Pore Scale Geochemical Processes 9781501502071, 9780939950966

Citation preview

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 C 0 2 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 MINERALOGICA!, SOCIETY OF AMERICA GEOCHEMICAL SOCIETY

DE GRUYTER

Reviews in Mineralogy and Geochemistry, Volume 80

Pore-Scale Geochemical Processes ISSN ISBN

1529-6466

978-0-939950-96-6

COPYRIGHT 2 0 1 5 BY-NC-ND THE M I N E R A L O G I C A L S O C I E T Y OF A M E R I C A 3 6 3 5 CONCORDE PARKWAY, SUITE 5 0 0 CHANTILLY, VIRGINIA, 2 0 1 5 1 - 1 1 2 5 , U . S . A . WWW.MINSOCAM.ORG

www.degruyter.com 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.

Pore-Scale Geochemical Processes 80

Reviews in Mineralogy and Geochemistry

80

FROM THE SERIES EDITOR As the 80th 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. All supplemental materials associated with this volume can be found at the MSA website. Errata will be posted there as well. Ian P. "Dwainson. 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 1529-6466/15/0080-0000S00.00

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 eminences 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 R0yne 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 isotopie 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 R0yne, 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

ν

Pore-Scale Geochemical Processes 80

Reviews in Mineralogy and Geochemistry

80

TABLE OF CONTENTS 1

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

INTRODUCTION FLUID-MINERAL INTERACTIONS: PRESSURE SOLUTION FLUID-MINERAL INTERACTIONS: MINERAL REPLACEMENT POROSITY AND FELDSPAR-FELDSPAR REPLACEMENT SECONDARY POROSITY ASSOCIATED WITH MINERAL REPLACEMENT— A UNIVERSAL PHENOMENON IMPLICATIONS OF MICROPOROSITY— SUPERSATURATION AND CRYSTAL GROWTH Critical and threshold supersaturation Crystallization experiments in confined media DISCUSSION CONCLUSIONS ACKNOWLEDGMENTS REFERENCES

2

1 2 3 5 7 8 9 11 16 19 19 19

Pore-Scale Controls on Reaction-Driven Fracturing Anja Roy ne, ISjórn Jamtveit

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

25 26 26 29 30 30 31 33 35 36 39

Pore-Scale Geochemical Processes - Table of

Contents

INTERFACE-DRIVEN TRANSPORT ON THE PORE SCALE CONCLUDING REMARKS ACKNOWLEDGMENTS REFERENCES

3

39 41 42 42

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 PORE-SIZE EVOLUTION DURING MINERAL PRECIPITATION AND DISSOLUTION PORE-SIZE EVOLUTION DURING MECHANICAL COMPACTION CHEMO-MECHANICAL COUPLING AND PORE-SIZE EVOLUTION CONCLUDING REMARKS ACKNOWLEDGMENTS REFERENCES

4

45 47 51 54 56 58 58

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

INTRODUCTION PETROPHYSICAL APPROACHES Direct methods Imaging methods Downhole porosity logs SCATTERING METHODS Theoretical basis of scattering experiments The two-phase approximation and its limitations Sample preparation Geometrical principles of small-angle scattering experiments Contrast matching Reduction and analysis of SAS data IMAGE ANALYSIS Sample preparation and image acquisition Combining imaging and scattering data Three-point correlations Monofrac tais and multifractals Lacunarity, succolarity, and other correlations Comparisons of multiple techniques CONCLUSIONS viii

61 63 63 72 83 87 91 93 94 99 109 Ill 120 120 123 125 129 132 138 140

Pore-Scale Geochemical Processes

- Table of

Contents

ACKNOWLEDGMENTS REFERENCES

5

142 143

Precipitation in Pores: A Geochemical Frontier Andrew G. Stack

INTRODUCTION RATIONALE PORE-SIZE-DEPENDENT PRECIPITATION Effects of precipitation on porosity and permeability Observations of precipitation in pores ATOMIC-SCALE ORIGINS OF A PORE SIZE DEPENDENCE Substrate and precipitate effects EFFECTS IN SOLUTION TRANSPORT CONCLUSIONS AND OUTLOOK ACKNOWLEDGMENTS REFERENCES

6

165 166 170 170 172 175 175 180 184 186 187 187

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

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

191 193 195 195 197 197 198 199 199 201 202 205 205 207 208 209 209

Pore-Scale Geochemical Processes - Table of

/

Contents

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

INTRODUCTION MAPPING OF MODEL PARAMETERS FROM IMAGE ANALYSIS Porosity Mineral volumes Mineral surface area Diffusivity Permeability Imaging issues impacting parameter estimation Image resolution MICRO-CONTINUUM MODELING APPROACHES Volume averaging of porosity and mineral volume fractions Multi-continuum approaches Resolution of nanoscale reaction fronts SUMMARY AND PATH FORWARD ACKNOWLEDGMENTS REFERENCES

8

217 218 219 220 221 222 226 228 229 229 230 234 237 242 243 243

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

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

247 248 249 252 252 253 253 253 254 256 257 258 260 261 262 264 264 264

Pore-Scale Geochemical Processes

- Table of

Contents

Rates of dissolution/precipitation reactions in porous rocks Rates of dissolution/precipitation reactions in fractures Porosity and permeability development in porous media and fractures Effects of texture and mineralogy on complex porosity-permeability relationships and transport Effects of pore-scale heterogeneity on permeability reduction CONCLUDING THOUGHTS ACKNOWLEDGMENTS REFERENCES

9

265 267 269 271 274 274 276 276

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

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

287 288 288 290 293 293 295 295 295 298 303 303 303 304 307 308

310 310 310 310 313 313 314 315 318 319 319 320 321 322

Pore-Scale Geochemical Processes - Table of

Contents

SUMMARY AND PERSPECTIVES ACKNOWLEDGMENTS REFERENCES

1 0

323 325 325

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

INTRODUCTION METHODS FOR POROSITY AND PORE-SIZE DISTRIBUTION QUANTIFICATION Sorption and intrusion techniques Electron and optical microscopy Neutron scattering Fractal nature of rocks CASE STUDIES Weathering of felsic to intermediate composition rocks Weathering of mafic rocks LINKING FRACTAL SCALING AND PORE-SCALE OBSERVATIONS TO WEATHERING MECHANISMS SUMMARY ACKNOWLEDGMENTS REFERENCES

11

331 333 333 334 335 337 339 339 344 346 349 350 351

Isotopie Gradients Across Fluid-Mineral Boundaries Jennifer L. Druhan, Shaun T. Brown, Christian Huber

INTRODUCTION A conceptual model of isotope partitioning at the pore scale Organization of article NOTATION A note on fractionation EXAMPLES OF ISOTOPIC ZONING ACROSS FLUID-SOLID BOUNDARIES Alpha recoil Diffusive fractionation Dissolution Precipitation TREATMENT OF TRANSIENT ISOTOPIC PARTITIONING AND ZONING ACKNOWLEDGMENTS REFERENCES

xii

355 356 358 358 361 362 362 366 370 376 381 384 384

Pore-Scale Geochemical Processes

IL

- Table of

Contents

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

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

1 3

393 396 396 397 398 399 401 401 403 407 409 412 412 412 413 414 416 416 418 420 423 424 424

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

INTRODUCTION PORE-SCALE MODELING Direct pore-scale modeling Pore-network models Network modeling of solute transport HYBRID MODELING CONCLUSIONS ACKNOWLEDGMENTS REFERENCES

433 434 434 437 439 449 453 454 454 xiii

Pore-Scale Geochemical Processes - Table of

¡q

Contents

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

INTRODUCTION PORE-SCALE PROCESSES Flow Multicomponent reactive transport Surface reactions DIRECT NUMERICAL SIMULATION Interface representation Micro-continuum and multiscale approaches SURFACE AREA ACCESSIBILITY AND EVOLUTION IN MINERAL REACTIONS Transport control on rates Surface area evolution PORE-SPACE EVOLUTION Level set method Phase-field method Continuum and multiscale approaches UPSCALING OF SURFACE REACTIONS BY VOLUME AVERAGING SUMMARY AND OUTLOOK ACKNOWLEDGMENTS REFERENCES

xiv

461 463 463 463 464 465 465 466 ..469 469 470 472 472 474 474 476 477 479 479

1

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) C.urtin University, Perth, Australia and Institut fiir Mineralogie, University of Münster Münster, Germany putnis@wìi-muenster.de;

[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. 1529-6466/15/0080-0001505.00

http://dx.doi.org/10.2138/rmg.2015.80.01

Ifani .'J'lJ'l.·! © 2015, P u t n i s . This w o r k is l i c e n s e d u n d e r the Creative C o m m o n s A t t r i b u t i o n - N o n C o m m e r c i a l - N o D e r i v s 3.0 License.

2

Putnis

The individual mineral grains in a rock may also have a micro structure 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 micro structure s can provide important information on how the rock formed and its subsequent thermal and deformational history. The micro structure 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 micro structure 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 micro structure 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 micro structure 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 micro structure s 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., Bj0rlykke 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

Transient

Porosity from Fluid—Mineral

Interaction

3

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-transportprecipitation, 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.

Putnis

4

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

j

a 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 dissolutionprecipitation 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

Transient Porosity from Fluid—Mineral

Interaction

5

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 Ca 2+ and C0 3 2 " ions 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 K B r - K C l - H 2 0 (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 fluidmineral 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

6

Putnis

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: NaAlSi 3 O s -CaAl 2 Si 2 O s + H 4 Si0 4 + Na + = 2 NaAlSi 3 O s + Ca2+ + Al3+ + 4 OH" where 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).

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.

Transient Porosity from Fluid—Mineral

Interaction

7

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 (NaCa 2 Si 3 O s OH) formed within the pores generated within the albite. These experiments were carried out at 600 °C and 2 kbars, and at those temperatures reaction rims up to 50 μπι wide formed within 14 days. Norberg et al. (2011) studied the micro structural evolution during experimental albitization of K-rich alkali feldspar, a reaction that only involved the ion-exchange of Κ 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; Pliimper 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

8

Putnis

(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 (R0yne et al. 2008; Jamtveit et al. 2009; Navarre-Sitchler et al. 2015, this volume; R0yne 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 (FeTi0 3 ) with acid to form rutile (Ti0 2 ), 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 (FeTi0 3 ) 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).

Transient Porosity from Fluid—Mineral

Interaction

9

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 Ω: (1) For nucleation to take place a certain critical supersaturation where Ω > 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:

(2) where δ is a shape related factor, σ 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, Τ is temperature and

10

Putnis

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 Γ 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 (σ) (and hence low solubility) and large molar volume (7) 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: Supers aturation rate oc (Threshold supersaturation)'" or Ra=K(nj"·

(3)

where Κ and m are empirical coefficients, and fìthis the threshold supersaturation.

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.

Transient Porosity from Fluid—Mineral

Interaction

11

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 °C and in some cases as low as -40 °C—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 μπι (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.

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 °C s 1 and the dashed line for a cooling rate of 1 "C.min The data for the lines are from a large number of experiments analyzed by Prupacher (1995).

12

Putnis

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 - 9 5 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 pm 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 oc (Threshold supersaturation)'". Note the wide range of threshold supersaturation depending on the solubility (and hence the surface energy) of the salt, consistent with the expectations f r o m 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 C a C 0 3 by counter-diffusion in gelatin hydrogels with different pore sizes and porosity distribution.

Crystallization 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 Ba 2 + and S 0 4 2 . Counter-diffusion of the components through the gel column eventually supersaturates the fluid and results in precipitation (of B a S 0 4 in this case).

Transient

Porosity from Fluid—Mineral

Interaction

13

Gypsum CaS04.2H20

6.5

7.0

7.5

8.0

8.5

4.5

5.0

6.5

7,0

7.5

Ln (threshold supersaturation, Ω ίή ) 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 t¡ is given by: t,=l/JV

(4)

where / is the nucleation rate and Vis 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 t¡. 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 torpore sizes of 0.1 pm and 1 pm; 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

Putnis

14

14000 12000 c ~ 10000 Π3

5

8000

^

6000

01 Q.

4000 2000 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 μπι and 1 μπι (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 Na 2 C0 3 , 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 μπι, 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

Transient Porosity from Fluid—Mineral

Interaction

15

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 CaC0 3 precipitation, Stack et al. (2014) introduced supersaturated solutions into a manufactured controlled-pore glass (CPG) which contained both macropores ( - 3 2 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. Miirmann 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; DeYoreo 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.

16

Putnis DISCUSSION

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.

Transient Porosity from Fluid—Mineral

Interaction

17

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 Miirmann 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.

18

Putnis

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 R0yne 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

Transient Porosity from Fluid—Mineral

Interaction

19

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 Τ, Ρ 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 Hâkon Austrheim and Bj0rn 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.

REFERENCES Bj0rlykke Κ (2014) Relationships between depositional environments, burial history and rock properties. Some principal aspects of diagenetic process in sedimentary basins. Sed Geol 3 0 1 : 1 - 1 4 Boles J R (1982) Active albitization of plagioclase, Gulf Coast Territory. Am J Sci 2 8 2 : 1 6 5 - 1 8 0 Bons PD, Elburg MA, Gomez-Rivas E (2012) A review of the formation of tectonic veins and their microstructures. J Struct Geol 4 3 : 3 3 - 6 2 Buss H, Sak P, Webb R M Brantley S (2008) Weathering of the Rio Blanco quartz diorite, Luquillo Mountains, Puerto Rico: Coupling oxidation, dissolution and fracturing. Geochim Cosmochim Acta 7 2 : 4 4 8 8 ^ - 5 0 7 Dalby KN, Anderson AJ, Mariano AN, Gordeon RA, Mayanovic RA, Wirth R (2010) An investigation of cathodoluminescence in albite from the A-type Georgeville granite, Nova Scotia. Lithos 1 1 4 : 8 6 - 9 4 Derluyn H, Moonen P, Carmeliet J (2014) Deformation and damage due to drying induced crystallization in porous limestone. J Mech Phys Solids 6 3 : 2 4 2 - 2 5 5 Desarnaud J, Derluyn H, Carmeliet J, Bonn D, Shahidzadeh Ν (2014) Metastability limit for the nucleation of NaCl crystals in confinement. J Phys Chem Lett 5 : 8 9 0 - 8 9 5 De Yoreo J (2013) Crystal nucleation: more than one pathway. Nature Mater 12:284—285

20

Putnis

Doehne E (2002) Salt weathering: a selective review. In: Siegesmund S, Weiss T, Vollbrecht A (eds) Natural Stones, Weathering Phenomena, Conservation Strategies and Case Studies. Geol Soc Spec Pubi 205:4356 Dohmen R, Milke R (2010) Diffusion in polycrystalline materials: grain boundaries, mathematical models and experimental data. Rev Mineral Geochem 72:921-970 Duft D, Leisner Τ (2004) Laboratory evidence for volume-dominated nucleation of ice in supercooled water microdroplets. Atmos Phys Chem 4:1997-2000 Earle ME, Kuhn Τ, Khalizov AF, Sloan JJ (2010) Volume nucleation rates for homogeneous freezing in supercooled water droplets: results from a combined experimental and modelling approach. Atmos Chem Phys 10:7945-7961 Edmunds WM, Bath AH (1976) Centrifuge extraction and chemical analysis of interstitial waters. Environ Sci Technol 10:467^172 Elburg MA, Bons PD, Foden J, Passchier CW (2002) The origin of fibrous veins : constraints from geochemistry. Geol Soc Spec Pubi 200:103-118 Emmanuel S, Ague JJ (2009) Modeling the impact of nano-pores on mineralization in sedimentary rocks. Water Resour Res 45:W04406 Emmanuel S, Berkowitz Β (2007) Effects of pore-size controlled solubility on reactive transport in heterogeneous rock. Geophys Res Lett 34:L06404 Emmanuel S, Ague JJ, Walderhaug O (2010) Interfacial energy effects and the evolution of pore size distributions during quartz precipitation in sandstone. Geochim Cosmochim Acta 74:3539-3552 Engvik AK, Putnis A, FitzGerald JD, Austrheim H (2008) Albitization of granitic rocks: the mechanism of replacement of oligoclase by albite. Can Mineral 46:1401-1415 Espinoza-Marzal RM, Scherer G (2010) Advances in understanding damage by salt crystallization. Accounts Chem Res 43:897-905 Fisher DM, Brantley SL (1992) Models of quartz overgrowth and vein formation: deformation and episodic fluid flow in an ancient subduction zone. J Geophys Res 97:20043-20061 Fisher DM, Brantley SL, Everett M, Dzvonik J (1995) Cyclic fluid flow through a regionally extensive fracture network within the Kodiak accretionary prism. J Geophys Res 100:12881-12894 Fletcher RC, Buss H, Brantley SL (2006) A spheroidal weathering model coupling porewater chemistry to soil thicknesses during steady-state denudation. Earth Planet Sci Lett 244:444—457 Folk RL (1955) Note on the significance of "turbid" feldpsars. Am Mineral 40:356-357 Gebauer D, Cölfen H (2011) Prenucleation clusters and non-classical nucleation. Nano Today 6:564—584 Grassmann O, Neder RB, Putnis A, Löbmann Ρ (2003) Biomimetic control of crystal assembly by growth in an organic hydrogel network. Am Mineral 88:647-652 Grader J-P, Dysthe DK and Renard F (2013) The role of pressure solution creep in the ductility of the Earth's upper crust. Adv Geophys 54:47-179 Guthrie GD, Veblen DR (1991) Turbid alkali feldspars from the Isle of Skye, northwest Scotland. Contrib Mineral Petrol 108:298-304 Harlov DE, Hansen EC, Bigler C (1998) Petrologie evidence for K-feldspar metasomatism in granulite facies rocks. Chem Geol 151:373-386 Henisch HK (1988) Crystals in Gels and Liesegang Rings. Cambridge University Press, Cambridge Hövelmann J, Putnis A, Geisler Τ, Schmidt Β, Golla-Schindler U (2010) The replacement of plagioclase feldspars by albite: observations from hydrothermal experiments. Contrib Mineral Petrol 159:43-59 Iyer K, Jamtveit B, Mathiesen J, Malthe-S0renssen A, Feder J (2008) Reaction-assisted hierarchical fracturing during serpentinization. Earth Planet Sci Lett 267:503-516 Jamtveit B, Yardley Β (1997) Fluid Flow and Transport in Rocks. Chapman & Hall, London Jamtveit B, Putnis CV, Malthe-S0renssen A (2009) Reaction induced fracturing during replacement processes. Contrib Mineral Petrol 157:127-133 Janssen A, Putnis A, Geisler Τ, Putnis CV (2010) The experimental replacement of ilmenite by rutile in HCl solutions. Mineral Mag 74:633-644 Kashchiev D (2000) Nucleation: Basic Theory with Applications. Butterworth-Heinemann, Oxford Kashchiev D, van Rosmalen GM (2003) Review: Nucleation in solutions revisited. Crystal Res Technol 38: 555-574 Kasioptas A, Geisler Τ, Perdikouri C, Trepmann C, Gussone N, Putnis A (2011) Polycrystalline apatite synthesised by hydrothermal replacement of calcium carbonates. Geochim Cosmochim Acta 75:34863500 Kubota N, Karasawa H, Kawakami Τ (1978) On estimation of critical supercooling from waiting times measured at constant supercooling. J Chem Eng Jpn 11:290-295 Kubota N, Kawakami T, Tadaki Τ (1986) Calculation of supercooling temperature for primary nucleation of potassium nitrate from aqueous solution by the two-kind active site model. J Cryst Growth 74:259-274

Transient Porosity from Fluid—Mineral

Interaction

21

Kubota N, Fujisawa Y, Tadaki Τ (1988) Effect of volume on the supercooling temperature for primary nucleation of potassium nitrate from aqueous solution. J Cryst Growth 8 9 : 5 4 5 - 5 5 2 Laporte D, Watson E B (1991) Direct observation of near-equilibrium pore geometry in synthetic quartzites at 6 0 0 - 8 0 0 0 °C and 2 - 1 0 . 5 kbar. J Geol 9 9 : 8 7 3 - 8 7 8 Lee VW, Mackwell SJ, Brantley S L (1991) The effect of fluid chemistry on wetting textures in novaculite. J Geophys Res 9 6 : 1 0 0 2 3 - 1 0 0 3 7 Lee JI, Lee Y I (1998) Feldspar albitization in Cretaceous non-marine mudrocks, Gyeongsang Basin, Korea. Sediment 4 5 : 7 4 5 - 7 5 4 Lee MR, Parsons I (1997) Dislocation formation and albitization in alkali feldspars from the Shap Granite. Am Mineral 8 2 : 5 5 7 - 5 7 0 Lee JRI, Han TY-J, Willey TM, Nielsen MH, Klivamsky L M , Liu Y, Chung S, Terminello LJ, van Buuren T, de Yoreo J J (2013) Cooperative reorganization of mineral and template during directed nucleation of calcium carbonate. J Phys Chem C 1 1 7 : 1 1 0 7 6 - 1 1 0 8 5 Martin RF, Bowden Ρ (1981) Peraluminous granites produced by rock-fluid interaction in the Ririwai nonorogenic Ting-complex, Nigeria: mineralogical evidence. Can Mineral 1 9 : 6 5 - 8 2 McDonald J E (1953) Homogeneous nucleation of supercooled water drops. J Meteorol 1 0 : 4 1 6 - 4 3 3 Merino E (1975) Diagenesis in Tertiary sandstones from Kettleman North Dome, California: I. Diagenetic mineralogy. J Sed Petrol 4 5 : 3 2 0 - 3 3 6 Milke R, Neusser G, Kolzer Κ, Wunder Β (2013) Very little water is necessary to make a dry solid silicate system wet. Geology 4 1 : 2 4 7 - 2 5 0 Montgomery CW, Brace W F (1975) Micropores in plagioclase. Contrib Mineral Petrol 5 2 : 1 7 - 2 8 Mukai H, Austrheim H, Putnis CV, Putnis A (2014) Textural evolution of plagioclase feldspar across a shear zone: implications for deformation mechanism and rock strength. J Petrol 5 5 : 1 4 5 7 - 1 4 7 7 Miirmann M, Kühn M, Pape H, Clauser C (2013) Numerical simulation of pore size dependent precipitation in geothermal reservoirs. Energy Procedía 4 0 : 1 0 7 - 1 1 6 Navarre-Sitchler A, Brantley SL, Rother G (2015) How porosity increases during incipient weathering of crystalline silicate rocks. Rev Mineral Geochem 8 0 : 3 3 1 - 3 5 4 Niederberger M, Cölfen H (2006) Oriented attachment and mesocrystals: Non-classical crystallization mechanisms based on nanoparticle assembly. Phys Chem Chem Phys 8 : 3 2 7 1 - 3 2 8 7 Niedermeier DRD, Putnis A, Geisler Τ, Golla-Schindler U, Putnis C V (2009) The mechanism of cation and oxygen isotope exchange in alkali feldspars under hydrothermal conditions. Contrib Mineral Petrol 157:65-76 Nielsen A E (1967) Nucleation in aqueous solution. J Phys Chem Solids suppl. 1 : 4 1 9 - 4 2 6 Nijland TG, Harlov DE, Andersen Τ (2014) The Bamble Sector, south Norway: A review. Geoscience Frontiers 5:635-658 Nindiyasari F, Fernández-Díaz L, Griesshaber E, Astilleros JM, Sánchez-Pastor Ν, Schmahl W W (2014) Influence of gelatin hydrogel porosity on the crystallization of C a C 0 3 . Cryst Growth Des 1 4 : 1 5 3 1 - 1 5 4 2 Norberg N, Neusser G, Wirth R, Harlov D (2011) Microstructural evolution during experimental albitization of K-rich alkali feldspar. Contrib Mineral Petrol 1 6 2 : 5 3 1 - 5 4 6 Norberg N, Harlov D, Neusser G, Wirth R, Rhede D, Morales L (2013) Experimental development of patch perthite from synthetic cryptoperthite: Microstructural evolution and chemical re-equilibration. Am Mineral 9 8 : 1 4 2 9 - 1 4 4 1 Nyvlt J (1968) Kinetics of nucleation in solutions. J Crystal Growth 3 : 3 7 7 - 3 8 3 Nyvlt J, Söhnel O, Matuchova M, Broul M (1985) The Kinetics of Industrial Crystallization. Elsevier, Amsterdam Ozawa H (1997) Thermodynamics of frost heaving: A thermodynamic proposition for dynamic phenomena. Phys Rev E 5 6 : 2 8 1 1 - 2 8 1 6 Parnell J (ed) (1998) Dating and duration of fluid flow and fluid-rock interaction. Geol Soc Spec Pubi 144 Parsons I (1978) Feldspars and fluid in cooling plutons. Mineral Mag 4 2 : 1 - 1 7 Parsons I, Lee M R (2009) Mutual replacement reactions in alkali feldpsars I: microtextures and mechanisms. Contrib Mineral Petrol 1 5 7 : 6 4 1 - 6 6 1 Perdikouri C, Kasioptas A, Geisler Τ, Schmidt BC, Putnis A (2011) Experimental study of the aragonite to calcite transition in aqueous solution. Geochim Cosmochim Acta 7 5 : 6 2 1 1 - 6 2 2 4 Perez R, Boles A R (2005) An empirically derived kinetic model for albitization of detrital plagioclase. Am J Sci 3 0 5 : 3 1 2 - 3 4 3 Pliimper O, Putnis A (2009) The complex hydrothermal history of granitic rocks: multiple feldspar replacement reactions under subsolidus conditions. J Petrol 5 0 : 9 6 7 - 9 8 7 Pollok K, Putnis CV, Putnis A (2011) Mineral replacement reactions in solid solution-aqueous solution systems: Volume changes, reactions paths and end-points using the example of model salt systems. Am J Sci 3 1 1 : 2 1 1 - 2 3 6

22

Putnis

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 C a S 0 4 . 2 H 2 0 . 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, ρ 139-167 Putnis A, John Τ (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 Κ (2004) A mechanism of mineral replacement: isotope tracing in the model system KC1K B r - H 2 0 . 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-SchindlerU, 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 Ρ, Grader 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 Κ (2004) Sodium NMR relaxation in porous materials. J Mag Res 167:25-30 Rijniers LA, Huinink HP, Pel L, Kopinga Κ (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 R0yne A, Jamtveit Β (2015) Pore-scale controls on reaction. Rev Mineral Geochem 80:25-44 R0yne A, Jamtveit B, Malthe-S0renssenA (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 254:329348 Shaw RA, Durant AJ, Mi Y (2005) Heterogeneous surface crystallization observed in undercooled water. J Phys Chem Β 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

Transient Porosity from Fluid—Mineral

Interaction

23

Steiger M (2005b) Crystal growth in porous materials II: Influence of crystal size on crystallization pressure. J Cryst Growth 2 8 2 : 4 7 0 - 4 8 1 Tabazadeh A, Djikaev Y S , Reiss H (2002) Surface crystallization of supercooled water in clouds. PNAS 99:15873-15878 Velbel M A (1993) Formation of protective surface layers during silicate-mineral weathering under wellleached, oxidizing conditions. Am Mineral 7 8 : 4 0 5 - 4 1 4 von Bargen Ν, Waff HS (1986) Permeabilities, interfacial areas and curvatures of partially molten systems: Results of numerical computations of equilibrium microstructures. J Geophys Res 9 1 : 9 2 6 1 - 9 2 7 6 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 4 2 : 1 0 7 0 - 1 0 8 1 Waldron KA, Parsons I, Brown W L (1993) Solution-redeposition and the orthoclase-microcline transformation: evidence from granulites and relevance to l s O exchange. Mineral Mag 5 7 : 6 8 7 - 6 9 5 Walker F D L (1990) Ion microprobe study of intragrain micropermeability in alkali feldspars. Contrib Mineral Petrol 1 0 6 : 1 2 4 - 1 2 8 Walker FDL, Lee MR, Parsons I (1995) Micropores and micropermeable texture in alkali feldspars: geochemical and geophysical implications. Mineral Mag 5 9 : 5 0 5 - 5 3 4 Walther JV, Wood B J (1984) Rate and mechanism in prograde metamorphism. Contrib Mineral Petrol 8 8 : 2 4 6 259 Watson E B , Brennan J M (1987) Fluids in the lithosphère, 1. Experimentally-determined wetting characteristics of C 0 2 - H 2 0 fluids and their implications for fluid transport, host-rock physical properties, and fluid inclusion formation. Earth Planet Sci Lett 8 5 : 4 9 7 - 5 1 5 Wellman HW, Wilson AT (1965) Salt weathering: neglected geological erosive agent in coastal and arid environments. Nature 2 0 5 : 1 0 9 7 - 1 0 9 8 Wheeler J (2014) Dramatic effects of stress on metamorphic reactions. Geology 4 2 : 6 4 7 - 6 5 0 Wood B J , Walther J V (1983) Rates of hydrothermal reactions. Science 2 2 2 : 4 1 3 - 4 1 5 Worden RH, Walker FDL, Parsons I, Brown W L (1990) Development of microporosity, diffusion channels and deuteric coarsening in perthitic alkali feldspars. Contrib Mineral Petrol 1 0 4 : 5 0 7 - 5 1 5 Yokoyama T, Nakashima S, Murakami T, Mercury L, Kirino Y (2011) Solute concentration in porous rhyolite as evaluated by sequential centrifugation. Appi Geochem 2 6 : 1 5 2 4 - 1 5 3 4

2

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

Pore-Scale Controls on Reaction-Driven Fracturing Anja R0yne and Bj0rn Jamtveit Physics of Geological Processes (PGP) Departments of Physics and Geoscience University of Osi o P.O. Box 1048 Blindem Ν-0316 Oslo Norway [email protected] bjom.jamh'eit@

geo. uio.no

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. 1529-6466/15/0080-0002505.00 ifani . ' i ' l j ' i a

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

© 2015, R0yne, J a m t v e i t .

This work is licensed under the Creative C o m m o n s Attribution-NonCommercial-NoDerivs 3.0 License.

26

Rey ne & Jamtveit FIELD-SCALE OBSERVATIONS

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; R0yne 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 (R0yne 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 R0yne et al. 2008].

Pore-Scale Controls on Reaction-Driven

Fracturing

27

Figure 2. Block of an andesitic intrusion subdivided into smaller domains (I-III) by internal fracturing. Domain III was further subdivided into IIIA and HIB, 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 (Φ), 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 (Γ oc 1/Φ), and the amount of the mobile reactant that the rock can contain (Θ calcite (Perdikouri et al. 2011), leucite —> analcime (Putnis et al. 2007; Jamtveit et al. 2009),

32

Rey ne & Jamtveit

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 (CaAl 2 Si 3 O 10 -3H 2 O) by tobermorite (Ca 5 Si 6 0 1 7 (0H) 2 -5H 2 0) after 3 days in a 2M-NaOH solution at 200 °C (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.]

Pore-Scale Controls on Reaction-Driven

Fracturing

33

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 (Pliimper 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; R0yne 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 subcriticai crack growth (Atkinson 1987). R0yne 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.

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 Β 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, R0yne A, Svensen Η (2011) Porosity evolution and crystallization-driven fragmentation during weathering of andesite. Journal of Geophysical Research-Solid Earth, Vol. 116, B12201, Fig. 5.]

Pore-Scale Controls on Reaction-Driven

Fracturing

35

f* 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, σ, 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 R0yne A, Meakin P. Malthe-S0renssen 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; R0yne et al. 2011a).

Rey ne & Jamtveit

36

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 NaCI was introduced in a micrometrie 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 airwater 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.

NaCI

NaCI

Figure 10. Halite crystals (dark grey), trapping a saturated solution of NaCI (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, γ, 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.

Pore-Scale Controls on Reaction-Driven

Fracturing

37

The intrinsic surface energy of a material, γ 0 , 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, γ 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 t/(°°) = γ 13 + γ 23 . Upon bringing the solids into dry contact, the energy becomes U(0) = γ 12 .

F i g u r e 11. S u r f a c e energies a n d s u r f a c e f o r c e s . L e f t : solid 1 a n d solid 2 s e p a r a t e d b y a t h i c k film of liquid (3), £ / ( » ) = γ 1 3 + γ 2 3 · R i g h t : s o l i d - s o l i d c o n t a c t , w i t h e n e r g y C / ( 0 ) = γ 1 2 . M i d d l e : solids s e p a r a t e d b y a thin film, U(h) = γ 1 3 + γ 2 3 +P(h).

As the film thickness decreases continually towards zero, the energy does not j u m p discontinuously from γ 13 + γ 23 to γ 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) = γ 13 + γ 23 + P(h), where P(°°) = 0 and P{0) = γ 12 - (γ 13 + γ 23 ). W h e n the surface separation changes, the change in interfacial energy gives rise to a measurable force per unit area:

F _ dUQi) _ dPQi) A ~

dh

~

(1)

dh

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, A h , 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

38

Rßyne

&

Jamtveit

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

u

=

(2)

\2nh2 '

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, E D L , between planar surfaces depends exponentially on the separation distance. For symmetric interfaces, monovalent electrolytes and surface potentials below about 25 mV, f/ EDL can be found in terms of the surface potential ψ 0 or surface charge σ as (Israelachvili 2011): Uedl =

XD

=

ε0ε

ν/λ».

(3)

Here, λ η is the Debye screening length, which is a function of the ionic strength of the electrolyte. In pure water, λ η is close to 1 μπι, 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

Pore-Scale Controls on Reaction-Driven

Fracturing

39

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 C 0 2 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

40

Rey ne & Jamtveit

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 micrometrie 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

Pore-Scale Controls on Reaction-Driven

Fracturing

41

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 μπι 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.

42

Rßyne & 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.

REFERENCES Alcantar N, Israelachvili J, Boles J (2003) Forces and ionic transport between mica surfaces: implications for pressure solution. Geochim Cosmochim Acta 67:1289-1304, doi:10.1016/s0016-7037(02)01270-x Anzalone A, Boles J, Greene G, Young K, Israelachvili J, Alcantar Ν (2006) Confined fluids and their role in pressure solution. Chem Geol 230:220-231, doi:10.1016/j.chemgeo.2006.02.027 Atkinson BK (ed) (1987) Fracture Mechanics of Rock. Academic Press, London Baimpos T, Shrestha BR, Raman S, Valtiner M (2014) Effect of interfacial ion structuring on range and magnitude of electric double layer, hydration, and adhesive interactions between mica surfaces in 0.05-3 M Li(+) and Cs(+) electrolyte solutions. Langmuir 30:4322-4332, doi:10.1021/la500288w Becker GF, Day AL (1905) The linear force of growing crystals. Proc Washington Acad Sci 8:283-288 Bergström L (1997) Hamaker constants of inorganic materials. Adv Colloid Interfac 70:125-169, doi:10.1016/ s0001-8686(97)00003-l Boström M, Williams D, Ninham Β (2001) Specific Ion Effects: Why DLVO theory fails for biology and colloid systems. Phys Rev Lett 87, doi:10.1103/PhysRevLett.87.168103 Bouzid M, Mercury L, Lassin A, Matray JM (2011) Salt precipitation and trapped liquid cavitation in micrometrie capillary tubes. J Colloid Interface Sci 360:768-776, doi:10.1016/j.jcis.2011.04.095 Bresme F, Cámara LG (2006) Computer simulation studies of crystallization under confinement conditions. Chem Geol 230:197-206, doi:10.1016/j.chemgeo.2006.02.025 Butt H-J, Cappella Β, Kappl M (2005) Force measurements with the atomic force microscope: Technique, interpretation and applications. Surf Sci Rep 59:1-152, doi:10.1016/j.surfrep.2005.08.003 Chapman RW, Greenfield M A (1949) Spheroidal weathering of igneous rocks. Am J Sci 247:407^-29 Clarke DR, Lawn BR, Roach DH (1986) The Role of Surface Forces in Fracture. In: Fracture Mechanics of Ceramics. Vol 8. Bradt RC, Evans AG, Hasselman DPH, Lange FF, (eds). Plenum Publishing Corporation, ρ 341-350 Correns CW (1949) Growth and dissolution of crystals under linear pressure. Discuss Faraday Soc 5:267-271 Dash JG, Rempel AW, Wettlaufer JS (2006) The physics of premelted ice and its geophysical consequences. Rev Mod Phys 78:695-741 de Gennes P-G, Brochard-Wyart F, Quéré D (2003) Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York Derjaguin B, Landau L (1941) Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes. Prog Surf Sci 43:30-59, doi:10.1016/00796816(93)90013-1 Donaldson SH, Jr., R0yne A, Kristiansen K, Rapp MV, Das S, Gebbie MA, Lee DW, Stock P, Valtiner M, Israelachvili J (2014) Developing a general interaction potential for hydrophobic and hydrophilic interactions. Langmuir, doi:10.1021/la502115g Duan C, Majumdar A (2010) Anomalous ion transport in 2-nm hydrophilic nanochannels. Nat Nanotechnol 5:848-852, doi:10.1038/nnano.2010.233 Dunkel KG, Putnis A (2014) Replacement and ion exchange reactions of scolecite in a high pH aqueous solution. Eur J Mineral 26:61-69 Emmanuel S, Berkowitz Β (2007) Effects of pore-size controlled solubility on reactive transport in heterogeneous rock. Geophys Res Lett 34:L06404, doi:10.1029/2006gl028962 Espinosa Marzal RM, Scherer GW (2008) Crystallization of sodium sulfate salts in limestone. Environ Geol 56:605-621, doi:10.1007/s00254-008-1441-7 Espinosa-Marzal RM, DrobekT, Balmer T, Heuberger MP (2012) Hydrated-ion ordering in electrical double layers. Phys Chem Chem Phys 14:6085-6093, doi:10.1039/c2cp40255f Flatt RJ, Steiger M, Scherer GW (2007) A commented translation of the paper by C.W. Correns and W. Steinborn on crystallization pressure. Environ Geol 52:221-237

