Reservoir Formation Damage (2007)(2nd ed.)(en)(1136s) [2 ed.] 9780750677387, 0750677384

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RESERVOIR FORMATION DAMAGE Second Edition

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RESERVOIR FORMATION DAMAGE F U N D A M E N TA L S , MODELING, A S S E S S M E N T, A N D M I T I G AT I O N

Second Edition

Faruk Civan

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Gulf Professional Publishing is an imprint of Elsevier

Gulf Professional Publishing is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA Linacre House, Jordan Hill, Oxford OX2 8DP, UK Copyright © 2007, Elsevier Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Recognizing the importance of preserving what has been written, Elsevier prints its books on acid-free paper whenever possible. Library of Congress Cataloging-in-Publication Data Civan, Faruk. Reservoir formation damage / Faruk Civan.—2nd ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-7506-7738-7 (acid-free paper) ISBN-10: 0-7506-7738-4 (acid-free paper) 1. Hydrocarbon reservoirs. 2. Petroleum—Geology. I. Title. TN870.57.C58 2007 622′ .338—dc22 2006036419 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 13: 978-0-7506-7738-7 ISBN 10: 0-7506-7738-4

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CONTENTS

Preface, xv About the author, xix

1 Overview of Formation Damage . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction, 1; Common Formation Damage Problems, Factors, and Mechanisms, 5; Team for Understanding and Mitigation of Formation Damage, 7; Objectives of the Book, 7

PART I

Characterization of Reservoir Rock for Formation Damage – Mineralogy, Texture, Petrographics, Petrophysics, and Instrumental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations . . . . . . . . . . . . . . . . . . . . . . . . . 13 Introduction, 14; Origin of Petroleum-Bearing Formations, 14; Constituents of Sedimentary Rocks, 15; Composition of Petroleum-Bearing Formations, 16; Mineral Sensitivity of Sedimentary Formations, 18; Mechanism of Clay Swelling, 28; Modeling Clay Swelling, 34; Cation Exchange Capacity, 56; Shale Swelling and Stability, 63

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3 Petrographical Characteristics of Petroleum-Bearing Formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Introduction, 78; Petrographical Characteristics, 79; Morphology of Dispersed Clays in Sandstones, 93; Rock Damage Tendency and Formation Damage Index Number, 95; Reservoir Characterization, 99

4 Petrophysics – Flow Functions and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Introduction, 101; Wettability Alteration, 102; Dependence of End-Point Saturations to Porosity and Permeability, 108; Alteration of Flow Functions: Capillary Pressure and Relative Permeability, 111; Temperature Dependency of the Rock Wettability, 116; Effect of Temperature on Formation Damage, 119; Effect of Morphology of Dispersed Clays on Capillary Pressure and Relative Permeability in Sandstones, 121

5 Porosity and Permeability Relationships of Geological Formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Introduction, 125; Basic Models for Permeability of Rocks, 126; Special Effects, 137; Advanced Applications, 139

6 Instrumental and Laboratory Techniques for Characterization of Reservoir Rock . . . . . . . . . . . . . . . . . . . 154 Introduction, 154; Formation Evaluation (FE), 155; Instrumental Laboratory Techniques, 158

PART II Characterization of the Porous Media Processes for Formation Damage – Accountability of Phases and Species, Rock–Fluid-Particle Interactions, and Rate Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7 Multiphase and Multispecies Transport in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Introduction, 177; MultiPhase and Species Systems in Porous Media, 178; Alternative Expressions of Various Species and Flow for Systems in Porous Media, 179; MultiSpecies and MultiPhase Macroscopic Transport Equations, 184 viii

8 Particulate Processes in Porous Media . . . . . . . . . . . . . . . 191 Introduction, 191; Particulate Processes, 193; Properties Affecting Particles, 194; Forces Acting Upon Particles, 195; Rate Equations for Particulate Processes in Porous Matrix, 203; Particulate Phenomena in Multiphase Systems, 220; Temperature Effect on Particulate Processes, 227

9 Crystal Growth and Scale Formation in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Introduction, 235; Types of Precipitation, 236; Solid–Liquid Equilibrium and Solubility Equation, 237; Crystallization Phenomena, 239; Particle Growth and Dissolution in Solution, 249; Scale Formation and Dissolution at the Pore Surface, 250; Crystal Surface Pitting and Displacement by Dissolution, 253

PART III

Formation Damage by Particulate Processes-Fines Mobilization, Migration, and Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10 Single-Phase Formation Damage by Fines Migration and Clay Swelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Introduction, 259; Algebraic Core Impairment Model, 260; Ordinary Differential Compartments-in-Series Core Impairment Model, 273; Simple Partial Differential Core Impairment Model, 277; Partial Differential Core Impairment Model Considering the Clayey Formation Swelling and both the Indigenous and the External Particles, 279; Plugging–Nonplugging Parallel Pathways Partial Differential Core Impairment Model, 285; Model-Assisted Analysis of Experimental Data, 292

11 Multiphase Formation Damage by Fines Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Introduction, 317; Formulation of a Multiphase Formation Damage Model, 318; Model-Assisted Analysis of Experimental Data, 331

12 Cake Filtration: Mechanism, Parameters, and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Introduction, 342; Incompressive Cake Filtration without Fines Intrusion, 345; Compressive Cake Filtration Including Fines Invasion, 374 ix

PART IV

Formation Damage by Inorganic and Organic Processes – Chemical Reactions, Saturation Phenomena, Deposition, and Dissolution . . . . . 405

13 Inorganic Scaling and Geochemical Formation Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Introduction, 407; Geochemical Phenomena—Classification, Formulation, Modeling, and Simulation, 410; Reactions in Porous Media, 412; Geochemical Modeling, 421; Graphical Description of the Rock–Fluid Chemical Equilibrium, 426; Geochemical Model–Assisted Analysis of Fluid–Fluid and Rock–Fluid Compatibility, 431; Geochemical Simulation of Rock–Fluid Interactions in Brine-Saturated Sedimentary Basins, 456

14 Formation Damage by Organic Deposition . . . . . . . . . . . 468 Introduction, 468; Characteristics of Asphaltenic Oils, 471; Mechanisms of the Heavy Organic Deposition, 477; Asphaltene and Wax Phase Behavior, 479; Prediction of Asphaltene Stability and Measurement (detection) of the Onset of Asphaltene Flocculation, 502; Algebraic Model for Formation Damage by Asphaltene Precipitation in Single Phase, 514; Plugging–Nonplugging Pathways Model for Asphaltene Deposition in Single Phase, 516; Two-Phase and Dual-Porosity Model for Simultaneous Asphaltene–Paraffin Deposition, 521; Single-Porosity and Two-Phase Model for Organic Deposition, 532

PART V Assessment of the Formation Damage Potential – Testing, Simulation, Analysis, and Interpretation . . . . . . . . . . . . . . . . . 557 15 Laboratory Evaluation of Formation Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 Introduction, 559; Fundamental Processes of Practical Importance for Formation Damage in Petroleum Reservoirs, 561; Selection of Reservoir-Compatible Fluids, 562; Experimental Setup for Formation Damage Testing, 564; Recommended Practice for Laboratory Formation Damage Tests, 577; Protocol for Standard Core Flood Tests, 588; Laboratory Procedures for Evaluation of Common Formation Damage Problems, 595; Evaluation of the Reservoir Formation Damage Potential by Laboratory Testing – a Case Study, 616 x

16 Formation Damage Simulator Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Introduction, 645; Description of Fundamental Model Equations, 647; Numerical Solution of Formation Damage Models, 649; Ordinary Differential Equations, 651; Partial Differential Equations, 655

17 Model-Assisted Analysis and Interpretation of Laboratory and Field Tests . . . . . . . . . . . . . . . . . . . . . . . 670 Introduction, 671; Measurement Error, 673; Model Validation, Refinement, and Parameter Estimation, 682; Formation Damage Potential of Stimulation and Production Techniques, 689; Reactive-Transport Simulation of Dolomitization, Anhydrite Cementation, and Porosity Evolution, 718; Impact of Scale Deposition in a Reservoir, 720; Simulation of fine particle mobilization, migration, and deposition in a core plug, 727

PART VI

Formation Damage Models for Fields Applications-Drilling Mud Invasion, Injectivity of Wells, Sanding and Gravel-Pack Damage, and Inorganic and Organic Deposition . . . . . . . . . . . . . . . . . . . . . 739

18 Drilling Mud Filtrate and Solids Invasion and Mudcake Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 Introduction, 741; Depth of Mud Damage Correlation, 746; Single-Phase Mud Filtrate Invasion Model, 747; Two-Phase Wellbore Mud Invasion and Filter Cake Formation Model, 751; Near-Wellbore Filtrate Invasion, 755; Dynamically-Coupled Mudcake Build-up and Immiscible Multiphase Filtrate Invasion, 758; Drilling Mud Loss into Naturally Fractured Reservoirs, 762

19 Injectivity of the Waterflooding Wells. . . . . . . . . . . . . . . . 775 Introduction, 775; Injectivity of Wells, 777; Water Quality Ratio (WQR), 780; Single-Phase Filtration Processes, 787; Diagnostic-Type Curves for Water Injectivity Tests, 800; Injection Rate Decline Function, 802; Field Applications, 802 xi

20 Reservoir Sand Migration and Gravel-Pack Damage: Stress-Induced Formation Damage, Sanding Tendency, and Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 Introduction, 814; Prediction of Sanding Conditions Using a Simple Model, 816; Prediction of Massive Sand Production Using a Differential Model, 817; Modeling Sand Retention in Gravel-Packs, 823; Reservoir Compaction and Subsidence, 824

21 Near-Wellbore Formation Damage . . . . . . . . . . . . . . . . . . . 829 Introduction, 829; Modeling Near-Wellbore Deposition and Its Effect on Well Performance, 831; Near-Wellbore Sulfur Deposition, 836; Near-Wellbore Calcite Deposition, 840; Near-Wellbore Asphaltene Deposition, 842

PART VII Diagnosis and Mitigation of Formation Damage—Measurement, Assessment, Control, and Remediation . . . . . . . . . . . . . . . . . . 857 22 Field Diagnosis and Measurement of Formation Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 Introduction, 859; Diagnosis and Evaluation of Formation Damage in the Field, 860; Pseudo-Damage vs. Formation Damage, 863; Measures of Formation Damage, 864; Model-Assisted Estimation of Skin Factor, 873; Model-Assisted Analysis of the Near-Wellbore Permeability Alteration Using Pressure Transient Data, 873; Productivity Decline Caused by Mud Invasion into Naturally Fractured Reservoirs, 878; Continuous Real Time Series Analysis for Detection and Monitoring Formation Damage Effects, 881; Formation Damage Expert System, 886

23 Determination of Formation Damage and Pseudo-Damage from Well Performance-Identification, Characterization, and Evaluation . . . . . . . . . . . . . . . . . . . . 889 Introduction, 890; Completion Damage and Flow Efficiency, 891; Formation Damage Assessment in the Field by Well surveillance, 902; Well-Testing Techniques, reservoir parameters, and interpretation methods, 904; Components of the Total Skin Factor, 915; Variable skin factor, 929

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24 Formation Damage Control and Remediation – Conventional Techniques and Remedial Treatments for Common Problems . . . . . . . . 937 Introduction, 938; Selection of Treatment Fluids, 941; Clay Stabilization, 942; Clay and Silt Fines, 948; Effect of Drilling Fluids on Shale Stability, 949; Bacterial Damage, 953; Inorganic Scales, 954; Organic Deposits, 956; Mixed Organic/Inorganic Deposits, 959; Formation Damage Induced by Completion-Fluids and Crude-Oil Emulsions, 959; Wettability Alteration and Emulsion and Water Blocks, 960; Intense Heat Treatment, 960; Sand Control, 960; Well Stimulation, 968; Recaputalization of the Methods for Formation Damage Mitigation, 969; Sandstone and Carbonate Formation Acidizing, 969; Water Injectivity Management, 981; Controlling the Adverse Side Effects of Remedial Treatments, 982

25 Reservoir Formation Damage Abatement – Guidelines, Methodology, Preventive Maintenance, and Remediation Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 Introduction, 986; Comprehensive Methodology for Mitigation of Formation Damage, 989; Treatment Fluid Application Methods, 1010; Thermal and Hydraulic Coupling of Wellbore with Reservoir During Remedial Fluid Treatments Illustrated for Hydraulically Fractured Well Acidizing, 1011

References, 1028 Index, 1091

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PREFACE

Formation damage is an undesirable operational and economic problem that can occur during the various phases of oil and gas recovery from subsurface reservoirs including production, drilling, hydraulic fracturing, and workover operations. Formation damage assessment, control, and remediation are among the most important issues to be resolved for efficient exploitation of hydrocarbon reservoirs. Such damage is caused by various adverse processes including chemical, physical, biological, and thermal interactions of formation and fluids, and deformation of formation under stress and fluid shear. Formation damage indicators include permeability impairment, skin damage, and decrease of well performance. The properly designed experimental and analytical techniques presented in this book can help understanding, diagnosis, evaluation, prevention, and controlling of formation damage in oil and gas reservoirs. This book provides an understanding of the fundamentals of the relevant processes causing formation damage and reducing the flow efficiency in the near-wellbore formation during the various phases of oil and gas production; an update review of the various approaches used in the modeling and simulation of formation damage for model-assisted analysis and interpretation of laboratory core tests, and for prediction and control of formation damage; and the techniques used for assessment, diagnosis, minimization, and control of formation damage in petroleum reservoirs. It focuses on the theory, modeling, and simulation of the rock, fluid, and particle interactions, fluid and particle invasion, filter cake, in situ mobilization, migration, and deposition of fines, organic and inorganic precipitation and scale formation, alteration of porosity, permeability, and xv

texture in laboratory cores and reservoir formations, and the effects of single- and multiphase fluid systems. Formation damage is evolving to be more science than art. Formation damage is an interesting interdisciplinary subject that attracts many researchers. Cost-effective mitigation of formation damage is still as much art as science. This book is a recapitulation of the present state-ofthe-art knowledge in the area of formation damage. It is intended to be a convenient source of information, widely spread over different sources. I have tried to cover the relevant material with sufficient detail, without overwhelming the readers. This book can be used by those who are engaged in the various aspects of formation damage problems associated with the production of hydrocarbons from subsurface reservoirs. It may serve as a useful reference and provides the knowledge of the theoretical and practical aspects of formation damage for various purposes, including model-assisted interpretation of experimental test data, prediction and simulation of various formation damage scenarios, evaluation of alternative strategies for formation damage minimization, and scientific guidance for conducting laboratory and field tests. Exhaustive effort has been made to gather, analyze, and systematically present the state-of-the-art knowledge accumulated over the years in the area of formation damage in petroleum reservoirs. This book is intended to provide a quick and coordinated overview of the fundamentals, and the experimental and theoretical approaches presented in selected publications. However, it should not be viewed as a complete encyclopedic documentation of the reported studies. It discusses processes causing formation damage and reducing the productivity of wells in petroleum reservoirs and systematically presents various approaches used in the diagnoses, measurement, evaluation, and simulation of formation damage. The techniques for assessment, minimization, control, and remediation of the reservoir formation damage are described. This book is intended for the petroleum, chemical, and environmental engineers, geologists, geochemists, and physicists involved in formation damage control, and for the undergraduate senior and graduate petroleum engineering students. The material presented in this book originates from my industry short courses and curriculum courses at the Mewbourne School of Petroleum and Geological Engineering at the University of Oklahoma. This book can be used in industry training courses and undergraduate senior and graduate level petroleum engineering courses. It is recommended for formation damage courses and as a companion for drilling, production, and stimulation courses. Readers will xvi

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learn the mechanisms and theoretical background of the common formation damage processes be familiar with the testing, modeling, and simulation techniques available for formation damage assessment be able to develop strategies for better management of the adverse processes to minimize and avoid formation damage in petroleum reservoirs.

I am indebted to the researchers who have contributed to the understanding and handling of the various issues and aspects of formation damage and mitigation. Their efforts have led to the accumulation of a substantial amount of knowledge and expertise on formation damage and helped develop techniques and optimal strategies for effective detection, evaluation, and mitigation of formation damage in subsurface reservoirs. Their works have been published in various literatures. I am pleased to have had the opportunity to analyze, integrate, transfer, and present the state-of-the-art knowledge of formation damage in a consistent manner in one source for the readers of this book, based on more than 870 references. I believe that most effective learning is through teaching. I have enjoyed such exercise as it provided tremendous opportunities to others to benefit from my teaching. Many of the figures, tables, and other relevant materials used in the preparation of this book were extracted from the literature published by various researchers, companies, and organizations. These include the following: Academic Press; AAPG – American Association of Petroleum Geologists; ACS – American Chemical Society; AGU – American Geophysical Union; AIChE – American Institute of Chemical Engineers; American Institute of Physics; API – American Petroleum Institute; ASME – American Society of Mechnical Engineers; A.A. Balkema Publisher; Baroid Drilling Fluids, Inc.; Canadian Institute of Mining, Metallurgy and Petroleum; Chemical Processing magazine; Chemicky Prumysl; Computational Mechanics, Inc.; Elsevier Science, The Geological Society Publishing House, IEEE – Institute of Electrical and Electronics Engineers, Inc.; ICheme-Institute of Chemical Engineers; International Institute for Geothermal Research, Italy; Ilinois State Geological Survey; John Wiley & Sons Limited; Marcel Dekker, Inc.; M-I L.L.C.; OSA – The Optical Society of America; Petroleum Society of CIM; Plenum Press; Routledge/Taylor & Francis Group LLC.; Sarkeys Energy Center at the University of Oklahoma; SPE – Society of Petroleum Engineers; Springer-Verlag; SPWLA – Society of xvii

Professional Well Log Analysts; Springer Science and Business Media, The American Oil & Gas Reporter; Transportation Research Board; National Academies, Washington, D.C.; Turkish Journal of Oil and Gas; and the U.S. Department of Energy. In addition, G. Atkinson, B. Bennett, T. Dewers, A. Hayatdavoudi, I. B. Ivanov, P. R. Johnson, P. A. Kralchevsky, R. Philip, T. S. Ramakrishnan, M. M. Reddy, M. Sahimi, G. W. Schneider, H. Tamura, K. J. Weber, and D. F. Zwager allowed the use of materials from their publications. B. Seyler of the Illinois State Geological Survey provided the photographs included in the book. The permission for use of these materials in this book is gratefully acknowledged. I am also grateful to Elsevier – Gulf Professional Publishing Company, Andrea Sherman and Julie Ochs, Integra Software Services Pvt. Ltd., India, and Kalpalathika Rajan for their support in the preparation and realization of the second edition of this book. Special thanks are due to Susan Houck for her care in typing the manuscript of the first edition of this book. I have typed all the additional materials included in the second edition. This book provides a broad background and knowledge on the practical and theoretical aspects of the various problems dealing with the processes and operations causing formation damage in subsurface geological porous formations. I wish that this book will be a convenient, informative, and useful companion for those involved in the reservoir formation damage issues at various capacities, from practitioners to academicians. Faruk Civan, Ph.D. Norman, Oklahoma, U.S.A.

While every effort was made to identify copyright holders and obtain permission for the re-use of all material with a third-party copyright that appears in this book, it is possible that there are items herein that are not correctly acknowledged, or for which we were unable to trace the copyright holder, or unable to obtain permission for re-use. If you are aware of any such case, please notify the publishers and the omission or error will be rectified in future printings of this book.

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ABOUT THE AUTHOR

Faruk Civan is an Alumni Professor of the Mewbourne School of Petroleum and Geological Engineering at the University of Oklahoma in Norman. Previously, he worked in the Chemical Engineering department at the Technical University of Istanbul, Turkey. Dr. Civan received an Advanced Engineering Degree from the Technical University of Istanbul, Turkey; an MS degree from the University of Texas at Austin, Texas; and a PhD degree from the University of Oklahoma, Norman, Oklahoma. All of his degrees are in chemical engineering. Dr. Civan specializes in petrophysics and reservoir characterization; formation and well damage modeling, diagnosis, assessment, and mitigation; reservoir and well analyses, modeling, and simulation; natural gas engineering, measurement, processing, hydrates, transportation, and storage; carbon dioxide sequestration; coalbed methane production; improved reservoir recovery techniques; corrosion protection in oil and gas wells; filtration and separation techniques; and air, water, and ground pollution modeling and control. He has published more than 200 technical articles in journals, edited books, handbook, and encyclopedia, and conference proceedings, and presented 85 invited seminars and/or lectures at various technical meetings, companies, and universities. He teaches industry short courses on xix

a number of topics worldwide. Additionally, he has written numerous reports on his funded research projects. Dr. Civan’s publications have been cited frequently in various publications, as reported by the Science Author Citation Index. Dr. Civan has received 20 honours and awards, including five distinguished lectureship awards and the 2003 SPE Distinguished Achievement Award for Petroleum Engineering Faculty. He is a member of the Society of Petroleum Engineers, the American Society of Mechanical Engineers, and the American Institute of Chemical Engineers. Dr. Civan serves as a member of the editorial boards of the Journal of Petroleum Science and Engineering, Turkish Oil and Gas Journal, Journal of Porous Media, and Journal of Energy Resources Technology. He has served on numerous petroleum and chemical engineering, and other related conferences and meetings in various capacities, including as committee chairman and member, session organizer, chair or co-chair, instructor, and as member of the editorial board of the Society of Petroleum Engineers Reservoir Engineering Journal and the special authors series of the Journal of Petroleum Technology.

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OVERVIEW OF FORMATION DAMAGE

Summary A comprehensive review of the various types of formation damage problems encountered in petroleum reservoirs is presented. The factors and processes causing these problems are described in detail. The design of a team effort necessary for understanding and controlling of the formation damage problems in the field is explained. The motivation for the writing of this book and the specific objectives are stated. The approach taken in the presentation of the materials in this book is explained. A brief executive summary of the topics covered in the book is given. The roles played by different professionals, such as the petroleum and chemical engineers, chemists, physicist, geologists, and geochemists, are described.

1.1 INTRODUCTION Formation damage is a generic terminology referring to the impairment of the permeability of petroleum-bearing formations by various adverse processes. Formation damage is an undesirable operational and economic problem that can occur during the various phases of oil and gas recovery from subsurface reservoirs including drilling, production, hydraulic fracturing, and workover operations (Civan, 2005). As expressed by Amaefule et al. (1988), “Formation damage is an expensive headache to the oil and gas industry.” Bennion (1999) described formation damage as, “The impairment of the invisible, by the inevitable and uncontrollable, resulting in an indeterminate reduction of the unquantifiable!” Formation damage assessment, control, and remediation are among the most important 1

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Overview of Formation Damage

issues to be resolved for efficient exploitation of hydrocarbon reservoirs (Energy Highlights, 1990). Formation damage may be caused by many factors, including physico-chemical, chemical, biological, hydrodynamic, and thermal interactions of porous formation, particles, and fluids, and the mechanical deformation of formation under stress and fluid shear. These processes are triggered during the drilling, production, workover, and hydraulic fracturing operations. Ordinarily, the mineral matter and fine particles loosely attached to the pore surface are at equilibrium with the pore fluids. However, variations in chemical, thermodynamic, and stress states may create nonequilibrium conditions and induce the salinity, velocity, and thermal shock phenomena and particle detachment and precipitate formation. When the equilibrium condition existing between the pore surface and the fluids is disturbed during reservoir production by primary and enhanced recovery processes, the mineral matter may dissolve and generate many different ions in the aqueous phase and the fine particles are unleashed from the pore surface into the fluid phases. Once these ions and particles are introduced into the fluid phases, they become mobile. Thus, a condition is created, like a bowl of soup of the mobile ions and fine particles in the pore space, which may interact freely with each other in many intricate ways to create severe reservoir formation damage problems. Formation damage indicators include permeability impairment, skin damage, and decrease of well performance. As stated by Porter (1989), “Formation damage is not necessarily reversible” and “What gets into porous media does not necessarily come out.” Porter (1989) called this phenomenon “the reverse funnel effect.” Therefore, it is better to avoid formation damage than to try to restore it. A verified formation damage model and carefully planned laboratory and field tests can provide scientific guidance and help develop strategies to avoid or minimize formation damage. Properly designed experimental and analytical techniques, and the modeling and simulation approaches can help understanding diagnosis, evaluation, prevention, remediation, and controlling of formation damage in oil and gas reservoirs. The consequences of formation damage are the reduction of the oil and gas productivity of reservoirs and noneconomic operation. Therefore, it is essential to develop experimental and analytical methods for understanding and preventing and/or controlling formation damage in oiland gas-bearing formations (Energy Highlights, 1990). The laboratory experiments are important steps in reaching an understanding of the physical mechanisms of formation damage phenomena. “From this

Overview of Formation Damage

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experimental basis, realistic models which allow extrapolation outside the scaleable range may be constructed” (Energy Highlights, 1990). These efforts are necessary to develop and verify accurate mathematical models and computer simulators that can be used for predicting and determining strategies to avoid and/or mitigate formation damage in petroleum reservoirs (Civan, 1994). Confidence in formation damage prediction using phenomenological models cannot be gained without field testing. Planning and designing field test procedures for verification of the mathematical models are important. Once a model has been validated, it can be used for accurate simulation of the reservoir formation damage and designing effective measures for formation damage mitigation. Current techniques for reservoir characterization by history matching do not consider the alteration of the characteristics of reservoir formation during petroleum production. In reality, formation characteristics vary (Civan, 2001, 2002a,b,e) and a formation damage model can help to incorporate this variation into the history matching process for accurate characterization of reservoir systems and, hence, an accurate prediction of future performance. Formation damage is an exciting, challenging, and evolving field of research. Eventually, the research efforts will lead to a better understanding and simulation tools that can be used for model-assisted analysis of rock, fluid, and particle interactions and the processes caused by rock deformation and scientific guidance for development of production strategies for formation damage control in petroleum reservoirs. In the past, numerous experimental and theoretical studies have been carried out for the purpose of understanding the factors and mechanisms that govern the phenomena involving formation damage. Although various results were obtained from these studies, a unified theory and approach still does not exist. In spite of extensive research efforts, development of technologies and optimal strategies for cost-effective mitigation of formation damage is still as much art as science. Civan (1996) explains A formation damage model is a dynamic relationship expressing the fluid transport capability of porous medium undergoing various alteration processes. Modeling formation damage in petroleum reservoirs has been of continuing interest. Although many models have been proposed, these models do not have the general applicability. However, an examination of the various modeling approaches reveals that these models share a common ground and, therefore, a general model can be developed, from which these models can be derived. Although modeling based on well accepted theoretical analyses is desirable and accurate, macroscopic formation damage

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Overview of Formation Damage

modeling often relies on some intuition and empiricism inferred by the insight gained from experimental studies.

As J. Willard Gibbs stated in a practical manner, “The purpose of a theory is to find that viewpoint from which experimental observations appear to fit the pattern” (Duda, 1990). Civan (1996) states The fundamental processes causing damage in petroleum-bearing formations are: (1) physico-chemical, (2) chemical, (3) hydrodynamic, (4) thermal, (5) mechanical, and (6) biological. Formation damage studies are carried out for (1) understanding of these processes via laboratory and field testing, (2) development of mathematical models via the description of fundamental mechanisms and processes, (3) optimization for prevention and/or reduction of the damage potential of the reservoir formation, and (4) development of formation damage control strategies and remediation methods. These tasks can be accomplished by means of a model assisted data analysis, case studies, and extrapolation and scaling to conditions beyond the limited test conditions. The formulation of the general purpose formation damage model describes the relevant phenomena on the macroscopic scale; i.e. by representative elementary porous media averaging (Civan, 2002f).

As stated by Civan (1996f), “Development of a numerical solution scheme for the highly nonlinear phenomenological model and its modification and verification by means of experimental testing of a variety of cores from geological porous media are the challenges for formation damage research.” As expressed by Porter (1989) and Mungan (1989), formation damage is not necessarily reversible. Thus, it is better to avoid formation damage than try to restore formation permeability using costly methods with uncertain successes in many cases. When a verified generalized formation damage model becomes available, it can be used to develop strategies to avoid or minimize formation damage.

Finally, it should be recognized that formation damage studies involve many interdisciplinary knowledge and expertise. An in-depth review of the various aspects of the processes leading to formation damage may require a large detailed presentation. Presentation of such encyclopedic information makes learning of the most important information difficult and, therefore, it is beyond the scope of this book. Instead, a summary of the well proven, state-of-the-art knowledges by highlighting the important features is presented in a concise manner for instructional purposes. The details can be found in the literature cited at the end of the book.

Overview of Formation Damage

5

1.2 COMMON FORMATION DAMAGE PROBLEMS, FACTORS, AND MECHANISMS Barkman and Davidson (1972), Piot and Lietard (1987), Amaefule et al. (1987, 1988), Bennion et al. (1991, 1993), and many others have described in detail the various problems encountered in the field, interfering with the oil and gas productivity of the petroleum reservoirs. Amaefule et al. (1988) listed the conditions affecting the formation damage in four groups: 1. Type, morphology, and location of resident minerals; 2. In situ and extraneous fluids composition; 3. In situ temperature and stress conditions and properties of porous formation; and 4. Well development and reservoir exploitation practices. Amaefule et al. (1988) classified the various factors affecting formation damage as the following: (1) Invasion of foreign fluids, such as water and chemicals used for improved recovery, drilling mud invasion, and workover fluids; (2) Invasion of foreign particles and mobilization of indigenous particles, such as sand, mud fines, bacteria, and debris; (3) Operation conditions such as well flow rates and wellbore pressures and temperatures; and (4) Properties of the formation fluids and porous matrix. Figure 1-1 by Bennion (1999) delineates the common formation damage mechanisms in the order of significance. Bishop (1997) summarized the seven formation damage mechanisms described by Bennion et al. (1991, 1993) and Bennion and Thomas (1991, 1994) as the following (after Bishop, ©1997 SPE; reprinted by permission of the Society of Petroleum Engineers): 1. Fluid–fluid incompatibilities, for example emulsions generated between invading oil-based mud filtrate and formation water. 2. Rock–fluid incompatibilities, for example contact of potentially swelling smectite clay or deflocculatable kaolinite clay by nonequilibrium water-based fluids with the potential to severely reduce near wellbore permeability. 3. Solids invasion, for example the invasion of weighting agents or drilled solids. 4. Phase trapping/blocking, for example the invasion and entrapment of water-based fluids in the near wellbore region of a gas well.

6

Formation damage

Chemical mechanisms

Mechanical mechanisms

Solids invasion

Foamy oils

Clay defloculation

Clay swelling

Oil-based fluids

Mechanical damage

Glazing

Emulsions

Solids

Precipitates

Scales

Hydrates

Dilatant Ionic

Dirty injection fluids

Paraffins

Adsorption

Geomechanical induced

Polymer Mud solids

Fluid–Fluid interactions

Wettability alterations

Mashing

Diamondoids

Asphaltenes

Compactive

Biological mechanisms

Corrosion

Polymer secretion Souring

Thermal mechanisms

Dissolution

Wettability changes

Mineral transformation

Figure 1-1. Classification and order of the common formation damage mechanisms (modified after Bennion, ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

Overview of Formation Damage

Water-based fluids

Rock–Fluid interactions

Perforation induced

Phase trapping

Fines migration

Overview of Formation Damage

7

5. Chemical adsorption/wettability alteration, for example emulsifier adsorption changing the wettability and fluid flow characteristics of a formation. 6. Fines migration, for example the internal movement of fine particulates within a rock’s pore structure resulting in the bridging and plugging of pore throats. 7. Biological activity, for example the introduction of bacterial agents into the formation during drilling and the subsequent generation of polysaccharide polymer slimes which reduce permeability.

1.3 TEAM FOR UNDERSTANDING AND MITIGATION OF FORMATION DAMAGE Amaefule et al. (1987, 1988) stated that formation damage studies require a cooperative effort between various professionals to combat the formation damage problems. These and their responsibilities are described in the following: (1) Geologist and geochemist on mineralogy and diagenesis and reservoir formation characterization and evaluation; (2) Chemist on inorganic/organic chemistry, physical chemistry, colloidal and interfacial sciences, and chemical kinetics; and (3) Chemical and petroleum engineers on transport phenomena in porous media, simulator development, interpretation of laboratory core tests, scaling from laboratory to field, interpretation of field tests, and development and implementation of strategies for formation damage control.

1.4 OBJECTIVES OF THE BOOK The focus of this book is to provide sufficient knowledge required for the following purposes: (1) Understand relevant processes by laboratory and field testing; (2) Develop theories and mathematical expressions for description of the fundamental mechanisms and processes, and phenomenological mathematical modeling and obtain numerical solutions for simulator development and computer implementation; (3) Predict and simulate the consequences and scenarios of the various types of formation damage processes encountered in petroleum reservoirs; (4) Optimize for prevention and/or reduction of the damage potential of the reservoir formation; and (5) Develop methodologies and optimal strategies for formation damage control and remediation.

8

Overview of Formation Damage

This book reviews and systematically analyzes the previous studies, addressing their theoretical bases, assumptions, and applications, and presents the state-of-the-art knowledge in reservoir formation damage in a systematic manner. Several exercise questions and problems are provided at the end of the chapters. The material is presented in seven parts: I. Characterization of Reservoir Rock for Formation Damage – Mineralogy, Texture, Petrographics, Petrophysics, and Instrumental Techniques II. Characterization of the Porous Media Processes for Formation Damage – Accountability of Phases and Species, Rock–FluidParticle Interactions, and Rate Processes III. Formation Damage by Particulate Processes – Fines Mobilization, Migration, and Deposition IV. Formation Damage by Inorganic and Organic Processes – Chemical Reactions, Saturation Phenomena, Deposition, and Dissolution V. Assessment of the Formation Damage Potential – Testing, Simulation, Analysis, and Interpretation VI. Formation Damage Models for Fields Applications – Drilling Mud Invasion, Injectivity of Wells, Sanding and Gravel-Pack Damage, and Inorganic and Organic Deposition VII. Diagnosis and Mitigation of Formation Damage – Measurement, Assessment, Control, and Remediation.

Exercises 1. What is formation damage? Give definitions of formation damage from various points of views. 2. Where does formation damage occur? 3. Classify and order the common formation damage mechanisms. 4. What are the consequences of formation damage? 5. What are the common adverse processes and mechanisms causing formation damage in petroleum reservoirs? 6. What are the typical indicators of formation damage? 7. Explain “the reverse funnel effect.” 8. Why is it better avoiding formation damage than attempting to alleviate it? 9. What are the typical factors and conditions affecting formation damage?

Overview of Formation Damage

9

10. What are the essential objectives and approaches involved in the formation damage studies? 11. What are the important requirements of development of strategies in order to avoid and/or minimize formation damage? 12. Which professionals are involved in what capacity in cooperative formation damage studies?

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PART I Characterization of Reservoir Rock for Formation Damage – Mineralogy, Texture, Petrographics, Petrophysics, and Instrumental Techniques

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MINERALOGY AND MINERAL SENSITIVITY OF PETROLEUM-BEARING FORMATIONS1

Summary The origin, mineralogy, and mineral sensitivity of petroleum-bearing formations are reviewed. The mechanisms and kinetics of mineral swelling, alteration, and fines generation are described. The models and rate equations for mineral-sensitive properties of rock, and the methods for interpretation and proper correlation of experimental data are presented. These models provide insight into the mechanism of the clay swelling process and a proper means of interpreting and analyzing the swelling-dependent characteristics of clayey formations. They allow for accurate determination of the swelling parameters of petroleum-bearing formations. They can be used to predict the conditions causing the wellbore stability problems and to alleviate the undesirable consequences of the swelling phenomenon in petroleum reservoirs.

1 Parts of this chapter have been reprinted with permission of the Society of Petroleum Engineers from Civan (1999a and 2001c).

13

14

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

2.1 INTRODUCTION Among others, Ohen and Civan (1993) point out that fines migration and clay swelling are the primary reasons for formation damage measured as permeability impairment. Poorly lithified and tightly packed formations having large quantities of authigenic, pore-filling clays sensitive to aqueous solutions, such as kaolinite, illite, smectite, chlorite, and mixed-layer clay minerals, are especially susceptible to formation damage (Amaefule et al., 1988). Formation damage also occurs as a result of the invasion of drilling mud, cements, and other debris during production, hydraulic fracturing, and workover operations (Amaefule et al., 1988). This chapter describes the mineral content and sensitivity of typical sedimentary formations, and the relevant formation damage mechanisms involving clay alteration and migration. Analytical models for interpretation and correlation of the effects of clay swelling on the permeability and porosity of clayey porous rocks are presented (Civan, 1999a, 2001c). The parameters of the models, including the swelling rate constants, and terminal porosity and permeability that will be attained at saturation, are determined by correlating the experimental data with these models. The swelling of clayey rocks is essentially controlled by absorption of water by a diffusion process, hindered by the water-exposed surface conditions. The swelling-dependent properties of clayey rocks vary proportionally with their values relative to their saturation limits and the water absorption rate. These models lead to practical and proper means of correlating and representing the various clayey rock properties.

2.2 ORIGIN OF PETROLEUM-BEARING FORMATIONS As described by Sahimi (1995), sedimentary porous formations are formed through two primary phenomena: (1) deposition of sediments, followed by (2) various compaction and alteration processes. Sahimi (1995) states that the sediments in subsurface reservoirs have undergone four types of diagenetic processes under the prevailing in situ stress, thermal, and flow conditions over a very long period of geological times: (1) mechanical deformation of grains, (2) solution of grain minerals, (3) alteration of grains, and (4) precipitation of pore-filling minerals, clays, cements, and other materials. These processes are inherent

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

15

in determining the characteristics and formation damage potential of petroleum-bearing formations.

2.3 CONSTITUENTS OF SEDIMENTARY ROCKS Many investigators, including Neasham (1977), Amaefule et al. (1988), Mancini (1991), and Ezzat (1990), presented detailed descriptions of the various constituents of oil and gas-bearing rocks. Based on these studies, the constituents of the subsurface formations can be classified in two broad categories: (1) indigenous and (2) extraneous or foreign materials. There are two groups of indigenous materials: (1) detrital materials, which originate during the formation of rocks and have restricted formation damage potential, because they exist as tightly packed and blended minerals within the rock matrix; and (2) diagenetic (or authigenic) materials, which are formed by various rock–fluid interactions in an existing pack of sediments, and located inside the pore space as loosely attached pore-filling, pore-lining, and pore-bridging deposits, and have greater formation damage potential because of their direct exposure to the pore fluids. Extraneous materials are externally introduced through the wells completed in petroleum reservoirs, during drilling and workover operations, and improved recovery processes applied for reservoir exploitation. A schematic, pictorial description of typical clastic deposits is given in Figure 2-1 by Pittman and Thomas (1979). Loosely packed authigenic clay in pores

Detrital clay-rich lamination

Detrital clay aggregate grains

Tightly packed detrital clay matrix fills pores

Figure 2-1. Disposition of the clay minerals in typical sandstone (after Pittman and Thomas, ©1979 SPE; reprinted by permission of the Society of Petroleum Engineers).

16

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

2.4 COMPOSITION OF PETROLEUM-BEARING FORMATIONS The studies of the composition of the subsurface formations by many, including Bucke and Mankin (1971) and Ezzat (1990), have revealed that these formations basically contain (1) various mineral oxides such as SiO2  Al2 O3  FeO Fe2 O3  MgO K2 O CaO P2 O5  MnO TiO2  Cl, Na2 O, which are detrital and form the porous matrix, and (2) various swelling and nonswelling clays, some of which are detrital, and the others are authigenic clays. The detrital clays form the skeleton of the porous matrix and are of interest from the point of mechanical formation damage. The authigenic clays are loosely attached to pore surface and are of interest from the point of chemical and physico-chemical formation damage. Typical clay minerals are described in Table 2-1 (Ezzat, 1990). However, the near-wellbore formation may also contain other substances, such as mud, cement, and debris, which may be introduced during drilling, completion, and workover operations, as depicted by Mancini (1991) in Figure 2-2.

Table 2-1 Description of the Authigenic Clay Minerals∗ Mineral

Chemical Elements∗ †

Kaolinite

Al4 Si4 O10 OH8

Chlorite

Mg Al Fe12 Si Al8 O20 OH16

Illite

K1–15 A14 Si7–65 A11–15 O20 OH4 

Smectite (or montmorillonite)

1/2Ca Na07 Al Mg Fe4 Si Al8 O20  • nH2 O

Mixed-Layer

Illite–Smectite and Chlorite–Smectite

∗ †

Morphology Stacked plate or sheets. Plates, honeycomb, cabbagehead rosette or fan. Irregular with elongated spines or granules. Irregular, wavy, wrinkled sheets, webby or honeycomb. Ribbons substantiated by filamentous morphology.

After Ezzat, ©1990 SPE; reprinted by permission of the Society of Petroleum Engineers. After J. E. Welton (1984).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

17

Sand-size detrital grain Isopachous rim cement

Oversize (dissolution) pore body

Intergranular pores

Pore throat Cement

Deformed mud fragment Intergranular pressure solution

Microporosity in clay

Intergranular pore body

Figure 2-2. Description of the constituents in typical sandstone (after Mancini, 1991; reprinted by permission of the U.S. Department of Energy).

There are various clay minerals in the sedimentary formations (Degens, 1965). Chilingarian and Vorabutr (1981) present a detailed review of the clays and their reactivity with aqueous solutions. Briefly, “clay” is a generic term, referring to various types of crystalline minerals described as hydrous aluminum silicates. Clay minerals occupy a large fraction of sedimentary formations (Weaver and Pollard, 1973). Clay minerals are extremely small, platy-shaped materials that may be present in sedimentary rocks as packs of crystals (Grim, 1942; Hughes, 1951). The maximum dimension of a typical clay particle is less than 0.005 mm (Hughes, 1951). The clay minerals can be classified into three main groups (Grim, 1942, 1953; Hughes, 1951): (1) Kaolinite group, (2) Smectite (or Montmorillonite) group, and (3) Illite group. In addition, there are mixed-layer clay minerals formed from several of these three basic groups (Weaver and Pollard, 1973). The morphology and the major reservoir problems of the various clay minerals is described in Table 2-2 by Ezzat (1990). The distribution of clays can be conveniently depicted by ternary line diagrams such as given in Figure 2-3 by Lynn and Nasr-El-Din (1998). They classified reservoir formations having less than 1 wt% total clay and permeability higher than 1 D as the high quality, and the low quality vice versa.

18

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations Table 2-2 Typical Problems Caused by the Authigenic Clay Minerals†

Mineral

Surface area (m2 /gm)∗ †

Major reservoir Problems

Kaolinite

20

Breaks apart, migrates and concentrates at the pore throat causing severe plugging and loss of permeability.

Chlorite

100

Extremely sensitive to acid and oxygenated waters. Will precipitate gelatineous FeOH3 which will not pass through pore throats.

Illite

100

Plugs pore throats with other migrating fines. Leaching of potassium ions will change it to expandable clay.

Smectite

700

Water sensitive, 100% expandable. Causes loss of microporosity and permeability.

Mixed-layer

100–700

Breaks apart in clumps and bridges across pores reducing permeability.

† ∗

After Ezzat, ©1990 SPE; reprinted by permission of the Society of Petroleum Engineers. After David K. Davies—Sandstone Reservoirs—Ezzat (1990).

2.5 MINERAL SENSITIVITY OF SEDIMENTARY FORMATIONS Among other factors, the interactions of the clay minerals with aqueous solutions are the primary culprit for the damage of petroleum-bearing formations. Amaefule et al. (1988) state that rock–fluid interactions in sedimentary formations can be classified in two groups: (1) chemical reactions resulting from the contact of rock minerals with incompatible fluids, and (2) physical processes caused by excessive flow rates and pressure gradients. Amaefule et al. (1988) point out that there are five primary factors affecting the mineralogical sensitivity of sedimentary formations: 1. Mineralogy and chemical composition determine the a. dissolution of minerals, b. swelling of minerals, and c. precipitation of new minerals. 2. Mineral abundance prevail the quantity of sensitive minerals.

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

19

Illite/Montmorillonite 10 20 30 40 50

60 70 80

10 90

90

20 80

80

N-1 30 70

70

N-4 40 60

60

50 50

50

60 40

40

70 30

30

80 20

20

90 10

90 10

100 0

100 0

Kaolinite

90 10

80 20

70 30

60 40

50 50

40 60

30 70

20 80

10 90

Chlorite

Figure 2-3. A ternary clay distribution chart (reprinted from Journal of Petroleum Science and Engineering, Vol. 21, Lynn, J. D., and Nasr-El-Din, H. A., “Evaluation of Formation Damage due to Frac Stimulation of Saudi Arabian Clastic Reservoir,” pp. 179–201, ©1998; reprinted with permission from Elsevier Science).

3. Mineral size plays an important role, because a. mineral sensitivity is proportional to the surface area of minerals, and b. mineral size determines the surface area to volume ratio of particles. 4. Mineral morphology is important, because a. mineral morphology determines the grain shape, and therefore the surface area to volume ratio, and b. minerals with platy, foliated, acicular, filiform, or bladed shapes, such as clay minerals, have high surface area to volume ratio. 5. Location of minerals is important from the point of their role in formation damage. The authigenic minerals are especially susceptible to alteration because they are present in the pore space as porelining, pore-filling, and pore-bridging deposits and they can be exposed directly to the fluids injected into the near-wellbore formation. Mungan (1989) states that clay damage depends on (1) the type and the amount of the exchangeable cations, such as K+  Na+  Ca2+ , and (2) the

20

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations Silicon-Oxygen tetrahedra sheet

6 (OH)

Gibbsite sheet

4 Al

7.2 A

4 O + 2(OH)

c-axis

Silicon-Oxygen tetrahedra sheet

4 Si 6O

b-axis Kaolinite (OH)8 Al4 Si4 O10

Figure 2-4. Schematic description of the crystal structure of kaolinite (after Gruner-Grim, 1942, and Hughes, 1951; reprinted courtesy of the American Petroleum Institute, 1220 L St., NW, Washington, DC 20005, Hughes, R. V., “The Application of Modern Clay Concepts to Oil Field Development,” pp. 151–167, in Drilling and Production Practice 1950, American Petroleum Institute, New York, NY, 1951, 344 p.).

layered structure existing in the clay minerals. Mungan (1989) describes the properties and damage processes of the three clay groups as the following: 1. Kaolinite has a two-layer structure (see Figure 2-4), K + exchange cation, and a small base exchange capacity, and is basically a nonswelling clay but will easily disperse and move. 2. Montmorillonite has a three-layer structure (see Figure 2-5), a large base exchange capacity of 90 to 150 meq/100 g and will readily adsorb Na+ , all leading to a high degree of swelling and dispersion. 3. Illites are interlayered (see Figure 2-6). Therefore, illites combine the worst characteristics of the dispersible and the swellable clays. The illites are most difficult to stabilize.

Sodium-montmorillonite swells more than calcium-montmorillonite because the calcium cation is strongly adsorbed compared to the sodium cations (Rogers, 1963). Consequently, when the clays are hydrated

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

21

Silicon-Oxygen tetrahedra sheet n H2O 9.6–21.4 A +

Silicon-Oxygen tetrahedra sheet

6O 4 Si

2 (OH) + 4 O

Gibbsite sheet

4 Al 2 (OH) + 4 O

c-axis

Silicon-Oxygen tetrahedra sheet

4 Si 6O b-axis

Montmorillonite (OH)4 Al4 Si8 O20 · n H2O

Figure 2-5. Schematic description of the crystal structure of montmorillonite (after Hoffman, Endell, and Wilm.-Grim, 1942, and Hughes, 1951; reprinted courtesy of the American Petroleum Institute, 1220 L St., NW, Washington, DC 20005, Hughes, R. V., “The Application of Modern Clay Concepts to Oil Field Development,” pp. 151–167, in Drilling and Production Practice 1950, American Petroleum Institute, New York, NY, 1951, 344 p.).

in aqueous media, calcium-montmorillonite platelets remain practically intact, close to each other, while the sodium-montmorillonite aggregates readily swell and the platelets separate widely. Therefore, water can easily invade the gaps between the platelets and form thicker water envelopes around the sodium-montmorillonite platelets than the calciummontmorillonite platelets (Chilingarian and Vorabutr, 1981) as depicted in Figure 2-7. Clay damage can be prevented by maintaining high concentrations of + K cation in aqueous solutions. At high concentrations of K + cation, clay platelets remain intact, because the small size K+ cation can penetrate the interlayers of the clay easily and hold the clay platelets together (Mondshine, 1973 and Chilingarian and Vorabutr, 1981) as depicted in

22

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations Silicon-Oxygen tetrahedra sheet yK

10.0 A

Silicon-Oxygen tetrahedra sheet

6O 4-ySi · yAI

2(OH) + 4 O

Gibbsite or brucite sheet

Al4 · Fe4 · Mg4 · Mg6 2(OH) – 4 O

c-axis

4-ySi · yAI

Silicon-Oxygen tetrahedra sheet

6O

yK

b-axis Illite (OH)4 Ky (Al4 · Fe4 · Mg4 · Mg6) (Si8–y · Aly) O20

Figure 2-6. Schematic description of the crystal structure of illite (after Grim, Bray, and Bradley-Grim, 1942, and Hughes, 1951; reprinted courtesy of the American Petroleum Institute, 1220 L St., NW, Washington, DC 20005, Hughes, R. V., “The Application of Modern Clay Concepts to Oil Field Development,” pp. 151–167, in Drilling and Production Practice 1950, American Petroleum Institute, New York, NY, 1951, 344 p.).

Figure 2-8. Many investigators, including Mungan (1965), Reed (1977), Khilar and Fogler (1983, 1985, 1987, 2000), Khilar et al. (1983), and Kia et al. (1987), have determined that some degree of permeability impairment occurs in clay-containing cores when aqueous solutions are flown through them. This phenomenon is referred to as the “water sensitivity.” Reed (1977) observed that young sediments are mostly friable micaceous sands and proposed a mechanism for damage. To justify this theory, Reed also conducted laboratory core tests by flowing various aqueous solutions through cores extracted from micaceous sand formations. The data shown by Figure 2-9 of Reed (1977) indicates permeability reduction. Based on the severeness of formation damage indicated by Figure 2-9, Reed concluded that mica alteration is a result of the exchange of K + cations with cations of larger sizes. Figure 2-9 shows

23

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations Calcium montmorillonite Cations

Ca++

Silica

SiO2 Al SiO2

++

Ca H2O

Alumina

Ca++ Ca++

Silica Hydration water Cations Montmorillonite

Silica Alumina Silica

Water Silica Alumina Silica

Silica Alumina Silica

Silica Alumina Silica

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Na+

Sodium montmorillonite

Sodium or Calcium montmorillonite

Figure 2-7. Expansion of the calcium and sodium montmorillonite by hydration (after Magcobar, ©1972, Figure 2, p. 2; reprinted by permission of the M-I L.L.C.).

that the deionized water caused the most damage, CaCl2 solution made the least damage, and damage by the NaCl solution is in between. Thus, the cations involved can be ordered with respect to the most to least damaging as H+ > Na+ > Ca++ . Whereas, Grim (1942) determined the order of replaceability of the common cations in clays from most to least easy cations as: Li+ > Na+ > K+ > Rb+ > Cs+ > Mg++ > Ca++ > Sr ++ > Ba++ > H+ . Hughes (1951) states, “hydrogen will normally replace calcium, which in turn will replace sodium. With the exception of potassium in illites, the firmness with which cations are held in the clay structure increases with the valence of the cation.” Reed (1977) postulated that formation damage in micaceous sands is a result of mica alteration and fines generation according to the process depicted in Figure 2-10 by Reed (1977) and later deposition in porous

24

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-8. Effect of the cation size on the cation migration into a clay interlayer (modified after Baroid Mud Handbook, 1975, Figure 12, p. 21; reprinted by permission of Baroid Drilling Fluids, Inc.).

100

3% CaCl2 Well 4290 core Ki = 1188 md

Permeability (% of original)

80

3% NaCl Well 1030 core Ki = 813 md

60 Deionized water Well 1030 core Ki = 1247 md

40

20

0

0

1

2 Volume throughput (liters)

3

4

Figure 2-9. Comparison of the permeability damages by the deionized water, and calcium chloride and sodium chloride brines in field cores (after Reed, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

25

Frayed edges Unaltered core Continued potassium extraction Flowing lowpotassium salt solutions

Partially altered

Completely altered

Effects of mica alteration: 1. 2. 3. 4.

Particles made smaller Expanded structure Particles more mobile Triggers instability in other minerals 5. Plugged pores and decresaed permeability

Unaltered mica particle Flowing solutions with added potassium Unaltered

Figure 2-10. Reed’s mechanism of mica alteration (after Reed, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

rock. As depicted in Figure 2-11, when clays are exposed to aqueous solutions containing no or small amounts of K+ cation or larger cations such as H+  Ca+2 , and Na+ , the K+ cation diffuses out of the clay platelets according to Fick’s law, because there are more K+ than the solution. In contrast, the larger cations present in the aqueous solution tend to diffuse

Figure 2-11. Schematic explanation of Reed’s (1977) mechanism for particle generation by mica alteration during exposure to low-potassium salt brine.

26

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

into clays because there are more of the larger cations in the solution compared to the clays. Because larger cations cannot fit into the interplanar gap depleted by the K+ cations, the edges of the friable mica flakes break off into small pieces as depicted in Figure 2-11. By a different set of experiments, Reed (1977) also demonstrated that dissolution of natural carbonate cement by aqueous salt solution can free mineral particles held by the cement. Reed’s reasoning is based on Figure 2-12, indicating increased concentrations of Ca+2 in the effluent while the permeability gradually decreases. The fine particles generated by mica alteration and unleashed by cement dissolution can, in turn, migrate with the flowing fluid and plug pore throats and reduce permeability. Mohan and Fogler (1997) explain that there are three processes leading to permeability reduction in clayey sedimentary formations: 1. Under favorable colloidal conditions, nonswelling clays, such as kaolinites and illites, can be released from the pore surface and then these particles migrate with the fluid flowing through porous formation.

Carbonate removed (meq)

Core weight – 33.6 gm

0.14 0.12

0.3

0.10 0.08

0.2

0.06 0.04

0.1

0.02 0.0

0

1

2

3

4

5

6

7

8

9

Effluent carbonate concentration (meq/l)

0.16

0.4

0.00 10

Volume 3.7% KCl throughput (liters)

Figure 2-12. Carbonate leaching from a field core by flowing a potassium chloride brine (after Reed, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

27

2. Whereas swelling clays, such as smectites and mixed-layer clays, first expand under favorable ionic conditions, and then disintegrate and migrate. 3. Also, fines attached to swelling clays can be dislodged and liberated during clay swelling. This phenomenon is referred to as fines generation by discontinuous jumps or microquakes. Consequently, formation damage occurs in two ways: (1) the permeability of porous formation decreases by reduction of porosity by clay swelling (Civan and Knapp, 1987; Civan et al., 1989; and Mohan and Fogler, 1997); and (2) the particles entrained from the pore surface by the flowing fluid are carried towards the pore throats and captured by a jamming process. Thus, the permeability decreases by obstruction and or plugging of pore throats (Sharma and Yorstos, 1987a,b,c; Wojtanowicz et al., 1987, 1988; Mohan and Fogler, 1997; Civan, 2001a,b, 2002b,d). Khilar and Fogler (1983) have demonstrated by the flow of aqueous solutions through Berea sandstone cores that there is a “critical salt concentration (CSC)” of the aqueous solution below which colloidally induced mobilization of clay particles is initiated and the permeability of the core gradually decreases. This is a result of the expulsion of kaolinite particles from the pore surface due to the increase of the double-layer repulsion at low salt concentration (Mohan and Fogler, 1997). The critical salt concentrations for typical sandstones are given by Mohan and Fogler (1997) in Table 2-3. The sodium Na+  cation concentration (meq/L) relative to the total concentration (meq/L) of the bivalent cations, such as calcium Ca2+  and magnesium Mg2+ , may provide a measure of the destabilization potential of the clay minerals exposed to an aqueous solution (Thomas et al., 1995). Table 2-3 Critical Salt Concentrations in Typical Sandstone Salt

Stevens M

Berea M

NaCl KCl CaCl2

0.50–0.25 0.3–0.2 0.3–0.2

0.07 0.03 None

After Mohan, K. K., and Fogler, H. S., ©1997; reprinted by permission of the AIChE, ©1997 AIChE. All rights reserved.

28

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

2.6 MECHANISM OF CLAY SWELLING Figure 2-13 by Hayatdavoudi (1999a) shows that all clays possess some degree of the work of swelling and therefore classification of different 3000

Smectite group & Vermiculite

Measured ∆H (cal/mole)

2500 2000

Illite group & Attapulgite

1500 1000 500

Kaolinite group & Chlorite

0 0.0

0.5

1.0

1.5

2.0

2.5

Ln [HHI] Scale (a)

Estimated ∆G (kcal/mole)

1000

Smectite group & Vermiculite

800

Sepiolite

Illite group & Attapulgite (Hydrated micah) 600

400

200

0 0.0

Kaolinite group & Chlorite 0.5

1.0

1.5

2.0

Ln [HHI] Scale (b)

Figure 2-13. Hayatdavoudi clay hydration charts: (a) measured enthalpy vs. hydration index and (b) theoretical free energy vs. hydration index for various clays (after A. Hayatdavoudi, ©1999a SPE; reprinted by permission of the Society of Petroleum Engineers).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

29

clays as swelling or nonswelling, as frequently done, has no significance. In this figure, the Hayatdavoudi hydration index (HHI=[O/OH]) is defined as the ratio of the number of the oxygen atoms to that of the hydroxyl groups in clays and it controls the enthalpy or free energy of the clays available for the work of swelling by hydration (Hayatdavoudi, 1999). HHI can be determined based on the chemical formula of the clays (see Table 2-1). For example, HHI = 10/8 = 125 for kaolinite, HHI = 20/16 = 125 for chlorite, and HHI = 20/4 = 50 for illite. The Gibbs free energy of formation for hydrated clay minerals can be predicted by an appropriate method, such as given by Vieillard (2002). Hence, higher hydration index is indicative of more clay swelling, according to Hayatdavoudi (1999a): G = H − TS = RT lnO/OH

(2-1)

where G, H, S, T, and R denote the free energy, enthalpy, entropy, absolute temperature, and the universal gas constant, respectively. In swelling clays, including smectite and illite, the negative charge of the clay platelets is balanced by means of the positive charge of the cations present in the interlayer locations, including K+  Na+  Ca2+ , or Mg2+ . This feature helps keep the stack of clay platelets together (see Figure 2-7) (de Siqueira et al., 1999; Xu et al., 2006). However, when exposed to low ionic-strength aqueous solutions, the interlayer cations adsorb the water molecules from the aqueous solution to form thick envelopes of water films over the clay platelets (see Figure 2-7) (Chilingarian and Vorabutr, 1981, Xu et al., 2006). This process causes the expansion of the interlayer and thus clay swelling. The clay shrinks by a reverse process when the swelling clay is exposed to high ionicstrength aqueous solutions. For example, this process for smectite can be represented according to the following equation (Xu et al., 2006): Smectite + nH2 O ⇄ Smectite • H2 On

(2-2)

Xu et al. (2006) resorted to a simple approach, expressing the relative variation of the bulk density of a swollen clay mineral as a linear function of the relative deviation of the aqueous solution ionic strength I from a critical minimum value Imin , required for clay swelling to occur, as:   Imin − I max − = fmax  I < Imin (2-3) max Imin = max 

I ≥ Imin

(2-4)

30

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Equation (2-3) could be generalized as a power-law empirical correlation. The ionic strength is expressed by I=

1 2 cz 2 i i i

(2-5)

where I denotes the various aqueous species, and ci and zi denote their concentrations (mole/kg H2 O) and electrical charges, respectively. A structural model of swelling clays having exchangeable cations, denoted by Mz+ , is shown by Zhou et al. (1996, 1997) in Figure 2-14. Zhou et al (1996) states, The structure layers are always deficient in positive charges due to cation substitution, and interlayer cations are required to balance the negative layer charge. Interlayer cations are exchangeable and the exchange is reversible for simple cations. The distance between two structure layers, that is (001) d-spacing, is dependent on the nature (type) of the exchangeable cation, composition of the solution, and the clay composition. Clay swelling is a direct result of the d-spacing increase and volume expansion when the exchangeable cations are hydrated in aqueous solution.

As stated by Zhou (1995), “clay swelling is a result of the increase in interlayer spacing in clay particles.” Clay swelling occurs when the clay is

Figure 2-14. Schematic structure of a swelling clay crystal containing an exchangeable MZ+ cation (after Zhou et al., ©1997 SPE; reprinted by permission of the Society of Petroleum Engineers).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

31

exposed to aqueous solutions having a brine concentration below the critical salt concentration (Khilar and Fogler, 1983). Therefore, Zhou (1995) concludes that “clay swelling is controlled primarily by the composition of aqueous solutions with which the clay comes into contact.” Norrish (1954) have demonstrated by experiments that clay swelling occurs by crystalline and osmotic swelling processes. Zhou (1995) explains that (1) crystalline swelling occurs when the clays are exposed to concentrated brine or aqueous solutions containing large quantities of divalent or multivalent cations. It is caused by the formation of molecular water layers on the surface of clay minerals. This leads to less swelling and less damage; and (2) osmotic swelling occurs when the clays are exposed to dilute solutions or solutions containing large quantities of Na+ cations. It is caused by the formation of an electric double layer on the surface of clay minerals. It leads to more swelling and more damage. These phenomena create repulsive forces to separate the clay flakes from each other. Mohan and Fogler (1997) conclude that crystalline swelling occurs at high concentrations above the critical salt concentration and osmotic swelling occurs at low concentrations below the critical salt concentrations. Mohan and Fogler (1997) measured the interplanar spacing as an indication of swelling of montmorillonite in various salt solutions. Thus, according to Figures 2-15 and 2-16 given by Mohan and Fogler (1997), the crystalline and osmotic swelling regions can be distinguished by a sudden jump or discontinuity in the value of the interplanar spacing which occurs at the critical salt concentration.

Interplanar spacing (Å)

200.0

Swy-1 – Norrish Stevens – This work Swy-2 – This work

150.0

100.0

Crystalline swelling Osmotic swelling

50.0

Critical salt concentration

0.0 0

2

4

6

8

10

c–0.5 (m–0.5)

Figure 2-15. Swelling of montmorillonite in sodium chloride brine (after Mohan, K. K., and Fogler, H. S., ©1997; reprinted by permission of the AIChE, ©1997 AIChE. All rights reserved).

32

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-16. Swelling of montmorillonite in various brine (after Mohan, K. K., and Fogler, H. S., ©1997; reprinted by permission of the AIChE, ©1997 AIChE. All rights reserved). Montmorillonite, NaCl/CaCl2 mixed

0.1

CaCl2(N)

Crystalline swelling 0.01

Osmotic swelling

0.001

0.0001 0.001

Formation damage zone

0.01

0.1

1

NaCl (N)

Figure 2-17. Swelling chart for montmorillonite exposed to sodium and calcium chloride brines (after Zhou et al., ©1996; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

Zhou et al. (1996a,b, 1997) suggest the use of clay swelling charts obtained by X-ray diffraction method similar to that given in Figures 2-17 and 2-18 to determine the compatibility of clays with mixed-electrolyte solutions. These charts indicate the cation concentrations of aqueous

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

33

Montmorillonite, NaCl/KCl mixed

KCl (N)

0.1

Crystalline swelling

0.01 Osmotic swelling Formation damage zone 0.001 0.001

0.01

0.1

1

NaCl(N)

Figure 2-18. Swelling chart for montmorillonite exposed to sodium and potassium chloride brines (after Zhou et al., ©1996; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

solutions that will cause crystalline or osmotic swelling. Consequently the cation compositions that will lead to formation damage can be identified readily in the region of the osmotic swelling, as shown in Figure 2-17, because osmotic swelling is the main cause of formation damage. Thus, Figure 2-17 provides some guidance as to the amount of Ca2+ necessary in the presence of Na+ cations to prevent montmorillonite swelling in NaCl/CaCl2 solutions. Figure 2-18 is a similar chart for montmorillonite in NaCl/KCl solutions given by Zhou et al. (1996). Civan (2001c) has drawn attention to some experimental data, which indicate a transition region characterized by distinct lower- and upperbound aqueous salt concentrations. The basal spacing does not vary over this region. Significant clay swelling and formation damage may occur when the aqueous solution salt concentration is below the lower-bound concentration of this region. Civan (2001c) describes, Swelling of clayey porous rocks is controlled by absorption of water by a water-exposed-surface hindered diffusion process. The characteristics of the swelling clayey formation, such as moisture content, volume, and permeability, vary at rates proportional to the water absorption rate and their values relative to their terminal values that would be attained at

34

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

the saturation limit. The rate laws of different properties allow for crosscorrelation between these properties.

2.7 MODELING CLAY SWELLING∗ This section describes the analytical models available for interpretation and correlation of measurements of swelling-dependent properties of reservoir formations containing swelling clays. These models can be used for representing these properties in the prediction and simulation of reservoir formation damage and in well-log interpretation. The laboratory studies by many researchers, including the ones by Zhou et al. (1997) and Mohan and Fogler (1997), have concluded that clay swelling primarily occurs by crystalline and osmotic swelling mechanisms. Civan and Knapp (1987) and Civan et al. (1989) recognized that water transfer through clayey porous media occurs by diffusion, and developed the phenomenological models for permeability and porosity reduction by swelling by absorption of water via the diffusion process. Ohen and Civan (1990, 1993) and Chang and Civan (1992, 1997) incorporated these models into the simulation of formation damage in petroleum reservoirs. Ballard et al. (1994) experimentally studied the transfer of water and ions through shales. They determined that diffusion controls the transfer process and osmosis does not have any apparent effect when pressure is not applied. Their findings reconfirm the mechanism proposed by Civan and Knapp (1987), Civan et al. (1989), and Civan (1999a, 2001c) that diffusion is the primary cause of water transfer through clayey porous formations. But, transfer rates tend to increase with pressure application. Ballard et al. (1994) observed that, beyond a certain threshold pressure, water and ions move at the same speed. This is because transfer by advection dominates and diffusion by concentration gradients becomes negligible. The Civan and Knapp (1987) and Civan et al. (1989) models for variation of porosity and permeability by swelling assume that the external surface of the swelling clay is in direct contact with water at all times and therefore they used a Dirichlet boundary condition in the analytic solution of the models. Civan (1999a, 2001c) developed improved models by ∗ After Civan, ©1999a, 2001c SPE, parts reprinted by permission of the Society of Petroleum Engineers from SPE 52134 and SPE 67293.

