorption, depending on the isotope involved. An isotope may absorb neutrons, characterized by the capture cross section, fission, characterized by the fission cross section, or scatter neutrons, characterized by the scatter cross section. As noted above, scattering can be split into coherent, and incoherent interactions, and the macroscopic scattering cross sections are just the product of the microscopic cross section per molecule and the number of molecules per unit volume. Typically neutron moderators (e.g., 1H) have large scattering cross sections, absorbers (e.g., 10 B, 113Cd) have large capture cross sections, and fuels have large fission cross sections (e.g., 235 U, 238U, 239Pu). These may then decay or not. The first is the origin of the radioactive activation observed for some materials that have been in a neutron beam, which is typically emitted in the form of gamma or beta radiation. In calculating the absorption cross section one typically assumes natural isotope abundances unless the sample has been specifically modified. Differences due to natural isotopic partitioning are typically too small to have much effect. For most materials the total cross section, then, is just the sum of the scattering and absorption cross sections. The latter also depends strongly on the energy of the neutron, and increases at low energies, typically as the inverse of the neutron velocity for lower energy neutrons. Thus cold neutrons are advantageous for studying many material properties as they interact more strongly with the sample. Absorption is also somewhat temperature-dependent, but this typically makes little difference for most (U)SANS studies. For most minerals the linear attenuation factor for a combination of absorption effects is relatively small. Thus it is multiple scattering, not absorption that is a primary limitation in the preparation of most samples for neutron studies. For electromagnetic radiation such as light and X-rays, however, absorption is primarily due to interactions with the electrons around each atom in a given material. Both scattering and absorption processes occur, but fission cross sections need not be considered. Energy level transitions in the electron orbitals can also be observed, permitting spectroscopic analysis as well (similar transitions can be observed using inelastic neutron techniques but, again, the absorption cross sections are very different, see Loong 2006). The interactions of light with matter have been considered extensively in a previous volume of this series (Fenter 2002) and will not be covered in detail here. However, in the context of sample preparation of SAXS/ USAXS/SALS studies increased absorption with thickness, especially at longer wavelengths, must be considered. Given the above definitions, we may now calculate the total scattering probability for a slab sample of thickness t at an angle  to the beam as: S

S 1  exp   T t sec      .  T 

(45)

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If we know the scattering probability, we can then solve for the thickness of a slab perpendicular to the beam as: t 

1  S T  ln 1  , T  S 

(46)

T . S

(47)

which has no solution if: S

(http://www.ncnr.nist.gov/instruments/dcs/dcs_usersguide/how_thick_sample/#cross_ sections) The second variable that must be considered is multiple scattering. If a neutron or electromagnetic wave can be scattered once during its transit through a sample, it stands to reason that it can be scattered more than once. Multiple scattering thus has two effects, both deleterious. It attenuates scattering that should be going in a given direction, and intensifies the signal at another angle. Typically this transfers intensity from low-Q to higher-Q without significantly changing the integrated intensity. The relationship between the differential cross section and the pair correlation function only holds in the limit of single-scattering. When an incident X-ray or neutron scatters multiple times from the sample, the straightforward relationship between the structure of the material and the measured scattering signal is lost. There are two basic approaches to dealing with this problem: minimize the effect, or correct for it. For a given sample the thinner the sample and the shorter the transmission path, the less likely this effect is to be significant. Although it is not always clear, a priori, what this sample thickness should be, a rule-of-thumb is that if transmissions are greater than 90 percent multiple scattering effects are small. Shorter wavelengths also reduce multiple scattering effects, as they are less absorbed by the sample. A second approach (Sears 1975) is to subdivide the sample into a series of smaller sample using absorbing spacers parallel to the incident beam. Alternatively, there have been several suggestions of data processing approaches to correcting for multiple scattering effects. These can be broken down into two groups, analytical approximations (e.g., Vineyard 1954; Blech and Averbach 1965; Sears 1975; Schelten and Schmatz 1980; Soper and Egelstaff 1980; Goyal et al. 1983; Berk and Hardman-Rhyne 1988; Andreani et al. 1989; Mazumdar et al. 2003) and Monte Carlo simulations (e.g., Copley 1988; Dawidowski et al. 1994; Rodríguez Palomino et al. 2007, Mancinelli 2012). In cases where the experimental design will require a thick sample where multiple scattering is likely it may also possible to correct for the effect, at least in part, using an empirical approach (e.g., Sabine and Bertram 1999, Connolly et al. 2006), in which measurements are made on pieces of the same sample of various thicknesses. These can be fitted, possibly using the equation described by Vineyard (1954), and multiple scattering from a sample of known thickness corrected for. Sabine and Bertram (1999) also suggest that measurements made at various thicknesses and wavelengths can be used to obtain absolute values for the scattering cross section for a material, but the reliability of this approach is uncertain. For the purposes of sample preparation, the results of the earliest of these studies (Vineyard 1954) provide a good starting point. Vineyard (1954) considered an infinite slab of some thickness. He assumed that only first and second order scattering were of importance, a monoenergetic beam, elastic scattering, and a quasi-isotropic approximation. Figure 16 shows the results of his model of the fraction of multiple scattering as a function of the angle of the neutron beam relative to the slab normal and a thickness parameter T t, where T is the

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Figure 16. Modeled multiple scattering. Ratio of second scattering event B to initial scattering event A as a function of scattering cross section times thickness. For most SANS experiments on geologic materials the value of  is at or near zero degrees. [Redrawn after Vineyard GH (1954) Multiple Scattering of Neutrons. Physical Review, Vol. 96, 93–98 Used with permission of the American Physical Society.]

total scattering cross section (scattering and absorption) and t is the thickness. He concludes that for modest scattering angles the ratio of the multiple scattering fraction to T t is almost independent of scattering angle, and that the value of T t must be smaller than approximately 0.05 if the fraction of multiple scattering is to be kept less than 10 percent. He also notes that, while the fraction of multiple scattering does decrease with thickness, it does so as T t ln(T t) and not linearly with t. Figure 17 shows the sample preparation strategies developed by Anovitz et al. (2009) for (U)SANS. These have also been used successfully for USAXS measurements at the APS, and thus probably form a reasonable starting point for those interested in neutron and X-ray small angle studies of geological and ceramic materials. The figures on the left in Figure 17 show the original technique in which samples were mounted on glass plates with superglue, ground to thickness, the floated off the glass using acetone to dissolve the glue and remounted on Cd masks. This was successful but difficult, as the thin samples tended to break. An alternative strategy of mounting the samples permanently on quartz glass plates is shown on the right of Figure 17. This is very simple to use and has been quite successful. In addition, as shown on the right-hand figure as well, powders or well cuttings can be cast in epoxy, then remounted on the quartz glass and ground to thickness. Initial experiments suggested that a thickness of approximately 150 m yielded significant scattering intensity with minimal multiple scattering. This is illustrated in Figure 18, which shows the transmission measurements for a series of shale samples from the Eagle Ford shale as a function of thickness. As can be seen, near a thickness of 150 m the transmission exceeds 90 percent, the value suggested by Vineyard (1954). Tests have shown this to be far superior to the alternative of filling 1-mm-wide quartz glass HelmaTM “banjo” or “lollipop” cells, in which multiple scattering, especially at the USANS scale, can be significant.

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Figure 17. Two methods of mounting samples for (U)SANS analysis. The images on the left shows a sample ground to 150 m, then floated off the glass slide (originally glued on using super glue) and attached to the Cd mask. The image on the right shows a sample mounted on a quartz glass slide that was then taped directly to the Cd mask. Unlike the samples to the left and middle, the sample on the right consists of drill cutting mounted in epoxy, rather than solid rock, but the method of mounting on quartz glass works similarly well for larger samples (Anovitz, unpb.).

Figure 17 also shows the samples mounted on Cd masks. This is necessary to define the beam in (U)SANS, but is not needed in USAXS where the beam can be focused or masked before the sample. Typically USAXS beam sizes are fairly small (< 1mm2 at APS), while those used at (U)SANS instruments are much larger (up to nearly 1 in2) to accommodate lower flux rates. However, in the latter cases specialized masks, such as rectangular, slit (cf. NavarreSitchler et al. 2013), or annular (Anovitz et al. 2015b) shapes can be used.

Figure 18. USANS transmission data given as the ratio of the rocking curve and wide transmissions for a series of clay and carbonate-rich samples from the Eagleford Shale. Tx. While there is significant scatter it is apparent that transmissions reach values near 0.9 at thicknesses near 150 mm (Anovitz, unpublished)

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Geometrical principles of small-angle scattering experiments There are five basic geometries/approaches used for small angle scattering (SAS) experiments: pinhole (SAXS, SANS, V-SANS, SALS), Bonse-Hart (USANS, USAXS), Kratky, spin-echo (SESANS) and time of flight (TOF-SANS). Depending on the wavelength of the incident energy each covers a specific size range. Thus, one or more are often used in combination to extend the range of pore scales interrogated. For neutron facilities a world directory of SANS instruments is maintained by the Large Scale Structures Group at the Institut Laue-Langevin at: http://www.ill.eu/instruments-support/instruments-groups/groups/ lss/more/world-directory-of-sans-instruments/ Pinhole SAS. The basic geometry of a two-dimensional pin-hole SAS system is shown in Figure 19. The scale of the instrumentation for pin-hole geometry instruments varies dramatically. SANS spectrometers can be as long as 80 m (D11 at the Institut Laue–Langevin), and laboratory-scale SAXS instruments may be only cabinet-sized. In addition, the type of detector must be selected for the energy type (X-rays, neutrons, light) of interest.

Figure 19. Schematic of a standard pinhole SAS instrument. The detectors may or may not be in a vacuum tank depending on the instrument type.

These instruments are, indeed, very similar in design to a standard pin-hole camera. As noted above, the scattering variable, Q, is defined as Q = (4/) sin(). Thus, like the more familiar X-ray diffraction (XRD), scattering data is measured in reciprocal space. However, unlike XRD data these are not derived from the absolute square of the Fourier transform of the structure, but rather of the density-density correlation function. For SAS instruments using a two-dimensional area detector some typical results of scattering experiment looks like those shown in Figure 20. Figure 20a shows an example of a sample of the Garfield Oil shale, which is typical of most patterns obtained for rocks. Such patterns may or may not be circular (this one is slightly ellipsoidal, reflecting bedding structure in the shale), and more complex features may occur that represent large-scale repeating structures in the material. However, for simple isotropic systems the results are typically circular, or nearly so, and can be radially integrated where the intensity I is often given as d c / d, the change in the macroscopic coherent scattering cross section with a change in angle. When normalized to an absolute scale (see below) this is given in units of inverse thickness (1/cm). Figure 20b, on the other hand, shows scattering from a powder sample of the synthetic zeolite MCM-41. The pores of this material are arranged in a regular lattice structure, and the first two Debye-Scherrer rings can be directly observed in the pattern. It is often useful to extend the range of a SANS measurement to lower Q in order to better overlap the USANS data. While the combined ranges of SANS (e.g., 0.008 nm-1 to 7.0 nm-1 for

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Figure 20. Examples of 2-D SAS scattering patterns. A) (top) A sample of the Garfield oil shale (Anovitz et al. unpb)., B) (bottom) synthetic zeolite MCM-41 [T. Prisk, pers. comm. Used with permission].

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NG7 SANS at NINS/NCNR) and USANS (e.g., 0.0003 nm-1 to ~ 0.1 nm-1 for BT5 USANS at NIST (Barker et al. 2005), Q > 0.0002 nm at HANARO/KIST, (M.H. Kim, pers. comm.), Q > 0.00014 nm-1 at Kookaburra/ANSTO, Rehm et al. 2013) techniques covers a wide range of scales from approximately 1 nm to 10 mm, Combination of data from the two approaches is, however, somewhat limited by the uncertainties in both instruments in the overlap range. For typical rock materials this is not a factor for USAXS, where this region is covered by the USAXS instrument itself, although even in that case there tends to be greater noise in the overlap region between the pinhole SAXS and USAXS at much smaller sizes (Q ~ 0.1–0.2). One method to extend the Q-range for the SANS instrument employs a set of biconcave MgF2 lenses placed in the beam before the sample (Eskildsen et al. 1998; Choi et al. 2000; Susuki et al. 2003; Littrell 2004; Oku et al. 2004; Mildner 2005; Hammouda and Mildner 2007). These have the effect of shrinking the neutron spot size on the detector, thus lowering the Q range and increasing the intensity at low Q. Unlike light, however, for most materials the refractive index for neutrons is less than one, but only by a few parts in 105, for cold (~10 Å) neutrons. Thus concave lenses, rather than convex lenses are convergent, but a number of them are needed for significant focusing to occur. Nonetheless, this technique has now become a successful method to extend the minimum Q range of pin hole SANS instruments. Another method that improves the quality of SANS data in the overlap region for neutron studies is the VSANS (Very Small Angle Neutron Scattering) instrument. There are several different designs for a stand-alone VSANS instrument: extremely long pinhole designs such as the 80-m D11 instrument at the Institut Laue–Langevin, Grenoble, France (Q > 0.005 nm-1, Lindner et al. 1992; Lieutenant et al. 2007); focusing instruments such as the KWS3 instrument at the Forschungs-Neutronenquelle Heinz Haier–Leibnitz (FRM II) near Munich, Germany (Alefeld et al. 1997, 2000a,b; Fig. 21), for which there are two sample positions, the 9.5 m position covers 0.001 nm-1 < Q < 0.03, and the 1.3 m position extends the high end of the Q range to 0.2 nm-1, and multiaperture converging pinhole collimator designs (Nunes 1978, Carpenter and Faber 1978, Glinka et al. 1986, Thiyagarajan et al. 1997; Barker 2006; Brûlet et al. 2007; Désert et al. 2007; Hammouda 2008) such as the V16 Instrument at BER II, Helmholtz-Zentrum Berlin (Clemens 2005; Vogtt et al. 2014; Q > 0.03 nm-1), the G 5-4 instrument (PAXE) at the Laboratoire Léon Brillouin, Saclay, France (Q > 0.01 nm-1), and the VSANS instrument under construction at NCNR/NIST, some of which are also combined time-of-flight instruments.

Figure 21. Schematic of the KWS-3 focusing mirror V-SANS and the Julich Centre for Neutron Science. Figure courtesy of JCNS. 1) Neutron guide NL3a, 2) velocity selector, 3) entrance aperture, 4) toroidal mirror, 5) mirror chamber, 6) sample positions, 7) detector. [V. Pipich, pers. comm. Used with permission]

As an example, the VSANS instrument at NCNR/NIST (Fig. 22) is 45 m in total length, and uses a high-resolution (1.2 mm fwhm) 2-D detector along with the longer flight path (45 m, as opposed to 30 m for the more standard NG7 SANS instrument at the NCNR) to cover 0.002 nm-1 ≤ Q ≤ 7 nm-1. To enhance the count rate at lower Q either larger samples using converging beam collimation, or relaxed resolution using slit collimation. The instrument has three detectors that can be placed independently at different distances from the sample allowing the full Q-range to be measured in one setting. The incident wavelength

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Figure 22. Schematic of the VSANS instrument under construction at NCNR/ NIST. [J. Barker, pers. comm. Used with permission]

and wavelength resolution are controlled over a wide range with either a standard resolution mechanical velocity selector (Δ/ = 12%), high resolution graphite monochromator (Δ/ = 2%), or low resolution filtered beam covering 0.4 nm ≤  ≤ 0.8 nm with Be filter and guide deflector. The instrument has a large 2-m sample area permitting large sample environments to be used, and full beam polarization using a 3He analyzer is also available (Barker et al. pers. comm.). A third type of pinhole instrument is a small angle light scattering (SALS) system, which uses a laser as the radiation source. While optical techniques have a long and honorable tradition in the analysis of geological materials, to our knowledge only one investigator (Liao et al. 2005) has, as yet, applied SALS to the analysis of porosity in rocks (coal), although it is well known in the study of aggregated particles, including soils. While some materials (black shales, sulfide ores) clearly will not lend themselves to such analysis others, especially those typically analyzed in thin section by transmitted light, would appear to be good candidates. The longer wavelength of light (relative to X-rays) will extend the low-Q range of available data (cf. Zhou et al. 1991; Weigel et al. 1996; Burns et al. 1997; Alexander and Hallett 1999; Cipelletti and Weitz 1999; Holoubek et al. 1999; Bushell and Amal 2000; Bushell et al. 2002; Gerson 2001; Stone 2002; Chou and Hong 2004, 2008; Nishida et al. 2008; Romo-Uribe et al. 2010). Liao et al. (2005) used several techniques, including SALS, light obscuration, settling, 2-D and 3-D imaging to estimate the mass fractal dimensions of coal aggregates. They conducted over 50 tests, and achieved results in reasonable agreement with 3-D structural analysis, while noting that SALS was a much faster method of analysis. This suggests that the potential applicability of this approach to analysis of geological materials needs to be more fully explored. Bonse–Hart. The Bonse–Hart instrument (Compton and Allison 1935; Fankuchen and Jellinek 1945; Bonse and Hart 1965; Shull 1973; Schwahn et al. 1985; Agamalian et al. 1997; Bellmann et al. 1997; Takahashi et al. 1999; Borbely et al. 2000; Hainbuchner et al. 2000; Treimer et al. 2001; Jericha et al. 2003; Villa et al. 2003; Barker et al. 2005; Hammouda 2008; M.H. Kim, pers comm.) is often referred to as a USANS or USAXS. The “U” in this case

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stands for ultra, and refers to the instruments’ ability to measure scattering patters at very low Q values. This range varies somewhat per instrument. As noted above the USANS instruments at NCNR (Barker et al. 2005) and HANARO (M.H. Kim, pers. comm.) can reach Q values down to 0.0003 and 0.0002 nm-1 respectively. The USAXS instrument with a combined pinSAXS for high-Q data at the APS (Ilavsky and Jemian 2009) covers a range from 0.001 to 12 nm-1 at 10–18 keV. That is, significantly larger scales than can be achieved with pinhole instruments of reasonably achievable lengths. Available USANS instruments include those at the NIST Center for Neutron Research (BT5, Barker et al. 2005), ANSTO, Australia (Kookaburra, Rehm et al. 2013), the Institute Laue–Langevin, Grenoble, France (S18, Hainbuchner et al. 2000), the Paul Scherrer Institute (ECHO), the Institute for Solid State Physics, Tokyo, Japan (C1-3 ULS, Aizawa and Tomimitsu 1995), and the Korean Institute of Science and Technology, South Korea (Kist-USANS, HANNARO Cold Guide Hall). USAXS instruments include those at the Stanford Linear Accelerator (BL4-2, Smolsky et al. 2007), the European Synchrotron Radiation Facility (ID02, Narayanan et al. 2001), and the Advanced Photon Source (Ilavsky et al. 2009). Figure 23 shows a schematic of the USANS image at the NCNR, which will be used as a general example of Bonse-Hart instruments (cf. Barker et al. 2005). The design of this instrument begins with sapphire and pyrolytic graphite prefilters and a pre monochrometer to remove higher energy components of the neutron spectrum and reduce radiation levels. The monochrometer and analyzer are channel-cut, triple-bounce silicon single crystals. The (220) reflection selects a neutron wavelength of 2.4 Å, and the triple bounce geometry dramatically

Figure 23. Schematic of the USANS instrument at NIST/NCNR [Hammouda B (2008) Probing nanoscale structures—The SANS toolbox. http://www.ncnr.nist.gov/staff/hammouda/the-SANS-toolbox.pdf. Used with permission]

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reduces the width of the reflection. It was this latter innovation (Schwahn et al. 1985), coupled with the addition of cutting the crystal and adding absorbers between reflectors (Agamalian et al. 1997) that permitted low background rocking curves to be obtained and successful scattering curves to be measured on more weakly scattering materials. Unlike a SANS/SAXS instrument where a range of Q values is measured simultaneously, in a Bonse–Hart instrument the Q value is varied during analysis by rotating the analyzer in small increments. If no sample is present the combined Bragg reflections require a very precise alignment for neutrons to pass through the instrument. If, however, a scattering sample is placed between the two crystals the alignment condition becomes satisfied for neutrons scattered at a given angle. Five end-window counters placed in the final reflection direction provide neutron detection. A key factor in understanding and analyzing data obtained using a Bonse–Hart instrument is the effect of slit geometry. A standard SANS instrument uses a two-dimensional detector, thus explicitly measuring the scattering pattern at all observable angles. A Bonse–Hart instrument, by contrast, uses a one-dimensional slit geometry. Under these conditions the twodimensional pattern is compressed, or “smeared” into one dimension. As noted by Hammouda (2008), for the NIST instrument the slit geometry provides very tight standard deviations in Q resolution (approximately 2.25 ×10-5 Å-1) in the horizontal direction, and much wider ones (approximately 0.022 Å-1) in the vertical direction. This is illustrated in Figure 24. While scattering from a sample is typically radial, if not necessarily circular, the slit geometry integrates the actual scattering over a narrow horizontal range, but a wide vertical range, thus including in that integration intensities from greater radial dimensions that the measured Q value (along the x-axis). In the latter case it may be possible to account for anisotropy using asymmetry values measured at the lowest Q values on the SANS instrument. This assumes, however, that this effect is not Q-dependent, which is unlikely. One limitation for most USAS instruments already mentioned is that the one-dimensional detector limits the ability to analyze non-isotropic scattering. The USAXS instrument at the

Figure 24. Binning caused by sli t geometry that leads to slit smearing. Scattering intensity us summed over the rectangular bin. [Redrawn after Hammouda B (2008) Probing nanoscale structures—The SANS toolbox. http://www.ncnr.nist.gov/staff/hammouda/the-SANS-toolbox.pdf]

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APS permits direct measurement of the asymmetry of the scattering pattern as low-Q (Ilavsky et al. 2009). This instrument adds a second set of channel-cut, two-reflection, Si (220) crystals before and after the sample. The main crystals are oriented vertically, and the second, inner set horizontally, thus both horizontally and vertically collimating the beam. The sample can be rotated, allowing measurements in multiple directions, either by fixing the sample angle () and varying Q, as in standard USAS experiments, or by fixing Q and varying . A two dimensional pattern (obtained point-by-point) can be obtained by varying both Q and . Beam-collimation reduces the intensity, however, which limits the practical range of Q to 10-4 Å-1 < Q < 0.1 Å-1. This integration is given as (Barker et al. 2005; Hammouda 2008): QV d  d  Q   1  d Q    y  d smeared Qv 0



Q 2  Qy2 d



(48)

The resulting data may either be fit as is, accounting for the observed smearing, or desmeared before fitting using one of several available algorithms (e.g. Kline 2006; Ilavsky and Jemian 2009). The latter makes association with SANS results at higher Q, and results obtained from image analysis at lower Q easier, but may introduce additional noise and uncertainties, especially if the scattering pattern is not circular. TOF-SANS. The difference between continuous-source SAS instruments and time-offlight (TOF) instruments lies less in the design of the instrument itself than in the nature of the source. For neutrons continuous sources are typically reactors (e.g. NCNR, HFIR), while pulsed sources are either spallation sources (WNR/LANSCE, SNS, ISIS, ILL), continuous sources to which a neutron chopper has been added or the pulsed IBR-2 reactor in Dubna, Russia.. As the name implies, In TOF-SAS instruments the initial flux of radiation hits the sample in a single pulse of some known time width and intensity. This usually uses a wide wavelength range simultaneously. Each pixel in the detector must, therefore, measure the intensity as a function of time relative to the time the pulse hits the sample, and the time signal for each neutron can be recorded (time-stamped). Continuous sources, by contrast, typically operate in an integrating mode. For most geological applications there is not much difference between continuous and TOF instruments, although the wide wavelength range can complicate the use of sample environment materials with a Bragg edge such as a sapphire window. However, the TOF instruments do provide the opportunity to measure kinetics of fast processes, and may be particularly useful for dynamic imaging. Examples of such instruments are the EQ-SANS and TOF_SANS instruments at the SNS at Oak Ridge National Laboratory, and LOQ and SANS2d at ISIS, REFSANS at the FRM-II, and LQD and LANSCE and Los Alamos National Laboratory. Kratky geometry. The Kratky geometry, often seen in commercial SAXS instruments, uses a line source and slit block collimation, rather than a pinhole (Kratky and Skala 1958). This allows for smaller laboratory-scale instruments, and often an increased sample flux. However, the line geometry induces smearing, much as does the Bonse–Hart. The scattering is highly collimated perpendicular to the slit direction, but allowed to broaden parallel to it, although some designs us a focused line geometry that minimizes smearing. GISAS. Unlike transmitted geometries, grazing incidence SAS is a surface-sensitive technique, commonly used for the analysis of nanostructured thin films. GISAS provides the opportunity to study surfaces using small angle techniques, where the intensities obtained from normal transmission geometries are typically very small. These measurements are performed in situ and, for GISAXS at least where the fluxes are suitably high can be done in a timeresolved manner to study reaction kinetics. They can also be used to study buried structures non-destructively (Naudon 1995). Most importantly, because the areas illuminated for both

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X-ray and neutron studies are fairly large, GISAS techniques probe a statistically relevant surface area of square millimeters or larger. As this technique has been recently reviewed in this series (De Yoreo et al. 2013) it will be only briefly discussed here. The first GISAS experiments were done using X-ray instruments (GISAXS, Levine et al. 1989, 1991; Müller-Buschbaum et al. 1997, 2003; Naudon and Thiaudiere 1997), and the technique has become well known for X-ray applications (cf. Rauscher et al. 1999; Lazzari 2002; Doshi et al. 2003; Forster et al. 2005; Henry 2005; Lee et al. 2005; Roth et al. 2006; Urban et al. 2006; Sanchez et al. 2008; Renaud et al. 2009). For neutron sources GISANS is in a more developmental stage. It was first reported in 1999 (Müller-Buschbaum et al. 1999a,b) but has become much more widely applied since, largely for polymer applications (MüllerBuschbaum et al. 2003, 2004, 2006, 2008; Wunnicke et al. 2003; Wolff et al. 2005; Ruderer et al. 2012). A time-of-flight version has also been developed (Forster et al. 2005; Kampmann et al. 2006; Müller-Buschbaum et al. 2009; Kaune et al. 2010; Müller-Buschbaum 2013). The general geometry of a GISAS experiment is shown in Figure 25. The beam is directed at the sample at a low incident angle (i), and the reflections are detected at both a final angle (f) and out-of-plane angle (2) which, as above, generates a momentum transfer vector (Q) with units of inverse distance per Bragg’s law. The scattering pattern typically contains a peak for specular reflection (where af intersects the detector in Figure 25), as well as a Yoneda peak defined by the critical angle for total external reflection of the material. In many cases both reflected and transmitted data are detected (cf. Lee et al. 2005). At angles less than the critical angle a certain amount of sample penetration occurs (the so-called evanescent wave), giving this technique the very limited depth penetration (typically only a few nanometer) needed for surface and near-surface analysis. This depth is sensitive to the incident angle. Form factors (defined by the shape of individual scatterers, see below) typically dominate the GISAS pattern for randomly oriented nanoparticles with well-defined shapes, while structure factors (defined by the relationship between the particles) tend to dominate scattering for ordered layers (e.g., polymer thin films, reacted surface layers). While reflectometry techniques have been extensively applied to analysis of mineral surfaces (e.g., Fenter et al. 2000a,b,c, 2001; Cheng et al. 2001a; Teng et al. 2001; Fenter 2002; Schlegel et al. 2002, 2006; Fenter and Sturchio 2004; Geissbuhler et al. 2004; Predota et al. 2004; Zhang et al. 2004, 2006, 2008; Park et al. 2006; Vlcek et al. 2007), experiments using GISAS for analysis of surface experiments and

Figure 25. Generalized geometry of a GISAS experiment. Figure from A. Meyer, Univ. Hamburg (pers. comm., used with permission)

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pore precipitation have been more limited (e.g. Jun et al. 2010; Fernandez-Martinez et al. 2012a,b, 2013; De Yoreo et al. 2013; Panduro et al. 2014), and we know of none for GISANS. However, these have shown that the potential applications of this technique for analysis of precipitation in pores or on mineral surfaces is significant. SESANS. The final type of small angle scattering instrument to be discussed here is the spin-echo SANS experiment (SESANS, Pynn 1980; Keller et al. 1995; Gähler et al. 1996; Rekveldt 1996; Bouwman and Rekveldt 2000; Bouwman et al. 2000, 2004, 2005, 2008; Krouglov et al. 2003a,b,c; Rekveldt et al. 2003, 2005; Uca et al. 2003; Pynn et al. 2005; Grigoriev et al. 2006; Plomp et al. 2007; Andersson et al. 2008a,b; Li et al. 2010; Washington et al. 2014). As with other SAS experiments the spin-echo technique also measures elastic scattering, but begins with a polarized neutron beam, and is based on the Larmor precession of neutron spins in a magnetic field (Mezei 1972, 1980). In a spin-echo instrument there are two identical magnetic fields with opposite orientations along the beam path: one before and one after the sample position. In the absence of a sample the neutron precesses at some angle 1 in the first field, which is reversed in the second so that d = 0 and the neutron polarization is returned to its original state. If a sample is present between the two fields, however, small angle scattering by the sample between the two fields breaks this symmetry, depolarizing the beam, because the path lengths in the second field are no longer equal to those in the first. This is measured using a second polarizer (an analyzer) after the second Larmor device. The polarization of the neutron beam P(z) is then a direct function of the projection G(z) of the autocorrelation function (r) of the density distribution of the sample (r), where z is the spin-echo length (in m). In SANS, by contrast, the intensity distribution I(Q) is the Fourier transform of the autocorrelation function (Andersson et al. 2008a,b). The relationships between these various functions are summarized in Figure 26. While SESANS typically covers a size range similar to that of USANS (typically from tens of nm up to several mm) it has several advantages. The flux is much higher, improving the counting statistics and shortening counting times. No desmearing is required, and multiple scattering is easily accounted for, allowing much thicker samples to be used. In addition, the results are obtained in real, rather than inverse space. Because of this, however, the data do not directly overlap with pinhole SANS at higher Q (cf. Rehm et al. 2013). To date, however, there has been very little work on rock materials using SESANS. Figure 27 shows preliminary data (Anovitz and Bouwman, unpb.) obtained from samples analyzed using (U)SANS by Anovitz et al. (2009). It is clear from these data that SESANS can be successfully applied to rock materials, and that there is significant opportunity to utilize this approach for geologic applications. Magnetic SANS. Another SANS technique that has received little attention for its potential geological applications is magnetic scattering. As mentioned above, neutrons interact not just with the nucleus of an atom, but with unpaired orbital electrons as well. Thus they are highly suited for studying the magnetic structure of materials, and there is a significant literature on this topic (e.g., Scharpf 1978a,b; Cebula et al. 1981; Dormann et al. 1997; Ohoyama et al. 1998; Wiedenmann 2005; Zhu 2005; Michels and Weissmüller 2008). As the focus of this article is pore structures, however, we will not discuss this approach in any further detail, except to comment that its utility in understanding geomagnetism (and possibly particle transport in porous media using magnetic test particles) has yet to be explored. Figure 28 shows an example of the use of SANS to investigate magnetic systems. In type II superconductors an externally-imposed magnetic field may form a flux lattice on the surface of the crystal. The magnetic field lines form filaments or vortices with a quantized magnetic flux that penetrates the superconductor in a regular lattice structure. The lattice constants of these vortex structures are on the order of a few nm. The sensitivity of neutrons to magnetic ordering implies that, in a small-angle neutron scattering experiment these flux lattices produce single

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Figure 26. Relationships between density distribution, autocorrelation, SESANS projection and scattering functions (redrawn after Andersson R, van Heijkamp LF, de Schepper IM, Bouwman WG (2008) Analysis of spin-echo small-angle neutron scattering measurements. Journal of Applied Crystallography, Vol 41, p. 869–885.

Figure 27. Test SESANS measurements on four carbonate samples (Anovitz and Bouwman, unpb.). Solid squares: MC88B94, open squares: Hueco ls, solid triangles: Solnhofen ls, open triangles: Solnhofen ls, heated to 700 ºC. (The first two samples are from Anovitz et al. 2009).

Figure 28. A vortex lattice diffraction pattern for YBa2Cu3O7- (YBCO) taken at 2 K in a 4T applied field after field cooling. Overlaid patterns indicate the different VL structures that make up the overall diffraction patterns, and the angles between certain Bragg spots.  is bisected by a×. White arrows indicate {110} directions. The diffraction patterns were constructed by summing detector measurements taken for a series of sample angles about the horizontal and vertical axes. The real space VL can be visualized by rotating the reciprocal space image by 90° about the field axis and adding an additional spot at the center [Reprinted from White JS, HinkovV, Heslop RW, Lycett RJ, Forgan EM, Bowell C, Strässle S, Abrahamsen AB, Laver M, Dewhurst CD, Kohlbrecher J, Gavilano JL, Mesot J, Keimer B, Erb A (2009) Fermi surface and order parameter driven vortex lattice structure transitions in twin-free YBa2Cu3O7. Physical Review Letters, Vol. 102, 097001 used with permission of the American Physical Society].

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crystal diffraction patterns. The magnetic diffraction pattern shown in Figure 28 shows the inverse-space lattice pattern for YBa2Cu3O7- (YBCO) taken at 2 K in a 4-T applied field after field cooling. Other examples of small-angle magnetic scattering include analysis of magnetic nanoparticles (e.g. Krycka et al. 2010). To our knowledge, however, small-angle scattering has yet to be advantageously used to study geomagnetism.

Contrast matching Contrast matching is a very useful technique in small angle scattering studies that provides a method to separate connected from unconnected porosity. In addition (Anovitz, unpb.) it can also be used in multiphase materials to explore the question of distinguishing bulk mineralogy from reactive mineralogy as a function of pore size. Figure 29 shows the basics of this approach. One of the key differences between techniques such as BET and MIP and scattering approaches is that scattering sees all of the porosity in the rock (as well, possibly, as effects from grain/grain boundaries, see above), while sorption/intrusion techniques interrogate only accessible porosity, which may be limited by intrusion pressures for nonwetting fluids as described by the Young–Laplace (Washburn’s) Equation (Eqn. 11 above). It is, therefore, of significant interest to separate connected from unconnected porosity in order to relate scattering measurements to phenomenon such as permeability and mass transport.

Figure 29. Schematic illustration of contrast matching. Left: Two-phase mineral/pore system. Middle: system with a contrast-matched fluid (grey) added, note accessible vs. in accessible porosity. Right system with matching fluid present at it appears to the scattering experiment.

As discussed above, the intensity of scattering is a function of the square of the scattering length density difference at an interface. Thus, assuming a two-phase system, if a rock is soaked in a wetting fluid (so that the fluid can be assumed to soak into all accessible pores) with a scattering length density equal to that of the matrix all of the accessible pores will “disappear” during the scattering experiment, and scattering will only be observed from unconnected pores. In many cases the contrast point may be unknown. This is especially true in the case where more than one phase is present in the system. In this case a series of fluid mixtures with different scattering length densities can be used. Because contrast is a function of the difference in scattering length density squared, to a first approximation the intensity should be a parabolic function of scattering length density. If more than one phase is present, however, and if these vary with pore size several parabolas may be needed to fit the data, and these may vary with Q. Alternatively, one can plot the square root of scattering length density, often either at a projected value at I(Q = 0) or a minimum value of Q, which should be composed of two linear trends. This approach has largely been applied in (U)SANS experiments (cf. Stuhrmann and Kirste 1965; Stuhrmann 1974, 2008, 2012; Ibel and Stuhrmann 1975; Stuhrmann et al. 1976,

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1977, 1978; Beaudry et al. 1976; Williams et al. 1979; Akcasu et al. 1980; Jahshan and Summerfield 1980; Koberstein 1982; Hadziioannou et al. 1982; Bates et al. 1983; Hasegawa et al. 1985, 1987; Allen 1991; Hua et al. 1994; Radlinski et al. 1999; Smarsley et al. 2001; Littrell et al. 2002; Connolly et al. 2006; Stuhrmann and Heinrich 2007; Clarkson et al. 2013; Ruppert et al. 2013; Thomas et al. 2014). This is because the scattering length density for neutrons depends on the isotope, rather than just the element involved, and there is a very large difference in scattering length density between hydrogen and deuterium, and thus between H2O (neutron sld = -0.56 × 1010 cm-2) and D2O (neutron sld = 6.392 × 1010 cm-2). Where useful hydrogenated/deuterated methanol, or other solvents can be used (cf. Allen et al. 2007). This range covers that of most minerals, allowing a range of compositions to be matched. While a similar approach is possible for USAXS by adding a highly soluble, high-Z material to water, molecular liquids, metals or other fluids (cf. Smith 1971; Tolbert 1971; Strijkers et al. 1999; Dore et al. 2002; Laszlo et al. 2005; Laszlo and Geissler 2006; Jahnert et al. 2009; Mter et al. 2009; Kraus 2010) it has not, to our knowledge, been tried for geological materials other than coal (Smith et al. 1995). An example of the utility of this approach is shown in Figure 30. Littrell et al. (2002) characterized a series of activated carbons produced from paper mill sludge using ZnCl2. They found that the surface area of the carbons increased as the concentration of ZnCl2 was increased. Contrast matching experiments were used to demonstrate the presence of two phases, a zing-rich particle and a nanoporous carbon, the relative sizes of which were determined from the Q-dependence of the contrast curves. Such an approach (Anovitz et al. 2015b) can also be used to analyze the pore surface mineralogy as a function of pore size, providing a link between porosity, overall mineralogy, and reactive mineralogy as a function of pore size and concentration.

Figure 30. Intensity as a function of scattering length density for activated carbons synthesized from paper mill sludge. Data at Q = 0.01 has a minimum at sld – 3.774 × 1010 cm-2, similar to ZnCl2, ZnO or metallic Zn. Data at Q = 0.1 has a minimum at 5.92 × 1010 cm-2, comparable to amorphous carbon. Replotted after Littrell et al. (2002).

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Reduction and analysis of SAS data Once SAS data have been obtained they must be processed prior to analysis (see Hammouda 2008; Ilavsky and Jemain 2009; for discussions of data reduction and analysis). The extent of this processing depends on the research goals of the project. For instance, if all that is of interest is the sizes of scatterers in some solution, then a simple radial averaging may be all that is needed to determine the Q value of peaks in the data. However, for most geological applications where the concentration of scatterers in a given rock volume is of interest (i.e. the pore fraction or absolute pore volume distribution), then the data must first be corrected for various effects and normalized to an absolute intensity scale (units of cm-1, Wignall and Bates 1987; Russell et al. 1988; Heenan et al. 1997; Glinka et al. 1998; Orthaber et al. 2000; Hu et al. 2011b; Wignall et al. 2012). This can either be done by referencing the results to a precalibrated sample or relative to the intensity of the direct beam. This must be decided before the SAS experiment is performed, as it requires that certain additional data be available, some of which must be acquired during the SAS experiment. The first corrections that must be made are for the effects of dead time, nonuniformities in the detector pixel efficiency, scattering from the empty cell, and blocked beam or dark current scattering which is a measurement made with the beam blocked by a strong absorber (e.g., boron nitride for neutrons) or by a closed shutter. Dead time corrections are made by normalizing the total counts to the beam monitor counts, and pixel efficiency corrections by dividing each pixel intensity by that for an isotropic scatterer such as water or plexiglass normalized to 1 count/pixel (Glinka et al. 1998). Then, for each pixel in the scattering data, one calculates   I sample+cell  I blocked beam     I cell  I blocked beam   I  Q sample     , Tsample+cell Tcell    

(49)

where I is the intensity of the scattering, and T is the transmission, the measurement of which is usually made at one detector distance for each wavelength used in the measurement. For samples mounted on quartz glass the “cell” is a quartz glass slide mounted on a mask of the same diameter (for neutron measurements) with no sample on the slide. This also corrects for other effects such as scattering from beam windows, aperture edges, air in the beam path and spillage of the direct beam around the beam stop (Glinka et al. 1998). Transmission measurements are measurements the fraction of the incident beam that is not scattered by the sample and are the ratio of the transmitted beam intensity, integrated only over the area of the beam spot, to that of the incident beam measured with no sample or cell present. They are often measured with an attenuator in place to avoid damaging the detector. If the sample is mounted on/in a cell transmissions must be measured for both the sample and the empty cell, as shown in Equation (49) above. The final step is to normalize the data to an absolute scale. For SANS the absolute scattering cross section (d/d(Q), Turchin 1965, Glinka et al. 1998, Wignall et al. 2012) is defined as the number of neutrons (n/s) scattered per second into a unit solid angle divided by the neutron flux (n/(cm2 s)). Normalized to sample volume this has units of cm2/cm3, or cm-1. The relationship between the cross section and the adjusted count rate I(Q) in 1/cm2.s is then: I Q  r 2 d Q  ,   d I 0 a AtT

(50)

where I0 is the incident intensity on the sample, a is the area of a detector element, r is the distance between the sample and the detector,  is the detector efficiency discussed above, A is the sample area, t is the sample thickness and T is the measured transmission. At high Q (small sample to detector distances) corrections for geometric effects may be needed as well.

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The result of the above calculations is a two-dimensional scattering pattern similar to that shown in Figure 20 in which all of the values are on an absolute scale. While the data may be analyzed in two-dimensional form using a program such as SASVIEW (SasView, http://www.sasview.org/) it is more common to convert it to a one-dimensional form by angularly averaging the results. This yields a single curve showing intensity as a function of Q. There are, however, a few caveats to this process. Examples of two of these are shown in Figure 31. The left hand image shows scattering (un-normalized) from a shale with the bedding oriented horizontally and parallel to the beam. The asymmetry of the result is clear (cf. Anovitz et al. 2014). A fully radial average of this sample would, therefore, smear out these differences. The figure on the right shows an example of asterisms. In this case these may be caused by oriented micas in the sample, but fractures, either natural or accidental, may cause similar results. In both cases the solution is to replace complete angular averaging with sector averaging. Data reduction packages include a function to allow averaging of only a selected angular range of the data. For oriented samples like shales this permits analysis of scattering perpendicular to, or parallel to bedding, shear planes, or other oriented structural fabrics in the sample. For samples with unwanted asterisms these angles can be avoided. An alternative approach for a sample with asterisms is to mask out the directional scattering, and radially average the remainder. Figure 32 shows an example of an integrated scattering curve (Anovitz et al. 2015a) for a sample of St. Peter sandstone with experimentally-generated quartz overgrowths. This presentation, log(I(Q)) plotted as a function of log(Q) is sometimes referred to as a Porod plot. The data show are a combination of data from three sources: SANS, USANS, and calculations from backscattered electron images taken on a scanning electron microscope (BSE/SEM). The approximate Q-ranges over which these data were obtained are shown, although these are approximate as results from the three techniques overlap. The data in Figure 32 show several typical features. The intensity at high Q is apparently independent of Q. This reflects the incoherent background, and in most geological samples is primarily a function of the hydrogen (water or hydroxyl) content of the sample. In the mid-Q range the data can be fitted to a power-law slope. While we have shown that, in many cases, there are actually significant details in this region (Anovitz et al. 2013a, 2015a), to a first approximation the log–log slope represents the fractal properties of the sample. Several scattering studies suggest that the length correlations of pore-grain interfaces can often be described by self-similar fractals with nonuniversal dimensions (2 < D < 3) (cf. Bale and Schmidt 1984; Mildner and Hall 1986; Wong et

Figure 32. Example radially integrated scattering data for a sandstone (sample 04Wi17b, 100 ºC, 8 weeks, Anovitz et al. 2015). The central part of the curve has a power-law slope of -3.538 (r2 = 0.99925). Ranges shown for data obtained from SEM/BSE imagery, USANS and SANS measurements are approximate as the data overlap.

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al. 1986; Radlinski et al. 1999; Connolly et al. 2006; Anovitz et al. 2009, 2011, 2013a,b, 2014, 2015a,b; Jin et al. 2011; Mastalerz et al. 2012; Melnichenko et al. 2012; Navarre-Sitchler et al. 2013; Wang et al. 2013; Swift et al. 2014). This leads to a non-integer power-law as a function of the scattering given by I(Q) = I0Q-x + B where B is the incoherent background. As summarized by Radlinski (2006) the magnitudes of these slopes are determined by the surface from which scattering occurs. Slopes between -2 and -3 are characteristic of mass fractal systems, those between -3 and -4 of surface fractal system, and those between -4 and -5 of non-fractal “fuzzy” interfaces. These may be interfaces in which the scattering length density varies monotonically between two phases, or ones in which this appears, on average, to be the case, such as needle-like scatterers imperfectly aligned towards the beam. For a volume or mass fractal scatterer, therefore, Dm = x, and for a surface fractal Ds = 6 - x (Bale and Schmidt 1984). Smooth interfaces give rise to scattering with a power-law slope of -4, which is referred to as Porod scattering. Such suggestions of fractal surface and mass scaling are common in scattering studies of rock materials. In general, a surface fractal is an object whose surface areas scales in a noninteger manner with its radius (or some other selected ruler length), as: S  kr Ds .

(51)

For a non-fractal, three-dimensional object Ds = 2 (as in the surface area of a sphere A = 4pr2) and for a surface fractal 2 < Ds < 3. As noted by Anovitz et al. (2013a), however, while the ranges for a two dimension surface fractal are one less than the range just given, the relationship between the fractal dimension of a three dimensional object and a two dimensional slice through it is uncertain. For a mass fractal, it is the mass (or volume) with non-integer scaling as: M  kr Dm ,

(52)

where Dm = 3 for a non fractal object, as in the volume or a sphere (V = 4/3r3) and for a three dimensional mass fractal object again 2 < Dm < 3. The question becomes, however, how both can co-exist in a given rock. Figure 33 shows one, deterministic example. In this case in Figure 33a (left) the individual particles are represented by a simple, three-level, twodimensional surface fractal object. Figure 33b (right) shows how these can be combined into a two-dimensional mass fractal, a Sierpinski carpet, with a mass fractal dimension of 1.8928.

Figure 33. Deterministic example of a combined surface fractal (left) and mass fractal (right). The three-level Sierpinski carpet at the right is composed of particles shaped like those on the left.

There may also be several inflection points in the data. These include a point, the surface fractal correlation length r, which forms the upper scaling limit of surface fractal behavior. Below this Q-range the scaling exponent is dominated by mass fractal behavior. A second point may separate the mass-fractal scattering region from a fuzzy-scattering region (this is sometimes found at high-Q as well, cf. Naudon et al. 1994). At yet lower Q-values

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corresponding to length scales greater than the largest aggregates, the mass-fractal correlation length, the slope of I vs. Q should flatten. This “Guinier region” is not commonly observed in direct scattering data for rock samples, but is present in data obtained from image analysis. This latter, however, may reflect the scale of the images, rather than any maximum in the samples themselves. The surface fractal dimension can also be used to determine the surface area to volume ratio as (Allen 1991): S V   r

S r       V 0  r0 

 2  Ds 

.

(53)

This equation is based on the assumption that the fractal surface is self-affine (i.e. the structure is invariant under an anisotropic scaling transformation). Because the surfaces of these materials are fractal, the magnitude of the surface area depends on the size of the “ruler” used to measure it. (S/V)0 is the surface area to volume ratio for a smooth particle, r is the fractal “ruler” length, and r0 is the correlation length representing the upper limit of surface fractal behavior, and Ds is the surface fractal dimension. Anovitz et al. (2009) noted, however, that these surface areas represent only those surfaces that scatter neutrons, and therefore this represents primarily pore/grain boundaries although, as noted above, large concentrations of two-mineral grain boundaries may contribute. Because the rocks under consideration consisted largely of calcite, they selected a value of 7.165 Å for r, from the cube root of the calcite unit cell volume (367.85Å). Values for r0 and Ds were taken from the SANS/USANS data. Wang et al. (2013) modified this approach slightly, again selecting the crossover length (called 2l there) between the regions of surface and mass fractal scattering for r0, and 7 Å, the size of an N2-gas molecule used for absorption studies, as r (called d in Wang et al. 2013). As can be seen in Figure 32, another inflection point occurs at high-Q. This is determined by the intensity of the incoherent background. In some cases Bragg peaks, typically due to the large d-spacings of clays are observed through the background, which complicates background subtraction, but no such peaks are observed in the data for the sample in Figure 32. The first step in analyzing the data is often to subtract this background. At high Q the Porod Law (Porod 1951, 1952) provides the relationship of the scattered intensity for an ideal two-phase system bounded by a smooth interface of area S, and the scattered intensity as Q goes to infinity: I Q  

2     S 2

Q4

 Ib ,

(54)

where Ib is the background and S is the specific surface area (surface area per unit volume, units of cm2/cm3 or 1/cm). Figure 34 shows one way to obtain the background value (Glatter and Kratky 1982). The slope of a plot of Q4I(Q) as a function of Q4 has units of 1/cm, and is dominated by the data at high Q. From Equation (54) this slope, therefore, defines the background. This plot also provides a convenient way to judge whether Bragg peaks, which are often rather broad in this region for SANS data, are present. If so this approach cannot be used to determine incoherent background values. The intercept of the line defined by Equation (54): Cp  2 ( Dr )2 S

(55)

is called the Porod constant. Multiplying by , and dividing by the invariant (Y, Eqn. 57 below) yields the surface area to volume ratio as Glatter and Kratky (1982):

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Figure 34. Porod transform of data for 04Wi17b, 100 ºC, 8 weeks (Anovitz et al. 2015a)

C S  p . V Y

(56)

Porod (1951, 1952) also showed that, for any sample, an integral of Q2I vs Q should be a constant, irrespective of details of the structure. If parts of the system are deformed the diffraction pattern may change, but the integral remains invariant (Glatter and Kratky 1982). The plot of this transform, shown in Figure 35 is know as the Kratky transform, and 

Y  Q 2 I  Q  dq  2 2     1    , 2

(57)

0

where Y is called the Porod invariant, and  is the volume fraction of scatterers or, in the case of a two-phase (rock-pore) system, the pore fraction. A critical factor in this calculation is the extent to which the Kratky transform is “closed”. The integral is extremely sensitive to values at high Q, and if this has not gone sufficiently to zero the results will be incorrect. Thus appropriate background subtraction is critical. Figure 35 shows the Kratky transform of the data in Figure 32. Another useful simple transform is the Guinier plot, which yields the radius of gyration of the scattering particles. At low Q, for smooth spherical, or at least isotropic scatterer (e.g., a polymer chain):  Q 2 Rg2  I  Q   I 0 exp   , 3  

where Rg is the radius of gyration of the scatterer. Therefore,

(58)

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Figure 35. Kratky transform of data for 04Wi17b, 100 ºC, 8 weeks, background subtracted (Anovitz et al. 2015a)

 Q 2 Rg2  ln  I  Q    ln  I 0     .  3 

(59)

As this is the equation of a straight line the values of I0 and Rg are easily determined from a plot of ln(I(Q)) vs. Q2. In the case of cylindrical objects, however (Glatter and Kratky 1982), of length L and radius R, this equation is valid at low Q with: Rg2 

L2 R 2  , 12 2

(60)

but at intermediate Q-values

I Q  

 Q 2 Rg2  I0 exp    Q 3  

(61)

and the appropriate plot is ln(QI(Q)) vs. Q2. For lamellar objects of thickness T, the appropriate equation becomes: I Q  

 Q 2 Rg2  I0 exp   , Q2 3  

(62)

where

Rg2 

T2 12

(63)

and the plot is ln(Q2I(Q)) vs. Q2. However, as noted above a Guinier region (flat at low Q in

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a plot of I(Q) vs Q) is seldom observed in scattering data from rock samples, and the fractal nature of the mineral/pore interfaces in most rock materials tends to make this approach less useful for geological purposes. Finally, as was discussed by Anovitz et al. (2013a), for many geological samples a simple Porod plot of I(Q) vs. Q is often not very convenient. This is because the near Q4 slope requires scaling of both the x and y axes in such a way that details of the data are often hard to discern. They suggested, therefore, that data be plotted on a semi-Porod transform (Q4I(Q) as a function of Q) instead. This has the effect of rotating the data so that a Q4 slope becomes horizontal, allowing significant magnification of the data and careful examination, both of changes between individual samples, and of the details for an individual sample previously hidden in the Porod plot. Figure 36 shows the semi-Porod transform for the same dataset as in the Porod plot in Figure 32. It is clear in this presentation that the data in the central part of the Q-range do not fall on a single fractal slope but, rather, are separated into several regions.

Figure 36. Semi-Porod transform of data for 04Wi17b, 100 ºC, 8 weeks, background subtracted (Anovitz et al. 2015a)

While the total porosity in the SAS size range can be calculated from the invariant as shown above, it is clearly of interest to derive the pore volume distribution or, if possible, the pore size distribution. There are, however, at least two caveats that must be considered. First, while a number of methods have been suggested for making these calculations the results may be model-dependent. Several are available as pre-programmed software, making them relatively easy to use, but the user is cautioned to understand the assumptions and methodologies of any technique adopted, and to consider these limitations in interpreting the results. Second, as noted by Anovitz et al. (2009, 2013a), conversions from volume distributions to pore distributions are highly problematic. To do so requires one or more assumptions about the shape of the pores involved. Figure 37 shows TEM images of a selection of pore images from the Marble Canyon contact aureole, west Texas (Anovitz et al. 2009). It is clearly evident that the pores are neither solely spheres, nor solely laminar, but vary significantly. Thus, interpretations of pore sizes based on pore volumes can be problematic.

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Figure 37. TEM images of pores from representative samples from the Marble Canyon contact aureole [Anovitz LM, Lynn GW, Cole DR, Rother R, Allard LF, Hamilton WA, Porcar L, Kim M-H (2009) A new approach to quantification of metamorphism using ultra-small and small angle neutron scattering. Geochimica et Cosmochim Acta, Vol. 73, p. 7303–7324, used with permission from Elsevier.] Note the strong variation in pore shapes.

A number of approaches have been suggested for calculating pore volume/size distributions. These include the smooth surface approach (Anovitz et al. 2013a), the polydisperse hard sphere model (PRINSAS, Hinde 2004, Radlinski 2006), Maximum Entropy approaches (Jaynes 1983; Skilling and Bryan 1984; Culverson and Clarke 1986; Potton et al. 1986, 1988a,b; Hansen and Pedersen 1991; Jemain et al. 1991; Semenyuk and Svergun 1991), regularization or maximum smoothness (Glatter 1977, 1979; Moore 1980; Svergun 1991; Pederson 1994), total non-negative least squares (Merrit and Zhang 2004; Ilavsky and Jemian 2009), Bayesian (Hansen 2000) and Monte Carlo (Martelli and Di Nunzio 2002; Di Nunzio et al. 2004; Pauw et al. 2013). There are also methods available based on Titchmarsh transforms for determining size distributions (Fedorova and Schmidt 1978; Mulato and Chambouleyron 1996; Botet and Cabane 2012), and the structure interference method (Krauthäuser et al. 1996). Maximum entropy, regularization and total non-negative least squares are available in the IRENA program (Ilavsky and Jemian 2009). While a detailed explanation of these techniques is beyond the scope of this review, it is worth illustrating the potential differences among them. Figure 38 shows the initial (U)SANS data (Anovitz unpb.) from a sample of dolostone from the Ordovician Kingsport formation and three pore distributions calculated using the total non-negative least squares, maximum entropy, and regularization approaches as implemented in IRENA. The overall similarities and detailed differences amongst the three approaches are apparent. The TNNLS and regularization approaches provide smoother estimates of the pore distributions, but all three suggest four, or maybe five subdistributions within the pore structure. In discussing these three approaches, however, Ilavsky and Jemian (2009) note that the regularization approach does not necessarily guarantee non-negative results for each bin. A similar multi-distribution pattern has been observed in sandstones (Anovitz et al. 2013a, 2015a) suggesting that modeling sandstones as a continuous fractal distribution is inappropriate. In the end, however, solutions such as those above are limited by any number of assumptions, including in most cases those of a single pore shape and contrast. More detailed

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Figure 38. Modeled pore distribution from USANS data for a dolostone sample from the Ordovician, Kingsport Fm., Knox group, Smith Co., TN, 1577’ deep. Assumed: spheroid, aspect ratio = 1, background = 0.0176, contrast = 29.33 × 1020 cm-4. Top) TNNLS, error multiplier = 1.6; middle) Maximum entropy, error multiplier = 1.6, sky bkgd = 3.12 × 10-8; bottom) regularization, error multiplier = 1.65. (Anovitz, unpb.)

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analysis of SAS data required modeling of the scattering results. For a dilute solution the intensity of a SAS pattern is described as: I  Q dilute  

2



 F Q, r 

2

V 2  r  NP  r  dr ,

(64)

0

where ||2 is the contrast, F(Q,r), the form factor, is an equation the represents the shape of the individual scatterers, V(r) is the particle volume, N is the total number of scatterers, and P(r) is the size distribution, the probability of a given particle of size r. For non-dilute solution the structure factor, which describes the interaction amongst the particles must be considered. For example, Anovitz et al. (2009) noted that modeling carbonates often requires both surface fractal (form factor) and mass fractal (structure factor) components, and Jin et al. (2011) obtained similar results from shales. Equations for the structure factor can be combined with form factor results as: I  Q   S  Q  * I  Q dilute ,

(65 )

or, combining the the various constants into a single empirical variable: I Q   A * F Q  * S Q  ,

(66)

although the application of this approach to polydispere systems can be more complex. Additional factors can also be added for backgrounds, Bragg peaks, fuzzy scattering or other factors as needed. Because there are a large number of possible scattering geometries, a large number of possible structure and form factors and size distributions, many derived for polymers, particles of known shapes, or complex fluids have been considered and are available in standard data fitting packages. These are described in more detail in several publications (see Kline 2006; Hammouda 2008; Ilavsky and Jemian 2009).

IMAGE ANALYSIS It is far beyond the scope of this review to even begin an analysis of the applications of image analysis to geological samples. However, in the context of analyzing and quantifying pore structures some discussion is appropriate, because analysis of low-magnification SEM/ BSE or X-ray computed tomographic images can be used to extend the scale range analyzed by SAS experiments, and thus imagery can be used for pore characterization beyond that provided by point counting. In addition, in the process of obtaining and processing these data one generates binary images of the pore structure of the rock, typically at scales greater than approximately 1 mm that can then be used to provide further quantification of the pores structure at these scales using other statistical techniques that require the two- or threedimensional data available in the images themselves.

Sample preparation and image acquisition A key requirement of many forms of pore structure image analysis is that they require binary images showing pore-space vs. non-pore space (mineral phases). These are typically obtained by thresholding grey scale SEM/BSE or X-ray tomographic images to separate the two phases. Figures 39 and 40 show a BSE and binary image pair for sample 04Wi17b, 100 ºC, 8 weeks from Anovitz et al. (2015a). A significant caveat should be mentioned at this point with respect to obtaining the binary images necessary for many of the image-based calculations discussed here. Even for a simple system (essentially just quartz and pores, with the pores filled with epoxy to yield a smoother, more two-dimensional result in the BSE images) the

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Figure 39. Backscattered electron image of sample 04Wi17b, 100 ºC, 8 weeks from Anovitz et al. (2015a). Image is 5.3 mm across.

Figure 40. Binary image (pores black, quartz white) of sample 04Wi17b, 100 ºC, 8 weeks from Anovitz et al. (2015a). Image is 5.3 mm across.

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process of image segmentation does not necessarily yield a unique solution. There are several reasons for this. First, the background, considered here as the grey-scale level for quartz or pores, may not be “flat” across the image. This is often a function of the instrument settings on the SEM and must be corrected for before thresholding/segmentation. More importantly, however, even within a given grain there are often variations in grey-scale level, adding to the noise level, and pixels at the boundaries between grains will have grey-scale values that average the values of both phases. Selecting the appropriate method for segmenting an image, either for simply choosing a threshold for a 2-D or 3-D image, the grey-level between the two phases, or using a more complex segmentation approach, however, may lead to significant undertainty. While threshold selection can be done manually, it is unlikely that this approach will lead to consistent results. Wildenschild and Sheppart (2013) note that, even in cases where simple thresholding is appropriate, selection by hand has been shown to be subject to significant operator subjectivity. On the other hand, while there is agreement that automated methods are preferable, it is quite common that thresolding-based methods as well do not provide consistent results as well if applied to slightly varied images of the same material. For thresholdable images we have often found that reasonable consistency can be obtained by trying a number of methods and selecting a threshold value near the median, but the statistical reliability of such an approach remains to be tested. Simple thresholding is also intolerant of image noise, and subject to uncertainties for pixels that straddle grain boundaries. This becomes even more complex in multiphase systems. Given the complexity of this issue, a careful examination and comparison of the various approaches is clearly beyond the scope of this review. There are, in fact, a very large number of algorithms for selecting a threshold. Sezgin and Sankur (2004) for instance, review forty different approaches in six categories: histogram shape, clustering, entropy, object attribute, spatial methods and local methods. Iassonov et al. (2009) reviewed segmentation methods, and provided some comparison with thresholding techniques, and Wildenschild and Sheppard (2013) summarized and referenced a number of approaches to thresholding and segmentations (see also Noiriel 2015, this volume). Exclusive of those aimed primarily at medical imaging, other reviews include: Pal and Pal (1993), Cheng et al. (2001b), Munoz et al. (2003), Udupa and Saha (2003), Cardoso and Corte-Real (2005), Cremers et al. (2007), Ilea and Whelan (2011), and Schülter et al. (2014). Readers are encouraged to evaluate these methodologies for their specific applications, but care must be taken in any event in order to obtain reasonable, consistent, and unbiased values. The materials from which the original rock is composed may also make it difficult to create suitable binaries showing the pore structure. The technique of impregnating the pores with epoxy, yielding a low backscatter contrast, flat material in the pores, works very well as long as material with a similar average atomic number is not already present. This is, however, not true for materials with a significant organic content such as coals or tight oil/gas shales which often contain kerogen or bitumen. An alternative approach, suggested by several authors (Swanson 1979; Dullien 1981; Hildenbrand and Urai 2003; Dultz et al. 2006; Kauffman 2009, 2010; Hu et al. 2012) is to impregnate the pores with Wood’s metal, an alloy of approximately 50% Bi, 25% Pb, 12.5% Zn and 12.5% Cd with a melting point of only 78 ºC yielding pores that are bright in backscattered imaging, and thus stand out from the dark organic matter. Wood’s metal does not wet silicates, however, and thus the minimum pore size that it will enter is a function of injection pressure (similar to MIP) as described by the Young–Leplace (Washburn’s) Equation. As scattering describes the smaller pores, however, it is not really necessary to inject the metal into pores smaller than about 1 mm in this case. Given a contact angle of 130° and a surface tension of 0.4 N/m (Darot and Reuschle 1999; Hu et al. 2012) this only requires a pressure of about 10 bars (145 psi).

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A difficulty with this approach is that, because of its Pb and Cd content, Wood’s metal is hazardous to use, and potentially difficult to dispose of correctly. A potentially safer alternative (Anovitz, unpb) is Field’s metal. Field’s metal is a fusible eutectic alloy of bismuth, indium, and tin (32.5 wt. % Bi, 51 wt. % In, 16.5 wt. % Sn. It melts at a lower temperature than Wood’s metal, becoming liquid at approximately 62 ºC (144 °F) and, as it contains no lead nor cadmium, is marketed as a non-toxic alternative to Wood’s metal.

Combining imaging and scattering data A key feature of all scattering approaches is that the range of pore sizes they can interrogate is inherently limited by the design of the instrument. While the combination of small angle, very- small angle, ultra-small angle and potentially light scattering and even wide angle techniques can cover a very wide range of scales, even this is inherently smaller that the real range of porosity in geological materials, which stretches from the structural pores in such phases as beryl and cordierite (cf. Anovitz et al. 2013c; Kolesnikov et al. 2014), dioptase, hemimorphite, zeolites, etc. (Ferraris and Merlino 2005) which may be as small as several angstroms, to many meters or even miles in length if the definition of a pore is extended, sensu lato, to cave systems. In order to extend the quantification of pore systems to larger scales, therefore, the results of another approach must be combined with those from the scattering data. To do so, we combine the results of imaging analysis, be it for two-dimensional images, usually obtained using backscattered electron imaging on an SEM, or three-dimensional X-ray computed tomographic images with the scattering data. This approach has the distinct advantage that it allows binary images of pore systems obtained at low magnifications using imaging techniques to be added to data obtained from scattering experiments. As backscattered electron images can easily be obtained that cover several square centimeters with mm- or sub-mm-resolution this allows the scales quantifiably analyzed using this extended “scattering” analysis to extend from the nanometer to the centimeter range—7 orders of magnitude. In this case the correlation function, the Fourier pair to the scattering function, becomes identical with the two-phase autocorrelation function, and can be described explicitly (cf. Anovitz et al. 2013a, Wang et al. 2013). To do so, following Berryman (1985), Berryman and Blair (1986) and Blair et al. (1996), we begin by defining a characteristic function f(x), which has values of either 0 or 1. This is equivalent to a binary image, and thus it is this same relationship that allows us to quantitatively connect backscattered electron, X-ray CT, or other imagery of the sample to the scattering data and, thereby, extend the range of the scattering data to cm scales. Torquanto (2002a,b) defined this in terms of an indicator function I(i)(x) where: 1, x Vi i I   x   , 0, x Vi

(67)

where Vi is the volume occupied by phase I, and Vi is the volume occupied by the other phase (rock). As summarized by Anovitz et al. (2013a, who used f(x) instead of I(i)x), if we then let f(x) = 1 for the pores, and 0 for the solid, then the first two void–void correlation functions (1- and 2-point) for an isotropic material are given by  (68) S1  f ( x )   and  S 2 (r )  f ( x ) f ( x  r ) ,

(69)

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where the brackets are a volume average over x, r is a lag distance and r = |r| for an isotropic material, and  is the pore fraction. Berryman (1985) showed that: S2  0   S1  ,

(70)

so that the zero intercept of the second correlation function is the porosity, and at the limit of large r lim S2 (r )  2 . r 

(71)

In addition, the specific surface area (s) defined as the ratio of the total surface of the poregrain interface to the total volume of the grains can be derived as: S2 (0) s  , r 4

(72)

4(1  ) , s

(73)

and the effective pore size is given as rc 

which is the intersection of a line tangent to the S2(r) curve at the zero intercept with S2(r) = 2. A quantitative estimate of the average grain size can also be obtained, but this depends on the sorting and arrangement of the grains in an individual sample (Blair et al. 1996). Alternatively, correlation probabilities can be represented using the related autocovariance and/or autocorrelation coefficient functions: X (r )   I (p) ( x )  p   I (p) ( x  r )  p   S2(p) (r )  2p ,

(74)

and c( r ) 

X (r ) X (r )  ,  p (1  p ) p g

(75)

where g is the volume fraction of grain phase, p + g = 1, and c(r) is a normalized version of X(r). In a statistically homogeneous two-phase system (isotropic or anisotropic), X(r) has limiting values of X(0) = pg and, X     0 and bounds in the range of  min(2p , g2 )  X (r )  p g . Normalizing X(r) by pg puts c(r) into a range between one and some negative value. Thus, in c(r) form, a value of one at a given r means perfect correlation, zero means no correlation, and negative values mean anticorrelation. Following Adler et al. (1990), Radlinsky (2006) associated c(r) with g(r). On the basis of this function Debye and Bueche (1949) and later studies (e.g., Guinier et al. 1955; Debye et al. 1957; Glatter 1980; Glatter and Kratky 1982; Adler et al. 1990; Lindner and Zemb 1991; Radlinksi et al. 2004; Radlinksi 2006) showed that small-angle scattering measurements can be used to obtain the autocorrelation coefficient of two-phase media. They showed that the normalized scattering intensity per unit sample volume V at wave number Q for a three-dimensional (3-D), isotropic, two-phase system comprised of solid and pore phases is proportional to the Fourier transform of the autocorrelation coefficient as:

Characterization and Analysis of Porosity and Pore Structures 

I (Q)  4 ( ) (   ) r 2 c(r ) 2

2

0

sin(Qr ) dr , Qr

125

(76)



1 sin(Qr ) c(r )  2 Q 2 I (Q) dQ, 2 2  Qr 2  ( ) (   ) 0

(77)

where ()2 is the scattering length density contrast, and Q is the scattering vector magnitude as defined above. The simplest method for calculating S2(r) is to calculate for each value of r the fraction of pixels for which both ends of a line segment of that length fall on the phase of interest. Alternatively, a Monte Carlo approach can be used to randomly select a suite of starting pixels and angles. The problem with such an approach, however, is that it is computationally slow. Anovitz et al. (2013a, 2015), therefore, used an alternative approach using the radial integration of the power spectrum of the Fourier Transform of the image (after extending the image size to avoid artifacts due to periodic boundary conditions) assuming that the image shows a random part of a much larger area having the same autocorrelation. This is based on the Wiener– Khinchin theorem (Weiner 1930, 1964; Khintchine 1934; Goodman 1985; Champeney 1987; Chatfield 1989; Hannan 1990; Couch 2001; Ricker 2003; Iniewski 2007). This shows that: FR  f   FFT  X  r   ,

(78)

S  f   FR  f  FR*  f  ,

(79)

c(r )  IFFT  S  f   .

(80)

and

Thus, the correlation function c(r) can be quickly calculated by calculating the Fourier transform of an image, multiplying by its complex conjugate, and than back transforming the result. The result is normalized by the autocorrelation of a function equal to 1 in the image area and 0 outside. This corresponds to the denominator in the usual equation for autocorrelation of discrete 1D data sets. The autocorrelation is scaled in such a way that zero means ‘no correlation’ and one means ‘perfect correlation’. Thus, at a distance of r = 0, the value is always one. This differs slightly from the function as defined by Berryman (1985) and Berryman and Blair (1986) in which the value at r = 0 is  and at large r is 2, but the scaling between the two results is linear. One limitation to either approach is that statistical noise necessitates truncation of the autocorrelation spectrum. The results at large r are often not a smoothly decreasing sinusoidal function. Because of these fluctuations at large radii, failure to truncate the data prior to calculation of the scattering intensity will introduce artifacts into the result. The noise in these results at high Q can be reduced by appropriate re-extrapolation of the truncated data (cf. Debye 1957; Anovitz et al. 2013a; Wang et al. 2013).

Three-point correlations The one-point and two-point correlation functions just described are the first and second moments of the probability distribution of the pore/grain system—the mean (porosity) and the variance. As in any such system, there are an infinite number of related correlations that can be applied to porosity analysis. These are the n-point correlation functions, of which the 1- and

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2-point correlations discussed above form a part. The reason to be concerned, at least with the 3-point correlation (cf. Beran 1968; Corson 1974a,b,c,d; Berryman 1985; Torquato 2002a; Jiao et al. 2007, 2008, 2009, 2010, 2013; Singh et al. 2012; Jiao and Chawla 2014), was stated by Berryman (1985): An elaborate theoretical machinery is available for calculating the properties of heterogeneous materials if certain spatial correlation functions for the materials are known. Formulas have been published for calculating bounds on dielectric constants, magnetic permeabilities, electrical and thermal conductivities, fluid permeabilities, and elastic constants if the two-point and three-point correlation functions are known. (Brown 1955; Prager 1961; Beran 1968)” This “theoretical machinery” has been even further defined since this time (cf. Berryman and Milton 1988; Bergman and Stroud 1992; Helsing 1995a,b; Blair et al. 1996; Torquato 2002a,b; Saheli et al. 2004; Prodanovic et al. 2007; Wang et al. 2007; Politis et al. 2008; Wang and Pan 2008; Yin et al. 2008; Deng et al. 2012; Wildenschild and Sheppard 2013), as have the facilities for two- and three-dimensional imaging (FIB, XCT, NCT) that provide the analyzable data. While these higher-correlation statistics are well-known in fields such as astronomy (e.g Baumgart and Fry 1991; Coles and Jones 1991; Gangui et al. 1994; Takada and Jain 2004; Seery and Lidsey 2005; Zehavi et al. 2005; Ade et al. 2014; Fu et al. 2014; Moresco et al. 2014) they have been less often applied to geological media, despite their potential for calculating important rock properties. Geometrically, the three-point correlation function is exactly equivalent to the two-point function described above. In this case it provides the probability that the three points that describe the corners of a triangle of a given size and orientation (i.e. two vectors r1 and r2 sharing and initial, moveable point) all fall on a single phase. A clear explanation of the three-point correlation function was provided by Berryman (1985, 1988, see also Velasquez 2010) and this discussion is summarized from there. Paralleling the definitions of the one- and two-point correlations, for a given phase in a homogeneous material in which only the differences in coordinate values are important, not the absolute locations: Sˆ3  r1r2   f  x  f  x  r1  f  x  r2  ,

(81)

where an oriented triangle is defined by the two vectors r1 and r2, and the triangular brackets represent a volume average over the range of x. If the material is further assumed to be isotropic, so that absolute angle is also unimportant (not necessarily true in geological materials, especially shales), then, letting r = |r|, so |r1| = x2 – x1 and |r2| = x3 – x1: Sˆ3  r1r2   S3  r1 , r2 , u12   S3  r2 , r1 , u12  ,

(82)

where: u12  cos 12 

r1.r2 . r1 r2

(83)

Therefore we have the three variables that define a triangle, the side lengths r1 and r2, and the angle  between them. This function has the following properties: lim S3  r1 , r2 , u12   S2  r2  ,

r1 0

(84)

Characterization and Analysis of Porosity and Pore Structures lim S3  r1 , r2 , u12   

r1 r2 

u12  1,

127 (85)

If there is no long-range order then: lim

r1 , r2 Fixed

S3  r1 , r2 , u12   S2  r2



(86)

and the three-point correlation function is bounded by: S3  r1 , r2 , u12   min[ S2  r1  , S2  r2  , S2  r3

  max S  r  , S  r  , S  r ]   2

1

2

2

2

3

(87)

where: 2

2

r3  r1  r2  2 r2 r1 u12 .

(88)

Berryman (1985) also suggested, and Berryman (1988) and Velasquez et al. (2010) modified a method for calculating the three-point correlation for an image. Berryman (1985) noted that, a minimal set of grid-commensurate triangles (ones in which all the corners fall on lattice points, or pixels in the case of an image), labeled with three integers (k, m, n) with k the length of the largest side, can be constructed as follows (note that Berryman used (l, m, n) rather than (k, m, n), This has been modified here for clarity). First, the longest axis is placed along the x-axis, defining a coordinate system with the intersection of the longest and shortest sides as (0,0) and the second vertex at (k, 0). The third vertex is then located in the first quadrant at (m, n). The shortest side shares the (0, 0) vertex, which places the third vertex at x ≤ k/2 within a circle of radius k from (k, 0). Berryman (1988) modified this to provide greater accuracy by considering all lattice points for the third vertex at:

 0,0 

 m 

 k / 2, 0  ,

0  n  k.

(89) (90)

This is described in Figure 41. For a homogeneous, isotropic system rotations of these triangles are not needed. One can then either calculate the correlation for a given triangle by testing every possible point (N) within the image (which is less than the total number of pixels as the size of the triangle will make a certain number of points on the right and top of the image inaccessible as (0, 0)), finding the number of times that all three corners of the triangle land on a single phase and calculating S3(k, m, n) as S3  k, m, n  

N"hits" N

(91)

or, as suggested by both Berryman (1988) and Velasquez et al. (2010), adopt a Monte Carlo scheme and randomly drop a given triangle on the image N number of times. While the size of their images is unclear, Velasquez et al. (2010) found that 100,000 drops was sufficient. Berryman (1988) discusses interpolation schemes to be used with a dataset of this type, for a triangle as suggested in Figure 41. Even with the simplifications achieved this this approach, however, there remain a large number of possibilities to consider due to the large number of possible triangles involved, each of which is characterized by the three parameters k, m, and n (or r1, r2 and ) and S3(k, m, n). While a full analysis seems optimally suited to parallelization, simpler schemes, such as that

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Figure 41. As shown by Berryman (1988): the lattice-commensurate triangles used by the modified algorithm for k < 8. The Monte Carlo integration scheme he suggests chooses triangles whose third vertex lies somewhere in one of the shaded regions. These vertices are surrounded by lattice points with known values, directly for k even and/or by symmetry around m = [k/2] for k odd. (Redrafted after Berryman JG (1988) Interpolating and integrating 3-point correlation-functions on a lattice. Journal of Computational Physics, Vol. 75, p. 86–102, used with permission from Elsevier).

adopted by Velasquez et al. (2010) of choosing one or a few triangle shapes and investigating the effect of scaling as:

 k, m, n 

*

 p  k, m, n 

(92)

where p is an integer, and plotting the resultant S3(k, m, n)* as a function of p, can provide more easily plotted and analyzed results.

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As with the two-point correlation, there are Fourier methods for calculating the three-point correlation. While the Fourier transform of the two-point correlation is the power spectrum, that for the three-point correlation is the bispectrum where, for two vectors r1 and r2 that define a triangle of given size and orientation: B  r1 , r2   F  r1  F  r2  F *  r1  r2  ,

(93)

where F* again refers to the complex conjugate. The bispectrum remains difficult to determine, however, because its parameter space, the set of all triangles, is very large. The two and three-point correlation approaches described by Berryman (1985, 1987, 1988), Berryman and Blair (1986, 1987) and Berryman and Milton (1988) although, described earlier in other contexts, have been cited in a number of studies of different materials including bone (e.g., Hwang et al. 1997; Wehrli et al. 1998; Wald et al. 2007) cements and concretes (e.g., Lange et al. 1994; Bentz 1997; Sumanasooriya and Neithalath 2009, 2011; Sumanasooriya et al. 2009, 2010; Erdogan 2013), asphalts (e.g Velasquez et al. 2010; Falchetto et al. 2012, 2013, 2014; Moon et al. 2013, 2014a,b,c), fuel cells (e.g., Mukherjee and Wang 2007; Mukherjee et al. 2011), composites (e.g., Torquato 1985; Smith and Torquato 1989; Helsing 1995b; Terada et al. 1997; Spowart et al. 2001; Reuteler et al. 2011), digital reconstruction of porous materials (e.g., Roberts 1997; Kainourgiakis et al. 2000) and others, including several studies of geological materials (Blair et al. 1996; Coker et al. 1996; Ioannidis et al. 1996; Meng 1996; Virgin et al. 1996; Berge et al. 1998; Masad and Muhumthan 1998, 2000; Quenard et al. 1998; Fredrich 1999; Lebron et al. 1999; Saar and Manga 1999; Ikeda et al. 2000; Schaap and Lebron 2001; Vervoort and Cattle 2003; Rozenbaum et al. 2007; Chen et al. 2009; Torabi and Fossen 2009; Anovitz et al. 2013a, 2015a; Wang et al. 2013; Nabawy 2014)

Monofractals and multifractals As has been noted above, SAS data suggest that pore structures in rocks exhibit both surface and mass fractal behavior. While the scattering data do not directly show what those structures look like, as noted above structure and form factor models such as those suggested by Beaucage (1995, 1996) and Beaucage et al. (1995, 2004) are based on models of this structure. Imaging data provides the opportunity to extend this analysis to a consideration of direct box-counting fractal (Block et al. 1990) and multifractal behavior based on actual observations. Monofractal analysis is essentially binary in nature. In the box-counting method of measuring the fractal dimension a binary image is subdivided into a series of boxes of size , and the number of boxes, n, that contain at least some of the image are counted. The box size is then reduced, and the procedure repeated. A plot of log(n), the number of “on” boxes, as a function of log( the box size, is then created, and the slope of the line, the scaling behavior of the system, is the fractal dimension. The limitation in monofractal analysis is that a significant amount of the available information is ignored. In counting each “on” box it does not account for the number of pixels that are “on” or, in another version of this metric, the relative grayscale of each box. Thus, nonuniform variations in the overall density of the image are not accounted for. The multifractal approach (Mandelbrot 1989; Evertsz and Mandelbrot 1992) is an expansion of the original fractal description (Mandelbrot 1977, 1983) that considers this additional information. In multifractal systems a single exponent is not sufficient to describe the system. Rather, an array of exponents, known as the singularity spectrum, is used. A number of studies have used this approach to study sandstones (Muller and McCauley 1992; Anovitz et al. 2013a, 2015a), soils (Perfect 1997; Grout et al. 1998; Posadas et al. 2001, 2003; Caniego et al. 2003; Martin et al. 2005, 2006, 2009; Bird et al. 2006; Dathe et al. 2006; Kravchenko et al. 2009; Paz Ferriero

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and Vidal Vázquez 2010; San José Martinez et al. 2007) chalk (Muller 1992, 1994, 1996; Muller et al. 1995), and others (Block et al. 1991). As summarized by Anovitz et al. (2013a), there are several, interrelated mathematical descriptors of multifractal structures, of which the most common are Hölder exponents () and Rényi dimensions D(q) (note that a lower case q is used in this case to separate it from the reciprocal space dimension Q in the neutron scattering data). As with the monofractal dimension we first begin by defining the length of one side of our measuring box as . We then define the total number of boxes of a given size as n(, and the “measure” of the box as , which can be any appropriate measure of its density, the number of “on” pixels, the grey scale, the fraction of all “on” pixels in the box, etc. We then further define, for each box  log    course   ,  log  

(94)

   .