Pore-Scale Controls on Reaction-Driven

Fracturing

43

Flatt RJ, Caruso F, Sanchez AM, Scherer G W (2014) Chemo-mechanics of salt damage in stone. Nat Commun 5:4823, doi:10.1038/ncomms5823 Fletcher RC, Brantley S L (2010) Reduction of bedrock blocks as corestones in the weathering profile: observations and model. Am J Sci 3 1 0 : 1 3 1 - 1 6 4 Fletcher RC, Merino E (2001) Mineral growth in rocks: Kinetic-rheological models of replacement, vein formation, and syntectonic crystallization. Geochim Cosmochim Acta 6 5 : 3 7 3 3 - 3 7 4 8 Fletcher RC, Buss HL, Brantley S L (2006) A spheroidal weathering model coupling porewater chemistry to soil thickness during steady-state denundation. Earth Planet Sci Lett 2 4 4 : 4 4 4 ^ - 5 7 Gibbs J W (1876) On the equilibrium of heterogeneous substances. Trans Connecticut Acad 3 : 1 0 8 - 2 4 8 Heidug W K (1995) Intergranular solid-fluid phase-transformations under stress - the effect of surface forces. J Geophys Res-Solid Earth 1 0 0 : 5 9 3 1 - 5 9 4 0 Horn RG, Smith DT, Haller W (1989) Surface forces and viscosity of water measured between silica sheets. Chem Phys Lett 162:404^108 Houben ME, ten Hove A, Peach CJ, Spiers CJ (2013) Crack healing in rocksalt via diffusion in adsorbed aqueous films: Microphysical modelling versus experiments. Phys Chem Earth, Parts A/B/C 6 4 : 9 5 - 1 0 4 , doi:10.1016/j.pce.2012.10.001 Hövelmann J, Austrheim H, Jamtveit Β (2012) Microstructure and porosity evolution during experimental carbonation of a natural peridotite. Chem Geol 3 3 4 : 2 5 4 - 2 6 5 , doi:10.1016/j.chemgeo.2012.10.025 Israelachvili J (2011) Intermolecular and Surface Forces. Academic Press, Amsterdam. Israelachvili JN, Adams G E (1978) Measurement of forces between two mica surfaces in aqueous-electrolyte solutions in range 0 - 1 0 0 nm. J Chem Soc, Faraday Trans I 7 4 : 9 7 5 - 1 0 0 1 Jamtveit B, Hammer 0 (2012) Sculpting of rocks by reactive fluids. Geochem Perspect 1 : 3 4 0 - 4 8 0 Jamtveit B, Malthe-S0renssen A, Kostenko O (2008) Reaction enhanced permeability during retrogressive metamorphism. Earth Planet Sci Lett 2 6 7 : 6 2 0 - 6 2 7 , doi:10.1016/j.epsl.2007.12.016 Jamtveit B, Putnis CV, Malthe-Sorenssen A (2009) Reaction induced fracturing during replacement processes. Contrib Mineral Petrol 1 5 7 : 1 2 7 - 1 3 3 Jamtveit B, Kobchenko M, Austrheim H, Malthe-Sorenssen A, R0yne A, Svensen H (2011) Porosity evolution and crystallization-driven fragmentation during weathering of andesite. J Geophys Res-Solid Earth 116:B12204, doi:10.1029/2011JB008649 Kelemen PB, Hirth G (2012) Force of crystallization during retrograde metamorphism: Olivine hydration and carbonation. Earth Planet Sci Lett 3 4 5 : 8 1 - 8 9 Lesko S, Lesniewska E, Nonat A, Mutin J-C, Goudonnet J-P (2001) Investigation by atomic force microscopy of forces at the origin of cement cohesion. Ultramicroscopy 8 6 : 1 1 - 2 1 Malvoisin B , Brunet F (2014) Water diffusion-transport in a synthetic dunite: Consequences for oceanic peridotite serpentinization. Earth Planet Sci Lett 4 0 3 : 2 6 3 - 2 7 2 , doi:10.1016/j.epsl.2014.07.004 Malvoisin Β, Brunet F, Carlut J, Roumejon S, Cannat M (2012) Serpentinization of oceanic peridotites: 2. Kinetics and processes of San Carlos olivine hydrothermal alteration. J Geophys Res-Solid Earth 117, doi: 10.1029/201 l j b 0 0 8 8 4 2 Megawati M, Hiorth A, Madland M V (2013) The impact of surface charge on the mechanical behavior of highporosity chalk. Rock Mech Rock Eng 4 6 : 1 0 7 3 - 1 0 9 0 Molnar P, Anderson R S , Anderson SP (2007) Tectonics, fracturing of rock, and erosion. J Geophys Res 112:F03014, doi:10.1029/2005jf000433 Murton J B , Peterson R, Ozouf J-C (2006) Bedrock fracture by ice segregation in cold regions. Science 3 1 4 : 1 1 2 7 - 1 1 2 9 , doi:10.1126/science.l 132127 Perdikouri C, Kasioptas A, Geisler Τ, Schmidt BC, Putnis A (2011) Experimental study of the aragonite to calcite transition in aqueous solution. Geochim Cosmochim Acta 7 5 : 6 2 1 1 - 6 2 2 4 Pliimper O, R0yne A, Magraso Sola A, Jamtveit Β (2012) The interface-scale mechanism of reaction-induced fracturing during upper mantle serpentinization. Geology 4 0 : 1 1 0 3 - 1 1 0 6 Prieto M, Cubillas Ρ, Fernandez-Gonzalez A (2003) Uptake of dissolved Cd by biogenic and abiogenic aragonite: a comparison with sorption onto calcite. Geochim Cosmochim Acta 6 7 : 3 8 5 7 - 3 8 6 9 Putnis A (2002) Mineral replacement reactions: from macroscopic observations to microscopic mechanisms. Mineral Mag 6 6 : 6 8 9 - 7 0 8 , doi:10.1180/0026461026650056 Putnis A (2015) Transient porosity resulting from fluid-mineral interaction and its consequences. Rev Mineral Geochem 8 0 : 1 - 2 3 Putnis CV, Geisler Τ, Schmid-Beurmann Ρ, Stephan Τ, Giampaolo C (2007) An experimental study of the replacement of leucite by analcime. Am Mineral 9 2 : 1 9 - 2 6 , doi: 10.2138/am.2007.2249 Renard F, Beauprêtre S, Voisin C, Zigone D, Candela Τ, Dysthe DK, Grader J-P (2012) Strength evolution of a reactive frictional interface is controlled by the dynamics of contacts and chemical effects. Earth Planet Sci Lett 3 4 1 - 3 4 4 : 2 0 - 3 4 , doi:10.1016/j.epsl.2012.04.048 Ricci M, Spijker Ρ, Stellacci F, Molinari JF, Voitchovsky Κ (2013) Direct visualization of single ions in the Stern layer of calcite. Langmuir 2 9 : 2 2 0 7 - 2 2 1 6 , doi:10.1021/la3044736

44

Rßyne & Jamtveit

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 R0yne A, Jamtveit B, Mathiesen J, Malthe-S0renssen 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 R0yne A, Bisschop J, Dysthe DK (2011a) Experimental investigation of surface energy and subcriticai crack growth in calcite. J Geophys Res 116:B04204, doi:10.1029/2010jb008033 R0yne A, Meakin P, Malthe-S0renssen A, Jamtveit B, Dysthe DK (201 lb) 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 Β (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 C 0 2 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 M M (1992) The influence of fluids on grain contact stiffness and frame moduli in sedimentary rocks. Geophysics 57:1571-1582 Ulven OI, Jamtveit B, Malthe-S0renssen A (2014a) Reaction driven fracturing of porous rocks. J Geophys Res:doi:10.1002/2014JB011102 Ulven OI, Storheim H, Austrheim H, Malthe-S0renssen A (2014b) Fracture initiation during volume increasing reactions in rocks and applications for C 0 2 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 13:18381848, doi:10.1021/cg301429g Verwey EJW, Overbeek JTG (1948) Theory of The Stability of Lyophobic Colloids. Elsevier, Amsterdam Walder J, Hallet Β (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 38:427^152

3

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 The Hebrew University Edmond J. Safra Givat Ram, Jerusalem,

Sciences of Jerusalem Campus 91904 Israel

simonem @ cc.huji.ac. it

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 metamorphismtoo, 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). 1529-6466/15/0080-0003505.00 I 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 'H are more readily accessed (e.g., 23 Na, 19F, 31P, 13C, 2 H) 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 'H resonance frequency of Vo = 2 MHz, corresponding to a magnetic field strength of BQ = 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 Τ 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 T 2 (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 fluidsolid 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 T 2 measurement is directly proportional to porosity, and the decay rate

Characterization

and Analysis

of Porosity and Pore

Structures

79

is related to the pore sizes and the fluid type and its viscosity in the pore space. Short T 2 times generally indicate small pores with large surface-to-volume ratios and low permeability. Conversely, longer T 2 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 T 2 time distributions. Figure 12A shows an example of T 2 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 T 2 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 T 2 distribution as follows: — =p - +—, T2 ν T2,B

(18)

where ρ is the relaxivity, A/Vis the ratio of the area to volume of a pore (an uncertain value for fractal pore surfaces), and T 2 B is the relaxation of the bulk fluid. For spherical pores with radius r, A/Vx 3Ir. Therefore, the NMR T2 distribution can be converted to pore-size r distribution by suitable selection of the relaxivity p. Typically for natural porous matrices the magnitude of ρ ranges between 1 and 10 μπι 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

80

Anoviiz

4.5 4 ν 3.5 3 2.5 "ο. 2 τ)Λ.

(37)

For isotropic media this becomes: /(β)=4π(η2)|γ(Γ)^ί^Γ. o Qr

(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

Characterization and Analysis of Porosity and Pore Structures

93

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 o f small angle neuton scattering Radlinski (2006) noted that "for a wide range of substances, the S A S 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 χ IO 10 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 o f 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 o f 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 o f those minerals relative to vacuum (empty pores), and dolomite. If we consider a calcite-dolomite-pore system with all three phases o f 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 xlO 1 0 cm"2, and contrast values are xlO 2 0 cm" 2 .

X-ray

Neutron

X-ray

Neutron

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

Mineral

ratio

Neutron

Contrast with Vacuum

Contrast with Dolomite

X-ray (10 keV)

Scattering Length Densities

94

Anovitz & Cole

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: Σ

β

=Σρλ,

(39)

k

Pk

Pw

(4°)

—

(41)

M

M=Yßkmk,

(42)

k

where mk is the atomic mass of element k, nk is the number of atoms of element k per scattering unit, iVAV is Avogadro's number, ρ is the mass density, p„ is the number density of the scattering units, ρ ι is the number density of atoms of element k, and 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 m 2 or 10"24 cm 2 . The macroscopic bound atom scattering cross section is not, however, equivalent to the macroscopic total scattering cross section E 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: Σ„ =?dZ?f f d Q - ^ , J0 J dQdFf 4π

(43)

U S Z U C

where (d 2 E/dQdE f ) is he macroscopic differential cross section for scattering into a solid angle

Characterization

and Analysis

of Porosity and Pore

Structures

95

Ω and energy Ef. E s may be significantly different from Eb, especially for hydrous materials. The macroscopic total cross section Σ τ is then defined as the sum of the macroscopic total scattering cross section E s and the macroscopic absorption cross section Σ Α . As with all transmission techniques absorption from this source may be described using the well-known Beer-Lambert law as: /=/0ex ρ(-ΣΑί),

(44)

where Σ Α 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., Ή ) 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 isotopie 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 α to the beam as: (45) 'Τ

96

Anovitz & Cole

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

Στ

In 1 _

: Es

(46)

which has no solution if: (47) (http://www.ncnr.nist.g0v/instruments/dcs/dcs_usersguide/h0w_thick_sample/#cr0ss_ 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-ß to higher-ß 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 σ τ Γ, where σ τ is the

Characterization

and Analysis

of Porosity and Pore

Structures

97

Figure 16. Modeled multiple scattering. Ratio of second scattering event Β to initial scattering event A as a function of scattering cross section times thickness. For most SANS experiments on geologic materials the value of θ 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 σ τ Γ is almost independent of scattering angle, and that the value of σ τ Γ 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 σ τ Γΐη(σ τ Γ) 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 μπι 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 μπι 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 Helma™ "banjo" or "lollipop" cells, in which multiple scattering, especially at the USANS scale, can be significant.

98

Anovitz & Cole

Figure 17. Two methods of mounting samples for (U)SANS analysis. The images on the left shows a sample ground to 150 μκα, 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)

Characterization

and Analysis

of Porosity and Pore

Structures

99

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 (Dl 1 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.

oveabieArea Detector In

Energy-selected X-rays or Neutrons

Sample

Momentum

Q = (4πΜ)είη(θ/2)

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 = (4π/λ) sin(0). 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 dEc / dQ, 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

100

Anoviiz & Cole

3.750 3.500 3.250 3.000 2.750 2.500 2.250 2.000 1.750 1.500 1.250 1.000 0.750 0.500 0.250 0.000 BO 256

100

120

140

160

ISO 192

¡(-pixel 1000,(

240

900.01

220 200

800.0(

ISO

700.0(

160

600. CK

"φ jg 140

500.01 400, Of

100 so

300,0(

60

200 .α

40

lOO.Of

20

0.000

0 80

100 x:-pixel

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

Characterization

and Analysis

of Porosity and Pore

Structures

101

NG7 SANS at NINS/NCNR) and US ANS (e.g., 0.0003 nm"1 to ~ 0.1 nm"1 for BT5 US ANS 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 (3-range for the SANS instrument employs a set of biconcave MgFo 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 A) 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 D i l 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 VI6 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 Jülich 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 (3-range to be measured in one setting. The incident wavelength

102

Anoviiz

Neutron Guides

& Cole

Focusing Lenses and Prisms RF Flipper Converging Apertures and Slits

guide Selector

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 (Δλ/λ = 12%), high resolution graphite monochromator (Δλ/λ = 2%), or low resolution filtered beam covering 0.4 nm < λ < 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 3 He 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-ζ? 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 Amai 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. Borne-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

Characterization

and Analysis

of Porosity and Pore

Structures

103

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 US ANS instruments at NC-NR (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-ζ? 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 (C-1.3 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 NC-NR, 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 A, and the triple bounce geometry dramatically

thermal neutrons sapphire filter graphite filter

.pre-monochromator beam η

monochromator

sample detector

isolation table transmission detector

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

104

Anoviiz

& Cole

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 S ANS/S AXS 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 xlO"5 A"1) in the horizontal direction, and much wider ones (approximately 0.022 A"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 Λ-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 ^-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

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

Characterization

and Analysis

of Porosity and Pore

Structures

105

APS permits direct measurement of the asymmetry of the scattering pattern as low-ß (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 (a) and varying Q, as in standard USAS experiments, or by fixing Q and varying a . A two dimensional pattern (obtained point-by-point) can be obtained by varying both Q and a . Beam-collimation reduces the intensity, however, which limits the practical range of Q to 10"4 A"1 < Q < 0.1 A"1. This integration is given as (Barker et al. 2005; Hammouda 2008): (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

106

Anoviiz & Cole

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 (a¡), and the reflections are detected at both a final angle (oif) and out-of-plane angle (2Θ) 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)

Characterization

and Analysis

of Porosity and Pore

Structures

107

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 θι in the first field, which is reversed in the second so that d0 = 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 y(r) of the density distribution of the sample p(r), where ζ is the spin-echo length (in μπι). 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

108

&

Anoviiz

Fourier

Cole

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

" Hankel

*.·.. • - . j .

"

J

o **

"" * * * * _

a

"

· Ί

•

*

i

* ι

e I È15

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

t

»

~

a

•

i···-,.

Il

L

?

*

...

4 Ç £ spin Ktw length [um]

7

H

9

ΙΠ

Figure 28. A vortex lattice diffraction pattern for YBajCujOy.g (YBCO) taken at 2 Κ 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, ρ is bisected by ax. 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 YBa 2 Cu 3 0 7 . Physical Review Letters, Vol. 102, 097001 used with permission of the American Physical Society].

Characterization

and Analysis

of Porosity and Pore

Structures

109

crystal diffraction patterns. The magnetic diffraction pattern shown in Figure 28 shows the inverse-space lattice pattern for YBaoCujOy.g (YBCO) taken at 2 Κ 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,

110

A n o v i i z&

Cole

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; Alien 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 H 2 0 (neutron sld = -0.56 χ IO10 cm"2) and D 2 0 (neutron sld = 6.392 χ IO10 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 ZnCl 2 . They found that the surface area of the carbons increased as the concentration of ZnCl 2 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 ^-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. 14

35

Solvent Scattering Length Density (IO 10 cm

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 χ IO10 cm 2 , similar to ZnCl2, ZnO or metallic Zn. Data at Q = 0.1 has a minimum at 5.92 χ 1010 cm 2 , comparable to amorphous carbon. Replotted after Littrell et al. (2002).

Characterization

and Analysis

of Porosity and Pore

Structures

111

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. 201 lb; 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, nonuniformi tie s 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 ( ^sample+cell

m

sample

^ b l o c k e d beam J

Τsample+cell

0'

(49)

where I is the intensity of the scattering, and Τ 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 (dE/dQ(g), 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 l/cm 2, s is then: V(Q)= ^ , d Q v ' ε/ 0 Δα AtT

(50)

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

112

Anovitz & Cole

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(/(2)) plotted as a function of log(2) is sometimes referred to as a Porod plot. The data show are a combination of data from three sources: SANS, US ANS, and calculations from backscattered electron images taken on a scanning electron microscope (BSE/SEM). The approximate (3-ranges o v e r 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-2 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 04Wil7b, 100 °C, 8 weeks, Anovitz et al. 2015). The central part of the curve has a power-law slope of -3.538 = 0.99925). Ranges shown for data obtained from SEM/BSE imagery, USANS and SANS measurements are approximate as the data overlap.

Characterization

and Analysis

of Porosity and Pore

Structures

113

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) = IQQ'X + Β where Β 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 - λ" (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*.

(51)

For a non-fractal, three-dimensional object Ds = 2 (as in the surface area of a sphere A = 4p;~) 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 = kr0",

(52) 3

where DM = 3 for a non fractal object, as in the volume or a sphere (V = 4/IM ) 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 (3-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-ζ? as well, cf. Naudon et al. 1994). At yet lower (3-values

114

Anovitz & Cole

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): (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 A for r, from the cube root of the calcite unit cell volume (367.85Â). Values for r 0 and Ds were taken from the SANS/USANS data. Wang et al. (2013) modified this approach slightly, again selecting the crossover length (called 21 there) between the regions of surface and mass fractal scattering for r 0 , and 7 A, the size of an N 2 -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-ß. This is determined by the intensity of the incoherent background. In some cases Bragg peaks, typically due to the large ¿-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: (54) where / b is the background and S is the specific surface area (surface area per unit volume, units of cm 2 /cm 3 or 1/cm). Figure 34 shows one way to obtain the background value (Glatter and Kratky 1982). The slope of a plot of ß 4 / ( ö ) as a function of β 4 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 =

2n(DrfS

(55)

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

Characterization and Analysis of Porosity and Pore Structures

115

Figure 34. Porod transform of data for 0 4 W i l 7 b , 100 °C, 8 weeks (Anovitz et al. 2015a)

— = π—.

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 = J ö 2 ^ ( ö ) d g = 2 π 2 (Δρ) 2 φ(ΐ - φ), ο

(57)

where Υ is called the Porod invariant, and φ 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): /(e)=/0exp

( (fK

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

(58)

116

Anoviiz

& Cole

Figure 35. Kratky transform of data for 04Wil7b, 100 °C, 8 weeks, background subtracted (Anovitz et al. 2015a)

f

ln(/(ß)) = ln(/0)-

ς> 2 ιξ Λ

(59)

As this is the equation of a straight line the values of / 0 and Rg are easily determined from a plot of ln(/(2)) 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: LL R: -12

(60)

EL 2

but at intermediate ^-values (61)

7(ß)=|« p

and the appropriate plot is In(QI(Q)) vs. Q2. For lamellar objects of thickness T, the appropriate equation becomes: /(2)=^ex p

(

Q2R;

λ

(62)

where R:=-

(63) 12

and the plot is In{Q 2 I{Q)) vs. Q2. However, as noted above a Guinier region (flat at low Q in

Characterization

and Analysis

of Porosity and Pore

Structures

117

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 λ" 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 (ö 4 /(ö) 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 2-range do not fall on a single fractal slope but, rather, are separated into several regions.

Figure 36. Semi-Porod transform of data for 04Wil7b, 100 °C, 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.

118

Anovitz & Cole

/ \ 50 nm

D

Àm E

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 (PRINS AS, 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

Characterization

and Analysis

io* 10 10® io T io 6

of Porosity and Pore

Structures

119

Particle diameter [Al ία*

VWume of seatterers = 0.015642 Mean diameter » 23987 Mode diameter = 68.791 Median diameter = 6568.6 Range of dia meters from 3 to 2e+05 [A]

io 5

ë 10* 10J

i

101 10v 10 1 10* io' 3 in 3

ΓΑ

¡Volume of s c a t t e r s = 0.014782 Mean diameter « 24050 Mode diameter « 60.711 Median diameter = 5920.4 jRajige of diameters from 3 ta2e+Q5

[A' -

.••^.-j.-.^^y.j.aas.ΆΆΧ. Volume of seatterers = 0.015B23 ¡Mean diameter = 24221 ¡Mode diameter = 38.089 Median diameter = 6236.4 ¡Range of diameters from 3 t o 2 e + 0 5 [A]

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 χ IO20 cm 4 . Top) TNNLS, error multiplier =1.6; middle) Maximum entropy, error multiplier =1.6, sky bkgd = 3.12 χ IO 8; bottom) regularization, error multiplier =1.65. (Anovitz, unpb.)

120

Anovitz & Cole

analysis of SAS data required modeling of the scattering results. For a dilute solution the intensity of a SAS pattern is described as: 7

(ßLte

=K

]\F{Q,r)\v\r)NP{r)àr, o

(64)

where ΙΔρΙ2 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, Ν 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: /(β)

(65)

or, combining the the various constants into a single empirical variable: /(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 04Wil7b, 100 °C, 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

Characterization

and Analysis

of Porosity and Pore

Structures

121

Figure 39. Backscattered electron image of sample 04Wil7b, 100 °C, 8 weeks from Anovitz et al. (2015a). Image is 5.3 mm across.

* *

^

. c • «α. ·

.ν

•

-

h

-

*

w - y

*

l·

/

μ

Figure 40. Binary image (pores black, quartz white) of sample 04Wil7b, 100 °C, 8 weeks from Anovitz et al. (2015a). Image is 5.3 mm across.

122

Anovitz & Cole

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), Muñoz et al. (2003), Udupa and Saha (2003), Cardoso and Corte-Real (2005), Cremers et al. (2007), Ilea and Whelan (2011), and Schlüter 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 °C 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).

Characterization

and Analysis

of Porosity and Pore

Structures

123

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 °C (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 fl\x) where: (67) where V¡ is the volume occupied by phase I, and V¡ is the volume occupied by the other phase (rock). As summarized by Anovitz et al. (2013a, who used/(x) instead of fl)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) and (69)

124

Anovitz & Cole

where the brackets are a volume average over x, r is a lag distance and r = Irl for an isotropic material, and φ is the pore fraction. Berryman (1985) showed that:

s2(0) = S^b

(70)

so that the zero intercept of the second correlation function is the porosity, and at the limit of large r l i m S ^ r ) = φ2.

(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:

dS2(0) _ s dr 4

(72)

and the effective pore size is given as 4φ(1 - φ)

(73)

K- =

which is the intersection of a line tangent to the S2(r) curve at the zero intercept with S2(r) = φ2. 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) = ( [ / < ρ ) ( χ ) - φ ρ ] [ / < ρ ) ( χ + r) - φ ρ ] ) = S'fir) - φ 2 ,

(74)

and X(r)

Φ Ρ (ΐ-Φ Ρ )

=

m

ΦΛ

where φΒ is the volume fraction of grain phase, φρ + φΒ = 1, and c(r) isa normalized version of X(r). In a statistically homogeneous two-phase system (isotropic or anisotropic), X(r) has limiting values of X(0) = φρφΒ and, X(co) =0 and bounds in the range of -πύη(φ ρ ,φ 2 ) < X(r) < φρφ£ . Normalizing X(r) by φρφΒ 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

/ ( β ) = 4π(Δρ) 2 (φ - Ψ2)|γ2£(γ) S l n ( g r ) d r , o Qr «0 =

,rA ' J e 2π (Δρ) (φ-φ

2

/ ( 0 ^ d g , ßr

125

(76)

(77)

where (Δρ) 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 WienerKhinchin theorem (Weiner 1930, 1964; Khintchine 1934; Goodman 1985; Champeney 1987; Chatfield 1989; Hannan 1990; Couch 2001; Ricker 2003; Iniewski 2007). This shows that: FK(/) = FFT[z(r)],

(78)

S(f)=FR(f)F¿(f),

(79)

c(r) = I F F T [ S ( / ) ] .

(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 ID 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 φ and at large r is φ2, 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 «-point correlation functions, of which the 1- and

126

Anovitz & Cole

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; Heising 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 r^ and r 2 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: Si{r1r2)=f(x)f(x

+ r1)f(x

+ r2),

(81)

where an oriented triangle is defined by the two vectors r^ and r 2 , 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 = Irl, so IrJ — X2 — X\ and lr2l — X3 — Χι : S3 (•riri ) =•S3 {η,r2,:u12) =,S3 (.r2,,:u12 ),

(82)

where: (83) Therefore we have the three variables that define a triangle, the side lengths angle θ between them. This function has the following properties: 12

and r 2 , and the

(84)

Characterization

and Analysis

lim

of Porosity and Pore

Structures

5 3 (Γ 1 ,Γ 2 ,Μ 1 2 )=Φ 3 ;

127

(85)

If there is no long-range order then:

and the three-point correlation function is bounded by: { R I' R 2' U N) - MIN[S 2 ( | t ¡ | ) , S 2 ( | r 2 | ) , S 2 ( | r 3 | ) ] < m a x [ 5 ' 2 (|r 1 |^,5' 2 ( | r 2 | ) , S 2 ( | r 3 | ) ] < φ

(87)

where: |r 3 | =·^/|Γ 1 | 2 +|Γ 2 | 2 -2|Γ 2 ||Γ 1 |Μ 1 2 .

(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, «) with k the length of the largest side, can be constructed as follows (note that Berryman used (/, m, «) rather than (k, m, «), 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 < kJ2 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 < {kl2,0),

(89)

0 < η < k. (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, «) as S

Jv k ,

m

,n)=^ ' Ν

or, as suggested by both Berryman (1988) and Velasquez et al. (2010), adopt scheme and randomly drop a given triangle on the image Ν number of times. of their images is unclear, Velasquez et al. (2010) found that 100,000 drops Berryman (1988) discusses interpolation schemes to be used with a dataset of triangle as suggested in Figure 41.

(91) a Monte Carlo While the size was sufficient. this type, for a

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 η (or r b r 2 and Θ) and S3(k, m, ή). While a full analysis seems optimally suited to parallelization, simpler schemes, such as that

128

Anoviiz

Ο

&

Cole

Ί

•

·

•

· 2

m

•

·

* •

• •

•

·

·

•

·

*

•

·

·

•m

m

·

*

a

· *

· *

m=k/2

m=k/2

8

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 = [A/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 ρ is an integer, and plotting the resultant S3(k, m, ri)* as a function oip, can provide more easily plotted and analyzed results.

Characterization

and Analysis

of Porosity and Pore

Structures

129

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 r^ and r 2 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; Falchetta 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; Heising 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 Lebrón 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 ε, 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(w), the number of "on" boxes, as a function of log(e), 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

130

Anovitz

& Cole

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 Holder exponents (a) 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 ε. We then define the total number of boxes of a given size as «(ε), and the "measure" of the box as μ ε , 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 μ log ε

(94)

(95)

μ=ε In the limit, as ε—>0 α =lim

log μ log ε

(96)

where α is the Holder exponent. Note that a collrse and α 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 ε, we can define Νε (acollrse) as the sum of the number of boxes with a given value of a collrse , and define the multifractal distribution (singularity exponent) as: / . ( 0

Pi

log

Pi

1=1

logs

(103)

In general where ρ < q.

(104)

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(l) 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, τ(q) can be analogized to the Free Energy of the system, %(q, s) to the partition function, and q~l 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 t(q) curve as: τ(9) = (q-l)D(q).

(105)

For q > 0, D(q) is dominated by large μ(/) and therefore by areas with a high density of the measure. For q < 0, D(q) is dominated by small μ(/) 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/(a) vs. a and the Rényi spectrum D(q) vs. q are not independent. They are related through τ(q) as: a(q) =

dq

(106)

132

Anovitz & Cole

and by the Legendre transformation

f a = qa(q)-% q),

(107)

so that: 0(0) = -τ(0) = / ( α ( 0 ) ) = /(α)] where/(α(0)) is the value oif(a(q))

(108)

at the maximum of the/(a(g)) vs a ( q ) curve,

dq

?->+» dq

(109)

(110)

At values of a(q) > a (0), q < 0, and for values of α ( q ) < a(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 stratigraphie 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

Characterization and Analysis of Porosity and Pore Structures

133

for fractal dimensions is again box counting. A box of size ε 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, ε, and grid orientation, g, the box-size specific lacunarity (λ) is calculated from the mean, σ, and standard deviation, μ, of the number of pixels "turned on" in each box as: (HI) The overall lacunarity of the image, Λ, is then the average of the single box size lacunarities (λ) 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 (ε) is given as: Ν = Αε°,

(112)

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

> =

—

(113)

π

exp^J

averaging over the (G) available orientations: ι

=

Σ

Α G

(114)

£

then the prefactor laccunarity (PA) can be defined from the prefactor as (Mandelbrot 1977):

PA =

Σ°=!

X " ^ G

1

—.

J

(115)

Figures 42^-5 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 S W 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

134

Anovitz & Cole

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 04Wil7bPRL, described in Anovitz LM, Cole DR, Rother G, Allard LF Jr, Jackson A, Littreil 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 (LTltra) 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.

Characterization

and Analysis

of Porosity and Pore

Structures

135

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)): andD(O) - 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, Littreil 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 L F Jr. Jackson A, Littrell KC (2013a) Diagenetic changes in macro-to nano-scale porosity in the St. Peter Sandstone: an (LTltra) small angle neutron scattering and backscattered electron imaging analysis. Geochimica et Cosmochimica Acta, Vol. 102, p. 280-305, used with permission from Elsevier.]

136

Anovitz & Cole

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 (ε), and for each box of a given size the percentage of "on" pixels, the occupation fraction (O(w)), 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:

_ΣίΡ(η)ρ(η)

(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

Characterization

and Analysis

of Porosity and Pore

Structures

137

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 (!) 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 (/), 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/gparticle 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 dr t around point ri ... and the center of another sphere in a volume Arq around point rqx, 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

Anovitz & Cole

138

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 C0 2 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, A

lllitïc shale

.

.

·' . γ - . " " " - : ' ^Y · ' ^

ν

ι U ) SANS (irci) ; Crtffi-r en ii »I lotJ ι po rvvo Ivm e d iî 11 i^u t ion Ml CP (blue]: Differential connect ed pore wolumc distribution

y* . I

SEMMB

Diameter (μιη)

Carbonate mudstone

Β

Glauconitic mudstone

C

Figure 46 a-c. Comparison of MIP (blue) and NS (red) data from the Eau Clair formation mudstone for three different lithologie 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 C 0 2 sequestration. Environmental Geoscience Vol. 21, p. 39-57 AAPG© [2014], reprinted by permission of the AAPG whose permission is required for further use.]

Characterization

and Analysis

of Porosity and Pore

Structures

139

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 A, and a broad, larger-scale regime centered around 10-20 μπι 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 μπι. 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-ζ? "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 μπι, which may have a shoulder around 115 μπι. The carbonate mudstone has a narrower peak at 30 μπι 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 pm 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 pm, 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)

140

Anovitz & Cole

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 (221300 μπι 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 μπι 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; Möllns 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 C 0 2 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

Characterization

and Analysis

of Porosity and Pore

Structures

141

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, U m 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.]

142

Anovitz & Cole

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 C0 2 . 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 Onderwij s 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

Characterization

and Analysis

of Porosity and Pore

Structures

143

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), Vitally 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).

REFERENCES Abiyev R, Kilic KI (2010) An efficient fractal measure for image texture recognition. 2009 Fifth International Conference On Soft Computing, Computing With Words and Perceptions. In: Sys Anal, Dec Control 161-164 Abramoff MD, Magalhaes PJ, Ram SJ (2004) Image processing with ImageJ. Biophoton Int 11:36-42 Ade Ρ, Aghanim Ν, Armitage-Caplan C, Arnaud M, Ashdown M, Atrio-Barandela F, Aumont J, Baccigalupi C, Banday AJ, and Barreiro RB et al. (2014) Constraints on primordial non-Gaussianity. Astron Astrophys 571:A24 Adesida AG, Akkutlu IY, Resasco DE, Rai CS (2011) Characterization of kerogen pore size distribution of Barnett shale using DFT analysis and Monte Carlo simulations. Soc Petrol Eng SPE 147397, doi. org/10.2118/147397-MS Adler PM, Jacquin CG, Quiblier JA (1990) Flow in simulated porous media. Int J Multiphas Flow 16:691-712 Agamalian M, Wignall GD, Triolo R (1997) Optimization of a Bonse-Hart ultra-small angle neutron scattering facility by eliminating the rocking curve wings. J Appi Crystallogr 30:345-352 Aizawa K, Tomimitsu H (1995) Design and use of a double crystal diffractometer for very small angle neutron scattering at JRR-3M. Physica Β 213:884-886 Akcasu AZ, Summerfield GC, Jahshan SN, Han CC, Kim CY, Yu Η (1980) Measurement of single chain neutron-scattering in concentrated polymer-solutions. J Polym Sci, Polym Phys Ed 18:863-869 Alefeld B, Hayes C, Mezei F, Richter D, Springer Τ (1997) High-resolution focusing SANS with a toroidal neutron mirror. Physica Β 234-236:1052-1054 Alefeld Β, Dohmen L, Richter D, Briickel Th (2000a) Space technology from X-ray telescopes for focusing SANS and reflectometry. Physica Β 276-278:52-54 Alefeld Β, Dohmen L, Richter D, Briickel Th (2000b) X-ray space technology for focusing small-angle neutron scattering and neutron reflectometry. Physica Β 283:330-332 Alexander, M Hallett, FR (1999) Small-angle light scattering: instrumental design and application to particle sizing. Appi Optics 38:4158-4163 Allen AJ (1991) Time-resolved phenomena in cements, clays and porous rocks. J Appi Crystallogr 24:624—634 Allen AJ, Thomas JJ, Jennings HM (2007) Composition and density of nanoscale calcium-silicate-hydrate in cement. Nat Mater 6:311-316 Allen D, Crary S, Freedmann R, Andreani M, Klopf W, Badry R, Flaum C, Kenyon W, Kleinberg R, Gossenberg Ρ, Horkowitz J, Logan D, Singer J, White J (1997) How to use borehole nuclear magnetic resonance. Oilfield Review, Summer 1997:34-57 Alnaimi SM, Mitchell J, Strange JH, Webber JBW(2004) Binary liquid mixtures in porous solids. J Chem Phys 120:2075-2077 Ambrose RJ, Hartman RC, Diaz-Campos M, Akkutlu IY, Sondergeld CH (2010) New pore-scale considerations for shale gas-in-place calculations: SPE Unconventional Gas Conference, Pittsburgh, Pennsylvania, February 23-25, 2010, SPE Paper 131772 Andersson R, Bouwman WG, Luding S, de Schepper IM (2008a) Stress, strain, and bulk microstructure in a cohesive powder. Phys Rev Β 77:051303 Andersson R, van Heijkamp LF, de Schepper IM, Bouwman WG (2008b) Analysis of spin-echo small-angle neutron scattering measurements. J Appi Crystallogr 41:869-885 Andreani C, Merlo V, Ricci M A (1989) A procedure for multiple scattering corrections in a neutron incoherent inelastic scattering experiment. Nucl Instrum Methods Phys Res Β 36:216-221 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. Geochim Cosmochim Acta 73 7 3 0 3 - 7 3 2 4 Anovitz LM, Rother G, Cole DR (2011) Characterization of rock pore features in geothermal systems using small angle neutron scattering (SANS). Proc 36th Workshop on Geothermal Res Eng Sgp-Tr-191:571-582

144

Anovitz & Cole

Anovitz LM, Cole DR, Rother G, Allard LF Jr, Jackson A, Littreil 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. Geochim Cosmochim Acta 102:280-305 Anovitz LM, Wang H-W, Cole DR, Sheets J, Rother G, Faulder DD, Walters M (2013b) Analysis of Multiscale Porosity at the Coso Geothermal Field. Proc. 38 th Workshop On Geothermal Res. Eng., SPG-TR-198 Anovitz LM, Mamontov E, Ishai PB, Kolesnikov AI (2013c) Anisotropic dynamics of water ultra-confined in macroscopically oriented channels of single-crystal beryl:A multi-frequency analysis. Phys Rev E 88:052306 Anovitz LM, Cole DR, Swift A, Sheets J, Elston H, Welch S, Chipera SJ, Littreil KC, Mildner DFR, Wasbrough MJ (2014) Multiscale (nano to mm) porosity in the Eagle Ford Shale: Changes as a function of maturity. Unconventional Resources Technology Conference Proceedings, Am Assoc Pet Geol Publication URTeC:1923519 Anovitz LM, Cole DR, Jackson AJ, Rother G, Littreil KC, Allard LF, Pollington AD, Wesolowski DJ (2015a) Effect of quartz overgrowth precipitation on the multiscale porosity of sandstone: a (U)SANS and imaging analysis. Geochim Cosmochim Acta 153:199-222 Anovitz LM, Cole DR, Sheets J, Swift A, Elston H, Welch S, Chipera SJ, Littreil KC, Mildner DFR, Wasbrough MJ (2015b) Effects of maturation on multiscale (nm to mm) porosity in the Eagle Ford Shale. Interpretations. (In submission) Archie GE (1942) The electrical resistivity log as an aid in determining some reservoir characteristics. Petrol Trans AIME 146:54-62 Archie GE (1947) Electrical resistivity an aid in core-analysis interpretation. Am Assoc Petrol Geol Bull 31:350-366 Archie GE (1950) Introduction to petrophysics of reservoir rocks. Am Assoc Petrol Geol Bull 34:943-961 Archie GE (1952) Classification of carbonate reservoir rocks and petrophysical considerations. Am Assoc Petrol Geol Bull 36:278-298 Arnéodo A, Bacry E, Muzy JF (1995) The thermodynamics of fractals revisited with wavelets. Physica A 213:232-275 Arthur M, Cole DR (2014) Unconventional hydrocarbon resources: prospects and problems. Elements 10:257264 Avdeev MV, Tropin TV, Aksenov VL, Rosta L, Garamus VM, Rozhkova NN (2006) Pore structures in shungites as revealed by small-angle neutron scattering. Carbon 44:954-961 Bahadur J, Melnichenko YB, Mastalerz M, Furmann A, Clarkson CR (2014) Hierarchical pore morphology of cretaceous shale:a small-angle neutron scattering and ultrasmall-angle neutron scattering study. Energy Fuels 28:6336-6344 Bale HD Schmidt PW (1984) Small-angle X-ray-scattering investigation of submicroscopic porosity with fractal properties. Phys Rev Lett 53:596-599 Barbera G, Barone G, Crupi V, Longo F, Malsano G, Majolino D, Mazzoleni P, Raneri S, Teixeira J, Valentina V (2014) A multi-technique approach for the determination of the porous structure of building stone. Eur J Mineral 26:189-198 Barker JG (2006) Performance review of various advanced optical designs for Very Small-Angle Neutron Scattering (VSANS) instrumentation, Report, NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, U S A Barker JG, Glinka CJ, Moyer JJ, Kim MH, Drews AR, Agamalian M (2005) Design and performance of a thermal-neutron double-crystal diffractometer for USANS at NIST. J Appi Crystallogr 38:1004-1011 Barrett EP, Joyner LG, Halenda PP (1951) The determination of pore volume and area distributions in porous substances. I. Computations from nitrogen isotherms, J Am Chem Soc 73:373-380 Bates FS, Berney CV, Cohen RE, Wignall GD (1983) Single-chain scattering in heterogeneous block copolymers. Polymer 24:519-524 Baumbart DJ, Fry JN (1991) Fourier spectra of 3-dimensional data. Astrophys J 375:25-34 Bazilevskaya E, Lebedeva M, Pavich M, Rother G, Parkinson DY, Cole D, Brantley SL (2013) Where fast weathering creates thinregolith and slow weathering creates thick regolith. Earth Surf Proc Land 38:847858 Beaucage G (1995) Approximations leading to a unified exponential/power-law approach to small-angle scattering, J Appi Crystallogr 28:717-728 Beaucage G (1996) Small-angle scattering from polymeric mass fractals of arbitrary mass-fractal dimension. J Appi Crystallogr 29:134-146 Beaucage G, Ulibarri TA, Black EP, Schaefer DW (1995) Multiple size scale structures in silica-siloxane composites studied by small-angle scattering. In: Hybrid Organic-Inorganic Composites. Mark, JE, Lee, C Y-C, Bianconi, PA (eds) ACS Symposium Series 585, 9:97-111 Beaucage G, Kammler HK, Pratsinis SE (2004) Particle size distributions from small-angle scattering using global scattering functions. J Appi Crystallogr 37:523-535