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

35

considering a water-exposed-surface-hindered-diffusion process and used a Neumann boundary condition in the analytical solution of the models involving the clay swelling effects. By means of a variety of experimental data reported in the literature, Civan (1999a, 2001c) demonstrated and verified that this boundary condition leads to improved analytic models which correlate the experimental data better. He has also shown that the various phenomenological parameters, such as the rate constants and the terminal porosity and permeability values that will be attained at water saturation, can be conveniently determined by fitting these models to experimental data. Civan (1999a) pointed out that the laboratory swelling tests are generally carried out using aqueous solutions of prescribed concentrations. Whereas, the composition of aqueous solutions in actual reservoir formations may vary, but this effect can readily be taken into account by incorporating a time-dependent clay surface boundary condition by applying Duhamel’s theorem. As a result, the effect of variable aqueous solution concentration can be adequately incorporated into the simulation of formation damage by clay swelling. As schematically depicted in Figure 2-19, swelling clay particles can absorb water and expand to enlarge the particle size, and the clayey porous formations containing swelling clays can absorb water and expand inward to reduce its porosity and permeability. The various models useful for interpretation of experimental data and representing formation damage are presented in the following sections.

Figure 2-19. Clay particle expansion and pore space reduction by swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers).

36

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

2.7.1

Osmotic Repulsive Pressure

Ladd (1960) explains that “The exchangeable cations are attracted to the clay particles by the negative electric field arising from the negative charge on the particles. Hence, the electric field acts as a semi-permeable membrane in that it will allow water to enter the double layer but will not allow the exchangeable cations to leave the double layer.” Thus, when the total ion (cations plus anions) concentration in the double layer between the clay particles is higher than that in the aqueous pore fluid, the water in the pore fluid diffuses into the double layer to dilute its ion concentration. This phenomenon creates an osmotic repulsive pressure between the clay particles. As a result, the interparticle distance increases causing the clay to expand and swell. Therefore, the driving force for osmotic pressure is the difference of the total ion concentrations between the clay double layer, cc , and the surrounding pore fluid, cf , as depicted by Figure 2-20 of Ladd (1960).

Double layers overlap

b

Imaginary semipermeable membrane surrounding clay particles

Figure 2-20. Mechanism of osmotic pressure generation between two clay particles in water (after C. C. Ladd, 1960; reprinted by permission of the Transportation Research Board, the National Academies, Washington, DC, from C. C. Ladd, “Mechanisms of Swelling by Compacted Clay,” in Highway Research Board Bulletin 245, Highway Research Board, National Research Council, Washington, DC, 1960, pp. 10–26).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

37

For only very dilute aqueous solutions, the van’t Hoff equation given below can be used to estimate the osmotic pressure (Ladd, 1960). posm = RT cc − cf 

(2-6)

However, nonideal models are required for concentrate solutions. 2.7.2

Clay Swelling Coefficient

The rate of clayey formation swelling is derived from the definition of the isothermal swelling coefficient given by (Collins, 1961)   V (2-7)

sw = Vw T V and Vw are the volumes of the solid and the water absorbed. Ohen and Civan (1990, 1993) used the expression given by Nayak and Christensen (1970) for the swelling coefficient:

sw =

1 CI m + 2 c

(2-8)

in which c is the water concentration in the solid and CI is the plasticity index. 1 and 2 are some empirical coefficients. m is an exponent. Chang and Civan (1997) used the expression given by Seed et al. (1962a,b):

sw =

k′ PI 244 Cc344 Cc − 10244

(2-9)

where Cc is the clay content of porous rock as weight percent, PI is the plasticity index, and k′ is an empirical constant. 2.7.3

Water Absorption Rate

Consider Figure 2-21 (Civan, 1994a, 1996a, 1999a) showing swelling of a solid by water absorption. Civan et al. (1989) assumed that water diffuses through the solid matrix according to Fick’s second law over a short distance near the surface of the solid exposed to aqueous solution,

38

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-21. Mechanism of formation swelling by water absorption (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers and after Civan, 1994; reprinted by permission of the U.S. Department of Energy).

because the coefficient of water diffusion in solid is small. Thus, the water absorption in the solid can be predicted by the one-dimensional transient-state diffusion equation c/t = D2 c/z2 

0 ≤ z <  t > 0

(2-10)

subject to the initial and boundary conditions given, respectively, by c = c0 

0 ≤ z <  t = 0

S˙ ≡ −Dc/z = k c1 − c  c = c0 

z = 0 t > 0

z →  t > 0

(2-11) (2-12) (2-13)

where c0 and c are the initial and instantaneous water concentrations in the solid, c1 is the water concentration of the aqueous solution, z is the distance from the pore surface, t is the actual exposure or contact time (time measured relative to the beginning of the water absorption process), k is the film mass transfer coefficient, and D is the diffusivity coefficient in the solid matrix. Equation (2-12) expresses that the water diffusion into clay is hindered by the stagnant fluid film covering the clay surface. Thus, Civan (1997, 1999a) used the analytical solution of

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

39

Eqs (2-10)–(2-13) to express the cumulative amount of water diffusing into the solid surface as given by Crank (1956) S≡−

t  o

 c dt −D z

 √

c − c0  2 √ = 1 exp h2 Dt erfc h Dt − 1 + √ h Dt h



(2-14)

and the rate of water absorption is given by differentiation of Eq. (2-14) as:  √

(2-15) S˙ = dS/dt = c1 − c0  hD exp h2 Dt erfc h Dt

where h = k/D is a derived parameter, representing the ratio of the film mass transfer coefficient to the diffusion coefficient. Civan et al. (1989) resorted to a simplified approach by assuming that the film mass transfer coefficient k in Eq. (2-12) is sufficiently large so that it becomes c = c1  z = 0 t > 0

(2-16)

and, therefore, an analytical solution of Eqs (2-10), (2-11), (2-16), and (2-13) according to Crank (1956) yields the expression for the cumulative and rate of water absorption, respectively, as: √ 2 S = √ c1 − c0  Dt

(2-17)

D 1 S˙ = √ c1 − c0  √

Dt

(2-18)

The analytical expressions given above consider constant water concentrations maintained in the aqueous pore fluid. Whereas, the rate of formation damage by clay swelling also depends on the variation of the water concentration in the aqueous solution flowing through porous rock. Therefore, they should be corrected for variable water concentrations by an application of Duhamel’s theorem. For example, if the time-dependent water concentration at the pore surface is given by c = c0 + c1 − c0  F t 

z = 0 t > 0

(2-19)

40

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

where Ft is a prescribed time-dependent function, the analytic solution can be obtained as illustrated by Carslaw and Jaeger (1959). Then, using Eq. (2-18), the rate of water absorption can be expressed by t

D c1 − c0   −F  S˙ = d √ 3/ 2 2 D t −  0

(2-20)

However, in the applications presented here the water concentrations involved in the laboratory experiments are essentially constant. The preceding derivations assume a plane surface as supposed to a curved pore surface. From the practical point of view, it appears reasonable because of the very short depth of penetration of the water from the solid–fluid contact surface due to the relatively low diffusion coefficient of water molecules through the solid material. The pore shape effect is included implicitly in the value of the model parameters when the following formulations are used to correlate the experimental data. 2.7.4 Kinetics of Swelling-Related Properties and Rate Equations The analyses of the various data indicate that the variation of the moisture, volume, and permeability of clayey formations during swelling by exposure to water is governed by similar rate (kinetic) equations, which can be generalized as (Civan, 1999a, 2001c) −

d f − ft  = kf S˙ f − ft  dt

(2-21)

subject to a prescribed initial condition as f = fo  t = 0

(2-22)

The porosity variation is also expected to follow the same trend as described by Eq. (2-21), because it is a result of solid expansion by water absorption. Let f denote the properties of clayey formations that vary by swelling. Further, fo and ft denote the initial and the final values of f t is time, kf is the rate constant for the property f , and S˙ is the rate of water absorption controlled by the surface-hindered diffusion of water into

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

41

the solid according to Eq. (2-14). The analytic solution of Eqs (2-21) and (2-22) can be written in the following form for a property f as:   fo − ft ln = kf S (2-23) f − ft Similarly, Eq. (2-23) can be written for another property g as:   go − gt ln = kg S g − gt

(2-24)

Thus, a property of f ∈ w   K can be related to another property of g ∈ w   K for f = g by eliminating the quantity S between Eqs (2-23) and (2-24) to obtain   k /k go − gt  f g  fo − ft = f − ft g − gt

(2-25)

Equation (2-25) is particularly useful to correlate between any pair of properties f and g without the involvement of the time variable. For example, applying Eq. (2-25), porosity and permeability variations can be correlated by the power-law equation (Civan, 2001c):   k /k o − t  K   Ko − Kt = K − Kt  − t

(2-26)

where kK and k are the rate coefficients for permeability and porosity reduction by swelling, respectively. The parameters of this kinetic model can be determined from experimental data conveniently by a straight-line Plotting Scheme as following. Substituting Eq. (2-14) into Eq. (2-23) yields   fo − ft ln = AtD (2-27) f − ft in which the dimensionless water diffusion time tD and the coefficient A are defined, respectively, by  √

2 √ tD = exp h2 Dt erfc h Dt − 1 + √ h Dt (2-28)

(2-29) A = kf c1 − co  h

42

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

√ The best estimates of the A and h D parameters can be determined by the linear least-squares regression of experimental data on a semi-log scale according to Eq. (2-27) from the best straight-line fit of data. For sufficiently large k values, considering Eq. (2-17), Eqs (2-28) and (2-29) are replaced by:  tD = t/to (2-30)  A = 2kf c1 − co  Dto /

(2-31)

where to is an arbitrarily or conveniently selected characteristic time scale, such as unity. The best estimate value of A can be determined by the linear least-squares regression of experimental data on a semi-log scale according to Eq. (2-27). 2.7.5

Basal Spacing of Clay

Figures 2-22–2-25 by Zhou et al. (1997) depict the effect of aqueous solution concentration, temperature, pressure, and net overburden stress on the 50 Saponite, NaCl solutions

35 30 25

Crystalline swelling

40 Osmotic swelling

D – Spacing (angstrom)

45

20 15 10 0.0

0.2

0.4

0.6

0.8

1.0

NaCl concentration (N)

Figure 2-22. Changes of (001) d-spacing of saponite as a function of NaCl concentration in aqueous solution (after Zhou et al., ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

43

Figure 2-23. Changes of (001) d-spacing of montmorillonite as a function of temperature (after Zhou et al., ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 2-24. Effect of pressure on clay swelling, overburden pressure equals fluid pressure (after Zhou et al., ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

44

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-25. Effect of differential pressure (overburden pressure minus fluid pressure) on clay swelling (after Zhou et al., ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

basal spacing of typical clays (saponite and montmorillonite). Figure 2-22 shows a significant increase in the saponite basal spacing at low aqueous solution concentrations. Figure 2-23 shows that the basal spacing of montmorillonite remains practically unchanged in the 20–100 C temperature range. Figure 2-24 shows that the basal spacing of montmorillonite does not vary with pressure when the fluid and overburden pressures are equal so that the fluid does not drain. Figure 2-25 shows that the basal spacing of montmorillonite decreases with increasing net overburden stress when the fluid is allowed to drain. The basal spacing of clay varies by crystalline and osmotic swelling when clay contacts with aqueous solutions (Himes et al., 1991; Mohan and Fogler, 1997; Norrish, 1954; and Zhou, 1995). The osmotic and crystalline swelling phenomena occur beyond the low- and high-concentration bounds of the transition salt concentration range, respectively. The clay is dispersed in an aqueous solution until a complete dispersion state is ˙ attained when the basal spacing exceeds a critical limit, for example 20 A for the smectite clay (Himes et al., 1991). Therefore, there is no terminal limit on the clay basal spacing. Civan (2001c) expressed that the rate of the basal spacing increases proportionally to the water absorption rate and the instantaneous basal spacing minus the clay platelet thickness (the gap

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

45

between the clay platelets). Thus, applying Eq. (2-21) without setting any limit on the basal spacing yields

dX − X∗  dt = kx S˙ X − X∗ 

(2-32)

where X and X∗ denote the instantaneous clay basal spacing and the clay platelet thickness, respectively, and kx is a rate coefficient. Civan (2001c) expressed the rate of decrease of the salt concentration of the aqueous solution by exchange of cations between the clay and the aqueous solution proportionally to the water absorption rate S˙ and the salt concentration C as: −

dC ˙ = kc SC dt

(2-33)

Hence, eliminating S˙ between Eqs (2-32) and (2-33) yields d X − X∗  dC = − kx /kc  X − X∗  C

(2-34)

The initial or reference basal spacing at a prescribed aqueous solution salt concentration is given by X = Xo  C = Co The analytical solution of Eqs (2-33) and (2-34) is given by   −k /k  kx /kc  x c X = X∗ + Xo − X∗  Co C

(2-35)

(2-36)

Equation (2-36) is a theoretical confirmation of the validity of the Mohan and Fogler (1997) straight-line plots of the measured basal spacing vs. the reciprocal square-root of the salt concentration for clays exposed to various aqueous salt solutions. Their plot of experimental data indicates that kx /kc = 05. Himes et al. (1991) facilitated the measurements of the variation of the basal spacing by water imbibition as an indication of the clay stabilization ability of various cations. The Himes et al. (1991) data for smectite basal spacing vs. the aqueous solution concentration of NaCl, CaCl2 , KCl, and NH4 Cl can be correlated linearly according to Eq. (2-36) by straight-line plots, as shown in Figures 2-26–2-29 by Civan (2001c), respectively. The ˙ (van Olphen, 1977). thickness of the clay platelet was taken as X∗ = 96 A

46

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-26. Basal spacing of smectite clay vs. aqueous solution NaCl concentration (Himes et al., 1991, data) (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

The previous studies, including Mohan and Fogler (1997), suggested the presence of a distinct value of the aqueous solution salt concentration, referred to as the critical salt concentration, below which swelling clay destabilization begins. The plots shown in Figures 2-26–2-29 reveal the existence of a transition concentration range over which the basal spacing remains unchanged. The data below and above the transition salt concentration range can be fitted separate straight lines, with different slopes, indicating different values for the rate coefficients ratio kx /kc . Simultaneously, the transition salt concentration range is identified by the shift from one line to the other. The best estimate values of kx /kc , that is the slopes of the straight lines obtained by the linear least-squares method, below and above the critical salt concentration range are 3.1 and 0.36 for NaCl, nondetermined (insufficient data) and 0.17 for CaCl2 , 0.34 and 0.091 for KCl, and 0.28 and 0.16 for NH4 Cl, respectively. As can be seen, these values are different than the 0.5 value of Mohan and Fogler (1997). Hence, the 0.5 value is specific to their data and is not generally applicable to other cases. The crystalline swelling phenomenon occurs at the high concentration side and causes less clay damage, as indicated by the low kx /kc values for all the salts. The osmotic swelling phenomenon occurs at the low concentration side and causes significant clay damage,

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

47

Figure 2-27. Basal spacing of smectite clay vs. aqueous solution CaCl2 concentration (Himes et al., 1991, data) (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 2-28. Basal spacing of smectite clay vs. aqueous solution KCl concentration (Himes et al., 1991, data) (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

48

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-29. Basal spacing of smectite clay vs. aqueous solution NH4 Cl concentration (Himes et al., 1991, data) (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

as indicated by the high kx /kc . Figure 2-27 indicates that the transition salt concentration range for CaCl2 extends to very low concentrations. This confirms the Himes et al. (1991) conclusion that the calcium ion is an effective clay stabilizer at low salt concentrations, but it is less effective at high concentrations. 2.7.6

Water Content during Clay Swelling

The rate of water retainment of clay minerals is assumed proportional to ˙ and the deviation of the instantaneous water the water absorption rate, S, content from the saturation water content as, according to Eq. (2-21), dw/dt = kw S˙ wt − w

(2-37)

subject to the initial condition w = wo  t = 0

(2-38)

where kw is a water retainment rate coefficient. w denotes the weight percent of water in clay and the subscripts o and t refer to the initial

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

49

t = 0 and terminal t →  conditions, respectively. An analytical solution of Eqs (2-37) and (2-38) yields w = wt − wt − wo  exp −kw S

(2-39)

Osisanya and Chenevert (1996) measured the variation of the water content of the Wellington shale exposed to deionized water. Figure 2-30 shows the correlation of their data using Eq. (2-39). The best √ fits were obtained using wo = 27 wt% wt = 33 wt% A = 020, and h D√= 1 for their Gage 1 data; wo = 277 wt% wt = 37 wt% A = 002, and h D = 1 for their √ Gage 2 data; and wo = 277 wt% wt = 39 wt% A = 0011, and h D = 1 for their Gage 3 data. Brownell (1976) reports the moisture content of a dried clay piece containing montmorillonite soaked in water. Figure 2-30 shows a correlation of the data using Eq. (2-39). The√best fit was obtained using wo = 0% wt = 142 wt% A = 017, and h D = 08. Note that Brownell (1976) only provides a smoothed curve of the experimental data without

Figure 2-30. Water pickup during swelling (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

indicating the actual data points on this curve. Therefore, the data points shown in Figure 2-30 are the discrete point readings of this curve.

2.7.7

Time-Dependent Clay Expansion Coefficient

By contact with water the swelling clay particles absorb water and expand. The rate of volume increase is assumed proportional to the water ˙ and the deviation of the instantaneous volume from absorption rate, S, the terminal swollen volume that will be achieved at saturation Vt − V ). Therefore, the rate equation is written as, according to Eq. (2-21),

dV dt = kV S˙ Vt − V 

(2-40)

V = Vo  t = 0

(2-41)

subject to

where kV is the rate coefficient of expansion. Thus, solving Eqs (2-40) and (2-41) yields V = Vt − Vt − Vo  exp −kV S

(2-42)

from which the expansion coefficient of a unit clay volume is determined as:

 ≡ V Vo − 1 = t 1 − exp −kV S

(2-43)

where t is the terminal expansion coefficient attained at saturation. Montes-H (2005) proposed a simplified kinetic model of the firstorder type for describing the swelling-shrinkage potential (in percentage) of expansive clays. Based on the experimental measurements on raw bentonite aggregate scale, Montes-H (2005) developed an exponential function as following: Vt − V  = Vt − Vo  exp −kV t

(2-44)

This equation can be derived from Eq. (2-42) by substitution of S = t. Thus, S˙ = 10 in Eq. (2-40).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

51

Alternatively, simply by letting the volume change as V and maximum volume change as VM : V = VM 1 − exp −kV t

(2-45)

However, Montes-H (2005) measured the surface area A variation instead of the volume variation by swelling of a particle. Note that the surface area and volume of a spherical particle are given, respectively, by A = D2

(2-46)

V = D3 /6

(2-47)

and

Thus eliminating the sphere diameter D between Eqs (2-46) and (2-47), and then inserting a shape factor C to account for the irregular particle shape, yields the following relationship between the surface area and the volume of an irregular shape particle:   C (2-48) V = √ A3/2 6 Thus, Eq. (2-45) was modified by substituting V ∼ A3/2 to obtain the following expression (different than that of derived by Montes-H, 2005): A3/2 = A3/2 M 1 − exp −kV t

(2-49)

In Eq. (2-49) the time is measured relative to the beginning of swelling. Seed et al. (1962a,b), Blomquist and Portigo (1962), Chenevert (1970), and Wild et al. (1996) measured the rates of expansion of the samples of compacted sandy clay, hydrogen soil, typical shales, and lime-stabilized kaolinite cylinders containing gypsum and ground granulated blast furnace slag, respectively. As shown in Figure 2-31, the least-squares linear regressions of the √ data were obtained using Eq. (2-43) with A = 0087 h D = 067, and t = 100V√ t − Vo /Vo = 37 vol% for the data of Seed et al. (1962), A = 25 h D = 1, and t = Vt − Vo /Vo = 95/Vo volume√fraction for the data of Blomquist and Portigo (1962), and A = 056 h D = 1, and t = 055% for the data of Chenevert (1970). Note that the initial sample volume Vo was not given in the original data. However, this value was

52

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-31. Volume change by swelling (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

not required for the plots of 1 − /t  because the Vo value cancels out in the ration of /t . Wild et al. (1996) tested lime-stabilized compacted kaolinite cylinders containing gypsum and ground granulated blast furnace slag. After moistcuring over prescribed time periods, they soaked these samples in water and measured the linear expansion of the samples. Figure 2-32 shows the representation of their three typical data sets using Eq. (2-43) referred to as Data #1, Data #2, and Data #3, respectively. The first set of data was obtained using a 7-day moist-cured kaolinite containing 6% lime and 4% gypsum. The second set of data is for a 28-day moist-cured kaolinite containing 6% lime and 4% gypsum. The third set of data represents a 28-day moist-cured kaolinite containing 2% lime, 4% gypsum and 8% ground granulated blast furnace slug. The least-squares linear regressions of Eq. (2-43) to the first, second, and third data sets were obtained with √ the best estimates of the parameters, given by A = 065 h D = 10, √ and t√ = 13 vol% A = 22 h D = 1, and t = 146 vol%; and A = 14 h D = 1, and t = 0655 vol%, respectively. Ladd (1960) measured the volume change and water content of the compacted Vicksburg Buckshot clay samples during swelling. For a linear

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

53

Figure 2-32. Volume change by swelling (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

plot of Ladd’s data first, the S term is eliminated between Eqs (2-39) and (2-43) to yield  kV / kw  wt − w  (2-50) 1− = t wt − wo Then, inferred by Eq. (2-50), Ladd’s data can be correlated on a log–log scale by a straightline as shown in Figure 2-33. The best linear fit of Eq. (2-50) was obtained using wo = 08 g wt = 32 g t = 132/Vo , and kV /kw = 1907. Note that the value of Vo is not given and not required because Eq. (2-50) employs the ratio of /t . 2.7.8

Porosity Reduction by Swelling

Based on the definition of the swelling coefficient, Civan and Knapp (1987) expressed the rate of porosity change by swelling of porous matrix as:

d dt = −k S˙ (2-51)

where  is porosity, t is time, k is a swelling rate coefficient, and S˙ is the rate of water absorbed per unit bulk volume of porous medium.

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-33. Volume change vs. water pickup during swelling (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

Civan (1996a) developed an improved equation assuming that the rate of porosity variation by swelling is proportional to the rate of water absorption and the difference between the instantaneous and the terminal or saturation porosities,  and t . Thus, applying Eq. (2-21) yields

−d dt = k S˙  − t  (2-52)

subject to

 = o  t = 0 Integrating Eqs (2-52) and (2-53) yields   o − t ln = k S  − t

(2-53)

(2-54)

from which the porosity variation by swelling can be expressed by 

 (2-55) sw =  − o = t − o  1 − exp −k S

where k is the formation swelling rate constant, t is the actual time of contact with water. Therefore, the swelling rate constant can be determined by fitting Eq. (2-54). It is difficult to measure porosity during swelling. Permeability can be measured more conveniently. Ohen and Civan (1993) used a permeability–porosity relationship to express porosity reduction in terms of permeability reduction.

55

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

2.7.9

Permeability Reduction by Swelling

Civan and Knapp (1987) assumed that the rate of permeability reduction by swelling of formation depends on the rate of the water absorption and the difference between the instantaneous permeability K and the terminal permeability Kt that will be attained at saturation as, according to Eq. (2-21), −

dK = kK S˙ K − Kt  dt

(2-56)

subject to the initial condition K = Ko  t = 0

(2-57)

where kK is the rate constant for permeability reduction by swelling. Thus, solving Eqs (2-56) and (2-57) yields 

Ko − Kt ln K − Kt



= kK S

(2-58)

from which the permeability variation by swelling is obtained as: Ksw = Ko − K = Ko − Kt  1 − exp −kK S

(2-59)

Civan and Knapp (1987) and Civan et al. (1989) have confirmed the validity of Eq. (2-58) using the Hart et al. (1960) data for permeability reduction in the outlet region of a core subjected to the injection of a suspension of bacteria. Because bacteria are essentially retained near the inlet side of the core, the permeability reduction in the near-effluent part of the core can be attributed to formation swelling by water absorption. The best linear, least-squares fit of Eq. (2-58) to Hart et al. (1960) data using Eq. (2-17) for S yields (Civan et al., 1989)    √ Kt Kt K = + 1− exp −B t Ko Ko Ko

(2-60)

√ with Kt /Ko  = 057 and B = 2kK c1 − co  D/ = 081 hr −1/2 as shown in Figure 2-34. However, the Hart et al. (1960) data can also be correlated using Eq. (2-14) for S as shown in Figure 2-34. In this case, the best√fit is obtained using the parameter values of A = kK c1 −co /h = 093 h D = 10, and Kt /Ko = 057.

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-34. Permeability reduction by swelling (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers).

Ngwenya et al. (1995) conducted core flood experiments by injecting artificial seawater into the Hopeman (Clashach) sandstone. The core samples used in their experiments contained trace amounts of clays. Their data plotted according to Eq. (2-58) in Figure 2-35 indicates that the swelling of the sandstone formation caused a permeability reduction. The linear leastsquares fit of Eq. (2-58) to data was obtained √ using the best estimates of the parameter values, given as A = 004 h D = 09, and Kt /Ko = 0087. Reed (1977) measured the permeability reduction in micaceous sand formations exposed to 3% CaCl2 and 3% NaCl aqueous solutions, and deionized water. The best least-squares linear fits of Eq. (2-58) to Reed’s (1977) data were obtained using the best estimates of the parameter √ values, given by A = 196 h D = 1, and Kt /Ko = 058 for CaCl √2  A = √ 187 h D = 1, and Kt /Ko = 039 for NaCl, and A = 606 h D = 1, and Kt /Ko = 029 for H2 O, as shown in Figure 2-36.

2.8 CATION EXCHANGE CAPACITY The total amount of ions (anions and cations) that are present at the clay surface and exchangeable with the ions in an aqueous solution in contact with the clay surface is referred to as the ion-exchange capacity (IEC) of the clay minerals and it is measured in meq/100 g (Kleven and Alstad,

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

57

Figure 2-35. Permeability reduction by swelling (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 2-36. Permeability reduction by swelling (after Civan, ©2001 SPE; reprinted by permission of the Society of Petroleum Engineers).

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

1996). The total IEC is therefore equal to the sum of the cation-exchange capacity (CEC) and the anion-exchange capacity (AEC): IEC = CEC + AEC

(2-61)

During reservoir exploitation, when brines of different composition than the reservoir brines enter the reservoir formation, an ion-exchange process may occur, activating various processes leading to formation damage (see Chapter 13). In the literature, more emphasis has been given to the measurement of the CEC, because it is the primary culprit, responsible for water sensitivity of clayey formations (Hill and Milburn, 1956; Huff, 1987; Khilar and Fogler, 1983, 1987; Muecke, 1979; Thomas, 1976). The mechanisms, by which aqueous ions interact with the clay minerals present in petroleum-bearing rock, have been the subject of many studies. Kleven and Alstad (1996) identified two different mechanisms: (1) lattice substitutions and (2) surface edge reactions. The first mechanism involves the ion-exchange within the lattice structure itself, by substitution of Al3+ for Si4+  Mg2+ for Al3+ , as well as other ions to a lesser degree, and does not depend on the ionic strength and pH of the aqueous solution (Kleven and Alstad, 1996). The second mechanism involves the reactions of the functional groups present along the edges of the silica–alumina units and it is affected by the ionic strength and pH of the aqueous solution (Kleven and Alstad, 1996). The relative contributions of these mechanisms vary by the clay mineral types. It appears that montmorillonite and illite primarily undergo lattice substitutions, and surface edge reactions are dominant for kaolinite and chlorite (Kleven and Alstad, 1996). Expansion of swelling clays, such as montmorillonite, increases their surface area of exposure and, therefore, their CEC (Kleven and Alstad, 1996). Theoretical description of the ion-exchange reactions between the aqueous phase and the sedimentary formation minerals is very complicated because of various effects, including ion composition, pH, and temperature (Kleven and Alstad, 1996). Because the ion-exchange reactions in petroleum-bearing rock are usually treated as equilibrium reactions for practical purposes, ionexchange isotherms relating the absorbed and the aqueous-phase ion contents in equilibrium conditions are desirable. For example, Kleven and Alstad (1996) determined the cation-exchange isotherms shown in Figures 2-37–2-39, respectively, for single cation-exchange reactions involving Ca2+ → Na+

(2-62)

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

59

Adsorbed calcium ions, meq/100g

80

60

40

20

0

0

200 400 600 Calcium ions in solution, meq/L

800

Figure 2-37. Calcium-sodium ion-exchange isotherms (circles = kaolinite, squares = montmorillonite, open figures = 20 C, and closed figures = 70 C) (reprinted from Journal of Petroleum Science and Engineering, Vol. 15, Kleven, R., and Alstad, J., “Interaction of Alkali, Alkaline-Earth and Sulphate Ions with Clay Minerals and Sedimentary Rocks,” pp. 181–200, ©1996, with permission from Elsevier Science).

Adsorbed barium ions, meq/100g

14 12 10 8 6 4 2 0

0

30 40 20 Barium ions in solution, meq/L

10

50

Figure 2-38. Barium–sodium ion-exchange isotherms (circles = kaolinite, squares = montmorillonite, open figures = 20 C, and closed figures = 70 C) (reprinted from Journal of Petroleum Science and Engineering, Vol. 15, Kleven, R., and Alstad, J., “Interaction of Alkali, Alkaline-Earth and Sulphate Ions with Clay Minerals and Sedimentary Rocks,” pp. 181–200, ©1996, with permission from Elsevier Science).

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Adsorbed calcium and barium ions, meq/100g

14 12

10

8

6

4

2 0

0

30 40 10 20 Calcium and barium ions in solution, meq/L

50

Figure 2-39. Calcium (open figures) and barium (closed figures) ion-exchange isotherms at 70 C (circles = kaolinite and squares = montmorillonite) (reprinted from Journal of Petroleum Science and Engineering, Vol. 15, Kleven, R., and Alstad, J., “Interaction of Alkali, Alkaline-Earth and Sulphate Ions with Clay Minerals and Sedimentary Rocks,” pp. 181–200, ©1996, with permission from Elsevier Science).

and Ba2+ → Na+

(2-63)

and binary cation exchange reactions involving Ca2+ + Ba2+ → Na+

(2-64)

Similarly, Figure 2-40 by Kleven and Alstad (1996) shows the typical anion-exchange isotherms for a single anion-exchange reaction involving − SO2− 4 → Cl . When more than one ions are present in the system, some are preferentially more strongly adsorbed than the others depending on the affinities of the clay minerals for different ions. This phenomenon is referred to as the selectivity expressing the competitive adsorption of ions. Kleven and Alstad (1996) have determined that the kaolinite and

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

61

Adsorbed sulphate ions, meq/100g

0.5

0.4

0.3

0.2

0.1

0

0

0.6 0.2 0.4 Sulphate ions in solution, meq/L

0.8

Figure 2-40. Sulfate–chloride ion-exchange isotherms at low sulfate concentrations (circles = kaolinite, squares = montmorillonite, open figures = 20 C, and closed figures = 70 C) (reprinted from Journal of Petroleum Science and Engineering, Vol. 15, Kleven, R., and Alstad, J., “Interaction of Alkali, Alkaline-Earth and Sulphate Ions with Clay Minerals and Sedimentary Rocks,” pp. 181–200, ©1996, with permission from Elsevier Science).

montmorillonite clays prefer Ba2+ over Ca2+ , as indicated by the normalized cation-exchange isotherms shown in Figure 2-41. Similarly, their Figure 2-42 showing the normalized anion-exchange isoterms indicate − that the kaolinite clay prefers SO2− 4 over Cl . Figure 2-41 also shows that the selectivity is also influenced by the swelling properties of clays. It is apparent that the affinity of divalent cations (such as Ca2+ ) over monovalent cations (such as Na+ ) is much higher for kaolinite (nonswelling clay) than montmorillonite (swelling clay). Petroleum-bearing formations contain various metal oxides, including Fe2 O3  Fe3 O4  MnO2 , and SiO2 . Tamura et al. (1999) propose a hydroxylation mechanism that the exposure of metal oxides to aqueous solutions causes water to neutralize the strongly base lattice oxide ions to transform them to hydroxide ions, according to O2− + H2 O → 2OH−

(2-65)

Figure 2-43 by Tamura et al. (1999) shows a typical isotherm for OH− ion for hematite. Figure 2-44 by Millan-Arcia and Civan (1992) show that the CEC of the cores extracted from reservoirs may vary significantly by the clay content.

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Eq. fractions of adsorbed calcium ions

1

0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1 Eq. fractions of calcium ions in solution at equilibrium

Figure 2-41. Normalized calcium–sodium ion-exchange isotherms (circles = kaolinite, squares = montmorillonite, open figures = 20 C, and closed figures = 70 C) (reprinted from Journal of Petroleum Science and Engineering, Vol. 15, Kleven, R., and Alstad, J., “Interaction of Alkali, Alkaline-Earth and Sulphate Ions with Clay Minerals and Sedimentary Rocks,” pp. 181–200, ©1996, with permission from Elsevier Science).

Eq. fractions of adsorbed sulphate ions

1

0.8

0.6

0.4

0.2

0

0

0.2 0.4 0.6 0.8 1 Eq. fractions of sulphate ions in solution

Figure 2-42. Normalized sulfate–chloride ion-exchange isotherms (circles = kaolinite, squares = montmorillonite, open figures = 20 C, and closed figures = 70 C) (reprinted from Journal of Petroleum Science and Engineering, Vol. 15, Kleven, R., and Alstad, J., “Interaction of Alkali, Alkaline-Earth and Sulphate Ions with Clay Minerals and Sedimentary Rocks,” pp. 181–200, ©1996, with permission from Elsevier Science).

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

63

Figure 2-43. Hydroxyl-hematite ion-exchange isotherm indicating the amount of hydroxyl ion consumed per unit surface area of hematite vs. the hydroxyl ion concentration in solution (after Tamura et al., 1999; reprinted by permission of the authors and Academic Press).

2.9 SHALE SWELLING AND STABILITY The physical and chemical interactions of clay-rich shales with aqueous solutions, such as contained in water-based drilling fluids, can cause swelling and weakening of shale formations, creating significant practical wellbore instability problems during drilling of wells into petroleum reservoirs (Chen et al., 2003; Chenevert, 1970, 1989; van Oort, 2003; Zhang et al., 2004). Table 2-4 by Zhang et al. (2004) presents an example composition and water activity for the Arco and Pierre-I shales. Essentially, the shale stability depends on three factors, as stated by van Oort (2003) as follows: 1) Transport processes (e.g. hydraulic flow, osmosis, diffusion of ions, and pressure), 2) Physical change (e.g. loss of hydraulic overbalance due to mud pressure penetration), and 3) Chemical change (e.g. ion exchange, alteration of shale water content, changes in swelling pressure).

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Cation exchange capacity (meq/100 gm)

6.0

4.0

2.0

0.0 0.0

2.0

4.0

6.0

Clay content (%)

Figure 2-44. Cation exchange capacity of the various Ceuta field core samples by Maraven S. A., Venezuela (Millan-Arcia and Civan, ©1992; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

An osmotically induced water transfer occurs across the shale and drilling mud interface when the activity of the water present in the shale is different than that in the mud (Chenevert, 1970; van Oort, 2003). However, the transfer of water and ions in shales occurs by diffusion when the shale and mud are maintained at the same pressure (Ballard et al., 1994). The water absorption causes the expansion of swelling clays in shales, weakens the interlayer-bonding, and reduces the shale strength (Zhang et al., 2004). The osmotic effects and ionic diffusion are the primary causes for water transport in shales. However, Zhang et al. (2004) point out that

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

65

Table 2-4 Mineralogical Composition and Water Activity of Arco and Pierre I Shales X-Ray Diffraction Analysis % by weight

Clay

Arco Shale

Pierre I Shale

Quartz Feldspar Calcite Dolomite Pyrite Siderite

236 40 — 12 24 41

19 40 3 7 2 1

Chlorite Kaolinite Illite Smectite Mixed-layer Total

36 57 150 110 294 647

4 11 190 17 49 64

Water Activity

078

098

After Zhang et al., ©2004 SPE; reprinted by permission of the Society of Petroleum Engineers.

the surface of the samples used in laboratory tests may be altered during sample preparation. Thus, the shale sample may entrain some air if it looses some of its water. As a result, the experimental measurements may require correction for the early water uptake in such samples owing to capillary effects (Zhang et al., 2004). Such capillary effects are inherent in the laboratory tests because of the contact of the shale samples with air during sample preparation and are not necessarily involved at the wellbore shale conditions. 2.9.1

Description of Shale Behavior

A review of the forces acting on shales and the swelling pressure is presented in this section according to van Oort (2003). As described schematically in Figure 2-45 by van Oort (2003), the forces acting on shales can be classified into two groups. The first group comprises the mechanical forces involving the pore pressure, the overburden (vertical) and lateral (horizontal) stresses, and the cementation bond

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-45. A schematic representation of downhole forces acting on a shale system, simplified as a single set of clay platelets connected to a pore. The forces include the in situ vertical and horizontal stresses, the pore pressure, the swelling pressure acting between the clay platelets, and tensile or compressive forces in the cementation developing upon compressive or tensile loading of the shale material, respectively. (Reprinted from Journal of Petroleum Science and Engineering, Vol. 38, Van Oort, E., “On the Physical and Chemistry Stability of Shales,” Nos 3–4, pp. 213–235, ©2003; with permission from Elsevier Science).

stresses at the intergranular contact points. The second group comprises the physico-chemical forces involving the electrostatic Born repulsion, the van der Walls attraction, and the hydration or swelling stress/pressure resulting from the hydration/solvation of the clay and ions contained in the clay interlayers present in shales. van Oort (2003) emphasizes that simple models cannot adequately describe the clay–shale swelling behavior because the nature of the swelling pressure is very complicated. For example, the results presented in Figure 2-46 by van Oort (2003), constructed by Karaborni et al. (1996) based on a molecular dynamics simulation study, indicates that swelling pressure in sodium montmorillonite fluctuates significantly depending on the layering of water between the clay platelets at various interlayer spacings. The shale system may experience different magnitudes of repulsive and attractive forces and, therefore, become unstable or stable, respectively, at different levels of water layering in between the clay platelets. van Oort (2003) draws attention that K+ is significantly more effective in reducing the montmorillonite swelling pressure than Ca2+ and Mg2+ probably because K+ ions undergo less hydration and create less repulsion in water than the other mentioned ions. Figure 2-47 by van Oort (2003) presents the swelling index of montmorillonite in various concentrations of KCl and KCOOH. As the fluid salinity (concentration)

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

67

Figure 2-46. Swelling pressure in Na-montmorillonite as a function of interplatelet distance/basal spacing d 100. Contribution of DLVO forces is not included. Stable states are indicated by arrows. Results are shown for the stable states with spacings at 9.7, 12.0, 15.5, 18.3 and 20.7 A. (Reprinted from Journal of Petroleum Science and Engineering, Vol. 38, Van Oort, E., “On the Physical and Chemistry Stability of Shales,” Nos 3–4, pp. 213–235, ©2003; with permission from Elsevier Science).

Figure 2-47. Oedometer test result for a shale containing 68% total clay, of which 76% montmorillonite, immersed in solutions of KCl and KCOOH of increasing salinity. The test shows an initial decrease in swelling for increase in salinity (note that the swelling index does not go to zero, i.e. there always is a residual swelling pressure), after which swelling increases again with the increase in salt content. (Reprinted from Journal of Petroleum Science and Engineering, Vol. 38, Van Oort, E., “On the Physical and Chemistry Stability of Shales,” Nos 3–4, pp. 213–235, ©2003; with permission from Elsevier Science).

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

increases, the swelling index first declines, then attains a minimum, and thereafter increases. van Oort (2003) cautions, however, that the results presented in Figure 2-47 are for the presence of high-salinity brines in the interplatelet spacing of clays, and such high salinity conditions are very rare in the field. The review of the various issues relevant to shale stability by Israelachvili (1991) and van Oort (2003) reveals that (1) the clay-rich shales naturally exhibit a swelling pressure, (2) the swelling pressure may vary either favorably or unfavorably when the shale is brought into contact with a water-based drilling mud because of chemical alterations, and (3) the magnitude of the swelling pressure is also strongly dependent on the type of clays present in shales. Therefore, van Oort (2003) states, “Whereas potassium has a strong effect on swelling of montmorillonite, it has hardly any effect on illite and may actually increase the swelling of kaolinite.” van Oort et al. (1995, 1996) described the transport behavior in the shale-fluid system as being similar to the “leaky osmotic membranes.”

2.9.2 Simplified Modeling of Processes Affecting Wellbore Stability Detailed analysis and modeling of the relevant processes and their effect on wellbore stability in shales can be found in numerous literatures. However, for the purposes of the present discussion, the simple modeling approach by van Oort (2003), emphasizing the fundamental mechanisms without overwhelming with details, is reviewed in the following. Consider the overbalanced drilling of a well into a shale formation. Ions diffuse into the shale owing to the chemical potential gradient when the ion concentration of the drilling fluid is greater than that of the pore fluid present in the shale. Figure 2-48 by van Oort (2003) shows a schematic of the invasion front locations for the mud-filtrate, solute/ion, and pressure, and the three consecutive zones formed, namely the filtrate invasion (FI), solute invasion (SI), and pore pressure (PP) zones, respectively. Neglecting the possibility of any type of coupling between the governing processes, van Oort (2003) estimated the effect of the changes in the pore fluid pressure, and ion and water concentrations on the shale stability based on the following formulation in the semi-infinite radial coordinate.

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

69

Figure 2-48. Schematic overview of the development of various fronts around a wellbore in a shale in time. From the central wellbore going out into the formation, the filtrate invasion front is preceded by a solute/ion invasion front, which in turn is preceded by the mud pressure invasion front. There is one to two orders of magnitude difference in penetration depth between the various invasion fronts. (Reprinted from Journal of Petroleum Science and Engineering, Vol. 38, Van Oort, E., “On the Physical and Chemistry Stability of Shales,” Nos 3–4, pp. 213–235, ©2003; with permission from Elsevier Science).

2.9.2.1

Pressure diffusion

The shale pore fluid pressure is determined using the pressure diffusion equation given by   p p 1  r  r ≥ rw  t > 0 = (2-66) t r r r The initial condition is p = po  r ≥ rw  t = 0

(2-67)

The boundary conditions are p = p m  r = rw  t > 0

(2-68)

p = po  r →  t > 0

(2-69)

where the initial pore fluid pressure and applied mud pressure are denoted by po and pm . The hydraulic diffusivity coefficient  is given by =

K ce

(2-70)

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Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

where K is the permeability and e is the effective porosity of the shale, and  is the fluid viscosity. The effective porosity is given by e =  +

cr − 1 + cs  c

(2-71)

Here,  denotes the actual porosity, and c cr , and cs denote the compressibility of the fluid, bulk formation, and individual grains. 2.9.2.2

Ion diffusion

The shale pore fluid pressure is determined using the pressure diffusion equation given by   C C 1  r  r ≥ rw  t > 0 (2-72) = t r r r The initial condition is r ≥ rw 

t=0

(2-73)

C = Cm 

r = rw 

t>0

(2-74)

C = Co 

r → 

t>0

(2-75)

C = Co  The boundary conditions are

where the initial pore fluid species concentration and applied mud species concentration are denoted by Co and Cm . The species diffusion coefficient is D. Typical ion diffusion coefficients are reported in Table 2-5 by van Oort (2003). 2.9.2.3

Front positions

Applying the analytical solution for these equations by Carslaw and Jaeger (1959), van Oort (2003) estimates the radius of the front positions shown in Figure 2-49 assuming 10 × 10−8 m2 /s pressure (hydraulic) diffusion coefficient and 10 × 10−10 m2 /s ion diffusion coefficient. The results presented in Figure 2-49 indicate that the pore-pressure front moves one to two orders of magnitude faster than the ion diffusion front in low permeability shale. The mud-filtrate front position movement calculated as the following is slower than the ion front movement.

71

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Table 2-5 Results of Transport Experiments Determining Permeability, Membrane Efficiency and Ion Diffusion Rates in Pierre Type I Shale Membrane efficiency (%)

Permeability (nD)

Test 35% CaCl2

2.0

50

21% NaCl

1.9

38

26% KCl

2.2

22

72% KCOOH

1.5

79

21% NaCl–7.5% Na–silicate mud

5.4a (before mud exposure), < 01 (after mud exposure)

Cation diffusion rate DCa2+  = 26 × 10−10 m2 /s

DNa+  = 29 × 10−10 m2 /s

DK+ = 19 × 10−10 m2 /s

DK +  = 13 × 10−10 m2 /s DNa+  = below detection limitb

61

a

The permeability of the shale was determined before and after exposure to the silicate drilling fluid—a dramatic drop in permeability was observed after exposure, consistent with the mechanism of pore blocking caused by silicate gellation and precipitation. b The diffusion coefficients were below the experimental detection limit of 05 × 10−10 m2 /s. Reprinted from Journal of Petroleum Science and Engineering, Vol. 38, Van Oort, E., “On the physical and chemical stability of shales,” Nos 3–4, pp. 213–235, ©2003, with permission from Elsevier Science.

2.9.2.4

Near-wellbore mud-filtrate invasion

The shale water or mud-filtrate content and front position movement with time are determined as the following. The volumetric rate of the invading fluid at the wellbore is given by q=−

KA p   r

r = rw 

t>0

(2-76)

Note that the minus sign indicates that the filtrate invasion occurs in the radial coordinate direction during overbalanced drilling. The cumulative volume of the invading fluid is given by Qt =

t 0

q t dt

t>0

(2-77)

72

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations 1000000

Time (sec)

10000

100

1

Pressure diffusion

0.01

Ion diffusion 0.0001 0.001

0.01 r/a - 1 (dimensionless)

1.0

Figure 2-49. Pressure penetration and ion diffusion in shale. Profiles were obtained by applying a short time approximation and using a diffusion constant of 10 ×10−8 m2 /s for pressure diffusion and 10 × 10−10 m2 /s for ion diffusion. (Reprinted from Journal of Petroleum Science and Engineering, Vol. 38, Van Oort, E., “On the Physical and Chemistry Stability of Shales,” Nos 3–4, pp. 213–235, ©2003; with permission from Elsevier Science).

The volume of the near-wellbore fluid invasion zone is given by   V = rf2 t − rw2 (2-78) Combining Eqs (2-76)–(2-78) yields    t K  p t   A rf t = rw2 + − dt

 r

t>0

(2-79)

0

Invoking the analytical solution of Eqs (2-66)–(2-69) for pt into Eq. (2-79), an estimate of the front position rf t with time can be obtained. 2.9.3 Physico-Chemical Sensitivity of Clayey Formation and Clay Reactivity Coefficient Fam et al. (2003) introduced a dimensionless parameter called the clay reactivity coefficient (CRC), denoted here by N , for clayey formations,

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

73

such as mud rocks (or mud stones), and frequently referred to as shale although their fissility is rather weak. This parameter is recommended as a measure of the sensitivity of the clay particles present in clayey formations, exposed to varying aqueous fluid properties and temperature gradients. It is defined in the following modified form: N = L



(2-80)

 where  L, and denote the mass density, double-layer thickness, and specific surface of the clay particles, respectively. Variations in the physical and chemical properties of the aqueous pore fluid may affect the thickness of the double-layer surrounding the clay particles. Fam et al. (2003) report that the typical double-layer thickness (10−8 to 10−9 m) is comparable to the fine clay size (10−7 to 10−9 m). Fam et al. (2003) emphasize that the macroscopic properties of fine-grained clayey formations vary with the double-layer thickness variations. They concluded that the water film thickness and particle size are important characteristic parameters of the clayey formations. For example, porosity may vary significantly if the water film thickness to particle size ratio exceeds about 0.05 for typical fine clay-aqueous liquid systems. This condition yields an approximate value of N = 01 for the reactivity coefficient. Consequently, they suggest this value as a means to distinguish the reactive fine-grained formations from the nonreactive coarse-grained formations. They confirmed the applicability of the double-layer theory by exposing three types of mud rocks to a concentrated 1.0 M NaCl solution by applying 1.6 MPa vertical stress increment for a period of 10 days. Their results presented in Figure 2-50 indicated that the rock sensitivity increases with increasing values of the reactivity coefficient.

Exercises 1. What is clay? 2. Which swelling mechanism, crystalline or osmotic swelling, causes significant formation damage? Explain the reasons. 3. What are the primary reasons for formation damage? 4. What types of sedimentary formations are particularly susceptible to formation damage? 5. What primary phenomena lead to the formation of subsurface sedimentary porous formations?

74

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

Figure 2-50. Sensitivity of three different mud rocks exposed to a concentrated 1.0 M NaCl solution under the 1.6 MPa vertical stress increment for 10 days. The sensitivity of the rocks measured by percent compression strain increased with increasing reactivity coefficient (A modified plot of the data of Fam et al. (2003); modified after Journal of Petroleum Science and Engineering, Vol. 38, Fam, M. A., Dusseault, M. B., and Fooks, J. C., “Drilling in mudrocks: rock behaviour issues,” Nos 3–4, pp. 155–166, ©2003; with permission from Elsevier Science).

6. What types of diagenetic processes do sediments in subsurface reservoirs undergo? 7. What conditions affect the diagenetic processes? 8. How are the typical constituents of the near-wellbore sedimentary formations generally categorized into two primary groups? 9. Describe the indigenous materials and their origin in sedimentary formations. 10. Describe the diagenetic (or authigenic) materials and their origin in sediments. 11. In what forms do authigenic materials present in the pore space? 12. Why do authigenic materials pose greater formation damage potential? 13. What is the general composition of the subsurface sedimentary formations? 14. What are the main groups of classification of the clay minerals? 15. Describe the mixed-layer clay minerals and their formation damage potential. 16. What are the two primary groups of rock–fluid interactions in sedimentary formations? 17. List the typical clay minerals in an ascending order of the specific surface area.

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

75

18. What is the significance of the specific surface area of clay minerals for formation damage? 19. Describe the major reservoir damage problems associated with various clay minerals. 20. What are the primary factors affecting the mineralogical sensitivity of sedimentary formations? 21. Describe the essential structural features of kaolinite, montmorillonite, and illite minerals. 22. Explain the primary mechanisms of clay damage. 23. What is the order of replaceability of the common cations in clays from the easiest to hardest? Point out the exceptions noted in the literature. 24. Describe the basic mechanisms of clay swelling. 25. What is the significance of the “critical salt concentration”? 26. What does swelling coefficient represent? 27. Why and when does water diffuse into clay? What causes the hindered diffusion of water into clay? 28. Prepare theoretical charts similar to Figures 2-30, 2-32, 2-33, and 2-35 showing ten lines in each for a typical range of parameter values. Indicate the selected parameter values with a key inserted into the figures. 29. Prepare a theoretical chart for porosity reduction by swelling similar to the previous question. 30. Determine the Hayatdavoudi hydration indices of the first four clay minerals listed in Table 2-1. 31. Estimate the values of the “critical salt concentrations” for swelling of montmorillonite in sodium chloride brine based on the data given in Figure 2-15. 32. Repeat the previous question for Figure 2-16 for other systems. 33. Based on Figure 2-17, would formation damage occur when montmorillonite is exposed to a brine solution containing 0.1 N NaCl and 0.001 N CaCl2 ? 34. Repeat previous question for a 0.1 N NaCl and 0.1 N KCl brine based on Figure 2-18. 35. Using the van’t Hoff equation, estimate the osmotic pressure when the total ion concentrations in the clay double-layer and the surrounding pore fluid are 0.1 m and 0.0 m, respectively. 36. Repeat the previous question for a 0.05 m total ion concentration fluid. 37. Based on Figure 2-17, would formation damage occur when the montmorillonite is exposed to a brine solution containing 0.01 N NaCl and 0.1 N CaCl2 ?

76

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

38. Consider the following problem designed by van Oort (2003). The effective porosity and permeability of a shale are 20% and 1.0 nD (approximately 10 × 10−21 m2 ). The shale temperature is 65 C. The viscosity and isothermal compressibility of the water are 434 × 10−4 Pas and 448 × 10−10 Pa−1 , respectively. a. What is the value of the hydraulic diffusion coefficient? (Answer: 25 × 10−8 m2 /s) b. What is the typical order of magnitude of the ion diffusion coefficient? (Answer: See Table 2-5 by van Oort (2003)) c. How different is the order of magnitude of the pressure diffusion coefficient compared to the ion diffusion coefficient? (Answer: About two orders of magnitude greater). 39. Describe the primary mechanisms of the aqueous ion interactions with the clay minerals in sedimentary formations. 40. Estimate the concentrations of the Ca2+ ions adsorbed on the montmorillonite mineral at equilibrium with a 400 meq/L Ca2+ containing brine at 20 and 70 C temperatures based on the data given in Figure 2-37. 41. For convenience, van Oort (2003) defined the following dimensionless pressure and concentration as pD =

p t − po pm − po

(2-81)

CD =

C t − Co Cm − Co

(2-82)

The dimensionless radial distance was defined by rD =

r rw

(2-83)

The dimensionless time for the pressure diffusion equation was defined as t tD = 2 (2-84) rw The dimensionless time for the species diffusion equation was defined as tD =

Dt rw2

(2-85)

Mineralogy and Mineral Sensitivity of Petroleum-Bearing Formations

77

a. Applying these dimensionless variables, show that both the sets of Eqs (2-66)–(2-69) and Eqs (2-72)–(2-75) yield the same dimensionless forms. b. Obtain the analytical solution of the resulting system from Carslaw and Jaeger (1959). 42. Starting with the definition of the clay expansion coefficient  expressed by Eq. (2-43), show that the density and volume variations by swelling can be related by Mo − M − Mo  Mo + 1/k  ln 1 − /t  = =  + 1 Mo  + 1 Mo 0       1 V 1  1− = 1− 0 1− (2-86)  +1 k t Mo V k t Mo

1−

where M V , and are the instantaneous mass, volume, and density of the swollen clay, and Mo  Vo , and o are their initial values before swelling. 43. Consider a porous rock sample having a prescribed pore-size distribution. Estimate the thickness of the immobile water film coating the pore surface and the connate-water saturation that will be attained in the rock sample as a function of the concentration of the dissolved salts in water under prescribed pressure and temperature conditions.

C

H

A

P

T

E

R

3

PETROGRAPHICAL CHARACTERISTICS OF PETROLEUM-BEARING FORMATIONS

Summary A review of petrographical characterization of petroleum-bearing formations, critical for formation damage analysis, is presented. The rock fabric and texture concepts are introduced. The petrographical properties are described in terms of the relevant parameters, including the spherical pore shape approximation, cross-sectional area open for fluid flow, pore connectivity and coordination number, pore and pore throat size distributions, and textural parameters. The dispersed clay morphology, formation damage tendency, and reservoir characterization issues are also discussed.

3.1 INTRODUCTION In situ fluids and particles transport processes occur in the pore space of the subsurface formations. The subsurface formations can be classified as follows (Collins, 1961; Kaviany, 1991): 1. 2. 3. 4.

Isotropic, anisotropic (directional dependency) Homogeneous, heterogeneous (spatial dependency) Consolidated, unconsolidated (cementation) Single or multiple porosity, naturally fractured, nonfractured (pore structure) 5. Ordered or disordered (grain packing). 78

Petrographical Characteristics of Petroleum-Bearing Formations

79

Description of petroleum-bearing formations by quantitative means is a difficult task and is presented in this chapter. A review of the petrophysical properties is presented in Chapter 4, and the various approaches available for development of the porosity–permeability transforms are described in Chapter 5.

3.2 PETROGRAPHICAL CHARACTERISTICS The petrographical parameters are facilitated to quantitatively describe the texture or appearance of the rock minerals and the pore structure. The fundamental parameters used for this purpose are described in the following sections. 3.2.1

Fabric and Texture

Lucia (1995) emphasizes that “Pore space must be defined and classified in terms of rock fabrics and petrophysical properties to integrate geological and engineering information.” Fabric is the particle orientation in sedimentary rock (O’Brien et al., 1994). Lucia et al. (2003) state, “Rock fabrics are geologic descriptors that characterize pore size according to particle size and sorting, interparticle porosity, and various types of vuggy porosity. Only rock fabrics, not pore throat size, permeability/porosity ratio, or flow zone indicators, have vertical and lateral continuity.” Défarge et al. (1996) defined, “Texture, i.e., the size, shape, and mutual arrangement of the constituent elements at the smaller scale of    sedimentary bodies, is a petrological feature that may serve to characterize and compare” them. Petrophysical classification of rock fabrics, such as shown in Figure 3-1 by Lucia (1995), distinguishes between depositional and diagenetic textures. Lucia (1995) points out that “The pore-size distribution is controlled by the grain size in grain-dominated packstones and by the mud size in mud-dominated pack-stones.” Lucia (1995) explains that “Touching-vug pore systems are defined as pore space that is (1) significantly larger than the particle size, and (2) forms an interconnected pore system of significant extent” (Figure 3-2). 3.2.2

Porosity

Porosity, , is a scalar measure of the pore volume defined as the volume fraction of the pore space in the bulk of porous media. The porous

80

Petrographical Characteristics of Petroleum-Bearing Formations

Interparticle pore space Particle size and sorting (Matrix interconnection) Mud-dominated fabric Grain-dominated fabric Grainstone Packstone Packstone Wackestone Mudstone

Percent interparticle porosity

Grain size controls Grain/mud size pore size controls pore size

Mud size controls connecting pore size

Limestone

Limestone

Intergranular pore space or cement

Intergranular pore space or cement

Dolomite

Dolomite crystal size controls connecting pore size Dolomite Crystal size 100 µm Intercrystalline pore space

Note: bar is µm

Intercrystalline pore space

Note: bar is 100 µm

QA 15773c

Figure 3-1. Geological and petrophysical classification of the carbonate rock interparticle pore structure (after F. J. Lucia, AAPG Bulletin, Vol. 79, No. 9, AAPG ©1995; reprinted by permission of the American Association of Petroleum Geologists whose permission is required for future use).

structure of naturally occurring porous media is quite complicated. The simplest of the pore geometry is formed by packing of near-spherical grains. When the formation contains different types of grains and fractured by stress and deformation, pore structure is highly complicated. For convenience in analytical modeling, the porous structure of a formation can be subdivided into a number of regions. Frequently, a gross classification as micropores and macropores regions according to Whitaker (1999) and Bai et al. (1993) can be used for simplification. However, in some cases, a more detailed composite description with multiple regions may be required (Guo and Evans, 1994; Cinco-Ley, 1996). Such descriptions may accommodate for natural fractures and grain-packed regions of different characteristics. The various regions are considered to interact with each other (Bai et al., 1993, 1995). Prince et al. (1999) emphasize that the porosity distribution in sandstones is not homogeneous or random. They describe that the depositional pore matrix in sandstones is composed of two distinct features: Packing flaws and close-packed domains. Packing flaws are oversized pores and

81

Petrographical Characteristics of Petroleum-Bearing Formations Vuggy pore space

Percent separate-vug porosity

Separate-vug pores (Vug-to-matrix-to-Vug connection) GRAIN-DOMINATED FABRIC

MUD-DOMINATED FABRIC

Example types

Example types

Moldic pores

Moldic pores

Composite moldic pores

Intrafossil pores

Intrafossil pores

Shelter pores

Touching-vug pores (Vug-to-vug connection) GRAIN-AND-MUDDOMINATED FABRICS

Example types

Cavernous

Breccia

Fractures

Intragranular microporosity

Solutionenlarged fractures

Fenestral QA 15762c

Figure 3-2. Geological and petrophysical classification of the rock vuggy pore structure (after F. J. Lucia, AAPG Bulletin, Vol. 79, No. 9, AAPG ©1995; reprinted by permission of the American Association of Petroleum Geologists whose permission is required for future use).

pore throats, which are well connected and thus provide continuity for flow through sandstones, whereas significantly small pores and pore throats and microporosity hold the irreducible fluids in sandstones.

3.2.3

Spherical Pore Space Approximation

For simplification and convenience, the shapes of the pore space and grains of porous media are approximated and idealized as spheres.

82

Petrographical Characteristics of Petroleum-Bearing Formations

The pore volume can be approximated in terms of the mean pore diameter, D, as VP =

D3 6

(3-1)

Then, given the bulk volume per one pore, VB , the porosity is expressed by =

VP  3 = D VB 6VB

(3-2)

The pore surface is given by A = D2 =  1/3 6VB 2/3 2/3

(3-3)

The specific pore surface in terms of the pore surface per pore volume is given by a=

A 6 = VP D

(3-4)

The expressions given above for a spherical shape can be corrected for irregular pore space, respectively, as (Civan, 1996a) VP = C1 D3

(3-5)

 = C2 D 3

(3-6)

A = C3 D2 = C4 2/ 3

(3-7)

a=

C5 D

(3-8)

where C1  C2      C5 are some empirical shape factors. Similarly, for the spherical idealization of a particle, the specific surface defined as the surface per volume of sphere is given by ap =

6 Dp

(3-9)

This can be corrected for irregular particle shape as ap = where C6 is a shape factor.

C6 Dp

(3-10)

Petrographical Characteristics of Petroleum-Bearing Formations

3.2.4

83

Area Open for Flow – Areosity

Areosity or areal porosity is the fractional area of the bulk porous media open for flow (Liu and Masliyah, 1996a,b). Liu and Masliyah (1996a,b) point out that, frequently, the areal porosity has been taken equal to the volumetric porosity  of porous media. Therefore, the area Af open for flow is estimated by Af = AB 

(3-11)

The symbol AB denotes the bulk surface area of porous media. They emphasize that Eq. (3-11) performs well for models considering a bundle of straight hydraulic flow pathways and nonconnecting constricted pathways. Whereas, for isotropic porous media, Liu et al. (1994) recommend that the areal porosity should be estimated using 2/3  Therefore, Af = AB 2/3

(3-12)

Civan (2001a, 2002a,b) employed a fractal representation. 3.2.5

Tortuosity

Tortuosity is defined as the ratio of the lengths, Lt and L, of the preferential tortuous fluid pathways and the porous media: =

Lt L

(3-13)

Liu and Masliyah (1996a,b) recommend the Bruggeman (1935) equation = 1/2

(3-14)

for random packs of grains of porosity  > 02 and the Humble equation (Winsauer et al., 1952) = 161115

(3-15)

for consolidated porous media of porosity  < 045. They point out that the latter may have a variable accuracy and, therefore, tortuosity should be measured.

84

Petrographical Characteristics of Petroleum-Bearing Formations

3.2.6 Interconnectivity of Pores – Coordination Number Based on their binary images shown in Figure 3-3, Davies (1990) classified the pore types into four groups (Davies, 1990, p. 74): Pore Type 1: Microspores, generally equant shape, less than 5 in diameter. These occur in the finest grained and shaly portions of the sand. Pore Type 2: Narrow, slot like pores, generally less than 15 in diameter, commonly slightly to strongly curved. These represent reduced primary

Figure 3-3. Thin-section images of various pore types (after Davies, ©1990 SPE; reprinted by permission of the Society of Petroleum Engineers).

Petrographical Characteristics of Petroleum-Bearing Formations

85

intergranular pores resulting from the reduction of original primary pores by extensive cementation. Pore Type 3: Primary intergranular pores, triangular in shape, 25–50

diameter. These are the original primary intergranular pores of the rock which have been affected only minimally by cementation. Pore Type 4: Solution enlarged primary pores: oversized primary pores, 50–200 diameter produced through the partial dissolution of rock matrix.

Frequently, for convenience, pore space is perceived to consist of pore bodies connecting to other pore bodies by means of the pore necks or throats as depicted in Figure 3-4. Many models facilitate a network of pore bodies connected with pore throats as shown in Figure 3-5. However, in reality, it is an informidable task to distinguish between the pore throats and the pore bodies in irregular porous structure (Lymberopoulos and Payatakes, 1992). Interconnectivity of pores is a parameter determining the porosity of the porous media effective in its fluid flow capability. In this respect, the Pore body

Pore neck

Figure 3-4. Description of the pore volume attributes (after Civan, ©1994; reprinted by permission of the U.S. Department of Energy).

86

Petrographical Characteristics of Petroleum-Bearing Formations

Real system

1

Sample 2

Flow

Comparisons

1) Laboratory data 2) Data filtering and smoothing 3) MarquardtLevenberg nonlinear optimization 4) Parameter estimation

Void space and flow lines

Macroscopic model

Multiple gradient model

Integration (boundary and initial conditions) 7

Volume averaging Network model 6

Modeling assumptions

3

8

Networking

4

Modeling assumptions Flow path

Microscopic model

5

Capillary orifice model

Figure 3-5. An integrated modeling approach to characterization of porous formation and processes (after Civan, ©1994; reprinted by permission of the U.S. Department of Energy).

pores of porous media, as sketched in Figure 3-6, are classified into three groups: 1. Connecting pores which have flow capability or permeability (conductor) 2. Dead-end pores which have storage capability (capacitor) 3. Nonconnecting pores which are isolated and therefore do not contribute to permeability (nonconductor).

Petrographical Characteristics of Petroleum-Bearing Formations

87

Figure 3-6. Interconnectivity of pores.

The interconnectivity is measured by the coordination number, defined as the number of pore throats emanating from a pore body. Typically, this number varies in the range of 6 ≤ Z ≤ 14 (Sharma and Yortsos, 1987). For cubic packing, Z = 6 and  = 1 − /6. The coordination number can be determined by nitrogen sorption measurements (Liu and Seaton, 1994). The coordination number expressed based on a representative elementary volume of porous media may not be an integer value (Civan, 2002e,f).

3.2.7

Pore and Pore Throat Size Distributions

Typical measured pore body and pore throat sizes, given by Ehrlich and Davies (1989), are shown in Figure 3-7. Figure 3-8 shows the pore throat size distribution measured by Al-Mahtot and Mason (1996). The mathematical representation of the distribution of the pore body and pore throat sizes in natural porous media can be accomplished by various statistical means. The frequently used approaches are described in the following sections.