(95)

 log     lim  ,  0 log   

(96)

or

In the limit, as →0

where  is the Hölder exponent. Note that course and  are not necessarily, or even likely to be, identical. If a given box containing four “on” pixels is divided into quarters the resultant four boxes might each contain one “on” pixel or all four might be in one box, etc., depending on their distribution. For any given box size , we can define N(course) as the sum of the number of boxes with a given value of course, and define the multifractal distribution (singularity exponent) as:

 log N    course   f ()     log   

(97)

 log N    course   f ()  lim    .  0 log   

(98)

and

In this description it is the singularity spectrum, f() as a function of , and especially the values of min, max, f()max,  at f()max, and the asymmetry of the spectrum:

A  ( max   at f ()max )  ( at f ()max   min )

(99)

that describe the statistical distribution of the measure in the image. In a monofractal the singularity spectrum is reduced to a single point. The alternative, but related approach is the Rényi dimension D(q). In this case we begin by defining a generating function: (q, )   i 1 iq , n()

(100)

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where q covers some range, usually approximately ± 10 to ± 15, and the sum represents the probability that q random points fall in the same box (Peitgen et al. 2004). We can then define the qth mass exponent (q )  lim

 log    q,     1 log   

 0

(101)

and the generalized or Rényi dimension n()   log  piq    1  i 1 , D(q )  lim    0 1  q  log        

(102)

 n()    pi log pi  . D(1)  lim  i 1 e0  log      

(103)

D  p   D  q  , where p  q.

(104)

except at q = 1 where

In general

A plot of D(q) vs. q is referred to as the Rényi spectrum. If D(q) strictly decreases with increasing q for q >0, the fractal is called inhomogeneous or multifractal (Peitgen et al. 2004). If D(q) as a function of q is constant, the system is a monofractal. In this description D(0), referred to as the capacity dimension, is equivalent to the monofractal dimension. D(1) has the same form as the microscopic description of entropy from statistical mechanics. It describes the entropy of the system, and is called the entropy or information dimension. Similarly, (q) can be analogized to the Free Energy of the system, (q, ) to the partition function, and q-1 to the temperature (Stanley and Meakin 1988; Arnéodo et al. 1995; Bershadskii 1998; Farge et al. 2004). D(2) is the correlation dimension, and gives the probability of finding pixels on an object within a given distance if you start at a pixel on the object. Thus it is related to the autocorrelation curve described above. These are related to the (q) curve as: q 

 q  1 D  q .

(105)

For q > 0, D(q) is dominated by large (i) and therefore by areas with a high density of the measure. For q < 0, D(q) is dominated by small (i) and therefore by areas of low density. It must be remembered, however, that the “measure” involved is the porosity, and thus “high density” refers to a high density of pores, not mass. The singularity spectrum f() vs.  and the Rényi spectrum D(q) vs. q are not independent. They are related through (q) as:

( q ) 

(q ) , q

(106)

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and by the Legendre transformation f ()  q  q  – (q ),

(107)

D  0     0   f    0    f ()max ,

(108)

so that:

where f((0)) is the value of f((q)) at the maximum of the f((q)) vs (q) curve, (q ) , q

(109)

(q ) . q  q

(110)

amin  lim

q 

amax  lim

At values of (q) >  (0), q < 0, and for values of (q) < (0) q > 0. While these descriptions are not independent they provide useful alternative approaches.

Lacunarity, succolarity, and other correlations While fractal and multifractal formalisms are excellent metrics to describe the scaling behavior of porous systems, they are not, in themselves, sufficient to fully quantify the pore structure. The reason for this is that they do not fully describe how a fractal structure fills space – the texture of the pore structure. Within a give area a fractal structure may be more, or less, heterogeneous, while still having the same scaling behavior. In his classic book “The Fractal Geometry of Nature” Mandelbrot (1977) began to address this limitation, originally noticed in his studies of galactic structures, by defining two additional parameters, the lacunarity, or gappiness, and the succolarity, or connectivity of the pore structure. As with fractal dimensions, these were originally proposed by Mandelbrot (1977, 1994, 1995) as a method of discerning amongst systems for which the fractal scalings are otherwise similar. The term lacunarity comes form the Latin word lacuna, meaning a gap or lake. The term should be generally familiar to geologists from its use to mean a gap in the stratigraphic record (Gignoux 1955; Wheeler 1958). Lacunarity is a quantitative measure of how clustered the pore structure is, and serves as an addition to the concept of fractal analysis (cf. Mandelbrot 1983, 1994, 1995). It can be seen as representing the homogeneity, or translational or rotational invariance of the system. It can also be viewed as a measure of the translational homogeneity of an image. From the point of view of understanding the relationship between porosity and permeability, therefore, this provides a quantification of how isolated each pore, or group of pores is from others. While much of the application of this approach has been in fields such as geography and organic/biological systems, several authors have investigated the utility of this measure for evaluating porosity and permeability in reservoir modeling (Garrison et al. 1993a,b; Cai et al. 2014), soils (Zeng et al. 1996; Millán 2004; Chun et al. 2008; Zamora-Castro et al. 2008; Luo and Lin 2009; Torres-Arguelles et al. 2010; Ulthayakumar et al. 2011), fractures (MirandaMartinez et al. 2006), porous silica (Denoyel et al. 2006), sediments (Bube et al. 2007), oil mobilization (Hamida and Babadagli 2008), and sandstones (Anovitz et al. 2013a, 2015a) in both two and three dimensions. There are a number of methods for calculating the lacunarity, several of which have been coded into the FracLac (Karperien 1999–2013) plugin for ImageJ (Abramoff et al. 2004; Rasband 1997–2014; Schneider et al. 2012). The simplest method, based on that used

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for fractal dimensions is again box counting. A box of size  is slid over the image, either sequentially in a fixed grid or in an overlapping pattern (sliding box counting). For a binary image divided into a given number of boxes of a given box size, , and grid orientation, g, the box-size specific lacunarity () is calculated from the mean, , and standard deviation, , of the number of pixels “turned on” in each box as:  e,g  (e,g /  e,g )2 .

(111)

The overall lacunarity of the image, , is then the average of the single box size lacunarities () over all box sizes and grid positions, although this can be done in terms of just box size or just orientation. Analysis of these values is, of course, limited by the resolution of the images in question. There are several other definitions of lacunarity. One is based on the prefactor in the boxcounting method for defining the fractal dimension. If the ln-ln scaling of the number of “on” boxes (N) as a function of the box size () is given as: N  A D ,

(112)

where A is calculated for a given orientation (g) of the image from the y-intercept of the ln–ln curve as: Ag 

1 , exp  yg 

(113)

averaging over the (G) available orientations:

 A

G

A

g 1 g

G

(114)

,

then the prefactor laccunarity (P) can be defined from the prefactor as (Mandelbrot 1977): 2

A   g 1  Ag  1 . P  G G

(115)

Figures 42–45 and Figure 43 show examples of the utility of both the multifractal and lacunarity approaches in the analysis of variations in pore structures (Anovitz et al. 2013a). These data were obtained from samples of the St Peter sandstone from SW Wisconsin originally collected and reported on by Kelly et al. (2007). Each sample contains both initial detrital quartz grains and optically continuous quartz overgrowths (although analysis by Anovitz et al. (2015a) suggests that there may be significant differences between the initial and overgrowth quartz). It is believed that these samples were never buried to any appreciable depth, and that the quartz overgrowths formed as silicretes, precipitation of dissolved silica. Figures 42 and 43 show examples of BSE/SEM images from low- and high-porosity samples, and Figures 44 and 45 show how the scattering (slope and subslope Ds) and imaging scale fractality (D(0)) as well as the correlation dimension (D(2)), the multifractality (D(0)–D(2)) and the lacunarity change with porosity. Since the primary geologic process here is overgrowth formation, which decreases porosity, the process variable increases to the left in these figures. It is clear from that there are distinctive, consistent changes in both the multifractal and lacunarity behavior of these sandstones as a function of overgrowth formation, as well as

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Figure 42. BSE/SEM image of a sample of St Peter sandstone from SW Wisconsin showing decreased fractality (D(0) = 1.5511) and increased multifractality (D(0) – D(2) = 0.1844) and lacunarity (0.5618) at decreased porosity. [Sample 04Wi17bPRL, described in Anovitz LM, Cole DR, Rother G, Allard LF Jr, Jackson A, Littrell KC (2013a) Diagenetic changes in macro-to nano-scale porosity in the St. Peter Sandstone: an (Ultra) small angle neutron scattering and backscattered electron imaging analysis. Geochimica et Cosmochimica Acta, Vol. 102, p. 280–305, used with permission from Elsevier.] Image pore fraction = 0.033. Image is 12.5 mm across.

Figure 43. BSE/SEM image of a sample of St Peter sandstone from SW Wisconsin showing increased fractality (D(0) = 1.8017): and decreased multifractality (D(0) – D(2) = 0.0539) and lacunarity (0.2224) at high porosity. [Sample 04Wi02(2): described in Anovitz LM, Cole DR, Rother G, Allard LF Jr, Jackson A, Littrell KC (2013a) Diagenetic changes in macro-to nano-scale porosity in the St. Peter Sandstone: an (Ultra) small angle neutron scattering and backscattered electron imaging analysis. Geochimica et Cosmochimica Acta, Vol. 102, 280–305, used with permission from Elsevier.] Image pore fraction = 0.228. Image is 12.5 mm across.

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Figure 44. Changes in the average slope and slope of only the fractal range obtained from the scattering data, in the box counting dimension (D(0): Rényi dimension for q = 0): Rényi dimension for q = 2 (D(2)): and D(0) – D(2): a measure of the multifractality for samples of the St. Peter sandstone from SW Wisconsin plotted as a function of pore fraction. Increasing overgrowth formation is, therefore, to the left in this diagram. [Anovitz LM, Cole DR, Rother G, Allard LF Jr, Jackson A, Littrell KC (2013a) Diagenetic changes in macro-to nano-scale porosity in the St. Peter Sandstone: an (Ultra) small angle neutron scattering and backscattered electron imaging analysis. Geochimica et Cosmochimica Acta, Vol. 102, p. 280–305, used with permission from Elsevier.]

Figure 45. Changes in lacunarity for samples of the St. Peter sandstone from SW Wisconsin plotted as a function of pore fraction. Increasing overgrowth formation is, therefore, to the left in this diagram[Anovitz LM, Cole DR, Rother G, Allard LF Jr, Jackson A, Littrell KC (2013a) Diagenetic changes in macro-to nano-scale porosity in the St. Peter Sandstone: an (Ultra) small angle neutron scattering and backscattered electron imaging analysis. Geochimica et Cosmochimica Acta, Vol. 102, p. 280–305, used with permission from Elsevier.]

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differences between changes observed at submicron (scattering), and supramicron (imaging) scales. At the imaging scale the overall fractality decreases with overgrowth formation, but the scale-dependence of the fractal behavior, the multifractality, and the inhomogeneity of the pore distribution increase. At scattering scales, however, the fractal dimension increases with overgrowth formation. These effects and their potential origins were described in more detail by Anovitz et al. (2013a). In his original descriptions of fractal systems Mandelbrot (1977) not only described fractal and multifractal behavior a lacunarity, but a fourth variable he called succolarity. This was originally defined by Mandelbrot (1977), as follows: “a succolating fractal is one that “nearly” includes the filaments that would have allowed percolation; since percolare means “to flow through” in Latin …, succolare (sub-colare) seems the proper neo-Latin for “to almost flow through.” While this clearly relates to a fractal description of such concepts as connectivity, tortuosity and percolation, and thus for our purposes to the relationship between porosity and permeability. However, at the time he did not further define this parameter. To our knowledge the first attempt to quantify the concept of succolarity was that of de Melo (2007, see also de Melo and Conci 2008, 2013). De Melo notes that succolarity is, in fact, a part of the very large field of percolation theory. This, generally speaking, asks what the probability is that a fluid pored on top of a porous medium will be able to reach the bottom. De Melo then defines succolarity as “the percolation degree of an image (how much of a given fluid can flow through this image)”. This process is, therefore, directional. For a rectangular image connected sections beginning at the top, right, left and bottom of an image are not necessarily identical. The calculation, therefore, begins by flooding all open pixels along one edge and determining all the open pixels connected to those across pixel edges. The image is then divided into (N) equal sized boxes of edge length (), and for each box of a given size the percentage of “on” pixels, the occupation fraction (O(n)), is calculated. Each box is then assigned a “pressure” (P(n)) equal to the number of pixels from the input side to the centroid (which may be the middle of a pixel) of the box. The sum of the product of the occupation percentage and pressure for each box is then calculated. This is then normalized to the sum when the occupation fraction of each box is 1, yielding the succolarity for a given flow direction as:

 

N

O  n P n

n 1

 n 1P  n  N

(116)

Interestingly, de Melo’s results (2007, de Melo and Conci 2008, 2013) suggest that this value is essentially independent of the box size, but not of flow direction. Despite de Melo’s quantification of the concept, to date succolarity as a measure of fractal texture has received much less attention than fractality and laccunarity, although there are studies of drainage systems (Shahzad et al. 2010, Mahmood et al. 2011), biomedical analysis (Ichim and Dobrescu 2013; Cattani and Pierro 2013; N’Diaye et al. 2013, 2015; Sangeetha et al. 2013) and image recognition (Abiyev and Kilic 2010). While there have, as yet, been no studies we know of quantifying the succolarity of inorganic porous materials and rocks, the potential utility of this approach is clear, and bears further testing in geological systems. While the succolarity as defined by de Melo (2007) may to have potential for defining percolation-related properties, other functions may also serve this purpose. Jiao et al. (2007 2008, 2009, 2010, 2013) and Jiao and Chawla (2014), for instance, have suggested that the twopoint cluster function (C2(r), see Torquato et al. 1988; Torquato 2002a,b) is also sensitive to

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topological connectedness. In fact, it should be noted that the approaches discussed above for quantifying the nature of a pore structure from imaging data (fractal, multifractal, lacunarity, succolarity, two-point autocorrelation) are only a fraction of the methods (cf. Heilbronner and Barrett 2014) and statistics (cf. Torquato 2002a) available for such analysis. Several other types of correlations have been derived for random, homogenous materials. Few, however, to our knowledge, have been applied to geological materials in a systematic way although Jiao et al. (2007, 2008) suggested that these may be a pragmatic alternative to the analysis of harder to calculate higher-order correlation functions. Torquato (2002a) discussed a number of these statistics and their relationships to deriving bulk material properties for porous materials. These include: surface correlation functions (between a point on a surface and a point in a pore or between a point on a surface and another point on the surface, applicable to trapping and flow problems), lineal path functions (the probability that a line of length (l) lies wholly in a single phase, provides linear connectedness information), chord-length density functions (probability of finding a chord, the line segments between phase boundaries along a line through the image, of length (l), defines discrete free paths for transport: see Thompson et al. (1987) for an application to sedimentary rocks), pore size functions (the probability that a randomly chosen point in a pore lies at a distance r from the pore/solid interface), percolation and cluster functions (the probability of finding two points in the same cluster of a phase or pores), nearest neighbor functions (for particles suspended in a medium the probability of finding the nearest-neighbor particle at a given distance from a reference particle), point/qparticle correlation functions (for particles in an inhomogeneous medium, the probability of finding a point in phase i at position x, the center of a sphere in some volume dr1 around point r1 … and the center of another sphere in a volume drq around point rq×, related to conductivity, elastic moduli, trapping and permeability), and surface/particle correlation functions (again for particles an inhomogeneous medium, the probability of the center of a particle being a distance r from a point on the surface, related to permeability through random beds of spheres). Further discussion of these correlation functions is beyond the scope of this review, and the interested reader is referred to the work of Torquato (2002a) for more information. A combination of these statistical techniques, together with the characterization approaches described above, should provide methods, not only to describe bulk properties of porous materials from those of the mineralogy of which it is composed but to provide real-world multiscale descriptions of porous materials that can be used in model, in silico, reservoir flow, oil and gas recovery, transport of heat and contaminants, aquifers, and other properties of porous reservoirs of significant interests. A proper statistical representation of meso- and micro-pore morphology in the form of a 3-dimensional, angstrom-to-millimeter scale model or rocks and other porous material is crucial for the study of fluids in porous rocks, as well as for upscaling atomic-scale mineral growth and dissolution rates to pore, hand sample and reservoir scales, as this will provide a realistic multiscale matrix for these efforts. There have been several attempts to provide such models. The simplest, based on geometrical idealizations (e.g., Thovert et al. 2001), provide a useful tool but are not sufficiently detailed representations of real rocks. The scattering and imaging data described above, however, provide direct input to more general but also more computationally demanding reverse Monte Carlo (RMC) techniques (McGreevy and Pusztai 1988). Salazar and Gelb (2007) showed that scattering and adsorption experiments provide complementary information that can yield more realistic RMC models. As just noted, Torquato (2002a) investigated various simple ‘structural descriptors’ to identify those containing the most information about pore structure and connectivity (Torquato 2002a; Jiao et al. 2010). Mariethoz et al. (2010; Mariethoz and Kelly 2011) showed how multiple-point statistics (MPS), based on experimental data or simple physically-based structural motifs and a Monte Carlo algorithm can be used to develop structure models that typically appear more realistic. ‘Soft’ data may be integrated within a Bayesian statistical scheme (Lu et al. 2009). Other approaches employ simulations of the

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physical processes generating a particular material, such as mimicking sedimentation followed by compaction and cementation of sandstone (Oren and Bakke 2002).

Comparisons of multiple techniques As noted above, it is valuable (and frequently necessary) to combine porosity and pore feature information from different complementary techniques. A case in point is the use of mercury intrusion porosimetry (MIP) and neutron scattering (e.g., Clarkson et al. 2012; Swift et al. 2014). MIP provides information on effective or accessible pore throat size distribution whereas (U)SANS reveals details about pore size, porosity, pore volume, surface area to pore volume ratios and fractal behavior. A good example of this was presented by Swift et al. (2014) who documented the relationship between mineralogy and porosity in the Eau Claire formation, a middle- to upper Cambrian regional mudstone located in the mid-continent of the U.S. The Eau Claire is the seal rock for the Mt. Simon formation used at the Decatur site in Illinois for CO2 storage demonstration. This study utilized MIP, (U)SANS and SEM to quantify the pore features of three subfacies in this formation, an illitic-shale facies (Fig. 46a), one rich in carbonate (Fig. 46b) and one enriched in glauconite (Fig. 46c). As a first step in the comparison between MIP and neutron scattering, one can use the cumulative pore volumes derived from the neutron scattering data as identified in Swift et al. (2014) to calculate pore size distributions. This should be done with caution, however, as it requires an assumption with regard to pore shape. As shown by Anovitz et al. (2009,

Figure 46 a-c. Comparison of MIP (blue) and NS (red) data from the Eau Clair formation mudstone for three different lithologic subfacies – an illitic shale, a carbonate-rich mudstone and a glauconitic mudstone (From Swift et al. 2014). The mineral maps on the left side of the figure were produced from QEMSCAN imaging using an FEI Quanta 250 Field Emission Gun SEM at Ohio State. [Reproduced from Swift AM, Anovitz L M, Sheets JM, Cole D R, Welch SA, Rother G (2014) Relationship between mineralogy and porosity in seals relevant to geologic CO2 sequestration. Environmental Geoscience Vol. 21, p. 39–57 AAPG© [2014], reprinted by permission of the AAPG whose permission is required for further use.]

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2011), TEM examination of nanoporosity in rocks suggests that pore shapes are extremely variable. In addition, the fractal nature of the pore/solid boundary further complicates the assumptions. However, by taking the derivative of the cumulative porosity curves the pore volume distribution can be obtained without assumptions as to pore shape. These distributions are shown for the three samples in Figure 46a-c along with examples of the mineral maps produced by the SEM QEMSCAN method. As can be seen, pore scales fall into several groups, not all of which are present in each material. At the nanoscale, two pore regimes near 25 and 135 Å, and a broad, larger-scale regime centered around 10–20 m occur in both of the mudstones. The porosity of the illitic shale is dominated by the first of the two nanoscale distributions. Microscale pores form only a small fraction of the total in this sample and are polydisperse, with a broad hump around 2 m. Although nanoscale pores appear to be present in both mudstones, only the glauconitic mudstone has a significant peak near 10 nm, mirroring the weaker pore size cluster at that scale in the shale. These clearly reflect the high-Q “humps” in the I(Q) versus Q plots given in Swift et al. (2014). In the microscale regime, the glauconiterich mudstone has a larger peak at 20 m, which may have a shoulder around 115 m. The carbonate mudstone has a narrower peak at 30 m that may correspond to part of the wider distribution observed in the glauconitic sample. By comparison, the MIP results indicate the connected porosity accessible by mercury through pore throats ranges between 4 nm and 50 μm in equivalent circular diameter is 2.2% of the rock volume for the shale, 0.2% for the carbonate-rich mudstone, and 4.5% for the glauconite-rich mudstone. As shown in Figure 46, the connected porosity of the illitic shale is dominated by pores having pore throats smaller than 0.1 μm, with a peak at or below 4 nm. The carbonate-rich mudstone has almost no connected porosity, a finding matching SEM observations of that sample. The glauconite-rich mudstone, by contrast, has substantial connected porosity and a bi-modal distribution of pore throats with clear peaks at roughly 10 nm and 700 nm. By combining the information provided by these two methods one can estimate the pore size to pore throat ratios, a proxy for pore accessibility. As we have observed in the case of the mudstone results there is some similarity between the pore size patterns and overlap in pore feature dimensions retrieved from scattering and those produced by MIP. This is due in large part to the fact the pores themselves are approaching the size of the pore throats especially at the smaller length scales. However, when one examines the pore to pore-throat dimensions in coarser grained clastic rocks like sandstone the difference can be more dramatic. A case in point is shown in Figure 47 where

Figure 47. Pore volume distribution (as %) from MIP (left data) and NS (right data) plotted against either pore throat size or pore size (in mm): respectively, for the Mt. Simon sandstone, Ohio (Swift, unpublished results). The insert is a BSE SEM image showing examples of pores and pore throats. (From A. Swift, unpublished)

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we show pore volumes from MIP data plotted with similar data from neutron scattering data against pore throat size and pore size, respectively, for the Mt. Simon sandstone from Ohio. It is clear that there is a significant difference in the dimensions of the pore throats compared to the actual pore sizes. In this example, the pore to pore-throat ratio is on the order of 100 which is a typical value reported for many sandstones (cf. Wardlaw and Cassan 1979; Nelson 2009; Anovitz et al. 2015a). The take-away message here is that by combining not just two methods such as MIP and neutron scattering, but adding another approach like the SEM imaging, one can begin to not only quantify the pore features but also visually identify and even classify what these look like in detail. Another excellent example of using multiple techniques to describe pores and pore throats was presented by Beckingham et al. (2013). In this study they described pore-network modeling of two kinds of samples, an experimental column of reacted coarse sediment (221– 300 μm diameter) from Hanford, WA and a sandstone from the Viking formation, the western Canadian sedimentary basin. The modeling was based on the analysis of 2-D SEM images of thin sections coupled with 3-D X-ray micro tomography (CMT) data (Fig. 48). X-ray CT imaging has the advantage of reconstructing a 3-D pore network while 2-D SEM imaging can easily analyze sub-grain and intragranular variations in mineralogy. Refer to Noiriel (2015) for details on the CMT technique. Pore network models informed by analyses of 2-D and 3-D images at comparable resolutions produced permeability estimates with relatively good agreement. For cases where there was less adequate overlap in resolution between methods orders of magnitude discrepency in permeability were observed. Comparison of permeability predictions with expected and measured permeability values showed that the largest discrepancies resulted from the highest resolution images and the best predictions of permeability will result from images between 2 and 4 μm resolution.

CONCLUSIONS The pore structures of natural materials (rocks, soils etc.), as well as those of many synthetics play a critical role in controlling the physical properties of and processes in rocks (Emmanuel et al. 2015; Navarre-Sitchler et al. 2015; Royne and Jamtveit 2015, this volume), and the interaction between them and the fluid that are stored, flow through, precipitate in (Stack 2015, this volume) and react with (Liu et al. 2015; Molins 2015; Putnis 2015, this volume) them. The better we understand and can quantify those porous structures, the better will be our ability to model, understand and predict the evolution of geological environments, either under natural conditions or those such as CO2 or radiological waste sequestration, or addition or removal of other fluids from geological reservoirs. The goal of this paper has been to present an overview of techniques for the measurement, description and quantification of pore structures in rocks and rock-like materials such as cements and ceramics. These make up the three-dimensional description of the critical pore/solid interface above atomic scales, and data on their structure provide a crucial basis for our understanding of permeation, transport and storage of fluids as well as various types of solid, liquid and gas contaminents. They also provide a link between the physical properties of the minerals that make up the rock, and those of the rock as a whole. To make this connection, however, we must understand not only the fraction of the rock occupied by pore space, but a number of its properties. These include the connectivity, surface area and roughness, size distribution, laccunarity, and other aspects of the texture of the pore structure. It is clear that, since the earliest attempts to quantify the porosity of geologic materials, there has been a significant increase in the complexity of analysis and quantification approaches. Early approaches required little that wasn’t readily available, or could be made available in a small laboratory. More recent approaches require expensive equipment that

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Figure 48. Conceptualization of the process of informing pore-network models with information from 3-D X-ray CMT or 2-D SEM imaging. [Reproduced from Beckingham LE, Peters CA, Um W, Jones KW, Lindquist WB (2013) 2-D and 3-D imaging resolution trade-offs in quantifying pore throats for prediction of permaebility. Advances in Water Resources, Vol. 62, p. 1–12, with permission from Advances in Water Resources.]

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may or may not be available at a given institution, or that requires travel to large, specialty user facilities. In many cases, however, far from supplanting the earlier methodologies, newer approaches frovide complementary data that extends our ability to quantitatively describe the pore structure of geologic materials. This quantification, in turn, is providing bridges to attempts to to describe the properties of macroscopic systems from those of the mineral grains of which they are composed, and detailed in-silico models of porous systems with statistical properties equivalent to those of real rocks and thus opportunities for finer scale and more realistic models of percolation and reactive and non-reactive transport. The availability of these new, multiscale quantification techniques has also opened up a number of new avenues for understanding fluid/rock interactions. Approaches that provide statistical analysis of relatively large rock volumes can be correlated with images of the pores themselves, as can the spatial relationships at many scales between pore types, especially accessible vs inaccessible porosity, and mineralogy. Newer techniques, as yet little or unexplored in geological contexts (e.g., SESANS, magnetic SANS) provide opportunities to probe such questions as the microscopic origins of geomagnetism and the nature of particulate transport, and to parallel inverse space with real space measurements. In many cases the basic theories and application methodologies of the techniques already exist for non-geological materials and problems, but it is also possible that these may require significant modification for applications to geological materials and problems. Thus, while it is our hope that this summary, and descriptions elsewhere in this volume (Noiriel 2015, this volume) will provide a useful reference for those interested in the analysis of porous materials, it is clear that future developments are likely to futher expand this toolkit of approaches to measuring, characterizing and quantifying the structure of natural and synthetic porous materials.

ACKNOWLEDGMENTS Effort was funded by the Department of Energy Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences through the Energy Frontier Research Center - Nanoscale Control of Geologic CO2. We acknowledge the support of the National Institute of Standards and Technology, Center for Neutron Research, U.S. Department of Commerce, which is supported in part by the National Science Foundation under agreement No. DMR-0944772; the High-Flux Isotope Reactor at the Oak Ridge National Laboratory, sponsored by the Scientific User Facilities Division, office of Basic Energy Sciences, US Department of Energy; the Manuel Lujan, Jr. Neutron Scattering Center at Los Alamos National Laboratory, which is funded by the Department of Energy’s office of Basic Energy Sciences. Los Alamos National Laboratory is operated by Los Alamos National Security LLC under DOE Contract DE-AC52-06NA25396; the Advanced Photon Source, a U.S. Department of Energy (DOE) office of Science User Facility operated for the DOE office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357; the Hoger Onderwijs Reactor, Delft University of Technology, The Netherlands, and the JCNS at the ForschungsNeutronenquelle Heinz Maier-Leibnitz, Garching, Germany, in providing the X-ray and neutron research facilities used in this work. We would especially like to thank our many other colleagues and co-authors at the above institutions for their help and guidance. Certain commercial equipment, instruments, materials and software may be identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, the Department of Energy, or the Oak Ridge National Laboratory, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. We would like to thank Timothy Prisk, Wim Bouwman, Jan Ilavsky, Roger Pynn, Bill Hamilton, Lauren Beckingham, and Alexis NavarreSitchler for their helpful reviews and suggestions. Prose on multifractal calculations and

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imaging/scattering calculations from Anovitz et al. (2013a), and Figures 37, 41, 42, 43, 44 and 45 from there, Berryman (1988) and Anovitz et al. (2015a) used with pemission from Elsevier. We would also like to thank colleagues who provided or agreed to our use of unpublished figures including: John Barker (Figure 22 and description of the NCNR VSANS), Vitaliy Pipich (Fig. 21), and Boualem Hammouda (Fig. 23 and original of Fig. 24) or measured data (Chris Duif and Wim Bouwman performed the SESANS measurements in Figure 27 at the Delft University of Technology, and Ken Littrell provided the original data from Littrell et al. (2002) used to plot Figure 30), and Alex Swift (Fig. 47).

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 165-190, 2015 Copyright © Mineralogical Society of America

Precipitation in Pores: A Geochemical Frontier Andrew G. Stack Chemical Sciences Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37831-6110, USA [email protected]

INTRODUCTION The purpose of this article is to review some of the recent research in which geochemists have examined precipitation of solid phases in porous media, particularly in pores a few nanometers in diameter (nanopores). While this is a “review,” it is actually more forwardlooking in that the list of things about this phenomenon that we do not know or cannot control at this time is likely longer than what we do know and can control. For example, there are three directly contradictory theories on how to predict how precipitation proceeds in a medium of varying pore size, as will be discussed below. The confusion on this subject likely stems from the complexity of the phenomenon itself: One can easily clog a porous medium by inducing a rapid, homogeneous precipitation directly from solution, or have limited precipitation occur that does not affect permeability or even porosity substantially. It is more difficult to engineer mineral precipitation in order to obtain a specific outcome, such as filling all available pore space over a targeted area for the purposes of contaminant sequestration. However, breakthrough discoveries could occur in the next five to ten years that enhance our ability to predict robustly and finely control precipitation in porous media by understanding how porosity and permeability evolve in response to system perturbations. These discoveries will likely stem (at least in part) from advances in our ability to 1) perform and interpret X-ray/neutron scattering experiments that reveal the extent of precipitation and its locales within porous media (Anovitz and Cole 2015, this volume), and 2) utilize increasingly powerful simulations to test concepts and models about the evolution of porosity and permeability as precipitation occurs (Steefel et al. 2015, this volume). A further important technique to isolate specific phenomena and understand reactivity is also microfluidics cell experiments that allow specific control of flow paths and fluid velocities (Yoon et al. 2012). An improved ability to synthesize idealized porous media will allow for tailored control of pore distributions, mineralogy and will allow more reproducible results. This in turn may allow us to isolate specific processes without the competing and obfuscatory effects that hinder generalization of observations when working with solely natural samples. It is likely that no one single experiment, or simulation technique will provide the key discoveries: to make substantive progress will require a collaborative effort to understand the interplay between fluid transport and geochemistry. Where rock fracturing and elevated pressures are of concern, an understanding and capability to model geomechanical properties are necessary (Scherer 1999). It is critical to understand not just how the precipitation reactions themselves occur, but how a given solution composition, net flow rate and porous substrate translate to macroscopic hydrologic parameters such as the evolution porosity and permeability that change in response to geochemical reactions. Predicting these macroscopic terms is prerequisite for extrapolating from laboratory-based or in silico (i.e., computational model) systems where every pore in 1529-6466/15/0080-0005$05.00

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the reactor/cell can be resolved or sampled to reservoir-scale simulations and field studies. In these larger length-scale studies, it is no longer practical to think about individual pores but instead one must consider pore distributions in aggregate. The current state-of-the-art is to use linear relationships where the porosity and permeability are calculated using empirically fit functions (Gibson-Poole et al. 2008). To improve the status quo will require us to develop new, up-scaling theories that can accurately approximate the richness of reactivity observed at the atomic- to pore-scales, but are still useful at the reservoir scale (Reeves and Rothman 2012). To verify and validate such models will require a strong connection between research performed at the nanometer- or micrometer-length scales and larger column- or field-scale studies.

RATIONALE A capability to predict and control precipitation in pores could result in more useful geochemistry in many situations in the subsurface, and in this section a few of them will be described. A good first example of where precipitation is important is the well known two order of magnitude discrepancy between field-based and laboratory-based rates of mineral weathering reactions (Drever and Clow 1995; White 2008; Stack and Kent 2015). There are abundant theories for the origin of the discrepancy, but two particularly important for this article are the existence of pore-size-dependent effects (Putnis and Mauthe 2001; Emmanuel and Ague 2009; Stack et al. 2014) and secondary mineral formation that reduces the reactive surface area (Drever and Clow 1995; Maher et al. 2009). Maher et al. (2009) in particular showed in a chronosequence of soils that laboratory-based dissolution rates can be consistent when precipitation of secondary phases is accounted for. The effect of secondary mineral precipitation on weathering will become even more significant if the extent of precipitation is large enough such that the permeability of the soil or rock is significantly degraded and flow of fluids through the rock is impeded. Mineral weathering itself is important for understanding the composition of ground and surface waters, and minerals act as the longer term buffer of carbon dioxide in the atmosphere (Berner et al. 1983) and acidity in rain (Drever and Clow 1995). In addition to natural processes, engineered precipitation in porous media is being explored as a contaminant sequestration and remediation strategy. It may be possible to take advantage of the low solubility of some minerals to remediate metal contaminants intentionally. An example includes radium, which readily substitutes for barium in barite (BaSO4) due to similar reaction behavior and size of radium and barium. Radium is an issue for spent nuclear fuel repositories that use bentonite as a sorbent since radium does not adsorb strongly to clay minerals (Curti et al. 2010). However, there is evidence that the mobility of radium is effectively controlled by barite solubility (Martin et al. 2003) because so much radium can incorporate into barite and barium is more common than radium. Already a method that takes advantage of the low solubility of Ba–Ra–Sr sulfate minerals has been proposed as an aboveground treatment strategy for hydraulic fracturing wastewater (Zhang et al. 2014), but it may be possible to utilize this for subsurface applications in porous media. Another example is that of uranium, which potentially could be remediated using hydroxyapatite which dissolves and causes a uranium phosphate precipitate to form (Arey et al. 1999; Fanizza et al. 2013). Instead of using abiotic hydroxapatite directly, less expensive bone meal has been considered (Naftz et al. 1998). When used as a permeable reactive barrier, the extent of this precipitation is such that the bone meal needs to be diluted with unreactive phases to avoid clogging the reactive barrier by reducing the permeability drastically (Naftz et al. 1998). To avoid the issue of clogging, other studies have examined the idea of using a soluble organo-phosphate that is degraded by bacteria in the subsurface to create dissolved inorganic phosphate, which can

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then react with the uranium (Wright et al. 2011; Beazley et al. 2007). This concept is attractive since an aqueous solution containing the organo-phosphate could be injected into a well, where it would presumably mix with ambient fluids and become dispersed. The microbially induced cleavage of the organophosphate would cause the dissolved inorganic phosphate concentration to increase over time and lead to precipitation of uranium phosphate. A final example metal contaminant is strontium-90, which is found sometimes as a legacy waste from nuclear weapons production (Riley et al. 1992). Due to the similar chemical behavior and size of strontium and calcium, strontium incorporates into the calcite (CaCO3) crystal lattice (Wasylenki et al. 2005; Bracco et al. 2012). If one could therefore induce precipitation of a strontium-rich calcite, the contaminant would be trapped as a solid phase and immobilized. Immobilization for approximately one hundred years would be sufficient time to allow the strontium-90 to decay to stable zirconium (Gebrehiwet et al. 2012), which in turn is incredibly insoluble (Wesolowski et al. 2004). It is not unreasonable to think that a calcite precipitate formed in the subsurface could last that long. As one of the authors in the Gebrehiwet et al. (2012) study suggested, however, one can only control three things in regards to induced precipitation in the subsurface: what solution one injects, where one injects it and how fast one does so (Redden G.; pers. commun.). Thus for a remediation scheme of this type to work in the subsurface, as opposed to in a soil or engineered reactive barrier, one would need to have a precise control over where precipitation takes place, how fast precipitation occurs and an understanding of how that precipitate changes the pore structure and communication of the fluid. This level of control of precipitation has not been demonstrated to my knowledge, but perhaps is not as far-fetched as one might initially guess. The issues involve being able to balance mixing of an injected fluid with the ambient groundwater and/or other injected fluids containing reactants, and the timing of precipitation reactions within the porous medium. However, as discussed above, permeability should be maintained while the precipitation reaction is ongoing if possible, i.e., until all of the contaminant is successfully sequestered in a solid phase. The precipitation should therefore not occur too quickly, otherwise the calcite will form too close to the injection well and clog, nor too slowly so that the strontium and injected fluids disperse prior to precipitation occurring. This might be described as a “Goldilocks” problem in mineral precipitation kinetics, get it to occur not too quickly, nor too slowly, but just right. By far the most common contaminant by mass that has been proposed to be sequestered by induced precipitation is carbon dioxide. There has been and continues to be intense research and pilot projects whose goal is to determine the feasibility of widespread sequestration of this contaminant. Despite the effort, there is still uncertainty about how much carbonate-containing mineral will precipitate (if any), the locales in which it will precipitate and how long the reactions will take. The conventional wisdom is that it will take something on the order of 1000 years to convert the carbon dioxide to a mineral (Metz et al. 2005). That is likely true in some situations, e.g., a storage reservoir that is a clean quartz sand and contains few highly reactive minerals such as the Sleipner field in the North Sea. However there are also some real world examples where significant mineral precipitation has either been directly observed, or evidence has been discovered that it may be occurring. In Nagaoka, Japan, some pore fluids have become supersaturated with respect to calcite only a few years after injection of CO2 started (Mito et al. 2008). In a site in west Texas where CO2 was injected over a period of 35 years for the purpose of enhanced oil recovery, fractures sealed with calcite can be observed in well casing cement (Carey et al. 2013). The latter example raises a particularly interesting possibility, which is that even if the net amount of carbon dioxide turned to a mineral is small, there could be enough precipitation to affect the storage security of a site. In this case, fractures in the well casing cement were sealed with calcite. One might expect that something similar could happen in a cap rock or seal that has had fractures open or form in it due to the increased pressure from the carbon dioxide injection. In what might be a maximum amount of precipitation observed

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thus far (for the available porosity) has been during CO2 injection in the Columbia River basalts. It appears that mineral precipitation begins immediately after injection, to the extent that pumps can get clogged and shut down (Fountain 2015). Substantial calcite precipitation is also suspected during CO2 injection tests in basalt in Iceland. One caution is that in the basalt injection tests have all been relatively small amounts of CO2 (kilotons), far short of the amount that would be needed to be sequestered to have an impact on the climate which are megatons to gigatons. In all three of the cases listed here where precipitation has been observed (or could occur), there has been substantial amounts of reactive mineral phases that buffer the pH and/or supply the cations necessary to cause carbonate minerals to precipitate. While some modeling studies have been undertaken to examine how much carbonate mineral can be precipitated for a given mineralogical composition (Zhang et al. 2013), what is lacking are direct measurements of how porosity and permeability evolve in a rock during the dissolution of pH-buffer and cation-source minerals and precipitation of carbonates. If precipitation occurs completely uniformly, it could be that residual bubbles of CO2 in the subsurface left over from plume migration would become surrounded by a self-limiting coating of carbonate mineral that prevents further reaction of the gas within the bubble (Cohen and Rothman 2015) (Fig. 1). This may not be a detrimental outcome since the precipitated material might also act as a protective casing surrounding the CO2 that prevents its migration, but it would slow and limit carbonate mineral precipitation. An important ongoing issue in industrial or municipal settings is the prevention and removal of scale, i.e., unintended mineral precipitates that form in the (porous) subsurface as well as within wells, pipes and equipment. Entire textbooks have been written on the subject of attempting to predict the formation of the most common scale-forming minerals and dealing with them after their formation (e.g., sparingly soluble salts such as barite, BaSO4, calcite, CaCO3, as well as covalently bonded phases such as silica or quartz, SiO2 and iron oxides, Fe2O3) (Frenier and Ziauddin 2008). It has been estimated that scale formation results in 1.4 billion USD costs annually due to lost production and removal (Frenier and Ziauddin 2008). New extraction technologies such as the combination of hydraulic fracturing and horizontal drilling are resulting in new scale formation and removal challenges as well as treatment of wastewater. An improved ability to predict and control precipitation reactions in porous media, i.e., prior to the fluids coming to the surface, could help to deal with these issues. If we could better predict and control reactions within porous media especially, it could help us understand

Figure 1. Model of mineral precipitation due to mixing of CO2-rich fluids with surrounding brine. a) The warmer (lighter gray) colors show where precipated carbonate minerals are predicted to occur, creating a zone of low permeability between the two liquids and self-limiting the precipitation reaction. Units on X and Y are characteristic lengths. b) Profile along the white arrow marked in part a). [Images slightly modified from Cohen and Rothman (2014), Proceedings of the Royal Society of London A, Vol. 471, 2010853. Creative Commons license, v.4.0]

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why scale does or does not form in a given scenario, and may guide us to treatment strategies, either above ground or directly in the subsurface prior to extraction. As a specific example, some of the wells in the Marcellus shale are producing elevated concentrations of radium, strontium and barium in their flowback or formation waters. All of these cations form low solubility sulfate minerals that clog porosity, wells, lines and equipment and have been found in effluents of wastewater treatment plants that accepted hydraulic fracturing wastewater and in sediments downstream (Ferrar et al. 2013; Warner et al. 2013). The economic and environmental ramifications of these dissolved species and their mineral forms have been significant. These findings were part of the rationale for the U.S. state of Pennsylvania’s request that municipal wastewater treatment plants stop accepting hydraulic fracturing wastewater (Boerner 2013). The uncertainties surrounding the public health effects of radium-containing scale were also cited in the U.S. State of New York Department of Health review that led to the state banning hydraulic fracturing completely (Zucker 2014). It is clear that we would benefit from an improved ability to predict why ambient fluids in the subsurface do not have precipitated minerals in them, and why precipitation only occurs after the fluids are brought up to the surface. Furthermore, if precipitation of scale forming minerals could be engineered in situ, without impacting the oil or gas production, it would prevent costly treatment strategies and obviate environmental impact of the oil and gas production. A sometimes overlooked set of effects of precipitation in pores are geomechanical. That is, when rocks, cements or other porous materials have fluids circulate through them that induce crystallization, the precipitated material exerts a pressure on the rock itself and vice versa. Over time, this pressure can be sufficiently large to crack or fracture a rock (Scherer 1999; Emmanuel and Ague 2009). The physics of this process are discussed in Emmanuel et al. (2015, this volume). This phenomonon has dramatic implications for weathering of rocks as it is a coupled chemical and physical process and the resulting fractures will act as conduits for new fluid that will further increase weathering rate (Jamtveit et al. 2011). Geology undergraduates can likely tell you the mechanisms of an every-day example of this process, which is pothole formation in asphalt concrete during frost wedging or heaving. When water freezes, its crystal structure is slightly less dense than the original water, and this causes exerts a force on the sand grains, wedging them apart. The crystallization pressure due to precipitation in pores may also play a role on a much larger scale than potholes or even weathering: it was recently suggested that precipitation of anhydrite (CaSO4) in pores may result in micro-seismicity near mid ocean ridges (Pontbriand and Sohn 2014). The evidence is that the seismic signature and locale of the earthquakes does not match those of a tectonic origin and are not correlated with any larger seismicity in the area. The proposed mechanism is that there is secondary circulation of seawater near the ridge, which causes anhydrite to precipitate as the seawater heats. This is because anhydrite has a retrograde solubility, meaning it becomes less soluble with increasing temperature. Upon seeing this argument, one is tempted to speculate about whether precipitation could be a contributing factor in other seismic events, such as during injection of wastewater (e.g., the Youngstown, Ohio earthquake was attributed to injection of hydraulic fracturing fluids; Funk 2014; Skoumal et al. 2015). This possibility has been raised for carbon sequestration as well (Melcer and Gerrish 1996), but there is no information at all on the potential for precipitation reactions to induce seismicity. In all the above examples, our understanding of precipitation in pores is poor in that we can define what we would like to have happen, but have trouble demonstrating that it is happening or can happen outside of some obvious indicator like the pump clogging or exhuming a reacted rock core to look for mineral precipitates. This leads to an inability to predict reliably the extent, timing and locale (e.g., pore-size distribution) of precipitation in reactions. In the remainder of this review, I summarize what I know about how precipitation in a porous medium will occur. While everything described in this article is reasonably plausible, some evidence and theories

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are contradictory and sorting through which effects are important at a given time and length scale will require the substantial interdisciplinary effort described in the introduction.

PORE-SIZE-DEPENDENT PRECIPITATION When reviewing the studies that have considered a pore size dependence for precipitation reactions, one can find a plausible argument in the literature for any trend in the dependence on pore size, or a lack thereof. That is, one can find models for uniform precipitation over all pore sizes with no intrinsic pore-size dependence (Borgia et al. 2012) (Fig. 2b), observations that precipitation in smaller pores is inhibited (Emmanuel et al. 2010) (Fig. 2c), theoretical predictions that precipitation should occur preferentially in smaller pores (Hedges and Whitelam 2012) (Fig. 2d) and observations of different behaviors depending on system chemistry (Stack et al. 2014). This lack of consensus arises from several factors, not the least of which is that this discrepancy may be real and the functional form of any pore-sizedependent behavior that is observed depends on the substrate and precipitate compositions and structures, or solution conditions. At this time we do not know precisely which processes are most important to determine the pore size range over which precipitation occurs, and further information is necessary to make reliable predictions. The length scale over which the precipitation is observed may be important as well—processes that occur at one length scale or mineral growth regime may not be significant in all cases. For example, processes important during the incipient stages of nucleation may not be important after aggregation and growth of the nuclei into larger crystals.

Effects of precipitation on porosity and permeability Prior to getting into the details, it is useful to discuss the reasons why a pore-size dependent precipitation should matter, especially for larger scale properties. As mentioned above, one

Figure 2. Schematic of pore-size dependence for precipitation reactions. a) Illustrative flow path through a series of mineral grains within a rock. b) Rock after precipitation that uniformly coats all grains. The pore throats close first, reducing permeability, but the larger pores are left mostly open. c) Preferential precipitation in large pores. d) Preferential precipitation in smaller pores. Pore throats will close first, strongly reducing permeability while having a minimal impact on porosity.

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area where advanced understanding is necessary is the link between porosity and permeability. If significant precipitation in a porous medium occurs, the effect of the precipitation on the porosity and permeability cannot be neglected. While one can estimate change in porosity based on the amount of precipitated material and its molar volume, the effect of that porosity change has on the permeability and the ultimate extent of reaction will vary depending on how the precipitation affects the flowpaths that dominate the transport of fluids through the medium. This in turn could potentially affect how the much material can be precipitated prior to the reaction becoming limited by the transport of new reactants to the site of precipitation. An example of this behavior is the CO2 bubble study above (Cohen and Rothman 2015), where the model indicated that the precipitation reaction was self-limiting because the change in permeability stopped the subsequent dissolution of CO2 gas into the aqueous phase (Fig. 1). This is despite the fact that the overall system is still very far from equilibrium. The textbook method to estimate permeability is to express it as proportional to the square of the grain size, such as observed using glass beads (e.g., Freeze and Cherry 1979): k  Cd 2 ,

(1)

where the permeability, k, has units of length squared, C is a fit parameter, and d is the average grain diameter. (The common units of permeability are in Darcy, where 1 Darcy  10-8 cm2). The larger the grain, the larger the pore size, and much larger is the permeability. While this simple expression is interesting to think about, in practice is only useful if mineralogy and other parameters such as grain shape remain more or less constant across samples. For example, an analysis of Gibson-Poole et al. (2008) find over three orders of magnitude variation in permeability for a given porosity for samples from a single formation across a potential CO2 storage basin (Fig. 3). Moreover, precipitation reactions may have non-linear effects on permeability: Tartakovsky et al. (2008) found that an impermeable layer of CaCO3 could form within a reactor with only a 5% reduction in porosity (see below). From these studies it is clear that porosity alone cannot be used to predict permeability, therefore one can incorporate additional empirical parameters that affect permeability, such as (Bloch 1991; Zhang et al. 2013):

Figure 3. Log permeability as a function of porosity in two different formations across a potential CO2 sequestration reservoir. Data from Gibson-Poole et al. (2008) for a single rock formation. Trendline is a best fit for log permeability as a function of porosity, dashed lines are the 95% prediction interval. The prediction intervals span more than three orders of magnitude in permeability.

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e  f  rigid grain content, sorting

(2)

where a–f are fit parameters. Here, the dependence of permeability on grain size is exponential, consistent with observations (Freeze and Cherry 1979; Gibson-Poole et al. 2008). Extrapolating Equations (1) and (2) to a distribution of pore sizes and assuming that the super-linear dependence of permeability on grain size (and hence pore size) is still valid, we can hypothesize a trend about the potential effects of a pore-size-dependent precipitation. If precipitation were to occur preferentially in the largest pores, it would have disproportionate effects on the permeability, since the largest pores are created by the largest grains and the largest grains will have the largest effect on the overall permeability (Fig. 2c). Contrarily, if precipitation occurs preferentially in the smallest pores, it might have minimal effects on the permeability since overall permeability might be, but this process may occlude pore throats, that is, the distance of closest approach between two mineral grains (Fig. 2d). This would strongly affect permeability. One behavior might be desirable over another in different situations. For example, in a cap-rock intended to restrain a plume of trapped CO2, if precipitation occurs in the largest pores or in fractures larger than the average pore size, a minimal amount of precipitation would be responsible for a self-sealing behavior and a more efficient cap rock. Alternately, if one wanted to precipitate as much material as possible within the available pore space, such as in the reservoir rock for carbon or other contaminant sequestration, one would prefer it if permeability of the rock was maintained until the reaction is completed to avoid a self-limiting behavior. This would allow continued reactant transport to the site of reaction and mixing of various reactants. Thus one would prefer it if precipitation occurred preferentially in the smallest pores and proceed to the larger ones. There does not necessarily need to be a pore-size-dependent precipitation to have an effect on the permeability of the medium prior to filling all the pore space. Precipitation reactions in reactive transport simulations have been modeled as uniformly coating the grains in the porous medium. As the precipitation proceeds, the pore throats become filled with precipitate but this leaves the largest pore spaces open (Fig. 2b). When this happens, permeability of the formation can also become reduced because communication and transport of fluids between pores is no longer possible. This behavior is conceptualized using the “Tubes in Series” theory (Verma and Pruess 1988), where the flow through the rock is approximated as a bundle of capillaries whose diameter is reduced in some portions, restricting the fluid flow. This technique has been used to model precipitation of evaporite minerals due to drying of the ambient fluids in a reservoir rock after CO2 is injected. The minerals, principally halite (NaCl), substantially clog permeability of the formation (Borgia et al. 2012).

Observations of precipitation in pores While the concept of examining the growth of minerals or other crystals from bulk solution or natural specimens has been around for a long time (see e.g., Stack 2014 for a review of calcium carbonate rates), examining how crystals grow in the middle of a porous medium has been a more difficult proposition to study because, by their nature, it is difficult to discern what is happening in the interior of rocks without irrevocably modifying and/or destroying the integrity of the sample. So until recently, most experiments and analysis of precipitation in pores have been done ex situ. See Putnis and Mauthe (2001) for excellent examples of mercury porosimetry analysis of dissolution experiments on precipitated material. Historically, in situ information has been obtained by analyzing the compositions of pore fluids (Morse et al. 1985). Sampling a pore fluid is often the only practical method to get information about the processes in the subsurface. As demonstrated in Steefel et al. (2014), the difficulty with this method is that it is inherently inferential. That is, one must infer which

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minerals are precipitating and where based on their solubilities or perhaps examining how permeability might change with reduction in solution flow rate or increase in pressure. The above-mentioned advances in synthesizing samples, measuring reactivity in situ and modeling the outcome have the potential to allow us to observe precipitation directly over the course of an experiment and to link precipitation rates, growth regime and flow rates with porosity and permeability evolution within a sample. It remains to be seen if these advances will lead to an ability to obtain a fine-grained control over precipitation reactions in porous media and contribute to the resolution of the discrepancy between laboratory- and field-based weathering rates. A further source of difficulty in measuring precipitation in pores is that many observations only cover a limited range of pore sizes, e.g., on thin sections or hand samples, and do not detect the smallest pore sizes. Pores size varies many orders of magnitude and the smallest pore are the interlayer spacings in clays (or their equivalent), which are on the nanometer scale. For example, a smectite clay has a d-spacing (or repeat distance) of 1–2 nm depending on how much water is in the interlayer (Bleam 2011). These smallest pores, termed nanopores, may dominate the overall porosity in a rock (Anovitz et al. 2013). Part of the difficulty is the limited range of pore sizes that can be analyzed using the standard tools for analyzing pore-size distributions: gas adsorption and mercury porosimetry. Gas adsorption relies on measurements of a powdered form of the rock sample, destroying the original rock fabric and may introduce a dependence of the pore size measurement on the particle size of the powder used in the experiment (Chen et al. 2015). To reach the smallest pore sizes, interpretation of mercury porosimetry relies on an assumption that the technique does not modify the sample in any way, yet to measure, e.g., a 3.5 nm pore size with this technique requires 400 MPa pressure applied to the sample (Giesche 2006). This is equivalent of burying the rock at 17 km depth using a lithostatic pressure gradient of 23 MPa/km (Bethke 1986). It is clear that compaction of the sample is a real danger in this type of measurement and restricts its typical use to larger pore sizes. Other methods to observe precipitation in pores have included optical microscopy, SEM, and microprobe analysis, but these tools also have a resolution limit of sub-micron or so at best, depending on the instrument and sample. Previous work on pore-size-dependent precipitation also focused on monitoring the supersaturation necessary prior to nucleate materials in idealized porous media such as silica aerogel (Prieto et al. 1990; Putnis et al. 1995). What they found is there exists a threshold supersaturation that is necessary to achieve prior to the nucleation of materials becoming favorable. This was attributed to a pore-size-dependent solubility (see below), stirring rate, interaction between substrate and precipitate. Classical nucleation theory calls for a critical supersaturation necessary prior to nucleation becoming energetically favorable, but this concept is a distinct modification of the surface energy (Fig. 4) (De Yoreo and Vekilov 2003; experimental evidence of this observed in Godinho and Stack 2015). This concept is supported by some ex situ work on sandstone thin sections that has shown naturally formed halite cements have a tendency to occlude larger pores and are not found in smaller ones (Putnis and Mauthe 2001). A recent Small Angle X-ray Scattering (SAXS) study on a nanoporous amorphous silica showed that nucleation in nanopores may or may not be inhibited, but depends on the surface chemistry between precipitate and substrate (Stack et al. 2014). In this work, we took the approach of measuring precipitation in both native 8-nm nanoporous amorphous silica (Controlled Pore Glass-75, or CPG-75), and CPG modified with a self-assembled monolayer containing an anhydride group, which presumably hydrolyzes in water to form a dicarboxylic acid. The idea was to have the same pore size distribution, but different surface functional groups and different surface reactivity at the substrate-water interface. Aqueous solution supersaturated with respect to calcite (CaCO3) was circulated past the CPG, and the small angle X-ray

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Figure 4. Classical nucleation theory, and potential effects of nanopores. a) For a growing nucleus, there are two energetic terms that determine its stability, a negative term from the bulk phase that grows with the cube of the radius and a positive surface area term that grows with the square of the radius. A nucleus growing in a nanopore will have its surface energy changed by the presence of the nanopore. b) The sum of the two terms in a), which determine the overall stability of the nucleus. The critical radius is determined by the position of the peak. If the presence of a pore increases the surface energy of the nucleus by increasing its interfacial energy, it will create a larger critical radius than in open solution. If a pore lowers the total surface energy, it will facilitate precipitation by creating a smaller critical radius, or alternately a lower supersaturation necessary for growth.

scattering was measured as the reaction proceeded. We found that where native CPG showed precipitation only in the spaces in between CPG grains (i.e., macropores ≥ 1 μm diameter), functionalizing the CPG with a polar self-assembled monolayer caused precipitation to occur in both nanopores and macropores (Fig. 5). At the time of publication, the results of this study were thought to suggest that the pore size in which the precipitation preferentially occurred was controlled by the favorability of nucleating the precipitating phase onto the substrate. While this is still a possibility, an alternate, solution-side explanation that relies on surface charge of the substrate is given below that is only briefly touched on in the previous work. Recent work has shown that small and ultra-small angle neutron and X-ray scattering can be combined with, e.g., traditional SEM-BSE analysis to obtain a measurement of pore sizes that range seven orders of magnitude (Wang et al. 2013). This series of techniques removes one of the issues with the pore size characterization techniques described above in that the sample is not ground to a powder. The X-ray and neutron sampling are non-invasive with respect to the integrity of the sample, although working on thicker samples (e.g., 1 mm thick) tends to create multiple scattering events that are more difficult to interpret. Thus far, these techniques have primarily been used to characterize rock samples ex situ. Each set of techniques, Small Angle X-ray/Neutron Scattering (SAXS, SANS), Ultra SAXS/SANS, and SEM-BSE each probe only a couple orders of magnitude of pore size, but overlapping ranges allows one to join the pore size distributions derived from each technique into one master porosity distribution. Because of this requirement, however, it may be difficult to obtain reasonable results in situ. Lastly, there are issues with interpreting the data from these methods, such as improper background subtraction can lead to anomalous changes in the apparent porosity distribution, etc. What has been observed thus far on rock samples the SANS/USANS/BSE-SEM techniques has been mixed. Wang et al. (2013) detected a reduced (relative) contribution to the total porosity from small scale pores in metamorphic rocks that underwent higher degrees of metamorphism. This implies that smaller pores tended to close first during the combustion and other metamorphic processes the rocks underwent over time. Alternately, Anovitz et al. (2015) found that despite being exposed to a solution supersaturated with respect to quartz

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Figure 5. Precipitation of calcium carbonate in controlled pore glass (CPG). a) Each CPG grain consists of amorphous silica filled with nanopores ~7–8 nm in diameter. b) The pore spaces in between grains form pores tens of micrometers in diameter. c) Small Angle X-ray Scattering (SAXS) intensity as a function of momentum transfer, Q, while a fluid supersaturated with respect to calcite is flowed past the CPG. The scattering shows large changes at small Q, indicating precipitation in the larger intergranular spaces. d) When the CPG is functionalized with a self-assembled monolayer (structure shown in inset), the precipitation behavior changes so that both the nanopores and macropores fill with precipitate. [Used by permission of American Chemical Society, from Stack AG, Fernandez-Martinez A, Allard LF, Bañuelos JL, Rother G, Anovitz LM, Cole DR, Waychunas GA (2014) Pore-size-dependent calcium carbonate precipitation controlled by surface chemistry. Environmental Science & Technology, Vol. 48, p. 6177–6183]

(SiO2), silica overgrowths initially dissolved in an arenite sandstone followed by precipitation in larger pores. This is consistent with an inhibition of precipitation in smaller pores and unstable precipitates in smaller pores. It is to be hoped that these techniques can be used for further, well controlled experiments that will systematically probe precipitation reactions in porous media. The source of the difficultly in interpreting the results of these experiments may be because these samples are natural ones in which multiple processes could be occurring at once, or at least that multiple effects could be the origin of the observed results. This ambiguity makes isolating specific processes and quantifying their effects difficult. However, measurements on idealized samples, while easier to interpret, are not as applicable. Ideally, work on idealized samples would then be compared to measurements on natural samples as a validation.

ATOMIC-SCALE ORIGINS OF A PORE SIZE DEPENDENCE Substrate and precipitate effects What kinds of things could happen that would drive a pore-size dependence for precipitation? Initially, one might imagine that the shape of the pore could result in a change in the ability of the substrate to induce nucleation of a precipitate, change the fluid composition

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inside a pore, changing the fluid’s ability to precipitate material and the ability of the fluid to transport reactants to the site of reaction. Many of these effects have their origin at the atomicscale. Here these will be discussed and an estimate of the maximum pore size where any effects could be important will be given. The reactivity of the substrate could change due to strain induced on the reactive sites on the pore wall due to the presence of the pore. That is, by forcing a curved pore wall, or other shape, the reactive sites at the fluid-substrate interface will be strained and this could change their reactivity, particularly towards dissociatively adsorbed water. This in turn would affect their acidity (pKa), and the surface charge of the substrate would change and the substrate’s affinity for dissolved ions that adsorb to nucleate the new phase. There are relatively few direct observations of the acidity of nanopores relative to planar substrates: FernandezMartinez (2009) fit the pKa’s of the octahedral aluminum surface sites on nanotubular imogolite (Al2SiO3(OH)4) and found they are one pKa unit more acidic than the equivalent sites on macroscopic gibbsite (Al(OH)3). However, Bourg and Steefel (2012) found that the difference in bond lengths on amorphous silica nanopores translated to only a 0.5-1 pKa unit difference using classical molecular dynamics simulations. Studies of the charge densities of nanoporous amorphous silica similarly found that silanol functional group densities are within a factor of two of a planar amorphous silica surface. There are 2.5–3 >SiOH groups/nm2 in a nanoporous amorphous silica, Mobil Composition of Matter No. 41 (MCM-41), versus 5–8 sites per /nm2 in typical amorphous silica (Sahai and Sverjensky 1997; Zhao et al. 1997). One strategy to understand the limits on possible effects of pore size on reactivity is to examine structural relaxation of near surface layers on bulk mineral phases and use their typical extent as a measure of the distances over which atomic-level strain is typically dissipated. Molecular simulation and X-ray reflectivity (XR) are the best methods to look at relaxation of a structure at a planar mineral–water interface. These typically show modifications of the average positions of atoms in a few of the top-most monolayers of a crystal surface due to the creation of the interface. For example in the barite {001} surface, in XR experiments (Fenter et al. 2001) and MD simulations (Stack and Rustad 2007), show a bulk-like structure after about three monolayers depth, which is about 1 nanometer (Fenter et al. 2001) (Fig. 6). XR on hematite (-Fe2O3) also shows about three monolayers that relax, or about 0.7 nm (Trainor et al. 2004), and XR/MD on calcite shows about 4 monolayers, or 1.1 nm (Fenter et al. 2013). If we take these measurements as a guide, it suggests that only the very smallest nanopores (less than a few nanometers) should show some change in localized atomic structure due to the presence of the pore and different reactivity. Any larger pores will likely show more or less planar-like reactivity of the substrate. An alternative guide might be taken from measurements on nanoparticles, which might be thought of as nanopores in reverse. Anatase (TiO2) nanoparticles show bulk-like points of zero charge and protonation constants when particle size is larger than 4 nm in diameter, but smaller than that they start to deviate (Ridley et al. 2013). Regarding other species besides water, Singer et al. (2014) measured sorption of strontium and uranium in a porous amorphous silica (MCM-41) with pore sizes of 4.7 nm. They found that the presence of the nanopores lead to the uranium and strontium desorption to be recalcitrant relative to bulk silica, i.e., it took a stronger solvent to labilize the uranium ions, but sorption occurred with a lower total adsorption density than the corresponding bulk phases. This was argued to be consistent with results from uranium contaminated-sediments containing a larger fraction of nanopores (Bond et al. 2008). However, as was discussed in the publication, it was not demonstrated that this is due to the reactivity of the pores themselves, but the possibility that the change in reactivity could be due to diffusion of the ions into pores themselves. This demonstrates a pervasive problem in understanding reactivity in pores: how does one separate transport from reactivity effects? In the Stack et al. (2014) study

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Figure 6. Atomic-scale electron density as a function of distance from a barite {001} mineral–water interface. Barite (negative numbers in x-axis) shows surface relaxation on the order of three monolayers. Water (positive numbers in x-axis) shows one or more ordered water peaks. Black line is X-ray Reflectivity (from Fenter et al., 2001), dashed line is from Molecular Dynamics (Stack and Rustad 2007). A 5% lattice mismatch has been corrected in the MD data. MD data has been broadened and weighted to atomic number courtesy of Sang Soo Lee (see acknowledgements).

described above, it was observed that neutron scattering originating from the nanopores in the native CPG responded to a solution change as rapidly as new scattering patterns could be measured (~20 min). This was thought to suggest that transport, at least of water, is relatively facile into the nanopores themselves. Additionally, CPG may contain differing reactivity from MCM-41. This is evidenced by SAXS measurements of MCM-41 and another nanoporous amorphous silica (SBA-15) that show dissolution and re-precipitation of silica in water at 60 ºC (Gouze et al. 2014), whereas we have not observed this with CPG (at least, at room temperature and pH ~8.5; Stack et al. 2014). In order to conceptualize some of how the presence of nanopores can affect precipitation reactions, it is useful to review some of classical nucleation theory (De Yoreo and Vekilov 2003). The free energy of a precipitating phase (ΔG) is the sum of two terms (De Yoreo and Vekilov 2003): G  Gbulk  Gsurface ,

(3)

where ΔGbulk is the contribution from the bulk volume of precipitated material and ΔGsurface is the contribution from the interfacial energy. For a heterogeneous nucleus, the free energy conserved for a given radius is: Gbulk

2 r 3 (2.303k BT  SI )  , 3Vm

(4)

where r is the radius of the nucleus, kB is Boltzmann’s constant, T is temperature, SI is saturation index, and Vm is the molar volume (in m3/mol). Saturation index is defined as the log of the activities of the constituent ions of the mineral divided by the solubility product. For calcite this is SI = log(aCaaCO3/Ksp). In Figure 4a, this term is calculated for SI = 0.76 and molar volume of calcite (3.69 × 10-5 m3/mol). For heterogeneous nucleation, the surface energy term is (Fig. 4a): Gsurface  r 2 (2  lc   sc   ls ),

(5)

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where lc is the interfacial energy between the precipitating crystal and the surrounding liquid, sc is the energy between the substrate and the crystal, and ls is that between the substrate and the liquid. The sum of these terms will be referred to as the apparent interfacial energy, . In Figure 4a, this term is calculate using an apparent interfacial energy for calcite of  = 0.036 J/m2 (Fernandez-Martinez et al. 2013). If a nanopore changes any of the terms in Equations (4) and (5), it would change how favorable nucleation is, and especially what the critical radius of the nucleus is, that is, the minimum size at which subsequent growth of the nucleus is energetically favorable (Fig. 4b). If the precipitate itself is the same as what would form on a planar substrate, ΔGbulk would not change. Within ΔGsurface, the interfacial energy between the substrate and liquid might change due to reactivity with respect to water or surface charge changes such as described above. The energy interfacial energy between the substrate and crystal might also change for the same reasons. These effects will in turn affect the critical nucleus size, or alternately, require a different saturation index to create a stable precipitated nucleus (adjusting SI in Equation (4) will change critical radius size as well). An additional possibility is that the close proximity of surface sites could enhance nucleation within a nanopore, even if reactivity of the surface sites is the same. Hedges and Whitelam (2012) ran a series of Ising model simulations to examine how pore geometry can affect nucleation rate. They found that pores of a specific size and shape could lower the free energy barrier to nucleation. This result is rationalized by the following train of logic: When the free energy of the interface between the precipitating phase and the substrate is lower than the free energy between the precipitating phase and the solution, heterogeneous nucleation onto the substrate will occur at a lower supersaturation than precipitation directly from solution. In this scenario, the precipitate nuclei will minimize the amount of surface area contacting solution and maximize the surface area contacting the substrate. If one were to think about this phenomenon in terms of pore size and shape, a particular size and geometry will minimize the amount of precipitate nuclei–solution interface (Fig. 4). For this phenomena to be an accurate description of what is occurring in pores, the size of the nuclei where significant savings in interfacial energy could be achieved will be similar to the size of the critical radius of the nuclei. This is something like a few nanometers or so (see below; Stack et al. 2014), that is, the pore must be a nanopore. Some evidence of this phenomenon has come from monitoring precipitation on planar substrates, where nuclei can be shown to form preferentially on steps on a surface (Stack et al. 2004), which might be thought of as sharing the structural characteristics of the nanopores as conceived of by Hedges and Whitelam (2012). To the contrary, nucleation in the controlled pore glass described above shows inhibited precipitation in nanopores in the native CPG but simultaneous precipitation in nanopores and macropores in the SAM-functionalized CPG. Note that this thought process is only valid when if interfacial energy is limiting the rate of nucleation. If the interfacial energy is a secondary effect because the kinetically viable pathways for precipitation are limited, the geometry of the nanopores may have no effect on nucleation. That is, despite that a reaction may be thermodynamically favorable, if there is no readily available reaction mechanism for that to happen, or if a competing reaction proceeds more rapidly than the most favorable one, it is likely the reaction will proceed to a metastable state rather than equilibrium. There is an alternate interpretation of the effects of pore size on surface energy. The argument runs that since the presence of the pore artificially limits the size of the nuclei that precipitate inside of them, it increases the proportion of the surface energy relative to the total energy. This is well known from the classical nucleation theory described above, where a critical radius is defined as the radius in which the energy conserved by creating

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the volume of the precipitated material is balanced by the energy penalty for creating the surface area of the nuclei-solution interface (Fig. 4b). Thus a smaller nuclei has a greater surface energy and is less stable (according to the theory). If the precipitation occurs, e.g., in a nanopore, the net effect is to increase the apparent solubility of a precipitate of restricted size. Thus, a higher saturation state is required to induce heterogeneous precipitation in the pore than would otherwise be required. This model is called the Pore Controlled Solubility (PCS) model. In its Equation form, it is:  2V   Sd  S0 exp  m  ,  RTr 

(6)

where Sd is the effective saturation state of the fluid inside a pore with respect to a mineral phase, S0 is the intrinsic saturation state in bulk fluid, Vm is the molar volume of the material (m3/mol),  is the interfacial energy of the precipitating phase (J/m2), R is the ideal gas constant (J/mol/K), T is temperature (K) and r is the radius of the pore (m). The evidence of this effect has been noted in silica aerogel (pore sizes of 100–400 nm) where increased saturation index is required above that normally necessary to nucleate barite (Prieto et al. 1990; Putnis et al. 1995). This effect has also been observed in halite (NaCl) cements in sandstones (Putnis and Mauthe 2001) and in porosity distributions of quartzcemented sandstone in proximity to a styolite (Emmanuel et al. 2010). In these studies at the micrometer scale, it was found the cements had filled more of the larger pores than the smaller pores, and Putnis and Mauthe (2001) found that the halite was preferentially leached from larger pores. In contrast, Stack et al. (2014) showed that the critical radius of a calcite grain is 1.5 nm using Vm = 3.69 × 105 m3/mol; sl = 0.094 J/m2 (Stumm and Morgan 1996) using Equation (6) in their system, smaller than the nanopores in the CPG. To put this into perspective, the critical diameter of the nanopore is equivalent to about twice the interlayer of a swelling clay—that is to say, the nuclei in all but the smallest nanopores should be larger than the critical radius of the calcium carbonate. For a pore that is e.g., 8 nm in diameter, the solubility should be increased by a factor of 2.1. This corresponds to a saturation index of 0.3 necessary to make these nuclei stable, which is a minor effect given that SI = 0 is equilibrium). The PCS model of Emmanuel and coworkers and the Ising model of Hedges and Whitelam are directly contradictory. The Ising model says that interfacial energy is reduced in a nanopore, promoting nucleation, whereas PCS says interfacial energy is increased in nanopores, inhibiting nucleation (Fig. 4b). The latter has significant empirical findings that correlate with it, but some of that work is in much larger, micrometer-sized pores, which are in regimes larger than what might be expected for these types of effects. Our work in nanopores has supported the PCS concept in the native CPG grains, but that nucleation can occur in nanopores (Stack et al. 2014). This was thought to occur by lowering the interfacial energy between the substrate and precipitate by introducing a SAM, something not accounted for in the PCS model. Nucleation density of the precipitate on the substrate may also be important. Using neutron diffraction, Swainson and Schulson (2008) found that ice nucleation in diatomite and chalk was inhibited and proceeded by a small number of nuclei that grew to fill neighboring pores. If nucleation density is low, one might expect what appeared to be a preference for precipitation in large pores, since these pores would contain the largest precipitates. A high nucleation density would appear more like a uniform coating of the substrate.

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In all of these studies, there are questions about the role of the composition of fluids. That is, one cannot rule out that the nanopores affect the near surface solution composition and affect the outcome of the precipitation reactions. We know that the presence of pores can induce changes in the structure and ion composition. How large are these effects? To get an idea, we will first examine the classical model of a charged mineral–water interface, a Gouy–Chapman diffuse layer, often coupled to a stern layer that includes discrete dissolved species. These are reviewed extensively in a previous volume of Reviews in Mineralogy and Geochemistry (Davis and Kent 1990; Parks 1990). Briefly, the concept is that the presence of the interface creates dissatisfied bonding environments for surface atoms, which leads dissociation of water as it adsorbs onto mineral surface sites whose charge creates an electrical potential that is attracts ions of opposite charge to adsorb onto the mineral surfaces themselves. Ions also collect in an area the electric double layer which extends into the solution until the electrical potential is negated. The extent to which the surface potential decays into the solution diffuse layer is approximately described as:    0 exp( d )

(7)

where  is the potential at some distance, d, from the surface, 0 is the potential at the surface (or Stern or beta layers if using a two or three layer model), and  is the Debye parameter:  2F 2 I  103  9 -1    3.29  10 I (m )  T R 0  

(8)

where F is the Faraday constant (96,485 C/mol), I is the ionic strength (in mol/L),  is the dielectric constant of water (78.4 at 298 K), 0 is the permittivity of a vacuum (8.854 × 10-12 C2/N/m2), R is the ideal gas constant (8.314 J/mol/K) and T is temperature (298 K). We can use the reciprocal of the Debye parameter as an estimate for where the overlap between diffuse layers becomes significant. This “Debye length” is 0.96 nm in solutions with I = 0.1 M electrolyte concentration, 3.1 nm for 0.01 M, and 9.6 nm for 0.001 M electrolyte concentrations (Fig. 7a). In pure water (I = 10-7 M), the Debye length is 960 nm. One might expect to observe significant changes in the concentrations of ions when the pore size is decreased sufficiently such that the diffuse layers on either side of the pore start to overlap. If we take the size of the area of significant interaction as roughly twice the Debye length, the pore size ranges from 2 nm up to nearly 2 μm in pure water depending on ionic strength. In concentrated solutions, the size of the pores where one would expect to observe electrolyte effects is quite small, basically the lower limit on the size of a nanopore. In the extreme case of dilute water however, this theory predicts that the pore sizes where electric double layer effects would be seen could be substantial. In natural systems pure water is not reasonably expected to be observed, but in experiments researchers will sometimes minimize the ionic strength since the composition and concentration of the electrolyte affects mineral precipitation reaction mechanisms and rates (e.g., Ruiz-Agudo et al. 2011; Kubicki et al. 2012; Bracco et al. 2013). Within the electric double layer, one would see elevated concentrations of the ions that are oppositely charged to the mineral surface and lower concentrations of the ions that are of the same charge. For example, at pH 7 in 0.1 M NaCl, amorphous silica is negatively charged (Sahai and Sverjensky 1997; Sverjensky 2006), so one would expect an excess of sodium cations and decreased amounts of chloride. This is significant in that differences in the reactivity of precipitation rates have been observed, depending on the cation-to-anion ratio of the solution. For example, with calcium carbonate, the calcium-to-carbonate ratio

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affects whether growth is observed at all (Stack and Grantham 2010), as well as whether growth proceeds through homogeneous nucleation directly from solution, or growth by preexisting seed crystals (De Yoreo and Vekilov 2003; Gebrehiwet et al. 2012; Stack 2014). How elevated/depleted could concentrations be inside a nanopore? Using the electric double layer formulation again to examine concentration of dissolved electrolyte:   zF    zF   c  c exp  c  c exp     RT   RT 

(9)

where c+ and c- are the concentrations of cations (c+) and anions (c-) within the electric double layer, c∞ is the concentration of that ion in bulk solution, z is the charge on the ion, and all the other symbols are defined above. Using this, we can say that if the surface potential is -50 mV and a monovalent cation concentration is 0.01 M, then the concentration of the cation at the surface is nearly 7× the bulk concentration (c+/c∞). This is substantial enough to make a large difference in if precipitation occurs or not (Fig. 7). This may not be valid however, since the diffuse layer model breaks down very close to the interface (see below). If the calculation is done at some distance from the mineral–fluid interface, e.g., at the Debye length (3.1 nm; Eqn. 5), the potential is -18 mV (Eqn. 4) and a diffuse layer concentration is therefore 2× the bulk concentration (Eqn. 6). This is much less significant and will not affect rates nearly as strongly, but the cation-to-anion ratio will still change by a factor of four.

Figure 7. Electric double layer (EDL) effects within a pore. a) Debye length, or double-layer thickness, as a function of ionic strength. Dashed lines are the example used to examine the data in Figure 5. b) Electrical potential decaying into solution for the ionic strength highlighed in a. c) Change in concentrations of calcium and carbonate (left axis) due to EDL as a function of distance from the pore wall. The right axis shows the aqueous calcium-to-carbonate ratio, which reaches extreme values closer to the pore wall as small radii.

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As a real world example of how the electric double layer may affect the concentrations of ions in nanopores, it can be applied to explain, perhaps, the results of Figure 5. Specifically, Stack and Grantham (2010) and Gebrehiwet et al. (2012) have shown that calcite growth rate depends in part on the calcium-to-carbonate ratio, so if the CPG in Stack et al. (2014) had significant surface charge, it could affect not just rate of precipitation, but even if it occurs or not. The ionic strength of the solution used in the experiments in Figure 5 was 8.2 × 10-3 M, whose constituent ions consist of chloride, calcium, sodium, carbonate and bicarbonate. Using Equation (8), the Debye length would be 3.3 nm, so we would expect a pore size dependence at 6.6-nm pore diameter (Fig. 7a), which is close to the fitted size of the pores, 6.9 nm. We would therefore expect that the nanopores would contain significant double layer effects that change the concentration of ions within the pores. The pH of the solution used is ~8.4, which would correspond a surface potential of -136 mV using a Stern–Graham model for silica that includes specific adsorption of calcium ions to surface sites (Sverjensky 2006). We therefore might expect substantial excess calcium adsorbed in the pore walls and within the pores and depleted carbonate and bicarbonate in those same areas. Using Equation (9) and a surface potential of -136 mV for amorphous silica leads to 6.5 × aqueous calcium concentration and 0.024 × carbonate concentration in the center of the pore. This creates an aqueous calcium-to-carbonate ratio at the center of the pore of 29,000 (using a 6.9-nm diameter pore), whereas in the bulk solution it is 107 (Fig. 7c). It is therefore conceivable that such a high calcium-to-carbonate ratio suppresses nucleation (Fig. 8). One caution is that Gebrehiwet et al. (2012) saw enhanced nucleation at calcium-to-carbonate ratios near 300, but Stack and Grantham (2010) saw reduced growth rates of single crystals at high calcium-to-carbonate ratios. The trend observed in Figure 5 is not consistent with increased nucleation rate, but it is conceivable that at such a high calcium-to-carbonate ratio, particularly close to the pore wall where the calcium-to-carbonate ratio would be expected to be more extreme than even in the center of the pore. In this scenario, the reason as to why the self assembled monolayer enhanced nucleation is that it modified the surface so it

Figure 8. Variation of growth rate of calcite with aqueous calcium-to-carbonate ratio. The peak rate occurs at some ratio slightly greater than one, but decreases substantially at ratios far from that. [Used by permission of John Wiley & Sons, from Stack AG (2014) Next-generation models of carbonate mineral growth and dissolution. Greenhouse Gas Science & Technology, Vol 4, p. 278–288]”

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contained a smaller surface charge than the native silica, or perhaps had not yet hydrolyzed to form a dicarboxylic acid (while surface charges were not measured in those experiments, the functionalized CPG was observed to clump more readily than the original material, suggesting that there may have been difference in surface charge). A smaller surface charge might allow the nanopore solution composition to reflect that of the bulk solution and create an environment more favorable for nucleation. One persistent doubt that remains is the physical plausibility of the electric double layer concept at the atomic scale. It is well known that the diffuse layer model alone overestimates concentrations at high concentrations, a failing which can be corrected by use of one or more Stern–Graham layers that accounts for adsorption of ions into planes of fixed height and capacitance (e.g., Davis and Kent 1990). However, recent evidence has suggested that the capacitance and thickness of a Stern–Graham layer are not necessarily constants, but vary with solution composition (Pařez and Předota 2012) and especially close to an interface (Pařez et al. 2014). A more difficult problem is that direct measurements of mineral–water interface structure using X-ray Reflectivity (XR) and Resonant Anomalous X-ray Reflectivity (RAXR) (Lee et al. 2010; Fenter and Lee 2014) have not shown increased concentrations of ions more than ~3 nm from an interface (Fig. 9). Distributions of ions that it does show tend to be localized in inner-sphere, outer-sphere or extended outer-sphere complexes and not a smoothly decaying concentration gradient as expected from the diffuse layer concept in

Figure 9. Average heights of distributions of ions adsorbed to a muscovite mica surface, measured using Resonant Anomalous X-ray Reflectivity. IS is inner-sphere, OSads is outer-sphere and OSext is an extended out-sphere complex. [Used by permission of the American Chemical Society, from Lee SS, Fenter P, Park C, Sturchio NC, Nagy NL (2010) Hydrated cation speciation at the muscovite (001)-water interface. Langmuir, Vol. 26, p. 16647–16651]

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Equation (7). This discrepancy may arise from a couple of possibilities, such as the detection limit of ions using the XR and RAXR techniques, and the fact that a fitted adsorption profile is not necessarily a unique fit to the scattering data. There is some recent evidence from molecular dynamics simulations that the assumption of a dielectric “constant” breaks down at the molecular level for a charged interface and that the effective dielectric constant can oscillate wildly depending on local water ordering and that this strongly affects decay of electrical potential into solution (Pařez et al. 2014). Typically in MD simulations ordered water is also observed less than one nanometer from the surface (Fig. 6) (Stack and Rustad 2007; Bourg and Sposito 2011; Fenter et al. 2013). This is consistent with the XR, although these two techniques are seldom in perfect agreement, likely stemming from the uncertainty in the MD models in addition to those of the XR. If one uses these more recent observations as a limit to what sized pores interfacial regions would start to overlap, the answer is much smaller than using the classical electric double layer.