Characterization

and Analysis

of Porosity and Pore

Structures

145

Beaudry P, Petersen HU, Grunberg-Manago M, Jacrot Β (1976) A neutron study of the 30 S-ribosome subunit and of the 30 S-IF3 complex. Biochem Biophys Res Comm 72:391-397 Beckingham LE, Peters CA, U m W, Jones KW, Lindquist WB (2013) 2D and 3D imaging resolution trade-offs in quantifying pore throats for prediction of permeability. Adv Water Resour 62:1-12 Bellmann D, Klatt M, Kampmann R, Wagner R (1997) Improvement of the double crystal diffractometer at the Geesthacht Neutron Facility (GeNF) by means of perfect channel-cut silicon crystals. Physica Β 241-243:71-73 Bentz DP (1997) Three-dimensional computer simulation of Portland cement hydration and microstructure development. J Am Ceram Soc 80:3-21 Beran MJ (1968) Statistical Continuum Theories. Interscience, New York Berge PA, Berryman JG, Blair SC, Pena C (1998) Scalar properties of transversely isotropic tuff from images of orthogonal cross sections. In: Imaging Technologies: Techniques and Applications in Civil Engineering. Second International Conference. Frost JD, McNeil S (eds) American Society of Civil Engineers ρ 185— 196 Bergman DJ, Stroud D (1992) Physical-properties of macroscopically inhomogeneous media. Solid State Physics: Advances in Research and Applications, Book Series: Solid State Physics—Advances In Research and Applications 46:147-269 Berk NF, Hardman-Rhyne KA (1988) Analysis of SAS data dominated by multiple scattering. J Appi Cry stallogr 21:645-651 Berryman JG (1985) Measurement of spatial correlation functions using image processing techniques. J Appi Phys 57:2374-2384 Berryman JG (1987) Relationship between specific surface-area and spatial correlation-functions for anisotropic porous-media. J Math Phys 28:244—245 Berryman JG (1988) Interpolating and integrating 3-point correlation-functions on a lattice. J Comp Phys 75:86-102 Berryman JG, Blair SC (1986) Use of digital image analysis to estimate fluid permeability of porous materials Application of two-point correlation functions. J Appi Phys 60:1930-1938 Berryman JG, Blair SC (1987) Kozeny-Carman relations and image processing methods for estimating Darcy's constant. J Appi Phys 62:2221-2228 Berryman JG, Milton GW (1988) Microgeometry of random composites and porous media. J Phys D: Appi Phys 21:87-94 Bershadskii A (1998) Multifractal specific heat. Physica A 253:23-37 Bird N, Cruz Diaz M, Saarn A, Tarquis AM (2006) Fractal and multifractal analysis of pore-scale images of soil. J Hydrol 322:211-219 Bitteiii M, Campbell GS, Flury M (1999) Characterization of particle-size distribution in soils with a fragmentation model. Soil Sci Soc Am J 63:782-788 Blair SC, Berge PA, Berryman JG (1996) Using two-point correlation functions to characterize microgeometry and estimate permeabilities of sandstones and porous glass. J Geophys Res-Solid Earth 101 (B9):2035920375 Blech ΙΑ, Averbach BL (1965) Multiple scattering of neutrons in vanadium and copper. Phys Rev A 137:1113— 1116 Block A, von Bloh W, Schellenhuber HJ (1990) Efficient box-counting determination of generalized fractal dimensions. Phys Rev A 42:1869-1874 Block A, von Bloh W, Klenke T, Schellnhuber HJ (1991) Multifractal analysis of the microdistribution of elements in sedimentary structures using images from scanning electron microscopy and energy dispersive X-ray spectrometry. J Geophys Res Β 96:16223-16230 Bonse U, Hart M (1965) Tailless x-ray single-crystal reflection curves obtained by multiple reflection. Appi Phys Lett 7:238-240 Borbely S, Heiderich M, Schwahn D, Seidl E (2000) Resolution of the USANS diffractometer at the FRJ-2 research reactor in Jülich. Physica Β 276-278:138-139 Borkovec M, Wu Q, Degovics G, Laggner P, Sticher Η (1993) Surface-area and size distributions of soil particles. Colloids Surf, A 73:65-76 Botet R, Cabane Β (2012) Simple inversion formula for the small-angle X-ray scattering intensity from polydisperse systems of spheres. J Appi Crystallogr 45:406^-16. Bouwman WG, Rekveldt M T (2000) Spin-echo small-angle neutron scattering calculations. Physica Β 2 7 6 278:126-127 Bouwman WG, van Oossanen M., Kraan WH, Uca O, Rekveldt M T (2000) Development of spin-echo smallangle neutron scattering. J Appi Crystallogr 33:767-770 Bouwman WG, Pynn R, Rekveldt M T (2004) Comparison of the performance of SANS and SES ANS. Physica B-Cond Matt 350 (1-3) Supplement 1:E787-E790

146

Anovitz & Cole

Bouwman WG, Kruglov TV, Plomp J, Rekveldt M T (2005) Spin-echo methods for SANS and neutron reflectometry. Physica B-Condens Matter 357:66-72 Bouwman WG, Plomp J, De Haan VO, Kraan WH, van Well AA, Habicht Κ, Keller Τ, Rekveldt M T (2008) Real-space neutron scattering methods. Nucl Instrum Methods Phys Res, Sect A 586:9-14 Brown W F Jr (1955) Solid mixture permittivities. J Chem Phys 23:1514-1517 Brûlet A, Lairez D, Lapp A, Cotton J-P (2007) Improvement of data treatment in small-angle neutron scattering. J Appi Crystallogr 40:165-177 Brunauer S, Emmett PH, Teller E (1938) Adsorption of gases in layers. J Am Chem Soc 60:309-319 Bryan J, Kantzas A, Mai A (2013) Heavy oil characterization using low field NMR. CSPG/CSEG/CWLS GeoConvention 2008, Calgary, Alberta, Canada, May 12-15, 2008. AAPG Search and Discovery Article #9017002013 Bube Κ, Klenke Τ, Freund JA, Feudel U (2007) Statistical measures of distribution patterns of silicon and calcium in marine sedimentary layers. Nonlinear Proc Geoph 14:811-821 Buckley ER (1898) Building and ornamental stones of Wisconsin:Wisconsin Geol Nat History SurvBull 4 Burns JL, Yan Y-D, Jameson GJ, Biggs S (1997) A light scattering study of the fractal aggregation behaviour of a model colloidal system. Langmuir 13:6413-6420 Bushell G, Amai R (2000) Measurement of fractal aggregates of poly disperse particles using small angle light scattering. J Coll Int Sci 221:186-194 Bushell GC, Yan YD, Woodfield D, Raper J, Amai R (2002) On techniques for the measurement of the mass fractal dimension of aggregates. Adv Coll Interfac Sci 95:1-50 Bustin RM, Bustin AMM, Cui X, Ross DJK, Murthy Pathi, VS (2008) Impact of shale properties on pore structure and storage characteristics. SPE Shale Gas production Conference, Fort Worth, Texas, 16-18 November 2008. Soc Petrol Eng SPE paper 119892, doi: 10.2118/119892-MS Cai JC, Martinez FSJ, Martin MA, Perfect E (2014) An introduction to flow and transport in fractal models of porous media: Parti. Fractals 22:402001, doi: 10.1142/S0218348X14020010 Caniego FJ, Martin, A, San José F (2003) Rényi dimensions of soil pore size distribution. Geoderma 112:205216 Cardoso JS, Corte-Real L (2005) Toward a generic evaluation of image segmentation. IEEE Trans Image Process 14:1773-1782 Carpenter JM, Faber J (1978) Design study of a time-of-flight small-angle diffractometer for a pulsed neutron source J Appi Crystallogr 11:464^-65 Cattani C, Pietro G (2013) On the fractal geometry of DNA by the binary image analysis. Bulletin Math Biol 75:1544-1570 Cebula DJ, Charles SW, Popplewell J (1981) The structure of magnetic particles studied by small-angle neutron-scattering. Coll Polym Sci 259:395-397 Chalmers GR, Bustin RM, Power IM (2012) Characterization of gas shale pore systems by porosimetry, pycnometry, surface area, and microscopy/transmission electron microscopy image analyses :examples from the Barnett, Woodford, Haynesville, Marcellus, and Doig units. Am Assoc Petrol Geol Bull 96:1099-1119 Champeney DC (1987) Power Spectra and Wiener's Theorems. A Handbook of Fourier Theorems. Cambridge University Press, Cambridge Chatfield C (1989). The Analysis of Time Series—An Introduction (fourth ed.). Chapman and Hall, London. ρ 94-95. ISBN 0-412-31820-2 Chayes F (1956) Pétrographie Modal Analysis. John Wiley & Sons, New York Chen C, Lau BLT, Gaillard JF, Packman AI (2009) Temporal evolution of pore geometry, fluid flow, and solute transport resulting from colloid deposition. Water Resour Res 45:W06416 Cheng L, Fenter P, Nagy KL, Schlegel ML, Sturchio NC (2001a) Molecular-scale density oscillations in water adjacent to a mica surface. Phys Rev Lett 87:156103 Cheng HD, Jiang XH, Sun Y, Wang JL (2001b) Color image segmentation:advances and prospects. Pattern Recog 34:2259-2281 Cherry R (1997) Magnetic resonance technology and its applications in the oil and gas industry, Part 1. Petrol Eng Inter 70:29-35 Choi SM, Barker JG, Glinka CJ, Cheng YT, Gammel PL (2000) Focusing cold neutrons with multiple biconcave lenses for small-angle neutron scattering. J Appi Crystallogr 33:793-796 Chou C-M, Hong P-D (2004) Nucleation, growth, fractal aggregation, and late-stage coarsening on structural development of polymer Physical Gels. Macromolecules 37:5596-5606 Chou CM, Hong P-D (2008) Light scattering from birefringent sphere and its aggregation. Macromolecules, 41:6147-6153 Chun HC, Gimenez D, Yoon S W (2008) Morphology, lacunarity and entropy of intra-aggregate pores : Aggregate size and soil management effects. Geoderma 146:83-93

Characterization

and Analysis

of Porosity and Pore

Structures

147

Cipelletti L, Weitz DA (1999) Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator. Rev Sci Instruments 70:3214—3221 Clarkson CR, Haghshenas Β (2013) Modeling of supercritical fluid adsorption on organic-rich shales and coal. Soc Petrol Eng SPE 164532 Clarkson CR, Jensen JL, Chipperfield S (2012a) Unconventional gas reservoir evaluation: What do we have to consider. J Nat Gas Sei Eng 8:9-33 Clarkson CR, Wood JM, Burgis SE, Aquini SD, Freeman M (2012b) Nanopore-structure analysis and permeability predictions for a tight gas siltstone reservoir by use of low-pressure adsorption and mercuryintrusion techniques. Soc Petrol Eng, Reservoir Eval Eng SPE 155537:648-661 Clarkson CR, Freeman M, He L, Agamalian M, Melnichenko YB, Mastalerz M, Bustin RM, Radlinski AP, Blach TP (2012c) Characterization of tight gas reservoir pore structure using USANS/SANS and gas adsorption analysis. Fuel 95:371-385 Clarkson CR, Solano N, Bustin RM, Bustin AMM, Chalmers GRL, He L, Melnichenko YB, Radlinski AP, Blach, TP (2013) Pore structure characterization of North American shale gas reservoirs using USANS/ SANS, gas adsorption, and mercury intrusion. Fuel 103(C):606-616 Clemens D (2005) Neutron Trajectory Definition by Multi-Pinhole Grids, International Conference on Neutron Scattering 2005, Sydney, Australia Coates G, Xiao L, Prammer M (1999) NMR logging principles and applications. Halliburton Energy Services, Houston TX et al., www.halliburton.com/public/lp/contents/Books_and_Catalogs/web/Nmr_textbook/ H02308.pdf Coker DA, Torquato S, Dunsmuir JH (1996) Morphology and physical properties of Fontainebleau sandstone via a tomographic analysis. J Geophys Res-Solid Earth 101 (B8):17497-17506 Cole DR, Ok S, Striolo A, Phan A (2013) Hydrocarbon behavior at nanoscale interfaces. Rev Mineral Geochem 75:495-545 Coles P, Jones Β (1991) A lognormal model for the cosmological mass-distribution. MNRAS 248:1-13 Compton AH, Allison SK (1935) X-rays in Theory and Experiment. 2nd ed Princeton:Van Nostrand Connolly J, Bertram W, Barker J, Buckley C, Edwards T, Knott R (2006) Comparison of the structure on the nanoscale of natural oil-bearing and synthetic rocks. J Petrol Sei Eng 53:171-178 Cook GH (1878) Report on the clay deposits of Woodbridge, South Amboy and other places in New Jersey. Geol Surv of New Jersey, Trenton, NJ Copley JRD (1988) The significance of multiple scattering in the interpretation of small-angle neutron scattering experiments. J Appi Crystallogr 21:639-644 Corson PB (1974a) Correlation-functions for predicting properties of heterogeneous materials . 1. Experimental measurement of spatial correlation-functions in multiphase solids. J Appi Phys 45:3159-3164 Corson PB (1974b) Correlation-functions for predicting properties of heterogeneous materials .2. Empirical construction of spatial correlation-functions for 2-phase solids. J Appi Phys 45:3165-3170 Corson PB (1974c) Correlation-functions for predicting properties of heterogeneous materials .3. Effective elastic-moduli of 2-phase solids. J Appi Phys 45:3171-3179 Corson PB (1974d) Correlation-functions for predicting properties of heterogeneous materials .4. Effective thermal-conductivity of 2-phase solids. J Appi Phys 45:3180-3182 Couch LW II (2001) Digital and Analog Communications Systems (sixth ed.). Prentice Hall, New Jersey. 406^109 Crandell LE, Peters CA, U m W, Jones KW, Lindquist WB (2012) Changes in the pore network structure of Hanford sediments after reaction with caustic tank wastes. J Contam Hydrol:89-99 Cremers, D, Rousson, M, Deriche, R (2007) Areview of statistical approaches to level set segmentati on:Integrating color, texture, motion and shape. Int J Comput Vision 72:195-215 Cui verweil ID, Clarke, GP (1986) Validation of the maximum entropy method for interpreting small-angle neutron scattering data. Inst Phys Conf Ser 81:87-96 Curtis ME, Sondergeld CH, Ambrose RJ, Rai CS (2012) Microstructural investigation of gas shales in two and three dimensions using nanometer-scale resolution imaging. Am Assoc Petrol Geol Bull 96:665-677 Dana JD (1875) Notice of the chemical and geological essays of T.S. Hunt. Am J Sci, Ser 3, 9:102-109 Darot M, Reuschle Τ (1999) Direct assessment of Wood's metal wettability on quartz. Pure Appi Geophys 155:119-129 Dathe A, Tarquis AM, Perrier E (2006) Multifractal analysis of the pore- and solid-phases in binary twodimensional images of natural porous structures. Geoderma 134:318-326 Davidowski J, Granada JR, Mayer RE, Cuello GJ, Gillette VH, Bellissent-Funel M-C (1994) Multiplescatering and inelasticity corrections in thermal-neutron scattering experiments on molecular systems. Physica Β 203:116-128 de Melo RHC (2007) Using Fractal Characteristics Such as Fractal Dimension, Laccunarity and Succolarity to Characterize Texture Patterns on Images. Universidade Federal Fluminense, Niterói, Brazil

148

Anovitz & Cole

de Melo RHC, Conci A (2008) Succolarity: Defining a method to calculate this fractal measure. In: Proceedings of IWSSIP 2008:15th International Conference On Systems, Signals and Image Processing. Rozinaj G, Cepko J, Truchly P, Vrabec J, Vojtko J (eds) ρ 291-294 de Melo RHC, Conci A (2013) How succolarity could be used as another fractal measure in image analysis. Telecommun Sys 52:1643-1655 De Yoreo JJ, Waychunas GA, Jun Y-S, Fernandez-Martinez A (2013) In situ investigations of carbonate nucleation on mineral and organic surfaces. Rev Mineral Geochem 77:229-257 Debye P, Beuche AM (1949) Scattering by an inhomogeneous solid. J Appi Phys 20:518-525 Debye P, Anderson Jr., HR, Brumberger Η (1957) Scattering by an inhomogeneous solid. II. The correlation function and its application. J Appi Phys 28:679-683 Deng H, Liu Y, Gai DH, Dikin DA, Putz KW, Chen W, Brinson LC, Burkhart C, Jiang B, Papakonstantopoulos GJ (2012) Utilizing real and statistically reconstructed microstructures for the viscoelastic modeling of polymer nanocomposites. Compos Sci and Tech 72:1725-1732 Denoyel R, Meneses JM, Armatas GS, Rouquerol J, Unger KK, Pomonis PJ (2006) Comparative study of morphometric properties characterizing the complexity of silicate pore networks probed by adsorption of nitrogen and methanol. Langmuir 22:5350-5357 Désert S, Thévenot V, Gabriel A, Permingeat P, Oberdisse J, Brûlet A (2007) A double-supermirror monochromator for neutron instrumentation at Laboratoire Leon Brillouin. J Appi Crystallogr 40:945949 Di Nunzio PE, Martelli S, Ricci Bitti R (2004) Use of Monte Carlo methods in characterizing nanostructured materials by wide- and small-angle x-ray scattering. J Disper Sci Technol. 25:491-501 Dickinson WR (1970) Interpreting detrital modes of graywacke and arkose. J Sed Petrol 40:695-707 Do DD, Do HD (2003) Pore characterization of carbonaceous materials by DFT and GCMC simulations:a review. Adsorp Sei Tech 21:389^123 Dore JC, Benfield RE, Grandjean D, Schmid G, Kroll M, Le Bolloc'h D (2002) Structural studies of mesoporous alumina membranes by small angle x-ray scattering. Stud Surf Sci Catal 144:163-170 Dormann JL, Fiorani D, Tronc E (1997) Magnetic relaxation in fine-particle systems Advances In Chemical Physics 98, Edited by:Prigogine I, Rice SA, Book Series:Adv Chem Phys 98:283-494 Doshi DA, Gibaud A, Goletto V, Lu MC, Gerung H, Ocko Β, Han SM, Brinker CJ (2003) Peering into the self-assembly of surfactant templated thin-film silica mesophases. J Am Chem Soci 125:11646-11655 Dullien FAL (1981) Wood's metal porosimetry and its relation to mercury porosimetry. Powder Tech 29:109116 Dultz S, Behrens H, Simonyan A, Kahr G, Rath T, (2006) Determination of porosity and pore connectivity in feldspars from soils of granite and saprofite. Soil Sci 171:675-694 Ehrlich R, Etris EL (1990) Pore types and core physics. In: Annual Technical Conference preprints. Soc Prof Well Log Analysts, Soc Core Analysts, Chapter-at-Large, Vol 3:SCA-9001. Ehrlich R, Kennedy SK, Crabtree SJ, Cannon RL (1984) Pétrographie image analysis. I. Analysis of reservoir pore complexes. J Sed Petrol 54:1365-1378 Eigmati MM, Zhang H, Bai Β, Fiori R (2011) Submicron-pore characterization of shale gas plays. Soc Petrol Eng SPE- 144050-MS Ellis DV, Singer JM (2008) Well Logging for Earth Scientists (Second ed.). Springer Ellis DV, Case CR, Chiaramonte JM (2003) Porosity from neutron logs I: measurement. Petrophys 44:383-395 Ellis DV, Case CR, Chiaramonte JM (2004) Porosity from neutron logs II: interpretation. Petrophys 45:73-86 Emmanuel S, Anovitz LM, and Day-Stirrat RJ (2015) Effects of coupled chemo-mechanical processes on the evolution of pore-size distributions in geological media. Rev Mineral Geochem 80:45-60 Erdman N, Drenzek Ν (2013) Integrated preparation and imaging techniques for the microstructural and geochemical characterization of shale from scanning electron microscopy. Am Assoc Petrol Geol Memoir 102:7-14 Erdogan ST (2013) Effect of clinker phase distribution within cement particles on properties of a hydrating cement paste. Constr Build Mater 38:941-949 Eskildsen MR, Gammel PL, Isaacs ED, Detlefs C, Mortensen K, Bishop DJ (1998). Compound refractive optics for the imaging and focusing of low-energy neutrons. Nature 391:563-566 Everstz CJG, Mandelbrot BB (1992) Multifractal measures. In: Chaos and Fractals. Peitgen H, Jürgens H, Saupe D (eds) Springer, Berlin, ρ 922-953 Falchette AC, Montepara A, Tebaldi G, Marasteanu M O (2012) Microstructural and rheological investigation of asphalt mixtures containing recycled asphalt materials. Construe and Buil Mater 35:321-329 Falchette AC, Montepara A, Tebaldi G, Marasteanu M O (2013) Microstructural characterization of asphalt mixtures containing recycled asphalt materials. J Mater Civil Eng 25:45-53 Falchette AC, Moon KH, Wistuba M P (2014) Microstructural Analysis and Rheological Modeling of Asphalt Mixtures Containing Recycled Asphalt Materials. Mater 7:6254—6280 Fankuchen I, Jellinek MH (1945) Low angle X-ray scattering. Phys Rev 67:201

Characterization

and Analysis

of Porosity and Pore

Structures

149

Farge M, Kevlahan NKR, Perrier V, Schneider Κ (2004) Turbulence analysis, modeling and computing using wavelets. In: Wavelets in Physics, van den Berg JC (ed) Cambridge University Press, Cambridge ρ 117— 200 Favvas EP, Sapalidis AA, Stefanopoulos KL, Romanos GE, Kanellopoulos NN, Kargiotis ΕΚ, Mitropoulos AC (2009) Characterization of carbonate rocks by combination of scattering, porosimetry and permeability techniques. Micropor Mesopor Mat 120:109-114 Fedorova IS, Schmidt PW (1978) General analytical method for calculating particle-dimension distributions from scattering data. J Appi Crystallogr 11:405^111 Fenter PA (2002) X-ray reflectivity as a probe of mineral-fluid interfaces: A user guide. Applications of Synchrotron Radiation. Rev Mineral Geochem 49:149-220 Fenter P, Sturchio NC (2004) Mineralwater interfacial structures revealed by synchrotron X-ray scattering. Prog Surf Sci 77:171-258 Fenter P, Cheng L, Rihs S, Machesky M, Bedzyk MJ, Sturchio NC (2000a) Electrical double-layer structure at the rutile-water interface as observed in situ with small-period X-ray standing waves. J Coll Interfac Sci 225:154-165 Fenter P, Geissbuhler P, DiMasi E, Srajer G, Sorensen LB, Sturchio NC (2000b) Surface speciation of calcite observed in situ by high-resolution X-ray reflectivity. Geochim Cosmochim Acta 64:1221-1228 Fenter P, Teng H, Geissbuhler P, Hanchar JM, Nagy KL, Sturchio NC (2000c) Atomic-scale structure of the orthoclase (OOl)-water interface measured with high-resolution X-ray reflectivity. Geochim Cosmochim Acta 64:3663-3673 Fenter P, McBride MT, Srajer G, Sturchio NC, Bosbach D (2001) Structure of barite (001)- and (210)-water interfaces. J Phys Chem Β 105:8112-8119 Fernandez-Martinez A, Hu Y, Jun YS, Waychunas GA (2012a) In situ GISAXS studies of carbonate mineral nucleation on mineral surfaces. 2012 Goldschmidt Conference in Montreal, Canada Fernandez-Martinez A, HuY, Lee B, Jun YS, Waychunas GA (2012b) Carbonate mineral nucleation on mineral substrates: in situ GISAXS and microscopy studies. 2012 Spring ACS Meeting, San Diego. Abstracts of Papers of the Am Chem Soc Volume:243 Meeting Abstract:509-COLL Fernandez-Martinez A, Hu Y, Lee B, Jun YS, Waychunas GA (2013) In situ determination of interfacial energies between heterogeneously nucleated C a C 0 3 and quartz substrates :thermodynamics of C 0 2 mineral trapping. Environ Sci Technol 47:102-109 Ferraris G, Merlino, S (2005) Micro- and Mesoporous Mineral Phases. Rev Mineral Geochem 57. Floriano MA, Venezia AM, Deganello G, Svensson EC, Root JH (1994) The structure of pumice by neutron diffraction. J Appi Crystallogr 27:271-277 Folk RL (1974) Petrology of Sedimentary Rocks. Second Edtion. Hemphill Press, Austin, Texas Forster S, Timmann A, Konrad M, Schellbach C, Meyer A, Funari SS, Mulvaney P, Knott R (2005) Scattering curves of ordered mesoscopic materials. J Phys Chem Β 109:1347-1360 Fredrich JT (1999) 3D imaging of porous media using laser scanning confocal microscopy with application to microscale transport processes. Phys Chem Earth, Part Α-Solid Earth Geod 24:551-561 Fricke S, Madio DP, Adolph R, Evans M, Leveridge R (2008) Thermal neutron porosity using pulsed neutron measurements. SPWLA 49 th Annual Logging Sym, May 25-28. SPWLA-2008-L, Society of Petrophysicists and Well-Log Analysts Fu LP, Kilbinger M, Erben Τ, Heymans C, Hildebrandt H, Hoekstra H, Kitching TD, Mellier Y, Miller L, Semboloni E, Simon P, Van Waerbeke L, Coupon J, Harnois-Deraps J, Hudson MJ, Kuijken K, Rowe B, SchrabbackT, Vafaei S, Velander M (2014) CFHTLenS: cosmological constraints from a combination of cosmic shear two-point and three-point correlations. MNRAS 441:2725-2743 Fukushima E, Roeder SBW (1981) Experimental Pulse NMR: A Nuts and Bolts Approach, Addison-Wesley Publishing Company, Advanced Book Program, Reading, Massachusetts, U.S.A Fuller M L (1906) Total amount of free water in the earth's crust: U.S. Geological Survey Water-Supply Paper 160:59-72 Gähler R, Golub R, Habicht Κ, Keller Τ, Felber J (1996) Space-time description of neutron spin echo spectrometry. Physica B, 229:1-17 Gangui A, LuccinF, Matarrese S, MollerachS (1994) The 3-point correlation-function of the cosmic microwave background in inflationary models. Astrophys J 430:447^-57 Garrison JR, Pearn WC, Vonrosenberg DU (1993a) The fractal Menger sponge and Sierpinski carpet as models for reservoir rock pore systems. 2. Image-Analysis of Natural Fractal Reservoir Rocks. In Situ 17:1-53 Garrison JR, Pearn WC, Vonrosenberg DU (1993b) The fractal Menger Sponge and Sierpinski Carpet as models for reservoir rock pore systems. 4. Relationship of fractal dimension to the measured permeability of natural fractal reservoir rocks. In Situ 17:331-362 Gary Ν (1898) Prüfung natürlicher Gesteine in den Betriebsjahren 1895/96 bis 1897/98: Königlichen technischen Versuchsanstalten zu Berlin, Mitt. 5:243-294

150

Anovitz & Cole

Gazzi Ρ (1966) Le arenarie del Flysch sopracretaceo dell'Appennino Modenese: correlazioni con il Flysch di Monghidoro. Mineralogica Petrográfica Acta 12:69-97 Geissbuhler Ρ, Fenter Ρ, DiMasi E, Srajer G, Sorensen LB, Sturchio NC (2004) Three-dimensional structure of the calcite-water interface by surface X-ray scattering. Surface Sci 573:191-203 Gerson AR (2001) The role of fuzzy interfaces in the nucleation, growth and agglomeration of aluminum hydroxide in concentrated caustic solutions. Prog Cryst Growth Ch Mater 187-220 Gignoux M (1955) Stratigraphie Geology. San Francisco: Freeman Gilchrist WA (2008) Compensated neutron log response issues - a tutorial. SPWLA 49 th Annual Logging Sym, May 25-28. SPWLA-2008-L, Society of Petrophysicists and Well-Log Analysts Giordano R, Teixeira J, Triscari M, Wanderlingh U (2007) Porosimetric and particle-size measurements by small-angle neutron scattering. Eur J Min 19:223-228 Glatter O (1977) New method for evaluation of small-angle scattering data. J Appi Crystallogr 10:415^-21 Glatter O (1979) Interpretation of real-space information from small-angle scattering experiments. J Appi Crystallogr 12:166-175 Glatter O (1980) Evaluation of small-angle scattering data from lamellar and cylindrical particles by the indirect transformation method. J Appi Crystallogr 13:577-584 Glatter O, Kratky O (1982) Small Angle X-ray Scattering. Academic Press, London, UK Glinka CJ, Rowe JM, LaRock JG (1986) The small-angle neutron-scattering spectrometer at the National Bureau of Standards. J Appi Crystallogr 19:427^139 Goldstein J, Newbury DE, Joy DC, Lyman CE, Echlin P, Lifshin E, Sawyer L, Michael JR (2003) Scanning Electron Microscopy and X-ray Microanalysis, 3rd edition. Springer, New York Goodman JW (1985) Statistical Optics. Wiley-Interscience Goyal PS, King JS, Summerfield GC (1983) Multiple scattering in small-angle neutron scattering measurements on polymers. Polymer 24:131-134 Grigoriev SV, Kraan WH, Rekveldt MT, Kruglov T, Bouwman WG (2006) Spin-echo small-angle neutron scattering for magnetic samples. J Appi Crystallogr 39:252-258 Grout H, Tarquis AM, Wiesner MR (1998) Multifractal analysis of particle size distributions in soil. Environ Sci Technol 32:1176-1182 Grubenmann U, Niggli P, Jeannet A, Moser R (1915) Die naturlichen Bausteine und Dachschiefer der Schweiz. Beitr Geologie Schweiz, Geotechnische Serie 5 Guinier A (1937) A dispositive permitting the obtention of diffraction diagramas of intense crystalline powders with monochromatic radiation. C R Hebd Seances Acad Sci 204:1115-1116 Guinier A, Fournet A (1955) Small angle scattering of X-rays. Wiley & Sons, New York Guinier A, Fournet G, Walker CB, Yudowitch CB (1955) Small-Angle Scattering of X-rays. Wiley, New York Hadziioannou G, Picot C, Scoulios A, Ionescu M-L, Mathis A, Duplessix R, Gallot Y, Lingelser J-P (1982) Low-angle neutron-scattering study of the lateral extension of chains in lamellar styrene isoprene block copolymers. Macromolecules 15:263-267 Haenel M (1992) Recent progress in coal structure research. Fuel 71:1211-1223 Hainbuchner M, Villa M, Kroupa G, Bruckner G, Baron M, Amenitsch H, Seidl E, Rauch H (2000) The new high resolution ultra small-angle neutron scattering instrument at the High flux reactor in Grenoble. J Appi Crystallogr 33:851-854 Hall PL, Mildner DFR, Borst RL (1983) Pore-size distribution of shaly rock by small-angle neutron-scattering. Appi Phys Lett 43:252-254 Hall PL, Mildner DFR, Borst RL (1986) Small-angle scattering studies of the pore spaces of shaly rocks. J Geophys Res 91(B2): 2183-2192 Hamida T, Babadagli Τ (2008) Displacement of oil by different interfacial tension fluids under ultrasonic waves. Colloid Surface A 316:176-189 Hammouda Β (2008) Probing nanoscale structures—The SANS toolbox, http://www.ncnr.nist.gov/staff/ hammouda/the-SANS-toolbox.pdf Hammouda B, Mildner DFR (2007) Small-angle neutron scattering resolution with refractive optics. J Appi Crystallogr 40:250-259 Hannan EJ (1990) Stationary Time Series. In: The New Palgrave: A Dictionary of Economics. Time Series and Statistics. Eatwell J, Milgate M, Newman Ρ (eds) Macmillan, London Hansen S (2000) Bayesian estimation of hyperparameters for indirect Fourier transformation in small-angle scattering. J Appi Crystallogr 33:1415-1421 Hansen S, Pedersen JS (1991) A comparison of three different methods for analysing small-angle scattering data J Appi Crystallogr 24:541-548 Hasegawa H, Hashimoto T, Kawai H, Lodge TP, Amis EJ, Glinka CJ, Han CC (1985) SANS and SAXS studies on molecular conformation of a block polymer in microdomain space. Macromolecules 18:67-78 Hasegawa H, Tanaka H, Hashimoto T, Han CC (1987) SANS and SAXS studies on molecular-conformation of a block polymer in microdomain space. 2. Contrast matching technique. Macromolecules 20:2120-2127

Characterization

and Analysis

of Porosity and Pore

Structures

151

Heath JE, Dewers TA, McPherson BJOL, Petrusak R, Chidsey TC, Rinehart AJ, Mozley PS (2011) Pore networks in continental and marine mudstones: Characteristics and controls on sealing behavior. Geosphere 7:429-454 Heenan RK, Penfold J, King SM (1997) SANS at pulsed neutron sources: Present and future prospects. J Appi Crystallogr 30:1140-1147 Heilbronner R, Barrett S (2014) Image Analysis in Earth Sciences. Springer, Berlin Heising J (1995a) Estimating effective properties of composites from cross-sectional photographs. J Comp Phys 117:281-288 Heising J (1995b) An integral-equation method for elastostatics of periodic composites. J Mech Phys Solids 43:815-828 Henry CR (2005) Morphology of supported nanoparticles. Prog Surf Sci 80:92-116 Hildenbrand A, Urai JL (2003) Investigation of the morphology of pore space in mudstones — first results. Mar Petrol Geology 20:1185-1200 Hinde AL (2004) PRINSAS - a Windows-based computer program for the processing and interpretation of small-angle scattering data tailored to the analysis of sedimentary rocks. J Appi Crystallogr 37:1020-1024 Hirschwald J (1912) Handbuch der bautechnischen Gesteinprüfung. Gebrüder Borntraeger, Berlin Holoubek J, Konak C, Stepanek Ρ (1999) Time-resolved small-angle light scattering apparatus. Part Part Syst Charact 16:102-105 Howard CV, Reed MG (2005) Unbiased Stereology (Garland Science/BIOS Scientific, New York) Hu, Y, Ray, JR, and Jun Y-S (201 la) Biotite-brine interactions under acidic hydrothermal conditions:Fibrous illite, goethite, and kaolinite formation and biotite surface cracking. Environ Sci Technol 45:6175-6180 Hu N, Borkar N, Kohls D, Schaefer DW (201 lb) Characterization of porous materials using combined smallangle x-ray and neutron scattering techniques. J Membrane Sci 379:138-145 Hu Q, Ewing RP, Dultz S (2012) Low pore connectivity in natural rock. J Contam Hydrol 133:76-83 Hua DW, DSouza JV, Schmidt PW, Smith DM (1994) Pore structure-analysis via small-angle X-ray-scattering and contrast matching. Characterization of Porous Solids III. Stud Surf Sci Catal 87:255-261 Huang J, Cavanaugh T, Nur Β (2013) An introduction to SEM operational principles and geologic applications for shale hydrocarbon reservoirs. Am Assoc Petrol Geol Mem 102:1-6 Hunt TS (1875) Chemical and Geological Essays. 490 pp. 8vo. James R. Osgood and Co., Boston Hwang SN, Wehrli FW, Williams JL (1997) Probability-based structural parameters from three-dimensional nuclear magnetic resonance images as predictors of trabecular bone strength. Med Phys 24:1255-1261 Iassonov P, Gebrenegus T, Tuller M (2009) Segmentation of X-ray computed tomography images of porous materials:a crucial step for characterization and quantitative analysis of pore structures. Water Resour Res 45:W09495 Ibel K, Stuhrmann HB (1975) Comparison of neutron and x-ray-scattering of dilute myoglobin solutions. J Mol Biol 93:255-265 Ichim L, Dobrescu R (2013) Characterization of tumor angiogenesis using fractal measures. In: 19th International Conference on Control Systems and Computer Science (CSCS 2013). Dumitrache I, Florea AM, Pop F (eds), ρ 345-349 Ikeda S, Nakano T, Nakashima Y (2000) Three-dimensional study on the interconnection and shape of crystals in a graphic granite by X-ray CT and image analysis. Mineral Mag 64:945-959 Ilavsky J, Jemian PR (2009) Irena: tool suite for modeling and analysis of small-angle scattering. J Appi Crystallogr 42:347-353 Ilavsky J, Jemian PR, Allen AJ, Zhang F, Levine LE, Long GG (2009) Ultra-small-angle X-ray scattering at the Advanced Photon Source. J Appi Crystallogr 42:469^-79 Ilea DE, Whelan PF (2011) Image segmentation based on the integration of colour-texture descriptors—A review. Pattern Recogn 44:2479-2501 Iniewski Κ (2007) Wireless Technologies: Circuits, Systems, and Devices. CRC Press Ioannidis MA, Kwiecien MJ, Chatzis I (1996) Statistical analysis of the porous microstructure as a method for estimating reservoir permeability. J Petrol Sei Eng 16:251-261 Jahnert S, Muter D, Prass J, Zickler GA, Paris O, Findenegg GH (2009) Pore structure and fluid sorption in ordered mesoporous silica. I. Experimental study by in situ small-angle x-ray scattering. J Phys Chem C 113:15201-15210 Jahshan SN, Summerfield GC (1980) Extracting single chain characteristics fron neutron and X-ray scattering experiments. J Polym Sci Polym Phys Ed 18:1859-1861 Javadpour F (2009) Nanopores and apparent permeability of gals flow in mudrocks (shales and siltstones). J Can Petrol Tech 48:16-21 Javadpour F, Fisher D, Unsworth M (2007) Nanoscale gas flow in shale gas sediment. J Can Petrol Tech 46:55-61 Javadpour F, Farshi MM, Amein M (2012) Atomic-force microscopy :a new tool for gas-shale characterization. J Can Petrol Tech 51:236-243

152

Anovitz & Cole

Jaynes E T (1983) Papers on Probability, Statistics and Statistical Physics. Rosenkrantz RD (ed) Dordrecht: Reidel Jemian PR, Weertman JR, Long GG, Spai RD (1991) Characterization of 9Cr-lMoVNb steel by anomalous small-angle X-ray scattering, Acta Metall Mater 39:2477-2487 Jennings J (1987) Capillary pressure techniques:application to exploration and development geology. Am Assoc Petrol Geol Bull 71:1196-1209 Jericha E, Baron M, Hainbuchner M, Loidl R, Villa M, Rauch H (2003) Ultra-small-angle neutron scattering studies of artificial lattices. J Appi Crystallogr 36:778-782 Jiao Y, Chawla Ν (2014) Modeling and characterizing anisotropic inclusion orientation in heterogeneous material via directional cluster functions and stochastic microstructure reconstruction. J Appi Phys 115:093511 Jiao Y, Stillinger FH, Torquato S (2007) Modeling heterogeneous materials via two-point correlation functions:Basic principles. Phys Rev E 76:031110. Jiao Y, Stillinger FH, Torquato S (2008) Modeling heterogeneous materials via twopoint correlation functions. II. Algorithmic details and applications. Phys Rev E 77:031135 Jiao Y, Stillinger FH, Torquato S (2009) A superior descriptor of random textures and its predictive capacity. PNAS 106:17634-17639 Jiao Y, Stillinger FH, Torquato S (2010) Geometrical ambiguity of pair statistics. II. Heterogeneous media. Phys Rev E 82:011106 Jiao Y, Pallia E, Chawla Ν (2013) Modeling and predicting microstructure evolution in lead/tin alloy via correlation functions and stochastic material reconstruction. Acta Mater 61:3370-3377 Jin L, Rother G, Cole DR, Mildner DFR, Duffy CJ, Brantley SL (2011) Characterization of deep weathering and nanoporosity development in shale - a neutron study. Am Mineral 96:498-512 Jin L, Mathur R, Rother G, Cole D, Bazilevskaya E, Williams J, Carone A, Brantley S (2013) Evolution of porosity and geochemistry in Marcellus Formation black shale during weathering. Chem Geol 356:50-63 Josh M, Esteban L, Piane CD, Sarout J, Dewhurst DN, Clennell MB (2012) Laboratory characterization of shale properties. J Petrol Sei Eng 88-89:107-124 Jun YS, Lee Β, Waychunas GA (2010) In situ observations of nanoparticle early development kinetics at mineral-water interfaces. Environ Sci Technol 44:8182-8189 Jung SJ, Amai R, Raper JA (1995) The use of small angle light scattering to study structure of floes. Part Part Syst Charact 12:274-278 Kahle A, Winkler B, Radulescu A, Schreuer J (2004) Small angle neutron scattering study of volcanic rocks. Eur J Mineral 16:407-417 Kahle A, Winkler B, Radulescu A (2006) Small-angle neutron scattering investigation of the effect of a heattreatment on the microstructure of porous basalt rocks. Mineral Mag 70:689-95 Kainourgiakis ME, Kikkinides ES, Steriotis TA, Stubos AK, Tzevelekos KP, Kanellopoulos NK (2000) Structural and transport properties of alumina porous membranes from process-based and statistical reconstruction techniques. J Colloid Interf Sci 231:158-167 Kampmann R, Haese-Seiller M, Kudryashov V, Nickel Β, Daniel C, Fenzl W, Schreyer A, Sackmann E, Rädler J. (2006) Horizontal ToF-neutron reflectometer REFSANS at FRM-II Munich/Germany :first tests and status. PhysicaB 385-386:1161-1163 Karastathis A (2007) Petrophysical Measurements on Tight Gas Shale. MS Thesis, University of Oklahoma, 117 ρ Karperien A (1999-2003) FracLac for ImageJ, http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Introduction. htm Kaufmann J (2009) Characterization of pore space of cement-based materials by combined mercury and Wood's metal intrusion. J Am Ceram Soc 92:209-216 Kaufmann J (2010) Pore space analysis of cement-based materials by combined nitrogen sorption — Wood's metal impregnation and multi-cycle mercury intrusion. Cem Concr Comp 32:514—522 Kaune G, Haese-Seiller M, Kampmann R, Moulin JF, Zhong Q, Müller-Buschbaum P. (2010) TOF-GISANS investigation of polymer infiltration in mesoporous T i 0 2 films for photovoltaic applications. J Polym Sci Part Β 48:1628-1635 Keller T, Gähler R, Kunze H, Golub R (1995) Features and performance of an NRSE spectrometer at BENSC. Neutron News 6:16-18 Kelly JL, Fu Β, Kita NT, Valley JW (2007) Optically continuous silcrete quartz cements of the St. Peter sandstone:high precision oxygen isotope analysis by ion microprobe. Geochim Cosmochim Acta 71:3812-3832 Kenyon W (1997) Petrophysical principles of applications of N M R logging. The Log Analyst 38:21^-3 Kessler DW (1919) Physical and chemical tests on the commercial marbles of the United States: US Bur Standards Tech Paper 123 Khintchine A (1934) Korrelationstheorie der stationären stochastischen Prozesse. Math Ann 109:604-615