3.2.7.1

Log-normal distribution

Because of its simplicity, the log-normal distribution function given below has been used by many, including Ohen and Civan (1993):  2   D/D ln 1 m  F D = 2sd D−1/2 exp − 2 sd 

0 ≤ D ≤  (3-16)

88

Petrographical Characteristics of Petroleum-Bearing Formations 0.01

Radius (microns)

0.1 Throats associated with intergranular pores

1.0

10

Intergranular pores Bodies Throats

100 100

60

80

20

40

Cumulative frequency (%)

Figure 3-7. Typical cumulative pore body and pore throat size distributions in porous formation (after Ehrlich and Davies, ©1989 SPE; reprinted by permission of the Society of Petroleum Engineers). 1 Mean flow radius = 0.35 µm

0.9

Pore size distribution (PSD)

0.8 0.7

Sample 475 Sample 220

0.6 0.5 0.4 0.3 0.2 0.1 0 0.001

0.01

0.1

1

10

100

1000

Pore throat radius, microns

Figure 3-8. Typical bimodal pore throat size distributions in porous formation (after Al-Mahtot and Mason, ©1996; reprinted by permission of the Turkish Journal of Oil and Gas).

Petrographical Characteristics of Petroleum-Bearing Formations

89

where sd is the standard deviation, D is the diameter of the pores approximated by spheres, Dm is the mean pore diameter calculated by Dmax

Dm =



DF D dD

(3-17)

Dmin

where Dmin and Dmax denote the smallest and the largest diameters, respectively.

3.2.7.2

-Distribution

Popplewell et al. (1989) used the -distribution function to represent the skewed size distribution because the diameters of the smallest and the largest particles are finite in realistic porous media. For convenience, they expressed the -distribution function in the following modified form: am

m

F x = x 1 − x

 1  0

xam 1 − xm dx

(3-18)

in which x denotes a normalized diameter defined by x = D − Dmin /Dmax − Dmin 

(3-19)

where Dmin and Dmax are the smallest and the largest diameters, respectively, and a and m are some empirical power coefficients. The mode, xm , and the spread, 2 , for Eq. (3-18) are given, respectively, by xm = a/a + 1

(3-20)

and 2 =

am + 1 m + 1

a + 1 m + 22 a + 1 m + 3

(3-21)

Chang and Civan (1991, 1992, 1997) used this approach successfully in a model for chemically induced formation damage.

90

Petrographical Characteristics of Petroleum-Bearing Formations

3.2.7.3

Fractal distribution

Fractal is a concept used for convenient mathematical description of irregular shapes or patterns, such as the pores of rocks, assuming selfsimilarity. The pore size distributions measured at different scales of resolution have been shown to be adequately described by empirically determined power-law functions of the pore sizes (Garrison et al., 1992, 1993; Verrecchia, 1995; Karacan and Okandan, 1995; Perrier et al., 1996). The expression given by Perrier et al. (1996) for the differential pore size distribution can be written in terms of the pore diameter as F D ≡ −

dV =  e − d De−d−1  dD

0 7, negligible filter cake involvement.

209

Particulate Processes in Porous Media

Pautz et al. (1989) point out that these rules-of-thumb have been derived based on experimental observations. The values 3 and 7 denote the critical values or cr . Note these values are very close to the values of 2 and 6 indicated by Figure 8-6 given by Gruesbeck and Collins (1982b) for bridging of particles in perforations. Civan (1990, 1996a) determined cr empirically by correlating between two dimensionless numbers. In the pore throat plugging process, the mean pore throat diameter, Dt , mean particle diameter, Dp , particle mass concentration, cp , viscosity of suspension, , and the interstitial velocity of suspension, v = u/ (Dupuit, 1863), are the important quantities. Therefore, a dimensional analysis among these variables leads to two dimensionless groups (Civan, 1996a). The first is an aspect ratio representing the critical pore throat to particle diameter ratio necessary for plugging given by

  (8-40)

cr = Dt Dp cr and the second is the particle Reynolds number given by  NRep = cp vDp 

(8-41)

The relationship between cr and NRep can be developed using experimental data. Inferred by the Gruesbeck and Collins (1982b) data for

Perforation diameter Average particle diameter

10

0

Maximum particle Concentration – Volume/volume 0.08 0.15 0.21 0.27 0.31

0.58

Tap water 100 cp Hydroxyethyl Cellulose solution

8 6 4

Bridging region 2 0

0

2 4 6 8 10 Maximum gravel content – LB /GAL

30

Figure 8-6. Chart for determination of the particle bridging conditions for perforations (Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

210

Particulate Processes in Porous Media

perforation plugging, and by Rushton (1985) and Civan (1990, 1996a), such a relationship is expected to obey the following types of expressions:   

cr = A 1 − exp −BNRep + C (8-42) or

 B

cr = A NRep + C

(8-43)

where A B, and C are some empirical parameters. The physical limit is C = 1 0 because the pores are clogged when = Dt /Dp < 1 0. However, the Iscan and Civan (2006) studies indicate that the value of C in Eq. (8-42) and (8-43) may be greater than 1.0. This is because, Iscan and Civan (2006) explain that “other factors, such as fluid viscosity and surface forces, may prevent a single particle to move through a pore throat even if the particle size is less than the pore throat size.” The author (see 1st edition, Civan, F., Reservoir Formation Damage, 2000) developed Figure 8-7 showing the correlation of the data of Gruesbeck and Collins (1982b) according to Eq. (8-42). Iscan and Civan (2006) correlated the various other data successfully using Eqs (8-42) and (8-43). However, the above formulation is a simplistic approach. In

Figure 8-7. Chart for determination of the particle bridging conditions using the aspect ratio and particle Reynolds number (modified after Civan, 1994; reprinted by permission of the U.S. Department of Energy).

Particulate Processes in Porous Media

211

Figure 8-8. Alteration of pore throat size distribution by formation damage and acid stimulation.

reality, the pore and pore throat sizes are distribution functions, which vary by damage or stimulation, as shown in Figure 8-8. This can be considered by the methods developed by Ohen and Civan (1993) and Chang and Civan (1997), as described in Chapter 5. For large particles, also the Froude and Archimedes dimensionless numbers should be considered (Valdes and Santamarina, 2006). 8.5.3

Filtration Coefficient

It can be shown that Eq. (8-30) can be expressed as following upon the substitution of  = o − t and rearranging: t = c1 + c2  up t

(8-44)

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Particulate Processes in Porous Media

where c1 and c2 are empirical parameters. Then, Eq. (8-44) can be generalized as t = up t

(8-45)

where is called the variable filtration coefficient (Civan and Rasmussen, 2005), given by (8-46)

= c1 + c2 

The filtration coefficient can be referred to as the overall deposition coefficient or the pore-filling deposition rate coefficient. Wennberg and Sharma (1997) point out that the filtration coefficient varies by particle deposition according to = F  o

(8-47)

in which o is the filtration coefficient with no deposited particles and F is a function of the volume fraction of particles deposited . Their review of the various expressions available for prediction of the filtration coefficient is summarized and presented in the following. Ives (1967) proposed a general multivariable power-law correlation expression as  x  y  z 

  = 1− 1+ 1− o M o o

(8-48)

in which x, y, z, and are some empirical parameters and M is the maximum volume fraction of the deposited particles necessary to render the filtration coefficient of porous media zero. This equation indicates that the filtration coefficient is equal to a constant value of o when there are no deposited particles in porous media, and the filtration coefficient becomes zero when the volume fraction of deposited particles reaches a certain characteristic maximum value of M . Chiang and Tien (1985) developed an empirical correlation as

 = 1 + 0 755 492 − 1 6 × 104 Nr + 1 46 × 105 Nr2  o

 < 10−2 (8-49)

213

Particulate Processes in Porous Media

where Nr is the suspended particle to porous media grain diameter ratio: Nr =

Dp Dg

(8-50)

Rajagopalan and Tien (1976) developed the following expression: 1 8 / 3  Nr < 0 18 = 0 72As NLo/ Nr15/ 18 + 2 4 × 10−3 As Ng1 2 Nr−0 4 + 4As1/ 3 Np−2 e (8-51)

in which As is Happel’s dimensionless geometric parameter, Ng is the gravity number given by Eq. (8-4), NPe is the Peclet number given by Eq. (8-7). Wennberg and Sharma (1997) analyzed the measurements of the filtration coefficient reported by various investigators and determined that these data mostly indicate power-law-type relationships to the volumetric flux, the suspended particle size, and the porous media grain size as ∼ u−  Dp  Dg−

(8-52)

where  , and  are some empirical exponents. They determined that 0 ≤  ≤ 2 > 0 for Dp > 1 m and < 0 for Dp < 1 m 0 6 < < 1 2 for Ottawa sand; and 0 9 ≤  ≤ 2. 8.5.4 Dislodgment and Redeposition of Particles at Pore Throats Gruesbeck and Collins (1982a) observed that the effluent particle concentration tended to fluctuate during constant flow rate experiments. Such phenomena did not occur during constant pressure difference experiments, which are more representative of the producing well conditions. They explain this behavior by consecutive dislodgment and formation of plugs at the pore throats. They postulate that, in heterogeneous systems, when a suspension of particles of various sizes flow through porous media made of a wide range of grain sizes, narrow pathways are likely to be plugged first, diverting the flow to wider pathways, which transfer the particles to the effluent more effectively. However, as the flow paths are plugged, the pressure difference across the porous media may exceed

214

Particulate Processes in Porous Media

the critical stress necessary to break some of the plugs. Therefore, these plugs break and release particles into the flowing media increasing its particle concentration. Subsequently, the deposition process progresses to form new plugs during which the flowing media particle concentration decreases. Gruesbeck and Collins (1982a) also observed a similar phenomena in systems of homogeneous grain sizes subjected to a constant rate injection of a suspension of particles. Millan-Arcia and Civan (1992) have reported frequent fluctuations in the effluent fluid concentrations and pH during injection of brine into sandstone due to continuous particle dislodgment and redeposition at the pore throats (Figure 8-9). Singurindy and Berkowitz (2003) concluded that “During dedolomitization, the inter play among flow, precipitation and dissolution processes can lead to oscillations in the temporal and spatial evolution of effective hydraulic conductivity and porosity.” For example, Figure 8-10 by Singurindy and Berkowitz (2003) provide an experimental evidence of the oscillatory behavior in permeability and porosity of typical dolomite sample. Figure 8-11 by Singurindy and Berkowitz (2003) describes the

Figure 8-9. Effect of frequent pore throat plugging and unplugging by particles on the effluent solution pH (Millan-Arcia and Civan, ©1992; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

Particulate Processes in Porous Media

215

Figure 8-10. Temporal variability in hydraulic conductivity and porosity (estimated from chemical concentrations in fluid samples) for different initial flow rates, with injection fluid pH = 36, and constant hydraulic head across the columns. Ko denotes initial hydraulic conductivity. (a) Evolution of hydraulic conductivity, initial Q = 1 cm3 /min, (b) evolution of hydraulic conductivity, initial Q = 52 cm3 /min, (c) evolution of porosity, initial Q = 1 cm3 /min, and (d) evolution of porosity, initial Q = 52 cm3 /min. The solid vertical lines indicate the approximate times at which the correlations between hydraulic conductivity and porosity becomes weak (Reproduced/modified by permission of American Geophysical Union from Singurindy, O. and Berkowitz, B., “Flow, Dissolution, and Precipitation in Dolomite,” Water Researches Research, Vol. 39, No. 6, pp. SBH (3-1)–(3-13), 2003. Copyright ©2003 American Geophysical Union).

regions of oscillatory/nonoscillatory behavior for ranges of injected fluid pH and volumetric flow rate Q conditions. Sigurindy and Berkowitz (2003) explain that open diamonds refer to experiments in which oscillations in hydraulic conductivity were not observed and dissolution dominated the reactions. Solid squares and open circles identify flow experiments in which oscillatory behavior in the column hydraulic conductivity was observed. Open circles denote the experimental conditions shown in Figure 8-11 (large columns), and solid squares denote small column experiments. The crosses and stars denote short-term experiments. The arrows show the direction of increasing amplitude of the oscillations. The solid line denotes the approximate region in which oscillatory behavior in the hydraulic conductivity and porosity occurs. The dashed line denotes the approximate region in which precipitation bands form.

216

Particulate Processes in Porous Media

Figure 8-11. Summary of flow experiments over a range of injected initial pH and flow rates Q. (Reproduced/modified by permission of American Geophysical Union from Singurindy, O. and Berkowitz, B., “Flow, Dissolution, and Precipitation in Dolomite,” Water Researches Research, Vol. 39, No. 6, pp. SBH (3-1)–(3-13), 2003. Copyright ©2003 American Geophysical Union).

8.5.5 Plugging of Fracture Entrances during Fines Invasion into Naturally Fractured Formations Several studies have addressed the fines invasion into naturally fractured formations, such as fractured carbonate formation, including Lietard et al. (1998, 1999), Civan and Rasmussen (2002), AbdelSalam and Chrysikopoulos (1994), and Salimi et al. (2004). Salimi et al. (2004) determined that the fines invasion through the drilling mud invasion depends on the drilling fluid properties, drilling conditions, and the fracture size. Figure 8-12 by Salimi et al. (2004) is a schematic illustration of the mechanism of drilling mud fines invasion into a typical fracture present in naturally fractured formations, such as fractured carbonate formations. As can be seen the application of the pore throat plugging criteria can be extended for naturally fractured formations. Based on their experimental investigations, Loeppke et al. (1990) conclude that “   particles outside an effective range for a fracture of interest do not contribute to a stable bridge, although they may act as filter material. Consequently, a tailored particle-size distribution for a narrow range of fracture widths should provide the best plugging capabilities.” Figure 8-13 by Loeppke et al. (1990) delineates the bridging capability

217

Particulate Processes in Porous Media 100 micron fracture 10 micron solids invasion

Matrix Fracture

Matrix 100 micron fracture 10–100 micron solids invasion

Matrix Fracture

Matrix 100 micron fracture 10–500 micron solids

Matrix Fracture

invasion

Matrix

Figure 8-12. Mechanism of solid invasion into naturally fractured formation (after Salimi et al., ©2004 SPE; reprinted by permission of the Society of Petroleum Engineers). 30 Thermoset rubber (S/E = 0.32) (L = 0.075 in)

Concentration (lbm/bbl)

25 20 15

Ground coal (S/E = 0.11) (L = 0.088 in)

Expanded aggregate (S/E = 0.15) (L = 0.11 in)

Thermoset rubber Thermoset plastic San – 1 flakes (L = 0.075 in)

10 5 0 0.00

Modified API tests Room temperature

0.50

1.00

1.50

2.00

2.50

3.00

Slot Size/Particle Size

Figure 8-13. Comparison of plugging performance curves for LCMs tested in the modified API tester (after Loeppke et al., ©1990 SPE; reprinted by permission of the Society of Petroleum Engineers).

of various types of particles at the fracture entrances depending on the slot-to-particle-size ratio and particle concentration. Loeppke et al. (1990) have thus demonstrated that a slot can be effectively plugged if its size is less than that of the largest particles present in a distribution of particles

218

Particulate Processes in Porous Media

and the ratio of the compressive strength-to-elastic modulus (S/E) of the particle material is sufficiently high. Jiao and Sharma (1996) determined that fiber particles instead of granular particles are effective in stable bridging across the fracture openings. 8.5.6

Colloidal Release and Mobilization

Colloidal mobilization is a result of the physico-chemical reactions that involve electro-kinetic forces, zeta potential, and ionic strength (Wojtanowicz et al., 1987). Let p denote the volume fraction of porous media occupied by the particles available for mobilization over the pore surface. The rate of colloidal expulsion or mobilization of particles at the pore surface owing to the salinity shock is proportional to the excess critical salt concentration ccr − c and the amount of the unblocked particles at the pore surface available for mobilization p e . r = − 1 +  kr p e 2/ 3 ccr − c t

(8-53)

r = ro  t = 0

(8-54)

subject to

 is the volumetric expansion coefficient for swelling clays as described in Chapter 2;  = 0 for nonswelling particles; ccr is the critical salt concentration; and e is the fraction of the unblocked particles approximated by (see Figure 8-2) (Civan et al., 1989; Civan, 1996a; Ohen and Civan, 1993)   e = exp − p (8-55) p

where is an empirical constant;

 p

p represents the total volume of various

types of particles retained within the pore space; kr is a particle release rate constant given by (Khilar and Fogler, 1983, 1987; Kia et al., 1987) kr = 0 when c < ccr

(8-56)

kr = 0 otherwise

(8-57)

and

Particulate Processes in Porous Media

8.5.7

219

Hydraulic Erosion and Mobilization

The rate of hydraulic mobilization of the particles at the pore surface owing to the velocity shock is proportional to the excess pore wall shear stress, w − cr , and the amount of the unblocked particles available for mobilization at the pore surface (Gruesbeck and Collins, 1982a; Khilar ˇ nanský and Široký, 1985; Civan, 1992, 1996a). and Fogler, 1987; Cerˇ e = − 1 +  ke p e 2/ 3  − cr  t

(8-58)

subject to e = eo 

t=0

(8-59)

cr is the critical shear stress, discussed earlier, and ke is an erosion rate constant given by (Khilar and Fogler, 1987) ke = 0 when w > cr

(8-60)

ke = 0 otherwise

(8-61)

and

There are several alternative ways of expressing the hydraulic force. The Rabinowitsch–Mooney equation for non-Newtonian fluid wall shear stress, w , in pipes is given by (Metzner and Reed, 1955)    n′ p 8 D ′ − =k w = 4 x D

(8-62)

Where D denotes the hydraulic tube diameter. The non-Darcy equation can be modified by applying the capillary tubes analogy and the procedure given by Ikoku and Ramey, Jr (1979):   p Nnd K n′ n′ u =  = − (8-63) e x where e is the effective viscosity given by   ′ ′ ′ e = Nnd 22+3n  k′ K D1+n  n

(8-64)

220

Particulate Processes in Porous Media

k′ and n′ are some empirical parameters, which assume the values of k′ =  and n′ = 1 for Newtonian fluids. Based on Eqs (8-62) and (8-63), the excess shear stress can be correlated for one-dimensional horizontal flow as    n′   p u e D D w − cr  ∼ − − cr ∼ − cr (8-65) 4 x 4Nnd K The previous studies are mostly limited to one-dimensional Newtonian fluid flow and they typically used (Gruesbeck and Collins, ˇ nanský and Široký, 1985; Khilar and Folger, 1987; Civan et al., 1982a; Cerˇ 1989; Ohen and Civan, 1989, 1990)    

w − cr  ∼ −p x − −p x cr ∼

   

u K − u K cr ∼ u − ucr 

(8-66)

It is better to use the excess of the critical interstitial velocity v − vcr  instead of the excess of the volumetric flux u − ucr  as the driving force because the fluid moves faster and the particle detachment is easier in longer tortuous paths (Civan, 2006c). In general, for multidimensional flow (Civan, 1996a)    −  cr ·   w −  cr  ·  ∼ −1 4D ·  (8-67)

where  is the flow potential and D is the hydraulic tube diameter tensor for anisotropic media.  is a unit vector.

8.6 PARTICULATE PHENOMENA IN MULTIPHASE SYSTEMS The behavior of particles in a multiphase fluid system is described in this section. 8.6.1

Effect of Wettability on Particle Behavior

Muecke (1979) explained that the wettability affects the behavior of the particles significantly in a multiphase fluid system. By means of

221

Particulate Processes in Porous Media

experimental investigations, Muecke (1979) has observed that particles tend to remain in the phases that can wet them. Figures 8-14a–e by Muecke (1979) schematically describe the behavior of fine particles in a multiliquid phase system based on the observations with linear flow tests conducted in a micromodel of porous media.

Oil Fines bridged at pore restriction

Connate water (immobile)

Fluid flow direction Mobile fines

(a)

Immobile water-wet fines

(b)

Mobile water Oil

Mobile oil

Connate water Mixed wettability fines

Mobile water

(d)

(c)

Oil

Water

Oil

Oil

Mutual solvent

Oil

(e)

Figure 8-14. Behavior of fine particles in the presence of water–oil fluid system in porous media: (a) when a single-fluid phase is present, fines move with the flowing fluid, unless bridged at pore restrictions, (b) water-wet fines are immobile when the water phase is immobile, (c) water-wet fines not bridged at pore restrictions are mobile when both water and oil are flowing, (d) fines of mixed wettability are constrained to move on∼ along the oil–water interface, and (e) mutual solvents release fines held by wetting and interfacial forces, causing them to migrate at high concentrations (after Muecke, ©1979 SPE; reprinted by permission of the Society of Petroleum Engineers).

222

Particulate Processes in Porous Media

Muecke (1979) arrived at the following conclusions (after Muecke, ©1979 SPE; reprinted by permission of the Society of Petroleum Engineers): 1. “When only a single-fluid phase is present and flowing rapidly enough to keep the particles suspended, formation fines move through the pores with the fluid, unless they mechanically bridge at pore restrictions. Bridges can be disrupted by pressure disturbances or flow reversals. (Figure 8-14a) 2. Besides mechanical bridging, particle wettability and surface/interfacial forces strongly influence particle mobility when multifluid phases are present. Particles will move only if the phase that wets them is moving. (Figure 8-14b) 3. Simultaneous flow of oil and water causes fines to migrate because the water is mobile enough to carry the fines and because the localized pressure disturbances caused by multiphase flow keep the fines agitated, which reduces their chances of developing permanent bridges. (Figure 8-14c) 4. Particles that were both oil- and water-wet were located at the interface between these fluids; their movement appeared to be confined to that interface. These particles stayed at the oil/water interface and followed its contour until reaching a region of low differential velocity between the oil and water. The particles then remained there until flow conditions changed. 5. Flow of water at residual oil saturations rapidly establishes an equilibrium bridged condition that has almost no fines movement associated with it, as long as the flow rate remains constant. (Figure 8-14d) 6. Injection of mutual solvent or surfactant solutions mobilizes fines that are held in place by wetting and interfacial forces.” (Figure 8-14e)

The core flooding tests in sand packs containing water-wet fine particles carried out by Muecke (1979) indicated that 1) fine particles were produced when the oil and water were flown simultaneously because the particles moved with the mobile water phase, 2) no measurable amount of fine particles was produced when only oil was flown at the connate water saturation conditions because the connate water held the water-wet fines at the pore surface, and 3) fine particles were not produced when only water was flown through the core because the fines formed bridges at the pore throats. These observations are commensurate with the above-mentioned observations in micromodels.

223

Particulate Processes in Porous Media

Tang and Morrow (1999) state that brine composition and fines migration may affect the rock–crude oil–brine interactions. They suggest a hypothesis that the adsorption of polar heavy components of crude oil on parts of fine particles may render them mixed-wet and the brine composition affect the force balance, and therefore influence the stripping of the fine particles from the pore surface during water flooding of oil reservoirs, as described schematically in Figure 8-15. Figure 8-16 by Gupta and Civan (1994a) schematically describes the behavior of particles depending on the wettability characteristics of the particles and pore surface. Obviously, particles tend to associate with the fluid phase that wet them (Muecke, 1979). Consequently, as stated by Gupta and Civan (1994a) that “the particle release potential and the corresponding damage are minimized if the pore surface and the particles have similar wettability characteristics. When the wettability of particle and surface are significantly different, then favorable conditions exist for bridging at the pore throats.” The latter causes the severest formation damage because the particles remaining in the mobile phase will approach and jam the pore throats. Also, the intermediately wet particles may cause severe damage because they are most stable at the oil–water interface (Ivanov et al., 1986) and move along the interface toward the pore throats (Figures 8-17 and 8-18). 8.6.2

Particle Transfer across Fluid–Fluid Interfaces

Ku and Henry (1987) described the transfer rate of water-wet particles from an aqueous phase to an oil phase by the first order kinetics equation as rp ≡

dcp = cp dt

(8-68)

where cp denotes the aqueous-phase particle concentration and the rate coefficient is given by =

6Dp fw I Dw

(8-69)

where Dp and Dw denote the particle and water-droplet diameters, fw is the fractional water (volume fraction of the water phase in the water–oil two-phase fluid system), I is the induction time required for movement

224

Particulate Processes in Porous Media

Oil

Adsorbed polar components of crude oil

Oil

Particle Fines with interstitial water

Rock grain

Water

Mobilized particle at oil-water interface

Adsorption onto potentially mobile fines at low intial water saturation

a. Adsorption of polar components from crude oil to form mixed-wet fines Mobilized mixed-wet fines

Transition towards increased water-wet

Oil

Oil

Oil Water Oil

Oil

Solid Mobilized water-wet fines

b. Partial stripping of mixed-wet fines from pore walls during waterflooding

Oil Dilute brine

Dilute brine Retained oil

Solid

Retained oil before injection of dilute brine

Partial mobilization of residual oil through detachment of fines

c. Mobilization of trapped oil

Figure 8-15. Role of potentially mobile fines in crude oil–brine–rock interactions and increase in oil recovery with decrease in salinity (Reprinted from Journal of Petroleum Science and Engineering, Vol. 24, Tang, G.-Q., and Morrow, N.R., “Influence of brine composition and fines migration on crude oil/brine/rock interactions and oil recovery,” Nos 2–4, pp. 99–111, ©1999, with permission from Elsevier Science).

225

Particulate Processes in Porous Media

W

W

O

O

W-Wet surface,

O-Wet surface,

W-Wet particles

W-Wet particles

(a)

(b)

W

O

W

O

W-Wet surface,

O-Wet surface,

O-Wet particles

O-Wet particles

(c)

(d)

Figure 8-16. Multiphase Transport of Fines (after Gupta and Civan, ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers). y (0,0) Fluid Phase 3

σ 13

Fluid Phase 2

ϒ

β

c

ψ

θ Solid particle Phase 1

x

σ 23 Fluid interface

σ 12

Figure 8-17. A particle stabilized at a fluid–fluid interface (modified after Ivanov et al., 1986; reprinted by permission of the author and Academic Press; after Civan, 1994; reprinted by permission of the U.S. Department of Energy).

226

Particulate Processes in Porous Media Solid-Fluid interface

Velocity profile

Wetting fluid (Phase 1) Non-wetting fluid (Phase 2) Solid matrix (Phase 3)

Fluid-fluid interface

Figure 8-18. Particle retention at solid–fluid and fluid–fluid interfaces and the velocity profiles in multiphase systems (after Civan, 1994; reprinted by permission of the U.S. Department of Energy).

of particles contained within the aqueous phase to the oil–water interface, from where they can be entrained by the oil phase. Gupta and Civan (1994a) explain that “the induction time is related to the thinning and rupture of the intervening wetting phase film between the bubble drop and the particle. Such film thinning dynamics is controlled by the film viscosity, interfacial tension, and the nature of interaction between the electrical double layers of two film interfaces.” The driving force for particle transfer between two fluid phases is the wettability of the fluid phases relative to the wettability of the particles. Particles prefer to be in the phase that wets them (Muecke, 1979) (see Figure 8-18 by Civan, 1994a). But, mixed-wet particles tend to remain at the interface where they are most stable (Ivanov et al., 1986) (see Figure 8-17). In the region involving the interface between wetting and nonwetting phases, it can be postulated that particles A in a weaker wettability phase-1 first move to the interface and then migrate from the interface to a stronger wettability phase-2 according to the following consecutive processes (Civan, 1996a): Nonwetting phase − 1 → Interface → Wetting phase − 2

(8-70)

Let m1  m12 , and m2 denote the mass of particles contained in Phase 1, at the interface region of Phases 1 and 2, and in Phase 2, respectively, expressed per unit bulk volume of porous media. Therefore, the following power-law rate expressions can be proposed. dm1 = − 1 m1 d t − t1 

(8-71)

Particulate Processes in Porous Media

227

dm12 = − 1 m1 − 2 m 12 d t − t12 

(8-72)

dm2 = − 2 m 12 d t − t2 

(8-73)

where 1 and 2 are some rate constants;  and are some empirical exponents of intensities; and t1  t12 , and t2 are the time delays due to the inertia of the respective transfer processes. The initial conditions are m1 = m1 

m12 = m12 

m2 = m2 

t=0

(8-74)

The particles captured at the interface are assumed to migrate at a speed determined by the relative speed of the fluid phases. When  = 1 0 and = 1 0, the analytical solution of Eqs (8-71)–(8-74) can be derived similar to Gupta and Civan (1994b).

8.7 TEMPERATURE EFFECT ON PARTICULATE PROCESSES Gupta and Civan (1994a) were the first to suggest that “temperature shock can cause formation damage in a manner similar to the salinity shock.” Hence, reservoir formation sensitivity to temperature plays an important role in formation damage. The existence of temperature shock criteria has been later confirmed by Schembre and Kovscek (2005). Gupta and Civan (1994a) caution that “the laboratory measurement of fines release, migration and trapping, and wettability at ambient temperatures are not directly applicable at subsurface reservoir temperatures.” “Various phenomena leading to formation damage, such as fines release, migration, bridging, clay swelling, mineral dissolution/precipitation, and relative permeability alteration, are controlled by temperature dependent characteristics of the porous medium, minerals, and associated fluids.” Schembre and Kovscek (2005) carried out theoretical and experimental studies with Berea sandstone, diatomite, and calcite samples for steam operations at moderate salinity (0.01–0.05 M) and alkaline pH (7–10) conditions. They demonstrated that when temperature was increased from 120 to 180 C, the increased fines mobilization and thermal shock effects caused an increase in permeability reduction from 35 to 95% in Berea

228

Particulate Processes in Porous Media

sandstone and diatomite rock samples. However, calcite is thermally stable under similar conditions. Schembre and Kovscek (2005) state that “Permeability reduction is observed with temperature increase and fines mobilization occurs repeatably at a particular temperature that varies with solution pH and ionic strength.” The particle detachment temperature under certain fluid salinity and pH conditions is defined as the temperature at which the total interactive potential between a particle and a pore surface changes from attractive to repulsive types (Schembre and Kovscek, 2005). Figure 8-19 by Schembre and Kovscek (2005) depicts the typical detachment temperature isotherms for Berea sandstone obtained by experimental studies and analytically based on the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory of colloidal stability on the salinity-pH charts (Derjaguin, 1940). The results vary with the particle and pore-surface geometry. Figure 8-20 by Schembre and Kovscek (2005) compares the detachment temperature isotherms for kaolinite–quartz, silica–silica, silica–kaolinite, and calcite– calcite systems, obtained analytically by considering cylindrical shape particles near a plate surface. Schembre and Kovscek (2005) emphasize that “the position of the isoelectric point has an effect on the sensitivity of the system to the fluid conditions and temperature.” Approximating the zeta potential with the following Nernst equation, Schembre and Kovscek (2005) defined the isoelectric point of a surface pHoi at which the zeta potential i (the electrical potential at the Stern and diffuse layer interface boundary) becomes zero.   kT pH − pHoi  (8-75) i ≡ oi = −2 3 e Figure 8-21 shows the correlation of the experimental data of Legens et al. (1999) by Schembre and Kovscek (2005), indicating that the isoelectric point is located at pH = 11 4 for calcite. However, the experimental results and therefore the isoelectric point reported by Legens et al. (1999) and Pierre et al. (1990) are somewhat different. Schembre and Kovscek (2005) conclude that “Materials with an isoelectric point located in the alkaline region demonstrate less temperature sensitivity,” such as calcite. Khilar and Fogler (1983) demonstrated that as temperature increases the colloidal particle release coefficient increases, correlating successfully with an Arrhenius type equation as   E (8-76) kr = A exp − RT

229

Particulate Processes in Porous Media 0.05 0.045 0.04

NaCl [M]

0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

7

7.5

8

8.5

9

9.5

10

pH

(a) 0.05

200

0.045

180

0.04

160

NaCl [M]

0.035

140

0.03 120 0.025 100

0.02 0.015

80

0.01

60

0.005

40

0

(b)

7

7.5

8

8.5

9

9.5

10

pH

Figure 8-19. Detachment temperature isotherms for Berea sandstone: (a) experimental results; and (b) analytical model (after Schembre and Kovscek, 2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

230

Particulate Processes in Porous Media 400

NaCl [M]

0.4

300

0.3 200

0.2 0.1 0

(a)

100 6

8

10

pH

12

14

NaCl [M]

400 0.4

300

0.3 200

0.2 0.1 0

100 6

8

(b)

10

pH

12

14 400

NaCl [M]

0.4

(c)

300

0.3 200

0.2 0.1 0

100 6

8

10

pH

12

14 400

NaCl [M]

0.4

200

0.2 0.1 0

(d)

300

0.3

100 8

10

pH

12

14

Figure 8-20. Detachment temperature obtained for a cylinder-plate geometry: (a) kaolinite–quartz, (b) silica–silica, (c) silica–kaolinite, calcite–calcite (after Schembre and Kovscek, 2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

For example, correlation of their experimental data with Eq. (8-76) yields the best estimate parameter values as A = 1 14 × 106 s−1 and E = 10 7 kcal/mole. Based on Eq. (8-32), Khilar and Fogler (1983) determined that, for all practical purposes, the particle capture coefficient kt does not depend on temperature. Gupta and Civan (1994a) provide some evidence for the temperature effect on permeability impairment as described in the following. As temperature increases, the porosity and permeability decreases (Civan, 2000b; see Chapter 5) and hence the probability of fines migration through porous

231

Particulate Processes in Porous Media 35 Experimental 0.1 M

30

Experimental 0.01 M

Zeta potential, mV

25

Model 0.1 M Model 0.01 M

20

Model 0.001 M

15

Model 0.0001 M

10 5 0 –5 –10

7

8

9

11

10

12

13

14

pH

Figure 8-21. Experimental and calculated zeta potential for calcite as a function of pH at different salt concentrations. Experimental values provided by Legens et al. (1999) (after Schembre and Kovscek, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

media without causing formation damage decreases. The reciprocal Debye length given by Eq. (8-26) decreases with increasing temperature, affecting the electrostatic force. The hydrodynamic drag given by Eq. (8-16) is proportional to the fluid viscosity, which usually decrease significantly with increasing temperature. Frequently, the fluid viscosity has been correlated with the Vogel–Tammann–Fulcher type asymptotic exponential functions (Civan, 2005):  =  exp−E/RT − Tc  !

(8-77)

where Tc is a characteristic temperature. Because the net normal force acting on particles is a function of temperature, the shear resistance of particle at the pore surface depends on temperature according to Eq. (8-17). Civan (2006c) shows that the temperature effect on the power needed for detachment of fine particles from pore walls in porous media decreases with temperature, which can be described using an Arrhenius-type asymptotic exponential function. The factors involving the interface particle transfer, Eqs (8-68) and (8-69), vary with temperature and affect the induction time required for particle movement to interface. For example, the viscosity and therefore the induction time are lower and hence the transfer of particles is faster at higher temperatures. As schematically depicted in Figure 8-22 by Gupta and Civan (1994a),

232

Particulate Processes in Porous Media

Figure 8-22. Interfacial Drag on Formation Fines (after Gupta and Civan, ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers).

the oil–water interface traversing through a pore space applies drag and mobilizes the particles attached to the pore surface and destabilizes the particle bridges formed across the pore throats owing to the interfacial forces. The capillary (interfacial) force is directly proportional to the interfacial tension  between the oil and the aqueous phases, which decreases with increasing temperature, as described by an asymptotic power-law equation (Rowlinson and Widom, 1982)  m T  = 1− o Tc

(8-78)

where the exponent is m = 1 26 o is a scaling value, and Tc is the critical absolute temperature value. Therefore, particle mobilization, bridges particle destabilization, and other relevant phenomena depending on interfacial tension, such as emulsion blocking, may be hindered at high temperatures. There is ample evidence (Gupta and Civan, 1994a) that the wettability of the reservoir formations and particles tend to vary toward more waterwetting conditions as temperature increases. This is supported based on Civan’s (2004) analysis of experimental data on the wettability of typical reservoir rocks that the wettability index can be satisfactorily correlated by an Arrhenius-type equation: WIUSBM = log10



 Edrainage − Eimbibition W drainage − log10 e W imbibition RT

(8-79)

Particulate Processes in Porous Media

233

where W drainage and W imbibition represent the pre-exponent coefficients, e denotes the natural logarithm base (2.71828...), and Edrainage and Eimbibition denote the activation energies associated with the forced-drainage and forced-imbibition processes, respectively. Hence, temperature affects the behavior of particles as depicted in Figure 8-16 by Gupta and Civan (1994a). Gupta and Civan (1994a) caution that particles may become intermediately wet and move along the interface and plug pore throats as transition between oil-wet and water-wet conditions occur during heating and cooling. When clays swell by absorbing water, they become larger and hence more susceptible to hydrodynamic mobilization owing to fluid shear (Ohen and Civan, 1993). Because the clay swelling phenomenon involves many temperature-dependent parameters, such as the water diffusion coefficient and swelling rate coefficient, clay swelling rate depends on temperature in a complicated manner (Gupta and Civan, 1994a).

Exercises 1. Using Einstein’s equation, Eq. (8-6), estimate the Brownian diffusivity of a 1 0 m diameter fine particle in water at 20 C temperature. (Answer: D = 4 3 × 10−9 cm2 /s; McDowell-Boyer et al., 1986) 2. Estimate the Stokes’ motion velocity of a particle using Eq. (8-3) in a fluid. The particle density is 2 5 g/cm3 , the mean spherical volume equivalent diameter is 0.005 mm, and the fluid density and viscosity are 1 0 g/cm3 and 1.0 cp, respectively. 3. Determine the mean spherical surface equivalent diameter of the different clay minerals listed in Table 2-2 by approximating their densities as being 2 5 g/cm3 . 4. Consider a suspension of clay particles in brine. The volume flux of the clay suspension is given as 1 5 × 10−4 mL/cm2 min. The clay particle volume fraction in the clay suspension is 0.10. The porosity of the rock sample is 25%. The particle deposition rate constant is 8 3 × 10−5 cm−1 . The stationary deposition factor value is 5 2 × 10−5 mL/cm2 min. Estimate the rate of deposition of the clay particles over the pore surface. 5. Consider the data given in Figure 8-7. Particle concentration of a drilling mud is 10 lbm /gallon. Answer the following questions: a) What is the critical pore throat-to-mud particle diameter ratio for bridging to occur at the sand face?

234

6.

7.

8.

9.

10.

Particulate Processes in Porous Media

b) Would pore throat bridging occur if the pore throat-to-mud particle diameter ratio had been equal to 4.0? Based on Eq. (8-62), estimate the wall shear stress for a nonNewtonian gel having k′ = 5 0 and n′ = 2 0, and flowing at a 0.05 cm/min interstitial velocity through a 0.0001 cm diameter capillary flow path in porous rock. Derive an analytical solution for the set of ordinary differential equations given by Eqs (8-71)–(8-73) for the special case of A = A = 1, subject to the condition given by Eq. (8-74) similar to Gupta and Civan (1994a). Prepare the plots of these analytical solutions for typical parameter values. Consider a suspension of clay particles in brine. The volume flux of the clay suspension is given as 1 2 × 10−4 ml/cm2 min. The clay particle volume fraction in the clay suspension is 0.15. The porosity of the rock sample is 20%. The particle deposition rate constant is 8 0 × 10−5 cm−1 . The stationary deposition factor value is 5 0 × 10−5 ml/cm2 min. Estimate the rate of deposition of the clay particles over the pore surface. Consider the pore throat plugging criterion given by the exponential equation Eq. (8-42). The parameter values are A = 4 6 B = 0 153, and C = 1 52. Particle concentration of a drilling mud is cp =√0 15 g/liter u = 0 1 cm/min Dp = 25 m  = 0 20  = 1 cp  = 2. Note that 1 m = 10−6 m. Answer the following questions: a) What is the critical pore throat-to-mud particle diameter ratio for bridging to occur at the sand face? b) Would pore throat bridging occur if the pore throat-to-mud particle diameter ratio had been equal to 3.0? Explain the salinity, velocity, and temperature shock phenomena, and their impact on formation damage. Discuss the differences owing to single-phase vs. multiphase pore fluid systems.

C

H

A

P

T

E

R

9

CRYSTAL GROWTH AND SCALE FORMATION IN POROUS MEDIA1

Summary In this chapter, the inorganic and organic precipitation–dissolution phenomena, solid–liquid equilibrium and solubility equation, crystallization kinetics, and their effect on the size of the suspended particles and porosity variation are discussed and formulated.

9.1 INTRODUCTION Civan (1996a) describes that Injection of fluids and chemicals for improved recovery, and liberation of dissolved gases, such as CO2 and light hydrocarbons from the reservoir fluids approaching the wellbore during production, and variation of fluid saturations can alter the temperature, pressure, and composition of the fluids in the near wellbore region and tubing. Consequently, the thermodynamic and chemical balance may change in favor of precipitate separation, aggregation of precipitates, crystal growth, and scale formation. Precipitates can cause formation damage by changing the wettability and permeability of petroleum bearing rock and cause scale formation and clogging in tubing and pore throats. 1 Parts reprinted by permission of the Society of Petroleum Engineers from Civan, ©1996a SPE, SPE 31101 paper.

235

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Crystal Growth and Scale Formation in Porous Media

9.2 TYPES OF PRECIPITATION 9.2.1

Inorganic Precipitation

Typical inorganic precipitates include anhydrate CaCO3 , gypsum CaSO4 ·2H2 O, hemihydrate CaSO4 ·H2 O, barite BaSO4 , celestite SrSO4 , magnesium sulfide MgSO4  originating from mixing seawater with brine, and rock and brine interactions (Oddo and Tomson, 1994; Atkinson and Mecik, 1997); iron hydroxide gel FeOH3  originating from the acid dissolution and precipitation of iron minerals such as pyrhotite (FeS), pyrite FeS2 , hematite Fe2 O3 , magnetite Fe3 O4 , and siderite FeCO3  (Rege and Fogler, 1989); silicium tetrahydroxide gel SiOH4  originating from the alkaline dissolution and precipitation of minerals in shaly sandstones such as quartz and kaolinite (Labrid, 1990); and polymeric substances produced by in situ gelation (Todd et al., 1993), alcohol-induced crystallization (Zhu and Tiab, 1993), separation of elemental sulfur (Roberts, 1997) and surfactant precipitation (Arshad and Harwell, 1985). Following Oddo and Tomson (1994), precipitation/dissolution reactions can be symbolically represented by 1 Me + 2 An ⇄ 3 Pr

(9-1)

where Me represents a cation or metal ion such as Sr +2  Ca+2  Mg+2 ; An −2 represents an anion such as CO−2 3  SO4 , and Pr represents a solid precipitate such as CaCO3  MgCO3  BaSO4  FeOH3  SiOH4  1  2 , and 3 are some stoichiometric coefficients. Oddo and Tomson (1994) correlated the saturation solubility product, Ksp , empirically as a function of temperature, T , pressure, p, and ionic strength, Si , for typical systems. Hence, the saturation ratio given by the following equation can be used to determine whether the conditions are favorable for precipitation (Oddo and Tomson, 1994). Fs =

Me1 An2 Ksp

(9-2)

Fs < 1 indicates an undersaturated solution, condition unfavorable for scaling, if Fs = 1, the solution is at equilibrium with the solid scale, and Fs > 1 indicates a supersaturated solution, condition favorable for scaling. The other relevant issues on inorganic precipitation are discussed in detail in Chapter 13.

Crystal Growth and Scale Formation in Porous Media

9.2.2

237

Organic Precipitation

Typical organic precipitates encountered in petroleum production are paraffins and asphaltenes. Paraffins are inert and asphaltenes are reactive substances. They both are sticky, thick, and deformable precipitates (Chung, 1992; Ring et al., 1994). Therefore, they can seal the pore throats and reduce the permeability to zero without needing to reduce the porosity to zero, and their deposition at the pore surface and tubing wall is irreversible unless a solvent treatment is applied (Leontaritis et al., 1992). The saturation ratio is given by Fs =

xA xA s

(9-3)

where Fs < 1 for undersaturated solution, Fs = 1 for saturated solution, and Fs > 1 for supersaturated solution; xA is the mole fraction of the dissolved organic in oil and xA s is the organic solubility at saturation conditions which is predicted using a thermodynamic model, such as given by Chung (1992). The other relevant issues on organic precipitation are discussed in detail in Chapter 14.

9.3 SOLID–LIQUID EQUILIBRIUM AND SOLUBILITY EQUATION The formulations of the solubility equation require a different treatment for electrolyte and nonelectrolyte (molecular) systems, and are presented in the following sections according to Wibowo et al. (2004) with some modifications. Description of the solid–liquid equilibrium (SLE) by a solubility equation is accomplished by equating the chemical potentials of all the components present in the solid phase to their chemical potentials in liquid phase. However, Wibowo et al. (2004) emphasize that the formulation of the solubility equation are different for nonelectrolytes (molecular systems) than electrolytes. The molecular systems dissolve in solution directly but the electrolytes first dissociate into their ions and then interact in a complicated manner, leading to a nonideal solution behavior.

238

9.3.1

Crystal Growth and Scale Formation in Porous Media

Solubility Equation for Molecular Solutions

The partitioning of component i between the liquid and the solid phases in molecular solutions is represented according to the following reaction: iS ⇄ iL

(9-4)

The equilibrium constant is given by Ki T  =

iL x T  xiL iS xiS

(9-5)

where the superscripts L and S denote the liquid and the solid phases, respectively; is the activity coefficient, calculated by an adequate thermodynamic model; and x is the component mole fraction in a phase. For pure components, such as pure solid phase, = 1 0 and x = 1 0. Then, iS xiS = 1 0 and Eq. (9-5) becomes Ki T  = iL x T  xiL

(9-6)

The solubility equation for nonelectrolyte (molecular) systems expresses the mole fraction of component i in a liquid phase in equilibrium with a solid phase at the saturation conditions according to    1 Hfi 1 (9-7) − Ki T  = exp − R T Tmi where Ki T is given by Eqs (9-5) or (9-6); Hf denotes the heat of fusion; T and Tm denote the absolute solution temperature and the melting point temperature of the solid, respectively; and R is the universal gas constant (8314 J/kmol K). Further developments on this subject are elaborated in Chapter 14 for applications concerning the asphaltene and paraffin in crude oil. 9.3.2

Solubility Equation for Electrolyte Solutions

An electrolyte P dissociates into the ions M and N in a solution (aq) according to the following reaction: PS ⇄ M M zM + aq + N N zN − aq

(9-8)

Crystal Growth and Scale Formation in Porous Media

239

where  denotes a stoichiometric coefficient. Alternatively, Eq. (9-8) can be written as 1 1 1 PS ⇄ M zM + aq + N zN − aq M N N M

(9-9)

The equilibrium constant is given by K T  =

∗  M mM 1/ N  N∗ mN 1/ M  P∗ mP 1/ M N 

(9-10)

where ∗ denotes the standard-state (infinite dilution) activity coefficient ands m denotes the ion concentration expressed as molality, that is kmole component/kg solvent. For pure electrolytes, P∗ = 1 0 and mP = 1 0. Then, Eq. (9-10) simplifies to the solubility product, expressed as ∗ Ksp T  =  M mM 1/ N  N∗ mN 1/ M

(9-11)

Thus, the solubility equation for electrolyte systems expresses the solubility product in a liquid phase in equilibrium with a solid phase at the saturation conditions according to    1 ∗M aq ∗N aq ∗P S K T  = Ksp T  = exp − + − (9-12) RT M N M N where Ksp T is given by Eq. (9-11). ∗ denotes the chemical potential of the ions or the solid electrolyte at unit molality. Further developments on this subject are elaborated in Chapter 14 for applications concerning the asphaltene and paraffin in crude oil.

9.4 CRYSTALLIZATION PHENOMENA Majors (1999) explains that “Crystallization is the arrangement of atoms from a solution into an orderly solid phase” and “Growth is simply the deposition of material at growth sites on an existing crystal face.” The process is called primary nucleation if there are no crystals present in the solution to start with and crystallization is occurring for the first time. Primary nucleation can be homogeneous or heterogeneous (Majors, 1999). Homogeneous nucleation occurs inside the solution without contact with any surface. Heterogeneous nucleation occurs over a solid surface. The process is called secondary nucleation if there are already some

240

Crystal Growth and Scale Formation in Porous Media

Concentration

Primary nucleation line

Metastable region Supersaturated

Saturation line

Undersaturated

Temperature

Figure 9-1. Concentration vs. temperature diagram for crystal formation (after Majors, 1999; reprinted by permission of the Chemical Processing Magazine).

crystals present in the system over which further deposition can occur. The schematic chart given in Figure 9-1 by Majors (1999) describes the concentration–temperature relationship for nucleation. As can be seen, the primary nucleation process requires a sufficiently high concentration of supersaturated solution, whereas secondary nucleation can occur at relatively lower concentrations above the saturation line. The metastable region represents the favorable conditions for crystal growth (Majors, 1999). The schematic chart given in Figure 9-2 by Majors (1999) describes the effect of the supersaturation ratio on the crystal growth and nucleation rates. Crystal growth rate is a low-order function of supersaturation and can be represented by a linear relationship, while nucleation rate is a highorder function of supersaturation and requires a more difficult nonlinear relationship (Majors, 1999). Majors (1999) explains that “Crystal growth is a dynamic process. While most of the crystals in the solution will grow, some may dissolve.”

9.4.1

Grain Nucleation, Growth, and Dissolution

The formation of crystalline particulates from aqueous solutions of salts involves a four-step phase change process (Dunning, 1969):

Growth rate, Nucleation rate

Crystal Growth and Scale Formation in Porous Media

241

Growth

Nucleation

Supersaturation ratio

Figure 9-2. Effect of saturation ration on the crystal growth and nucleation rates (after Majors, 1999; reprinted by permission of the Chemical Processing Magazine).

1. Alteration of chemical and/or physical conditions to lead to supersaturation of the solution 2. Initiation of the first small nuclei of the crystals 3. Crystal growth 4. Relaxation leading to coagulation of crystalline particles. The process is called homogeneous or heterogeneous crystal nucleation depending on the absence or presence, respectively, of some impurities, seed crystals, or contact surfaces, called substrates (Figure 9-3 by Leetaru, 1996). As stated by Schneider (1997), “Nucleation commonly occurs at sites of anomalous point defects on the grain surface, at structural distortions caused by edge or screw dislocations, or at irregular surface features produced by dissolution and etching.” Because, Schneider (1997) adds, “When nucleation occurs at one of these sites, the free energy of the defect, dislocation, or surface irregularity can contribute to help overcome any energy barrier to nucleation.” Also, the mineral grain surfaces serve as seed for nucleation if the mineral crystal lattice structure matches that of the precipitating substance (Schneider, 1997). The free energy change associated with heterogeneous nucleation at a surface is expressed by (Schneider, 1997) G = Gvolume + Gsurface + Gstrain

(9-13)

where Gstrain is the change of the strain volume free energy of shrinking of a nucleus.

242

Crystal Growth and Scale Formation in Porous Media

Figure 9-3. SEM photomicrograph of a calcite cement nucleating at a site in an Aux Vases sandstone sample (after Leetaru, 1996; reprinted by permission of the Illinois State Geological Survey).

Stumm and Morgan (1996) expressed the interface free energy change by Ginterface = Acw cw + Acs  cs − sw 

(9-14)

Thus Eq. (9-13) becomes G = VGv + Acw cw + Acs  cs − sw  + Ve

(9-15)

where cw, cs, and sw denote the deposit–water, deposit–substrate, and substrate–water interfaces, respectively; A denotes the surface area and

denotes the interfacial free energy; cs and cw denote the surface energies per unit surface area of the deposited particle–substrate interface and the deposited particle–solution interface, respectively; e is the strain energy per unit volume; V is the volume of particle formed by precipitation; is the surface energy per unit particle surface; and Gv is the change of volume free energy from solution to solid phases per unit particle volume, given by (Stumm and Morgan, 1996)

Crystal Growth and Scale Formation in Porous Media

Gv = −

kb T ℓn v



a ao



243

(9-16)

where kb is the Boltzmann constant, T is absolute temperature, v is the molar volume, and a and ao are the activity of the mineral dissolved in solution and its theoretical activity at saturation, respectively. Considering a semi-spherical deposition of radius r over a planar substrate surface as an approximation, such that (see Figure 9-3)   1 4 3 (9-17) r V= 2 3 Acw =

1  2 r 2

Acs = r 2

(9-18) (9-19)

By combining the various efforts, Eq. (9-15) can be expressed as (Walton, 1969; Putnis and McConnell, 1980; Richardson and McSween, 1989; Schneider, 1997; Stumm and Morgan, 1996)      1 1 4 3 2 2 Gv + r 4r cw + r  cs − sw  G r = 2 3 2    1 4 3 + r e (9-20) 2 3 The depositing substance and the substrate surface match well when

cs < cw , and cs is negligible and sw = cw for perfect matching (Schneider, 1997). Thus, the critical minimum radius necessary for formation of stable particles can be determined by equating the derivative of Eq. (9-20) with respect to the radius to zero as rc = −

cs + 2 cw − sw Gv + e

(9-21)

Then, substituting Eq. (9-21) into Eq. (9-20) yields the expression for the activation energy necessary for formation of stable particles as Gcr =

  cs + 2 cw − sw 3 3Gv + e 2

(9-22)

244

Crystal Growth and Scale Formation in Porous Media

For homogeneous nucleation, Acs = 0 and Eq. (9-15) simplifies to G = VGv + Acw cw + Ve

(9-23)

Approximating the shape of the deposit by a sphere of radius r such that 4 V =  r3 3

(9-24)

Acw = 4 r 2

(9-25)

4 G r =  r 3 Gv + e  + 4 r 2 cw 3

(9-26)

Eq. (9-23) can be written as

The minimum critical particle radius for a homogeneous nucleus to form a stable deposit at a given super saturation state can be estimated by equating the derivative of Eq. (9-26) to zero: rcr = −

2 cw Gv + e

(9-27)

Thus, the activation energy necessary for starting homogeneous nucleation can be estimated by substituting Eq. (9-27) into Eq. (9-26) as Gcr =

9.4.2

3 16 cw

3Gv + e 2

(9-28)

Crystallization Kinetics

The time necessary to initiate nucleation of crystals from a supersaturated solution is called “induction time” (Reddy, 1995). It is a function of the solution supersaturation, that is, the ratio of the ion activity product to the solubility product of the precipitating crystalline matter as demonstrated in Figure 9-4 by Reddy (1995) for calcium carbonate nucleation in the presence of magnesium ions. Figure 9-4 indicates that the induction time is lower for higher supersaturation. Below the supersaturation value of about 10, the induction time for calcium carbonate nucleation is very long.

245

Crystal Growth and Scale Formation in Porous Media 9000 8000

Mg2+/Ca2+

7000

0.0

tI (sec)

6000

0.1

5000

0.5

4000

1.0

3000

2.0

2000 1000 0 10

15

20

25

30

35

40

45

30000

Mg2+/Ca2+

25000

0.0

tI (sec)

20000

0.1 0.5

15000 1.0 10000

2.0 5.0

5000

0 5

10

15

20

25

30

35

40

45

Ωc (IAP/Ksp)

Figure 9-4. Calcium carbonate induction time vs. the supersaturation for solutions containing magnesium-to-calcium ion concentrations ratios of (a) 0 to 2, and (b) 0 to 5 (after Reddy, ©1995; reprinted by permission of the author and Plenum Press).

In this region, the solution is at a “metastable” condition and, therefore, calcium carbonate crystals cannot be formed without the aid of a matching growth surface or substrate (Reddy, 1995). It can also be observed that the presence of magnesium ions increases the induction time for calcium carbonate nucleation and therefore has a retardation and/or inhibition

246

Crystal Growth and Scale Formation in Porous Media

effect. Reddy (1995) explains the magnesium ion inhibition of calcium carbonate nucleation by adsorption of the magnesium ions and, thus, the occupation of some crystal growth sites on the calcium carbonate crystal surface. For a quantitative interpretation of this phenomenon, Reddy (1986, 1995) resorts to a growth rate analysis and a Langmuir adsorption isotherm model using experimental data obtained by a seeded growth method. Reddy expressed the crystal growth rate as being proportional to the surface available for crystal growth and the square of the driving force for precipitation: dN = ksN 2 dt

(9-29)

where N represents the calcium carbonate crystal concentration in the solution in mol/L, t denotes the time measured from the time of initiation of the crystallization by seeding, s is the concentration of the seed added to provide the surface area for growth in mg/liter, and k is the crystal growthrate constant. If No denotes the initial theoretical crystal concentration that would be produced by precipitation from a stable supersaturated solution at the time of seeding, the integration of Eq. (9-29) yields (Reddy, 1986) −N −1 − No−1  = ks t

(9-30)

The plot of the calcium carbonate growth data given by Reddy (1995) in Figure 9-5 confirms the validity of Eq. (9-30) and indicates that the presence of magnesium ions reduces the slopes of the straight lines and thus the crystallization rate constant and inhibits the calcite formation. Figure 9-6 by Reddy (1995) shows a rapid decline of the crystallization rate constant by the increasing magnesium ions’ presence. The plot of data according to the Langmuir model −1  ko = 1 + kd ka TMg2+ ko − k

(9-31)

given in Figure 9-7 by Reddy (1995) clearly indicates that the mechanism of the inhibition of the calcite crystal growth is the magnesium ion adsorption on the growth sites, where ka and kd denote the rate constants for adsorption and desorption of the magnesium ions at the growth sites, ko and k are the crystallization growth-rate constants without and with the presence of magnesium ions, respectively, and TMg2+ is the total concentration of the magnesium ions present in the system.

Crystal Growth and Scale Formation in Porous Media

247

Figure 9-5. Effect of the magnesium ions on the calcite crystal growth rate for different magnesium ion concentrations indicated at the end of each line (after Reddy, ©1995; reprinted by permission of the author and Plenum Press).

Figure 9-6. Effect of the magnesium ion concentration on the calcite precipitation rate constant (after Reddy, ©1995; reprinted by permission of the author and Plenum Press).

9.4.3 Prediction and Correlation of the Nucleation Time The classical nucleation theory predicts the nucleation time tI in seconds by the following equation (Söhnel and Mullin, 1988): log10 tI = a +

4 3 2 3 ′ 2 Vm/ NA/ 2

2 303 4 RT  SI

+

 3 Vm2 NA 2 303 4 2 RT 3 SI2

(9-32)

248

Crystal Growth and Scale Formation in Porous Media

Figure 9-7. Langmuir isotherm plotted against the reciprocal added magnesium ion concentration for calcite precipitation (after Reddy, ©1995; reprinted by permission of the author and Plenum Press).

where

a = 0 25 −0 32 + log10



5 3 Vm/ 8 3 1 2 NA/ D4 Ksp/



+ log10



Fs Fs − 1 02

 (9-33)

where D is the diffusion coefficient, NA is the Avogadro number. SI is the saturation index. Vm is the molar volume of the crystal. The symbols  and ′ are some geometric shape factors (dimensionless). The symbol denotes specific surface energy. Symbol T denotes the absolute temperature. The number of ions in a molecule is indicated by . Fs represents the supersaturation of the solution defined by Eq. (9-2). Ksp is the solubility product. For example, applying Eq. (9-32), Xiao et al. (2001) correlated the barite induction time in seconds as 1087 − 0 30T 2 1087 − 0 30T 3 − 0 12 log10 tI = −2 24 + T 2 SI T 3 SI2

(9-34)

Cosolvents used as inhibitors, such as methanol (MeOH) and monoethylene glycol (MEG), can prevent the hydrate formation but induce the formation of scales, such as barite, because the solubility of such mineral salts is low in the presence of these solvents. Let xMeOH and xMEG denote the mole fractions of methanol and monoethylene glycol, respectively. Tomson et al. (2005) modified Eq. (9-34) as following,

Crystal Growth and Scale Formation in Porous Media

249

respectively, by correlating the experimental data for the effect of MeOH and MEG on the barite induction time in seconds:  2 2 1087 − 0 30T + 1695 2xMeOH − 7564 2xMeOH log10 tI = − 2 24 + T 2 SI  3 2 1087 − 0 30T + 1695 2 xMeOH − 7564 2 xMeOH − 0 12 T 3 SI2 (9-35) and  2 2 1087 − 0 30 T + 1397 5xMEG − 6350 3 xMEG log10 tI = − 2 24 + T 2 SI  3 2 1087 − 0 30 T + 1397 5 xMEG − 6350 3xMEG (9-36) − 0 12 T 3 SI2

9.5 PARTICLE GROWTH AND DISSOLUTION IN SOLUTION The particle growth is assumed to occur at a rate proportional to the surface, Ac , available for growth and the deviation of the saturation ratio from unity (Chang and Civan, 1992; Civan, 1996a): dmc = kc Ac Fs − 1 dt

(9-37)

for which the initial amount of crystals present per unit bulk medium is given by mc = moc  t = 0

(9-38)

Relating the crystal shape to spherical shape, the mass and surface area of the crystalline particle is given, respectively, by m c = c



C1 Dc3 6



(9-39)

250

Crystal Growth and Scale Formation in Porous Media

and Ac = C2 Dc2

(9-40)

in which C1 and C2 are the shape factors, c is the density, and Dc is the diameter. kc is a crystallization rate constant. Thus, Eqs (9-37)–(9-40) lead to the following model in which the shape factors and the constant (1/2) have been incorporated into the constant kc′ : a dDc = kc′ Fs − 1 dt

(9-41)

Dc = Dco  t = 0

(9-42)

c subject to

The saturation ratio is given by ⎧ ⎪ ⎨> 1 for crystal growth Fs = 1 at equilibrium ⎪ ⎩ < 1 for crystal dissolution

(9-43)

For constant saturation, Eqs (9-41) and (9-42) yield Dc = Dco + kt

(9-44)

The crystal size increases linearly with time. For example, Dco = 5 0 m, and k = kc′ Fs − 1 0/c is equal to 1.4 and 10 3 m/s for calcium carbonate crystal growth at 25 and 50 C, respectively, using the Dawe and Zhang (1997) data. Hence, the crystallization rate is higher at the high temperature.

9.6 SCALE FORMATION AND DISSOLUTION AT THE PORE SURFACE The effects of various conditions on dissolution rates, including lithologic variation, hydrodynamics, ionic strength, saturation state, mixed-kinetic control, and surface treatment, have been investigated by Raines and Dewers (1997a,b), Hajash Jr et al. (1998), and Merino and Dewers (1998).

Crystal Growth and Scale Formation in Porous Media

251

The rate of scale formation at the pore surface can be expressed similarly to crystal growth. The pore volume relates to the diameter of an equivalent spherical shape pore space by (Civan, 1996a)  = C3 D 3

(9-45)

indicating that D=



 C3

1 / 3

(9-46)

Thus, the pore surface relates to the porosity according to 2

A = C 4 D = C4



 C3

2 / 3

= C5 2/ 3

(9-47)

C3 and C4 are some shape factors and C5 = C4 /C32/3 . The initial porosity of the solid porous matrix o can be expressed as a sum of the instantaneous porosity, , and the pore space occupied by the scales, s , as o =  + s

(9-48)

Thus, the mass of the scale formed over the pore surface is given by mc = s s = s o − 

(9-49)

Therefore, substituting Eqs (9-47) and (9-49) into Eq. (9-37) yields (Civan, 1996a) −s

d = kc′ 2/ 3 Fs − 1 dt

(9-50)

subject to  = o  t = 0 The saturation ratio is given by ⎧ ⎪ ⎨> 1 for scale formation Fs = 1 at equilibrium ⎪ ⎩ < 1 for scale dissolution

(9-51)

(9-52)

252

Crystal Growth and Scale Formation in Porous Media

and kc′ is a scale formation rate constant incorporating the abovementioned shape factors and some constants. The minus sign in Eq. (9-50) is for the reduction of porosity by scale formation at the pore surface. Thus, assuming the rock porosity, o , remains constant and substituting Eq. (9-48) into (9-50) leads to an equation similar to Ortoleva et al. (1987): s

ds = kc′ o − s 2/ 3 Fs − 1 dt

(9-53)

subject to s = 0

t=0

(9-54)

Assume that the pore surface area available for crystal growth can be expressed empirically by Ac = mc



(9-55)



 in which  is the specific surface of the mineral–fluid contact area (surface area per unit mineral mass) expressed as a function of porosity . Thus, substituting Eq. (9-55) into Eq. (9-37) leads to Holstad’s (1995) equation:  d mc = kc mc  Fs − 1 dt

(9-56)

Holstad (1995) expressed the temperature dependency of the crystallization rate constant by the Arrhenius (1889) equation:   E kc = FM AM exp − M RT

(9-57)

where FM  AM , and EM denote an empirical mineral property factor, an Arrhenius pre-factor, and the activation energy. Liu et al. (1997) used a similar equation kc =

kco exp



E − M RT



(9-58)

where kco is the high-temperature T →  limit of the rate constant.

Crystal Growth and Scale Formation in Porous Media

253

9.7 CRYSTAL SURFACE PITTING AND DISPLACEMENT BY DISSOLUTION Civan (2006d) presents a comprehensive review of the various aspects of crystal dissolution and etching. Briefly, crystal dissolution occurs by surface retreat following etch-pit formation (Lasaga and Luttge, 2001, 2003; Duckworth and Martin, 2004). The dissolution and etching of a crystalline matter in contact with an undersaturated solution can be studied by measuring the progress of the crystal surface as a function of time. Hunkeler and Bohni (1981) and Dunn et al. (1999) used this technique. Civan (2000a, 2002c) determined that the position of the progressing etch-pit depth from the initial crystal surface could be correlated by  xt − xo = kM ln xt − x 

(9-59)

for which x xo , and xt are the instantaneous, initial, and final surface positions, respectively, k is a rate constant, and M is the amount of solute precipitated or dissolved, given by √ 2 M = √ c1 − c0  Dt 

(9-60)

where t is time, co and c1 are the solute concentrations of the solution at the beginning and equilibrium, respectively, and D is the diffusion coefficient of the solute. Civan (2000a) verified this model using the Dunn et al. (1999) measurements of the pit depth during barite dissolution. The etch pits can develop into various shapes at the mineral surface depending on the mineral types (Duckworth and Martin, 2004; Lasaga and Luttge, 2003; Luttge et al., 2003). Civan (2006d) described the etch-pit dissolution rate adequately by the empirical power-law function, given as  dVpit dt =kAmpit

(9-61)

where Apit and Vpit denote the surface area and volume of an average etch pit, respectively, t is time, and k and m are the etching-rate coefficient and reaction order, respectively. Stocker et al. (1998) demonstrated that the temperature dependence of the etching rate obeys the Arrhenius (1889)

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Crystal Growth and Scale Formation in Porous Media

equation. Zhao et al. (1996) correlated the number of etch pits formed over a crystal surface by   (9-62) nt = neq 1 − e−t

where neq denotes the number of pits formed at equilibrium conditions, t is time, and  is a rate coefficient. Zhao et al. (1996) considered that the area containing a pit decreases as the number of etch pits increases over a crystal surface. This area is proportional to the square of the distance l between the pits. Therefore, Zhao et al. (1996) assumed that 1 n t ∝ √

l

(9-63)

Thus, substituting Eq. (9-63) into Eq. (9-62) yields the following expression:  leq = 1 − e−t (9-64) lt Equation (9-64) is different than that of Zhao et al. (1996) because they applied a number of practical considerations for convenient correlation of their experimental data.