TRANSPORT The last subject that will be addressed here is that of transport. This is the subject of other articles in this volume (Steefel et al. 2015, this volume), so this discussion will only revolve around those aspects that specifically involve precipitation. The classical concept for precipitation reactions is that they are either surface chemistry controlled or transport controlled, depending on the mixing rate of the solution. For example, Plummer et al. (1978) found that below pH ~5.0, the dissolution rate of calcite depended on how rapidly an impellor stirred the solution in the reactor. This is quantified as the Damköhler number, which is the reaction rate divided by the convective mass transport rate (Fogler 2006). In a natural porous system, i.e., in groundwater, it is not clear if sufficiently high flow rates are ever reached to make the system entirely free of a transport constraint. This was demonstrated recently by Molins et al. (2012) who showed the dissolution rate of calcite as a function of darcy velocity, or net fluid velocity (Fig. 10). Molins et al. (2012) show that the transition from a transportlimited to a surface chemistry-limited reaction is not sharp, but is a gradient. Furthermore,

Figure 10. Dissolution rate of calcite as a function of specific discharge (velocity). As the solution moves through the porous medium more quickly, transport of the fluid plays more of a role in determining the rate of reaction, but there is a broad range of flow velocities where the rate is transport limited in some pores, but limited by reaction kinetics in others. Adapted from Molins et al. (2012).

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they observed that the volume-averaged rate of dissolution was slower than the open solution rate because not all pores had the same fluid velocities. Liu et al. (2013) also found that while chemical reactions may be quite rapid on the near atomic scale, once the tortuous nature of diffusion in a porous system is accounted for, the apparent rate of diffusion is much slower. This is not an insignificant effect: the difference between the microscopic rates of adsorption/ desorption or diffusion and the pore- or even grain-scale rates are many orders of magnitude (Liu et al. 2013). Reeves and Rothman (2012) also addressed this issue by highlighting the need for functions for chemical reactivity that scale with time and length. We also know that for precipitation reactions, the rate of reaction in these systems cannot be described accurately using a single rate constant if solution composition and precipitate surface site concentration vary: one should fit rates using rate constants that reflect the mechanisms of attachment and detachment of ions to known surface sites on the growing crystals, that is, use a kinetics model that reflects the chemical processes observed to occur on a mineral surface (Stack 2014). The rate constants and mechanism for attachment of these ions vary depending on the ion and the mineral. For example, the rate constants for attachment of calcium and carbonate to calcite are ~6.7 × 106 s-1 and 3.6 × 107 s-1 (Bracco et al. 2013), whereas those for attachment of magnesium and carbonate ions to magnesite (MgCO3) are 4.0 × 105 s-1 and 2.0 × 106 s-1 (Bracco et al. 2014). From this, one would expect that the zone over which surface kinetics and transport are both important depends not just on the flow rate and the reactivity of the mineral, but the identity of the constituent ions will affect how much of a reaction is transport controlled. That is, for calcite growth one would expect that one could have a condition where calcium attachment is surface-kinetics-limited and carbonate is transport-limited. Due to the relatively small difference in rate constants in these systems, the range of solution conditions over which this might exist is limited but this may not always be true. In fact, mineral growth kinetics measurements such as these may actually reflect this condition already since possible transport limits are often poorly controlled or verified; in Bracco et al. (2012), the step velocity was measured as a function of flow rate under one condition, but not under all calcium-to-carbonate ratios so the rate constant for carbonate quoted above may reflect a partial transport control over carbonate attachment. Another significant issue due to transport effects are due to the mixing of solutions. Because the grid cell size used in conventional reactive transport models is larger than the scale over which precipitation is typically observed, they have a tendency to overestimate the amount of fluid mixing and precipitation. In Tartakovsky et al. (2008) and Yoon et al. (2012), the precipitation of calcium carbonate phases was observed and modeled in a sandpacked reactor where solutions of dissolved CaCl2 and Na2CO3 were injected along parallel flowpaths. They observed precipitation where the two streams mixed, but at a much finer scale than what would have been captured by a conventional grid-cell approach (Fig. 11a). They found that a smoothed-particle hydrodynamic model was able to capture the localization of the precipitation reaction well (Fig. 11b), or additionally a Darcy-type simulation with smaller grid sizes in the zone where precipitation was observed also was accurate. These studies highlight an ongoing research problem, which is how create a model that scales dynamically to capture the microscopic reactions well, but also is practical to use at much larger scales. One is not able run the molecular dynamics simulation over an entire reservoir or watershed, and never will be, so models that can capture atomic-scale reactions well, but also scale upwards in time and space, are necessary. As mentioned in the discussion above, Tartakovsky et al. (2008) found that the relationship between porosity and permeability is not as straightforward as it may seem in that the precipitation could create an impermeable barrier at only a 5% reduction in porosity. This result demonstrates that empirical relationships between porosity and permeability built from analysis of natural samples will not necessarily be applicable to anthropogenically induced precipitation. Lastly, one must also account for dissolution. During

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the mixing experiments just described, the sequence of events is that as the two solutions mix, they become saturated and eventually supersaturated sufficiently for precipitation to become favorable. Precipitation occurs and blocks communication between the two mixed fluids. Once the mixing is shut down, the solution surrounding the precipitate is then undersaturated and the precipitate should start to dissolve. Yoon et al. (2012) found that while they could model the precipitation relatively well by adjusting conventional precipitation models (Chou et al. 1989), but the subsequent dissolution was not well captured.

Figure 11. Calcium carbonate precipitation in a sand-packed reactor. Two solutions, CaCl2 and Na2CO3 are circulated side-by-side. a) Photograph of results; the white vertical stripe is a thin layer of calcium carbonate phases that have precipitated. b) Smoothed Particle Hydrodynamic model results that captures the localized nature of the precipitation well. The green (lighter gray) color shows the precipitated material. Red (left) and blue (right) are the two different injected solutions. [Used by permission of John Wiley & Sons, from Tartatakovsky AM, Redden G, Lichtner PC, Scheibe TD, Meakin P (2008) Mixing-induced precipitation: Experimental study and multiscale numerical analysis. Water Resources Research, Vol. 44, W06S04]

CONCLUSIONS AND OUTLOOK From these studies, it is clear that precipitation within a porous medium is a complex process that is a challenge to observe and model accurately and even in idealized systems, there are multiple effects that potentially explain the results. One must consider mineral precipitation kinetics (which are very complex themselves), substrate reactivity, surface charges and ion adsorption affinities that are possibly different from the bulk phase, geometric factors that inhibit or enhance nucleation, surface energy effects, and last but not least, solvent transport. In natural systems multiple minerals and other phases (such as organic carbon), gradients in pore size distributions and other components create potentially other complicating factors that reduce our ability to discern what is occurring in these systems. Nanopores may contain the largest deviations from bulk-like reactivity, and at the same, may constitute the majority of pores in a rock. Yet, due to the difficulty in quantitatively measuring these, the relative importance of nanopores to the net reactivity of the rock, and their reactivity in this context are just beginning to be examined. The precipitation itself is fairly difficult to observe since it is occurring in the middle of three dimensional network of solids, leaving one to either interpret data from thin sections or utilize newer methods of X-ray and neutron scattering that can allow one to gather statistical averages of the porosity distribution. To interpret and understand how the presence of a porous medium affects mineral precipitation will find application in multiple areas of scientific, environmental and industrial interest. These include metal contaminant treatment, carbon sequestration, scale formation, mineral/rock weathering, perhaps seismicity,

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etc. The potential to examine reactivity in pores is positive, however, in that new experimental probes such as X-ray and neutron scattering are being adapted to probe these reactivity in porous media, and coupled enhanced reactive transport modeling capabilities. Understanding derived from these combined methods may result in predictive theories that can accurately account for atomic-scale reactivity and structure, but are useful at larger scales where it is no longer practical to resolve individual pores.

ACKNOWLEDGMENTS The author wishes to thank both Sang Soo Lee at Argonne National Laboratory for his translation of the MD probability curves to be more comparable to the XR (Fig. 6), and Michael L. Machesky of the Illinois State Water Survey for sanity-checking the calculations of the surface potential of amorphous silica (Fig. 7). Research on CO2 sequestration sponsored by the Center for Nanoscale Control of Geologic CO2, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number (DE-AC02-05CH11231). Research on barite and metal contaminants was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division. The author is grateful for the insightful comments by Alejandro Fernandez-Martinez and Qingyun Li that significantly improved the manuscript.

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 191-216, 2015 Copyright © Mineralogical Society of America

Pore-Scale Process Coupling and Effective Surface Reaction Rates in Heterogeneous Subsurface Materials Chongxuan Liu, Yuanyuan Liu, Sebastien Kerisit and John Zachara Pacific Northwest National Laboratory Richland, Washington, USA [email protected]

INTRODUCTION Heterogeneity in pore structure and reaction properties including grain size and mineralogy, pore size and connectivity, and sediment surface area and reactivity is a common phenomenon in subsurface materials. Heterogeneity affects transport, mixing, and the interactions of reactants that affect local and overall geochemical and biogeochemical reactions. Effective reaction rates can be orders of magnitude lower in heterogeneous porous media than those observed in well-mixed, homogeneous systems as a result of the pore-scale variability in physical, chemical, and biological properties, and the coupling of pore-scale surface reactions with mass-transport processes in heterogeneous materials. Extensive research has been performed on surface reactions at the pore-scale to provide physicochemical insights on factors that control macroscopic reaction kinetics in porous media. Mineral dissolution and precipitation reactions have been frequently investigated to evaluate how intrinsic reaction rates and mass transfer control macroscopic reaction rates. Examples include the dissolution and/or precipitation of calcite (Bernard 2005; Li et al. 2008; Tartakovsky et al. 2008a; Flukiger and Bernard 2009; Luquot and Gouze 2009; Kang et al. 2010; Zhang et al. 2010a; Molins et al. 2012, 2014; Yoon et al. 2012; Steefel et al. 2013; Luquot et al. 2014), anorthite and kaolinite (Li et al. 2006, 2007), iron oxides (Pallud et al. 2010a,b; Raoof et al. 2013; Zhang et al. 2013a), and uranyl silicate and uraninite (Liu et al. 2006; Pearce et al. 2012). Adsorption and desorption at the pore-scale have been investigated to understand the effect of pore structure heterogeneity on reaction rates and rate scaling from the pore to macroscopic scales (Acharya et al. 2005; Zhang et al. 2008, 2010c, 2013b; Zhang and Lv 2009; Liu et al. 2013a). Microbially mediated reactions have also been studied at the pore-scale including denitrification (Raoof et al. 2013; Kessler et al. 2014), sulfate reduction (Raoof et al. 2013), organic matter and nutrient transformation (Knutson et al. 2007; Gharasoo et al. 2012; Raoof et al. 2013), and biomass growth (Dupin and McCarty 2000; Dupin et al. 2001; Nambi et al. 2003; Knutson et al. 2005; Zhang et al. 2010b; Tartakovsky et al. 2013). These studies indicate that pore-scale heterogeneity and coupling of reaction and transport have major impacts on the macroscopic manifestation of reactions and reaction rates. Various experimental and numerical approaches have been developed and applied for pore-scale investigation of surface reactions in porous media. A micromodel is one of the experimental systems most widely used for studying pore-scale reactions under flow conditions. It is a 2-dimensional (2-D) flow cell system typically fabricated with silicon materials, into which a desired pore structure can be imprinted to form a pore network (Zhang et al. 2010a). The interfaces between pores and solids in the pore network can be coated with certain redoxsensitive materials such as hematite (Zhang et al. 2013a) to provide reactive sites for surface 1529-6466/15/0080-0006$05.00

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reactions. The micromodel allows real time observation of surface reactions at the pore-scale using a suite of spectroscopic and microscopic techniques including optical microscopy (Dupin and McCarty 2000; Nambi et al. 2003; Knutson et al. 2005; Zhang et al. 2010a; Yoon et al. 2012), synchrotron X-ray microprobe (XMP) and X-ray absorption spectroscopy (XAS) (Pearce et al. 2012), and Raman spectroscopy (Zhang et al. 2013a). Other experimental systems for pore-scale research include flow-cell reactors (Fredd and Fogler 1998; Panga et al. 2005; Pallud et al. 2010a,b), capillary tubes (Molins et al. 2014), and columns (Bernard 2005; Luquot and Gouze 2009; Fridjonsson et al. 2011; Noiriel et al. 2012; Luquot et al. 2014). These experimental apparatus complemented with non-destructive imaging techniques, such as X-ray microtomography (XMT) (Bernard 2005; Luquot and Gouze 2009; Navarre-Sitchler et al. 2009; Noiriel et al. 2012; Luquot et al. 2014; Molins et al. 2014) and nuclear magnetic resonance (NMR) (Fridjonsson et al. 2011), provide powerful tools for studying pore-scale surface reactions, reaction-induced pore structure and permeability changes, and feedback between surface reactivity and reactive transport in porous media (Dupin and McCarty 2000; Nambi et al. 2003; Knutson et al. 2005; Willingham et al. 2008; Zhang et al. 2010b; Edery et al. 2013). Numerical approaches are an important component of pore-scale geochemical research. Pore-scale flow and reactive transport models are used to simulate the temporal and spatial evolution of surface reactions in coupling with transport, to analyze and interpret experimental results, and to complement experimental approaches by extending experimental conditions. Commonly used numerical approaches include pore-network models (Acharya 2005; Li et al. 2006, 2007; Mehmani et al. 2012; Raoof et al. 2013; Varloteaux et al. 2013), lattice Boltzmann models (Kang et al. 2006, 2010, 2014; Knutson et al. 2007; Lichtner and Kang 2007; Zhang et al. 2008; Huber et al. 2014), smoothed particle hydrodynamics models (Tartakovsky et al. 2008b), and conventional finite-difference or finite-volume methods (van Duijn and Pop 2004, 2005; Willingham et al. 2008; Orgogozo et al. 2010; Porta et al. 2012, 2013; Yoon et al. 2012; Liu et al. 2013a; Steefel et al. 2013; Molins et al. 2014; Trebotich et al. 2014). These models can explicitly incorporate pore geometry, reactive surface area, and distribution, and even molecular reaction rates to simulate coupled transport and reactions at the local scale and to provide insights into effective rates at the macroscopic scale. Both experimental and numerical studies indicate that effective reaction rates can decrease by orders of magnitude from the molecular to macroscopic scale, with the magnitude decrease dependent on specific reactions and the structures of porous media (Swoboda-Colberg and Drever 1993; Malmström et al. 2000; Acharya et al. 2005; Li et al. 2006; Meile and Tuncay 2006; Navarre-Sitchler and Brantley 2007; Zhang et al. 2008; Bi et al. 2009; Flukiger and Bernard 2009; Zhang and Lv 2009; Miller et al. 2010; Liu et al. 2013a; Zhang et al. 2013b; Raoof et al. 2013; Salehikhoo et al. 2013). Various factors have been identified that may individually or collectively contribute to the scale-dependent behavior of geochemical reaction rates (Alekseyev et al. 1997; White and Brantley 2003; Meile and Tuncay 2006; Lichtner and Kang 2007; Steefel and Maher 2009; Maher 2010; Miller et al. 2010; Dentz et al. 2011; Liu et al. 2013a, 2014a). These factors can be grouped into two categories: 1) scale-dependent variations in mineral surface properties, such as surface roughness that controls reactive surface area and secondary mineralization that alters surface reactivity (White 1995; Alekseyev et al. 1997; Benner et al. 2002; White and Brantley 2003), and 2) heterogeneous distributions of flow, transport, and chemical properties in subsurface environments (Malmström et al. 2004; Li et al. 2006; Meile and Tuncay 2006; Lichtner and Kang 2007; Maher 2010; Shang et al. 2011; Liu et al. 2013a, 2014a; Zhang et al. 2013a). Previous research into the existence and cause of scale-dependent reaction rates has targeted specific examples with relatively constrained physical and chemical conditions.

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General theories and/or approaches for systematically describing and conceptually predicting the scaling effect are under-developed. In this chapter, we begin with the development of a theoretical base for predicting the scale-dependent behavior of effective surface reaction rates in heterogeneous porous media, including explicit linkage of the effective reaction rate constant with its intrinsic rate constant and the important influence of pore-scale variations in reactant concentrations (see the Section Theoretical Consideration of Effective Reaction Rates). Approaches to estimate intrinsic reaction rates and rate constants are then discussed with emphasis on molecular simulations (see Intrinsic Rates and Rate Constants). We follow with examples of how the coupling of pore-scale subsurface reaction and transport processes creates variations in reactant concentrations, and how these, in turn, cause the effective rate constants to deviate from the intrinsic reaction rate constant as a function of time and space in: i) grain-scale reactive diffusion systems (see Grain-scale reactions, sub-grain process coupling, and effective rates) and ii) in reactive transport systems with fluid flow (see Subgrid variations in reactant concentrations and effective rate constants under flow conditions). We conclude with a summary of the results and discussion in this chapter, a concept to minimize the scale-dependent behavior of effective surface reaction rates in macroscopic models through consideration of subgrid heterogeneity, and a brief discussion of other potential important factors affecting reaction rate scaling.

THEORETICAL CONSIDERATION OF EFFECTIVE REACTION RATES Geochemical reaction rates on mineral surfaces can be generally described using the mass action law: ri  kins Ai    j c j 

vij

,

(1)

j

where ri is the reaction rate, kins is the intrinsic rate constant, Ai is the reactive surface area, cj and j are the concentration and activity coefficient of chemical species j, vij is the reaction order with respect to chemical species j. Rigorously, the mass action law (Eqn. 1) is only applicable at the pore scale, and concentrations and reaction rates in Equation (1) are defined locally at specific reactive surface locations or sites. As described in the following sections, the reactive surface locations and chemical species concentrations are generally heterogeneously distributed in subsurface materials. The observed and simulated chemical reaction rates in subsurface materials are effective reaction rates that are either explicitly or implicitly defined within a numerical grid cell volume that contains many pores and mineral surfaces. The grid cell volume can be the representative elemental volume (REV) to define Darcy-scale physical properties (Bear 1979), but most often it is a volume of porous media with its size constrained by computational resources and resolution need (Mo and Friedly 2000; Chiogna and Bellin 2013). Grid cells may be selected at different scales depending on applications (Fig. 1) including: 1) Earth system models (Harrison et al. 2009; Poudel et al. 2013) where an entire watershed may represent one numerical grid cell; 2) field scale numerical problems with numerical grid sizes ranging from meters to hundred meters (Scheibe et al. 2006; Maher 2010; Molins et al. 2010; Li et al. 2011; Bao et al. 2014); 3) core scale numerical simulations with numerical grid size from centimeter to meters (Zinn and Harvey 2003; Shang et al. 2011, 2014; Liu et al. 2014a); and 4) pore-scale research at the scales of m to mm (Willingham et al. 2008; Zhang et al. 2013a).

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Figure 1. Schematic diagrams showing likely numerical grid scales for observations, experiments, and modeling.

Subgrid heterogeneity ranging from pore to large scales is a common feature of all these applications, indicating that Equation (1) is not directly applicable to describe effective reaction rates in heterogeneous subsurface materials. To derive the effective reaction rate, Equation (1) is averaged over a porous medium volume (V) to yield: ri  kins Ai    j c j  , vij

(2)

j

where the variables with a bar are the average variables over the pore space within V. Using the average variables, pore-scale variables in Equation (1) can be written as the follows: Ai  Ai  Ai ',

(3)

c j  c j  c j ',

(4)

 j   j   j ',

(5)

where Ai ', c j ', and  j ' are the deviations from their corresponding average values within V. Note that the average of Ai ', c j ', or  j ' within V is zero. Replacing Equations (3–5) into Equation (2) yields:



ri  kins Ai   j c j j



vij

vij

 Ai '     j '  c j '    1    . 1     1  Ai  j    j  c j    

(6)

In subsurface systems, the relative deviation of the activity coefficient from its average value is typically small (Lasaga 1979). However, the relative deviations of pore-scale reactive

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surface area and chemical species concentrations from their average values within V can be significant in heterogeneous materials. Equation (6) reveals that the effective reaction rate defined for V is not only a function of average surface area and chemical species concentrations, but also depends on the pore-scale variations of these variables within V. The calculation of the pore-scale variations in Equation (6) requires detailed knowledge of pore structure, pore-scale chemical species concentrations, and reactive surface area or site concentration distributions. Such information is difficult to obtain in natural subsurface environments. Consequently, the pore-scale variations in Equation (6) are lumped into an effective rate constant: vij

keff

 A '     '  c '    kins 1  i     1  j  1  j   , Ai  j    j  c j    

(7)

and the effective reaction rate in V can be expressed using the average variables:



ri  keff Ai   j c j j



vij

.

(8)

Equation (8) has the same form as Equation (1). However, the variables and rate parameters are different. Equation (8) equals Equation (1) if the spatial variability of all variables is minimal in V, such as in a well-mixed system. Well-mixed systems are rare, and consequently Equation (8) is generally different from Equation (1). However, Equation (8) is used interchangeably with Equation (1) in most subsurface reactive transport studies and simulations. In these cases, an effective rate constant must be used instead of the intrinsic rate constant. The effective rate constant is influenced by the relative deviations of pore-scale concentrations from their average values within V (Eqn. 7). The relative deviation of reactive surface area depends on reactive mineral distribution and accessibility, as well as geochemical/ biogeochemical reactions that act on the surfaces and their potential to alter surface reactivity as reactions proceed. The concentrations of dissolved species will depend on not only surface reactions, but also transport processes that supply reactants at the pore scale and remove reaction products. Consequently, the effective rate constant (keff) will be strongly affected by the coupling of reaction and transport, and will generally vary with space and time. Next we present several generic scenarios to demonstrate potential changes of keff from its intrinsic value. In sections that follow, we discuss reactive transport studies to derive effective rate constants in heterogeneous porous media.

Well-mixed conditions Under well-mixed conditions, the spatial variations in dissolved species concentrations are negligible. Although reactive surface area or reactive surface site concentration can be spatially variable, the effective reaction rate constant is the same as the intrinsic reaction rate constant: vij

keff

  A '     '  c '   A '  kins 1  i     1  j  1  j    kins 1  i   kins . Ai  j   Ai   j  c j     

(9)

Mass transport limited conditions For the convenience of demonstration, we first consider a case with only one dissolved species limiting the rate of a surface reaction (i.e., j = 1 and ij = 1 in Eqn. 1). Equation (7) can then be simplified after ignoring the pore-scale variation in activity coefficient:

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(10)

kHIIkLQW

The pore-scale distribution of reactive area (or site concentration) is often negatively correlated with that of dissolved reactant concentrations under mass transport limited conditions in porous media. For example, finer-grained materials typically have a higher surface area and more reactive sites. These finer materials can form lenses, aggregates, coatings, or spots, which are less accessible to advective fluid due to the limitation of local hydraulic conductivity or pore connectivity. Consequently these regions may have lower concentrations of dissolved reactants supplied through fluid flow. Under this condition, Equation (10) predicts a lower keff than kins, because the second term on the right hand site of Equation (10) is a negative value. With increasing size of V, large scale hydraulic heterogeneity, such as macroscopic flow channels, can affect the spatial distributions of reactants. This may further increase the range of dissolved reactant concentrations within V, and thus increase the negative magnitude of the second term on the right hand side of Equation (10). The net result is that the effective rate constant decreases with increasing scale (domain I in Fig. 2). Equation (10), however, also predicts an alternative scenario when the variations in reactive surface site and dissolved reactant concentration are positively correlated. When this happens, keff will be larger than kins (domain II in Fig. 2). This scenario requires that more reactive sites reside at locations where dissolved reactants are rapidly supplied.

GRPDLQ,,



ZHOOPL[HG

GRPDLQ, 

ǻ9

Figure 2. Effective rate constant as a function of numerical grid volume in heterogeneous porous media. Under well-mixed conditions, keff = kins. Under mass transfer limited conditions, keff can be smaller (domain I) or larger than kins (domain II).

More complex scenarios can develop when two or more chemical reactants limit the rate of a surface reaction, or when a reaction is not first order with respect to a reactant. For example, when a second-order rate expression is considered with respect to dissolved species j (i.e., j = 1, i1 =2 in Eqn. 1), then keff

2   c ' 2 A'i c1' A'i  c1'   1  kins 1     2      c1  Ai c1 Ai  c1    

(11)

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The bracketed value in Equation (11) not only depends on the pore-scale correlation between the variations in reactive surface site and dissolved reactant concentrations, but also on the relative strength of the self-correlation for variations in reactant concentration. A large, self-correlation in the variations of dissolved reactant concentration can decrease the effect of the negative correlation between dissolved species and reactive surface site concentrations, leading to less scale-dependent behavior. Similar complex scenarios can exist for reactions involving multiple reactants. For these cases, the effective rate constants in Figure 2 may not uniformly decrease or increase as scale increases.

INTRINSIC RATES AND RATE CONSTANTS The effective rate constant is proportional to the intrinsic rate constant (Eqn. 7). Wellmixed experimental systems such as batch and stirred flow-cell reactors are often used to characterize reaction mechanisms and to estimate intrinsic rate constants that are independent of transport processes. Recently, molecular-scale simulations have been used to estimate kinetic rates and intrinsic rate constants. In this section, the most commonly used molecular simulation methods to calculate reaction rates are briefly described. The application of these methods for determining molecular-scale rates and rate constants of surface reactions are then discussed using several surface reaction examples.

Approaches to calculate molecular-scale reaction rates Molecular dynamics (MD) is a powerful technique to study the molecular-scale behavior of geochemical systems (Stack et al. 2013). However, because the integration time step in MD simulations is constrained by the timescale for molecular vibrations that is usually on the order of 1 femtosecond, the simulated time that can be achieved routinely with MD simulations is limited to tens to hundreds of nanoseconds. Therefore, a range of techniques, referred to as rare-event techniques hereafter, have emerged with the goal of extending the timescale of MD simulations to investigate kinetic progress with rates that are lower than those manageable with standard MD algorithms. Examples of rare-event methods include transition path sampling (Dellago et al. 1998), metadynamics (Laio and Parrinello 2002), and reactive flux (Chandler 1987). A frequent application of the reactive flux approach to geochemically relevant systems is the study of water exchange reactions around aqueous ions (Rey and Guardia 1992; Rey and Hynes 1996a,b; Spångberg et al. 2003; Kerisit and Parker 2004b; Loeffler et al. 2006; Rustad and Stack 2006; Stack and Rustad 2007; Kerisit and Rosso 2009; Dang and Annapureddy 2013; Kerisit and Liu 2013). This same approach was subsequently applied to quantify the rates of surface complexation of aqueous ions on mineral surfaces (Kerisit and Parker 2004b) and eventually to look at attachment and detachment reactions at kink sites to simulate the elementary steps of mineral dissolution and growth (Stack et al. 2012). In the reactive flux approach, the reaction rate is written as the product of the transition state rate constant, kTST, and the transmission coefficient, . The former corresponds to the rate at which the system reaches the transition state and is defined as k

TST

2 k BT  * exp  W   *    , 2  * 2 exp  W     d   

(12)

0

where μ is the reduced mass of the atoms involved in the reaction,  is 1/kBT where kB is the Boltzmann constant and T is the temperature,  is the reaction coordinate, * is the value of the reaction coordinate at the transition state, and W() is the free energy of the system at , which can be obtained using a number of techniques, including the potential of mean force approach (PMF). In the PMF approach, separate MD simulations are performed with the

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system constrained at specific values of , and the mean force along the reaction coordinate is integrated to obtain W(). The transmission coefficient accounts for the departure from ideal transition state behavior due to barrier re-crossing (i.e., not all trajectories that reach the transition state are successful transitions between the reactants and products free energy wells) and is determined from the plateau value of the normalized reactive flux, which can be computed as (t ) 

 (0) (t )   *  (0)  (0) 

c

,

(13)

c

where [x] is the Heaviside function, which is 1 if x is larger than 0 and 0 otherwise, and (0) is the initial velocity along the reaction coordinate. The subscript ‘c’ denotes that the initial configurations are generated in the constrained reaction coordinate ensemble. The reactive flux is usually computed by running a large number of short trajectories forward and backward in time from initial configurations constrained at the transition state.

Molecular-scale rates of uranyl sorption reactions at mineral surface sites Molecular simulation techniques have been used to study uranyl sorption at mineral surfaces (Kremleva et al. 2010; Geckeis et al. 2013). Mineral surfaces have included silica (Greathouse et al. 2002; Patsahan and Holovko 2007; Boily and Rosso 2011), alumina (Moskaleva et al. 2004; Glezakou and deJong 2011; Tan et al. 2013), iron and titanium oxides (Steele et al. 2002; Perron et al. 2006a,b, 2008 ; Drot et al. 2007; Sherman et al. 2008; Roques et al. 2009; Skomurski et al. 2011; Pan et al. 2012; Sebbari et al. 2012) , phyllosilicates (Zaidan et al. 2003; Greathouse and Cygan 2005, 2006; Greathouse et al. 2005; Kremleva et al. 2008, 2011, 2012; Veilly et al. 2008; Hattori et al. 2009; Martorell et al. 2010; Lectez et al. 2012; Liu et al. 2013b; Yang and Zaoui 2013a,b; Teich-McGoldrick et al. 2014); feldspars (Kerisit and Liu 2012, 2014), and carbonates (Doudou et al. 2012). The majority of these studies focused on energy minimizations of adsorbed uranyl complexes at various mineral surfaces. As a whole, they have shown that significant variations exist in surface complex configurations and binding energies for different surfaces and sites of a given mineral. Although adsorption rates were not calculated in most cases, this body of work strongly suggests that molecularscale heterogeneities should influence the adsorption/desorption rates of uranyl for a given mineral. For example, static density functional theory (DFT) calculations of uranyl surface complexation on the basal and edge surfaces of kaolinite (Al2Si2O5(OH)4) (Kremleva et al. 2008, 2011; Martorell et al. 2010) revealed significant differences between the tetrahedral Si layer and the octahedral Al layer, as well as a wide range of binding energies on different sites om the (010) edge surface. Similar conclusions were drawn from simulations of uranyl surface complexation on pyrophyllite (Al2Si4O10(OH)2) edge surfaces (Kremleva et al. 2012), where it was concluded that multiple adsorbed complexes likely exist on any given surface, and that different edge surfaces exhibit different preferred surface complex coordination environments. A small subset of these studies investigated the kinetics of uranyl sorption. For example, MD simulations by Boily and Rosso (2011) using PMF calculations showed significantly different barrier heights for uranyl adsorption/desorption on three quartz (-SiO2) surfaces, providing further evidence that sorption rates are heterogeneous at the molecular scale even for a given mineral. MD simulations of uranyl sorption on the most stable surfaces of orthoclase (KAlSi3O8 - tectosilicate) and kaolinite (phyllosilicate) (Kerisit and Liu 2012, 2014) revealed large quantitative differences in the free energy barriers for adsorption/desorption as well as in the affinity of uranyl for these two types of silicate minerals. The formation of uranyl carbonate complexes affected desorption free energy barriers on the two minerals to different extents, indicating that variations among mineral phases can be compounded by solution

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conditions that impact aqueous speciation. Desorption rates of adsorbed uranyl orthoclase were calculated to be on the order of 106 hr-1 (Kerisit and Liu 2012), a rate several orders of magnitude faster than the grain-scale rate constant (Liu et al. 2013a).

Molecular-scale rates of elementary mechanisms of mineral growth and dissolution Initial applications of rare-event techniques such as reactive flux to calculate molecularscale rates of mechanisms relevant to mineral growth and dissolution focused on adsorption of ions on ideal, planar surfaces (i.e., formation of adatoms) (Piana et al. 2006; Kerisit and Parker 2004a,b). For example, adsorption of alkaline-earth cations on the {10 14} calcite surface was found to be very fast (~108 s-1) and desorption was calculated to be only marginally slower (~106–107 s1) (Kerisit and Parker 2004b), demonstrating the presence of labile species on flat surfaces. The exact values of these rates as well as the extent of the driving force for adsorption is dependent on the details of the force-field parameters (Raiteri and Gale 2010). Similar free-energy barriers for adsorption/desorption were obtained by Piana et al. (2006) for Ba2+ on a pure, flat {001} barite (BaSO4) surface. In contrast, detachment rates of Ba2+ from a step edge on the same surface were determined by Stack et al. (2012) to be much slower (~104 s-1). Therefore, orders of magnitude differences in adsorption and desorption rates can exist between different sites and surface structures on the same mineral surface. The work of Piana et al. (2006) on Ba2+ sorption on three barite surfaces also demonstrated that the free-energy barrier for adsorption is strongly surface specific and can be significantly reduced by the presence of anions adsorbed on the surface, indicating that the sequence of sorption events plays an important role in determining the overall reaction rate. MD simulations of Ca2+ and CO32- dissolution from calcite surfaces also resulted in free-energy barriers that were heavily surface specific (Spagnoli et al. 2006). In addition, detailed rare-event simulations of Ba2+ attachment and detachment to/from a stepped surface by Stack et al. (2012) highlighted that, even for this seemingly simple surface reaction, multiple intermediates were involved with a wide range of transition rates between intermediates (104 to 1010 s-1). Rareevent techniques have also improved understanding of mineral nucleation. For example, MD simulations demonstrated importance of precursor amorphous clusters in CaCO3 homogenous nucleation by demonstrating the presence of small free energy barriers and favorable reaction free energies for the addition of CaCO3 ion pairs to a growing nucleus (Tribello et al. 2009; Raiteri et al. 2010). The studies described above made use of classical MD techniques due to the large size of the simulated systems. Another approach to computationally probe the rates of mineral growth and dissolution involves the use of small clusters that are treated quantum mechanically to represent surface sites and that can be immersed in a continuum dielectric model to simulate solvation effects. With this approach, potential energy surfaces can be obtained by progressively elongating bonds within the clusters to evaluate the energetics of mineral dissolution. Although the effects of the extended crystal structure are lost with this approach, it allows for the making/breaking of covalent bonds and investigating the effects of pH. In particular, this approach has been applied, for example, to study the dissolution of silicate minerals such as olivine (Morrow et al. 2010) and feldspar (Criscenti et al. 2005, 2006).

GRAIN-SCALE REACTIONS, SUB-GRAIN PROCESS COUPLING, AND EFFECTIVE RATES Subsurface materials contain microscopic pores and/or fractures within individual minerals, grains, and grain-aggregates (Anbeek 1993; Berkowitz and Scher 1998; Lee et al. 1998; White et al. 2001; Berkowitz 2002; Zachara et al. 2004; Liu et al. 2006; Luquot and Gouze 2009; Hay et al. 2011). These intra-granular porous domains have pore sizes ranging

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from nanometers to tens of micrometers, and have a high interfacial surface area to pore volume ratio (Anbeek 1993; Hurlimann et al. 1994). Various geochemical reactions can occur in the intra-granular pore domains that affect bulk solution chemistry in external pore regions (Barber Ii et al. 1992; McGreavy et al. 1992; Anbeek 1993; Lee et al. 1998; Liu et al. 2006; McKinley et al. 2006, 2007). The transport in these intra-granular domains is dominated by diffusion that is strongly affected by pore connectivity (Ewing et al. 2012). Slow diffusion process allows intra-granular solutions to evolve close to mineral equilibrium that can deviate from that in bulk solutions. Geochemical reactions such as sorption/desorption (Ball and Roberts 1991; Qafoku et al. 2005; Liu et al. 2008; Ginn 2009) and mineral precipitation and dissolution (McGreavy et al. 1992; Lee et al. 1998; Anderson et al. 2002; Liu et al. 2006; McKinley et al. 2006, 2007; Pallud et al. 2010a) can occur in defiance of thermodynamic conditions in bulk solutions. The slow diffusion, on the other hand, decreases mass exchange rate with external pore waters, leading to non-ideal behavior of chemical transport even for non-reactive species (Beven and Germann 1982; Ball and Roberts 1991; Ewing et al. 2010, 2012; Hay et al. 2011). Micromodel experimental systems have been used to evaluate the relationship between the intrinsic and effective rate constants (Zhang et al. 2013a; Liu et al. 2015). Micromodels can be fabricated to simulate the complex physical structure of natural sediment grains and grainaggregates. Figure 3 shows an example of micromodel systems. The four corner regions in the micromodel (Fig. 3) are composed of grain aggregates and intra-aggregate pores. Surrounded by the aggregates is a macropore domain where advection dominates. The micromodel system has been used to provide insights into the scale-dependent variation in the effective rate of uranyl desorption that is over 3–5 orders of magnitude slower than the intrinsic rate estimated from molecular simulations (Liu et al. 2013a). It has also been used to examine hematite reduction in heterogeneous media under flow conditions and to investigate the difference in effective rate constants between well-mixed and heterogeneous transport systems (Zhang et al. 2013a). These examples display the importance of the coupling of transport and reaction at the pore-scale in controlling the grain-scale reaction rates. Accurate simulations of grain-scale reaction rates are, however, challenged by the difficulties in the estimation of effective reaction rates in both intra- and inter-granular domains (Liu et al. 2006; Zhang et al. 2013a). We now demonstrate how a pore-scale reactive diffusion model can be used to simulate the variability in reactant concentrations in intra-granular domains. The simulated concentrations are then



Figure 3. A micromodel system abstracted from imaging analysis of grain aggregates (4 corner regions) (from Zhang et al. 2013a).

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used by Equation (7) to calculate effective rate constants for grain-scale reactive transport models. Grain-scale models based on mass transfer and diffusion will be compared to assess how different physical models influence the deviation between effective and intrinsic rate constants.

Pore-scale variability in reactant concentrations at the sub-grain scale The evolution of reactant concentrations in intra-granular domains can be described using a reactive diffusion equation: Nr c j     D j c j    a jiri , t i 1

j  1,2,...N ,

(14)

where Di is the molecular diffusion coefficient of aqueous chemical species j, aji is the stoichiometric coefficient of species j in reaction i, Nr and N are the total numbers of kinetic reactions and species in the system, respectively, and the other variables are defined before for Equation (1). Various numerical approaches exist for solving the multi-species diffusion problem described by Equation (14) using implicit or explicit numerical scheme, or their combination (Yeh and Tripathi 1991; Press et al. 1992; Liu et al. 2011). The reactive diffusion equation (Eqn. 14) was used to simulate pore-scale variations in reactive surface site and dissolved species concentrations during hematite reduction in the intra-granular pore spaces as shown in the four corner regions in the micromodel (Fig. 3). In this example, hematite is assumed to be uniformly distributed on all the pore surfaces (bottom and side walls) in both intra- and inter-granular domains in the micromodel. The experimental design, micromodel fabrication, and hematite deposition on the pore surfaces for this example have been detailed elsewhere (Zhang et al. 2013a). To initiate hematite reduction in the intragranular domain for the modeling purpose, reduced flavin mononucleotide (FMNH2) with a constant concentration (100 M) was provided in the external pore space (center white region in Fig. 3). FMNH2 is a biogenic organic molecule capable of reducing iron oxides (Marsili et al. 2008; Shi et al. 2012). Driven by the concentration gradient, FMNH2 diffuses into the intragranular domains where it reacts with hematite on pore surfaces. Hematite (Fe2O3) undergoes reductive dissolution to yield Fe(II) and FMN, which is an oxidized form of FMNH2. The kinetics of hematite reduction in a well-mixed system follows the rate expression (Liu et al. 2007; Shi et al. 2012; Zhang et al. 2013a):

r  kinsc

Fe 2 O3

cFMNH2 ,

(15)

where cFe2O3 is the pore-scale hematite concentration on the mineral surfaces (), cFMNH2 is the pore-scale aqueous concentration (M), and kins is the rate constant at the pore-scale (M-1h-1), which is equivalent to the intrinsic rate constant for this study (Eqn. 7). Equation (14) and Equation (15) were solved using the finite-volume method with a constant numerical node size of 10 × 10 m2. The initial hematite concentration on each pore surface is 0.73 μmol/cm2 in the micromodel. The surface based hematite concentration was converted to volumetric concentration using a surface area of 10 × 28 m2 for the side wall and 10 × 10 m2 for the bottom surface wall. The statistical distributions of FMNH2 and residual hematite concentrations in the intraaggregate domains were calculated to vary with time (Fig. 4). Initially, FMNH2 concentrations were zero at all diffusion nodes (Fig. 4A). Initial hematite concentration was one of five possible values after normalizing to local pore node volume (5 red bars in Fig. 4B), a consequence of the fact that each pore node can contain 1 bottom pore wall, and additional zero to four side walls of hematite depending on the topology of water-grain interfaces in a grid node. The corresponding five possible hematite concentrations were 0.26, 0.99, 1.72, 2.45,

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1RGHV

1RGHV



K K

 



 

%  K K

$ 





  )01+P0



  

       +HPDWLWH 0

Figure 4. A) Pore-scale FMNH2 concentrations in the intra-granular domain, and B) residual hematite concentrations in the intra-granular domain. Dash lines are the average concentrations. A red line denotes the results at time zero, and the blue line denotes the results after 23 hours of coupled FMNH2 diffusion and hematite reduction.

and 3.18 M. The high molar concentration of hematite resulted from a high surface to pore volume ratio at the pore-scale. After 23 hours of FMNH2 diffusion and hematite reduction, the FMNH2 concentration distribution shifted toward higher concentrations (Fig. 4A), while the hematite concentration distribution shifted toward lower concentration (Fig. 4B). The hematite concentrations in nodes with four side pore walls did not change, because those nodes were not connected to external pore regions. Nodes with the larger decrease in hematite concentrations were those containing higher concentrations of FMNH2, which lead, in turn, to the negative correlation between hematite and FMNH2 concentrations at later times. This correlation affected the effective rate constant in the intra-granular domain as discussed in the next subsection.

Effective reaction rates and rate constants The intra-granular diffusion domain is often termed as the immobile domain and external pore regions where water flows is often termed as the mobile domain in hydrology. The mass transfer between mobile and immobile domains is often described using the first-order mass exchange equation (Brusseau et al. 1992; Haggerty and Gorelick 1995; Šimunek et al. 2003):

c im j  km (c mj  c im j ), t

(16)

im m where c j is the average concentration of species j in the immobile domain, c j is the average concentration in the mobile domain, and km is the mass transfer coefficient. In this model, the spatial change in species concentrations in the immobile domain is no longer considered. Using this model to include reactions in the immobile domain, Equation (16) becomes: Nr c im j im  km (c mj  c im j )   a jiri , i 1 t

(17)

im

where ri is the average rate of reaction i in the immobile domain. Using terminology for Equation (7), the reaction rate in Equation (17) can be expressed as the follows for the case of hematite reduction in the micromodel (Fig. 4),

riim  ri  keff cFe2 O3 cFMNH2 .

(18)

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From Equation (7), we have







cFe2 O3  cFe2O3 cFMNH2  cFMNH2 keff .  1 kins cFe2O3 cFMNH2

(19)

The ratio of the effective rate constant to intrinsic rate constant can be calculated using the pore-scale concentration results in Figure 4. The calculated effective rate constant (Fig. 5) was lower than the intrinsic rate constant and decreased with increasing time. This was because residual hematite concentration was increasingly negatively correlated with the FMNH2 concentration distribution in the diffusion domain as hematite reductively dissolved. Note that the simulation was only run for 23 hours in this case, and the effective rate constant is expected to further decrease with time, before it eventually increases back to 1 when all hematite is reduced and dissolves.

Figure 5. Deviation of effective hematite-reduction rate constant from the intrinsic value in the intragranular domain. Solid line showing the effective rate constant for the entire intra-granular domain; dashed lines are the effective rate constants for three numerical nodes within the intra-granular domain.

Diffusion models, such as one-dimensional slab or spherical diffusion models (Crank 1975; Ball and Roberts 1991) represent an alternative approach to describe intra-granular reactions and mass exchange with the mobile domain: Nr c im j im     Da c im j    a jiri , i 1 t

(20)

where Da is the effective diffusivity. The diffusion model is macroscopic without considering details in pore structure and connectivity in the intra-granular domain. However, Figure 6

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shows that the effective reaction rate constants become closer to the intrinsic rate constant using the diffusion model. In this specific case, a slab reactive diffusion model was used, and the intra-granular macroscopic diffusion domain was divided into 3 numerical nodes (exterior, which is close to external pore space; interior, which is the farthest from external pore space; and middle, which is between exterior and interior nodes). The effective reaction rate constant in each node was calculated using the simulated pore-scale concentrations of FMNH2 and residual hematite within a specific numerical node. The calculated effective reaction rate constants still deviate from the intrinsic value in each numerical node with a largest deviation observed in the exterior node (Fig. 5). The deviation correlates with the concentration change that was the greatest in the exterior node. It is expected that the effective rate constant in each node will become closer to the intrinsic rate constant when more numerical nodes are used. However, complete elimination of the deviation between the effective and intrinsic rate constants cannot be achieved regardless of the number of numerical nodes used in discretizing Equation 20, because of the macroscopic nature of the grain-scale diffusion model. The effective rate constant also changed with numerical grid size when a macroscopic model is used to describe the reaction in the intra-granular domain (Fig. 6). Figure 6 only shows the results for two types of grain-scale grid node sizes: 1) the intra-granular domain was divided into 3 numerical nodes (grid node size = 700 mm, subgrain scale), and 2) the intra-granular domain was treated as one numerical grid node (grid node size = 2000 mm, grain scale) in corresponding to the results in Figure 5. The effective rate constant generally decreased with increasing grid size (Fig. 6); however, its value varied significantly at each grid size scale, reflecting spatial and temporal changes in pore-scale concentrations of reactants and their correlation (Fig. 4). The effective rate constants in Figure 6 were calculated using the pore-scale concentrations at time 5, 15, and 23 hours of FMNH2 diffusion and hematite reduction. It is expected to have a wider range of effective rate constant distribution when other time results are also included.

Figure 6. Effective hematite-reduction rate constant as a function of numerical grid node size at three different observation times (5, 15, and 23 h) after hematite reduction.

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SUBGRID VARIATIONS IN REACTANT CONCENTRATIONS AND EFFECTIVE RATE CONSTANTS UNDER FLOW CONDITIONS Subsurface materials are heterogeneous in many ways (Fig. 1) in addition to intragranular complexity in pore structure and reactivity (Figs. 3 and 4). A numerical grid system used for simulating large scale reactive transport typically contains heterogeneity in both intraand inter-granular domains within individual grid nodes. This heterogeneity is referred to as subgrid heterogeneity, which has been attributed to 5–7 orders of magnitude decrease in the effective rate of U(VI) desorption in sediments from the molecular to field scales (Liu et al. 2013a, 2014a; Ma et al. 2014), and four orders of magnitude decrease for microbial reduction of uranium from laboratory to field systems (Bao et al. 2014). Here the micromodel system (Fig. 3) was used again to demonstrate how to: 1) calculate the variations in reactant concentrations in systems with subgrid heterogeneity, and 2) use the calculated results to estimate effective rate constant as a function of space and time. Macroscopic reactive transport models with and without considering subgrid heterogeneity are compared to evaluate their effect on the estimated effective rate constants.

Pore-scale concentration variations under flow conditions Simulations of pore-scale reactive transport under flow conditions require calculating fluid flow and chemical reactive transport. Fluid flow at the pore-scale can be generally described using the Navier–Stokes (N-S) equation (Bird et al. 2007).

u  u u   gh   2 u  g, t

(21)

where u is the velocity vector, g is the gravitational constant, g is the gravitational force, h is the pressure or water head, and  is the effective kinematic viscosity ( = μ/, where μ is the dynamic viscosity and  is the fluid density). The simulation of fluid flow using Equation (21) requires knowledge of the pore geometry. However, soil and sediment systems are heterogeneous, consisting of various sizes of pores, minerals, organic matter, and organisms, often exhibiting hierarchical structures spanning seven or more order of magnitude. Pore size distribution can be quantified to the nm scale by direct and indirect methods including mercury porosimetry and neutron scattering, but the important pore network is much more difficult to define, especially at scale of < 1m (Ball et al. 1990; Hay et al. 2011). This creates a scenario that pore geometry for large pores can be spatially resolved, while that for small pores is below spatial resolution. The soil and sediment domains containing small pores will therefore have to be treated as porous media (mixed pores and solids with unknown pore geometry) in numerical discretization. This will prevent the direct application of Equation (21) to describe fluid flow in soils and sediments. An alternative model has recently been developed to describe fluid flow in mixed media containing both pores and porous domains (Yang et al. 2014). The model becomes N–S in the pore regions, and becomes a Darcy-law-based model for fluid flow in the porous domain. When the fluid velocity field is specified, the reactive solute transport can be simulated using the pore-scale multi-species advection and diffusion equations coupled with reactions: N c     D j c   u c j   a ji ri , j  1,2, N . i 1 t r

j

j

(22)

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Equations (21) and (22) were used to simulate fluid flow, hematite reduction, and reactive transport in the micromodel (Fig. 3) to calculate reactant variations with time. FMNH2 (100 M) was injected from the left side of the micromodel with a constant flow rate (270 L/h). The injected FMNH2 reacts with hematite on pore surfaces in both inter- and intra-granular domains. Hematite was reduced and dissolved quickly in the advection domain (red region) as FMNH2 carried by fluid from the left side reacted with the hematite on the bottom pore wall in the advection domain (Fig. 7). There were regions near the grain–grain contacts in the inter-granular pore domain (center up and bottom, Fig. 7), where hematite reduction was much slower than that in the advection domain. These regions mimic the wedge spaces or dead end pores created by grain-grain contacts in porous media. This slow reduction region is termed as the macropore diffusion domain, as the advection bypasses these regions (Zhang et al. 2013a; Liu et al. 2015).



$ 

%  Figure 7. Snapshots of pore-scale distributions of FMNH2 (plot A) and residual hematite (plot B) concentrations in the micromodel after 23 hours of FMNH2 injection and hematite reduction (From Liu et al. 2015).

The hematite-reduction front migrated from the advection domain to the macropore and intra-granular diffusion domains through FMNH2 diffusion. Figure 7 only shows one snapshot. The hematite-reduction fronts moved further into the diffusion domains with continual FMNH2 injection. Eventually all hematite in the macropore diffusion domain was reduced. The reduction rate in the intra-granular domain was, however, much slower as discussed previously. Note that the initial FMNH2 concentration was zero and initial hematite concentration was one of 5 possible values depending on topology of water–grain interfaces in the grid node. After 23 hours of reduction, the FMNH2 concentration in most pores in the advection domain was equal to the FMNH2 concentration in the injection solution (= 100 M); but its concentration was still zero in the interior of the intra-granular domain. The net result is a wide distribution of FMNH2 concentration (Fig. 8). The average concentration of FMNH2

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increased in the micromodel, while the residual hematite concentration decreased with time (dashed lines in Fig. 8). Those nodes with a higher FMNH2 concentration have a lower concentration of hematite than their average concentrations, leading to a negative value for the second right-hand term in Equation (7).

K K



$ 

% 



K K





1RGHV

1RGHV







  





   )01+P0





       +HPDWLWH0

Figure 8. Statistics of pore-scale concentrations of FMNH2 (plot A) and residual hematite (plot B) in the micromodel at time zero and after 23 hours of FMNH2 injection and reaction. Dash lines are the average concentrations. The red line denotes the results at time zero, and the blue line denotes results at time 23 hours.

Effective rate constants The effective overall rate constant for the micromodel (calculated using the results in Fig. 8 and Eqn. 7) decreased by two orders of magnitude during the first 23 hours of the hematite–FMNH2 reaction (Fig. 9A). Ratio of keff/kins decreased immediately from 1 to about 0.75 within a few seconds of FMNH2 injection because of a fast hematite reduction in the advection domain. After that, the effective rate decreased steadily with time. The effective rate

Figure 9. A): Deviation of effective hematite-reduction rate constant from the intrinsic rate constant as a function of time for the entire micromodel (solid line), and for subgrid domains: advection domain (short dashed line), macropore diffusion domain (dash-dot line), and intra-granular domain (dashed line). B) predicted Fe(II) mass accumulated in effluent as a result of hematite reduction in the micromodel. The solid line is the result simulated using the calculated effective rate constant for the micromodel, and the dashed line is the result simulated using the intrinsic rate constant. Triangles are the measured effluent Fe(II) (red triangles) (Liu et al. 2015).

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constant for the micromodel was lower than the effective rate for the intra-granular domain (Fig. 6), as a result of larger concentration variations in the micromodel than those in the intragranular domain only. The effluent concentration of Fe(II) produced from hematite reduction can be predicted using the effective rate for the micromodel under the assumption that the micromodel was a well-mixed system (Fig. 9B). By comparison, the model significantly overpredicted effluent Fe(II) if an intrinsic rate constant was used in the well-mixed model. The above analysis of the effective rate constant treats the entire micromodel as a wellmixed reactor, or as one numerical grid volume in a reactive transport systems. The estimated effective rate constant reflects the pore-scale deviations in FMNH2 and hematite concentrations from their average values in the micromodel pore network (Fig. 6). Effective rate constants can also be calculated for each subgrid domain (advection domain, macropore diffusion domain, and intra-granular domain, Fig. 9A). The effective rate constants within individual subgrid domains deviate much less from the intrinsic value. The result implied that macroscopic models would improve the prediction by explicitly considering subgrid reactive domains if effective rate constants are unknown, and the intrinsic rate constant is used for model prediction in heterogeneous materials. However, Figure 9A demonstrates that the effective rate constants for individual domains still deviate from the intrinsic value. This deviation is expected to decrease with the application of more subgrid domains, but it cannot be completely eliminated. Model prediction using the intrinsic rate constant will always contain errors regardless how many subgrid domains are considered.

CONCLUSIONS Geochemical and biogeochemical reactions occur fundamentally at the pore scale, and they are coupled and modified by transport processes in subsurface materials. These coupled pore-scale processes are, however, below the spatial resolution of most core and field scale observations. Consequently solute and solid phase concentrations and reaction rates observed at the macroscopic scales reflect the manifestation of coupled pore-scale reactions and transport that in most cases cannot be explicitly separated. Pore-scale reactive transport models can theoretically be used to simulate pore-scale processes and their coupling. These models, however, require pore-scale properties that are difficult to obtain for practical problems at the macroscopic scales. Consequently, medium-based reactive transport models with porous media reaction and transport properties defined on a numerical grid volume of porous media have to be adopted based on the mass conservation law. The theoretical analysis and results in this chapter demonstrate that reaction rates and rate constants defined on a numerical grid volume in heterogeneous materials are generally different from the intrinsic reaction rate and rate constants. The effective rate constants for the grid volume depend on the statistical distributions of reactants and their spatial correlation, which are expected to vary with both temporal and spatial scales, causing scale-dependent behavior of effective reaction rates and rate constants. Importantly, the deviation of the effective rate constants from the intrinsic value can be reduced if a macroscopic model considers reactive transport in subgrid domains. The subgrid domains may be derived from non-reactive tracer transport behavior that can be independently measured for a subsurface system at the appropriate scales of interest. However, subgrid domain models cannot completely eliminate the deviation of the effective rate constants from the intrinsic value, and thus contain uncertainty in model prediction. Other factors beyond pore-scale process coupling can also affect the scaling of reaction rates from the molecular to macroscopic scales in heterogeneous subsurface materials. For example, the reaction rate at the molecular scale can be heterogeneous, even on a single mineral phase (Piana et al. 2006; Spagnoli et al. 2006; Boily and Rosso 2011; Lüttge et al. 2013); the rates of reactions occurring in nano-pores can be influenced by water ordering,

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surface curvature, and overlapping double layers (Kerisit and Liu 2009, 2012; Argyris et al. 2010; Bourg and Steefel 2012); and mineral surface reactivity can be modified as reactions proceed (Alekseyev et al. 1997; Benner et al. 2002; White and Brantley 2003; Bernard 2005; Noiriel et al. 2005; Luquot and Gouze 2009). The pore geometry and connectivity can also be changed by mineral precipitation (Steefel and Lichtner 1994; Davis et al. 2006; Zhang et al. 2010a; Yoon et al. 2012; Fanizza et al. 2013; Boyd et al. 2014) and dissolution (Noiriel et al. 2004, 2005, 2009; Bernard 2005; Flukiger and Bernard 2009; Luquot and Gouze 2009; Kang et al. 2010), microbial growth (Taylor and Jaffe 1990; Nambi et al. 2003; Thullner et al. 2004, 2007; Knutson et al. 2005; Seifert and Engesgaard 2007, 2012; Brovelli et al. 2009; Zhang et al. 2010b; Kirk et al. 2012; Tang et al. 2013) and biomineralization (Zhang and Klapper 2010; Cunningham et al. 2011; Pearce et al. 2012; Phillips et al. 2012; Hommel et al. 2013). These and other factors can be superimposed on each other. The effective rate equation (Eqn. 6), the effective rate constants (Eqn. 7), and the numerical results and data presented in this chapter represent a starting point for a systematic analysis of scale-dependent behavior of geochemical and biogeochemical reactions. Given the ubiquity of heterogeneity in subsurface materials, understanding, predicting, and minimizing the deviation of the effective reaction rate and rate constant from the intrinsic rate and rate constant is critically important in applying mechanism-based kinetic parameters for calibrating and predicting larger scale reactions and reactive transport.

ACKNOWLEDGMENT This research is supported by the U.S. DOE, Office of Science, Biological and Environmental Research (BER) as part of the Subsurface Biogeochemical Research (SBR) Program through Pacific Northwest National Laboratory (PNNL) SBR Science Focus Area (SFA) Research Project. This research was performed using Environmental Molecular Science Laboratory (EMSL), a DOE Office of Science user facility sponsored by the DOE’s Office of BER and located at PNNL. PNNL is operated for DOE by Battelle Memorial Institute under contract DE-AC05-76RL01830.

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van Duijn CJ, Pop IS (2004) Crystal dissolution and precipitation in porous media: pore-scale analysis. J Reine Angew Math 577:171–211 van Duijn CJ, Pop IS (2005) A microscopic description of crystal dissolution and precipitation, In: Huyghe, JM, Raats, PAC, Cowin, SC (eds) IUTAM Symposium on Physicochemical and Electromechanical Interactions in Porous Media. Springer, p 343–348 Varloteaux C, Bekri S, Adler PM (2013) Pore network modelling to determine the transport properties in presence of a reactive fluid: From pore to reservoir scale. Adv Water Resour 53:87–100 Veilly E, Roques J, Jodin-Caumon M-C, Humbert B, Drot R, Simoni E (2008) Uranyl interaction with the hydrated (001) basal face of gibbsite: A combined theoretical and spectroscopic study. J Chem Phys 129:244704 White AF (1995) Chemical weathering rates of silicate minerals in soils. Rev Mineral Geochem 31:407–461 White AF, Brantley SL (2003) The effect of time on the weathering of silicate minerals: why do weathering rates differ in the laboratory and field? Chem Geol 202:479–506 White AF, Bullen TD, Schulz MS, Blum AE, Huntington TG, Peters NE (2001) Differential rates of feldspar weathering in granitic regoliths. Geochim Cosmochim Acta 65:847–869 Willingham TW, Werth CJ, Valocchi AJ (2008) Evaluation of the effects of porous media structure on mixingcontrolled reactions using pore-scale modeling and micromodel experiments. Environ Sci Technol 42:3185–3193 Yang W, Zaoui A (2013a) Behind adhesion of uranyl onto montmorillonite surface: A molecular dynamics study. J Hazard Mater 26:224–234 Yang W, Zaoui A (2013b) Uranyl adsorption on (001) surfaces of kaolinite: A molecular dynamics study. Appl Clay Sci 80–81:98–106 Yang X, Liu C, Shang J, Fang Y, Bailey VL (2014) A unified multiscale model for pore-scale flow simulations in soils. Soil Sci Soc Am J 78:108–118 Yeh G-T, Tripathi VS (1991) A Model for simulating transport of reactive multispecies components: Model development and demonstration. Water Resour Res 27:3075–3094 Yoon H, Valocchi AJ, Werth CJ, Dewers T (2012) Pore-scale simulation of mixing-induced calcium carbonate precipitation and dissolution in a microfluidic pore network. Water Resour Res 48:W02524 Zachara JM, Kukkadapu RK, Gassman PL, Dohnalkova A, Fredrickson JK, Anderson T (2004) Biogeochemical transformation of Fe minerals in a petroleum-contaminated aquifer. Geochim Cosmochim Acta 68:1791– 1805 Zaidan OF, Greathouse JA, Pabalan RT (2003) Monte Carlo and molecular dynamics simulation of uranyl adsorption on montmorillonite clay. Clays Clay Miner 51:372–381 Zhang T, Klapper I (2010) Mathematical model of biofilm induced calcite precipitation. Water Sci Technol 61:2957–2964 Zhang X, Lv M (2009) The nonlinear adsorptive kinetics of solute transport in soil does not change with porewater velocity: Demonstration with pore-scale simulations. J Hydrol 371:42–52 Zhang X, Crawford JW, Young LM (2008) Does pore water velocity affect the reaction rates of adsorptive solute transport in soils? Demonstration with pore-scale modelling. Adv Water Resour 31:425–437 Zhang C, Dehoff K, Hess N, Oostrom M, Wietsma TW, Valocchi AJ, Fouke BW, Werth CJ (2010a) Porescale study of transverse mixing induced CaCO3 precipitation and permeability reduction in a model subsurface sedimentary system. Environ Sci Technol 44:7833–7838 Zhang C, Kang Q, Wang X, Zilles JL, Müller RH, Werth CJ (2010b) Effects of pore-scale heterogeneity and transverse mixing on bacterial growth in porous media. Environ Sci Technol 44:3085–3092 Zhang X, Qi X, Qiao D (2010c) Change in macroscopic concentration at the interface between different materials: Continuous or discontinuous. Water Resour Res 46:W10540 Zhang C, Liu C, Shi Z (2013a) Micromodel investigation of transport effect on the kinetics of reductive dissolution of hematite. Environ Sci Technol 47:4131–4139 Zhang X, Jiang B, Zhang X (2013b) Reliability of the multiple-rate adsorptive model for simulating adsorptive solute transport in soil demonstrated by pore-scale simulations. Transp Porous Media 98:725–741 Zinn B, Harvey CF (2003) When good statistical models of aquifer heterogeneity go bad: A comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields. Water Resour Res 39:SBH 1–4

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 217-246, 2015 Copyright © Mineralogical Society of America

Micro-Continuum Approaches for Modeling Pore-Scale Geochemical Processes Carl I. Steefel and Lauren E. Beckingham Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California 94720, USA [email protected]

Gautier Landrot Synchrotron SOLEIL 91192 Gif sur Yvette, France INTRODUCTION The recent profusion of microscopic characterization methods applicable to Earth Science materials, many of which are described in this volume (Anovitz and Cole 2015, this volume; Noiriel 2015, this volume), suggests that we now have an unprecedented new ability to consider geochemical processes at the pore scale. These new capabilities offer the potential for a paradigm shift in the Earth Sciences that will allow us to understand and ultimately quantify such enigmas as the apparent discrepancy between laboratory and field rates (White and Brantley 2003) and the impact of geochemical reactions on the transport properties of subsurface materials (Steefel and Lasaga 1990, 1994; Steefel and Lichtner 1994; Xie et al. 2015). It has only gradually become apparent that many geochemical investigations of Earth materials have suffered (perhaps inadvertently) from the assumption of bulk or continuum behavior, leading to volume averaging of properties and processes that really need to be considered at the individual grain or pore scale. For example, a relationship between reactioninduced porosity and permeability change can perhaps be developed based on bulk samples, but ultimately a mechanistic understanding and robust predictive capability of the associated geochemical and physical processes will require a pore-scale view. The question still arises: Do we need pore-scale characterization and models in geochemistry and mineralogy? The laboratory–field rate discrepancy (or enigma) is a good example of where a pore-scale understanding may provide insights not easily achievable with bulk characterization and models. If the reasons for this apparent discrepancy between laboratory and field rates cannot be explained, then it appears unlikely that scientifically defensible and quantitative models for a number of important Earth Science applications ranging from chemical weathering and its effects on atmospheric CO2, to subsurface carbon sequestration, to nuclear waste storage, to contaminant remediation and transport, can be fully developed and applied. The reasons for the discrepancy (apparent or real) have been widely discussed, and over time the number of possibilities for explaining it have narrowed. One potentially important effect that contributes to this apparent laboratory–field rate discrepancy is geochemical in origin and has to do with the fact that most laboratory studies do not consider mineral dissolution as regulated by the precipitation of a secondary phase, that is, as an incongruent reaction. As proposed by Zhu and co-workers and as investigated further by Maher and co-workers, the slow precipitation of secondary clay minerals as a result of primary silicate 1529-6466/15/0080-0007$05.00

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mineral dissolution (e.g., feldspar) can result in an approach to thermodynamic equilibrium with respect to the dissolving primary phase, and thus a slowing of the rate of reaction (Zhu et al. 2004; Maher et al. 2006, 2009). A second potentially important effect is related to the heterogeneous nature of natural porous media, which can result in bypassing of reactive zones by groundwater and surface water flow (Malmström et al. 2000). At the pore scale, this effect can occur when pores and/or micro-environments are not connected to the macropores within which most of the flow occurs. The result can be that minerals lining these pores contribute little or nothing to the overall reactivity of the formation (Peters 2009; Landrot et al. 2012). In addition, diffusion boundary layers can form around reactive grains, further reducing the rate of reaction relative to the experimental surface reaction rates determined in the absence of these transport effects (Li et al. 2008; Noiriel et al. 2012; Molins et al. 2014). One can argue that the ability to demonstrate a predictive capability for geochemical processes, including those operating at the pore scale, is the ultimate test of understanding (Steefel et al. 2005). The development of new characterization techniques for the pore scale has important implications for the models that can be applied to these systems. One possibility is to import the microscopic characterization or mapping as initial or final conditions into true porescale models (Molins et al. 2012, 2014, 2015, this volume; Steefel et al. 2013; Yoon et al. 2015, this volume). Another option that typically allows for larger spatial domains to be considered is the use of the characterization data in pore network models (Mehmani and Balhoff 2015, this volume). A third possible approach that is summarized in this chapter is to make use of “microcontinuum” models informed by high-resolution geochemical, mineralogical, and physical data to describe geochemical pore-scale processes. Micro-continuum geochemical models are typically coarser than either the true pore-scale or pore-network models and thus cannot resolve pore-scale interfaces between mineral, liquid, and gas. The approach suffers from most of the same limitations that apply to larger scale continuum descriptions of porous media, namely the inability to resolve pore-scale solid–liquid–gas interfaces and the requirement that many parameters and properties (e.g., permeability or reactive surface area) need to be averaged or upscaled in some fashion. However, the approach is capable of improving on coarsely resolved (meter-scale) models by assigning differing mineralogical/geochemical and physical properties (porosity, permeability, and diffusivity) values to the domain, thus making it possible to calculate larger scale (bulk) reaction rates and transport properties.

MAPPING OF MODEL PARAMETERS FROM IMAGE ANALYSIS An important first step in developing micro-continuum pore-scale geochemical models is the collection and interpretation of data on the mineralogical, geochemical, and transport properties at a fine (< mm) scale. Detailed reviews are provided elsewhere on the range of characterization techniques available to describe pore-scale geochemical processes (Anovitz and Cole 2015, this volume; Navarre-Sitchler et al. 2015, this volume; Noiriel 2015, this volume). Here we focus on the approaches that are specifically suited for the development of pore-scale parameter distributions for micro-continuum modeling, although many of the techniques could also be used for direct pore-scale or pore-network modeling where the resolution is sufficiently high. The pore-scale parameters of interest for micro-continuum modeling include the porosity, mineral volume fractions, and mineral reactive surface area, along with the more challenging transport-related parameters of permeability and diffusivity. While porosity is typically considered as a scalar quantity and therefore relatively easy to quantify with a variety of mapping/characterization techniques, the more important quantity for the purposes of reactive transport modeling is the connected porosity (Navarre-Sitchler et al. 2009; Peters 2009; Landrot et al. 2012). As demonstrated with modeling of basalt weathering over hundreds of thousands of years, the connected porosity is the parameter that controls reactivity under open

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system conditions (Navarre-Sitchler et al. 2011). Or even where the pores are connected, the reactive phases within the pore are coated with other (typically secondary) phases. The fluids in the connected pores then might “see” only the secondary phases (e.g., clay or iron hydroxide) rather than the reactive feldspar. In this respect, both the impacts of the porosity and the reactive mineral surface area cannot be completely separated from the transport properties of the medium under consideration. Similarly, the physical surface area of reactive minerals is another parameter that may be relatively straightforward to determine on bulk samples, whether using a grain size geometric analysis or using gas adsorption isotherms, e.g., BET methods (Brunauer et al. 1938). But the physical surface area of the minerals may not translate to a unique reactivity without consideration of the reactive site density, which can vary significantly between natural samples (Beig and Lüttge 2006; Bracco et al. 2013). Two-dimensional and three-dimensional images can be analyzed to extract reactive transport model parameters. This includes the sample specific parameters of porosity, mineral volume fractions and surface areas, diffusivity and even permeability. Details regarding the collection and underlying principles of various imaging methods, including 2-D Scanning Electron Microscopy (SEM), and 3-D Focused Ion Beam-Scanning Electron Microscopy (FIB-SEM), X-ray Computed Micro-Tomography (X-ray microCT), optical petrology and Small Angle Neutron Scattering (SANS) of nanoscale porosity have been discussed in other articles (Anovitz and Cole 2015, this volume; Noiriel 2015, this volume) and thus will only be mentioned briefly here. In this article, we will discuss the data processing approaches that can be used to extract model parameters from two- and three-dimensional SEM, FIB-SEM, and X-ray microCT images. In addition, some of the outstanding issues, such as image segmentation and resolution, will be discussed in the context of their effect on parameter estimation. Image segmentation refers to the partitioning of the histogram of pixel intensities (SEM imaging) or voxel attenuation values (X-ray CT imaging) into one or more categories such as pores and grains. Comparison of parameters from bulk samples (e.g., reactivity) and from image-derived micro-continuum samples, however, should provide insight into the scaling relationships in reactive porous media. Finally, it should be noted that several of these parameters can be measured on bulk samples in the laboratory. Parameter estimation from images allows for more discrete parameter evaluation, including the ability to map parameters at multiple scales or associated model grid cell sizes.

Porosity Sample porosity can be easily determined from imaging techniques. Given that porosity is an intensive property, it can be computed from either 2-D or 3-D images. Perhaps the simplest approach is determining porosity from 2-D SEM images of a polished section. Polished sections, including thin sections, can be easily prepared by impregnating samples with epoxy, curing, and then cutting and polishing the samples to the desired thickness. Numerous commercial companies also offer inexpensive sectioning services, removing the limitations of experience or facilities. Further guidance and extensive details on sample preparation for SEM imaging can be found in several existing texts (Echlin 2011). Scanning electron microscopes, now widely available, use an electron beam to capture electron-sample interactions. Most SEM instruments are also equipped with a backscattered electron (BSE) detector. In this mode, the degree of backscatter is proportional to the mean atomic number, producing an image with varying grayscale intensities (Krinsley et al. 2005). Before imaging, polished samples are typically coated with a thin layer of conductive material such as carbon or gold using an ion beam sputterer to prevent surface charging when imaging. This is unnecessary if the instrument is operated in environmental mode or under low vacuum. Pores are easily distinguishable in SEM BSE images of most geologic samples in which the pore space shows a significant contrast with the minerals.

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Benchtop or synchrotron CT images can provide a three-dimensional depiction of a coherent geologic sample. In this method, a series of radiographs or projections are collected over a range of angles (Cnudde and Boone 2013; Wildenschild and Sheppard 2013) and a 3-D image. Several variations in image acquisition as well as complexities in image reconstruction exist and are covered in more detail in existing reviews (Wildenschild and Sheppard 2013; Noiriel 2015, this volume). In X-ray CT imaging, voxel attenuation is proportional to the energy of incident X-rays, material density, and atomic number (Cnudde and Boone 2013; Wildenschild and Sheppard 2013). The contrast in attenuation allows pores to be distinguished from grains, although varying degrees of phase retrieval may be required to make the approach fully quantitative (Wildenschild and Sheppard 2013). Micron-scale resolution is possible with both SEM and microCT imaging, but this resolution is not sufficient to capture the abundant sub-micron-scale pores in carbonates and shales. FIB-SEM imaging, however, can be used to characterize nanoscale porosity (Curtis et al. 2012; Landrot et al. 2012). Using FIB-SEM, high-resolution (nm) 3-D images are developed from a series of 2-D SEM images reconstructed to a 3-D volume. Before imaging, a trench is milled in front of the area of interest using the ion beam. A SEM image is then captured before the ion beam is used to mill away a small layer from the sample. This sequence of image capturing and milling is cycled, generating a series of 2-D images. Given the close spacing of the slices, these 2-D images can be reconstructed into a 3-D volume (Curtis et al. 2012; Landrot et al. 2012). Small angle neutron scattering or SANS can also be used to investigate nanoscale pore distributions and processes, but it is a statistical rather than mapping technique and is not discussed further here, although interested readers can find discussion in other articles (Anovitz and Cole 2015, this volume; Navarre-Sitchler et al. 2015, this volume). The porosity in SEM, X-ray CT, and FIB-SEM images can be determined by computing the ratio of pore pixels or voxels to total pixels or voxels in the 2-D or 3-D image. This first requires segmentation of pore and grain pixels/voxels. There are a variety of existing segmentation techniques that have been used with varying success in the literature. In general, the pore–grain threshold occurs at a minimum in the histogram of grayscale intensities between the individual intensity distributions for pores and grains (Peters 2009). The choice of thresholding technique should be carefully made so as to be optimal for the sample of interest, as further discussed below. In addition, some samples will require extensive pre-segmentation filtering and even manual correction to remove image artifacts. Once the segmented image is produced, pore and grain pixels can be easily summed using commercially available image processing software or using individually developed computer programs. While porosities can be determined from either 2-D or 3-D images, a sufficient number of images are required in order to ensure that the volume used is representative of the sample, thus obtaining reliable porosity, mineral volume fractions, and mineral surface areas. The representativeness of the area or volume can be determined by computing the porosity on smaller volumes subsampled from the original image. As the sampled volume is increased, the computed porosity should approach a uniform value as a representative elementary volume (REV) is reached.

Mineral volumes Despite recent efforts based on 3-D imaging (Mutina and Koroteev 2012), mineral volume fractions in mineralogically complex systems can be reliably determined only from 2-D SEM imaging. Where only a single mineral is present, as in the study by Noiriel et al. (2012), or where there is a significant contrast in density between the minerals present, 3-D X-ray synchrotron mapping may be able to provide quantitative determinations of mineral volumes. Mineral volume determination at the microscopic scale is possible on SEMs equipped with Energy-Dispersive X-ray Spectroscopy (EDS or EDX) capabilities. The 2-D BSE imaging

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approach described above makes it possible to distinguish between quartz, clay, and other reactive minerals (Peters 2009) with a few hundreds of nm per pixel length resolution. EDX imaging further allows for mineral identification by determining the elements present at the microscopic scale. Two-D elemental maps can be captured and processed to determine the mineral volume percentages within each pixel (Landrot et al. 2012). Processing of EDX signals can be carried out with commercial software, such as QEMSCAN, or though customized thresholding and processing codes such as those presented in Landrot et al. (2012). These methods couple information on BSE intensities with elemental intensities so as to identify individual minerals. In either method, knowledge of the bulk mineralogy is needed to aid in mineral characterization, as many minerals contain similar elemental compositions and BSE intensities. This can be obtained from X-ray Diffraction (XRD) or X-ray Fluorescence (XRF) on bulk samples. Identification of minerals can be challenging in highly heterogeneous samples even when commercial software like QEMSCAN is used. In addition, grain edge artifacts at grain–epoxy boundaries can alter BSE intensities (Dilks and Graham 1985), potentially causing misidentification of minerals at grain boundaries. Following mineral segmentation, computation of mineral volume fractions can be carried out by counting the number of pixels corresponding to each mineral and dividing by the total grain pixels. An additional advantage of determining mineral volumes from imaging is the ability to assess explicitly the accessibility of minerals to reactive fluids if the porosity is imaged as well, and if the imaging is at sufficiently high resolution that the connectivity of the pore network can be quantified. As observed in recent studies (Peters 2009; Landrot et al. 2012), mineral abundance alone may not always accurately reflect mineral accessibility and pore connectivity. At least in continuum models, it is typically more accurate to discretize minerals based on pore accessibility fractions rather than based on total volume fractions. Mineral accessibilities can be computed by first identifying the grain-pore boundary and then by counting the number of associated pixels for each mineral that are present at the interface, or adjacent to the pore space. Recent studies have also considered the connectivity of the pore space as well, since this is a primary control on the transport of ions to and from reactive surfaces (Navarre-Sitchler et al. 2009; Landrot et al. 2012).

Mineral surface area There are several complexities and intricacies involved in obtaining and interpreting mineral surface area, a full review of which is beyond the scope of this chapter. Instead, we will briefly give a few examples of ways to interpret surface area from 2-D and 3-D images. It should be noted that it is uncertain what the appropriate mineral surface areas are for reactive transport modeling of porous media. A range of surface area estimates have been used without evaluation of their impact or success. This includes liberally interchanging geometric surface area (GSA), specific surface area (SSA), and reactive surface area (RSA). Mineral GSA typically refers to a surface area computed from an average grain size and assuming a particular geometric grain shape, typically perfectly smooth spheres (White et al. 2005; Alemu et al. 2011). Mineral SSA refers to the total or rough surface area per gram mineral, often measured via the Brunauer, Emmett, Teller (BET) analysis (Brunauer et al. 1938). RSA accounts (or attempts to account) for the distribution of reactive sites on a mineral surface and is usually estimated by applying a scaling factor of one to three orders of magnitude to SSA or GSA (Zerai et al. 2006), although the basis for applying this factor is not clear. Geometric, specific, and reactive mineral surface areas can be approximated from 2-D and 3-D images using a variety of approaches. Geometric surface areas can be approximated from the grain sizes observed in 2-D or 3-D images assuming typical grain geometry and either an average grain size or range of grain sizes and corresponding surface areas. An alternative approach is to assume mineral-specific geometries and use image-observed grain dimensions to compute surface areas based on the ideal geometries for each mineral (Bolourinejad et al.

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2014). This approach is somewhat limited as real minerals often deviate from ideal geometries or do not have perfectly smooth surfaces (Bolourinejad et al. 2014). It should also be noted that these 2-D approaches additionally require bias correction (Weibel 1979; Crandell et al. 2012). Specific surface areas that reflect surface roughness can be estimated from GSA determinations by applying a roughness factor, although the choice of a factor is difficult to justify and only an average at best (Zerai et al. 2006). Alternatively, the BET surface area can be measured in the laboratory and distributed among minerals as identified in 2-D images. In 3-D images, specific surface areas can be successfully computed by creating a triangulated or polygonized surface mesh, such as through the marching cubes algorithm (Lorensen and Cline 1987; Landrot et al. 2012), then summing the area of the polygons. This is necessary as it has been shown that simply counting surface voxel faces in 3-D images overestimates surface area, as is explained in the review by Wildenschild and Sheppard (2013). This approach can be used to compute surface areas from both FIB-SEM images and X-ray CT images.

Diffusivity Diffusivity from bench-top experiments. Diffusivity within homogeneous porous samples may be determined experimentally with a range of techniques, some of which do not require pore-scale imaging. The most common experimental techniques are based on a diffusion cell in which one end of the cell is held at a constant concentration (effectively a Dirichlet or fixed boundary condition), while a reservoir on the other side of the diffusion cell is monitored for solute breakthrough, either under transient (e.g., where the reservoir is stagnant) or steady-state conditions (where the reservoir is subject to flow). For example, Figure 1 shows a schematic for an experimental setup that has been used to determine ion diffusivity in bentonite clay (Tachi and Yotsuji 2014). Another possible experimental approach for porous medium samples relies on chemical mapping of the diffusion profile. For example, Navarre-Sitchler et al. (2009) used micro-X-ray synchrotron mapping of bromide fluorescence (XRF) to determine the diffusivity of samples of unweathered and weathered basaltic andesite. An effective, upscaled diffusion coefficient

Figure 1. A) Example of a through diffusion cell setup: (a) inlet reservoir, (b) peristaltic pump, (c) throughdiffusion cell, and (d) outlet reservoir. Arrow heads indicate the circulation of water from the reservoir to the filter in order to homogenize the inlet and outlet solutions compositions. B) Plot of concentrations in inlet and outlet reservoirs versus time. [Reproduced from Tachi Y, Yotsuji K (2014) Diffusion and sorption of Cs+, Na+, I− and HTO in compacted sodium montmorillonite as a function of porewater salinity: Integrated sorption and diffusion model. Geochimica et Cosmochimica Acta, Vol. 132, p. 75-93, Figs. 1 and 3, with permission from Elsevier.]