Characterization and Analysis of Porosity and Pore Structures

153

Kleinberg RL (1999) Nuclear Magnetic Resonance. In: Methods in the Physics of Porous Media, No. 35, Wong, Ρ (ed) Academic Press, San Diego ρ 337-385 Kleinberg R, Vinegar Η (1996) NMR properties of reservoir fluids. The Log Analyst 37:20-32 Kleinberg RL, Jackson JA (2001) An introduction to the history of NMR well logging. Concept Magnetic Res 13:340-342 Klimentidis R, Lazar OR, Bohacs KM, Esch WL, Pededrsen Ρ (2010) Integrated petrography of mudstones. AAPG Ann Conv New Orleans, April 11-14 Kline SR (2006) Reduction and analysis of SANS and US ANS data using Igor Pro. J Appi Crystallogr 39:895900 Knudsen KD, Fossum JO, Helgesen G, Haakestad MW (2004) Small-angle neutron scattering from a nanolayered synthetic silicate. Physica Β 352:247 258 Koberstein J (1982) Small-Angle Neutron Scattering from Two-Phase Polymeric Systems J Polym Sci Polym Phys Ed 20:593-602 Kolesnikov AI, Anovitz LM, Mamontov E, Podlesnyak AA, Ehlers G (2014) Strong anisotropic dynamics of ultra-confined water. J Phys Chem Β 47:13414-13419 Kopycinska-Miiller M, Prasad M, Rabe U, Arnold W (2007) Elastic properties of clay minerals by atomic force microscopy technique. Acoustic Imaging 28:409^-16 Kratky O (1938) Die Berechnung der Micelldimensionen von Faserstoffen aus den unter kleinsten Winkeln abgebeugten Interferenzen. Naturwiss 26:94 Kratky O, Skala Ζ (1958) Neues Verfahren zur Herstellung von blendenstreuungsfreien Röntgenkleinwinkelaufnahmen. Ζ Electrochem 62:73-77 Kraus MA (2010) Fundamental building blocks of nanoporous networks from ultra-small-angle X-ray scattering (USAXS) and small-angle X-ray scattering (SAXS) experiments. Dissertation, University of Missouri-Columbia Krauthäuser HG, Lennartz W, Nimtz G (1996) Real-space distributions from small-angle scattering data: structure interference method versus indirect transformation method. J Appi Crystallogr 29:7-15 Kravchenko AN, Martin MA, Smucker AJM, Rivers ML (2009) Limitations in determining multifractal spectra from pore-solid soil aggregate images. Vadose Zone J 8:220-226 Krouglov T, de Schepper IM, Bouwman WG, Rekveldt MT (2003a) Real-space interpretation of spin-echo small-angle neutron scattering. J Appi Crystallogr 36:117-124 Krouglov T, Kraan WH, Plomp J, Rekveldt MT, Bouwman WG (2003b) Spin-echo small-angle neutron scattering to study particle aggregates. J Appi Crystallogr 36:816-819 Krouglov T, Bouwman WG, Plomp J, Rekveldt MT, Vroege GJ, Petukhov AV, Thies-Weesie DME (2003c). Structural transitions of hard-sphere colloids studied by spin-echo small-angle neutron scattering. J Appi Crystallogr 36:1417-1423 Krycka KL, Booth RA, Hogg CR, Ijiri Y, Borchers JA, Chen WC, Watson SM, Laver M, Gentile TR, Dedon LR, Harris S, Rhyne JJ, and Majetich SA (2010) Core-shell magnetic morphology of structurally uniform magnetite nanoparticles. Phys Rev Lett 104:207203 Kuila U (2013) Measurement and interpretation of porosity and pore size distribution in mudrocks: The hole story of shales. PhD Dissertation, Colorado School of Mines Landrot G, Ajo-Franklin JB, Yang L, Cabrini S, Steefel CI (2012) Measurement of accessible reactive surface area in a sandstone, with applicatiobs to C 0 2 mineralization. Chem Geol 318-319:113-125 Lange DA, Jennings HM, Shah SP (1994) Image-analysis techniques for characterization of pore structure of cement-based materials. Cem Concr Res 24:841-853 Laszlo K, Geissler E (2006) Surface chemistry and contrast-modified SAXS in polymer-based activated carbons. Carbon 44:2437-2444 Laszlo K, Czakkel O, Josepovits K, Rochas C, Geissler E (2005) Influence of surface chemistry on the SAXS response of polymer-based activated carbons. Langmuir 21:8443-8451 Lazzari R (2002) IsGISAXS:A program for grazing-incidence small-angle X-ray scattering analysis of supported islands. J Appi Crystallogr 35:406^-21 Lebedev VM, Lebedev VT, Orlov SP, Pevzner BZ, Tolstikhin IN (2004) Small-angle neutron scattering study of the structure of quartz irradiated with fast neutrons. Kristallografiya 49 (Suppl. 1):89S Lebedev VM, Lebedev VT, Orlov SP, Pevzner BZ, Tolstikhin IN (2006) Small-angle neutron scattering study of radiation defects in synthetic quartz. Phys Solid State 48:678-685 Lebedev VM, Lebedev VT, Orlov SP, Pevzner BZ, Tolstikhin IN (2007) Nature of radiation defects in synthetic quartz according to the small-angle neutron scattering data. Crystallogr Rep 52:456-459 Lebron I, Schaap MG, Suarez DL (1999) Saturated hydraulic conductivity prediction from microscopic pore geometry measurements and neural network analysis. Water Resour Res 35:3149-3158 Lee B, Park I, Yoon J, Park S, Kim J, Kim KW, Chang T, Ree M (2005) Structural analysis of block copolymer thin films with grazing incidence small-angle X-ray scattering. Macromol 38:4311-4323

154

Anovitz & Cole

Levine JR, Cohen JB, Chung YW, Georgopoulos Ρ (1989). Grazing-Incidence Small-Angle X-ray Scattering:New Tool for Studying Thin Film Growth. J Appi Crystallogr 22:528-532 Levine JR, Cohen JB, Chung YW (1991) Thin-film island growth-kinetics - a grazing-incidence small-angle x-ray-scattering study of gold on glass. Surf Sci 248:215-224 Li X, Shew C-Y, Liu Y, Pynn R, Liu E, Herwig KW, Smith GS, Robertson JL, Chen W-R (2010) Theoretical studies on the structure of interacting colloidal suspensions by spin-echo small angle neutron scattering. J C h e m P h y s 132:174509 Liao JYH, Selomulya C, Bushell G, Bickert G, Amai R (2005) On different approaches to estimate the mass fractal dimension of coal aggregates. Part Part Syst Charact 22:299-309 Lieutenant K, Lindner P, Gahler R (2007) A new design for the standard pinhole small-angle neutron scattering instrument D l 1. J Appi Crystallogr 40:1056-1063 Lindner P, Zemb Τ (1991) Neutron, X-Ray and Light Scattering. Elsevier, Amsterdam Lindner P, May RP, Timmins PA (1992) Upgrading of the SANS instrument D i l at the ILL. Physica B, 180-181:967-972 Lindquist WB, Lee S-M, Coker DA, Jones KW, Spanne Ρ (1996) Medial axis analysis of three-dimensional tomographic images of porous media. J Geophys Res 101:8297-310 Littreil KC (2004) A comparison of different methods for improving flux and resolution on SANS instruments. Nucl Instrum Methods Phys Res Sect A 529:22-27. Littrell KC, Khalili NR, Campbell M, Sandi G, Thiyagarajan Ρ (2002) Microstructural analysis of activated carbons prepared from paper mill sludge by SANS and BET. Chem Mater 14:327-333 Liu C, Liu Y, Kerisit S, Zachara J (2015) Pore-scale process coupling and effective surface reaction rates in heterogeneous subsurface materials. Rev Mineral Geochem 80:191-216 Loong C-K (2006) Inelastic scattering and applications. Rev Mineral Geochem 63:233-254 Loucks RG, Reed RM, Ruppel SC, Jarvie DM (2009) Morphology, genesis, and distribution of nanometerscale pores in siliceous mudstones of the Mississippian Barnett Shale. J Sed Res 79:848-861 Loucks RG, Reed RM, Ruppel SC, Hammes U (2012) Spectrum of pore types and networks in mudrocks and a descriptive classification for matrix-related mudrock pores. Am Assoc Petrol Geol Bull 96:1071-1098 Lu D, Zhang T, Yang J, Li D, Kong X (2009) A reconstruction method of porous media integrating soft data with hard data. Chin Sci Bull 54:1876-1885 Lucido G, Caponetti E, Triolo R (1985) Preliminary small angle neutron scattering experiments on magmatic rocks to detect critical phenomena. Mineral Petrog Acta 29:133-138 Lucido G, Triolo R, Caponetti E (1988) Fractal approach in petrology: Small-angle neutron scattering experiments with volcanic rocks. Phys Rev Β 38:9031-9034 Lucido G, Caponetti E, Triolo R (1991) Fractality as a working tool for petrology: small angle neutron scattering experiments to detect critical behavior of magma. Geologica Carpahica 42:85-91 Luffel DL, Guidry FK (1992) Evaluation of Devonian shale with new core and log analysis methods. J Petrol Technol 44:1192-1197 Luo LF, Lin H (2009) Lacunarity and fractal analyses of soil macropores and preferential transport using micro-x-ray computed tomography. Vadose Zone J 8:233-241 Mahmood SA, Gloaguen R (2011) Fractal measures of drainage network to investigate surface deformation from remote sensing data: A paradigm from Hindukush (NE-Afghanistan) J Mountain Sci 8:641-654 Mancinelli R (2012) Multiple neutron scattering corrections. Some general equations to do fast evaluations. J Phys, Conf Ser 340:012033 Mandelbrot Β (1977) Fractals:Form, Chance and Dimension. W.H. Freeman and Co. Mandelbrot BB (1983) The Fractal Geometry of Nature. New York: W.H. Freeman Mandelbrot BB (1989) Multifractal measures, especially for the geophysicist. Pageoph 131:5^-2 Mandelbrot BB (1994) A fractal's lacunarity, and how it can be tuned and measured. In: Fractals in Biology and Medicine. Nonnenmacher TF, Losa GA, Weibel ER (eds) Birkhäuser Verlag, Basel Mandelbrot BB (1995) Measures of fractal lacunarity: Minkowski content and alternatives. In: Fractal Geometry and Stochastics. Bandt C, Graf S, Zähle M (eds) Birkhäuser Verlag, Basel Manger GE (1963) Porosity and Bulk Density of Sedimentary Rocks. Geol Surv Bull 1114—E Mares TE, Radlinski AP, Moore TA, Cookson D (2009) Assessing the potential for C 0 2 adsorption in a subbituminous coal, Huntly Coalfield, New Zealand, using small angle scattering techniques. International J Coal Geol 77:54-68 Mariethoz G, Kelly BF (2011) Modeling complex geological structures with elementary training images and transform-invariant distances. Water Resour Res 47:W07527 Mariethoz G, Renard Ρ, Straubhaar J (2010) The direct sampling method to perform multiple-point geostatistical simulations. Water Resour Res 46:W11536 Martelli S, Di Nunzio PE (2002) Particle size distribution of nanospheres by Monte Carlo fitting of small angle X-ray scattering curves. Part Part Syst Charact 19:247-255

Characterization

and Analysis

of Porosity and Pore

Structures

155

Martin MA, Perfect E, Pachepsky Y (2005) Scaling fractals and diversity in soil and ecohydrology. Ecol Model 182:217-384 Martin MA, Perfect E, Pachepsky Y (2006) Fractal geometry applied to soil and related hierarchical systems. Fractals, complexity and heterogeneity, Geoderma 134:237^-56 Martin MA, García-Gutiérrez C, Reyes M (2009) Modeling multifractal features of soil particle size distributions with Kolmogorov fragmentation on algorithms. Vadose Zone J 8:202-208 Masad E, Muhunthan Β (1998) Simulation of three-dimensional anisotropic soil microstructure. In: Imaging Technologies: Techniques and Applications in Civil Engineering.Second International Conference. Frost JD, McNeil S (eds) American Society of Civil Engineers ρ 265-285 Masad E, Muhunthan Β (2000) Three-dimensional characterization and simulation of anisotropic soil fabric. J Geotech Geoenvironm Eng 126:199-207 Mastalerz M, He LL, Melnichenko YB, Rupp JA (2012) Porosity of Coal and Shale:Insights from Gas Adsorption and SANS/USANS Techniques. Energy Fuels 26:5109-5120 Mastalerz M, Schimmelmann A, Drobniak A, Chen YY (2013) Porosity of Devonian and Mississippian New Albany Shale across a maturation gradient:Insights from organic petrology, gas adsorption, and mercury intrusion. Am Assoc Petrol Geol Bull 97:1621-1643 Mazumder S, Bhagwat KV, Sequeira A (2003) Estimation of particle size distribution in multiple small-angle scattering. J Phys-Condens Matter 7:9737-9746 McGreevy RL, Pusztai L (1988) Reverse Monte Carlo simulation: A new technique for the determination of disordered structures. Mol Simul 1:359-367 McMahon PJ, Snook I, Treimer W (2002) The pore structure in processed Victorian brown coal. J Colloid Interf Sci 251:177-183 Melnichenko YB, Radlinski AP, Mastalerz M, Cheng G, Rupp J (2009) Characterization of the C 0 2 fluid adsorption in coal as a function of pressure using neutron scattering techniques (SANS and USANS). Int J Coal Geol 77:69-79 Melnichenko YB, He L, Sakurovs R, Kholodenko AL, Blach T, Mastalerz M, Radlinski AP, Cheng G, Mildner DFR (2012) Accessibility of pores in coal to methane and carbon dioxide. Fuel 91:200-208 Meng Β (1996) Determination and interpretation of fractal properties of the sandstone pore system. Mater Struct 29:195-205 Merrit M, Zhang Y (2004) Technical report TR04-08, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, 77005, USA. http://www.caam.rice.edu/caam/trs/2004/TR04-08.pdf Mezei F (1972) Neutron spin-echo - new concept in polarized thermal-neutron techniques. Ζ Physik 255:145160 Mezei F (1980) Neutron Spin-Echo. Lecture Notes in Physics 128, Mezei F (ed) Springer Verlag, Berlin Michels A, Weissmüller J (2008) Magnetic-field-dependent small-angle neutron scattering on random anisotropy ferromagnet. Rep Prog Phys 71:066501 Mildner DFR (2005) Resolution of small-angle neutron scattering with a refractive focusing optic. J Appi Crystallogr 38:488^192 Mildner DFR, Hall PL (1986) Small-angle scattering from porous solids with fractal geometry. J Phys D, Appi Phys 19:1535-1545 Mildner DFR, Rezvani R, Hall PL, Borst RL (1986) Small-angle scattering of shaly rocks with fractal pore interfaces. Appi Phys Lett 480:1314-1316 Millán H (2004) Fragmentation of soil initiators: Application of the pore-solid fractal model. Fractals 12:357363 Miranda-Martínez ME, Oleschko K, Parrot J-F, Castrejón-Vacio F, Taud H, Brambila-Paz F (2006) Porosity in naturally fractured media: A fractal classification. Rev Mex Ciencias Geológicas 23:199-214 Mitchell J, and Fordham EJ (2014) Contributed review:Nuclear magnetic resonance core analysis at 0.3 T. Rev Sei Instrum 85:111502 Mitchell J, Webber JBW, Strange JH (2008) Nuclear magnetic resonance cryoporometry. Phys Rep 461:1-36 Mitchell J, Chandrasekera TC, Johns ML, Gladden LF, Fordham EJ (2010) Nuclear magnetic resonance relaxation and diffusion in the presence of internal gradients: The effect of magnetic field strength. Phys Rev E 81:026101 Molins S (2015) Reactive interfaces in direct numerical simulation of pore scale processes. Rev Mineral Geochem 80:461^181 Moon KH, Falchetta AC, Marasteanu M O (2013) Rheological modelling of asphalt materials properties at low temperatures:from time domain to frequency domain. Road Materials and Pavement Design 14:810-830 Moon KH, Falchetto AC, Jeong JH (2014a) Microstructural analysis of asphalt mixtures using digital image processing techniques. Can J Civil Eng 41:74-86 Moon KH, Falchetto AC, Marasteanu M O (2014b) Investigation of limiting criteria for low temperature cracking of asphalt mixture. KSCE J Civil Eng 18:172-181

156

Anovitz & Cole

Moon KH, Falchette AC, Yeom WS (2014c) Low-temperature performance of asphalt mixtures under static and oscillatory loading. Arab J Sei Eng 39:7577-7590 Moore CC (1904) The study of the volume composition of rocks, and its importance to the geologist. Geol Soc Liverpool Proc 9:129-162 Moore PB (1980) Small-angle scattering - information-content and error analysis. J Appi Crystallogr 13:168175 Moresco M, Marnili F, Baldi M, Moscardini L, Cimatti A (2014) Disentangling interacting dark energy cosmologies with the three-point correlation function. MNRAS 443:2874—2886 Mter D, Jhnert S, Dunlop JWC, Findenegg GH, Paris O (2009) Pore structure and uid sorption in ordered mesoporous silica. II. modeling. J Phys Chem C 113:15211-15217 Mukherjee PP, Wang C-Y (2007) Direct numerical simulation modeling of bilayer cathode catalyst layers in polymer electrolyte fuel cells. J Electrochem Soc 154:B1121-B1131 Mukherjee PP, Qinjun K, Wang C-Y (2011) Pore-scale modeling of two-phase transport in polymer electrolyte fuel cells-progress and perspective. Energ Environ Sci 4:346-369 Mulato M, Chambouleyron I (1996) Small-angle X-ray and neutron scattering of polydisperse systems: determination of the scattering-particle-size distribution J Appi Crystallogr 29:29-36 Muller J (1992) Multifractal characterization of petrophysical data. Physica A 191:284-288 Muller J (1994) Characterization of the North Sea chalk by multifractal analysis. J Geophys Res 99(B4):72757280 Muller J (1996) Characterization of pore space in chalk by multifractal analysis. J Hydrol 187:215-222 Muller J, Huseby OK, Saucier A (1995) Influence of multifractal scaling of pore geometry on permeabilities of sedimentary rocks. Chaos Solitons Fractals 5:1485-1492. Muller J, McCauley JL (1992) Implication of fractal geometry for fluid flow properties of sedimentary rocks. Transp Porous Media 8:133-147 Müller-Buschbaum Ρ (2013) Grazing incidence small-angle neutron scattering: challenges and possibilities. Polym J 45:34^12 Müller-Buschbaum Ρ, Vanhoorne Ρ, Scheumann V, Stamm M (1997) Observation of nano-dewetting structures. Europhys Lett 40:655-660 MUller-Buschbaum Ρ, Gutmann JS, Cubitt R, Stamm M (1999a) Probing the in-plane composition of thin polymer films with grazing-incidence small-angle neutron scattering and atomic force microscopy. Colloid Polym Sci 277:1193-1199 MUller-Buschbaum P, Gutmann JS, Stamm M (1999b) Dewetting of confined polymer films: an X-ray and neutron scattering study. Phys Chem Chem Phys 1:3857-3863 MUller-Buschbaum P, Cubitt R, Petry W (2003) Nanostructured diblock copolymer films: A grazing incidence small-angle neutron scattering study. Langmuir 19:7778-7782 MUller-Buschbaum P, Hermsdorf N, Roth SV, Wiedersich J, Cunis S, Gehrke R (2004) Comparative analysis of nanostructured diblock copolymer films. Spectrochim Acta Β 59:1789-1797 MUller-Buschbaum P, Maurer E, Bauer E, Cubitt R (2006) Surface versus confinement induced morphology transition in triblock copolymer films: A grazing incidence small angle neutron scattering investigation. Langmuir 22:9295-9303 MUller-Buschbaum P, Schulz L, Metwalli E, Moulin JF, Cubitt R (2008) Lateral structures of buried interfaces in ΑΒΑ-type triblock copolymer films. Langmuir 24:7639-7644 MUller-Buschbaum P, Metwalli E, Moulin JF, Kudryashov V, Haese-Seiller M, Kampmann R (2009) Time of flight grazing incidence small angle neutron scattering. Eur Phys J Special Topics 167:107-112 Munoz, X, Freixenet, J, Cufi, X, Marti, J (2003) Strategies for image segmentation combining region and boundary information. Pattern Recogn Lett 24:375-392 Murphy DP (1995) NMR logging and core analysis—simplified. World Oil 216:65-70 Nabawy BS (2014) Estimating porosity and permeability using Digital Image Analysis (DIA) technique for highly porous sandstones. Arab J Geosci 7:889-898 Narayanan T, Diat O, Bösecke Ρ (2001) SAXS and USAXS on the high brilliance beamline at the ESRF. Nucl Instrum Meth A 467^168:1005-1009 Naudon A (1995) Grazing-incidence small-angle x-ray scattering, application to layers and surface layers. In: Modern Aspects of Small-Angle Scattering. Brumberger Η (ed) NATO Advanced Science Institutes Series, Series C, Mathematical and Physical Sciences Vol 451. Springer, Berlin ρ 181-202 Naudon A, Thiaudiere D (1997) Grazing-incidence small-angle scattering. Morphology of deposited clusters and nanostructure of thin films. J Appi Crystallogr 30:822-827 Naudon A, Goudeau P, Halimaoui A, Lambert B, Bomchil G (1994) Small-angle x-ray scattering study of anodically oxidized porous silicon layers. J Appi Phys 75:780-784 Navarre-Sitchler AK, Cole DR, Rother G, Jin L, Buss HL, Brantley SL (2013) Porosity and surface area evolution during weathering of two igneous rocks. Geochim Cosmochim Acta 109:400^-13

Characterization and Analysis of Porosity and Pore Structures

157

Navarre-Sitchler A, Brantley SL, Rother G (2015) How porosity increases during incipient weathering of crystalline silicate rocks. Rev Mineral Geochem 8 0 : 3 3 1 - 3 5 4 Nelson PH (2009) Pore-throat sizes in sandstones, tight sandstones, and shales. Am Assoc Petrol Geol Bull 93:329-340 N'Diaye M, Degeratu C, Bouler J M , Chappard D (2013) Biomaterial porosity determined by fractal dimensions, succolarity and lacunarity on microcomputed tomographic images. Mat Sei Eng C-Mat Biol Appi 3 3 : 2 0 2 5 - 2 0 3 0 N'Diaye M, Terranova L, Mallet R, Mabilleau G, Chappard D (2015) Three-dimensional arrangement of ß-tricalcium phosphate granules evaluated by microcomputed tomography and fractal analysis. Acta Biomater 1 1 : 4 0 4 - 4 1 1 Newsham KE, Rushing J A (2001) An integrated work-flow to characterize unconventional gas resources: Part 1 - geological assessment and petrophysical evaluation. Soc Petrol Eng S P E 71351 Nishida K, Ogawa H, Matsuba G, Konishi T, Kanaya Τ (2008) A high-resolution small-angle light scattering instrument for soft matter studies. J Appi Crystallogr 4 1 : 7 2 3 - 7 2 8 Noiriel C (2015) Resolving time-dependent evolution of pore-scale structure, permeability and reactivity using X-ray microtomography, Rev Mineral Geochem 8 0 : 2 4 7 - 2 8 5 Nunes AC (1978) Converging-beam small-angle neutron diffractometer - calculated resolution. J Appi Crystallogr 1 1 : 4 6 0 - 4 6 4 Ohoyama K, Kanouchi T, Nemoto K, Ohashi M, Kajitani T, Yamaguchi Y (1998) The new neutron powder diffractometer with a multi-detector system for high-efficiency and high-resolution measurements. Jpn J Appi Phys 1 3 7 : 3 3 1 9 - 3 3 2 6 Oku T, Susuki J, Sasao H, Hadachi T, Shinohara T, Ikeda K, Morishima T, Kiyanagi Y, Furusuka M, Shimizu H (2004). Feasibility study on application of a magnetic neutron lens to SANS experiments. Nucl Instrum Meth A 5 2 9 : 1 1 6 - 1 1 9 Oren PE, Bakke S (2002) Process based reconstruction of sandstones and prediction of transport properties. Transp Porous Med 4 6 : 3 1 1 - 3 4 3 Ortega C, Aguilera R (2014) a complete petrophysical evaluation method for tight formations from drill cuttings on in the absence of well logs. S P E J A u g : 6 3 6 - 6 4 7 Orthaber D, Bergmann A, Glatter O (2000) S A X S experiments on absolute scale with Kratky systems using water as a secondary standard. J Appi Crystallogr 3 3 : 2 1 8 - 2 2 5 Pal NR, Pal S K (1993) A review on image segmentation techniques. Pattern Recogn 2 6 : 1 2 7 7 - 1 2 9 4 Panduro EAC, Granlund H, Sztucki M, Konovalov O, Breiby DW, Gibaud A (2014) Using Three-Dimensional 3 D Grazing-Incidence Small-Angle X-ray Scattering ( G I S A X S ) Analysis To Probe Pore Deformation in Mesoporous Silica Films. Appi Mater Interfaces 6 : 2 6 8 6 - 2 6 9 1 Park C, Fenter PA, Nagy KL, Sturchio NC (2006) Hydration and distribution of ions at the mica-water interface. Phys Rev Lett 97:016101 Passey QR, Bohacs KM, Esch R, Klimentidis R, Sinha S (2010) From oil-prone source rock to gas-producing shale reservoir - geologic and petrophysical characterization of unconventional shale-gas reservoirs. Soc Petrol Eng S P E 131350 Pauw B R , Pedersen J S , Tardif S, Takata M, Iversen B B (2013) Improvements and considerations for size distribution retrieval from small-angle scattering data by Monte Carlo methods. J Appi Crystallogr 46:365-371 Paz Ferriero J, Vidal Vázquez E (2010) Mulifractal analysis of soil porosity based on mercury injection and nitrogen adsorption. Vadose Zone J 9 : 3 2 5 - 3 3 5 Pedersen J S (1994) Determination of size distributions from small-angle scattering data for systems with effective hard-sphere interactions. J Appi Crystallogr 2 7 : 5 9 5 - 6 0 8 Peitgen H-O, Jürgens H, Saupe D (2004) Chaos and Fractals: New Frontiers of Science. Second Edition. Springer, New York, Perfect E (1997) Fractal models for the fragmentation of rocks and soils: A review. Eng Geol 4 8 : 1 8 5 - 1 9 8 Peters E J (2012) Advanced Petrophysics: Volume 1: Geology, Porosity, Absolute Permeability, Heterogeneity and Geostatistics. Live Oak Book Co, Austin, Texas Pittman E (1992) Relationship of porosity and permeability to various parameter derived from mercury injection-capillary pressure curves for sandstone. Am Assoc Petrol Geol Bull 7 6 : 1 9 1 - 1 9 8 Plomp J, de Haan VO, Dalgliesh RM, Langridge S, van Well A A (2007) Time-of-flight spin-echo small-angle neutron measurements. Physica B-Condensed Matter 3 9 7 : 7 6 - 7 8 Politis M G , Kikkinides ES, Kainourgiakis ME, Stubos A K (2008) A hybrid process-based and stochastic reconstruction method of porous media. Micropor Mesopor Mat 1 1 0 : 9 2 - 9 9 Porod G (1951) Die Röntgenkleinwinkelstreuung von dichtgepackten kolloiden Systemen. Kolloid Ζ 1 2 4 : 8 3 114 Porod G (1952) Die Röntgenkleinwinkelstreuung von dichtgepackten kolloiden Systemen II. Kolloid Ζ 125:108-122

158

Anovitz & Cole

Posadas AND, Giménez D, Bitteiii, M, Vaz, CMP, Flury, M (2001) Multifractal characterization of soil particlesize distributions. Soil Sci Am J 65:1361-1367 Posadas AND, Giménez D, Quiroz R Protz R (2003) Multifractal characterization of soil pore systems. Soil Sei Soc Am J 67:1361-1369 Potton JA, Danieli GJ, Rainford BD (1986) Application of maximum entropy to small-angle neutron scattering data and liquid and amorphous material diffraction data analysis. Inst Phys Conf Ser 81:81-86 Potton JA, Danieli GJ, Rainford BD (1988a) Particle-size distributions from sans data using the maximumentropy method. J Appi Crystallogr 21:663-668 Potton JA, Danieli GJ, Rainford BD (1988b) A new method for the determination of particle-size distributions from small-angle neutron-scattering measurements. J Appi Crystallogr 21:891-897 Prager S (1961) Viscous flow through porous media. Phys Fluids 4:1477-1482 Predota M, Zhang Ζ, Fenter Ρ, Wesolowski DJ, Cummings PT (2004) Electric double layer at the rutile (110) surface. 2. Adsorption of ions from molecular dynamics and X-ray experiments. J Phys Chem Β 108:12061-12072 Prinz D, Pyckhout-Hintzen W, Littkea R (2004) Development of the meso- and macroporous structure of coals with rank as analysed with small angle neutron scattering and adsorption experiments. Fuel 83:547-556 Prodanovic M, Lindquist WB, Seright RS (2007) 3D image-based characterization of fluid displacement in a Berea core. Adv Water Resour 30:214-226 Putnis A (2015) Transient porosity resulting from fluid-mineral interaction and its consequences. Rev Mineral Geochem 80:1-23 Pynn R (1980) Neutron spin echo and three-axis spectrometers. In: Neutron Spin Echo. Mezei F (ed) SpringerVerlag, Berlin Pynn R, Fitzsimmons MR, Fritzsche H, Gierlings M, Major J, Jason A (2005) Neutron spin echo scattering angle measurement (SESAME). Rev Sei Instrum 76:053902 Quenard DA, Xu K, Kunzel UM, Bentz DP, Martys NS (1998) Microstructure and transport properties of porous building materials. Mater Struct 31:317-324 Radlinski AP (2006) Small-angle neutron scattering and rock microstructure. Rev Mineral Geochem 63:363397 Radlinski AP, Radlinska EZ (1999) The microstructure of pore space in coals if different rank: a small angle scattering and SEM study. In: Coalbed Methane: Scientific, Environmental and Economic Evaluation. Mastalerz M, Glikson M, Golding SD (eds) Kluvier Scientific Publishers ρ 329-365 Radlinski AP, Boreham CJ, Wignall GD, Lin J-S (1996) Microstructural evolution of source rocks during hydrocarbon generations small-angle-scattering study. Phys Rev Β 53:14152-14160 Radlinski AP, Radlinska EZ, Agamalian M, Wignall GD, Lindner P, Randl OG (1999) Fractal geometry of rocks. Phys Rev Lett 82:3078-3081 Radlinski AP, Boreham CJ, Lindner P, Randl O, Wignall GD, Hinde A, Hope JM (2000a) Small angle neutron scattering signature of oil generation in artificially and naturally matured hydrocarbon source rocks. Org Geochem 31:1-14 Radlinski AP, Radlinska EZ, Agamalian M, Wignall GD, Lindner P, Randl OG (2000b) The fractal microstructure of ancient sedimentary rocks. J Appi Crystallogr 33:860-862 Radlinski A, Ioannidis M, Hinde A, Hainbuchner M, Baron M, Rauch H, Kline S (2004) Angstrom-tomillimeter characterization of sedimentary rock microstructure. J Colloid Interf Sci 274:607-612 Ramakrishnan TS, Wasan DT (1986) Effect of capillary number on the relative permeability function for two phase flow in porous media. Powder Technol 48:99-124 Ramakrishanan TS, Rabaute A, Fordham EJ, Ramamoorthy R, Herron M, Matteson A, Raghuraman B, Mahdi A, Akbar M, Juchuk F (1998) A petrophysical and pétrographie strudy of carbonate core from Thamama Formation. Soc Petrol Eng SPE 49502:14 ρ Rasband WS (1997-2014) ImageJ, U. S. National Institutes of Health, Bethesda, Maryland, USA, http:// imagej.nih.gov/ij/ Rauscher M, Paniago R, Metzger H, Kovats Z, Domke J, Peisl J, Pfannes HD, Schulze J, and Eisele I (1999) Grazing incidence small angle x-ray scattering from free-standing nanostructures. J Appi Phys 86:67636769 Rehm C, Brûlé A, Freund AK, Kennedy SJ (2013) Kookaburra: the ultra-small-angle neutron scattering instrument at OPAL. J Appi Crystallogr 46:1699-1704 Reich MH, Russo SP, Snook IK, Wagenfeld HK (1990) The application of SAXS to determine the fractal properties of porous carbon-based materials. J Colloid Interf Sci 135:353-362 Rekveldt M T (1996) Novel SANS instrument using neutron spin echo. Nucl. Instrum. Methods Phys Res A, 114:366-370 Rekveldt MT, Bouwman WG, Kraan WH, Uca O, Grigoriev S, Habicht S, Keller Τ (2003) Elastic neutron scattering measurements using Larmor precession of polarized neutrons. In: Neutron Spin Echo, Lecture Notes in Physics, Vol. 601, Mezei F, Pappas C, Gutberlet Τ (eds) Springer, Berlin ρ 87-99

Characterization and Analysis of Porosity and Pore Structures

159

Rekveldt M, Plomp J, Bouwman WG, Kraan WH, Grigoriev S, Blaauw M. (2005) Spin-echo small angle neutron scattering in Delft. Rev Sei Instrum 76:033901 Renaud F, Lazzari R, Leroy F (2009) Probing surface and interface morphology with grazing incidence small angle X-ray scattering. Surf Sci Reports 6 4 : 2 5 5 - 3 8 0 Reuteler J, Hutter M, Gauckler L J (2011) Backbone of conductivity in two-dimensional metal-insulator composites. J Appi Phys 110:024909 Ricker DW (2003) Echo Signal Processing. Springer Rider M H (1999) The Geological Interpretation of Well Logs. Second edition. Whittles Publishing Services. Riseman J (1952) Particle-size distribution from small-angle X-ray scattering. Acta Crystallogr 5 : 1 9 3 - 1 9 6 Roberts AP (1997) Statistical reconstruction of three-dimensional porous media from two-dimensional images. Phys Rev E 5 6 : 3 2 0 3 - 3 2 1 2 Rodríguez Palomino LA, Dawidowski J, Blostein J J , Cuello G J (2007). Data processing method for neutron diffraction experiments. Nucí Instrum Meth Β 2 5 8 : 4 5 3 - 4 7 0 Romo-Uribe A, Alvarado-Tenorio Β, Romero-Guzmán M E (2010) A small-angle light scattering instrument to study soft condensed matter. Rev Lat Am Metalurgia Materiales 3 0 : 1 9 0 - 2 0 0 Ross DJK, Bustin R M (2009) The importance of shale composition and pore structure upon gas storage potential of shale gas reservoirs. Marine Petrol Geol 2 6 : 9 1 6 - 9 2 7 Roth SV, Dohrmann R, Dommach M, Kuhlmann M, Kroger I, Gehrke R, Walter H, Schroer C, Lengeler B, Müller-Buschbaum Ρ (2006) Small-angle options of the upgraded ultrasmall-angle x-ray scattering beamline B W 4 at H A S Y L A B . Rev Sci Instrum 7 7 : 0 8 5 1 0 6 Rouquerol J, Rodriguez Reinoso F, Sing KSW, Unger K K (1994) Characterization of Porous Solids III. Book Series:Stud Surf Sci Catal 8 7 : 2 5 5 - 2 6 1 Rouquerol J, Rouquerol F, Sing Κ (1999) Adsorption By Powders & Porous Solids: Principles, Methodology and Applications. Academic Press, San Diego R0yne A, Jamtveit Β (2015) Pore-scale controls on reaction. Rev Mineral Geochem 80:25-44 Rozenbaum O, Le Trong E, Rouet JL, Bruand A (2007) 2-D image analysis:A complementary tool for characterizing quarry and weathered building limestone. J Cultural Heritage 8 : 1 5 1 - 1 5 9 Ruderer MA, Meier R, Porcar L, Cubitt R, Müller-Buschbaum Ρ (2012) Phase separation and molecular intermixing in polymer-fullerene bulk heterojunction Thin Films. J Phys Chem Letters 3 : 6 8 3 - 6 8 8 Ruppert LF, Sakurovs R, Blach TP, He L L , Melnichenko Y B , Mildner DFR, Alcantar-Lopez L (2013) A USANS/SANS Study of the Accessibility of Pores in the Barnett Shale to Methane and Water. Energy and Fuels 2 7 : 7 7 2 - 7 7 9 Russell TP, Lin J S , Spooner S, Wignall G D (1988) Intercalibration of small angle X-ray and neutron scattering data. J Appi Crystallogr 2 1 : 6 2 9 - 6 3 8 Saar MO, Manga M (1999) Permeability-porosity relationship in vesicular basalts. Geophys Res Lett 2 6 : 1 1 1 114 Sabine TM, Bertram W K (1999) The use of multiple-scattering data to enhance small-angle neutron scattering experiments. Acta Crystallogr A 5 5 : 5 0 0 - 5 0 7 Saheli G, Garmestani H, Adams B L (2004) Microstructure design of a two phase composite using two-point correlation functions. J Comput A i d M a t Des 1 1 : 1 0 3 - 1 1 5 Sakurovs R, Radlinski AP, Melnichenko Y B , Blach T, Cheng G, Lemmel H, Rauch H (2009) Stability of the bituminous coal microstructure upon exposure to high pressures of helium. Energy Fuels 2 3 : 5 0 2 2 - 5 0 2 6 Salazar R, Gelb L D (2007) A computational study of the reconstruction of amorphous mesoporous materials from gas adsorption isotherms and structure factors via evolutionary optimization. Langmuir 2 3 : 5 3 0 - 5 4 1 San José Martinez F, Caniego FJ, García-Guitiérrez C, Espejo R (2007) Representative elementary area for multifractal analysis of soil porosity using entropy dimension. Nonlinear Proc Geoph 1 4 : 5 0 3 - 5 1 1 Sanchez C, Boissiere C, Grosso D, Laberty C, Nicole L (2008) Design, synthesis, and properties of inorganic and hybrid thin films having periodically organized nanoporosity. Chem Mater 2 0 : 6 8 2 - 7 3 7 Sangeetha S, Sujatha CM, Manamaiii D (2013) Characterization of trabecular architecture in femur bone radiographs using succolarity. 39th Annual Northeast Bioengineering Conference (NEBEC 2013):Annual I E E E Northeast Bioengineering Conference, 2 2 5 - 2 2 6 Sastry PU, Sen D, Mazumder S, Chandrasekaran K S (2000) Structural variations in lignite coal:a small angle X-ray scattering investigation. Solid State Commun 1 1 4 : 3 2 9 - 3 3 3 Schaap M G , Lebron I (2001) Using microscope observations of thin sections to estimate soil permeability with the Kozeny-Carman equation. J Hydrol 2 5 1 : 1 8 6 - 2 0 1 Scharpf O (1978a) Theory of magnetic neutron small-angle scattering using dynamical theory of diffraction instead of born approximation. 1. Determination of scattering angle. J Appi Crystallogr 1 1 : 6 2 6 - 6 3 0 Scharpf O (1978b) Theory of magnetic neutron small-angle scattering using dynamical theory of diffraction instead of born approximation. 2. Behavior of intensity of neutrons after transition through a helical magnetic-structure. J Appi Crystallogr 1 1 : 6 3 1 - 6 3 6