Exercises 1. Consider the calcium carbonate precipitation reaction given by Ca2+ + HCO−3 ⇄ CaCO3s + CO2g + H2 O

(9-65)

a) Present an expression for the chemical equilibrium constant for this reaction. b) Obtain the values of the chemical equilibrium constant at 1 atm and 50 C conditions from a data reference book. c) Determine the saturation ration value for a solution containing 0.1 M (mol/L) Ca2+ and 0.5 M HCO−3 ions at 1 atm and 50 C conditions. What can be said as to the saturation condition of this solution? 2. Based on the data given in Figure 9-4, estimate the calcium carbonate induction time for a solution containing a ratio of 2.0 magnesiumto-calcium ions concentrations and a Fs = 10 0 saturation ration.

Crystal Growth and Scale Formation in Porous Media

255

3. The values of the rate constants of the calcium carbonate crystal growth at 25 and 50 C temperature conditions are 1.4 and 10 3 m/s, respectively. Determine the values of the activation energy and the high-temperature limit of the rate coefficient. 4. Assuming a set of representative values for s  kc′  o , and Fs a) Derive an analytical solution for Eqs (9-50) and (9-51). b) Prepare a plot of porosity variation by scale deposition as a function of time.

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PART III Formation Damage by Particulate Processes – Fines Mobilization, Migration, and Deposition

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SINGLE–PHASE FORMATION DAMAGE BY FINES MIGRATION AND CLAY SWELLING1

Summary A review of the primary considerations and formulations of the various single-phase fluid models for formation damage by fines migration and clay swelling effects is presented. The applicability and parameters of these models are discussed.

10.1 INTRODUCTION The majority of the formation damage models were developed for singlephase fluid systems. This assumption is valid only for very specific cases such as the production of particles with single-phase flow of either oil or water, and for special core tests. Nevertheless, it is instructive to understand these models before looking into the multiphase effects. Therefore, the various processes involving single-phase formation damage are discussed and the selected models available are presented along with some modifications and critical evaluation as to their practical applicability and limitations. The methodology for determination of the model parameters 1

Parts reprinted by permission of the Society of Petroleum Engineers from Civan, ©1992 SPE, SPE 23787 paper, and by permission of the U.S. Department of Energy from Civan, 1994a.

259

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Single-Phase Formation Damage by Fines Migration and Clay Swelling

is presented. The parameters that can be measured directly are identified. The rest of the parameters are determined by means of a history matching technique. The applications of the models and the parameter-estimation method are demonstrated using several examples. An evaluation and comparison of the outstanding selected models bearing direct relevance to formation damage prediction for petroleum reservoirs are carried out. The modeling approaches and assumptions are identified, interpreted, and compared. These models are applicable for special cases involving single-phase fluid systems in laboratory core tests. However, they can be readily expressed in radial coordinates and multidirectional Cartesian coordinates for field applications. Porous media is considered in two parts: (1) the flowing phase, denoted by the subscript f, consists of a suspension of fine particles flowing through with the fluid and (2) the stationary phase, denoted by the subscript s, consists of the porous matrix and the particles retrained.

10.2 ALGEBRAIC CORE IMPAIRMENT MODEL Wojtanowicz et al. (1987, 1988) considered a core length average representation of the preferential fluid path through a porous material using a bundle of capillary tubes realization. They analyzed the various formation damage mechanisms assuming that one distinct mechanism dominates at a time under a certain condition. 10.2.1

Bundle of Capillary Tubes Model

Porous medium is visualized as having tortuous pathways represented by Nh tubes of the same mean hydraulic equivalent diameter, Dh , located between the inlet and the outlet ports of the core as depicted in Figure 10-1. The cross-sectional area of the core is A and the length is L. The tortuosity factor for the tubes is defined as the ratio of the actual tube length to the length of the core. =

Lh L

(10-1)

The cross-sectional area of the hydraulic tubes is approximated by Ah = C1 Dh2

(10-2)

261

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Figure 10-1. Bundle of hydraulic tubes realization of flow paths in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy and after Civan 1992 SPE; reprinted by permission of the Society of Petroleum Engineers).

in which C1 is an empirical shape factor that incorporates the effect of deviation of the actual perimeter from a circular perimeter. 10.2.2

Permeability Impairment Model

As a suspension of fine particles flows through the porous media, tubes having narrow constrictions are plugged and put out of service. If the number of nonplugged tubes at any given time is denoted by Nnp and the plugged tubes by Np , then the total number of tubes is given by Nh = Np + Nnp

(10-3)

The area open for flow is given by Af = Nnp Ah

(10-4)

The Darcy and Hagen–Poiseuille equations given, respectively, by qL KA

(10-5)

32qLh Af Dh2

(10-6)

p = and p =

are considered as two alternative forms of the porous media momentum equations. q is the flow rate of the flowing phase and p is the pressure

262

Single-Phase Formation Damage by Fines Migration and Clay Swelling

differential across the thin core slice. Thus, equating Eqs (10-5) and (10-6) and using Eqs (10-1) and (10-2) the relationship between permeability, K, and open flow area, Af , is obtained as K=

Af Ah C22

(10-7)

in which the new constant is defined by C2 2 ≡ 32AC1 10.2.3

(10-8)

Damage Mechanisms

The permeability damage in porous media is assumed to occur by three basic mechanisms (1) gradual pore reduction (pore narrowing, pore lining) by surface deposition, (2) single pore blocking by screening (pore throat plugging), and (3) pore volume filling by straining (internal filter cake formation by the snowball effect). 10.2.3.1 Gradual pore size reduction by surface deposition (interception or diffusion)

Gradual pore reduction is assumed to occur by deposition of particles smaller than pore throats on the pore surface to reduce the cross-sectional area, A, of the flow tubes gradually as depicted in Figure 10-2. Thus, the number of tubes open for flow, Nnp , at any time remains the same as the total number of tubes, Nh , available. Hence, Nh = Nnp 

Np = 0

(10-9)

Then, using Eq. (10-9) and eliminating Ah between Eqs (10-4) and(10-7) leads to the following equation for the permeability to open-flow area relationship during the surface deposition of particles: Af = C3 K 1/2

(10-10)

in which the new constant is defined by C3 = C2 Nh1/2

(10-11)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

263

Figure 10-2. Pore surface deposition in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy and after Civan 1992 SPE; reprinted by permission of the Society of Petroleum Engineers).

10.2.3.2

Elimination of flow tubes by single pore blocking

Single pore blocking is assumed to occur by elimination of flow tubes from service by plugging of a pore throat or constriction, that may exist somewhere along the tube, by a single particle to interrupt the flow through that particular tube. Therefore, the cross-sectional areas of the individual tubes, Ah , do not change. But, the number of tubes, Nnp , open for the flow is reduced as depicted in Figure 10-3. The area of the tubes eliminated from service is given by Ap = N p Ah

(10-12)

Figure 10-3. Pore throat plugging in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy and after Civan 1992 SPE; reprinted by permission of the Society of Petroleum Engineers).

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Single-Phase Formation Damage by Fines Migration and Clay Swelling

The number of tubes plugged is estimated by the ratio of the total volume of pore throat blocking particles to the volume of a single particle of the critical size.

Np =

t qfcr cp dt/p o

D3 /6

(10-13)

The critical particle size, Dcr , is defined as being comparable to the average size of the critical pore constrictions in the core. fcr is the volume fraction of particles in the flowing phase, having sizes comparable or greater than Dcr . p is the particle grain density. cp is the mass concentration of particles in the flowing suspension of particles. Because Ah is a constant, Eq. (10-7) leads to the following permeability to open-flow area relationship: A f = C4 K

(10-14)

in which the new constant is given by C4 =

C2 2 Ah

(10-15)

10.2.3.3 Cake formation (straining or size exclusion and/or pore plugging and filling)

External cake formation occurs near the inlet face of the core when a suspension of high concentration of particles in sizes larger than the size of the pore throats is injected into the core as depicted in Figure 10-4. Internal cake formation occurs when smaller than pore throat size particles at sufficiently high concentrations approach the pore throats (Gruesbeck and Collins, 1982a; Iscan and Civan, 2006). The permeability, Kc , of the particle-invaded region decreases by accumulation of particles. But, in the uninvaded core region near the outlet, the permeability of the matrix, Km , remains unchanged. The harmonic mean permeability, K, of a core section (neglecting the external cake which may have formed at the inlet face) can be expressed in terms of the permeability, Kc , of the Lc long pore-filling region and the permeability, Km , of the Lm long uninvaded region as L L Lc + m = K t Kc t Km

(10-16)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

265

Figure 10-4. Filter cake formation in a core including the combined effects of external and internal filter cakes (after Civan, 1994; reprinted by permission of the U.S. Department of Energy and after Civan 1992 SPE; reprinted by permission of the Society of Petroleum Engineers).

which can be written as K t =

L Lc Rc t + Lm Rm

(10-17)

where Rc t and Rm are the resistances of the pore-filling and uninvaded regions defined by Rc t = Rm =

1 Kc t

(10-18)

1 Km

(10-19)

The rate of increase of the filtration resistance of the pore-filling particles is assumed proportional to the particle mass flux of the flowing phase according to    dRc q kc c (10-20) = dt Lc A p where kc is the pore-filling particles resistance rate coefficient, subject to the initial condition R c = Rm 

t=0

(10-21)

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Single-Phase Formation Damage by Fines Migration and Clay Swelling

Then, solving Eqs (10-20) and (10-21) yields t  k  q  c Rc t = Rm + c dt Lc A p

(10-22)

0

10.2.4 Coupling Formation Damage and Flow through Capillary Tubes The instantaneous porosity of a given cross-sectional area is given by

= o − p

(10-23)

where o and denote the initial and the instantaneous porosity values, respectively, p is the fractional bulk volume of porous media occupied by the deposited particles, given by p =

mp p

(10-24)

where mp is the mass of particles retained per unit volume of porous media and p is the particle grain density. For convenience, these quantities can be expressed in terms of initial and instantaneous open-flow areas, Afo and Af , and the cross-sectional area of the flow tubes covered by the particle deposits, Ap , as

=

Af A

(10-25)

o =

Afo A

(10-26)

p =

Ap A

(10-27)

Substituting Eqs (10-25)–(10-27), Eqs (10-23) and (10-24) become, respectively, Afo = Af + Ap Ap =

Amp p

(10-28) (10-29)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

267

The particle mass balance for a thin core slice is given by



d  AL cp + mp + q cp out − q cp in = 0 dt

subject to the initial conditions:



cp + mp = cp + mp o 

t=0

(10-30)

(10-31)

where cp in and cp out are the particle mass concentrations in the flowing phase at the inlet and the outlet of the core, respectively. Wojtanowicz et al. (1987, 1988) omitted the accumulation of particles in the thin core slice and simplified Eq. (10-30) to express the concentration of particles leaving a thin section by



AL dmp cp out = cp in − q dt

(10-32)

The rate of particle retention (deposition) on the pore surface is assumed proportional to the particle mass concentrations in the flowing phase according to   dmp = kd cp (10-33) rd ≡ dt d where kd denotes the rate coefficient for pore surface deposition of particles. The rate of entrainment (mobilization, pore surface sweeping) of the surface-deposited particles by the flowing phase is assumed proportional to the mass of particles available on the pore surface according to   dmp re ≡ = k e mp (10-34) dt e where ke denotes the rate coefficient for entrainment of particles from the pore surface. Then, the net rate of deposition is given as the difference between the retention and the entrainment rates as dmp = kd cp − ke mp dt subject to the initial condition given by

mp = mp o  t = 0

(10-35)

(10-36)

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Single-Phase Formation Damage by Fines Migration and Clay Swelling

10.2.5

Diagnostic Equations for Typical Cases

Wojtanowicz et al. (1987, 1988) have analyzed the typical formation damage scenarios and developed the diagnostic equations for two special cases of practical importance: (1) Deposition of the externally introduced particles during the injection of a suspension of particles and (2) Mobilization and subsequent deposition of the indigenous particles of porous medium during the injection of a particle-free solution. 10.2.5.1

Deposition of externally introduced particles

Three distinct permeability damage mechanisms are analyzed for a given injection fluid rate and particle concentration. As depicted in Figure 10-4, particles are retained mainly in the thin core section near the inlet face. In this region the concentration of the flowing phase is assumed the same as the injected fluid (i.e., cp  cp in . Case 1: Gradual Pore Reduction

Gradual pore reduction by surface deposition occurs when the particles of the injected suspension are smaller than the pore constrictions. Assume that the surface deposition is the dominant mechanism compared to the entrainment, that is, kd ≫ ke . Then, the solution of Eqs (10-35) and (10-36) yields (10-37)



mp = kd cp in t

Substitution of Eq. (10-37) into Eq. (10-29) leads to the following expression for the area occupied by the surface deposits: Ap =



Akd cp in p

t

(10-38)

Substitution of Eqs (10-10) and (10-38) into Eq. (10-28) yields the following diagnostic equation: 

K Ko

1 / 2

= 1 − C5 t

(10-39)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

in which the empirical constant is given by

Akd cp in C5 = 1 2 C Ko /  3

269

(10-40)

p

Case 2: Single Pore Blocking

Single pore blocking occurs when the size of the particles present in the injected fluid are comparable or bigger than the size of the pore constrictions. Substitution of Eqs (10-12), (10-13), and (10-14) into Eq. (10-28) yields the following diagnostic equation: K = 1 − C6 t Ko in which the empirical constant is given by

6Ah qfcr cp in C6 = C4 Ko p d3

(10-41)

(10-42)

Case 3: Cake Formation

Cake formation near the inlet face of the porous media occurs when the particles in the injected solution are larger relative to the pore size or small in size but at a sufficiently high concentration. Combining Eqs (10-22) and (10-17) yields the following diagnostic equation: Ko = 1 + C7 t K

(10-43)



kc q cp in

(10-44)

in which C7 =

ALRm

10.2.5.2 Mobilization and subsequent deposition of indigenous particles

This case deals with the injection of a clear (particle-free) solution into a porous media. A core is visualized as having two sections designated

270

Single-Phase Formation Damage by Fines Migration and Clay Swelling

as the inlet and outlet sides. The particles of the porous media entrained by the flowing phase in the inlet part are recaptured and deposited at the outlet side of the core. Near the inlet port, the mobilization and entrainment of particles by the flowing phase is assumed to be the dominant mechanism compared to the particles retention (i.e., ke ≫ kd . Thus, dropping the particle retention term, Eqs (10-35) and (10-36) yield the following solution for the mass of particles remaining on the pore surface: mp = mpo exp −ke t

(10-45)

Substituting Eq. (10-45) and cp in = 0, Eq. (10-32) yields the following expression for the particle concentration of the flowing phase passing from the inlet to the outlet side of the core:   ke ALmpo cp = exp −ke t (10-46) q Depending on the particle concentration and size of the flowing phase entering the core, the outlet side diagnostic equations for three permeability damage mechanisms mentioned previously are derived next. Case 1: Gradual Pore Reduction by Surface Deposition and Sweeping

Assume that the mass of the indigenous or previously deposited particles on the pore surface is m∗p . Then, the cross-sectional area of flow tubes occupied by these particles is given by Eq. (10-29) as A∗p =

Am∗p p

(10-47)

and the area open for flow is given by Eq. (10-28) as Afo = A∗f + A∗p

(10-48)

where Afo denotes the open-flow area when all the deposits are removed. If simultaneous, gradual pore surface deposition and sweeping are occurring near the outlet region, then both the entrainment and the retention terms are considered equally important. Thus, substituting

Single-Phase Formation Damage by Fines Migration and Clay Swelling

271

Eq. (10-46), Eq. (10-35) yields the following ordinary differential equation:   kd ke ALmpo dmp exp −ke t (10-49) + k e mp = dt q The solution of Eq. (10-49), subject to the initial condition mp = m∗p (previously deposited particles), is obtained by the integration factor method as

mp = m∗p + C11 t exp −ke t (10-50) in which

kd ke ALmpo q

C11 =

(10-51)

Then, the area occupied by the remaining particles is given by Eq. (10-29) as   Am∗p C11 Ap = 1+ t exp −ke t (10-52) p m∗p and the area open for flow is given by Eq. (10-28) as Afo = Af + Ap

(10-53)

Eliminating Afo between Eqs (10-48) and (10-53), substituting Eqs (10-47) and (10-52) for A∗p and Ap , and then applying Eq. (10-10) for Af and A∗f yields the following diagnostic equation: 

K K∗

1/ 2

= 1 + C12 − C12 + C8 t exp −ke t

(10-54)

in which C12 =

Am∗p C3 K ∗1/2 p

(10-55)

and C8 =

kd ke A2 Lmpo q C3 K ∗1/2 p

(10-56)

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Single-Phase Formation Damage by Fines Migration and Clay Swelling

Normally, mpo = m∗p . The time required to reach the minimum permeability when the deposition and erosion rates are equal is given by setting the derivative of Eq. (10-54) equal to zero as tK min =

1 C12 − ke C8

(10-57)

Wojtanowicz et al. (1987) simplified Eq. (10-54) by substituting C12 = 0 when the mass of the particles initially available on the pore surface is small compared to the mass of particles deposited later (i.e., m∗p  0 as follows. 

K Ko

1/ 2

 1 − C8 t exp −ke t

(10-58)

If only pore sweeping occurs, then kd ≪ ke . Thus, substitute kd = 0 in Eq. (10-56) to obtain C8 = 0 and Eq. (10-54) becomes 

K K∗

1 / 2

= 1 + C12 1 − exp −ke t

(10-59)

If only gradual surface deposition is taking place in the outlet region, then kd ≫ ke . Therefore, dropping the particle retention term and substituting Eq. (10-46), Eqs (10-35) and (10-36) for mpo = 0 are solved to obtain the amount of particles retained as mp =



kd ALmpo q



1 − exp −ke t

(10-60)

Then, substituting Eqs (10-60), (10-29), and (10-10) into Eq. (10-28) yields the following diagnostic equation: 

K Ko

1/ 2

= 1 − C9 1 − exp −ke t

(10-61)

in which C9 =

kd A2 Lmpo 1 2 C Ko / q  3

p

(10-62)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

273

Case 2: Single Pore Blocking

If the permeability damage is solely due to single pore blocking, then substituting Eqs (10-46), (10-12), (10-13), and (10-14) into Eq. (10-28) yields the following diagnostic equation: K = 1 − C7 1 − exp −ke t Ko

(10-63)

in which C7 =

6fcr ALmpo D3 p C4

(10-64)

Case 3: Cake Formation (Pore Throat Plugging Followed by Pore Filling)

If the permeability damage is by cake formation, then substituting Eqs (10-46) and (10-22) into Eq. (10-17) yields the following diagnostic equation Ko = 1 + C10 1 − exp −ke t K

(10-65)

in which C10 =

k c mp o Rm

(10-66)

A list of the diagnostic equations derived in this section are summarized in Table 10-1 for convenience.

10.3 ORDINARY DIFFERENTIAL COMPARTMENTS-IN-SERIES CORE IMPAIRMENT MODEL Khilar and Fogler (1987) divided a core into n-compartments as depicted in Figure 10-5. The contents of these compartments are assumed well-mixed. Therefore, the composition of the flow stream leaving the compartments should be the same as the contents of the compartments.

Damage Mechanism

Particulate Suspension

Pore surface deposition Pore throat plugging (screening) Cake Forming (straining, pore filling)

Particle-free solution

Diagnostic Equation  K Ko 1/2 = 1 − C5 t

 K Ko = 1 − C6 t

 Ko K = 1 + C7 t

Straight-Line Plotting Scheme  K Ko 1/2 vs t

T1–2

 Ko K vs t

T1–3 T1–4a

 K Ko 1/2 = 1+C12 −C12 + C8 t e−ke t

Least squares fit

Simplified pore surface deposition and sweeping for negligible initial particle content

 K Ko 1/2  1 − C8 t e−ke t

ln

Pore surface sweeping

 K Ko 1/2 = 1 + C12 1 − e−ke t

 K Ko 1/2 = 1 − C9 1 − e−ke t

 K Ko = 1 − C13 1 − e−ke t

     ln 1 − K Ko 1/2 − 1 C12 vs t      ln 1 + K Ko 1/2 − 1 C9 vs t      ln 1 + K Ko − 1 C13 vs t

Pore surface deposition

Cake Forming (pore filling)

 Ko K = 1 + C10 1 − e−ke t

T1–1

K Ko vs t 

Simultaneous pore surface deposition and sweeping

Pore throat blocking (screening)

Equation Number

    1 − K Ko 1/2 t vs t

  ln1 − Ko K − 1 C10  vs t

T1–4b (Eq. T1–4a for C12 = 0) T1–5 T1–6 T1–7 T1–8

After Wojtanowicz et al. 1987, 1988; Civan, ©1992 SPE, reprinted by permission of the Society of Petroleum Engineers, and Civan, 1994a; reprinted by permission of the U.S. Department of Energy.

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Injection Fluid

274

Table 10-1 Diagnostic Equations for Typical Permeability Damage Mechanisms

Single-Phase Formation Damage by Fines Migration and Clay Swelling

275

Figure 10-5. Continuously stirred compartments in series realization in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy and after Civan, ©1992 SPE; reprinted by permission of the Society of Petroleum Engineers).

However, because particles having sizes comparable or larger than the pore throats are trapped within the porous media, the particle concentration of the stream leaving a compartment will be a fraction, , of the concentration of the fluid in the compartment.  is called the particle transport efficiency factor. Pore surfaces are considered as the source of in situ mobilized particles and the pore throats are assumed the locations of particle capture. A particle mass balance over a thin slice yields





d  AL cp + mp + m∗p j + q cp out − q cp in = 0 dt

(10-67)

where mp and m∗p denote the mass of particles captured at the pore throats and the indigeneous particles remaining on the pore surfaces, respectively. In Eq. (10-67)



cp in = cp j−1 (10-68) (10-69)





cp out = cp j

Substituting Eqs (10-68) and (10-69) and rearranging, Eq. (10-67) becomes



d cp j



d mpj ALj = q  cp j−1 − q  cp j − ALj (10-70) dt dt subject to the initial condition given by



cp j = o cpo  j = 1 2     n

t=0

(10-71)

and the inlet boundary condition given by cpj = cpo 

j = 0

t>0

(10-72)

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Single-Phase Formation Damage by Fines Migration and Clay Swelling

The mass balance of particles captured at the pore throats is given by dmp = rt dt

(10-73)

mp = 0 t = 0

(10-74)

subject to

The mass balance of indigeneous particles remaining on pore surfaces is dm∗p dt

= −re

(10-75)

subject to m∗p = m∗po  t = 0

(10-76)

where m∗po is the mass of particles initially available on pore surface. The rate of particle entrainment by the flowing phase or particle removal from the pore surface is assumed both colloidally and hydrodynamically induced. re = kr m∗p + ke  − c as

(10-77)

where  is shear stress, as is the specific pore surface area, kr is the colloidally induced particle release rate coefficient given by (Khilar and Fogler, 1983) kr = 0

for Cs < Csc

(10-78)

kr = 0

for Cs ≥ Csc

(10-79)

where Cs is the prevailing salt concentration, Csc is the critical salt concentration required for particle expulsion, ke is the hydrodynamically induced release coefficient given by (Gruesbeck and Collins, 1982a) ke = 0

for  > c

(10-80)

ke = 0

for  ≤ c

(10-81)

where c is the critical shear stress required to mobilize particles present on pore surface.

Single-Phase Formation Damage by Fines Migration and Clay Swelling

277

The rate of capture of particles at pore throats is assumed proportional to the flowing-phase particle concentration: rt = kt cp

(10-82)

where kt is the particle capture rate coefficient. Let cpc be the critical particle concentration above which bridging across pore throats occurs and particles cannot travel between pore bodies. If the particle concentration is below cpc , then no trapping at pore throats takes place. Therefore, set  = 1 kt = 0

for cp < cpc

(10-83)

 = 0 kt = 0

for cp > cpc

(10-84)

The correlation between entrapment and permeability reduction is based on the Hagen-Poiseuille theory of flow through the pore throat, given by (see Chapter 5)   p 2 K (10-85) = 1− Ko

o where  = 10 − Kf /Ko 1/2 is a characteristic constant, Ko is the initial permeability, and Kf is the final permeability, o is the initial porosity, p is the volume fraction of the deposited particles present in porous media, given by Eq. (10-24).

10.4 SIMPLE PARTIAL DIFFERENTIAL CORE IMPAIRMENT MODEL ˇ nanský and Široký (1985) considered injection of a low particle conCerˇ centration suspension at a constant rate into porous media made of a bed of filaments. Neglecting the diffusion of particles and the contribution of the small amount of particles in the flowing suspension, they expressed the total mass balance of particles similar to the simplified mass balance equation of Gruesbeck and Collins (1982a). Thus, for incompressible liquid and particles and constant injection rate, the total volumetric particle balance equation is given by   +u =0 t x

(10-86)

278

Single-Phase Formation Damage by Fines Migration and Clay Swelling

subject to  = o 

 = o = 0

0 ≤ x ≤ L

t=0

(10-87)

and  = in  x = 0  t > 0

(10-88)

ˇ nanský and Široký (1985) expressed the net rate of particle deposition Cerˇ in porous media as the difference between the deposition by pore filling following throat plugging and the entrainment by hydrodynamic mobilization. Considering the critical shear stress necessary to mobilize the deposited particles in porous media, Civan et al. (1989) modified their rate equation as  = kp u − ke   − cr t in which the shear stress is expressed as   p D − = 4 x

(10-89)

(10-90)

where D is the hydraulic tube diameter, and the pressure gradient is represented by Darcy’s law: −

p u = x K

(10-91)

The instantaneous porosity is given by

= o − 

(10-92)

where o is the initial porosity. Thus, substituting Eqs (10-90)–(10-92) into Eq. (10-89) yields   u  = kp u  o −  − ke′  − cr′ t K

(10-93)

in which ke′ = ke D/4 and cr′ = 4cr /D are the redefined entrainment coefficient and critical shear stress, respectively. ke′ = 0 when u/K < cr′ . At equilibrium, /t = 0.

Single-Phase Formation Damage by Fines Migration and Clay Swelling

279

ˇ nanský and Široký (1985) Based on their experimental studies, Cerˇ proposed an empirical permeability–porosity relationship as       

K 1−E K  = 1− = E 1− + exp G 1 − −1

o

o Ko exp G − 1 Ko (10-94) where E and G are some empirical constants.

10.5 PARTIAL DIFFERENTIAL CORE IMPAIRMENT MODEL CONSIDERING THE CLAYEY FORMATION SWELLING AND BOTH THE INDIGENOUS AND THE EXTERNAL PARTICLES Civan et al. (1989) and Ohen and Civan (1989, 1990, 1993) considered the formation damage by clayey formation swelling and migration of externally injected and indigenous particles. They assumed constant physical properties of the particles and the carrier fluid in the suspension. They also considered the effect of fluid acceleration during the narrowing of the flow passages by formation damage. Ohen and Civan (1993) classified the indigenous particles that are exposed to solution in the pore space into two groups: lump of total expansive (swelling, i.e., total authigenic clay that is smectitic) and lump of total nonexpansive (nonswelling) particles, because of the difference in their rates of mobilization and sweepage from the pore surface. They considered that the particles in the flowing suspension are made of a combination of the indigenous particles of porous media entrained by the flowing suspension and the external particles introduced to the porous media via the injection of external fluids. They considered that the particles of the flowing suspension can be redeposited and reentrained during their migration through porous media and the rates of mobilization of the redeposited particles should obey a different order of magnitude than the indigenous particles of the porous media. Further, they assumed that the deposition of the suspended particles over the indigenous particles of the porous media blocks the indigenous particles and limits their contact and interaction with the flowing suspension in the pore space. They considered that the

280

Single-Phase Formation Damage by Fines Migration and Clay Swelling

swelling clays of the porous media can absorb water and swell to reduce the porosity until they are mobilized by the flowing suspension. They assumed that permeability reduction is a result of the porosity reduction by net particle deposition and formation swelling and by formation plugging by size exclusion. The Ohen and Civan (1993) formulation is applicable for dilute and concentrated suspensions. The mass balance equations for the total water (flowing plus absorbed) in porous media and the total particles (suspended plus deposited) in porous media are given, respectively, by    u = 0  w + w w + t x w w

   

p + p + ∗p p + p up = 0 t x

(10-95) (10-96)

Thus, adding Eqs (10-95) and (10-96) yields the total mass balance equation for water and particles in porous media as



  

w w + p p + w w + p + ∗p p t +

   w w + p p u = 0 x

(10-97)

In Eqs (10-95)–(10.5), is the instantaneous porosity, w and p are the densities of water and particles, u is the volumetric flux of the flowing suspension of particles, w  p , and ∗p represent the volume fraction of porous media containing the absorbed water, particles deposited from the flowing suspension, and the indigenous particles in the pore space, respectively, and w and p denote the volume fractions of water and particles, respectively, in the flowing suspension. Thus, w + p = 1

(10-98)

According to Eq. (10.95) the density of the flowing suspension is given as a volumetric-weighted sum of the densities of the water and particles by

 = w w + p p = w + p − w p

(10-99)

For simplification purposes, assume constant densities for water and particles. However, note that the density of suspension is not a constant,

Single-Phase Formation Damage by Fines Migration and Clay Swelling

281

because it is variable by the particle and water volume fractions based on Eq (10-99). Therefore, Eqs (10-96) and (10.5) can be expressed, respectively, as



p ∗p  + =0

p + u + t x p t t   p ∗p      + u + p + w w = 0 + t x t t t

(10-100) (10-101)

Considering the rapid flow of suspension as the flow passages narrow during the formation damage, the Forchheimer equation is used as the equation of motion:  −1   Nnd K u=−  x

(10-102)

where  is the flow potential defined as: =

 p dp

po



+ g z − zo sin 

(10-103)

in which  is the inclination angle and zo is a reference level. Nnd is the non-Darcy number given by 

uK Nnd = 1 + 

−1

(10-104)

The inertial flow coefficient, , can be estimated by the Liu et al. (1995) correlation: =

891 × 108 

K

(10-105)

where is fraction, K is mD, and  is ft−1 . Brinkman’s application of Einstein’s equation is used to estimate the viscosity of the suspension:

25  = w 1 − p

where w is the viscosity of water.

(10-106)

282

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Substituting Eq. (10-102) into Eq. (10-101) yields the following equation for the flow potential:  x

   2 Nnd K  x   p ∗p   + w w + =   + p t t t t 

(10-107)

The particle volume fraction and the flow potential can be calculated by solving Eqs (10-100) and (10-107) simultaneously, using an appropriate numerical method such as the finite-difference method used by Ohen and Civan (1993), subject to the initial and boundary conditions given by p = po x 

p = p in 

 = o x 

0 ≤ x ≤ L

 = in or u = uin   = out 

x = L

x = 0 t>0

t=0

(10-108)

t>0

(10-109) (10-110)

The volumetric rate of water absorption is estimated by (Civan et al., 1989) w = B t−1/ 2 t

(10-111)

where t is the actual contact time of flowing water with the porous media and B is an absorption rate constant. The porosity change by clayey formation swelling is estimated by (Civan and Knapp, 1987; Ohen and Civan, 1990, 1993) sw  = w t t

(10-112)

where  is the swelling coefficient determined by an appropriate empirical correlation such as by those given by Seed et al. (1962a,b) and Nayak and Christensen (1970) (see Chapter 2). The volume balance of particles (indigenous and/or external types) of the flowing suspension deposited in porous media is given as the difference of the deposition by the pore surface and pore throat deposition

Single-Phase Formation Damage by Fines Migration and Clay Swelling

283

processes and the reentrainment rates by the colloidal and hydrodynamic processes as (Civan, 1996a,b): p = kd a + u p 2/ 3 + kp up

t −kr p e ccr − c − ke p e 2/ 3  − cr

(10-113)

where kp = 0

for t < tp

(10-114)

kr = 0

for c > cr

(10-115)

ke = 0

for  < cr

(10-116)

The initial condition is given by p = po 

0 ≤ x ≤ L

t=0

(10-117)

Let single and double primes denote the nonswelling and swelling clays, respectively. The volume balances of the nonmobilized indigenous nonswelling and swelling clays remaining in porous media is given in terms of the colloidal and hydrodynamic mobilization rates, respectively, by ∗p

′

′

t ′′

∗p t

′

= −kr′ ∗p ′e c′cr − c − k′e ∗p ′e 2/3  −  ′cr

(10-118)

  ′′ ′′ = 1 +  −k′′r ∗p e′′ ccr′′ − c − ke′′ ∗p ′′e 2/3  −  ′′cr (10-119)

where  is the expansion coefficient of a unit clay volume, estimated by (see Chapter 2)  √  = s − s − 1 exp −2k4 B t (10-120) in which s is the expansion coefficient at saturation. The initial conditions are given by ′

′

′′

0 ≤ x ≤ L

∗p = ∗po  ∗p = ′′po 

t=0

(10-121)

Therefore, ∗p t

=

∗p t

′

+

∗p t

′′

(10-122)

284

Single-Phase Formation Damage by Fines Migration and Clay Swelling

and ′

∗p = ∗p + ∗p

′′

(10-123)

Note kr′ = 0 for c > ccr′  kr′′ = 0 for c > ccr′′  ke′ = 0 for  < cr′ , and ke′′ = 0 for  < cr′ . ′

(10-124)

′′

(10-125)

e′ = e−k5 p e′′ = e−k5 p

where k5′ and k5′′ denote some empirical coefficients. The instantaneous porosity is given by

= o − sw − p + ∗p

(10-126)

where o is the initial porosity. The instantaneous permeability is estimated by means of the modified Kozeny–Carman equation (see Chapter 5 for derivation).  3

K = Ko

o

(10-127)

where Ko and o are the initial permeability and porosity, respectively and  is the flow efficiency factor, which is a measure of the fraction of the pore throats remaining open (see Figure 10-3). Thus, when all the pore throats are closed, then  = 0 and K = 0, even if = 0. The cumulative volume of fluid injected at x = 0, expressed in terms of the initial undamaged pore volume, is given by Q0 = L o

−1

t

u0 dt

(10-128)

0

u0 is the injection fluid volumetric flux. The cumulative fines production at x = L in the effluent is QpL = A

t

pL uL dt

(10-129)

0

uL and pL are the effluent volumetric flux and particle volume fraction.

285

Single-Phase Formation Damage by Fines Migration and Clay Swelling

The harmonic mean average permeability of the core of length L is calculated by Km =

L L

(10-130)

1/K dx

0

The linear flow model presented above can be converted to the radial flow model by the application of the transformation given by (Ohen and Civan, 1990)   r x = ln (10-131) rw r and rw denote the radial distance and the wellbore radius, respectively.

10.6 PLUGGING–NONPLUGGING PARALLEL PATHWAYS PARTIAL DIFFERENTIAL CORE IMPAIRMENT MODEL Gruesbeck and Collins (1982a) developed a partial differential model based on the concept of parallel flow of a suspension of particles through plugging and nonplugging pathways of porous media, as depicted in Figure 10-6. Relatively smooth and large diameter flowpaths mainly involve surface deposition and are considered nonplugging. Flowpaths that are highly tortuous and having significant variations in diameter are considered plugging. In the plugging pathways, retainment of particles

Figure 10-6. Nonplugging and plugging paths realization in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy and after Civan, ©1992 SPE; reprinted by permission of the Society of Petroleum Engineers).

286

Single-Phase Formation Damage by Fines Migration and Clay Swelling

occurs by jamming and blocking of pore throats when several particles approach narrow flow constrictions. Sticky and deformable deposits usually seal the flow constrictions (Civan, 1994a, 1996a). Therefore, conductivity of a flow path may diminish without filling the pore space completely. Thus, the fluid seeks alternative flow paths until all the flow paths are eliminated. Then the permeability diminish even though the porosity may be nonzero. Another important issue is the criteria for jamming of pore throats. As demonstrated by Gruesbeck and Collins (1982b) experimentally for perforations, the probability of jamming of flow constrictions strongly depends on the particle concentration of the flowing suspension and the flow constriction-to-particle diameter ratio. Gruesbeck and Collins (1982a) assumed that the carrier liquid and particles present in the flowing particle suspension have constant physical properties. The porous media is incompressible, homogeneous, and isotropic. There is hydraulic communication through the interconnectivity of the plugging and nonplugging pathways and therefore the pressure gradients and the particle concentrations of the suspension flowing through the plugging and nonplugging pathways are the same. The volume flux through the core is constant and only the external particle invasion is considered. The flow through porous media was assumed to obey the Darcy Law. In this section, the Gruesbeck and Collins (1982a) model is presented with the modifications and improvements made by Civan (1995a) and Civan and Nguyen (2005). The initial pore volume fractions of the plugging and nonplugging pathways of the porous media are denoted by po and npo , respectively. (Civan, 1994a, 1995a). These values can be determined experimentally for a given porous media and the particle size distribution. p and np represent the fractions of the bulk volume occupied by the deposits. Thus, the instantaneous porosities are

p = po − p

(10-132)

np = npo − np

(10-133)

The fractions of the bulk volume containing the plugging and nonplugging pathways can be approximated, respectively, by (Civan, 1995a) fp =

Ap p =  A

(10-134)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

fnp =

Anp np =  A

287

(10-135)

However, Gruesbeck and Collins (1982a) assume a constant value for fp (and therefore fnp = 1 − fp ), which is a characteristic of the porous medium and the particles. Total instantaneous and initial porosities are given, respectively, by

= p + np

(10-136)

o = po + npo

(10-137)

The total deposit volume fraction and the instantaneous available porosity are given, respectively, by  = p + np

(10-138)

= o − 

(10-139)

The rate of deposition in the plugging pathways is given by assuming the pore filling after pore throat plugging mechanism (Civan and Nguyen, 2005):

p = kt + kp p up p t p = po  t = 0

(10-140) (10-141)

where kt is a pore throat plugging rate coefficient, kp is a pore filling rate coefficient, p is the volume fraction of the particles flowing through the plugging pathways, kp = 0 when t < tp  tp is the time at which the pore throats are blocked by forming particle bridges and jamming. This occurs when the pore throat to particle diameter ratio decreases to approach its critical value. Civan (1990, 1994a) recommended the following empirical correlation:   

Dt /Dp > Dt /Dp cr  Dt /Dp cr = A 1 − exp −BNRep + C (10-142)

which is determined empirically as a function of the particle Reynolds number: NRep = p p vp Dp /

(10-143)

288

Single-Phase Formation Damage by Fines Migration and Clay Swelling

in which the interstitial velocity, vp , is related to the superficial velocity, up , by duPuit (1863): vp =

up p

p

(10-144)

where p represents the tortuosity of the plugging pathways. This quantity is the actual length of the tortuous flow paths divided by the length of porous media. The symbol Dp represents the mean particle diameter. The rate of deposition in the nonplugging tubes is given as the difference between the rates of surface deposition and sweeping of particles (Civan, 1994a):

np 2/3 = kd unp np 2/3 np − ke np np e wnp − wcr t

(10-145)

subject to the initial condition

np = npo 

t=0

(10-146)

where np is the volume fraction of particles in the suspension of particles flowing through the nonplugging pathways. kd and ke are the surface deposition and mobilization rate constants, respectively. ke = 0 when wnp < wcr  e is the fraction of the uncovered deposits that can be mobilized from the pore surface, estimated by

e = exp −knp

(10-147)

where k is an empirical factor. wcr is the minimum shear stress necessary to mobilize the surface deposits. wnp is the wall shear stress in the nonplugging tubes, given by the Rabinowitsch–Mooney equation (Metzner and Reed, 1955): wnp = k

′



8vnp Dnp

n′

(10-148)

in which the interstitial velocity, vnp , is related to the superficial velocity, unp , by duPuit (1863): vnp =

unp np

np

(10-149)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

289

where np represents the tortuosity of the nonplugging pathways. This quantity is the actual length of the tortuous flow paths divided by the length of porous media. The mean pore diameter is given by   √ Knp 1/2 Dnp = C4 2

np

(10-150)

where C is an empirical shape factor. It can be shown that Eqs (10-140) and (10-145) simplify to the deposition rate equations given by Gruesbeck and Collins (1982a):

p = c1 + c2 p up  t

(10-151)

np = c3 np − c4 unp − uc np t

(10-152)

where c1  c2  c3 , and c4 are some empirically determined coefficients. This requires that the effects of the permeability and porosity changes be negligible, the fraction of the uncovered deposits be unity, the suspension of particles be Newtonian, and the particle volume fractions of the suspensions flowing through the plugging and nonplugging pathways be the same, that is, (10-153)

p = np = 

Note that the coefficient appearing in Eq. (10-151) can be referred to as the variable filtration coefficient  as (Civan and Rasmussen, 2005): (10-154)

 ≡ c1 + c2 p

The permeabilities of the plugging and nonplugging pathways are given by the following empirical relationships (Civan, 1994a,b):  

n   n Kp = Kpo exp − po − p 1 = Kpo exp −p 1 (10-155)

and

Knp = Knpo



np

npo

n 2

= Knpo



np 1−

npo

n2

(10-156)

290

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Then, the average permeability of the porous medium is given by (10-157)

K = fp Kp + fnp Knp

Note that Eq. (10-157) was derived independently by Civan (1992) and Schechter (1992) and is different than the expression given by Gruesbeck and Collins (1982a). The superficial flows in the plugging and nonplugging pathways are given, respectively, by up =

uKp K

(10-158)

unp =

uKnp K

(10-159)

Thus, the total volumetric flux is given by (10-160)

u = fp up + fnp unp

Considering that the physical properties of the particles and the carrier liquid are constant, the volumetric balance of particles in porous media is given by     +  + u = 0 t x

(10-161)

Substituting Eq. (10-139) into Eq. (10-161), and then rearranging, an alternative convenient form of Eq. (10-161) can be obtained as  o − 

   + u + 1 −  =0 t x t

(10-162)

Following Gruesbeck and Collins (1982a), Eq. (10-162) can be simplified for cases where  and  are small compared to o and unity, respectively, and for constant injection rate, as

o

   +u + =0 t x t

(10-163)

The initial particle contents of the flowing solution and porous media are assumed to be zero:  = 0 = 0

 = 0 

0 ≤ x ≤ L

t=0

(10-164)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

291

where L is the length of porous medium. The particle content of the suspension of particles injected into the porous media is prescribed as x = 0

 = in 

t>0

(10-165)

Alternatively, the pressures of the inlet and the outlet ends of the porous media instead of the flow rate can be specified. Then, the volumetric flux can be estimated by the Darcy law:    K p u= −  x

(10-166)

Substituting Eq. (10-166) into the volumetric equation of continuity  u + =0 t x

(10-167)

results in the pressure equation  x







t

(10-168)

p = pin 

x=O

(10-169)

p = pout 

x=L

(10-170)

K p  x

=

subject to the boundary conditions

Then, the pressure obtained by solving Eqs (10-168)–(10-170) is substituted into Eq. (10-166) to determine the volume flux. The preceding formulation of Eq. (10-161) or (10-162) applies to the overall system following Gruesbeck and Collins’ (1982a) assumption that the particle concentrations in the plugging and nonplugging pathways are the same according to Eq. (10-153). When different concentrations are considered, Eq. 10-120 should be applied separately for the porous media regions Ap and Anp containing the plugging and nonplugging paths, respectively, modified after Civan (1995a):

p 

p



po − p + = k up p − unp np (10-171) p up + 1 − p t x t

292

Single-Phase Formation Damage by Fines Migration and Clay Swelling

np





np  np unp + 1 − np + = −k up p − unp np

npo − np x t t

(10-172)

subject to p = 0

np = 0 p = np = in 

0 ≤ x ≤ L

t=0

(10-173)

x = 0

t>0

(10-174)

where k is a particle exchange rate coefficient. A solution of Eqs (10-171)–(10-174) along with the particle deposition rate equations, Eqs (10-140) and (10-145), yields the particle volume fractions in the plugging and nonplugging flow paths.

10.7 MODEL-ASSISTED ANALYSIS OF EXPERIMENTAL DATA Without the theoretical analysis and understanding, laboratory work can be premature, because the analyst may not exactly know what to look for and what to measure. The theoretical analysis of various processes involved in formation damage provides a scientific guidance in designing the experimental tests and helps in selecting a proper, meaningful set of variables that should be measured. Having studied the various issues involving formation damage by fines migration, we are prepared to conduct laboratory experiments in a manner to extract useful information about the mechanisms causing formation damage. In the following, the analysis of experimental data by means of the mathematical models developed in this chapter is illustrated by several examples. 10.7.1

Applications of the Wojtanowicz et al. Model

In general, formation damage may be a result of a number of mechanisms acting together with different relative contributions. But the Wojtanowicz et al. (1987, 1988) analysis of experimental data is based on the assumption that one of the potential formation damage mechanisms is

Single-Phase Formation Damage by Fines Migration and Clay Swelling

293

dominant under certain conditions. Therefore, by testing the various diagnostic equations given in Table 10-1 derived by Wojtanowicz et al. for possible mechanisms involving the laboratory core damage, the particular governing damage mechanism can be identified. They have demonstrated that portions of typical laboratory data can be represented by different equations, indicating that different mechanisms are responsible for damage. For example, as indicated by Figures 10-7 and 10-8, the initial and later portions of the experimental data for damage by foreign (external) particles invasion with low particle concentration drilling muds (0.2%, 0.5%, and 1.0% by weight) can be represented by Eqs T1-1 and T1-2, successfully, revealing that the pore surface deposition and pore throat plugging mechanisms are dominant during the early and late times, respectively. Figure 10-9 shows that Eq. T1-3 provides an accurate straight-line representation of the core damage with injection of suspensions of high-concentration drilling muds (2% and 3% by weight) of foreign particles, revealing that the dominant formation damage mechanism should be the pore-filling and internal-cake formation. The data plotted in Figure 10-10 shows that the sizes and concentrations of the particles of the injected suspension significantly affect the durations and extent of

Figure 10-7. Diagnostic chart for gradual pore blockage by external particles invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

294

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Figure 10-8. Diagnostic chart for gradual pore blockage by external particles invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

Figure 10-9. Diagnostic chart for cake forming by external particles invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

Single-Phase Formation Damage by Fines Migration and Clay Swelling

295

Figure 10-10. Diagnostic chart for transition from gradual pore blockage to single pore blockage during external particles invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

the initial formation damage by pore surface deposition (Eq. T1-1) and later formation damage by pore throat plugging (Eq. T1-2) mechanisms. Figure 10-11 shows that the damage of the core by a particle-free calcium chloride–based completion fluid is due to the plugging of pore throats by particles mobilized by brine incompatibility, because Eq. T1-7 can represent the data satisfactorily by a straight line. Figures 10-12 and 10-13 show that the damage of the core by particle-free ammonium nitrate/alcohol-based completion fluids are due to pore surface deposition and pore surface sweeping, because the data can be satisfactorily fitted by straight lines according to Eqs T1-6 and T1-5, respectively. Figure 10-14 showing the plot of data for the combined effects of pore surface deposition and sweeping according to Eq. T1-4 indicates the effect of the flow rate on damage. As can be seen, the rate of formation damage increases by the flow rate. Wojtanowicz et al. (1987) explain this increase due to about a fivefold increase in the value of the release coefficient, ke , as a result of about a threefold increase in the flow rate from 3 to 10 mL/min. The best estimates of the intercept and slope values obtained by the least-squares regression analysis for the cases analyzed are presented by Wojtanowicz et al. (1987) in Figures 10-7–10-14. Using these parameters

296

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Figure 10-11. Diagnostic chart for cake forming by calcium chloride-based completion fluid invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

Figure 10-12. Diagnostic chart for gradual pore blocking by ammonium nitrate/alcohol-based completion fluid invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

Single-Phase Formation Damage by Fines Migration and Clay Swelling

297

Figure 10-13. Diagnostic chart for gradual pore sweeping by ammonium nitrate/alcohol-based completion fluid invasion (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

Figure 10-14. Diagnostic chart for combined effects of gradual pore blocking and sweeping (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

298

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Figure 10-15. Comparison of the effects of gradual blocking, screening, and straining on permeability reduction (after Wojtanowicz et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers and after Wojtanowicz et al., ©1988; reprinted by permission of the ASME).

in the relevant equations representing these cases, the relative retained permeability curves vs. time are plotted in Figure 10-15 for comparison. As can be seen, pore filling by cake formation causes the most severe damage.

10.7.2 Model

ˇ nanský and Široký Applications of the Cerˇ

ˇ nanský and Široký The objectives of the experimental studies by Cerˇ (1982, 1985) were threefold: 1. Determine an empirical relationship between permeability and porosity in the form of Eq. (10-94). 2. Determine the values of the deposition and entrainment rate constants, kp and ke′ . 3. Study the effects of the length of porous media, and the rate and concentration of the particle suspension injected into the porous media. For this purpose, a porous material was prepared by using nonwoven felt of filaments of polypropylene. The porous material samples of 4.0 cm diameter and 0.5, 1.0, 1.5, and 2.0 cm lengths were

Single-Phase Formation Damage by Fines Migration and Clay Swelling

299

used. The particle suspension was prepared using finely ground limestone of 2825 kg/m3 density in water. The mean diameter of particles is 202 m and the estimated dimension of pores is 50 m. The porosity is o = 875 kg/m3 /2825 kg/m3 = 031 (fraction). The concentration of the injected suspension is cin = 01 kg/m3 or in = 01 kg/m3 /2825 kg/m3 = 354×10−5 m3 /m3 . The pressure difference across the porous media and the particle concentration of the effluent were measured as functions of time during the injection of a suspension of finely ground limestone particles at a given concentration and rate. The porosity was determined by the weighting method. The initial time is denoted by i = 1. The discrete times at which measurements are taken are denoted by the subscripts i = 2 3     N . The permeability was determined by Darcy’s equation by neglecting the effect of gravity for short samples: Ki =

uL  i = 2 3     N pi

(10-175)

The volume of particles deposited per unit volume of porous media was calculated by integrating Eq. (10-86) and applying the mean value theorem: 

 t



avg

≡

L  − out u 1   dx = in L t L

(10-176)

O

from which t u  in − out dt  = o + L

(10-177)

o

where o = 0 for an initially particle-free porous material. For a constant injection suspension particle concentration, Eq. (10-177) is evaluated numerically by applying the trapezoidal rule of integration as  N  out i−1 + out i u i = o + ti  i = 2 3     N (10-178)  − L i=2 in 2 Equations (10-175) and (10-178) were applied at different times and the data were plotted in Figure 10-16. As shown in Figure 10-16, virtually

300

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Figure 10-16. Correlation of the deposit fraction and permeability for material 3 ˇ nanský and Široký, POP 1, using a 5 mm thick porous material and co = 01 kg/m (Cerˇ 1985; reprinted by permission of the AIChE, ©1985 AIChE, all rights reserved, and ˇ nanský and Široký, 1982; reprinted by permission). after Cerˇ

the same results were obtained for different injection velocities of u = 05 and 1.0 cm/s. The E and G parameter values of Eq. (10-94) were determined as a function of the length of porous media by nonlinear regression to the data given in Figure 10-16. Plots of E and G vs. the length are given in Figure 10-17. Exponential regressions of these data indicate that E  014 and G  53 in the limit as the length approaches zero, although the data are of low quality, as indicated by the coefficients of regressions R2 = 078 and R2 = 018, respectively. To determine the deposition and entrainment rate constants, in Eq. (10-93) cr′ = 0 was substituted and the derivative with respect

Single-Phase Formation Damage by Fines Migration and Clay Swelling

301

ˇ nanský and Široký (1985) data for variation of Figure 10-17. Correlation of the Cerˇ the E and G parameters by the length of porous media for material POP 1. Eavg = 0179 and Gavg = 566.

to time was evaluated numerically using the central and backward finite-difference approximations given below, respectively, for the interior and the final points (see Chapter 16 for derivation): i i+1 − i−1  i = 2 3     N − 1  t 2ti N  − 4N −1 + 3N  N −2 t 2tN

(10-179) (10-180)

The average concentration was estimated as the logarithmic mean value of the injection and the effluent suspension concentrations according to i =

in − out i   i = 2 3     N  ln in

(10-181)

out i

The rate parameters, kp and ke′ , in Eq. (10-93) were determined for different injection velocities using the method of least squares with the values calculated by Eqs (10-175) and (10-179)–(10-181). The results presented in Figure 10-18 indicate that the retention rate coefficient, kp , decreases and the entrainment rate coefficient, ke , increases with the injection velocity.

302

Single-Phase Formation Damage by Fines Migration and Clay Swelling

ˇ nanský and Široký (1985) data for variation Figure 10-18. Correlation of the Cerˇ of the rate coefficients by volumetric flux.

ˇ nanský and Široký (1985) data for variation Figure 10-19. Correlation of the Cerˇ of the rate coefficients by the thickness of porous media.

These calculations were repeated for different length porous media and the results are summarized in Figure 10-19. These values can be extrapolated to zero core length; however, again the quality of data is not good. The effect of the suspension particle concentration and particle size on the pressure drop is shown in Figure 10-20. For a given injection suspension particle concentration and rate, at equilibrium, Eq. (10-93) for cr′ = 0 yields  (10-182) kp in  o − ∗ − ke′ ∗  K ∗ = 0

in which the equilibrium state permeability is determined by K∗ =

uL p∗

(10-183)

Single-Phase Formation Damage by Fines Migration and Clay Swelling

303

ˇ nanský and Široký (1985) data for the effect of susFigure 10-20. Plot of the Cerˇ pension concentration and particle size on pressure drop.

ˇ nanský and Široký (1985) data for variation Figure 10-21. Correlation of the Cerˇ 3

of the limiting saturation values for co = 01 kg/m by the volumetric flux.

Figure 10-21 shows the equilibrium ∗ and K ∗ values calculated by Eqs (10-182) and (10-183), which are attained during the injection of cin = 01 kg/m3 or in = 354 × 10−5 m3 /m3 concentration suspension at various constant injection rates.

10.7.3 Model

Applications of the Gruesbeck and Collins

Gruesbeck and Collins (1982a) conducted core flow tests under constant rate and pressure difference conditions and studied the formation damage effects and determined the relevant rate constants. They used synthetic

304

Single-Phase Formation Damage by Fines Migration and Clay Swelling

unconsolidated and natural consolidated core samples. Their data analysis methods and results are presented and reviewed in the following sections. 10.7.3.1

Unconsolidated core tests at constant flow rate

Assuming that and  essentially remained constant with time during their experiments, Gruesbeck and Collins (1982a) simplified Eq. (10-161) for a constant injection rate as   +u =0 t x

(10-184)

and designed special experiments to verify their model as described below. Case 1: Particle Deposition and Mobilization in Nonplugging Pathways

First they considered a case where particle mobilization does not occur, and the particle deposition rate is proportional to the particle concentration of the suspension according to  = kd  t

(10-185)

Then, Eqs (10-184) and (10-185) can be solved subject to the conditions  = o 

0 ≤ x ≤ L

t=0

(10-186)

and  = in 

x = 0

t>0

(10-187)



(10-188)

to obtain the following analytic solutions: 

−kd x  = in exp u  = o + kd t

(10-189)

305

Single-Phase Formation Damage by Fines Migration and Clay Swelling

An application of Eq. (10-188) over the length of the porous media yields the deposition rate coefficient estimated as   out u (10-190) kd = − ln L in Ignoring the pore volume occupied by the immobile fluids, the interstitial velocity of fluids is given by the duPuit (1863) equation as v=

u

(10-191)

where  is the tortuosity of porous media, that is, the ratio of the actual flow length to the length of porous media. For very coarse porous media, it is reasonable to assume that   10 and  i , where i denotes the initial porosity. Gruesbeck and Collins (1982a) injected a low concentration suspension of CaCO3 particles into a column of clean, unconsolidated sand pack. Under these conditions, they assumed that removal of deposited particles was negligible. They measured the concentration of particles in the effluent. Applying Eq. (10-190) with their data given in Figure 10-22, Gruesbeck and Collins determined the same kd value on the average for different flow rates. Therefore, they concluded that the rate law postulated by Eq. (10-185) is valid.

Concentration ratio, Co /Ci

1.0

u/φ i = 0.95 cm/s

0.8

u/φ i = 0.66 cm/s

0.6

u/φ i = 0.32 cm/s

8 µ m CaCO3 Fines

0.4

Ci = 50 × 10–6

u/φ i = 0.21 cm/s

0.2

0 0

5

10

15

20

25

30

35

40

45

50

55

60

Pore volumes

Figure 10-22. Deposition of fines in a porous medium of 30.48 cm pack of 840–2000 m diameter sand grains (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

306

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Second, Gruesbeck and Collins neglected the deposition rate and considered that the particle mobilization and entrainment phenomena are dominant. Thus, integrating Eq. (10-184) over the length of the porous media and applying the mean-value theorem yields 

 t



L

 − out u 1   ≡ dx = in L t L avg

(10-192)

o

This expression can be modified as: 

  i  − out u = in t L i

(10-193)

avg

where in = 0 for injection of a particle-free solution. In order to test the validity of Eq. (10-192), Gruesbeck and Collins prepared a sand pack by mixing sand and a suspension of CaCO3 particles into a column. Then, a particle-free KCl solution was injected into the column. In Figures 10-23 and 10-24 of Gruesbeck and Collins (1982a), their experimental data and plots of Eq. (10-192) are given, respectively. Figure 10-24 indicates that the particle mobilization rate can be represented by 

    i uc u − (10-194) = ke  t

i i or simply by  = ke  u − uc t

(10-195)

where the entrainment rate coefficient is taken as ke = 0 when u < uc

(10-196)

where uc is the critical flux below which mobilization does not occur. Case 2: Particle Deposition in Plugging Pathways

To verify their plugging rate equation, Gruesbeck and Collins considered sand packs and suspension of glass particles of various sizes. They estimated the effective pore diameter of the passages between the closest

Single-Phase Formation Damage by Fines Migration and Clay Swelling

307

130 120

100

80 70 60 50

30 20 10 00

u/φ i = 0.36 cm/sec

40

u/φ i = 0.92 cm/sec

u/φ i = 0.75 cm/sec

90

u/φ i = 0.29 cm/sec

Effluent concentration, Ce106, cm3/cm3

110

10

20

30

40

50

60

Pore volumes

Figure 10-23. Entrainment of 0.5%, initially deposited, 8 m CaCO3 fines in a porous medium of 60.96 cm pack of 840–2000 m diameter sand grains (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

packing of sand grains as the grain diameter, Dg , divided by 6.5, that is, Dg /65. The data of sand grain and glass particle sizes as well as the estimated effective pore diameter to particle diameter ratio are presented in Table 10-2. They measured the pressure difference across the sand pack as a function of the pore volume of the suspension of glass beads injected at a constant rate. They classified the deposition type as indicated in Table 10-2 based on the observed variation of the pressure difference. The deposition process was considered mainly as the pore surface deposition when the variation of the pressure difference was small, and indicated by “S” in Table 10-2. The deposition was considered due to the pore throat plugging, when the pressure difference indicated a monotonic increase and indicated by “P” in Table 10-2. A rise in pressure difference to a plateau was an indication of simultaneous pore surface deposition and

308

Single-Phase Formation Damage by Fines Migration and Clay Swelling

40 (b)

8

sec–1

50

( δδ σt (avg × 10

Net rate of entrainment of fines.

60

(a)

30

20

10

00

0.2

0.4

0.6

0.8

Interstitial velocity, u /φ i, cm/s

Figure 10-24. Net entrainment rate of fines in a porous medium: Line (a) from data given in Figure 10-23, and Line (b) similar data obtained using a 10 mPa s viscosity fluid (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

Table 10-2 Deposition of 5–10 m Glass Beads in Sandpacks Sand Grain Diameter (Dg  m 840–2000 840–2000 420–840 420–840 250–297 250–297 177–210 104–124

Effective Pore Size (Ds  m

Glass Bead Diameter (Df  m

Ds /Df

Deposition Type∗∗

218† 218 97 97 42 42 30 18

5 − 10 8 − 25 5 − 10 8 − 25 5 − 10 8 − 10 5 − 10 5 − 10

291 132 129 59 56 26 40 24

S S S S and P S and P FC P FC

After Gruesbeck and Collins, ©1982a SPE; reprinted by permission of the Society of Petroleum Engineers. ∗∗ S = surface, P = plugging, FC = filter cake † = 1/2 840 + 2000 /65 = 218 for closest packing of spheres  = 218/ 1/2 5 + 10 = 291

Single-Phase Formation Damage by Fines Migration and Clay Swelling

309

pore throat plugging, and indicated by “S and P” in Table 10-2. Filter cake deposition was indicated by “FC” in Table 10-2. Gruesbeck and Collins carried out experiments under conditions favorable for simultaneous surface deposition and pore throat plugging. For this purpose, suspensions of class beads were injected into columns of clean sand packs. The effluent glass beads concentration, pressure difference across the sand pack, and cumulative class bead deposit in the sand pack were measured as a function of the pore volume of suspension of glass beads injected at constant rates. The experimental data are presented in Figure 10-25. Next, they have solved their model equations, Eqs. (10-151), (10-152), (10-155)–(10-157), (10-163), numerically by assuming trial values for the various phenomenological

Figure 10-25. Deposition and entrainment of 5–10 m diameter glass beads in a porous medium of 15.24 cm pack of 250–297 m diameter sand grains for 95 × 10−4 injection suspension concentration (after Gruesbeck and Collins, ©1982a SPE; reprinted by permission of the Society of Petroleum Engineers).

310

Single-Phase Formation Damage by Fines Migration and Clay Swelling

Figure 10-26. Numerical simulation of the deposition and entrainment experimental results presented in Figure 10-25 (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

parameters to match the measurements. The simulation results are shown in Figure 10-26. The fact that their model reasonably predicts the experimental observations indicates that their model based on the plugging and nonplugging pathways concept is valid for their experimental systems. When the plugging pathways are eliminated by particle retention, the flow is diverted to the nonplugging pathways. Then the particle detainment continues in the nonplugging pathways by pore surface deposition until a dynamic equilibrium is attained. At this condition, Eq. (10-152) yields the following expression for the equilibrium amount of the deposits accumulating in the nonplugging pathways as ∗np =

c 3 np

c4 unp − uc

(10-197)

311

Single-Phase Formation Damage by Fines Migration and Clay Swelling

in which (10-198)

np =  unp = u

because all flow goes through the plugging pathways. The fact that the cumulative amounts of deposits reach certain limiting values as shown in Figure 10-25 is indicative of attainment of such equilibrium conditions. Note, however, that the amounts shown in Figure 10-25 are the cumulative amounts including the amount of deposits present in the plugging pathways. Therefore at equilibrium, (10-199)

∗ = ∗p + ∗np

10.7.3.2 Unconsolidated core tests at constant pressure difference

1.0 (b)

0.8

Volume fines/pore volume, σ

Permeability ratio, K/Ki

Constant pressure tests are more representative of the producing well conditions. Therefore, Gruesbeck and Collins (1982a) flowed suspensions of glass particles through sand packs at constant pressure differences by applying relatively high pressure difference to a column of fine sand pack and relatively low pressure difference to a column of coarse sand pack. The results are reported in Figure 10-27. In the fine sand packs, they observed more deposition near the injection side, and the mean permeability of the sand pack decreased to zero because almost all the pathways are of the plugging type in the fine sand pack. The deposition occurred almost

(a)

0.6 0.4

Ci = 9.5 × 10–4

0.2 0.0

0

100 200 Pore volumes

300

8 (a)

6 4

(b)

2 0

0

4

8 12 16 20 Pack length, cm

Figure 10-27. Constant pressure deposition and entrainment of 5–10 mm diameter glass beads in a pack of (a) 177–210 m diameter sand grains subjected to 900 kPa/m pressure gradient and (b) 250–297 m diameter sand grains subjected to 450 kPa/m pressure gradient (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

312

Single-Phase Formation Damage by Fines Migration and Clay Swelling

uniformly along the coarse sand pack and the mean permeability of the sand pack decreased to an equilibrium value. This is because most of the pathways are of the nonplugging types in coarse sand pack. Gruesbeck and Collins (1982a) state that their computer simulation produced results similar to the measurements reported in Figure 10-27. They assumed equal particle concentrations in the plugging and nonplugging pathways. Civan et al. (1989), Ohen and Civan (1990, 1993), and Civan and Nguyen (2005) also simulated these experiments successfully. For example, refer to the exercise problem given in Chapter 17 for results generated by Civan and Nguyen (2005). Civan and Rasmussen (2005) present a radial fines migration model for simulating the near-wellbore region damage, as described in Chapter 19. 10.7.3.3

Consolidated core tests at constant flow rate

Gruesbeck and Collins (1982a) tested the natural consolidated Berea and field cores. Case 1: Berea core tests

The Berea cores were tested using 1. 2% KCl brine in a dry core (single phase system) 2. 2% KCl brine and white oil at a 50/50 ratio in a dry core (two phase system) 3. white oil in a dry core (single phase system) 4. white oil in a core at connate 2% KCl brine saturation (two phase). Cores were tested at various constant injection rates over a period of time determined by a prescribed, cumulative pore volume amount of the injection fluid. During each test, the pressure difference was measured and the permeability was calculated using Darcy’s law. Typical results obtained using a 2% KCl brine in a Berea core are presented in Figure 10-28. As can be seen, the permeability remained unchanged at the low flow rate of 00367 cm3 /s, while it decreased further at each of the increased high flow rates of 0.0682, 0.1002, 0.1310, and 01702 cm3 /s. The final permeability values attained after each of the high flow rates are used to calculate the permeability reductions from the initial state, which

Single-Phase Formation Damage by Fines Migration and Clay Swelling

φ i = 0.29

0.0367

0.1702

0.0367

0.1310

0.0367

20

0.0682

30

0.1002

40

0.0367

50

q = 0.0367 cm3/s

Effective permeability, µ m2

60

313

10 0

0

100

200

300

400

500

600

700

800

900

1000 1100

Pore volumes

Figure 10-28. Effect of fluid velocity on the entrainment and redeposition of fines in a 3.81 cm diameter and 3.0 cm long Berea core during a 2% KCl solution injection (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

30 2% KCl Brine % Permeability reduction

25 20

50/50 2% KCl Brine Isopar m Isopar m µ = 2.6 mPa⋅s

15 10 Isopar m at connate 2% KCl Brine saturation

5 0

0

.02

.04

.14 .08 .12 .10 .06 Interstitial velocity, u/φ i, cm/s

.16

.18

.20

Figure 10-29. Permeability reduction as a function of the interstitial velocity determined using the Figure 10-28 data (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

314

Single-Phase Formation Damage by Fines Migration and Clay Swelling

% permeability reduction

25

ki = 100 µ m2 φ i = 0.25

20

Isopar m µ = 2.6 mPa⋅s

15

Isopar m at connate 2% KCl Brine saturation

10 5 0

0

0.10

0.20

0.30

0.40

Interstital velocity, u /φ i, cm/s

Figure 10-30. Permeability reduction as a function of the interstitial velocity determined using a 3.81 cm diameter and 3.0 cm long field core sample (after Gruesbeck and Collins, ©1982 SPE; reprinted by permission of the Society of Petroleum Engineers).

are then plotted against these high flow rates as shown in Figure 10-29. The results shown in Figure 10-29 are indicative of surface particle removal, similar to Figure 10-24. They stated that the removal of indigenous particles in the cores from the pore surface and subsequent redeposition at the pore throats caused the permeability reduction. Case 2: Oil-field core tests

The core samples were taken from an oil field, indicating an abnormal decline of productivity in some wells. These cores were tested using 1. White oil in a dry core 2. White oil in a core at connate 2% KCl brine saturation. The experimental results presented in Figure 10-30 indicate a trend similar to Figure 10-29. The results presented in Figures 10-29 and 10-30 indicate that the indigenous particles of Berea and field cores are water-wet. This is apparent by the effect of the two phases on the critical velocity values required to initiate particle mobilization. The implication of this is that variation of the fluid system from oil to oil/water can reduce the critical velocity, induce surface particle mobilization, and increase permeability damage in the near-wellbore formation.