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was determined for the porous andesite by fitting the profile with a numerical solution of the diffusion equation

Ci    Di*Ci  t

(1)

where Di* is the diffusion coefficient for species i in porous media that incorporates the effects of tortuosity, Ci is the tracer concentration,  is the divergence operator,  is the porosity, and t is time (Steefel et al. 2015). For example, a XRF map of a sample of weathered basalt from a Costa Rican chronosequence shows the distribution of bromide tracer (blue) after 7 days resulting from diffusion from left to right, as shown in Figure 2 (Navarre-Sitchler et al. 2009). Navarre-Sitchler et al. (2009) took the additional step of comparing these results to estimates of the diffusivity from pore-network modeling (see discussion below).

Figure 2. Bromide tracer (blue) diffusion profile after 7 days based on XRF mapping at the Advanced Light Source, Lawrence Berkeley National Laboratory. Diffusion is from left to right. See Navarre-Sitchler et al. (2009) for discussion of simulations.

Diffusivity from numerical experiments. Experiments and/or diffusion profile mapping work well for determining diffusivity where the properties of the porous material are largely homogeneous, since in this case a single parameter value can be fitted to the data. In some cases, it may also be possible to estimate diffusivities for heterogeneous samples if the distribution of properties (e.g., grain size) is known, although the likelihood of a non-unique fit increases with the number of different properties in a sample. Alternatively, if the distribution of properties is unknown, it is possible to estimate diffusivity even in highly heterogeneous samples using numerical modeling based on 2-D or 3-D pore structure characterization. Navarre-Sitchler et al. (2009) made use of X-ray synchrotron microtomography to map the pore structure of samples of weathered basalt similar to that shown in Figure 2. Using a simple implementation of thresholding to map basalt versus pores, they were able to delineate chemical weathering related macroporous zones (> 4.4 m voxel resolution) that were connected in 3-D. NavarreSitchler et al. (2009) then carried out numerical tracer diffusion experiments in 3-D cubes of weathered basalt by assuming a low diffusivity of 1.75 × 10-14 m2 s-1 for the largely unconnected pore structure of unaltered basalt and a free ion diffusivity of 10-9 m2 s-1 (corresponding to a tortuosity of 1.0) for connected pores that can be fully resolved with the 4.4 m discretization. Implemented in this way, the numerical tracer diffusion simulations with the code CrunchFlow are similar to what could be done with a pore network model. A 2-D slice through the skeletonized pore structure of one of the weathered zones is shown in Figure 3A (basalt in blue, pores in red). Results of the numerical tracer experiment are shown in Figure 3B, with diffusion of the tracer from the bottom of the Figure 3B towards the top. Note that in these 3-D tracer diffusion simulations, only two distinct tortuosities (or diffusion coefficients) are

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Figure 3. A) Segmented X-ray synchrotron microtomographic data collected at the Advanced Light Source at Lawrence Berkeley National Laboratory with a voxel resolution of 4.4 m. Macropores developed as a result of chemical weathering in the basalt and are connected primarily in the third dimension (into the page), with red indicating pores, blue indicating basalt. B) Tracer diffusion simulation results using the pore structure shown in Figure 3A. Results are shown after 7 days of diffusion of the tracer from bottom to top. Simulation assumed a Dirichlet boundary condition at the bottom with a fixed tracer concentration of 0.01. See Navarre-Sitchler et al. (2009) for description of experiments.

used and they are based on the segmented 3-D porosity map of the weathered sample. The upscaled effective diffusion coefficient, in contrast, is determined by fitting a 1-D profile to the 3-D simulation results. The diffusion of the bromide tracer was assumed to follow a modified Archie’s Law model that incorporates a critical porosity threshold value of 9%, below which the porosity is considered to be largely unconnected: De  Dp  D0  e  , 2

(2)

where Dp is the diffusivity in the unweathered parent basalt, D0 is the diffusion coefficient in pure water, and De is the effective diffusion coefficient in porous media and where the effective porosity, e, is defined as: e  a   T  c  ;  T  c 

e  0;

 T  c .

(3)

The parameter a in Equation (3) is taken as 1.3 while the parameter  is assumed to be 1.0 for 3-D volumes measuring 220 m on a side (Navarre-Sitchler et al. 2009), while T refers to the total porosity as mapped with the X-ray synchrotron microtomography with a 4.4 m voxel resolution, and c is the critical porosity estimated as 9% based on the same data. A summary of the 3-D tracer diffusion simulation results for the Costa Rican basalts are given in Table 1. Samples with less than 10% porosity are essentially unweathered. Note that the results are to some extent dependent on sample size, and also on the porosity of the volume considered as expected from Equation (2). The results of the tracer diffusion simulations agree broadly with the laboratory tracer diffusion experiments using bromide (Fig. 4). Using numerical modeling based on pore-scale imaging for the purposes of estimating diffusivity offers practical advantages when the material of interest within a sample is too small to easily investigate with laboratory tracer experiments. As an example, consider the chloritefilled pores within the Lower Tuscaloosa Formation that hosts the Cranfield CO2 sequestration site investigated by Landrot et al. 2012. One such pore of approximately 2 m diameter imaged with scanning electron microscopy (SEM) is shown in Figure 5A. The Figure shows a quartz grain on the right side partly milled with focused ion beam techniques, while Figure 5B shows

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Table 1. Results of fitting of tracer distribution from 3-D simulations with a 1-D diffusion model. Sample size (mm3)

De (m2 s-1)

% Porosity

1.38

1.2 × 10-14

3

1.38

-12

14

-15

2

-12

10

0.17

-12

6.1 × 10

14

0.17

1.9 × 10-11

21

0.17

4.8 × 10-11

30

0.17

-11

33

0.17 0.17

1.0 × 10

3.7 × 10 1.7 × 10

7.1 × 10

Figure 4. Comparison of results from laboratory and numerical (simulated) diffusion experiments. A) The XRF image of Br concentrations measured in basalt samples after 7 days (high concentration in white, low concentration in darker shades). B) Contour plot of simulated Br (tracer) distribution after 7 days in the same sample based on pore structure determined with X-ray synchrotron microtomography. C) 1-D effective diffusion coefficient fit to the 3-D data. [Reproduced from Navarre-Sitchler A, Steefel CI, Yang L, Tomutsa L, Brantley SL (2009) The evolution of dissolution patterns: Permeability change due to coupled flow and reaction. In: Chemical Modeling of Aqueous Systems II. Vol 416. Melchior D, Bassett RL (eds)American Chemical Society, Washington, p 212–225, Fig. 9 with permission from the American Geophysical Union.]

Figure 5. A) FIB-SEM excavation of a single quartz grain (right side of image) and nano-crystalline chlorite filling a pore (left side of image). B) 3-D reconstruction of chlorite and associate nanopore structure based on FIB-SEM results. [Reproduced from Landrot G, Ajo-Franklin J, Yang L, Cabrini S, Steefel CI (2012) Measurement of accessible reactive surface area in a sandstone, with application to CO2 mineralization. Chemical Geology, Vol. 318–319, p. 113–125, Fig. 7, with permission from Elsevier.]

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the nano-crystalline chlorite as reconstructed in 3-D with the techniques described above. Carrying out a laboratory tracer experiment on such a small zone would be quite difficult, but determining an approximate upscaled diffusion coefficient is relatively simple using a true porescale model (Molins 2015, this volume), or the approach as described in Figures 3 and 4. As in the Costa Rican basalts described above, the Tuscaloosa Formation sandstone is divided into pores and chlorite mineral and these are represented directly in the pore-scale model using the same grid resolution as the image resolution (14 nm). A non-reactive tracer with a concentration of 0.01 M is released from the left hand side of the cube (Dirichlet boundary) and allowed to diffuse to the right. All other boundaries are treated as no-flux. Based on the average diffusion profile, one can estimate an upscaled, effective diffusion coefficient of 7 × 10-12 m2/s (Fig. 6). The upscaled diffusion coefficient can then be used to describe diffusivity in the chlorite-rich zones within micro-continuum representations, as described more fully below.

Figure 6. A) Simulated 2-D slice of tracer concentration from a 3-D cube of chlorite and pores from the Lower Tuscaloosa Formation at the Cranfield CO2 sequestration site, which was characterized with FIBSEM (Landrot et al. 2012). The code CrunchFlow (Steefel et al. 2015) is used to carry out a 3-D tracer diffusion simulation, with release of the tracer at the left boundary at a concentration of 0.01 M. B) Fit of 1-D diffusion profile assuming a single homogeneous diffusivity of 7 × 10-12 m2/s (line) versus concentration from 3-D simulation.

Permeability Imaging methods have been increasingly used as the basis for the prediction of permeability (Caubit et al. 2009; Algive et al. 2012; Beckingham et al. 2013). In addition to empirical relationships such as the Kozeny–Carmen equations that compute permeability from experimental or image-computed porosity (Kozeny et al. 1927; Carman 1939), permeability can also be estimated from pore networks extracted from 2-D and 3-D images. Pore and porethroat size distributions as well as connectivities are needed to recreate a representative pore network. A suite of different approaches have been used to characterize pore and pore-throat size distributions from 2-D and 3-D images. This includes, for example, multiple point statistics (Okabe and Blunt 2004), image erosion-dilation (Crandell et al. 2012), maximum inscribed spheres (Baldwin et al. 1996), and watershed segmentation (Beucher and Lantuéjoul 1979; Silin and Patzek 2003). It should be noted that some of these 2-D methods require bias correction (Crandell et al. 2012) and may not be able to determine pore connectivities without relying on information from 3-D images (Beckingham et al. 2013), or determined by some other means. From these statistical distributions, simple pore network models can be created and used to compute continuum-scale permeability. In these models, a series of pores are defined on a regular cubic lattice and connected by pore throats. These statistical distributions of pore sizes also provide information on pore connectivity for the models. Fixed fluid pressures are then applied at the inlet and outlet (Beckingham et al. 2013). Using Poiseuille’s law to describe

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pore throat conductance, and assuming an incompressible fluid, the pressure in each node can be determined (Li et al. 2006). At the continuum scale, Darcy’s law can then be applied to solve for the permeability (Li et al. 2006). While this method has been shown to successfully predict permeability (Beckingham et al. 2013), image resolution and segmentation effects can affect predictions, as discussed further below. Estimating permeability using pore-scale numerical simulation. As an alternative to these statistical methods, pore network structures have also been directly extracted and simulated from 3-D images. Direct 3-D network extraction from 3-D images has been carried out using a variety of techniques, such as skeletonization based on the medial axis transform as in seminal work by Lindquist and co-workers (Lindquist et al. 1996). Direct modeling using Lattice Boltzmann based on the image-derived pore structure is often used to estimate flow and network properties (Blunt et al. 2013). The 3-D imaging could be X-ray synchrotron microtomography, or for potentially higher resolution, FIB-SEM techniques. For example, Oostrom and coworkers (Oostrom et al. 2014) determined permeability for micro-models based on the pressure drop across the model for a range of imposed flow rates using Darcy’s Law. While the micromodels investigated by Oostrom et al. (2014) were based on idealized geometries, the approach can be generalized to more complex natural pore structures, for example a capillary tube filled with crushed calcite grains (Molins et al. 2014), or in 3-D to fractured shales imaged with high resolution FIB-SEM techniques (Trebotich and Graves 2015). In Molins et al. (2014) and in Trebotich and Graves (2015), the pore structure is fully resolved and the Navier–Stokes equation is solved for the domain, in the case of the fractured shale (Fig. 7) at a resolution of 48 nm. The approach in order to determine an upscaled permeability in the case of multidimensional heterogeneous samples requires an averaging technique for the pressure, since there is no single pressure value at the downstream side of the volume.

Figure 7. Steady-state flow through fractured shale based on Navier–Stokes equation. The surface plot shows the interface between pores and solid (shale). The surface is colored according to velocity magnitude in the pore, with blue representing lower velocities, shades of green, yellow, and red representing higher velocities. Resolution 48.4 nm with 1920 × 1600 × 640 total grid cells. Original pore structure obtained from FIB-SEM mapping, which indicates a bulk porosity of 18%. [Reproduced from Trebotich D, Graves DT (2015) An adaptive finite volume method for the incompressible Navier–Stokes equations in complex geometries Communications in Applied Mathematics and Computational Science, Vol. 10(1), p. 43–82, Fig. 13, with permission from Mathematical Sciences Publishers.]

Imaging issues impacting parameter estimation Image segmentation. To date, a variety of segmentation and interpretation methods have been applied to 2-D and 3-D images with varying success (Sezgin and Sankur 2004; Dong and Blunt 2009; Iassonov et al. 2009; Peters 2009; Porter and Wildenschild 2010; Bhattad et al. 2011; Wildenschild and Sheppard 2013). Despite the range of automated thresholding methods, there remains a lack of consistency in results even on images of the same sample (Wildenschild and Sheppard 2013). Typically, operator input remains necessary (Cnudde and Boone 2013). Several image artifacts and issues continue to make image segmentation challenging. One complexity that is particularly common in 3-D images is partial volume or edge effects

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that occur, for example, when a single voxel contains multiple substances or straddles a pore-mineral boundary. In these cases, the corresponding composite voxel is assigned an intermediate X-ray attenuation value that reflects some combinations of the neighboring voxel compositions (Ketcham and Carlson 2001). Additional difficulties result from image artifacts, such as noise, surface charging, and ring artifacts. Also, variations in intensities across images are common and occur in SEM images when surfaces are not completely flat, or as a result of beam hardening in X-ray CT imaging (Wildenschild and Sheppard 2013). Filtering of some of these artifacts can be achieved through relatively simple approaches involving noise removal by pixel flipping of isolated misidentified phases Peters (2009). Others artifacts are not easily correctable and require additional more complex filtering processes (Blunt et al. 2013; Cnudde and Boone 2013; Wildenschild and Sheppard 2013) or even manual image correction (Crandell et al. 2012) before or after thresholding to correctly segment images. Given the range and extent of segmentation methods and pre-processing procedures, a detailed review is not included here, but the interested reader is referred to other reviews on the subject (Cnudde and Boone 2013; Wildenschild and Sheppard 2013). Imprecision and errors in image segmentation can have several impacts on parameter estimation. Small-scale features may be misinterpreted as noise and thus erroneously removed pre- or post-thresholding, altering the total porosity, mineral volume fractions, and mineral surface areas. This may also impact the estimation of permeability if these features are removed in pore throats and result in erroneously increasing pore throat size(s). A recent review in Iassonov et al. (2009) evaluated porosities determined from a range of segmentation methods applied to 3-D X-ray CT images of porous media. They found large discrepancies of one to over two orders of magnitude in porosity resulting from different segmentation methods, even on simple samples such as glass beads (Iassonov et al. 2009). Given that this initial segmentation step is needed to define the grain-pore boundaries, segmentation error directly impacts analysis of pore and pore-throat sizes, and thus the permeability that is predicted. Similar pore–grain segmentation methods have been used to process BSE and EDS images to identify minerals (Peters 2009; Landrot et al. 2012). It is thus likely that similar potentially significant errors may result in the case of mineral volumes and surface areas as well. Care should be taken to reduce segmentation error whenever possible. This can be achieved by evaluating the appropriate segmentation method for each sample. In some cases, different images from the same sample may require different segmentation procedures (Wildenschild and Sheppard 2013). Additionally, image determined parameters should be compared with measured parameters whenever possible. Successful segmentation should produce porosity estimates that agree with laboratory-measured values, taking into account the fraction of porosity that is accounted for in each method. Similarly, image determined mineral volume fractions should agree with laboratory-measured mineralogy from XRD or XRF, keeping in mind that the sensitivity of the instruments may not be sufficient to characterize the minor phases that can be captured in high-resolution imaging.

Image resolution High resolution images typically require small sample sizes. There are several methods that have been used with some success to increase the sample size of high-resolution images. In one approach that is often used with 2-D SEM images, a series of overlapping highresolution images are captured and stitched together into a larger, high-resolution, composite image (Crandell et al. 2012). Similar mosaic techniques have been used with 3-D images as well (Mokso et al. 2012). In addition, alternative reconstruction and transform approaches have been successful in some cases in increasing the sample size of high resolution images (Defrise et al. 2006; Cnudde and Boone 2013). The use of multi-scale imaging that relies on a range of imaging techniques or resolutions has recently been gaining interest as well. These include, for example, combining X-ray CT images at different resolutions or fusing X-ray CT

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images with FIB-SEM and SEM BSE images (Sok et al. 2010; Landrot et al. 2012). These studies used high-resolution FIB-SEM imaging to evaluate the porosity and surface area of clay minerals that could not be characterized with the lower resolution pore-scale imaging approaches (Sok et al. 2010; Landrot et al. 2012). Regardless of the approach, features smaller than the voxel or pixel size cannot be distinguished with the approaches discussed here. These features, however, may be present via the partial volume effect (Cnudde and Boone 2013). In CT imaging for example, this occurs where a single voxel contains more than one phase, with the result that an intermediate attenuation value is assigned to the voxel. This results in image imprecision, particularly at interfaces between materials (Cnudde and Boone 2013), and can introduce potentially large errors when small features and pores are characterized. Imprecision in small-scale features may have little impact on overall porosity and mineral volume fractions, but large impacts on other image-estimated parameters. For example, there could be a large effect on the determination of mineral surface areas given that small-scale features often account for a large portion of the surface area. This is in addition to the impact that lower image resolutions have on correctly capturing surface roughness and thus surface area. Misclassifying smallscale features in either 2-D or 3-D images also introduces error in permeability and diffusivity predictions when using image-informed network models (Caubit et al. 2009; Beckingham et al. 2013). Accurate permeability predictions may still be possible from models informed by lower resolution images in the case where smaller pores (below the resolution of the technique) do not have a large impact on permeability and flow (e.g., Blunt et al. 2013).

MICRO-CONTINUUM MODELING APPROACHES If the imaging methods are sufficiently high resolution that individual grains and pores and most importantly their interfaces can be resolved, then the 2-D and 3-D maps can be used directly in high resolution pore-scale modeling (e.g., Molins et al. 2014). However, this requires access to pore-scale reactive transport modeling software (still a specialized field) and to high performance computer hardware. Thus it may not be practical for all researchers approach, even if the full pore-scale approach is arguably the most rigorous. If the domain size is too large and/or spatial resolution too low, or if a true pore-scale code is not available, then micro-continuum modeling is another possibility. The data can be used at the spatial resolution at which is collected, or volume averaged to a coarser discretization so as to handle a larger domain, or simply to make the simulations computationally feasible. Scalar quantities like the porosity and mineral volume fractions can be volume averaged directly, vectorial quantities like permeability or diffusivity are more likely scale-dependent and require special treatment (see discussion below).

Volume averaging of porosity and mineral volume fractions Given an initial map of porosity and mineral volume fractions at some resolution, it is a relatively simple procedure to calculate equivalent quantities at coarser resolutions by using volume averaging. However, these scalar quantities are influenced by the connectivity and thus the transport properties of the medium. If the connectivity and the accessibility of reactive surface area are scale-dependent, then volume averaging as an upscaling procedure may introduce errors into the simulations. How accurate the volume-averaged quantities are at various scales is a future research area, one that has been neglected to date perhaps because researchers interested in upscaling in porous media have focused primarily on the physical properties, or because the routine use of high-resolution chemical and mineralogical mapping is still in its infancy.

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If pores and individual mineral grains are fully resolved (i.e., no voxel includes more than a single mineral, or a mixture of pore space and minerals), then producing coarser representations of the medium consists of adding up the various image voxels corresponding to porosity and the individual minerals. The percentage of porosity per unit volume porous medium, for example, is just the number of voxels made up completely of pores divided by the total number of voxels in the volume of interest. The volume procedure, however, is equally straightforward where individual voxels consist of a mix of either different minerals or minerals and porosity, since the porosity (connected or unconnected) will be just the weighted average of the porosity in the individual voxels. An example of volume averaging is given by choosing an image of the porosity and mineral distribution from the Lower Tuscaloosa Formation at the Cranfield CO2 sequestration site with a resolution of 331 nm (Fig. 8), which is adequate to resolve all but the nano-crystalline chlorite-filled pores (Landrot et al. 2012). With sufficient computational resources (software and hardware), it should be possible to simulate the porescale geochemical processes at the original resolution of 4 m, that is, use the information on porosity (accessible and inaccessible) and mineralogy shown in Figure 8 directly. In order to make the reactive transport simulations tractable for our purposes, however, we assume a 256 by 256 2-D section with 16-m grid resolution that produces a section measuring 4.1 mm by 4.1 mm. As an example, volume averaging of porosity and mineral abundance produces data like that shown in Table 2, which represents a small portion of the Lower Tuscaloosa Formation sandstone sample investigated in Landrot et al. (2012). The volume averaging to 16 m2 produces a porosity map (Fig. 9A) and mineral abundances for quartz, chlorite (chamosite), and illite as shown in Figures 9B, 9C, and 9D respectively. These maps are used as initial conditions in the reactive transport modeling of the 4.1 mm by 4.1 mm 2-D section discussed below.

Pixel

Porosity

Quartz

Chlorite

Illite

Ti-Oxide

Kaolinite

Smectite

Fe-oxide

Table 2. Example of volume averaging of porosity and mineral percentages from Lower Tuscaloosa Formation (Cranfield) sandstone. Volume averaging to 16 m is based on an original image resolution of 4 m (Landrot et al. 2012). Each pixel in the table corresponds to a 2-D section measuring 16 m2.

24

0

100

0

0

0

0

0

0

25

11

89

0

0

0

0

0

0

26

5

95

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0

0

0

0

0

27

1

97

2

0

0

0

0

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28

14

8

77

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1

29

9

24

60

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6

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15

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31

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32

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7

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72

8

10

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34

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18

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36

25

18

7

47

0

0

3

0

37

48

13

23

11

0

14

5

0

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Figure 8. Connected porosity mapping in a 29 mm2 region of the Lower Tuscaloosa Formation sandstone (Cranfield), considering the nanoporosity within chlorite as 100% connected. The porosity is mapped with a 331 nm/pixel resolution. The white color represents the connected pore network that starts from the edges of the image and propagates inward through the map, and the turquoiseblue color represents the chlorite fraction that is linked to the connected pore network measured in the 29 mm2 region. [Reproduced from Landrot G, Ajo-Franklin J, Yang L, Cabrini S, Steefel CI (2012) Measurement of accessible reactive surface area in a sandstone, with application to CO2 mineralization. Chemical Geology, Vol. 318–319, p. 113–125, Fig. 8, with permission from Elsevier.]

Figure 9. A: Porosity distribution in 2-D section of the Lower Tuscaloosa Formation sandstone (Cranfield sequestration CO2 site) volume averaged from 4 m2 to 16 m2. B: Volume fraction of quartz. C: Volume fraction of chlorite (chamosite). D: Volume fraction of illite.

Micro-continuum reactive transport simulations of fractured tuff. The first microcontinuum reactive transport modeling described in the literature that we are aware of was presented by Glassley and co-workers (Glassley et al. 2002). This mysteriously unrecognized and largely uncited publication was the first to make use of mineralogical and chemical data collected with modern synchrotron techniques at the micro-scale and then used as initial conditions for high resolution reactive transport simulations. This study developed spatially distributed representations of porosity and mineralogy based on a combination of optical mineralogy and -XRF mapping at a resolution of approximately 1 m in 2-D. The rock samples consisted of fractured tuffaceous rock from Yucca Mountain, Nevada. A sample area of 106 m2 was mapped in detail and the resulting element and porosity maps were digitized, thus creating a domain decomposed into 12,208 grid cells that were 8.77 m on

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a side (Fig. 10). A bulk porosity of about 6% was estimated based on averaging of the entire sample. Simulations were conducted in which a dilute fluid enters the discretized porous medium at two different flow rates of 0.1 and 1.0 m3 m-2 yr-1, assumed to be uniform across the domain. The fluid is reacted with the rock at 90 ºC. Simulations involving the slower flow rate, in which the fluid residence times is approximately 3.65 days, provide fluid composition results at the downstream end that are very similar to those obtained from homogeneous mineral distribution representations. At the higher flow rate of 1.0 m3 m-2 yr-1 (residence time of approximately 8.76 hours), however, the fluid composition differs between the heterogeneous and homogeneous cases along the entire length of the flow path. The authors concluded that the simulation results demonstrate that the fluid composition characteristics in the homogeneous and discrete mineral representations will be similar only when the bulk average contact times for the individual mineral phases along the flow paths are approximately equivalent (within a few percent).

Figure 10. Discretized mineralogy of tuff sample generated from digitized element maps. Gray regions represent pure silica (treated as cristobalite in the simulations), and the black regions are K-feldspar. The white regions map fine-grained groundmass, which was treated as inter-grown cristobalite+alkali feldspar+Ca-smectite+hematite. The gray sinuous band running from top to bottom of the figure is a silicafilled fracture. Area is 1000 × 1000 m. [Reproduced from Glassley WE, Simmons AM, Kercher JR (2002) Mineralogical heterogeneity in fractured, porous media and its representation in reactive transport models. Applied Geochemistry, Vol. 17, p. 699–708, Fig. 2, with permission from Elsevier.]

Micro-continuum reactive transport simulations in Lower Tuscaloosa Formation (Cranfield) sandstone. The Cranfield Oil Field in Mississippi has been used as a subsurface CO2 injection pilot site, with super-critical CO2 injected into the lower Tuscaloosa Formation at about 300 m depth. The Tuscaloosa Formation is a 15-m-thick heterogeneous fluvial sandstone that was the subject of an experimental study by Lu and co-workers (Lu et al. 2012), who reported low reactivity for the sandstone in contact with CO2-infused brine. The question arises as to whether the low reactivity is due primarily to the limited availability of reactive surface area? This can be evaluated more quantitatively with the micro-continuum approach. The bulk reactivity can be estimated by carrying out 2-D diffusion-reaction simulations using the volume-averaged porosity and mineral distributions presented in Figure 9, with the left hand boundary set as a fixed or Dirichlet boundary condition. This effectively makes the CO2 reservoir (5 bars, 25 ºC) infinite. The simulations are carried out over the 4 mm by 4 mm domain for a period of 365 days, which

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is a sufficient amount of time for the system to come to a steady state. An effective diffusion coefficient of 10-10 m2 s-1 is assumed for the domain, with the exception of zones containing 50% or more chlorite where the effective diffusion coefficient is assumed to be 7 × 10-12 in agreement with 1-D simulation shown in Figure 6B. No correction is made for the 3-D connectivity in the 2-D simulations, so access to some reactive phases is likely underestimated. How to incorporate 3-D tortuosity into 2-D modeling domains is a future research topic. In the case where the local porosity is zero, the diffusive flux will be zero because of the use of harmonic means to calculate properties at grid cell interfaces. Thus, unconnected porosity is automatically accounted for within the resolution of the discretization. Sub-grid connectivity, however, is not accounted for, although this should be possible with a more rigorous treatment of the data. The spatial distribution of chlorite dissolution rates in the 2-D Tuscaloosa Sandstone section is shown in Figure 11. The sparse distribution of accessible chlorite certainly contributes to the low bulk reactivity of the material, so perhaps the observations Lu et al. (2012) of low reactivity are understandable. Bulk rates calculated from the micro-continuum simulations are given in Table 3. The rates are normalized to a cubic centimeter of Tuscaloosa Sandstone, which is close to the size of the rock sample used in the Lu et al. (2012) experiments. The low reaction rates of the lower Tuscaloosa Formation sample in contact with the CO2-infused brine is thus a consequence of both the low volume fractions of the reactive phases (quartz dominates) and the poor accessibility of some of the phases.

Figure 11. Spatial distribution of chlorite dissolution rates in units of mol L-1 fluid s-1 after 365 days of diffusion–reaction (no flow) simulation. Given these results, the low reactivity of the Tuscaloosa Sandstone samples investigated by Lu et al. (2012) is perhaps understandable.

Multi-continuum approaches There are many examples of soils, sediments, and rocks that are characterized by multiple length scales, each with its own set of physical and/or chemical properties. Probably the best known example is that of fractured rock. Here flow in the fractures is described by meter or larger length scales, while transport in finer-grained, unfractured material within the same volume is dominated by diffusion length scales (mm–cm). Other examples of hierarchical

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Table 3. Bulk rates calculated in 2-D section of Lower Tuscaloosa Formation sandstone using porosity and mineral abundance distributions shown in Figure 9. Intrinsic rates per unit surface area mineral used are standard literature values. Bulk porosity = 0.13.

Mineral Concentration (mol/cm3)

Specific Surface Area (m2/g)

Bulk Rate (mol/cm3/s)

Quartz

3.1 × 10-2

0.024

8.5 × 10-20

Chlorite

7.1 × 10-4

1.0

-1.2 × 10-17

Illite

8.1 × 10-5

1.0

-1.4 × 10-18

Kaolinite

2.3 × 10-4

1.0

-3.4 × 10-18

Smectite

1.6 × 10-4

1.0

-5.3 × 10-18

Fe-hydroxide

2.8 × 10-4

1.0

-2.9 × 10-15

Calcite

0

0.04

0

Magnesite

0

0.04

0

Amorphous Silica

0

0.024

0

Mineral

porous media can be mentioned, as for example where pore or smaller scale parameters and processes affect larger macroscale behavior. The models used to describe these systems are typically referred to as multi-continuum models. In some cases, these scales are separated in space and can be represented as discrete spatial zones within a flow or reactive transport model—an example might be a discrete fracture located in what is otherwise an unfractured, low permeability material. In other cases, however, the domains may overlap within the same representative elementary volume (REV). In these cases, the domains are typically represented as two or more distinct continua with their own set of mass balance equations and physical-chemical properties (Barenblatt et al. 1960; Pruess and Narisimhan 1985). Functions describing exchange between the various continua may be included as well. Multi-continuum models had their origin primarily in fractured rock systems where it has been difficult or impossible to represent all of the fractures discretely with a single representative elementary volume. The general approach was first introduced by Barenblatt and co-workers in 1960 (Barenblatt et al. 1960) and it has since been implemented in various forms, including: (1) the equivalent continuum model, or ECM (Wu 2000), (2) the dual permeability model (DPM), dual or multi-porosity model (Warren and Root 1963), and (3) multiple interacting continua or MINC approach (Pruess and Narisimhan 1985; Aradóttir et al. 2013). Among these three commonly used approaches, the dual-continuum approach has been most extensively applied in different subsurface environments (Arora et al. 2011), perhaps because it is relatively simple compared to the other approaches and because it is capable of describing many natural subsurface materials. The dual-continuum approach considers two interacting regions, one associated with the less permeable soil or rock matrix, and the other characterized by flow and/or diffusion in macropores or discrete fractures. In this approach, the representative elementary volume (REV) is partitioned into sub-volumes of each domain such that: VREV  Vmacro  Vmicro ,

(4)

where VREV refers to the total volume of the porous medium, Vmacro and Vmicro refer to the volume of the macro and micro (or equivalently, fracture and matrix) domains, respectively.

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The fraction, e, of volume occupied by the macropores and micropores, then, can be described respectively as (Lichtner 2000) macro 

Vmacro V , micro  micro . VREV VREV

(5)

These relations can be easily extended to include multiple interacting domains, as in the MINC approach (Pruess and Narisimhan 1985). The dual or multi-domain conceptualizations can be distinguished by their different formulations of the governing equations of flow in the fracture domain and/or by their different approaches to establish exchange between the two overlapping continua (or multiple domains). Various reviews of the different multi-continuum approaches, including the governing equations and exchange functions, are available elsewhere (Berkowitz and Balberg 1992; Lichtner 2000; Šimůnek et al. 2003; MacQuarrie and Mayer 2005; Aradóttir et al. 2013). Multi-continuum models applied to micro-continuum systems. While the multicontinuum approach has been applied to various problems in which two or more domains with contrasting permeability and/or diffusivity are identifiable, the approach has less commonly been used to capture micro-continuum scale effects, for example, interactions between grain and pore scale processes (e.g., nm to mm scale) and the larger domain within which flow and reactive transport occurs (e.g., m scale). Perhaps the earliest contributions that considered interactions between microscopic and macroscopic domains via diffusion were those of Ortoleva and co-workers (Dewers and Ortoleva 1990; Sonnenthal and Ortoleva 1994). The case of diffusive exchange between a macroscopic melt and microscopic discrete crystals was considered by Wang (1993). Wanner and Sonnenthal presented a three region model for kinetic Cr isotopic exchange (shown schematically in Fig. 12) that considered a mobile region within which advective flow occurs, an immobile region within which transport is only via diffusion, and a “mineral region” in which all of the reactions take place (Wanner and Sonnenthal 2013). The advantage of the MINC approach is that it is able to handle diffusive fluxes, JD, between the minerals and

Figure 12. Schematic representation of multiple-interacting continua (MINC) representation of kinetic Cr isotopic fractionation (Wanner and Sonnenthal 2013). By treating the mineral where the reduction occurs as a discrete domain, mixed diffusion–surface reaction rates can be considered. JD1 and JD2 refer to the fluxes of Cr(VI) from the mobile and immobile continua to the mineral surfaces, respectively.

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both the other domains within which transport occurs, thus allowing for an explicit treatment of diffusive limitations to the rate (Xu 2008). Providing a surface reaction-controlled rate at the mineral surface in combination with the MINC approach allows one to consider a mixed diffusion-surface reaction control on the rate, as in the discrete model presented by Noiriel et al. (2012). In a study by Aradóttir et al. (2013), the method of ‘multiple interacting continua’ (MINC) was applied to include microscopic rate-limiting processes operating at the grain scale within continuum (cm to m) scale reactive transport models of basaltic glass dissolution. In contrast with the nanometer-scale resolution model for glass dissolution discussed below, the approach taken by Aradóttir et al. (2013) allows for the use of a coarse numerical grid while capturing the interaction with the microscopic grains via the multi-continuum approach. The MINC method involves dividing the system up to ambient fluid and grains, using a specific surface area to describe the interface between the two (Fig. 13A). The various grains and regions within grains are then described by dividing them into continua separated by dividing surfaces. Millions of grains can thus be considered within the method without the need to explicitly discretize them individually. Four continua were used for describing a dissolving basaltic glass grain; the first one describes the ambient fluid around the grain, while the second, third and fourth continuum refer to a diffusive leached layer, the dissolving part of the grain and the inert part of the grain, respectively (Fig. 13B).

Figure 13. A) A four-dimensional MINC interpretation of basaltic glass dissolution. The left figure shows a zoom-in of real grains in the simulated column, which is packed with basaltic glass grains of size 125–250 m, yielding a porosity of 0.45. The middle figure shows a blow up of several grain clusters within the column and their interpretation as four interacting continua within the MINC approach. Each grain cluster consists of approximately 25,000 individual grains. B) A schematic illustration of elements and connections in the four-dimensional MINC setup, with each column representing a different continuum. [Reproduced from Aradóttir ESP, Sigfússon B, Sonnenthal EL, Björnsson G, Jónsson H (2013) Dynamics of basaltic glass dissolution–Capturing microscopic effects in continuum scale models. Geochimica et Cosmochimica Acta, Vol. 121, p. 311–327, Figs. 2 and 5 with permission from Elsevier.]

Resolution of nanoscale reaction fronts In addition to their incorporation in MINC, micro-continuum models also have important application to the simulation of nanoscale reaction fronts, particularly where there has been interest in the long term performance of the engineered or natural materials (Grambow 2006; Gin et al. 2013b). Unlike the multi-continuum approach in which a relatively coarse numerical discretization is used (that approach relies on the multiple interacting continua to capture microscopic behavior), very high resolution gridding is used in this section to capture microscopic effects.

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There has been increasing interest in this topic in recent years as various characterization methods have dramatically improved the spatial resolution of reaction fronts that can be achieved. In particular, the higher resolution of the newer chemical profiling techniques, which include atom probe tomography (APT), scanning transmission electron microscopy (STEM), energy filtered transmission microscopy (EFTEM) , and time-of-flight secondary ion mass spectrometry (ToF-SIMS), has called into question the long-standing model of glass and mineral dissolution in which diffusion and hydration lead to the selective release of cations from the surface-altered zone (Geisler et al. 2010; Hellmann et al. 2012, 2015; Gin et al. 2013a, 2015). The higher resolution techniques demonstrate convincingly that the broad sigmoidal profiles interpreted as inter-diffusion cation profiles are largely an effect of the low resolution techniques that average elemental concentrations across a broad region. The broad inter-diffusion profiles in glass and minerals that investigators thought they saw in the past had led to models for dissolution that involved selective leaching of elements from the glass or mineral structure. In contrast, Hellmann et al. (2015) report nm to sub-nm scale reaction front widths in altered borosilicate glass for all ions except for H+, the measurement of which they suspect to be subject to too much error for high resolution mapping. The sharp nm-scale fronts indicated by higher resolution profiling, along with isotopic studies targeting the gel layers formed from glass corrosion (Geisler et al. 2010), have led to reinterpretation of these as dissolution–precipitation rather than inter-diffusion fronts (Hellmann et al. 2012, 2015). It should be pointed out that some high resolution studies like that on the nuclear glass altered for 25 years (reported by Gin et al. 2013a, see Fig. 14A) still indicate ~20-nm fronts for H+ and Li+, suggesting that at longer time scales, it may still be possible for the inter-diffusion fronts to develop, even if they are much narrower than previously thought. It is noteworthy that even in the case of the 25-year glass investigated by Gin et al (2013a), however, the reaction front for boron and sodium are narrower than the fronts for Li+ and H+, arguing that a dissolution– precipitation mechanism controls the release of B and Si (Fig. 14B).

Figure 14. A) High resolution atom probe tomography (APT) profile across glass alteration front. Note sharper front for B and Na as compared to Li and H. B) Schematic representation of distribution of fronts for 25-year altered glass shown on left. [Modified slightly after Gin S, Ryan JV, Schreiber DK, Neeway J, Cabié M (2013a) Contribution of atom-probe tomography to a better understanding of glass alteration mechanisms: Application to a nuclear glass specimen altered 25 years in a granitic environment. Chemical Geology, Vol. 349, p. 99–109, Figs. 3 and 6, reproduced with permission of Elsevier.]

Analytical and numerical models for reaction fronts. A number of models for diffusion and reaction have been presented over the years, with noteworthy contributions by Thompson, Korzhinskii, and Weare (Korzhinskii 1959; Thompson 1959; Weare et al. 1976). A special class of analytical models have been developed for the case of inter-diffusion of cations applied primarily to the problem of nuclear glass corrosion (Doremus 1975; Hellmann 1997). Lichtner et al. (1986) presented perhaps the first numerical reaction–diffusion model that could accommodate kinetic models for mineral (or glass) reaction (Lichtner et al. 1986). The

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first comprehensive numerical study of a geological diffusion–kinetic reaction system may have been that presented by Steefel and Lichtner in 1994, a study that highlights some of the advantages of this approach over those relying on analytical solutions to the diffusion or diffusion-reaction system (Steefel and Lichtner 1994). The advantages of the numerical versus analytical models is their ability to couple multicomponent diffusion and kinetically controlled mineral reaction while considering aqueous and surface complexation. In addition, the grid-based numerical formulation allows one to consider changes in porosity, diffusivity, and permeability resulting from the chemical reactions. Numerical models for glass alteration—the GRAAL model. While numerical diffusionreaction models are now becoming more common, in part because their advantages in coupling multicomponent transport and reaction as discussed above (see, for example, Marty et al. 2015), applications to the micro- or nanoscale are more rare. The applications to date have primarily been to glass corrosion, although the approaches should be applicable to mineral dissolution as well. A noteworthy first effort in this regard are the series of papers by Frugier and co-workers that describe the basis for their glass corrosion model (referred to as the GRAAL model) and its application to the problem of glass corrosion (Frugier et al. 2008). The key features of glass corrosion implemented in the GRAAL (glass reactivity with allowance for the alteration layer) model include (Fig. 15): • Relatively rapid exchange and hydrolysis reactions involving the mobile glass constituents (alkalis, boron, etc.); • Slower hydrolysis involving silicon, which results in the formation of an amorphous silica-rich layer at the glass/solution interface; • The amorphous layer itself dissolves as long as the external solution is undersaturated with respect to silica; • The amorphous layer becomes a barrier to diffusion (referred to as a Passive Reactive Interface, or PRI), which at steady-state becomes the rate-limiting mechanism. The PRI is assumed to form at the interface between the gel layer and the inter-diffusion zone and

Figure 15. Simplified diagram of predominant mechanisms accounted for in the GRAAL model for glass corrosion. The Passive Reactive Interface (or PRI) is interpreted to form at the interface between gel and the ion inter-diffusion zone (not shown) and is assumed to be the rate-limiting step in the overall glass corrosion process. [Reproduced from Frugier P, Gin S, Minet Y, Chave T, Bonin B, Godon N, Lartigue J-E, Jollivet P, Ayral A, De Windt L (2008) SON68 nuclear glass dissolution kinetics: Current state of knowledge and basis of the new GRAAL model. Journal of Nuclear Materials, Vol. 380, p. 8–21, Fig. 13, with permission from Elsevier.]

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it represents a densification of a portion of the gel layer; • Crystallization of secondary phases may occur in the broader gel layer. The principal simplying assumptions in the GRAAL model are: • The rate-limiting step for glass corrosion is water diffusion within the PRI; • Water diffusion in the glass and proton/alaklin ion exchange are ignored, and; • The reactivity of the PRI is described by its thermodynamic state relative to the leaching solution. The model equations that are solved, then, include: 1.

An equation describing the kinetics of dissolution of the PRI as a result of undersaturation with respect to the solution;

2.

An equation describing the kinetics of formation of the PRI as a result of water diffusion;

3.

An equation describing the kinetics of formation of secondary phases in the gel;

4.

Mass balance for silicon, and:

5.

Mass balance for boron.

Numerical models for glass alteration—the KC model. As an alternative to the GRAAL model in which diffusion through the Passive Reactive Interface (PRI) is assumed to be rate-limiting, it is possible to develop a more general model that makes no a priori assumptions about the rate-limiting step in the overall glass alteration process. The Kinetic Micro-Continuum (KC) model that has been developed is based on the reactive transport software CrunchFlow (Steefel et al. 2015) and includes the following processes: • Diffusion of water through the pristine glass and its alteration products; • Ion exchange between water and the cations in the glass; • Kinetically controlled hydrolysis reactions resulting in breaking of glass network bonds (Si, B, Al, etc.). The rate may be described by either a linear or a nonlinear transition state theory (TST) law with an affinity control supplied by a specific phase (e.g., amorphous silica), or with an irreversible rate law with no affinity control. In either case, far from equilibrium dependencies of the rate on other dissolved (e.g., pH, Al, silica) or sorbed species can be included; • Multicomponent diffusion of ions through the glass corrosion products; • Precipitation reactions for amorphous and/or crystalline phases of variable composition that are kinetically and thermodynamically controlled; • Kinetically controlled ripening and/or densification reactions that can reduce the porosity and/or pore connectivity (and thus the diffusivity) of the corrosion products; • Kinetically and thermodynamically controlled formation of new crystalline phases (e.g., smectite, zeolite), with possible consequences for the transport properties of the corrosion layer; • Flow and diffusion in the aqueous phase adjacent to the glass surface. The KC model incorporates the possibility (unlike the requirement in the GRAAL model) of diffusion-limited glass corrosion by considering explicitly the kinetically-controlled densification of either (1) a residual silica-rich glass network in which other important

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components (e.g., the cations and network former boron) have been leached, or (2) of a newly precipitated silica-rich gel layer. Whether a passivating layer (i.e., defined as the Passivating Reactive Interface (PRI) by Frugier et al. 2008) forms in the model depends on the relative rates of (1) silica recrystallization and densification, (2) leaching of the glass constituents, and (3) dissolution and/or recrystallization of the corrosion products. Application to the 25-year glass alteration test. As an application of the KC model described above, the 25-year glass alteration experiment as described by Guittoneau and coworkers (Guittonneau et al. 2011; Gin et al. 2013a) is simulated. The principal objective of the modeling is to capture the width of the various reaction zones close to the pristine glass surface and their relative positions as recorded by Atom Probe Tomography (APT) rather than to match the long term corrosion rate. We assume a pure diffusion-controlled regime and a constant grid spacing of 1 nm. The assumption that a continuum model applies at this spatial scale is a severe approximation given that pore sizes are close to this value, but the approach allows us to compare results with elemental profiles and avoids the additional requirement of a full computationally expensive and chemically simplified atomistic treatment (Bourg and Steefel 2012). As a boundary condition at one end of the reactor-glass specimen system, we consider a Dirichlet or fixed concentration condition corresponding to the mineral water used to replenish periodically the experimental reactor (Guittonneau et al. 2011). The fixed concentration boundary condition in 1-D is probably a good approximation to a flow-through or continuously flushed system. At the other end of the 1-D system, we assume a no-flux condition, which is reasonable as long as the corrosion front does not fully penetrate the glass specimen. Within the first 50 nm of the reaction, the system is characterized by a porosity of 0.41 (as in the experimental system reported by Guittonneau et al. 2011) and a mixture of quartz sand and granite upon which the borosilicate glass coupon rests. A diffusivity of 10–12 m2 s-1 for all ions was assumed for the sand-granite mixture. The alloy specimens included in the experiments were not considered in the modeling. From 50 nm out to 200 nm, the system was assumed to consist of a borosilicate glass with a porosity of 1%. The diffusivity of the borosilicate glass was assumed to follow a threshold type of model (Navarre-Sitchler et al. 2009), with a value of 5 × 10-24 m2 s-1 (in approximate agreement with the value proposed by Gin et al. 2013a) for values of the porosity below 50% and a value of 10–12 m2 s-1 for porosity values above 50%. Modeling carried out on weathered basalts (Navarre-Sitchler et al. 2011) indicate that a simple porosity dependence (as in an unmodified Archie’s Law formulation) cannot replicate the observed concentration profiles, since the reaction front continuously widens due to the simulated porosity and diffusivity enhancement. Some form of a threshold model, based on the idea that dissolution and porosity enhancement increase the rate of diffusivity by increasing connectivity (Navarre-Sitchler et al. 2009, 2011), appears to be required. In the modeling, the dissolution of the glass is assumed to follow a Transition State Theory (TST) rate law with a dependence on the saturation state (departure from equilibrium, or affinity) with respect to amorphous silica (Grambow 2006). Since a linear TST dependence on the departure from equilibrium does not capture the sharp B front and places the B (and Na) dissolution fronts too close to the Li–H inter-diffusion front, the dissolution of the glass is assumed to have a cubic dependence on the departure from equilibrium with respect to amorphous silica. The rate of glass corrosion is also assumed to depend on the hydration state of the glass: before the H2O diffusion front has penetrated the pristine glass and hydrates it, the rate is effectively zero. A higher-order dependence on the concentration of hydrated sites in the borosilicate glass is also required so as to locate the boron release front further from the Li–H inter-diffusion front. The rate law used is therefore

Micro-Continuum Modeling of Pore-Scale Geochemical Processes

Rcorr  ka

10 H-hydrated

 Qam-silica  1    K am-silica 

241

3

(6)

where k is the rate constant and aH-hydrated is the concentration of hydrated sites in the glass, and Qam-silica and Kam-silica are the ion activity products and equilibrium constants with respect to amorphous silica. This formulation could be reconciled with a model in which the number of hydrated sites needs to reach some (high) threshold value before the dissolution of the glass accelerates appreciably. Here we present a semi-quantitative comparison of the simulation results from the KC model with nanometer discretization to the data from the Gin et al. (2013) study. The focus is on the relative position of the fronts, and in general, the width of the fronts as they evolve over time rather than on the total extent of alteration or even the rate of alteration. Schematically, the geometry that we wish to capture in the modeling is given in Figure 14 above (Gin et al. 2013a). The model results for the 1-D run after three years are shown in Figure 16A. The simulations predict that the Li–H inter-diffusion front maintains a relatively constant width of about 20 nm over three years (time evolution not shown). The B release (dissolution) front is even sharper and is located further from the pristine glass interface than is the Li–H interdiffusion front, a result that can be justified based on the assumption that the dissolution of the glass occurs rapidly when hydration of the glass is nearly complete. Thus, the key component of the model is the coupling of glass dissolution to the extent of hydration, which is driven by diffusion into the pristine glass (not the PRI discussed by Frugier et al. 2008). According to the simulations, the rate-limiting step for the overall glass alteration process is the diffusion into the pristine glass—once hydrated, the borosilicate glass dissolves quickly, as indicated by its sharp front. The simulations also predict an early time period when the steady-state 20-nm inter-diffusion zone is not fully developed (results not shown), which might help to reconcile the observations by Hellmann et al. (2015) of a nanometer to even sub-nanometer Li+ front in their shorter term experiments (recall that the Gin et al. 2013 study was based on experiments conducted over nearly 25 years, as described in Guittoneau et al. 2011).

Figure 16. A) Simulation results using the KC model, with alteration proceeding from left to right. The original glass wafer edge is located at 50 nm. Note the position of the boron release front further from the pristine glass than the Li–H inter-diffusion front, in qualitative agreement with the observations shown in Figure 14. The simulations predict that the Li–H inter-diffusion front maintains a relatively constant width of about 20 nm over three years (time evolution not shown). B) Position of borosilicate glass corrosion front as a function of time. The linear advance rate is only compatible with a model in which diffusivity is enhanced within the altered zone, resulting in a nearly constant width reaction zone over time.

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In addition, the simulations predict a linear rate of front propagation over time once the initial period (less than 1 year) is passed (Fig. 16B). This can be explained by a constant-width zone over which diffusion is limiting, in agreement with earlier results on weathering of basalt (Navarre-Sitchler et al. 2011). In the case of the nuclear glass altered for 25 years, the constant width is a result of the development of an approximately 20-nm-wide inter-diffusion zone at the edge of the pristine glass. Between this zone and the dissolving gel at the outer boundary, the increase in porosity is sufficient that diffusion is not limiting (even if the zone were to grow with time). If diffusion through a continuously growing silica-rich gel layer was limiting the rate, the dependence on time should be parabolic. A constant width PRI, as discussed by Frugier et al (2008), could also result in a linear front advance rate.

SUMMARY AND PATH FORWARD While true pore-scale models are arguably the most rigorous way to treat geochemical processes operating at the pore-scale, micro-continuum modeling approaches offer some advantages in terms of their relative ease of use, ability to apply well-tested software (e.g., Steefel et al. 2015), and computational efficiency. This approach offers the additional advantage that micron to even nm-scale mineralogical, chemical, and physical heterogeneities can be incorporated into the simulations. The disadvantage of the approach is that one still faces many of the standard limitations of continuum representations of the pore scale, namely the need to average geochemical, mineralogical, and physical properties and the inability to explicitly resolve interfaces between solids, gases, and fluids. Many important parameters and processes still operate at the sub-grid scale (e.g., nanopore connectivity) and these must be accounted for in order to achieve a realistic simulation of pore-scale geochemical processes. The challenge of dealing with reactive surface area, for example, persists in any continuum treatment of the pore-scale, in contrast to the more rigorous geometric methods in true pore-scale models where the fluid–mineral interface is resolved explicitly (Molins 2015, this volume). Certainly the advent of new microscopic characterization techniques, including increasingly higher resolution X-ray microtomography, BSE-SEM, and FIB-SEM, are motivating the search for novel and complementary modeling methods. The longer time and space scales that achievable with the micro-continuum models is another reason why they will not soon be replaced by either molecular dynamics (MD) modeling approaches or even true pore-scale models. The true pore-scale and MD approaches have an important role to play here, however, since they can be used to provide upscaled parameters for the micro-continuum models. Eventually, one expects the development of a new class of hybrid models that link the molecular, true pore-, and micro-continuum scales within a single dynamic, multi-scale framework. Much remains to be done in the field of micro-continuum modeling of pore-scale geochemical processes. In fact, the field is still in its infancy. Since the mineralogical mapping is predominantly carried out in 1-D or 2-D, we require an improved treatment of how 3-D effects and parameters are incorporated. Tortuosity and permeability are two of the most obvious examples. We also need improved representations of the correlation between mineralogy and physical transport properties like diffusivity and permeability at the nanoand micro-scale. Consistent upscaling strategies are required so that it is possible to change model resolution without undue loss of process fidelity. Ultimately, it is clear that the microcontinuum approaches will need to incorporate a more formal multi-scale framework, particularly where there is interest in capturing nanoscale features like pore connectivity and reaction fronts within larger scale domains.

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ACKNOWLEDGMENTS This work was supported as part of the Center for Nanoscale Control of Geologic CO2, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Contract No.  DE-AC02-05CH11231 to Lawrence Berkeley National Laboratory. This work was also supported in part by the Director, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, of the U.S. Department of Energy under the same contract to Lawrence Berkeley National Laboratory. We are grateful to Qingyun Li, Jennifer Druhan, and Bhavna Arora for their careful reviews of the chapter.

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 247-285, 2015 Copyright © Mineralogical Society of America

Resolving Time-dependent Evolution of Pore-Scale Structure, Permeability and Reactivity using X-ray Microtomography Catherine Noiriel Géosciences Environnement Toulouse Observatoire Midi-Pyrénées Université Paul Sabatier, CNRS, IRD Toulouse, France [email protected]

INTRODUCTION Dissolution and precipitation reactions are the primary mechanisms that occur when a rock (i.e., a mineral assemblage) is in contact with a fluid out of equilibrium. They play a critical role in natural processes (e.g., weathering, compaction, meteoric and marine diagenesis) and anthropogenic processes (e.g., reservoir acidizing, CO2 sequestration, acid mine drainage, contaminant mobility, bioremediation). Such fluid–rock interactions result in complex changes in pore structure and mineral composition, leading in turn to changes in flow, mechanical, and transport properties, such as permeability, dispersivity, strength, and pore accessibility. Consequently, geochemical disequilibrium can lead to complex modifications of hydrodynamic and transport properties in porous and fractured rocks. Porous rocks are often characterized by complex textures and mineral compositions that are derived from their depositional and diagenetic environments. They typically have heterogeneous structures, the macroscopic physical properties of which depend on microscopic characteristics. Permeability, for example, is closely related to the microstructure, in particular the size and the spatial distribution of pore throats, pore roughness, and presence of fine clogging particles. The coupled hydrological, mechanical, and chemical (HMC) processes are highly non-linear and minor changes at the pore scale in one property can result in large modifications of the others properties. Prediction of system response to chemical conditions requires understanding how individual processes that occur at the microscopic scale contribute to the observed large-scale flow and transport distribution patterns. Predictive modeling remains challenging for the time and spatial scales involved in geological processes and because of the lack of information about how the physical properties of the porous medium evolve as a result of chemical reactions. In particular, the role of microstructures and their possible effects on flow and transport have long been neglected. Consequently, upscaling the flow and transport properties remains poorly constrained by porescale observations despite a multitude of experiments, field observations, and modeling efforts. Pore-scale processes have received increasing attention over the last decade, particularly in the context of subsurface CO2 injection and sequestration programs (Crawshaw and Boek 2013; Steefel et al. 2013). Advances in 3-D imaging techniques, such as X-ray microtomography (XMT) (Elliott and Dover 1982; Flannery et al. 1987; Sasov 1987a,b) have improved the capability for imaging porous media at the pore scale (e.g., Spanne et al. 1994; Coles et al. 1996; Lindquist et al. 2000; Noiriel et al. 2004; Bernard 2005; Blunt 1529-6466/15/0080-0008$05.00

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et al. 2013). The scale of observation from a few nanometers to centimeters can be used to link micro-scale properties of porous rocks to effective parameters at the aquifer or reservoir scale. The technique is particularly suited for 3-D rendering of structural details. In addition, the 3-D geometry derived in this way is then available for discrete calculations of porous media properties such as permeability, transport, electrical conductivity or fracture–matrix fluid transfer, and observations at the pore-scale can bring new information to improve predictability and reliability of numerical modeling (e.g., Fredrich et al. 2006; Flukiger and Bernard 2009; Noiriel et al. 2012; Bijeljic et al. 2013b; Blunt et al. 2013; Molins et al. 2014 and references below). The fact that XMT is a non-invasive, non-destructive technique allows for time-dependent observations and analyses, thus providing insights into HMC dynamic processes. This article is concerned with the application of X-ray microtomography for the purpose of improving our understanding of dynamic geochemical processes at the pore scale. The techniques will be detailed and physical properties derived from imaging of porous rocks and fractures will be presented. The article will then examine fluid–mineral properties at the pore scale when changes of rock or material geometry are involved. Dissolution and precipitation reactions leading to particle displacement and migration are included in this analysis. Applications to dissolution and precipitation in porous and fractured rock will be presented to illustrate the capacity of XMT for investigating the effect of small-scale geometry modifications on flow and transport properties. This article is focused primarily on experimental investigations, and is not intended to cover the pore-scale modeling approaches and investigations, which are fully detailed in other articles (Liu et al. 2015; Mehmani and Balhoff 2015; Molins 2015; Steefel et al. 2015; Yoon et al. 2015, all in this volume).

Resolving pores using X-ray microtomography The development of XMT relies on 3-D reconstruction of a sample from a series of 2-D radiographs. The technique has added a new dimension to the understanding of fluid–rock interactions in fractures and porous media through the direct visualization of pores, pore space, mineral distribution, fluid–rock and fluid–fluid interfaces, as well as fluid flow. Such porescale observations should be helpful in unraveling the processes that occur when a reactive fluid comes into contact with minerals. One advantage of XMT over several other techniques is that it is non-invasive and nondestructive, and therefore can be used to characterize geometry changes during dynamic processes or experiments over time. In some cases, the imaging can be carried out in situ over the course of a dynamic experiment, in others it is conducted ex situ, before and/or after an experiment. The technique can be used as complementary to other techniques that provide additional physical, chemical, mineralogical and structural descriptions of materials, either while the experiment is running or post-mortem (e.g., measurements of fluid chemistry, permeability, acoustic velocity, electrical conductivity and resistivity, Hg-intrusion porosity, BET surface area and microscopy observations (e.g., scanning electron microscopy (SEM), focused ion beam coupled with SEM (FIB-SEM), confocal microscopy). When combined with other characterization techniques, direct comparisons can be made between the 3-D porestructure and the different physical and chemical properties measured during experiments. The increase in computational capabilities has reduced the time needed for reconstruction, image processing and visualization of large data sets, greatly enhancing the size, resolution, and complexity of the porous materials that can be handled. The application of XMT imaging to the field of geosciences and chemically reactive environments has recently increased thanks to the development of lab-based systems complementary to synchrotron-based systems, allowing for a micrometer resolution, and even nanometer resolution with the recent development of X-ray nanotomography (e.g.,

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Brisard et al. 2012). XMT is based on the measurement of attenuation of an X-ray beam passing through a material. Most of the measurements are carried out using X-ray absorption properties (absorption contrast XMT, dual-energy XMT), but some specific setups also rely on coherent X-ray propagation (phase-contrast XMT) or secondary emission (X-ray fluorescence microtomography) properties. Technological evolution of instrumentation and detectors as well as the development of new reconstruction procedures continues to increase the capacities and utility of XMT. For example, the development of ultra-fast tomography allows for tracking of very rapid dynamic processes (e.g., Mokso et al. 2011; Armstrong et al. 2014a,b), as time needed to obtain a full 3-D data set can be less than 1 s. Techniques based on the energy dependence of the attenuation coefficient, such as dual-energy microtomography, can reveal the distribution of distinctive chemical compositions within a sample, as well as X-ray fluorescence microtomography (e.g., Van Geet et al. 2000; Egan et al. 2014). Detailed reviews of XMT development and applications are provided in Van Geet et al. (2001), Wildenschild et al. (2002), Carlson et al. (2003), Carlson (2006), Blunt et al. (2013), Cnudde and Boone (2013) and Wildenschild and Sheppard (2013).

Absorption contrast X-ray microtomography Theory. When interacting with matter, X-ray incident intensity is progressively reduced. Through a homogeneous material, the attenuation is linear and depends on X-ray energy and sample atomic composition and density (in relation to its chemical composition, lattice parameter, and micro-porosity). In the energy range 1–100 keV, interactions between X-rays and matter include elastic scattering (Rayleigh scattering), inelastic scattering (Compton scattering), and the photoelectric effect. In the range 5–30 keV, within which most of the experiments are performed, the photoelectric effect is dominant. The linear attenuation coefficient characterizes the attenuation of incident X-rays through a material, and it can be theoretically calculated for elements and mixtures, based on the NIST XCOM database (Berger et al. 2011; Fig. 1). For a single material, the total attenuation of a material over a distance x and at a given photon energy (E) follows the Beer–Lambert law: I  I 0 exp( Ti ( E ) x ),

(1)

with  Ti ( E ) the linear attenuation coefficient of material i, and I 0 and I the incident and transmitted beam intensities, respectively. Attenuation coefficients are additive over length, and a composite material crossed by an X-ray beam results, at a given photon energy (E), in a linear average attenuation coefficient,  T( E ) :  T( E )    Ti ( ,E ) d,

(2)

with the length of the crossed composite material. Materials with high attenuation coefficients will absorb X-rays more efficiently, while materials with low attenuation coefficients allow more efficient transmission of X-rays. Experimental setup. The best experimental configuration depends on the X-ray source and beam specifications, which typically differ between a laboratory-based and a synchrotronbased setup. In a laboratory-based setup, X-rays are generated from low- or high-flux X-ray tubes resulting in a polychromatic cone beam. For low-flux sources, magnification is set by adjusting the sample position between the X-ray source and the detector, and a spatial resolution of ~1 μm can be reached. Alternatively, X-ray optics can be used with high-flux X-ray sources to achieve a higher magnification, leading to a spatial resolution as fine as 50 nm for recently developed systems. In synchrotrons, a parallel beam of highly spatially coherent polychromatic X-rays is generated using insertion devices such as bending magnets (typical case for a X-ray imaging

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Figure 1. X-ray attenuation for calcite between 1 and 100 keV, data from XCOM software (Berger et al. 2011).

beamline), wigglers or undulators inserted in the storage ring. A monochromatic X-ray beam can be obtained using monochromators, which are pairs of crystals (Si(111) or multilayers) operating according to the Bragg Law of diffraction to select specific energies. Generally, the detector system operates through a scintillator converting the X-rays into visible light and an optical system (objective lenses and mirrors) ending with a detector recording the signal. Advantages in using a synchrotron source are that the data is of higher quality in terms of noise and artifact reduction (see below) and the ability to take advantage of the coherence of the X-ray beam when needed. It is also worth remarking that a pink beam (i.e., a polychromatic filtered white beam) can also be used, for example when an extremely high photon flux is needed (e.g., for ultrafast imaging). A synchrotron source may be preferred to a polychromatic cone beam, as the higher the signal-to-noise ratio, the more accurate and precise is data segmentation and quantification. Image acquisition. Image acquisition consists of recording a series of 2-D radiographic projections of the transmitted photons at different angular positions of the sample placed on a mechanical stage rotating over 180° (parallel beam) or 360° (cone beam) around an axis perpendicular to the beam. In some setups, the sample is fixed and the X-ray source describes a helical trajectory relative to the sample (spiral cone beam). In absorption contrast tomography, the sample is set immediately in the front of the scintillator to avoid phase contrast (see further discussion below). Between one and three thousand radiographs are usually recorded. The X-ray beam energy is adjusted to a value which allows for a high absorption contrast between the different materials forming the composite sample (e.g., mineral and air), while a reasonable transmission of X-rays (~20–30%) in the most absorbing zone of the sample is still available (to minimize noise). The exposure time can be increased to increase the transmission of X-rays, but should remain short enough to avoid saturation of the detectors when bright field images (i.e., radiographs without the sample in the field of view) are acquired. Phasecontrast imaging or holotomography can substitute for absorption modes when contrast between two phases of interest is not high enough. Phase contrast in the near-field region can also be used to highlight small features at or just below the resolution of the experimental setup, such as grain boundaries or cracks (e.g., Marinoni et al. 2009). For samples rich in heavy elements, higher energies are generally required, as increasing energy increases X-ray penetration. On a synchrotron source, the brightness (i.e., the photon flux) of a beam derived

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from an insertion device decreases at the higher energies, meaning that samples rich in heavy elements will generally require a longer acquisition time. This effect is in addition to the fact that the exposure time must also generally be increased for heavy elements, as X-ray linear attenuation is higher. Radiographs (i.e., projections) of approximately 2000 × 2000 pixels are recorded. The intensity of transmitted photons is detected using a charge coupled device (CCD) or a flat panel detector coupled to scintillators converting X-rays to visible light. Before reconstruction (the procedure converting the set of radiographs in a 3-D volume), the radiographs must be corrected for background noise, beam structure, defects in the detector, and beam intensity fluctuation during acquisition. Fluctuations in beam intensity are usually corrected by normalizing the projections over a region of interest (ROI) where the sample is not present. Other corrections are carried out by subtracting dark field and flat field from original radiographs, using the following formula: I cor I  I dark  , cor I0 I flat  I dark

(3)

cor with I 0 and I cor the corrected intensities of the incident and transmitted beam, respectively, I dark the intensity of the dark field image, which is a projection collected without the beam (to record the background noise), and I flat the intensity of the flat field, which is a radiograph without the sample in the field of view.

Image reconstruction and correction. 3-D image reconstruction producing data sets of approximately 2000 × 2000 × 2000 voxels (i.e., volumetric pixels) can be achieved using either analytic (i.e., filtered back projection) or iterative (i.e., algebraic or simultaneous algebraic) reconstruction methods. The most popular is the filtered back-projection method based on the inverse Radon transform (Herman 1980). The set of projections over all the angular positions is first converted line by line in the frequency domain using the Radon transform. The set of projections for a specific line over all the angular positions in the frequency domain is called a sinogram. 3-D reconstruction, which corresponds to the Cartesian coordinate back transformation in the spatial domain, is then performed using a filtered back-projection algorithm. The reconstruction is performed slice by slice, and the 3-D volume is obtained by stacking the 2-D slices. Several artifacts can result from XMT acquisition, depending on the beam characteristics, setup and sample composition, such as hot pixels, streak artifacts, ring artifacts, beam hardening and cone-beam effect. They appear in the 3-D reconstructed data sets as either a significant number of distinct geometrical features (e.g., lines, streaks, rings) or globally spread noise. A number of algorithms exist for improving the quality of 3-D reconstructed data sets (Fig. 2).

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Figure 2. XMT slice of a sintered limestone, showing correction of hot points and ring artifacts using a conditional median filter on 2-D radiographs and sinogram filtering before reconstruction. [Reprinted from Noiriel (2005).]

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Presently, most of the corrections are made routinely before, during or after reconstruction, and are included in reconstruction software (e.g., Dierick et al. 2004; Vlassenbroeck et al. 2007) without any end-user intervention. Registration issues. Observations of geometry changes at different stages of a dynamic process require 3-D image registration of the data sets in the same coordinate system in order to map a 3-D volume to a reference one. Digital image correlation is often used to quantify local strain distribution when mechanical deformation of the sample has occurred (e.g., Lenoir et al. 2007; Hall et al. 2010). When there is no mechanical deformation, registration consists of a rigid transformation involving 3 rotations and 3 translations around and in the x-, y- and z-directions, combined with interpolation. Many optimization algorithms based on image intensity or feature extraction can be used to perform 3-D image registration in either spatial or frequency domains as, for example, after the calculation of the centre of gravity of intensities, the moment of inertia in the x-, y- or z- direction, or normalized cross-correlation (e.g., Maintz and Viergever 1998). However, when studying dynamic processes, automatic registration using iterative optimization algorithms often fails because geometry changes lead locally to modifications of features that change the image intensity distribution. In particular, it can be difficult to establish a correspondence between several distinct features. If this is the case, a manual registration remains the only alternative. Manual registration is based on the selection of several control points (or landmarks), which generally correspond to isolated pores or very absorbing minerals (e.g., iron oxides or sulfides). A minimum of four points are required to calculate the six unknown parameters of the rigid-transformation matrix, but a statistically representative collection of control points is preferred for error minimization. Different interpolation techniques can be then used to recalculate the grayscale of every pixel in the registered data sets, e.g., tri-linear interpolation (Gonzales and Woods 1992).

Phase contrast X-ray microtomography The absorption contrast between materials of different densities (e.g., air and mineral) is generally large enough for them to be distinguished on 3-D data sets. However, XMT in the absorption contrast mode can fail to distinguish between different materials (or phases) when the absorption contrast is poor (e.g., between liquid and vapor), or when their densities are similar (e.g., water and organic material, calcite and dolomite, or quartz and feldspar). In these cases, propagation-based imaging, including phase contrast microtomography, can be used to enhance contrast at the interface of two phases (e.g., Snigirev et al. 1995). Phase-contrast imaging relies on the fact that two isotropic materials of different density have a different refractive index, i.e., different X-ray propagation speeds. As a result of the difference in density, not only the intensity but also the phase of X-rays is modified when interacting with different materials, leading to a phase shift. Weak perturbation of the wave front generates interference patterns. The interference fringes are not directly linked to the phase itself, but rather to the second derivative of the phase, with the result that the method is more sensitive to abrupt changes in the decrement of the refractive index. Near-field propagation phase imaging enhances phase contrast in the regions of large changes in the refractive index between two phases, i.e., at their edges. For low-absorption materials or when absorption is similar between two phases, an alternative is to increase the sample–detector distance to allow for recording of the interference patterns (Cloetens et al. 1999). In this case, a high-resolution detector is required, although it limits spatial resolution. Retrieval of the phase distribution of the sample using specific algorithms can also be useful to obtain the 3-D distribution of the complex refraction index and thereby enhance the contrast between phases (e.g., Arzilli et al. 2015).

X-ray fluorescence microtomography (XFMT) The photoelectric effect involves atom ionization (i.e., ejection of an electron) resulting

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from the interaction between X-rays and the inner shells of an atom. Filling the inner-shell vacancy in the atom is accompanied by either the emission of a second electron (Auger effect) or by emission of a secondary fluorescence X-ray photon (X-ray fluorescence). Fluorescence is emitted isotropically from all excited atoms, and the use of an energy-dispersive detector (e.g., a Si-drift diode detector) allows for recording integrated 2-D projections of the elemental distributions within the sample. The reconstruction of the 3-D elemental distributions is then possible from the 2-D projections. Combining XMT with XFMT means that both geometric and chemical mapping can be achieved. The technique is useful for distinguishing between minerals with similar densities (i.e., silicates, solid solutions), measuring element concentration ratios (Lemelle et al. 2004), and analyzing the distribution of radionuclides and metals in soils (Lind et al. 2013). However, the spatial resolution is limited (micrometer resolution) and acquisition time can take several days. Tracking of several ions (Cs, Ba, La) was also performed during diffusion experiments in porous media (Betson et al. 2005) and the technique appears promising for environmental studies, i.e., for quantifying trace elements in samples, although there are several technical challenges and analytical complexities (e.g., the absorption of the fluorescence X-ray photons by the sample itself) with its use.

Dual-energy X-ray microtomography An XMT data set is a 3-D image of the X-ray attenuation of the sample. Dual-energy XMT is based on the dependence of the linear attenuation coefficient of an element or a material to the X-ray energy (E). The technique requires scanning the sample twice at two different energies. Van Geet et al. (2000) and Remeysen and Swennen (2010) used the technique with a Skyscan 1072 lab-source at 100 and 130 kV to recover densities and effective atomic numbers within carbonate samples. Nevertheless, due to the noise inherent in a polychromatic source, segmentation was required in order to calculate the repartitioning of the different rock-forming minerals. Due to the ability to select specific energies, synchrotron source provides the best experimental setup for examining a specific element with dual-energy XMT, as a large difference in absorption exists in the absorption edge region of an element (see Fig. 1). In this case, the acquisition is performed at two energy settings, just above and below the absorption edge energy of the considered element, a range over which the attenuation coefficients of the other elements do not significantly change. The difference between the two XMT data sets therefore allows a 3-D chemical mapping of the considered element, which can in turn be used to obtain 3-D mineral distribution in materials such as rock specimens (Tsuchiyama et al. 2013). The technique is suitable for mapping elements with K- or L-absorption edges in the same energy range as that set up to image rock samples (e.g., Fe, As, Pb, Cs, U).

PORE-SCALE CHARACTERIZATION Image segmentation Image segmentation, in which voxels are classified into two (i.e., pore and solid) or more phases of interest (e.g., the different constituents of the solid phase, such as minerals), is an essential step for extracting geometrical properties of rocks. Digitization of the pore space and mineral matrix is also required for performing direct numerical simulations of flow and transport (e.g., Oren et al. 2007; Flukiger and Bernard 2009; Narsilio et al. 2009; Menon et al. 2010; Bijeljic et al. 2013a; Molins et al. 2014). A 3-D XMT data set of rock is assumed to consist of a fixed number of distinct mineral phases (solid phase) and fluid and/or gas phases (pore space). The dynamic range (i.e., the distribution of grayscale data) of several data sets can vary for different acquisitions of the same sample due to fluctuations in the X-ray beam or additional noise, even if similar acquisition and reconstruction parameters were chosen. As

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a consequence, normalization of every data set of a sample is necessary before segmentation. Segmentation involves converting the grayscale data into several sets of voxels corresponding to the distinct phases of interest based on their different X-ray absorption properties. In most cases, the different phases cannot be clearly separated on the grayscale histograms as the data is noisy and can be complicated by partial volume effects (see further explanation below), with the result that applying a simple threshold method to the whole image would lead to misclassification. Different methods have been developed to improve segmentation accuracy (see Iassonov et al. 2009 and Schlüter et al. 2014 for reviews), although there is no clear test to evaluate their performance. For example, edge-preserving noise reduction filters, such as a 3-D median filter or an anisotropic diffusion filter, can be used before segmentation to improve the signal-to-noise ratio without altering information at the interface between two phases. Methods that incorporate local spatial information (i.e., based on the intensity of the neighboring voxels) are usually more appropriate over global threshold methods, e.g., indicator kriging (Oh and Lindquist 1999), watershed transform (Beucher 1992; Sheppard et al. 2004), or region growing (Pitas 2000). After segmentation, each phase is assigned a fixed value (e.g., 0, 1, 2, etc.). In some cases, segmentation can also produce image artifacts, even when applied to high quality XMT data sets, and filtering of the segmented data can be necessary. In addition to the algorithm used, segmentation efficiency depends on the initial image quality (i.e., signal-to-noise ratio, which directly impacts the distribution of the grayscale values of the different phases on the histogram), and on whether the image resolution is sufficient to resolve the objects of interest. For example, a porosity difference of less than 0.5% was obtained by XMT consistent with chemical measurement (Noiriel et al. 2005). Uncertainties, however, are usually higher, particularly when the pore network is complex or there is a insufficient resolution in the image. A comparison between different segmentation methods by Andrä et al. (2013a) gave a maximum porosity difference of about 2% for Berea sandstone. In contrast, the difference was approximately 8% for Grosmont dolostone where the histogram of grayscale values was much more complex. In some cases, contrast agents can be added to enhance contrast of the fluid phase or to study fluid partitioning in the pore space (e.g., Seright et al. 2002). Nevertheless, the presence of a contrast agent of intermediate grayscale between initial fluid and solid can affect how the other phases can be segmented (Prodanovic et al. 2006, 2007). It should be mentioned that segmentation is a critical step in image processing, as the quality of segmentation directly impacts the computed effective properties (Andrä et al. 2013b).

Porosity determination, pore geometry and pore-space distribution Once segmentation has been completed, porosity, , can be calculated as simply the ratio of pore space to the total number of voxels, i.e.,   N void / ( N void  N solid ), with N void and N solid the number of voxels identified as pore and solid, respectively. As reactivity and other properties may change along the flow path, it may be important to calculate porosity as a function of sample depth (z-direction) or sub-sampled in every direction of space. When partial volume effects are too large to resolve pores, it may still be possible to estimate porosity without segmenting the data. In this case, either references of attenuation of the different materials forming the sample (e.g., solid the attenuation of the solid and  void the attenuation of air) or a porosity reference (e.g., determined from Hg-intrusion) are required (Fig. 3). Porosity of a voxel, in which attenuation is , is defined as:   (solid  ) / (solid   void )

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Figure 3. (a) Porosity field map resulting from a compaction experiment on (b) a diatomite rock sample (2-D horizontal cross-section obtained with XMT with a resolution of 0.7 μm). Due to partial volume effects (median pore diameter is 0.5 μm for the rock, and only pores of 3–7 μm in size can be distinguished on the image) porosity was set to the initial average porosity of the rock (i.e., 68%, determined from Hg-injection) in non-deformed areas. An area of porosity decrease by more than 50% between two sub-horizontal fractures is observed under the indenter. Data courtesy of Jean-Pierre Gratier, ISTERRE, University of Grenoble.

idealized representation of the complex pore space geometry and a description of topology (see Thovert et al. 1993; Spanne et al. 1994). Extraction of the pore-space structure is commonly based on medial-axis skeletonization, which can be used to partition the void space into pore-bodies and pore-throats. Lindquist et al. (1996, 2000) have developed algorithms to calculate pore connectivity, tortuosity, pore-body size, pore-throat size and surface area based on studies of the Berea and Fontainebleau sandstones. Al-Raoush and Willson (2005) calculated the spatial correlation between pore-body sizes for several unconsolidated porous media using semi-variograms and integral scale concepts. Bauer et al. (2012) developed a dual-pore skeletonization to obtain a separate description of macro-porosity and microporosity in limestone samples with bimodal pore-size distribution. Their model combines electrical transport properties of micro-porosity with those of the interconnected macro-pore network. Skeletonization is, however, sensitive to image resolution and noise, as small details or topological artifacts (e.g., edge-connected voxels of two non-connected objects) can lead to errors when estimating the medial axis. Increasing resolution does resolve this problem, but Plougonven and Bernard (2011) proposed a method based on the sequential detection of these topological artifacts associated with their removal to improve the reliability of skeletonization without adding resolution. To overcome partial volume effects and solve for non-Fickian dispersive transport in porous limestone, Gouze et al. (2008) developed an alternative approach based on a combination of superimposed grayscale and segmented data sets. First, they combined a burning algorithm with a down-gradient search in order to obtain a dual-pore description based on the mobile– immobile fraction of connected porosity. Combined with the non-connected porous network (considering a voxel resolution of 5.01 μm) and the micro-porous matrix, three different domains can be identified: one is a mobile fraction composed of connected porosity where advection is dominant; the second is an immobile fraction made up of connected porosity in which flow is negligible and has both unconnected porosity and a micro-porous matrix, the latter of which is accessible for diffusion; and a third consisting of a solid fraction below which a critical porosity threshold for diffusion is effectively zero (Fig. 4).

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Figure 4. (a) XMT cross-section and (b) sub-volume rendering of different connected porosity types in a porous limestone. The mobile and immobile fractions of the connected porosity are displayed in cyan and blue, respectively. The non-connected porosity is displayed in yellow, the solid fraction with no diffusion in orange, and the porosity of the micro-porous matrix accessible for diffusion in shades of gray (dark and light grayscales denote low and high porosity, respectively). The interface between the mobile and immobile domains is underlined (red line). Coordinates are given in pixels (pixel size 5.01 m). Reprinted from Gouze et al. (2008), with additional data courtesy of Philippe Gouze, Géosciences Montpellier, University of Montpellier. [Used by permission of AGU, from Gouze P, Melean Y, Le Borgne T, Dentz M, Carrera J (2008) Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resources Research, Vol. 44, W11416, Fig. 7].