160

Anovitz & Cole

Schelten J, Schmatz W (1980) Multiple scattering treatment for small-angle scattering problems. J Appi Crystallogr 13:385-390 Schlegel ML, Nagy KL, Fenter P, Sturchio NC (2002) Structures of quartz (100)- and (lOl)-water interfaces determined by X-ray reflectivity and atomic force microscopy of natural growth surfaces. Geochim Cosmochim Acta 66:3037-3054 Schlegel ML, Nagy KL, Fenter P, Cheng L, Sturchio NC, Jacobsen SD (2006) Cation sorption on the muscovite (001) surface in chloride solutions using high-resolution X-ray reflectivity. Geochim Cosmochim Acta 70:3549-3565 Schlüter, S, Sheppard A, Brown K, Wildenschild D, (2014) Image processing of multiphase images obtained via X- ray microtomography: A review. Water Resour Res 50:3615-3639 Schmidt PW (1989). Use of scattering to determine the fractal dimension. In: The Fractal Approach to Heterogeneous Chemistry. Avnir D (ed) Wiley, Chichester, England, ρ 67-79 Schneider CA, Rasband WS, Eliceiri KW (2012) NIH Image to ImageJ: 25 years of image analysis. Nat Methods 9:671-675 Schwahn D, Miksovsky A, Rauch Η, Seidl E, Zugarek G (1985) Test of channel-cut perfect crystals for neutron small-angle scattering experiments. Nucl Instrum Meth A 239:229-234 Schwarz (1870-1871) quoted in Prinz E, 1923, Handbuch der Hydrologie 2d ed.:Berlin, Julius Springer Sears VF (1975) Slow-neutron multiple scattering. Adv Phys 24:1-45 Seery D, Lidsey JE (2005) Primordial non-Gaussianities in single-field inflation. J Cosmol Astroparticle Phys 6:003 Semenyuk AV, Svergun DI (1991) GNOM - a program package for small-angle scattering data processing. J Appi Crystallogr 24:537-540 Sen D, Mazumder S, Chitra R, Chandrasekaran KS (2001) Pore morphology in pressurized lignite coal:A small angle x-ray scattering investigation. J Mater Sci 36:909-912 Sen D, Mazumder, S, Tarafdar S (2002a) Pore morphology and pore surface roughening in rocks:a small-angle neutron scattering investigation. J Mater Sci 37:941-947 Sen D, Mazumder S, Tarafdar S (2002b) SANS investigation on pore surface roughening in rocks. Appi Phys Α-Mater 74:S 1049-S1051 Sezgin M, Sankur Β (2004) Survey over image thresholding techniques and quantitative performance evaluation. J Electro Imaging 13:146-165 Shahzad F, Mahmood SA, Gloaguen R (2010) Nonlinear analysis of drainage systems to examine surface deformation:an example from Potwar Plateau (Northern Pakistan). Nonlinear Proc Geoph 17:137-147 Shao H, Ray JR, Jun Y-S (2011) Effects of salinity and the extent of water on supercritical C0 2 -induced phlogopite dissolution and secondary mineral formation. Environ Sei Tech 45:1737-1743 Shull CG (1973) Perfect crystals and imperfect neutrons. J Appi Cryst 6:257-266 Shull CG, Smart JS (1949) Detection of antiferromagnetism by neutron diffraction. Phys Rev 76:1256-1257 Sigal RF, Odusina E (2011) Laboratory NMR measurements on methane saturated Barnett Shale samples. Petrophys 52:32-49 Silin D, Kneafsey Τ (2012) Shale gas:nanometer-scale observations and well modelling. J Can Petrol Tech 51:464-475 Sing K, Everett D, Haul R, Moscou L, Pierotti R, Rouquérol J Siemieniewska Τ (1985). IUPAC Commission on colloid and surface chemistry including catalysis. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984). Pure Appi Chem 57:603-619 Singh SS, Williams JJ, Jiao Y, Chawla Ν (2012) Modeling anisotropic multiphase heterogeneous materials via directional correlation functions:Simulations and experimental verification. Metall Mater Trans A 43:4470^1474 Skilling J, Bryan RK (1984) Maximum-entropy image-reconstruction - general algorithm. MNRAS 211:111— 124 Smarsly B, Goltner C, Antonietti M, Ruland W, Hoinkis E (2001) SANS investigation of nitrogen sorption in porous silica. J Phys Chem Β 105:831-840 Smith WS (1971) An X-ray scattering study of the smaller ribosomal subunit of E. coli. Ph.D. Thesis, University of Wisconsin Smith PA, Torquato S ( 1989) Computer-simulation results for bounds on the effective conductivity of composite media. J Appi Phys 65:893-900 Smith DM, Earl W, Hua DW (1995) Advanced NMR-based techniques for pore structure analysis of coal. Final Project Report, March 1995. DOE/PC/91296-15 Smolsky IL, Liu P, Niebuhr M, Ito Κ, Weiss TM, Tsuruta Η (2007) Biological small-angle x-ray scattering facility at the Stanford synchrotron radiation laboratory. J Appi Crystallogr 40 Supplì:S453-S458 Sondergeld CH, Newsham KE, Comisky JT, Rice MC, Rai CS (2010) Petrophysical considrations in evaluating and producing shale gas. Soc Petrol Eng SPE 131768

Characterization

and Analysis

of Porosity and Pore

Structures

161

Soper AK, Egelstaff PA (1980) Multiple-scattering and attenuation of neutrons in concentric cylinders. 1. isotropic first scattering. Nucl InstrumMeth 178:415-425 Sorby HC (1908) On the application of quantitative methods to the study of the structure and history of rocks. Q J Geol Soc London 64:171-233 Spowart JE, Maruyama B, Miracle DB (2001) Multi-scale characterization of spatially heterogeneous systems implications for discontinuously reinforced metal-matrix composite microstructures. Mat Sci Eng A-Struct 307:51-66 Stack AG (2015) Precipitation in pores: A geochemical frontier. Rev Mineral Geochem 80:165-190 Stanley HE, Meakin Ρ (1988) Multifractal phenomena in physics and chemistry. Nature 335:405 Stone S, Bushell G, Amai R, Ma Ζ, Merkus HG, Scarlett Β (2002) Characterization of large fractal aggregates by small-angle light scattering. Meas Sci Technol 13:357-364 Straley C, Rossini D, Vinegar P, Tutunjian P, Morriss C (1997) Core analysis by low field NMR. The Log Analyst 38:84-93 Strange JH, Webber JBW (1997) Spatially resolved pore size distributions by NMR. Meas Sci Technol 8:555561 Strange JH, Rahman M, Smith EG (1993) Characterization of porous solids by NMR. Phys Rev Lett 71:35893591 Strange JH, Mitchell J, Webber JBW (2003) Pore surface exploration by NMR. Mag Res Imaging 21:221-226 Strijkers GJ, Dalderop JHJ, Broeksteeg MAA, Swagten HJM, de Jonge WJM (1999) Structure and magnetization of arrays of electrodeposited Co wires in anodic alumina. J Appi Phys 86:5141-5145 Stuhrmann HB (1974) Neutron small-angle scattering of biological macromolecules in solution. J Appi Crystallogr 7:173-178 Stuhrmann HB (2008) Small-angle scattering and its interplay with crystallography, contrast variation in SAXS and SANS. Acta Cryst A 64:181-191 Stuhrmann HB (2012) Contrast variation application in small-angle neutron scattering experiments. 2nd International Workshop on SANS-YUMO user meeting at the start-up of scientific experiments on the IBR-2M reactor - devoted to the 75th anniversary of Yu M Ostanevich's birth location: Frank Lab Neutron Phys (Joint Inst Nucl Res), Dubna, Russia. J Phys Conf Ser 35:012002 Stuhrmann HB, Heinrich Β (2007) Contrast variation in X-ray and neutron scattering. In: 13th International Conference on Small-Angle Scattering. Kyoto, Japan, July 9-13,2006. J Appi Crystallogr 4 0 S u p p : l : S 2 3 S27 Stuhrmann HB, Kirste RG (1965) Elimination der intrapartikularen Untergrundstreuung bei der rontgenkleinwinkelstreuung an kompakten Teilchen (proteinen). Ζ Physik Chem Neue Folge 46:247-50 Stuhrmann HB, Haas J, Ibel K, De Wolf B, Koch MHJ, Parfait, R (1976) New low resolution model for 50s subunit of Escherìchia-coli ribosomes. PNAS 73:2379-2383 Stuhrmann HB, Koch MHJ, Parfait R, Haas J, Ibel Κ, Crichton RR (1977) Shape of 50s subunit of Escherìchiacoli ribosomes. PNAS 74:2316-2320 Stuhrmann HB, Koch MHJ, Parfait R, Haas J, Ibel Κ, Crichton RR (1978) Determination of distribution of protein and nucleic-acid in 70 s-ribosomes of Escherìchia-coli and their 30 s-subunits by neutronscattering. J Mol Biol 119:203-212 Sumanasooriya MS, Neithalath Ν (2009) Stereology- and morphology-based pore structure descriptors of enhanced porosity (pervious) concretes. ACI Mater J 106:429^-38 Sumanasooriya MS, Neithalath Ν (2011) Pore structure features of pervious concretes proportioned for desired porosities and their performance prediction. Cem Concr Comp 33:778-787 Sumanasooriya MS, Deo O, Neithalath Ν (2009) Predicting the elastic moduli of enhanced porosity (pervious) concretes using reconstructed 3D material structures. Brittle Matrix Composites 9:275-289 Sumanasooriya MS, Bentz DP, Neithalath Ν (2010) Planar image-based reconstruction of pervious concrete pore structure and permeability prediction. ACI Mater J 107:413-421 Suzuki J, Oku T, Adachi T, Shimizu HM, Hirumachi T, Tsuchihashi T, Watanabe I (2003) Cold neutron beam focusing by a superconducting sextupole magnet. J Appi Crystallogr 36:795-799 Svergun DI (1991) Mathematical-methods in small-angle scattering data-analysis J Appi Crystallogr 24:485492 Swanson BF (1979) Visualizing pores and nonwetting phase in porous rock. J Petrol Technol 31:10-18 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 C 0 2 sequestration. Environ Geosci 21:39-57 Takada M, Jain Β (2004) Cosmological parameters from lensing power spectrum and bispectrum tomography. MNRAS 348:897-915 Takahashi T, Hashimoto M, Nakatani S (1999) Ultra small angle neutron scattering instrument at C I - 3 in JRR3M. J Phys Chem Solids 60:1591-1594. Tchoubar D, Méring J (1969) Interpretation of central scattering of x-rays by porous systems. 2. examples of application. J Appi Crystallogr 2:128-138

162

Anovitz & Cole

Teng HH, Fenter P, Cheng LW, Sturchio NC (2001) Resolving orthoclase dissolution processes with atomic force microscopy and X-ray reflectivity. Geochim Cosmochim Acta 65:3459-3474 Terada K, Miura T, Kikuchi Ν (1997) Digital image-based modeling applied to the homogenization analysis of composite materials. Comput Mech 20:331-346 Thiyagarajan P, Epperson JE, Crawford RK, Carpenter JM, KlippertTE, Wozniak DG (1997) The time-of-flight small-angle neutron diffractometer (SAD) at IPNS, Argonne National Laboratory. J Appi Crystallogr 30:280-293 Thomas JJ, Valenza II JJ, Craddock PR, Bake KD, Pomerantz AE (2014) The neutron scattering length density of kerogen and coal as determined by CH 3 OH/CD 3 OH exchange. Fuel 117:801-808 Thompson AH, Katz AJ, Krohn, CE (1987) The microgeometry and transport properties of sedimentary rock. Adv Phys 36:625-694 Thovert J-F, Yousefiean F, Spanne Ρ, Jacquin CG, Adler PM (2001) Grain reconstruction of porous media:application to low-porosity Fontainebleau Sandstone. Phys Rev E 63:061307 Tiab D, Donaldson EC (2004) Petrophysics: Theory and Practice of Measuring Reservoir Rock and Fluid Transport Properties, 2ntl Edition. Elsevier Tiab D, Donaldson EC (2012) Petrophysics :Theory and Practice of Measuring Reservoir Rock and Fluid Transport Properties. 3rd Edition. Elsevier Tolbert WR (1971) A small angle X-ray scattering study of 50S ribosomal subunits for E. coli. Ph.D. Thesis, University of Wisconsin Torabi A, Fossen H (2009) Spatial variation of microstructure and petrophysical properties along deformation bands in reservoir sandstones. AAPG Bull 93:919-938 Torquato S (1985) Effective electrical-conductivity of 2-phase disordered composite media. J Appi Phys 58:3790-3797 Torquato S (2002a) Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer. 701 pp Torquato S (2002b) Statistical description of microstructures. Ann Rev Mater Res 32:77-111 Torquato S, Beasley JD, Chiew YC (1988) Two-point cluster function for continuum percolation. J Chem Phys 88:6540-6547 Torres-Arguelles V, Oleschko K, Tarquis AM, Korvin G, Gaona C, Parrot JF, Ventura-Ramos E (2010) Fractal metrology for biogeosystems analysis. Biogeosci 7:3799-3815 Treimer W, Strobl M, Hilger A (2001 ) Development of a tuneable channel cut crystal. Phys Lett A 289:151-154 Triolo F, Triolo A, Agamalian M, Lin J-S, Heenan RK, Lucido G, Triolo R, (2000) Fractal approach in petrologyxombining ultra small angle, small angle and intermediate angle neutron scattering. J Appi Crystallogr 33:863-866 Triolo R, Lo Celso F, Gorgoni C, Pallante Ρ, Schwahn D, Baron M (2006) Mesoscopic structure of marble determined by combined USANS and SANS. J Neutron Res 14:59-67 Turchin VF (1965) Slow Neutrons. Sivan Press, Jerusalem Uca O, Bouman WG, Rekveldt M T (2003) Model calculations for the spin-echo small-angle neutron-scattering correlation function. J Appi Crystallogr 36:109-116 Udupa JK, Saha PK (2003) Fuzzy connectedness and image segmentation. Ρ IEEE 91:1649-1669 Urban JJ, TalapinDV, Shevchenko EV, Murray CB (2006) Self-assembly of PbTe quantum dots into nanocrystal superlattices and glassy films. J Am Chem Soc 128:3248-3255 Uthayakumar R, Prabakar GA, Azis SA (2011) Fractal analysis of soil pore variability with two dimensional binary images. Fractals 19:401-406 Velasquez R, Marasteanu MO, Labuz JF (2010) Microstructure characterization of asphalt mixtures with 2-and 3-Point correlation functions. Road Materials and Pavement Design 11:251-272 Venkataramanan L, Hurlimann MD, Tarvin JA, Fellah K, Acero-Allard D, Seleznev NV (2014) Experimental study of the effects of wettability and fluid saturation on muclear magnetic resonance and dielectric measurements in limestone. Petrophys 55:572-586 Vervoort RW, Cattle SR (2003) Linking hydraulic conductivity and tortuosity parameters to pore space geometry and pore-size distribution. J Hydrol 272:36^-9 Villa M, Baron M, Hainbuchner M, Jericha E, Leiner V, Schwahn D, Seidl E, Stahn J, Rauch H. (2003) Optimisation of a crystal design for a Bonse-Hart camera. J Appi Crystallogr 36:769-773 Vineyard GH (1954) Multiple scattering of neutrons. Phys Rev 96:93-98 Virgin B, Haslund E, Hilfer R (1996) Rescaling relations between two- and three-dimensional local porosity distributions for natural and artificial porous media. Physica A 232:1-20 Vlcek L, Zhang Z, Machesky ML, Fenter P, Rosenqvist J, Wesolowski DJ, Anovitz LM, Predota M, Cummings PT (2007) Electric double layer at metal oxide surfaces: Static properties of the cassiterite-water interface. Langmuir 23:4925^1937

Characterization

and Analysis

of Porosity and Pore

Structures

163

Vogtt K, Siebenbürger M, Clemens D, Rabe C, Lindner P, Russina M, Fromme M, Mezei F, Ballauff M (2014) A new time-of-flight small-angle scattering instrument at the Helmholtz-Zentrum Berlin:V16/VSANS. J Appi Crystallogr 47:237-244 Vonk CG (1976) On two methods of determination of particle size distribution functions by means of smallangle X-ray scattering. J Appi Crystallogr 9:433^-40 Wald MJ, Vasilic B, Saha PK, Wehrli FW (2007) Spatial autocorrelation and mean intercept length analysis of trabecular bone anisotropy applied to in vivo magnetic resonance imaging. Med Phys 34:1110-1120 Wang M, Pan Ν (2008) Predictions of effective physical properties of complex multiphase materials. Mater Sei Eng R 63:1-30 Wang MR, Pan N, Wang JK, Chen SY (2007) Mesoscopic simulations of phase distribution effects on the effective thermal conductivity of microgranular porous media. J Colloid Interf Sci 311:562-570 Wang H-W, Anovitz LM, Burg A, Cole DR, Allard LF, Jackson AJ, Stack AG, Rother G (2013) Multi-scale characterization of pore evolution in a combustion metamorphic complex, Hatrurim basin, Israel: Combining (ultra) small-angle neutron scattering image analysis. Geochim Cosmochim Acta 121:339362 Wardlaw, NC, Cassan, JP (1979) Oil recovery efficiency and the rock-pore properties of some sandstone reservoirs. Bull Can Petrol Geol 27:117-138 Warne STJ (1981) Staining Techniques. In: Encyclopedia of Mineralogy. Frye, Κ (ed). Springer ρ 484^-89 Washington AL, Li X, Schofield AB, Hong Κ, Fitzsimmons MR, Dalgliesh R, Pynn R (2014) Inter-particle correlations in a hard-sphere colloidal suspension with polymer additives investigated by Spin Echo Small Angle Neutron Scattering (SESANS): Soft Matter 10:3016-3026 Waxman MH, Smits LJM (1968). Electrical conductivities in oil-bearing shaly sands. SPE J 8:107-122 Webb PA (2001) An Introduction to the Physical Characterizations of Materials by Mercury Porosimetry with Emphasis on Reduction and Presentation of Experimental Data. Micromeritics Instrument Corp, Norcross, GA Webb PA, Orr C (1997) Analytical Methods in Fine Particle Technology. Micromeritics Instrument Corp Norcross, GA Wehrli FW, Hwang SN, Ma JF, Song HK, Ford JC, Haddad JG (1998) Cancellous bone volume and structure in the forearm: Noninvasive assessment with M R microimaging and image processing. Radiology 206:347357 Weigel R, Läuger J, Richtering W, Lindner Ρ (1996) Anisotropic small angle light and neutron scattering from a lyotropic lamellar phase under shear. J Phys II France 6:529-542 Welton JE (2003) SEM Petrology Atlas, 2nd Edition. Methods in Exploration Series, Am Assoc Petrol Geol Method Explor Ser 237 Wheeler HA (1896) Clay Deposits. Missouri Geological Survey, Vol 11, Jefferson City, M O Wheeler HE (1958) Time-stratigraphy. AAPG Bull, 42:1047-1063 White JS, Hinkov V, 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 YBa 2 Cu 3 0 7 . Phys Rev Lett 102:097001 Wiedenmann A (2005) Polarized SANS for probing magnetic nanostructures. Physica Β 356:246-253 W i e n e r N (1930) Generalized Harmonic Analysis. Acta Math 55:117-258 Wiener Ν (1964) Time Series. MIT Press, Cambridge, Massachusetts. Wignall GD, Bates FS (1987) Absolute calibration of small-angle neutron scattering data. J Appi Crystallogr 20:28^10 Wignall GD, Littreil KC, Heller WT, MelnichenkoYB, Bailey KM, Lynn GW, Myles DA, Urban VS, Buchanan MV, Selby DL, Butler PD (2012) The 40 m general purpose small-angle neutron scattering instrument at Oak Ridge National Laboratory. J Appi Crystallogr 45:990-998 Wildenschild D, Sheppard AP (2013) X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv Water Resour 51:217-246 Williams CE, Nierlich M, Cotton JP, Jannink G, Boue F, Daoud M, Farnoux B, Picot C, de Gennes P-G, Rinaudo M, Moan M, Wolff C (1979) Poly electrolyte solutions - intrachain and interchain correlations observed by SANS. J Polym Sci Polym Lett Ed 17:379-384 Wolff M, Magerl A, Zabel H (2005) Structure of polymer micelles close to the solid interface - A grazing incidence small angle neutron scattering study. Eur Phys J E 16:141-145 Wong PZ, Howard J, Lin J-S (1986) Surface roughening and the fractal nature of rocks. Phys Rev Lett 57:637640 Wu Q, Borkovec M, Sticher H (1993) On particle-size distributions in soils. Soil Sci Soc Am J 57:883-890 Wunnicke O, Müller-Buschbaum Ρ, Wolkenhauer M, Lorenz-Haas C, Cubitt R, Leiner V, Stamm M (2003) Stabilization of thin polymeric bilayer films on top of semiconductor surfaces. Langmuir 19:8511-8520 Xia F, Zhao J, Etschmann BE, Brugger J, Garvey CJ, Rehm C, Lemmel H, Ilvasky J, Han YS, Pring A (2014) A characterization of porosity in sulfide ore minerals: A US ANS/SANS study. Am Mineral 99:2398-2404

164

Anovitz & Cole

Yin XL, Chen W, To A, McVeigh C, Liu W K (2008) Statistical volume element method for predicting micro structure-constitutive property relations. Comput Method Appi Mech Eng 197:3516-3529 Zamora-Castro SA, Oleschko Κ, Flores L, Ventura E, Parrot JF (2008) Fractal mapping of pore and solid attributes. Vadose Zone J 7:473-492 Zehavi I, Zheng Z, Weinberg DH, Frieman JA, Berlind AA, Blanton MR, Scoccimarro R, Sheth RK, Strauss MA, Kayo I, SutoY, FukugitaM, NakamuraO, Bahcall NA, Brinkmann J, Gunn JE, Hennessy GS, Ivezic E, Knapp GR, Loveday J, Meiksin A, Schlegel DJ, Schneider DP, Szapudi I, Tegmark M, Vogeley MS, York DG (2005) The luminosity and color dependence of the galaxy correlation function. Astrophys J 630:1-27 Zeng Y, Gantzer CJ, Payton RL, Anderson SH (1996) Fractal dimension and lacunarity of bulk density determined with X-ray computed tomography. Soil Sci Soc Am J 60:1718-1724 Zhang S, Klimentidis RE (2011) Porosity and permeability analysis on nanoscale FIB_SEM 3D imaging of shale rock. Presentation at the International Symposium of the Society of Core Analysts held in Austin, Texas, U S A 18-21 September, 2011, http://www.scaweb.org/assets/papers/2011_papers/SCA2011-30. pdf Zhang Z, Fenter P, Cheng L, Sturchio NC, Bedzyk MJ, Predota M, Bandura A, Kubicki JD, Lvov SN, Cummings Ρ, Chialvo AA, Ridley MK, Benezeth P, Anovitz L, Palmer DA, Machesky ML, Wesolowski DJ (2004) Ion adsorption at the rutile-water interface:Linking molecular and macroscopic properties. Langmuir 20:4954^1969 Zhang Z, Fenter P, Cheng L, Sturchio NC, Bedzyk MJ, Machesky ML, Anovitz LM, Wesolowski DJ (2006) Zn 2+ and Sr 2+ adsorption at the T i 0 2 (110)-electrolyte interface:Influence of ionic strength, coverage, and anions. J Colloid Interf Sci 295:50-64 Zhang Z, Fenter P, Wesolowski DJ, Anovitz LM (2008) Probing the cassiterite (110)-water interface structure with X-ray scattering method. Geochim Cosmochim Acta 72:A1091-A1091 Supplement: 1 Zhou Z, Wu P, Chu, Β (1991) Cationic surfactant induced fractal silica aggregatesi light scattering study. J Coll Int Sci 146:541-555 Zhu Y (2005) Modern Techniques for Characterizing Magnetic Materials, Kluwer Academic Publishers, Boston, 65-103

5

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 stackag @ornl. gov

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 microfiuidics 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 hydrologie 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 1529-6466/15/0080-0005505.00

http://dx.doi.org/10.2138/rmg.2015.80.05

l a a i ^ ü J ! ! . · ©2015, Stack. This w o r k is licensed u n d e r the Creative C o m m o n s A t t r i b u t i o n - N o n C o m m e r c i a l - N o D e r i v s 3.0 License.

166

Stack

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). Mäher 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 (BaS0 4 ) 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

Precipitation

in Pores: A Geochemical

Frontier

167

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 (CaC0 3 ) 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 C 0 2 started (Mito et al. 2008). In a site in west Texas where C 0 2 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

168

Stack

thus far (for the available porosity) has been during C 0 2 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 C 0 2 injection tests in basalt in Iceland. One caution is that in the basalt injection tests have all been relatively small amounts of C 0 2 (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 C 0 2 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 C 0 2 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, BaS0 4 , calcite, CaC0 3 , as well as covalently bonded phases such as silica or quartz, Si0 2 and iron oxides, Fe 2 0 3 ) (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 C0 2 -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]

Precipitation

in Pores: A Geochemical

Frontier

169

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 (CaS0 4 ) 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

Stack

170

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

0

V

/--"Λ /

• \\ '. era.,

Λ

3 r a l n

W

..// T í grain) φ ( grain

d)

X f ""

y

r

era

, Ι9™,1 K ^ J

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.

Precipitation

in Pores: A Geochemical

171

Frontier

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 C0 2 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 C0 2 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 = Cd2,

(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 S 10"s 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 C0 2 storage basin (Fig. 3). Moreover, precipitation reactions may have non-linear effects on permeability: Tartakovsky et al. (2008) found that an impermeable layer of CaC0 3 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):

0.00

0,05

• ι 0,10

ι 0.15

ι 0.20

ι 0,25

ι 030

-H 0,35

Porosity Figure 3. Log permeability as a function of porosity in two different formations across a potential C 0 2 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.

Stack

172

log10 k = d + ex grain size H 1- / χ 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 C0 2 , 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 C0 2 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

Precipitation

in Pores: A Geochemical

Frontier

173

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 ¿-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 tookthe 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 (CaC0 3 ) was circulated past the CPG, and the small angle X-ray

Stack

174 a) 3000

b) 200

2000

E 100 3

IDOO O -11)00

Sí -100

-2000

-3000 0.0

-200

1.0

2.0

Nucleus Radius (nrn)

3.

0.Q

Nanoporeeffect: 'lílií - - - prCirPiCHiQrt nhibltfcn 1.0

2.0

3.C

Nucleus Radius (nm)

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 pm 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

Precipitation in Pores: A Geochemical Frontier

c)

N a t i v e P o r o u s Silica

175

Functionalized changes in micropores

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]

(Si0 2 ), 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

176

Stack

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 (pKJ, 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 pK a 's of the octahedral aluminum surface sites on nanotubular imogolite (Al 2 Si0 3 (0H) 4 ) and found they are one pK a 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 pK a 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/nm 2 in a nanoporous amorphous silica, Mobil Composition of Matter No. 41 (MCM-41), versus 5 - 8 sites per/nm 2 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 (a-Fe 2 0 3 ) 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 (Ti0 2 ) 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

Precipitation

in Pores: A Geochemical

Frontier

177

Distance (nanometer) Figure 6. Atomic-scale electron density as a function of distance from a barite {001} mineral-water interface. Barite (negative numbers in .v-axis) shows surface relaxation on the order of three monolayers. Water (positive numbers in .v-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 M D data. M D 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 °C (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 (AG) is the sum of two terms (De Yoreo and Vekilov 2003):

where AGblllk is the contribution from the bulk volume of precipitated material and AGsllrface is the contribution from the interfacial energy. For a heterogeneous nucleus, the free energy conserved for a given radius is: ^Q

=

2ro~3(2.303kBr • SI) 3v;„

(4)

where r is the radius of the nucleus, kB is Boltzmann's constant, Τ 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(ac¡¡acoJKsp). In Figure 4a, this term is calculated for SI = 0.76 and molar volume of calcite (3.69 x 10"5 m3/mol). For heterogeneous nucleation, the surface energy term is (Fig. 4a): AGsmface = ro-2(2ylc +y sc -y l s ),

(5)

178

Stack

where ylc is the interfacial energy between the precipitating crystal and the surrounding liquid, ysc is the energy between the substrate and the crystal, and yls is that between the substrate and the liquid. The sum of these terms will be referred to as the apparent interfacial energy, γ'. In Figure 4a, this term is calculate using an apparent interfacial energy for calcite of γ' = 0.036 J/m 2 (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, AGblllk would not change. Within AGsllrfilce, 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 S AM-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

Precipitation

in Pores: A Geochemical

Frontier

179

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: (6) where S¿ 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 (m 3 /mol), γ is the interfacial energy of the precipitating phase (J/m2), R is the ideal gas constant (J/mol/K), Τ 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 x 105 m 3 /mol; ysl = 0.094 J/m 2 (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.

Stack

180

EFFECTS IN SOLUTION 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:

ψ = ψ0 exp(-Kif)

(7)

where ψ is the potential at some distance, d, from the surface, ψ 0 is the potential at the surface (or Stern or beta layers if using a two or three layer model), and κ is the Debye parameter: ^2F2/x103A ss 0 RT

= 3.29xl0 9 V7

(m 1 )

(8)

where F is the Faraday constant (96,485 C/mol), I is the ionic strength (in mol/L), ε is the dielectric constant of water (78.4 at 298 Κ), ε 0 is the permittivity of a vacuum (8.854 χ IO"12 C 2 /N/m 2 ), R is the ideal gas constant (8.314 J/mol/K) and Τ 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 ( / = IO"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 μπι 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

Precipitation

in Pores: A Geochemical

Frontier

181

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: c+ =c^exp f - t f V ) c_ =c^exp ί ^ ψ ί RT J yRT,

C9Ì (?)

where c + and c. are the concentrations of cations (c + ) and anions (c_) within the electric double layer, c^ is the concentration ofthat ion in bulk solution, ζ 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 7x the bulk concentration (cjcj). 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 2x 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.

Stack

182

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 χ IO"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 χ aqueous calcium concentration and 0.024 χ 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

0.5

I I llllllj—I I I I'lll^—I llllllij

"TTTTTTj—I I llllllj

SI = 0.75

0.001

[Ca3+]/[COa2 ] 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 f r o m that. [Used by permission of John Wiley & Sons, f r o m Stack AG (2014) Next-generation models of carbonate mineral growth and dissolution. Greenhouse Gas Science & Technology, Vol 4, p. 278-288]"'

Precipitation in Pores: A Geochemical Frontier

183

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 (Parez and Predota 2012) and especially close to an interface (Parez 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

t Ηβ",

\

1

os e

Sr"Ä

Off

ψ

sr"J® à, - . . HB"

os*,

Β»·'* Pb"

ïHb

\

Ì

>

Ca1

?

t

s v

IS

H.O

0

_J 0.5

I

I

I 1.0

I

I

I

J

I

I

1.5

I

I

L_ 2.0

Effective ionic radius (Â) 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, OS atls is outer-sphere and OS ext 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 (OOl)-water interface. Langmuir, Vol. 26, p. 16647-16651]

184

Stack

Equation (7). This discrepancy may arise from a couple o f possibilities, such as the detection limit o f ions using the X R and R A X R 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 o f a dielectric " c o n s t a n t " 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 o f electrical potential into solution (Parez et al. 2 0 1 4 ) . Typically in M D simulations ordered water is also observed less than one nanometer from the surface (Fig. 6 ) ( S t a c k and Rustad 2 0 0 7 ; Bourg and Sposito 2 0 1 1 ; Fenter et al. 2 0 1 3 ) . This is consistent with the X R , although these two techniques are seldom in perfect agreement, likely stemming from the uncertainty in the M D models in addition to those o f the X R . I f 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 T h e last subject that will be addressed here is that o f transport. This is the subject o f other articles in this volume (Steefel et al. 2 0 1 5 , 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 o f the solution. For example, Plummer et al. ( 1 9 7 8 ) found that below pH - 5 . 0 , the dissolution rate o f 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 2 0 0 6 ) . 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 o f a transport constraint. This was demonstrated recently by Molins et al. ( 2 0 1 2 ) who showed the dissolution rate o f calcite as a function o f darcy velocity, or net fluid velocity (Fig. 10). Molins et al. ( 2 0 1 2 ) show that the transition from a transportlimited to a surface chemistry-limited reaction is not sharp, but is a gradient. Furthermore,

Specific Discharge (m/s) 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).

Precipitation

in Pores: A Geochemical

Frontier

185

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 χ IO6 s"1 and 3.6 χ IO7 s"1 (Bracco et al. 2013), whereas those for attachment of magnesium and carbonate ions to magnesite (MgC0 3 ) are 4.0 χ 105 s"1 and 2.0 χ IO6 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 CaCl 2 and Na 2 C0 3 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. 1 lb), 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

186

Stack

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, CaCl 2 and N a 2 C 0 3 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 Ρ (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,

Precipitation

in Pores: A Geochemical

Frontier

187

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 C 0 2 sequestration sponsored by the Center for Nanoscale Control of Geologic C0 2 , 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.

REFERENCES Anovitz LM, Cole DR (2015) Characterization and analysis of porosity and pore structures. Rev Mineral Geochem 80:61-164 Anovitz LM, Cole DR, Rother G, Allard LF, Jackson AJ, Littrel KC (2013) Diagenetic changes in macro- to nano-scale porosity in the St. Peter Sandstone: An (ultra) small angle neutron scattering and backscattered electron imaging analysis. Geochim Cosmochim Acta 102:280-305 Arey JS, Seaman JC, Bertsch PM (1999) Immobilization of uranium in contaminated sediments by hydroxyapatite addition. Environ Sci Technol 33:337-342 Beazley MJ, Martinez RJ, Sobecky PA, Webb SM, Taillefert M (2007) Uranium biomineralization as a result of bacterial phosphatase activity: insights from bacterial isolates from a contaminated subsurface. Environ Sci Technol 41:5701-5707 Berner RA, Lasaga AC, Garrels RM (1983) The carbonate-silicate geochemical cycle and its effect on atmospheric carbon dioxide over the past 100 million years. Am J Sci 283:641-683 Bethke CM (1986) Inverse hydrologie analysis of the distribution and origin of Gulf Coast-type geopressured zones. J Geophy Res 91:6535-6545 Bleam W F (2011) Soil and Environmental Chemistry. Academic Press, Amsterdam. Bloch S (1991) Empirical prediction of porosity and permeability in sandstones. Am Assoc Pet Geol Bull 75:1145-1160 Boerner LK (2013) Sewage plants struggle to treat wastewater produced by fracking operations. Chemical & Engineering News, March 18, 2013 Bond DL, Davis JA, Zachara JM (2008) Uranium(VI) release from contaminated vadose zone sediments: estimation of potential contributions from dissolution and desorption. In: Barnett, MO, Kent, DB (eds), Adsorption of Metals by Geomedia II. Academic Press, San Diego, CA, ρ 379-420 Borgia A, Pruess K, Kneafsey TJ, Oldenburg CM, Pan L (2012) Numerical simulation of salt precipitation in the fractures of a C0 2 -enhanced geothermal system. Geothermics 44:13-22 Bourg IC, Sposito G (2011) Molecular dynamics simulations of the electrical double layer on smectite surfaces contacting concentrated mixed electrolyte (NaCl-CaCl 2 ) solutions. J Coll Int Sci 360:701-715 Bourg IC, Steefel C (2012) Molecular dynamics simulations of water structure and diffusion in silica nanopores. J Phys Chem C 116:11556-11564 Bracco JN, Grantham MC, Stack AG (2012) Calcite growth rates as a function of aqueous calcium-tocarbonate ratio, saturation index, and inhibitor concentration: Insight into the mechanism of reaction and poisoning by strontium. Cryst Growth Des 12:3540-3548 Bracco JN, Stack AG, Steefel CI (2013) Upscaling calcite growth rates from the mesoscale to the macroscale. Environ Sci Technol 47:7555-7562

188

Stack

Bracco JN, Stack AG, Higgins S R (2014) Magnesite step growth rates as a function of the aqueous magnesiumxarbonate ratio. Cryst Growth Des 1 4 : 6 0 3 3 - 6 0 4 0 Carey, J W (2013) Geochemistry of wellbore integrity in C 0 2 sequestration: Portland cement-steel-brine-C0 2 interactions. Rev Mineral Geochem 7 7 : 5 0 5 - 5 3 9 Chen Y, Wei L, Mastalerz M, Schimmelmann A (2015) The effect of analytical particle size on gas adsorption porosimetry of shale. Intl J Coal Geol. 1 3 8 : 1 0 3 - 1 1 2 Chou L, Garrels RM, Wollast R (1989) Comparative study of the kinetics and mechanisms of dissolution of carbonate minerals. Chem Geol 7 8 : 2 6 9 - 2 8 2 Cohen Y, Rothman DH (2015) Mechanisms for mechanical trapping of geologically sequestered carbon dioxide. Proc. R. Soc. A 471:2010853 Curti E, Fujiwara K, Iijima K, Tits J, Cuesta C, Kitamura A, Glaus MA, Müller W (2010) Radium uptake during barite recrystallization at 23 ± 2 °C as a function of solution composition: An experimental 1 3 3 Ba and 2 2 6 Ra tracer study. Geochim Cosmochim Acta 7 4 : 3 5 5 3 - 3 5 7 0 Davis, JA, Kent D B (1990) Surface complexation modeling in aqueous geochemistry. Rev Mineral 2 3 : 1 7 7 - 2 6 0 De Yoreo JJ, Vekilov PG (2003) Principles of Crystal Nucleation and Growth. Rev Mineral Geochem 5 4 : 5 7 - 9 3 Drever JI, Clow DW (1995) Weathering rates in catchments. Rev Mineral 3 1 : 4 6 3 - 4 8 3 Emmanuel S, Ague J J (2009) Modeling the impact of nano-pores on mineralization in sedimentary rocks. Water Resour Res 4 5 : W 0 4 4 0 6 Emmanuel S, Anovitz LM, Day-Stirrat R J (2015) Effects of coupled chemo-mechanical processes on the evolution of pore-size distributions in geological media. Rev Mineral Geochem 80:45-60 Emmanuel S, Ague J J , Walderhaug O (2010) Interfacial energy effects and the evolution of pore size distributions during quartz precipitation in sandstone. Geochim Cosmochim Acta 7 4 : 3 5 3 9 - 3 5 5 2 Fanizza MF, Yoon H, Zhang C, Oostrom M, Wietsma TW, Hess NJ, Bowden M E , Strathmann TJ, Finneran KT, Werth CJ (2013) Pore-scale evaluation of uranyl phosphate precipitation in a model groundwater system. Water Resour Res 4 9 : 8 7 4 - 8 9 0 Fenter P, Lee SS (2014) Hydration Layer Structure at Solid-Water Interfaces. M R S Bulletin 3 9 : 1 0 5 6 - 1 0 6 1 Fenter P, McBride MT, Srajer G, Sturchio NC, Bosbach D (2001) Structure of Barite (001)- and (210)-Water Interfaces. J Phys Chem Β 1 0 5 : 8 1 1 2 - 8 1 1 9 Fenter P, Kerisit S, Raiteri P, Gale J D (2013) Is the Calcite-Water Interface Understood? Direct Comparisons of Molecular Dynamics Simulations with Specular X-ray Reflectivity Data. J Phys Chem C 1 1 7 : 5 0 2 8 - 5 0 4 2 Fernandez-Martinez A (2009) Physics of natural nanoparticles - water interfaces: chemical reactivity and environmental implications. PhD Dissertation, Observatoire de Grenoble et Laboratoire de Géophysique Interne et Tectonophysique, Grenoble, France Fernández-Martínez A, Hu Y, Lee B, Jun Y-S, Waychunas G A (2013) In situ determination of interfacial energies between heterogeneously nucleated C a C 0 3 and quartz substrates: Thermodynamics of C 0 2 mineral trapping. Environ Sci Technol 4 7 : 1 0 2 - 1 0 9 Ferrar KJ, Michanowicz DR, Christen CL, Mulcahy N, Malone SL, Sharma R K (2013) Assessment of effluent contaminants from three facilities discharging marcellus shale wastewater to surface waters in Pennsylvania. Environ Sci Technol 4 7 : 3 4 7 2 - 3 4 8 1 Fogler S (2006). Elements of Chemical Reaction Engineering (4th ed.). Pearson Education, Upper Saddle River, NJ. Fountain H (2015) Turning carbon dioxide into rock, and burying it. New York Times, February 9 th , 2015. Freeze RA, Cherry J A (1979) Groundwater. Prentice Hall, Englewood Cliffs, NJ. ρ 604 Frenier WW, Ziauddin M (2008) Formation, removal, and inhibition of inorganic scale in the oilfield environment. Society for Petroleum Engineers, Richardson, T X . Funk J (2014) Ohio acknowledges connection between hydraulic fracturing and Youngstown quakes, will require seismic testing near known fault lines. The Plain Dealer, April 11th, 2014. Hedges LO, Whitelam S (2012) Nucleation barriers in the Ising model suggest a strategy for patterning a surface to maximally enhance nucleation. Soft Matter 8:8624—8635 Gebrehiwet TA, Redden GD, FujitaY, B e i g M S , Smith R W (2012) The effect of the C 0 3 2 to Ca 2 + ion activity ratio on calcite precipitation kinetics and Sr 2 + partitioning. Geochem Trans 13:1 Gibson-Poole CM, Svendsen L, Underschultz J, Watson MN, Ennis-King J, van Ruth PJ, Nelson EJ, Daniel RF, Cinar Y (2008) Site characterisation of a basin-scale C 0 2 geological storage system: Gippsland Basin, southeast Australia. Environ Geol 5 4 : 1 5 8 3 - 1 6 0 6 Giesche H (2006) Mercury porosimetry: a general (practical) overview. Part Part Syst Charact 2 3 : 9 - 1 9 Godinho J R A , Stack AG (2015) Growth kinetics and morphology of barite crystals derived from face-specific growth rates. Cryst Growth Des 1 5 : 2 0 6 4 - 2 0 7 1 doi: 10.1021/cg501507p Gouze B, Cambedouzou J, Parres-Maynadie S, Rébiscoul, D (2014) How hexagonal mesoporous silica evolves in water on short and long term: Role of pore size and silica wall porosity. Microporous Mesoporous Mater. 183:168-176