Single-Phase Formation Damage by Fines Migration and Clay Swelling

315

Exercises 1. Prepare the plots of the experimental data of Wojtanowicz et al. (1987, 1988) given in Figure 10-15 according to the plotting schemes of Wojtanowicz et al. (1987, 1988) given in Table 10-1 for various permeability damage mechanisms. Identify the governing mechanisms and the values of their associated parameters for each data set by means of the plots resulting in straight lines. 2. Prepare the plots of the experimental data of Gruesbeck and Collins (1982a) given in Figure 10-27 according to the plotting schemes of Wojtanowicz et al. (1987, 1988) given in Table 10-1 for various permeability damage mechanisms. Identify the governing mechanisms and the values of their associated parameters for each data set by means of the plots resulting in straight lines. 3. Determine the kd values and their units using Eq. (10-190) with the experimental data of Gruesbeck and Collins (1982a) given in Figure 10-22 for the four interstitial velocity values. Correlate the kd values with the interstitial velocity. Determine the average value of the kd coefficient. 4. Prepare a plot similar to Figure 10-24 using the experimental data of Gruesbeck and Collins (1982a) given in Figure 10-23. a. How many data points can be generated from Figure 10-23 for construction of a plot similar to Figure 10-24? b. What are the value and unit of the particle entrainment (mobilization) rate constant and the critical interstitial velocity necessary for particle mobilization according to Eq. (10-195)? 5. Particle-free water is injected into a core plug containing mobilizable particles present at its pore surface. The permeability impairment is assumed to occur as a result of the plugging of the pore throats by the particles mobilized from the pore surface. In an effort to develop a simple mathematical model to describe the relevant phenomena leading to formation damage in the core plug, carry out the necessary formulations in the following steps: a) Present the total particle material balance equation. b) Express the initial condition for the total particle material balance equation. c) Express the core-injection-end boundary condition for the total particle material balance equation. d) Is a core-effluent-end boundary condition required for the total particle material balance equation? If your answer is yes, then

316

Single-Phase Formation Damage by Fines Migration and Clay Swelling

express the effluent-end boundary condition. If your answer is no, then justify it with reasoning. e) Express the rate of particle deposition/mobilization equations. f) Express the initial conditions for the rate of particle deposition/mobilization equations. g) Had the mathematical model derived by steps (a) through (f) been solved numerically, for example using a finite-difference method, what could have been calculated from this exercise? 6. Consider the plots of the experimental data given in Figure 10-13 according to the plotting schemes of Wojtanowicz et al. (1987, 1988) given in Table 10-1 for various permeability damage mechanisms. a) Identify the governing damage mechanism(s) b) Estimate the values of the associated entrainment (or erosion) rate constants for each data set plotted by two straight lines in Figure 10-13. 7. Civan and Nguyen (2005) emphasize that “In order for the model to be predictive, carefully designed experiments should be designed and conducted for a priori determination of the set of 17 model parameters.” Some examples of determining various parameters by specially designed experiments have been mentioned at various places in this chapter for determination of the critical shear stress or critical interstitial fluid velocity according to Gruesbeck and Collins (1982a), and the parameters of permeability impairment by particle deposition, the particle deposition and mobilization rate coefficients, and their ratio at the dynamic equilibrium which is attained between the deposition ˇ nanský and Široký (1982, 1985), the and mobilization according to Cerˇ other parameters using the special algebraic relationships according to Wojtanowicz et al. (1987, 1988), and the initial fraction of the plugging and nonplugging pathways, and their initial porosity and permeability by means of petrographical analysis (Civan and Nguyen, 2005). Develop experimental methods supported by theoretical description for determination of the values of the various parameters mentioned in this chapter.

C

H

A

P

T

E

R

11

MULTIPHASE FORMATION DAMAGE BY FINES MIGRATION

Summary Most reservoirs contain multiphase fluid systems. Formation damage processes in such reservoirs are more complicated because of the effects of the relative wettabilities of fine particles and formation, interface transport, relative permeability, and capillary pressures, and the alteration of these properties by various processes. There are only a few models available for multiphase systems. These models have been developed for and tested with two-phase laboratory core flow data. This chapter discusses the additional processes on top of those involving single-phase formation damage that need to be considered for multiphase formation damage by fines migration. A systematic analysis and formulation of the relevant processes involving fines migration and formation damage during twophase fluid flow through sedimentary formations as well as applications to typical laboratory core damage tests are presented. The formulation can be readily extended for the multiphase and multidimensional systems and the actual fluid conditions existing in reservoir formations.

11.1 INTRODUCTION Several investigators including Muecke (1979), Sarkar (1988), and Sarkar and Sharma (1990) have determined that fine particles behave differently in a multiphase fluid environment and formation damage follows a different course than the single-phase systems. However, the majority of the 317

318

Multiphase Formation Damage by Fines Migration

reported experimental studies are limited to two-phase formation damage. Sutton and Roberts (1974) and Sarkar and Sharma (1990) have experimentally observed that formation damage in two-phase is less severe than in single-phase. Liu and Civan (1993a,b, 1995, 1996) have shown that two-phase formation damage requires the consideration of other factors, such as the wettability effect and the partitioning of particles between various phases. In this chapter, mutual interactions and effects between the two-phase flow systems, fine particles, and porous matrix are described in a phenomenological manner to develop a predictive model for formation damage by fines migration in two-phase systems flowing through porous formations. The formulation is carried out by extending the Liu and Civan (1993a,b, 1994, 1995, 1996) model for more realistic applications. The tests and case studies used by Liu and Civan (1995, 1996) are presented for demonstration and verification of the model. Although the model presented here involves some simplifications pertaining to the laboratory core damage experiments, it can be readily modified and generalized for the actual conditions encountered in petroleum reservoirs.

11.2 FORMULATION OF A MULTIPHASE FORMATION DAMAGE MODEL It is safe to assume that the liquid phases (oil and water) can capture the fine particles readily and therefore the gas phase does not carry any solid particles (i.e., it is nonwetting for all particles). The equations describing the various aspects for formation damage by fines migration during two-phase fluid flow through porous formations are formulated here. However, the formulation can be extended readily to multiphase fluid systems. For convenience in modeling, the bulk porous media is considered in four phases as schematically depicted in Figure 11-1: (1) the solid matrix, (2) the wetting fluid, (3) the nonwetting fluid, and (4) the interface region. These phases are indicated by S W N , and I, respectively. The species present in the wetting and nonwetting phases are denoted by w and n. The porous matrix is assumed nondeformable. Therefore, it is stationary and its volumetric flux is zero. The wetting and nonwetting phases flow at the volumetric fluxes denoted, respectively, by uW and uN . The interface region is located between the wetting and nonwetting phases and is assumed to move at a flux equal to the absolute value of

Multiphase Formation Damage by Fines Migration

319

Figure 11-1. Multiphase system in porous media.

the difference between the fluxes of the wetting and nonwetting phases (i.e., its flux is uI = uW − uN ). The relative motion of the various phases create a mixing action along the fluid interface. The various particles involving the formation damage are classified as (1) the foreign particles introduced externally at the wellbore, (2) the indigenous particles existing in the porous formation, and (3) the particles generated inside the pore space by various processes, such as the wettability alteration considered in this chapter. Another classification of particles is made with reference to the wettability as (1) the wetting particles, (2) the nonwetting particles, and (3) the intermediately wetting particles. These particles are identified, respectively, by wp, np, and ip. The latter classification is more significant from the modeling point of view, because, as explained by Muecke (1979), the wettability affects the behavior of these particles in a multiphase fluid system. By means of experimental investigations, Muecke (1979) has observed that particles tend to remain in the phases that can wet them. Ku and Henry, Jr (1987) have shown that intermediately wet particles accumulate at the interface of the wetting and nonwetting phases, because they are most stable there (see Chapter 8). Therefore, in the following formulation, an interface region containing the intermediately wet particles is perceived to exist in between the wetting and nonwetting phases as schematically indicated in Figure 11-1. Further, it is reasonable to consider that the wettability of some particles may be altered by various processes, such as asphaltene, paraffin, and inorganic precipitation, or by other mechanisms such as

320

Multiphase Formation Damage by Fines Migration

the turbulence created by rapid flow due to convective acceleration in the near-wellbore region. Consequently, these altered particles tend to migrate into the phases that wet them as inferred by the experimental studies of Ku and Henry, Jr (1979). In addition to the particles, the various phases may contain a number of dissolved species. The salt content of the aqueous phase is particularly important, because it can lead to conditions for colloidally induced release of clay particles when its salt concentration is below a critical salt concentration (Khilar and Fogler, 1983). Further, the monovalent (Na+ ) and divalent (Ca2+ ) ions present in the aqueous phase may react with the naphthenic acids present in the oilphase to form sodium carboxylate soap emulsion and calcium naphthenate soap scale (Shepherd et al., 2006). For convenience in formulation, the locations for particles retention can be classified in three categories: (1) the wetting pore surface, (2) the nonwetting pore surface, and (3) the pore space available behind the plugging pore throats. These regions are denoted by wS, nS, and tS, respectively, as indicated schematically in Figure 11-1. The areal fractions of the wetting and nonwetting sites can vary as a result of the various rock, fluid, and particle interactions during formation damage, such as by asphaltene, paraffin, and inorganic deposition. Therefore, a parameter fkS indicating the fraction of the pore surface that is wetting for species k is introduced in the formulation. Because the applications to describe and interpret the laboratory core damage data, conducted at mild temperature and pressure conditions, are intended, the formulation is carried out for one-dimensional flow in homogeneous core plugs, isothermal conditions, and incompressible particles and fluids. This allows the use of a simplified formulation based on volumetric balances and a fractional flow concept. However, the derivation can be readily extended for compressible systems encountered at the prevailing elevated pressure conditions of the reservoir formations.

11.2.1

Fluid and Species Transport

Assuming incompressible species, the volumetric balance of species j transported via phase J through porous media is given by:         jJ    D   + u  + qjJ = t J jJ x J jJ x J J jJ x J

Multiphase Formation Damage by Fines Migration

321

J = various phases (W, N, I, wS, nS, tS) and j = fluid species (w, n) or particles (wp, np, ip)

(11-1)

where J indicates the volume fraction of phase J in porous media, jJ is the volume fraction of species j in phase J, uJ is the volumetric flux of phase J through porous media, and qjJL represents the volume rate of transfer of species j from phase J to phase L. DjJ denotes the coefficient of dispersion of species j in phase J, and J is the density of phase J, which varies by its composition even if the individual constituent species may be considered incompressible. x and t denote the distance along the flow direction and time. The dispersion term for relatively large particles is usually neglected. The volumetric rate of particle lost per unit bulk media by various processes is given by   qjJ = qjlJ + qjJL (11-2) J=L

l=j

in which qjlJ denotes the volume rate of transformation of species j type to species l type in phase J expressed per unit bulk volume. Summing Eq. (11-1) over all species j in phase J and considering that the dispersion terms of various species j cancel each other out in a given phase, the volumetric equation of continuity for phase J is obtained as J uJ + + qJ = 0 t x

J = W N wS nS tS

(11-3)

in which the volumetric loss of all types of particles from phase J is given by  qJ = qjJ (11-4) j

Finally, by summing Eq. (11-3) for all phases J, the total equation of continuity for the multiphase fluid system is obtained as   u + +q = 0 t x

(11-5)

where the total volumetric flux and all types of particles lost by the multiphase fluid system are given, respectively, by  u = uJ (11-6) J

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Multiphase Formation Damage by Fines Migration

q=



qJ

(11-7)

J

Considering the possibility of the generation of inertial effects by rapid flow due to the narrowing of pores during formation damage, the volumetric flux of phase J is represented by the non-Darcy flow equation given by   KkrJ NndJ pJ + J g sin  J = W N (11-8) uJ = − J x where is the angle of inclination of the flow path and pJ and J are the pressure and viscosity of phase J. krJ is the relative permeability of phase J, and K is the permeability of porous media. NndJ is the phase J non-Darcy number given according to the Forchheimer equation as −1  (11-9) NndJ = 1 + NReJ  0 ≤ NndJ ≤ 1

in which NReJ is the phase J Reynolds number given by (Ucan and Civan, 1996) NReJ = KJ uJ / J

(11-10)

where  is the inertial flow coefficient given by a suitable correlation, such as by Liu et al. (1995) (see Chapter 7). Obviously, uJ = 0 for the immobile phases wS, nS, and tS. 11.2.2 Determination of Fluid Saturations and Pressures Two alternative convenient formulations can be considered for solution of the equations of continuity and motion given by Eqs (11-3) and (11-8) for pressures and saturations of the various phases flowing through porous media. In the first approach, Eq. (11-8) is substituted into Eq. (11-3) to obtain     1 pJ  NndJ krJ K + J g sin = J + qJ  J = W N (11-11) x J x t The capillary pressure is defined as the difference between the nonwetting and wetting phase pressures according to pcNW = pN − pW

(11-12)

Multiphase Formation Damage by Fines Migration

323

The phase J volume fraction is given by J =  SJ

(11-13)

where  is porosity and SJ is the saturation of phase J. Thus, substituting Eqs (11-12) and (11-13) into Eq. (11-11) yields the following equations for the wetting and nonwetting phases, respectively,     SW  NndW krW K  pW + W g sin = + qW (11-14) x W x t     NndN krN K  pW dpcNW SW + + N g sin

x N x dSW x =

 SN + qN t

(11-15)

Note that the saturations add up to one: SW + SN = 1

(11-16)

Therefore, adding Eqs (11-14) and (11-15) yields the following equation:    pW NndW krW NndN krN  + K x W N x    NndN krN K dpcNW SW  + + W + N g sin = +q x N dSW x t (11-17)

where the total volumetric loss of particles from the two-phase system is given by q = qW + qN

(11-18)

Equations (11-14) and (11-17) can be solved simultaneously to determine the wetting phase pressure and saturation, pW and SW , using an appropriate numerical solution method, such as the finite-difference method. A second and more convenient approach facilitates the fractional flow formulation. This is especially suitable for incompressible systems described by the equation of continuity given by Eq. (11-3). Civan’s

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Multiphase Formation Damage by Fines Migration

(1996d) formulation based on Richardson’s (1961) formulation can be modified as follows to include the loss terms and inertia flow affect in the saturation equation:   SW  SW dFW SW u DW NndW K + −  t dSW x x x −

  TW NndW K sin

− FW q + qW + SW − FW =0 x t

(11-19)

for which the capillary dispersion and gravity transport coefficients are given, respectively, by DW = −FW TW = FW

krN dpcNW N dSW

krN  − N g N W

(11-20) (11-21)

The zero capillary pressure and zero gravity fractional flow term is given by 

k FW = 1 + rN W krW N

−1

(11-22)

In the fractional flow formulation, the saturation of the wetting phase is calculated by solving Eq. (11-19). But the pressure of the wetting phase is still determined by solving Eq. (11-17). As explained by Civan (1996d), the solution of equations presented above requires the capillary pressure and relative permeability data for the two-phase system. These data continuously vary during formation damage, and empirical models, such as those given in Chapter 4, are required to incorporate these affects in the solution. This problem can be alleviated in a practical manner by resorting to an end-point mobility ratio formulation similar to Civan (1996c,d) and Luan (1995), by extending and generalizing the unit-mobility ratio formulations given by Craig (1971), Collins (1961), and Dake (1978). In view of the uncertainties in determining the exact nature of the variations of these data, it is reasonable to make the following assumptions. First, similar to Liu and Civan (1996), the capillary pressure effect can be neglected. Second, the relative permeability can be approximated by

Multiphase Formation Damage by Fines Migration

325

linear relationships with respect to the phase saturations as (Yokoyama and Lake, 1981) krJ = krJo S J  J = W N

(11-23)

where krJo is the end-point relative permeability. Third, the end-point mobility ratio parameter as defined below can be implemented M=

o N krW o W krN

(11-24)

Under these conditions, Eqs (11-20) and (11-22), respectively, become DW = 0

(11-25)

and FW =

MS W 1 + M − 1 S W

(11-26)

Consequently, Eq. (11-19) can be simplified significantly by substituting Eqs (11-25) and (11-26). In addition, the non-Darcy effect can be neglected by substituting NndJ = 10 J = W N

(11-27)

The end-point relative permeability and fluid densities may be replaced by average values as

 o  o o o krW kro krN 1 krW krN    + (11-28) W N 2 W N   W  N = W + N /2

(11-29)

As a result of substituting Eqs (11-27) and (11-28), Eq. (11-5) can be simplified as (Civan, 1996d)

kro



  pW    K + g sin

= =q x x x t

(11-30)

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Multiphase Formation Damage by Fines Migration

11.2.3 Determination of Species Concentrations in Various Phases Once the phase saturations are determined, then the species concentrations can be calculated by solving the following equation obtained by combining Eqs (11-1) and (11-3):    jJ jJ  jJ  J D  + uJ + jJ qJ + qjJ = t x x J J J x J J = W N I wS nS iS and j = w n wp np ip

(11-31)

The dispersion term is considered for dissolved species, such as those contained in the aqueous phase, but it is usually neglected for the particles. In accordance with the experimental observations by Muecke (1979), Liu and Civan (1993, 1995, 1996) have assumed that wettable particles remain in the wetting phase and nonwettable particles remain in the nonwetting phase and the intermediately wet particles are situated along the interface. They did not consider the possibility of wettability alteration of the particles and the pore surface by rock–fluid interactions in porous media and they assumed that the dispersion terms are negligible for the particles. They considered that the porous media has uniform wetting properties. Under these circumstances, Eq. (11-31) simplifies significantly because qjlJ = 0 and the particle loss only occurs from the fluid phases to the solid matrix. Liu and Civan (1996) considered a water–oil system flowing through a homogeneous (i.e., one type – water-wet or oil-wet) porous media. They assumed that the wettability of the porous medium does not change during the short period of time involving the typical laboratory core tests.

11.2.4 Wettability Transformation and Interface Transfer of Particles The literature on studies of the mechanisms of wettability alteration and interface particle transfer is rather limited and insufficient to formulate these processes accurately and rigorously. Therefore, Liu and Civan (1996) have resorted to a simplified approach, which yielded reasonably good results. They have combined the rate processes of the wettability transformation and the phase-to-phase particle transfer into one step

Multiphase Formation Damage by Fines Migration

327

assuming that the particles would immediately migrate into the phases, which wet them once their wettabilities change from one type to another. Based on the experimental observations and the studies of the mechanisms of interface particle transfer of Ku and Henry (1987), Liu and Civan (1996) assumed that the rate of the combined processes of wettability transformation and interface transfer of particles can be expressed as being proportional to the particle concentration according to qjlJL = kjlJL jJ

(11-32)

However, a plausible mechanism of interface particle transfer is presented in Chapter 8. 11.2.5

Particle Retention in Porous Media

Although particle retentions may occur at various locations in porous media by various mechanisms, only the most likely mechanisms are considered here. The wetting and nonwetting particles preferentially deposit over the similar wettability–type pore surfaces. They can also be captured at and detained behind the pore throats under favorable conditions leading to pore filling. The intermediately wet particles most likely move directly toward the pore throats and are captured there under certain conditions, because they migrate along the interface. 11.2.5.1

Surface deposition

The volumetric rate of deposition of the particle species j from phase J over a similar wetting pore surface can be expressed by (Civan, 1996a) qjJjS ≡ djJjS /dt = kdjJjS J + uJ jJ 2/3 fjS

(11-33)

subject to the initial condition jJS = ojJS 

t=0

(11-34)

In Eq. (11-33), kdjJjS is a deposition rate constant, J is a stationary deposition constant,  is a porosity, and fjS is the fraction of the pore surface of the same wettability type of the particle species j. Similar to Gruesbeck and Collins (1982a), Liu and Civan (1996) assumed that the porosity variation by deposition of small amounts of

328

Multiphase Formation Damage by Fines Migration

particles is negligible (i.e.,  is a constant). Liu and Civan (1996) considered a homogeneous wettability porous media, hence fjs = 10, and neglected the stationary deposition, therefore J = 0. 11.2.5.2

Pore throat plugging

The volumetric rate of retention of particles in the pore space (pore filling) following the pore throat plugging can be expressed by (Civan, 1996a, Gruesbeck and Collins, 1982a) qjJtS ≡ djJtS /dt = ktjJtS uJ jJ 

(11-35)

subject to the initial condition jJtS = ojJtS  t = 0

(11-36)

Liu and Civan (1996) assumed the porosity change is negligible in Eq. (11-35) (i.e.,  = constant). In Eq. (11-35), ktjJtS denotes the rate constant for deposition by pore throat plugging. Civan (1990, 1996a) proposed a dimensionless correlation to determine the conditions favorable for pore throat plugging in single-phase fluid media. This equation determines the critical ratio of the pore throat to particle diameters below which pore throat plugging by jamming of particles occurs. Thus,   (11-37) ktjJtS = 0 when Dt /Dp ≤ Dt /Dp cr For multiphase flow, this equation can be modified as    cr = A 1 − exp −B NReP + C

(11-38)

where the pore throat to particle diameter ratio and the particle Reynolds numbers are given, respectively, by cr = Dt /Dp NReP =

cp uDp 

(11-39) (11-40)

in which  is the tortuosity. The total particle mass flux is given by  cp u = j uJ jJ (11-41) J

j

329

Multiphase Formation Damage by Fines Migration

and the saturation weighted average particle diameter is given by Dp =



SJ

J



jJ DjJ

(11-42)

j

Liu and Civan (1993, 1995, 1996) have resorted to a simplified approach in an ad hoc manner and demonstrated by comparison of the calculated results with experimental data that it works. They assumed that the fraction of the plugged pore throats is proportional to the amount of particles detained behind the plugged pore throats. Therefore, their expression for the fraction of the nonplugged pore throats can be written as ft = 1 −

 J

ktjJ jJtS

(11-43)

j

where ktjJ are some empirical coefficients. They considered that there is a minimum characteristic value of ft min for which the pore throat blocking happens. Thus, ktjJtS = 0 when f ≤ ft min 11.2.5.3

(11-44)

Colloidal mobilization

The volumetric rate of colloidally induced surface particle release can be expressed, by modifying the formulations by Khilar and Fogler (1983) and Civan (1996a), as qrjJjS = drjJjS /dt = −krjJjS jS e 2/3 fjS ccrJ − cJ

(11-45)

subject to the initial condition rjJjS = orjJjS 

t=0

(11-46)

where e denotes the fraction of particles that can be mobilized. Liu and Civan (1996) assumed e = 10 and fjJ = 10. Note that the colloidal mobilization rate coefficient is given by: krjJjS = 0 when cJ < ccrJ

(11-47)

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Multiphase Formation Damage by Fines Migration

11.2.5.4

Hydraulic mobilization

The volumetric rate of surface particle mobilization by the fluid shearing force can be expressed by modifying the formulations by Khilar and Fogler (1983) and Civan (1996a) as qejJjS ≡ dejJjS /dt = −kejJjS jS e 2/3 fjS J − crJ

(11-48)

subject to the initial condition ejJjS = oejJ jS 

t=0

(11-49)

Liu and Civan (1993, 1995, 1996) assumed e = 10 fjS = 10, and  ≈ constant, and used J −crJ ∼ uJ −ucrJ . Note the hydraulic mobilization rate coefficient is given by: kejJjS = 0 when J < crJ or uJ < ucrJ 11.2.6

(11-50)

Porosity and Permeability Variation

The porosity is expressed by (Liu and Civan, 1996)  = o +

 J

S

JSj

(11-51)

j

where JSj denotes the variation of the pore volume by deposition of fine particles. The permeability is estimated by (Liu and Civan, 1996)  3 K  = 1 − ft kf + ft Ko o

(11-52)

where Ko and o are the reference permeability and porosity, respectively, kf is a parameter accounting for the residual permeability of plugged formation, and ft is a flow efficiency factor expressing the fraction of the unplugged porous media available for flow, given by ft = 1 − kt tl

(11-53)

where ki is a rate constant and tl is the quantity of the pore throat deposits.

Multiphase Formation Damage by Fines Migration

11.2.7

331

Filter Cake Formation at the Injection Face

The details of the development of the filter cake models are presented in Chapter 12. When the suspended particles existing in the injected fluid are large enough, they cannot invade the formation, or when a sufficient amount of fine particles are deposited in the porous formation, preventing further particle invasion, a filter cake formation begins over the injection face. Liu and Civan (1995, 1996) applied a simple rate equation for the filter cake buildup similar to (Peng and Peden, 1992) 1 − c p dLc /dt = uin cin − ke 

(11-54)

where ke  0 in their applications.

11.3 MODEL-ASSISTED ANALYSIS OF EXPERIMENTAL DATA1 In this section, the application of the model to the analysis of formation damage in a variety of core tests is demonstrated. The model is validated and model parameters are determined using the data of core tests. 11.3.1

Damage by Formation Fines Migration

Sarkar and Sharma (1990) examined fines migration in two Berea core samples, one of them containing residual oil saturation (ROS). Data for the two core tests are given in Table 11-1. 11.3.1.1

Single- and two-phase flow tests

The core samples were first saturated with 3% NaCl brine. Formation damage due to fines migration took place upon freshwater injection. Values of some model parameters were gathered from Khilar and Fogler (1983) while the others were obtained by matching the model responses to the measured data, as summarized in Table 11-2. Figure 11-2 shows 1 Reproduced by permission from Liu and Civan, © 1996 SPE; reprinted by permission of the Society of Petroleum Engineers.

332

Multiphase Formation Damage by Fines Migration Table 11-1 Core Test Data for Fines Migrations

Data

Core without ROS

Core diameter (cm) Core length (cm) Initial porosity (fraction) Initial permeability (D) Residual oil saturation End-point relative permeability Injection velocity (cm/s) Water viscosity (cp)

2.54 8.30 0.21 0.0654 0.0 1.0 43 × 10−4 1.0

Core with ROS 2.54 8.30 0.21 0.0825 0.367 0.038 43 × 10−4 1.0

Information extracted from Sarkar and Sharma, 1990. After Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers.

Table 11-2 Model Parameters for Fines Migrations Parameter

Core without ROS

cfpo gm/cm3 Ccr (mole/liter) krjJjS sliter/mole kejJjS cm−1 ktjJtS cm−1 kdjJjS cm3 /gm fmin kf

0.025 70 × 10−3 0.435 0.0 5.25 35.4 0.0 0.0

Core with ROS 0.02 70 × 10−3 0.28 0.0 5.25 35.4 0.0 0.0

After Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers.

that the simulation results favorably represent the experimental data for the two core tests. The simulation study also confirms that formation damage in the presence of oil is less pronounced. As can be seen in Table 11-2, the amount of formation fines that can be released from the pore surface, cfpo , is 20% less, and the rate constant for fines release due to colloidal effects, krjJjS , is 35% lower in the presence of residual oil. 11.3.1.2

Two-phase flow test

Sarkar (1988) conducted a laboratory test using a Berea core of 8.27 cm in length to investigate fines migration in two-phase flow. The core porosity and permeability initially were 0.21 fraction and 0.122 D, respectively.

Multiphase Formation Damage by Fines Migration

333

Figure 11-2. Simulation of the instantaneous-to-initial-permeability ratio data of Sarkar and Sharma (1990) (or permeability alteration factor) vs. pore volume during formation fines migration in single-phase flow (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

The core saturated with crude oil was displaced with 3% NaC1 brine. Berea sandstones generally do not suffer from permeability reduction during a brine flood. Neglecting the effects of capillary pressure, the model was used to simulate the two-phase flow test. Relative permeability data were obtained by matching the simulated results with the measured pressure drop across the core as shown in Figure 11-3. An oil flood was then carried out to reestablish the connate water saturation. The core was finally displaced with freshwater, and formation damage took place due to fines migration in two-phase flow. Using the relative permeabilities obtained from the two-phase flow test without formation damage, simulation was carried out to match the measured pressure drop as shown in Figure 11-4. Alteration in the rock permeability, predicted in Figure 11-5, indicates that formation damage due to fines migration in two-phase flow of oil and freshwater is similar to that of single-phase flow of freshwater in the presence of residual oil. Detailed information on core data and model parameters are presented elsewhere (Liu and Civan, 1995). 11.3.2

Damage by Particle Invasion

Experimental data of two similar core samples conducted by Eleri and Ursin (1992) were used to analyze formation damage due to particle

334

Multiphase Formation Damage by Fines Migration

Figure 11-3. Simulation of the pressure drop across an undamaged core vs. pore volume during two-phase flow data of Sarkar (1988) (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 11-4. Simulation of the pressure drop across a damaged core vs. pore volume during two-phase flow data of Sarkar (1988) (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

invasion. The two samples were labeled as Core #26 and Core #27 in the Eleri and Ursin (1992) study. Prior to flow tests, the core samples were treated to eliminate formation fines migration. Latex particles of less than 3 in size suspended in water were injected into Core #26 at the

Multiphase Formation Damage by Fines Migration

335

Figure 11-5. Predicted instantaneous-to-initial-permeability ratio (or permeability alteration factor) vs. pore volume during formation fines migration in two-phase flow (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

concentration of 05 × 10−4 g/cm3 and into Core #27 at the concentration of 20 × 10−4 g/cm3 . Simulations were performed to examine the two tests. Permeability alteration vs. cumulative volume of injected fluid is illustrated in Figure 11-6 including a comparison between the experimental and simulated results. Detailed information on core data and model parameters is presented by Liu and Civan (1993). All model parameters for the two core tests are the same except that fmin = 058 for Core #26 and fmin = 041 for Core #27. The difference reveals that higher particle concentration causes more pores being plugged. Both experimental and simulation results indicate that particle concentration is a major factor for formation damage caused by particle invasion.

11.3.3 11.3.3.1

Damage by Mud Filtration Single-phase flow tests

Rahman and Marx (1991) studied formation damage by mud filtration. A core sample was contaminated by circulating a drilling fluid over the surface of core inlet under a constant differential pressure of 34.5 atm across the core. Before mud filtration, the core was saturated with 1.5% KCl water to prevent formation fines migration. Permeability alteration

336

Multiphase Formation Damage by Fines Migration

Figure 11-6. Simulation of the instantaneous-to-initial-permeability ratio data of Eleri and Ursin (1992) (or permeability alteration factor) vs. pore volume during external fines invasion (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

along the core was measured after one hour of mud contamination. Data for the core test and values of model parameters for simulation are presented elsewhere by Liu and Civan (1993). Experimental and simulated results for drilling fluid loss vs. time and permeability alteration vs. core distance after one hour of mud contamination are illustrated in Figures 11-7 and 11-8. Simulation results indicate that the model can favorably represent the process of mud filtration. Another laboratory test involving dynamic mud filtration was conducted by Jiao and Sharma (1992). A freshwater-based mud was circulated over the surface of core inlet and infiltrated into a Berea core under an average differential pressure of 6.3 atm across the system. This Berea core sample was previously saturated with 3% NaCl brine. Formation damage in this test is caused by external solid invasion and formation fines migration. Pressure taps were placed at different locations along the core of 20.3 cm in length to measure permeability change during the test. Experimental and simulated mud filtration volumes are in good agreement, as presented in Figure 11-9. As shown in Figure 11-10, experimental results of permeability alteration in the core section between 6.4 and 11.4 cm from core inlet compare quite well with simulation results. Further discussion on the simulation of this test is presented elsewhere (Liu and Civan, 1993).

Multiphase Formation Damage by Fines Migration

337

Figure 11-7. Simulation of the cumulative fluid loss vs. filtration time during mud filtration data of Rahman and Marx (1991) (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 11-8. Simulation of the instantaneous-to-initial-permeability ratio (or permeability alteration factor) vs. core length after one hour of filtration time during mud filtration data of Rahman and Marx (1991) (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

338

Multiphase Formation Damage by Fines Migration

Figure 11-9. Simulation of the cumulative fluid loss vs. filtration time during dynamic mud filtration data of Jiao and Sharma (1992) (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 11-10. Simulation of the instantaneous-to-initial-permeability ratio (or permeability alteration factor) vs. filtration time during dynamic mud filtration data of Jiao and Sharma (1992) (after Liu and Civan, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

Multiphase Formation Damage by Fines Migration

11.3.3.2

339

Two-phase flow test

Prediction of formation damage due to dynamic mud filtration in twophase flow was also carried out to demonstrate the capacity and application of the model and to provide a comparison with single-phase flow results. If the core studied by Jiao and Sharma (1992) was saturated by oil prior to mud filtration, the invasion of the water-based mud would lead to two-phase flow of oil and water in the core. The same model parameters determined above for mud filtration in single-phase flow were used. Additional data necessary for simulation, including connate water saturation, residual oil saturation, and relative permeabilities, are adapted from the case of fines migration in two-phase flow that was also simulated above. The predicted mud filtration volume and permeability alteration in two-phase flow are also plotted in Figures 11-9 and 11-10, as denoted by the dashed lines. These results indicate that filtration volume and formation damage are significantly less when a water-based mud invades an oil-bearing formation. This is because the total mobility for simultaneous two-phase flow of water and oil is usually less than that of single phase of water in formations, especially in Berea sandstones, which are generally strongly water-wet and have a very low permeability for water phase with the presence of oil in the formations.

Exercises 1. Write the expressions similar to Eqs (11-1)–(11-8) for radial flow around a well. Hint: Refer to a mathematical handbook or other appropriate sources for transformation from linear to cylindrical coordinates. 2. Write the expressions similar to Eqs (11-11)–(11-19) for radial flow around a well. Hint: Refer to a mathematical handbook or other appropriate sources for transformation from linear to cylindrical coordinates. 3. Write an expression similar to Eq. (11-31) for radial flow around a well. Hint: Refer to a mathematical handbook or other appropriate sources for transformation from linear to cylindrical coordinates. 4. Do the rate equations given in Chapter 11 for various particulate processes apply for both linear, radial, and other types of flows in petroleum reservoirs? Explain and justify your answer. 5. Prepare a chart showing the K/Ko vs /o plots of Eq. (11-52) for various values of ft and Kp . Note that the range of variation is 0 ≤ ft  K/Ko  /o ≤ 10.

340

Multiphase Formation Damage by Fines Migration

6. Based on the data given in Figure 11-2, determine the overall percent permeability reduction 1−K/Ko ×100 at the infinite limit of the pore volumes fluid injected into the cores with and without the presence of the residual oil. 7. Estimate the spurt loss based on the data given in Figure 11-7. 8. Why is the cumulative filtrate volume significantly less in the twophase flow case than that of the single-phase flow case based on the data given in Figure 11-9? Explain and justify your answer. 9. Why is the permeability reduction significantly less in the two-phase flow case than that of the single-phase flow case based on the data given in Figure 11-10? Explain and justify your answer.

C

H

A

P

T

E

R

12

CAKE FILTRATION: MECHANISM, PARAMETERS, AND MODELING1

Summary Models for interpretation and prediction of incompressible and compressible filter cake thickness, and filtrate volume and rate data for linear and radial filtration cases, under static and dynamic filtration conditions, are presented. Effects of compressibility and small particle invasion and deposition inside the cake and formation, as well as the Darcy vs. nonDarcy flow regimes, are considered. Methods and diagnostic charts for determining the model parameters from experimental filtration data are reviewed. Applications for radial and linear filtration cases are presented, and the results are compared for constant rate and constant pressure drive filtration. Model-assisted analyses of experimental data demonstrate the diagnostic and predictive capabilities of the models. The parametric studies indicate that the particle screening efficiency of the formation is an important factor on the filter cake properties and filtration rate, the differences between the linear and the radial cake filtration performances are more pronounced, and the cake thickness and filtrate volume are smaller, for constant pressure filtration than constant rate filtration. The present thickness-averaged ordinary differential models are shown to reproduce the predictions of the previous partial differential model rapidly with 1

Parts of this chapter have been reprinted with permission of the American Institute of Chemical Engineers and the Society of Petroleum Engineers from Civan (©1998b,c AIChE, and ©1999b,c SPE).

341

342

Cake Filtration: Mechanism, Parameters, and Modeling

significantly less computational effort. Because of the simplicity of the equations and reduction of computational effort, the thickness-averaged linear and radial filter cake formation models offer significant advantages over the partial differential models for the analysis, design, and optimization of the cake filtration processes involving the wellbore and hydraulically created fracture surfaces. Simplified models considering incompressible particles and carrier fluids, and analytical solutions for incompressible cakes without fines invasion are also presented. These models provide insight into the mechanism of cake filtration and offer practical means of interpreting experimental data, estimating the model parameters, and simulating the linear and radial filtration processes.

12.1 INTRODUCTION Cake filtration occurs inherently in many in situ hydrocarbon reservoir exploitation processes. For example, hydraulic fracturing of petroleumbearing rock and overbalanced drilling of wells into petroleum reservoirs usually cause a cross-flow filtration, which leads to a filter cake buildup over the face of the porous rock and filtrate invasion into the reservoir (Civan, 1994a,c, 1996a,b). When the slurry contains particles of different sizes, the larger particles of the slurry form the skeleton of the filter cake and the smaller particles can migrate into and deposit within the porous cake formed by the large particles. Simultaneously, the cake may undergo a compaction process by the effect of the fluid drag as the suspension of smaller particles flow through the cake (Tien et al., 1997). Consequently, the porosity, permeability, and thickness of the cake vary, which in turn affect the performance of the filtration process. Static filtration occurs when slurry is applied to a filter without cross-flow. Therefore, the particles are continuously deposited to form thicker filter cakes until the space available is full of the filter cake. Dynamic filtration involves some cross-flow. Therefore, the filter cake thickness varies until the particle deposition and erosion rates equal. Model-assisted analyses and interpretation of experimental data, and optimization and simulation of the filtration processes are of continuing interest. The majority of the previous modeling efforts have been limited to linear filtration applications, in spite of the fact that many filtration processes involve radial filtration applications. Linear filtration models can closely approximate radial filtration only when the thicknesses of the filter and filter cake are sufficiently small relative to the radius of the

Cake Filtration: Mechanism, Parameters, and Modeling

343

filter surface exposed to slurry. Otherwise, radial models should be used for radial filtration. Because of their simplicity, empirical correlations such as those reviewed by Clark and Barbat (1989) are frequently used for static and dynamic filtration. Xie and Charles (1997) have demonstrated that the use of a set of properly selected dimensionless groups leads to improved empirical correlations. Simple models are preferred in many applications because of their convenience and the reduced computational effort. The applicability of the majority of the previously reported simple analytical models, such as by Collins (1961), Hermia (1982), and de Nevers (1992), are usually limited to linear and constant rate filtration. However, models for constant pressure filtration are also required for certain applications, such as during drilling of wells. Civan (1998b) developed and verified improved linear and radial filtration models applicable for incompressibles cake filtration without fines invasion into porous rock at static and dynamic conditions. Simplified models omit the internal details of the filtration processes and, therefore, may lead to incorrect results if applied for conditions beyond the range of the experimental data used to obtain the empirical correlations. In many applications, the phenomenological models describing the mechanisms of the cake formation, based on the conservation laws and rate equations, are preferred for filter cake buildup involving small particle migration and deposition and cake compaction, because these models allow for extrapolation beyond the range of data used to test and calibrate the models. Chase and Willis (1992), Sherman and Sherwood (1993), and Smiles and Kirby (1993) presented partial differential models for compressible filter cakes without particle intrusion into filter media. Liu and Civan (1996) developed a partial differential model for incompressible filter cake buildup, and filtrate and fine particle invasion into petroleumbearing rock at dynamic condition. Tien et al. (1997) have developed a partial differential model for compressible filter cakes considering small particle retention inside the cake at static condition. The solutions of such partial differential models require complicated, time-consuming, and computationally intensive numerical schemes. To alleviate this difficulty, Corapcioglu and Abboud (1990), Abboud (1993), and Civan (1994c) have resorted to formulations facilitating cake thickness averaging. Consequently, the partial differential filtration models have been reduced to ordinary differential equations requiring much less computational effort. Such mathematically simplified models are particularly advantageous because ordinary differential equations can be solved rapidly, accurately,

344

Cake Filtration: Mechanism, Parameters, and Modeling

and conveniently by readily available and well-established numerical methods. The thickness-averaged models developed by Corapcioglu and Abboud (1990) and Abboud (1993) consider a constant porosity and linear cake filtration at static condition. The constant porosity assumption was justified by their filtration experiments because they used very dilute suspensions of particles and low pressure filtration, near the atmospheric pressure. Their models would not be applicable for high-pressure filtration of thick slurries considered by Tien et al. (1997). Further, they assumed the same values for the rates of deposition of the small and large particles over the progressing filter cake surface. This assumption is invalid for most applications. Civan (1998c) developed improved ordinary differential, linear, and radial filtration models incorporating the effects of filter cake compaction, small particle invasion, and retention at static and dynamic conditions. He applied filter cake thickness averaging by extending the methodology by Corapcioglu and Abboud (1990) and Civan (1994c, 1996a). The new models alleviate the aforementioned problems associated with the previous models. Civan (1998b) also derived the simplified models, considering that the particles and carrier fluid can be assumed incompressible for many practical applications. He presented the applications to radial and linear filtration processes and compared the results. The thicknessaveraged ordinary differential filter cake model reproduced the predictions of the Tien et al. (1997) partial differential model rapidly with less computational effort. In most filtration models, the flow through porous media is represented by Darcy’s law. Consequently, the applicability of these models is limited to filtration undergoing at low flow rate or low pressure difference conditions. Civan (1999b,c) also developed linear and radial filtration models incorporating a non-Darcy flow behavior and applicable under static and dynamic filtration conditions by extending Civan’s (1998b,c) model considering Darcy behavior. The non-Darcy behavior is represented by Forchheimer’s (1901) law. Civan (1998b) also developed and verified several methods for determining the parameters of these incompressible cake filtration models from experimental data by constructing diagnostic charts of linear types. However, some parameters should be either directly measured or determined by a least-squares regression of experimental data with the filtration models as demonstrated by Civan (1998b,c). In this chapter, Civan’s (1998b,c, 1999b,c) filtration models are reviewed. The filtration models are presented by including the

Cake Filtration: Mechanism, Parameters, and Modeling

345

non-Darcy effects. However, these models also apply for Darcy flow because the non-Darcy effects diminish at low flow rates.

12.2 INCOMPRESSIVE CAKE FILTRATION WITHOUT FINES INTRUSION In this section, models for interpretation and prediction of incompressible filter cake thickness, and filtrate volume and rate data for linear and radial filtration cases, at static and dynamic conditions, are presented. Methods for determining the model parameters from experimental filtration data are developed. Model-assisted analyses of three sets of experimental data demonstrate the diagnostic and predictive capabilities of the model. These models provide insight into the mechanism of incompressible cake filtration and offer practical means of interpreting experimental data, estimating the model parameters, and simulating the linear and radial filtration processes.

12.2.1 12.2.1.1

Linear Filter Cake Dynamic linear filter cake model

A schematic of the formation of a filter cake over a surface, such as a hydraulically created fracture, is shown in Figure 12-1. Figure 12-2 shows the simplified, one-dimensional linear cake filtration problem considered in this section. The locations of the mud slurry side cake surface, and

Figure 12-1. Filter cake buildup over a hydraulically created fracture surface (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers).

346

Cake Filtration: Mechanism, Parameters, and Modeling

Figure 12-2. Linear filter cake over a flat surface of a core plug (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers).

the slurry and effluent side surfaces of the porous medium are denoted, respectively, by xc  xw , and xe . Consistent with laboratory tests using core plugs, the cross-sectional area is denoted by a and the core length by L ≡ xe − xw . The mass balance of particles in the filter cake is given by (Civan, 1996a, 1998b)    − p s c dxc dt = Rps

(12-1)

where p is the particle density, t is time, s is the volume fraction of particles of the cake that can be expressed as a function of the porosity c of the cake as s = 1 − c

(12-2)

and Rps is the net mass rate of deposition of particles of the slurry to form the cake given by (Civan, 1998c, 1999b,c)   Rps = kd uc cp − ke s p c s − cr  U s − cr 

(12-3)

The first term on the right side of Eq. (12-3) expresses the rate of particle deposition as being proportional to the mass of particles carried toward the filter by the carrier fluid filtration volumetric flux uc , normal to the filter surface, given by  uc = q a

(12-4)

Cake Filtration: Mechanism, Parameters, and Modeling

347

where q is the carrier fluid filtration flow rate and a is the area of the cake surface. cp is the mass of particles contained per unit volume of the carrier fluid in the slurry. kd and ke denote the deposition and erosion rate coefficients, respectively. The deposition and erosion rate constants depend on the properties of the particles and carrier fluid, and the conditions of the slurry, such as particle concentration, flow rate, and pressure. The second term on the right side of Eq. (12-3) expresses the rate of erosion of the cake particles from the cake surface on the slurry side. Erosion takes place only when the shear stress s applied by the slurry to the cake surface exceeds a minimum critical shear stress cr necessary for detachment of particles from the cake surface. The shear stress is given by (Metzner and Reed, 1955) ′

s = k′ 8 n

(12-5)

where k′ and n′ are the consistency dyne/cm2 /sn  and flow (dimensionless) indices, which are equal to the fluid viscosity, , and unit for Newtonian fluids, respectively, and v is the tangential velocity of the slurry over the filter cake surface. The critical shear stress is dependent on various factors, including surface roughness and particle stickiness on the particle detachment (Civan, 1998b,c) and aging (Ravi et al., 1992). Ravi et al. (1992) have determined that the following equation proposed by Potanin and Uriev (1991) predicts the critical shear stress with the same order of magnitude accuracy of their experimental measurements:   (12-6) cr = H 24Dl2 ′

where H = 30 × 10−13 erg is Hamaker’s constant, D (cm) is the average particle diameter, and l (cm) is the separation distance between the particle surfaces in the filter cake. However, the values calculated from Eq. (12-6) are only a first-order accurate estimate because Eq. (12-6) has been derived from an ideal theory. The ideal theory does not take into account the effect of the above-mentioned other factors. Therefore, the actual value of the critical shear stress may be substantially different than that predicted by Eq. (12-6) using the particle size and separation distance data. Hence, Ravi et al. (1992) recommend experimental determination of the critical shear stress. Us −cr  is the Heaviside step function [Us −cr  = 0 when s < cr , and Us − cr  = 1 when s ≥ cr . s p c is the mass of particles contained per unit bulk volume of the slurry side cake surface. The erosion rate is related also to the particle

348

Cake Filtration: Mechanism, Parameters, and Modeling

content of the cake s p c and erosion cannot occur if there is no cake, that is if s p c = 00. Here, the cake properties are assumed constant.   s p c = s p = constant

(12-7)

Therefore, ke and s p c can be combined into one coefficient as (Civan, 1999b)    ke ≡ ke s p c

(12-8)

Then, Eq. (12-3) can be simplified to Civan’s (1999b) equation as Rps = kd uc cp − ke s − cr  U s − cr  H s 

(12-9)

 = xw − xc

(12-10)

in which Hs  = 0 when s = 0 (no cake) and Hs  = 1 when s > 0. The function Hs  can be expressed in terms of the cake thickness, , as H  = 0 when = 0 and H  = 1 when > 0, because s = 0 when = 0. The filter cake thickness is given by

Note the slurry side filter surface position xw is fixed. Substituting Eqs (12-2), (12-4), (12-9), and (12-10), Eq. (12-1) can be written as (Civan, 1998b)

where

 d dt = Aq − B 0 ≤ A = kd cp



(12-11)

 1 − c  p a

B = ke s − cr  U s − cr  H  



= ke s − cr  U s − cr  H  

1 − c  p

(12-12) 

(12-13)

The initial condition for Eq. (12-11) is

= 0 t = 0

(12-14)

Cake Filtration: Mechanism, Parameters, and Modeling

349

The rapid filtration flow of the carrier fluid through the cake and filter can be expressed by Forchheimer’s (1901) equation −

p

= u + u2 x K

(12-15)

The inertial flow coefficient is given by the Liu et al. (1995)   = 292 × 104  K

(12-16)

where  is the inertial flow coefficient in cm−1  K is the permeability in Darcy, and  is the tortuosity (dimensionless). Substituting Eq. (12-4) into Eq. (12-15) yields −

p

 = q + 2 q2 x aK a

(12-17)

As explained by Civan (1998b, c), the instantaneous carrier fluid filtration flow rate q is the same everywhere in the cake and filter irrespective of whether the process is undergoing a constant pressure or a constant rate filtration. In the following, the formulations for variable and constant rate filtration processes are derived. For variable rate filtration occurring under an applied pressure difference, integrating Eq. (12-17) for conditions existing prior to and during the process of formation of a filter cake leads to, respectively, pc − pe =

qo Lf f Lf q02 + aKf a2

(12-18)

and pc − pe = pc − pw  + pw − pe  

K = 1+ f Kc Lf



 c f Lf q 2 q Lf + 1+ aKf f Lf a2

(12-19)

Consequently, eliminating pc − pe  between Eqs (12-18) and (12-19), and then solving for q, yields for Darcy flow f = c = 0:

q = −   

(12-20)

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Cake Filtration: Mechanism, Parameters, and Modeling

and for non-Darcy flow:

q= in which

  2 − 4 a y − + 

(12-21)

2 a

 a = f Lf + c 

 a2

(12-22)



Lf Kf   = 1+ Kc Lf aKf  qo Lf f Lf qo2  + =− aKf a2

(12-23) (12-24)

Alternatively, eliminating pc − pe  between Eqs (12-18) and (12-19) and then solving for yields   2

Lf qo q f Lf qo2 q 1− + 1− 2 aKf qo a qo (12-25)

= c q 2

q + aKc a2 Notice that Eq. (12-25) yields = 0 when q = qo . Differentiating Eq. (12-25) with respect to time and then substituting into Eq. (12-11) yields   c q 2

q

Lf 2f Lf q + + − aKf a2 aKc a2    

Lf qo q f Lf qo2 q2 + 1− + 1− 2 (12-26) aKf qo a2 qo 

2c q

+ aKc a2



dq = dt



 q 2

q + c2 aKc a

2

Aq − B

The initial condition for Eq. (12-26) is q = qo  t = 0

(12-27)

Cake Filtration: Mechanism, Parameters, and Modeling

351

Substituting Eq. (12-21) and considering the initial condition given by Eq. (12-14), Eq. (12-11) can be solved using a numerical scheme, such as the Runge–Kutta–Fehlberg four (five) method (Fehlberg, 1969). Equations (12-26) and (12-27) can also be solved numerically using the same method. The relationships between filtrate flow rate and cumulative filtrate volume are given by Q=

t

qdt

(12-28)

o

 q = dQ dt

(12-29)

Note that Eqs (12-25) and (12-26) simplify to Eqs (12-30) and (12-34) (Civan, 1998b), respectively, when the inertial effects are neglected, that is, for f = c = 0.

where

and

subject to



= C q−D C = qo D  D = Lf Kc Kf  uc = q a     dq dt = − 1 C q 2 Aq − B q = qo  t = 0

(12-30)

(12-31) (12-32) (12-33)

(12-34)

(12-35)

Then, the analytical solutions for the filtrate flow rate and cumulative volume as well as the filter cake thickness can be derived as demonstrated by Civan (1998b, 1999b).

352

Cake Filtration: Mechanism, Parameters, and Modeling

The analytical solution of Eqs (12-34) and (12-35) is (Civan, 1998b) ⎧ ⎪ ⎪ ⎨

⎫ ⎤ A 1 ⎪ ⎪ − ⎥ 1 1⎬ C A ⎢ B q ⎢ ⎥ + − ln ⎣ t=− A 1 ⎦ q qo ⎪ B⎪ ⎪B ⎪ ⎩ ⎭ − B qo ⎡

(12-36)

Eliminating q between Eqs (12-30) and (12-36) yields another expression as ⎧ ⎫ ⎡ A +D ⎤ ⎪ ⎪ ⎨ ⎬ − C A ⎢ B ⎥ − o C ln ⎣ t=− ⎦+ A o + D B⎪ C ⎪ ⎩B ⎭ − B C

(12-37)

in which, usually, o = 0 at t = 0 (i.e., no initial filter cake). Filtrate invasion rate decline function Makardij et al. (2002) developed a semi-theoretical model for the unsteady-state filtrate invasion flux during dynamic filtration. Their derivation is presented with modifications for the purpose of this chapter. Considering the decline of invasion rate by particle deposition to form a cake and increase of invasion rate by cake erosion owing to fluid shear, the rate of change of the filtrate flow through the system of the filter cake and porous medium is given by  −dq dt = kd cp q − ke Ren 

t>0

(12-38)

Here kd and ke are the particle deposition and erosion rate coefficients, cp is the particle concentration of the fluid injected at the wellbore, and b = q is the limiting flow rate. This equation considers that the injection rate decline with time is directly related to the particle concentration of the injected fluid and the deviation of the instantaneous injection rate from the limiting injection rate. Thus, Eq. (12-38) can be rearranged as

where

 −dq dt = aq − b a = kd cp  b = ke Ren

t>0

  kd cp 

t>0

(12-39)

(12-40)

Cake Filtration: Mechanism, Parameters, and Modeling

353

The initial flow rate is given by t=0

q = qo 

(12-41)

At dynamic equilibrium  dq dt = 0

q=b

(12-42)

Thus, the analytic solution is obtained as q t = b + qo − b exp−at 12.2.1.2

(12-43)

Static Linear Filter Cake Model

Equation (12-37) is different from Eq. 7-96 of Collins (1961) because Collins did not consider the filter cake erosion. Therefore, Collins’ equation applies for static filtration. To obtain Collins’ result, ke = 0 or B = 0 must be substituted in Eq. (12-11). Thus, eliminating q between Eqs (12-30) and (12-11), and then integrating, yields the following equation for the filter cake thickness (Civan, 1998b):    2 1 2 + D = ACt

(12-44)

which results in Eq. 7-96 of Collins (1961) by invoking Eq. (12-18) for f = 0, Eqs (12-31), (12-32), and (12-12) and expressing the mass of suspended particles per unit volume of the carrier fluid in terms of the volume fraction, p , of the particles in the slurry according to cp = p p

  1 − p

(12-45)

Civan (1998b) derived the expressions for the filtrate flow rate and the cumulative filtrate volume by integrating Eq. (12-34) for B = 0 and applying Eq. (12-28), respectively, as q = qo and



   1 + 2Aqo2 C t

    Q = C A q −1 − qo−1

(12-46)

(12-47)

354

Cake Filtration: Mechanism, Parameters, and Modeling

Alternatively, substituting Eq. (12-46) into Eq. (12-47) yields          1 + 2Aqo2 C t − 1 Q = C qo A

(12-48)

q = qo exp −t

(12-49)

Equation (12-46) expresses that the filtrate rate declines with time due to static filter cake buildup. Donaldson and Chernoglazov (1987) used an empirical decay function:

in which  is an empirically determined coefficient. For constant rate filtration, Eq. (12-11) subject to Eq. (12-14) can be integrated numerically for varying shear stress, s . When the shear stress is constant or does not vary significantly, an analytical solution can be obtained as (Civan, 1999b)

= Aq − Bt

(12-50)

in which B = 0 because  = 0 for static conditions and B = 0 because  = 0 for dynamic conditions. The cumulative filtrate volume is given by, for both the static and the dynamic filtration, (12-51)

Q = qt

Then, the pressure difference pc − pe  or the slurry injection pressure pc , when the back pressure at the effluent side of the porous filter media pe is prescribed, can be calculated by Eq. (12-19). The following conventional filtration equation (Hermia, 1982; de Nevers, 1992) can be derived by invoking Eq. (12-51) into Eq. (12-47): A 1 t = Q+ Q C qo

(12-52)

Another useful data interpretation scheme can be developed as follows. First, integrating Eq. (12-11) subject to Eq. (12-14) yields

= AQ −

t 0

Bdt

(12-53)

Cake Filtration: Mechanism, Parameters, and Modeling

Second, Eq. (12-19) is rearranged as   c q

Lf f Lf q pc − pe = + + + 2

q aKf a2 aKc a

355

(12-54)

Equations (12-53) and (12-54) are applicable regardless of whether the flow rate is constant or variable. A plot of pc − pe /q vs. of the experimental data can be used to determine the values of the unknown parameters. Alternatively, a special form of Eq. (12-54) can be derived by substituting Eq. (12-53) into Eq. (12-54) for constant rate filtration q = ct, applying Eq. (12-28), and assuming c = ct. as:   pc − pe c q

Lf f Lf q A − B/q Q (12-55) + + 2 + = q aKf a2 aKc a A plot of pc − pe /q vs. Q of the experimental data can be used to determine the values of the unknown parameters. 12.2.2 12.2.2.1

Radial Filter Cake Dynamic radial filter cake model

A schematic of the formation of a filter cake over the sand face during over-balanced mud circulation in a wellbore is shown in Figure 12-3. Figure 12-4 is a quadrant areal view of the problem. The radii of the mud slurry side cake surface, the sand face over which the cake is built up, and the external surface considered for the region of influence are denoted by rc  rw , and re , respectively. The formation thickness is h. The particle mass balance equation is given by (Civan, 1994c, 1998b)    − p s c drc dt = Rps

(12-56)

The filter cake thickness is given by

= rw − rc

(12-57)

Rps is given by Eq. (12-9). The slurry shear stress at the cake surface is given by the Rabinowitsch–Mooney equation (Metzner and Reed, 1955)   n′ s = k′ 4 rc

(12-58)

356

Cake Filtration: Mechanism, Parameters, and Modeling

Figure 12-3. Mud-cake buildup over a wellbore sandface (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 12-4. Radial filter cake over a wellbore sandface (after Civan, ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers).

Cake Filtration: Mechanism, Parameters, and Modeling

357

The carrier fluid filtration flux uc at the filter cake surface in terms of the carrier fluid filtration flow rate is given by uc =

q 2rc h

(12-59)

Substituting Eqs (12-2), (12-9), (12-57), and (12-59) into Eq. (12-56) results in (Civan, 1999b) d q =A − B 0 ≤ ≤ rw dt rw −

(12-60)

where kd cp 2h 1 − c  p

A=

(12-61)

and B is given by Eq. (12-13). The initial condition for Eq. (12-60) is

= 0 t = 0

(12-62)

Forchheimer’s (1901) equation for radial flow of the carrier fluid reads as −

p

= u + u2 r K

(12-63)

The radial volumetric flux of the carrier fluid is given by u=

q 2rh

(12-64)

Thus, substituting Eq. (12-64) into Eq. (12-63) results in −

p

q  ! q "2 = + r 2hK r 2h2 r

(12-65)

Integration of Eq. (12-65) for conditions prevailing prior to and during filter cake formation leads to the following expressions, respectively (Civan, 1999b) pc − pe =

qo

ln 2hKf



 1 f qo2 1 re + − rw 2h2 rw re

(12-66)

358

Cake Filtration: Mechanism, Parameters, and Modeling

and

    Kf re r q

ln + ln w pc − pe = 2hKf rw Kc rc    1 c 1 1 f q 2 1 − + − + 2h2 rw re f rc rw

(12-67)

Thus, eliminating pc − pe  between Eqs (12-66) and (12-67), substituting Eq. (12-57), and then solving for q, yields for Darcy flow f = c = 0:

q = −    (12-68) and for non-Darcy flow:

q= in which



  2 − 4  y − +  2 



  1 1 1 1  f = − + c − rw re rw − rw 2h2    

Kf re rw  = ln + ln 2hKf rw Kc rw −     f qo2 1 1 re qo

 ln + − =− 2hKf rw 2h2 rw re 

(12-69)

(12-70) (12-71) (12-72)

Substituting Eq. (12-69) and considering the initial condition given by Eq. (12-62), Eq. (12-60) can be solved using a numerical scheme, such as the Runge–Kutta–Fehlberg four (five) method (Fehlberg, 1969). The cumulative filtrate volume is given by Eq. (12-28). The pressure difference pc − pe , or the slurry injection pressure pc when the back pressure pe is prescribed, can be calculated by Eq. (12-67). When the inertial flow terms are negligible, equating Eqs (12-66) and (12-67) and rearranging leads to (Civan, 1998b)             ln rw rc = q0 q − 1 Kc Kf ln re rw (12-73) Equation (12-73) can be written as     rc rw = exp −C q + D

(12-74)

Cake Filtration: Mechanism, Parameters, and Modeling

359

where  C = q0 D Kf

(12-75)

where q0 is the injection rate given by Eq. (12-66) for f = 0 before the filter cake buildup and       D = Kc Kf ln re rw

(12-76)

Thus, substituting Eqs (12-57) and (12-74) into Eq. (12-60) and rearranging yield the filtration flow rate equation as (Civan, 1998b) dq/dt = −1/Cq 2 Aq expC/q − D − B expC/q − D

(12-77)

subject to the initial condition given by q = qo  t = 0

(12-78)

The wall shear stress is calculated by Eq. (12-58) for the varying cake radius, rc = rc t. The filter cake thickness is calculated by means of Eqs (12-57) and (12-74). Equations (12-77) and (12-78) can be solved numerically using an appropriate method such as the Runge–Kutta method. However, for thin cakes, it is reasonable to assume that the wall shear stress is approximately constant, because rc  rw . Then, Eq. (12-77) can be integrated as (Civan, 1998b) t = −C

q #

qo

       $−1 q 2 exp C q − D Aq exp C q − D − B dq (12-79)

For constant rate filtration, Eq. (12-60) subject to Eq. (12-62) can be integrated numerically for varying shear stress s . When the filter cake is thin, the variation of the shear stress s by the cake radius rc can be neglected and an analytical solution can be obtained as for dynamic filtration conditions B = 0 (Civan, 1999b):

Aq t = − + 2 ln B B



rw − − Aq B rw − Aq B

(12-80)

360 12.2.2.2

Cake Filtration: Mechanism, Parameters, and Modeling

Static radial filter cake model

The solution for static filtration conditions B = 0 is obtained as (Civan, 1999b):  1 2 1 r − t= Aq w 2

(12-81)

Equations (12-80) and (12-81) apply irrespective of whether the flow is Darcy or non-Darcy. Other relationships can be derived by a similar procedure to those presented in the static linear filter cake model.

12.2.3 Determination of Model Parameters and Diagnostic Charts The majority of the reported filtration studies have not made attempts at measuring a full set of measurable parameters. The filtration models presented in this chapter may provide some guidance for the types of parameters needed for simulation. As listed in Table 12-1, Civan’s (1998b, 1999b) filtration models require the values of 20 parameters for simulation. Only five of these parameters may not be directly or conveniently measurable with the conventional techniques. These are the permeability Kc and porosity c of the filter cake, and the deposition and erosion rate constants kd and ke , respectively, and the critical shear stress cr for the particles. However, given the experimental measurements of the filtrate volume Qcm3 , or rate qcm3 /s, and the filter cake thickness as functions of the filtration time t, some of these parameters can be determined by means of the diagnostic charts constructed as described in the following. These are presented separately for the linear and radial filtration processes obeying Darcy’s law according to Civan (1998b).