Solid-phase distribution and quantification The identification and characterization of the different minerals or materials forming porous media can be carried out in a similar manner to the characterization of pore space after segmentation. Smith et al. (2013b), for example, combined XMT with SEM analysis to quantify calcite and dolomite abundance in a vuggy limestone and marly dolostone based on grayscale intensities. They quantified the reactivity of both minerals during dissolution over the entire core samples as well as within the regions where wormholes developed later in the experiment. Labeling of individual materials implies that they can be distinguished in the 3-D images, either based on their attenuation properties or on their morphology. Morphological criteria can be used to distinguish between materials, particularly when three-phase or more segmentation is not possible. However, the technique is often restricted to idealized materials with regular shapes, e.g., glass beads or mineral aggregates. Starting with binary images, the centroids of the different grains can be found by computing a distance map, the maxima regions of the distance maps corresponding to the centroids. Once every centroid has been identified, a fast watershed algorithm based on topographic distance can be applied, inverting the distance map in order to find out the separation lines that correspond to the boundaries between grains. However, it is worth mentioning that the efficiency of watershed separation depends on the degree of consolidation of the medium and the shape of individual grains, and partitioning of clusters into individual grains is often not trivial (Thompson et al. 2006). Once all the grains have been separated, they can be labeled. Morphological criteria, such as grain size or sphericity, can be used to classify different grain populations (e.g., Dann et al. 2011). Figure 5 illustrates the procedure, with a comparison with three-phase segmentation and labeling (Fig.6). Once the different grains have been labeled, it is possible to calculate individual and statistical parameters based on their morphology. Fonseca et al. (2012), for example, characterized the morphology of the grains forming Reigate sand, including the number of particles, particle size and surface area, orientations of the major, minor, and intermediate axes

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Figure 5. Glass beads and calcite spar crystals separated and labeled based on material morphology criterion after two-phase segmentation. (a) XMT slice (4.56 × 4.56 mm2) of a column packed with glass beads (intermediate grayscale) and Iceland spar crystals (high grayscale); the third phase is composed of air (low grayscale). (b) A morphology-based watershed algorithm was used to separate all the grains after twophase segmentation (i.e., air and solid). (c) Labeling of the glass beads relying on morphological criterion (sphericity > 0.83); the Iceland spar crystals (in white) have not been labeled yet; note that the procedure results in some mislabeling near the edges, due to the presence of truncated spheres of sphericity < 0.83. For more information about the 3-D data set, see Noiriel et al. (2012). (a)

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Figure 6. Glass beads calcite spar crystals separation and labeling based on three-phase segmentation. (a) XMT cross-section (1.37 × 1.37 mm2) of a column packed with glass beads (intermediate grayscale) and Iceland spar crystals (high grayscale); the third phase is composed of air (low grayscale). (b) Three-phase segmentation of the different materials. (c) Labeling of the individual calcite crystals (different colors) relies only on material classification after segmentation; glass beads have not yet been labeled. For more information about the 3-D data set, see Noiriel et al. (2012).

for each particle, and the distributions of particle aspect ratio, sphericity, and convexity. The authors observed that calculations of particle size and shape made by analysis of 3-D images differed appreciably from the values obtained from 2-D images.

Specific and reactive surface area measurements Rates of fluid–mineral interactions are directly controlled by the mineral surface area available for sorption and dissolution/precipitation. The measurement of the surface area (Sr), which is often not simply related to reactive surface area, is essential for interpreting experimental results and for numerical modeling of reactive transport. Reactive surface area is generally determined from BET measurement (Brunauer et al. 1938) or a geometrical estimate of the specific surface area. However, imaged material suffers from resolution limitation compared to BET measurements, and different methods generally lead to different estimates of the geometric surface area (Seth and Morrow 2007). Estimation of geometric surface area is possible from grayscale XMT data sets using a marching cube algorithm (Lorensen and Cline 1987), but is mostly carried out using segmented data sets. At some point, improving resolution does not improve the estimate (Dalla et al. 2002; Porter and Wildenschild 2010). Although geometric surface area scales directly with the observation scale, XMT is a powerful tool for resolving surface area changes at the pore scale, particularly when calculations are performed

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on the same sample at different stages of experiment. For example, by superimposing XMT volumes (Fig. 7), Noiriel et al. (2005) were able to locally quantify and distinguish between reactive and non-reactive surface areas during dissolution of a porous limestone sample. The shifting (and non-shifting) of the fluid–mineral interface was directly linked to reactive (and non-reactive) mineral surfaces. XMT can also be combined with 2-D microscopic analyses for improved estimates of the contribution of each mineral to the total specific surface of the rock (Landrot et al. 2012; Smith et al. 2013a). Geometrical estimates of the specific surface area changes were also carried out using XMT during dissolution (Noiriel et al. 2009; Gouze and Luquot 2011; Qajar et al. 2013; Molins et al. 2014) and precipitation (Noiriel et al. 2012) experiments. Although XMT makes it possible to determine specific surface area, a poor correlation between reactive and specific surface areas is often reported (see, for example, Anbeek 1992, who reports a larger specific surface for weathered minerals compared to unweathered minerals). The weathered surfaces actually correspond to low-reactivity surfaces, similar to what is observed when different dissolution rates between minerals contribute to the increase of specific surface area (e.g., Gouze et al. 2003). To complicate matters further, dissolution–precipitation reactions may result in changes of reactive surface area in addition to the effects on specific surface area (Noiriel et al. 2009; Gouze and Luquot 2011). Unfortunately, the changes are not necessarily correlated. For example, Noiriel et al. (2009) measured the changes in reactive surface (from chemistry) and specific surface (from XMT images) areas, and observed different trends during a dissolution experiment in a porous limestone containing two different calcites (micrite pellets and sparite cement), with different chemical signatures, particularly in Sr and Ba. However, their geometric model of spherical pore growth and micrite sphere reduction failed to reproduce observations of fluid chemistry. A semi-empirical sugar-lump model based on observations carried out at a finer scale using SEM was more successful in describing the increase of sparite crystal boundaries exposed to the fluid in this example.

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Figure 7. (a) Observation of fluid–rock interface shift in a limestone sample during three stages of dissolution (t1: 1.4 h, t2: 13.9 h, t3: 22.4 h; the grayscale slice corresponds to the initial state), reprinted from Noiriel et al. (2004). [Used by permission of Wiley, from Noiriel C, Gouze P, Bernard D (2004) Investigation of porosity and permeability effects from microstructure changes during limestone dissolution. Geophysical Research Letters, Vol. 31, L24603, Fig. 3]. (b) Pore volume comparison between the end of the experiment and the previous stage of dissolution (size 2 × 2 × 2 mm3), showing the dissolved volume and the non-reactive surface (where no dissolution is observed). See also Noiriel et al. (2005) and Bernard (2005) for more details.

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Fracture characterization Flow in single fractures is closely related to aperture distribution, fracture wall roughness, tortuosity and asperities. Peyton et al. (1992), Jones et al. (1993), Keller (1998), and Vandersteen et al. (2003) used different estimates based on grayscale levels to define and measure fracture aperture, am, from XMT images with partial volume effects. Gouze et al. (2003) used segmented data to obtain the void structure, then extracted fracture walls and aperture. However, their method has some limitations, particularly when altered fracture walls exhibit some overlap or when secondary fracture branching occurs. Characterization of the void structure (Fig. 8) permits computation of aperture distribution and related statistics, e.g., fracture volume, tortuosity, and contact areas (e.g., Gouze et al. 2003; Karpyn et al. 2007; Noiriel et al. 2013). Methods of describing fracture wall roughness can also be applied, e.g., determination of the roughness factor (Patir and Cheng 1978), roughness coefficient (Myers 1962), or fractal dimension (Brown and Scholz 1985). Either the 3-D void structure or the 2-D

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maps of aperture or fracture wall elevation can be used as an input for flow modeling (e.g., Noiriel et al. 2007a, 2013; Crandall et al. 2010a,b). Crandall et al. (2010a) generated different meshes from the same data set on a fractured Berea sandstone core, by changing the fracture wall roughness properties through different refinement procedures. Given the different results for tortuosity and transmissivity, Crandall et al. (2010a) established a relationship between the method by which a fracture is meshed and the flow properties calculated by numerical models.

Multi-resolution imaging Rocks typically have complex structures over a wide range of length scales. XMT is unable to resolve pores at a scale smaller than the spatial resolution, with the result that there is insufficient resolution to capture the complete geometry of features, leading to the so-called partial volume effects. When this occurs, voxel intensity balances intensities of several phases (e.g., water and solid matrix at the fluid–rock interface), and this can result to misleading representations of the pore size and connectivity. In contrast, a higher resolution is better for describing the fluid–rock interface, pore shape and size, but it can fail to provide a reasonable estimate of the size of the representative elementary volume (REV), which is the smallest volume which allows for a continuum description of a representative property, for example permeability. In addition, the resolution achievable in XMT is not only limited by the beamline and detector properties, but also by the sample size, as the sample must usually remain within the field of view during data acquisition (although local XMT is possible, see below). Although mosaic scanning can overcome part of the problem, the sample size can remain a limiting factor for investigating porous media, particularly when the pore structure is heterogeneous and a REV is sought for upscaling the flow and transport properties. In other words, although scales much smaller than the resolution used for imagery can play an important role in flow and transport, these features cannot be seen, leading to erroneous parameter estimation or poor process modeling in reactive transport. The most sensitive parameter is certainly the mineral surface area, the measurement of which depends on scale, and thus on the ability to resolve micro-porosity. Permeability can also be poorly estimated, especially when its value is constrained primarily by small pore throats or narrow channels. To overcome this problem, Prodanovic et al. (2015) derived a two-scale network from their XMT data sets to estimate flow and transport properties: a macro-network that mapped the inter-granular porosity that was clearly resolved with XMT; and a micro-network that mapped the micro-porous regions unresolved with XMT, the properties of which could be derived from SEM observations at a higher resolution. Other studies include XMT investigations coupled with SEM or FIB-SEM analyses (e.g., Sok et al. 2010; Landrot et al. 2012; Beckingham et al. 2013). Multi-resolution XMT, which consists of data acquisition at various resolutions, can be used to obtain a detailed description over several different scales. It sometimes involves local XMT, which is a technique used when the sample is larger than the field of view and requires a specific reconstruction method of the region of interest to compensate for the incomplete projection data. Several studies have examined the multi-scale characterization of rock samples (Bera et al. 2011; Dann et al. 2011; Peng et al. 2012, 2014; Hébert et al. 2015). Hébert et al. (2015) have investigated the intrinsic variability and hierarchy of the connected pore space of limestone and dolostone samples at different XMT resolutions ranging from 0.42 to 190 μm. The distribution of porosity values between 0.42, 1.06, and 2.12 μm resolutions highlights partial volume effects (i.e., more porosity details are visible when resolution is increased, Fig. 9), and challenges the ability of XMT to provide reliable representations of pore networks and permeability estimates for low-porosity and highly heterogeneous samples. However, Peng et al. (2014) observed that the contribution of small pores to the permeability was minor in Berea Sandstone, where the pore network appears to be well connected at a resolution higher than 5.92 μm. However, the micro-porous phase that was resolved at a resolution of 1.85 μm (but not at 5.92 μm) was shown to increase porosity, surface area, and pore network connectivity estimates (Fig. 10).

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Figure 9. XMT slice of a limestone at different resolutions: (a) 2.12 μm, (b) 1.06 μm and (c) 0.42 μm and (d, e, f) corresponding slice after segmentation. Data courtesy of Vanessa Hébert, VOXAYA, University of Montpellier. For more information, see Hébert et al. (2015).

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Figure 10. Medial axis for two data sets of Berea sandstone showing different pore network connectivity estimates depending on the image resolution, i.e., 5.92 μm (a) and 1.85 μm (b). [Used by permission of Elsevier, from Peng S, Marone F, Dultz S (2014) Resolution effect in X-ray microcomputed tomography imaging and small pore’s contribution to permeability for a Berea sandstone. Journal of Hydrology, Vol. 510, 403–411, Fig. 4].

COMBINING EXPERIMENTS, 3-D IMAGING AND NUMERICAL MODELING The first investigations of 3-D fractures and porous media at the micrometer scale combined with experimentation in the geosciences were focused on single-phase flow experiments and modeling of macroscopic flow and transport properties (Coles et al. 1996; Arns et al. 2001, 2005; Enzmann et al. 2004a,b; Fredrich et al. 2006; Bijeljic et al. 2013b; Kang et al. 2014). Colloidal deposits and their effects on permeability reduction were also investigated (Gaillard

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et al. 2007; Chen et al. 2008, 2010). Studies of multi-phase flow and gas trapping during imbibition or drainage experiments for the purpose of enhancing oil recovery or for better understanding of groundwater contamination by non-aqueous phase liquid (NAPL) or CO2trapping processes have also benefited from XMT (e.g., Coles et al. 1996; Seright et al. 2002; Wildenschild et al. 2002; Arns et al. 2003; Culligan et al. 2004; Al-Raoush and Willson 2005; Karpyn et al. 2007; Karpyn and Piri 2007; Prodanovic et al. 2007, 2010; Al-Raoush 2009, 2014; Iglauer et al. 2011; Silin et al. 2011; Armstrong et al. 2012; Blunt et al. 2013; Ghosh and Tick 2013; Kneafsey et al. 2013; Smith et al. 2013a,b; Andrew et al. 2014; Celauro et al. 2014; Landry et al. 2014; see also Wildenschild and Sheppard 2013 for a review). Several studies have also examined multi-phase flow and transport in soil aggregates (Carminati et al. 2008; Koestel and Larsbo 2014), including evaporation and salt crystallization in the form of discrete efflorescence or desiccation cracks (Shokri et al. 2009; Rad et al. 2013; DeCarlo and Shokri 2014).

X-ray microtomography for monitoring reactive transport Reactive transport studies include dissolution or precipitation experiments, biofilm growth, or weathering processes. Most of the dynamic experiments performed in the lab are based on flow-through reactors (e.g., Noiriel et al. 2012; Kaszuba et al. 2013). The apparatus is generally composed of cm-scale columns packed with solid material (e.g., glass beads, crushed crystals, or rock aggregates), or core holders containing rock samples. In some cases, the experimental setup can be mounted directly on the X-ray beamline. However, the core holder must be made of material transparent or weakly attenuating to X-rays, or the samples must be removed from the core holder before imaging. Some studies have used XMT to visualize changes in pore structure, while other parameters can be recorded to better clarify the evolution of flow and transport properties and processes, e.g., permeability measurement, chemistry analyses, pH, or tracer experiments. Further measurements can also be made at different stages of the experiment (e.g., Vialle et al. 2014), including geophysical monitoring (e.g., P-wave velocity, electrical conductivity), BET surface area measurement, or determination of the porosity distribution by Hg-intrusion. XMT images can also be used directly or indirectly to estimate solute transport properties in low-porosity materials or to follow the propagation of fronts in reactive transport experiments. Polak et al. (2003), Altman et al. (2005), and Agbogun et al. (2013) have used X-ray absorbent tracers (NaI, CsCl, or KI) to directly visualize their displacement within the porous network. Nakashima and Nakano (2012) combined 3-D image analysis of porosity and surfaceto-volume ratio with a tracer experiment (KI) to determine tortuosity. Mason et al. (2014) developed a method that relies on the linear attenuation coefficients to estimate the volume percentage of different materials (i.e., carbonates, amorphous components, and porosity) and the Ca-concentration profiles in cement during alteration by a CO2-rich brine. Burlion et al. (2006) followed the alteration front displacement in a mortar during an accelerated leaching process by an ammonium nitrate solution (Fig. 11). They observed an exponential decrease in the rate of advance of the propagation front through the altered cement paste over time, while aggregates remained unaltered. The experimental results can be compared with a theoretical or numerical solution of the diffusion or reaction–diffusion equation. Noiriel et al. (2007b) also observed that the rate of propagation of a calcite dissolution front across fracture walls in an argillaceous limestone decreased exponentially due to modification of transport mechanisms. Initially, the transport mechanisms were advection-dominant within the fracture. Then, they evolved to a combination of advection in the fracture and diffusion in the newly formed microporous clay coating which grows over time. The propagation rate of a reaction front in diffusion-dominated systems does not necessarily result in a parabolic (t1/2) dependence on time, particularly when feedback between porosity and the effective diffusion coefficient is involved. Navarre-Sitchler et al. (2009) quantified

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Figure 11. 3-D XMT sub-volume rendering of a mortar sample (a) before experiment, and after (b) 24 h and (c) 61 h of leaching. Aggregates are in light grey, sound cement in white and leached cement has been made transparent in order to visualize the complex front displacement. Reprinted from Burlion et al. (2006). [Used by permission of Elsevier, from Burlion N, Bernard D, Chen D (2006) X-ray microtomography: Application to microstructure analysis of a cementitious material during leaching process. Cement and Concrete Research, Vol. 36, 346–357, Fig. 12].

porosity changes associated with weathering processes in a basalt clast from Costa Rica as a result of mineral dissolution and precipitation of secondary phases. They combined 3-D image analysis with laboratory and numerical diffusion experiments to examine changes in total and effective porosity and effective diffusion coefficient across the weathering interface. Due to an increase in the porosity as a result of chemical weathering, the diffusive transport of aqueous weathering products away from the core/rind interface was enhanced (Navarre-Sitchler et al. 2015). This occurs at a critical porosity of about 9%, beyond which the number of connected pathways for diffusive transport increases dramatically. Reactive transport modeling further demonstrates that the rate of advance of the weathering front can be constant over time, (i.e., linear) even if the dominant transport mechanism is diffusion, in the case where an increase of the effective diffusion coefficient occurs due to porosity enhancement (Navarre-Sitchler et al. 2011). A number of studies of porosity and permeability evolution associated with injection or sequestration of CO2-rich fluids in reservoir rocks, caprocks, or cements have been carried out (e.g., Gouze et al. 2003; Noiriel et al. 2004, 2005; Luquot and Gouze 2009; Gouze and Luquot 2011; Luquot et al. 2012, 2013, 2014a,b; Jobard et al. 2013; Smith et al. 2013a; Ellis et al. 2011, 2013; Abdoulghafour et al. 2013; Deng et al. 2013; Jung et al. 2013; Luhmann et al. 2013; Mason et al. 2014; Walsh et al. 2014a,b). Complementary to the experiments, numerical modeling has also been used to evaluate mass transfer at the pore scale (e.g., Flukiger and Bernard 2009; Molins et al. 2014). Direct simulation on 3-D segmented images appears to be the modeling approach of choice for single-phase flow and transport, since the complexity of the pore space is preserved with this approach (Blunt et al. 2013). Most experiments involving reactive transport have focused primarily on mineral dissolution, but mineral precipitation experiments are also reported in the literature, under either abiotic or biotic conditions. Precipitation experiments involving biofilm growth (Davit et al. 2010; Iltis et al. 2011) or biomineralization (Armstrong and Ajo-Franklin 2011; Wu et al. 2011) were conducted to improve in situ degradation of organic pollutants or sequestration of radionuclide or trace metals into solid phases. Noiriel et al. (2012) evaluated upscaling of precipitation rates from an integrated experiment and modeling approach to the study of calcite precipitation in columns packed with glass beads and calcite spar crystals. Although upscaling was possible using kinetic data determined from well-stirred reactor experiments, the study highlights that nonlinear, time dependence of reaction rates, as related to evolving surface area and/or reactivity, can be difficult to assess in natural contexts. In other cases, a more complex suit of reactions are involved, including dissolution of primary minerals by the reactive fluid that leads to precipitation of secondary phases. Cai et al. (2009) and Crandell et al. (2012) quantified the changes in porosity and pore size

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characteristics in Hanford sediments exposed to simulated caustic tank waste. The authors found that secondary precipitation of sodalite and cancrinite, following the dissolution of quartz and alumino-silicates, resulted in a decrease in both the abundance and size of large pore throats.

X-ray microtomography for monitoring geomechanical evolution Evaluation of mechanical or mechano-chemical properties of rocks, concrete, and aggregates have also benefited from XMT observations. Dautriat et al. (2009) measured the dependence of porosity and permeability on the stress path in elastic, brittle, and compaction regimes in Estaillades limestone samples. The permeability drop was linked to pore collapse associated with the propagation of micro-cracks through the dense calcite (i.e., bioclasts and cement) and filling of the pores by some fine calcite fragments. In this case, the less dense calcite (i.e., micro-porous red algal debris and altered micro-sparitic cement) appeared to accommodate strain in a more diffusive manner. By comparing the digital images of the specimen in the reference and the deformed states, Higo et al. (2013) were able to apply digital image correlation to obtain the full-field surface displacements in Toyoura sand during triaxial tests. They were able to observe the rotation of sand particles and their displacement in shear bands as a result of the expansion of the voids associated with dilation. Cilona et al. (2014) investigated the effects of rock heterogeneity on the localization of compaction in porous carbonates. They performed a micro-structural characterization of deformation bands under different strain and stress conditions and determined the crack distribution and density. Compaction bands were found to be localized in the laminae where porosity was initially higher. Renard et al. (2004) performed a series of uniaxial stress-driven dissolution experiments on halite aggregates for the purpose of studying their deformation as a result of pressure solution. They measured directly the vertical shortening of the sample on the radiographs, determining that the reduction in permeability from 2.1 to 0.15 Darcy (after 18.2% compaction) was linked to grain indentation and pore connectivity reduction by precipitation on the free surface of pore throats. Peysson et al. (2011) investigated permeability alteration due to salt precipitation during drying of brine, and reported the accumulation of salt near the sample, which led to a reduction in permeability. In this study, complete pore sealing due to the precipitation of salt resulted in very low permeability. In contrast, a drying experiment conducted by Noiriel et al. (2010), in which intense fracturing was induced by halite precipitation, involved an increase in permeability. Rougelot et al. (2009) were able to link micro-crack formation in a cementitious material containing 35% glass beads to the leaching of calcium from the cement paste by ammonium nitrate on the basis of the observation of a higher density of micro-cracks around the glass beads. Using numerical simulations complementary to their observations, the authors showed that cracking was inherent to the initial pre-stressing of the composite, inducing tensile stress to develop around glass beads as the mechanical properties of the leached cement paste deteriorated. Dewanckele et al. (2012) quantified porosity changes in a building rock sample during weathering processes by SO2. Despite a reduction in pore size, the authors observed an increase in porosity as a result of the formation of micro-cracks in the rock caused by efflorescence.

EMERGING APPLICATIONS Effect of mineral reaction kinetics on evolution of the physical pore space The dissolution and precipitation kinetics of minerals, combined with the effect of transport of species to and from the fluid–mineral interface, exerts an important control on the processes

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of mass transfer between fluids and minerals along a flow path. The relative importance of surface versus transport control during dissolution/precipitation has been extensively discussed (e.g., Plummer et al. 1978; Berner 1980; Rickard and Sjöberg 1983; Brantley 2008; Steefel 2008). The overall reaction rate integrates both the effects of geochemical reactions and transport of species, which is difficult to access in transport-controlled conditions, as transport processes can be highly heterogeneous in rocks. The transition state theory (TST) is generally considered appropriate to describe geochemical reaction rates (Lasaga and Kirkpatrick 1981). A general equation to describe dissolution/precipitation reaction rates of a mineral by TST is given by: rmin  

j dnmin   kr Sr   kr ' Sr [ ai ] (1   n )n ' , dt i

(5)

with kr and kr ' the reaction and intrinsic kinetic rate constants, respectively (mol∙m-2∙s-1), Sr the reactive surface area, ai the activity of the inhibitor and catalyst species i of the reaction, and  the saturation index (   IAP / K sp ), with Ksp the solubility and IAP the ion activity product. j, n and n’ are semi-empirical constants the values of which depend on the kinetic behavior involved in the chemical reaction. Rates of mineral–water interaction have been conventionally determined from bulk measurement during well-stirred flow-through experiments, after determination of the mineral surface area using the BET method; this assumes that the surface area SBET scales linearly with the reactive surface area Sr. However, the generally observed increase in geometric surface area that occurs without a corresponding change of reactivity (Gautier et al. 2001), along with the large discrepancies in observed roughness factors between fresh and weathered minerals (Anbeek 1992), suggests that (i) the density of reactive sites is poorly correlated with geometric surface area and (ii) the reactive surface may change during reactions as observed in numerous studies (Arvidson et al. 2003; Lüttge et al. 2003; Hodson 2006; Noiriel et al. 2009). More recently, the determination of bulk rates of dissolution/precipitation have been augmented by studies of surface processes at the microscopic scale using various techniques such as atomic force microscopy (AFM) (e.g., Hillner et al. 1992; Stipp et al. 1994; Dove and Platt 1996; Shiraki et al. 2000; Teng et al. 2000; Teng 2004) and optical interferometry (e.g., Sjöberg and Rickard 1985; Lüttge et al. 1999; Arvidson et al. 2003; Fischer and Luttge 2007; Colombani 2008; Cama et al. 2010; King et al. 2014). With these characterization techniques, experiments can be performed under continuous flow conditions and the dynamics of the mineral surface can be resolved with a lateral and vertical resolution of ~50 nm and ~0.1 nm, respectively. Although they have also been applied to the polished surfaces of fine-grained rocks (e.g., Emmanuel 2014; Levenson and Emmanuel 2014), these methods are mostly limited to the study of micrometer-scale oriented crystal faces due to the limited depth of field possible. XMT can be used in complement to these methods to provide full 3-D measurements of precipitation or dissolution rates. In addition, coupling flow-through experiments with XMT observation gives access to overall reactions rates that integrate both chemical reactions and transport effects over time and space.

Rates of dissolution/precipitation reactions in porous rocks As a result of mineral dissolution/precipitation associated with reactive fluid injection or mixing, the fluid–rock interface can evolve over time, thus modifying the flow and transport properties of rocks. The velocity of the moving interface is generally non-uniform through space and time as a result of the difference in reaction rates between minerals and the modification of transport processes at the pore scale. XMT, after segmentation and registration of 3-D data sets, offers an alternative method for calculating rates in such dynamic systems at the pore scale. Rates integrate the effects of both chemical reactions and local transport

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conditions; they can be measured through time-lapse positioning of the fluid–solid interface. Mapping of the displacement velocity of the fluid–mineral (or fluid–solid) interface is possible from image subtraction. The surface-normal velocity (m∙s-1) is defined by: dI fs  n , (6) dt with I fs the vector position of the fluid–solid interface and n the normal to the surface. By combining a distance transform (e.g., Pitas 2000) of the original data set with the image difference, it possible to evaluate the displacement of the interface over time and thus calculate the velocity. Integration of the velocity over a surface allows for calculation of the local dissolution/precipitation rate, r (mol∙s-1): vfs 

1 1 dV vfs dS   , (7)  S V V V dt with S the surface area (m2), V the local volume dissolved/precipitated (m3) and V the mineral molar volume (m3∙mol-1). The kinetic rate kr (mol∙m-2∙s-1) can also be calculated, assuming that the reactive surface area is known, i.e., kr  r / Sr . r

An example of a pore-scale rate calculation is shown in Figure 12, which shows the calcite precipitation rates determined in a flow-through experiment in a column initially packed with glass beads and aragonite grains and injected with a supersaturated solution. Image subtraction after 3-D registration of the data sets makes it possible to localize the newly formed crystals (Fig. 12a) that precipitated primarily around the aragonite grains (a few crystals also developed locally on glass beads). The sizes of the new crystals indicate that precipitation rates were heterogeneous over the duration of the experiment. Different rates are linked to chemical conditions at the fluid–rock interface at the micrometer or smaller scale (e.g., the local saturation index loc), growth rate according to the different spar crystal faces, growth competition between crystals, and nucleation stage. Note that there is no indication that crystals have nucleated and grown on glass beads from the very beginning of the experiment. Combining a distance transform of the original material with the newly formed crystals along with the determination of their local maxima and propagation of these maxima allows for determination of the growth velocity of every new crystal, from Equation (6) (Fig. 12b). The growth velocity is shown to be between 0 (i.e., no precipitation) and 4.5 μm∙d-1.

0

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Figure 12. (a) XMT truncated volume (1.56 × 1.56 × 1.56 mm3) showing precipitation of calcite crystals (dark blue) on glass beads (orange) and aragonite ooids (white-yellow); the experimental conditions were similar to that described in Noiriel et al. (2012). (b) Determination of the local velocities of precipitation (greyscale) of the newly formed crystals.

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Rates of dissolution/precipitation reactions in fractures Application of Equation (7) to dissolution within a planar fracture reduces to: r

S am . V t

(8)

An example of a calcite dissolution rate calculation from a 95-hour flow-through experiment in an artificially fractured limestone is shown in Figure 13. The experiment consisted of injecting deionized water equilibrated with a partial pressure of CO2 of 1 bar at a (a)

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Figure 13. Map of fracture aperture (a) before (theoretical) and (b) after experiment (obtained from XMT, see Noiriel et al. (2013) for details about the aperture mapping procedure). (c) Determination of the local rate of dissolution averaged over the experiment duration kr ' (mol∙m-2∙s-1) and (d) corresponding theoretical pH using a simplified approach and kinetic data from Plummer et al. (1978) and a local reactive surface area equal to the surface area of the pixel (4.91 × 4.91 μm). Reprinted from Noiriel (2005).

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constant flow rate Q of 10 cm3·h-1. Solution pH was determined to an average of 3.9 and 5.5 at the inlet and outlet, respectively. A trapezoid fracture of 15 mm long and 7.5 mm wide was obtained after stitching two flat (polished) fracture walls together. To induce a heterogeneous flow field over the fracture width, the fracture aperture was varied from 20 μm (shorter base) to 45 μm (longer base). As the rock used was a very fine-grained and pure limestone, the reactive surface area can be assumed to be almost constant over space and time. A map of fracture aperture at the end of the experiment shows the development of a principal flowpath within which dissolution has been enhanced, and this corresponds to the portion of the fracture with a larger initial aperture. Assuming that the fracture can be represented as a continuum, dissolution is steadystate, and reactive surface area scales with the XMT geometric surface area (see further discussion), it is possible to recalculate the apparent kinetic rate ( kr ' ) and even a theoretical pH based on kinetic law. Taking S = Sr , for example, and the kinetic law obtained by Plummer et al. (1978) (kr '  k1aH  k2 aCO2  k3) yields:

kr ' 

S 1 a   pix t Vcalcite Sr

(9)

and Spix k a  k3   a   2 CO2 pH cal   log( aH )   log  . k1  t calcite k1Sr 

(10)

Results presented in Figure 13 demonstrate that kr ' measured locally in the fracture void integrates transport effects, the highest values being obtained within the main flowpath, where residence time is shorter and the average flow velocity higher. Formation of preferential flow pathways is often associated with a reduction in the flux of dissolved species at the outlet (Noiriel et al. 2005; Luquot and Gouze 2009). This could be interpreted as the result of modification of the reactive surface, although it appears more likely that it is actually the result of a reduction in the “transport efficiency” in areas where the flow rate was reduced. The result is a situation in which transport progressively shifts from advective– dispersive-dominant to diffusion-dominant, as a result of the decrease in fluid velocity. Comparison with the experimental results shows that the average pH is overestimated by the calculation at the inlet (pH cal ~ 4.1) and underestimated at the outlet (pH cal ~ 4.4). These differences can arise from the kinetic formulation used, the estimation of the surface area, the assumption of a continuum (i.e., constant pH values across the fracture walls) and steady-state dissolution. For the calculation, the dissolution rate was taken to be constant everywhere within the fracture over the course of the experiment, although flowfield calculations indicated that the flow velocity decreased by one order of magnitude. However, only a direct or inverse fully coupled modeling approach of reactive transport, taking into account feedback between chemistry, flow, and transport at the micro-scale, associated with upscaling from the pore scale to the sample-scale, could allow for a proper interpretation of the XMT observations. As shown in Figure 13, the initial heterogeneous flow field within fractured (or porous) media can quickly lead to localization of the flow through the primary flow paths (see also Fig. 16) as a result of heterogeneous transport along the different flow paths. However, incomplete transverse mixing across the fracture walls may also result in heterogeneous rates of reaction, particularly when there are differences between areas of higher and smaller fracture aperture, and for which velocity profiles vary. Li et al. (2008) showed that development of concentration gradients within single pores or fractures can lead to scale-dependent reaction rates. The highest discrepancies between pore-scale and continuum-scale models were observed for a mixed kinetic control (i.e., similar characteristic times for transport and surface-reaction) due to comparable rates of

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chemical reaction and advective transport. Molins et al. (2014) also compared pore-scale with continuum-scale simulations of calcite dissolution inside a capillary. They pointed out the effect of larger diffusive boundary layers formed around grains and in slow-flowing pore spaces that exchanged mass by diffusion with fast flow paths. Overall, the assumption of well-mixed conditions—the conceptual basis of the continuum scale model—did not apply in the capillary tube because of the local importance of transport limitations to the bulk (or effective) rates.

Porosity and permeability development in porous media and fractures Porosity and permeability development in porous media. The relationship between porosity and permeability in rocks is complex and varies according to the porous and mineral networks and the geological processes involved (Bourbié and Zinszner 1985). Several models linking porosity to permeability have been proposed (e.g., Carman 1937; Bear 1972), but their predictive capacity is usually limited due to the lack of in situ observations of the parameters that control the dynamic of the porosity–permeability relationship. Combining XMT observation and determination of porosity, along with the measurement of permeability, allows us to examine more closely the relationship between these two major reservoir properties. Noiriel et al. (2004), for example, linked two distinct porosity–permeability power law relationships (Fig. 14a) to different processes that occurred successively during a flow-through dissolution

(a)

(b)

(c)

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Figure 14. (a) Porosity-permeability relationship within a porous limestone during dissolution; n is the exponent of the power-law relationship between porosity and permeability: ~ kn. [Used by permission of Wiley, Noiriel C, Gouze P, Bernard D (2004) Investigation of porosity and permeability effects from microstructure changes during limestone dissolution. Geophysical Research Letters, Vol. 31, L24603, Fig. 4]. (b) Porous network before experiment, and visualization of the porous network connectivity increase after 1.5 h (c) and 22.5 h (d) of the flow-through experiment, respectively (reprinted from Noiriel 2005). The volumes are approximately 10 × 10 × 6.9 mm3.

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experiment. First, a permeability increase of an order of magnitude for a porosity change of only 0.3% was linked to micro-crystalline particle migration within the rock (Fig. 7a). Only a small fraction of the pore space was involved at that stage of the experiment. Permeability increased later in the experiment as both the pore wall roughness decreased and the pore network connectivity increased, as shown in Figure 14. Porosity–permeability evolution in reactive rocks results from the physical evolution of the pore space, which in turn results from interplay between reaction kinetics and advective and diffusive transport. Possible feedback between the flow regime and geochemical alteration can lead to instabilities and localization or divergence of the flow, depending on whether dissolution or precipitation is involved. The two parameters used for characterizing and predicting these phenomena are the Péclet (Pe) and Damköhler (Da) numbers (e.g., Steefel and Lasaga 1990; Hoefner and Fogler 1998), defined locally as: Pe  v L* / Dm and Da  kr L*2 / Dm , where v is the fluid velocity (m·s-1), Dm is molecular diffusion (m2·s-1), kr is a first order kinetic constant (s-1), and L* is a characteristic length (m), e.g., the fracture aperture or pore size. Starting from a very homogenous rock in terms of structure and mineralogy, the changes in the rock geometry resulting from dissolution generally follow a pattern determined by the couple of the Damköhler and Péclet numbers. When reaction kinetics are slow compared with transport (low Da), dissolution is rather uniform and ramified reaction fronts result (Hoefner and Fogler 1998; Golfier et al. 2002), whereas compact and channelized dissolution can be observed at high Da, with a dissolution front advancing from the point of injection. Between these values, conical, dominant, or ramified wormholes can form. The patterns are directly linked to dissolution instabilities: as the reactive fluid infiltrates areas of higher permeability, a positive feedback between transport and chemical reactions develops, and leads to the growth of the wormholes (Ortoleva et al. 1987; Steefel and Lasaga 1990; Daccord et al. 1993). Several experiments have been carried out to investigate the effect of dissolution regime on porosity development in the context of CO2 sequestration. Different values of Da were explored by either adjusting the fluid velocity (Luquot and Gouze 2009; Luhmann et al. 2014) or reactivity (Carroll et al. 2013; Smith et al. 2013a). Luquot and Gouze (2009) observed homogeneous dissolution over the length of a core (named D3) at the highest flow rate (lowest Da), whereas gradients in reaction developed at lower flow rates (higher Da), resulting in wormholes associated with highly heterogeneous porosity development within two other cores (named D1 and D2, Fig. 15). Smith et al. (2013a) investigated the effects of pore-space heterogeneity on the development of dissolution fronts in a vuggy limestone and marly dolostone. They observed that a homogeneous pore-space distribution (90% of pore sizes differing by only one order of magnitude) resulted in stable, uniformly advancing dissolution fronts associated with porosity increase and only minor changes in permeability. Conversely, heterogeneous pore space distributions (90% of pore sizes spanning at least 3.5 orders of magnitude) resulted in greater variability in local fluid velocity and mass transfer rates, leading to the formation of unstable dissolution associated with dramatic permeability increases of several orders of magnitude. Although the Péclet and Damköhler numbers can provide some useful information about the evolution of a particular system, these non-dimensional parameters must be used with care. First, the existence of feedback that enhances permeability implies that the fluid velocity is not constant with time. Second, the dissolution kinetics of a reactive fluid through a rock is not constant and can vary over one or two orders of magnitude between far-from-equilibrium and close-to-equilibrium states. As a result, both Da and Pe numbers evolve with space and time. Finally, heterogeneities present initially in rocks can exert a first-degree influence on the type of dissolution front and resulting relationship between porosity and permeability (Smith et al. 2013a), thus competing with dissolution instabilities linked purely to the couple (Pe, Da).

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Figure 15. Porous network obtained with XMT within 3 limestone mini-cores (diameter is 9 mm and length 4.5 mm) during dissolution experiments at different Damköhler number, leading to different porosity-permeability evolution, reprinted from Luquot and Gouze (2009). [Used by permission of Elsevier, from Luquot L and Gouze P (2009) Experimental determination of porosity and permeability changes induced by injection of CO2 into carbonate rocks. Chemical Geology, Vol. 265, 148–159, Fig. 7].

Porosity and permeability development in fractures. The development of wormholes can also be observed in fractures, even in very flat artificial ones. The following example illustrates how dissolution kinetics can lead to different dissolution patterns in fracture, even though Da and Pe values were similar at the inlet of the samples at the beginning of each experiment (i.e., at t = 0 and z = 0). The effects of reaction kinetic law have been explored through two flow-through dissolution experiments. Two different inlet fluids at pH 3.9, either deionized water + HCl or deionized water equilibrated with PCO2 = 1 bar, were injected at a flow rate Q = 100 cm3∙h-1 in artificial, almost flat, limestone fractures of initial aperture ~25 μm. While the fracture injected with HCl shows the formation of dominant wormholes (Fig. 16a), the fracture injected with CO2 does not exhibit any particularly localized wormholes (Fig. 16b), despite the flow beginning to localize due to slight heterogeneities in the initial aperture field. The difference arises from different kinetic paths taken by the two different inlet fluids during calcite dissolution. The dissolution rate of calcite is influenced by pH, PCO2 and surface reaction (Plummer et al. 1978). As the fluid reacts with calcite, pH increases and dissolution rates decrease. However, the decrease is higher (~2 orders of magnitude) and more abrupt for HCl than for CO2 (~1 order of magnitude) (Fig. 17), as a result of the buffering effect of the carbonated species. This example shows clearly that very different dissolution patterns can develop despite similar initial Da and Pe values, due to the kinetic reaction path followed by the reactive fluid.

Effects of texture and mineralogy on complex porosity–permeability relationships and transport Even as 3-D mineralogical mapping of rocks remains a challenge, advances in XMT have illuminated the role of rock mineral composition and spatial distribution during geochemical processes. Primary rock texture and mineralogy have an important influence on the evolution of flow and transport properties, as pointed out in recent studies at the pore scale. For example, opposite trends of the porosity and permeability evolution have been observed, despite the fact

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Figure 16. Maps of fracture aperture (μm) obtained with XMT (resolution 4.91 μm). Initial aperture is indicated on the fracture edges, where glue resin was added to avoid mechanical closure. Different wormhole regimes are observed in the planar fracture after injection of (a) water + HCl (inlet pH = 3.9) or (b) water equilibrated with PCO2 = 1bar (inlet pH = 3.9). Reprinted from Noiriel (2005).

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that net dissolution occurred in the system (e.g., Noiriel et al. 2007a; Ellis et al. 2013; Luquot et al. 2014a). Ellis et al. (2013) even reported opposite outcomes with respect to fracture permeability evolution for two very similar experiments performed on nearly identical rock samples. In the experiment of Noiriel et al. (2007a), the heterogeneity of the dissolution rates (differing by about one order of magnitude for calcite and dolomite (Chou et al. 1989) and ten orders of magnitude for clays and carbonates (Köhler et al. 2003)) of the minerals comprising the rock matrix led to the development of dissolution heterogeneities at the fluid–rock interface. Increase of tortuosity and fracture roughness led in turn to a reduction in permeability, despite net dissolution. Roughness and tortuosity increases associated with large differences in the dissolution rate of minerals were also reported by Gouze et al. (2003), Noiriel et al. (2007a, 2013), and Ellis et al. (2011) (Fig. 18; see also Fig. 8).

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Substantial reductions in permeability were also observed or implied in several experiments after the matrix-forming minerals had lost their cohesion and individual grains were being removed and transported by the fluid, thus contributing to pore clogging (Noiriel et al. 2007b; Ellis et al. 2013; Mangane et al. 2013; Qajar et al. 2013; Sell et al. 2013). An increase in the roughness of the fluid–rock interface due to spatially and mineralogically heterogeneous reaction rates affects the transport of reactants and dissolved species to and from the fluid–mineral interface. For example, Noiriel et al. (2007b) observed an exponential decrease with time of the flux of dissolved species in a fractured argillaceous limestone during dissolution. By increasing diffusion compared to advection close to a mineral surface, dissolution of calcite grains combined with the development of a micro-porous clay coating nearby the fluid–mineral interface resulted in a decrease of transport “efficiency” (Fig. 19).

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The scale dependence and the effects of mineral spatial distribution on reaction rates is also an important topic, as demonstrated in various experimental and numerical modeling studies (Li et al. 2006, 2007, 2013; Kim and Lindquist 2013; Salehikhoo et al. 2013). Such effects originate from the differences in the rates of mass transport between reactive and nonreactive pores and also depend on the spatial distributions of reactive minerals (Li et al. 2007).

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Figure 18. (a) XMT image difference emphasizing the development of an altered zone along the fracture surface and (b) SEM observation of surface roughness increase in the area highlighted by box 2 in (a). Reprinted from Ellis et al. (2011) [Used by permission of Wiley, from Ellis BR, Peters C, Fitts J, Bromhal G, McIntyre D, Warzinski R, Rosenbaum E (2011) Deterioration of a fractured carbonate caprock exposed to CO2-acidified brine flow. Greenhouse Gas Science and Technology, Vol. 1, 248–260, Figs. 5 and 6]. (c) SEM observation of fracture morphology changes in a limestone due to heterogeneous dissolution rates between minerals. (a)

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Figure 19. (a) XMT observation of fracture void (low grayscale) geometry evolution in argillaceous limestone (high grayscale) during dissolution, showing the development of a micro-porous clay coating (intermediate grayscale). (b) Schematic representation of the transport phenomena in the fracture, initially (left) and since the clay coating has started to form (right). Reprinted from Noiriel C, Madé B, Gouze P (2007b) Impact of coating development on the hydraulic and transport properties in argillaceous limestone fracture. Water Resources Research, Vol. 43, W09046, Fig. 11].

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Geometrical changes accompanying dissolution of anorthite in a sandstone also resulted in continuously changing upscaled reaction rates, as shown by Kim and Lindquist (2013) using a pore-network model that took into account changes in pore volume, reactive surface area, and topological changes.

Effects of pore-scale heterogeneity on permeability reduction As discussed above, permeability is closely linked to pore-scale heterogeneity. In the case of mineral precipitation, the location of the newly formed crystals is constrained by the induction time for nucleation (i.e., the time required before growth can initiate) and the degree of epitaxy between pre-existing mineral surfaces and newly formed crystals. In addition, the growth rate of newly formed crystals depends on their orientation (linked to the orientation of pre-existing mineral faces when growth is epitaxial), the different facets exhibiting different growth rates (e.g., Hilgers and Urai 2002; Nollet et al. 2006). Growth competition between neighboring crystals also involves variations of growth rate over time. Geometry changes resulting from growth at the fluid–mineral interface can have a significant effect on the hydrodynamics and transport fluid within a porous network. An example is shown here for different nucleationand-growth scenarios derived from an experimental study of precipitation in columns packed with glass beads and calcite spar crystals (Noiriel et al. 2012). The growth of newly formed calcite crystals was epitaxial on calcite spar, leading to a smooth surface; conversely sparse rhombohedra of calcite developed on glass beads, leading to a rougher surface (Fig. 6). The experiment illustrates, that under similar geochemical and flow rate conditions, the variability of location, shape, and size of the new crystals depend on the pre-existing mineral substrate (i.e., either calcite spar or glass beads). To examine the effects of evolving surface roughness on pore hydrodynamics, different growth scenarios were generated and combined with flow modeling. Starting from an initial sub-volume of 300 × 300 × 300 voxels, an algorithm was used to nucleate and grow cubic crystals at the calcite spar surface to mimic experimental observations (Fig. 20; see also Fig. 6) so as to compare the evolution of permeability between uniform (case 1) and heterogeneous (case 2) case of precipitation. Permeability was calculated by solving Stokes’ equations based on a volume averaging technique (Batchelor 1967; Quintard and Whitaker 1996). The pores with heterogeneous precipitation (case 2) exhibited a higher reduction in permeability in the simulations compared with the uniform case (case 1, Fig. 21a). Reduction in permeability is linked to the decrease in porosity that induces pore-throat size reduction. However, this example illustrates than a similar reduction in porosity can lead to different degrees of permeability reduction, as it also depends significantly on the fluid–mineral interface roughness, and so the location and shape of the newly formed crystals. Quantification of the fluid–solid surface roughness factor illustrates that trend (Fig. 21b). The effect of roughness development on the transport of reacting species and precipitation rate near the interface is far more complex to assess, as the dynamics of growth are much more difficult to unravel in systems where flow velocity and species concentration fields evolve continuously along with the surface area of the crystal facets. Nollet et al. (2006) noted changes of growth rates of crystal faces through time, which they attributed to growth of individual facets, growth competition, flow direction, and changes of flow velocity. Local modifications of hydrodynamics and transport near the interface are likely involved in this case.

CONCLUDING THOUGHTS Rocks and other subsurface materials exhibit both a structural and behavioral complexity during hydro-mechano-chemical processes, which makes predicting flow and reactive transport in them challenging. Advances in XMT have begun to provide a better mechanistic

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understanding of the HMC coupling through direct rendering of the flow, transport, and chemical processes at the pore scale. XMT used together with physico-chemical measurements during experiments and finer-scale observations offers the opportunity to better access and understand reaction-induced changes of pore structure, permeability and surface reactivity, as a result of the interplay between geochemical reactions, changing the rock geometry, and the hydrodynamic and transport properties. In particular, the ability offered by 4D XMT to follow the movement of the fluid–rock interface and to measure local rates of dissolution/ precipitation through time gives new insights about rock and mineral reactivity in open hydrological systems. The potential of combining XMT experiments with numerical simulations to study porescale processes in rocks is far from mature, but shows encouraging developments. Upscaling experimental laboratory data to reservoir scale is one of the most challenging issues for predicting the long-term evolution of reservoirs in the presence of reactive fluids. Refining of macro-scale modeling will be possible once pore-scale modeling has reached maturity. Even if direct simulation and network modeling of flow and solute transport are now used routinely to calculate averaged properties at different scales, the coupling between chemical reaction, rock geometry modifications, and hydrodynamic and transport properties remains insufficiently constrained, particularly in physically and mineralogically heterogeneous rocks. Indeed, experimental observations have established some of the relationships between microscale features and macro-scale properties and this can lead in certain cases to potentially

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unexpected (emergent) effects. Our knowledge of fluid–rock interaction in chemically-reactive environments will have improved significantly once it is possible to integrate 4D XMT data with fully coupled modeling of reactive transport that include multicomponent chemistry, fluid velocity distribution and transport of solute species in fluid phase.

ACKNOWLEDGMENTS I am grateful for the constructive reviews of the manuscript and helpful comments provided by Maša Prodanovič, Marco Voltolini and Carl Steefel.

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 287-329, 2015 Copyright © Mineralogical Society of America

Ionic Transport in Nano-Porous Clays with Consideration of Electrostatic Effects Christophe Tournassat Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California 94720, USA and Water, Environment and Ecotechnology Division French Geological Survey (BRGM) Orléans, France [email protected]

Carl I. Steefel Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California 94720, USA INTRODUCTION The mineralogical and chemical properties of clays have been the subject of longstanding study in the research community—in fact, entire journals are devoted to the topic. In the field of hydrology where transport behavior is more routinely considered, clays and clay-rich rock were largely relegated to a minor role because of their low hydraulic conductivity. However, this very property explains in part the renewed interest in the behavior of clays and clay rocks in several important subsurface energy-related applications, including the long-term disposal of nuclear wastes in geological repositories and the storage of CO2 in subsurface geological formations. In these applications and environments, the low permeability of the clay-rich formations or engineered barriers provides at least part of the safety functions for radionuclide contaminants confinement and subsurface CO2 sequestration. From a geochemical and mineralogical point of view, the high adsorption capacity of clay minerals adds to the effect of low hydraulic conductivities by greatly increasing the retardation of radionuclides and other contaminants, making clays ideal where isolation from the biosphere is desired. The low permeability of clay-rich shales also explains why hydrocarbon resources are not easily exploited from these formations, thus requiring in many cases special procedures like hydraulic fracturing in order to extract them. Clay properties remain also topic of intensive research in the oilfield industry in connection with their swelling behavior, which has an adverse impact on drilling operations (Anderson et al. 2010; Wilson and Wilson 2014; De Carvalho Balaban et al. 2015). While the low permeability and high adsorption capacity of clay minerals are widely acknowledged, it is clear nonetheless that there is a need for an improved understanding of how the chemical and mineralogical properties of clay rocks impacts transport through them. It is at the pore-scale that the chemical properties of clay minerals become important since their electrostatic properties can play a large role. For all above examples, it is necessary to predict gas, water and oil transport properties in clay materials porosity as a function of a range of physical and chemical contexts. For example, the prediction of diffusion and retention properties of clay-rocks and engineered bentonite barriers are of paramount importance for waste storage 1529-6466/14/0080-0009$05.00

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applications in order to assess the long-term safety of the storage systems under consideration. In these systems, advective flux is negligible as compared to diffusive flux since the clays and clayrich rock have permeability values as low as 10-21–10-19 m2 (Delay et al. 2006, 2014). Examples include the clay-rock from the Callovian-Oxfordian layers in Bure (France) (Homand et al. 2006) and the clay-rock from the Opalinus Clay formation in Mont-Terri (Pearson et al. 2003). Safety assessment calculations must be carried out over periods ranging from a few hundreds to hundreds of thousand years (Altmann 2008). This is despite the fact that the interpretation of laboratory results are typically obtained i) on time periods ranging from days to years and ii) on samples with centimeter dimension. This suggests the need for a good knowledge of the fundamental properties of clays at the pore-scale is required in order to extend these analyses to the longer time and space scales. Clay transport properties are however not simple to model, as they deviate in many cases from predictions made with models developed previously for “conventional” porous media such as permeable aquifers (e.g., sandstone). For example, water flow in clay cannot be described consistently with Darcy’s law (Neuzil 1986). In addition, model predictions must also take into account spatial heterogeneities, which can be time-dependent due to physical and chemical perturbations. In this respect, a significant advantage of modern reactive transport models is their ability to handle complex geometries and chemistry, heterogeneities and transient conditions (Steefel et al. 2014). Indeed, numerical calculations have become one of the principal means by which the gaps between current process knowledge and defensible predictions in the environmental sciences can be bridged (Miller et al. 2010). In this regard, the present article approaches the topic of pore-scale transport through clays and clay-rich rocks by adopting a reactive transport approach (Steefel and Maher 2009). This article begins with a discussion of the classic theory of diffusion in porous media, the limitations of which for clay media are highlighted with selected diffusion experiment results from the literature. The second section introduces specifics of clay minerals and clay materials that explain some of the inconsistencies found in using the classical diffusion theory. In a third section, constitutive equations for diffusion in clay porous media are proposed and the link between the predictions made with these equations and the experimental results is achieved through simple examples. In a fourth section, reactive transport (RT) model applications to the selected examples provide the basis for a discussion on diffusion process understanding and the current limitations of the proposed approach. In a final section, a summary is provided, together with perspectives on RT models and further code development that is needed.

CLASSICAL FICKIAN DIFFUSION THEORY Diffusion basics Diffusion processes are most often treated in terms of Fick’s laws. Fick’s first law states that the diffusive flux of a species i in solution (Ji) is proportional to its concentration (ci) gradient (here in 1-D along x) (Steefel et al. 2014):

J i   De,i

ci , x

(1)

where De,i is the effective diffusion coefficient that is specific to the chemical species i. The diffusion coefficient includes a correction for the tortuosity () and the porosity () of the porous media: De,i    D p,i      D0,i ,

(2)

where D0,i is the diffusion coefficient of species i in water (or self-diffusion coefficient), and D p,i is the pore diffusion coefficient (D p,i    D0,i ). The tortuosity is defined as the square of the ratio of the path length the solute would follow in water alone, L, relative to the tortuous

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 289 path length, it would follow in porous media, Le (Bear 1972):

   L / Le  . 2

(3)

Note that the terminology of the diffusion coefficient terms is very diverse (Shackelford and Moore 2013). The terminology presented here is the most commonly used in geosciences. In particular the effective diffusion coefficient is defined here to include the porosity. Fick’s second law is derived from the mass conservation law that includes the divergence of the flux:

Ctot,i J  i, t x

(4)

where Ctot,i is the concentration of species i in the porous media (i.e., the amount of species i in the solution and in the solid normalized to the solution and solid volumes). If the species i is in the solution only then: ci   c    De,i i  . t x  x 

(5)

If the species i is also adsorbed on or incorporated into the solid phase, then it is possible to define a rock capacity factor  that relates the concentration in the porous media to the concentration in solution: i 

Ctot,i . ci

(6)

The quantification of adsorption processes is commonly translated into distribution ratio values, Rd (L·kg-1): Rdi 

csurf ,i , ci

(7)

where csurf, i is the concentration on the surface of the element of interest (mol·kg-1). If the concentration of species i on the solid is only due to adsorption processes, then Equation (6) can be combined with Equation (7), yielding:

i    d Rdi ,

(8)

where d is bulk dry density of the material. In that case, Equation (4) transforms into: i ci   c    De,i i  . t x  x 

(9)

When interpreting diffusion data, the distribution ratio is commonly assumed to be constant (the adsorption is linearly dependent on the concentration) and representative of an instantaneous and reversible adsorption process. Under these conditions, the Rd value is designated as the distribution coefficient, KD. If it is further assumed that the media is homogeneous, Equation (9) reduces to:

De,i  2ci ci .  t   d K Di x 2

(10)

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Anion, cation and water diffusion in clay materials Diffusion parameters for cations, anions and water in clay materials have been extensively studied in the literature. Various experimental setups have been used to determine porosity, diffusion coefficients, tortuosity and –KD values. In the following, results obtained by Tachi and Yotsuji (2014), who performed through-diffusion experiments in order to study the diffusion of HTO, I-, Na+ and Cs+ in montmorillonite samples compacted at a bulk dry density of d  0.8 kg·dm-3, are discussed. In the context of the geometry of their experimental setup (Fig. 1), all parameters of Equation (10) can be determined simultaneously by fitting experimental points to the analytical solution: C  x, t  

C  0,0     1

 2C  0,0  

 De  2m   Lx L  x    exp   t     cos  m     m sin  m    2  L  L      L   ,  2 m 1    m        1  cos m        2   m  sin m

 

(11)

 

where: 

 AL , Vin

(12)

Vout , Vin

(13)



and where m fulfills:

 

tan m 

     1  m .   2m  2

(14)

C(0,0) is the initial concentration in the inlet reservoir with volume Vin (mol m-3); Vout is the volume of the outlet reservoir (mol m-3), A is the cross-sectional area of the sample (m2) and L is the thickness of the sample (m). Diffusion parameters derived from fitting of the data presented in Figure 2 with Equation (11) are given in Table 1. The rock capacity factor for water (~0.77) is consistent with the total porosity value that can be obtained by considering the clay mineral “grain” density (g~2.7–2.8 kg dm-3) and the bulk dry density of the material according to:

  1

d . g

(15)

This result is in agreement with the case where the entire porosity is assumed (or treated) as fully connected for diffusion of water as a tracer (tritiated water, or HTO). The higher  value for cations than for water can be related to their adsorption to the solid surfaces (KD > 0). The lower  values for anions than for water indicate that anions do not have access to all of the porosity. This result is a first direct evidence of the limitation of the classic Fickian diffusion theory when applied to clay porous media: it is not possible to model the diffusion of water and anions with the same single porosity model. The observation of a lower  value for anions than for water led to the development of the important concept of anion accessible porosity (sometimes also improperly named ‘geochemical’ porosity) to be compared to the

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 291

Figure 1. Example of a through diffusion cell setup: (a) inlet reservoir, (b) peristaltic pump, (c) throughdiffusion cell, and (d) outlet reservoir. Arrow heads indicates the circulation of water from the reservoir to the filter in order to homogenize the inlet and outlet solutions compositions. [Figure from Tachi and Yotsuji (2014) Diffusion and sorption of Cs+, Na+, I- and HTO in compacted sodium montmorillonite as a function of porewater salinity: Integrated sorption and diffusion model. Geochimica et Cosmochimica Acta Vol. 132, p 75–93. Reproduced with the permission from Elsevier.]

water saturated or total porosity (Pearson 1999). The ratio of the anion accessible porosity to the total porosity depends not only on the nature of the clay minerals in the material, but also on the chemical conditions, particularly the ionic strength (Glaus et al. 2010; Tournassat and Appelo 2011). Thus, changes in chemical conditions can lead to significant modifications of anion diffusion properties (Fig. 3). The effective diffusion coefficients normalized to the self-diffusion coefficient values depend on the nature of the aqueous species, with effective diffusion coefficient values in the order (De/D0)Cs+ > (De/D0)Na+ > (De/D0)HTO > (De/D0)I- when measured under similar experimental conditions (Table 1). Higher effective diffusion coefficient values for cations than for neutral species and higher values for neutral species than for anions have been reported repeatedly in the literature for clay materials (Nakashima 2002; Van Loon et al. 2003, 2004a,b, 2005; García-Gutiérrez et al. 2004; Appelo and Wersin 2007; Glaus et al. 2007, 2010; Descostes et al. 2008; Wersin et al. 2008; Birgersson and Karnland 2009; Melkior et al. 2009; Appelo et al. 2010; Gimmi and Kosakowski 2011; Wittebroodt et al. 2012).

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Figure 2. Flow-through experiment results for HTO, I-, Na+, and Cs+ diffusion in a montmorillonite sample (symbols). The continuous lines correspond to the fit of the data with Equation (11) and parameters listed in Table 1. [Figure adapted from Tachi and Yotsuji (2014) Diffusion and sorption of Cs+, Na+, I- and HTO in compacted sodium montmorillonite as a function of porewater salinity: Integrated sorption and diffusion model. Geochimica et Cosmochimica Acta Vol. 132, p 75–93, Figs. 2 and 3. Reproduced with permission from Elsevier.]

Figure 3. Diffusion data from Glaus et al. (2010) illustrating the changes in 36Cl- diffusion parameters with ionic strength. [Figure reproduced from Glaus MA, Frick S, Rosse R, Van Loon LR (2010) Comparative study of tracer diffusion of HTO, 22Na+ and 36Cl- in compacted kaolinite, illite and montmorillonite. Geochimica et Cosmochimica Acta, Vol. 74, p 1999–2010, Fig. 2. Reproduced with permission from Elsevier.]

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 293 A second fundamental problem related to the classic Fickian diffusion theory arises when the pore diffusion coefficient, Dp,i, Equation (2), is estimated for cations based on their effective diffusion coefficient values and a porosity value. Since  cannot be greater than the total porosity, it follows that D p,i  De,i /  . For Cs+ diffusion in an experiment conducted at 0.01 mol·L-1 NaCl, Dp,Cs+ values were higher than 3.1 × 10-9 m2·s-1, a value that is higher than the self-diffusion coefficient of Cs+ in pure water (2.07 × 10-9 m2·s-1, Li and Gregory, 1974). This result, also reported repeatedly in the literature (Van Loon et al. 2004b; Glaus et al. 2007; Wersin et al. 2008; Appelo et al. 2010; Gimmi and Kosakowski 2011), is not physically correct and points out the inconsistency of the classic Fickian diffusion theory for modeling diffusion processes in clay media. Again, the large changes of Cs+ diffusion parameters as a function of chemical conditions (De,Cs+ decreases when the ionic strength increases, Table 1, Figure 2) highlight the need to couple the chemical reactivity of clay materials to their transport properties in order to build reliable and predictive diffusion models. Table 1. Diffusion parameters for HTO, I-, Na+ and Cs+ in a montmorillonite sample compacted at 0.8 kg·dm-3 (sample thickness L = 10 mm, and diameter d = 20 mm). Parameters were obtained from Tachi and Yotsuji (2014) and correspond to the fitting lines shown in Figure 2 and obtained with Equation (11). D0,i values were taken from Li and Gregory (1974).

Vin

Vout



(mol·L-1)

(L)

(L)

(-)

10-11m2·s-1

(-)

HTO

0.1

0.6

0.6

0.775 ± 0.025

6.62 ± 0.11

(3.26 ± 0.005)•10-2

Na+

0.1

2

2

8.12 ± 0.20

24.4 ± 0.5

(1.83 ± 0.04)•10-1

I-

0.1

2

0.2

0.421 ± 0.017

0.01

2.5

2.5

1460 ± 50

405 ± 9

(1.96 ± 0.04)•102

0.1

2

0.2

316 ± 4

73.3 ± 4

(3.54 ± 0.02)•10-1

0.5

0.6

0.2

65.5 ± 0.8

23.4 ± 0.3

(1.13 ± 0.01)•10-1

Tracer i

Cs+

NaCl conc.

De,i

De,i / D0,i

0.694 ± 0.037 (3.47 ± 0.02)•10-3

Diffusion under a salinity gradient Most of reported diffusion experiments have been performed under spatially constant ionic strength conditions. Recently, Glaus et al. (2013) reported experimental results of 22 Na+ diffusion under a gradient of NaCl concentration. The experimental setup was similar to the one depicted in Figure 1: the inlet and outlet reservoirs contained a 0.5 mol·L-1 and 0.1 mol·L-1 NaCl solution respectively. At time t = 0, both solutions were spiked with the same concentration of 22Na+ and the concentrations in both reservoir were monitored. From Fick’s diffusion equation, it would have been expected that 22Na+ diffuses from both reservoirs at an equal rate into the clay material, eventually producing a zero concentration gradient inbetween the reservoirs (dashed lines on Figure 4). However, the experimental observations were completely different: 22Na+ accumulated in the high NaCl concentration reservoir as it was depleted in the low NaCl concentration reservoir, evidencing non-Fickian diffusion processes.

KD values obtained from static and diffusion experiments The adsorption properties of a material can be evaluated using batch (static) experiments. Batch KD values can be evaluated independently from diffusion experiments and then compared with  parameters derived from diffusion results. Unfortunately, this comparison often leads to KD values that differ from the diffusion experiment-based values, calling into question the usefulness of batch KD measurements to predict transport parameters of adsorbing

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species. For some reasons these KD discrepancies have often been attributed to the fact that the conditions of batch adsorption tests “have long been known to be unrepresentative of those existing in compacted clays”, i.e., the surface properties and/or the adsorption site accessibility depend on the compaction (Shackelford and Moore 2013). This statement finds its origin in the batch and diffusion KD discrepancies observed primarily for Cs+ in a range of published studies, with batch KD values typically higher than diffusion KD values (Miyahara et al. 1991; Tsai et al. 2001; Jakob et al. 2009; Aldaba et al. 2010). Studies of Cs+ that compared adsorption in loose and compacted clay material, without relying on diffusion experiments, led to apparently contradictory results. Oscarson et al. (1994) found that KD values for Cs+ on bentonite decreased with increasing compaction. In contrast, Montavon et al. (2006) did not observe any significant differences in KD values for differing degrees of compaction under otherwise similar experimental conditions. Van Loon et al. (2009) reached the same conclusion when comparing Cs+ adsorption on crushed clay-rock from the Opalinus Clay formation (Mont-Terri, Switzerland) dispersed in water with Cs+ adsorption on intact samples. Chen et al. (2014) concluded also that there was no effect of compaction on Cs+ adsorption on i) the clay mineral fraction of a natural clay-rock (Callovian-Oxfordian clay-rock from Bure, France) and ii) on samples of the clay-rock itself. Whether the samples were i) powdered clay-rock samples dispersed in water, or ii) re-compacted powdered clay-rock samples, or iii) intact clay-rock samples had no impact on the measured KD values. Altogether, all recent Cs+  adsorption experiments showed no effect of compaction on the KD values, thus it is necessary to find a different reason for the discrepancy between KD values derived from batch and diffusion experiments.

Figure 4. 22Na+ diffusion under a gradient of salinity. The dashed lines indicate the expected concentration profiles as a function of time in inlet and outlet reservoirs. Symbols indicate the measured concentrations. [Reprinted with permission from Glaus MA, Birgersson M, Karnland O, Van Loon LR (2013) Seeming steady-state uphill diffusion of 22Na+ in compacted montmorillonite. Environmental Science & Technology, Vol. 47, p 11522–11527, Fig. 1. Copyright 2013.]

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 295 From classic diffusion theory to process understanding The limitations of the classic Fickian diffusion theory must find their origin in the fundamental properties of the clay minerals. In the next section, these fundamental properties are linked qualitatively to some of the observations described above.

CLAY MINERAL SURFACES AND RELATED PROPERTIES Electrostatic properties, high surface area, and anion exclusion Crystallographic origin of clay mineral electrostatic properties. The fundamental structural unit of phyllosilicate clay minerals consists of layers made of a sheet of edgesharing octahedra fused to one sheet (1:1 or TO layers) or two sheets (2:1 or TOT layers) of corner-sharing tetrahedra (Fig. 5). The metals in the octahedral sheet of clay minerals consist predominantly either of divalent or trivalent cations. In the first case, all octahedral sites are occupied (trioctahedral clay minerals) whereas in the second case, only two-thirds of the octahedral sites are occupied (dioctahedral clay minerals). The clay minerals smectite and illite may constitute ~30 % of the material making up sedimentary rocks (Garrels and Mackenzie 1971). Those minerals have 2:1 layer structures and they are frequently dioctahedral. Kaolinite is also a very common clay mineral, the structure of which is dioctahedral and made up of 1:1 layers. For these three minerals, tetrahedral and octahedral cations are primarily Si4+ and Al3+ respectively. Their ideal structural formulae can be written Si2Al2O5(OH)4 for kaolinite and Si4Al2O10(OH)2 for dioctahedral illite and smectite. In the following, most examples consider illite, the principal constituent of most clay-rocks, and montmorillonite, a smectite that is the main constituent of bentonite, which is the most studied material in diffusion experiments. Illite and montmorillonite layers differ by the nature and amount of the isomorphic substitutions taking place in their octahedral and tetrahedral sheets: in montmorillonite, most

Figure 5. Structure of a (dioctahedral) TOT layer and scheme of TOT layer and compensating cations in a clay mineral particle. [Figure adapted from Tournassat C, Bizi M, Braibant G, Crouzet C (2011) Influence of montmorillonite tactoid size on Na-Ca cation exchange reactions. Journal of Colloid and Interface Science, Vol. 364, p 443–454, Fig 1. with permission from Elsevier.]

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of the substitutions occur in the octahedral sheet where Al3+ is replaced by Fe3+ or a cation of lower charge (Mg2+, Fe2+); in illite, a significant amount of the substitutions occur also in the tetrahedral sheet with Al3+ or Fe3+ replacing Si4+. Isomorphic substitutions by cations of lower charge results in negative layer charge. Smectites have negative layer charges ranging between 0.2 and 0.6 molc·mol-1 (the layer charge of montmorillonite is commonly in the range 0.2–0.45  molc·mol-1), while illite has negative layer charges values between 0.6 and 0.9 molc·mol-1 (Brigatti et al. 2013). Morphology of illite and montmorillonite particles. Clay mineral particles are made of layer stacks and the space between two adjacent layers is named the interlayer space (Fig. 5). Illite particles typically consist of 5 to 20 stacked TOT layers (Sayed Hassan et al. 2006). The interlayer spaces of illite particles are occupied by non-solvated cations (K+ and sometimes NH4+). In contrast, the interlayer spaces of montmorillonite particles are occupied by cations and variable amounts of water. The number of layers per montmorillonite particle depends on the water chemical potential and on the nature and external concentration of the layer charge compensating cation (Banin and Lahav 1968; Shainberg and Otoh 1968; Schramm and Kwak 1982a; Saiyouri et al. 2000). The variable amount of interlayer water in montmorillonite leads to variations in interlayer distance, i.e., swelling, with discrete basal spacing (crystalline swelling) from 11.8–12.6 Å (one-layer hydrate), 14.5–15.6 Å (two-layer hydrate), and up to 19–21.6 Å (four-layer hydrate) (Holmboe et al. 2012; Lagaly and Dékány 2013). In the case of Na+- and Li+-smectites, swelling can result in even larger basal spacing values in a continuous manner (osmotic swelling) (Méring 1946; Norrish 1954). The TOT layer thickness from the center-to-center of oxygen atoms is approximately 6.5 Å. As a rough estimation, the total thickness of the TOT layer can be obtained by summing this value to twice the ionic radius of external oxygen atoms: hTOT = 6.5 + 2 × 1.5 = 9.5 Å. This dimension can be compared with the lateral dimension of the TOT layers: from 50 to 100 nm for illite (Poinssot et al. 1999; Sayed Hassan et al. 2006) and from 50 to 1000 nm for montmorillonite (Zachara et al. 1993; Tournassat et al. 2003; Yokoyama et al. 2005; Le Forestier et al. 2010; Marty et al. 2011). Consequently, illite and montmorillonite particles have high aspect ratios (from 2.5 to 1000) and their surface area is dominated by the contribution of the basal surfaces corresponding to the plane of TOT layer external oxygen atoms. The contribution of the surface area corresponding to the layer terminations (the edge surface) to the total surface area is minor (in the case of illite) if not negligible (in the case of montmorillonite). The specific area of basal surfaces (SSAbs) does not depend on the TOT layer lateral dimension, and it can be calculated from the structural formula and the crystallographic parameters, amounting to approximately 750 m2·g-1 (Tournassat and Appelo 2011). From the specific surface area and the layer charge, it is possible to calculate a specific surface charge. For a montmorillonite with a typical layer charge of 0.32 molc·mol-1, the specific surface charge is -0.11 C·m-2. With illite particles, only part of this surface, corresponding to the outer basal surfaces, is in contact with the porosity and interacts with water, while the other part that corresponds to the interlayer basal surfaces has no contact with water because of the collapse of the interlayer space. The specific outer basal surface area (SSAbs_outer) depends on the average number of stacked layer (nst) in a single illite particle: SSAbs_outer = SSAbs / nst. The outer basal surface area of illite particles can be measured by atomic force microscopy (AFM) and by gas adsorption experiments using the derivative isotherm summation (DIS) method, with a typical value of 100 m2·g-1 considered as representative of SSAbs_outer (Sayed Hassan et al. 2006). With montmorillonite particles, all the basal surfaces are in contact with water because interlayer spaces are hydrated. However, the number of stacked TOT layers in montmorillonite particles dictates the distribution of water in two distinct types of porosity: the interlayer porosity in contact with a specific surface area equal to SSAbs_inter = SSAbs ×(1-1/nst) and the inter-particle

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 297 porosity in contact with a specific surface area equal to SSAbs_outer. It is not possible to give generic and representative values of SSAbs_inter and SSAbs_outer, since the value nst changes as a function of conditions, such as the degree of compaction, salt background nature and concentration (Banin and Lahav 1968; Schramm and Kwak 1982a,b; Bergaya 1995; Benna et al. 2002; Melkior et al. 2009). From particle structure and morphology to anion exclusion. The negative charge of the clay layers is responsible for the presence of a negative electrostatic potential field at the clay mineral basal surface–water interface. The concentrations of ions in the vicinity of basal planar surfaces of clay minerals depend on the distance from the surface considered. In a region known as the electrical double layer (EDL), concentrations of cations increase with proximity to the surface, while concentrations of anions decrease. This leads to a progressive screening of the surface charge by a solution having an opposite charge. At infinite distance from the surface, the solution is neutral and is commonly described as bulk or free solution (or water). This spatial distribution of anions and cations gives rise to the anion exclusion process that is observed in diffusion experiments. Measurements of anion exclusion and electrophoretic mobility in aqueous dispersions of clay mineral particles indicate that the EDL has a thickness on the order of several nanometers with a strong dependence on ionic strength (Sposito 1992). As the ionic strength increases, the EDL thickness decreases, with the result that the anion accessible porosity increases as well. The EDL thickness, where more than 90 % of the surface charge is screened, is commensurable with 2–3 Debye lengths, -1: 

2F 2 1000 I , RT

(16)

where I is the non-dimensional ionic strength (Solomon 2001),  is the water dielectric constant (78.3 × 8.85419 × 10-12 F·m-1 at 298 K), F is the Faraday constant (96 485 C·mol-1), R is the gas constant (8.3145 J·mol-1·K-1) and T is the temperature (K). The ionic charge distribution in the EDL is related to the potentials of mean force for the various ions and those potentials are for the most part related to the local magnitude of the electrostatic potential. Unfortunately, there is no experimental method to measure directly the electrostatic potential: the values derived from experiments such as electrophoretic measurements (Delgado et al. 1986, 1988; Sondi et al. 1996) are always model-dependent. The EDL can be conceptually subdivided into a Stern layer containing inner- and outersphere surface complexes, in agreement with spectroscopic results (Lee et al. 2010, 2012), and a diffuse layer (DL) containing ions that interact with the surface through long-range electrostatics (Leroy et al. 2006; Gonçalvès et al. 2007), in agreement with direct force measurements (Zhao et al. 2008; Siretanu et al. 2014). Molecular dynamics (MD) calculations can also provide information on the Stern layer and diffuse layer structure at the clay mineralwater interface (Marry et al. 2008; Tournassat et al. 2009a; Rotenberg et al. 2010; Bourg and Sposito 2011). The ion distribution in the diffuse layer obtained from MD simulation is in close agreement with the prediction of the simple modified Gouy-Chapman (MGC) model prediction, where this model is applicable. In the MGC model, ion concentrations (DL ci ) at a position y from the starting position of the diffuse layer follow a Boltzmann distribution: DL

  zi F DL  y   ci  y   ci 0 exp   RT  

(17)

that is related to the charge of the ions (zi) and the electrostatic potential ( DL) calculated from the Poisson equation:

298

Tournassat & Steefel d 2  y  dy

where

2



1 zi F DL ci  y  ,   i

(18)

ci 0 is the concentration of species i in the bulk water.

An equivalent anion accessible porosity can be estimated from the integration of the anion concentration profile (Fig. 6) from the surface to the bulk water (Sposito 2004). In compacted clay material, the pore sizes may be small as compared to the EDL size. In that case, it is necessary to take into account the EDLs overlap between two neighboring surfaces. If the pores have all the same size, this calculation is straightforward. Clay mineral particles are, however, often segregated into aggregates delimiting inter-aggregate spaces whose size is usually larger than inter-particle spaces inside the aggregates. In clay-rocks, the presence of non-clay minerals (e.g., quartz, carbonates, pyrite) also influences the structure of the pore network and the pore size distribution. Pores as large as few micrometers are frequently observed (Keller et al. 2011, 2013) and these co-exist with pores having a width as narrow as the clay mineral interlayer spacing, i.e one nanometer. In practice, this complex distribution of pore sizes makes it difficult to calculate the anion accessible porosity from bulk sample data (e.g., specific surface area, total porosity and pore water ionic strength) (Tournassat and Appelo 2011).

Adsorption processes in clays Adsorption processes on basal surfaces. The high specific basal surface area and their electrostatic properties give rise to adsorption processes in the diffuse layer, but also in the Stern layer. The composition of the Stern layer can be calculated according to various models such as the double layer model (DDL), the Triple layer—or plane—model (TLM or TPM), and the charge distributed model (CD) etc., depending on the required level of details. In the following, the double layer model is selected. In the DDL model, all specifically adsorbed species are located on the same plane that corresponds also to the start of the diffuse layer. The quantification of the adsorbed cationic species Mei on basal surfaces sites, >B-, is calculated

Figure 6. Electrostatic potential and chloride concentration in the diffuse layer calculated according to the MGC model, as a function of the distance from a clay mineral surface (x-axis), for two ionic strengths (left: I = 0.015; and right: I = 0.15) and with two different electrolytes (NaCl: blue plain lines; CaCl2: red dashed line). The specific surface charge is -0.1 C·m-2.

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 299 according to the surface complexation reactions (Dzombak and Hudson 1995; Appelo and Wersin 2007; Appelo et al. 2010; Tournassat et al. 2011): zi >B- + Meizi (>B)zi Mei

(19)

and their related equilibrium constants: bs

Ki 

a(  B)zi Mei a  B zi aMezi





,

(20)

i

where a is an activity term. The definition of surface species activity has always been problematic (Kulik 2009). Here, we consider that the activity of surface species is related to their concentrations on the surface and to the surface potential () experienced by the species i. We do not consider any non-electrostatic surface activity coefficient term, in accordance with most of geochemical codes conventions. The surface concentration term is calculated based on a coverage mole fraction convention in order to avoid thermodynamic inconsistencies with heterovalent reactions (Parkhurst and Appelo 1999; Kulik 2009; Tournassat et al. 2013; Wang and Giammar 2013): a  B 





c B

c B  F 0  c B  F 0  exp  exp    ,   izi c(  B)zi Mei  RT  c BTOT  RT 

a(  B)zi Mei 

zi c(  B)zi Mei c BTOT

F 0   exp   zi , RT  

(21)

(22)

where c is a concentration term (mol·L-1) and c BTOT is the total concentration of adsorption sites >B on the basal surface. The surface potential  0 is calculated according to:  F 0 0  8000RTI sinh   2RT

 . 

(23)

Equation (23) corresponds to the exact analytical solution of the modified Gouy–Chapman model for an infinite and flat surface in contact with an infinite reservoir of 1:1 electrolyte. As such, it must be remembered that Equation (23) is only an approximation for systems where the solution contains multi-valent species, for which the surface potential is lower, all other parameters (surface charge, ionic strength) remaining equal (Fig. 6). Interlayer basal surface–solution interaction. The interlayer space can be seen as an extreme case where the diffuse layer vanishes leaving only the Stern layer of the adjacent basal surfaces. For this reason, the interlayer space is often considered to be completely free of anions (Tournassat and Appelo 2011), although this hypothesis is still controversial (Rotenberg et al. 2007c; Birgersson and Karnland 2009). The reactions between the species in the interlayer space can be accounted for in the framework of the ion exchange theory (Vanselow 1932; Gapon 1933; Gaines and Thomas 1953; Sposito 1984). In this theory, negatively charged sites (X-) are fully compensated by counter cations (Mei) in the vicinity of the sites according to the reaction and species on the exchanger sites can be exchanged with other species in solution: zj Xzi-Mei + zi Mejzj

zj Meiz + zi Xz - Mej, i

j

(24)

The sum of the sites X- is referred as the cation exchange capacity (CEC). As for the adsorbed surface species, the activity of exchangeable cations is an ill-defined thermodynamic parameter.

300

Tournassat & Steefel

There is no unifying theory to calculate the activity coefficients of surface or exchange species. In the following, we will rely on the equivalent ratio form of activity in cation exchange theory, where activity is related to the site molar ratio that is occupied by the species i (Ei) and a surface activity coefficient (i,exch), following the Gaines and Thomas convention (Gaines and Thomas 1953): exch



i 

exch

exch

i0  RT ln exch ai 

exch

i0  RT ln

 i ,exch zi exch ci  jz j exch c j

(25)

i0  RT ln  i ,exch Ei ,

where exchci is the concentration of species i one the exchanger in mol·L-1 of interlayer water. Note that there is no electrostatic potential term in Equation (25); the surface charge is considered to be fully compensated by the exchangeable cations leaving no surface potential. Usually, the equivalent fraction convention of Gaines and Thomas is preferred over the mole fraction convention of Vanselow (1932) in geochemical and reactive transport codes because the term z j exch c j remains constant and is equal to the CEC (cCEC in mol·L-1 of interlayer j

water) whatever the composition of the exchanger. This choice is not dictated by any theoretical reasons and it must be seen as being arbitrary, the Gaines and Thomas convention being easier to implement numerically. In addition the activity coefficient term  i ,exch is often, if not always, dropped owing to the difficulty to quantify it (Chu and Sposito 1981). Consequently, the chemical potential of exchanged species becomes: exch

i 

exch

i0  RT ln Ei .

(26)

The equivalent fraction of species Mei on the exchanger sites are calculated according to the selectivity coefficients, exch K i  j , of binary reactions (Eqn. 24): exch

Ki j 



E j zi aMezi Ei

zj

i



zj

a 

zi

.