Precipitation

in Pores: A Geochemical

Frontier

189

Hedges LO, Whitelam S (2012) Patterning a surface so as to speed nucleation from solution. Soft Matter 8:8624-8635 Jamtveit, B, Kobchenko M, Austrheim H, Malthe-S0rensen A, R0yne A, Svensen H (2011) Porosity evolution and crystallization-driven fragmentation during weathering of andesite. J Geophys Res 116:B12204 Kubicki JD, Sofo JO, Skelton AA, Bandura AV (2012) A New hypothesis for the dissolution mechanism of silicates. J Phys Chem C 116:17479-17491 Lee SS, Fenter P, Park C, Sturchio NC, Nagy NL (2010) Hydrated cation speciation at the muscovite (001)-water interface. Langmuir 26:16647-16651 Liu C, Shang J, Kerisit S, Zachara JM, Zhu W (2013) Scale-dependent rates of uranyl surface complexation reaction in sediments. Geochim Cosmochim Acta 105:326-341 Mäher K, Steefel CI, White AF, Stonestrom DA (2009) The role of reaction affinity and secondary minerals in regulating chemical weathering rates at the Santa Cruz Soil Chronosequence, California. Geochim Cosmochim Acta 73:2804-2831 Martin AJ, Crusius J, McNee JJ, Yanful EK (2003) The mobility of radium-226 and trace metals in pre-oxidized subaqueous uranium mill tailings. Appi Geochem 18:1095-1110. Melcer A, Gerrish HW (1996) Effects of formation damage on injection operations and on pressure transient tests. In: Deep Injection Disposal of Hazardous and Industrial Waste; Apps JA, Tsang C (eds) Academic Press: San Diego, CA, ρ 277-286 Metz Β, Davidson O, de Coninck Η, Loos M, Meyer L (2005) IPCC Special Report on Carbon Dioxide Capture and Storage. Cambridge University Press: Cambridge, U.K. Mito S, Xue Ζ, Ohsumi Τ (2008) Case study of geochemical reactions at the Nagaoka C 0 2 injection site, Japan. Int J Greenh Gas Control 2:309-318 Molins S, Trebotich D, Steefel CI, Shen C (2012) An investigation of the effect of pore scale flow on average geochemical reaction rates using direct numerical simulation. Water Resour Res 48:W03527 Morse JW, Zullig JJ, Bernstein LD, Millero FJ, Milne P, Mucci A, Choppin GR, (1985) Chemistry of calcium carbonate-rich shallow water sediments in the Bahamas. Am J Sci 285:147-185 Naftz DL, Feltcorn EM, Fuller CC, Wilhelm RG, Davis JA, Morrison SJ, Freethey GW, Piana MJ, Rowland RC, Blue JE (1998) Field demonstration of permeable reactive barriers to remove dissolved uranium from groundwater, Fry Canyon, Utah. EPA Report 402-C-00-001 Parez S, Predota M (2012) Determination of the distance-dependent viscosity of mixtures in parallel slabs using non-equilibrium molecular dynamics. Phys Chem Chem Phys 14:3640-3650 Parez S, Predota M, Machesky M (2014) Dielectric properties of water at rutile and graphite surfaces: effect of molecular structure. J Phys Chem C 118:4818-4834 Parks GA (1990) Surface Energy and Adsorption at Mineral/Water Interfaces: An Introduction. Rev Mineral 23:133-176 Plummer LN, Wigley TML, Parkhurst DL (1978) The kinetics of calcite dissolution in C0 2 -water systems at 5° to 60 °C and 0.0 to 1.0 atm C 0 2 . Am J Sci 278:179-216 Pontbriand CW, Sohn RA (2014) Microearthquake evidence for reaction-driven cracking within the TransAtlantic Geotraverse active hydrothermal deposit. J. Geophys Res Solid Earth 119:822-839 Prieto M, Putnis A, Fernandez-Diaz L (1990) Factors controlling the kinetics of crystallization: supersaturation evolution in a porous medium. Application to barite crystallization. Geol Mag 6:485^-95 Putnis A, Mauthe G (2001) The effect of pore size on cementation in porous rocks. Geofluids 1:37-41 Putnis A, Prieto M, Fernandez-Diaz L (1995) Fluid supersaturation and crystallization in porous media. Geol Mag 1:1-13 Reeves D, Rothman DH (2012) Impact of structured heterogeneities on reactive two-phase porous flow. Phys Rev E 86:031120 Ridley MK, Machesky ML, Kubicki JD (2013) Anatase nanoparticle surface reactivity in NaCl media: A C D - M U S I C model interpretation of combined experimental and Density Functional Theory studies. Langmuir 29:8572-8583 Riley RG, Zachara JM, Wobber FJ ( 1992) Chemical Contaminants on DOE Lands and Selection of Contaminant Mixtures for Subsurface Science Research. U.S. Department of Energy, D O E / E R - 0 5 4 7 T DE92 014826 Ruiz-Agudo E, Putnis CV, Wang L, Putnis A (2011 ) Specific effects of background electrolytes on the kinetics of step propagation during calcite growth. Geochim Cosmochim Acta 75:3803-3814 Sahai N, Sverjensky DA (1997) Evaluation of internally consistent parameters for the triple-layer model by the systematic analysis of oxide surface titration data. Geochim Cosmochim Acta 61:2801-2826 Scherer GW (1999) Crystallization in Pores. Cem Concr Res 29:1347-1358 Singer DM, Guo H, Davis JA (2014) U(VI) and Sr(II) batch sorption and diffusion kinetics into mesoporous silica (MCM-41). Chem Geol 390:152-163 Skoumal RJ, Budzinski MR, Currie BS (2015) Earthquakes induced by hydraulic fracturing in Poland Township, Ohio. Bull Seismol Soc Am, doi: 10.1785/0120140168

190

Stack

Stack AG (2014) Next generation models of carbonate mineral growth and dissolution. Greenhouse Gas Sci Technol 4:278-288 Stack AG, Rustad JR (2007) Structure and Dynamics of Water on Aqueous Barium Ion and the {001} Barite Surface. J Phys Chem C 111:16387-16391 Stack AG, Grantham M (2010) Growth rate of calcite steps as a function of aqueous calcium-to-carbonate ratio: Independent attachment and detachment of calcium and carbonate ions. Cryst Growth Des 10:1409-1413 Stack AG, Kent PRC (2015) Geochemical reaction mechanism discovery from molecular simulation. Environ Chem 12:20-32 Stack AG, Erni R, Browning ND, Casey W H (2004) Pyromorphoite growth on lead-sulfide surfaces. Environ Sci Technol 38:5529-5534 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. Environ Sci Technol 48:6177-6183 Steefel CI, Druhan JL, Maher Κ (2014) Modeling coupled chemical and isotopie equilibration rates. Proc Earth Planet Sci 10:208-217 Steefel CI, Beckingham LE, Landrot G (2015) Micro-Continuum Approaches for Modeling Pore-Scale Geochemical Processes. Rev Mineral Geochem 80:217-246 Stumm W, Morgan JJ (1996) Aquatic Chemistry, Chemical Equilibria and Rates in Natural Waters. John Wiley and Sons, Inc., New York. Sverjensky DA (2006) Prediction of the speciation of alkaline earths adsorbed on mineral surfaces in salt solutions. Geochim Cosmochim Acta 70:2427-2453 Swainson IP, Schulson EM (2004) Invasion of ice through rigid anisotropic porous media. J Geophys Res 109:B12205 Tartatakovsky AM, Redden G, Lichtner PC, Scheibe TD, Meakin Ρ (2008) Mixing-induced precipitation: Experimental study and multiscale numerical analysis. Water Resour Res 44:W06S04 Trainor TP, Chaka AM, Eng PJ, Newville M, Waychunas GA, Catalano JG, Brown GE Jr. (2004) Structure and reactivity of the hydrated hematite (0001) surface. Surf Sci 573:204-224 Verma A, Pruess Κ (1988) Thermohydrologic conditions and silica redistribution near high-level nuclear wastes emplaced in saturated geological formations. J. Geophys Res. 93:1159-1173 Warner NR, Christie CA, Jackson RB Vengosh A (2013) Impacts of shale gas wastewater disposal on water quality in western Pennsylvania. Environ Sci Technol 47:11849-11857. Wang H-W, Anovitz LM, Burg A, Cole DR, Allard LF, Jackson AJ, Stack AG, Rother G (2013) Multi-scale characterization of pore evolution in a combustion metamorphic complex, Hatrurim basin, Israel: Combining (ultra) small-angle neutron scattering and image analysis. Geochim Cosmochim Acta 121: 339-362 Wasylenki LE, Dove PM, Wilson DS, De Yoreo JJ (2005) Nanoscale effects of strontium on calcite growth: An in situ AFM study in the absence of vital effects. Geochim Cosmochim Acta 69:3017-3027 Wesolowski DJ, Ziemniak SE, Anovitz, LM, Machesky ML, Bénézeth Ρ, Palmer DA (2004) Solubility and surface adsorption characteristics of metal oxides. In: Palmer DA, Fernandez-Prini R., Harvey AH (eds) Aqueous Systems at Elevated Temperatures and Pressures Elsevier, London, ρ 493-595 White AF (2008) Quantitative approaches to characterizing natural chemical weathering rates. In: Kinetics of Water-Rock Interaction. Brantley, SL, Kubicki JD, White AF (eds) Springer: New York ρ 469-543 Wright KE, Hartmann Τ, Fujita Y (2011) Inducing mineral precipitation in groundwater by addition of phosphate. Geochem Trans 12:8 Yoon H, Valocchi AJ, Werth CJ, Dewers Τ (2012) Pore-scale simulation of mixing-induced calcium carbonate precipitation and dissolution in a microfluidic pore network. Water Resour Res 48:W02524 Zhang S, DePaolo DJ, Xu Τ, Zheng L (2013) Mineralization of carbon dioxide sequestered in volcanogenic sandstone reservoir rocks. Int J Greenh Gas Control 18:315-328 Zhang T, Gregory K, Hammack RW, Vidic RD (2014) Co-precipitation of radium with barium and strontium sulfate and its impact on the fate of radium during treatment of produced water from unconventional gas extraction. Environ Sci Technol 48:4596-4603 Zhao XS, Lu GQ, Whittaker AK, Millar GJ, Zhu HY (1997) Comprehensive study of surface chemistry of MCM-41 using 29 Si CP/MAS NMR, FTIR, Pyridine-TPD, and TGA. J Phys Chem Β 101:6525-6531 Zucker, HA (2014) A Public Health Review of High Volume Hydraulic Fracturing for Shale Gas Development. New York State Department of Health.

6

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 Chongxuan.¡[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 redox sensitive materials such as hematite (Zhang et al. 2013a) to provide reactive sites for surface 1529-6466/15/0080-0006505.00 I c Φε = 0:

(3)

Φτ 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).

Figure 6. Glass beads calcite spar crystals separation and labeling based on three-phase segmentation, (a) X M T cross-section (1.37 χ 1.37 mm 2 ) 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 (S,.), 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

258

Noiriel

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.

Figure 7. (a) Observation of fluid-rock interface shift in a limestone sample during three stages of dissolution (/]! 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 x 2 x 2 mm 3 ), 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.

Resolving

Time-dependent

Evolution

using X-ray Microtomography

259

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

M e c h a n i c a l aperture a m ( μτη )

Figure 8. (top) Detail of fracture void geometry ( 2 x 2 mm 2 ) obtained after image segmentation of different stages of a flow-through dissolution experiment, leading to an increase of the fracture wall surface roughness and aperture, (bottom) Fracture aperture distribution and comparison with a normal distribution (dotted line) at the different stages of experiment. Reprinted from Noiriel et al. (2013). [Used by permission of Elsevier, from Noiriel C, Gouze P. Made Β (2013) 3-D analysis of geometry and flow changes in a limestone fracture during dissolution. Journal of Hydrology, Vol. 486, 211-223, Fig. 3].

260

Noiriel

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 μτη. The distribution of porosity values between 0.42, 1.06, and 2.12 μτη 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 μτη. However, the micro-porous phase that was resolved at a resolution of 1.85 μτη (but not at 5.92 μτη) was shown to increase porosity, surface area, and pore network connectivity estimates (Fig. 10).

Resolving

Time-dependent

a)

Evolution

using X-ray Microtomography

b)

S

s

'•Λ '

» "

c)

•·'.

· ' ,

*

'

*

• 3 *

*'

;

"

-

. " ¿a' j * ' L- U ' •.'•£:. -; uf •• Λ>:

·

' î

•••

*

"

r.f ,> •* ' V . V T. um I t e gi ,•. e) - '

' 1 0 0 " .

' f

d)

Λ

* • Λ

'

i 4 ι

'

'

t

-

:

*-..

'.••".

.

, .

• - • ' Λ

•

261

•y

' t >

V '3

Vi.'·

v

' " · · *

'..

'

•- '•'•.••.> - / . 'i . ·"

' ·

'r :

-v •;· ft.'

,".,:.····>·.

y viy

j

•

.J'

. * 'ν

fe Λ-

"i . iti

n • : - » t-.

' iff S ì >

Figure 9. X M T slice of a limestone at different resolutions: (a) 2.12 μπι, (b) 1.06 μπι and (c) 0.42 μπι 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).

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 μπι (a) and 1.85 μπι (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, 4 0 3 ^ 1 1 , 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

262

Noiriel

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 C0 2 trapping 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 (Nal, 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 C0 2 -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 areaction front in diffusion-dominated systems does not necessarily result in a parabolic (t m ) dependence on time, particularly when feedback between porosity and the effective diffusion coefficient is involved. Navarre-Sitchler et al. (2009) quantified

Resolving Time-dependent Evolution using X-ray Microtomography

263

Figure 11. 3-D X M T 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 f r o m 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 COo-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 bio tic 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

264

Noiriel

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-spari tic 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 S0 2 . 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

Resolving Time-dependent Evolution using X-ray Microtomography

265

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: rmm =

= ±kr Sr=±

k, ^ Π [ α ι ] ' ( 1 - Ω Ύ ' ,

(5)

with kr and kr· the reaction and intrinsic kinetic rate constants, respectively (mol-m^-s"1), Sr the reactive surface area, a¡ the activity of the inhibitor and catalyst species i of the reaction, and Ω the saturation index ( Ω = IAP / Κ ), with Ksp the solubility and IAP the ion activity product, j, η 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 S B E T 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 - 5 0 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. X M T 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

266

Noiriel

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: v

f s

=%A at

(6)

with I (s the vector position of the fluid-solid interface and η 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): γV is Js

ts

yν Jv 'v dt Jt

(7)

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^-s"1) can also be calculated, assuming that the reactive surface area is known, i.e., kr =r / Sr. 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 Ω1ο(.), 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 pm-d"1.

Figure 12. (a) X M T truncated volume (1.56 χ 1.56 χ 1.56 mm 3 ) 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.

Resolving

Time-dependent

Evolution

using X-ray Microtomography

267

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

(8)

V At '

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 C0 2 of 1 bar at a

a (μιη) Ρ ι

α (μιη)

44 42

m 40

—38

---2000

4000

6000

—

36 34 32 30 28 26 24 22 20

y (μιη)

y (μιη)

Κ (mol.m 2 s-1) I 3.5 10-" 3.25 10-" 3.0 10-» ι 2.75 10-» 2.5 10-» m 2.25 10-» 2.0 10-» Ρ 1.75 10-» i 1.5 10-» 1.25 10-» — 1.0 io-» — 7.5 10-» — 5.0 10-» 1 — 2.5 10- » —0

•

y (μιη)

y (μιη)

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 ') 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 χ 4.91 μκα). Reprinted from Noiriel (2005).

268

Noiriel

constant flow rate Q of 10 cm3·!!"1. Solution p H was determined to an average of 3.9 and 5.5 at the inlet and outlet, respectively. A trapezoid fracture of 15 m m long and 7.5 m m 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 μτη (shorter base) to 45 μπι (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 X M T geometric surface area (see further discussion), it is possible to recalculate the apparent kinetic rate (k r ') and even a theoretical p H based on kinetic law. Taking S = Sr, for example, and the kinetic law obtained by Plummer et al. (1978) (kr ' = kfi^ + k2aco + k3) yields: S

pix

S,

(9)

and (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 f r o m advectivedispersive-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 f r o m the kinetic formulation used, the estimation of the surface area, the assumption of a continuum (i.e., constant p H 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 f r o m the pore scale to the sample-scale, could allow for a proper interpretation of the X M T 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

Resolving Time-dependent

Evolution using X-ray Microtomography

269

chemical reaction and advective transport. Molins et al. (2014) also compared pore-scale with continuum-sc ale 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 X M T 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

Figure 14. (a) Porosity-permeability relationship within a porous limestone during dissolution; η is the exponent of the power-law relationship between porosity and permeability: φ ~ k". [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 χ 10 χ 6.9 mm 3 .

270

Noiriel

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 = vL* / Dm and Da = kr L*2 / Dm, where ν 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 C 0 2 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).

Resolving Dì

Time-dependent Before

After

Evolution D2

Before

After

using X-ray Microtomography D3

271

Before

After

Figure 15. Porous network obtained with X M T 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 Ρ (2009) Experimental determination of porosity and permeability changes induced by injection of C 0 2 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 ζ = 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 PCOi = 1 bar, were injected at a flow rate Q = 100 cm3-h_1 in artificial, almost flat, limestone fractures of initial aperture - 2 5 pm. While the fracture injected with HCl shows the formation of dominant wormholes (Fig. 16a), the fracture injected with C0 2 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, PCOiand 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 C0 2 (~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

272

Noiriel Aperture (μιη)

Aperture (μιη)

800 750 700 650 600 550 500 450 400 350 300 250 200 E 150 E 100 = 50

Figure 16. Maps of fracture aperture (μπι) obtained with X M T (resolution 4.91 μπι). 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 Pco = lbar (inlet pH = 3.9). Reprinted from Noiriel (2005).

Figure 17. (a) Kinetics of calcite dissolution by HCl and P c 0 i = 1 bar as a function of pH, starting from an initial pH = 3.9, using equation from Plummer et al. 1978 far from equilibrium, (b) Relationship between kinetic rate and saturation index Ω. The results were obtained with CHESS (van der Lee 1998) assuming dissolution of calcite in open system.

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).

Resolving

Time-dependent

Evolution

using X-ray Microtomography

273

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 mineralogie ally 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). 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).

-Dp,=4>-T-DOJ,

(2)

where Doi is the diffusion coefficient of species i in water (or self-diffusion coefficient), and Dp ¡ is the pore diffusion coefficient (D ¡ = τ · D0¿). 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): τ = ( L / Lef.

(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:

dCtoti dt

dJj dx

=

(4)

where C t o t ¡ 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: 3φc . d f

dt

d^)

dx \

(5)

dx

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

(6)

The quantification of adsorption processes is commonly translated into distribution ratio values, Rd (L-kg"1): Rd,=^L,

Ci

(7)

where csllrf ¡ 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: α , = φ + ρd Rdj,

(8)

where p¿ is bulk dry density of the material. In that case, Equation (4) transforms into:

dt

dx I

' dx

(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:

dt

φ + ρdKD¡ dx2

Tournassat & Steefel

290

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 α-K D 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 pd = 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)= V ' exp

C ( 0 , ° ) —2C(0,0)x V δ+β+l '

D. λ2 \ a-L2

δ-cosl θ„· —

i-ß-0„sink

L

X

(H)

[ β · θ ^ - δ ( δ + β + ΐ ) ] ε ο 8 ( θ „ ) + (δ·β + δ + 2 β ) · θ „ · 8 ί η ( θ „ ) where: δ=

a-A-L K,

ν out v

' Κ,

(12)

(13)

and where 0m fulfills: tan

δ·(β + ΐ)·θ„

(14)

β·θ*

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 (p g ~2.7-2.8 kg dm"3) and the bulk dry density of the material according to: >=1 —

(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 α value for cations than for water can be related to their adsorption to the solid surfaces (KD > 0). The lower α 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 α 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, ρ 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 (Ό/Ο 0 ) & + > (De/D0)Na+ > ( Ό / Ο 0 ) η τ ο > (De/Do)!- 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).

292

Tournassat

20

40 60 Time (days)

80

& Steefel

100

20

40

60 Time (days)

80

100

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, ρ 75-93, Figs. 2 and 3. Reproduced with permission from Elsevier. 1

60

80

a - 2.0 - θ - 1.0 β-0.5 β-0,1

M M M M

NaC!0 4 NaC!0 4 NaCI0 4 NaCÌOi

100

120

140

160

Time (d) Figure 3. Diffusion data from Glaus et al. (2010) illustrating the changes in 36C1 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, 22 Na + and 36C1 in compacted kaolinite, illite and montmorillonite. Geochimica et Cosmochimica Acta, Vol. 74, ρ 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,¡, Equation (2), is estimated for cations based on their effective diffusion coefficient values and a porosity value. Since φ cannot be greater than the total porosity, it follows that Dpi> Dei / φ . For Cs+ diffusion in an experiment conducted at 0.01 mol-L"1 NaCl, Dp Cs+ values were higher than 3.1 χ IO"9 m2-s_1, a value that is higher than the self-diffusion coefficient of Cs+ in pure water (2.07 χ IO"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 C + 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-, values were taken from Li and Gregory (1974).

Tracer i

NaCl conc.

vout

α

De,i

De,i/Do,i

(mol-L 1 )

(L)

(L)

(-)

10 n m 2 -s 1

HTO

0.1

0.6

0.6

0.775 ± 0.025

6.62 ±0.11

Na +

0.1

2

2

8.12 ±0.20

24.4 ± 0.5

I

0.1

2

0.2

0.421 ±0.017

0.694 ± 0.037

(3.47±0.02)·10- 3

2.5

2.5

1460 ± 5 0

405 ± 9

(1.96 ± 0.04)·10 2

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

0.01 Cs

+

(-) (3.26 ±0.005)· 10 (1.83 ±0.04)·10 >

(1.13 ±0.01)·10 >

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 α 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

294

Tournassat

& Steefel

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.

O E c ca a O co -I—'

c

ω

o c o O

•

0.5 M NaCI04

O

o.t M NaCi0 4 Aqueous phase diffusion (0.5 M) Aqueous phase diffusion (0.1 M) 20

40

60

80

100

120

140

Time (d) Figure 4. 2 2 Na + 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 22 Na + in compacted montmorillonite. Environmental Science & Technology, Vol. 47, ρ 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 T O 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 - 3 0 % 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 Si 4+ and Al 3+ respectively. Their ideal structural formulae can be written SÌ2A1 2 0 5 (0H) 4 for kaolinite and Si 4 Al 2 O 10 (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

T O T layer

cn ο )

Tetrahedral substitution

Prolonated ;d oxygen

E c

edge sites

λ

substitutions

Particle - 5 0 - 2 0 0 nm E e O O

Τ

/

Outer basal surface Edge surfaces Interlayer basal surface

Cations on outer/basal surfaces compensating permanent layer charge Cations in interlayer space compensating permanent layer charge

Figure 5. Structure of a (dioctahedral) TOT layer and scheme of TOT layer and compensating cations in a clay mineral particle. [Figure adapted f r o m 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, ρ 443-454, Fig 1. with permission f r o m Elsevier.]

296

Tournassat & Steefel

of the substitutions occur in the octahedral sheet where Al 3+ is replaced by Fe 3+ or a cation of lower charge (Mg 2+ , Fe 2 + ); in illite, a significant amount of the substitutions occur also in the tetrahedral sheet with Al 3+ or Fe 3 + replacing Si 4+ . 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 NH 4 + ). 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 A (one-layer hydrate), 14.5-15.6 A (two-layer hydrate), and up to 19-21.6 A (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 A. 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 χ 1.5 = 9.5 A. 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 (SSA b s ) 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 m 2 -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_ollter) depends on the average number of stacked layer (wst) in a single illite particle: SSAhs_ollter = SSAhs / 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 m 2 -g _1 considered as representative of SSAhs_ollter (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 SSAbsjnter= SSAbs x ( l - l / « s t ) and the inter-particle

Ionic Transport in Nano-Porous

Clays Considering

Electrostatic Effects

297

porosity in contact with a specific surface area equal to SSAhs_outer. It is not possible to give generic and representative values of SSAbs_inter and SS!Abs_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 f r o m 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 E D L 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 E D L thickness, where more than 90 % of the surface charge is screened, is commensurable with 2 - 3 Debye lengths, κ"1: (16)

where / i s the non-dimensional ionic strength (Solomon 2001), ε is the water dielectric constant (78.3 χ 8.85419 x l O ^ F - m " 1 at 298 K), F is the Faraday constant (96 485 C-mol" 1 ), R is the gas constant (8.3145 J-moH-K" 1 ) and Τ is the temperature (K). The ionic charge distribution in the E D L 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 f r o m M D simulation is in close agreement with the prediction of the simple modified Gouy-Chapman (MGC) model prediction, where this model is applicable. In the M G C model, ion concentrations ( DL c,) at a position y f r o m the starting position of the diffuse layer follow a Boltzmann distribution: (17)

that is related to the charge of the ions (z¡) and the electrostatic potential (\|/DL) calculated f r o m the Poisson equation:

298

Tournassat

& Steefel

=

(18)

where CjQ is the concentration of species / in the bulk water. An equivalent anion accessible porosity can be estimated from the integration of the anion concentration profile (Fig. 6) f r o m the surface to the bulk water (Sposito 2004). In compacted clay material, the pore sizes may be small as compared to the E D L 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 ort 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 D D L 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 Me¡ 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 (.v-axis), for two ionic strengths (left: 1= 0.015; and right: 1= 0.15) and with two different electrolytes (NaCl: blue plain lines; CaCl 2 : 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): z¡ >B" + Me¡zi < » (>B) Z1 Me¡

(19)

and their related equilibrium constants: b

,„

(>B).

fl

Ki=

Me,

(20)

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 (Ψ 0 ) 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):

M

Ρ C>B_„ίΡΨθ , exp ^ Ψ^ θ ί _=—ïS—exp C>B- + Z , ^ c c > B ) ^ U r J C>BTOT U Τ

Z¡C(>B) Me¡ e a{>B)aMe, = —

x

[ FV|/0 I P["z'^J'

(21)

(22)

where c is a concentration term (mol-L"1) and c > B is the total concentration of adsorption sites > B on the basal surface. The surface potential ψ 0 is calculated according to: σ0 = V 8 0 0 0 £ R r / s i n h í - ^ i .

UrΤ J

(23)

Equation (23) corresponds to the exact analytical solution of the modified Gouy-Chapman model for an infinite and fiat 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 (Me¡) 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 xz -Me¡ + Zi Me/¡ «

z¡ Me¡z. + z¡ Xz¡ - Me;·,

(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.

Tournassat & Steefel

300

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 (E¡) and a surface activity coefficient (y¡>exch), following the Gaines and Thomas convention (Gaines and Thomas 1953): exc

V, =

exc

V«° + t f r i n e x c y =

exc

exch V,° + R r i n ^, x c h Z ' c, Jj ¿ Γ

(25)

where exchc¡ 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 ¡ syLchCj 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 y¿ 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: exc

V, = e x c V ° + R T l n £ ; .

(26)

The equivalent fraction of species Me¡ on the exchanger sites are calculated according to the selectivity coefficients, , of binary reactions (Eqn. 24):

'

"%,·

(27)

E-'ia zi ' I Me*' 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, SSAeige, 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~l. 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., Ni 2 + ), 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 + M e ; a > B M e ® " 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): (29)

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

is responsible, in a large part, for the observed

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: 2 NaX + Ca 2+

CaX 2 + 2 Na + log ATNa/Ca = 0.5.

(30)

The concentration of Ca 2+ on the exchanger is obtained from: 2[CaX 2 ]-C C E C [Na + ] 2 Y ^

(31)

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

302

Tournassat

& Steefel

ψ · [NaX]2 · KNa/Ca ~ η2·Ρ,-C r r[ N a + ] γ, ^ · n CEC

D

( il

->

Under the experimental conditions considered here, the Na+ concentration is constant and thus yCa2+ /[Na+]2yNa+ 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: g

^Ca2+

_ Ψ ' CCEC ' ^~Na/Ca 2

+

·Ρ"

^Ca2+

_ ICEC ' ^~Na/Ca 2

|>a ]VN.-~

+

(33)

2

[Na ] y¿/

where gcEcis the CEC. expressed in mol-kg" 1 ^. 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 χ 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 χ IO"10 nr-s"1. According to Equation (33), and considering a qce.c value of 1 mol-kg" 1 ^ 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 χ 10"3 mol-L"1 (Fig. 7). At 5 χ 10"3 mol-L"1, the two breakthrough curves could only be

4.010"3.5-10 "

/δ

Cation exchange, [Ca 2+ ] in | et =10 3 mol L"1 Cation exchange, [Ca 2+ ] in | et =5 10

3.0-10 " ^

-Δ 1

'

Cation exchange, [Ca 2+ ] in | et =10 7 mol L"1

—

0

•

2.5-10"'

Δ

^ g 2.0-10:

1

2+

Kd = 100 L kg" , [Ca ] ¡n | et =10

7

mol L"1

;δ

mol L"1

Α

Kd = 100 L kg"1, [Ca 2+ ] in | et =10" 3 mol L"1 1

2+

3

Kd = 81 L kg" , [Ca ] ¡n | et =5 IO" mol L"

A

k :

1

Δ

^ CM

1.5-10 ΠΙ

-5

U 1.0-10

-

k

-

fe

,-6 5.0-10 0.0-10

/ ,

;Δ

'

/

,'?

I ψ ι ti I Τ >4 ! ή

'

' J ' r α J* /

Δ /

/ir

ι

0

5

10

15

20

25

30

Time (days) Figure 7. Effect of the non-linearity of adsorption on the KD parameter derived f r o m 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 Ca 2+ at a concentration of 5 χ IO"3 rnol-L"1 is 42 L-kg" 1 according to Equation (32) and thus a Constantino 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

μ, = b μ° + RT In b a, = b μ° + RT In Y, + RT In b γ,.

(34)

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

j

b b -Λ b "i c'- d Ι-1' |z¿|F dx

"Λ

b ^b., i d Ψ-üff dx

c

(35)

where hu¡ is the mobility in bulk water (m2-s"1-V) and where h y ^ is the diffusion potential that arises because of the diffusion of charged species at different rates. The gradient in chemical potential along χ is:

304

Tournassat & Steefel ö V _RTdbc, b dx c, dx

,

b Rr _Öln Yl .

(36)

dx

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

bι =

_

DL ι *

b

j

>

(

7

!

)

316

Tournassat & Steefel

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 ι'φ1 = b2 and ϋΙ 'φ 1 = ϋΙ "φ 2 :

J

-

( 7 2 )

2φ

!

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: ;

.

b Λ.

=

, b ι

i i i b 2φ

DL ι

/

;

+

J ,

D

L

.

(73)

2φ

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:

Consequently, Equation (72) lacks a condition of continuity. This problem can be solved by considering the more complex equation: j '

=

b ι 2 b j *

DL ι Φΐ

DL j

>

|

ι Φ -

DL ι Φΐ -

b ι Ψ2 b to D L j

*

( 7 5 )

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. A s 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

Solid 2

Solid 1 Diffuse layer 1 οι φ1 CD o.. Bulk 1

317

•Ê Β c

Diffuse layer 2 DL ch Bulk 2 b

2

Φι

Porous medium 1

Porous medium 2

Figure 14. Scheme of an interface between two numerical domains. Simple case with no gradient in porosity properties.

Solid 1

Solid 2

Diffuse layer 1 g

£ Bulk 1 b

4>i

Porous medium 1

Β ~

Diffuse layer 2 DL

ch

Bulk 2 b

2

Porous medium 2

Figure 15. Scheme of an interface between two numerical domains. Complex case 1.

318

Tournassat & Steefel

Solid 1

Solid 2

Diffuse layer 1 Db

ch

8

£

Bulk 1

Bulk 2 1

Β c ~

1

Φι

Porous medium 1

Φ2

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 Λ,,, the numerical equivalent of Equation (65) is given by: f ι HC'. ! I ' 111 ι " . _!_(./ Ι"ιι·h . 2~ ./ "ir '+1/ At Δ*

j.I)

c

(76)

where superscripts new and old refer to two consecutive time steps, and where Λ'„+ι/2 and λ'„_ι/2 are the positions at the interface between cells η and n +1, and η and η-1, respectively. v

and

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 , v the equations are represented numerically by finite differences as: 1,1. I)

-_

b

Din A Ui.ν Λ , ι !

_ b

C'"'.+l

" C'"'. (77)

~I 2jΙ ;

D; Ci v ι,χ"η+\Ω. Αv hXn+m b 1+112 '> I. 2.J b

c

While the term — — Λ

»+1

—

_

b

c

Λ

»

—

J _

J J'xn+m 2 I i/I 11 , ,

b

J>„1I2 C./-V„+l,2

Λ

2

is a well-defined quantity, the terms

c•v

(the

concentration in the bulk water at the interface between cells η and n+1)7 and A, ,. 1/2 v(the

Ionic Transport in Nano-Porous

Clays Considering Electrostatic Effects

319

Solid 2

1

Diffuse layer 2 Diffuse layer 1 φ 0ί φ. g

d l 4>2

Bulk 2

φ

Bulk 1

"Φι 1 Porous medium 1

b

2

Porous medium 2

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 η and /7+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: t>c

+

(78)

Alternatively, a harmonic mean can be used if the numerical grid is heterogeneous. The averaging of the diffuse layer enrichment factor A¡ v 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 CrunehFlowMC 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.

320

Tournassat & Steefel 0.1 „0.08 ^ 0.06

"So.04 Ï 0.02

f

¿

^

*

°

#

f

Distance χ (m)

f

^

*

Distance χ (m)

Figure 18. Averaging methods for the bulk concentration (left) and the diffuse layer enrichment factor, A, (right) for a monovalent cationic tracer at the interface between two grid cells whose centers are located at xn = 0 and x„+1 = 0.01 m. The variation of A, 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 KN¡1 = 0.7 (in agreement with the MD results from Tournassat et al. 2009) and log KCs = 2.5: >Surf" + Na + >SurfNa KNa =

>Surf" + Cs+ >SurfCs KN =

C>SmfNa

exp f ^

«WW-

yRT

C>SmfCs

a Cst - c>Suif

J

expiai.

{RT )

v

],

(79)

(80)

y

Half of the total porosity was attributed to the diffuse layer, so that the mean CI" (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 t r < τΗτο < î 2 2 Nï + < t c s > 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 ti 37 Cs + , 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). 137

+

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

10 20 30 40 50 60 70 80 90100

321

10 20 30 40 50 60 70 Í

Time (days)

Time (days)

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 NaC104 solution in the experimental conditions fromTachi andYotsuji (2014). See Figure 2 for the reference data and Table 1 for the experimental conditions.

Diffusion u n d e r a salinity g r a d i e n t The experiments conducted and described by Glaus et al. ( 2 0 1 3 ) were also modeled using CrunchFlowMC. The resistance to the diffusion by the filters was explicitly taken into account by assuming a tortuosity o f 0 . 2 5 and a porosity o f 0 . 3 2 for the filters. This corresponds to an effective diffusion coefficient o f ~10" 1 0 nr-s" 1 for Na + in the filter, a value in agreement with published values (Glaus et al. 2 0 0 8 ) . The total porosity in the montmorillonite plug was set to a value o f 0 . 3 , in agreement with the 1.9 kg-dm" 3 dry bulk density o f the material (Glaus et al. 2 0 1 0 ) . At this high degree o f compaction, the actual presence o f bulk water is not certain. However, it is necessary to have a non-zero bulk water volume to run C r u n c h F l o w M C (or P H R E E Q C ) . This bulk water volume was set to a very low porosity value ( 0 . 0 2 ) , 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 o f Tachi and Yotsuji's data (see previous section) was used in this case. The tortuosity parameter was set to a value o f 0 . 0 1 4 in the montmorillonite, in close agreement with the value measured for water, whose effective diffusion coefficient is about ( 1 . 5 - 1 . 7 ) - 1 0 " u nr-s" 1 in similar conditions (Glaus et al. 2 0 1 0 , 2 0 1 3 ) . T h e modeling results are shown on Figure 2 0 and agree almost perfectly with the experimental data. Table 2. Tortuosity values calculated with CrunchFlow for each species in the diffusion experiment from Tachi andYotsuji (2014), and according to a bulk + diffuse layer water diffusion model. Tracer i

τ

HTO

0.047

Na+

0.071

I Cs+

0.09 0.136

322

Tournassat

& Steefel

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 fiat 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 NaC-1 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.

Time (days)

Figure 20. Modeling of the diffusion of 22 Na + 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/C 0 of 22 Na + , while the dashed curves are the NaC10 4 concentrations in the reservoirs (blue: 0.5 mol-L 1 NaC10 4 : red: 0.1 mol-L 1 NaC10 4 ). 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 χ IO"11 m V 1 , together with an interlayer Na + diffusion coefficient of 4 χ 10"11 nr-s"1, 1 χ 10"11 m V 1 , 0.1 χ IO"11 m V 1 , and 0.1 χ ΙΟ"11 m V 1 , for the experiments at NaCI concentration of 1, 0.1, 0.05 and 0.003 mol-L"1 respectively. The decrease of the Na+ interlayer diffusion coefficient with NaCI concentration is in agreement with MD results from Tertre et al. (2015), which show a decrease of its value with an increase of the Ca 2+ /Na + occupancy ratio in the interlayer (Fig. 21c). At 1 mol-L"1 NaCI, the PHREEQC and MD results fully agree. At 0.05 and 0.003 mol-L"1 NaCI, 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).

-¿U Stainless steel plates clamping screw

10" 5

1

2

5

10

Time (days) Figure 21. Experiments f r o m Tertre et al. (2015): a) setup; b) modeling of the Ca 2+ out-diffusion results using the interlayer diffusion option of P H R E E Q C (this study); c) self-diffusion coefficients of Na + and Ca 2+ in the vermiculite interlayer as a function of the equivalent fraction of Ca on the surface (X Ca ). and obtained f r o m 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,

324

Tournassat & Steefel

the classical Fickian diffusion model applied in the framework of a single pore diffusion model with linear adsorption processes (K D 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.