12.2.3.1

Linear filtration

Case 1: Dynamic Linear Filtration

A plot of Eq. (12-11) for d /dt vs. q yields a straight line. Substituting the slope A and intercept −B of this line into Eqs (12-12) and (12-13)

361

Cake Filtration: Mechanism, Parameters, and Modeling Table 12-1 Data for the Laboratory Filtration Applications

Parameters

Suspension type

Radial Flow (Fisk et al., 1991) Seawater-based partially hydrolyzed polyacrylamide drilling mud

Linear Flow (Jiao and Sharma, 1994) Freshwater bentonite suspension

Linear Flow (Willis et al., 1983)

Lucite in water suspension

Filter permeability, Kf (D)

6∗

0.104†

–

Cake permeability, Kc (D)

135 × 10−6

21 × 10−4

–

Cake porosity, c

0.40‡

0.40‡

0.388§

Filter length, Lf (cm)

–

20.34†

–

Filter diameter, D (cm)

–

2.54

–

Slurry injection side filter radius, rw (cm)

2.5∗

–

–

Filtrate outlet side filter radius, re (cm)

3.8∗

–

–

Filter width, h (cm)

1.9∗

–

–

Filtrate density, w g/cm3 

1.0

1.0

0.997§

Particle density, p g/cm3 

2.5‡

2.5‡

1.18§

Particle mass per carrier fluid volume, cp g/cm3 

0.56∗

0.04†

0.055§

Deposition rate constant, kd

1.1

4.3

–

Erosion rate constant, ke (s/cm)

3 × 10−6

74 × 10−7

–

Critical shear stress, cr dyne/cm2 

0.5

5.0

–

Filtrate (water) viscosity, (cp)

1.0

1.0

0.969§

Consistency constant, k′ n′ dyne/cm2 /s 

8.0†

8.0†

–

Flow index, n′

0.319†

0.319†

–

†

(table continued on next page)

362

Cake Filtration: Mechanism, Parameters, and Modeling Table 12-1 (Continued)

Parameters

Suspension type

Radial Flow (Fisk et al., 1991)

Linear Flow (Jiao and Sharma, 1994)

Seawater-based partially hydrolyzed polyacrylamide drilling mud

Freshwater bentonite suspension

Linear Flow (Willis et al., 1983)

Lucite in water suspension

Slurry tangential velocity, v (cm/s)

125∗

8.61†

–

Slurry application pressure, pc (atm)

34∗

6.89†

1.7§

Filter outlet side back pressure, pe (atm)

1

1

1§

∗

Data from Fisk et al. (1991). Data from Jiao and Sharma (1994). ‡ Data assumed. § Data from Willis et al. (1983). After Civan, F., 1998b; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved. †

yields, respectively, the following expressions for the particle deposition and erosion rate constants:  kd = Aa 1 − c  p cp  ke = B 1 − c  p  − cr 

(12-82) (12-83)

In dynamic filtration, the filter cake thickness attains a certain limit value,

 , when the particle deposition and erosion rates equate. Simultaneously, the filtration rate also reaches a limiting value, determined by Eq. (12-11) as  q = B A

(12-84)

At this condition, Eq. (12-30) yields the limiting value of the filter cake thickness as 

 = C q − D

(12-85)

Cake Filtration: Mechanism, Parameters, and Modeling

363

Consequently, substituting Eqs (12-31), (12-32), (12-12), and (12-13) for A, B, C, and D into Eqs (12-84) and (12-85) leads to the following relationships for the cake permeability and the ratio of the erosion and deposition rate constants, respectively, as     Kc =  Kf Lf qo q − 1   ke kd = cp q a  − cr 

(12-86) (12-87)

Equation (12-34) can be rearranged in a linear form as d − dt

 1 dq 1 A B = 2 =− q+ q q dt C C

(12-88)

Thus, the intercept B/C and slope −A/C of the straight-line plot of Eq. (12-88) can be used with Eqs (12-31), (12-32), (12-12), and (12-13) to obtain the following expressions:    B C cp ke =   kd A C  − cr  a    A C 1 − c  p aqo q  kd = cp qo − q 

(12-89)

(12-90)

Comparing Eqs (12-87) and (12-89) yields an alternative expression for determination of the limit filtrate rate as      q = B C A C

(12-91)

Equation (12-91) can be used to check the value of q obtained by Eq. (12-84). Equation (12-86) can be used to determine the filter cake permeability, Kc . Equations (12-82) and (12-87) or (12-89) and (12-90) can be used to calculate the particle deposition and erosion rate coefficients kd and ke , if the cake porosity c and the critical shear stress cr are known. c can be measured. cr can be estimated by Eq. (12-6), but the ideal theory may not yield a correct value as explained previously in this chapter. Therefore, Ravi et al. (1992) suggested that cr should be measured directly.

364

Cake Filtration: Mechanism, Parameters, and Modeling

Case 2: Static Linear Filtration

A further simplified form of Eq. (12-55) can be obtained as following when the inertial flow effects are neglected c = f = 0, a static filtration is considered (B = 0 and kd = 10), and Eq. (12-12) for A is substituted (Khatib, 1994):  

cp p pc − pe Lf Q (12-92) + 2 ≡ = q q aKf a Kc 1 − c  p The intercept and the slope of the straight-line fit to p/q vs. Q experimental data yields the following relationships (Khatib, 1994):

Lf = intercept aKf

cp a2 Kc 1 − c  p

= slope

(12-93)

(12-94)

Equation (12-93) allows the determination of the formation permeability Kf . Equation (12-94) is an equation containing two unknowns Kc and c . If a constitutive equation relating Kc and c is available, such as those given in Chapter 5, or the cake porosity c can be measured by other means, then Kc can be calculated as illustrated by Khatib (1994). 12.2.3.2

Radial filtration

Case 1: Dynamic Radial Filtration

Given the filter cake thickness , the progressing cake surface radius rc can be calculated by Eq. (12-57). Then a straight-line plot of lnrc /rw  vs. 1/q data according to Eq. (12-74) yields the values of C and D as the slope and intercept of this line, respectively. A straight-line plot of d /dt vs. q/rw −  data according to Eq. (12-60) yields the values of A and B as the slope and intercept of this line, respectively. At static filtration conditions, = 0 and  = 0 according to Eq. (12-58). Therefore, B = 0 according to Eq. (12-13). Consequently, substituting B = 0 and Eq. (12-75), Eq. (12-77) can be expressed in the following linear form:    #     $ = ln −q −3 dq dt ln d dt 1 2q 2        (12-95) = ln A C − 2CKf qo + 2C q

Cake Filtration: Mechanism, Parameters, and Modeling

365

Thus, a straight-line plot of ln−q −3 dq/dt vs. 1/q yields the values of 2C and ln A/C − 2CKf /qo as the slope and intercept of this line, respectively. This allows for determination of the A and C coefficients only. The determination of a full set of A B C, and D from Eqs (12-60) and (12-77) requires both the filtrate flow rate (or volume) and the cake thickness vs. the filtration time data. Once these coefficients are determined, then their values can be used in Eqs (12-61), (12-13), (12-75), and (12-76) to determine the values of the deposition and erosion rate constants kd and ke . The discussion of the linear filtration about the determination of cr by Eq. (12-6) is valid also in the radial filtration case. At dynamic equilibrium, the filter cake thickness and the filtrate flow rate attain certain limiting values  and q . Then, substituting Eq. (12-57) into Eqs (12-60) and (12-74) yields the following relationships, respectively:    Aq = B 1 −  rw     1 −  rw = exp −C q + D

(12-96) (12-97)

The filter cake permeability is determined by Eq. (12-76) as     Kc = DKf ln re rw

(12-98)

The equations and the linear plotting schemes developed in this section allow for determination of the parameters of the filtration models, mentioned at the beginning of this section, from experimental filtrate flow rate (or volume) and/or filter cake thickness data. The remaining parameters should be either directly measured or estimated. In the following applications, the best estimates of the missing data have been determined by adjusting their values to fit the experimental data. This is an exercise similar to several other studies, including Liu and Civan (1996) and Tien et al. (1997). They have resorted to a model-assisted estimation of the parameters because there is no direct method of measurement for some of these parameters. Case 2: Static Radial Filtration

The relationships for static radial filtration are similar to those presented for static linear filtration and can be derived in a similar manner.

366

12.2.4

Cake Filtration: Mechanism, Parameters, and Modeling

Applications

The numerical solutions of the present models require the information on the characteristics of the slurries, particulates, carrier fluids, filters and filter cakes, the actual conditions of the tests conducted, and the measurements of all the system parameters and variables. The reported studies of the slurry filtration have measured only a few parameters and the filtrate volumes or rates, and do not offer a complete set of suitable data that is needed for full-scale experimental verification of the present models. Civan (1998b) used the Willis et al. (1983) and Jiao and Sharma (1994) data for linear filtration, and the Fisk et al. (1991) data for radial filtration after making reasonable assumptions for the missing data. The data is presented in Table 12-1 in consistent Darcy units, which are more convenient for flow through porous media. 12.2.4.1

Linear filtration applications

Jiao and Sharma (1994) carried out linear filtration experiments using concentrated bentonite suspensions. They only measured the filtrate volume and predicted the filter cake thickness using a simple algebraic model. In Figures 12-5–12-7, their data are plotted according to the linear plotting schemes presented in Section 12.2.3 for determination of parameters. As

Figure 12-5. Correlation of Jiao and Sharma (1994) experimental data (after Civan, 1998; reprinted by permission of the AIChE, ©1998a AIChE. All rights reserved).

Cake Filtration: Mechanism, Parameters, and Modeling

367

0.2

Cake thickness, δ, cm

y = 0.0034x – 0.0076 0.15

R 2 = 0.949

0.1 Predicted 0.05

0

Linear (Predicted)

10

20

30 40 50 Recriprocal filtrate flow rate, 1/q, min/ml

60

Figure 12-6. Correlation of Jiao and Sharma (1994) predicted filter cake thickness data (after Civan, 1998; reprinted by permission of the AIChE, ©1998a AIChE. All rights reserved).

Variation of cake thickness, dδ/dt, cm/min.

0.002

0.0015 y = 0.0229x – 0.0003 R 2 = 0.9873 0.001

Predicted Linear (Predicted)

0.0005

0

0

0.02

0.04 0.06 Filtrate flow rate, q, ml/min

0.08

0.1

Figure 12-7. Correlation of Jiao and Sharma (1994) predicted filter cake thickness data (after Civan, 1998a; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

368

Cake Filtration: Mechanism, Parameters, and Modeling

can be seen from these figures, the coefficients of Eqs (12-88), (12-30), and (12-11) obtained by the least-squares regression method and the corresponding coefficients of regression are given, respectively, by    A C = 8297 min cm6  B C = 01136 cm−3  R2 = 08713 (12-99)  C = 00034 cm4 min D = 00076 cm, R2 = 0949  A = 00229 cm−2  B = 00003 cm min R2 = 09873

(12-100)

(12-101)

The coefficients of regressions very close to 1.0 indicate that the present equations closely represent the data. The coefficient of regression R2 = 08713 indicated by Figure 12-5 and Eq. (12-99) is lower than those indicated by Figures 12-6 and 12-7 and Eqs (12-100) and (12-101), inferring the possibility of larger measurement errors involved in the filtrate volume data. Another source of errors may be due to the threepoint finite difference numerical differentiation of the filtrate volume data considered to obtain the filtrate flow rate data used to construct Figure 12-5. The data necessary for Figure 12-5 were obtained by a series of numerical procedures, first to calculate q = dQ/dt from the filtrate volume Q data, and then l/q and d/dt l/q . The initial filtrate volume rate is obtained as q0 = 0096 mL/min by a three-point forward differentiation of the measured, initial filtrate volume data. This data is expected to involve a larger error because of the possibility of relatively larger errors involved in the early filtrate volume data. The noisy data had to be smoothed prior to numerical differentiation, which may have introduced further errors. Because of the propagation of the significantly larger measurement errors involved in the early filtrate volume data, the first two of the d/dt l/q values degenerated and deviated significantly from the expected straight-line trend. Therefore, these two data points formed the outliers for linear regression and had to be discarded. Substituting the values given in Eq. (12-99) into Eq. (12-91) yields the limiting filtrate flow rate as q = 0014 mL/min. On the other hand, substituting the values given in Eq. (12-101) into Eq. (12-84) yields q = 0013 mL/min. These two values obtained from the filtrate flow rate and

Cake Filtration: Mechanism, Parameters, and Modeling

369

cake thickness data, respectively, are very close to each other. The limiting filtrate volume rate q estimated by an extrapolation of the derivatives of the filtrate volume data beyond the range of the experimental data is q = 0017 mL/min and close to the values obtained by the regression method. This is an indication of the validity of the filtration model. Using q = 0014 mL/min in Eq. (12-30) yields the limiting filter cake thickness as  = 024 cm. The predicted cake thickness data of Jiao and Sharma (1994) indicates a value of approximately 0.17 cm. Therefore, their prediction of the limiting filter cake thickness appears to be an underestimate compared to the 0.24 cm value obtained by Civan (1998b). The above-obtained values can now be used to determine the values of the model parameters as follows. The filter cake permeability can be calculated by Eq. (12-86). Equations (12-82), (12-83), (12-87), (12-89), and (12-90) form a set of alternative equations to determine the deposition and erosion rate constants, kd and ke , respectively. Here, Eqs (12-82) and (12-87) were selected for this purpose. However, Jiao and Sharma (1994) do not offer any data on the cake porosity c and the critical shear stress cr necessary for detachment of the particles from the progressing cake surface. Therefore, the c and cr parameters had to be estimated and used with Eqs (12-82) and (12-87) to match the filtration data over the period of the filtration process. Then, the c and cr values obtained this way were used in Eqs (12-82) and (12-87) to calculate the kd and ke values. Using the slurry tangential velocity of v = 861 cm/s, the typical particle diameter of D = 25 × 10−4 cm, and the particle separation distance of l = 2 × 10−7 cm in Eq. (12-5), the critical shear stress for particle detachment is estimated to be cr = 125 × 103 dyne/cm2 . Whereas, the prevailing shear stress calculated by Eq. (12-5) is only  = 16 dyne/cm2 . Under these conditions, theoretically the cake erosion should not occur because  cr and the cake erosion occurred in the actual experimental conditions of Jiao and Sharma (1994). In view of this discussion, it becomes apparent that the theoretical value obtained by Eq. (12-6) is not realistic. The Jiao and Sharma (1994) data and the missing parameter values, which have been approximated by fitting the experimental data, are given in Table 12-1. The results presented in Figure 12-8 indicate that the model represents the measured filtrate volumes over the complete range of 600 min of filtration time closely. However, they did not measure the cake thickness, but predicted it using a simple algebraic model. As shown in Figure 12-9, the cake thicknesses predicted by Jiao and Sharma (1994) and Civan (1998b) are close to each other. Willis et al. (1983) conducted linear filtration experiments using a suspension of lucite in water. As shown in Table 12-1, they reported only a few parameter values. They only provide some measured filtrate flow rate and cake thickness data. However, the filtration time data is missing. Therefore, a full-scale simulation of their filtration process as a function of time could not be carried out. Only the linear plotting of the measured data according to Eq. (12-30) could be accomplished. As indicated by Figure 12-10, the best linear fit of Eq. (12-30) with the

Figure 12-8. Comparison of the predicted and measured filtrate volumes for linear filtration of freshwater bentonite suspension (after Civan, 1998; reprinted by permission of the AIChE, ©1998a AIChE. All rights reserved).

371

Cake Filtration: Mechanism, Parameters, and Modeling

Figure 12-9. Comparison of the predicted cake thicknesses for linear filtration of freshwater bentonite suspension (after Civan, 1998a; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Cake thickness, δ, m

0.2

0.15

y = 0.0655x – 0.0006 R2 = 0.9921

0.1 Experimental data

0.05

0

0.5

Linear (Experimental data)

0.5 0.5 2 Reciprocal filtrate flow rate, 1/q, 104 s/m3

2.5

Figure 12-10. Correlation of Willis et al. (1983) measured filter cake thickness data (after Civan, 1998; reprinted by permission of the AIChE, ©1998a AIChE. All rights reserved).

372

Cake Filtration: Mechanism, Parameters, and Modeling 450

Pressure drop/rate

400 350 300

µ Lf

250

A Kf Slope =

200

µρ L W A2ρs (1 – φc) Kc

150 100 50

0

100

200 300 Cum. injected volume, cc

400

Figure 12-11. Correlation of core flood test data obtained using an iron hydroxide/ bentonite system (after Khatib, ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers).

least-squares method has been obtained with a coefficient of regression of R2 = 09921, very close to 1.0. This reconfirms the validity of the filtration model. Khatib (1994) conducted static linear filtration experiments in a compression-permeability cell using various solid particles, including the iron hydroxide/bentonite mixtures, in water. Figure 12-11 by Khatib (1994) demonstrates the application of Eqs (12-92), (12-93), and (12-94) for interpretation of the core flood data obtained using a mixture of 75% iron hydroxide and 25% bentonite by volume in the injected water. For illustration purposes, Khatib (1994) assumed a cake porosity of c = 025 and calculated the permeability of the filter cake using Eq. (12-94) with the slope value obtained in Figure 12-11. 12.2.4.2

Radial filtration applications

Fisk et al. (1991) conducted radial filtration experiments using a seawaterbased partially hydrolyzed polyacrylamide mud. They provide the measured dynamic and static filtrate volumes vs. filtration time data. Their static filtration data contains only three distinct measured values. This data is insufficient to extract meaningful information on the values of the A and C coefficients by regression of Eq. (12-95), because the calculation of ln−q −3 dq/dt requires a two-step, sequential numerical differentiation – first to obtain the filtrate flow rate q = dQ/dt by differentiating

Cake Filtration: Mechanism, Parameters, and Modeling

373

Figure 12-12. Comparison of the predicted and the measured filtrate volumes for radial filtration of a seawater-based partially hydrolyzed polyacrylamide drilling mud (after Civan, 1998; reprinted by permission of the AIChE, ©1998a AIChE. All rights reserved).

the filtrate volume Q, and then differentiating q to obtain dq/dt. On the other hand, their dynamic filtration data is limited to the filtrate volume. As explained in Section 12.2.3 on the determination of parameters, the determination of all coefficients of A B C, and D by means of Eqs (12-60) and (12-77) requires both the filtration volume and the filter cake thickness measurements. Therefore, the Fisk et al. (1991) radial filtration data has several missing parameter values, which had to be approximated as given in Table 12-1. Figure 12-12 shows that the model predicts the measured dynamic and static filtrate volumes with reasonable accuracy in view of the uncertainties involved in the estimated values of the missing data. Fisk et al. (1991) did not report any results on the filter cake thickness and therefore a comparison of the cake thicknesses cannot be made in the radial filtration case. 12.2.5 Comments on Models for Incompressible Cakes without Fine Particle Intrusion The models presented in this section offer practical means of interpreting experimental data, estimating the model parameters, and simulating the linear and radial incompressive cake filtration processes at static

374

Cake Filtration: Mechanism, Parameters, and Modeling

and dynamic filtration conditions. The simplified forms of these models conform with the well-recognized simplified models reported in the literature. These models are capable of capturing the responses of typical laboratory filtration tests while providing insight into the governing mechanisms.

12.3 COMPRESSIVE CAKE FILTRATION INCLUDING FINES INVASION The applicability of the majority of the previous models, such as those by Corapcioglu and Abboud (1990), Liu and Civan (1996), Tien et al. (1997) and Civan (1998c), is limited to low rate or low pressure difference filtration processes because these models facilitate Darcy’s law to describe flow through porous media. However, filtration at high flow rates and high overbalance pressure differences may involve some inertial flow effects, especially during the initial period of the filter cake formation. The initial nonlinear relationships of the filtrate volume vs. the square root of time is mostly attributed to invasion and clogging of porous media by fine particles during filtrate flow into porous media prior to filter cake formation. The cumulative volume of the carrier fluid (filtrate) lost into porous media during this time is usually referred to as the spurt loss (Darley, 1975). Based on an order of magnitude analysis of the relevant dimensionless groups of the general mass and momentum balances of the multiphase systems involving the cake buildup, Willis et al. (1983) concluded that nonparabolic filtration behavior is not caused by non-Darcy flow. Instead, it is a result of the reduction of the permeability of porous media by clogging by fine particles. This conclusion is justified for their experimental conditions; however, some reported experimental data appear to involve a non-Darcy flow effect during the initial period of filter cake buildup depending on the magnitude of the filtration flow rate and/or the applied pressure difference. In the following, linear and radial compressible cake filtration models are presented according to Civan (1998c, 1999c). These models are more generally applicable because of the following salient features: 1. A cake-thickness-averaged formulation leads to a convenient and computationally efficient representation of the filtration processes by means of a set of ordinary differential equations.

Cake Filtration: Mechanism, Parameters, and Modeling

375

2. The nonhomogeneous-size particles of the slurry are classified into the groups of the large and fine particles, and the large particles form the cake matrix and the fine particles deposit inside the cake matrix. 3. The flow through porous cake and formation, which acts as a filter, is represented by Forchheimer’s (1901) law to account for the inertial flow effects encountered during the early filtration period. 4. The dynamic and static filtration conditions encountered with and without the slurry flowing tangentially over the cake surface, respectively, are considered. 5. The variation of the filter cake porosity and permeability by compaction due to the drag of the fluid flowing through the cake matrix and deposition of fine particles within the cake matrix is considered. 6. An average fluid pressure is used to determine the fluid drag force applied to the cake matrix. 7. The formulations are presented for general purposes, but applied for commonly encountered cases involving incompressible particles and carrier fluids. 8. The constant and variable rate filtration processes can be simulated. The models presented in this section incorporate empirical constitutive relationships for the permeability and porosity variations of compressible cakes retaining fine particles. The simulation of a series of filtration scenarios demonstrates the parametric sensitivity of the model. It is shown that permeability impairment by fine particles reentrainment and pore throat clogging in the filter cake is increasingly induced by cake compression. Further, constant pressure filtration limits the filtrate invasion more effectively than constant rate filtration, and the non-Darcy flow effect is more significant during the initial period of the filter cake formation. The cake formation models developed in this section can be used for predicting the effects of the compressible filter cakes involving the drilling muds and fracturing fluids. 12.3.1

Radial Filtration Formulation

Consider that slurry is applied over the inner surface of a hole drilled through a porous disk of certain thickness, representing the region of influence in the near-wellbore formation, and the filtrate leaves from its outer surface (see Figure 12-4). The filter cake is located between the inner hole surface radius rw (cm), over which the cake is formed, and

376

Cake Filtration: Mechanism, Parameters, and Modeling

the slurry side cake surface radius rc (cm), and its thickness is denoted by = rw − rc . The external surface radius of the disk from which the filtrate leaves is re (cm) and the disk thickness is indicated by h(cm), such that the area of the inner hole surface over which the cake is formed is 2rw h. The slurry flows over the cake surface at a tangential or cross-flow velocity of vf (cm/s) and the filtrate flows into the formation at a filtration velocity of uf cm3 /cm3 s normal to the hole face due to the overbalance of the pressure between the slurry and the effluent sides of the porous disk. The flowing suspension of particles and the filter cake (solid) are denoted by the subscripts f and s, respectively. The carrier phase (liquid) and the particles are denoted, respectively, by l and p. Following Tien et al. (1997), the slurry is considered to contain particles larger than the pore size that form the filter cake and the particles smaller than the pore sizes of the filter cake and the porous disk, which can migrate into the cake and the formation to deposit there. All particles (small plus large) are denoted by p, and the large and small particles are designated by p1 and p2, respectively. Civan (1998c, 1999c) developed the filtration models by considering the cake-thickness-averaged volumetric balance equations for 1. the total (fine plus large) particles of the filter cake 2. the fine particles of the filter cake 3. the carrier fluid of the suspension of fine particles flowing through the filter cake 4. the fine particles carried by the suspension of fine particles flowing through the filter cake. The radial mass balances of all particles forming the cake, the small particles retained within the cake, the carrier fluid, and the small particles suspended in the carrier fluid are given, respectively, by (Civan, 1998c)     d  2 rw − rc2 s p = 2rc Rps + rw2 − rc2 Rp2s dt

    d  2 rw − rc2 s cp2s = 2rc Rp2s + rw2 − rc2 Rp2s dt   d  2 dr 2 rw − rc2 l l + l l slurry c dt dt

= 2rc l ul slurry − 2rw l ul filter

(12-102) (12-103)

(12-104)

Cake Filtration: Mechanism, Parameters, and Modeling

    dr 2 d  2 rw − rc2 l cp2l + l cp2l slurry c dt dt     = 2rc cp2l ul slurry − 2rw cp2l ul filter   − 2rc Rp2s − rw2 − rc2 Rp2s

377

(12-105)

Equations (12-102)–(12-105) apply over the cake, located within rc ≤ r ≤ rw , and for t > 0. s and l denote the volume fractions of the bulk cake system occupied by the cake forming particles and the carrier fluid, respectively. p and l are the densities of the particles and the carrier fluid g/cm3 , respectively. us and ul are the volumetric fluxes of the compressing filter cake and the carrier fluid flowing through the cake cm3 /cm2 s, respectively. cp2s and cp2l denote the small particle masses contained per unit volume of the cake-forming particles and the carrier fluid flowing through the cake g/cm3 , respectively. t and r denote the time and radial distance (cm), respectively. Rps is the mass rate of particle deposition from the slurry over to the moving cake surface g/s/cm3  given by Rps = Rp1s + Rp2s

(12-106)

where Rp1s and Rp2s denote, respectively, the mass rates of large and small particles deposition from the slurry over the cake surface g/s/cm3 . Rp2s is usually negligible unless the small particles are retained by a process of jamming of small particles across the pores of the large particles, such as described by Civan (1994c, 1996d) and Liu and Civan (1996). The variation of the filter cake thickness = rw − rc (cm) can be calculated using the variable radius, rc = rc t, of the slurry side filter cake surface. For many practical applications, it is reasonable to assume that the particles and the carrier fluid are incompressible. The volumetric retention rates of the large and small particles are given, respectively, by  (12-107) Nis = Ris p  i = p p1 p2  (12-108) Np2s = Rp2s p

The volumetric concentration (or fraction) of species i in phase j, the volume fraction of species i of phase j in the bulk of the cake system, and the superficial velocity of species i of phase j are given, respectively, by  (12-109) ij = cij i

378

Cake Filtration: Mechanism, Parameters, and Modeling

ij = j ij

(12-110)

uij = uj ij

(12-111)

Here, t denotes the time;  p2s , and p2l are the cake-thickness-average porosity, the fine particle volume fractions of the cake matrix, and the suspension of fine particles flowing through the cake matrix, respectively; pl slurry is the volume fraction of the total (fine plus large) particles in the slurry; and ul slurry and ul filtrate denote the volume fluxes of the carrier fluid entering and leaving the filter cake, respectively. Substituting Eqs (12-107)–(12-111) into Eqs (12-102)–(12-105) leads to the following volumetric balance equations, respectively (Civan, 1999c): "&  !   d % 2 rw − rc2 1 −  = 2rc Nps + rw2 − rc2 N p2s dt     d  2  rw − rc2 p2s = 2rc Np2s + rw2 − rc2 N p2s dt & dr 2 "& %   ! d % 2 c rw − rc2  − p2l + 1 − pl slurry dt dt = 2rc ul slurry − 2rw ul filtrate

      dr 2 d  2 rw − rc2 p2l + p2l slurry c = 2rc up2l slurry dt dt      − 2rw up2l filtrate − 2rc Np2s − rw2 − rc2 N p2s

(12-112) (12-113)

(12-114)

(12-115)

Equations (12-112)–(12-115) can be solved numerically subject to the initial conditions given by rc = rw  p2s = p2l = 0 t = 0 12.3.2

(12-116)

Linear Filtration Formulation

The radial filter cake equations derived above can be readily converted to linear filter cake equations by means of the transformation given by x = r 2  = xw − xc

(12-117)

Cake Filtration: Mechanism, Parameters, and Modeling

379

Thus, application of Eq. (12-117) to Eqs (12-102)–(12-105) yields, respectively, the following cake-thickness-averaged mass balance equations for the linear cake formation (Civan, 1998c):  d 

s p = Rps + Rp2s dt

 d 

s cp2s = Rp2s + Rp2s dt

d d  l l  − l l slurry = l ul slurry − l ul filter dt dt    d d 

l cp2l − l cp2l slurry dt dt     = cp2l ul slurry − cp2l ul filter − Rp2s − Rp2s

(12-118) (12-119) (12-120)

(12-121)

Similarly, Eqs (12-112)–(12-115), respectively, become (Civan, 1999c) "& ! d % xw − xc  1 −  = Nps + xw − xc  N p2s dt  d   xw − xc  p2s = Np2s + xw − xc  N p2s dt

& dx "& % !   d % c xw − xc   − p2l + 1 − pl slurry dt dt = ul slurry − ul filtrate

(12-122) (12-123)

(12-124)

dx d xw − xc p2l + p2l slurry c dt dt  = up2l slurry − up2l filtrate − Np2s − xw − xc N p2s

(12-125)

Equations (12-122)–(12-125) can be solved numerically, subject to the initial conditions given by xc = xw  p2s = p2l = 0 t = 0

(12-126)

380

Cake Filtration: Mechanism, Parameters, and Modeling

The volume fractions of the filter cake solids and pore fluid can be expressed in terms of the cake porosity, respectively, as s = 1 − 

(12-127)

f = l + p2l = 

(12-128)

where  is the average cake porosity cm3 /cm3 . The following expressions for the small particle volume flux and mass per carrier fluid volume can be written according to Eqs (12-109)–(12-111), respectively, as   up2l = ul p2l l = ul cp2l p (12-129)  (12-130) cp2l = p p2l l

Note that Eqs 18, 28, and 24 of Corapcioglu and Abboud (1990) correspond to the present Eqs (12-118), (12-124), and (12-121), respectively, with some differences. Equation (12-118) simplifies to their Eq. 18, assuming p is constant and substituting Eq. (12-127). The present Eqs (12-121) and (12-124) simplify to their Eqs 24 and 28, substituting Eq. (12-128) for p2l > Rp2s and, thus, Rps  Rp1s ). However, it is more accurate to use Eq. (12-106). 12.3.3

Pressure–Flow Relationships

The slurry carrier fluid flow rate ul can be expressed using the effluent fluid pressure pe (atm) at the outlet side of the porous formation, the

Cake Filtration: Mechanism, Parameters, and Modeling

381

pressure pc (atm) at the slurry side cake surface, and the harmonic average permeability of the cake and formation, kc and kf , respectively. Forchheimer’s (1901) law of flow through porous media for the linear case is given by −

p

= u + u2 x K

(12-131)

The pressure differences over the filter cake and the porous media can be expressed by integrating Eq. (12-131), respectively, as (Civan, 1999c)  

2 pc − pw = uc + c c uc xw − xc  (12-132) Kc  

2 uf + f f uf xe − xw  (12-133) pw − pe = Kf The instantaneous volumetric fluxes and densities of the suspensions of fine particles flowing through the cake matrix and porous formation are assumed the same. Then, adding Eqs (12-132) and (12-133), and rearranging and solving, yields (Civan, 1999c) for Darcy flow f = c = 0:

uc = uf  uc = −    (12-134) and for non-Darcy flow: uc = uf  uc = in which

ul slurry   = 1 − pl slurry

  2 − 4   − +  2 

% &   =  c xw − xc  + f xe − xw    x − xc xe − xw  = w + Kc Kf   = − pc − pe 

(12-135)

(12-136) (12-137) (12-138)

Although the preceding approach yields a reasonably good accuracy, a more rigorous treatment should facilitate uc = uc − xw − xc 

duc dxc

(12-139)

382

Cake Filtration: Mechanism, Parameters, and Modeling

Forchheimer’s law (1901) for the radial flow case is given by −

p

= u + u2 r K

(12-140)

The volumetric flux and flow rate are related by u=

q 2rh

(12-141)

where h is the formation thickness. Thus, invoking Eq. (12-141) into Eq. (12-140) yields −

!q " p  ! q "2 = + r 2hK r 2h2 r

(12-142)

The pressure differences over the filter cake and porous media can be expressed by integrating Eq. (12-142), respectively, as (Civan, 1999c)

q c ln pc − pw = 2hK c



 rw 1 c q 2 1 + − rc 2h2 rc rw

(12-143)

q f ln pw − pe = 2hK f



 re 1 f q 2f 1 + − rw 2h2 rw re

(12-144)

The densities and instantaneous flow rates of the suspensions of fine particles flowing through the cake matrix and porous formation are assumed the same. Then, adding Eqs (12-143) and (12-144), and rearranging and solving, yields (Civan, 1999c) for Darcy flow f = c = 0:

and for non-Darcy flow:

q c = q f  qc = −   

q c = q f  qc =

=

(12-145)

ul  uc   slurry = 2rc h 2rc h 1 − pl slurry

 2 − 4   − +  2 

(12-146)

Cake Filtration: Mechanism, Parameters, and Modeling

383

in which 



  1 1 1 1  c − + f − = rc rw rw re 2h2   r r 1 1

w  ln ln e + = 2h K c rc rw Kf 

  = − pc − pe 

(12-147)

(12-148) (12-149)

Although this approach leads to a reasonably good estimate, a more rigorous approach should employ   2 rw − rc2 dq c qc = q c − (12-150) 2rc drc The inertial flow coefficient is estimated by the Liu et al. (1995) correlation given by Eq. (12-16). 12.3.4

Particle Deposition Rates

The rate of deposition of the particles of the slurry over the slurry side cake surface is assumed proportional to the particle mass flux approaching the filter cake. The rate of erosion of the particles from the slurry side cake surface is assumed proportional to the tangential, excess shear stress above the critical stress necessary for particle mobilization. Therefore, for dynamic filtration involving cross-flow, the net mass rate of all particles (large plus small) deposition per unit area of the slurry side cake surface is given by the difference between the deposition and the erosion rates as (Civan, 1996a, 1998c)     Rps = kd ul cpl slurry − ke s p c  − cr  U  − cr  (12-151)

where ul is the carrier fluid filtration flux normal to the cake surface cm3 /cm2 s and cpl is the slurry particle concentration expressed as the particle mass per unit volume of the carrier fluid in the slurry. For small particles retention over the slurry side of the cake, an expression similar to Eq. (12-151) can be written as       Rp2s = kd2 (12-152) s cp2s c  − cr2  U  − cr2  ul cp2l slurry − ke2

384

Cake Filtration: Mechanism, Parameters, and Modeling

in which cp2l slurry denotes the mass of small particles per volume of the carrier fluid in the slurry. The net mass rate of deposition of small particles within the filter cake is given by (12-153)

Rp2s = kd l cp2l − ke s cp2s

which is similar to the equation of Corapcioglu and Abboud (1990), but the deposition and the mobilization terms are more consistently expressed. The rate expressions given by Eqs (12-151)–(12-153) for the deposition of the total (fine plus large) and fine particles of the slurry over the progressing cake surface and the retention of the fine particles of the flowing suspension within the cake matrix can be expressed in terms of the volumetric rates, respectively, as (Civan, 1999c) Nps

=

kd



ul pl 1 − pl

 Np2s

 =kd2





− ke 1 − c  s − cr  U s − cr  (12-154) slurry

ul p2l 1 − pl 



slurry



 − ke2 p2s c s − cr2  U s − cr2 

(12-155) (12-156)

N p2s = kd p2l − ke p2s

In Eqs (12-10)–(12-12), the slurry shear stress, s , acting over the progressing cake surface is estimated using the Rabinowitsch–Mooney equation (Metzner and Reed, 1955). This equation can be expressed for linear and radial flow cases, respectively, as follows: ′

(12-157)

s = k′ 8vn

s = k′ 4v/rc n

′

(12-158)

where k′ and n′ are the consistency dyne/cm2 /sn  and flow (dimensionless) indices, which are equal to the fluid viscosity, , and unit for Newtonian fluids, respectively, and v is the tangential velocity of the slurry over the filter cake surface. For static filtration, v = 0 and therefore  = 0, and the second term on the right side of Eq. (12-151) drops out, ′

Cake Filtration: Mechanism, Parameters, and Modeling

385

leading to an expression similar to Corapcioglu and Abboud (1990) and Tien et al. (1997). The minimum slurry shear stress necessary for detachment of particles from the progressing cake surface is cr . Following Ravi et al. (1992), the critical shear stress necessary for detachment of the deposited particles from the progressing cake surface can be estimated according to Potanin and Uriev (1991) by Eq. (12-6). However, the actual critical stress can be substantially different than predicted by Eq. (12-6), because the ideal theory neglects the effects of the other factors, including aging (Ravi et al., 1992), surface roughness, and particle stickiness (Civan, 1996a) on the particle detachment. Therefore, Ravi et al. (1992) recommend that the critical shear stress be determined experimentally. Us − cr  is the Heaviside unit step function. It is equal to zero when s < cr and one for s ≥ cr . kd and ke are the rate coefficients for the total (fine plus large) particles deposition and detachment at the progressing   and ke2 are the rate coefficients for the cake surface, respectively. kd2 fine particles deposition and detachment at the progressing cake surface, respectively. kd and ke are the cake-thickness-average rate coefficients for the deposition and mobilization of the fine particles within the filter cake matrix, respectively. When the particulate suspension contains various size particles, described by a particle size distribution function fD where D denotes the particle diameter, progressively the smaller particles deposit to form a filter cake because of the gradual reduction of the filtration rate as a result of the increasing filter cake thickness (Lu and Ju, 1989; Dewan and Chenevert, 2001). Thus, the particle deposition rate coefficient kd varies, given by (Dewan and Chenevert, 2001) ⎡ ⎤' Dc t Dmax   fDdD⎦ fDdD (12-159) kd t = ⎣ 0

0

where Dmax denotes the maximum particle size. Equation (12-159) represents the adhesion fraction of the particles of a suspension that are smaller than a critical particle diameter Dc t (cm) for adhesion (cake surface deposition). The critical particle diameter Dc t is given below by Fisher et al. (1998), the coefficient of which has been modified from 0.21 to 0.45 by Dewan and Chenevert (2001):

5   045 1044 qt fe (12-160) Dc t = 20 Kc04

386

Cake Filtration: Mechanism, Parameters, and Modeling

where qt is the filtration rate (cm/s), fe is the friction coefficient for the filter cake buildup (dimensionless),  is the shear rate at the filter cake surface (1/s), and Kc is the filter cake permeability (mD). Filter cake erosion occurs when the shear stress of the particulate suspension s dynes/cm2  flowing over the cake can overcome the filter cake shear strength c dynes/cm2 , that is s > c . The shear stress of the particulate suspension s is given by Eqs (12-157) or (12-158). Dewan and Chenevert (2001) expressed the shear strength of a filter cake by the sum of the interparticle (or intermolecular) forces of the filter cake co dynes/cm2  and the strength owing to the pressure applied to the particles p dynes/cm2  as (12-161)

c = co + p

The p dynes/cm2  term is expressed as a product of the filter cake friction coefficient fe and the grain–grain stress g dynes/cm2 . The latter is given by Outmans (1963) as   1 1−  g = 68 950 pc − pw  d /

(12-162)

where pc –pw  is the pressure differential applied across the filter cake (psi), d is the distance into the filter cake measured from the suspension side cake surface (cm), is the filter cake thickness (cm), and is a pressure-up compressibility exponent. 12.3.5

Porosity and Permeability Relationship

Tiller and Li (2000) expressed the solidity s = 1 − c  or the solid volume fraction in a filter cake and the cake permeability Kc , respectively, by s = so



K c = Kco

p 1+ s pa





(12-163)

−

(12-164)

p 1+ s pa

where the subscript “o” denotes a reference or initial condition, and the exponents  and roughly relate by  5.

Cake Filtration: Mechanism, Parameters, and Modeling

387

Incorporating the effects of fine particles packing in porous media during deposition according to Arshad (1991) and cake compaction according to Tien et al. (1997), Civan (1998c) estimates the cake-thickness-average porosity by the following constitutive equation:       p2s n  ps 1 1 = 1− − −1 1+ (12-165) o o o o pa Considering the fine particles deposition and cake compaction, Civan (1998c) estimates the cake-thickness-average permeability by the Tien et al. (1997) constitutive equation:

  − − −1  (12-166) K c Kco = 1 + 1 p2s2 1 + ps pa

In Eqs (12-165) and (12-166), o and Kco represent the fine particlesfree and noncompacted cake porosity and permeability, respectively,  n pa   1  2 , and are the empirically determined parameters. 12.3.6 Cake-Thickness-Averaged Fluid Pressure and Cake Porosity The average fluid pressure in the filter cake for linear filtration can be expressed similar to Dake (1978) as p=

xw

pdx

xc

' xw 

dx

(12-167)

xc

The average cake porosity is given by =

xw

xc

dx

' xw 

dx

(12-168)

xc

The following expression can be derived from Eqs (12-167) and (12-168): p =

xw

xc

pdx

' xw  xc

dx

(12-169)

388

Cake Filtration: Mechanism, Parameters, and Modeling

Note xw is a constant, but xc = xc t varies by time. Similarly, the following three expressions can be written for radial flow: ' rw  rw rdr (12-170) p = prdr rc

rc

=

rw

rdr

p =

rw

rdr

(12-171)

rc

rc

rw

'

prdr

'

rw

rdr

(12-172)

rc

rc

Equation (12-170) is given by Dake (1978). Note that rw is a constant, but rc = rc t varies by time. Equations (12-169) and (12-172) define the average fluid pressure, but they cannot be used directly because the pressure distribution over the cake thickness is not a priori known. Civan (1998c, 1999c) circumvented this problem by applying a procedure similar to Jones and Roszelle (1978) to express a local function value in terms of its average, as follows. The local cake porosity at the slurry side of the cake can be expressed in terms of the cake-thickness-average porosity. For linear filtration, Civan (1998c) differentiated Eq. (12-168) to obtain c =  − xw − xc 

d dxc

(12-173)

For radial filtration, Eq. (12-171) yields (Civan, 1999c) 

 rw2 − rc2 d c =  − 2rc drc

(12-174)

Similar to Tiller and Crump (1985), the cake-thickness-average viscous drag force, ps , created by the flow of the suspension of fine particles through the filter cake is determined using p s = pc − p

(12-175)

Cake Filtration: Mechanism, Parameters, and Modeling

389

in which pc is the pressure of the slurry applying at the progressing filter cake surface and p is the cake-thickness-average pressure of the suspension of fine particles flowing through the cake. For linear filtration, pc and p can be related by differentiating Eq. (12-169) and then substituting Eq. (12-173) to obtain (Civan, 1998c) ! " ( ) d p d = pc  − xw − xc  p − xw − xc  (12-176) dxc dxc For computational convenience, Eq. (12-176) can be reformulated in a form of an ordinary differential equation as (Civan, 1999c)

−1 dxc 1 d dp (12-177) = pc − p + dt xw − xc dt  dt Differentiating Eq. (12-172) and then substituting Eq. (12-174) for radial flow, Eqs (12-176) and (12-177) are replaced, respectively, by (Civan, 1998c): ! " + *  2  2   2 d p rw − rc rw − rc2 d p − = pc  − (12-178) 2rc drc 2rc drc dp = pc − p dt



−2rc drc 1 d + rw2 − rc2 d t  d t

(12-179)

Equation (12-177) or (12-179) can now be solved numerically subject to the initial condition p = pc  t = 0

12.3.7

(12-180)

Applications

The applications of the linear and radial filter cake buildup models are illustrated using the data given in Table 12-2. Corapcioglu and Abboud (1990) obtained a numerical solution for the linear constant rate filtration problem involving small particle invasion

390

Cake Filtration: Mechanism, Parameters, and Modeling Table 12-2 Model Input Parameters

Parameter

Symbol

Data I

Data II

Cake porosity without compaction and small particle retention, cm3 pore volume/cm3 bulk volume

o = 1 − os

0.39b

0.73c

Cake particle volume fraction cm3 particle/cm3 bulk volume

os = 1 − o

0.61c

0.27d

Particle density, g/cm3

p

1.18b

1.18a

Carrier fluid (water) density, g/cm3 Slurry total particle mass fraction, g particles/g slurry

l

0.97b

0.97a

0.101b

–

–

0.2d

Slurry total particle volume fraction, cm3 particles/cm3 slurry Slurry total particle mass per carrier fluid volume, g particle/cm3 carrier fluid Slurry carrier fluid volume fraction, cm3 carrier fluid/cm3 slurry



wpf



slurry

  pf slurry

wpf slurry l 0.109c 1 − wpf slurry   1 = p −1 l slurry %

&−1 l slurry = 1 + cpl slurry p 0.915c cpl slurry =

0.295c

0.8c

= 1 − pf slurry

Slurry carrier fluid volumetric flux, cm3 carrier fluid/(cm2 cake surface s)

ul slurry

20 × 10−3b 20 × 10−3d

Slurry injection pressure, atm

pc

8.9a

8.9d

Filter outlet pressure, atm

pe

1.0a

1.0d

Slurry small particles mass per carrier fluid volume, g small particles/cm3 carrier fluid

cp2l slurry = p p2l slurry

0.049a

0.059a

p2l slurry = cp2l p slurry

0.415c

0.05d

Slurry small particles volume per carrier fluid volume, cm3 small particles/cm3 carrier fluid

391

Cake Filtration: Mechanism, Parameters, and Modeling Table 12-2 (Continued) Parameter

Symbol

Data I

Data II

Filtrate small particle mass per carrier fluid volume, g small particle/cm3 carrier fluid

cp2l filter

0, 0.005a

0, 0.005a

Filter thickness, cm

Lf

05a

05a

Slurry side filter radius, cm

rw

5.08a

–

Filtrate side filter radius, cm

re

2.54a

–

Rate constant for small particle deposition within the cake, s−1

kd

65 × 10−3b

10 × 10−3ar 10 × 10−6ap

Rate constant for small particle entrainment within the cake, s−1

ke

435 × 10−5

50 × 10−5ar 50 × 10−7ap

Rate constant for total particle deposition over the slurry side cake surface, dimensionless

kd

1.0a

14ar 75ap

Rate constant for total particle erosion over the slurry side cake surface, cm−1 s

ke

0s

0s

Rate constant for small particle deposition over the slurry side cake surface, dimensionless

 kd2

0.1a

0.05ar 0ap

Rate constant for small particle erosion over the slurry side cake surface, cm−1 s

 ke2

0s

0s

Parameter in Eq. 60, dimensionless

1

30a

300d

Parameter, dimensionless

2

1a

10d

Parameter, dimensionless



009a

Parameter, dimensionless

a

049

Cake permeability without compaction and small particle deposition, D

Kco

35 × 10

007a 047a −3a

35 × 10−3d

(table continued on next page)

392

Cake Filtration: Mechanism, Parameters, and Modeling Table 12-2 (Continued)

Parameter

Symbol

Data I

Data II

Filter permeability, D

Kf

10 × 10−4a

10 × 10−4a

Viscosity of carrier fluid (water), cp



10a

10a

Parameter, dimensionless



10f

10f

Parameter, dimensionless

n

1/2

f

1/2f

An empirical constant, atm n′ A constant, dyne/cm2 /s

pa

a

k′

1.0a

1.0a

′

a

10a

00118d

A constant, dimensionless

n

10

Tangential velocity of the injected slurry, cm/s



0, 0.01a

0, 0.01a

a

Data assumed Corapcioglu and Abboud (1990) c Data calculated d Tien et al. (1997) f Adin (1978) p Data for the constant pressure case r Data for the constant rate case s Static filtration After Civan, F., 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved. b

at static condition, assuming that the cake is incompressible, the cake porosity remains constant, and all particles are filtered. Abboud (1993) repeated a similar calculation and also considered the effect of small particles migration into the filter. Tien et al. (1997) considered both constant rate and constant pressure-driven compressive cake filtrations in a linear and static case only. In the following, the applications by Civan (1998c, 1999c) to constant rate and constant pressure-driven filtration processes in linear and radial cases are presented and compared. The data considered are composed from the data used by Corapcioglu and Abboud (1990), Tien et al. (1997), and the missing data estimated by Civan (1998c), given in Table 12-2. Civan obtained the best estimates of the missing data by fitting the model to data as practiced by Liu and Civan (1996) and Tien et al. (1997). The numerical solutions of the ordinary differential equations, Eqs (12-122)–(12-125) for the linear model and Eqs (12-112)–(12-115) for the radial model, are obtained using the Runge–Kutta–Fehlberg four (five) method (Fehlberg, 1969) to determine the filter cake thickness,

Cake Filtration: Mechanism, Parameters, and Modeling

393

= xw − xc for the linear and = rw − rc for radial cases, and the volume fractions of the small particles retained in the cake and suspended in the flowing slurry, p2s and p2l , respectively. Equations (12-124) and (12-114) are used to determine the filtrate carrier fluid volumetric flux, ul filter , for the linear and radial cases, respectively. First, using the data given in Table 12-2, identified as Data I, the numerical solutions are carried out with the present, improved model for both linear and radial constant rate filtrations. The results for all particles filtered, for which cp2l filter = 0, as expected from an efficient filter, are compared and the effect of fine particle invasion into an inefficient filter is demonstrated by assuming a value of cp2l filter = 0005 g/cm3 . Civan’s (1998c) results presented in Figures 12-13–12-17 have similar trends, but different values than the results of Corapcioglu and Abboud (1990) and Abboud (1993), because of the simplifying assumptions involved in the latter calculations, such as incompressible cake and constant cake porosity and the use of the same rates of deposition for small and all (large plus small) particles over the progressing cake surface. Also, the average porosity of the filter cake can vary significantly in actual cases as described by Tien et al. (1997). Next, Civan (1998c) obtained the numerical solution for the constant pressure drive filtration. Corapcioglu and Abboud (1990) and Abboud (1993) did not present

Figure 12-13. Comparison of the cake thickness for linear and radial constant rate filtration (after Civan, 1998b; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

394

Cake Filtration: Mechanism, Parameters, and Modeling

Figure 12-14. Comparison of the small particle deposition for linear and radial constant rate filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Figure 12-15. Comparison of the suspended small particles for linear and radial constant rate filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Cake Filtration: Mechanism, Parameters, and Modeling

395

Figure 12-16. Comparison of the cake porosity for linear and radial constant rate filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998c AIChE. All rights reserved).

Figure 12-17. Comparison of the filtrate volume for linear and radial constant rate filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998c AIChE. All rights reserved).

396

Cake Filtration: Mechanism, Parameters, and Modeling

any results for this case. The flow rate is allowed to vary according to Eqs (12-146) and (12-135) for the radial and linear cases, respectively. In Figures 12-18–12-22, Civan’s (1998c) results for the linear and radial cases are compared. The results presented in Figures 12-13–12-22 indicate that fine particle invasion into the filter plays an important role. The differences between the radial and the linear filtration results are more pronounced and the cake thickness and filtrate volume are less for the constant pressure filtration. Tien et al. (1997) have solved their partial differential model numerically for linear filtration at static condition and reported numerical solutions along the filter cake only at the 100- and 1000-seconds times. Their model generates numerical solutions over the thickness of the filter cake, whereas Civan’s (1998c, 1999c) models calculate the thickness-averaged values. Therefore, Civan averaged the profiles predicted by Tien et al. (1997) over the cake thickness and used for comparison with the solutions obtained with the thickness-averaged filter cake model. Because Tien et al. (1997) reported numerical solutions at only two time instances, this resulted in only two discrete values. Civan generated the numerical solutions with the linear filtration model using the data identified as Data II in Table 12-2 for constant rate and constant pressure filtrations. As can be seen by Civan’s (1998c) results presented in Figures 12-23–12-26, his

Figure 12-18. Comparison of the cake thickness for linear and radial constant pressure filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Cake Filtration: Mechanism, Parameters, and Modeling

397

Figure 12-19. Comparison of the small particle deposition for linear and radial constant pressure filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Figure 12-20. Comparison of the suspended small particles for linear and radial constant pressure filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

398

Cake Filtration: Mechanism, Parameters, and Modeling

Figure 12-21. Comparison of the cake porosity for linear and radial constant pressure filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Figure 12-22. Comparison of the filtrate volume for linear and radial constant pressure filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Cake Filtration: Mechanism, Parameters, and Modeling

399

Figure 12-23. Comparison of the cake thickness for constant rate filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Figure 12-24. Comparison of the cake porosity for constant rate filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

400

Cake Filtration: Mechanism, Parameters, and Modeling

Figure 12-25. Comparison of the cake thickness for constant pressure filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Figure 12-26. Comparison of the cake porosity for constant pressure filtration (after Civan, 1998c; reprinted by permission of the AIChE, ©1998 AIChE. All rights reserved).

Cake Filtration: Mechanism, Parameters, and Modeling

401

ordinary differential model, described above, can closely reproduce the results of the Tien et al. (1997) partial differential model. Note that, as indicated in Table 12-2, the values of the parameters at the present cake thickness-averaged level should be different than those for the formulation at the local level considering the spatial variations, such as by Tien et al. (1997). 12.3.8 Comments on Models for Compressible Cakes Involving Fine Particle Intrusion Because of the improved phenomenological description and convenient cake thickness-averaged formulation, the ordinary differential models can provide faster numerical solutions with reduced computational effort and, therefore, offer certain practical advantages over the partial differential models for the analysis, design, and optimization of the cake filtration processes. The applicability of the models by Corapcioglu and Abboud (1990) and Abboud (1993) is limited to static and low pressure filtration of dilute suspensions and their assumption of the same rates for the deposition of the small and large particles over the progressing cake surface is not reasonable. The Tien et al. (1997) model can alleviate these problems but it is computationally intensive and also limited to static filtration. These models are for linear filtration and may sufficiently approximate radial filtration only when the cake and the filter media are much thinner compared to the radius of the filter surface exposed to the slurry. However, the radial model developed by Civan (1998b,c, 1999b,c) can better describe the radial filtration involving thick filter cake and large near-wellbore formation. The filtration models presented in this section provide insight into the mechanism of compressive cake filtration and a convenient means of simulation of the oil-field filter cake problems with additional features.

Exercises 1. Using the parameter values given by Eqs (12-99), (12-100), and (12101), and making reasonable assumptions for missing data, prepare the plots of a.  q, and Q vs. t according to Eqs (12-36) and (12-37).

402

Cake Filtration: Mechanism, Parameters, and Modeling

b. c. d. e.

2. 3. 4. 5.

 q, and Q vs. t according to Eqs (12-44), (12-46), and (12-47).

 q, and Q vs. t according to Eqs (12-50), (12-51), and (12-52). q and Q vs. t according to Eq. (12-79).

vs. t for dynamic and static filtration conditions according to Eqs (12-80) and (12-81). Estimate the spurt losses based on the data given in Figures 12-8 and 12-12. Making reasonable assumptions for missing data, construct the threedimensional plots of  and kc vs. p2s and ps according to Eqs (12-165) and (12-166). Based on the data given in Figure 12-2, compare the various effects involving the linear and radial filtration processes occurring under constant rate and constant pressure conditions. Consider an incompressible, linear, and dynamic filtration of a mud slurry over a core plug. The mud particle concentration is 055 g/cm3 . The particle density is 25 g/cm3 . The cross-sectional surface area of the core plug is 10 cm2 . The porosity of the mud cake formed over the core plug is 35%. The actual shear stress of the mud fluid flowing over the core surface is 15 dyne/cm2 and the critical shear stress necessary for filter cake particle erosion is 05 dyne/cm2 . The limiting filter-cake thickness to be attained at dynamic equilibrium between the particle deposition and the erosion rates for the mud cake is 0.2 cm. The limiting filtrate flow rate into the core plug at dynamic equilibrium is one-fourth of the initial filtrate flow rate prior to any cake formation. A straight-line plot of experimental data according to Eq. (12-88) is given in Figure 12-27. Calculate the following.

–d(1/q)/dt, 1/ml

0.14 y = 2.04x + 0.016 R 2 = 0.96

0.12 0.10 0.08 0.06 0.04 0.02 0.00

0

0.02 0.04 Filtrate flow rate, q, ml/min

0.06

Figure 12-27. Straight-line plot of measured variation of reciprocal filtrate flow rate (Plot prepared by the author).

Cake Filtration: Mechanism, Parameters, and Modeling

403

a. The values of the mud cake particle deposition and erosion rate constants with correct units. b. The limiting filtrate flow rate into the core plug to be attained at dynamic equilibrium between the particle deposition and the erosion rates for the mud cake. 6. Dewan and Chenevert (2001) recommend the use of the “slowness” S (s/cm) rather than the filtrate rate q cm3 /s in the diagnostic plots of experimental data because of the convenience of using the initial close-to-zero values of the slowness corresponding to infinityapproaching values of the initial flow rate. The relationship between the slowness S and the flow rate q is defined as S = a/q

(12-181)

where a denotes the cake surface area cm2 . Reformulate the diagnostic filtration equations presented in this chapter in terms of the slowness variable. 7. Consider the filtration model given by (Luckert, 1994)  q dt d2 t (12-182) =K 2 dQ dQ where t and Q denote the filtration time and the filtrate volume, respectively. Determine the parameters K and q of the filtration model. For this purpose, first linearize this equation by taking a logarithm as  2  dt dt log (12-183) = log K + q log 2 dQ dQ Then, construct a straight-line plot of Eq. (12-183) using a leastsquares fit to obtain the best estimate values of log K and q as the intercept and slope of this line, respectively. Demonstrate this procedure using typical data given in this chapter. 8. Derive an analytical solution to Eq. (12-182) after a variable transformation given by dt =y dQ

(12-184)

dy = Kyq dQ

(12-185)

so that Eq. (12-182) becomes

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Cake Filtration: Mechanism, Parameters, and Modeling

Apply the initial and final conditions given, respectively, by Q = 0 t = 0

(12-186)

and  dQ dt = ct. t → 

(12-187)

PART IV Formation Damage by Inorganic and Organic Processes – Chemical Reactions, Saturation Phenomena, Deposition, and Dissolution

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13

INORGANIC SCALING AND GEOCHEMICAL FORMATION DAMAGE

Summary Various processes leading to inorganic scaling and formation damage are discussed. Special attention is given to formation damage caused by the adverse reactions encountered during acid stimulation, brine incompatibility in seawater injection for water flooding, precipitation caused by CO2 and light hydrocarbon gases near wellbore, and sulfur deposition. The alterations of thermodynamic and chemical balances in favor of precipitation, precipitate aggregation, crystal growth, and inorganic scale formation are discussed and mathematically formulated. The rate processes governing the cation exchange, adsorption/desorption, and dissolution/precipitation reactions are formulated. The criteria for precipitate forming conditions are derived. Typical applications to above mentioned inorganic precipitation and scale formation processes are presented.

13.1 INTRODUCTION Inorganic scaling is a process of deposition of scales from aqueous solutions of minerals, referred to as brines, when they become supersaturated as a result of the alteration of the state of their thermodynamic and chemical equilibrium (Amaefule et al., 1988). Inorganic scaling can occur in the well tubings and near well bore formations of the production and injection wells. 407

408

Inorganic Scaling and Geochemical Formation Damage

Amaefule et al. (1988) explain that conditions leading to supersaturation can be created by various mechanisms at different stages of reservoir exploitation. Scaling is caused essentially by mixing incompatible fluids during well development operations, such as drilling, completion, and workover, such as acidizing. Scaling is caused by a decrease of pressure and temperature during the production of reservoir fluids. Scaling associated with the enhanced recovery processes, such as water, carbonated water, alkaline water, and carbon dioxide injection, may be caused by mixing incompatible fluids and/or pressure and temperature variations. The scale formation mechanisms can be classified as (1) natural scaling and (2) induced scaling (Amaefule et al., 1988). These mechanisms can be best explained by means of scale precipitation charts, such as those given by Shaughnessy and Kline (1982, 1983), who developed practical charts depicting the relationships between dissolved calcium Ca+2  and bicarbonate HCO−3  ions, calcium carbonate CaCO3  precipitate, CO2 partial and total pressures, and temperature, based on the equilibrium relationship for the calcium carbonate scale formation by the reaction Ca+2 + 2HCO−3 −→ ←− CaCO3 s + CO2 g + H2 Ol

(13-1)

Dissolved Bicarbonate, mmole/l

The chart given in Figure 13-1 by Shaughnessy and Kline (1982) shows the calcium carbonate precipitation regions located above the equilibrium curves of the 2.8 MPa (400 psi) and 3.4 MPa (500 psi) CO2 partial 46

44 42

40

200° F (93° C)

50 0 (3 psi .4 MP CO a) 2 40 0 (2 psi .8 MP CO a) 2

Reservoir Brine l”

“Induced”

ura “Nat

Precipitated Calcium

38 5

6 Dissolved Calcium, mmole/l

7

Figure 13-1. Natural and induced scale damage mechanisms (Shaughnessy and Kline, ©1983 SPE; reprinted by permission of the Society of Petroleum Engineers).

Inorganic Scaling and Geochemical Formation Damage

409

pressures at 93 C 200 F temperature. Natural scaling occurs mostly in the near production wellbore regions as a result of the liberation of dissolved light gases from the formation brine by high drawdown (Amaefule et al., 1988). Consequently, the loss of the CO2 gas from the brine promotes calcium carbonate precipitation. Amaefule et al. (1988) explained this phenomenon by the Le Chatelier’s principle. Because H2 O is in abundance, its concentration can be taken as a constant. Then, the chemical equilibrium constant for Eq. (13-1) reads as CaCO  CO2  K eq =  +2  3 Ca HCO−3 2

(13-2)

Therefore, applying Le Chatelier’s cause-and-effect principle, when CO2 is liberated and removed by pressure reduction, the CO2 concentration will reduce. To compensate this effect, more CaCO3 will be produced to maintain the constant K eq . Amaefule et al. (1988) explain that induced scaling occurs by mixing of formation brine with extraneous incompatible fluids invading the reservoir formation during drilling, cementing, completion, and workover operations. The same may occur by injection of fluids for enhanced recovery purposes. Any increase of the dissolved calcium Ca+2  ion concentration caused by these operations is compensated by calcium carbonate CaCO3  precipitation according to Eq. (13-1), following Le Chatelier’s cause-and-effect principle. Civan et al. (2000) draw attention to some important phenomena that have been overlooked for the most part. First, the presence of oil and gas phases may effect the thermodynamics and chemistry of aqueous phases. Second, scale formation in the near wellbore formation is more kinetically controlled than thermodynamically controlled because the rapid flow that occurs around wellbores compared to the rest of the reservoir does not allow for sufficient time to attain equilibrium. This may result with an incomplete release of the light dissolved gases, such as carbon dioxide, from the aqueous phase. Hence, the saturation conditions at the actual near-wellbore fluid pressure may not be attained and the partitioning of various light gases between the liquid and the gas phases may not reach the equilibrium condition. Civan et al. (2000) caution these phenomena should be considered for accurate scale predictions. Civan (2006a) points out that a similar phenomenon can occur during rapid flow in wells. Geochemical interactions of the aqueous phase and the solid porous matrix result in alterations of minerals and the texture, porosity,

410

Inorganic Scaling and Geochemical Formation Damage

and permeability of porous formation. As stated by Lichtner (1985), geochemical systems involve various reversible and irreversible chemical interactions, such as oxidation–reduction reactions, ion complexing, mineral dissolution/precipitation, and adsorption/desorption. Dissolution of solid minerals is a slow process, and complete dissolution cannot occur within the convection time scale of the flow in the near wellbore (Nordstrom and Munoz, 1994). However, alteration of the composition and saturation of the aqueous phase, and the fluid shear can induce the entrainment, migration, and redeposition of fine mineral particles and therefore cause formation damage (Chang and Civan, 1991, 1992, 1997). Formation damage resulting from the injection of incompatible waters into reservoirs can be avoided if the initial rock–fluid equilibria and, hence, the initial reservoir quality can be maintained (Schneider, 1997). Injecting oxygenated waters into reservoirs can oxidize the reduced Fe and S species present in the pore water and can cause precipitation and plugging of pores (Schneider, 1997). Geochemical models are important for predicting the complications that will result from the interactions of the various drilling and production fluids with the reservoir formation (Schneider, 1997). Yeboah et al. (1993) draw attention to the fact that most models use limited solubility or thermodynamic data and ignore the effects of ion pairs and presence of other ions (such as magnesium) on the solubility, and the kinetic and transport phenomena factors. Therefore, Yeboah et al. (1993) caution that “the available models predict only scaling tendency and with a high degree of uncertainty,” but “a positive scaling potential does not necessarily imply that scale will form.”