(27)

z

Me j j

It can be emphasized that Equation (27) is equivalent to the combination of two Equations (22) for species i and j with the consideration of an electrostatic potential value of 0. This last condition corresponds indeed to the condition of full compensation of the surface charge by adsorbed cations. In this respect, the exchange model can be seen as a limiting case of the surface complexation model given above. Adsorption processes at edge surfaces. Unlike basal surface area, the specific edge surface area, SSAedge, depends on the lateral dimension of the TOT layers. The edge specific surface area can also be measured by AFM and DIS methods (Bickmore et al. 2002; Tournassat et al. 2003; Le Forestier et al. 2010; Marty et al. 2011; Reinholdt et al. 2013). Average reported values of SSAedge value ranges from 5 m2·g-1 to 30 m2·g-1. Although the edge surface area is quantitatively less important than the basal surface area, edge surfaces dominate in determining the surface complexation properties of clay minerals (Tournassat et al. 2013) and they cannot be ignored for the modeling of strongly interacting species such as divalent and trivalent metallic cations, lanthanides or actinides that interact with amphoteric sites at the clay mineral layer edges, or Cs+ that interacts with size specific adsorption sites at frayed-edge sites on illite. For metallic cations (e.g., Ni2+), at least two adsorption sites—a high energy and a low energy amphoteric adsorption sites—are necessary for modeling adsorption isotherms as a function of pH and as function of concentration. Adsorption reactions on layer edge

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 301 amphoteric sites may be modeled with surface complexation models, for which a generic reaction stoichiometry is written as: >SOH + Meizi 

>B Meizi-1 + H+,

(28)

where >SOH is an amphoteric surface site. A range of surface complexation models are available for adsorption processes on clay mineral layer edges (Bradbury and Baeyens 2005; Tertre et al. 2009; Gu et al. 2010). The most successful models in terms of range of conditions of applicability are the non-electrostatic models despite the abundant evidence of the presence of an electrostatic field at clay mineral layer edges (Tournassat et al. 2013). For Cs+, frayed edge sites reactivity is commonly modeled with a cation exchange model with high values of selectivity coefficients for Cs+ and K+ as compared to Na+ (Brouwer et al. 1983; Poinssot et al. 1999; Bradbury and Baeyens 2000; Zachara et al. 2002; Steefel et al. 2003; Gaboreau et al. 2012; Chen et al. 2014). Effect of nonlinearity of adsorption processes on experimental diffusion parameters. The superposition of various adsorption processes can result in highly non-linear adsorption isotherms. In such cases, adsorption needs to be described with multi-site models that include multiple cation exchange sites (as described above) and/or another set of more selective adsorption sites. However, interpretations of diffusion data for adsorbing species almost invariably rely on Equation (10), which assumes that adsorption is linear and rapid relative to the time-scale of diffusion. The consideration in Equation (6) of a non-linearity in the adsorption process yields the following alternative to Equation (9):    ci   c    De,i i  .    ci  ci  t x  x  

(29)

 is responsible, in a large part, for the observed ci discrepancy between KD values obtained from batch and diffusion experiments, and for which a difference in Cs+ adsorption capacities from dispersed to compact system is often invoked (Miyahara et al. 1991; Tsai et al. 2001; Van Loon et al. 2004b; Maes et al. 2008; Wersin et al. 2008). It is also possible to interpret diffusion data with numerical reactive transport software, in which case there is no special restriction to linear and/or equilibrium treatments of adsorption. Non-linear adsorption is not restricted to the case where a range of adsorption sites are present with different affinities for the tracer of interest. Even in the case of a single adsorption site, a non-linear adsorption isotherm can occur if the tracer concentration is high enough that the ions occupy a significant fraction of the adsorption sites. The following simple example illustrates this problem. Na+ / Ca2+ cation exchange is considered to be the only reaction taking place at the clay mineral surface:

According to Appelo et al. (2010), the term

2 NaX + Ca2+ CaX2 + 2 Na+ log KNa/Ca = 0.5.

(30)

The concentration of Ca2+ on the exchanger is obtained from:  2 2  CaX 2   CCEC  Na   Na+  , 2  NaX  Ca 2   Ca2+ 2

K Na/Ca

(31)

where CCEC is the cation exchange capacity of the clay mineral in mol·L-1pore water. Equation (31) can be combined with the KD equation and yields:

302

Tournassat & Steefel    NaX   K Na/Ca

 Ca2+

2 d  CCEC

 Na    2Na+

2

KD 

2

.

(32)

Under the experimental conditions considered here, the Na+ concentration is constant and thus Ca2+ /[Na+]2 Na+ is constant. If the concentration of Ca2+ on the exchanger is negligible as compared to the concentration of Na+, then  NaX   CCEC and Equation (32) transforms into: KD 

 Ca2+  Ca2+   CCEC  K Na/Ca q K  CEC Na/Ca , 2 2 2  2 d 2  Na   Na+  Na    2Na+

(33)

where qCEC is the CEC expressed in mol·kg-1clay. All the terms in Equation (33) are constant and Ca2+ adsorption is linear. As soon as [NaX] deviates from the CCEC value, however, Ca2+ adsorption becomes non-linear. In the following, the effect of the non-linearity of adsorption is explored by comparing the simulation of diffusion breakthrough curves using reactive transport (RT) modeling with PHREEQC (Parkhurst and Appelo 1999, 2013) in combination with i) a KD model or ii) a cation exchange model. The system modeled is similar to the one depicted in Figure 1. The Ca2+ concentration was held constant in the inlet reservoir, either at 10-7 mol·L-1, at 10-3 mol·L-1, or at 5 × 10-3 mol·L-1, and the outlet concentration was simulated as a function of time. The inlet and outlet reservoir had a volume of 1 L and the diameter and length of the sample were set at 2 cm and 1 cm, respectively. The dry bulk density was set at 0.8 kg·dm-3, and the porosity value was 0.72. The reference Ca2+ pore diffusion coefficient was set at 2 × 10-10 m2·s-1. According to Equation (33), and considering a qCEC value of 1 mol·kg-1clay and a NaCl background concentration of 0.1 mol·L-1, the KD value for Ca2+ at trace concentration is KD = 100 L·kg-1. While the KD model and the cation exchange models yielded identical results for the case with a Ca2+ concentration of 10-7 mol·L-1 in the inlet reservoir, the diffusion breakthrough curves were very different for the case with a Ca2+ concentration of 10-3 mol·L-1 and 5 × 10-3 mol·L-1 (Fig. 7). At 5 × 10-3 mol·L-1, the two breakthrough curves could only be

Figure 7. Effect of the non-linearity of adsorption on the KD parameter derived from diffusion breakthrough curves.

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 303 matched by decreasing the KD value to KD = 81 L·kg-1. Note that the KD value of Ca2+ at a concentration of 5 × 10-3 mol·L-1 is 42 L·kg-1 according to Equation (32) and thus a constantKD model calibrated on this value would have led to an incorrect position of the breakthrough curve. This simple example highlights the need to take into account the non-linearity of adsorption for the interpretation of experimental diffusion data. This result echoes the previous findings of various authors (Melkior et al. 2005; Jakob et al. 2009; Appelo et al. 2010) who showed that Cs+ diffusion breakthrough curves can only be adequately modeled when the non-linearity of its adsorption isotherm obtained from batch experiment is correctly taken into account. Non-linear adsorption isotherms are not easily introduced in analytical solutions of the diffusion equations, whereas RT codes can handle easily various and complex adsorption models such as cation exchange and surface complexation reactions (Steefel et al. 2014).

CONSTITUTIVE EQUATIONS FOR DIFFUSION IN BULK, DIFFUSE LAYER, AND INTERLAYER WATER From real porosity distributions to reactive transport model representation The pores in clay material and clay rocks are usually fully saturated as evidenced by the agreement between the porosity values derived from water loss measurements, density measurement (wet, dry, and grain densities), and water diffusion-accessible porosity measurements (Fernández et al. 2014). While these results imply that the pore network is fully connected, a full characterization of the connected pore geometry (particularly the pore throats) is still beyond the scope of what is possible, at least in the case of clay materials. Continuum reactive transport codes do not handle in full this complexity and average properties of the porosity must be considered (Steefel et al. 2014). Still it is possible to define three porosity domains, or water domains, that can be handled separately: the bulk water, the diffuse layer water and the interlayer water, the properties for which can be each defined independently. One limitation of reactive transport models currently is that it is necessary to consider a non-zero volume for the bulk water and that the bulk water volumes are connected from one cell to the other. This representation of the system can be at variance with a real system in which the macropores (with bulk water) are inter-connected with only small pores with only EDL or interlayer water, i.e., a system where there is a discontinuity of the diffusion path in macropores.

Diffusive flux in bulk water Fick’s law as presented in Equation (1) is a strictly phenomenological relationship that is more rigorously treated with the Nernst–Planck equation. In the following, we will consider a pseudo 2-D Cartesian system in which diffusion takes place along the x axis only (Fig. 8). In absence of an external electric potential, the electrochemical potential in the bulk water can be expressed as (Ben-Yaakov 1981; Lasaga 1981): b

i  b i0  RT ln b ai  b i0  RT ln b ci  RT ln b  i .

(34)

The gradient in chemical potential along the x axis is the driving force for diffusion and the flux of ions i in the bulk water bJi can be written as: b

Ji  

b

ui b ci  b i ui zi b ci  b  diff ,  zi F x zi x

(35)

where b ui is the mobility in bulk water (m2·s-1·V) and where b  diff is the diffusion potential that arises because of the diffusion of charged species at different rates. The gradient in chemical potential along x is:

304

Tournassat & Steefel  b i RT  b ci  ln b  i  b  RT . x ci x x

(36)

Combining Equation (36) with Equation (35), we obtain the Nernst–Planck equation: b

J i   b Di

 b ci b b  ln b  i zi F b Di b ci  b  diff ,  Di ci  x x RT x

(37)

with b Di  b ui RT / zi F , the diffusion coefficient of species i in the bulk water (Steefel et al. 2014). In the absence of an external electric field, there is no electrical current and so:

z

b j

J j  0.

(38)

j

The combination of Equations (37) and (38) provides an expression for the gradient of the diffusion potential along the x-axis in the bulk water:

 b  diff x

  b c j b  ln b  j  b z D  j j j  x  c j x  RT  .  2 b b F z D c j j j j

(39)

Consequently, it is possible to express the Nernst-Planck equation with known parameters only, i.e., concentrations and activity coefficients (Boudreau et al. 2004).

  b c j b  ln b  j  b z D  j j j  x  c j x    b ci b  ln b  i  b b b b   J i   Di   ci   zi Di ci 2 b b x   x  jz j D j c j

(40)

Diffusive flux in the diffuse layer Flux equation. According to Equation (17), it is possible to write:

ln DL ci  x, y   ln DL ci  x,   

zi F zF  DL  x, y   ln b ci  x   i  DL  x, y  . RT RT

(41)

The profiles of concentrations obtained from the Poisson–Boltzmann equation are representative of equilibrium between the solution in the diffuse layer and the solution at an infinite distance from the diffuse layer, i.e., the bulk water: DL

 i  x, y   b  i  x  ,

DL

i0  RT ln DL ci  x, y   RT ln DL  i  x, y   zi F DL  x, y 

(42)

 b i0  RT ln b ci  x   RT ln b  i  x  .

The combination of Equations (41) and (42) gives: DL

i0  RT ln DL  i  x, y   b i0  RT ln b  i  x  .

(43)

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 305

Figure 8. Pseudo-2-D Cartesian system with diffusion along the x-axis and electrostatic potential developing along the y-axis due to the negative charge at clay mineral surfaces.

And so, by following the MGC model we implicitly assume: i0  b i0 ,

(44)

 i  x, y   b  i  x  .

(45)

DL

DL

The flux along x in the diffuse layer can be expressed in the same way as in the bulk water (Appelo and Wersin 2007): DL

J i  x, y   

DL

ui DL ci  x, y   DL i  x, y  ui zi DL ci  x, y   DL  diff  x, y   . zi F x zi x

(46)

Since the chemical potential in the diffuse layer is the same as in the bulk water, it follows that: DL

J i  x, y   

DL

ui DL ci  x, y   b i  x  ui zi DL ci  x, y   DL  diff  x, y   . zi F x zi x

(47)

Again, in absence of an external electric field and thus electrical current in the diffuse layer along the axis x, we obtain:

306

Tournassat & Steefel

 DL  diff  x, y  RT  F x

 jz j DL D j e z

zj

F DL  x , y  RT

2 DL

j j

  b c j b  ln b  j   cj   x   x ,

Dj bc j  x  e

zj

(48)

F DL  x , y  RT

DL

u j RT , the diffusion coefficient of species i in the diffuse layer water along the zj F x-axis. The fluxes of species i in the diffuse layer along the x-axis become: with

DL

Dj 

DL

J i  x, y    Di e DL

 zi DL Di b ci  x  e

 zi

 zi

  b ci  x  b  ln b  i   ci  x    x   x F DL  x , y    bcj  x  b zj  ln b  j  DL RT  z D e c x   j  j j j x   x .

F DL  x , y  RT

F DL  x , y  RT

 jz j 2 DL D j b c j  x  e

zj

(49)

F DL  x , y  RT

It can be noted that Equations (40) and (49) have the same form (Appelo and Wersin 2007; Appelo et al. 2010):   b ci b  ln b  i   ci J i   b/DL Di Ai   x   x   b c j b  ln b  j  b/DL  cj z D A  j j  j j x x  b/DL b  zi Di ci Ai  jz j 2 b/DL D j b c j Aj

b/DL

with the ‘DL enrichment factor’ Ai  e Ai  1 for the flux in the bulk water (b).

 zi

(50)

F DL x , y 

for the flux in the diffuse layer (DL) and

RT

According to Equation (50), the diffusive flux can be split in three contributions for both diffuse layer and bulk waters:  the concentration gradient:

 b/DL Di Ai

 the activity coefficient gradient:

 the diffusion potential:



z

j j

zi

b/DL

b

Di ci Ai

b/DL

 b ci . x

 ln b  i . Di Ai ci x b

  b c j b  ln b  j   cj D j Aj   x   x .  jz j 2 b/DL D j b c j Aj

b/DL

Calculation of DL. A numerical resolution of the full Poisson–Boltzmann (PB) equation is possible, but it is often too demanding computationally for RT codes applications that aim to solve the Nernst–Planck equation together with a full geochemical reaction network. If the details of the ion distribution in the diffuse layer as a function of the position along y are not needed, a simplified model can be considered where only average ion concentrations in the diffuse layer, DL ci , are calculated:

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 307

DL

ci 

1 LDL

LDL

b  zi cie

 zi

F DL  y  RT

dy  b ci e

 zi

F M RT

,

(51)

y 0

where LDL is the length of the DL along the y axis and  M is the ‘mean potential’ in the diffuse layer. This model is often referred as a Donnan model (Muurinen et al. 2004; Appelo and Wersin 2007; Birgersson and Karnland 2009; Appelo et al. 2010; Tournassat and Appelo 2011) although it is based on an approximation of the PB equation and its underlying hypotheses. Note that Equation (51) can be solved for each individual ion, but that the resulting value of M that is calculated can be different depending on the ions considered. Since only a single value of M is desired, the right side of Equation (51) can only be an approximation of DL ci . The average ion concentrations in the EDL are related to the surface charge,  (in C·m-2), according to: F M  zi   zi DL ci  zi b ci e RT . FhDL i i

(52)

Note that hDL, the considered width of the diffuse layer, can be different from LDL, the actual width of the diffuse layer, in order to have a better agreement between the left and right terms of Equation (51). Re-substituting Equation (52) into Equation (49) makes it possible to express the species flux in the diffuse layer along the x-axis independently of the position within the diffuse layer along the y-axis, since an average electrostatic potential is used.

Interlayer diffusion According to Equation (26), the gradient of chemical potential in the interlayer is:  exch i RT Ei  , x Ei x

(53)

and the diffusive flux in the interlayer can be expressed as: exch

J i   exch Di

ci Ei zi F exch Di exch ci  exch  diff  . Ei x RT x

exch

(54)

The concentration of species i in the interlayer is related to the cation exchange capacity and Ec to the equivalent fraction of species i on the exchanger, exch ci  i CEC , and so: zi exch exch c E F Di Ei cCEC   diff exch (55) J i   exch Di CEC i  . x zi x RT The diffusion potential term in the interlayer, charge flux condition: 

exch

J i   exch Di

cCEC E j z j x , exch  j D j E j cCEC

 RT

 diff  x F

exch

 exch  diff , is obtained by considering the zero x exch

j

Dj

cCEC Ei exch  Di Ei cCEC zi x

cCEC E j z j x . exch  j D j E j cCEC



exch

j

Dj

(56)

(57)

308

Tournassat & Steefel

Approximations for Nernst-Planck equation for bulk and EDL water Ion activity coefficient gradients. Ions activity coefficients are often calculated according to an extended Debye-Huckel model:

log10 b  i  

ADH zi 2 I  bi I , 1  Bai0 I

(58)

with I the ionic strength:

I

0.5   bcjz j2. c0 j

(59)

It follows: b

   ln b  i b 0.5  ln 10  I  ADH zi 2 Bai0 ADH zi 2 I ci  ci    2bi I  . 2 0  x x 1  Bai I I 1  Bai0 I  





(60)

In principle, this expression can be substituted into Equation (50), but in practice the consideration of the gradient of ionic strength can be difficult as a function of the numerical scheme used for solving the transport and chemistry equations because the ionic strength is a function of all concentrations terms. Nonetheless, it is possible to do so within the standard RT numerical frameworks and there are some cases where it is necessary to do so (Molins et al. 2012; Steefel et al. 2014). In most reactive transport codes, ionic strength and activity gradients are neglected and Equation (50) reduces to:  bcj  ci b/DL x .  zi b/DL Di b ci Ai J i   b/DL Di Ai 2 b/DL b x z D c j j Aj j j b

 jz j b/DL D j Aj

(61)

Alternatively, by considering:

 ln b  i  ln b  i  b ci 1  ln b  i  b ci  b  b x  ci x ci  ln b ci x

(62)

it is possible to further simplify Equation (50) into:   ln b  j   b c j b /DL z D A  j j 1  j j b b b      ln c   ln c j  x (63) b /DL b /DL b /DL b i i  zi Ji   Di Ai 1  Di ci Ai .  b   ln ci  x  jz j 2 b /DL D j b c j Aj

Equation (63) is for example the flux equation embedded in PHREEQC. It must however be remembered that Equation (62) is a simplification which neglects the cross-coupling b  ln b  i  ln b  i  c j terms since . The simplification made in Equation (62) leads to  b x  c j x j calculation results strictly equivalent to Equation (50) as long as it is evaluated numerically

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 309  b ci  b ci does not go to zero. If  0 , it is not x x b  ln  i possible to evaluate numerically , the denominator for which is zero when discretized  ln b ci   ln b  i   b ci  0 . In this very between two numerical cells and one must assume that 1   b   ln ci  x special case Equation (63) is equivalent to Equation (61) and the activity gradient is neglected,

and as long as the concentration gradient

ci  ln b  i is not equal to zero if there is a gradient of salinity. x If Equation (62) is solved analytically, one obtains: b

even though the term

b

ci

 ln b  i b  ln b  i  b ci  ci b x  ci x

   b ci 0.5  ln 10  I  ADH zi 2 Bai0 ADH zi 2 I  ci    2bi I  2 b 0  0 x  c I 1  Bai I i 1  Bai I     2  b ci 0.25  zi  ln 10   ADH zi 2 Bai0 ADH zi 2 I  b  ci  2bi I  . 2   1  Ba 0 I  0 x c0 I i 1  Bai I   b







(64)



From Equation (64), it is clear that the approximation made in Equation (62) leads to activity coefficient gradient terms which are underestimated (in absolute value). As an example let’s consider tracers whose charges are the same as the salt background ions, but whose concentrations are 1000 times lower than the background salt concentration. Their activity

c  ln   ln  . However  background 1000 x x  ln  tracer cbackground  ln  background  applying Equation (64), it follows that ctracer because 106 x x c cbackground c ctracer tracer  background . This result implies that for tracers, Equation (62) neglects 6 10 x x the activity gradient terms and thus is very similar to Equation (63).

coefficients are the same and thus if follows that ctracer

Diffuse layer and interlayer effective diffusion coefficients. In PHREEQC, the diffusion coefficients in the diffuse layer are set proportional to the diffusion coefficients in the bulk water. This simplifying assumption is however not strictly necessary, and diffusion coefficients in the diffuse layer could be set to values which are not correlated with the diffusion coefficients values in the bulk water, as done in CrunchFlowMC. The latter case would imply that the tortuosity, and thus the effective diffusion coefficient, need not be the same in the bulk and EDL porosity. One can however justify the PHREEQC simplification by noting that the decrease of the diffusion coefficient in the diffuse layer is similar for all species according to molecular dynamics calculations (Tournassat et al. 2009b; Bourg and Sposito 2011; Holmboe and Bourg 2014), although such calculations are incapable of taking into account the heterogeneous, potentially hierarchical structure of natural porous media.

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RELATIVE CONTRIBUTIONS OF CONCENTRATION, ACTIVITY COEFFICIENT AND DIFFUSION POTENTIAL GRADIENTS TO TOTAL FLUX Model system An important consideration is that the relative contributions of the three terms of Equation (50) to the total diffusive flux are not the same in the diffuse layer as in the bulk water. In the following sections, we explore the coupling of these different terms using simple examples in which the flux equations are solved numerically so as to explain experimental observations which have been reported repeatedly for diffusion in clay materials. We consider a one centimeter long system with a constant salt background concentration along the x-axis (0.1 mol·L-1 NaCl), but with a linear gradient of 22Na and 36Cl tracer concentrations along x from 10-9 mol·L-1 at x = 0 to 0 mol·L-1 at x = 1 cm (Fig. 9). The pore diffusion coefficients are set to values ten times lower in the porous medium than in pure water, i.e., D p,Na+ = 1.33 × 10-10 m2·s-1 and D p,Cl- = 2.03 × 10-10 m2·s-1 in both bulk water and diffuse layer porosity. For simplicity, the Davies activity coefficient model is used (ADH = 0.5, Bai0 = 1, bi = -0.3 for all ions). The surface charge value is chosen to be representative of a clay plug packed at a bulk density of dry = 1 kg·dm-3 and with a cation exchange capacity (CEC) of 0.1 molc·kg-1, that is to say, values representative of an illite-rich clay material. The specific surface area is set to SSA = 100 m2·g-1, a value in good agreement with the properties of illite particles. The chosen thickness of the diffuse layer, hDL, is assumed = 3 × 10-9 m. The total porosity  1  0.64 , the diffuse layer porosity is DL  hDL  SSA  dry  0.3, and is tot  1  dry  1  illite 2.8  the volumetric charge compensated in the diffuse layer is q  CEC  dry  0.33 mol·L-1. DL Based on these parameters, it is possible to calculate the mean potential in the diffuse layer using Equation (52) as M = -0.033 V.

Example 1: Constant ionic strength In the first example, the ionic strength is taken as constant across the length of the system, which is equilibrated with a 0.1 mol·L-1 NaCl solution. The tracers concentrations are assumed to show a linear gradient along x from 10-9 mol·L-1 at x = 0 to 10-11 mol·L-1 at x = 1 cm. The contribution of the tracers to the ionic strength is negligible and thus, the activity coefficient gradient is negligible. Under these conditions, the diffusion potential term is also negligible in the bulk water and in the diffuse layer water because the term z j 2 b/DL D j b c j Aj is high as j   b c j b  ln b  j  36 . While the Cl fl ux is higher than the 22Na+ compared to z j b/DL D j Aj   cj   x  x j   flux in the bulk water because of the higher of diffusion coefficient values of the former, the reverse is true in the diffuse layer (Fig. 9). This effect is due to the term A in Equation (50) which is related to the accumulation of 22Na+ and the depletion of 36Cl- in the diffuse layer water as compared to the bulk water.

Example 2: Gradient in ionic strength and tracer concentration In a second example, we add a linear gradient of NaCl concentrations along the x-axis from 0.1  mol·L-1 at x = 0 to 0.001 mol·L-1 at x = 1 cm, i.e., the tracers gradient and the NaCl gradient are now the same once scaled to their respective maximum concentrations. The diffusion fluxes in the bulk water are dominated by the concentration gradient term (Fig. 10). In this case, the diffusion potential term is not negligible. The Na+ diffusive flux in the bulk water is increased as a result of this term, while the Cl- diffusive flux is decreased because of the higher diffusivity of Cl- as compared to Na+ coupled to the electroneutrality condition in the bulk water. In this example, the activity coefficient term is minor as compared to the two other terms, but it represents a negative contribution of up to 12% of the total 36Cl- flux in the

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 311

Figure 9. Example 1. Calculation of 22Na+ and 36Cl- concentrations and fluxes in the bulk and in the diffuse layer waters according to Equation (50). Ionic strength is constant along the x-axis (I = 0.1). The dashed line and the open green circles in the upper left Figure correspond to the gradient of the activity coefficient calculated according to Equation (60) or Equation (62), respectively.

bulk water (up to 8% for 22Na+, Fig. 11). Moreover the contribution of this term is not constant along the x-axis and its negative contribution to the total fluxes decreases over the length of the system as the total salt background concentration decreases. Consequently, the initial linear gradient of NaCl concentration is not truly representative of steady state conditions. In the diffuse layer water, the contributions of the different terms are radically different compared to the bulk water. While 22Na+ and 36Cl- diffusive fluxes remain equal in the diffuse layer, the 22Na+ diffusive flux is dominated by the concentration gradient (positive) term that is counteracted primarily by the diffusion potential (negative) term (recall that, in contrast, the diffusion potential term is positive in the bulk water for Na+) and to a lesser extent by the activity coefficient term. For 36Cl-, the concentration gradient and the diffusion potential terms have almost the same positive contribution to the diffusive flux. At x = 1 cm, Na+ and Cl- diffusive fluxes in the diffuse layer drop to a very low value because of the low ionic strength (I  = 0.001), which in turn is responsible for the very low value of the enrichment factor A for Cl- and 36Cl- in the EDL. Because of the surface charge compensation condition, the enrichment factor A remains high for Na+ in the diffuse layer, but the charge conservation conditions and the related diffusion potential term cancel its effect on the total flux intensity. Similar to the bulk water, the calculated 22Na+ and 36Cl- fluxes in the diffuse layer are not constant as a function of the position along the x-axis and thus the system is not representative of steady state conditions because of the activity gradient term in the flux Equation (50). If this term is dropped as in Equation (61), then the fluxes increase by up to 4% for the conditions considered here (Fig. 11).

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Figure 10. Example 2. Calculation of 22Na+ and 36Cl- concentrations and fluxes in the bulk and in the diffuse layer waters. The salt background NaCl concentration follows a linear gradient from 0.1 to 0.001 mol·L-1 along the x-axis. The dashed line and the open green circles in the upper left Figure correspond to the gradient of activity coefficient calculated according to Equation (60) or Equation (62) respectively. The blue open circles on the bottom-right Figure corresponds to the J22Na+ values in the diffuse layer for comparison with the J36Cl- values.

Figure 11. Example 2. Calculation of 36Cl- fluxes in the bulk and in the diffuse layer waters according to Equation (50) (full equation, lines) and Equation (61) (no activity gradient term, open circles). Results for 22 Na+ fluxes are identical to those for 36Cl-.

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 313 Example 3: Gradient in ionic strength and no tracer gradient In this last example, we consider a system where the NaCl concentration follows the same linear concentration gradient as in Example 2. However, in contrast to Example 2, we consider 22 Na+ and 36Cl- concentrations gradients that are initially zero. This last condition cancels the concentration gradient term in the diffusion equation for the tracers in both bulk and diffuse layer waters, and the activity coefficient gradient and diffusion potential terms are the only remaining contributors to the total diffusive flux. In those conditions, the diffusive flux is positive for 22Na+ in the bulk water and negative for 36Cl- while the reverse is true in the diffuse layer (Fig. 12). In these conditions, the diffusion potential term dominates over the activity coefficient gradient term (Fig. 13).

Links to experimental diffusion results In Example 1 above, the fluxes are independent of the position along the x-axis. Therefore, this system is representative of a through-diffusion experiment that has reached steady state conditions. The presence of a diffuse layer at a negatively charged surface increases the diffusivity of cationic species and decreases the diffusivity of anionic species as compared to bulk water, at least in the case where the diffusion coefficients are the same in the EDL and bulk water. Under the same conditions, the diffusive flux of a neutral species (such as HTO, a water tracer) is not impacted by the presence of a diffuse layer (the enrichment term A is equal to 1 and there is no diffusion potential term for neutral species). As such, this very simple example captures essential features of diffusion experiments performed on clay samples where the observed fluxes at steady state are in the order JCations > JHTO > JAnions (Glaus et al. 2010; Tachi and Yotsuji 2014).

Figure 12. Example 3. Calculation of 22Na+ and 36Cl- concentrations and fluxes in the bulk and in the diffuse layer waters. The salt background NaCl concentration follows a linear gradient from 0.1 to 0.001 mol·L-1 along the x-axis in this case where there is no tracer concentration gradient.

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Figure 13. Example 3. Calculation of 36Cl- fluxes in the bulk and in the EDL porosity according to Equation (50) (full equation, lines) and Equation (61) (no activity gradient term, open circles).

Example 2 points out the extent of the approximation made in the calculation of the diffusive flux if the activity coefficient gradient is dropped in the solution of the Nernst–Planck equation. The relative error made in the calculation of the diffusive flux is less in the EDL water than in the bulk water, thereby justifying this simplification if the contribution of the EDL to the total flux dominates the flux in the bulk water. (Fig. 11) Example 3 shows that the presence of a salt background concentration gradient can lead to up-gradient as well as to down-gradient diffusion of charged tracers in the bulk and in the EDL water. If the contribution of the diffuse layer to the total volume of water is not negligible, cationic tracers experience up-gradient diffusion, because the diffusive flux is two orders of magnitudes higher in the EDL than in the bulk water (Fig. 12). For anions, the direction of total diffusive flux depends on the relative contributions of bulk and EDL waters. Diffusion does not follow the activity gradients, since the diffusion potential term governs the direction of the diffusive flux for the tracers in this case (Fig. 13). Again, this observation helps to justify the simplifying assumption of ignoring the activity coefficient gradient term in the Nernst–Planck equation. Up-gradient diffusion of 22Na has been recently reported in an experimental system similar to that described in Example 3 (Glaus et al. 2013).

FROM DIFFUSIVE FLUX TO DIFFUSIVE TRANSPORT EQUATIONS The analysis presented above of the different contributions of the terms in the flux equation highlights the feasibility of explaining qualitatively experimental observations of water and ion fluxes with the Nernst Planck model developed from Equations (34) to (64). Quantitative simulations of those fluxes including transient conditions can be achieved only by implementing the flux equation in transport equations.

Diffusive transport equation for porous medium with interlayer and EDL water The 1D Cartesian diffusion equation in a saturated porous media has the form:

Ci J i ,  t x

(65)

where Ci is the concentration of species i in the porosity (in mol·dm-3). If the species i is not adsorbed on a surface, and if it does not participate in dissolution-precipitation reactions or other processes which would reduce its concentration in solution (for example, radioactive

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 315 decay), then Ci  ci , where  is the total porosity and ci is the total concentration in the porosity:

ci J i .  t x

(66)

In the presence of a diffuse layer and of an interlayer space, the term ci can be split into three contributions (bulk – b, diffuse layer – DL, and interlayer, or exchangeable, – exch): ci  b  b ci  DL  DL ci  exch  exch ci .

(67)

The same approach (and nomenclature) can be used for the flux term J i . Thus, the transport equation becomes: b  b ci  DL  DL ci  exch  exch ci J i  DL J i  exch J i .   t x b

(68)

The solution of Equation (68) can pose several problems. Firstly, the porosity terms on the left hand side of the equation are not constant if the salinity changes as a function of time, and in this case the derivative of the porosity terms as a function of time must be solved (for example, an increase of salinity contracts the diffuse layer, and thus the bulk porosity value increases proportionally to the decrease of the diffuse layer porosity). Secondly, each of the flux terms includes porosity and tortuosity contributions within their effective diffusion coefficient terms: b/DL/exch

Di 

b/DL/exch

 b/DL/exch i b/DL/exch D0,i .

(69)

 b/DL/exch   0 ), then it is difficult to define x unambiguously summation rules at the interface for the different terms. In particular, if one part of the modeled system contains only bulk water and the other part contains bulk and diffuse layer (and a fortiori interlayer water), it is not clear whether or not a flux between the bulk water and the diffuse layer water should be considered. This problem is exemplified in the following section by considering simple illustrative examples.

If the porous media is not homogeneous (

Summation of bulk and diffuse layer diffusive fluxes over an interface Normalized flux term. We define here the terms, i.e., fluxes normalized to a porosity value of one:

J i* 

J i b * b J i DL * ; Ji  b ; Ji   

DL

Ji b * ; Ji  DL 

exch

Ji . 

exch

(70)

Simple case. We will first consider an interface between two adjacent numerical domains (porous media 1 and 2) whose bulk water porosities are the same, and whose diffuse layer porosities are also the same (Fig. 14). In this particular case, we can safely assume that the surface available for bulk and DL water diffusion are proportional to the bulk and diffuse layer water porosities. Consequently, J i* can be expressed as the weighed sum of b J i* and DL * Ji :

J i* 

 b * DL  DL * Ji  Ji .  

b

(71)

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Complex case 1: Same total porosity, different diffuse layer porosity. In this first complex case (Fig. 15), the calculation of the diffuse layer and bulk water contributions to the overall diffusion flux is not straightforward. A simple method can be the use of an arithmetic mean, which is also consistent with Equation (71) if b 1  b 2 and DL 1  DL 2 :

J i* 

b

1  b 2 b * Ji  2

DL

1  DL 2 2

DL

J i* .

(72)

Complex case 2: Same total porosity, one zone without diffuse layer porosity. In the second complex case exemplified in Figure 16, the calculation of the diffuse layer and bulk water contributions to the overall diffusion flux is again not straightforward. One can see this case as a limiting case of Complex case 1, and so one should write:

J i* 

b

1  b 2 b * DL 1 Ji  2 2

DL

(73)

J i* .

However, if we consider that the bulk water of the porous medium 2 is in contact with the diffuse layer water of porous medium 1, the scheme of Figure 16 is equivalent to the scheme in Figure 14, but with the consideration that the diffuse layer 2 volume has the same properties as bulk 2 water (i.e., the electrostatic potential is zero: this condition can be easily considered for the solution of Equation (63)). In that case the flux summation becomes: J i*  b 1 b J i*  DL 1 DL J i* .

(74)

Consequently, Equation (72) lacks a condition of continuity. This problem can be solved by considering the more complex equation:

J i* 

b

2 b * DL 1 Ji   

DL

J i* 

  DL 1  b 2 

b to DL

J i* ,

(75)

where b to DL J i* is a flux term that is calculated with Equation (63) with the consideration of an electrostatic potential of zero on one site of the interface and a non-zero electrostatic potential on the other side of the interface. As such, Equation (75) fulfills the continuity conditions. Complex case 3. Variable total porosity, differing diffuse layer and bulk porosities. A still more complex case where the total porosity is not constant along the x-axis (Complex case 3, Fig. 17) cannot be unambiguously treated with Equation (75). In this case, the priority that should be given to the connectivity between diffuse layer or bulk water volumes (or interlayer porosity) is not clear and the convention that is chosen must then be seen as arbitrary. Summary of cases. The calculation of the fluxes in the diffuse layer and in the bulk water can be obtained theoretically from the consideration of the Nernst–Planck equation coupled to the modified Gouy–Chapman model (or its simplified form, the mean potential model). In contrast, the summation of the flux at complex interfaces between two numerical cells does not follow rules that are dictated by any firmly grounded theory. The choices of summation rules which are given above, therefore, must be considered as intuitive choices. One should note that the complex case 2 illustrates conditions similar to a boundary condition between a filter (without diffuse layer) and a clay plug (with a diffuse layer), so it is far from an academic scenario. The Complex case 1 can be seen as representative of a medium with homogeneous surface charge properties, but one with a gradient in salinity.

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 317

Solid 2

Diffuse layer 1 DL ϕ1

Diffuse layer 2 DL ϕ2

Bulk 1 b

ϕ1

Interface

Solid 1

Porous medium 1

Bulk 2 b

ϕ2

Porous medium 2

Figure 14. Scheme of an interface between two numerical domains. Simple case with no gradient in porosity properties.

Diffuse layer 1 DL ϕ1 Bulk 1 b

ϕ1

Solid 2 Interface

Solid 1

Porous medium 1

Diffuse layer 2 DL ϕ2 Bulk 2 b ϕ2 Porous medium 2

Figure 15. Scheme of an interface between two numerical domains. Complex case 1.

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Diffuse layer 1 Db ϕ1 Bulk 1 L

ϕ1

Solid 2 Interface

Solid 1

Bulk 2 L ϕ2

Porous medium 1

Porous medium 2

Figure 16. Scheme of an interface between two numerical domains. Complex case 2.

Differentiation of the flux at interface between two numerical grid cells The numerical differentiation of the flux poses another challenge. For a grid cell n, at position xn, the numerical equivalent of Equation (65) is given by: Ci,xnewn  Ci,xoldn 1    J i,xn +1/2  J i,xn -1/2  , t x

(76)

where superscripts new and old refer to two consecutive time steps, and where xn+1/2 and xn-1/2 are the positions at the interface between cells n and n + 1, and n and n - 1, respectively. J i , xn1/2 and J i , xn1/2 can be calculated according to Equations (57) (for the interlayer

contribution) and (61) (for the bulk and diffuse layer contributions and neglecting the gradient of activity coefficient). For the bulk and diffuse layer contributions of J i , xn1/2 , the equations are represented numerically by finite differences as: b/DL

J i,xn +1/2 b

b

  DL Di,x 1 Ai,x n+

2

n+

1 2

ci,xn+1 - b ci,xn xn+1 - xn b

c j,xn+1 - b c j,xn  j z j D j Aj,xn+1/2 x - x n+1 n . 2 b/DL b  z j D j,xn+1/2 c j,xn+1/2 Aj,xn+1/2

(77)

b/DL

 z i b/DL Di,xn+1/2 b ci,xn+1/2 Ai,xn+1/2

j

ci , xn1  ci , xn is a well-defined quantity, the terms b c j , xn1/2 (the xn 1  xn concentration in the bulk water at the interface between cells n and n+1) and Aj , xn1/2 (the b

While the term

b

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 319

Figure 17. Scheme of an interface between two numerical domains. Complex case 3.

diffuse layer enrichment factor water concentration related to the electrostatic potential of the diffuse layer at the interface between cells n and n+1) are not known quantities, and their value must be approximated. A simple arithmetic averaging is frequently used for the bulk concentration that is strictly valid for an equally spaced grid: b b

c j , xn1/2 

ci , xn1  b ci , xn . 2

(78)

Alternatively, a harmonic mean can be used if the numerical grid is heterogeneous. The averaging of the diffuse layer enrichment factor Aj , xn1/2 is more problematic if there is a gradient of electrostatic potential from one cell to the next, as in the case of a gradient of ionic strength. A linear gradient of ionic strength results in a highly non-linear gradient of electrostatic potential as exemplified in Figure 18. In that case an arithmetic or harmonic mean may be inaccurate for estimating the Ai value at the interface. The same kind of problem may occur for a gradient of surface charge between two grid cells.

APPLICATIONS Code limitations While a range of reactive transport codes handle the Nernst–Planck equations for diffusive fluxes in the bulk water, very few of them can handle transport processes in the diffuse layer (Steefel et al. 2014). Available publications with reactive transport simulations considering diffusion in the diffuse layer are limited to PHREEQC and CrunchFlowMC application studies (Appelo and Wersin 2007; Appelo et al. 2008, 2010; Alt-Epping et al. 2014). To our knowledge, interlayer diffusion processes are handled by PHREEQC only, and the interlayer diffusion option has been applied in only two published studies (Appelo et al. 2010; Glaus et al. 2013). In the following PHREEQC and CrunchFlowMC are used to illustrate the importance of considering coupled diffusion/surface reaction effects in order to understand and to predict migration processes and associated parameters in charged porous media, especially clays.

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Figure 18. Averaging methods for the bulk concentration (left) and the diffuse layer enrichment factor, Ai (right) for a monovalent cationic tracer at the interface between two grid cells whose centers are located at xn = 0 and xn+1 = 0.01 m. The variation of Ai is due to a linear gradient of ionic strength from 0.1 (left of the system) to 0.001 (right of the system).

Simultaneous diffusion calculations of anions, cations, and neutral species The data from Tachi and Yotsuji (2014), which were presented previously (Fig. 2), were modeled with CrunchFlowMC (Steefel et al 2014). The total porosity of the montmorillonite plug was set at 0.71, in agreement with the bulk dry density of the material (0.8 kg·dm-3). A typical total specific surface area of 750 m2·g-1 was assumed for montmorillonite. The total surface charge was set at 1 mol·kg-1. Constants of the surface adsorption reactions were set at log KNa = 0.7 (in agreement with the MD results from Tournassat et al. 2009) and log KCs = 2.5: >Surf- + Na+  >SurfNa K Na 

cSurfNa  F  exp  0  , aNa+  cSurf   RT 

(79)

>Surf- + Cs+  >SurfCs K Na 

cSurfCs  F  exp  0  . aCs+  cSurf   RT 

(80)

Half of the total porosity was attributed to the diffuse layer, so that the mean Cl- (or I-) accessible porosity was ~0.41, in close agreement with the value given in Table 1. The tortuosity values were fitted for each species and are reported in Table 2. Results are plotted in Figure 19. The tortuosity values follow the order I- < HTO < 22Na+ < 137Cs+, indicating that the tortuous diffusion pathways are not the same for all of these species. Or alternatively, that the adsorbed species in the Stern layer, considered in the calculations as immobile, are in fact mobile. Nevertheless, the contribution of the diffuse layer to the diffusion flux calculation make it possible to derive a physically feasible value (< 1) for 137Cs+ , in contrast with the results obtained with a single porosity diffusion model. Note also that the assumed adsorption constant for Cs+ is responsible for a KD value of 430 L·kg-1 at a ionic strength of 0.1, i.e., a value in agreement with KD values obtained from batch experiments from Tachi and Yotsuji (2014). The bulk + diffuse layer water diffusion model presented here makes it possible to calculate the diffusive flux of neutral species, anions and cations with the same conceptual model and with physically feasible parameters. In this respect, the advantage of the RT codes is their ability to test these kinds of models under transient conditions and in the context of complex geometries in order to derive porosity and tortuosity values as well as adsorption parameters. Model benchmarking is thus not restricted to the comparison of data under steadystate conditions and/or the estimation of apparent diffusion coefficients where the adsorption and the diffusion parameters are lumped together. The effectiveness of this type of integrated approach has previously been put forward in the geochemical literature (Appelo and Wersin 2007; Appelo et al. 2008, 2010), but still remains the exception rather than the rule for the interpretation of diffusion data.

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 321

Figure 19. Modeling with CrunchFlowMC and a bulk + diffuse layer water diffusion model of the diffusion of 22Na+ , Cs+, HTO and I- through a montmorillonite plug equilibrated with a 0.1 mol·L-1 NaClO4 solution in the experimental conditions from Tachi and Yotsuji (2014). See Figure 2 for the reference data and Table 1 for the experimental conditions.

Diffusion under a salinity gradient The experiments conducted and described by Glaus et al. (2013) were also modeled using CrunchFlowMC. The resistance to the diffusion by the filters was explicitly taken into account by assuming a tortuosity of 0.25 and a porosity of 0.32 for the filters. This corresponds to an effective diffusion coefficient of ~10-10 m2·s-1 for Na+ in the filter, a value in agreement with published values (Glaus et al. 2008). The total porosity in the montmorillonite plug was set to a value of 0.3, in agreement with the 1.9 kg·dm-3 dry bulk density of the material (Glaus et al. 2010). At this high degree of compaction, the actual presence of bulk water is not certain. However, it is necessary to have a non-zero bulk water volume to run CrunchFlowMC (or PHREEQC). This bulk water volume was set to a very low porosity value (0.02), so that the overall fluxes were not impacted by the diffusion in this volume. The same surface adsorption model as was used for the modeling of Tachi and Yotsuji’s data (see previous section) was used in this case. The tortuosity parameter was set to a value of 0.014 in the montmorillonite, in close agreement with the value measured for water, whose effective diffusion coefficient is about (1.5–1.7)·10-11 m2·s-1 in similar conditions (Glaus et al. 2010, 2013). The modeling results are shown on Figure 20 and agree almost perfectly with the experimental data. Table 2. Tortuosity values calculated with CrunchFlow for each species in the diffusion experiment from Tachi and Yotsuji (2014), and according to a bulk + diffuse layer water diffusion model. Tracer i



HTO

0.047

Na+

0.071

I

-

0.09 +

Cs

0.136

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Glaus et al. (2013) have also modeled their data using two types of model. The first model they considered was an interlayer diffusion model implemented in PHREEQC. The second model was based on that of Birgersson and Karnland (2009) (BK-model), in which the entire porosity was represented by a diffuse layer with a homogeneous mean electrostatic potential. The authors were able to obtain an equally good fit of the data with both models. In Figure 20, a third alternate and equally good model (from the point of view of how well it fits the data) is given. It should be noted that the three models are totally different from a conceptual point of view: i) the interlayer diffusion model makes the assumption of a complete screening of the surface charge by the exchanged cations and the macroscopic driving force for cations mobility is the gradient of activity of the exchanged cations; ii) the BK-model makes the assumption of no screening of surface charge other than in the diffuse layer, and the macroscopic driving force for the diffusion is the gradient of concentration in the diffuse layer; and iii) the present model makes the assumption of partial screening of the surface charge by cations adsorbed in the Stern layer, where the mobility of cations is assumed to be negligible, and the macroscopic driving force of the diffusion is the diffusion potential in the diffuse layer, as shown in Figure 12.

Interlayer diffusion Recently, Tertre et al. (2015) published data from their diffusion experiments carried out with centimeter-size mono-crystal of vermiculite. Vermiculite is a swelling clay mineral that has a high CEC (~1.8 mol·kg-1) originating primarily from isomorphic substitutions in the tetrahedral sheet. The size of the sample and the experimental setup (Fig. 21a) are ideal for probing self-diffusion processes in the interlayer porosity, as the setup makes it possible to eliminate the tortuosity parameter in the diffusion equation (the interlayer spaces are sandwiched between two flat surfaces). The exchange sites in the vermiculite were saturated with Ca2+. The interlayer width in the vermiculite corresponded to a bi-layer hydrate. Following contact with a NaCl solution, a release of Ca2+ was observed in solution. This experiment showed unambiguously that interlayer diffusion exists, and it made it possible also to quantify the diffusion coefficient for Ca2+ in the interlayer. By immersing the vermiculite in a NaCl solution with a high salinity (0.1 mol·L-1), the authors were able demonstrate that the Ca2+ interlayer diffusion coefficient was in good agreement with the value that they obtained from MD simulations. For experiments performed at lower ionic strength, they observed a discrepancy between the two values that they attributed to the diffusion of the solute species at the interface between the NaCl reservoir and the vermiculite interlayers. Their conclusion was based on the results of Brownian dynamics calculations.

Figure 20. Modeling of the diffusion of 22Na+ under a salinity gradient for the experimental conditions considered by Glaus et al. (2013) using CrunchFlowMC. The blue plain curves correspond to the ratio C/C0 of 22 Na+, while the dashed curves are the NaClO4 concentrations in the reservoirs (blue: 0.5 mol·L-1 NaClO4; red: 0.1 mol·L-1 NaClO4). Left: Experiment with a 5-mm thick clay plug and reservoir volumes of 250 mL. Right: Experiment with a 10-mm thick clay plug and reservoir volumes of 100 mL.

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 323 The data from Tertre et al. (2015) were reinterpreted using the interlayer diffusion option of PHREEQC (Fig. 21b). It was possible to reproduce the data with the same level of quality as the Brownian dynamics calculations by considering an interlayer Ca2+ diffusion coefficient value of 0.8 × 10-11 m2·s-1, together with an interlayer Na+ diffusion coefficient of 4 × 10-11 m2·s-1, 1 × 10-11 m2·s-1, 0.1 × 10-11 m2·s-1, and 0.1 × 10-11 m2·s-1, for the experiments at NaCl concentration of 1, 0.1, 0.05 and 0.003 mol·L-1 respectively. The decrease of the Na+ interlayer diffusion coefficient with NaCl concentration is in agreement with MD results from Tertre et al. (2015), which show a decrease of its value with an increase of the Ca2+/Na+ occupancy ratio in the interlayer (Fig. 21c). At 1 mol·L-1 NaCl, the PHREEQC and MD results fully agree. At 0.05 and 0.003 mol·L-1 NaCl, the interlayer diffusion coefficient fitted with PHREEQC correspond to the lowest value obtained with MD (within the range of the error bands). Consequently, the apparent decrease in Ca2+ interlayer diffusion coefficient can also be interpreted as a result of the decrease of the Na+ interlayer diffusion coefficient that arises from the coupling between the diffusion of these two species through the diffusion potential term in Equation (57).






   
  
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Figure 21. Experiments from Tertre et al. (2015): a) setup; b) modeling of the Ca2+ out-diffusion results using the interlayer diffusion option of PHREEQC (this study); c) self-diffusion coefficients of Na+ and Ca2+ in the vermiculite interlayer as a function of the equivalent fraction of Ca on the surface (XCa), and obtained from molecular dynamics calculations (from Tertre et al. 2015).

SUMMARY AND PERSPECTIVES Diffusion processes through clay materials is the result of a complex interplay of transport and non-linear adsorption processes under the influence of electrostatic fields. In this context,

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the classical Fickian diffusion model applied in the framework of a single pore diffusion model with linear adsorption processes (KD model) appears to be inappropriate for describing diffusion data without making less than satisfying modeling assumptions, such as i) different definitions of the porosity as a function of the nature of the tracer of interest (see anion accessible porosity) and as a function of the conditions (anion accessible porosity change with time; ii) change of the adsorption parameters from batch to diffusion experiment; and iii) physically unrealistic pore diffusion coefficients (or tortuosity values). It is clear that RT codes have been capable of solving the problem of the transfer of KD values from batch experiments to compact porous media systems for some time. Much of the difficulty stems from the non-linearity of the adsorption process for strongly adsorbed species on clay mineral surfaces, which can be handled readily by numerical reactive transport codes (Steefel et al 2014). Perhaps surprisingly, this has been done only recently for Cs+ diffusion data (Appelo et al. 2010), although it was done some time in the past for column percolation experiments (Steefel et al. 2003). Recent developments of selected RT codes that can handle diffusion processes in diffuse layer and interlayer porosities made it possible to model the diffusion data of neutral, anionic and cationic species within the same conceptual framework. Also, the contribution of the diffuse layer and/or the interlayer to the overall diffusion of ions makes it possible to explain the origin of the apparent acceleration of cation diffusion as compared to water, which otherwise would require unrealistic tortuosity values for cations in classical Fickian diffusion models. RT modeling has also helped to improve our understanding of anomalous diffusion behavior such as that of up-hill diffusion. Despite the successes of these new RT modeling approaches, it must be stressed that the model and its parameters derived from diffusion experiments are not always unique. Two examples given above highlight the fact that several different conceptual models can provide equally good fits of the data. As such, the modeling effort is typically under constrained, a fact that explains the multitude of conceptual and numerical models available in the literature that describe the ionic transport properties of clay media (Leroy et al. 2006; Appelo and Wersin 2007; Gonçalvès et al. 2007; Birgersson and Karnland 2009; Gimmi and Kosakowski 2011; Tachi et al. 2014). The transport properties of clay nanopores have been the matter of intensive research using advanced computational methods such as molecular dynamics, lattice-Boltzmann etc. (Bourg and Sposito 2010; Rotenberg et al. 2010; Obliger et al. 2013). The input of these techniques is clearly needed to constrain the macroscopic diffusion models (Fig. 21). Upscaling strategies have been developed to derive macroscopic diffusion parameters from microscopic information (Rotenberg et al. 2007a,b, 2014; Jardat et al. 2009; Bourg and Sposito 2010; Churakov and Gimmi 2011; Churakov et al. 2014) and these can be complemented by RT modeling that takes into account the complexity of the chemical reactivity of the material in finer details (e.g., adsorption processes, activity coefficients) However, it is noteworthy that the interpretation of diffusion data is complicated by complex microstructures that are dependent on physical and chemical conditions, and that these microstructures have not been yet characterized down to the scale of the smallest pores, i.e., the interlayer pores. The connectivity of the pore network connectivity across the full range of pore sizes has not been successfully determined for any of the investigated systems as well. These properties cannot be probed easily. For example, microscopic and tomography techniques still fail at imaging the porous network at the correct resolution (Hemes et al. 2013). The development of new observation techniques is thus necessary to make a step forward in the understanding of transport processes in clay materials.

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 325 ACKNOWLEDGMENTS This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and by the European Atomic Energy Community Seventh Framework Programme [FP7 – Fission – 2009] under grant agreement n°624 249624 Collaborative Project Catclay. C. Tournassat acknowledges funding from L’Institut Carnot BRGM for his visit to the Lawrence Berkeley National Laboratory.

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Parkhurst DL, Appelo CAJ (1999) User’s guide to PHREEQC (Version 2) - A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations (No. Denver, CO. U.S. Geological Survey. Water Resources Investigations Report p 99–4259.) Parkhurst DL, Appelo C (2013) Description of Input and Examples for PHREEQC Version 3–a Computer Program for Speciation, Batch-reaction, One-dimensional Transport, and Inverse Geochemical Calculations Pearson FJ (1999) What is the porosity of a mudrock? In: Muds and Mudstones: Physical and Fluid Flow Properties. Aplin AC, Fleet AJ, Macquaker JH (eds) Geol Soc Spec Pub, London, p 9–21 Pearson FJ, Arcos D, Boisson J-Y, Fernandez AM, Gäbler H-E, Gaucher E, Gautschi A, Griffault L, Hernan P, Waber HN (2003) Mont Terri project - Geochemistry of water in the Opalinus clay formation at the Mont Terri Rock Laboratory. Geology Series No. 5 Swiss Federal Office for Water and Geology, Bern Poinssot C, Baeyens B, Bradbury MH (1999) Experimental and modelling studies of caesium sorption on illite. Geochim Cosmochim Acta 63:3217–3227 Reinholdt MX, Hubert F, Faurel M, Tertre E, Razafitianamaharavo A, Francius G, Prêt D, Petit S, Béré E, Pelletier M, Ferrage E (2013) Morphological properties of vermiculite particles in size-selected fractions obtained by sonication. Appl Clay Sci 77:18–32 Rotenberg B, Marry V, Dufrêche J-F, Giffaut E, Turq P (2007a) A multiscale approach to ion diffusion in clays: Building a two-state diffusion-reaction scheme from microscopic dynamics. J Colloid Interface Sci 309:289–295 Rotenberg B, Marry V, Dufreche J-F, Malikova N, Giffaut E, Turq P (2007b) Modelling water and ion diffusion in clays: A multiscale approach. C R Chim 10:1108–1116 Rotenberg B, Marry V, Vuilleumier R, Malikova N, Simon C, Turq P (2007c) Water and ions in clays: Unraveling the interlayer/micropore exchange using molecular dynamics. Geochim Cosmochim Acta 71:5089–5101 Rotenberg B, Marry V, Malikova N, Turq P (2010) Molecular simulation of aqueous solutions at clay surfaces. J Phys : Condens Matter 22:284114 Rotenberg B, Marry V, Salanne M, Jardat M, Turq P (2014) Multiscale modelling of transport in clays from the molecular to the sample scale. C R Geosci 346:298–306 Saiyouri N, Hicher PY, Tessier D (2000) Microstructural approach and transfer water modelling in highly compacted unsaturated swelling clays. Mech Cohes-Frict Mat 5:41–60 Sayed Hassan M, Villieras F, Gaboriaud F, Razafitianamaharavo A (2006) AFM and low-pressure argon adsorption analysis of geometrical properties of phyllosilicates. J Colloid Interface Sci 296:614–623 Schramm LL, Kwak JCT (1982a) Influence of exchangeable cation composition on the size and shape of montmorillonite particles in dilute suspension. Clays Clay Miner 30:40–48 Schramm LL, Kwak JCT (1982b) Interactions in clay suspensions: The distribution of ions in suspension and the influence of tactoid formation. Colloid Surface 3:43–60 Shackelford CD, Moore SM (2013) Fickian diffusion of radionuclides for engineered containment barriers: Diffusion coefficients, porosities, and complicating issues. Eng Geol 152:133–147 Shainberg I, Otoh H (1968) Size and shape of montmorillonite particles saturated with Na/Ca ions (inferred from viscosity and optical measurements). Isr J Chem 6:251–259 Siretanu I, Ebeling D, Andersson MP, Stipp SS, Philipse A, Stuart MC, Ende D van den, Mugele F (2014) Direct observation of ionic structure at solid-liquid interfaces: a deep look into the Stern Layer. Sci Rep 4:4956 Solomon T (2001) The definition and unit of ionic strength. J Chem Educ 78:1691–1692 Sondi I, Biscan J, Pravdic V (1996) Electrokinetics of pure clay minerals revisited. J Colloid Interface Sci 178:514–522 Sposito G (1984) The Surface Chemistry of Soils. Oxford University Press, New York Sposito G (1992) The diffuse-ion swarm near smectite particles suspended in 1:1 electrolyte solutions: modified Gouy-Chapman theory and quasicrystal formation. In: Clay–Water Interface and Its Rheological Implications. Güven N, Pollastro RM (eds) Clay Minerals Society, p 127–156 Sposito G (2004) The Surface Chemistry of Natural Particles. Oxford University Press, New York Steefel CI, Carroll S, Zhao P, Roberts S (2003) Cesium migration in Hanford sediment: a multisite cation exchange model based on laboratory transport experiments. J Contam Hydrol 67:219–246 Steefel CI, Maher K (2009) Fluid-rock interaction: A reactive transport approach. Rev Mineral Geochem 70:485–532 Steefel CI, Appelo CAJ, Arora B, Jacques D, Kalbacher T, Kolditz O, Lagneau V, Lichtner PC, Mayer KU, Meeussen J. C. L., Molins S, Moulton D, Shao H, Šimunek J, Spycher N, Yabusaki SB, Yeh GT (2014) Reactive transport codes for subsurface environmental simulation. Computat Geosci:1–34, doi=10.1007/ s10596-014-9443-x

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects 329 Tachi Y, Yotsuji K (2014) Diffusion and sorption of Cs+, Na+, I- and HTO in compacted sodium montmorillonite as a function of porewater salinity: Integrated sorption and diffusion model. Geochim Cosmochim Acta 132:75–93Tachi Y, Yotsuji K, Suyama T, Ochs M (2014) Integrated sorption and diffusion model for bentonite. Part 2: porewater chemistry, sorption and diffusion modeling in compacted systems. J Nucl Sci Techn 51:1191-1204 Tertre E, Beaucaire C, Coreau N, Juery A (2009) Modelling Zn(II) sorption onto clayey sediments using a multi-site ion-exchange model. Appl Geochem 24:1852–1861 Tertre E, Delville A, Prêt D, Hubert F, Ferrage E (2015) Cation diffusion in the interlayer space of swelling clay minerals—A combined macroscopic and microscopic study. Geochim Cosmochim Acta 149:251–267 Tournassat C, Appelo CAJ (2011) Modelling approaches for anion-exclusion in compacted Na-bentonite. Geochim Cosmochim Acta 75:3698–3710 Tournassat C, Neaman A, Villiéras F, Bosbach D, Charlet L (2003) Nanomorphology of montmorillonite particles: Estimation of the clay edge sorption site density by low-pressure gas adsorption and AFM observations. Am Mineral 88:1989–1995 Tournassat C, Chapron Y, Leroy P, Boulahya F (2009a) Comparison of molecular dynamics simulations with Triple Layer and modified Gouy–Chapman models in a 0.1 M NaCl - montmorillonite system. J Colloid Interface Sci 339:533–541 Tournassat C, Gailhanou H, Crouzet C, Braibant G, Gautier A, Gaucher EC (2009b) Cation exchange selectivity coefficient values on smectite and mixed-layer illite/smectite minerals. Soil Sci Soc Am J 73:928–942 Tournassat C, Bizi M, Braibant G, Crouzet C (2011) Influence of montmorillonite tactoid size on Na–Ca cation exchange reactions. J Colloid Interface Sci 364:443–454 Tournassat C, Grangeon S, Leroy P, Giffaut E (2013) Modeling specific pH dependent sorption of divalent metals on montmorillonite surfaces. A review of pitfalls, recent achievements and current challenges. Am J Sci 313:395–451 Tsai S-C, Ouyang S, Hsu C-N (2001) Sorption and diffusion behavior of Cs and Sr on Jih-Hsing bentonite. Appl Radiat Isot 54:209–215 Van Loon LR, Baeyens B, Bradbury MH (2009) The sorption behaviour of caesium on Opalinus Clay: A comparison between intact and crushed material. Appl Geochem 24:999–1004 Van Loon LR, Soler JM, Bradbury MH (2003) Diffusion of HTO, 36Cl- and 125I- in Opalinus Clay samples from Mont Terri: Effect of confining pressure. J Contam Hydrol 61:73–83 Van Loon LR, Soler JM, Muller W, Bradbury MH (2004a) Anisotropic diffusion in layered argillaceous rocks: a case study with Opalinus Clay. Env Sci Tech 38:5721–5728 Van Loon LR, Wersin P, Soler JM, Eikenberg J, Gimmi T, Hernan P, Dewonck S, Savoye S (2004b) In-situ diffusion of HTO, 22Na+, Cs+ and I- in Opalinus Clay at the Mont Terri underground rock laboratory. Radiochim Acta 92:757–763 Van Loon LR, Müller W, Iijima K (2005) Activation energies of the self-diffusion of HTO, 22Na+ and 36Cl- in a highly compacted argillaceous rock (Opalinus Clay). Appl Geochem 20:961–972 Vanselow AP (1932) The utilization of the base-exchange reaction for the determination of activity coefficients in mixed electrolytes. J Am Chem Soc 54:1307–1311 Wang Z, Giammar DE (2013) Mass Action Expressions for Bidentate Adsorption in Surface Complexation Modeling: Theory and Practice. Env Sci Tech 47:3982–3996 Wersin P, Soler JM, Van Loon L, Eikenberg J, Baeyens B, Grolimund D, Gimmi T, Dewonck S (2008) Diffusion of HTO, Br-, I-, Cs+, 85Sr2+ and 60Co2+ in a clay formation: Results and modelling from an in situ experiment in Opalinus Clay. Appl Geochem 23:678–691 Wilson MJ, Wilson L (2014) Clay mineralogy and shale instability: an alternative conceptual analysis. Clay Miner 49:127–145 Wittebroodt C, Savoye S, Frasca B, Gouze P, Michelot J-L (2012) Diffusion of HTO, 36 Cl- and 125 I- in Upper Toarcian argillite samples from Tournemire: Effects of initial iodide concentration and ionic strength. Appl Geochem 27:1432–1441 Yokoyama S, Kuroda M, Sato T (2005) Atomic force microscopy study of montmorillonite dissolution under highly alkaline conditions. Clays Clay Miner 53:147–154 Zachara JM, Smith SC, McKinley JP, Resch CT (1993) Cadmium sorption on specimen and soil smectites in sodium and calcium electrolytes. Soil Sci Soc Am J 57:1491–1501 Zachara JM, Smith SC, Liu C, McKinley JP, Serne RJ, Gassman PL (2002) Sorption of Cs+ to micaceous subsurface sediments from the Hanford site, USA. Geochim Cosmochim Acta 66:193–211 Zhao H, Bhattacharjee S, Chow R, Wallace D, Masliyah JH, Xu Z (2008) Probing surface charge potentials of clay basal planes and edges by direct force measurements. Langmuir 24:12899–12910

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 331-354, 2015 Copyright © Mineralogical Society of America

How Porosity Increases During Incipient Weathering of Crystalline Silicate Rocks Alexis Navarre-Sitchler Department of Geology and Geological Engineering and Hydrologic Sciences and Engineering Program Colorado School of Mines Golden, Colorado 80401, USA [email protected]

Susan L. Brantley Earth and Environmental Systems Institute and Department of Geosciences The Pennsylvania State University University Park, Pennsylvania, USA [email protected]

Gernot Rother Chemical Sciences Division Oak Ridge National Laboratory Oak Ridge, Tennessee, USA [email protected]

INTRODUCTION Weathering of bedrock to produce porous regolith, the precursor to biologically active soil and soluble mineral nutrients, creates the life-supporting matrix upon which Earth’s Critical Zone—the thin surface layer where rock meets life—develops (Ollier 1985; Graham et al. 1994; Taylor and Eggleston 2001). Water and nutrients locked up in low porosity bedrock are biologically inaccessible until weathering helps transform the inert rock into a rich habitat for biological activity. Weathering increases the water-holding capacity and nutrient accessibility of rock and regolith by increasing porosity and mineral surface area, affecting the particle-size distribution, and enhancing ecosystem diversity (Cousin et al. 2003; Certini et al. 2004; Zanner and Graham 2005). Especially in areas where soils are thin and climate is dry, the water stored in weathered rock is essential to ecosystem productivity and survival (Sternberg et al. 1996; Zwieniecki and Newton 1996; Hubbert et al. 2001; Witty et al. 2003). Removal of soluble material during weathering decreases the concentrations of major elements such as Ca, Na, and Mg and the overall mass of the solid, decreasing the bulk density and increasing porosity. These chemical and physical changes result in decreased uniaxial compressive strength and elastic moduli of the rock and increased infiltration of water through the weathered rock (Tugrul 2004). Porosity in intact bedrock is comprised of inter- and intra-granular pores developed during (re-) crystallization in igneous and metamorphic rocks or diagenesis in sedimentary rocks. As 1529-6466/14/0079-0010$05.00

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we conceptualize it, the conversion of low-permeability bedrock to regolith generally begins due to the transport of meteoric water into protolith through the large-scale fractures that are present as a result of regional tectonic factors or exhumation (Wyrick and Borchers 1981; Molnar et al. 2007). In zones near the fractures, water can infiltrate into the low-porosity rock matrix. This infiltrating meteoric water contains dissolved oxygen and is acidified by CO2 and organic acids, promoting chemical reactions with primary minerals in the rock (e.g., feldspars, pyroxenes, and micas). This ultimately leads to increased porosity through mineral dissolution and weathering-induced fracturing (WIF). As weathering increases both matrix and fracture porosity, more water can infiltrate the rock, leading to more weathering in a positive feedback loop that drives long-term regolith production. Dissolution and fracturing lead eventually to disaggregation of the rock. In this respect, weathering of primary minerals can be an autocatalytic reaction, as described previously (Brantley et al. 2008). Here, the reaction product of the autocatalytic reaction that accelerates the reaction rate is the surface area of the reacting mineral, which increases due to opening of internal porosity to meteoric water infiltration, roughening of the surface, or WIF. Of course, weathering also promotes precipitation of secondary minerals that can occlude porosity and armor dissolving grains. Eventually, the dissolution of the primary mineral grains cannot be compensated by increases in wetted surface area, and the overall surface area of dissolving primary minerals decreases. A long history of geochemical research has helped elucidate the macroscale behavior of weathering processes, including quantification of rock weathering rates and soil production and factors that influence these rates (e.g., Merrill 1906; Berner 1978; Pavich 1986; Nahon 1991; Blum et al. 1994; Drever and Clow 1995; Clow and Drever 1996; Anderson et al. 2002, 2011; Gaillardet et al. 2003; Amundson 2004; Bricker et al. 2004; Burke et al. 2007; White 2008; Brantley and Lebedeva 2011; Hausrath et al. 2011). However, until recently the very earliest pore-scale physical changes associated with incipient weathering were largely unstudied. Recent evidence shows how weathering begins the evolution of porosity by affecting even the smallest pores (pores with diameters < 100 nm) in crystalline igneous rocks (Navarre-Sitchler et al. 2009). Ultimately, feedback between weathering and porosity creation transforms bedrock to regolith. A better understanding of these pore-scale changes that occur in rock during incipient weathering will help link micro-scale behavior and processes to those that can be observed and predicted in numerical simulation of macro-scale changes (e.g., Kang et al. 2007; Li et al. 2008; Jamtveit and Hammer 2012; Molins et al. 2012; Emmanuel et al. 2015, this volume; Molins 2015, this volume). For example, in a meta-analysis of data in the literature, Bazilevskaya et al. (2013) concluded that regolith on granitic rocks worldwide tends to be thicker than on basaltic rock compositions when measured at ridgetops under similar climate regimes (Fig. 1). These differences have been attributed to lithological controls on WIF, which can ultimately open a rock to deep infiltration of meteoric water. Simple modeling exercises document that the depth interval over which a mineral reacts from parent concentration to 0%—the reaction front—is wider and the depth of regolith itself is thicker for a rock where advection contributes to solute transport as opposed to one where solute transport occurs only by diffusion (Brantley and Lebedeva 2011; Bazilevskaya et al. 2013). Specifically, Bazilevskaya et al. (2013) argued that larger volumes of weathering fluids infiltrate granitic rocks than basaltic rocks because WIF occurs when biotite in granites oxidizes at depth. In this article we review studies of porosity development during incipient weathering of igneous rocks—the rock type in which the bulk of this type of research has been performed to date and we explore the presence or absence of WIF in this context. Throughout the next sections we discuss aspects of the pore network, such as pore-size distribution, connectivity, and pore morphology as they have been quantified or observed at nanometer to micron length scales. For simplicity, we refer to

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Figure 1. Regolith thickness shown in box and whisker plots for mafic rocks (14, dark boxes) versus granitic material (13, light boxes) from ridgetop systems reported in the literature and summarized by Bazilevskaya et al. (2013) for different climate regimes as indicated (all systems had precipitation > potential evapotranspiration). Each box represents the envelope for 50% of the reported measurements. The median is shown as the solid line and the mean as the dotted line in each box. Whiskers show one standard deviation. Outliers are shown as symbols. [Used by permission of John Wiley & Sons, Ltd. from Bazilevskaya EA, Lebedeva M, Pavich M, Rother G, Parkinson D, Cole DR, Brantley SL (2013) Where fast weathering creates thin regolith and slow weathering creates thick regolith. Earth Surface Processes and Landforms, Vol. 38, Fig 1, p. 848.]

these porosities as nanoporosity and microporosity, respectively. Although this terminology is loose, it is operationally useful because different techniques are used in the measurement of the differently sized pores, as discussed below. The International Union of Pure and Applied Chemistry (IUPAC) defines micropores as pores with width smaller than 2 nm, mesopores have pore widths of 2–50 nm, and macropores have widths larger than 50 nm (Sing et al. 1985; Rouquerol et al. 1994).

METHODS FOR POROSITY AND PORE-SIZE DISTRIBUTION QUANTIFICATION Detailed analysis and characterization of natural pore systems requires a multitude of techniques that are capable of interrogating different aspects of the pore network across many orders of magnitude length scale. Some of the most widely used techniques include gas sorption, fluid intrusion (including mercury porosimetry), various microscopy and image analysis approaches, and X-ray and neutron scattering.

Sorption and intrusion techniques Gas sorption techniques, especially nitrogen sorption measurements at 77 K, are routinely used for the measurement of internal surface area and pore size in the region of 2–200 nm. Nitrogen gas sorption analysis (often referred to as BET analysis) of an intact sample yields the internal surface area and pore size distribution of accessible, connected pore spaces, while analysis of finely powdered samples yields the total porosity and surface area, plus the surface

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area produced by grinding (Brunauer et al. 1938). Analysis of sorption isotherms with density functional theory (DFT) methods yields the most accurate information about the pore size distribution, because it considers the molecular structure of the absorbate (Thommes and Cychosz 2014). However, several important uncertainties often arise during the interpretation of nitrogen sorption data even using advanced DFT techniques. It is often difficult to differentiate between pore adsorption and pore surface roughness using BET data, limiting the utility of the technique for the study of very fine pores (Thommes and Cychosz 2014). Uncertainties in calculated pore size distributions arise from heterogeneity in mineral identity and organic materials in the rock that possess component-specific interaction potentials and wetting properties for nitrogen and an implicit assumption of exclusively cylindrical or slit pore geometries. Thus, the model-dependent analysis of the adsorption isotherm may yield a distorted pore size distribution. Nitrogen sorption analysis often relies on the evaluation of the desorption branch to eliminate pore geometry effects on pore size distribution. However several possible hysteresis effects can obscure the pore size distributions obtained from both the adsorption and desorption branches of the isotherms (Lowell et al. 2004; Thommes and Cychosz 2014). Current research of fluid adsorption and pore condensation to well-defined synthetic mesoporous materials aims at understanding and quantitative description of these effects. Decoupling of the pore shape and confinement effects of the porous medium is possible, for example, using the hydraulic pore radius, rh, for the characterization of pore size. The hydraulic pore radius is defined as

rh 

2Vp , As

(1)

with Vp the pore volume and As the specific surface area, and describes confinement effects independent of pore geometry. For cylindrical pores, pore radius and hydraulic pore radius are identical, while for slit pores the hydraulic radius equals two times the pore width (Rother et al. 2004; Woywod et al. 2005). The hydraulic pore radius allows characterization of irregularly shaped pore systems. However, this model has not yet been extended to eliminate pore geometry effects on the pore size distribution obtained from gas adsorption measurements. Mercury porosimetry is a complementary technique to nitrogen sorption, yielding information about the sizes and size distributions of pore throats and associated pore volumes at length scales from ca. 2 nm to 200 m. Mercury porosimetry relies on the forced intrusion of liquid mercury into the pore spaces. The high surface tension of mercury of ca. 486 mN/m leads to wetting angles between 90° and 180°, i.e., partial wetting of non-reactive surfaces is found. Therefore, mercury intrudes the larger pores and pore throats at lower pressure, and fills pores of decreasing size with increasing pressure. The Washburn equation gives the relationship between the pressure P and the pore size r into which mercury will intrude:

Pr  2  cos ,

(2)

with  the surface tension, and  the wetting angle. Commonly, wetting angles of ca. 130–160° are found for mercury at mineral surfaces. The mercury intrusion curve is commonly not reversible, and uncertainties about the results can arise with respect to possible damages to the sample imposed by the high pressures of up to several kbar involved in the process. A comprehensive review of sorption and intrusion techniques for the characterization of porous solids with numerous examples is given in (Lowell et al. 2004).

Electron and optical microscopy Electron and optical microscopy are important tools for the determination of pore shapes and their associations with individual mineral phases. Electron micrographs provide direct information about pore shapes and orientations in 2-D for traditional imaging and 3-D for imaging combined with focused ion beam (FIB) slice and view techniques. However, these

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techniques sample only small volumes. Considering the strong spatial heterogeneities typically found in natural rocks, this is a severe limitation that significantly reduces the statistical relevance of the results. Combined spectroscopic imaging techniques provide unique local structural and chemical information that cannot be obtained otherwise. Electron imaging paired with energy dispersive spectroscopy (EDS) mineral mapping and tomography techniques provide a powerful tool to identify exposed mineral surfaces in pores and correlate pore types with mineralogy as demonstrated by Landrot et al. (2012). While X-ray and neutron tomography methods image 3-D volumes of rock and partially overcome sample size limitations of electron imaging, the datasets are at micron to millimeter length-scale resolution and thus do not capture nanometer scale details of the pore network. A complement to imaging techniques, neutron scattering is uniquely suited to characterize pore size distributions and surface roughness at length scales from single nanometers to 10’s of microns (Anovitz and Cole 2015, this volume). An advantage of neutron scattering techniques over many others is the ability to interrogate both the connected and unconnected fractions of the pore network without physical destruction of the sample. Ultimately, each of these techniques has its unique mix of strengths and limitations, and quite often only the combined use of several techniques allows a detailed and informed analysis. In the following section we will focus on the utility of neutron scattering techniques in the context of rock porosity characterization. Emphasis will be put on the utility of small-angle and ultra- small-angle neutron scattering (SANS and USANS) to comprehensively quantify the nano- to micrometer porosity and surface area, both of which control the initial stages of rock weathering.

Neutron scattering Jin et al. (2011) were the first to use neutron scattering to show that the fraction of nanoporosity increases within a rock undergoing weathering. A small-angle neutron scattering experiment is carried out by measuring the scattering signal from a target sample illuminated by a collimated neutron beam with known wavelength, defined beam geometry, and calibrated neutron flux. The majority of neutrons are transmitted through the sample with no interactions, making neutron scattering suitable for the study of much larger samples compared to X-rays. A percentage (up to 10–15%) are coherently scattered upon interactions with interfaces between regions with contrasts in chemistry and density, resulting in contrasts in scattering length density within the target sample. (Radlinski 2006). The coherent scattering length density j* (SLD) for a solid phase j is given by Equation (3):

( i 1 bci ) n

j 

Vm

.

(3)

Here, bci is the bound coherent scattering length of atom i of n atoms of a molecule and Vm is the molecular volume (g mol-1). The neutron SLD does not change monotonically with atomic number, and can be very different for different isotopes of the same element. Therefore, neutrons are sensitive to certain light elements, and isotope contrast variation (especially H/D) can be utilized to highlight or suppress certain structural features in neutron scattering. The angle at which the neutrons scatter is a function of the size of the scattering structures, with an inverse relationship between particle size and scattering angle. Typical small-angle neutron scattering (SANS) and ultra- small-angle neutron scattering (USANS) instrument configurations used to study rocks interrogates length scales of approximately 1 nm–30 μm. The intensity of scattered neutrons at each angle is a function of the number of scattering particles and the scattering contrast. Sample sizes of tens to hundreds of mm3 can be efficiently interrogated with neutron scattering methods, making the measurements useful for studies of larger volumes than microscopy methods that give comparable data (e.g., focused ion beam scanning electron microscopy).

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Beam sizes are on the order of 5–200 mm2, and samples can be interrogated with step width on the order of mm. The required sample and beam sizes depend largely on the scattering power of the sample, which is proportional to the difference in the SLD squared of the nanodispersed phases. SLD values of various minerals are generally in the range of 4 x 1010 cm-2, while voids (i.e., empty pores) have a scattering length density of zero. Thus, while neutrons also scatter from mineral grain interfaces, the intensity of scattered neutrons that arises from interfaces between minerals and pores is usually an order of magnitude higher, and rocks can often be treated as a two-phase system when analyzing neutron scattering data, i.e., minerals + pores (e.g., Radlinski 2006; Anovitz et al. 2009; Navarre-Sitchler et al. 2013). The 2-D pattern of scattered neutrons contains a wealth of information about the internal structure of the sample (Fig. 2). Once corrections are applied for empty beam or cell scattering, transmission and detector efficiency, the 2-D scattering data are analyzed for radial symmetry and averaged to obtain scattered intensity as a function of scattering angle (Fig. 2D). Radially isotropic patterns result from rock samples without preferred orientation of scattering objects, for example, pores in many rock types such as sandstone, limestone, and crystalline igneous rocks. For these rocks, the pore size distribution, internal surface area, and fractal dimensions show no dependence on the sample orientation in the neutron beam. Clays and shales, as well as other rocks with elongated pores with preferential pore orientation, show radial variations in the scattering intensity. Anisotropies of the scattering intensity occur if the sample is oriented in the neutron beam such both the long and the short pore axes are exposed (Fig. 2). These patterns have been explored in more detail in samples of Marcellus shale cut parallel and perpendicular to bedding by Gu et al. (2015). A unique capability of neutron scattering is its ability to interrogate the entire, undisturbed pore system comprised by both accessible and inaccessible pores and differentiate between the connected and unconnected pore fractions using contrast-matching techniques. In samples saturated with an H2O/D2O mixture that have the same scattering contrast as the bulk rock, scattering from connected pores is eliminated, with the result that only unconnected pores are sampled (Navarre-Sitchler et al. 2013; Bazilevskaya et al. 2015). Analysis of SANS from dry and contrast-matched water soaked samples then allows for analysis of the properties of connected and unconnected pore fractions. This technique helps delineate the pore-scale changes that result in creation of a connected pore network for fluid infiltration in weathered rocks. Following the evolution of connected and unconnected porosity fractions in rockweathering fronts has provided detailed insight into weathering processes in a number of recent studies (Jin et al. 2011; Bazilevskaya et al. 2013, 2015; Buss et al. 2013; NavarreSitchler et al. 2013; Xin et al. 2015). While neutron scattering is a powerful tool for analysis of rock pore networks, it is best when confirmed and complemented by additional information from sorption studies and detailed electron microscopy (Radlinski et al. 2000, 2004; Kahle et al. 2004; Radlinski 2006; Anovitz et al. 2009, 2013; Jin et al. 2011; Bazilevskaya et al. 2013; Navarre-Sitchler et al. 2013). The reduction and analysis pathway of neutron scattering data depends on the characteristics of the sample. First, the 2-D detector data are corrected for background and detector sensitivity, and the scattered intensity is normalized to the sample thickness and neutron transmission. Then the intensity is normalized to the empty beam or a calibration standard, and the data are radially averaged, either over the entire detector or azimuthal sections to produce 1D curves of scattering intensity I(Q) as a function of momentum transfer (Q), which is a function of the scattering angle. Rock nanopores are commonly disordered assemblages of pores with different shapes and sizes, which are partly interconnected. Polydisperse hard sphere models can be applied in that case, which permit calculation of parameters such as pore volume, internal surface area and surface roughness, and the pore size distribution from the scattering data (e.g., Hinde 2004). Particle scattering from pores of

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Figure 2. Scattering intensity in 2-D (detector counts shown in color from low in blue to high in white) plotted as x-position versus time of flight in 100s of nanoseconds from a shale sample analyzed at the Lujan Neutron Scattering Center at Los Alamos National Laboratory on a time-of-flight SANS instrument configuration. Data from a sample cut perpendicular to bedding giving an anisotropic scattering pattern that indicates preferentially oriented pores (A). The same sample broken into chips and packed into a cuvette to randomize the pore orientations give an isotropic scattering pattern (B). Scattering data from a thick section of unweathered basaltic andesite from Costa Rica (C) show isotropic scattering from random pore orientations in the rock. Pixel position on the detector ranges from 0 to 120 on x- and y-axes. Correlating Q values range from 0.04 to -0.04 Å-1. Scattering intensity is shown from low in black to high in yellow. Isotropic 2-D scattering data are radially averaged to obtain scattering intensity as a function of scattering angle data for analysis. An example of small-angle scattering shown in radially averaged (1D) is from the unweathered core of a basaltic andesite clast from Costa Rica (D). Small-angle neutron scattering data (black circles) are combined with ultra- small-angle neutron scattering data (white circles) to obtain scattering over three orders of magnitude in length scale (from < 10 nm to > 30 mm).

characteristic size R is observed at Q ≈ 2.5/R. Surface scattering from these structures is found at Q  > 1/R, which can be delineated from particle scattering to assess surface roughness associations with pore size. In many rocks and soils, pore sizes span several orders of magnitude, and scattering data can often be modeled using fractal models. Fractal dimensions and their associated size ranges can give important statistical insight into the morphological properties of rock porosity and its changes with reaction progress. As described in the next section, the surface or mass fractal dimensions of the rock can be directly obtained from the slope of the scattering curves in the applicable range of momentum transfers (Teixeira 1988).