REFERENCES Aldaba D, Rigol A, Vidal M (2010) Diffusion experiments for estimating radiocesium and radiostrontium sorption in unsaturated soils from Spain: Comparison with batch sorption data. J Hazard Mater 181:1072— 1079 Alt-Epping P, Tournassat C, Rasouli P, Steefel CI, Mayer KU, lenni A, Mäder U, Sengor SS, Fernández R (2014) Benchmark reactive transport simulations of a column experiment in compacted bentonite with multispecies diffusion and explicit treatment of electrostatic effects. Computat Geosci 1 - 1 6 Altmann S (2008) Geochemical research: A key building block for nuclear waste disposal safety cases. I Contam Hydrol 102:174-179 Anderson RL, Ratcliffe I, Greenwell HC, Williams PA, Cliffe S, Coveney PV (2010) Clay swelling—a challenge in the oilfield. Earth-Sci Rev 98:201-216 Appelo CAI, Wersin Ρ (2007) Multicomponent diffusion modeling in clay systems with application to the diffusion of tritium, iodide, and sodium in Opalinus clay. Env Sei Tech 41:5002-5007 Appelo CAI, Vinsot A, Mettler S, Wechner S (2008) Obtaining the porewater composition of a clay rock by modeling the in- and out-diffusion of anions and cations from an in-situ experiment. I Contam Hydrol 101:67-76 Appelo CAI, Van Loon LR, Wersin Ρ (2010) Multicomponent diffusion of a suite of tracers (HTO, CI, Br, I, Na, Sr, Cs) in a single sample of Opalinus Clay. Geochim Cosmochim Acta 74:1201-1219 Banin A, Lahav Ν (1968) particle size and optical properties of montmorillonite in suspension. Isr I Chem 6:235-250 Bear I (1972) Dynamics of Fluids in Porous Media. Courier Dover Publications Ben-Yaakov S (1981) Diffusion of seawater ions: Significance and consequences of cross-coupling effects. Am I Sci 281:974-980 Benna M, Kbir-Ariguib Ν, Clinard C, Bergaya F (2002) Card-house microstructure of purified sodium montmorillonite gels evidenced by filtration properties at different pH. In: Adsorption and Nanostructure. Springer, ρ 204-210 Bergaya F (1995) The meaning of surface area and porosity measurements of clays and pillared clays. lournal of Porous Materials 2:91-96 Bickmore B, Nagy KL, Sandlin PE, Crater TS (2002) Quantifying surface areas of clays by atomic force microscopy. Am Mineral 87:780-783 Birgersson M, Karnland O (2009) Ion equilibrium between montmorillonite interlayer space and an external solution-Consequences for diffusional transport. Geochim Cosmochim Acta 73:1908-1923 Boudreau BP, Meysman FI, Middelburg II (2004) Multicomponent ionic diffusion in porewaters: Coulombic effects revisited. Earth Planet Sci Lett 222:653-666 Bourg IC, Sposito G (2010) Connecting the molecular scale to the continuum scale for diffusion processes in smectite-rich porous media. Env Sei Tech 44:2085-2091 Bourg IC, Sposito G (2011) Molecular dynamics simulations of the electrical double layer on smectite surfaces contacting concentrated mixed electrolyte (NaCl-CaCl 2 ) solutions. I Colloid Interface Sci 360:701-715 Bradbury MH, Baeyens Β (2000) A generalised sorption model for the concentration dependent uptake of caesium by argillaceous rocksbae. I Contam Hydrol 42:141-163 Bradbury MH, Baeyens Β (2005) Modelling the sorption of Mn(II), Co(II), Ni(II), Zn(II), Cd(II), Eu(III), Am(III), Sn(IV), Th(IV), Np(V) and U(VI) on montmorillonite: Linear free energy relationships and estimates of surface binding constants for some selected heavy metals and actinides. Geochim Cosmochim Acta 69:875-892 Brigatti MF, Galán E, Theng BKG (2013) Structure and Mineralogy of Clay Minerals. In: Handbook of Clay Science, Developments in Clay Science. Bergaya F, Lagaly G (eds) Elsevier, ρ 21-81 Brouwer E, Baeyens B, Maes A, Cremers A (1983) Cesium and Rubidium ion equilibria in illite clay. I Phys Chem 87:1213-1219

326

Tournassat & Steefel

Chen Ζ, Montavon G, Ribet S, Guo Ζ, Robinet J, David Κ, Tournassat C, Grambow B, Landesman C (2014) Key factors to understand in-situ behavior of Cs in Callovo-Oxfordian clay-rock (France). Chem Geol 387:47-58 Chu SY, Sposito G (1981) The thermodynamics of ternary cation exchange systems and the subregular model. Soil Sci Soc Am J 45:1084-1089 Churakov SV, Gimmi Τ (2011) Up-scaling of molecular diffusion coefficients in clays: A two-step approach. J Phys Chem C 115:6703-6714 Churakov SV, Gimmi T, Unruh Τ, Van Loon LR, Juranyi F (2014) Resolving diffusion in clay minerals at different time scales: Combination of experimental and modeling approaches. Appi Clay Sci 96:36-44 De Carvalho Balaban R, Vidal ELF, Borges MR (2015) Design of experiments to evaluate clay swelling inhibition by different combinations of organic compounds and inorganic salts for application in water base drilling fluids. Appi Clay Sci 105-106:124-130 Delay J, Trouiller A, Lavanchy J-M (2006) Propriétés hydrodynamiques du Callovo-Oxfordien dans l'Est du bassin de Paris: comparaison des résultats obtenus selon différentes approches. C R Geosci 338:892-907 Delay J, Bossart Ρ, Ling LX, Blechschmidt I, Ohlsson M, Vinsot A, Nussbaum C, Maes Ν (2014) Three decades of underground research laboratories: what have we learned? Geological Society, London, Special Publications 400, ρ 7 - 3 2 Delgado A, Gonzalez-Caballero F, Bruque JM (1986) On the zeta potential and surface charge density of montmorillonite in aqueous electrolyte solutions. J Colloid Interface Sci 113:203-211 Delgado A, Gonzalez-Caballero F, Salcedo J, Cabrerizo MA (1988) A study of the electrophoretic properties of montmorillonite particles in aqueous electrolyte solutions. Mater Chem Phys 19:327-340 Descostes M, Blin V, Bazer-Bachi F, Meier P, Grenut B, Radwan J, Schlegel ML, Buschaert S, Coelho D, Tevissen E (2008) Diffusion of anionic species in Callovo-Oxfordian argillites and Oxfordian limestones (Meuse/Haute-Marne, France). Appi Geochem 23:655-677 Dzombak DA, Hudson RJM (1995) Ion-exchange: the contributions of diffuse layer sorption and surface complexation. In: Aquatic Chemistry. Interfacial and Interspecies Processes. Advances in Chemistry Series 244. Huan CP, O'Melia C, Morgan J J (eds) Am Chem Soc, Washington, ρ 59-94 Fernández AM, Sánchez-Ledesma DM, Tournassat C, Melón A, Gaucher EC, Astudillo J, Vinsot A (2014) Applying the squeezing technique to highly Consolidated clayrocks for pore water characterisation: Lessons learned from experiments at the Mont Terri Rock Laboratory. Appi Geochem 49:2-21 Gaboreau S, Claret F, Crouzet C, Giffaut E, Tournassat C (2012) Caesium uptake by Callovian-Oxfordian clayrock under alkaline perturbation. Appi Geochem 27:1194—1201 Gaines GLJ, Thomas HC (1953) Adsorption studies on clay minerals. II. A formulation of the thermodynamics of exchange adsorption. J Chem Phys 21:714-718 Gapon EN (1933) On the theory of exchange adsorption in soils. J. Gen. Chem. USSR (Engl. Transi.) 3:144160 García-Gutiérrez M, Cormenzana JL, Missana Τ, Mingarro M (2004) Diffusion coefficients and accessible porosity for HTO and 36C1 in compacted F E B E X bentonite. Appi Clay Sci 26:65-73 Garrels RM, Mackenzie FT (1971) Evolution of sedimentary rocks. WW Norton & Co, New York Gimmi T, Kosakowski G (2011) How mobile are sorbed cations in clays and clay rocks? Env Sei Tech 45:1443-1449 Glaus M, Baeyens B, Bradbury MH, Jakob A, Van Loon LR, Yaroshchuk A (2007) Diffusion of 22 Na and s 5 Sr in montmorillonite: evidence of interlayer diffusion being the dominant pathway at high compaction. Environ Sci Technol 41:478-485 Glaus MA, Rosse R, Van Loon LR, Yaroshchuk AE (2008) Tracer Diffusion in Sintered Stainless Steel Filters: Measurement of Effective Diffusion Coefficients and Implications for Diffusion Studies with Compacted Clays. Clays Clay Miner 56:677-685 Glaus MA, FrickS, Rosse R, Van Loon LR (2010) Comparative study of tracer diffusion of HTO, Na-22(+) and Cl-36(-) in compacted kaolinite, illite and montmorillonite. Geochim Cosmochim Acta 74:1999-2010 Glaus MA, Birgersson M, Karnland O, Van Loon LR (2013) Seeming steady-state uphill diffusion of 22 Na + in compacted montmorillonite. Env Sei Tech 47:11522-11527 Gonçalvès J, Rousseau-Gueutin P, Revil A (2007) Introducing interacting diffuse layers in TLM calculations: A reappraisal of the influence of the pore size on the swelling pressure and the osmotic efficiency of compacted bentonites. J Colloid Interface Sci 316:92-99 Gu XY, Evans LJ, Barabash SJ (2010) Modeling the adsorption of Cd (II), Cu (II), Ni (II), Pb (II) and Zn (II) onto montmorillonite. Geochim Cosmochim Acta 74:5718-5728 Hemes S, Desbois G, Urai JL, De Craen M, Honty M (2013) Variations in the morphology of porosity in the Boom Clay Formation: insights from 2D high resolution BIB-SEM imaging and Mercury injection Porosimetry. Neth J Geosci 92:275-300 Holmboe M, Bourg IC (2014) Molecular dynamics simulations of water and sodium diffusion in smectite interlayer nanopores as a function of pore size and temperature. J Phys Chem C 118:1001-1013

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects

327

Holmboe M, Wold S, Jonsson M (2012) Porosity investigation of compacted bentonite using XRD profile modeling. J Contam Hydrol 128:19-32 Homand F, Shao J-F, Giraud A, Auvray C, Hoxha D (2006) Pétrofabrique et propriétés mécaniques des argilites. C R Geosci 338,:882-891 Jakob A, Pfingsten W, Van Loon LR (2009) Effects of sorption competition on caesium diffusion through compacted argillaceous rock. Geochim Cosmochim Acta 73:2441-2456 Jardat M, Dufreche JF, Marry V, Rotenberg Β, Turq Ρ (2009) Salt exclusion in charged porous media: a coarsegraining strategy in the case of montmorillonite clays. Phys Chem Chem Phys 11:2023-2033 Keller LM, Holzer L, Wepf R, Gasser Ρ (2011) 3D geometry and topology of pore pathways in Opalinus clay: Implications for mass transport. Appi Clay Sci 52:85-95 Keller LM, Holzer L, Schuetz P, Gasser Ρ (2013) Pore space relevant for gas permeability in Opalinus clay: Statistical analysis of homogeneity, percolation, and representative volume element. J Geophys Res: Solid Earth 118:2799-2812 Kulik DA (2009) Thermodynamic concepts in modeling sorption at the mineral-water interface. Rev Mineral Geochem 70:125-180 Lagaly G, Dékány I (2013) Chapter 8 - Colloid Clay Science. In: Handbook of Clay Science, Developments in Clay Science. Bergaya, F, Lagaly G (eds) Elsevier ρ 243-345 Lasaga AC (1981) The treatment of multicomponent diffusion and ion pairs in diagenetic fluxes; further comments and clarification; reply. Am J Sci 281:981-988 Le Forestier L, Muller F, Villieras F, Pelletier M (2010) Textural and hydration properties of a synthetic montmorillonite compared with a natural Na-exchanged clay analogue. Appi Clay Sci 48:18-25 Lee SS, Fenter P, Park C, Sturchio NC, Nagy KL (2010) Hydrated cation speciation at the muscovite (001)water interface. Langmuir 26:16647-16651 Lee SS, Fenter P, Nagy KL, Sturchio NC (2012) Monovalent ion adsorption at the muscovite (OOl)-solution interface: Relationships among ion coverage and speciation, interfacial water structure, and substrate relaxation. Langmuir 28:8637-8650 Leroy Ρ, Revil A, Coelho D (2006) Diffusion of ionic species in bentonite. J Colloid Interface Sci 296:248-255 Li Y-Η, Gregory S (1974) Diffusion of ions in sea water and in deep-sea sediments. Geochim Cosmochim Acta 38:703-714 Maes N, Salah S, Jacques D, Aertsens M, Van Gompel M, De Canniere P, Velitchkova Ν (2008) Retention of Cs in Boom Clay: Comparison of data from batch sorption tests and diffusion experiments on intact clay cores. Phys Chem Earth, Parts A/B/C 33:S149-S155 Marry V, Rotenberg Β, Turq Ρ (2008) Structure and dynamics of water at a clay surface from molecular dynamics simulation. Phys Chem Chem Phys 10:4802-4813 Marty NCM, Cama J, Sato T, Chino D, Villiéras F, Razafitianamaharavo A, Brendlé J, Giffaut E, Soler JM, Gaucher EC, Tournassat C (2011) Dissolution kinetics of synthetic Na-smectite. An integrated experimental approach. Geochim Cosmochim Acta 75:5849-5864 Melkior T, Yahiaoui S, Motellier S, Thoby D, Tevissen E (2005) Cesium sorption and diffusion in Bure mudrock samples. Appi Clay Sci 29:172-186 Melkior T, Gaucher EC, Brouard C, Yahiaoui S, Thoby D, Clinard C, Ferrage E, Guyonnet D, Tournassat C, Coelho D (2009) Na+ and HTO diffusion in compacted bentonite: Effect of surface chemistry and related texture. J Hydrol 370:9-20 Méring J (1946) On the hydration of montmorillonite. Trans Faraday Soc 42:B205-B219 Miller AW, Rodriguez DR, Honeyman BD (2010) Upscaling sorption/desorption processes in reactive transport models to describe metal/radionuclide transport: A critical review. Env Sei Tech 44:7996-8007 Miyahara K, Ashida T, Kohara Y, Yusa Y, Sasaki Ν (1991) Effect of bulk-density on diffusion for cesium in compacted sodium bentonite. Radiochim Acta 52-3:293-297 Molins S, Trebotich D, Steefel CI, Shen CP (2012) An investigation of the effect of pore scale flow on average geochemical reaction rates using direct numerical simulation. Water Resour Res 48:WR011404 Montavon G, Alhajji E, Grambow Β (2006) Study of the interaction of Ni 2+ and Cs + on MX-80 bentonite; Effect of compaction using the "capillary method". Environ Sci Technol 40:4672^-679 Muurinen A, Karnland O, Lehikoinen J (2004) Ion concentration caused by an external solution into the porewater of compacted bentonite. Phys Chem Earth, Parts A/B/C 29:119-127 Nakashima Y (2002) Diffusion of H 2 0 and I in expandable mica and montmorillonite gels: contribution of bound H 2 0. Clays Clay Miner 50:1-10 Neuzil CE (1986) Groundwater flow in low-permeability environments. Water Resour Res 22:1163-1195 Norrish Κ (1954) The swelling of montmorillonite. Faraday Discuss Chem Soc 18:120-134 Obliger A, Duvail M, Jardat M, Coelho D, Békri S, Rotenberg Β (2013) Numerical homogenization of electrokinetic equations in porous media using lattice-Boltzmann simulations. Phys Rev E 88:013019 Oscarson DW, Hume HB, King F (1994) Sorption of cesium on compacted bentonite. Clays Clay Miner 42:731-736

Tournassat & Steefel

328

Parkhurst DL, Appelo CAJ (1999) User's guide to P H R E E Q C (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 ρ 99^-259.) 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, ρ 9 - 2 1 Pearson FJ, Arcos D, Boisson J-Y, Fernandez AM, Gabler Η-E, Gaucher E, Gautschi A, Griffault L, Hernán Ρ, Waber ΗΝ (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 M H (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. Appi Clay Sci 77:18-32 Rotenberg Β, Marry V, Dufrêche J-F, Giffaut E, Turq Ρ (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 Β, Marry V, Dufreche J-F, Malikova N, Giffaut E, Turq Ρ (2007b) Modelling water and ion diffusion in clays: A multiscale approach. C R Chim 10:1108-1116 Rotenberg Β, Marry V, Vuilleumier R, Malikova N, Simon C, Turq Ρ (2007c) Water and ions in clays: Unraveling the interlayer/micropore exchange using molecular dynamics. Geochim Cosmochim Acta 71:5089-5101 Rotenberg Β, Marry V, Malikova N, Turq Ρ (2010) Molecular simulation of aqueous solutions at clay surfaces. J Phys : Condens Matter 22:284114 Rotenberg Β, Marry V, Salanne M, Jardat M, Turq Ρ (2014) Multiscale modelling of transport in clays f r o m 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) A F M 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^-8 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 f r o m 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 Τ (2001) The definition and unit of ionic strength. J C h e m 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. Giiven N, Pollastro R M (eds) Clay Minerals Society, ρ 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 Κ (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, Simunek J, Spycher N, Yabusaki SB, Yeh GT ( 2 0 1 4 ) Reactive transport codes for subsurface environmental simulation. Computat Geosci:l-34, doi=10.1007/ S10596-014-9443-X

Ionic Transport in Nano-Porous Clays Considering Electrostatic Effects

329

Tachi Y, Yotsuji Κ (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 5 1 : 1 1 9 1 - 1 2 0 4 Tertre E, Beaucaire C, Coreau N, Juery A (2009) Modelling Zn(II) sorption onto clayey sediments using a multi-site ion-exchange model. Appi Geochem 2 4 : 1 8 5 2 - 1 8 6 1 Tertre E, Delville A, Prêt D, Hubert F, Ferrage E (2015) Cation diffusion in the interlay er space of swelling clay minerals—A combined macroscopic and microscopic study. Geochim Cosmochim Acta 1 4 9 : 2 5 1 - 2 6 7 Tournassat C, Appelo C A J (2011) Modelling approaches for anion-exclusion in compacted Na-bentonite. Geochim Cosmochim Acta 7 5 : 3 6 9 8 - 3 7 1 0 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 A F M observations. Am Mineral 8 8 : 1 9 8 9 - 1 9 9 5 Tournassat C, Chapron Y, Leroy Ρ, 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 3 3 9 : 5 3 3 - 5 4 1 Tournassat C, Gailhanou H, Crouzet C, Braibant G, Gautier A, Gaucher E C (2009b) Cation exchange selectivity coefficient values on smectite and mixed-layer illite/smectite minerals. Soil Sci Soc Am J 7 3 : 9 2 8 - 9 4 2 Tournassat C, Bizi M, Braibant G, Crouzet C (2011) Influence of montmorillonite tactoid size on N a - C a cation exchange reactions. J Colloid Interface Sci 3 6 4 : 4 4 3 ^ - 5 4 Tournassat C, Grangeon S, Leroy Ρ, 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 3 1 3 : 3 9 5 - 4 5 1 Tsai S-C, Ouyang S, Hsu C-N (2001) Sorption and diffusion behavior of Cs and Sr on Jih-Hsing bentonite. Appi Radiat Isot 5 4 : 2 0 9 - 2 1 5 Van Loon LR, Baeyens B, Bradbury M H (2009) The sorption behaviour of caesium on Opalinus Clay: A comparison between intact and crushed material. Appi Geochem 2 4 : 9 9 9 - 1 0 0 4 Van Loon L R , Soler JM, Bradbury MH (2003) Diffusion of HTO, 36C1 and 1 2 5 I in Opalinus Clay samples from Mont Terri: Effect of confining pressure. J Contam Hydrol 6 1 : 7 3 - 8 3 Van Loon L R , Soler J M , Muller W, Bradbury MH (2004a) Anisotropic diffusion in layered argillaceous rocks: a case study with Opalinus Clay. Env Sei Tech 3 8 : 5 7 2 1 - 5 7 2 8 Van Loon LR, Wersin P, Soler JM, Eikenberg J, Gimmi T, Hernán Ρ, Dewonck S, Savoye S (2004b) In-situ diffusion of HTO, 2 2 Na + , Cs + and I in Opalinus Clay at the Mont Terri underground rock laboratory. Radiochim Acta 9 2 : 7 5 7 - 7 6 3 Van Loon L R , Müller W, Iijima Κ (2005) Activation energies of the self-diffusion of HTO, 22Na+ and 36C1- in a highly compacted argillaceous rock (Opalinus Clay). Appi Geochem 2 0 : 9 6 1 - 9 7 2 Vanselow AP (1932) The utilization of the base-exchange reaction for the determination of activity coefficients in mixed electrolytes. J Am Chem Soc 5 4 : 1 3 0 7 - 1 3 1 1 Wang Z, Giammar D E (2013) Mass Action Expressions for Bidentate Adsorption in Surface Complexation Modeling: Theory and Practice. Env Sei Tech 4 7 : 3 9 8 2 - 3 9 9 6 Wersin P, Soler JM, Van Loon L, Eikenberg J, Baeyens B, Grolimund D, Gimmi T, Dewonck S (2008) Diffusion of HTO, Br , I , Cs + , ^ S r 2 4 and 6 0 Co 2 + in a clay formation: Results and modelling from an in situ experiment in Opalinus Clay. Appi Geochem 2 3 : 6 7 8 - 6 9 1 Wilson MJ, Wilson L (2014) Clay mineralogy and shale instability: an alternative conceptual analysis. Clay Miner 4 9 : 1 2 7 - 1 4 5 Wittebroodt C, Savoye S, Frasca B, Gouze Ρ, Michelot J - L (2012) Diffusion of H T O , 3 6 CI a n d 1 2 5 1 in Upper Toarcian argillite samples from Tournemire: Effects of initial iodide concentration and ionic strength. Appi Geochem 2 7 : 1 4 3 2 - 1 4 4 1 Yokoyama S, Kuroda M, Sato Τ (2005) Atomic force microscopy study of montmorillonite dissolution under highly alkaline conditions. Clays Clay Miner 5 3 : 1 4 7 - 1 5 4 Zachara JM, Smith SC, McKinley JP, Resch C T (1993) Cadmium sorption on specimen and soil smectites in sodium and calcium electrolytes. Soil Sci Soc Am J 5 7 : 1 4 9 1 - 1 5 0 1 Zachara JM, Smith SC, Liu C, McKinley JP, Seme RJ, Gassman P L (2002) Sorption of Cs + to micaceous subsurface sediments from the Hanford site, USA. Geochim Cosmochim Acta 6 6 : 1 9 3 - 2 1 1 Zhao H, Bhattacharjee S, Chow R, Wallace D, Masliyah JH, X u Ζ (2008) Probing surface charge potentials of clay basal planes and edges by direct force measurements. Langmuir 2 4 : 1 2 8 9 9 - 1 2 9 1 0

10

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 Hydrologie Sciences and Engineering Program Colorado School of Mines Golden, Colorado 80401, USA asitchle @ mines.edu

Susan L. Brantley Earth and Environmental Systetns Institute and Department of Geosciences The Pennsylvania State University University Park, Pennsylvania, USA brantley @ essc.psu.edu

Gernot Rother Chemical Sciences Division Oak Ridge National Laboratory Oak Ridge, Tenn essee, USA rotherg @ ornl.gov

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 (Oilier 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 1529-6466/14/0079-0010505.00 |(cc)l .'j'l-i'l.g

http://dx.doi.org/10.2138/rmg.2015.80.10

© 2015, Navarre-Sitchler, Brantley, Rother.

This w o r k is l i c e n s e d u n d e r the Creative C o m m o n s A t t r i b u t i o n - N o n C o m m e r c i a l - N o D e r i v s 3.0 License.

332

Navarre-Sitchler, Brantley & Rother

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 C 0 2 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 < 1 0 0 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

Porosity Increases in Crystalline

Temperate

333

Silicate Rocks

Humid tropie/subtropic

data

Figure 1. Regolith thickness shown in box and whisker plots for mafic rocks (14, dark boxes) versus granitic material ( 13, light boxes) f r o m ridgetop systems reported in the literature and summarized by Bazilevskaya et al. (2013) for different climate regimes as indicated (all systems had precipitation > potential évapotranspiration). 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. f r o m 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 Landfoims, 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

334

Navarre-Sitchler,

Brantley

& Rother

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 absórbate (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

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; Woy wod 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 μπι. 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 Ρ and the pore size r into which mercury will intrude: Pr = - 2ycos0,

(2)

withy the surface tension, and θ 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

Porosity Increases in Crystalline Silicate Rocks

335

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 pj* (SLD) for a solid phase j is given by Equation (3): Ρ

'

LLAI.

Vm

(3)

Here, bci is the bound coherent scattering length of atom i of η 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 μπι. 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).

336

Navarre-Sitchler,

Brantley

&

Rother

Beam sizes are on the order of 5 - 2 0 0 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 χ IO10 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 H 2 0 / D 2 0 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 ID 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

Porosity Increases in Crystalline

Silicate

Rocks

337

Figure 2. Scattering intensity in 2-D (detector counts shown in color from low in blue to high in white) plotted as .v-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 .v- and y-axes. Correlating Q values range from 0.04 to -0.04 À 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 ( ID) 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 < 1 0 nm to > 30 mm).

characteristic size R is observed at Q ~ 2.5IR. Surface scattering from these structures is found at Q > 1 IR, 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

338

Navarre-Sitchler,

Brantley

&

Rother

many rocks, the intensity (/) of scattered neutrons is observed to be inversely proportional to Qm o v e r ^ t e n d e d regions in Q, where m is termed the Porod exponent: H Q ) ~ + B.

(4)

Here Β 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, -«, 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 /-log β 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 η 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 η between 3 and 4 characterizes a surface fractal. In this case the surface fractal dimension Ds equals (6 - «). 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 poly disperse (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 /-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).

Porosity Increases in Crystalline Silicate Rocks

339

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 - 1 5 % 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 C 0 2 (Li et al. 2013; Mäher 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

340

Navarre-Sitchler,

Brantley & Rother

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 S 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,

Porosity Increases in Crystalline Silicate Rocks

341

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 < 1 0 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 1 2 - 1 6 % 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 - 1 4 % in samples with weathering ages of 120 ka. The porosity increase was attributed to an increased number of pores of dimension > 1 0 0 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

342

Navarre-Sitchler,

a

Completely developed profile

θ

Brantley & Rother

Incompletely developed profile

( M i n e r a l 1 0 0 % d e p l e t e d at s u r f a c e )

(Mineral partially depleted at surface)

Local equilibrium regime

Kinetic regime

Minimal fracturing Limited pore connectivity

Significant fracturing High pore

^^•connectivity

Minimal advection t h r o u g h rock

'aibite

aibite

Figure 3. Schematic showing 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 (Β). τ„ Μ Β = 0 when 0 % of the aibite has dissolved away and τ ^ κ = -1 when all the aibite 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 f o r m on crystalline rocks where weathering is not rate-limiting. More mafic crystalline rocks generally develop thinner reaction fronts (/ = 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 f r o m mass + surface fractal to one or more surface fractals well before disaggregation into saprolite. For any given set of weathering conditions, both I 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. f r o m 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 Landfoims, Vol. 38, Fig 5, p. 854.]

Porosity Increases in Crystalline

Silicate

Rocks

343

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 < 2% in the unweathered sample to > 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 S 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 (J am tve it 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 μτη) or relatively large (10-300 μπι). 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

344

Navarre-Sitchler,

Brantley & Rother

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 - 5 0 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 un weathered 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 > 1 0 0 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 μπι), 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 bio ti te. When exposed at the surface or in samples recovered from drill cores, corestones were observed to be characterized by weathering rinds that are usually

Porosity Increases in Crystalline

Silicate

Rocks

345

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 Κ 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 saprofite. 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

346

Navarre-Sitchler,

Brantley & Rother

neutron scattering that exhibited non-isotropic scattering targets showing azimuth al 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 μπι. As porosity increases with weathering, the pore throats associated with that porosity are characterized by diameters < 10 μπι, 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 Na 2 0, K 2 0, 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 C 0 2 (i.e., sum of the base cation oxide concentrations), the high content of ferrous iron may deplete weathering fluids in 0 2 at depth so that the deepest weathering reaction is C0 2 - rather than 0 2 -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).

Porosity Increases in Crystalline

Silicate

347

Rocks

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).

2.5 0.0

3.0

mass fractal

3.5

2.5

3.0

3.5

SOIL

surface fractal

0.5' lowO \

1.0

highQ 1•

•

A £

Ω. Φ Q

WR

1.5'

•

•

2.0'

•

•

2 5

•

3.0 3.5 porod exponent, m

10

•

•

UWR .

2.5 30

•

SAP

S" 15·

Q

WR

20·

UWR

A

mass fractal

25 2.5

\ · A • \•

surface fractal

3.0 3.5 porod exponent, m

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, W R = 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, lin 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 tea in Fig. 6). Both the basaltic andesite and the quartz diorite contain internal

348

Navarre-Sitchler,

Brantley

a - C o s t a Rica Basalt

&

Rother

b - Puerto Rico Granite

IE-«

-3.6 -3.5 -3.4 -3,3 -3.2 -3.1 -3

• " •

•

•

28

•

• •

>

•

LowQ

Surface Fractal

R

•

g

• Mass Fractal

40

Ρ

•

i

RF

34 36

•

\

1E-3 0.01 0.1 Scattering Vector. Q (A1)

Slope -4.00 -3.75 -3.50 -3.25 -3 00 -2.75 -2.50 -1.

0

•

1B4

.

-c 25· Ή. O 30· Ω 35

D

t α• α !

· ·· •

15

i

•• Surface Fractal

ν

s U

• .

-0.5 0.0 0.6 1.

z

25 •

30 ••

35

\

Ρ

Mass Fractal

Figure 6. log (/ χ (34) 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-(3 data (plotted as squares) and low-(3 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. xCa 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: D,„ = 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 μπι. 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 D s 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

Porosity Increases in Crystalline

T o t a l P o r o s i t y , % o r S p e c i f i c S u r f a c e area, m 2 g '

0.5 η Ε

2

4

Rocks

349

B - Q u a r t z Diorite

Α-Basaltic Andesite

0

Silicate

6

8

10

12

14

16

18

T o t a l Porosity. % o r S p e c i f i c S u r f a c e area, m g

20

22

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

04 -

Ι-

•

ΟΙΟ

1 03δ u E 2 0.2 s ¡5 0.1-

• • •

O O

O

0.0

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 (Β) 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 micro fracturing 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 lithologie 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

350

Navarre-Sitchler,

Brantley

& Rother

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; R0yne 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 0 2 and C0 2 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

Porosity Increases in Crystalline

Silicate

Rocks

351

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 R0yne for their very helpful reviews of this chapter.

REFERENCES Aharonov E, Rothman DH (1996) Growth of correlated pore-scale structures in sedimentary rocks: A dynamical model. J Geophys Res-Solid Earth 101:2973-2987 Amundson R (2004) Soil formation. In: Treatise in Geochemistry: Surface and Ground Water, Weathering, and Soils. Drever JI (ed) Elsevier Pergamon, Amsterdam, ρ 1 - 3 5 Anderson SP, Dietrich WE, Brimhall GH (2002) Weathering profiles, mass-balance analysis, and rates of solute loss: Linkages between weathering and erosion in a small, steep catchment. Geol Soc Am Bull 114:1143-1158 Anderson SP, Anderson RS, Hinckley E-LS, Kelly P, Blum A (2011) Exploring weathering and regolith transport controls on critical zone development with models and natural experiments. Appi Geochem 26:S3-S5 Anovitz LM, Cole DR (2015) Analysis of porosity and pore structures. Rev Mineral Geochem 80:61-164 Anovitz LM, Lynn GW, Cole DR, Rother G, Allard LF, Hamilton WA, Porcar L, Kim MH (2009) A new approach to quantification of metamorphism using ultra-small and small angle neutron scattering. Geochim Cosmochim Acta 73:7303-7324 Anovitz LM, Cole DR, Rother G, Allard LF, Jackson AJ, Littreil KC (2013) Diagenetic changes in macro- to nano-scale porosity in the St Peter Sandstone: An (ultra) small angle neutron scattering and backscattered electron imaging analysis. Geochim Cosmochim Acta 102:280-305 Balleseros E, Garcia Talegon J, Inigi Inigo AC, Gonzalez Sanchez M, Herrero Fernandez H (2011) Importance of porosity and transfer of matter in the rock weathering processes: two examples in central Spain. Environmental Earth Sciences 64:1741-1754 Bandstra J, Buss HL, Campen RK, Liermann LJ, Moore J, Hausrath E, Navarre-Sitchler AK, Jang J-H, Brantley SL (2008) Appendix: Compilation of Mineral Dissolution Rates. In: Brantley SL, Kubicki JD, White AF (eds) Kinetics of Water-Rock Interaction. Springer, New York, ρ 735-822 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 Surf Process Landf 38:847-858 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 Sci Soc Am J 79:55-73 Berner RA (1978) Rate control of mineral dissolution under Earth surface conditions. Am J Sci 278:1235-1252 Bisdom EBA (1967) The role of micro-crack systems in the spheroidal weathering of an intrusive granite in Galicia (NW Spain). Geologie en Mijnbouw 46:333-340 Blum JD, Erel Y, Brown Κ (1994) (87)Sr/(86)Sr ratios of Sierra Nevada stream waters: Implications for relative mineral weathering rates. Geochim Cosmochim Acta 58:5019-5025 Brantley SL, Lebedeva M (2011) Learning to read the chemistry of regolith to understand the critical zone. Ann Rev Earth Planet Sci 39:387^116 Brantley SL, Bandstra J, Moore J, White AF (2008) Modelling chemical depletion profiles in regolith. Geoderma 145:494-504 Brantley SL, Holleran M, Jin L, Bazilevskaya EA (2013a) Probing deep weathering in the Shale Hills Critical Zone Observatory, Pennsylvania (USA): the hypothesis of nested chemical reaction fronts in the subsurface. Earth Surf Process Landf 38:1280-1807 Brantley SL, Lebedeva M, Bazilevskaya E A (2013b) Relating weathering fronts for acid neutralization and oxidation to p C 0 2 and p 0 2 , In: Treatise of Geochemistry, The Atmosphere—History, 2nd edition. Farquhar J, Kasting J, Canfield D (eds) Elsevier, Amsterdam, The Netherlands, ρ 327-352 Brunauer S, Emmett PH, Teller E (1938) Adsorption of gases in multimolecular layers. J Am Chem Soc 60:309-319 Bricker O, Bowser C, Jones Β (2004) Mass balance approach to interpreting weathering reactions in watershed systems. In: Treatise on Geochemistry: Surface and Ground Water, Weathering, and Soils. Drever JI (ed), Elsevier, Amsterdam, ρ 119-132

352

Navarre-Sitchler,

Brantley & Rother

Burke BC, Heimsath AM, White AF (2007) Coupling chemical weathering with soil production across soilmantled landscapes. Earth Surf Process Landf 32:853-873 Buss HL, Sak P, Webb RM, Brantley S (2008) Weathering of the Rio Blanco quartz diorite, Luquillo Mountains, Puerto Rico: Coupling oxidation, dissolution and fracturing. Geochim Cosmochim Acta 72:4488-4507 Buss HL, Brantley SL, Scatena FN, Bazilevskaya EA, Blum A, Schultz MJ, White AF, Rother G, Cole DR (2013) Probing the deep critical zone beneath the Luquillo Experimental Forest, Puerto Rico. Earth Surf Process Landf 38:1170-1186 Certini G, Campbell CD, Edwards AC (2004) Rock fragments in soil support a different microbial community from the fine earth. Soil Biol Biochem 36:1119-1128 Chatterjee A, Raymahashay BC (1998) Spheroidal weathering of Deccan Basalt: a three-mineral model. Q J Eng Geol 31:175-179 Clow DW, Drever JI (1996) Weathering rates as a function of flow through an alpine soil. Chem Geol 132:131— 141 Cousin I, Nicoullaud B, Coutadeur C (2003) Influence of rock fragments on the water retention and water percolation in a calcareous soil. Catena 53:97-114 Drever JI, Clow DW (1995) Weathering rates in catchments. Rev Mineral 31:463^-81 Eggler DH, Larson EE, Bradley W C (1969) Granites, grusses, and the Sherman erosion surface, southern Laramie Range, Colorado-Wyoming. Am J Sci 267:510-522 Emmanuel S, Anovitz LM, Day-Stirrat RJ (2015) Effects of coupled chemo-mechanical processes on the evolution of pore-size distributions in geological media. Rev Mineral Geochem 80:45-60 Fletcher RC, Buss HL, Brantley SL (2006) A spheroidal weathering model coupling porewater chemistry to soil thicknesses during steady-state denudation. Earth Planet Sci Lett 244:444—457 Gaillardet J, Millot R, Dupre Β (2003) Chemical denudation rates of the western Canadian orogenic belt: the Stikine terrane Chem Geol 201:257-279 Graham R, Rossi A, Hubbert Κ (2010) Rock to regolith conversion: Producing hospitable substrates for terrestrial ecosystems. GSA Today 20:4—9 Graham RC, Guertal WR, Tice KR (1994) The pedologie nature of weathered rock. In: Cremeens DL, Brown RB, Huddleston JH (eds), Whole Regolith Pedology, SSSA Special Publication Number 34, Soil Science Society of America, Madison WI, ρ 21-40 Gu X, Cole DR, Rother G, Mildner DFR, Brantley SL (2015) Pores in Marcellus Shale: A neutron Scattering and FIB-SEM Study Energy & Fuels, doi: 101021/acsenergyfuels5b00033 Hausrath E, Navarre-Sitchler A, Sak PB, Williams J, Brantley SL (2011) Soil profiles as indicators of mineral weathering rates and organic interactions for a Pennsylvania diabase. Chem Geol 290:89-100 Hinde AL (2004) PRINSAS - a Windows-based computer program for the processing and interpretation of small-angle scattering data tailored to the analysis of sedimentary rocks. J Appi Crystallogr 37:1020-1024 Hubbert KR, Beyers JL, Graham RC (2001) Roles of weathered bedrock and soil in seasonal water relations of Pinus jeffreyi and Arcto staphylo s patula. Can J Forest Res 31:1947-1957 Jackson CT (1840) Report on the geological and agricultural survey of the State of Rhode Island, B. Cranston and Co., Providence, ρ 327 Jamtveit B, Hammer O (2012) Sculpting of Rocks by Reactive Fluids. Geochem Perspect 1:341^-80 Jamtveit B, Kobchenko M, Austrheim H, Malthe-S0renssen A, R0yne A, Svensen Η (2011) Porosity evolution and crystallization-driven fragmentation during weathering of andesite. J Geophys Res 116:B12204 Jin L, Brantley SL (2011) Soil chemistry and shale weathering on a hillslope influenced by convergent hydrologie flow regime at the Susquehanna/Shale Hills Critical Zone Observatory. Appi Geochem 26:S51-S56 Jin L, Ravella RBK, Bierman PR, Heaney P, White T, Brantley SL (2010) Mineral weathering and elemental transport during hillslope evolution at the Susquehanna/Shale Hills Critical Zone Observatory. Geochim Cosmochim Acta 74:3669-3691 Jin L, Rother G, Cole DR, Mildner DFR, Duffy CJ, Brantley SL (2011) Characterization of deep weathering and nanoporosity development in shale - a neutron study. Am Mineral 96:498-512 Jin L, Mathur R, Rother G, Cole DR, Bazilevskaya EA, Williams JZ, Carone A, Brantley SL (2013) Evolution of porosity and geochemistry in Marcellus Formation black shale during weathering. Chem Geol 356:14 Kahle A, Winkler B, Radulescu A, Schreuer (2004) Small-angle neutron scattering study of volcanic rocks. Eur J Mineral 16:407^117 Kang QJ, Lichtner Ρ, Zhang DX (2007) An improved lattice Boltzmann model for multi-component reactive transport in porous media at the pore scale. Water Resour Res 43:W12S14 Katsura S, Kosugi K, Mizutani T, Mizuyama Τ (2009) Hydraulic properties of variously weathered granitic bedrock in headwater catchments. Vadose Zone J 8:557-573