13.2 GEOCHEMICAL PHENOMENA — CLASSIFICATION, FORMULATION, MODELING, AND SIMULATION Fluids and minerals present in petroleum-bearing formations may undergo various interactive chemical reactions in response to the alteration of the in situ conditions by various operations, including drilling, workover, and enhanced recovery. Geochemical models provide scientific guidance for controlling adverse reactions that may result from rock–fluid interactions. Excellent treaties of the geochemical reaction modeling are available from several sources, including Melchior and Bassett (1990), Ortoleva (1994),

Inorganic Scaling and Geochemical Formation Damage

411

and Bethke (1996). Only the fundamentals of this extremely complex subject are outlined here. The readers are encouraged to resort to literature for details and to use ready-made software available from various sources. Petroleum-bearing formations can be generally viewed as being geochemical systems in which fluids consisting of oil, gas, water, and solid phases formed from an assemblage of minerals interact through various chemical reactions. Lichtner (1985) classified such reactions into four categories: (1) aqueous ion complexing, (2) oxidation and reduction, (3) mineral precipitation and dissolution, and (4) ion exchange and adsorption reactions. As stated by Kharaka et al. (1988) and Amaefule et al. (1988), such reactions occur in response to changing temperature, pressure, and fluid composition by various factors, including the addition of incompatible fluids during drilling, workover and enhanced recovery processes, and liberation of light gases, such as CH4  CO2  H2 S, and NH3 , during pressure-drawdown. Changes in temperature and pressure often cause the variation of the pH of the reservoir aqueous phase, which in turn induces adverse processes such as the precipitation of iron and silica gels (Kharaka et al., 1988; Rege and Fogler, 1989; Labrid and Bazin, 1993). Geochemical reactions can also be classified as homogeneous and heterogeneous depending on whether the reaction occurs inside a phase or with another phase, respectively. Geochemical reactions can also be classified as reversible and irreversible. As explained by Lichtner (1985), the rates of reversible reactions are independent of the surface area. Reversible reactions can attain local equilibrium over a sufficiently long period of time, at which time the reaction rates terms vanish in the transport equations. However, Lichtner (1985) adds that irreversible reactions require kinetic or rate expressions, in terms of the pertinent driving forces, that is chemical affinity, and/or the surface available for reactions. Detailed geochemical description is a very cumbersome task and often unnecessary and unjustified in view of the lack of the basic thermodynamic and kinetic data required for description. Rather, geochemical models are constructed to emphasize the chemical reactions of the important aqueous species and minerals, which are essential for adequate description of the rock–water interactions, and neglect all other reactions. This is done to compromise between the quality of description and the effort necessary to gather basic thermodynamic and kinetic data and to carry out the numerical computations. Among the various alternatives, the kinetic and equilibrium models are extensively utilized. The kinetic models describe the rate of change of the amount of mineral and aqueous species in porous media in terms

412

Inorganic Scaling and Geochemical Formation Damage

of the relevant driving forces and factors, such as deviation from equilibrium concentration and mineral–aqueous solution contact surface. The proportionality constant is called the rate constant. The equations formed in this way are called the rate laws or kinetic equations. The equilibrium models assume geochemical equilibrium between the pore water and the minerals of porous formation. Because equilibrium can be reached over a sufficiently long time, equilibrium models represent the closed systems at steady-state conditions. Mathematically, the equilibrium models can be derived from kinetic models in the limit of infinitely large rate constants. Hence, rapid reactions reach equilibrium faster. Therefore, the equilibrium models represent the limiting conditions and yield conservative predictions (Schneider, 1997). Equilibrium models are particularly advantageous for determining the mineral stability and graphical representations of the mineral and aqueous species interactions (Bjørkum and Gjelsvik, 1988; Stumm and Morgan, 1996; Schneider, 1997). Because of the highly intensive numerical computations involved, the geochemical models of the rock–water interactions are usually implemented by computer-coded software. The geochemical computer software is constantly evolving and becoming more robust and accurate as a result of the advancement in computer technology, availability of accurate thermodynamic data, and development of efficient numerical solution methods. The engineers responsible for developing operational strategies and procedures for scale control in petroleum reservoirs should rely on such software. However, intelligent and efficient use of the ready-made software requires some familiarity with the fundamental concepts, theories, and methods involved in the treatment and formulation of geochemical reactions. This information is usually provided with the user’s guide and/or by relevant publications. In the following sections, a brief review of the description and graphical representation of aqueous and mineral species reactions and various approaches to geochemical modeling are presented.

13.3 REACTIONS IN POROUS MEDIA The various chemical reactions occurring in the pore space can be classified into the groups of homogeneous and heterogeneous reactions (Lichtner, 1992). The reactions occurring within the aqueous fluid phase are called the homogeneous or aqueous reactions. The reactions of the aqueous phase species with the solid minerals of porous formation,

Inorganic Scaling and Geochemical Formation Damage

413

occurring at the pore surface, are called the heterogeneous or mineral reactions. A convenient treatment of the geochemical reactions can be achieved by grouping the various reacting solute species into primary and secondary sets of species (Kandiner and Brinkley, 1950; Lichtner, 1992). The primary set of species is formed by selecting a minimum, critical number of reacting aqueous species, S , necessary for adequate description of the homogeneous and heterogeneous reactions. Thus, all other species form the set of the secondary species. The secondary species are derived from the primary species by means of the equations of the relevant chemical reactions. 13.3.1

Aqueous Phase Reactions

Lichtner (1992) classifies the homogeneous reactions into three categories: (a) ion pairing/exchange reactions, (b) complexing reactions, and (c) redox reactions. The aqueous phase reactions are generally rapid relative to the mineral reactions (Liu et al., 1997). The rapid rates of aqueous phase reactions require kinetic descriptions with significantly large rate constants. Thus, for all practical purposes, these reactions can be assumed instantaneous and a transport-controlled, local chemical equilibrium assumption is usually considered reasonable (Walsh et al., 1982; Lichtner, 1992; Liu et al., 1997; Liu and Ortoleva, 1996a,b). Consider an aqueous phase undergoing a total of Nrf chemical aqueous reactions, where r = 1 2     Nrf denotes the index for the various aqueous reactions. Nr represents the total number of aqueous species involved in the rth aqueous or homogeneous reaction. S  = 1 2     Nr denotes the various aqueous species involved in the rth aqueous reaction. Then, the aqueous reactions can be typically represented by (Walsh et al., 1982; Liu et al., 1997) f krf

f f f f

r1 S1 + r2 S2 + · · · −→ ←− rn Sn + r n+1 Sn+1 + · · ·  f krb

r = 1 2     Nrf

(13-3)

or simply as Nr 

=1

f

r S = 0

r = 1 2     Nrf

(13-4)

414

Inorganic Scaling and Geochemical Formation Damage

f where r denotes the stochiometric coefficient of species  involved f is negative for the reactants and in the rth aqueous reaction. Note r positive for the products. Applying the mass action law of Prigogine and DeFay (1954), the chemical equilibria between the products and reactants of the rth reaction can be expressed as (Walsh et al., 1982; Liu et al., 1997)

Krf

=

Nr 

f

a r 

r = 1 2     Nrf

(13-5)

=1

Krf denotes the thermodynamic equilibrium constant for the rth aqueous f of the forward reaction given as the ratio of the rate constants krff and krb and backward reactions represented by Eq. (13-3): Krf =

krff f krb

(13-6)

The chemical activity of the aqueous species  can be expressed in terms of the molal concentration, C , of species  as a =  C

(13-7)

in which  is the activity coefficient (dimensionless) determined by the Debye–Hückel theory (Helgeson et al., 1970). The concentration and therefore the activity of species are expressed in molarity, defined as the moles of species per unit volume of solution. 13.3.2

Mineral Reactions

The reactions of the aqueous phase species with the solid mineral matter of the porous matrix are referred to as the mineral reactions. Most mineral reactions are typically hydrolysis reactions. The interactions of minerals and aqueous species are generally slow relative to the aqueous phase reactions (Lichtner, 1992). Their reaction kinetics are controlled by the external mineral surface area contacting the aqueous phase. The mineral surface area is determined by the sizes of the grains of the porous formation. The rates of mineral reactions are gradual and, therefore, require kinetic descriptions with finite reaction rate constants. Consider a porous formation containing a total of Ns different mineral species and undergoing a total of Nrs different chemical reactions between

Inorganic Scaling and Geochemical Formation Damage

415

its mineral species and the aqueous phase species. s = 1 2     Ns denotes the index for the participating mineral species of the porous formation. Nr denotes the total number of aqueous species involved in the rth heterogeneous reaction. Mrs denotes the sth mineral species undergoing the rth heterogeneous reaction. S  = 1 2     Nr denotes the various aqueous species involved in the rth mineral reaction. Then, the reactions between porous formation minerals and aqueous species can be typically represented by (Lichtner, 1992; Liu et al., 1996) s s s s s s s

r1 Sn−1 −→ S1 + r2 S2 +· · ·+ rn−1 ←− Mr + rn Sn+ r n+1 Sn+1 +· · ·+ rNrs SNr

r = 1 2     Nrs and s = 1 2     Ns

(13-8)

or simply by Mrs +

Nr 

=1

s

r S = 0

r = 1 2     Nrs and s = 1 2     Ns (13-9)

s where r denotes the stochiometric coefficients associated with the s aqueous phase species per one participating mineral species. Note that r is negative for the reactants and positive for the products. For example, Eq. (13-9) can be applied to the calcium carbonate precipitation reaction given by Eq. (13-1) as following:

CaCO3 s + CO2 g + H2 Ol − Ca+2 − 2HCO−3 = 0

(13-10)

Applying the mass action law (Prigogine and DeFay, 1954), and assuming that the activity of the solid minerals is equal to one, the chemical dynamics between the minerals of the porous formation and the aqueous species of the rth reaction can be expressed as follows. The forward reaction rate can be expressed by rf = kf

n−1 

s

r  a



=1



actual

(13-11)

where kf is the rate constant for the forward reaction. Similarly, the reverse reaction rate is given by rr = kr

Nr   s  ar actual

=n

(13-12)

416

Inorganic Scaling and Geochemical Formation Damage

where kr is the rate constant for the reverse reaction. Thus, the effective forward reaction rate is given by rrseff = rf − rr =

krsf

n−1 

=1

s r  a

− krsr  actual



Nr   s  ar actual

(13-13)

=n

At equilibrium, rrseff = 0, and therefore the following expression, referred to as the equilibrium constant Krseq , is obtained (Walsh et al., 1982): Krseq

≡

krsf /krsr

Nr   s  = ar equil =n

n−1 Nr    s     s  a r equil = ar equil  =1

=1

r = 1 2     Nrs and s = 1 2     Ns minerals

(13-14)

The expression Krseq has a constant value at prescribed temperature and s pressure conditions. Note that r is negative for reactants and positive for products. For example, Eq. (13-10) can be applied to the calcium carbonate precipitation reaction expressed by Eq. (13-1) as a−2 − aCaCO3 s aCO2 g aH2 Ol K eq = a−1 Ca+2 HCO 3

(13-15)

Applying the definition of the equilibrium constant given by Eq. (13-14) to (13-13), for a mineral s undergoing a dissolution/precipitation reaction r under nonequilibrium saturation conditions, the effective forward reaction rate or driving force can be expressed as (Liu et al., 1996)

Nr n−1       s s r  a

− 1/Krseq  a r actual rrseff = krsf  actual =n

=1

for

r < 0

for

s

r >0

(13-16)

The first term inside the brackets is associated with the reactants  = 1 2     n − 1 and the second term inside the brackets is associated with the products  = n n + 1    Nr  rrseff > 0 for mineral dissolution and rrseff < 0 for mineral precipitation. krsf denotes the dissolution rate constant of the rth reaction of the sth mineral.

417

Inorganic Scaling and Geochemical Formation Damage

When the chemical system is not at equilibrium, then the actual instantaneous reaction quotient can be expressed similar to Eq. (13-14) as Qrs ≡

krsf /krsr

Nr   s  ar actual = =n

r = 1 2     Nrs

n−1 Nr    s     s  a r actual = ar actual  =1

=1

and s = 1 2     Ns minerals

(13-17)

In the absence of a denominator, the equilibrium constant Krseq is called the equilibrium saturation ion activity product or the solubility product Krssp and the reaction quotient Qrs is called the actual ion activity product Krsap (Brown et al., 1991). The instantaneous actual saturation ratio is defined as Frs =

Qrs Krseq

(13-18)

The chemical system will undergo a forward reaction when Qrs Krseq and therefore Frs >1 0. Approximating the shape of the mineral grains by a sphere, and assuming that the mineral reactions are kinetic reactions, Liu et al. (1997) express the rate of dissolution or precipitation of the sth mineral by the rth reaction by Wrs = ns s 4R2s r eff rs

(13-19)

as being proportional to the number of the sth mineral grains per formation bulk volume, ns , the surface area of the sth mineral grain, 4R2s , and the reaction driving force, rrseff . The grain mass density of the sth mineral, s , is inserted to express the mass rate of dissolution or precipitation of the ith mineral grain. Rs represents the radius of the sth mineral grain. Thus, Liu et al. (1997) express the rate of change of the sth grain radius by dissolution and/or precipitation by various reactions as Nr Rs  = rrseff t r=1

(13-20)

The mass conservation equation for species  undergoing transport through porous media by various mechanisms is given in Chapter 7.

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Inorganic Scaling and Geochemical Formation Damage

The rate of generation of the th aqueous species per bulk formation volume, required for this equation, is given by Liu et al. (1997) as f

r =

Nr  r=1

s

f

r Wrf

+

Nr 

s

r Wrs + q

 = 1 2     N

(13-21)

r=1

where N is the total number of aqueous species involved; Nrf and Nrs denote the total number of aqueous and mineral reactions, respectively; Wrf and Wrs represent the rates of the rth aqueous and mineral reactions, f s respectively; r and r denote the stochiometric coefficients of the species  in the aqueous and mineral reactions, respectively, and q represents the rate of species  addition per bulk formation volume by means of direct injection of fluids through wells completed in the reservoir.

13.3.3

Ion Exchange and Adsorption Reactions

Ion exchange and adsorption are surface chemical or surface complexation processes leading to the exchange of chemical species between the aqueous solution and the mineral surfaces present in geological porous formations (Jennings and Kirkner, 1984; Lichtner, 1985; Kharaka et al., 1988). Kharaka et al. (1988) explain the difference between ion exchange and adsorption as following: “The ion exchange model treats the exchange of cations or anions on a constant charge surface” and “the adsorption model simulates the exchange process on a surface where the surface charge is developed due to the ionization of surface sites at the solution–surface interface.” Therefore, adsorption is a more general concept and ion exchange is a special case of adsorption (Lichtner, 1985; Sahai and Sverjensky, 1998). Among the various surface complexation models, Sahai and Sverjensky (1998) facilitate the triple-layer model (Yates et al., 1974) for describing the electrical charge near mineral surfaces, as described in Figure 13-2 (Sahai and Sverjensky, 1998) according to Westall (1986). As indicated in Figure 13-2, this model considers the mineral surface, referred to as the O-plane, for adsorption of hydroxide ions and protons and at a short distance near the mineral surface, referred to as the -plane, for adsorption of electrolyte ions, and the surface charge is generated by adsorption of the electrolyte ions and protons (Sahai and Sverjensky, 1998).

419

Inorganic Scaling and Geochemical Formation Damage

Compact layer of adsorbed ions

Potential, Ψ (Volts)

Ψ0

Diffuse layer of counterions

Ψβ Ψd

ε1

ε2

C1

C2

0-plane (mineral surface)

Bulk solution

Distance, x (meters) d-plane

β-plane (electrolyte adsorption)

(After Westall, 1986)

Figure 13-2. Triple-layer description of the potential vs. distance from the mineral surface (Reprinted from Journal of Computers and Geosciences, Vol. 24, Sahai, N., and Sverjensky, D. A., “GEOSURF: A Computer Program for Modeling Adsorption on Mineral Surfaces from Aqueous Solution,” pp. 853–873, ©1998, with permission from Elsevier Science).

Clays present in geological porous formations have many active ion exchange sites, , occupied by various cations, and cation exchange takes place for replacement of the cations in the order of the replacing tendency of Ca+2 > Mg+2 > K+ > Na+ (Liu et al., 1997). The cation exchange capacity (CEC) of rocks can be expressed as the total number of moles of exchange sites  per unit mass of rock, Qex , or per unit volume of rock, w , which are related by (Lichtner, 1985) w = 1 −  s Qex

(13-22)

Lichtner (1985) points out that “precipitation/dissolution reactions can alter the exchange capacity of the porous medium by creating or destroying exchange sites,” but this effect has not been taken into account in most reported studies. In Eq. (13-22),  and s denote the porosity and the grain density of the rock, respectively. Represent the exchange sites

420

Inorganic Scaling and Geochemical Formation Damage

by , the total number of different exchange sites by N , an exchange site of type  with unit charge by E , the ith cation species with valence zi by Si , and the concentration of the ith species attached to the exchange sites  by Ci , expressed in moles per unit bulk volume. Lichtner (1985) then describes the chemical reactions at mineral surfaces by zj Si + zi Sj E zj −→ ←− zi Sj + zj Si E zi

 = 1 2     N

(13-23)

and the conservation of the ion exchange sites by  =

N 

(13-24)

zk Ck

k=1

where N is the total number of chemically reacting species. Sj E zj and Si E zi represent the cations attached to the active exchange sites. Sears and Langmuir (1982) report that ion exchange and adsorption reactions in soils typically require a time of seconds to days to attain equilibrium. Therefore, Jennings and Kirkner (1984) describe these reactions by rate equations for full kinetic modeling. Applying their approach to Eq. (13-23) according to Chang and Civan (1997) yields the following kinetic expression for the rates of consumption of Sj E zj and production of Si E zi , respectively: S −zi i = −zj t

  Sj E zj t

z

   Si E zi Sj = zj = zi t t

z

z

z

= kfij Si j Sj i E zj − krij Sj i Si j E zi

(13-25)

where  is the porosity and kfij and krij denote the rate coefficients for the forward and reverse reactions, respectively. If Iij is the reaction rate for the exchange of the ith cation present in aqueous solution with the jth cation attached to the th site on the mineral surface, and Ir is the rate of the reactions of the ith cation of the aqueous solution other than adsorption, the transport equation for the ith cation present in aqueous solution is given by (Lichtner, 1985) N N M      a ci  +  · Ji = − zj Iij + ir Ir t =1 j=1 r=1 j=i

i = 1 2     N (13-26)

Inorganic Scaling and Geochemical Formation Damage

421

where a denotes the volume fraction of the aqueous phase in the bulk of porous formation and ci denotes the concentration of the ith cation in the aqueous solution, expressed in moles per unit volume of the aqueous phase. The balance of the ith cation adsorbed on the th site of the mineral surface is given by (Lichtner, 1985) N Ci  = zj Iij t j=1

i = 1 2     N and  = 1 2     N

(13-27)

j=i

where Ci is the concentration of the ith species attached to the exchange sites  expressed in moles per unit bulk volume. As Iij ≡

Si t

(13-28)

Iji ≡

Sj t

(13-29)

so, zj Iij = −zi Iji

(13-30)

Thus, Lichtner (1985) combined Eqs (13-26) and (13-27) into the following convenient form by summing Eq. (13-27) over all the exchange sites , adding the resultant equation to Eq. (13-26), and eliminating the exchange reaction rates by means of Eq. (13-30): 

N M    a ci + Ci +  · Ji = ir Ir i = 1 2     N (13-31) t =1 r=1

13.4 GEOCHEMICAL MODELING As stated by Plummer (1992),∗ “Geochemical modeling attempts to interpret and (or) predict chemical reactions of minerals, gases, and organic matter with aqueous solutions in real or hypothetical water–rock Reprinted from “Water–Rock Interaction,” Proceedings of the 7th international symposium, WRI-7, Park City, Utah, 13–18 July 1992 Kharaka, Y. K. & A. S. Maest (eds), 90 5410 075 3, 1992, 25 cm, 1730 pp., 2 vols, EUR 209.00/US$246.00 GBP147. Please order from: A. A. Balkema, Old Post Road, Brookfield, Vermont 05036 (telephone: 802-276-3162; telefax: 802-276-3837; e-mail: [email protected]).

∗

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Inorganic Scaling and Geochemical Formation Damage

Figure 13-3. Common elements of aqueous chemical models (Reprinted with permission from Basset, R. L., and Melchior, D. C., “Chemical Modeling of Aqueous Systems – An Overview,” Chapter 1, pp. 1–14, in Chemical Modeling of Aqueous Systems II, Melchior, D. C. and Basset, R. L. (eds), ACS Symposium Series 416, ACS, Washington, 1990, Figure 2, page 6; ©1990 American Chemical Society).

systems.” Figure 13-3 by Bassett and Melchior (1990) outlines the basic constituents and options of most geochemical models. Plummer (1992)∗ classified the various geochemical modeling efforts into four groups: ∗

Reprinted from “Water-Rock Interaction.”

Inorganic Scaling and Geochemical Formation Damage

423

1. Aqueous speciation models for geochemical applications 2. Inverse geochemical modeling techniques for interpreting observed hydrochemical data 3. Forward geochemical modeling techniques for simulating the chemical evolution of water–rock systems 4. Reaction–transport modeling for the coupling of geochemical reaction modeling with equations describing the physics of fluid flow and solute transport processes.

Brief descriptions of these models are presented in the following sections, according to Plummer (1992).

13.4.1

Aqueous Speciation Models

Aqueous speciation models describe the thermodynamic properties of aqueous solutions and they are an integral part of the geochemical models. Plummer (1992)∗ summarizes the constituents of these models as follows. 1. Mass balance equations for each element 2. Mass action equations and their equilibrium constants for complex-ion formation 3. Equations that define individual ion-activity coefficients.

Two types of aqueous specification models are popular: (1) ion-association models and (2) specific interaction models. The ion association and the specific interaction models facilitate, respectively, the extensions and a complex expansion of the Debye–Hückel theory to estimate the individual ion activity coefficients of aqueous species (Plummer, 1992). The specific interaction models are preferred for highly concentrated solutions of mixed electrolytes (Plummer, 1992). As pointed out by Plummer (1992), aqueous geochemical models can be used for forward and inverse geochemical modeling.

13.4.2

Geochemical Modeling – Inverse and Forward

Plummer (1992)∗ summarizes that “Two approaches to geochemical modeling have evolved – ‘inverse modeling,’ which uses water and ∗

Reprinted from “Water-Rock Interaction.”

424

Inorganic Scaling and Geochemical Formation Damage

rock compositions to identify and quantify geochemical reactions, and ‘forward modeling,’ which uses hypothesized geochemical reactions to predict water and rock compositions.” However, the application of these models is rather difficult because the basic data necessary for these models are often incomplete and/or uncertain (Plummer, 1992). Plummer (1992)∗ describes the most essential information necessary for geochemical modeling and its applications as following: 1. The mineralogy and its spatial variation in the system 2. The surface area of reactants in contact with aqueous fluids in ground-water systems 3. The chemical and isotopic composition of reactants and products in the system 4. The hydrology of the system 5. The extent to which the system is open or closed 6. The temporal variation of these properties 7. The fundamental knowledge on the kinetics and mechanisms of important water–rock reactions 8. The kinetics of sorption processes 9. The degradation pathways of organic matter.

13.4.2.1

Inverse geochemical modeling

Plummer (1992)∗ explains that “Inverse geochemical modeling combines information on mineral saturation indices with mass-balance modeling to identify and quantify mineral reactions in the system.” The mass-balance modeling requires the following (Plummer, 1992). 1. 2. 3. 4. 5.

Element mass-balance equations Electron conservation equations Isotope mass-balance equations, when applicable Aqueous compositional and isotopic data Mineral stochiometry data for all reactants and products.

Plummer (1992)∗ warns that “The inverse-modeling approach is best suited for steady-state regional aquifers, where effects of hydrodynamic dispersion can often be ignored.” ∗

Reprinted from “Water-Rock Interaction.”

Inorganic Scaling and Geochemical Formation Damage

13.4.2.2

425

Forward geochemical modeling

The objective of the forward geochemical modeling is to predict mineral solubilities, mass transfers, reaction paths, potentiometric-activity (pH) and electron-activity (pe) by using available solid-aqueous data in aqueous specification models (Plummer, 1992). Some of the important features of the advanced forward geochemical models are cited by Plummer (1992)∗ as 1. Access to a large thermodynamic data base 2. Generalized reaction-path capability 3. Provision for incorporation of reaction kinetics in both dissolution and precipitation 4. A variety of activity coefficient models 5. Treatment of solid solutions 6. Calculation of pH and pe 7. Calculation of mineral solubility with and without accompanying irreversible reaction 8. Calculation of boiling, cooling, wall-rock alteration, ground-water mixing with hot waters and evaporation, and 9. Equilibrium or partial equilibrium states in gas–solid-aqueous systems.

Plummer (1992)∗ states that forward geochemical modeling can be used “in developing reaction models that can account for the observed compositional–mineralogical relations in the deposit, if there are no aqueous or solid data for the system.”

13.4.3

Reaction–Transport Geochemical Modeling

The reaction–transport models describe the geochemical reactions under the influence of fluid flow and convective and dispersive transport of various species in geological porous media. These models couple the geochemical reaction and the fluids and species transport submodels to accomplish temporal and spatial prediction of the evolution of geochemical reactions in compositionally complex geological systems (Plummer, 1992). These models are more applicable in most petroleum reservoir exploitation and scale formation studies. ∗

Reprinted from “Water-Rock Interaction.”

426

Inorganic Scaling and Geochemical Formation Damage

13.5 GRAPHICAL DESCRIPTION OF THE ROCK–FLUID CHEMICAL EQUILIBRIUM Properly designed charts provide convenient means of describing the equilibrium chemical reactions of the rock–fluid systems. Frequently, the saturation index, activity–activity, and electron activity–potentiometric activity (pe–pH) charts are facilitated for convenient description of equilibrium chemical systems. The construction of these charts are based on the description of chemical systems at thermodynamic equilibrium. In this section, the theoretical bases, characteristics, and utilization of these charts are described according to Schneider (1997). 13.5.1 Effect of Conditions on Species Proportionate at Equilibrium The composition of the various species in aqueous solutions undergoing dissolution/precipitation processes depends on various factors including pressure, temperature, and pH. For example, Figure 13-4, generated by Schneider (1997) using the SOLMINEQ.88 software (Kharaka et al., 1988), depicts the effect of pH on the composition of the typical carbonate − species, namely H2 CO03  HCO3 , and CO−2 3 . Similarly Figure 13-5 indicates the effect of pH on the composition of typical aqueous aluminum species, namely Al+3  AlOH+2  AlOH−4 , and AlOH+2 , generated by Schneider (1997) using SOLMINEQ.88.

Percentage of each species

pH Control for Carbonate Species 25 and 100 degrees Centigrade 100

100° C

100° C –

80

HCO3

25° C

25° C

60

H2CO3

40 20 0

CO3–2

25° C 100° C

4

6

8

10

12

pH

Figure 13-4. Effect of pH on distribution of carbonate Schneider, ©1997; reprinted by permission of G. W. Schneider).

species

(after

427

Inorganic Scaling and Geochemical Formation Damage

Percentage of each species

100 +3

AI

80

–

AI(OH)4

60 40 +

+2

AI(OH)2

AI(OH)

20 0 2

3

4

5

6

8

7

9

10

pH

Figure 13-5. Effect of pH on distribution of aluminum Schneider, ©1997; reprinted by permission of G. W. Schneider).

13.5.2

species

(after

Saturation Index Charts

The reactions for electrolyte dissolution in water can be represented by (Schneider, 1997) +n −m Am Bnsolid −→ ←− mAaq + nBaq

(13-32)

Substituting unity for the activity of the solid phase, the expression of the reaction quotient leads to the actual or instantaneous ion activity product, given by ap

K =



amA+n aq



actual

 anB−n aq

(13-33) actual

At equilibrium saturation, Eq. (13-33) yields the equilibrium saturation ion activity product constant given by   m K = aA+n sp

aq

eq

 anB−n aq

(13-34) eq

Thus, a saturation index can be defined as SI = log10



K ap K sp



(13-35)

428

Inorganic Scaling and Geochemical Formation Damage

and is used to determine the state of saturation of an aqueous solution by a mineral as follows: ⎧ ⎪ ⎨< 0 undersaturated SI = 0 saturated (13-36) ⎪ ⎩ > 0 supersaturated

The saturation index charts described later in this chapter provide information about the saturation conditions of aqueous solutions. 13.5.3

Activity–Activity or Mineral Stability Charts

Mineral stability charts are convenient means of representing the various equilibrium reactions of the solid minerals and aqueous solutions in geological porous media in terms of the saturation concept. Mineral stability charts can be more meaningfully developed by considering the incongruent equilibrium reactions of various solid phases including the igneous and metamorphic reactions (Schneider, 1997). Incongruent reactions represent the direct relationships of the various solid minerals involved in aqueous solution systems. The expressions of the incongruent reactions are derived from a combination of the relevant mineral dissolution/precipitation reactions in a manner to conserve certain key elements of the solid minerals so that the aqueous ionic species of these elements do not explicitly appear in the final equation. For example, the incongruent reactions of the alumino silicate minerals, including clay minerals, feldspars, and chlorites, are usually expressed to conserve the aluminum element (Fletcher, 1993; Schneider, 1997). Aluminum is a natural choice as the conserved element because this element is mostly immobile and the activities of the aqueous aluminum species are relatively low (Hayes and Boles, 1992; Schneider, 1997). Consequently, the incongruent mineral reaction equations do not involve the potential dissolved aluminum species such as Al+3  AlOH+2  AlOH−4  AlOH+2 , and AlOH03 (Schneider, 1997). Thus, the aluminum element conserving incongruent reaction to form the chlorite mineral from the kaolinite mineral reads as (Schneider, 1997) 1 4Al2 Si2 O5 OH4 +2 15Mg+2 + 2 25Fe+2 +5 8H2 0 Kaolinite

0 + −→ ←− Mg2 15 Fe2 25 Al2 8 Si2 7 O10 OH8 + 0 1H4 SiO4 + 8 8H Chlorite

(13-37)

Inorganic Scaling and Geochemical Formation Damage

429

The activity–activity charts depict the regions of precipitation of various solid mineral phases. The equations of the lines separating these regions are obtained by rearranging the logarithmic expression of the equilibrium constant in a linear form to relate the saturation products of the various mineral phases. For example, the equilibrium constant for Eq. (13-37) is given by (Schneider, 1997)    a8 8 a0 1 sp 0 H+ eq H4 SiO4 eq Kchlorite   K eq =  ≡ sp  Kkaolinite a2 25 a2 15 Mg+2 Fe+2 eq 

(13-38)

eq

in which the activities of the water and the solid kaolinite and chlorite phases were taken unity. A logarithm of Eq. (13-38) yields the linear equation for the kaolinite–chlorite phase boundary as sp sp = log 10 Kchlorite − log10 K eq log10 Kkaolinite

(13-39)

sp sp A plot of Kchlorite vs. Kkaolinite according to Eq. (13-39) yields the kaolinite–chlorite stability chart (Schneider, 1997) using SOLMINEQ.88 (Kharaka et al., 1988). Schneider (1997) points out that the determination of the aqueous species activities is particularly complicated in highly concentrated oilfield brines because of the complexing of cations with inorganic and organic anions, and can be better accomplished by means of a simulator such as SOLMINEQ.88 by Kharaka et al. (1988).

13.5.4 Electron Activity–Potentiometric Activity (pe–pH) Charts The electron activity–potentiometric activity or the so called pe–pH charts are constructed to describe the redox state of reservoirs (Stumm and Morgan, 1996; Schneider, 1997). Considering the electrons, e− , and protons, H+ , involved, chemical equilibrium reactions, such as oxidation–reduction (redox) and acid–base reactions, are represented by aA + bB + · · · ne− + mH+ −→ ←− cC + dD + · · ·

(13-40)

430

Inorganic Scaling and Geochemical Formation Damage

The electron activity (pe) and potentiometric acidity (pH) can be conveniently expressed by the following equations, respectively. pe = − log 10 e− 

(13-41)

pH = − log 10 H+ 

(13-42)

and

The electrode potential (Eh) and electron activity (pe) are related by (Schneider, 1997)   2 30259RT pe (13-43) Eh = F in which T denotes the absolute temperature in K, R = 8 31441 JK−1 mol−1 is the universal gas constant, and F = 9 64846 × 104 Coloumb/mol is the Faraday constant. The electrode potential can be measured directly. Equations (13-41)–(13-43) form the convenient mathematical bases for constructing the pe–pH or Eh–pH charts. However, the pe–pH charts are preferred to the Eh–pH charts because, while the sign of pH does not change and the slopes of the stability boundaries are independent of temperature, the sign of the Eh potential depends on the direction of the reaction and the slopes of the stability boundaries are temperature dependent (Schneider, 1997). For example, consider (Schneider, 1997) 3Fe2 O3 + 2e− + 2H+ −→ ←− 2Fe3 O4 + H2 O

(13-44)

Magnetite

Hematite

Assigning unity for the activities of the water and solid mineral phases, the equilibrium constant expression reads as (Schneider, 1997) K eq =

1 a2H+

a2e−

(13-45)

Hence, a logarithm of Eq. (13-45) yields the equation for the hematite–magnetite boundary as pe =

1 log Keq − pH 2

which can be used to construct a pe–pH chart (Schneider, 1997).

(13-46)

Inorganic Scaling and Geochemical Formation Damage

431

13.6 GEOCHEMICAL MODEL-ASSISTED ANALYSIS OF FLUID–FLUID AND ROCK–FLUID COMPATIBILITY Because of the highly complicated nature of the interactions of solid minerals and aqueous solution in geological porous media, it is most convenient to facilitate appropriate geochemical models for the analysis of the potential chemical interactions and formation damage affects. A typical example of such studies has been carried out by Schneider (1997) in an effort to quantify the potential formation damage problems, which would result from the invasion of incompatible foreign water, such as by drilling and workover fluids and water flooding, into the Lower Spraberry sandstone reservoir formation of Texaco’s Jo Mill Unit (JMU) field. Schneider (1997) used SOLMINEQ.88 to simulate the potential interactions and adverse affects of the formation minerals and aqueous phase. He assumed equilibrium conditions for conservative predictions of the rock–fluid interactions and water compatibility. In the following, a step-by-step description of the case studies carried out by Schneider (1997) is presented. The results demonstrate the scenarios of the potential fluid–fluid and rock–fluid interactions. 13.6.1

Description of Formation Properties

Based on the analyses using a scanning electron microscope, electron microprobe, and X-ray diffraction, Schneider (1997) determined the properties of the sandstone as “Fine-grained, immature sandstones that contain considerable detrital clay minerals and carbonate clasts, along with quartz, plagioclase, and minor volcanic rock fragments and K-feldspar grains. Observed accessory minerals were muscovite, glauconite, hornblende, zircon, and pyrite. Authigenic minerals are dominated by carbonate cements and filamentous lathes of pore-lining and pore-filling illite. Some authigenic chlorite and overgrowths of quartz and feldspar are also present.” 13.6.2

Mineral Composition

Schneider (1997) reports that the formation porosities and permeability are in the ranges of 10–20% and 0.5 to several mD, having considerable natural fracture permeability in certain regions and possibly some systematic fractures. He reports that this sandstone formation contains

432

Inorganic Scaling and Geochemical Formation Damage

6–10 volume % illite, 1–2 volume % chlorite, and negligible amounts of kaolinite (Scott, 1988). Typical minerals present in porous rocks and subject to dissolution in contact with aqueous phase include the various types of carbonates such as calcite, CaCO3 , magnesite, MgCO3 , dolomite, CaMgCO2 2 , strontianite, SrCO3 , witherite, BaCO3 , and siderite, FeCO3 , and various types of sulfates such as anhydrite, CaSO4 , gypsum, CaSO4 · 2H2 O, celestine, SrSO4 , and barite, BaSO4 (Schneider, 1997). Schneider (1997) points out that the kaolinite compositions remain close to the Al2 Si2 O5 OH4 formula, but the illite and chlorite formulae may vary as indicated by Aja et al. (1991a,b). He considered the typical mean compositions of the Bothamsall (Pennsylvanian), Rotliegendes (Permian), and Gulf Coast illites given, respectively, by (Kaiser, 1984; Warren and Curtis, 1989) K0 80 Mg0 13 Fe0 07 Al1 80  Al0 60 Si3 40  O10 OH2

(13-47)

K Mg0 15 Fe0 15 Al1 70  Al0 70 Si3 30  O10 OH2

(13-48)

K0 6 Mg0 25 Al1 80  Al0 5 Si3 5  O10 OH2

(13-49)

Schneider considered the typical mean compositions of the Gulf Coast (Kaiser, 1984) and the North Sea (Curtis et al., 1984, 1985) chlorites given, respectively, by

13.6.3 13.6.3.1

Mg2 3 Fe2 3 Al1 4 Al1 4 Si2 6  O10 OH8

(13-50)

Mg2 15 Fe2 25 Al1 5 Al1 30 Si2 70  O10 OH8

(13-51)

Water Characterization Water analysis

Although the analyses of the various waters produced by the Jo Mill Unit were available from the Inorganic Laboratory of Texaco EPTD, Schneider (1997) considered only the analyses of waters from the wells at five locations that did not make any appreciable amount of water. Therefore, for all practical purposes, these locations preserved their original water compositions. He also considered the analyses of the Mule Shoe Ranch and Canyon Reef waters that can be used for drilling and waterflooding operations. The analyses of these waters are presented in Table 13-1 by Schneider (1997).

Table 13-1 Ionic Species Concentrations (mg/L) of the Jo Mill Unit Reservoir Waters∗ Ionic Species

58,650 2,847 663 962 2.41 780 10.3 0.5 1,06,400 835 283 0 240 2.2 6.83 1,70,837 ND ND 24.6 ND 23.3 304 22 5.7 ND

JMU #7231 60,430 3,037 753 412 2.18 812 0.16 0.6 1,10,000 892 246 0 240 4.3 7.23 1,75,932 ND ND 26.2 ND 8.3 290 8.9 5.4 0: Supersaturated

0

0

20

40

60

Water infiltration, %

Figure 13-8. Illite saturation for mixtures of connate and invasion waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

Figure 13-9a reveals that the Mule Shoe Ranch water becomes incompatible with the JMU Connate water with respect to the chlorite stability upon large volumes of water injection, whereas the Canyon Reef water is compatible, as indicated by Figure 13-9b. Schneider (1997) constructed the kaolinite saturation index curves shown in Figure 13-10a and b based on the following dissolution/ precipitation reaction: − + Kaolinite + 7H2 O −→ ←− 2A l OH4 + 2H2 SiO4 + 2H

(13-54)

A comparison of Figure 13-10a and b reveals that the Canyon Reef water is compatible with the JMU connate water with respect to kaolinite, whereas the Mule Shoe water becomes incompatible upon large volumes of water injection into the reservoir.

438

Inorganic Scaling and Geochemical Formation Damage Chlorite Saturation in JMU Reservoir Mule Shoe Water Infiltration (8.4 pH)

Saturation index, Log (IAP/Ksp)

6 4

SI > 0: Supersaturated

2 0 –2

SI < 0: Undersaturated

–4 –6

0

20

40 60 Water infiltration, %

80

100

(a) Chlorite Saturation in JMU Reservoir Infiltration by Amoco SWD #28 Water

Saturation index, Log (IAP/Ksp)

6 4 2

SI < 0: Undersaturated

–2 –4 –6

(b)

SI > 0: Supersaturated

0

0

20

40 60 Water infiltration, %

80

100

Figure 13-9. Chlorite saturation for mixtures of connate and invasion waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

13.6.4.2 Saturation index charts for carbonates and sulfates

Using SOLMINEQ.88, Schneider (1997) generated the saturation index curves for the mixing of waters for calcite, CaCO3 , magnesite, MgCO3 , dolomite, CaMgCO3 2 , witherite, BaCO3 , strontianite, SrCO3 , anhydrite, CaSO4 , gypsum, CaSO4 ·2H2 O, barite, BaSO4 , and celestine, SrSO4 given in Figures 13-11–13-14. Figure 13-11a and b show the barite saturation index charts for mixing the JMU connate water with the Mule Shoe Ranch and Canyon Reef waters, respectively. It is apparent that mixing sufficient volumes of the Mule Shoe Ranch water with the JMU connate water will induce barite precipitation. But, the mixtures of the Canyon Reef and JMU connate waters will not lead to any precipitation. Figure 13-12a and b show the celestine saturation index charts for mixing the JMU connate water with the Mule Shoe Ranch and Canyon Reef waters. Figure 13-11a

439

Inorganic Scaling and Geochemical Formation Damage Kaolinite Saturation in JMU Reservoir Mule Shoe Water Infiltration (10.5 pH)

Saturation index, Log (IAP/Ksp)

6 4 2

SI > 0: Supersaturated

0

SI < 0: Undersaturated

–2 –4 –6

0

20

40 60 Water infiltration, %

80

100

(a) Kaolinite Saturation in JMU Reservoir Infiltration by Amoco SWD #28 Water

Saturation index, Log (IAP/Ksp)

6 4 2

SI < 0: Undersaturated

–2 –4 –6

(b)

SI > 0: Supersaturated

0

0

20

40 60 Water infiltration, %

80

100

Figure 13-10. Kaolinite saturation for mixtures of connate and invasion waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

shows that around 80% volume Mule Shoe Ranch water content, the water mixture is nearly saturated with celestine. However, the mixtures of the Canyon Reef and the JMU connate waters do not yield any precipitation. As indicated by the water analysis presented in Table 13-1, the JMU connate water is undersaturated by anhydrite and gypsum. Therefore, the saturation index charts shown in Figure 13-13a and b indicate no possibility of anhydrite or gypsum precipitation as a result of mixing JMU connate water with any portions of the Canyon Reef water. The water analysis given in Table 13-1 indicate that the JMU connate water is supersaturated by calcite and dolomite and undersaturated by witherite. Consequently, as indicated by Figure 13-14a, b, and c, mixing of this water with any portions of the Mule Shoe Ranch and Canyon Reef waters will not result in appreciable calcite and dolomite dissolution and any witherite precipitation.

440

Inorganic Scaling and Geochemical Formation Damage Barite Saturation in JMU Reservoir Mule Shoe Water Infiltration (8.4 pH)

Saturation index, Log (IAP/Ksp)

6 4 2

SI > 0: Supersaturated

0

SI < 0: Undersaturated

–2 –4 –6

0

20

(a)

40 60 Water infiltration, %

80

100

80

100

Barite Saturation in JMU Reservoir Infiltration by Amoco SWD #28 Water

Saturation index, Log (IAP/Ksp)

6 4 2

SI < 0: Undersaturated

–2 –4 –6

(b)

SI > 0: Supersaturated

0

0

20

40 60 Water infiltration, %

Figure 13-11. Barite saturation for mixtures of connate and invasion waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

13.6.5 13.6.5.1

Activity–Activity or Mineral Stability Charts Stability fields of clay minerals

Schneider (1997) equilibrated the activities of the five connate water compositions given in Table 13-1 to the 135 F temperature of the Jo Mill Unit reservoir using SOLMINEQ.88. He then plotted these activity values on the activity–activity charts. As can be seen in Figures 13-15–13-18, all points appear inside the mineral stability fields of the types of clay minerals present in the sandstone formation of the Jo Mill Unit reservoir. Hence, this confirmed the validity of the geochemical model and the accuracy of the mineral stability field charts generated by SOLMINEQ.88.

441

Inorganic Scaling and Geochemical Formation Damage SrSO4 Saturation in JMU Reservoir Mule Shoe Water Infiltration (8.4 pH)

Saturation index, Log (IAP/Ksp)

6 4

SI > 0: Oversaturated

2 0 –2

SI < 0: Undersaturated

–4 –6

0

20

(a)

40 60 Water infiltration, %

80

100

80

100

SrSO4 Saturation in JMU Reservoir Infiltration by Amoco SWD #28 Water

Saturation index, Log (IAP/Ksp)

6 4 0 –2 SI < 0: Undersaturated

–4 –6

(b)

SI > 0: Oversaturated

2

0

20

40 60 Water infiltration, %

Figure 13-12. Celestine saturation for mixtures of connate and invasion waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

Case 1 – Stability of Aluminosilicate Minerals

Schneider (1997) explains that the formula K Mg0 15 Fe0 15 Al1 7  Al0 7 Si3 3  O10 OH2

(13-55)

of the Rotliegendes illite is somewhat similar to the formula KAl2 AlSi3 O10 OH2 of the muscovite, which is an end-member composition illite. Because the JMU reservoir contains a high amount of illite (6–10 volume %), the JMU reservoir connate water compositions should appear within the muscovite stability region as indicated by Figure 13-15 by Schneider (1997). Case 2 – Illite–Chlorite Stability

Schneider (1997) constructed the illite–chlorite mineral stability charts shown in Figure 13-16 based on the following illite to chlorite incongruent

442

Inorganic Scaling and Geochemical Formation Damage Anyhydrite Saturation in JMU Reservoir Infiltration by Amoco SWD #28 Water

Saturation index, Log (IAP/Ksp)

6 4

SI > 0: Supersaturated

2 0 –2

SI < 0: Undersaturated

–4 –6

0

20

(a)

40 60 Water infiltration, %

80

100

Gypsum Saturation in JMU Reservoir Infiltration by Amoco SWD #28 Water

Saturation index, Log (IAP/Ksp)

6 4

0 –2 SI < 0: Undersaturated

–4 –6

(b)

SI > 0: Supersaturated

2

0

20

40 60 Water infiltration, %

80

100

Figure 13-13. Saturation of various sulfate minerals for mixtures of connate and invasion waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

reactions using the proper stoichiometric coefficients according to the compositional formulae of the illites and chlorites mentioned above: + + Illite + Mg+2 + Fe+2 + H2 O −→ ←− Chlorite + K + H4 SiO4 + H (13-56)

Again, as indicated by the mineral stability charts shown in Figure 13-16 by Schneider (1997), all the JMU reservoir connate water composition appear inside the mineral stability regions of the illites. Case 3 – Illite–Kaolinite Stability

Schneider (1997) constructed the illite–kaolinite mineral stability charts shown in Figure 13-17 based on the following illite to kaolinite incongruent reactions using a proper set of stochiometric coefficients according

443

Inorganic Scaling and Geochemical Formation Damage

Calcite Saturation in JMU Reservoir Infiltration by Amoco SWD #28 Water

Saturation index, Log (IAP/Ksp)

6 4

SI > 0: Supersaturated

2 0

SI < 0: Undersaturated

–2 –4 –6

0

20

40

60

80

100

Water infiltration, %

(a)

Dolomite Saturation in JMU Reservoir Infiltration by Amoco SWD # 28 Water

Saturation index, Log (IAP/Ksp)

6 4

SI > 0: Supersaturated

2 0

SI < 0: Undersaturated

–2 –4 –6

0

20

40

60

80

100

Water infiltration, %

(b) BaCO3 Saturation in JMU Reservoir Infiltration by Amoco SWD # 28 Water

Saturation index, Log (IAP/Ksp)

6 4

SI > 0: Supersaturated

2 0

SI < 0: Undersaturated

–2 –4 –6

0

20

40

60

80

100

Water infiltration, %

(c)

Figure 13-14. Saturation of various carbonate minerals for mixtures of connate and invasion waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

444

Inorganic Scaling and Geochemical Formation Damage 8 Muscovite

Log {K+}/{H+}

6

4

JMU Connate Water Compositions

2

Gibbsite

K-feldspar

Kaolinite Pyrophyllite

0

–2

–6

–5

–3 –4 Log {H4SiO4}

–2

–1

Figure 13-15. Stability chart for aluminosilicate minerals (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

to the compositional formulae of the illites and kaolinites considered for the study: +2 +2 + Illite + H+ + H2 O −→ ←− Kaolinite + K + Mg + Fe + H4 SiO4 (13-57)

Because of the existence of a large quantity of illite (6–10 volume %) and a negligible amount of kaolinite in the JMU sandstone reservoir formations, all the JMU connate waters appear inside the illite stability region. Case 4 – Chlorite–Kaolinite Stability

Schneider (1997) constructed the chlorite–kaolinite mineral stability charts shown in Figure 13-18 based on the following chlorite to kaolinite incongruent reactions using the proper set of stoichiometric coefficients according to the compositional formulae of the chlorites considered for the study: +2 +2 Chlorite + H+ + H4 SiO4 −→ ←− Al2 Si2 O5 OH4 + Mg + Fe + H2 O (13-58)

Because of a relatively larger quantity of the chlorite (1–2 volume %) compared to the negligible amount of kaolinite present in the JMU

445

Log {Mg}1.64 × {Fe}1.89/{K+}.6

Inorganic Scaling and Geochemical Formation Damage

10 0 –10

GULF COAST CHLORITE

–20 –30 –80

JMU Connate Water Compositions

GULF COAST ILLITE –70

–60

Log

–50

–40

{H4SiO4}1.37 × {H+}6.46

Log {Mg}1.7129 × {Fe}1.8586/{K+}.8

(a)

10 0 –10

NORTH SEA CHLORITE JMU Connate Water Compositions

BOTHAMSALL ILLITE

–20 –30 –70

–60

–50

–40

–30

Log {H4SiO4}1.0857 × {H+}6.3428

Log {Mg}1.6929 × {Fe}1.7786/{K+}

(b)

10 0 –10

NORTH SEA CHLORITE JMU Connate Water Compositions

–20 –30 –70

ROTLIEGENDES ILLITE –60

–50

–40

–30

Log {H4SiO4}.9857 × {H+}5.9429 (c)

Figure 13-16. Illite–chlorite mineral stability chart (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

446

Inorganic Scaling and Geochemical Formation Damage

Log {Mg++}2.15 × {Fe++}2.25

2 0

GULF COAST ILLITE

–2 JMU Connate Water Compositions

–4

KAOLINITE

–6 –8

–4

–6

–8

–2

0

2

Log {H4SiO4}.1 × {H+}8.8 (a)

Log ({Mg}.13 {Fe}.07 {K+}.8)

2

BOTHAMSALL ILLITE

0 –2

JMU Connate Water Compositions

–4

KAOLINITE –6 –8

–6

–8

Log

0

–2

–4

2

({H+}1.2/{H4SiO4})

(b)

Log {Mg}.15 × {Fe}.15 × {K+}

2.5 0

ROTLIEGENDES ILLITE

–2.5 JMU Connate Water Compositions

KAOLINITE

–5 –7.5 –12.5

–10

–7.5

–5

–2.5

Log ({H+}1.6/{H4SiO4}.9) (c)

Figure 13-17. Illite–kaolinite mineral stability chart (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

447

Inorganic Scaling and Geochemical Formation Damage

Log {Mg++}2.3 × {Fe++}2.3

–10

JMU Connate Water Compositions

–20

–30

GULF COAST CHLORITE KAOLINITE

–40

–50 –90

–70

–80

–60

–50

Log {H4SiO4}.2 × {H+}9.2

(a)

Log {Mg++}2.15 × {Fe++}2.25

0

–10

–20 JMU Connate Water Compositions

KAOLINITE

–30

–40 –80

(b)

NORTH SEA CHLORITE

–70

–60

Log

–50

–40

{H4SiO4}.1 × {H+}8.8

Figure 13-18. Chlorite–kaolinite mineral stability chart (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

sandstone formation, all the JMU connate waters appear inside the chlorite stability region. 13.6.5.2 Mineral stability during mixing of foreign and reservoir waters

Effects of mixing foreign and reservoir waters on mineral stability are best realized by constructing mixing paths on the mineral stability charts. Schneider (1997) investigated the compatibility of the JMU#7231 well connate water with the Mule Shoe Ranch and Canyon Reef waters considered for potential use in drilling and/or water flooding operations. The analyses of these waters are given by Schneider (1997) in Table 13-1. − It is apparent that the Mule Shoe water has a higher CO−2 2 and HCO3

448

Inorganic Scaling and Geochemical Formation Damage

content and, therefore, higher alkalinity than the other waters reported in Table 13-1. He used SOLMINEQ.88 and simulated the consequences of mixing the JMU #7231 connate water with 10% volume increments of the Mule Shoe Ranch and Canyon Reef waters. Schneider (1997) constructed an illite–chlorite mineral stability chart based on the following reaction equation with proper stoichiometric coefficients for the Rotliegendes illite formula: Illite + 1 6929Mg+2 + 1 7786Fe+2 + 7 3714H2 O 0 + + −→ ←− Chlorite + K + 0 9857H4 SiO4 + 5 9429H

(13-59)

The equilibrium constant for this reaction is given by eq

K =

a1 6929 a1 7786 a7 3714 H2 O Mg+2 Fe+2 aK+ a0 9857 a5 9429 H SiO0 H+ 4

4

=

sp KIllite sp KChlorite

(13-60)

The equation for the illite–chlorite phase boundary is given by the logarithm of Eq. (13-60) as      0 9857 5 9429 1 7786 7 3714 log a1 6929 a a a = log a a + log K eq (13-61) + + K H2 O Mg+2 Fe+2 H SiO0 H 4

4

Case 1 – Effect of pH on Illite Stability

Schneider (1997) then plotted the curves for mixing the JMU connate water with the Mule Shoe water on this chart for the 8.4, 9.5, and 10.5 pH values, as shown in Figure 13-19a,b, and c, respectively. Figure 13-19 indicates that illite becomes less stable at higher pH. Case 2 – Effect of K+ on Illite Stability

Schneider (1997) also investigated the effect of the K+ activity on the illite stability. The mixing curves for 0, 2, and 5 weight % KCl solutions at the 10.5 pH level are shown in Figure 13-20a,b, and c, respectively. Clearly, adding KCl increases the illite stability. However, K + activity has a relatively smaller effect than pH, in view of the comparison of Figures 13-19 and 13-20.

449

Log {Mg}1.6929 × {Fe}1.7786 / {K+}

Inorganic Scaling and Geochemical Formation Damage

10

JMU Connate Mixed with Mule Shoe Water (8.4 pH) Connate water Mixed water

0

CHLORITE –10

ILLITE

–20 –30 –70

–50

–60

–40

–30

Log {H4SiO4}.9857 × {H+}5.9429

Log {Mg}1.6929 × {Fe}1.7786 / {K+}

(a) 10

JMU Connate Mixed with Mule Shoe Water (9.5 pH) Connate water Mixed water

0

CHLORITE –10

ILLITE

–20 –30 –70

–50

–60

–40

–30

Log {H4SiO4}.9857 × {H+}5.9429

Log {Mg}1.6929 × {Fe}1.7786 / {K+}

(b)

10

JMU Connate Mixed with Mule Shoe Water (10.5 pH) Connate water Mixed water

0

CHLORITE –10 –20 –30 –70

ILLITE

–60

Log

–50

–40

–30

{H4SiO4}.9857 × {H+}5.9429

(c)

Figure 13-19. Effect of pH on illite stability for mixtures of connate and Mule Shoe waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

450

Log {Mg}1.6929 × {Fe}1.7786 / {K+}

Inorganic Scaling and Geochemical Formation Damage

10

JMU Connate & Mule Shoe Water (10.5 pH & 0% KCI) Connate water Mixed water

0

CHLORITE –10 –20

ILLITE

–30 –70

–60

–50

–40

–30

Log {H4SiO4}.9857 × {H+}5.9429

Log {Mg}1.6929 × {Fe}1.7786 / {K+}

(a) 10

JMU Connate & Mule Shoe Water (10.5 pH & 2% KCI) Connate water Mixed water

0

CHLORITE

–10 –20

ILLITE

–30 –70

–60

–50

–40

–30

Log {H4SiO4}.9857 × {H+}5.9429

Log {Mg}1.6929 × {Fe}1.7786 / {K+}

(b) 10

JMU Connate & Mule Shoe Water (10.5 pH & 5% KCI) Connate water Mixed water

0

CHLORITE –10 –20 –30 –70

ILLITE –60

–50

–40

–30

Log {H4SiO4}.9857 × {H+}5.9429 (c)

Figure 13-20. Effect of KCl on illite stability for mixtures of connate water and Mule Shoe waters (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

Inorganic Scaling and Geochemical Formation Damage

451

13.6.6 Electron Activity–Potentiometric Activity (pe–pH) Charts 13.6.6.1

Aqueous species of the Fe–O–H2 O system

Schneider (1997) constructed the pe–pH charts for the aqueous species involving the following half-reactions of the Fe–O–H2 O system: +2 Fe+3 + e− −→ ←− Fe

(13-62)

+3 FeOH3 + 3H+ −→ ←− Fe + 3H2 O

(13-63)

+2 FeOH3 + 3H+ + e− −→ ←− Fe + 3H2 O

(13-64)

+2 FeOH+ + H+ −→ ←− Fe + H2 O

(13-65)

+ − FeOH+ + 2H2 O −→ ←− FeOH3 + 2H + e

(13-66)

for which he wrote the following pe–pH relationships:

  Fe+2 eq pe = log K − log Fe+3    1 1 log Keq − log Fe+3 3 3   pe = log Keq − log Fe+2 − 3pH

   Fe+2 eq pH = log K − log FeOH+  pH =

  pe = − log K eq − 2pH − log FeOH+

(13-67)

(13-68) (13-69) (13-70) (13-71)

Schneider (1997) then applied SOLMINEQ.88 and constructed the charts shown in Figure 13-21a by plotting Eqs (13-67)–(13-71) to represent the stability boundaries. For this purpose, he used a mean activity value of 10−5 6 m for Fe+2 , even though the actual activity values for Fe+2 in the JMU connate waters vary between 10−4 3 and 10−6 4 m. All JMU connate water compositions appear in the FeOH+ solution region. But, FeOH3 can precipitate if the pe or pH is varied to move into the FeOH3 stability region.

452

Inorganic Scaling and Geochemical Formation Damage pe–pH DIAGRAM: Aqueous iron species

20

Fe (+3)

16

O2 fugacity > 1

12

pe

8

Fe(OH)3

Fe (+2)

4

JMU Connate Water Compositions

0 –4

H2 fugacity > 1

FeOH+

–8 –12 0

2

4

(a)

6

8

16

Fe (+3)

O2 fugacity > 1

12

Fe2O3 (Hematite)

8

pe

12

pe–pH DIAGRAM: Hematite–Siderite–Magnetite

20

4

JMU Connate Water Compositions

Fe (+2)

0

H2 fugacity > 1

–4 –8

FeCO3 (Siderite)

–12 0

(b)

10

pH

2

4

Fe3O4 (Magnetite) 6

8

10

12

pH

Figure 13-21. pe–pH chart for Fe–O–H2 O system (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

13.6.6.2

Mineral species of the Fe–O–H2 O system

Schneider (1997) also constructed the pe–pH charts for the minerals involving the half-reactions of the Fe–O–H2 O system: +2 Fe+3 + e− −→ ←− Fe

(13-72)

+3 Fe2 O3 + 6H+ −→ ←− 2Fe + 3H2 O

(13-73)

Inorganic Scaling and Geochemical Formation Damage +2 Fe2 O3 + 6H+ + 2e− −→ ←− 2Fe + 3H2 O

Fe2 O3 + 2HCO−3 + 4H+ + 2e− −→ ←− 2FeCO3 + 3H2 O − +2 FeCO3 + H+ −→ ←− Fe + HCO3

Fe3 O4 + 3HCO−3 + 5H+ + 2e− −→ ←− 3FeCO3 + 4H2 O

453

(13-74) (13-75) (13-76) (13-77)

+2 Fe3 O4 + 8H+ + 2e− −→ ←− 3Fe + 4H2 O

(13-78)

3Fe2 O3 + 2H+ + 2e− −→ ←− 2Fe3 O4 + H2 O

(13-79)

from which he wrote the following pe–pH relationships: pe =

(13-80)

1 log Keq + log HCO−3  − 2 pH 2

(13-81)

1 5 3 log Keq + log HCO−3  − pH 2 2 2

(13-82)

pe = pe =

  1 log Keq − log Fe+2 − 3 pH 2

Schneider (1997) then applied SOLMINEQ.88 and constructed the charts shown in Figure 13-21b by plotting Eqs (13-80)–(13-82) to represent the stability boundaries. The JMU connate water compositions yield somewhat different pe values for the redox reaction couples of Fe2 O3 –Fe+2  Fe2 O3 –FeCO3 , and Fe3 O4 –FeCO3 , as indicated by triangles, squares, and circles, respectively, in Figure 13-21b. Overall, Schneider concludes that “the JMU reservoir is a highly-reduced environment” based on Figure 13-21a and b. 13.6.6.3

Aqueous species of the Fe–O–H2 O–S system

Schneider (1997) considered the following half-reactions for the sulfate and sulfide species present in an aqueous system: 0 + − −→ SO−2 4 + 10H + 8e ←− H2 S + 4H2 O

(13-83)

− + − −→ SO−2 4 + 9H + 8e ←− HS + 4H2 O

(13-84)

454

Inorganic Scaling and Geochemical Formation Damage − + H2 S0 −→ ←− H + HS

(13-85)

− + −→ SO−2 4 + 8H + 6e ←− Ss + 4H2 O

(13-86)

0 Ss + 2H+ + 2e− −→ ←− H2 S

(13-87)

Of the relationships that could be written for the half-reactions given by Eqs (13-83)–(13-87), Schneider (1997) considered only the following relationship for the SO−2 4 –Ssolid redox reaction couple: pe =

  4 1 1 log Keq − pH + log SO−2 4 6 3 6

(13-88)

because only the sulfate concentrations were determined as indicated by the water analyses given in Table 13-1 and the H2 S0 and HS− species activities determined by SOLMINEQ.88 are dependent on the SO−2 4 activities. Figure 13-22a shows that all the JMU connate water compositions appear in the reduced stability region. 13.6.6.4

Mineral species of the Fe–O–H2 O–S system

Schneider (1997) also considered the following half-reactions for the minerals involving the Fe–O–H2 O–S system: +2 2SO−2 + 16H+ + 14e− −→ ←− FeS2 + 8H2 O 4 + Fe

(13-89)

+ − −→ Fe2 O3 + 4SO−2 4 + 38H + 30e ←− 2FeS2 + 19H2 0

(13-90)

+2 Fe2 O3 + 6H+ + 2e− −→ ←− 2Fe + 3H2 0

(13-91)

from which he wrote the following relationships. pe =

   8  1 1 1 log Keq + log SO−2 + log Fe+2 − pH 4 14 7 14 7 pe =

  38 1 4 log Keq + log SO−2 pH − 4 30 30 30

(13-92) (13-93)

These equations were used to construct the hematite and pyrite stability regions shown in Figure 13-22b for the SO−2 4 –FeS2solid redox reaction couple. The JMU connate water compositions appear in the reduced stability region.

455

Inorganic Scaling and Geochemical Formation Damage pe–pH DIAGRAM: Sulfate and Sulfide Species

20

Jo Mill Unit Reservoir Temperature

16

O2 fugacity > 1

12 8 SO4 (–2)

pe

4 0

JMU Connate Water Compositions

–4

H2 fugacity > 1

–8 –12

S (c) 2

H2S (aq)

3

4

5

6

HS– 7

8

9

10

pH

(a)

pe–pH DIAGRAM: Pyrite–Hematite Stability Fields

20

JMU Reservoir Temperature

16

O2 fugacity > 1

12

Fe2O3

pe

8 [Fe++]

4

[SO4 (–2)]

JMU Connate Water Compositions

0 –4 –8 –12 (b)

[H2S]

H2 fugacity > 1

2

3

[HS–]

FeS2

S (c) 4

5

6

7

8

9

10

pH

Figure 13-22. pe–pH chart for Fe–O–H2 O–S system (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

13.6.6.5

The aqueous O–H2 O system

Schneider (1997) points out that the waters pumped from the surface to the JMU reservoir are likely to contain some dissolved oxygen gas. Therefore, Schneider (1997) constructed a pe–pH chart for the oxygenated waters at the JMU reservoir temperature as shown in Figure 13-23, based on the stability boundaries developed for the reduction of oxygen according to the following reactions given by Drever (1988):

456

Inorganic Scaling and Geochemical Formation Damage pe–pH DIAGRAM: Oxygenated Water

20

Jo Mill Unit Reservoir Temperature

16

O2 fugacity > 1

12

Range of values for oxygenated waters

pe

8 4 0 H2 fugacity > 1

–4 –8 –12

0

2

4

6 pH

8

10

12

Figure 13-23. pe range for oxygenated waters determined by the reaction model of Drever (1988) (after Schneider, ©1997; reprinted by permission of G. W. Schneider).

O2 + 2H+ + 2e− −→ ←− H2 O2

(13-94)

H2 O2 + 2H+ + 2e− −→ ←− 2H2 O

(13-95)

Therefore, the following relationship can be written: pe = −pH −

1 log Keq 2

(13-96)

Figure 13-23 shows the range of the pe values for the oxygenated waters at the JMU reservoir temperature.

13.7 GEOCHEMICAL SIMULATION OF ROCK–FLUID INTERACTIONS IN BRINE-SATURATED SEDIMENTARY BASINS Xu et al. (2006) state that carbon dioxide may be retained in subsurface formations essentially by three mechanisms: a) Hydrodynamic trapping below a low-permeability cap rock either at gaseous or at supercritical fluid states

Inorganic Scaling and Geochemical Formation Damage

457

b) Solubility trapping by dissolving into the ground water c) Mineral trapping by precipitation of secondary carbonates as a result of reactions with the mineral and organic substances present in the geological formation. The consequences of the chemically reactive nonisothermal multiphase fluid flow in geological formations may be investigated by means of a suitable numerical simulation program such as TOUGHREACT (Xu et al., 2006). In this section, two examples are presented, involving CO2 sequestration in saline subsurface aquifers, for illustration of the geochemical simulation.

13.7.1

Case 1 – Zhao et al. (2001) Simulation

Zhao et al. (2001) simulated the alteration of the rock and fluid properties by carbon dioxide CO2  injection into a two-dimensional aquifer of 3000 m× 1000 m size, as depicted in Figure 13-24. Their mathematical model and simulation results are reviewed here. The aquifer rock is assumed isothermal and homogeneous. The initial rock porosity is 10%. The rock contains 99% quartz and 1% K-feldspar by weight. Hence, their generalized concentrations (expressed per unit rock bulk volume) are 36.82 and 0 87 kmol/m3 -rock, respectively. The horizontal 2 volumetric flux of the brine through the aquifer is 1 0 × 10−8 m3 /m -s. + + The initial concentrations of H and K in the brine at equilibrium 3000 m

CO2

99% Quartz + 1% K-feldspar

1000 m

φ 0 = 0.1, u x 0 = 10–8 m/s

Figure 13-24. Geometry and initial conditions for the fluid–rock interaction problem in a pore-fluid-saturated aquifer (Reprinted from Computer Methods in Applied Mechanics and Engineering, Vol. 190, Zhao, C. B., Hobbs, B. E., Walshe, J. L., Mühlhaus, H. B., and Ord, A., “Finite element modeling of fluid-rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins,” Nos. 18–19, pp. 2277–2293, ©2001, with permission from Elsevier Science).