Fractal nature of rocks As shown by neutron scattering analysis, the pore size distribution in many rocks can be described by a mass fractal (Radlinski et al. 2000; Anovitz et al. 2009). Similarly, the roughness of internal pores and cracks often shows surface fractal behavior. Specifically, for

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many rocks, the intensity (I) of scattered neutrons is observed to be inversely proportional to Qm over extended regions in Q, where m is termed the Porod exponent: I (Q) 

A  B. Qm

(4)

Here B is the background, observed to be independent of scattering vector. The background arises typically from incoherent scattering from scatterers such as hydrogen nuclei or smallscale nuclear density inhomogeneities (Radlinski 2006). This background scattering, which may lead to a constant scattering intensity at large Q, does not contain structural information, and is usually subtracted following standard procedures. When a plot of log I versus log Q is linear over more than one order of magnitude of Q, the object causing scattering is considered a fractal where the slope, -n, is related to the fractal dimension. Research during the 1980s and 1990s showed that unweathered shale, sandstone, and igneous rocks can yield plots of log I–log Q that are linear over several orders of magnitude (Mildner et al. 1986; Aharonov and Rothman 1996; Radlinski et al. 1999). In fact, in the case of shales these plots can be linear over ten orders of magnitude (Jin et al. 2013). Unweathered rocks can show evidence of both mass and surface fractals. A value of n between 2 and 3 indicates presence of a mass fractal. In this case, the mass fractal dimension, Dm, is equivalent to the value of n. In contrast, a value of n between 3 and 4 characterizes a surface fractal. In this case the surface fractal dimension Ds equals (6 – n). These relationships, which have been discussed by many authors (Mildner et al. 1986; Wong and Bray 1988; Schmidt 1991; Radlinski 2006), allow straightforward classification of the multi-scale, complex disordered pore systems found in rocks. Interpretation of the obtained fractal dimensions relies on the concepts developed in fractal systems analysis. One definition of a surface fractal is that it is an object of dimension L with a surface area that varies as LDs, where Ds lies between 2 and 3 but is non-integral. Similarly, a mass fractal has a mass that varies as LDm, where Dm lies between 2 and 3, and is non-integral (Radlinski 2006). As the fractal dimension of an object approaches a value of 3, the object becomes either more space-filling (surface fractal) or more polydisperse (mass fractal). In weathering rocks, neutrons scatter from pores and bumps on surfaces that vary in dimension from nanometers to tens of microns. In the simplest case, the internal porosity of the unweathered rock comprises only one fractal. For example, in both the Rose Hill shale (Jin and Brantley 2011) and Marcellus shale (Jin et al. 2013) sampled from central Pennsylvania (U.S.A.), neutron scattering reveals that the internal porosity is a fractal with dimension near 3. However, for crystalline rocks, the parent material is often comprised of two fractals, i.e., both a mass and surface fractal. For example, internal porosity has been characterized by neutron scattering as a mass plus surface fractal for the following crystalline rock protoliths: Costa Rica andesitic basalt (Navarre-Sitchler et al. 2013), Virginia diabase (Bazilevskaya et al. 2013, 2015), Puerto Rico quartz diorite (Navarre-Sitchler et al. 2013), Puerto Rico volcaniclastic sedimentary rock (Buss et al. 2013), and Virginia metagranite (Bazilevskaya et al. 2013, 2015). These rock systems are described in more detail in the next sections. In each case, the authors have argued that the mass fractal at low Q can be conceptualized as an object comprised of pores ranging in size from hundreds of nanometers to tens of microns. These pores are likely positioned mostly at grain boundaries and triple junctions. In contrast, the surface fractal found at large Q is the distribution of smaller scatterers (1 to 300 nm) that can be conceptualized as bumps on the pore surfaces. Obviously, some spatial dimension describes the point where bumps grade into pores: in fact, the break in slope on the log I–log Q curve (the size delineation between the mass and surface fractal) varies from rock to rock, but generally is related to the average grain size (Radlinski 2006).

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CASE STUDIES Studies of incipient nano- and microporosity development during terrestrial weathering have spanned various rock lithologies from granitic to ultramafic compositions in igneous rocks to shales in sedimentary rocks. Here we review study results and methods used to quantitatively analyze the development of porosity during the initial stages of weathering of the crystalline igneous rocks with a focus on compositions from granitic to basaltic. We do not review the recent suite of papers that treat carbonation of ultramafic rocks (e.g., Keleman et al. 2011), but rather focus on systems without significant precipitation of carbonate minerals. In the reported case studies, incipient weathering and development of fine-scale porosity has been studied in the context of different spatial scales. For example, incipient weathering has been investigated at the pedon scale by probing the regolith-bedrock interface at the base of undisturbed weathering profiles. On the other hand, it has also been probed at a smaller scale such as the boundary between weathering rinds and unweathered cores of clasts found weathering in alluvial or glacial deposits. This review includes studies at both scales. At the pedon scale, weathering processes are often studied at ridgetop locations to allow investigators to model the natural processes as one-dimensional systems, i.e., minimizing the influence of lateral fluid flow and sediment translocation that can add material to the top of the profile through downslope transport (e.g., Jin et al. 2010). However, given the small number of studies of incipient porosity development, we do not limit this review to studies of ridgetop locations only. While there are numerous studies of bulk porosity in regolith and soils, to the authors’ knowledge the following studies of the initiation of weathering represent the only published papers where total porosity was quantified along with information on the morphology or pore size distribution of the earliest nano- or microporosity created. We limited our review to these studies for the purpose of gaining insight into the physical changes that occur in rocks during weathering at the pore scale. In some studies chemical analysis was used to define the degree of weathering of each sample. In others, more descriptive means were used to define the extent of weathering. Thus, it is not possible to link the degree of porosity development directly to the degree of chemical weathering for all the studies. Nonetheless, the thickness and advance rate of weathering profiles on unweathered bedrock are highly dependent on the rates of mineral weathering and the flux of water through the weathering front and some generalizations can be made (Lichtner 1988; White 2002). Laboratory-measured mineral dissolution rates of minerals common in basaltic and andesitic rocks (Ca-rich plagioclase, augite, and actinolite for example) are generally faster than those common in the more felsic granite and granodiorite rocks (quartz, Na-rich feldspar, and alkali feldspar). Related to this, regolith on more felsic rocks tends to be thicker than regolith developed on more mafic rocks (Fig. 1). To explore these observations we have separated the case studies into felsic and mafic rocks in the next sections.

Weathering of felsic to intermediate composition rocks In this section we review porosity changes during incipient weathering on rocks ranging in composition from granites and granodiorites to quartz diorite to diorite. These compositions are globally important: granite and granodiorite are exposed over ~15% of the global land surface and underlie many mountain watersheds (Twidale and Vidal Romani 2005). Therefore, granitic weathering is also an important sink of atmospheric CO2 (Li et al. 2013; Maher and Chamberlain 2014). The more felsic rocks differ during weathering compared to the more mafic rocks discussed in the next section because of the common occurrence of WIF. Our summary of felsic rocks includes weathering reported in seven separate settings. Rocks of this lithology typically have very low primary porosity (less than a few percent) and correspondingly low permeability, but generally have large-scale fractures and joints related to tectonic processes such as mountain building and exhumation (Twidale and Vidal

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Romani 2005; Molnar et al. 2007). The predominant mineralogy of the rocks described here includes quartz, alkali feldspar, and Na-rich plagioclase feldspar with muscovite and/ or biotite micas. Accessory minerals include hornblende, magnetite, ilmenite, pyrite, sphene, and zircon. Biotite mica is the predominant Fe-bearing mineral in the samples reported here. In the following paragraphs, we summarize each study setting and its weathering and porosity characteristics. Puerto Rico quartz diorite. A study of samples of quartz diorite from the Luquillo Critical Zone Observatory in the Luquillo Experimental Forest provides the most detailed published information on porosity development during initial weathering of felsic rocks in a setting where many other water and soil fluxes have been measured (Navarre-Sitchler et al. 2013). Navarre-Sitchler et al. investigated an unweathered corestone of quartz diorite, several meters in diameter and surrounded by onionskin-like spheroidal fractures that had been previously described in the literature (Turner et al. 2003). The spheroidal fractures subtend rindlets that are each roughly 2 or 3 cm in width (Buss et al. 2008). The entire set of rindlets around the corestone is 40 cm thick. Samples of rindlets and unweathered corestone were analyzed by SANS and USANS along with transmission electron microscope (TEM) imaging (NavarreSitchler et al. 2013). The low porosity (< 2%) corestone transforms to saprolite across the rindlet zone, and the concentration of plagioclase decreases from parent values to near zero (where most of the loss is in the outermost rindlets): in other words, the rindlet set comprises the plagioclase reaction front. The individual rindlets are intact but increasingly fractured pieces of rock that are hypothesized to have been created by WIF (Fletcher et al. 2006; Buss et al. 2008). Although bulk density decreases from the corestone outward to the outermost rindlets (Buss et al. 2008), the nanoporosity—i.e., porosity measured by neutron scattering— increases markedly in the outermost rindlet to 9.4%. TEM images show microfracture development throughout the rindlet set. These microfractures inside individual rindlets are documented in the neutron scattering data and TEM images at length scales of 60–600 nm, i.e., length scales that are consistent with fracture aperture. The major spheroidal fractures that subtend rindlets have been attributed to volume expansion driven by Fe oxidation during biotite alteration (Fletcher et al. 2006; Buss et al. 2008). Presumably, the microfractures form within individual rindlets both due to WIF as well as relaxation of the rindlets after spheroidal fracture formation. In the innermost rindlets, nanoporosity increases mostly through microfracturing, with little mass removal though mineral dissolution. Once a connected pore network in a rindlet is established through microfractures, however, primary minerals begin to dissolve and the rock matrix nanoporosity increases to > 9% in the last 3 cm of the 40-cm rindlet zone. Analysis of nanopore surface area distribution in the innermost rindlets suggests that the largest surface area increases occurs in pores ≲ 50 nm in the innermost (youngest) rindlets (Navarre-Sitchler et al. 2013). Weathering in the outer rindlets is thought to be enhanced by the presence of Fe-oxidizing organisms that further accelerate weathering by production of organic acids that mobilize Fe and Al. Virginia Piedmont metagranite. Three recent papers report the growth of porosity during incipient weathering of metagranite at ridgetops in the Virginia Piedmont, U.S.A. (Bazilevskaya et al. 2013, 2015; Brantley et al. 2013b). In that setting, the slow rate of erosion and long timescale of exposure to weathering have allowed the weathered regolith profile to putatively reach a steady-state thickness. The authors point out that both the regolith itself and the plagioclase weathering front are 20-times thicker on the metagranite (a metamorphosed quartz monzonite comprised of quartz, albite, orthoclase, muscovite, and biotite) than on a nearby diabase. The diabase is a basaltic composition rock that is discussed in the next section. A combination of methods was used to probe this difference, including neutron scattering,

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X-ray micro-computed tomography, TEM, SEM, and chemical analysis. Total porosity of the metagranite increased from ~2.5% in the unweathered samples to a maximum of 30% in some small sections of weathered material sampled at 20 m depth in intact but weathered rock beneath the saprolite. These large porosities were measured using X-ray micro-computed tomography. This high porosity in the weathered rock was attributed to microfractures observed to be associated with biotite. Nanoporosity (measured by neutron scattering) increased from < 1% in the unweathered rock to 7% in the saprolite, but much of the increases were in the unconnected portions of the pore network. Therefore, although pore opening was occurring, precipitation during weathering was presumably impacting the pore network connectivity. The dominant pore size in the unweathered rock analyzed by neutron scattering was < 10 nm in diameter, but porosity was documented to be present at length scales up to 1 mm and greater. With weathering, the fraction of pores at length scales from 10 nm to 100 mm decreased, either through closure due to precipitation or enlargement out of the range of neutron scattering length scales (Bazilevskaya et al. 2015). The authors concluded that the greater thickness of the regolith and the reaction front on the metagranite compared to the diabase was due to differences in solute transport during early weathering: WIF allowed solute transport by advection that in turn led to thicker regolith. In contrast, on the nearby diabase (described in the next section), microfracturing was not observed and solute transport was limited to diffusive processes. Numerical modeling of weathering systems also supports the conclusion that fracturing leads to greater regolith thickness and reaction front compared to weathering in a non-fractured system (Fig. 3). Turkish granodiorite. Our third case example reports porosity estimates as a function of weathering based on methods other than neutron scattering (electron imaging and fluid or gas intrusion techniques). In this example, Tugrul (2004) studied the effect of weathering on pore geometry in granodiorite and basalt (described in the next section) using pycnometer and Hg porosimetry tests to determine total and connected porosity of variably weathered samples, respectively. The degree of weathering of the samples was classified by strength characteristics and not by geochemical analysis. These authors reported that the primary porosity in the Cavusbasi granodiorite in Istanbul was ~5% and consisted mostly of microfractures. The total porosity of the weathered samples increased with each stage of alteration and at all length scales analyzed (diameters from 0.01 to 100 mm) to a total porosity of 12–16% in the most weathered samples. These highly weathered samples were described as weathered rock where the majority of microfractures are open and the original texture of the rock is still visible. Showing similar behavior to the Puerto Rico quartz diorite, the weathered samples were not characterized by chemical leaching or mineral dissolution until the later stages of weathering. The effective porosity (4% in the unweathered rock) also did not increase until the last stage of weathering where it increased to 8%. California granitic corestones. In our fourth case example—granite corestones weathering in the Bishop Creek Moraines, CA—Rossi and Graham (2010) measured porosity to understand water movement and storage in soil and rock fragments. Porosity of these granite samples was determined by the difference between bulk and particle density measurements made by 3-D laser scanning. Porosity was observed to increase with increasing exposure age of the moraine material (determined by Cl-36 cosmogenic dating). They reported a porosity growth rate of ~0.1% total porosity per kyr: i.e., porosity increased from ~2% (with low connectivity) in the youngest samples to ~14% in samples with weathering ages of 120 ka. The porosity increase was attributed to an increased number of pores of dimension > 100 mm (i.e., pores observed in SEM images) without significant change in the extent of weathering as quantified by bulk chemistry. They concluded that microfracturing was the predominant mechanism driving the early porosity increases. In the oldest samples, porosity was observed

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Figure 3. Schematic showing albite values plotted versus depth for two types of weathering profiles that can be observed at ridgetops where the rate of erosion equals the rate of weathering advance: completely developed profiles (A) and incompletely developed profiles (B). albite = 0 when 0% of the albite has dissolved away and albite = -1 when all the albite has dissolved away. Incompletely developed profiles occur for protoliths that weather so slowly that the system becomes weathering-limited. In this case, the feldspar is still present in the soil at the land surface. Shales can be characterized by incompletely developed weathering profiles (Jin et al. 2010). Completely developed profiles often form on crystalline rocks where weathering is not rate-limiting. More mafic crystalline rocks generally develop thinner reaction fronts (l = reaction front thickness) and regolith (L = regolith thickness) because solute transport across the reaction front is by diffusion. In contrast on felsic rocks, microfracturing in the weathering rock commonly transforms the rock from mass + surface fractal to one or more surface fractals well before disaggregation into saprolite. For any given set of weathering conditions, both l and L therefore tend to grow thicker. In this case the mass + surface fractal transforms to a surface fractal deep in the weathered rock (see Fig. 5). [Used by permission of John Wiley & Sons, Ltd. from Bazilevskaya EA, Rother G, Mildner DFR, Pavich M, Cole DR, Bhatt MP, Jin L, Steefel C, Brantley SL (2015) How oxidation and dissolution in diabase and granite control porosity during weathering. Earth Surface Processes and Landforms, Vol. 38, Fig 5, p. 854.]

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to increase to a maximum of ~24% in concert with chemical weathering. They argued that a connected pore network in the matrix adjacent to the microfractures allowed weathering to proceed in these later stages. Spanish granodiorite. In the fifth case example, Ballesteros et al. (2011) analyzed samples from a 55–60-m deep weathering profile developed on a two-mica Hercynian granodiorite in central Spain using Hg porosimetry, gas adsorption, and density measurement techniques. Porosity increases from 10% in the upper portion of the profile, ~45 m from the unweathered bedrock. Hg-porosimetry data was reported to show no measureable pore volume accessible in pore throats < 5 mm diameter. This was inferred to indicate very low connectivity of small pores in the sample. As the granodiorite weathered, the investigators documented increasing pore connectivity and fluid transport pathways particularly through small pore throats. While no direct evidence of microfracturing was derived from the Hg-porosimetry data, photomicrographs of the weathered granodiorite show microfracturing at micron length scales that likely contribute to the connected pore network. Japanese granite. The sixth case example is a study of the hydraulic properties of the Tanakami Granite in Japan reported as a function of weathering. This study provides information on porosity development based on macroscale core analyses. Katsura et al. (2009) collected cores of weathered granite in the Kiryu Experimental Watershed and performed infiltration experiments to determine permeability as a function of depth and appearance of rock alteration. Water retention curves from the samples are consistent with the inference that initial alteration of the granite increased the density of micropores (pore diameter ≲ 1.5 mm) while the density of macropores only increased in the latest stages of weathering when the rock began to lose most of its internal structure. The development of macropores also corresponded to the largest increases in effective porosity and permeability. Argentina andesite. Weathering of an andesitic sill in Argentina resulted in complex patterns that include evidence of WIF. Specifically, the andesites show evidence of Liesegang banding, spheroidal weathering fractures (also referred to as onionskin spallation), and hierarchical fracturing (Jamtveit et al. 2011, 2012). Unlike the spheroidal weathering in quartz diorite in Puerto Rico described above where fractures form at outer edges of unweathered corestones, the Argentina andesite was described as showing fractures throughout the weathered zone. Fractures were attributed to precipitates that formed throughout the pore space. Like the Puerto Rico system, weathering fluids were inferred to have entered the unaltered rock (8% porosity) along perpendicular joints formed prior to weathering. The porosity, investigated by Jamtveit et al. (2011) with porosimetry and X-ray microtomography, consists of pores that are either small (< 10 μm) or relatively large (10–300 μm). The larger pores are subtended by very narrow pore throats observed in the range of 20–200 nm. The authors argue that high solubilities are maintained within the small pore throats because of pore-size-related solubility effects (Emmanuel et al. 2015, this volume). These solubilities were inferred to have driven transport of solutes into the larger pores where secondary minerals precipitated. Specifically, the authors argue that dissolution of actinolite + ilmenite + plagioclase in the oxidizing, C-containing meteoric fluids caused precipitation of quartz, calcite, and ferrihydrite. Jamtveit et al. (2011) emphasize that weathering in the andesites is autocatalytic because of the production of protons during Fe oxidation. The precipitates are localized in the Liesegang bands that are spaced at 2–5 mm. Based on a textural argument, they propose that the rate and position of ring formation are controlled by feedbacks between dissolution localized in pore throats and supersaturation that drives nucleation in the large pores: between the bands, the nucleation barrier is not crossed and no precipitates form. As a result, large pores fill with precipitates, while small pores remain precipitate-free. He and Hg injection, used to measure porosity, showed that porosity remains high in the andesites, largely because precipitation also causes dilation of the rock at grain boundaries around the larger pores. After 5 to 10 Liesegang

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bands form, the rock tends to fracture spheroidally to form a rindlet. Repetition of these ongoing processes creates well banded and fractured corestones. In addition to banding and onionskin fracturing, an additional phenomenon, hierarchical fracturing, was also identified to occur. Hierarchical fracturing of corestones into smaller fragments was attributed to stress buildups due to weathering.

Weathering of mafic rocks In this section we review the available literature on porosity changes with incipient weathering on intermediate to mafic rocks: basalt, basaltic-andesite, and diabase reported in five separate studies. The basalt, basaltic-andesite, and diabase in three of these studies are comprised mainly of plagioclase and augite with little to no olivine, while the andesite examples contain plagioclase and hornblende or actinolite with other minor phases. In general, these rocks are more Mg and Fe rich (more mafic) than the rocks described in the previous section. In contrast to the felsic rocks described above, WIF is less common on these rocks. Costa Rica basaltic andesite. Navarre-Sitchler et al. (2009, 2013) analyzed porosity growth with incipient weathering in weathering rinds on basaltic andesite clasts from Costa Rica using neutron scattering and X-ray computed tomography. The clasts, comprised of plagioclase phenocrysts in a plagioclase and augite matrix, were deposited in alluvial terraces during periods of marine highstands ranging from ~50 k to 250 k years ago (Sak et al. 2004). Weathering of the primary minerals in the basaltic andesite to secondary Fe and Al oxides occurs over a narrow (~4 mm) reaction front that was imaged in thin sections. Neutrons were scattered from sections cut across the narrow reaction front between the core and weathering rind of three clasts. By situating a screen with a slit for neutron passage in front of the rock section and moving it between neutron scattering measurements, Navarre-Sitchler et al. (2013) measured scattering intensity as a function of distance from the unaltered core of the clast. As expected, the intensity of scattering was observed to increase with distance from the core. Total porosity in these samples (combined porosity measured with neutron scattering and X-ray computed tomography) was observed to increase from ~3% to > 30% across that narrow zone and the abundances of the weathering primary minerals (e.g., plagioclase, pyroxene) were observed to decrease from parent concentration to near 0%. Pores in the unweathered basalt were observed to mostly be < 100 nm in diameter and were observed primarily along grain boundaries and at triple grain junctions. As the basalt weathers, these pores were observed to increase in size and intra-grain porosity with diameters > 100 nm developed as minerals dissolved in heterogeneous patterns (Fig. 4). The connectivity of porosity was observed to be very low in the unweathered basalt and nanoporosity was observed to be important in the connecting of larger pores. Effective porosity (the porosity contained in a connected pore network) increased with weathering at all analyzed length scales once the total porosity increased to > 9% (Navarre-Sitchler et al. 2009). The increases in total and connected porosity were inferred to promote important feedbacks in solute transport through the weathered material related to mineral dissolution and weathering rind growth (Navarre-Sitchler et al. 2011). Volcaniclastic sedimentary rocks in Puerto Rico. Like in the Costa Rica basaltic andesites, no evidence of WIF was commonly observed in a set of weathered andesitic corestones from the volcaniclastic sedimentary rocks of the Fajardo formation in the Bisley watershed in Puerto Rico (Buss, H., pers. comm.). These corestones had about 8 + 4% total porosity (for pores < 10 μm), and they generally remained angular during weathering (Buss et al. 2013). The porosities of two corestones were analyzed with neutron scattering and described by Buss et al. (2013). These samples are more evolved than the basalts from Costa Rica or Turkey, with plagioclase phenocrysts in a quartz and alkali feldspar matrix with minor amphibole content but no biotite. When exposed at the surface or in samples recovered from drill cores, corestones were observed to be characterized by weathering rinds that are usually

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B

A Pore

Rind Pore

200 microns

200 nm

C Core

Pore

500 nm

Figure 4. Scanning electron microscope (A) and transmission electron microscope (B, rind and C, core) of weathered basalt clasts from Costa Rica. Pores along grain boundaries (C) enlarge and connect during weathering and promote additional mineral dissolution that leads to nm and mm scale pores in heterogeneous patterns at the core rind interface (A) and in the rind (C).

millimeters in thickness. When measured with neutron scattering, the nanoporosity increased slightly with incipient weathering from < 1 to 2–4% over a distance of ~4 mm across the rind. In addition, Fe was enriched and Ca and K depleted in the rind compared to Al. The rind material typically reached thicknesses no greater than 1 cm: this observation is consistent with significant increases in porosity in rind material that allowed spalling off from the corestones. Such thin reaction fronts and rinds on corestones are considered diagnostic of solute transport by diffusion (Navarre-Sitchler et al. 2009, 2011, 2013; Ma et al. 2011; Lebedeva and Brantley 2013). Virginia Piedmont diabase. Samples from a diabase in the Piedmont region of Virginia (USA) were analyzed with neutron scattering, X-ray computed tomography, and microscopy to evaluate porosity development with weathering (Bazilevskaya et al. 2015). Once again, microfracturing was not observed. The mineralogy of the diabase was dominantly plagioclase and pyroxene. Porosity in the diabase increased from < 2% in the unweathered diabase to a maximum of ~25% in the saprolite. In the unweathered rock, most of the porosity was comprised of nanoporosity (diameters < 1 mm). Approximately half of the nanoporosity was in a connected network with dominant pore dimensions from 10–20 nm with additional contributions from pores in the 30–70 nm diameter range. Neither the total (connected + unconnected) nor the connected nanoporosity increased significantly with weathering. However, pore size distributions constructed from neutron scatter data show the nanoporosity that developed was in the 100–500 nm diameter range. Most of the porosity increases were in pores > 1 mm diameter, the size range analyzed with X-ray computed tomography. Using SEM analysis, Brantley et al. (2013b) observed that the first reaction in the diabase was dissolution of Fe(II)-containing pyroxene lamellae without any observable Fe(III)-oxide precipitation (Bazilevskaya et al. 2015). This phenomenon correlated with unusual patterns of

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neutron scattering that exhibited non-isotropic scattering targets showing azimuthal orientations of high scattering intensity. These patterns were inferred to document the dissolution of thin, Fe-containing lamellae in the pyroxenes, probably below the water table. Turkish basalt. Results from pycnometry and Hg-porosimetry analysis of samples from the Hasanseyh Basalt Formation in Turkey also reveal patterns of porosity growth during incipient basalt weathering that were not characterized by WIF (Tugrul 2004). These samples are similar petrologically to the Costa Rica basalts with plagioclase phenocrysts embedded in a plagioclase and augite matrix. Analysis by Hg-porosimetry revealed that pores in the unweathered basalt are mostly accessible through pore throats < 10 m. As porosity increases with weathering, the pore throats associated with that porosity are characterized by diameters < 10 m, with many < 10 nm. The authors attribute narrow pore throats in these samples to mineral neoformation in pore spaces, similar to those observed in andesite weathering by Mulyanto and Stoopes (2003). It is likely that the development of narrow pore throats leads to similar effects related to effective porosity during weathering as observed in the Costa Rica basalts (Navarre-Sitchler et al. 2009). Specifically, as porosity increases with incipient weathering, the effective (connected) porosity remains roughly constant until total porosity is > 8%. In the basalt studies by Tugrul (2004), this marked increase in effective porosity does not occur until the sample had weathered enough to be broken apart by hand (weathering category IV as defined in that paper). This relationship between total and effective porosity was shown to have important implications for solute transport through weathered basaltic andesite rocks by Navarre-Sitchler et al. (2009).

LINKING FRACTAL SCALING AND PORE-SCALE OBSERVATIONS TO WEATHERING MECHANISMS Brantley et al. (2013b) recently argued that the ratio of ferrous oxides to base cation oxides (i.e., FeO concentration / the sum of the concentrations of Na2O, K2O, CaO, and MgO) in protolith may be a predictor of whether WIF occurs (Brantley et al., 2013b). Here, the term WIF is restricted the fracturing driven by chemical weathering, and does not include fracturing driven by freeze–thaw, root pressure, or other such processes. In this ratio of FeO to base cation oxides, the numerator and denominator summarize the relative capacity of the rock to consume oxygen versus the capacity to consume carbon dioxide during weathering, respectively. In rocks with low FeO concentrations, oxygen may not be consumed at shallow depths, and oxidation may therefore be the deepest weathering reaction. When oxidation leads to reaction products with larger volume than reactants, as inferred for biotite oxidation, the expansion is likely to crack the rock and promote infiltration of meteoric fluids. For example, the volume expansion that occurs during oxidation of biotite has long been well known (Jackson 1840; Eggler et al. 1969; Graham et al. 2010; Rossi and Graham 2010). Thus, for felsic rocks that contain biotite, it may be common that deep oxidation causes WIF that promotes deep infiltration of fluids (Buss et al. 2008; Bazilevskaya et al. 2013, 2015; Navarre-Sitchler et al. 2013). In contrast, Brantley et al. (2013b) argue that for mafic rocks with more capacity to consume oxygen (i.e., high FeO content) than CO2 (i.e., sum of the base cation oxide concentrations), the high content of ferrous iron may deplete weathering fluids in O2 at depth so that the deepest weathering reaction is CO2- rather than O2-promoted. Without a deep oxidation reaction, the rock may not microfracture. Deep porosity that grows during weathering is therefore developed exclusively though dissolution of minerals (Fig. 4). Without significant increases in effective porosity from microfracturing to create a connected pore network, thin regolith and thin reaction fronts may develop as described for the Costa Rica basaltic andesites (NavarreSitchler et al. 2009), the Virginia diabase (Bazilevskaya et al., 2013), and the Hasanseyh Basalt (Tugrul 2004).

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It is interesting to note that these different weathering mechanisms correlate with differences in the transformations of the mass + surface fractals during weathering. For example, Bazilevskaya et al. (2013) showed that the mass + surface fractal that comprises the unweathered granitic rock in the Piedmont of Virginia transforms during weathering to a surface fractal at 20 m of depth (e.g., Fig. 5, left) where biotite oxidation commences. In contrast to that biotitecontaining rock, in the nearby Virginia diabase, the deepest reaction is non-oxidative dissolution of Fe-containing pyroxene. This reaction occurs at a depth of only a few meters beneath the land surface. Therefore, the pore network in the weathered diabase rock was observed to remain a mass + surface fractal until disaggregation into saprolite (Fig. 5, right).

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Figure 5. Plots of the values of Porod exponents versus depth plots for diabase (left) and metamorphosed granitic rock (right) weathering in the Piedmont of Virginia. Depth axis shows depth below the mineralorganic soil interface (SOIL = bioturbated soil, SAP = saprolite, WR = cohesive rock showing visual signs of weathering, UWR = cohesive rock showing little to no weathering features). The granitic rock transforms from a mass + surface fractal to a surface fractal deeper than 20 m depth, where biotite oxidation initiates in weathered rock. In contrast, the diabase never transforms to a surface fractal in the weathered rock. [Used by permission of Soil Science Society of America from Bazilevskaya EA, Rother G, Mildner DFR, Pavich M, Cole DR, Bhatt MP, Jin L, Steefel C, Brantley SL (2015) How oxidation and dissolution in diabase and granite control porosity during weathering. Soil Science Society of America Journal, Vol. 79, Fig. 7b, p. 858.]

Retention of the mass + surface fractal during weathering of still intact rock as observed in the diabase may therefore be a signature of a rock weathering without WIF where solute transport is dominated by diffusion. In contrast, transformation of intact rock porosity to a surface fractal such as in the granitic rocks may be diagnostic of WIF. This comparison between the Virginia diabase and metagranite is similar to the fractal characteristics for basalt and quartz diorite shown in Figure 6 (data from Navarre-Sitchler et al. 2013). On the left, neutron scattering data are summarized for the clast of basaltic andesite that weathered in a fluvial terrace in Costa Rica for 125,000 y (see earlier section). In contrast, scattering data are summarized for the quartz diorite weathering in Puerto Rico on the right of Figure 6. The increases in nanoporosity and associated surface area during weathering are accompanied by a large depletion in plagioclase in the basaltic andesite but not in the quartz diorite (compare the plots of Ca in Fig. 6). Both the basaltic andesite and the quartz diorite contain internal

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Figure 6. log (I × Q4) versus log Q (i.e., Porod) plots for Costa Rica basaltic andesite (a) and Puerto Rico quartz diorite (b). Porod plots are stacked so that distance from parent (P) increases upward into the rind (R) or saprolite (S) for (a) basaltic andesite and (b) quartz diorite, respectively. The reaction front is labeled RF for the basalt and RZ for rindlet zone in the quartz diorite. Data fit with a surface fractal are shown in black squares and data fit with a mass fractal are shown as grey squares on the Porod plots. In the lower graphs, the slope of the high-Q data (plotted as squares) and low-Q data (plotted as circles) indicate the mass and surface fractal behaviors of the pore networks as a function of depth and weathering progress. Note that in (b) a new set of features is indicated by the region of shallow slope (hatched) in the RZ, plotted as open squares on the graph. This region is attributed to scattering from smooth surfaces on micro-cracks that develop during weathering. Portions of scattering curves that are non-linear (not fit to fractal model) are shown in open squares. Ca values plotted versus depth for each of the systems show depletion of Ca through plagioclase dissolution across the weathering front in the basalt, but little depletion of Ca in the quartz diorite.

porosity that is characterized by a mass + surface fractal: Dm = 2.9 or 2.8 and Ds = 2.7 or 2.5 before weathering (Navarre-Sitchler et al. 2013). The delineation between the mass and surface fractals in both rocks is approximately 1 m. As weathering proceeds across the plagioclase reaction front, the mass + surface fractal is retained in the basaltic andesite (Fig. 6a) but is transformed in the quartz diorite to multiple surface fractals, including a surface fractal region at length scales from 60 to 600 nm with Ds ranging from 2.2–2.3 (Fig. 6b) attributed to relatively smooth surfaces of microfractures. This transformation to surface fractals is accompanied by only minor change in specific surface area and porosity until late in the transformation of rock to saprolite (Fig. 7). Many lines of evidence document that solute transport during weathering of the basaltic andesite clasts is due to diffusion (Navarre-Sitchler et al. 2009, 2011). For example, a decrease

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Figure 7. Plots showing the variation in total porosity (closed squares) and specific surface area (open circles) for sample locations within the plagioclase reaction front in the weathering basaltic andesite (A) and quartz diorite (B) shown in Figure 6. In each case, 0 represents the interface between weathered and unweathered material, and weathering intensity increases as distance increases. Between 6 and 37 cm in (B), the rock has spheroidally fractured into centimeter-thick rindlets. Total porosity and specific surface area were calculated using PRINSAS (Hinde 2004). The surface area does not include contributions from pores with radius larger than 30 mm. [Used by permission of Elsevier from Navarre-Sitchler AK, Cole D, Rother G, Jin L, Buss HL, Brantley SL (2013) Porosity and surface area evolution during weathering of two igneous rocks. Geochimica et Cosmochimica Acta, Vol. 109, Fig. 5, p. 408.]

in the surface fractal dimension across the reaction front in the basaltic andesite is consistent with the internal porosity becoming less space-filling as the reaction proceeds. Smoothing of mineral surfaces during weathering is generally associated with transport limitation and is thus consistent with diffusion as the solute transport mechanism. In contrast, the transformation of mass + surface fractal to multiple surface fractals during spheroidal weathering is accompanied by precipitation of Fe and Mn in some rindlets in the quartz diorite, both of which are consistent with infiltration of meteoric fluids into fractures. The fracturing is inferred to break apart the rock into subzones that are still limited by diffusion of solutes (like the basaltic andesite). However, these subzones become increasingly smaller as microfracturing proceeds. The difference in solute transport through the reaction front—diffusion for the non-fractured basaltic andesite versus advection in the spheroidally weathered felsic rock—is considered the explanation for the orders of magnitude difference in weathering advance rates between the two rock types: 0.24 versus 100 mm kyr-1, respectively (Fletcher et al. 2006; Pelt et al. 2008). Like the Piedmont rocks, the difference in mechanism of solute transport for the andesitic basalt and the quartz diorite is documented in the fractal dimensions during weathering.

SUMMARY In this article we have reviewed the relatively few papers that have been published concerning nano- and micro-scale porosity in incipiently weathering crystalline rocks. Growth of such porosity is diagnostic of the very first stages of regolith formation. To date, few models of porosity formation are available to quantify how fast regolith forms and what environmental or lithologic variables control the rate. For example, the intrinsic dissolution rates of many Fe(II)- and Mg-containing minerals and the more calcic plagioclases in mafic rocks are faster than the equivalent Fe- and Mg-poor minerals and more sodic plagioclase in felsic rocks (Bandstra et al. 2008). While some might predict thicker regolith on rocks with faster mineral reaction rates, in general the regolith thickness and reaction front thickness on mafic rocks are thinner than on felsic rocks when developed at ridgetops under similar climate (Bazilevskaya

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et al. 2013; Fig. 1) . The observations indicate that chemical weathering cannot be surface reaction (kinetically) controlled alone. It was proposed in the case of the Costa Rican basaltic andesites that the rate was controlled by the rate of porosity generation at the grain (micron) scale (i.e., local-scale surface reaction control leading to enhancement of transport rates within the weathering front). This would suggest that basalts should weather faster, expect for the fact that fracturing intervenes in the case of granitic weathering. The presence or absence of fracturing during weathering may explain why regolith developed on mafic rocks at ridgetops tends to be thinner than regolith on granitic rocks when other variables are held constant. We have shown several examples where felsic rocks fracture during weathering but mafic have lower occurrences of weathering induced fracturing (WIF). In the felsic rocks the mass + surface fractal that comprises the protolith transforms to one or more surface fractals due to this microfracturing associated with biotite in felsic rocks. This WIF then allows infiltration of advecting meteoric fluids, which in turn widens the reaction front and contributes to formation of thicker regolith (Bazilevskaya et al. 2013; Brantley et al. 2014). Thus weathering of granitic material tends to develop thicker reaction fronts and thicker regolith for a given set of conditions. In contrast, for mafic compositions—perhaps especially those that do not contain biotite—the mass + surface fractal that comprises the protolith is maintained throughout much of the weathering, and the main solute transport mechanism is diffusion. In mafic rocks that weather without WIF, solutes are transported into the reacting low-porosity rock mainly by diffusion, and the reaction front and the regolith that develops tends to be thinner. Even though several of the studies summarized here for incipient porosity development did not report WIF in mafic rocks, WIF sometimes occurs in the more Fe- and Mg-rich rocks (e.g., Chatterjee and Raymahashay 1998; Patino et al. 2003; Røyne et al. 2008; Hausrath et al. 2011). It remains to be explained why WIF occurs in some cases but not in others. Presence or absence of biotite may be part of the answer, but a one-to-one correspondence between the presence of biotite and the observation of WIF has not been documented. Another possible explanation is that WIF may be controlled not only by the reduction and acid-neutralizing capacity of the protolith, but also by the relative concentrations of O2 and CO2 in the soil atmosphere (Brantley et al. 2013b). Furthermore, many different mechanisms of WIF have also been proposed (Bisdom et al. 1967; Ollier 1971; Chatterjee and Raymahashay 1998; Fletcher et al. 2006; Jamtveit et al. 2011) and more work is needed to test and understand these and other mechanisms driving fracturing during weathering. A full understanding of regolith formation will only be possible when we can quantitatively describe the changes in porosity and surface area at the pore scale that occur during weathering, especially deep in the Critical Zone.

ACKNOWLEDGMENTS The small-angle neutron scattering at the National Institute of Standards and Technology was supported in part by the National Science Foundation under Agreement No DMR-0944772. S. Brantley acknowledges DOE OBES funding DE-FG02-OSER15675 for work using neutron scattering and NSF Critical Zone Observatory Funding for work in the Luquillo Critical Zone Observatory. We thank T. Clark and M. Yashinski at Material Characterization Laboratory at the Pennsylvania State University for FIB-SEM. We thank the Appalachian Basin Black Shales Group at the Pennsylvania State University and PA Topographic and Geologic Survey for providing shale samples. The small-angle neutron scattering at the National Institute of Standards and Technology was supported in part by the National Science Foundation under Agreement No DMR-0944772. S. Brantley acknowledges NSF grant OCE 11-40159 for support for working on Marcellus shale, DOE OBES funding DE-FG02-OSER15675 for work on porosity using neutron scattering. Work by G. Rother was supported by the U.S. Department of Energy, Office

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of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division. Research was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U. S. Department of Energy. The identification of commercial instruments in this paper does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the equipment used are necessarily the best available for the purpose. We would like to thank Carl Steefel and Anja Røyne for their very helpful reviews of this chapter.

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Reviews in Mineralogy & Geochemistry Vol. 80 pp. 355-391, 2015 Copyright © Mineralogical Society of America

Isotopic Gradients Across Fluid–Mineral Boundaries Jennifer L. Druhan Department of Geology University of Illinois at Urbana-Champaign Urbana, Illionois 61801, USA and Department of Geological and Environmental Sciences Stanford University Stanford, California 94305, USA [email protected]

Shaun T. Brown Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California 94720, USA and Department of Earth and Planetary Science University of California Berkeley Berkeley, California 94720, USA [email protected]

Christian Huber School of Earth and Atmospheric Sciences Georgia Institute of Technology Atlanta, Georgia 30332, USA [email protected]

INTRODUCTION The distribution of stable and radiogenic isotopes within and among phases provides a critical means of quantifying the origin, residence and cycling of materials through terrestrial reservoirs (Wahl and Urey 1935; Epstein and Mayeda 1953; Johnson et al. 2004; Eiler 2007; Porcelli and Baskaran 2011; Wiederhold 2015). While isotopic variability is globally observable, the mechanisms that govern both their range and distribution occur largely at atomic (e.g., radioactive decay), molecular (e.g., the influence of mass on the free energy of atomic bonds) and pore (e.g., diffusive transport to reactive surface) scales. In contrast, the vast majority of isotope ratio measurements are based on sample sizes that aggregate multiple pathways, species and compositions. Inferring process from such macroscale observations therefore requires unraveling the relative contribution of a variety of potential mechanisms. In effect, the use of isotopes as proxies to infer a specific parameter, such as temperature (Urey 1947) or residence time (Kaufman and Libby 1954), carries the implicit requirement that one mechanism is the primary influence on the measured isotopic composition of the composite sample. 1529-6466/15/0080-0011$05.00

http://dx.doi.org/10.2138/rmg.2015.80.11

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In the present chapter, we consider a wide variety of macro-scale observations of isotope partitioning across fluid–solid phase boundaries. For this purpose we define the continuum scale as a representation in which interfaces are averaged over elementary volumes, as opposed to the pore scale in which these interfaces are explicitly resolved. Throughout this review it will be demonstrated that observations of isotope partitioning across fluid–solid boundaries require some representation of the isotopic composition of the solid surface and surrounding fluid distinct from ‘bulk’ or ‘well mixed’ reservoirs. For example, this distinction is necessary in order to (1) quantify the partitioning of radioactive and radiogenic species, (2) describe transport limitations that may impact the macroscopic partitioning of isotope ratios, (3) explain observations of transient fractionation due to dissolution through preferential release at the solid surface, and (4) parameterize apparently variable fractionation factors during precipitation. The ability to describe isotopic partitioning specific to the phase boundary then influences the accuracy of simulations from highly variable, field scale systems (e.g., Druhan et al. 2013) to highly controlled laboratory experiments (e.g., Tang et al. (2008)). These observations imply that quantifying the composition of solids and fluids as averages across a given representative volume carries some inherent loss of information that isotopes appear to be sensitive to. This conclusion leads us to consider mechanistic descriptions of isotope partitioning that could be significantly improved by modeling approaches that are capable of resolving spatial zoning within individual solids (e.g., Li et al. 2006; Tartakovsky et al. 2008; Molins et al. 2012; Molins 2015, this volume; Yoon et al. 2015, this volume). The suite of experimental data and quantitative approaches described herein support a conceptual framework in which macroscopic observables, such as an apparent fractionation factor, are the emergent result of multiple interacting processes that are strongly influenced by the physical and chemical characteristics of the fluid–solid boundary. In this sense we consider the pore scale as a characteristic length over which the unique physical and chemical properties of the phase interface may be described as distinct from bulk or aggregate values. This definition of the pore scale provides a critical reference frame over which molecular scale mechanisms combine to yield the macroscopic observables (Steefel et al. 2013).

A conceptual model of isotope partitioning at the pore scale The flux of solutes in porous media is influenced by the development of spatial and temporal gradients resulting from the combined effects of transport and reactivity. How the pore-scale nature of these processes emerges into continuum scale observations of fluid–solid interaction is yet unresolved and requires multiscale (e.g., fractal) upscaling methods that remain a challenge. For example, Darcy scale fluid flow may occur along a fixed gradient, but the size, shape and distribution of individual solid grains is such that at the pore scale velocity vectors vary in both magnitude and direction. As a result, the principle mechanism of solute transport in some areas is advective, while in others it is diffusive. Mixing between these distinct regimes is approximated at the continuum scale by either a heterogeneous conductivity field (Li et al. 2010; Sudicky et al. 2010), a hydrodynamic dispersion coefficient (Gelhar and Axness 1983; Steefel and Maher 2009) or a non-uniform fluid travel time distribution (Maloszewski and Zuber 1982; Bellin and Tonina 2007). Reactions that occur between fluid and solid phases are influenced by these local porescale transport regimes. For example, where transport of solutes between a solid surface and a surrounding fluid is accomplished primarily by diffusion, the rate of reaction across that interface may be limited by either the delivery of solute to the reactive surface or the approach to equilibrium. When advection is dominant, the same process may be governed by the relative rates of reactivity and flow (Rolle et al. 2009; Maher 2010; Hochstetler et al. 2013). The factors influencing isotopic partitioning are then equally variable across the pore scale (Fig. 1). In areas of the domain governed by diffusion, the partitioning of isotopes may reflect a transport limitation or diffusive fractionation, whereas in areas where flow is relatively fast, the isotopic

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fluid

solid

Isotopic abundance

Figure 1. Conceptual model of isotopic partitioning at the pore scale during fluid–rock interaction. Darcyscale flow occurs from left to right, whereas at the pore scale local variations in both magnitude and direction lead to areas of the domain that are either advection or diffusion dominated. The bulk fluid isotopic composition also varies from left to right as a result of reaction progress, but the isotopic composition of fluid and the solid surface is subject to multiple fractionating processes and thus variable. The precipitation of new solid or dissolution of existing solid is then also isotopically variable and distinct from the bulk.

composition is likely governed by the partitioning associated with a reaction (Lemarchand et al. 2004; Fantle and DePaolo 2007; DePaolo 2011). The accumulation of newly formed solids is also anticipated to vary with the local transport regime, which in turn exerts an influence on the isotopic composition of the fluid–solid interface. This relationship between the dominant mechanisms of transport and fractionation suggests that over a representative volume of porous media a variety of fractionation factors may be observed. For example a fractionation factor characteristic of diffusive transport between the solid–fluid surface and the well-mixed fluid, or a fractionation factor associated with the difference in isotope composition between the fluid and solid at each reactive surface. The continuum-scale or observable fractionation factor over the timescale of the reaction (e.g., seconds to years) is then subject to the location and volume of material sampled. Over much longer periods of time, processes like recrystallization and solid-state diffusion homogenize spatial zoning within the solid. This implies that transient isotopic partitioning should be observable over significant periods of time, and that the observed macroscopic fractionation factor is potentially (1) a combination of multiple, distinct mechanisms and (2) variable as a result of parameters such as saturation state and flow rate. From this perspective, variability in the magnitude of an observed fractionation factor is in some ways analogous to the discrepancy in rate constants observed across natural systems (Malmstrom et al. 2000; White and Brantley 2003; Maher et al. 2006b). This wide range in what should in principle be a fixed value has been attributed to the relative influence of a variety of processes, such as changes in reactive surface area (VanCappellen 1996), transport limitation (Steefel and Lichtner 1998; Maher 2011; Li et al. 2014); and even climate variations (Kump et al. 2000; Maher and Chamberlain 2014). Similarly, the apparent isotopic partitioning observed in flux-weighted or homogenized natural samples is often distinct from that obtained under controlled experimental conditions. These effects have been noted in a

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variety of contexts, including oxygen and nitrogen isotope fractionation in marine sediments (Brandes and Devol 1997), selenium isotope fractionation in wetlands (Clark and Johnson 2008), compound-specific stable isotope analysis (CSIA) of organics (Van Breukelen 2007) and chromium isotope fractionation in a contaminated aquifer (Berna et al. 2010). To the extent that pore structure influences the distribution of isotope ratios in reactive systems, using volume-averaged sample measurements to quantify related parameters, such as reaction progress or mixing, can result in significant uncertainty. For example, both analytical and numerical solutions have demonstrated that neglecting the effects of hydrodynamic dispersion on observed stable isotope ratios can lead to an underestimation of reactivity in through-flowing systems (Abe and Hunkeler 2006; Van Breukelen and Prommer 2008). As noted above, dispersion is one method of parameterizing mixing at the continuum scale, and thus approximating the contribution of distinct fluid isotopic compositions at the pore scale (Fig. 1). Improved accuracy should then be achievable by describing distributed porescale isotopic compositions as the result of a variety of mechanisms, e.g., multiple fractionating reactions, a difference in the diffusion coefficient of the isotopologues of a compound or the dampening of a kinetic fractionation as a result of transport limitation. In the current article, we consider observations of isotopic partitioning across fluid–solid boundaries associated with a variety of fractionating mechanisms. In reviewing these examples we emphasize the aspects of each process that lead to pore-scale heterogeneity and thus a disconnect in behavior between the scale of mechanism and the scale of observation. These categories are by no means an exhaustive list of all processes that result in isotopic partitioning across reactive interfaces, but serve as examples in which interpreting the response of continuum-scale isotopic values to variations in external parameters is improved by consideration of pore-scale isotopic distributions.

Organization of article The structure of the article is broadly divided into three sections. First, we provide a brief explanation of the notation used to describe isotope partitioning, with an accompanying discussion of primary mechanisms and models for fractionation. Second, we describe experimental observations for four examples of isotopic partitioning across fluid–mineral interfaces: -recoil, diffusion, dissolution, and precipitation. Across this wide range of processes, a common observation is that the interface between fluid and solid phases (1) governs material transfer between reservoirs and (2) displays isotopic compositions distinct from bulk values. Associated models for the mass balance at phase boundaries are described with particular emphasis on the mechanisms necessary in order to quantify macroscale observations. This leads to the second section in which current modeling techniques for the description of transient isotopic final and the development of zoned mineral grains are discussed, concluding with a novel application of pore-scale modeling techniques to describe the distribution of stable isotopes across a fluid–mineral boundary as an initially oversaturated system establishes equilibrium. Throughout this review the principal intent is to describe experimental and modeling studies of isotopic partitioning with reference to the ways in which the composition and gradients across fluid–mineral boundaries is distinct from bulk-averaged values and uniquely influences continuum-scale observations.

NOTATION The exchange of material across fluid–mineral surfaces influences both radiogenic and stable isotope distributions. Unlike (most) stable isotope fractionation, variations in radiogenic isotopes are mass-independent and arise due to the radioactive decay of a parent nuclide to an intermediate radioactive nuclide or a stable, radiogenic nuclide. The uranium decay series (Fig. 2) illustrates the relationship amongst parent and daughter isotopes, and the terminology is applicable to other radioactive-radiogenic systems.

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234

U

t1/2 2.45e5 Yr

238

U

t1/2 4.55e9 Yr

234

Pa

t1/2 6.7 h

α 234

Th

β-

α 230

Th

β-

t1/2 24.1 Days

t1/2 24.1 Days

α 226

Ra

t1/2 1600 Yr

α 222

Rn

t1/2 3.7 Days

β-,α 206

Pb

Stable daugter

Figure 2. Simplified illustration of the 238U decay series. The half-life and primary decay path are indicated for each isotope.

The 238U isotope is radioactive and for N nuclei at time t in a closed system the number of decaying nuclei dN in the time interval dt follows a homogeneous Poisson statistical law and is proportional to N: 1 dn (1) N dt where lambda () is the decay constant. The expression on the right side is a probability per unit time of a radioactive decay event. Integration of Equation (1) yields an exponential equation for radioactive decay: 

N  t   N 0 e t

(2)

where the integration constant (N0) is equal to the initial number of nuclei at t = 0. The decay constant is related to the nuclide half-life t1  2

ln  2  

For intermediate daughter products such as commonly discussed in terms of activity (A) where:

(3)

234

A  t   N  t 

U and

226

Ra, isotopic abundance is (4)

and thus the initial activity at t = 0 is A0 = N0. In a closed system that starts with no daughter isotopes, for example the 238U- 234U system, the initial activity of 234U is zero and it will increase until A238U  A234U . At this point the supply of 234U from 238U decay is balanced by the decay of 234U to 230Th, representing a steady state where the ratio of  limits the maximum possible intermediate daughter activity. When A1 = A2 this is a special condition

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known as secular equilibrium. In the remainder of the chapter we will use the notation 234 U / 238 U AR  A234U / A238U where the subscript AR denotes the activity ratio. Finally, for stable radiogenic daughter isotopes (e.g., 206Pb, in the 238U series), accumulation of isotope N2 is related to the initial abundance of the parent isotope (N0) and time:



N 2  N 0 1  e 1t



(5)

such that as t goes to infinity N2 goes to N0. Krane (1988) presents an exhaustive discussion of radioactive decay including considerations for branched decay, and very short-lived intermediate daughter isotopes. The subsequent discussion will use the 238U–234U system to consider the influence of pore-scale effects in quantifying -recoil damage and associated alterations to solid surfaces and solution chemistry. As noted above, there can be ambiguity in the notation used to describe the partitioning of stable isotopes. In the subsequent text we primarily adhere to the guidelines put forth by Coplen (2011). The isotope ratio (R) of a particular reservoir is defined as



R iE / jE



P

 

 N ( i E )P / N j E

(6)

P

where N is the number of atoms of iE and jE, the isotopes i and j of the element E in substance P. This value is commonly reported relative that of a standard ratio (std), referred to as the delta value (), and defined as

 

 i EP  R i E / j E



 R iE / jE



 P

P



 R iE / jE



/ R iE / jE



  / R E / E i

j

std

std

1

std

(7)

This delta value is nondimensional, but is commonly multiplied by 1000 to report values as parts per thousand or per mil (‰). The difference between the isotope ratios (Δ) of two compounds or phases (P and Q) is then  i EP /Q   i EP   i EQ

(8)

The apparent or net isotopic fractionation factor does not make use of delta notation, but is defined as



 i EP /Q  R i E / j E



P



/ R iE / jE



Q

(9)

Conversion between ΔiEP/Q and iEP/Q may be approximated as ΔiEP/Q ≈ ln iEP/Q. From these relationships it is noted that both the difference between the isotope ratios (Eqn. 8) and the apparent fractionation factor (Eqn. 9) between two reservoirs are obtained from direct measurement regardless of the variety of reaction pathways necessary to produce them, and are therefore a function of the representative volume of the sample in all but the most simplified systems. In contrast, a fractionation factor associated with a specific reaction pathway may be derived in reference to the rate law for that reaction. These types of fractionating processes are discussed in the subsequent section. Finally, an isotopic mole fraction (X) may be defined as the ratio of the amount of a particular isotope in a given species, compound, or reservoir divided by the total amount of that element in the same group

  /  nE 

X i EP  n i E

P

(10)

This value is used in the derivation of isotope-specific reversible rate expressions that involve a solid phase.

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A note on fractionation In this chapter, stable isotope fractionation will be discussed in terms of equilibrium and kinetic processes. For this purpose chemical equilibrium is described as a dynamic state that occurs when two elementary reactions, the forward reaction from reactants to products and the backward reaction from products to reactants, are in balance. For example, the hydroxylation of dissolved CO2: CO2  OH -  HCO3-

(11)

which is at equilibrium when in a closed system the distribution of species is invariant in time. The ratio of the elementary forward and backward rates is then termed the equilibrium constant (Keq). Isotopic equilibrium may be described similarly, for example considering carbon isotope equilibrium between the carbon-bearing species in Equation (11): 12

CO2  H13CO3-  13 CO2  H12 CO3-

(12)

leading to a separate equilibrium constant to account for this isotopic partitioning. A comparable relationship could be written for the partitioning of oxygen isotopes between CO2 and HCO3-, or between HCO3- and OH-. While all of these equilibria are described using the same law of mass action, it is important to note that they are not predicated on one another. In other words, these descriptions allow for the establishment of chemical equilibrium without necessarily requiring isotopic equilibrium. For reactions involving simple stoichiometric relationships, where one atom of the element of interest occurs in all reactant and product species, the isotopic equilibrium constant is equivalent to the fractionation factor (Eqn. 9); however, the relationship is often more complex (Schauble 2004). In general, the equilibrium fractionation factor is temperature dependent and larger for low mass elements and for isotopes of the same element that have large differences in mass. Typically, the partitioning of stable isotopes between two phases at equilibrium preferentially incorporates the heavier isotope in the phase with lower bond energy. An imbalance between the forward and backward rates leads to a net accumulation of either the product or reactant, and the rate of this mass transfer is described by kinetics. Many reactions take place under conditions in which the reverse reaction is in some way prohibited, or the system is very far from equilibrium, such that isotopic partitioning is entirely kinetic. A kinetic fractionation factor is expressed as the ratio of the isotopic composition of the instantaneously generated product species (Pinst) and the residual reactant (Q) through a single reaction pathway:



 kin i EP /Q  R i E / j E



Pinst



/ R iE / jE



Q

(13)

This kin can be related to the kinetic rate constant of the reaction (k), depending on the order of the reaction. For example, a first order dependence on concentration (e.g., dP/dt = kQ), leads to an expression for kin = ik/jk (Mariotti et al. 1981). From an observational perspective it is often difficult to categorically identify equilibrium vs. kinetic effects on isotope partitioning. In low-temperature systems equilibration can be extremely slow, and, in addition, open system conditions may support the establishment of a steady state that appears balanced but is not necessarily equilibrated. In this sense the distinction between a specific state that is dynamic equilibrium, and an observable net rate of precipitation or dissolution does not imply an exclusive influence of equilibrium vs. kinetic fractionation. The approach towards an equilibrated system implies instead a continuum between pure kinetic fractionation and equilibrium fractionation. Later sections of this chapter explore the consequences of such a model and the extent to which pore scale treatments are capable of improving upon current approaches.

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Under certain conditions, it is possible to develop theoretical models to predict the evolution of the isotopic composition of products and reactants. These models offer the advantage that they are simple to use, but are generally limited by particular assumptions and do not reflect the complexity of the reaction pathway or the relationship between transport and reaction in the isotopic mass balance description. Rayleigh fractionation, for instance, assumes an open system distillation process, where the reactant is progressively consumed, such that the isotope ratio of the reactant follows (Rayleigh 1902): R  R0 f ( 1)

(14)

where the isotope ratio (Eqn. 6) is equal to the product of the initial isotope ratio (R0) and the fraction of reactant remaining relative to the initial concentration (f) raised to the power (1). This relationship requires a constant fractionation factor , and produces an exponential relationship between reaction progress and isotopic partitioning. This model is only strictly intended for systems in which the reactant is continually supplied (f cannot go to zero) and the product of the reaction is instantaneously removed or segregated from the reactive system (Criss 1999). In practice the Rayleigh model is used in a wide variety of systems because it offers a simple relationship between reaction progress and fractionation without requiring knowledge of the reaction pathway or transport mechanisms. As a result, several studies have pointed out limitations to the Rayleigh model in application to hydrogeochemical systems (Brandes and Devol 1997; Abe and Hunkeler 2006; Van Breukelen and Prommer 2008). A goal of the current chapter is to describe a variety of common fractionating mechanisms that often undermine the assumptions of such simplified models and promote new quantitative methods for explicit treatment of reactivity and transport in the description of isotope partitioning.

EXAMPLES OF ISOTOPIC ZONING ACROSS FLUID–SOLID BOUNDARIES Alpha recoil Prior to the development of modern radioactive decay counting and mass spectrometry techniques it was widely assumed that the 234U/238UAR could not deviate from secular equilibrium. This was assumed to be the case because the small mass difference (~1.7%) of the two isotopes would not produce stable isotope effects like those observed for hydrogen and oxygen. Careful study of natural rocks and minerals in the 1950’s, however, revealed variations from secular equilibrium (Chalov 1959). Since the chemical behavior of 234U and 238U should be nearly identical, and yet variations in 234U/238UAR can exceed 500%, researchers suggested that the energy associated with 238U decay could directly eject the 234Th daughter isotope from the mineral surface (10’s of nm) or damage the mineral lattice allowing preferential leaching of the daughter isotope (Rosholt et al. 1963; Kigoshi 1971). Kigoshi (1971) carried out a pioneering study where fine-grained zircon sand was suspended in dilute aqueous solutions and the addition of 234Th and 234U to the solution was quantified. The 234Th activities of the solutions were consistent with predicted -recoil injection to the solution based upon a spherical grain model and an -recoil distance of 55 nm. Earlier inferences (Turkowsky 1969) and later laboratory studies (Fleischer 1988) suggest that the recoil distance is closer to 30–40 nm. Both the 234Th and 234U in the fluid increased with time at a rate greater than the increase in 238U, demonstrating that 234Th has a recoil distance of tens of nm and can be ejected from the mineral structure or preferentially leached from lattice defects (Kigoshi 1971; Fleischer and Raabe 1978). Kigoshi (1971) calculated the expected rate of addition of -recoil ejection from a mineral grain based on:

234

Th to a fluid (Q) due to

Isotopic Gradients Across Fluid–Mineral Boundaries Q

1 LSN 238  238 4

363 (15)

where L is the -recoil range, S is the surface area,  is the solid density, and N238 and 238 are the number of 238U atoms per gram of solid and the decay constant, respectively. The activity of 234Th in solution is a production–decay equation (combining Eqns. 4 and 5) of the form: A234Th 





1  t LSN 238  238  1  e 234Th /  234Th 4

(16)

All of the terms on the right side of the equation are known or can be measured independently, however, in natural systems the grain surface area (S) and its influence in directing -recoil to the fluid phase is arguably the most important variable. For example, Kigoshi (1971) assumed spherical grain geometry for their 1–10 μm diameter zircon sand and then fit the measured 234 Th activities to calculate a characteristic recoil distance of 55 nm. Once L is known from similar experiments or determined by other methods, the probability of an -particle (f) being ejected from a mineral grain assuming a spherical geometry can be approximated:

3 L L3  f     4  r 12r 3 

(17)

Subsequent papers have explored the evolution of the solid phase and the effects of nonideal grain geometry on the production of daughter products to the fluid phase (Kigoshi 1971; Fleischer and Raabe 1978; Fleischer 1980; 1988; Maher et al. 2004; DePaolo et al. 2006, 2012; Lee et al. 2010; Handley et al. 2013). A general conclusion from these studies, particularly those of Lee et al. (2010) and Handley et al. (2013), is that the chemical treatment of sediments and the model of grain surface structure need to be carefully considered and normalized across multiple laboratories in order for results to be broadly useful. Similar observations are noted, with important caveats for the differing chemical behavior, in the elements Th, Ra and Rn (Torgersen 1980; Semkow 1990; Sun and Semkow 1998; Porcelli and Swarzenski 2003). For example Ra is strongly adsorbed to mineral surfaces at low ionic strength but soluble at high ionic strength, thus the solution chemistry is critical for interpreting Ra activities (Moore 1976). The general equations describing the evolution of uranium series isotopic ratios in pore fluids and minerals are presented by Ku et al. 1992; Porcelli et al. 1997; Henderson et al. 2001; Tricca et al. 2001; Porcelli and Swarzenski 2003; Maher et al. 2004. The evolution of pore fluid composition is related to primary mineral dissolution, secondary mineral precipitation reactions, -recoil and preferential leaching of daughter isotopes to the pore fluid, sorption– desorption reactions and diffusion–advection of fluid in and out of a pore. Alpha-recoil loss, preferential leaching near the mineral surface, and solid-state diffusion, will all affect the solid composition with respect to time. However, these processes operate at different timescales, allowing mineral grains to evolve distinct 234U/238UAR domains. There is considerable discussion in the uranium series literature about the relative roles of preferential leaching and direct -recoil leading to daughter isotope accumulation in the pore fluid (Rosholt et al. 1963; Vigier et al. 2005; DePaolo et al. 2006; Dosseto et al. 2006). Preferential leaching may occur due to mineral lattice damage associated with 234Th recoil (Rosholt et al. 1963) or due to preferential oxidation of the 234U in damaged mineral lattice by aqueous fluids (Kolodny and Kaplan 1970). Regil et al. (1989), Roessler (1983, 1989) and Adloff and Roessler (1991) present detailed models of 234U oxidation due to -recoil.

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It is difficult to ascertain the exact mechanism that transfers 234U preferentially from the solid phase to the pore fluid, but this process has important implications for the interpretation of uranium series isotopes in mineral–fluid systems. DePaolo et al. (2006) suggest that based on fine-grained alluvial sediments, the leaching depth into grains is not appreciably greater than the recoil distance. For the purposes of this discussion we proceed under the assumption that direct -recoil is the primary mechanism of 234U transfer, but acknowledge this may be unwarranted, particularly in coarser-grained or uranium rich minerals. The mass conservation equations presented below are applied to the 234U and 238U isotopes and illustrated schematically for distinct pore fluid, solid surface, and solid interior compositions in Figure 3. This formulation can be applied to other intermediate daughter products with additional consideration for differing chemical behavior (e.g., strong sorption/ secondary mineral partitioning of thorium and radium under certain conditions). At the scale of a single pore surrounded by mineral grain surfaces, the 234U/238UAR of the fluid will evolve based on the dissolution–precipitation reactions and -recoil ejection to the fluid. The 234U/238UAR evolution of mineral grains with time is: 234

U / 238 U AR  1  f    A0  1  fa   e t

(18)

It is apparent from Equation (18) that probability of -particle ejection (f) and time are the critical variables that affect the activity ratio of the solid grains (DePaolo et al. 2006). Estimating f requires knowledge of the mineral volume surface area and recoil distance. DePaolo et al. (2006, 2012) calculated f as a function of grain diameter and shape (surface area), demonstrating, for example, that mineral grains of 10-m diameter could have greater than a factor of 10 variability in f (Fig. 4). Porewater (234U/238U)a > 1.0 Recoil Loss Pa

234 234

U

Rim (234U/238U)a < 1.0

Th

234

238

U

Depleted surface layer

Bulk Solid (234U/238U)a = 1.0

L
>F/x

0

kCo1 m). The former is the fundamental scale in which physical processes (flow, transport, and geochemistry) take place, and the porous medium is regarded as discrete in nature (void space vs. grain space). The latter is a more practical scale, where we would ultimately like to have a reliable description of flow and reactive transport, and the porous medium is regarded as a continuum. The macroscopic parameters appearing in the description of continuum models, such as permeability or dispersion coefficient, are typically extracted from experiments or stand-alone pore-scale simulations. While such a “hierarchical” upscaling approach is often appropriate, there are many practical problems for which this is not the case. In these problems, perturbations of state variables (e.g., concentration) at the pore scale are tightly coupled to their averaged quantities at the continuum scale. In other words, a clear separation between the temporal scales and/or the spatial scales does not exist. An often encountered scenario is solute transport in the presence of strong fluid–mineral reactions (Kechagia et al. 2002; Battiato and Tartakovsky 2011). Any improvement to the continuum description of these processes requires a fundamental understanding of the relevant physics at the micro scale. In this effort, pore-scale models are an essential asset. 1529-6466/15/0080-0013$05.00

http://dx.doi.org/10.2138/rmg.2015.80.13

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This chapter is divided into two parts. In the first, we review various pore-scale modeling approaches for simulating flow and reactive transport, and discuss their advantages and limitations. We focus primarily on single-phase flow and transport, and give a brief discussion on multiphase flow for completeness. Several review articles on the latter are provided as reference for the interested reader. Among the various pore-scale models, we place the greatest emphasis on pore-network models. Pore-network models are “intermediate-scale”, or “mesoscale”, models bridging the gap between the pore scale (1–100 μm) and the core scale (10 cm–1 m). This is an important scale because it incorporates several thousand pores, where most macroscopic behaviors begin to emerge. Network models are also attractive because of their simplicity, which is why they can handle domain sizes much larger than other pore-scale models. However, their simplicity can also be the source of inaccuracy and ambiguity. In some cases, one can modify the simplifying assumptions of traditional network models (at the expense of increasing their complexity) and substantially increase their accuracy, while preserving computational efficiency. In other cases, such a modification may be inherently limited, and a different pore-scale approach has to be sought. We give examples of both scenarios in this chapter. In the second part, we turn our attention to the problem of transferring information obtained from pore-scale models to larger continuum models. This is an essential step, since the latter is the most practical scale of interest to most geochemists, hydrologists, and subsurface engineers. This chapter focuses only on a review of modeling strategies that are designed for scenarios in which the pore and macro scales are tightly coupled, and are not amenable to hierarchical upscaling. Instead these approaches seek to establish a dynamic “two-way communication” between the pore and the continuum scales, typically over a small region of the porous medium. In doing so, both pore-scale and continuum models are used within the same computational domain, which is why they are commonly referred to as “hybrid” models. The hope is that the following material familiarizes beginning researchers with current advances in pore-scale as well as hybrid modeling, and aids them in adopting and improving upon those methods that best fit their research interests.

PORE-SCALE MODELING Direct pore-scale modeling Over the past few decades modeling fluid flow and solute transport at the pore scale has seen the adoption and development of various computational methods. The first and perhaps most critical step prior to any modeling effort is the accurate characterization of the pore-space geometry/topology. Imaging techniques such as X-ray microtomography (XMT) (Wildenschild and Sheppard 2013; Noiriel 2015, this volume) have made it possible to obtain accurate 3-D characterizations of the complex pore-space geometry of rock samples. These techniques have also been used to visualize and quantify pore structural changes as a result of biofilm growth (e.g., Iltis et al. 2011) and reactive transport (e.g., Noiriel et al. 2005) as shown in Figure 1. For granular media, Monte Carlo (Maier et al. 2003), cooperative rearrangement (Thane 2006), and sequential sedimentation (Coelho et al. 1997; Øren and Bakke 2002) algorithms are commonly used to digitally reconstruct representations of various grain packs, in which the pore space geometry is well-defined. Such reconstructions, although approximations of actual porous media, provide valuable insights into the link between depositional processes (e.g., cementation and compaction) and hydraulic/transport properties (e.g., permeability) of granular media (Bryant et al. 1993a,b; Bakke and Øren 1997). Modeling can then proceed either by simulating directly on the complex void geometry, or on a simplification thereof. The former approach is typically referred to as direct modeling, while the latter is closely associated with pore-network modeling (see the Pore-network models section). In this section, we review a handful of direct modeling approaches recently used at

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Figure 1. (i) 3-D XMT image of biofilm growth on glass beads [Used by permission of John Wiley & Sons, Inc., from Iltis GC, Armstrong RT, Jansik DP, Wood BD, Wildenschild D (2011) Imaging biofilm architecture within porous media using synchrotron-based X-ray computed microtomography. Water Resources Research, Vol. 47, no. 2, doi: 10.1029/2010WR009410, Fig. 1. Copyright © 2011 by the American Geophysical Union]. Cross-sectional XMT image of a limestone core sample at (iia) initial state and (iib) difference between initial and final states of a reactive transport experiment (i.e., dissolution by acidic carbon-dioxide-saturated water); (dark) porosity, (high grey level) matrix, (low grey level) dissolved matrix (i.e., increased porosity) [Used by permission of Oil and Gas Science Technology, from Noiriel C, Bernard D, Gouze P, Thibault X (2005) Hydraulic properties and microgeometry evolution accompanying limestone dissolution by acidic water, Oil & Gas Science and Technology, Vol. 60, p. 177–192, Fig. 4].

the pore scale. These include computational fluid dynamics (CFD) (Wendt 2009; Molins et al. 2014; Trebotich et al. 2014; Molins 2015, this volume), Lattice-Boltzmann (LB) (Chen and Doolen 1998; Yoon et al. 2015, this volume), and smoothed particle hydrodynamics (SPH) (Monaghan 1992; Tartakovsky et al. 2008b). These methods can be used to solve the governing Equations (1–2) for single-phase, incompressible, Newtonian flow on the pore space.  v     v v   p   2 v  fb  t 

(1a)

v  0

(1b)

v |fs  0

(1c)

c  v c  Dm 2c  rhm t

(2a)

 Dmc |fs  rht

(2b)

Equations (1a) and (1b) represent the Navier–Stokes and continuity equations, respectively. v, p, , μ, c denote fluid velocity, pressure, density, viscosity, and solute concentration. fb denotes body forces exerted on the fluid. Equation (1c) is the no-slip boundary condition at   the rock–fluid interface, fs. Since typical Reynolds numbers (Re = d/μ; d is a characteristic length of the medium typically the grain size) in the subsurface are low (1 m). The reliability of such intermediate-scale models, however, hinges upon their underlying simplifying assumptions. There are two levels of approximations made in every pore-network model. The first is geometric, and occurs in replacing the complex void space geometry with an “equivalent” pore network. The other is physical, and occurs when we make simplifying assumptions to describe the pore-scale physics on this idealized geometry. The key to constructing predictive network models is to identify and capture the most essential geometric and physical features associated with a given phenomenon (as alluded to in Direct pore-scale modeling). These essential features are not invariant from one problem to the next. In multiphase flow, for example, it is very important to include angular geometries in the description of the pore network. Excluding this feature ignores the possibility of corner flow of the wetting phase, and snap-off of the non-wetting phase because of it. On the other hand, including inertial effects in the description of two-phase flow in a pore-network model seems to be of little importance if residual saturation trapped behind an advancing drainage front is to be predicted (Moebius and Or 2014). In single-phase flow, angularities are unimportant and only accurate values for hydraulic conductivities of the network elements are required. When geochemical reactions are present, neglecting to accurately quantify fluid–mineral interfacial areas will lead to large errors in macroscopic predictions. In passive solute transport, while assuming perfect mixing conditions within individual pores is inappropriate for ordered media, it appears to be a good assumption for disordered media (see Network modeling of solute transport). In identifying these essential geometric and physical features, direct pore-scale modeling as well as controlled (e.g., micromodel) experiments play an indispensable role. Once an essential feature is identified, traditional network models may be modified to attain improved predictive accuracy. Although such a modification may not always be possible, in which case a different mesoscale modeling strategy has to be sought. In the following section, we review network modeling of single-phase solute transport and provide two examples where direct porescale modeling was used to establish the need for modifying traditional mesoscale assumptions. In the first, modifications lead to substantial improvements in the predictive capacity of the traditional network model, while in the second improvements appear to be inherently limited.