Porosity Increases in Crystalline Silicate Rocks

353

Kelemen PB, Matter J, Streit EE, Rudge JF, Curry WB, Blusztian J (2011) Rates and mechanisms of mineral carbonation in peridotite: Natural processes and receipts for enhanced, in situ C 0 2 capture and storage. Ann Rev Earth Planet Sci 39:545-576 Landrot G, Ajo-Franklin JB, Yang L, Cabrini S, Steefel CI (2012) Measurement of accessible reactive surface area in a sandstone, with application to C 0 2 mineralization. Chem Geol 318-319:113-125 Lebedeva M, Brantley SL (2013) Exploring geochemical controls on weathering and erosion of convex hillslopes: beyond the empirical regolith production function. Earth Surf Process Landf 38:1793-1807 Li L, Steefel C, Yang L (2008) Scale dependence of mineral dissolution rates within single pores and fractures. Geochim Cosmochim Acta 72:360-377 Li D, Jacobson AJ, Mclnerney DJ (2013) A reactive-transport model for examining tectonic and climatic controls on chemical weathering and atmospheric C 0 2 consumption in granitic regolith. Chem Geol 365:30^12 Lichtner Ρ (1988) The quasi-stationary state approximation to coupled mass transport and fluid-rock interaction in a porous medium. Geochim Cosmochim Acta 52:143-165 Lowell S, Shields JE, Thomas MA, Thommes M (2004) Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density. Kluwer Academic Publishers, Dordrecht Ma L, Jin L, Brantley S L (2011) How mineralogy and slope aspect affect REE release and fractionation during shale weathering in the Susquehannah/Shale Hills Critical Zone Observatory. Chem Geol 290:31-49 Mäher K, Chamberlain CP (2014) Hydrologie Regulation of Chemical Weathering and the Geologic Carbon Cycle. Science 343:1502-1504 Merrill GP (1906) A Treatise on Rocks, Rock Weathering, and Soils. MacMillan Ine, New York Mildner DFR, Rezvani R, Hall PL, Borst RL (1986) Small-angle scattering of shaly rocks with fractal pore interfaces. Appi Phys Lett 48:1314-1316 Molins S (2015) Reactive interfaces in direct numerical simulation of pore scale processes. Rev Mineral Geochem 80:461^181 Molins S, Trebotich D, Steefel C, Shen C (2012) An investigation of the effect of pore scale flow on average geochemical reaction rates using direct numerical simulation. Water Resour Res 48:W03527 Molnar P, Anderson RS, Anderson SP (2007) Tectonics, fracturing of rocks, and erosion. J Geophys Res-Solid Earth 112:1-12 Mulyanto B, Stoops G (2003) Mineral neoformation in pore spaces during alteration and weathering of andesitic rocks in humid tropical Indonesia. Catena 54:385-391 Nahon D (1991) Introduction to the Petrology of Soils and Chemical Weathering. John Wiley & Sons, New York Navarre-Sitchler A, Steefel CI, Yang L, Tomutsa L (2009) Evolution of porosity and diffusivity associated with chemical weathering of a basalt clast. J Geophys Res - Earth Surf 114:1-14 Navarre-Sitchler A, Steefel CI, Sak PB, Brantley SL (2011) A reactive-transport model for weathering rind formation on basalt. Geochim Cosmochim Acta 75:7644-7667 Navarre-Sitchler AK, Cole D, Rother G, JinL, Buss HL, Brantley SL (2013) Porosity and surface area evolution during weathering of two igneous rocks. Geochim Cosmochim Acta 109:400-413 Oilier CD (1971) Causes of spheroidal weathering. Earth-Sci Rev 7:127-141 Oilier CD (1985) Deep weathering, groundwater, and climate. Geografiska Annaler Series A, Phys Geog 70:285-290 Patino LC, Velbel MA, Price JR, Wade JA (2003) Trace element mobility during spheroidal weathering of basalts and andesites in Hawaii and Guatemala. Chem Geol 202:343-364 Pavich M (1986) Process and rates of saprolite production and erosion on a foliated granitic rock of the Virginia Piedmont. In: Colman S, Dethier D (eds), Rates of Chemical Weathering of Rocks and Minerals. Academic Press, London, ρ 552-590 Pelt E, Chabaux F, Innocent C, Navarre-Sitchler A, Sak P, Brantley S (2008) Uranium-throium chronometry of weathering rinds: Rock alteration rate and paleo-isotopic record of weathering fluids. Earth Planet Sci Lett 276:98-105 Radlinski AP (2006) Small-angle neutron scattering and the microstructure of rocks. Rev Mineral Geochem 63:363-397 Radlinski AP, Radlinska EZ, Agamalian M, Wignall GD, Lindner P, Randl OG (1999) Fractal geometry of rocks. Phys Rev Lett 82:3078-3081 Radlinski AP, Radlinska EZ, Agamalian M, Wignall GD, Lindner P, Randl OG (2000) The fractal microstructure of ancient sedimentary rocks. J Appi Crystallogr 33:860-862 Radlinski AP, Ioannidis MA, Hinde AL, Hainbuchner M, Baron M, Rauch H, Kline SR (2004) Angstromto-millimeter characterization of sedimentary rock microstructure. J Colloid Interface Sci 274:607-612 Rouquerol J, Avnir D, Fairbridge CW, Everett DH, Haynes JH, Pernicone N, Ramsay JDF, Sing KSW, Unger KK (1994) Recommendations for the characterization of porous solids. Pure Appi Chem 66:1739-1758

354

Navarre-Sitchler, Brantley & Rother

Rossi A, Graham RC (2010) Weathering and porosity formation in subsoil granitic clasts, Bishop Creek Moraines, California. Soil Sci Soc Am J 74:172-185 Rother G, Woywod D, Schoen M, Findenegg G H (2004) Confinement effect on the adsorption from a binary liquid system near liquid/liquid phase separation. J Chem Phys 120:11864—11873 R0yne A, Jamtveit B, Mathiesen J, Malthe-Sorenssen A (2008) Controls on rock weathering rates by hierarchical weathering-induced fractures. Earth Planet Sci Lett 275:364—369 Sak PB, Fisher DM, Gardner TW, Murphy K, Brantley S L (2004) Rates of weathering rind formation on Costa Rican basalt. Geochim Cosmochim Acta 68:1453-1472 Schmidt PW (1991) Small-angle scattering studies of disordered, porous and fractal systems. J Appi Crystallogr 24:414-435 Sing KSW, Everett DH, Haul RW, Moscou L Pierotti, RA Rouquerol J, Siemieniewska Τ (1985) Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity. Pure Appi Chem 57:603-619 Sternberg PD, Anderson MA, Graham RC, Beyers JL, Tice KR (1996) Root distribution and seasonal water status in weathered granitic bedrock under chaparral. Geoderma 72:89-98 Taylor G, Eggleston RA (2001) Regolith Geology and Geomorphology. John Wiley and Sons, Chichester Teixeira J (1988) Small-angle scattering by fractal systems. J Appi Crystallogr 21:781-785 Thommes M, Cychosz KA (2014) Physical adsorption characterization of nanoporous materials: progress and challenges. Adsorption 20:233-250 Tugrul A (2004) The effect of weathering on pore geometry and compressive strength of selected rock types from Turkey. Eng Geol 75:215-227 Turner BF, Stallard RF, Brantley SL (2003) Investigation of in situ weathering of the quartz diorite bedrock in the Rio Icacos basin, Luquillo Experimental Forest, Puerto Rico. Chem Geol 202:313-341 Twidale CR, Vidal Romani J R (2005) Landforms and geology of granite terrains. AA Balkema Publishers, Leiden, The Netherlands White AF (2002) Determining mineral weathering rates based on solid and solute weathering gradients and velocities: application to biotite weathering in saprolites. Chem Geol 190:69-89 White AF (2008) Quantitative approaches to characterizing natural chemical weathering rates. In: Brantley SL, Kubicki JD, White AF (Eds), Kinetics of Water Rock Interaction. Springer, New York, ρ 469-544 Witty JH, Graham RC, Hubbert KR, Doolittle JA, Wald JA (2003) Contributions of water supply from teh weathered bedrock zone to forest soil quality. Geoderma 114:389^-00 Wong P-z, Bray AJ (1988) Porod scattering from fractal surfaces. Phys Rev Lett 60:1344-1344 Woywod D, Schemmel S, Rother G, Findenegg GH, Schoen M (2005) Phase behavior and local structure of a binary mixture in pores: Mean-field lattice model calculations for analyzing neutron scattering data. J Chem Phys 122:124510 Wyrick GG, Borchers JW (1981) Hydrologie effects of stress-relief fracturing in an Appalacian valley, United States Geological Survey. Water-Supply Paper 2177 Zanner CW, Graham RC (2005) Deep regolith: exploring the lower reaches of soil. Geoderma 126:1-3 Zwieniecki MA, Newton M (1996) Seasonal pattern of water depletion from soil-rock profiles in a mediteranean climate in southwestern Oregon. Can J Forest Res 26:1346-1352

11

Reviews in Mineralogy & Geochemistry Vol. 80 pp. 355-391.2015 Copyright © Mineralogical Society of America

Isotopie 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 jd mhan@ iiiinois. edu

Shaun T. Brown Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, California 94720, USA and Departmen t of Earth and Planetary Science University of California Berkeley Berkeley, California 94720, USA stbrown @ lbl.gov

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 isotopie 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 isotopie composition of the composite sample. 1529-6466/15/0080-0011505.00 Ifani . ' i ' l J ' i a

http://dx.doi.org/10.2138/rmg.2015.80.ll

© 2015, D r u h a n , B r o w n , Huber.

This w o r k is l i c e n s e d u n d e r the Creative C o m m o n s A t t r i b u t i o n - N o n C o m m e r c i a l - N o D e r i v s 3.0 License.

356

Druhan, Brown & Huber

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 isotopie composition of the solid surface and surrounding fluid distinct f r o m '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 isotopie partitioning specific to the phase boundary then influences the accuracy of simulations f r o m 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 andfluids 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; M o h n s 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 f r o m 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. W h e n 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 isotopie 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 isotopie

Isotopie Gradients Across Fluid-Mineral

Boundaries

357

Isotopie abundance

Figure 1. Conceptual model of isotopie 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 isotopie composition also varies from left to right as a result of reaction progress, but the isotopie 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 isotopie 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 isotopie 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 isotopie 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

358

Druhan, Brown & Huber

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 Frommer 2008). As noted above, dispersion is one method of parameterizing mixing at the continuum scale, and thus approximating the contribution of distinct fluid isotopie compositions at the pore scale (Fig. 1). Improved accuracy should then be achievable by describing distributed porescale isotopie 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 isotopie 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 isotopie partitioning across reactive interfaces, but serve as examples in which interpreting the response of continuum-scale isotopie values to variations in external parameters is improved by consideration of pore-scale isotopie 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 isotopie partitioning across fluid-mineral interfaces: α-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 isotopie 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 isotopie 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 isotopie 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.

Isotopie Gradients Across Fluid-Mineral

Boundaries

359

Figure 2. Simplified illustration of the 23S U decay series. The half-life and primary decay path are indicated for each isotope.

The 238U isotope is radioactive and for Ν nuclei at time Γ in a closed system the number of decaying nuclei diV in the time interval At follows a homogeneous Poisson statistical law and is proportional to N:

Ν dt where lambda (λ) 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) = N0e-x'

(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

ty=^T/2

(3)

λ

For intermediate daughter products such as commonly discussed in terms of activity (A) where:

234

Α(ί) = λΛΤ(ί)

U and

226

Ra, isotopie abundance is

(4)

and thus the initial activity at t = 0 is A0 = λΝ0. In a closed system that starts with no daughter isotopes, for example the 238 U- 234U system, the initial activity of 234U is zero and it will increase until A238 = A234 . At this point the supply of 234U from 238U decay is balanced by the decay of 234U to 230 Th, representing a steady state where the ratio of λ limits the maximum possible intermediate daughter activity. When A¡ = A2 this is a special condition

360

Druhan, Brown & Huber

known as secular equilibrium. In the remainder of the chapter we will use the notation 234 U /238 U ^ = A234d / A238d 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: iV2=iV0(l-e-M)

(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 238 U- 234 U system to consider the influence of pore-scale effects in quantifying α-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 Copien (2011). The isotope ratio (R) of a particular reservoir is defined as

R('E/jE)

NÇE) /iV( É)

(6)

where Ν is the number of atoms of Έ and;Z?, 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 (δ), and defined as

VEp = [R{Έ/ >É)p -R^EI

'Ε)

ω

j/ R[Έ/ 'E) ^

= R(^EIiE)pIR(iEliE)a-1

^

This delta value is nondimensional, but is commonly multiplied by 1000 to report values as parts per thousand or per mil (%c). The difference between the isotope ratios (Δ) of two compounds or phases (Ρ and Q) is then A'E p / q = δ'Ερ - δ'Εβ

(8)

The apparent or net isotopie fractionation factor does not make use of delta notation, but is defined as

a'Ep/Q =R[iEliE)pIR[iEliE)a

(9)

Conversion between Δ'Ζ?Ρ/β and α Έ Ρ / β may be approximated as &Ep/Q ~ In α'E p/ 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 isotopie 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 χ·ΕΡ=η(·Ε)ρ/Ση(Ε)

(10)

This value is used in the derivation of isotope-specific reversible rate expressions that involve a solid phase.

Isotopie Gradients Across Fluid-Mineral

Boundaries

361

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 C0 2 : C 0 2 + 0 H HC03

(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 (STcq). Isotopie equilibrium may be described similarly, for example considering carbon isotope equilibrium between the carbon-bearing species in Equation (11): 12

C0 2 + H 13 C0 3 1 3 C0 2 + H 12 C0 3

(12)

leading to a separate equilibrium constant to account for this isotopie partitioning. A comparable relationship could be written for the partitioning of oxygen isotopes between C 0 2 and HC0 3 ", or between HC0 3 " 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 isotopie equilibrium. For reactions involving simple stoichiometric relationships, where one atom of the element of interest occurs in all reactant and product species, the isotopie equilibrium constant is equivalent to the fractionation factor (Eqn. 9); however, the relationship is often more complex (Schäuble 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 isotopie partitioning is entirely kinetic. A kinetic fractionation factor is expressed as the ratio of the isotopie composition of the instantaneously generated product species (Pinst) and the residual reactant (Q) through a single reaction pathway:

αUn%,e=R('E/JE)^

IR('EI'E)Q

(I3)

This α Η π 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 α^π = 'kl'k (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.

362

Druhan, Brown & Huber

Under certain conditions, it is possible to develop theoretical models to predict the evolution of the isotopie 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 isotopie 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 = R0f(a-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 ( α - I ) . This relationship requires a constant fractionation factor a, and produces an exponential relationship between reaction progress and isotopie partitioning. This model is only strictly intended for systems in which the reactant is continually supplied (/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 234 U/ 238 U AR 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 α-recoil injection to the solution based upon a spherical grain model and an α-recoil distance of 55 nm. Earlier inferences (Turkowsky 1969) and later laboratory studies (Fleischer 1988) suggest that the recoil distance is closer to 30^-0 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 α-recoil ejection from a mineral grain based on:

234

Th to a fluid (Q) due to

Isotopie Gradients Across Fluid-Mineral 1

Q=-LSN23S

Boundaries

ρλ'238 :

363 (15)

where L is the α-recoil range, S is the surface area, ρ is the solid density, and N2¡g and λ238 are the number of 238U atoms per gram of solid and the decay constant, respectively. The activity of 234 Th in solution is a production-decay equation (combining Eqns. 4 and 5) of the form: A234II

=^LSN2ìs

ρλ238 χ ( ΐ - β - λ — ' ) / λ :

(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 α-recoil to the fluid phase is arguably the most important variable. For example, Kigoshi (1971) assumed spherical grain geometry for their 1-10 μπι 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 α-particle ( f a ) being ejected from a mineral grain assuming a spherical geometry can be approximated: (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 ofLee et al. (2010) andHandley 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 isotopie 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, α-recoil and preferential leaching of daughter isotopes to the pore fluid, sorptiondesorption 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 234 U/ 238 U AR domains. There is considerable discussion in the uranium series literature about the relative roles of preferential leaching and direct α-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 234 Th 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 α-recoil.

Drulian, Brown & Huber

364

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 α-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 23SU 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 234 U/ 23S U AR of the fluid will evolve based on the dissolution-precipitation reactions and α-recoil ejection to the fluid. The 2 3 4 U/ 2 3 S U a r evolution of mineral grains with time is: (18) It is apparent from Equation (18) that probability of α-particle ejection (fa) and time are the critical variables that affect the activity ratio of the solid grains (DePaolo et al. 2006). Estimating f a requires knowledge of the mineral volume surface area and recoil distance. DePaolo et al. (2006, 2012) calculated f a as a function of grain diameter and shape (surface area), demonstrating, for example, that mineral grains of 10-μπι diameter could have greater than a factor of 10 variability i n / a (Fig. 4). Porewater (234U/238U)a > 1.0 Rim (234U/238U)a < 1.0 Recoil J " "

234-JJ

Bulk Solid (234U/238U)a = 1.0

Depleted surface layer

grain diameter (d ) P'

Figure 3. Mineral grains in contact with pore fluids lose 234 U by alpha recoil, preferential leaching along recoil tracks, dissolution-precipitation and diffusion. The recoil distance of Th (L) defines the depleted surface area depth. Adapted from Maher et al. (2006a).

Isotopie Gradients Across Fluid-Mineral

Boundaries

365

A subsequent study by Bourdon et al. 2009 suggests employing a combined B E T sorption surface area measurements and a fractal dimension model to overcome the uncertainty of mineral surface areas in natural sediments.

/α

4

a 4-D{ L

D—2

(19)

where S B E T is the measured surface area, a is the molecule diameter for the adsórbate and D is the fractal dimension of the grain surface, which is measured independently. Non-ideal (spherical) surface area is particularly important at the pore scale, where roughness (Lee et al. 2010) can increase the surface area of a mineral grain by a factor of 17. The sorption data of Bourdon et al. (2009) have been successfully applied to studies of soils (Oster et al. 2012) and to date ice using trapped fine-grained sediments (Aciego et al. 2011). The evolution of 2 3 4 U/ 2 3 8 U A R of pore fluids is described by Equation (20): Rf ^

= RdMs^(A

- A ) + (λ 238 - λ 234 ) Λ Λ + λ 234 ^

+ faMs ^ j

(20)

where C is the concentration and A is the 2 3 4 U/ 2 3 8 U A R of the solid (s) and fluid (f), is the rate of dissolution in inverse time, R{ is the retardation factor accounting for sorbed uranium and Ms is the solid/fluid volume ratio (Tricca et al. 2001; Mäher et al. 2004). The conservation equation can be rearranged to yield the following approximation for the pore fluid at steady state (Af(ss)): _ RdA + λ 2 3 4 / α

n u

This relationship can then be manipulated to use the activity ratios to calculate the solid dissolution rate:

f 1234

Maher et al. (2004) used this relationship to calculate the dissolution rates of deep-sea sediments over timescales longer than those accessible by laboratory experiments. The models described above are a small sample of the work on uranium series isotopes in hydrogeologic systems. At the pore scale additional complexities may be considered, particularly when the steady-state assumption is not valid. For example, rapid changes in the dissolution rate could dissolve the 2 3 4 U depleted layer of mineral grains, lowering the 2 3 4 U/ 2 3 8 U a r in the pore fluid. Additionally, where mineral grains are in direct contact with one another the ejected daughter isotope may be implanted into an adjacent mineral grain and not accumulated in the pore fluid. The net effect of significant implantation would be to underestimate the sediment dissolution rate (Eqn. 22) or the age of fine-grained sediments (DePaolo et al. 2006). In general the aforementioned studies demonstrate that uranium series disequilibria is largely controlled by the surface structure of individual grains that affect the isotopie composition of fluids considered at the continuum scale. Uncertainty in the relative contributions of α-recoil and preferential leaching to isotopie fractionation in the uranium series and precise measurements of mineral surface area and recoil tracks currently limit the fidelity of existing models. Future directions in this area may include incorporating daughter isotopes with different recoil distances such as 2 3 0 Th and 2 2 6 Ra to better constrain the fa

366

Druhan, Brown & Huber

0.1

0.01

0.001 0.1

10

1

100

1000

Mean grain diameter (μιη) Figure 4. Estimates oí fa as a function of grain size modified from DePaolo et al. (2006), Mäher et al. (2006a) and DePaolo et al. (2012). Values of λΓ represent a roughness factor accounting for deviations in grain shape and surface area from an ideal sphere. Grey shaded zone indicates the region of measured values from the King's River Fan (Lee et al. 2010), dust from ice cores (Aciego et al. 2011), marine sea core sediment (DePaolo et al. 2006) and sediment from Hanford, WA (Maher et al. 2006a).

parameter and more sophisticated imaging techniques to quantify mineral surface structure. Additionally, precise measurements of the α-recoil distance (L) and the relationship between surface area a n d / a will be necessary to improve the application of pore scale 234 U/ 23S U AR to diverse problems such as sediment dating, mineral dissolution rates and vadose zone transport. Diffusive fractionation The change in concentration (C) of a solute due to a net divergence of the flux (/) in an arbitrary volume ^

dt

= -V·/

(23)

is often approximated with the linear transport theory that relates the diffusive flux to the gradient in chemical potential or concentration, i.e., Fick's first law: J =-DVC

(24)

Here the transport coefficient D is the molecular diffusivity of the solute and has units of length squared per time.

Isotopie Gradients Across Fluid-Mineral

Boundaries

367

Under ideal conditions, kinetic theory provides an expression for the diffusion coefficient of molecules in a gas (25) as the product of the mean free path (/) between collisions, and the square root of the product of the gas constant (R) and temperature (7) divided by the mass (m) of the molecule or atom. This relationship suggests that the mass of the particle influences the transport coefficient and by extension the flux to the reactive surface. Therefore the lighter the particle the larger the diffusion coefficient, which leads to the expectation that if two particles differ in their masses (mt and m2), then for the same conditions the difference in their diffusion coefficients would be (26) However, this relationship is specific to an ideal gas, and in liquids the factors that contribute to the value of an individual species diffusion coefficient become much more complex (Moller et al. 2005; Yamaguchi et al. 2005). For instance, in electrolyte solutions, the influence of electrochemical forces on the interaction of diffusing species cannot be neglected, and thus requires accounting for cross-diagonal terms representing the influence of one species' or isotopie activity gradient or electrochemical potential gradient on another. This effect may be quantified in the absence of charged surfaces by including treatment of electrochemical migration using the Nernst-Plank equation (Steefel and Maher 2009; Steefel et al. 2015; Rasouli et al. 2015), while treatment of surface charge requires further inclusion of a complete electrical double layer model (Tournassat and Steefel 2015, this volume). In the context of stable isotope fractionation in aqueous solutions, this complexity has only recently been considered. Richter et al. (2006) conducted a series of experiments to determine the fractionation of isotopes associated with diffusion in comparison with previously estimated values. Their experimental design allowed a small volume of aqueous solution containing a dissolved salt (V t ) to be suspended inside of a much larger volume of dilute fluid (V2). The length and aperture of the tube connecting the two reservoirs was selected such that the solution inside of would remain well mixed over the timescale necessary for dissolved solutes to diffuse into V2. Richter et al. (2006) considered departure from Equation. (26) by recasting the power as a parameter (β), which they then fit to their experimental data. (27) The results of their study determined β values less than 0.5, but greater than 0.0, thus providing evidence that kinetic fractionation due to diffusion could contribute to observed variations in stable isotope ratios in aqueous solutions. However, their method required the implicit assumption that an inverse power law (Eqn. 27) is an accurate description of the dependence of a diffusion coefficient on the mass of the solute. Bourg and Sposito (2007) sought to directly test this assumption by utilizing molecular dynamics (MD) simulations. The use of a numerical method allowed them to obtain the diffusion coefficient for more than two isotopes of the same species, and thus test the generality of the β values reported by Richter et al. (2006). Their results showed good agreement with the Richter et al. (2006) values, demonstrating that the value of β for a given mass pair in aqueous solutions is typically less the value of 0.5 predicted

368

Druhan, Brown & Huber

by kinetic theory for an ideal gas (Eqn. 26). Following on the agreement of these results, Bourg et al. (2010) extended both the experimental and numerical results to include additional ions. The results of these studies are summarized in Table 1. Table 1. Experimentally and numerically determined values of β for solutes at 348 K. Summarized from Richter et al. (2006); Bourg and Sposito (2007); Bourg et al. (2010). Solute +

Measured

Measured β

MD modeled β a

0.0148 ± 0.0017

0.0171 ± 0.0159b

Mrl, / D22m = 0.99800

0.023 ± 0.023e

0.029 ± 0.022·1

/D V k = 0.99790'

0.042 ± 0.002e1

0.042 ± 0.017d

Li

D7>L· / D6l = 0.99772

Na+

D

K+ +

0.003 ±0.018·1

Cs

37Q /D35ci = 0.99857°

0.0258 ± 0.0144a

Ö

37q /D35ci = 0.99857°

0.0320 ± 0.0097f

Mg+

DZ 253 /DmΙΆ = 1.00003° Mg Mg

0 ± 0.0015a

0.006 ± 0.018b

Ca2+

°44C,, /Z)40c- = 0.99957'

0.0045 ± 0.0005e1

0.0000 ± 0.0108e1

CI

Ö

Br

0.034 ±0.018b

Notes: "Richter et al. (2006); b Bourg and Sposito (2007); Calculated by Richter et al. (2006) from measurements at 298 Κ from Kunze and Fuoss (1962); d Bourg et al. (2010); Calculated based on difference in masses of measured isotopes and reported β value; Calculated by Bourg et al. (2010) from measurements made between 275-353 Κ by Eggenkamp and Coleman (2009)

Two key aspects of these observations are that the monovalent solutes show a larger contrast in the diffusivity of their isotopes than divalent solutes, and this difference does not follow a direct correlation with the ratio of the masses. Bourg et al. (2010) demonstrated that this relationship occurs as a result of (1) the size of the solute radius, and (2) the strength of attractive interactions between the solute and solvating water (effective mass of the diffusing species), both of which influence coupling of motion between the solute and solvent. Using the residence time of water (τ) in the first solvation shell of the ion obtained from the M D simulations as a proxy for this coupling, Bourg et al. (2010) showed a clear correlation between the inverse of τ and β (Fig. 5). The aforementioned experimental observations and M D simulations indicate that pore scale gradients in both concentration and isotopie abundance between a reactive surface and the surrounding fluid (Fig. 1) are influenced by diffusion coefficients that differ as a result of the size and charge of ions and the difference in masses of their isotopes. The effects of ion size and charge are explicitly treated at the resolution of M D simulations, but must be approximated at larger scales. Therefore accurate representation of these diffusive effects on isotopie partitioning at the pore and continuum scale requires independent knowledge of the magnitude of fractionation associated with the isotopes of each ion of interest (Table 1). Furthermore, the extent to which observed differences in diffusivity relate to fractionation associated with dispersion at larger scales is unknown. From the results of Richter et al. (2006), Bourg and Sposito (2007) and Bourg et al. (2010), the value of β relating the ratio of the masses to the ratio of the diffusion coefficients is highly variable among ions and solution compositions. In principle, the ratio of the diffusion coefficients could be directly obtained for two isotopologues, specific to a given solution

Isotopie Gradients Across Fluid-Mineral

Boundaries

369

Ρ 0.07

Q

f ÛMD simulation • Experiment l/T 0 25 !PS'J

Figure 5. β values as a function of inverse residence time of water in the first solvation shell for ions where both experimental and M D simulation data are available. Figure modified from Bourg et al. (2010).

composition, through a simple diffusion experiment. For example, implementing a generic tracer in a discretized domain using the CrunchFlow (Steefel et al. 2015) reactive transport code such that at the start of the simulation an elevated concentration exists in the center of a closed system, one may obtain the spatial partitioning of the tracer as it diffuses through the system (Fig. 6A). If this total tracer is broken into two components, where one 'isotope' composes the majority of the tracer concentration and the other is a trace amount, then a slight difference in the diffusion coefficients of the two species results in a spatial profile of their ratio through time (Fig. 6B). We emphasize that this simple modeling example demonstrates a proof of concept wherein an experimental system could provide a time series of the tracer concentration (Fig. 6C) and isotope ratio (Fig. 6D) at a fixed observation point. For any system in which variation in the isotope ratios are large enough to be detected beyond measurement error, a time series dataset of this nature could be used to obtain unique values for the diffusion coefficients. If the difference in the masses of the isotopologues is known, these values could then be related to obtain direct estimates of β for a given system. The sensitivity of isotope ratio measurements are such that diffusive fractionation may influence the interpretation of observed fractionation, but these effects have only recently begun to be considered in hydrologie studies. Analytical and numerical models have demonstrated that even in the absence of true diffusive fractionation, neglecting the effects of diffusion and dispersion leads to a lower overall estimate of reaction progress based on stable isotope ratios than when these factors are considered (Abe and Hunkeler 2006; Van Breukelen and Frommer 2008). These results have lead to a recent focus on the effects of dispersive mixing on stable isotope ratios, with particular emphasis on compound-specific stable isotope labels used to track contaminant degradation. Hydrodynamic dispersion is a fundamentally distinct process from molecular diffusion, arising from the fluctuations in velocity within and among connected pores during flow. Under certain assumptions, dispersion has been shown to display a diffusive behavior. In practice, most efforts to directly simulate fractionation due to dispersion in through-flowing systems have made use of estimated relationships between the ratio of the masses and the ratio of the dispersion coefficients. Dispersive isotope fractionation has been simulated at the continuum scale for both hydrogen and carbon isotopologues of solutes during flow through heterogeneous porous media using a variety of approximations for the difference in isotope-specific dispersivities. LaBolle et al. (2008) calculated isotope-specific

370

Drulian,

Brown & Huber

Figure 6. Numerical simulation of a tracer diffusing through time in a discretized domain demonstrating how experimental diffusion data could be used to constrain relative diffusion coefficients. The system is closed, and initially an elevated concentration exists in the center of the domain. (A) the concentration of the tracer across the domain through arbitrary time points 0.25, 1 and 5; (B) the isotope ratio relative to an arbitrary standard in delta notation (%o) for the same time points. In (A) and (B) the vertical dashed line corresponds to the location over which time series of concentration (C) and isotope ratio (D) are monitored.

values of dispersivity by substituting the reduced masses of the solutes into Equation (26). Rolle et al. (2009), Eckert et al. (2012) and Van Breukelen and Rolle (2012) used an empirical correlation from Worch (1993) close to Equation (26) but with a power of 0.53. Thus far it is not evident what, if any relationship may exist between diffusive and dispersive fractionation, or even if the power law relationship between the ratio of the masses and ratio of the diffusion coefficients validated by Bourg and Sposito (2007) holds true for the ratio of two isotopie dispersion coefficients. Dissolution Fractionation directly associated with the dissolution of material from the solid phase is often considered negligible over the reactive timescales represented by many natural samples. This assumption is justified in that many of the mechanisms thought to contribute to isotopie partitioning, such as transport (discussed above), and changes in solvation (Hofmann et al. 2012), are restricted within a solid phase. Furthermore, where apparent shifts in isotopie ratio are observed during net dissolution of natural samples, it is difficult to distinguish preferential mobilization of individual, isotopically distinct mineral phases from true fractionation (Ryu et al. 2011). However, a wide variety of studies now appear to demonstrate observable fractionation during dissolution, usually during the initial stages. These observations are significant in that they implicitly require that the solid become isotopically differentiated as material is removed from the reactive surface. This implies the development of a surface that is compositionally distinct from the interior of the grain, a process that is not readily compatible with bulk or volume fraction representations of solids.

Isotopie Gradients Across Fluid-Mineral

Boundaries

371

The earliest observations of fractionation during dissolution involved dissimilatory reduction of ferric iron by anaerobes (Beard et al. 1999, 2003; Brantley et al. 2001, 2004), which naturally led to the issue of whether observed iron isotope partitioning was a vital effect or if it could be reproduced in abiotic systems. From this starting point, a wide range of experimental conditions, including both pure mineral phases and whole rocks, as well as a variety of stable isotope ratios, have shown evidence for partitioning of isotopes during net dissolution (Table 2). The compilation in Table 2 is limited to low temperature and ambient pressure experimental conditions in which efforts have been made to minimize fractionation of fluid phase isotopes due to additional effects, such as secondary mineral precipitation. Despite wide variability in experimental conditions, a common observation is that during dissolution the isotopie composition of the fluid phase tends to be lighter than that of the initial solid, i.e. the light isotope tends to be preferentially dissolved. Furthermore, in studies where the isotopie value of the fluid is monitored as a function of time, the magnitude of this fractionation appears to vary. When a solid is introduced to an aqueous solution such that the system is undersaturated, the fractionation between the fluid and solid phase tends to exhibit a maximum value relatively soon after the experiment is initiated (Fig. 7). For example, the dissolution of goethite at pH 3 (Wiederhold et al. 2006) promoted by ligand complexation leads to a maximum fractionation between dissolved Fe and the bulk solid of -1.83%c, but this difference lessens as dissolution continues (Fig. 7A). A similar temporal trend is noted in iron isotopes during proton-promoted dissolution of granite (Kiczka et al. 2010)( Fig. 7b), in zinc isotopes during both proton and ligand promoted dissolution of biotite (Weiss et al. 2014) and in silicon isotopes during proton promoted dissolution of Hawaiian basalt (Ziegler et al. 2005). At higher temperatures and pressures, the same trend is noted in lithium isotopes during the dissolution of basalt (Verney-Carron et al. 2011) and in magnesium isotopes during the dissolution of basalt and forsterite (Wimpenny et al. 2010), but magnesium isotopes show the opposite behavior during magnesite dissolution (Pearce et al. 2012). A low-temperature exception to this general trend may be the fractionation of copper isotopes associated with oxidative dissolution of sulfide minerals. Mathur et al. (2005) reported A65Cu values of +2.74%c and +1.32%c for the abiotic dissolution of chalcocite and chalcopyrite, respectively. They suggested this isotopie enrichment of the solute was associated with the accumulation of secondary phases, however, a subsequent study demonstrating +2.0%c values for dissolution of whole rock containing chalcopyrite argued that secondary mineral formation was negligible (Fernandez and Borrok 2009). The general observation of an initially negative ASllid_blllksolid value that trends towards zero with further reaction progress is common to a variety of reactive pathways, minerals and isotope systems, and is observable even during dissolution of natural samples. For example, the biotite and chlorite enriched component of a granite sample dissolved in the presence of hydrochloric and oxalic acids demonstrated negative ASldd_solid values that were most pronounced early in the reaction (Kiczka et al. 2010) (Fig. 7B). This study explored a range of dissolution rates by varying the potassium concentration, which served to slow down the rate of iron release in the HCl experiments, and observed a greater enrichment in the light isotopes of the fluid phase with decreasing dissolution rate. In contrast, addition of potassium to the oxalic acid experiments only made a small difference in the Fe release rates, and little difference in the ASllid_blllksolid values were observed. Reaction of a biotite granite with the same acids yielded similar temporal trends, though the maximum ASllid_blllksolid values showed lighter isotope composition for the fluid phase with more acidic solutions (Chapman et al. 2009). These observations suggest that the partitioning of stable isotopes may potentially be utilized as an indicator of the mechanisms of mineral weathering, leading to development of a variety of conceptual and quantitative models for fractionation during congruent dissolution.

372

Druhan, Brown & Huber

Table 2. Compilation of measured fractionation between dissolved and initial solid phases during net dissolution at ambient temperature and pressure. Where fractionation is measured as a function of time, the maximum value is reported. Values are divided into abiotic and microbially mediated experimental conditions. Values reported in italics are considered within error of 0.0%c. Isotope ratio

Solid phase

Reactant or bacterium

Stirred/ shaken

(°C)

Τ

Initial

Max

pH

^fluid-solid %o

Ref.

57Li

forsterite

HCl

yes

25

2-3

+1.6

[1]

57Li

basalt glass

HCl

yes

25

3

-3.3

[1]

5 26 Mg

forsterite

HCl

yes

25

2-3

-0.37

[1]

5 26 Mg

basalt glass

HCl

yes

25

3

-0.21

[1]

530Si

Hawaiian basalt

HCl

no

*

3

-2.5

[2]

856Fe

granite

HCl

yes

22

4

-0.55

[3]

856Fe

granite

HCl

yes

22

2

-0.51

[3]

856Fe

granite

HCl + 0.5 mM Κ

yes

22

4

-1.15

[3]

856Fe

granite

HCl + 5 mM Κ

yes

22

4

-1.25

[3]

856Fe

granite

oxalic acid

yes

22

4.5

-0.40

[3]

856Fe

granite

oxalic + 0.5 mM Κ

yes

22

4.5

-0.49

[3]

8 Fe

granite

oxalic + 5 mM Κ

yes

22

4.5

-0.37

[3]

856Fe

biotite granite

HCl

**

22

0.34

-1.85

[4]

856Fe

biotite granite

oxalic acid

**

22

2.18

-1.33

[4]

856Fe

tholeiite basalt

HCl

**

22

0.34

-1.52

[4]

8 Fe

tholeiite basalt

oxalic acid

**

22

2.18

-0.82

[4]

857Fe

goethite

HCl

yes

*

0.3

-0.17

[5]

56

56

57

8 Fe

goethite

oxalic (dark)

yes

*

3

-1.86

[5]

857Fe

goethite

oxalic (light)

yes

*

3

-2.44

[5]

856Fe

hornblende

acetic acid

yes

*

n.r.

-0.96a

[6]

856Fe

hornblende

citric acid

yes

*

n.r.

-1.22a

[6]

856Fe

goethite

siderophore

no

*

n.r.

+0.21a

[6]

856Fe

hornblende

oxalic acid

no

*

n.r.

-0.25a

[7]

856Fe

hornblende

siderophore

no

*

n.r.

-0.36a

[7]

δ 66 Ζη

biotite granite

HCl

yes

*

0.34

-1.24

[8]

66

δ Ζη

biotite granite

oxalic acid

yes

*

2.18

-1.15

[8]

ô56Fe

hematite

Shewanella

n.r.

30

6.8

-2.04

[9]

ô56Fe

goethite

Shewanella

n.r.

30

6.8

-0.95

[9]

ô Fe

hematite

Geobacter

n.r.

30

6.8

-2.09

[9]

ô56Fe

goethite

Geobacter

n.r.

30

6.8

-0.93

[9]

ô56Fe

hematite

Shewanella

no

*

n.r.

-1.3a

[10]

ô56Fe

goethite

Bacillus

no

*

n.r.

-1.44a

[6]

ô Fe

hornblende

Bacillus

no

*

n.r.

-0.56a

[7]

ô56Fe

hornblende

Streptomyces

no

*

n.r.

-0.48a

ô56Fe

ferrihydrite

Shewanella

no

*

n.r.

-1.3a

56

56

[7] [11]

Notes: ^reported as ambient r o o m temperature; **shaken once every 2 4 h o u r s ; a no time series reported [1] W i m p e n n y et al. (2010) [2] Ziegler et al. (2005) [3] Kiczka et al. (2010) [4] C h a p m a n et al. (2009) [5] Wiederhold et al. (2006) [6] Brantley et al. (2004) [7] Brantley et al. (2001) [8] Weiss et al. (2014) [9] Crosby et al. (2007) [10] B e a r d et al. (2003) [11] Beard et al. (1999)

Isotopie Gradients Across Fluid-Mineral

Boundaries

373

The preferential removal o f light isotopes during the initial stages o f a dissolution reaction is commonly considered a kinetic fractionation associated with enrichment o f the solid surface in the heavy isotope. S o m e studies have considered whether or not the initial stage in which this surface layer is established could be described as a distillation or Rayleigh process. Wiederhold et al. ( 2 0 0 6 ) discussed this conceptual model in the context o f iron isotopes and Weiss et al. ( 2 0 1 4 ) in the context o f zinc isotopes. B o t h studies point out that a Rayleigh model would carry the implicit assumption that the isotopes o f the solid phase are well mixed, and thus homogeneously released. However, the development o f either a depth into the solid that is isotopically zoned or a non-uniform distribution o f reactive surface sites along a monoatomic surface layer both contradict this requirement. The consistent observation o f a maximum A Sllid _ blllksolid early in the reaction progress that b e c o m e s less depleted with time (Fig. 7) then requires consideration o f an additional mechanism. T h e development o f an isotopically distinct solid surface has been modeled in terms o f the propagation o f a reacted surface layer into the unreacted solid. This is based on the idea o f a preferentially reacted or 'leached' layer as commonly used to describe non-stoichiometric dissolution o f silicates, e.g., Casey et al. ( 1 9 9 3 ) ; Casey and Ludwig ( 1 9 9 6 ) ; Hellmann et al. ( 1 9 9 0 ) ; Oelkers ( 2 0 0 1 ) . Brantley et al. ( 2 0 0 4 ) derived a model for isotopie fractionation

ι 0.5

0 -0.5

-1

(

|

1

-1.5 -2

-2.5

0

0.1

1

10

100

1000

10000

-0.8

-0.9

-1 r -i-i