Network modeling of solute transport Pore-network modeling of solute transport has received special interest from many authors in the past few decades and several methodologies have been proposed. A prerequisite to simulating transport is the computation of the velocity field within throats. The procedure

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is quite standard and involves: a) imposing pressure boundary conditions on the pore network, b) describing flow rates within throats via a constitutive equation e.g., Hagen–Poiseuille for Newtonian fluid in a cylindrical throat, c) enforcing mass balance at each pore e.g., Equation (3), where gij is the throat hydraulic conductivity, pij the pressure drop across the throat, and N ith the number of throats connected to pore i, d) solving the resultant system of (linear or nonlinear, depending on fluid rheology and/or flow regime) equations for pore pressures, and e) computing throat flow rates/velocities from the aforementioned constitutive throat equation. Nith

gij

  p

ij

0

(3)

j 1

The simulation of transport then resumes using computed throat flow rates/velocities. Below in this section, we review several Eulerian and Lagrangian approaches developed in the literature for modeling transport on pore networks. Advantages and limitations of each method are outlined and discussed. We then focus on the main sources of error and difficulty within each modeling class, and the possibility of minimizing them. Finally, we provide a brief review of modeling homogeneous/heterogeneous reactions on pore networks. Eulerian network models. Bryntesson (2002), Acharya et al. (2005, 2007b), Li et al. (2006), Kim et al. (2011), Mehmani et al. (2012), and Nogues et al. (2013) are among those who have adopted the popular mixed-cell method (MCM), in which solute-balance equations are written for each pore, i.e., Equation (4). th ,q 0

th, q 0

Ni Ni c dc Ni Vpi i   ci qij   c j qij   Dm aij ij  R(ci ) dt lij j 1 j 1 j 1 th

(4)

In Equation (4), ci and Vpi are the concentration and volume of pore i; qij, aij, and lij are the throat flow rate, cross-sectional area, and length, respectively; R(ci) is the reaction term. In this method, throats are assumed to be volumeless and the solute within the pores perfectly mixed (i.e., single concentration value assigned to each pore). Solute flow rates within throats are formulated as the algebraic sum of an upwinded advection term (first two terms on the RHS of Eqn. 4) and a linearly varying diffusion term (third term on the RHS of Eqn. 4). Essentially, MCM can be regarded as a low-order, finite-volume method on an unstructured grid, i.e., the pore network. The advantage of MCM is that it is very computationally efficient and highly adaptable to various transport scenarios. For example, Acharya et al. (2005) used MCM to study non-linearly adsorbing solute transport, and determined that more than a million pores were required for their results to be statistically representative. Li et al. (2006) and Kim et al. (2011) studied complex geochemical reaction kinetics of anorthite and kaolinite precipitation/ dissolution relevant to CO2 sequestration. Nogues et al. (2013) studied porosity/permeability evolutions in carbonates due to carbonic-acid-driven precipitation/dissolution reactions. They considered 18 aqueous species and 5 mineral species undergoing 14 independent reactions. The flexibility and computational efficiency of MCM is why such complex systems acting on sufficiently large pore-scale domains can even be considered. A number of variants and/or modifications of MCM have also been developed in the literature. For example, Raoof et al. (2013) assign volume to both pores and throats (solute still perfectly mixed in both) and sub-discretize the wetting filaments in the corners of partially drained pores to account for the partial mixing of solute within them (Fig. 4iv). Van Milligen and Bons (2014) proposed a modification to the throat rate expressions used in MCM (i.e., first three terms on the RHS of Eqn. 4) by deriving analytical expressions based on a steadystate plug-flow assumption within throats. They occasionally further sub-discretized throats into smaller “pores” for increased modeling resolution. A generalization of this method

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Figure 4. (i) Evolution of a solute plume (three snapshots per case) through a 2-D domain for different Péclet numbers and degrees of heterogeneity simulated using DPT. Fluid injection is from a small region to the left side of the domain marked by a thick segment [Used by permission of John Wiley & Sons, Inc., from Bruderer C, Bernabé Y (2001), Network modeling of dispersion: Transition from Taylor dispersion in homogeneous networks to mechanical dispersion in very heterogeneous ones, Water Resources Research, Vol. 37, p. 897–908, Fig. 6. Copyright © by the American Geophysical Union]. (ii) Comparison of predicted longitudinal (upper curve) and transverse (lower curve) dispersion coefficients using DPT with experiments for unconsolidated granular media, for various Péclet numbers [Used by permission of the Society of Petroleum Engineers, from Bijeljic and Blunt (2006), A Physically-Based Description of Dispersion in Porous Media, In: SPE Annual Tech. Conf. and Exhibition, Society of Petroleum Engineers, SPE-102869-MS, Fig. 3. Copyright 2013, Society of Petroleum Engineers. Further reproduction prohibited without permission]. (iiia) Schematic of partial mixing at the 2-D intersection of two fractures, and (iiib) its conceptualization by a unidirectional flow/transport geometry amenable to analytical treatment [Used by permission of John Wiley & Sons, Inc., modified after Park YJ, Lee KK (1999), Analytical solutions for solute transfer characteristics at continuous fracture junctions, Water Resources Research, Vol. 35, Issue 5, p. 1531–1537, Figs. 1, 2, and 3. Copyright 1999 by the American Geophysical Union]. (iva) Schematic of a drained cubic pore by a non-wetting phase. (ivb) The wetting filaments in the crevices of the pore are subdiscretized into 8 pores (corresponding to the corners) and 12 throats (corresponding to the edges) [Used by permission of Elsevier, Ltd., modified after Raoof A, Nick HM, Hassanizadeh SM, Spiers CJ (2013), PoreFlow: A complex pore-network model for simulation of reactive transport in variably saturated porous media, Computers & Geosciences, Vol. 61, p. 160–174, Fig. 4].

for non-uniform velocity profiles was given by Mehmani (2014) (referred to as the ratemodified MCM), but it was shown that these modifications offered little improvement in model predictions compared to MCM. Algive et al. (2010, 2012) and Varloteaux et al. (2013) similarly modified throat solute rate expressions in the context of reactive transport. These works employed moment theory to derive corrected macroscopic parameters (i.e., solute mean velocity, dispersion coefficient, reaction source term) for each pore/throat element in the long-time asymptotic regime. The model was used to study the effects of dissolution/ precipitation reactions on macroscopic single- and two-phase flow properties in the contexts of CO2 sequestration and diagenetic alterations in carbonate rocks. Suchomel et al. (1998b) developed a model in which pores were assumed volumeless and throats were sub-discretized into finite-difference grids. Interpore diffusion (equivalent of the third term in the RHS of

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Eqn. 4) was implicitly incorporated by adjusting numerical diffusion via grid and time-step sizes (although this does not account for diffusion countercurrent to the flow direction), and perfect mixing was assumed within pores. The model was used to study permeability/porosity alterations during bacterial biofilm growth in porous media. A simple but interesting model was developed by Martins et al. (2009), in which the solute balance equations at the pores were formulated as a system of delay-differential equations, i.e., Equation (5). th, q 0

Vpi

th, q 0

Ni dci (t ) Ni   ci (t )qij   c j (t  ij )qij dt j 1 j 1

(5)

In Equation (5), ij is the throat residence time, which accounts for the delay in transport from one pore to the next. In essence, the model uses upstream concentrations from earlier time steps to compute solute flow rates flowing into pores at later times. This is implemented by sub-discretizing the throats and marching pore concentrations forward within the subdiscretized segments (akin to a traveling wave). However, several limiting assumptions were made including the neglect of diffusion, plug-flow within throats, and perfect mixing within pores. A different set of models formulate transport equations in the Laplace transform domain with respect to the time variable. There is a certain extent of elegance and convenience associated with working in the Laplace domain, mainly due to the fact that convolutions of transit-time probabilities are converted to simple algebraic multiplications. Another advantage is that computation of temporal moments becomes rather straightforward (the Laplace transform of transit-time distributions act as moment generating functions). De Arcangelis et al. (1986) developed first-passage-time probabilities for tracer particles that move through throats connecting two neighboring pores. They assumed perfect mixing at the (volumeless) pores and plug flow with molecular diffusion at the throats. Under these conditions, they derived exact transit probabilities in the Laplace domain for particle motions in a network. They then used a “probability propagation” algorithm to determine the first-passage-time distribution of a 10 × 10 diamond lattice network and computed longitudinal dispersion coefficients for various Péclet numbers (= advection/diffusion). At no point was the timedomain concentration field computed. Koplik et al. (1988) used the same set of equations as de Arcangelis et al. (1986) but diverged in their analysis by writing species balance equations for each pore in the Laplace domain. The resulting linear system of equations was then solved and numerically inverted into the time domain using the Stehfest (1970) algorithm. A strategy for computing higher-order moments of first-passage-time distributions of the network was further outlined. This method was later extended by Alvarado et al. (1997) to reversible adsorption/reaction scenarios, where they arrived at the interesting conclusion that dispersion coefficients depend on the degree of spatial heterogeneity of reactive sites in a porous sample and scale non-linearly with Péclet number. However, they noted that the numerical Laplace inversion step was prohibitive for networks larger than 20 × 20 pores and inaccurate for large Péclet numbers (>10). Indeed numerical inversion of the Laplace transform is known to be notoriously difficult (often unstable) and ill-posed in computational and applied mathematics. For this reason, although valuable for performing moment analyses, time-domain predictions via these methods on representative sample sizes seem impractical and unlikely. Lagrangian network models. A more natural description of transport is provided by Lagrangian models, among which particle tracking (PT) is almost exclusively employed. In this method the steady-state flow equation is first solved on the network, as described in the beginning of this section, to obtain mean fluid velocities within each throat. Then, depending on the specific throat geometry, a (rectilinear) velocity profile is assumed and used to track particles from pore to pore, subject to simultaneous convection and molecular diffusion (Fig. 4i). Lagrangian network models are generally more computationally expensive than

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Eulerian network models, although more accurate. The limiting factor is the large number of solute particles often required in Lagrangian network models to obtain statistically converged results. PT methods on pore networks can be divided into two categories: a) those that trace particle motions in detail within throats following a discrete-time random walk process (Bruderer and Bernabé 2001; Bijeljic et al. 2004; Acharya et al. 2007a; Jha et al. 2011), and b) those that perform continuous-time random hops from one pore to the next (without explicit throat-level simulations) using throat transit-time distributions (Sahimi et al. 1986; Sorbie and Clifford 1991; Rhodes and Blunt 2006; Bijeljic and Blunt 2006; Picard and Frey 2007). We refer to the first class as DPT (discrete-time particle tracking) and to the second as CPT (continuous-time particle tracking). Compared to CPT, DPT simulations are more time consuming since computational performance is limited by the time step size (controlled by the minimum throat residence time throughout the network). However, DPT can be quite accurate and has been used to successfully predict dispersion coefficients in unconsolidated granular media (e.g., Bijeljic et al. 2004; Jha et al. 2011) (Fig. 4ii). It is also very flexible in the sense that various velocity profiles (e.g., parabolic or plug-flow) within the throats can be considered, and one has substantial control on how to reassign particles to new outlet throats upon their arrival at the pores. Specifically, particles can be reassigned to any desired throat connected to the arrival pore, and even any desired location on the cross-section of that throat. In contrast, CPT is computationally more efficient but comparatively less flexible and less accurate especially in ordered media, as discussed in Difficulties and Sources of Error. The efficiency is due to the fact that particle motions within throats are not explicitly simulated, but are instead imbedded in the throat transit-time distributions. The reduced flexibility and accuracy are due to the loss of control in reassigning incoming particles to accurate cross-sectional locations of the outlet throats of a pore. This means that once an outlet throat is chosen (based on some probability), a transit-time distribution is used to determine the time the particle requires to exit the throat. But the transit-time distribution is typically stationary in time and is derived with uniform inlet conditions for concentration. Therefore, there is no way to specify the cross-sectional location an incoming particle should be assigned to in an outlet throat of a pore. This is particularly important in simulating dispersion in ordered media, for which CPT will not yield the expected (e.g., Edwards et al. 1991) DL ~ Pe 2d scaling, where DL is the longitudinal dispersion coefficient, and Pe d is the Péclet number defined as the ratio of advection to diffusion (Mehmani 2014). CPT methods can be further subdivided into those that use deterministic transit-time distributions (e.g., Sorbie and Clifford 1991; Picard and Frey 2007), and those that use ensemble-averaged transit-time distributions (e.g., Bijeljic and Blunt 2006; Rhodes et al. 2009). Bijeljic and Blunt (2006) derived an ensemble-averaged transit-time distribution for Berea sandstone by fitting a truncated power-law distribution to their simulation results. They provided physically meaningful interpretation of the distribution variables and fitted their simulated data with a single adjustable parameter. On the other hand, deterministic transittime distributions are often derived based on similar mathematics and assumptions as those already discussed in the context of Laplace-domain Eulerian network models (e.g., Rhodes and Blunt 2006; Picard and Frey 2007). These assumptions include perfect mixing within volumeless pores and plug flow within throats. Deterministic transit-time distributions are often numerically inverted from the Laplace domain to the time domain in order to draw particle transit times within the throats. An exception to this is Sorbie and Clifford (1991) who derived deterministic transit-time distributions based on rigorous single throat simulations assuming non-uniform velocity profiles. Difficulties and sources of error. The most common sources of ambiguity and error in both Eulerian and Lagrangian network models of single-phase solute transport are in the

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descriptions of partial mixing conditions within pores (Fig. 5a), and shear dispersion (i.e., spreading of the solute due to non-uniform velocity profiles) within throats (Fig. 5b). Accurate modeling of these fundamental transport physics could have a significant impact on quantitative macroscopic predictions of solute dispersion and effective reaction rates. Almost all Eulerian network models developed in the literature seem to assume perfect mixing within pores and neglect shear dispersion within throats. Models assuming Taylor–Aris shear dispersion coefficients within throats are unrealistic because throat lengths are typically not long enough for an asymptotic regime to be reached (Mehmani 2014). On the other hand, the incorporation of shear dispersion into Lagrangian network models (e.g., PT) is quite straightforward (see the discussion of these models above), but accurate description of pore-level partial mixing is still a source of ambiguity in these models. In reality, perfect mixing occurs when diffusive forces are much more dominant than either advective or reactive forces. In the absence of reactions, increasing the Péclet number changes pore-level mixing conditions from perfect mixing to partial mixing. Figure 5a shows direct numerical simulations of this scenario in a 2-D pore (Mehmani et al. 2014). When reactions are present, increasing the Damköhler number (= reaction/diffusion), while keeping the Péclet number constant, causes a high-concentration-gradient boundary layer to develop near the fluid–solid interface. In pore-network models, the effect of reaction-induced boundary layers may be implicitly taken into account in the calculation of pore concentrations, although this scenario unfortunately has not been explored enough in the literature. In the following, we shall focus on shear dispersion and partial mixing in the absence of chemical reactions, and defer further discussion on the effects and modeling approaches of reactions until the very end of this section. The ambiguity in accurately describing partial mixing within pores stems from the difficulty in approximating flow streamlines within pores and the extent of diffusive mixing occurring therein (Fig. 5a). Sahimi et al. (1986) and Bruderer and Bernabé (2001) developed simple intuitive rules for the redistribution of solute particles from the inlet to the outlet throats of a 2-D cross-shaped volumeless pore. Both ignored diffusive mixing within pores, in the sense that the particles could not randomly “hop” between outlet throats. Sorbie and Clifford (1991) introduced general heuristic, and admittedly approximate, rules for redistributing particles arriving at a pore among its outlet throats. These rules, and variants thereof, were subsequently

Figure 5. (a) Schematic of partial mixing within a sample pore and its dependence on local Péclet number (underlying streamlines also shown) [Used by permission of John Wiley & Sons, Inc., modified after Mehmani Y, Oostrom M, Balhoff MT (2014), A streamline splitting pore-network approach for computationally inexpensive and accurate simulation of transport in porous media, Water Resources Research, Vol. 50, Issue 3, pp. 2488–2517, Fig. 1. ©2014. American Geophysical Union. All Rights Reserved]. (b) Schematic of shear dispersion of solute within the throats of a 1D pore-network, due to the non-uniform velocity profile (shown by curved solid line). Figures obtained through CFD simulations on respective pore-space geometries using COMSOL© [Used by permission of author, modified after Mehmani (2014), Modeling of single-phase flow and solute transport across scales, PhD dissertation, University of Texas at Austin, Fig. 4.5].

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applied and analyzed in later publications (e.g., Acharya et al. 2004, 2007a,b; Bijeljic et al. 2004; Bijeljic and Blunt 2007). At high Péclet numbers, these rules reduce to redistribution of solute particles based on flow-rate-averaged probabilities; at low Péclet numbers, they reduce to redistribution based on throat cross-sectional-area-averaged (multiplied by other corrective parameters) probabilities. These rules, however, are quite inaccurate at moderate-to-high Péclet regimes as they completely ignore the underlying streamlines that map the inlets to the outlets of a given pore (Mehmani et al. 2014). Jha et al. (2011) proposed a deterministic mapping of incoming particles to the outlet throats, intendedly along the streamlines, in an attempt to generalize the rules developed by Bruderer and Bernabé (2001). The approach was limited to pores with a coordination number of four or less, and ignored mixing due to molecular diffusion. Despite being attractive due to its simplicity, the mapping neglected spatial orientations of the inlet throats, and consequently the interaction between inflowing streams, and was therefore less predictive (Mehmani et al. 2014). The mixing problem within pores has additionally received quite a bit of attention from the field of fracture-network modeling (e.g., Berkowitz et al. 1994, Park and Lee 1999; Park et al. 2001a,b; Johnson et al. 2006). Although in this context, the “pores” correspond to the juncture at which two fractures intersect. Park and Lee (1999) derived physically sound, efficient, and accurate transition probabilities for mapping particles from the inlets to the outlets of fracture junctions. In doing so, they conceptualized partial mixing at fracture junctions as a onedimensional problem amenable to analytical treatment (Fig. 4iii). While accurate for fracture networks, the mixing equations of Park and Lee (1999) are not applicable to 3-D networks of porous media. This is because fracture junctions are essentially 2-D cross-shaped “pores”, which are geometrically simpler than the 3-D pores with non-planar throat orientations found in porous media. In an effort to accurately capture partial mixing within pores and shear dispersion within throats, Mehmani et al. (2014) and Mehmani (2014) developed two pore-network models, respectively: the streamline splitting method (SSM) and superposing transport method (STM). These models were developed with the aim of exploring the possibility of capturing the porescale physics discussed above under an Eulerian framework. The insistence on the models being Eulerian was because, in the context of pore-network modeling, such a framework is generally computationally more efficient than a Lagrangian counterpart, albeit possibly less accurate. Therefore, the goal was to find an appropriate balance between acceptable predictive accuracy and computational efficiency. To satisfy the latter, both models used MCM as the starting point for their development. In MCM, a single concentration value is assigned to every pore, which implicitly assumes perfect mixing within them, independent of the local Péclet regime. Speciesbalance equations are then written in every pore, and solved for the pore concentrations. To circumvent the perfect mixing assumption, SSM divides the volume of the pores into smaller compartments (or pockets) and assigns different concentration values to each compartment. The number of compartments within a pore is equal to the number of inlets of that pore, as each inlet may carry with it fluids with different concentrations that need to be kept separate. However, mass transfer (or mixing) between two adjacent compartments within the same pore is allowed, which takes place purely due to molecular diffusion (perpendicular to the streamlines). Unlike Park and Lee (1999), pore-wall effects on the mixing process were also taken into account. The outlets of each compartment are determined by “splitting” the inflowing “streams” between the outlet throats, using a constrained optimization algorithm; which was shown to be also applicable in Lagrangian network models (Mehmani et al. 2014). The compartments are then regarded as control volumes and species balance is enforced in all of them. Figure 6 provides a conceptual picture underlying SSM and contrasts it to that of MCM. It was shown that SSM predictions were in very good agreement with direct CFD simulations as well as micromodel experiments, and computational costs were only marginally higher than MCM.

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Figure 6. Schematic of a single pore p0 connected to four neighbors (whose own neighbors are not depicted). (a) Conceptual picture of MCM, where pores are considered as perfectly mixed reactors (to which single concentration values are assigned). (b) Conceptual picture of SSM, where pore volumes are divided into smaller compartments (or pockets denoted by pk) in order to account for partial mixing within pores (the depiction is an abstract, rather than literal, representation of the splitting of streamlines) [Used by permission of John Wiley & Sons, Inc., modified after Mehmani Y, Oostrom M, Balhoff MT (2014), A streamline splitting pore-network approach for computationally inexpensive and accurate simulation of transport in porous media, Water Resources Research, Vol. 50, Issue 3, p. 2488–2517, Fig. 2. ©2014. American Geophysical Union. All Rights Reserved].

On the other hand, the focus in STM is placed on shear dispersion due to the non-uniform velocity profiles within throats. This important pore-scale process is ignored in MCM as it assumes throats to be volumeless, which causes the solute to instantaneously transport from one pore to the next. In STM, elementary species rate expressions are constructed for each throat, as a function of their aspect ratio (i.e., diameter over length). These elementary rate expressions are then used to perform space-time superpositions across the pore network. This is accomplished by recording pore concentrations as they dynamically evolve during a simulation, which makes it similar to, but much more general than, the model developed by Martins et al. (2009) (the latter ignores shear dispersion and molecular diffusion). The superpositions in STM have the effect of performing convolutions of the elementary rate expressions across the pore network. This makes STM useful in other fields such as signal transmission in electrical engineering. The obvious limitation of STM is that it is applicable to linear transport problems only. This excludes scenarios with non-linear fluid–mineral reactions such as those encountered in CO2 sequestration. But, for linear problems, the method was verified against direct CFD simulations and simple convolution integral expressions. It was additionally validated against longitudinal dispersion experiments for unconsolidated bead packs from the literature. Yet a final question remains to be answered: what is the actual magnitude of the improvements imparted to the prediction of a parameter of interest by incorporating these additional pore-scale details into our network models? After all, modeling efforts should ideally seek ultimate simplicity while disposing of unnecessary complexity. Mehmani et al. (2014) showed that the impact of pore-level mixing assumptions, partial vs. perfect, is very significant in ordered media (e.g., micromodels), but comparatively insubstantial in 3-D disordered granular media (e.g., sandstones). In fact, the average difference between the concentration fields obtained via SSM and the relatively less accurate MCM for disordered granular media was ~6% of the maximum concentration value (although the impact on upscaled transverse

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dispersion coefficients is yet to be studied). A similar conclusion was drawn by Park et al. (2001a,b), who used PT to study the impact of various mixing assumptions (perfect mixing vs. no mixing) at fracture junctions on overall macroscopic transport behavior in ordered and disordered (2-D) fracture networks. They found that mixing assumptions have a larger impact in ordered media compared to random media (i.e., less than 5% of fracture junctions were affected in random media), and they attributed this to the lower effective coordination number and higher inlet flux ratios, at fracture junctions, in random networks. Despite the various differences between the two media (porous vs. fractured), the conclusions appear to be the same: for disordered (granular/fractured) media the model with the simpler description of pore-level mixing is the one that is preferred (e.g., MCM). On the other hand, Mehmani (2014) showed that the incorporation of shear dispersion within throats in network models of disordered granular media is rather important at high Péclet numbers (>100). Namely, longitudinal dispersion coefficients predicted by STM undergo a ~2.5-fold improvement compared to MCM. This result corroborates earlier works that used PT to simulate longitudinal dispersion (Acharya et al. 2007a; Jha et al. 2011). A more important conclusion drawn using STM was that all Eulerian network models, including STM, are inherently limited at sufficiently high Péclet numbers when applied to ordered media (e.g., micromodels). This was demonstrated by simply conceptualizing a straight tube under purely advective transport, as an equivalent 1D “network” (or string) of shorter tubes (Fig. 7a). The concentration profiles obtained from STM and MCM were shown to be representative of neither of the analytically known flux-averaged or cross-sectional-averaged concentration profiles (Fig. 7b). Even worse was the fact that simulated profiles were shown to converge towards a Gaussian distribution with an increase in the number of tube segments traveled. This is known to be impossible under a purely advective regime (Fig. 7c). It is known from Taylor–Aris theory that in the presence of molecular diffusion (no matter how small its contribution) a DL ~ Pe 2d asymptotic scaling is expected from this system (DL is the longitudinal dispersion coefficient, and Pe d is the Péclet number); similar to ordered media (Edwards et al. 1991). Instead, an Eulerian network model (e.g., STM) would yield a scaling that is closer to DL ~ Pe d at high Pe d , since Gaussian profiles are reached not because of molecular diffusion, but because of cross-sectional smearing of concentrations at the “pores”

Figure 7. (a) Schematic of a circular tube under pure advection (constant inlet concentration), divided into segments along the dashed lines. (b) Concentration profiles (at different times) along the duct, including: true cross-sectional average (dashed-dotted line), true flux-averaged (dashed-plus line), simulated STM (thick solid line), and simulated MCM (thin solid line). Arrow of time annotated, and  is x-normalized against the length of 50 segments. (c) Convergence of STM profiles towards a normal distribution, with an increase in the number of tube segments traveled (annotated by NJ). x is normalized by the distance travelled by the (tube) centerline velocity (=V0t ) [Used by permission of author, from Mehmani (2014), Modeling of single-phase flow and solute transport across scales, PhD dissertation, University of Texas at Austin, Fig. 4.12].

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(denoted by dashed lines in Fig. 7a). In other words, solute particle “memories” are effectively erased upon their arrival at the pores (i.e., Markovian process), whereas in reality they are retained over a given distance that depends on the local Péclet regime. This distance is larger than the length of a single throat at sufficiently high Péclet numbers. The foregoing problem is not just specific to Eulerian network models. Lagrangian methods that draw throat transit times from the same distribution, that is stationary in time, are equally limited. An example includes the CPT model developed by Sorbie and Clifford (1991). The implication is that the “simplest” model, capable of accurately capturing shear dispersion in ordered media, must preserve particle memories in passing them from the inlet to the outlet throats of a pore. This means that it is not just sufficient to know how long a particle takes to transit a given throat, but also the cross-sectional position of that throat whence it exits. Such detailed information seems only resolvable in a pore network, with adequate simplicity, under a Lagrangian framework. Reactions. Comprehensive reviews on modeling reactive transport can be found in Steefel and MacQuarrie (1996), Steefel et al. (2005, 2013). Reactions at the pore-scale can be divided into those occurring within the fluid bulk, referred to as homogeneous reactions, and those that take place at the fluid-mineral interface, referred to as heterogeneous reactions. Homogeneous reactions themselves consist of zero-order decay processes independent of species concentrations, and higher-order concentration-dependent reactions that occur due to the mixing of two or more solute species. Homogeneous reactions are commonly modeled as instantaneous processes at equilibrium, while heterogeneous reactions as kinetically controlled processes. In direct pore-scale models, the former is described via source terms in the solute balance equation (Eqn. 2a), and the latter via appropriate boundary conditions at the fluid–mineral interface (Eqn. 2b). In Eulerian network models, where pores are viewed as continuously stirred tank reactors, all reaction types (as well as adsorption) are almost invariably described via source terms in the mesoscale solute balance equation (Acharya et al. 2005, 2007b; Li et al. 2006, 2007a,b; Kim et al. 2011, 2012; Mehmani et al. 2012; Raoof et al. 2013; Nogues et al. 2013). This effectively ignores concentration gradients and transportlimited effects on reaction rates at the scale of individual pores. Depending on the application, this may or may not be a good assumption. Li et al. (2008) showed that for typical flow and chemical/mineralogical conditions relevant to geologic CO2 sequestration, pore-level perfect mixing appears, for all practical purposes, appropriate. Their conclusion was based on molecular diffusion coefficient and activity of the hydrogen ion among other assumptions. In problems where these parameters assume very different values, transport-limited effects can become important and the perfect mixing assumption inappropriate. The network models developed by Algive et al. (2010, 2012) and Varloteaux et al. (2013) are the only known exception, in which concentration gradients due to fluid–mineral reactions are implicitly taken into account by calculating effective transport parameters for the pores and throats. However, their model is limited to long-time asymptotic regimes, where changes in the concentration field are slow with time. Moreover, none of the above mentioned models accounts for diffusion-limited mixing reactions that occur when two fluids with differing chemical compositions come into contact within a pore. In SSM, discussed in Difficulties and sources of error, the rate of mass transfer between two adjacent compartments within a pore (pk1 and pk2 in Fig. 6b) is computed via solving a local bounded Riemann problem. Mehmani et al. (2014) proposed the possibility of modifying this local problem (via appropriate initial/boundary conditions) to derive mass transfer rates in the presence of mixing-induced reactions. For reactions at the fluid–solid interface within the throats, one may derive modified elementary rate expressions that can be used in the STM model discussed in Difficulties and sources of error (although limitations for ordered media still persist). This would complement the long-time asymptotic method of Algive and coworkers, for short-time pre-asymptotic regimes. Finally, particle tracking (PT) can be used to describe all foregoing

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reactions types at the pore scale in high detail. This requires the development of non-trivial stochastic particle interaction criteria and collision probabilities between fluid–fluid and solid– fluid species (Gillespie 1977; Benson and Meerschaert 2008; Dentz et al. 2011). However, the adoption and implementation of PT in pore networks for accurate simulations of geochemical reactions have unfortunately been scarce.

HYBRID MODELING Flow and reactive transport phenomena in the subsurface often span a wide range of spatial scales (nanometer to kilometer), which renders the development of predictive models capable of accurately bridging them a formidable task. Macroscopic parameters (e.g., permeability) or closure relations (e.g., capillary pressure vs. saturation) are commonly extracted from pore-scale modeling or experiments on representative samples of the real medium, followed by their direct implementation into continuum-scale simulators. While such an approach is appropriate for situations in which a clear separation between scales exists, it may lead to large errors otherwise (Kechagia et al. 2002; Tartakovsky et al. 2008b; Battiato and Tartakovsky 2011). Mixing induced precipitation ensuing form a bimolecular reaction is one such example (Battiato et al. 2009). In these cases a dynamic communication between the pore and the continuum scales seems to be required. Li et al. (2006, 2007a,b) used a pore-network model to study reaction kinetics of kaolinite and anorthite in the context of geologic carbon sequestration. They demonstrated that reaction rates obtained from continuum-scale representations of transport and/or using volume/fluxaveraged concentrations in reaction rate expressions leads to large errors, sometimes even wrongly predicting the direction of the reactions (i.e., precipitation vs. dissolution). Although their study was qualitative in nature, using 3-D regular lattice networks, it provided an explanation for the commonly reported discrepancy between reaction rates observed at the field scale and those obtained from well-mixed batch experiments on crushed samples. Namely, transport limitations at the pore scale control overall reaction rates, which are non-existent under well-mixed conditions in batch experiments. Effects of flow rate and reactive cluster size/ abundance were also studied, and it was concluded that the higher the degree of incomplete mixing (i.e., spatial variability of concentration) the higher the scaling error. Incomplete mixing was found to be strongest at medium flow rates (or Péclet numbers). Kim et al. (2011) and Kim and Lindquist (2012) extended the work of Li and coworkers using networks extracted from XMT images of real sandstones. They were able to determine surface mineral distributions from the images allowing for better quantitative analysis. Similar conclusions were drawn regarding the “lab–field discrepancy”, and an approximately power-law scaling of reaction rates vs. flow rate was reported for anorthite, while a more complex scaling emerged for kaolinite. In the context of filtration combustion in porous media, Lu and Yortsos (2005) similarly observed that spatially averaged macroscopic reaction rates were very different (discrepancies of a factor of two or higher) than those determined from using averaged variables in microscopic reaction rate expressions. They attributed this to the strong influence of microscopic heterogeneities on macro scale behavior. Recently, Molins et al. (2012) conducted sophisticated direct pore-scale simulations of calcite dissolution, and showed that pore-scale heterogeneities can result in an underestimation of reaction rates, due to mass transport limitations, even when total reactive surface area and porosity are held constant between samples. Kechagia et al. (2002) demonstrated that for reactive transport with fast/finite kinetics, homogenization of microscopic equations via volume averaging does not hold except at the limit of macroscopic equilibrium. They further showed that even under these circumstances an eigenvalue problem enforces a coupling between the micro and the macro scales. A systematic study was performed by Battiato and Tartakovsky (2011) to identify transport regimes

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(characterized by the Damköhler (Da) and Péclet (Pe) numbers), in which a purely continuum description of advection-diffusion with nonlinear surface reactions breaks down. They used multiple-scale expansions to upscale the pore-scale equations, and presented their results in the form of a Pe–Da phase diagram. The results were in agreement with an earlier work that used volume averaging (Battiato et al. 2009), which substantiated their independence from the specific upscaling method employed. A recent work by Boso and Battiato (2013) extended this analysis to three-component systems undergoing two homogeneous and one heterogeneous reactions (all reversible). It is interesting to note, that Molins et al. (2012) located their simulations on the aforementioned Pe–Da phase diagram and determined their correspondence to a case in which spatial scales were coupled. While all foregoing examples focused on geochemical reactions, a coupling between scales is also observed in problems involving viscous and density-driven instabilities in multiphase and miscible displacement among others (Tartakovsky et al. 2008c). The implications of the foregoing studies (among others) have given rise to a new class of modeling approaches referred to as “hybrid multiscale methods” (Scheibe et al. 2007), in which micro- and macro-scale simulations are simultaneously performed on the same computational domain. Balhoff et al. (2007) were one of the first to couple a pore-scale model to the continuum. However, their iterative coupling strategy had strong limitations in terms of flexibility and efficiency. These limitations were later lifted by Balhoff et al. (2008) through the introduction of mortars. Mortars are finite-element function spaces that ensure the (weak) continuity of flux at the interface between two coupled models (e.g., pore-scale and continuum-scale), and are the essential component of a highly flexible, efficient, and accurate non-overlapping domain decomposition method (Bernardi et al. 1994; Arbogast et al. 2000; Peszynska et al. 2002). Subsequently, Sun et al. (2012a) showed that mortars can be used as accurate upscaling tools for pore-scale models in obtaining macroscopic properties (e.g., permeability). They demonstrated that a large heterogeneous pore-scale domain can be decomposed along structural discontinuities and coupled via mortars to closely approximate the true permeability. Sun et al. (2012b) developed a single-phase reservoir simulator, in which Darcy grids in the near-well region were substituted with pore-scale models (Fig. 8ii). The study focused on upscaling strategies for the permeability field of the near-well region. In the foregoing studies, application of mortars was limited to linear, single-phase Newtonian flow without species transport and computational aspects were left unexamined. These issues were later addressed by Mehmani et al. (2012) and Mehmani and Balhoff (2014), who extended the use of mortars to non-linear (power-law) flow and (passive/reactive) solute transport and demonstrated computational efficiency and parallel scalability of their new algorithms (Fig. 8iii). The application of the above hybrid mortar methods is most appropriate when a tight coupling between spatial/temporal scales exists and a characterization of the pore-scale subdomain is available. This scenario is most likely to occur in a “skin-deep" region around wellbores (e.g., matrix acidization, and CO2 leakage through wellbore cement), in which fluids are furthest away from equilibrium and direct access to porescale samples for geometric characterization is available. Tartakovsky et al. (2008a) non-iteratively coupled the pore scale and the continuum in a diffusion-reaction system using an SPH formulation for both scales. They were able to show that their hybrid method produced reliable predictions, compared to modeling purely at the pore scale, of mixing-induced precipitation, which is typically not amenable to a purely continuum description as the reactions occur over the length of a few grain diameters (Fig. 8i). While the non-iterative nature of the method makes it very attractive from a computational standpoint, the method lacks flexibility in the sense that it requires both scales to be formulated using SPH. Advection was additionally ignored in that work. A novel overlapping method for coupling porescale inclusions to the surrounding continua was developed by Battiato et al. (2011) and was successfully verified for the case of Taylor dispersion in a reactive fracture. While the method

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Figure 8. (i) Mixing-induced mineral precipitation (represented by black particles) obtained from SPH simulations on a (ia) fully pore-scale domain, and (ib) a hybrid domain with a pore-scale region sandwiched between two continuum regions [Used by permission of the Society for Industrial and Applied Mathematics, from Tartakovsky AM, Tartakovsky DM, Scheibe TD, Meakin P (2008), Hybrid simulations of reaction-diffusion systems in porous media, SIAM Journal on Scientific Computing, Vol. 30, p. 2799–2816, Fig. 5. Copyright ©2008 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved]. (iia) Pressure, and (iib) flow fields obtained from the coupling of a nearwell pore-scale region to the surrounding continuum using mortars [Used by permission of the American Chemical Society, from Sun et al. (2012b), Hybrid Multiscale Modeling through Direct Substitution of Pore-Scale Models into Near-Well Reservoir Simulators, Energy & Fuels, Vol. 26, p. 5828–5836, Fig. 6 and 7. Copyright © 2012, American Chemical Society]. (iiia) Pressure, and (iiib) concentration fields obtained on a (3-D) hybrid domain (results collapsed onto 2-D) with non-matching continuum grids [Used by permission of the Society for Industrial and Applied Mathematics, from Mehmani Y, Balhoff MT (2014), Bridging from pore to continuum: A hybrid mortar domain decomposition framework for subsurface flow and transport, Multiscale Modeling & Simulation, Vol. 12, p. 667–693, Fig. 11. Copyright © 2014 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved]. (iv) Schematic of the heterogeneous multiscale method (HMM) for performing hybrid two-phase drainage simulations [Used by permission of Springer and Advances in Applied Mathematics, Modeling, and Computational Sciences, from Chu J, Engquist B, Prodanovi M, Tsai R (2013), A multiscale method coupling network and continuum models in porous media II—Single-and two-phase flows. In: Advances in Applied Mathematics, Modeling, and Computational Sciences, p. 161–185, Fig. 4. With kind permission from Springer Science and Business Media]. Shaded rectangles represent network models, which are used to compute the pressure field (those depicted at the boundaries of macroscopic control volumes) and advance the saturation front (those oriented normal to the displacement front).

is more general compared to the previous work (it included advection and was not limited to SPH), it appears to be limited to rather small pore-scale point inclusions that are dependent on the underlying macro-grid structure of the domain. The specific coupling algorithm used can also be rather computationally expensive, since the differential-algebraic system was solved by iterating between the differential and algebraic parts, rather than solving the system as a whole. Roubinet and Tartakovsky (2013) built upon the work of Battiato et al. (2011) and developed a non-overlapping hybrid method in 1D, in which the computational performance was enhanced by solving a global (instead of a sequential) differential-algebraic system. The method used finite volumes to discretize both the continuum- and the pore-scale subdomains, and introduced

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additional unknowns for each grid on either side of the interface. This dependence of interface unknowns on the number of interface grids can render the approach quite expensive for large problems. It is also noteworthy, that the approach seems to be a special case of the Global Jacobian methods of Ganis et al. (2012) and Mehmani and Balhoff (2014). Chu et al. (2012) proposed an approach based on the heterogeneous multiscale method (HMM), in which macroscopic conservation equations are written assuming unavailability of constitutive flow equations (e.g., Darcy’s law) at the macro scale. The missing data were instead supplied from locally sampled pore-network simulations across the domain. Specifically, pore-scale models were used as providers of accurate in/out-fluxes at the boundaries of macroscopic control volumes. Chu et al. (2013) extended this work to twophase flow problems, where pore-network models were used to track macroscopic two-phase drainage fronts (Fig. 8iv). While the method is a superior alternative to pure continuum-scale models in cases where local effects are dominant, its extension to problems with strong fluid– mineral reactions (where a macroscopic source/sink term is unknown and highly coupled to transport itself) is not obvious. Sheng and Thompson (2013) developed a dynamic coupling algorithm for two-phase flow, in which network models were embedded at the center of 1D macroscopic control volumes. The network models provided relative permeability values to the continuum model, while the continuum model provided fractional flow rates as boundary conditions to the network model. The network model was specially designed to handle these non-trivial boundary conditions. The large time-scale discrepancy between the pore scale and the continuum scale was addressed by performing successive steady-state simulations at the pore scale. The work showed that macroscopic and pore-scale saturations do not necessarily agree, and highlighted the difficulty in enforcing such an agreement. Tomin and Lunati (2013) developed a hybrid algorithm based on the multiscale finite volume (MsFV) method. In this approach the macroscopic domain is discretized into primal control volumes, upon which a complementary dual grid is superposed. Local pore-scale problems are then solved on the dual grid, in order to construct a set of bases as well as correction functions; these can be thought of as pore-scale closures for the continuum model. Global balance equations are then solved by forming a linear combination of these bases/correction functions and computing their corresponding multipliers. The continuum solution is then used to construct a conservative velocity field at the pore scale for advancing tracer concentrations and/or phase saturations. The hybrid method was successfully verified against fully pore-scale simulations for tracer transport and stable/unstable drainage problems. The essence of all hybrid methods discussed is a “two-way communication” between the pore scale and the continuum. Hybrid methods are a relatively recent development compared to single-scale modeling strategies such as molecular dynamics (MD), pore-scale modeling, and reservoir simulation. Naturally, their development has been based upon the vast diversity of single-scale methods and various combinations thereof. Consequently, they themselves are very diverse in the ways they approach the problem, which might raise the obvious question as to which hybrid method is most appropriate for a given problem. Scheibe et al. (2015) have attempted to provide a general road map for choosing an appropriate hybrid strategy, referred to as the Multiscale Analysis Platform (MAP), depending on the degree of complexity of the hydrological problem at hand (i.e., degree of separability between temporal/spatial scales). MAP classifies various hybrid methods into separate “motifs”, and leads its user (with a specific application in mind) towards a suitable motif by asking a series of questions regarding the nature of the problem to be solved (Fig. 9). Such a classification additionally provides a useful context, within which the ever-increasing multiscale methods developed in the literature can be categorized.

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Figure 9. Flowchart of the multiscale analysis platform (MAP) [Used by permission of John Wiley and Sons, from Scheibe TD, Murphy EM, Chen X, Rice AK, Carroll KC, Palmer BJ, Tartakovsky AM, Battiato I, Wood BD (2015), An analysis platform for multiscale hydrogeologic modeling with emphasis on hybrid multiscale methods, Groundwater, Vol. 53, p. 38–56, Fig. 3. © National Ground Water Association].

CONCLUSIONS Recent advances in rock imaging technology (e.g., XMT) have made it possible to resolve the complex pore-space geometry in sufficient detail, to the point that direct numerical simulations of various pore-scale processes are becoming commonplace and an area of active research. In addition, novel micromodel fabrication and measurement techniques (Werth et al. 2010; Karadimitriou and Hassanizadeh 2012) are allowing for an unprecedented possibility of quantitative comparison between modeling and experiments. However, our recent capability to resolve much of the pore-scale details with high accuracy, both experimentally and numerically, does not detract from the importance of mesoscale (or pore-network) models and the role they play in bridging the pore scale to the continuum. On the contrary, we can now use observations from such detailed experimental and direct numerical results to increase the accuracy of traditional network models and remove some of their ambiguity. The significance for doing so is threefold: a) it allows us to translate our detailed observations to an intermediate scale that is simple enough to understand, b) forces us to determine what features of the problem are the most essential so we can discard unnecessary details, and c) regardless of advances in computational performance, mesoscale models will always be one scale ahead of direct porescale methods. While improving network models based on detailed numerical/experimental observations is often successful, there are however limitations to this process. In the first part of the chapter, we tried to emphasize this point by providing two examples in the context of single-phase solute transport. In the first example, we demonstrate that modifications to a traditional Eulerian network model allows it to capture partial mixing within pores with higher accuracy, while preserving computational efficiency and relative simplicity. In the same example, we also show that the need for accurately capturing partial mixing within pores depends on the porous medium. If the porous medium is ordered, partial mixing is important. But in disordered media (e.g., sandstones), partial mixing does not seem to be as significant, which means that we do not need

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to add this additional complexity to our mesoscale model. In the second example, we similarly show that modifications of a traditional Eulerian network allows it to capture shear dispersion within throats, and improves predictions of macroscopic dispersion in disordered (granular) media. However, when the porous medium is ordered, no amount of modification appears to prevent it from producing the wrong DL vs. Ped scaling due to the Eulerian nature of the network model. Thus, a Lagrangian network model seems to be the next simplest mesoscale description. Similar conclusions may or may not hold for problems involving multiphase flow, geochemical reactions, adsorption, etc. However, the current state of computational power and experimental advances are allowing for the possibility of investigation into these questions more than ever. On the other hand, the practical need to perform large-scale predictions requires pore-scale information to be translated to the continuum scale. While traditional upscaling approaches are successful in various scenarios, their applicability is rather limited when insufficient separation between scales exists. Overwhelming evidence in the literature hints towards this being quite common in the presence of strong fluid-mineral reactions; rendering averaged reaction rates transport-limited at the pore scale. Another example involves viscous and density-driven instabilities in multiphase and miscible displacement scenarios (Tartakovsky et al. 2008c). In recent years, various “hybrid” models have been developed as a means of simulating such scenarios, with the typical premise that the break-down of macroscopic continuum equations occurs locally. The essence of all such methods is a “two-way communication” between the pore and the macro scales, accomplished by incorporating models of both scales into the same computational domain. A useful classification given by Scheibe et al. (2015) facilitates one in choosing a suitable hybrid method, from a large selection developed in the literature, depending on the particular application at hand. As developments in both pore-scale and hybrid models continue into the future, the hope is for all these efforts to converge towards a deeper understanding of the scale transition problem and arrive at a predictive description (modeling and/or theory) of the relevant processes. Such an understanding would undoubtedly be indispensable in tackling currently pressing and highly challenging problems such as the safe sequestration and storage of anthropogenic CO2, increasing hydrocarbon recovery from mature formations, and clean-up and remediation of nuclear waste repositories.

ACKNOWLEDGMENTS This material is based upon work supported as part of the Center for Frontiers of Subsurface Energy Security, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences under Award Number DESC0001114.

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Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes Sergi Molins Lawrence Berkeley National Laboratory Earth Sciences Division Berkeley, California 94720, USA [email protected]

INTRODUCTION Darcy-scale simulation of geochemical reactive transport has proven to be a useful tool to gain mechanistic understanding of the evolution of the subsurface environment under natural or human-induced conditions. At this scale, however, the porous medium is typically conceptualized as a continuum with bulk parameters that characterize its physical and chemical properties based on the assumption that all phases coexist in each point in space. In contrast, the pore scale can be defined as the largest spatial scale at which it is possible to distinguish the different fluid and solid phases that make up natural subsurface materials. Because the pore scale directly accounts for the pore-space architecture within which mineral reactions, microbial interactions and multi-component transport play out, it can help explain biogeochemical behavior that is not understood or predicted by considering smaller or larger scales (Fig. 1). Specifically, the nonlinear interaction between the coupled physical and geochemical processes may result in emergent behavior, including changes in permeability, diffusivity, and reactivity that is not captured easily by a Darcy-scale continuum description. Reactive processes in porous media such as microbially mediated reduction–oxidation (Fig. 1) or mineral dissolution–precipitation (Fig. 2) take place at interfaces between fluid and solid phases. Because the different phases are distinguishable at the pore scale, experimental and modeling studies need to consider these interfaces so as to accurately determine reaction rates. An interface is the surface between two phases that differ in their physical state or chemical composition. Depending on the scale of observation, the appearance of the interface can vary. Sharp interfaces are those in which the physical and chemical characteristics change abruptly across the interface. Diffuse interfaces are those in which the characteristics change smoothly over a layer of varying thickness. Reactive processes themselves can change the appearance of the interface. For example, mineral heterogeneity can result in the creation of degraded zones (Fig. 2), where dissolution of a faster-dissolving mineral (e.g., calcite) from within a matrix of relatively insoluble minerals (e.g., dolomite and silicates) leaves behind a porous continuum Deng et al. (2013). As advances in experimental and imaging techniques allow for improved characterization of pore-scale processes, modeling approaches are being challenged to incorporate the textural and mineralogical heterogeneity of natural porous media, in particular with regard to their treatment of interfaces. Simulation of the evolution of reactive interfaces is critical to capture the processes that lead to emergent behavior, including the reactive infiltration instability (Ortoleva et  al. 1987; Hoefner and Fogler 1988; Steefel and Lasaga 1990) or reactivity evolution (Luquot and Gouze 2009; Noiriel et al. 2009). The focus of this chapter is on the 1529-6466/15/0080-0014$05.00

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Figure 1. Conceptual sequence of length scales associated with (microbially mediated) interfacial reactions in porous media. [Reprinted from Wood BD, Radakovich K, Golfier F (2007) Effective reaction at a fluid–solid interface: Applications to biotransformation in porous media. Advances in Water Resources, Vol. 30, p. 630–1647, Fig. 1 with permission.]

Figure 2. SEM backscattered-electron images showing fracture surface geometries resulting from reaction with CO2-acidified brine: (a) comb-tooth wall geometry resulting from receding calcite bands and persistence of less soluble minerals and (b) porous “degraded zone” along the fracture wall created by preferential dissolution of calcite. [Reprinted from Deng H, Ellis BR, Peters CA, Fitts JP, Crandall D, Bromhal GS (2013) Modifications of Carbonate Fracture Hydrodynamic Properties by CO2-Acidified Brine Flow. Energy Fuels, Vol. 27, p. 4221–4231, Fig 1 with permission. © 2013 American Chemical Society.]

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 463 direct numerical simulation of reactive processes at the pore scale, with an emphasis on the role of fluid–solid interfaces (Kang et  al. 2007; Tartakovsky et  al. 2007). Direct numerical simulation (DNS) employs mesh-based methods that often require an explicit representation of these reactive interfaces. Many concepts discussed in this chapter, however, are applicable more generally, including to other modeling approaches such as the lattice Boltzmann or particle methods (Kang et al. 2007; Tartakovsky et al. 2007). In this chapter, the equations for pore-scale processes of flow, transport and geochemical reactions are succinctly presented, followed by a description of the methods for their direct numerical simulation within pore-scale domains. Next, the representation of reactive surfaces in DNS applications is reviewed, with an emphasis on surface reactivity evolution and transport limitations to reactive surfaces. This is followed by a review of methods to capture the physical evolution of the pore space with a focus on reactive instabilities. To end the chapter, an approach for upscaling interfacial reactions from the pore scale to the Darcy scale is presented.

PORE-SCALE PROCESSES Flow Given the low compressibility of water, the incompressible Navier–Stokes equations can be used to accurately describe the flow of water in the pore space via the conservation of momentum and mass, respectively, for most subsurface conditions:  u     u   u   p   2 u,  t 

(1)

  u  0,

(2)

where the left-hand side of Equation (1) describes the inertial forces, and the right-hand side includes the pressure gradient (p) and the viscous forces, with u being the fluid velocity,  the fluid density, and μ the dynamic viscosity. Inertial forces have their origin in the convective acceleration of the fluid as it flows through the tortuous pore space. Viscous forces originate in the friction between water molecules and are responsible for the dissipation of energy. When viscous forces dominate (i.e., at very small Reynolds number) the above equations can be simplified to the steady-state Stokes equations (Steefel et al. 2013): 0  p   2 u,

(3)

  u  0.

(4)

Equations (1–2) and (3–4) are to be solved within the pore space occupied by the fluid phase and delineated by the fluid–solid interfaces (Fig. 1).

Multicomponent reactive transport Transport and reaction of dissolved species in the aqueous phase can be described by the following conservation equation:   MH2Oci     uMH2Oci  Di  MH2Oci   Ri , t





(5)

where ci is the molal concentration of species i in solution (mol kg-1H2O), M H2O is the mass

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fraction of water (kg H2O kg-1), Di is the diffusion coefficient (m2 s-1) and Ri is the reaction term (mol m3 s-1). In Equation (5), transport takes place by advection (the translation in space of dissolved or suspended material at the rate of movement of a bulk fluid phase) and diffusion (mixing of solutes in the multicomponent mixture driven by concentration gradients). For simplicity, electrochemical migration associated with diffusion of charged species (Steefel et al. 2013) has not been included in Equation (5), although its contribution may be significant under acidic conditions (Molins et  al. 2012; Ovaysi and Piri 2013). The reaction term Ri is expressed in volumetric terms and includes homogeneous reactions such as aqueous complexation. Aqueous complexation reactions are typically fast enough that they can be considered as locally at equilibrium, and Equation (5) can be written in the canonical form in terms of components (Steefel et al. 2014). Interfacial reactions such as mineral dissolution– precipitation take place at solid–fluid boundaries and are not included in Equation (5).

Surface reactions Interfacial reactions can instead be expressed as a boundary condition at the solid–fluid boundaries  Di  M H2Oci   imrm ,

(6)

where rm is the surface reaction rate (expressed in units of mass per unit time per unit surface) and im is the stoichiometric coefficient of the i-th component in each surface reaction m. A wide range of rate expressions can be considered. For mineral dissolution–precipitation reactions, the transition state theory law is often employed (e.g., Molins et al. 2014) m1

   G  m3    E  rm  km exp   a   ain 1  exp  m2   ,   RT     RT    

(7)

where km is the intrinsic rate constant (mol m-2 reactive surface s-1), Ea is the activation energy (kcal mol-1),  ain is a product representing the inhibition or catalysis of the reaction by various ions in solution raised to the power n , with ai being the activity of species I, G is the Gibbs free energy (kcal), with m1, m2 , and m3 being three parameters that affect the affinity dependence, R is the ideal gas constant (kcal K-1 mol-1), and T is the temperature (K). In the case of far-from-equilibrium dissolution, the affinity or G term can often be neglected, as in the studies of chlorite dissolution under high-pCO2 conditions (Smith et al. 2013). Other rate expressions have been used for interfacial reactions. For example, Wood et al. (2007) used Michaelis–Menten kinetics for the enzyme-mediated reaction rate applicable at the fluid–solid interface, assuming microbial cells were uniformly distributed on the surface (Fig. 1.I): rm  im km

ci , K m  ci

(8)

where km is the surface reaction rate, which includes the effect of the enzyme or biomass present on the mineral surface (mol m-2 reactive surface s-1), and Km is the half-saturation constant associated with the redox species i that defines the kinetic rate. In addition to kinetic models, equilibrium models, which do not require an explicit calculation of the rate, have also been employed at the pore scale. For example, Zaretskiy et  al. (2012) used an equilibrium Langmuir model to calculate surface concentrations of adsorption species (si mol m-2 surface) as a function of species concentrations:

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 465 si  smax

ci , 1  ci

(9)

where smax and  represent the maximum surface concentration and the equilibrium constant, respectively (Zaretskiy et al. 2012).

DIRECT NUMERICAL SIMULATION Direct numerical simulation involves the use of conventional discretization methods to solve the flow, transport, and geochemical equations. These include Eulerian, meshbased methods such as finite-difference, finite-element, or finite-volume methods, in which the differential equations are discretized by defining the value of the unknowns at fixed points in space. Finite differences have been applied to the simulation of flow and transport in sandstones (Sallès et  al. 1993; Bijeljic et  al. 2011a,b; Blunt et  al. 2013) and carbonates (Bijeljic et al. 2013). The finite-element method has been used for the simulation of flow and transport in sphere packs (Cardenas 2008, 2009) and as a mixed finite-element–finite-volume method for flow and reactive transport in Fontainebleau sandstone (Zaretskiy et al. 2012). A finite-volume method has been applied for flow and reactive transport in computer-generated and experimentally derived simulation domains (Molins et al. 2012, 2014). Direct numerical simulation has also been used in combination with other approaches. For example, Yoon et al. (2012) used finite-volume methods to simulate reactive transport with a flow solution obtained with a lattice Boltzmann method, and a random-walk method was used for reactive transport by Sadhukhan et al. (2012) while a direct finite-difference solver was employed for Navier– Stokes flow. Both structured (e.g., Molins et al. 2014) and unstructured (e.g., Zaretskiy et al. 2012) meshes have been used in these applications. To perform direct numerical simulation at the pore scale, experimental images of porous media resolved at the micrometer scale (e.g., from X-ray computed microtomography or XCMT) need to be converted to computational domains. Typically, segmentation is used to identify the discrete materials in an image (Wildenschild and Sheppard 2013). A number of techniques are used to segment images to obtain the morphology and topology of the pore space (Wildenschild and Sheppard 2013), frequently by binarization (e.g., Noiriel et al. 2009) but more recently using ternary segmentation methods (e.g., Deng et al. 2013; Scheibe et al. 2015). Binary reconstructed domains consist of voxels that are classified as either pore space or solid, while ternary reconstructed domains may allow for assigning a porosity value to areas of the domain that are under resolved by the characterization method (Scheibe et al. 2015) or may have been subject to reaction (Deng et al. 2013).

Interface representation Binary domains can be directly incorporated into numerical models. In this scenario, fluid–solid interfaces can take the shape of a staircase in the case of structured meshes, where complex surfaces are represented as perpendicular walls locally at the grid cell level. Most structured-grid applications to simulation of flow and transport use this approach (e.g., Sallès et al. 1993; Bijeljic et al. 2011a,b; Blunt et al. 2013). However, for reactive applications the approach can lead to an overestimation of the actual interfacial area available for surface reaction (Fig. 3b). Finite-element and finite-volume unstructured methods are not affected by this issue because they can capture arbitrary surfaces by appropriate meshing strategies (e.g., Cardenas 2008, 2009; Zaretskiy et al. 2012). However, this advantage may be lost when the mineral surface evolves due to dissolution and precipitation, as re-meshing may be required.

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Figure 3. Example of an irregular geometry on a Cartesian grid (left), in which shaded areas represent volume of cells excluded from domain: (a) Embedded boundary representation with interfaces “cutting” regular cells (left) and single-cut cell showing boundary fluxes (right); (b) Binary “staircase” representation of the interface [Adapted from Trebotich et al 2014].

To remedy the issue of reactive surface overestimation for a structured mesh, Molins et al. (2011) used the marching cubes algorithm to evaluate the interfacial area specifically for surface reactions while still solving for flow and transport within a binary, staircase domain. Another approach, the embedded boundary method, uses cut cells that result from intersecting the irregular fluid–solid interfaces with the structured Cartesian grid (Fig. 3a) to discretize the equations using the finite-volume method (Trebotich et al. 2014; Trebotich and Graves 2015). The embedded boundary method is thus a more rigorous approach in that flow, transport, and reactions are solved within the same domain. This method also makes it possible to leverage the advantages of structured methods while capturing complex surficial geometries (Molins et al. 2014). In Molins et al. (2014), a level set was used on the segmented microtomographic image to obtain the implicit function representing the calcite surface on the Cartesian grid. With a level set, the fluid–solid interface,   x  is represented by a contour of a function   x,t  such that:   {x |   x, t   c},

(10)

where c is a constant, and the level-set function  is greater than c for one phase, and less than c for the other phase. The so-called level-set method builds on this description of the interface to track its motion (in particular, as a result of dissolution–precipitation reactions) and is discussed in the section devoted to pore-space evolution.

Micro-continuum and multiscale approaches Binary representations of porous media as discussed above may not be sufficient where reactive processes result in degraded porous zones due to mineral heterogeneity (Deng et al. 2013) (see Fig. 2) or in porous precipitates that allow for diffusion (Yoon et al. 2012). The porosity in these reacted porous regions may be under-resolved at a given pore-scale model resolution; thus, the interface can be better described as discontinuous and diffuse rather than

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 467 continuous and sharp. In fact, one could argue that binary representations do not represent natural porous materials where porosity is distributed—as a fractal or otherwise—over a very large range of spatial scales (Anovitz et al. 2013), or where reactivity may be preferentially found in connected nanoporosity, e.g., within chlorite in the Cranfield sandstone (Landrot et al. 2012). The micro-continuum approach assumes the existence of a porous medium continuum at very small spatial scales. This assumption is valid over porous volumes in which the properties of the medium are continuously distributed (region II in Fig. 4). Darcy’s law is the governing equation for flow in the micro-continuum, while reactive transport is described by the mass balance equation of each species. Micro-continuum equations are parameterized using bulk parameters such as porosity and reactive surface area. For more details, the reader is kindly referred to the chapter of this volume devoted to the micro-continuum approach (Steefel et al. 2015, this volume). A well-established example of the use of the continuum approach for interfacial reactive processes is that of reactive transport in fractures (Steefel and Lichtner 1998a,b; Noiriel et al. 2007). In fractured media, there is a sharp contrast—with a clear separation of scales—between the porosity in the fracture and in the rock matrix. As a result, the fracture is modeled as a fast flow path, where transport is dominated by advection, and the rock matrix, where transport is dominated by diffusion (Fig. 5). Heterogeneous reactions take place within the porous rock matrix or the layer of precipitate that coats the surface (Fig. 5). To simulate flow in discrete fractures in Darcy-scale models, the cubic law is often used to obtain a fracture permeability under the assumption of parallel smooth walls, e.g., Steefel and Lichtner (1998a,b) and Noiriel et al. (2007). However, this model does not typically provide an accurate estimation of fracture hydrodynamic properties at realistic fracture roughness, e.g., Deng et al. (2013). Pore-scale flow in multiscale domains, such as fractured or vuggy media, can instead be simulated using the Stokes–Darcy or the Stokes–Brinkman equations (Golfier et al. 2002; Popov et al. 2009; Gulbransen et al. 2010; Yang et al. 2014). In the Stokes–Darcy approach, Darcy’s law and mass conservation are applied in the porous subdomains, and the Stokes equations in the free-flow subdomains. To close the model, mass- and momentum-conservation equations are specified at the interfaces between domains (Popov et al. 2009). When the geometry of these interfaces is very complex, however, the Stokes–Brinkman model is advantageous, in that a single set of equations is used over the entire domain. In the Stokes–Brinkman model, the Navier–Stokes equations (Yang et al. 2014) or the Stokes equations (Golfier et al. 2002; Popov et al. 2009; Gulbransen et al. 2010) are modified to add a Darcy term, e.g.:

 2 V  0, K 1V  P  

(11)

  V  0,

(12)

where V is the effective velocity, K is the permeability tensor, P is the pressure, and  is an effective viscosity. The uppercase notation indicates here that these quantities apply to a porous medium continuum, where both fluid and solid phases are assumed to coexist at each point in space, rather to each discrete phase separately as in a pore-scale description. The process of obtaining these quantities from the microscale (pore-scale) description, formally referred to as upscaling, is discussed in a separate section below. In the pore space, permeability tends to infinity, the Darcy term becomes negligible and Equations (11-12) simplify to the Stokes equations (Eqn. 3-4). In the porous regions, on the other hand, both viscous and inertial terms become negligible due to slow velocities and Darcy’s law governs fluid flow (Popov et al. 2009; Yang et al. 2014). A very similar approach has been recently used by Scheibe et al. (2015) to simulate flow and tracer transport in a soil column imaged by XCMT and segmented using a ternary approach making it possible to explain the breakthrough curves observed experimentally.

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Figure 4. Porosity () variation over growing porous medium volumes, indicating the range of applicability of the continuum assumption. Reprinted from Borges et al. (2012) based on Bear (1988).

Figure 5. Schematic representation of the transport phenomena in an idealized fracture

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 469 For reactive transport problems, multiscale hybrid (Battiato et  al. 2011; Roubinet and Tartakovsky 2013) and mortar methods (Mehmani et al. 2012) have been proposed. In these, Darcy-scale and pore-scale models are used in different subdomains, with the appropriate continuity of mass fluxes being enforced at the interfaces between subdomains (Battiato et al. 2011; Mehmani et al. 2012; Roubinet and Tartakovsky 2013). While these studies have simulated the macroscale treating the porous medium as a continuum (i.e., at the Darcy-scale, region II in Fig. 4) and the microscale with a pore-scale representation (region I in Fig. 4), the methods are general enough that they could be applied in multiscale problems in which a pore-scale representation were used for the macroscale (inhomogeneous region III in Fig. 4) and a micro-continuum representation for unresolved nanoscale porous regions (region II in Fig. 4). In this sense, they would be conceptually equivalent to the Stokes–Darcy approach for multiscale flow.

SURFACE AREA ACCESSIBILITY AND EVOLUTION IN MINERAL REACTIONS Mineral rates measured in laboratory experiments are often several orders of magnitude faster than those estimated from natural systems, e.g., Malmström et  al. (2000). These differences in rates have been attributed to a variety of factors including, among others, reactive surface area accessibility in natural porous media (Peters 2009; Landrot et al. 2012), limitations on flow and transport in heterogeneous material (Drever and Clow 1995; Malmström et al. 2000; Salehikhoo et  al. 2013; Li et  al. 2014), or transport, rather than interface control of rates (Drever and Clow 1995; Steefel and Lichtner 1998b). Pore-scale modeling can be used to address some of these hypotheses by explicitly accounting for the rate-limiting effect of transport, and by incorporating mechanistic descriptions for the evolution of reactive surface area.

Transport control on rates Transport to mineral surfaces, especially in physically heterogeneous media, can lead to poorly mixed conditions at the pore scale. As a result, effective rates cannot be determined using an average concentration in the pore space. Instead concentrations need to be resolved within the pore space, and rates calculated at each mineral surface. The pore-scale description can thus be used to evaluate the impact on the effective rates of the departure from the assumption of well-mixed conditions. Molins et al. (2014) used a combination of experimental, imaging, and modeling techniques to investigate the pore-scale transport and surface reaction controls on calcite dissolution under elevated-pCO2 conditions. The laboratory experiment consisted of the injection of a solution at 4-bar pCO2 into a capillary tube packed with crushed calcite. A high-resolution, pore-scale, direct numerical model was used to simulate the experiment based on a computational domain consisting of reactive calcite, pore space, and the capillary wall, constructed from volumetric X-ray microtomography images. Part of the reported discrepancy between simulated Darcyscale and pore-scale effluent concentrations was apparently due to mass-transport limitations to and from reactive surfaces. These were most pronounced near the inlet where larger diffusive boundary layers formed around grains. Transport limitations resulted from the heterogeneity of the pore structure: in slow-flowing pore spaces that exchanged mass by diffusion with fast flow paths, the assumption of well-mixed conditions did not apply (Fig. 6). Although the difference between pore- and Darcy-scale results due to transport controls was discernible with the highly accurate methods employed, this difference is expected to be more significant in more heterogeneous porous media, such as in natural subsurface materials.

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Figure 6. Results from reactive transport simulations of a crushed calcite capillary experiment (Molins et al. 2014). pH values are shown on calcite grain surfaces while vectors (black arrows) indicate direction and magnitude of fluid velocities in the pore space. Values of pH along fast flow paths are lower than in slow-flow paths, resulting in heterogeneous dissolution rates on minerals surfaces located in close proximity. The diameter of the capillary tube is 524 μm. [Modified from Molins S, Trebotich D, Yang L, Ajo-Franklin JB, Ligocki TJ, Shen C, Steefel CI (2014) Pore-scale controls on calcite dissolution rates from flow-through laboratory and numerical experiments. Environmental Science and Technology, Vol. 48, p. 7453–7460 Abstract Art with permission. © 2014 American Chemical Society.]

Surface area evolution In the study of Molins et al. (2014) it was assumed that the surface and pore-space geometry did not evolve over the 16 s of simulated time. However, as a result of reactive processes, normally both the reactive surface area and pore structure evolves. Here, two studies are presented that considered evolution of reactive surface area using a micro-continuum approach. Methods to track pore-space evolution are introduced in the next section. In a study by Noiriel et al. (2009) of limestone infiltrated by CO2-rich solution, Sr and Ca release rates were used to assess the relative dissolution rates of the sparitic and microcrystalline phases in the limestone subjected to infiltration of CO2-rich solution. The results demonstrated that the reactive surface area (RSA) of the sparite increased greatly, as recorded by the rate of dissolution of that phase over time. In contrast, its geometric surface area, as recorded by XCMT, decreased slightly. To describe the time-dependent behavior, Noiriel et al. (2009) proposed a “sugar cube” model in which disaggregation of the granular network (presumably resulting in the large increase in RSA of the sparitic phase, which is now exposed to more of the reactive infiltrating solution) precedes dissolution of the individual grains of sparite (Fig. 7). Noiriel et al. (2009) described mathematically the time-dependent RSA, Ar, with the expression n2      n1      n3  Ar  Ar 0  Arm  1        ,    0     0     

(13)

where Ar 0 is the initial surface area, Arm is the maximum surface area given by the sum of all of the surface areas of the individual particles,  is the concentration of calcite over time, 0 is its initial concentration, and n1, n2, and n3 are empirical coefficients that depend on the geometry of the aggregate.

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 471

Figure 7. (top) Dissolution process according to the sugar-lump model. The matrix is composed of (a) spherical grains of surface area Ar 0 (b) which dissociate into smaller grains, thus increasing the waterexposed surface area. (c) Subsequently, the individual particles dissolve, which reduces the surface area. (bottom) Thin section of the rock observed with optical microscopy in transmitted light. (b) Scanning electron microscopy (SEM) observations of the sparite and the micrite. [Reprinted from Noiriel C, Madé B, Gouze P (2007) Impact of coating development on the hydraulic and transport properties in argillaceous limestone fracture. Water Resources Research, Vol. 43, W09406, Fig 1, 2 with permissions.]

Dissolution rates can also be affected by the precipitation of secondary mineral or transformation products, as it can limit access to the surface of the dissolving mineral. This is the case of the carbonation of wollastonite (CaSiO3) studied by Daval et al. (2009):

CaSiO3  CO2  CaCO3  SiO2 .

(14)

Experimental results showed that under circumneutral pH conditions dissolution rates were inhibited by the formation of a dense calcite coating. The passivation effect was successfully accounted for by the use of an effective reactive surface area

Aeff  t   a0 CaSiO3  t   k p' CaCO3  t  . p

(15)

This equation links the true reactive surface area of wollastonite to the amounts of wollastonite and neo-formed calcite (CaCO3 ), with a0 being the specific surface area in units of mol m-2, k p' is a proportionality factor that can be obtained assuming that when t→∞, Aeff(∞) = 0, and p  2 / 3 for monodisperse spherical particles, and p  2 / 3 for particle size distributions with a non-spherical geometry. Although the model slightly overestimated the rate for intermediate values of the reaction extent, the overall behavior of the extent of carbonation as a function of time was reproduced closely (Fig. 8). This showed that secondary calcite precipitation can play an important role as coatings of reactive surfaces (Daval et al. 2009).

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Figure 8. (a) Backscattered SEM images of a cross-section of a carbonated wollastonite grain after 2 days of reaction in circumneutral pH conditions. Note the succession of the inner intact core of reacting wollastonite, a fractured layer composed with calcite and silica, and the compact continuous and poorly permeable coating of calcite. (b) Normalized extents of wollastonite carbonation as a function of time in circumneutral pH conditions. The lower two curves correspond to fits of the data assuming the progressive formation of a passivating coating of calcites. [Reprinted from Daval D, Martinez I, Corvisier J, Findling N, Goffé B, Guyot F (2009) Carbonation of Ca-bearing silicates, the case of wollastonite: Experimental investigations and kinetic modeling. Chemical Geology, Vol. 265, p. 63–78, Figs. 3c, 6, with permission.]

PORE-SPACE EVOLUTION Dissolution and precipitation of minerals and biogeochemical transformations such as bacterial growth change the geometry of the pore space. Although this evolution takes place at each fluid–mineral interface, the resulting changes have impacts at larger scales. For example, (bio)precipitates can significantly reduce the macroscopic porosity and permeability by plugging pore throats, e.g., Oelkers et al. (2008), Englert et al. (2009). Similarly, dissolution driven by injection of CO2 in the subsurface leads to an increase in porosity and permeability as well as changes in the mineral reactive surface area (Luquot and Gouze 2009). Mathematically, solute dissolution and/or precipitation can be formulated as a moving boundary, or Stefan problem. Assuming uniform dissolution–precipitation of a single-mineral solid phase (m), the velocity of the moving solid–fluid interface (un ) can be described by, e.g., Li et al. (2010): un  u   n  Vm rm ,

(16)

where Vm is the molar volume of the mineral. Equation (16) needs to be solved along with Equations (1, 2, 5, and 6) or Equations (3–6). The front-tracking (Glimm et  al. 1998), volume-of-fluid (Hirt and Nichols 1981), and level-set (Osher and Sethian 1988) methods have been used to solve moving boundary problems for tracking or capturing sharp interfaces. The advantage of these methods is that one can perform direct numerical simulations involving surfaces using the Eulerian approach on fixed Cartesian grids without having to parameterize these surfaces. Diffuse interface models, such as the phase-field method (Langer 1986), in which the interface is captured with a continuous variation of an order parameter, rather than explicitly represented as a sharp boundary, can also be used to solve the moving boundary problem. The level-set method and the phase-field method are described below in the context of recent applications to modeling pore-scale dissolution and precipitation processes.

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 473 Level set method In the level-set method, the fluid–solid interface (  x ) is represented by a contour level of the level-set function   x,t  , Equation (10). The evolution of the level set is governed by the following advection equation, which describes the motion of the fluid–solid interface (Li et al. 2010):    un n   0, t

(17)

where n is the norm of the level-set function (n   /  ) at  = c. Li et al. (2010) used the level-set method for capturing interface evolution in simulations of dissolution and precipitation in a three-dimensional, single-pore throat (e.g., Fig. 9). The simulations were performed for a single-component system, where the reaction rate was described with both a zero-order kinetic term and an affinity term to account for nearequilibrium conditions. The three-dimensional effects of flow conditions and reaction rates were explored quantitatively. The simulation showed that the evolution of the permeability– porosity relationship depended on particular parameter values used (i.e., flow rates, rate constants). Further, the empirical Carman–Kozeny constitutive model did not capture evolution of permeability–porosity, especially for the conditions that led to non-uniform dissolution or precipitation patterns (Li et al. 2010).

Figure 9. Evolution of the three-dimensional solid surface together with the velocity and concentration fields for precipitation in a pore throat with an initially sinusoidal-shaped aperture under different conditions expressed in the form of the dimensionless Damköhler (Da) and Péclet (Pe) numbers. [Reprinted from Li X, Huang H, Meakin P (2010) A three-dimensional level set simulation of coupled reactive transport and precipitation/dissolution. International Journal of Heat and Mass Transfer, Vol. 53, p. 2908–2923, Fig, 6 with permission.]

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Phase-field method Phase-field methods are based on the idea that the free energy depend on an order parameter (the phase-field variable) that acts as a function indicating what phase a point in space is in. The method replaces the boundary conditions at the interface (e.g., Eqn. 6) with a partial differential equation for the evolution of the phase field. The field changes smoothly in a diffuse zone around the interface that has a finite width. A sharp-interface asymptotic analysis of the phase-field equations was developed by Xu and Meakin (2008) for the precipitation–dissolution problem considering diffusion as the only transport process. In this model, the evolution of the concentration field is a function of the evolution of the phase field:    D 2    c   2 t  D c    1   km   t t     

(18)

where  is proportional to the molar volume of the mineral. The evolution of the phase field is described with 

  2 2   1  2    c   2   , t





(19)

where is the phase-field characteristic time parameter,  is closely related to the interface thickness, and  controls the strength of the coupling between the phase field  and the concentration c. The relationship between   and  is then obtained from an asymptotic analysis, which ensures that the phase-field equations converge to the sharp interface solution:   

2  5 2D    . D  3 km  

(20)

Two-dimensional dendritic growth due to solute precipitation was simulated using this phase-field model (Xu and Meakin 2011). The simulations were performed under diffusionlimited conditions by setting the chemical reaction rate much larger than the rate of diffusion— i.e., the kinetics of the reaction at the interface was not relevant. The Mullins–Sekerka instability (Mullins and Sekerka 1964)—the same responsible for the dendritic growth of the snowflake—caused the growth process of a small circular nucleus placed in the center of a square domain to be very sensitive to perturbations (Xu and Meakin 2011). In the phase-field simulations of Xu and Meakin (2011), the perturbation was provided by the discretization of Equations (18–20) on a square lattice. As a result, diffusion-limited precipitation was observed to take place as an unstable dendritic growth (Fig. 10). The resulting solid–fluid interfacial pattern displayed a fractal geometry, whose fractal dimension agreed well with that estimated independently with a diffusion aggregation model (Meakin and Deutch 1984).

Continuum and multiscale approaches As apparent from the results presented previously, the application of interface tracking methods to direct numerical simulation of pore-scale processes has been limited to relatively small, ideal problems. Instead direct simulation of pore-space evolution has often relied on treating the porous medium as a continuum at the Darcy scale. An example of pore-space evolution that has received wide attention in the literature is the well-known reactive infiltration instability, which results in the formation in wormholes

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 475

a

b

c

d

Figure 10. Snapshots of interfacial patterns of unstable dendritic growth obtained with the phase field method of Xu and Meakin (2008) for precipitation under diffusion-limited conditions, at various simulation times (dimensionless) : a) t = 0.002 b) t = 0.004 c) t = 0.006 d) t = 0.008. [Reprinted from Xu Z and Meakin P (2011) Phase-field modeling of two-dimensional solute precipitation/dissolution: Solid fingers and diffusion-limited precipitation. Journal of Chemical Physics, Vol. 134, 044137, Fig 3. with permission.]

(Ortoleva et  al. 1987; Hoefner and Fogler 1988; Steefel and Lasaga 1990). Similar to the unstable dendritic-growth process simulated by Xu and Meakin (2011), in the reactive infiltration instability, mineral dissolution takes places under transport-controlled conditions. In this instability, however, dissolution is limited by the rate of advection, which leads to the more rapid dissolution in high-flow-velocity pathways (wormholes) over slower pathways. Direct numerical simulation at the Darcy scale has shown that the wormholes become more highly ramified and ultimately diffuse when the Damköhler number is small (surfacereaction-controlled) for a given system (Steefel and Lasaga 1990; Steefel and Maher 2009). More recently, Golfier et al. (2002) used the multiscale Stokes–Brinkman equations (11–12) to simulate the formation of wormholes. The reactive transport equations were formulated by upscaling the pore-scale equations to the Darcy scale with the volume-averaging method (cf. Appendix A Golfier et al. 2002). This upscaling approach is discussed in the next section. As a result, the surface reaction (i.e., dissolution) was represented at the Darcy scale in its upscaled form, via a mass-transfer coefficient (Golfier et al. 2002), thus avoiding the direct representation of the surface as in the level set or phase-field method examples presented above. The model was able to reproduce the dissolution regimes observed experimentally with increasing flow rates (Fig. 11): (a) face dissolution (diffusion dominates over advection and instabilities do not develop), (b) conical wormholes (instabilities are present but still strongly influenced by diffusion), (c) dominant wormholes (advection-dominated instabilities), (d) ramified wormholes, and (e) uniform dissolution. Golfier et al. (2002) pointed out that for (a)– (c) very rapid dissolution led to local equilibrium conditions at the sharp interfaces, while for (d)–(e) faster flow rates led to local non-equilibrium dissolution, with more diffuse interfaces, and even stable displacement observed for (e). The Stokes–Brinkman model was able to capture flow in the multiscale domain, with Stokes flow being prominent in the free-flow regions (i.e., wormholes) and Darcy flow dominant in the partially dissolved porous medium.

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Figure 11. (top) Example of dissolution patterns observed experimentally in the work of Golfier et  al. (2002). (bottom) Porosity fields obtained with the multiscale direct simulation of Golfier et  al. (2002): (a) face dissolution (b) conical wormhole (c) dominant wormhole (d) ramified wormhole and (e) uniform dissolution. [Reprinted from Golfier F, Zarcone C, Bazin B, Lenormand R, Lasseux D, Quintard M (2002) On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium. Journal of Fluid Mechanics, Vol. 457, p. 213–254, Figs. 9, 10 with permission.]

UPSCALING OF SURFACE REACTIONS BY VOLUME AVERAGING While numerical simulation of pore-scale processes can help explain biogeochemical behavior not easily predicted at larger scales, it can also be used to derive the parameters that apply to models at larger scales. For this purpose, the pore-scale problem, in which interfaces are resolved explicitly and thus the variation of medium properties is not continuous, needs to be upscaled to the Darcy-scale continuum, in which they vary continuously (Fig. 4). Homogenization by volume averaging is one of a number of upscaling methods. Volume averaging has been successfully used as an upscaling method for porous-media processes (Whitaker 1999; Golfier et  al. 2002; Wood et  al. 2007). In particular, upscaling by volume averaging can be used to generate a mechanistic description of the effective parameters from the microscale representation of the physical and biogeochemical properties as well as the geometry of the pore space (Wood et al. 2007). In the approach of Wood et al. (2007), the pore-scale equations (5–6) are first averaged over a representative control volume and then the pore-scale concentrations and velocities are decomposed into volume averages and deviations from these averages. As a result of this process, a macroscale mass balance equation is obtained that provides a continuum representation of the pore-scale processes. However, in this form the solution still depends on the concentration deviations, which are pore-scale quantities, through the hydrodynamic dispersion, macrodiffusion and the macroscale reaction rate terms. A closure problem for the concentration deviations needs to be posed to model their behavior. Golfier et al. (2002) and

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 477 Wood et al. (2007) assume that simple periodic unit cells (e.g., solid cubes or spheres) capture the essential features of the system for the computation of the effective rate. Substitution of the solution of the closure problem makes it possible to write the macroscopic equations as a function of averaged concentrations, now using effective parameters, e.g., Wood et al. (2007): ci   V ci     D* ci   1 Av Ri ,eff , t





(21)

where ci  is the spatially averaged concentration of the reactant in the fluid portion of the pore space (), V is an effective pore water velocity, defined as the intrinsic average of the fluid velocity vector in the averaging volume ( u  ), D* is the effective hydrodynamic dispersion tensor,  is the porosity, Av is the surface area per unit volume of porous medium, and Ri ,eff is the macroscale effective reaction rate (specifically, the contribution of the rate of the microbially mediated reaction to the mass of reactant i). Essentially, the effective parameters contain the pore-scale information that has been averaged in the upscaling process. While Wood et  al. (2007) obtained a closure problem solution for the zero- and first-order limits of the Michaelis–Menten reaction rate, it was not possible for the authors to use the nonlinear kinetics in Equation (8) for the solution of the closure problem without coupling the closure to the value of the macroscale concentration, ci. This would have significantly increased the complexity of the problem. Rather, they proposed the following semi-empirical nonlinear form for the macroscale effective reaction rate term: Rm ,eff  im km

ci  , K m ,eff  ci 

(22)

where Km,eff is now an effective half-saturation constant for the reaction. Comparison of results of a direct numerical simulation of pore-scale reactive transport (taken as ground truth) with the macroscopic, upscaled version suggested that the variance of the concentration field had a dramatic impact on the effective reaction rate (Fig. 12). As a result, the accuracy of Equation (22) as a predictor of the effective reaction rate decreased as the variance increased. These discrepancies occurred mostly because of the errors induced by the proposed average of the nonlinear term; more specifically because the order of averaging operations cannot be interchanged for nonlinear processes such as Michaelis–Menten kinetics.

SUMMARY AND OUTLOOK Applications of direct numerical simulation of pore-scale processes in subsurface materials are growing in part due to the computational advantages of well-established computational fluid dynamics (CFD) methods, e.g., Bijeljic et al. (2011a,b), Zaretskiy et al. (2012), Blunt et al. (2013). Availability of high-level, open-source libraries such as OpenFOAM (OpenFOAM 2015) or Chombo (Adams et al. 2014; Colella et al. 2014) have also facilitated implementation of models into new simulation capabilities. In reactive systems, the ability to calculate the heterogeneous reaction rates at the fluid–solid interfaces makes DNS a suitable tool to mechanistically model processes that are not captured by models at larger scales, especially when well-mixed assumptions are made. In this chapter, selected applications to evaluate transport limitations to reactive surfaces and to capture the evolution of both the surface area and the pore space have been reviewed. Explicit representation of reactive interfaces as they evolve remains, however, a significant challenge for direct numerical simulation approaches. The level-set and the phase-field

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Figure 12. Comparison of results obtained from an upscaled model using an empirical macroscale nonlinear Michaelis–Menten-type kinetic rate (Eqn. 22) with results obtained from a direct numerical pore scale simulation: (Top) The approximation proposed by Wood et al. (2007) is sensitive to the size of the variance of the concentration field. (Top and bottom) The empirical equation (Eqn. 22) becomes a more accurate predictor of the macroscale reaction rate (as determined by direct numerical simulation) as the concentration variance decreases. [Reprinted from Wood BD, Radakovich K, Golfier F (2007) Effective reaction at a fluid–solid interface: Applications to biotransformation in porous media. Advances in Water Resources, Vol. 30, p. 1630–1647, Fig. 6 with permission.]

method have been successfully applied to dissolution–precipitation reactions in rather simple computational domains (Li et al. 2010; Xu and Meakin 2011). In these simulations, a single mineralogy is assumed. However, natural reactive systems are characterized by physical and mineralogical heterogeneity at a variety of different scales. As a result, reactive interfaces are often discontinuous and diffuse, e.g., Noiriel et al. (2009) and Deng et al. (2013). In multiscale problems, direct numerical simulation has typically been performed using continuum methods rather than strictly using pore-scale methods. In these methods, however, surface reactions (as well as transport processes) are not explicitly calculated at the interface and are upscaled from the pore scale. Upscaling of strongly coupled non-linear processes is not straightforward and the solution of macro- and micro-scale problems is coupled. Applying the same conceptual model for the reaction rate expression (e.g., Eqn. 8) at the pore scale and at the Darcy scale with averaged quantities is in general not warranted. In this context, pore-scale modeling has the potential to challenge the conceptual models that are routinely applied at the Darcy scale, in particular for heterogeneous reactions. To accomplish this objective, however, direct numerical simulation needs to be able to incorporate the physical and mineralogical heterogeneity of the subsurface environments at different scales that are captured by advanced characterization techniques, e.g., Landrot et al. (2012). Further, methods developed for the evolution of pore spaces, such as those of Li et al. (2010) and Xu and Meakin (2011), need to be applied to complex subsurface porous media, as characterized by these advanced techniques. It is ultimately the combination of new microscopic characterization, experimental, and modeling approaches that presents the opportunity to provide mechanistic explanations for many of the long-standing questions in geochemistry in porous media.

Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes 479 ACKNOWLEDGMENTS The author would like to thank Li Li and Carl Steefel for reviews of the manuscript. This material is based upon work supported as part of the Center for Nanoscale Control of Geologic CO2, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-AC02-05CH11231.

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RiMG Volume 80 CONTENTS 1–23 Transient Porosity from Fluid—Mineral Interaction

Putnis

25–44 Pore-Scale Controls on Reaction-Driven Fracturing

Røyne & Jamtveit

45–60 Effects of Coupled Chemo-Mechanical Processes

Emmanuel et al.

61–164 Characterization and Analysis of Porosity and Pore Structures

Anovitz & Cole

165–190 Precipitation in Pores: A Geochemical Frontier

Stack

191–216 Pore-Scale Process Coupling and Effective Surface Reaction Rates

Liu et al.

217–246 Micro-Continuum Modeling of Pore-Scale Geochemical Processes

Steefel et al.

247–285 Resolving Time-dependent Evolution using X-ray Microtomography Noiriel 287–329 Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects

Tournassat & Steefel

331–354 Porosity Increases in Crystalline Silicate Rocks

Navarre-Sitchler

355–391 Isotopic Gradients Across Fluid–Mineral Boundaries

Druhan et al.

393–431 Lattice Boltzmann-Based Approaches for Pore-Scale Reactive Transport

Yoon et al.

433–459 Mesoscale and Hybrid Models of Fluid Flow and Solute Transport

Mehmani & Balhoff

461–481 Reactive Interfaces in Direct Numerical Simulation of Pore-Scale Processes

Molins

ISBN 9780939950966

9 780939 950966 80_Cover_degruter.indd 3

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