458

Inorganic Scaling and Geochemical Formation Damage

before CO2 injection are 1 6 × 10−5 and 0 1 kmol/m3 -brine. The H+ concentration in the brine injected from the left boundary is taken as 6 4 × 10−3 kmol/m3 -brine. The dispersion coefficient values were taken as 2 0 × 10−6 m2 /s for K + and 2 0 × 10−4 m2 /s for the injected H+ (same as for the injected CO2 ). The injected CO2 rapidly reacts with water H2 O in brine (reaction rate constant k → ) to produce H+ . The reaction attains a quasi-instantaneous chemical equilibrium condition according to the following reaction: Fast

CO2 + H2 O −→ HCO−3 + H+

(13-97)

Consequently, K-feldspar dissolves and muscovite precipitates according to the following heterogeneous chemical reaction: k1 + 3KAlSi3 O8 + 2H+ −→ ←− 2K + KAl3 Si3 O10 OH2 + 6SiO2

(13-98)

The dissolution rate constant is k1 = 5 03 × 10−12 kmol/m2 -s and the chemical equilibrium constant is K1 = 3 89 × 10−7 . Simultaneously, muscovite dissolves and pyrophyllite precipitates according to the following heterogeneous chemical reaction: k2 + 2KAl3 Si3 O10 OH2 + 2H+ + 6SiO2 −→ ←− 2K + 3Al2 Si4 O10 OH2 (13-99)

The dissolution reaction rate constant is k2 = 4 48 × 10−12 kmol/m2 -s and the chemical equilibrium constant is K2 = 8318 0. Inferring from the above-mentioned three chemical reactions, Zhao et al. (2001) considered six primary chemically reactive species. In the brine, two aqueous species were considered, namely H+ and K + . Four minerals were considered in the rock, namely K-feldspar, muscovite, pyrophyllite, and quartz. The transient-state convection–dispersion–reaction equation for the aqueous species is given by 

C C Ci + ux i + uy i = Di  2 Ci + Ri t x y

(13-100)

Note that this equation neglects the variation of porosity  and species dispersion coefficient Di in the transport by dispersion term. For computational convenience, this equation has been reformulated as shown

Inorganic Scaling and Geochemical Formation Damage

459

below. Multiply by o / and then add ±uxo and ±uyo to the convective transport terms to obtain     o o Ci Ci Ci o + ux ± uxo + uy ± uyo = o Di  2 Ci + o Ri t  x  y (13-101) Then, rearrange so that the terms on the left of the equation will have constant coefficients: o

C C Ci + uxo i + uyo i = o Di  2 Ci + o Ri + Rei t x y

(13-102)

   o Ci Ci o + uyo − uy = uxo − ux  x  y

(13-103)

Where Rei



Only a transient-state reaction equation is considered for the mineral species as Ci = −Ri t

(13-104)

The instantaneous porosity accounting for the effect of the heterogeneous chemical reactions is calculated by  = 1 0 −

m  M i Ci i=1

i

(13-105)

Where m denotes the number of minerals involved, Mi denotes the molecular weight, Ci is the generalized concentrations (rock bulk volume based), and i is the density of mineral i. The reaction rate term Ri is given by Ri = −

N 

ij rj

(13-106)

j=1

Where i and j denote the various minerals/species, N denotes the number of heterogeneous mineral chemical reactions, ij the participitation factor of a mineral/species j to the reaction rate of mineral/species i, and rj represents the dissolution rate of mineral j, given by (Lasaga et al., 1994)       (13-107) rj = ±Aj kj 1 − Qj Kj 

460

Inorganic Scaling and Geochemical Formation Damage

Where the positive and negative values indicate dissolution and precipitation, respectively, kj is the reaction rate constant, Kj is the reaction equilibrium constant, Qj is the chemical affinity of the heterogeneous chemical reaction j, and  and  are empirical parameters (Zhao et al., 2001, took  = 1 0 and  = 1 0). Aj is the surface area of mineral j expressed per unit brine volume, given by an empirical power-law function of the generalized concentration of a mineral, as Aj = j Cjq

(13-108)

Where j and q are empirically determined parameters, whose values depend on various factors, such as composition, size, packing of mineral grains. However, q = 2/3 if uniform packing of same size spherical grains is considered. In a two-dimensional case considered in Figure 13-24, q = 1/2 for uniform packing of circles of the same size. Zhao et al. considered j = 1 0 and q = 0 5 in the numerical simulation of the rock–fluid alteration over a period of ten years. They simulated the system as an initial value problem and not as a boundary value problem. Thus, no boundary condition was specified. Their results are presented in a different form in Figure 13-25, depicting the propagation of the front locations of the K-feldspar dissolution, muscovite precipitation and dissolution, and pyrophyllite precipitation.

x DF, Dimensionless front position from injection side

0.9 0.8 0.7 0.6 0.5 0.4 0.3 K-feldspar dissolution Muscovite precipitation Muscovite dissolution Pyrophyllite precipitation

0.2 0.1 0

0

1000

2000

3000

4000

5000

t , yr

Figure 13-25. Dissolution/precipitation front positions for various minerals inferred by the simulation results of Zhao et al. (2001) (plot prepared by the author).

Inorganic Scaling and Geochemical Formation Damage

13.7.2

461

Case 2 – Xu et al. (2006) Simulation

Xu et al. (2006) simulated the rock and fluid alterations following the carbondioxide injection into a saline aquifer in the near-wellbore formation. They injected CO2 at a rate of 100 kg/s (8640 tons/day) through a vertical well located at the center of a 10,000-m radius, uniform thick circular region for 100 years and simulated the geochemical transport processes and alteration of rock and fluid properties using the TOUGHREACT computer simulation program. They considered the porosity and permeability alteration and clay swelling effects. Table 13-3 presents the hydrogeological parameters of the simulated problem. Table 13-4 presents Table 13-3 Hydrogeological Parameters for 1-D Radial CO2 Injection Problem Aquifer thickness Permeability Porosity Compressibility Temperature Pressure Salinity CO2 injection rate Relative permeability: Liquid√(van Genuchten, 1980): krl = S ∗ 1 − 1 − S ∗ 1/m m 2 Irreducible water saturation Exponent

100 m 10−13 m2 0.30 4 5 × 10−10 Pa−1 75 C 200 bar 0.06 (mass fraction) 100 kg/s

S ∗ = Sl − Slr /1 − Slr  Slr = 0 30 m = 0 457

Gas (Corey, 1954): krg = 1 − S2 1 − S 2  Irreducible gas saturation Capillary pressure: Van Genuchten (1980): Pcap = −P0 S ∗ −1/m − 11−m Irreducible water saturation Exponent Strength coefficient

Sl − Slr  Sl − Slr − Sgr  Sgr = 0 05 S=

S ∗ = Sl − Slr /1 − Slr  Slr = 0 00 m = 0 457 P0 = 19 61 kPa

Reprinted from Computers & Geosciences, Vol. 32, Xu, T., Sonnenthal, E., Spycher, N., and Pruess, K., “TOUGHREACT – A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration,” Issue 2, pp. 145–165, ©2006, with permission from Elsevier Science.

462

Parameters for kinetic rate law Neutral mechanism Mineral

Vol% of solid

A (cm2 /g) k25 (mol/m2 /s)

Ea (KJ/mol)

Acid mechanism Ea

k25

n H+ 

Base mechanism k25

Ea

n H+ 

Primary: Quartz

57 888

9 8

1 023 × 10−14

87 7

Kaolinite

2 015

151 6

6 918 × 10−14

22 2

4 898 × 10−12 65 9

0 777

8 913 × 10−18 17 9 −0 472

1 660 × 10−13

35

1 047 × 10−11 23 6

0 34

3 020 × 10−17 58 9 −0 4

Calcite

1 929

Assumed at equilibrium

Illite

0 954

151 6

Oligoclase

19 795

9 8

1 445 × 10

69 8

2 138 × 10

K-feldspar

8 179

9 8

3 890 × 10−13

38

8 710 × 10−11 51 7

0 5

6 310 × 10−12 94 1 −0 823

Na-smectite

3 897

151 6

1 660 × 10

−13

35

1 047 × 10

0 34

3 020 × 10−17 58 9 −0 4

Chlorite

4 556

9 8

3 02 × 10−13

88

7 762 × 10−12 88

0 5

Hematite

0 497

12 9

66 2

4 074 × 10

1

−12

2 512 × 10

−15

−10

−11

−10

65 23 6 66 2

0 457

Inorganic Scaling and Geochemical Formation Damage

Table 13-4 Initial Mineral Volume Fractions, Possible Secondary Mineral Phases, and their Kinetic Properties

9 8 9 8 9 8 9 8 9 8 9 8 151 6 12 9

4 571 × 10−10 2 951 × 10−8 2 754 × 10−13 1 260 × 10−9 1 260 × 10−9 1 260 × 10−9 1 660 × 10−13 k25 = 2 818 × 10−5 Ea = 56 9 nO2 aq = 0 5

23 5 52 2 69 8 62 76 62 76 62 76 35

4 169 × 10−7 6 457 × 10−4 6 918 × 10−11 6 457 × 10−4 6 457 × 10−4 6 457 × 10−4 1 047 × 10−11 k25 = 3 02 × 10−8 Ea = 56 9 nH+  = −0 5 nFe3+  = 0 5

14 4 36 1 65 36 1 36 1 36 1 23 6

1 0 5 0 457 0 5 0 5 0 5 0 34

2 512 × 10−16

71

−0 572

3 020 × 10−17

58 9

−0 4

Notes: (1) all rate constants are listed for dissolution; (2) A is reactive surface area, k25 is kinetic constant at 25 C Ea is activation energy, and n is power term; (3) power term n for both acid and base mechanisms are with respect to H + ; (4) for pyrite, neutral mechanism has an n with respect to O2 aq, acid mechanism has two species involved: one n with respect to H + and another n with respect to Fe3+ ; (5) dolomite, Ca-smectite, and pyrite were included in the list of possible secondary mineral phases in input but they were not formed during simulation.

Inorganic Scaling and Geochemical Formation Damage

Secondary: Magnesite Dolomite Low-albite Siderite Ankerite Dawsonite Ca-smectite Pyrite

Reprinted from Computers & Geosciences, Vol. 32, Xu, T., Sonnenthal, E., Spycher, N., and Pruess, K., “TOUGHREACT – A simulation program for nonisothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration,” Issue 2, pp. 145–165, ©2006, with permission from Elsevier Science.

463

464

Inorganic Scaling and Geochemical Formation Damage

the initial mineral abundances and relevant data for three mechanisms (neutral, acid, and base), which they obtained from various literatures, reported in their paper. Figures 13-26, 13-27, and 13-28 present the simulation results, such as the water saturation (remaining part is the

0.010

Calcite

–0.005 0.000 1000 yr –0.005

5000 10,000

–0.010 –0.015

(a)

0

6000 2000 4000 Radial distance (m)

8000

Change of abundance (volume fraction)

Change of abundance (volume fraction)

Figure 13-26. Water saturation and pH at different times for I-D radial CO2 injection problem (Reprinted from Computers & Geosciences, Vol. 32, Xu, T., Sonnenthal, E., Spycher, N., and Pruess, K., “TOUGHREACT – A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration,” Issue 2, pp. 145–165, ©2006, with permission from Elsevier Science). 0.010

–0.005

K-feldspar

–0.010 1000 yr –0.015

5000 10,000

–0.020

(b)

0

2000 4000 6000 Radial distance (m)

8000

Figure 13-27. Change in mineral abundance (negative values indicate dissolution and positive precipitation) after different times for I-D radial CO2 injection problem (Reprinted from Computers & Geosciences, Vol. 32, Xu, T., Sonnenthal, E., Spycher, N., and Pruess, K., “TOUGHREACT – A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration,” Issue 2, pp. 145–165, ©2006, with permission from Elsevier Science).

465

Oligoclase 1000 yr 2000 5000

–0.150 0

Change of abundance (volume fraction)

(c)

0.030

1000 yr 0.020

5000 yr 10,000

0.010 0.000 –0.010 0

Change of abundance (volume fraction)

2000 4000 6000 Radial distance (m)

8000

Kaolinite 0.010

1000 yr 5000 yr 10,000

–0.010 0

2000 4000 6000 Radial distance (m)

8000

1000 yr 5000 yr 10,000 0.020 Ankerite

0.010

0.000 0

2000 4000 6000 Radial distance (m)

1000 yr

5000

–0.030

10,000 –0.040 0

8000

2000 4000 6000 Radial distance (m)

8000

0.050 1000 yr 0.040

5000 yr

Illite

10,000

0.030 0.020 0.010 0.000 –0.010 0

(j)

2000 4000 6000 Radial distance (m)

8000

0.10

Na-smectite

0.05

0.00

1000 yr 5000 yr 10,000

–0.05 0

(h)

0.040

0.030

–0.020

(f)

0.020

0.000

Chlorite 0.010

(d)

Quartz

(g) Change of abundance (volume fraction)

8000

0.040

(e)

(i)

2000 4000 6000 Radial distance (m)

Change of abundance (volume fraction)

–0.100

0.000

Change of abundance (volume fraction)

–0.050

Change of abundance (volume fraction)

0.000

Change of abundance (volume fraction)

Change of abundance (volume fraction)

Inorganic Scaling and Geochemical Formation Damage

2000 4000 6000 Radial distance (m)

8000

0.040 Dawsonite 0.030 1000 yr 0.020

5000 yr 10,000

0.010

0.000 0

2000 4000 6000 Radial distance (m)

Figure 13-27. (Continued )

8000

Inorganic Scaling and Geochemical Formation Damage

0.004

1000 yr 5000

0.003

10,000

0.002 Siderite 0.001

0.000 0

2000

4000

6000

Change of abundance (volume fraction)

Change of abundance (volume fraction)

466

0.004 Magnesite

0.003

1000 yr

0.002

5000 10,000

0.001

0.000 0

8000

2000

(k)

4000

6000

8000

Radial distance (m)

Radial distance (m)

(l)

Figure 13-27. (Continued )

CO2 sequestrated (kg/m3 medium)

70 60 1000 yr 50

5000

40

10,000

30 20 10 0 0

2000 4000 6000 Radial distance (m)

8000

Figure 13-28. Cumulative CO2 sequestration by carbonate precipitation for different times. Positive values in background region x > 4000 m are due to calcite precipitation (Reprinted from Computers & Geosciences, Vol. 32, Xu, T., Sonnenthal, E., Spycher, N., and Pruess, K., “TOUGHREACT – A simulation program for nonisothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration,” Issue 2, pp. 145–165. ©2006, with permission from Elsevier Science).

gas saturation), pH, change of abundance of various minerals, and CO2 sequestrated along the radial distance from the well for 1000, 5000, and 10,000 years after the initiation of CO2 injection.

Inorganic Scaling and Geochemical Formation Damage

467

Exercises 1. Estimate the percentage of the various species at pH = 6 based on the data given in Figure 13-4. 2. Estimate the percentage of the various species at pH = 6 based on the data given in Figure 13-5. 3. Consider the following solution–mineral reaction: + − −→ Fe2 O3 + 4SO2− 4 + 38H + 30e ←− 2FeS2 + 19H2 O (13-109)

a. Determine the mathematical expression for the pe–pH relationship. b. Sketch a pe–pH chart for thisreaction.  c. Determine the value of pe if SO2− = 0 1 moles/Liter, pH = 3 0, 4 and Keq = 1 7. 4. Consider the calcium carbonate precipitation reaction given by Ca2+ + 2HCO−3 −→ ←− CaCO3s + CO2g + H2 O

(13-110)

The value of the chemical equilibrium constant at 1 atm and 25 C conditions is given as 5 0 × 10−9 . a) Present an expression for the chemical equilibrium constant for this reaction. b) Determine the saturation ration value for a solution containing 0.1 M (mol/L) Ca2+ and 0.5 M HCO−3 ions at 1 atm and 25 C conditions. c) What can be said about the saturation condition of this solution? Is the solution understurated, saturated, or supersaturated? 5. Carry out a numerical simulation of the rock and fluid alteration processes resulting from carbon dioxide injection into a brine-saturated aquifer similar to that considered by Zhao et al. (2001). Compare the results with those reported by Zhao et al. (2001).

C

H

A

P

T

E

R

14

FORMATION DAMAGE BY ORGANIC DEPOSITION

Summary Paraffins, asphaltenes, and resins are the primary sources of organic deposition in wells, pipelines, and reservoir formation during petroleum production. As a generic term, “wax” refers to deposits of paraffins, asphaltenes, and resins, mixed with some inorganic matter, such as clays, sand, and other debris. Organic deposition can occur both on the surfaces of well tubing and reservoir formation pores to reduce the flow efficiency and eventually to clog the flow paths completely. This chapter presents a review of the thermodynamic and physicochemical foundations of organic precipitation and scale formation as well as the governing phase behavior and rate equations. The criteria for precipitation are derived. The single-phase and multiphase formation damage models are modified to accommodate organic precipitation at below and above bubble point conditions. The outstanding models available for prediction of formation damage by organic deposition are reviewed. Applications are presented for the formation damage in petroleum reservoirs by organic deposition.

14.1 INTRODUCTION Organic scaling can be classified in two groups: (1) natural and (2) induced (Houchin and Hudson, 1986; Amaefule et al., 1988). Invasion of the near-wellbore formation by high pH filtrates, and injecting low surface tension fluids, such as light paraffins including pentane, hexane, 468

Formation Damage by Organic Deposition

469

diesel, gasoline, and naphta, and gas condensates into asphaltenic oil reservoirs can cause asphaltene precipitation (Amaefule et al., 1988). Asphaltenic/parafinic sludges can be formed with the spent acid at low pH conditions that can be created during acidizing (Amaefule et al., 1988). In contrast, paraffins deposit primarily by cooling. Generally, the organic deposits encountered along the production string and surface facilities contain larger proportions of paraffins, some asphaltenes and resins co-precipitated with the paraffins, some oil trapped within the deposits, and various inorganic substances, including clays, sand, and other materials (Khalil et al., 1997). The paraffin deposition primarily occurs by temperature decrease, whereas asphaltene and resin deposition occur because of a number of complicated phenomena, including the polydispersivity, steric colloid formation, aggregation, and electrokinetic deposition processes (Mansoori, 1997). Leontaritis et al. (1992) state, “Probable causes of asphaltene flocculation are: (1) Drop in the reservoir pressure below the pressure at which asphaltenes flocculate and begin to drop out; (2) Mixing of solvents, CH4  CO2 with reservoir oil during EOR.    After flocculation asphaltenes exhibit an intrinsic change, which is usually positive. As a result, they show a strong tendency to attach to negatively charged surface, such as clays and sand.” As soon as the wells in asphaltenic reservoirs begin to produce, the organic deposition begins within the upper section of the wells over which the pressure drops to below the asphaltene flocculation pressure, and then the organic deposition zone gradually progresses toward the bottomhole and eventually enters the near wellbore formation (Minssieux, 1997). Especially, the reservoir formations containing clays of large specific surfaces, such as Kaolinite, can initially adsorb and retain the polar asphaltenes and resins rapidly (Minssieux, 1997). As a result, multilayer molecular deposits are formed over the pore surface (Acevedo et al., 1995). However, as the asphaltene precipitates suspended in the oil phase combine and form sufficiently large aggregates, these particles cannot pass through and are captured at the pore throats (Minssieux, 1997). The pore throat plugging causes the severest permeability loss because the gates connecting the pores are closed and/or an in situ cake is formed by pore filling if the plugged pore throat still allows some flow through the jammed particles. Simultaneously, the flow is diverted toward larger flow paths (Wojtanowicz et al., 1987, 1988; Civan, 1995a; Chang and Civan, 1997; Minssieux, 1997). “Organic deposits usually seal the flow constrictions because they are sticky and deformable. Therefore, the

470

Formation Damage by Organic Deposition

conductivityof a flow path may diminish without filling the pore space completely” (Civan, 1994a, 1995a). Leontaritis (1998) stresses that the organic damage in oil reservoirs is primarily caused by asphaltene deposition and the region of asphaltene deposition may actually extend over large distances from the wellbore, especially during miscible recovery. Wang and Civan (2005a,b,c) have confirmed that asphaltene deposition is not only limited to the near-wellbore region, but it can occur throughout the reservoir formation, whereas the wax deposition is rather limited to a short distance (less than 1 feet) from the wellbore, because wax deposition in the near wellbore region usually occurs by the cooling of the oil caused either by high perforation pressure losses during oil production or by invasion and cooling of the hot oil saturated with the wax dissolved from the well walls as a result of the overbalanced, hot oiling treatments of the wells. The decline of productivity of wells in asphaltenic reservoirs is usually attributed to the reduction of the effective mobility of oil by various factors (Amaefule et al., 1988; Leontaritis et al., 1992; Leontaritis, 1998). The effective mobility of oil is a convenient measure of oil flow capability because it combines the three relevant properties in one group as 0 =

Kkr0 0

(14-1)

where K is the permeability of the reservoir formation, and kr0 and 0 are the relative permeability and viscosity of the oil, respectively. Hence, Leontaritis (1998) states that the asphaltene-induced damage can be explained by three mechanisms. The first is the increase of the reservoir fluid viscosity by formation of a water-in-oil emulsion if the well is producing oil and water simultaneously. The oil viscosity may also increase by the increase of the asphaltene particle concentration in the near-wellbore region as the oil converges radially toward the wellbore. But, experimental measurements indicate that the viscosity increase by asphaltene flocculation is negligible. The second mechanism is the change of the wettability of the reservoir formation from water-wet to oil-wet by the adsorption of asphaltene over the pore surface in the reservoir formation. However, this phenomenon is less likely because, usually, the asphaltenic reservoir formations are already mixed-wet or oil-wet, due to the fact that asphaltenes have already been adsorbed over the pore surface during the long periods of geological times prior to opening the wells for production. The third and most probable mechanism is the impairment of

Formation Damage by Organic Deposition

471

the reservoir formation permeability by the plugging of the pore throats by asphaltene particles. The problems associated with organic deposition from the crude oil can be avoided or minimized by choosing operating conditions such that the reservoir oil follows a thermodynamic path outside the deposition envelope and, therefore, the deposition envelope concept can provide some guidance in this respect (Leontaritis et al., 1992). For example, Wang and Civan (2005a,b,c) accomplished this condition by an early water injection process. However, mathematical models implementing the deposition phase charts are also necessary in developing optimal strategies for optimal mitigation of the deposition problems during the exploitation of the petroleum reservoirs. In the following sections, the characteristics, adsorption and phase behavior, and deposition and formation damage modeling of organic precipitates are presented.

14.2 CHARACTERISTICS OF ASPHALTENIC OILS As indicated by Figure 14-1 by Philp et al. (1995), the boiling and melting points of hydrocarbons increase by the carbon number. Heavy crude oils contain large quantities of higher boiling components, which create problems during oil production (Speight, 1996). Speight and Long (1996) point out that chemical and physical alteration of oils may affect the dispersibility and compatibility of their higher molecular weight fractions and create various problems such as phase separation, precipitation, and sludge formation during the various phases of petroleum production, transportation, and processing. Speight (1996) classified the constituents of the crude oil into four hydrocarbon groups: (1) volatile saturates (paraffins) and aromatics, (2) nonvolatile saturates (waxes) and aromatics, (3) resins, and (4) asphaltenes. The determination of saturates, aromatics, resins, and asphaltenes present in oil is referred to as the “SARA analysis.” Speight (1996) explains that the nomenclature of the petroleum fractions, such as given in Figure 14-2, is based on the techniques of separation of the crude oil into its fractions. Figure 14-3 by Leontaritis (1997) describes the various steps and techniques involved in the analysis of the crude oil, including cryoscopic distillation (CD), solvent extraction (SE), gas chromatography (GC), high performance liquid chromatography (HPLC), and gel permeation chromatography (GPC). Table 14-1 by Srivastava and

472

Formation Damage by Organic Deposition 600 500

Temperature - °C

400 300 200 100 0 Boiling point –100

Melting point

–200 0

10

20

30 40 Carbon number

50

60

70

Figure 14-1. Effect of n-alkane carbon number on boiling and melting points (after Philp, R. P., Bishop, A. N., Del Rio, J.-C., and Allen, J., Cubitt, J. M., and England, W. A. (eds), Geological Society Special Publication, No. 86, pp. 71–85, ©1995; reprinted by permission of R. P. Philp and the Geological Society Publishing House).

Feedstock n-Heptane

Deasphaltened oil

Insoluble Benzene or Toluene

Insolubles

Asphaltenes

Carbon disulfide or Pyridine

Carboids (insolubles)

Silica or Alumina

3. BenzeneMethanol

Carbenes (solubles)

Resins (polars)

2. Benzene or 1. Heptane Toluene

Aromatics

Saturates

Figure 14-2. Classification of petroleum constituents based on laboratory fractionation (reprinted from Journal of Petroleum Science and Engineering, Vol. 22, Speight, J. G., “The Chemical and Physical Structure of Petroleum: Effects on Recovery Operations”, pp. 3–15, ©1999, with permission from Elsevier Science).

473

Formation Damage by Organic Deposition Live sample

Cryoscopic distillation C7+ fraction

C6– fraction

nC6 Aspaltene separation

nC6 Resins

nC6 Asphaltenes

Insoluble fraction nC6 Asphaltenes

GPC

Soluble fraction nC6 Maltenes

HPLC

GC, GPC

Heterocyclics

Aromatics

Paraffins-Wax

Heterocyclics

Aromatics

Paraffins-Wax

GC Analysis

Pure components

Pseudo-components

Figure 14-3. Steps of oil analysis and characterization for paraffin, aromatic, resin, and asphaltene (after Leontaritis, ©1997 SPE; reprinted by permission of the Society of Petroleum Engineers).

Huang (1997) presents data on the chemical and physical properties of typical oil samples taken from Weyburn wells. Leontaritis (1996)∗ described the heavy fractions of petroleum as the following: Asphaltenes These are highly condensed polyaromatic structures or molecules, containing heteroatoms (i.e., S, O, N) and metals (e.g., Va, Ni), that exist in petroleum in an aggregated state in the form of suspension and are surrounded and stabilized by resins (i.e., peptizing agents). They are known to carry an electrical charge, and thought to be polydisperse. Asphaltenes ∗

Reprinted from Leontaritis ©1996, p. 14, by courtesy of Marcel Dekker, Inc.

474

Formation Damage by Organic Deposition Table 14-1 Chemical and Physical Properties of Weyburn Dead-Oils∗ Oil W1a

Temperature 

C

Density kg/m3

15 20 59 61 63

8789 8759 8461

Pressure MPa

Density @59 Cd

0.1 3.54 6.99 10.44 17.33

8461 8492 8524 8580 8609

Oil W2b

Viscosity mPa•s

Oil W3c

Density kg/m3

Viscosity mPa•s

Density kg/m3

Viscosity mPa•s

8549 8424 – 8131 –

– 460 – 235 –

8692 8644 – – 8394

1176 940 – – 315

Viscosity @59 Cd

Density @61 Cd

Viscosity @61 Cd

Density @63 Cd

Viscosity @63 Cd

42 – – – –

8131 8164 8196 8229 8293

235 249 262 276 304

8394 8424 8452 8484 8547

– 128 42 – –

315 326 337 349 371

BS&W, vol%

01

0.2

0.5%

Molecular Weight, g/g-mol

230

203

215

Component

wt.%

wt.%

wt.%

Saturates Aromatics Resins Asphaltenes

48.5 33.5 13.2 4.8

55.3 31.1 9.6 4.0

48.4 33.5 13.2 4.9

a b c d ∗

Collected from Weyburn well 14-17-6-13 W2M. Collected from Weyburn well 3-11-7-13 W2M. Collected from Weyburn well Hz 12-18-6-13 W2M. Reservoir temperature for the oil samples. After Srivastava and Huang, ©1997 SPE; reprinted by permission of the Society of Petroleum Engineers.

are a solubility class; hence, they are not pure, identical molecules. Pentane and Heptane are the two most frequently used solvents for separating asphaltenes from crude oil. The prefix n-Pentane or n-Heptane in asphaltenes refers to the solvent used for their separation. The composition of n-Pentane asphaltenes is different from that of n-Heptane asphaltenes. Resins These are aromatic and polar molecules, also often containing heteroatoms and metals, that surround the asphaltene structures and are dissolved in the

Formation Damage by Organic Deposition

475

oil and help keep the asphaltenes in suspension. They are surface active and, at some thermodynamic states, form their own reversible micelles. They are polydisperse and have a range of polarity and aromaticity. Resins are considered to be precursors to asphaltenes. Paraffin Waxes These are primarily aliphatic hydrocarbons (both straight and branched chain) that change state from liquid to solid during conventional oil production and processing operations. In addition to aliphatics, field deposits usually contain aromatic, naphthenic, resin, and asphaltenic molecules as well. The combined mass is called wax. Paraffin waxes usually melt at about 110–160 F. Field waxes contain molecules that can have melting points in excess of 200 F. Asphalt This is the residual (nondistillable) fraction of crude oil that contains suspended asphaltenes, resins, and the heaviest aromatic and paraffinic components of oils. Although propane has been traditionally a very efficient and convenient solvent for separating asphalt from petroleum, the latest commercial processes use other more efficient solvents for asphalt separation.

Leontaritis (1997) describes that “Since waxes, asphaltenes, and most resins are solid in their pure form and the other oil molecules are in liquid form, the overall crude oil mixture is a liquid solution of waxes, asphaltenes, and resins in the remaining liquid oil.    In general, the waxes and resins are dissolved in the overall crude oil. Whereas the asphaltenes are mostly undissolved in colloidal state.” Andersen et al. (1997) state, “Petroleum asphaltenes are defined as the solids precipitating from a crude oil upon addition of an excess of a light hydrocarbon solvent, in general n-heptane or n-pentane.” Therefore, for practical purposes, the crude oil is considered in two parts. The first part consists of the high-boiling-point and polar asphaltic components. This fraction of the crude oil creates various deposition problems during the exploitation of petroleum reservoirs. The second part is the rest of the crude oil, referred to as the deasphaltened oil or the maltenes. This fraction of the crude oil acts as a solvent and maintains a suspension of the asphaltenes in oil as illustrated in Figure 14-4 by Leontaritis (1996). However, ordinarily, the asphaltenes do not actually disperse in the maltene unless some resins are also present in the crude oil. The resins help asphaltenes to disperse in oil as a suspension by means of the hydrogen-bonding process and the irreversible acid-base reactions of

476

Formation Damage by Organic Deposition

Liquid phase

Asphaltene phase

Figure 14-4. A proposed model for asphaltenic oils (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

the asphaltene and resin molecules (Speight, 1996; Chang and Fogler, 1994, 1996; Speight and Long, 1996). Therefore, Leontaritis et al. (1992) point out that: “An oil that contains asphaltenes will not necessarily cause asphaltene problems during recovery and processing.” Leontaritis et al. (1992) draw attention to the fact that the Boscan crude of Venezuela has not created any asphaltene problems, although it has a large fraction (over 17% by weight) of asphaltenes (Lichaa, 1977). Whereas, the HassiMessaoud oil of Algeria has created severe asphaltene problems, although it has only a small fraction (0.1% by weight) of asphaltenes (Haskett and Tartera, 1965). In fact, de Boer et al. (1995) have concluded that light to medium crudes containing small amounts of asphaltenes may create more asphaltene precipitation problems during primary production. Nghiem and Coombe (1997) explain: “Heavier crudes that contain a larger amount of

Formation Damage by Organic Deposition

477

asphaltene have very little asphaltene precipitation problems as they can dissolve more asphaltene.” Leontaritis et al. (1992) state that: “Asphaltene flocculation can be prevented by addition of resins and aromatics.” The investigations of Chang and Fogler (1994, 1996) using model chemicals for resins have verified this statement. Leontaritis (1996) describes that “   asphaltene particles or micelles aggregate or flocculate into larger aggregates or flocs.    Asphaltene flocculation can be both reversible and irreversible.    Paraffin waxes, on the other the hand,    exhibit the phenomenon of crystallization.    Wax crystallization is generally a reversible process. However, paraffin waxes more than often precipitate together with resins and asphaltenes (which are said to be responsible for the observed irreversible thermodynamic phenomena). Hence, some wax precipitation is occasionally reported as irreversible.” Leontaritis (1996) points out that temperature and composition have a large effect and pressure has a small effect on the solubility of wax in oil. Leontaritis (1996) explains that the behavior of wax in oils can be determined by means of the cloud and pour points. Ring et al. (1994) defined the cloud point as “the equilibrium temperature and pressure at which solid paraffin crystals begin to form in the liquid phase.” Leontaritis (1996) states that the flow or “pour point is defined as the lowest temperature at which the fuel will pour and is a function of the composition of the fuel.”

14.3 MECHANISMS OF THE HEAVY ORGANIC DEPOSITION In this section, the mechanisms of the heavy organic deposition according to Mansoori (1997) are described. Mansoori (1997) states that organic deposition during petroleum production and transportation may occur by one or several of the following four mechanisms: 1. Polydispersivity effect: As depicted in Figure 14-5a by Mansoori (1997), a stable state of a polydispersed oil mixture can be attained for a certain proper ratio of the polar to nonpolar and the light to heavy constituents in the crude oil at given temperature and pressure conditions. Thus, when the composition, temperature, or pressure is varied, the system may become unstable and undergo several processes. Figure 14-5b by Mansoori (1997) depicts the formation of micelle-type aggregates of asphaltene when polar miscible compounds

478

Formation Damage by Organic Deposition

(a)

(b)

(c)

(d)

(e)

(f)

Figure 14-5. (a) Heavy organics in petroleum crude (straight/curved line = paraffin molecules, solid ellipse = aromatic molecules, open ellipse = resin molecules, and solid blocky forms = asphaltene molecules). (b) Colloidal phenomenon activated by addition of a polar miscible solvent (solid ellipse = an aromatic hydrocarbon). (c) Flocculation and precipitation of heavy components by addition of a nonpolar miscible solvent (dashed line = a paraffin hydrocarbon). (d) Steric colloidal phenomenon activated by addition of paraffin hydrocarbons. (e) Migration of peptizing molecules (solid arrows) by change of chemical potential balance. (f) Flocculation and deposition (big arrow) of large heavy organic particles (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Mansoori, G. A., “Modeling of Asphaltene and Other Heavy Organic Depositions”, pp. 101–111, ©1997, with permission from Elsevier Science; after Mansoori ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers).

Formation Damage by Organic Deposition

479

are added into the system. Figure 14-5c by Mansoori (1997) describes the separation of the asphaltenes as a solid aggregate phase when more paraffinic hydrocarbons are added into the system. 2. Steric colloidal effects: At high concentrations, asphaltenes tend to associate in the form of large particles, as depicted in Figure 14-5d by Mansoori (1997). In the presence of some peptizing agents, such as resins, these particles can adsorb the peptizing agents and become suspended in the oil. 3. Aggregation effect: When the concentration of the peptizing agent is low and its adsorbed quantity is not sufficient to occupy the particle surface completely, several particles can combine to form bigger particles as depicted in Figure 14-5e by Mansoori (1997). This phenomenon is called flocculation. When the particles become sufficiently large and heavy, they tend to deposit out of the solution as depicted in Figure 14-5f by Mansoori (1997). 4. Electrokinetic effect: As explained by Mansoori (1997), during the flow of oil through porous media and pipes, a “streaming current” and a potential difference are generated because of the migration of the charged particles of the asphaltene colloids. The asphaltene particles are positively charged but the oil phase is negatively charged, as depicted in Figure 14-6 by Mansoori (1997). Therefore, the negative upstream and positive downstream potentials are generated along the pipe to resist the flow of the colloidal particles, as depicted in Figure 14-7 by Mansoori (1997). This, in turn, induces a back diffusion of the colloidal asphaltene particles. Mansoori (1997) points out that the deposition of the polar asphaltene by the electrokinetic effect and the nonpolar paraffins by the dynamic pour point crystallization effect could occur simultaneously when the oil contains both asphaltenes and paraffins.

14.4 ASPHALTENE AND WAX PHASE BEHAVIOR 14.4.1 14.4.1.1

Deposition Envelopes Description of deposition envelopes

In this section, a brief summary of the review of the asphaltene and wax phase behavior by Leontaritis (1996) is presented.

480

Formation Damage by Organic Deposition Flowing crude oil

Charged heavy organic particles

Conduit

Figure 14-6. Streaming potential generated by oil flow in a pipe (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Mansoori, G. A., “Modeling of Asphaltene and Other Heavy Organic Depositions”, pp. 101–111, ©1997, with permission from Elsevier Science; after Mansoori ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 14-7. Electrokinetic deposition in a pipeline (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Mansoori, G. A., “Modeling of Asphaltene and Other Heavy Organic Depositions”, pp. 101–111, ©1997, with permission from Elsevier Science).

481

Formation Damage by Organic Deposition

Accurate measurement of the asphaltene and wax phase behavior is expensive and requires sophisticated techniques for proper handling of the reservoir fluid samples and laboratory testing of the recombined reservoir fluids. Therefore, Leontaritis (1996) suggests that phase diagrams can be more economically and rapidly developed by simulation with a limited number of actual data required for tuning and calibration. Leontaritis (1996) demonstrated this exercise by applying the Thermodynamic– Colloidal model by Leontaritis (1993). Nghiem and Coombe (1997) state, “Above the saturation pressure, the precipitation is solely due to pressure, while below the saturation both pressure and composition affect the precipitation behavior.” Leontaritis (1996) points out that wax crystallization and asphaltene flocculation phenomena occur at low and high temperatures, respectively. Then, he hypothesizes that these two phenomena should, therefore, represent the two extreme cases of the phase behavior and there should be continuously varying intermediate phase behavior in between these two extremes depending on the composition of the heavy fractions of the crude oils, as schematically shown in Figure 14-8 by Leontaritis (1996). The schematic Figures 14-9 and 14-10 by Leontaritis (1996) depict the features of typical asphaltene deposition envelope (ADE) and wax

WDE behavior

Pressure

ADE behavior

Low T

High T

Figure 14-8. Unification of the wax deposition envelope (WDE) and the asphaltene deposition envelope (ADE) (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

482

Formation Damage by Organic Deposition

(Pres, Tres) Liquid phase ADE up

per bou

Bubble-point line

ndary

Pressure

Liquid +Asphaltene phases

Liquid + Vapor + Asphaltene phases Liquid + Vapor phases nd

ou

E

AD

ary

b er low

Temperature

Figure 14-9. Typical asphaltene deposition envelope (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

Pressure

Bubble-point line Solid + Liquid phases

Liquid phases

Liquid + Vapor phases Solid + Liquid + Vapor phases

Temperature

Figure 14-10. Typical wax deposition envelope (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

483

Formation Damage by Organic Deposition

deposition envelope (WDE), respectively. As explained by Leontaritis (1996), the phase diagrams of the asphaltenic fluids typically do not have a critical point, because the asphaltenic fluids can only have bubblepoint lines and no dew-point lines, as they cannot vaporize. Leontaritis (1996, 1998) refers to the locus of the thermodynamic conditions for asphaltene flocculation as the ADE, as shown in Figure 14-9. Typically, the pressure–temperature phase diagrams of the asphaltenic oils are characterized by several phase quality lines and a saturation (bubble-point) line in between the upper and the lower boundaries of the ADE as indicated in Figure 14-11 by Leontaritis (1996) for a South-American oil. Leontaritis (1996) estimated the intersection of the upper ADE with the bubble-point line at around 370 F for this oil. Figures 14-12–14-14 by Leontaritis (1996) are typical simulated charts showing the ADE, the asphaltene phase volume vs. temperature, and the asphaltene phase volume vs. pressure for a North-American oil, respectively. Leontaritis (1996) refers to the locus of the thermodynamic conditions for wax crystallization as WDE. The sketch of a typical WDE is given in Figure 14-10 by Leontaritis (1996). Figures 14-15 and 14-16 by Leontaritis (1996) depict the effect of the light-ends and the

7000

Upper ADE boundary 6000

1.0*

Pressure, psig

2.0 3.0

5000

4.0

e

tion lin

Satura

4000

3.0

3000

E r AD

e

Low

ry

nda

bou

2000 140

180

220

260

300

Temperature, °F * Mls of asphaltene phase per mole of reservoir fluid.

Figure 14-11. Asphaltene deposition envelope for a South American reservoir oil (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

484

Formation Damage by Organic Deposition

400

No solids

350

Pressure, atm

300

Reservoir pressure, 350.0 atm Reservoir temperature, 344.27° K No solids

Solids

250 200

Lower onset P Bubble P

Solids

150

Upper onset P

100 50 0

280

300

320

340 360 Temperature, °K

380

400

420

Figure 14-12. Asphaltene deposition envelope for Asph Wax Oil Company live-oil (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.). 400

Temperature, ° K

380

Reservoir pressure, 350.0 atm Reservoir temperature, 344.27° K 360 340 320 300

0

0.5

1

2 2.5 1.5 Asphaltene phase volume, cc

3

3.5

Figure 14-13. Asphaltene phase volume vs. temperature for an Asph Wax Oil Company live-oil at 200 atm pressure (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

pressure–temperature relationship on the onset of wax crystallization (cloud point) of oils. The effect of the pressure on the onset of wax crystallization is demonstrated for three live oils in Figure 14-17 by Leontaritis (1996). The typical WDE of North American recombined live-oil developed by laboratory measurements is given in Figure 14-18 by Leontaritis

485

Formation Damage by Organic Deposition 400 350

Bubble point pressure, 279.16 atm at 340° K

Pressure, atm

300 250

Reservoir pressure, 350.0 atm Reservoir temperature, 344.27° K

200 150 100 50 0 0

1

2

3

4

5

6

7

8

Asphaltene phase volume, cc

Figure 14-14. Asphaltene phase volume vs. pressure for Asph Wax Oil Company live-oil at 340 K temperature (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.). 130

Onset of wax crystallization, ° F

120 110

100 90 80 70

60 0

1000

2000

3000

4000

5000

Bubble point pressure, psig (at 195° F)

Figure 14-15. Onset of wax crystallization vs. the bubble-point temperature (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

(1996). He also developed the WDE given in Figure 14-19 for North Sea live-oil by using a Wax Model. Using the same Wax Model, Leontaritis (1996) has predicted the effect of temperature on the fraction of the wax crystallized at 200, 50, and 1 atm pressures as shown in Figures 14-20,

486

Formation Damage by Organic Deposition

6000

Onset pressure, psia

5000 4000 3000 2000 1000 0

68

70

72

74

76

78

80

82

84

Onset temperature, °F

Figure 14-16. Pressure–temperature effects on onset of wax crystallization in a synthetic mixture of kerosene and candle wax (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.). 140

Temperature, ° F

130 120 Oil A

110

Oil B Oil C

100 90 80 70 2000

3000

4000

5000 6000 Pressure, psig

7000

8000

Figure 14-17. Upper wax deposition envelope boundaries for three different reservoir oils (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

14-21, and 14-22, respectively, as well as the effect of the pressure on the fraction of the wax crystallized at 280 K as shown in Figure 14-23. 14.4.1.2

Representing the asphaltene deposition envelope

For convenience, Wang and Civan (2005a,b) described the asphaltene deposition envelope using a truncated bi-variate power series expansion as

487

Formation Damage by Organic Deposition 3000

Pressure, psig

2500

Onset pressure, psig BP pressure, psig

2000 1500 1000 500 0

30

40

50

60

70

80

90

100

Temperature, °F

Figure 14-18. Wax deposition envelope for a North American recombined reservoir oil (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.). 250

Pressure, atm

200

No solids Solids

Reservoir pressure, 280.0 atm Reservoir temperature, 338.0° K

150 100 50 0 250

Onset pressure

No solids

Bubble point pressure

Solids

260

270

280

290 300 310 Temperature, °K

320

330

340

350

Figure 14-19. Wax deposition envelope for an Asph Wax Oil Company live-oil (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

xA = 1 + 2 p + 3 p2 + 4 p3 + 5 T + 6 pT + 7 p2 T + 8 T 2 + 9 pT 2 + 10 T 3

(14-2)

where xA is the concentration of precipitated asphaltene, i , where i = 1 2     10, denotes fitting coefficients, and T and p denote the temperature and pressure of the oil. Wang and Civan (2005a,b) show that Eq. (14-2) represents the Leontaritis (1996) asphaltene deposition

488

Formation Damage by Organic Deposition 0.100

Wax weight fraction

0.080 0.060

Reservoir pressure, 280.0 atm Reservoir temperature, 338.0° K

0.040 0.020 0.000 240

250

260 270 Temperature, °K

280

290

Figure 14-20. Wax weight fraction vs. temperature for an Asph Wax Oil Company live-oil at 200 atm pressure (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.). 0.15

Wax weight fraction

0.12 0.09

Reservoir pressure, 280.0 atm Reservoir temperature, 338.0° K

0.06 0.03 0 250

260

270

280 290 300 Temperature, °K

310

320

330

Figure 14-21. Wax weight fraction vs. temperature for an Asph Wax Oil Company live-oil at 50 atm pressure (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

envelope satisfactorily. Wang and Civan (2005c) obtained the polynomial coefficients yielding the best fit of the asphaltene deposition data of Leontaritis (1996) as 1 = −121 × 101 ml/moleoil  2 = 9026 × 10−3 ml/moleoil − psi 3 = −2402 × 10−6 ml/moleoil − psi2  4 = 2574 × 10−10

489

Formation Damage by Organic Deposition 0.3

Wax weight fraction

0.25 Reservoir pressure, 280.0 atm Reservoir temperature, 338.0° K

0.2 0.15 0.1 0.05 0 250

260

270

280 290 300 Temperature, °K

310

320

330

Figure 14-22. Wax weight fraction vs. temperature for an Asph Wax Oil Company stock-tank-oil at 1 atm pressure (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.). 0.200

Wax weight fraction

0.160 0.120

Reservoir pressure, 280.0 atm Reservoir temperature, 338.0 °K

0.080 0.040 0.000

0

50

100 Pressure, atm

150

200

Figure 14-23. Wax weight fraction vs. pressure for an Asph Wax Oil Company live-oil at 280 K temperature (after Leontaritis, ©1996; reprinted by courtesy of Marcel Dekker, Inc.).

ml/moleoil − psi3  5 = −5444 × 10−3 ml/moleoil −  F 6 = 3786 × 10−5 ml/moleoil − psi −  F 7 = −1035 × 10−8 ml/moleoil − psi2 −  F 8 = −3402 × 10−4 ml/moleoil −  F2  9 = 1701 × 10−7 ml/moleoil − psi −  F2  and 10 = −9675 × 10−7 ml/moleoil −  F3 

490

Formation Damage by Organic Deposition

14.4.1.3 Pressure–composition phase diagrams for miscible gas injection

Mansoori (1997) mentions that experimental measurement of the pressure–composition phase diagrams involving heavy organic deposition by miscible gas injection, at reservoir temperatures, is very costly. Therefore, he has suggested generating these charts by simulation. Figure 14-24 produced by Mansoori (1997) is an example of a typical chart for asphaltenic oils dissolving carbon dioxide. Figures 14-25 and 14-26 by Mansoori (1997) indicate the electrokinetics affect on asphaltene deposition in pipelines from typical asphaltenic oils dissolving a miscible component at various temperatures. These figures contain two charts. The upper chart shows the static onset of deposition of asphaltene on a pressure vs. composition relationship. The lower chart shows the dynamic, Q (defined below) vs. pressure relationship for asphaltenic oils flowing in wells or pipelines for different miscible component and oil ratio. The Q function is given by (Mansoori, 1997) Q=

U 175 D075

(14-3)

3500

Pressure (psig)

3000 2500 2000 1500

1Φ

3Φ

1000 500

2Φ

0 0

10

20

30

40

50

60

70

80

90

100

CO2 mole%

Figure 14-24. Static pressure vs. composition (P–x) phase diagram of a crude oil mixed with a miscible injectant (MI) at 60 F (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Mansoori, G. A., “Modeling of Asphaltene and Other Heavy Organic Depositions”, pp. 101–111, ©1997, with permission from Elsevier Science, after Mansoori ©1994 SPE, reprinted by permission of the Society of Petroleum Engineers).

Formation Damage by Organic Deposition T = 120 F

3000

Pressure, psig

L 2000

L-A

L-V

1000 P = 1520 psig

L-V 0

491

0

L-V-A

x = 43.46 20

40

L-V 60

80

100

X, Mole % of miscible injectant

7

x E-6

MI/crude oil T = 120 F MI/Oil mole % 0.0 25.5 40.6 63.1 77.4 87.2 93.2

6

Q 5 4 3

0

1000

2000

3000

Pressure, psig

Figure 14-25. Static pressure vs. composition (P–x) and dynamic (P–Q) phase diagrams of the same crude oil in Figure 14-24, mixed with the same miscible injectant at 120 F (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Mansoori, G. A., “Modeling of Asphaltene and Other Heavy Organic Depositions”, pp. 101–111, ©1997, with permission from Elsevier Science, after Mansoori ©1994 SPE, reprinted by permission of the Society of Petroleum Engineers).

in which U is the average oil velocity in the pipe, D is the pipe diameter, and is the conductivity of the oil. The regions above and below these curves express the flow conditions leading to deposition and no deposition of asphaltenes, respectively. Hence, these charts help determine the proper operating conditions of pipes to avoid precipitation.

14.4.2

Solubility Theory

The theoretical relationships describing the paraffin and asphaltene solubility in crude oil are presented below.

492

Formation Damage by Organic Deposition T = 160 F

3000 L

Pressure, psig

L-A 2000

L-V P = 1770 psig

L-V-A

1000 L-V X = 44.0

L-V

0 0

20

40

60

80

100

X, mole % of miscible injectant xE-6 7

Ml / Crude oil T = 160 F Ml / Oil mole % 0.0 25.5

6

40.6 Q 5

63.1 77.4 87.2 93.2

4

3

0

1000

2000

3000

Pressure, psig

Figure 14-26. Static pressure vs. composition (P–x) and dynamic (P–Q) phase diagrams of the same crude oil in Figure 14-24, mixed with the same miscible injectant (MI) at 160 F (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Mansoori, G. A., “Modeling of Asphaltene and Other Heavy Organic Depositions, pp. 101–111, ©1997, with permission from Elsevier Science, after Mansoori ©1994 SPE, reprinted by permission of the Society of Petroleum Engineers).

14.4.2.1

Solubility of paraffin in oil

Essentially, the solubility of paraffin in crude oil depends on temperature and less on presseure. The solubility of paraffin in crude oil can be estimated by applying the following real solution model (Chung, 1992):

Formation Damage by Organic Deposition

  V 1 HP 1 − P  m − P 2 − xPL = xPS exp − R T TPM RT    VP V − ln −1+ P Vm Vm

493



(14-4)

where, xpL denotes the dissolved paraffin mole fraction in oil, xpS is the paraffin precipitate mole fraction in oil, HP is the latent heat of fusion of the paraffin, R is the universal gas constant, T is temperature, TPM is the melting point or fusion temperature for paraffin, VP and Vm are the molar volumes of paraffin and oil, and P and m are the solubility parameters of paraffin and oil. 14.4.2.2

Solubility of asphaltene in crude oil

The experimental approaches used in an effort to determine the existence state of asphaltene in crude oil can be summarized in four categories (Wang and Civan, 2005a,b): 1) Electron microscope (Prechshot et al., 1943; Katz and Beu, 1945) 2) Electrical effect (Dykstra et al., 1944; Katz and Beu, 1945) 3) Reversibility (Hirschberg et al., 1984; Thawer et al., 1990; Danesh et al., 1987; Danesh et al., 1989) 4) Molecular weight (Ray et al., 1957; Witherspoon and Munir, 1958; Speight and Long, 1996; Speight, 1999). Reviewing the existing literature, Wang and Civan (2005a,b) conclude that “the existence state of asphaltene in crude oil is not very clear. Nevertheless, the following facts can be expressed. Crude oil containing asphaltene is a uniform solution, asphaltene precipitation and dissolution processes are reversible, and molecular weight of asphaltene ranges within 1500–2500 g/mole.” There are two essential theoretical approaches available concerning the physical description of the existence state of asphaltene in crude oil: 1) Colloidal theory. The colloidal theory assumes that asphaltene is suspended as a colloid in crude oil and therefore can better describe the asphaltene peptization/flocculation phenomena (Leontaritis and Mansoori, 1987; Mansoori, 1994a,b, 1997).

494

Formation Damage by Organic Deposition

2) Real-solution theory. The real-solution theory assumes that asphaltene is completely dissolved in crude oil (Hirschberg et al., 1984; Burke et al., 1990; Novosad and Costain, 1990; Kawanaka et al., 1991; Chung, 1992; Thomas et al., 1992; Nghiem et al., 1993; Nor-Azian and Adewumi, 1993; Boer et al., 1995; Cimino et al., 1995; MacMillan et al., 1995; Yarranton and Masliyah, 1996; Zhou et al., 1996; Nghiem and Coombe, 1997; Nghiem et al., 1998; Wang et al., 1999; Wang and Civan, 2001, 2005a,b; Kohse and Nghiem, 2004). The real-solution theory has been implemented as either a regular solution or a polymer solution. However, the polymer solution theory is preferable because the asphaltene molecules are large molecules (Hirschberg et al., 1984; Wang and Civan, 2005a,b). Hirschberg et al. (1984) combined the Flory–Huggins theory with the Hildebrand solubility concept to express the volume fraction of asphaltene dissolved in the crude oil A as   VA VA 2  − L (14-5) −1−

A = exp VL RT A where VA denotes the asphaltene molar volume, VL is the oil molar volume, R is the universal gas constant, and T is the absolute temperature. The symbol A denotes the solubility parameter of asphaltene, calculated by A = 20041 − hT

(14-6)

where h is a characteristic constant value for a given oil. The symbol L denotes the solubility parameter of oil, calculated by (Hildebrand, 1929)  L = Uvaporization /VL (14-7)

where Uvaporization represents the change of internal energy per mole of oil by vaporization. Wang and Civan (2005a,b) represented the precipitation and dissolution of asphaltene in crude oil by means of the vapor–oil and asphaltene– oil equilibriums. They first calculated the composition of oil based on the vapor–oil equilibrium using the Peng–Robinson equation and then the solubility of asphaltene in oil using the polymer solution theory for the asphaltene–oil equilibrium. The crude oil is the solvent-rich phase and asphaltene precipitate is the solute-rich phase (Hirschberg et al., 1984).

Formation Damage by Organic Deposition

495

Wang and Civan (2005a,b) assumed that the asphaltene molecules present in the oil are identical so that Eq. (14-5) formulated for monodisperse polymer solutions can be used and that the second equilibrium does not effect the first equilibrium. Wang and Civan (2005a,b) applied the shift parameter concept to improve the predicted value of the molar volume of oil (Jhaverl and Youngren, 1988). They determined the critical properties and eccentric parameter of C7+ to match a characteristic property of the oil, such as the bubble-point pressure, and calculated the vapor–oil equilibrium using the Peng–Robinson equation and the molar volume of oil using the modified Peng–Robinson equation (Jhaverl and Youngren, 1988). 14.4.3 14.4.3.1

Asphaltene Adsorption Bilinear adsorption model

Nonequilibrium adsorption in porous media may be described by a bilinear adsorption model according to Gupta and Greenkorn (1973), given by     = k1 + k2 c + k3  + k4 c = k1 + k2 c 1 + + k3  (14-8) t k2 /k4 subject to the initial condition that  = 0  t = 0 At equilibrium, Eq. (14-8) becomes   k1 1 k k c− = − 3 − 4c  k2 k2 k2

(14-9)

In Eqs (14-8) and (14-9), t is time,  is the concentration of species adsorbed in porous media, c is the concentration of the species in solution, and k1  k2  k3 , and k4 are some empirical coefficients. Manoranjan and Stauffer (1996) used simplified forms of Eqs (14-8) and (14-9). The first is referred to as the nonequilibrium sorption equation given by  = kf c s −  − kb s t

(14-10)

subject to the initial condition that  = 0 

t=0

(14-11)

496

Formation Damage by Organic Deposition

The second is the Langmuir isoterm, which applies at local equilibrium (/t = 0 in Eq. (14-10)), 1 c c = +  s s K

(14-12)

80 30 28 60

24 20

40

16 Toluene/n-Dodecane (1.75:1.0 w/w)

12

Toluene

8

Nitrobenzene

4

0

20

Chloroform

400

800

1200

1600

2000

2400

0 2800

Weight asphaltene adsorbed/weight kaolin, mg/g

Weight asphaltene adsorbed/weight kaolin, mg/g

where kf and kb denote the rate constants for the forward, sorption, and backward, desorption, rate processes, K = kf /kb denotes the equilibrium constant, s is the saturation concentration of the adsorbed species at complete monolayer coverage of the pore surface. Dubey and Waxman (1991) have shown that the adsorption of asphaltene from anhydrous nonpolar solvents and toluene on common minerals followed the monolayer, Langmuir Type I adsorption mechanism according to Eq. (14-12). However, the adsorption of asphaltene from nitrobenzene solution followed a multilayer, Langmuir Type II adsorption mechanism (Figures 14-27 and 14-28 by Dubey and Waxman (1991)). They have also shown that there is an adsorption/desorption hysteresis for asphaltene as indicated by Figure 14-29. Figures 14-30 and 14-31 reported by Acevedo et al. (1995) also indicate monolayer and multilayer adsorption mechanisms, respectively.

Equilibrium concentration of asphaltenes, ppm

Figure 14-27. Adsorption isotherms for asphaltenes on kaolin from various solvents (after Dubey and Waxman, ©1991 SPE; reprinted by permission of the Society of Petroleum Engineers).

Formation Damage by Organic Deposition

497

40

mg Asphaltenes adsorbed/g substrate

35

30

25

20

15

10

Berea sandstone (>100 mesh) Dickite (Wisconsin) Dolomite (Dolocron) Ottawa sand (Super x, >325 mesh) Calcite (Dover chalk) Kaolin mineral Illite (Beaver’s bend) Alumina

5

0

1000

2000

Equilibrium asphaltene concentration, ppm Figure 14-28. Adsorption isotherms for asphaltenes on clay and mineral surfaces from toluene (after Dubey and Waxman, ©1991 SPE; reprinted by permission of the Society of Petroleum Engineers).

14.4.3.2

Surface excess theory

Ali and Islam (1997, 1998) used the following model for adsorption of asphaltene according to the application of the surface excess theory by Sircar et al. (1972). The asphaltenic oil is considered to have an asphaltene and an oil (maltene) pseudo-species, denoted respectively by the indices i = 1 and i = 2. Let xi and xi′ denote the mass fractions of

498

Weight asphaltene adsorbed/ Weight kaolin, mg/g

Formation Damage by Organic Deposition

30

20

10

1000 500 1500 2000 Asphaltene equilibrium concentration, ppm

0

2500

Figure 14-29. Hysteresis of adsorption/desorption isotherms for asphaltenes on kaolin from toluene (after Dubey and Waxman, ©1991 SPE; reprinted by permission of the Society of Petroleum Engineers).

Equilibrium concentration in solid (mg/g)

3.5 3 2.5 2 1.5 1 0.5 0

0

900 1200 3000 300 600 Equilibrium concentration of asphaltenes (mg/l)

Figure 14-30. Adsorption isotherm for Cerro Negro asphaltenes on inorganic material surface from toluene at 26 C (reprinted from Journal of Fuel, Vol. 74, Acevedo, S., Ranaudo, M. A., Escobar, G., Gutiérrez, L., and Ortega, P., “Adsorption of Asphaltenes and Resins on Organic and Inorganic Substrates and Their Correlation with Precipitation Problems in Production Well Tubing”, pp. 595–598, ©1995, with permission from Elsevier Science).

Equilibrium concentration in solid (mg/g)

Formation Damage by Organic Deposition

499

24 21 18 15 12 9 6 3 0

0

200

400

600

800

1000 1200 1400 1600 1800 2000 2200 2400 2600

Equilibrium concentration of asphaltenes (mg/L)

Figure 14-31. Adsorption isotherm for Ceuta asphaltenes on inorganic material surface from toluene at 26 C (reprinted from Journal of Fuel, Vol. 74, Acevedo, S., Ranaudo, M. A., Escobar, G., Gutiérrez, L., and Ortega, P., “Adsorption of Asphaltenes and Resins on Organic and Inorganic Substrates and Their Correlation with Precipitation Problems in Production Well Tubing”, pp. 595–598, ©1995, with permission from Elsevier Science).

species i dissolved in the oil phase and adsorbed in the porous medium, respectively. n′i is the mass of species i adsorbed per unit mass of porous media. n′ is the total mass of species (oil plus asphaltene) adsorbed per unit mass of porous media given by n′ =

2 

n′i

(14-13)

i=1

Then, assuming that all oil is in contact with the porous media, the surface excess of species i can be expressed as nei = n′ xi′ − xi 

i = asphaltene or oil

(14-14)

They assume that the theory is applicable for both monolayer and multilayer adsorption. A balance of the oil and asphaltene adsorbed over the pore surface yields 1 x2′ x1′ + = n′ m1 m2

(14-15)

500

Formation Damage by Organic Deposition

In Eq. (14-15), m1 and m2 denote the monolayer coverage of asphaltene and carrier oil, respectively, expressed as mass of species adsorbed per unit mass of porous solid. Then, a selectivity parameter, as defined below, is introduced: S=

x1′ /x2′ x1 /x2

(14-16)

Therefore, from Eqs (14-15) and (14-16), they obtained the following expression for the amount of asphaltene adsorbed: ne1 = n′ x1′ =

m1 x1 S Sx1 + m1 /m2 x2

(14-17)

Using Eqs (14-14) and (14-16), they derived the following expression for the surface excess amount of the asphaltene: nea 1 =

m1 x1 x2 S − 1 Sx1 + m1 /m2 x2

(14-18)

As a result, the rates of adsorption or desorption are expressed according to nea 1 = kj ne1 − nea 1  t

j = adsorption, desorption

(14-19)

where ne1 and nea denote the amount of species 1 (asphaltene) 1 adsorbed/desorbed and the actual surface excess of species 1 per unit mass of porous formation. The initial condition is given as ea nea 1 = n10 

14.4.4

t=0

(14-20)

Asphaltene Aggregation Kinetics

Asphaltene melocules form miscelles and then the miscelles combine to form aggregates (Yen and Chilingarian, 1994). In this section, the description of the Asphaltene aggregation kinetics according to the approach developed by Burya et al. (2001) is presented. Burya et al. (2001) explain that essentially the asphaltene particle stability in crude oil depends on the aromatics-to-saturates ratio and resins-to-asphaltenes ratio, reduction of which may cause the coalescence

Formation Damage by Organic Deposition

501

of the asphaltene particles in oil to form larger aggregates. Burya et al. (2001) demonstrate that the aggregation kinetics of colloidal asphaltene particles involves the diffusion-limited aggregation (DLA) and reactionlimited aggregation (RLA) mechanisms with a crossover between them. The average number of particles N forming fractal aggregates is given by N=



R Ro

df

(14-21)

Where Ro and R denote the initial particle radius and the instantaneous mean aggregate radius, respectively, and df is the fractal dimension. Equation (14-21) can be applied for both the diffusion- and the reaction-limited fractal aggregates with different values of the fractal dimension df . Therefore, N denotes the average number of asphaltene molecules forming an aggregate for the DLA. Alternatively, N denotes the average number of miscelles forming an aggregate for the RLA and crossover. Let the subscripts D and R refer to the DLA and RLA mechanisms, respectively. Burya et al. (2001) described the crossover behavior from the RLA to DLA by N dN = D N + R dt

(14-22)

subject to N = No 

t = to

(14-23)

 d f

(14-24)

Where  is a constant, given by =



R Ro

D

The symbols D and R denote the characteristic times char for the DLA and RLA mechanisms, respectively. Thus, a scaled time can be defined as t∗ =

t char

(14-25)

R Ro

(14-26)

A scaled radius can be defined as: R∗ =

502

Formation Damage by Organic Deposition

Thus, the analytical solution of Eq. (14-22) subject to Eq. (14-23) is given by D  N − No + R ln N/No = t − to

(14-27)

Note that No = 1 to = 0 at the beginning of the aggregation process. Two special solutions of the Smoluchowski equation can be readily generated from Eq. (14-27) in terms of the scaled time and scaled radius defined by Eqs. (14-25) and (14-26). The solutions obtained for these special cases can be expressed in terms of the aggregate radius by substituting Eq. (14-21) for N . These special solutions are given below (Burya et al., 2001): Solution #1: For diffusion-limited aggregation, D >> R and thus the diffusion-controlled aggregation kinetics is described by simplifying Eq. (14-27) by substituting R = 0. N = 1 + t/D 

R = 1 + t/D 1/df Ro

(14-28)

Solution #2: For reaction-limited aggregation, D Poap

P < Poap

Air bath

Figure 14-34. Schematic diagram of gravimetric concept (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM).

505

Formation Damage by Organic Deposition

Asphaltene content (%, w/w)

n -Heptane insolubles

n -Pentane insolubles

1.5 Poap (lower) = 13.5 Mpa

Pb – 22 Mpa

1.0

0.5 Poap (upper) – 43 MPa

0.0

0

10

20

30

40

50

60

70

Pressure (MPa)

Figure 14-35. Results of gravimetric method (isothermal depressurization using Oil A) (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM).

is based on the measurement of the precipitating asphaltene particles, separating and settling at the bottom of a pressure–volume–temperature (PVT)-cell under the gravity effect when the system is depressurized isothermally to allow the separation of dissolved gas. Figure 14-35 by Jamaluddin et al. (2002) shows the separated asphaltene weight percent as a function of the pressure at a prescribed temperature of 116 C for oil, referred to as Oil A. As can be seen, the upper and lower asphaltene precipitation conditions can be readily identified from the inflection points. The two data sets given in Figure 14-35 are for the n-pentane and n-heptane insolubles. However, Jamaluddin et al. (2002) did not indicate the specific data shown for the n-pentane and n-heptane in this figure. 14.5.1.2

Acoustic resonance technique

Jamaluddin et al. (2002) conclude that the acoustic resonance technique is rapid and can determine the bubble-point and upper asphaltene envelope, requiring only a small amount of oil sample. But, it could not determine the lower asphaltene deposition envelope for the crude oils used in their study. The operation principle of the acoustic resonance technique is as following. As shown in Figure 14-36 by Jamaluddin et al. (2002), an oil sample is placed in between two piezoelectric elements. An electric voltage is applied to the first piezoelectric element (source) causing it to vibrate and apply an acoustic stimulation to the oil sample. Then,

506

Formation Damage by Organic Deposition

the stimulated fluid oscillations cause the second piezoelectric element (receiver) to vibrate and generate a voltage. As shown in Figure 14-37 by Jamaluddin et al. (2002), the normalized measurent of the acoustic response (sonic frequency measured in Hertz) as a function of pressure allows the determination of the bubble-point and upper asphaltene Temperature probe

Acoustic

Acoustic receiver Air bath

Normalized acoustic response

Figure 14-36. Schematic diagram of acoustic resonance technology (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM).

1.00 0.75 0.50 0.25 Poap (upper) = 43 MPa

Pb = 23 MPa

0.00

0

10

20

30

40

50

60

70

Pressure (MPa)

Figure 14-37. A typical acoustic response (isothermal depressurization using Oil A) (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM).

Formation Damage by Organic Deposition

507

deposition condition during the isothermal depressurization tests. The sudden appearances of gas and asphaltene phases are detected by two sharp variations, respectively, as shown in Figure 14-37. Jamaluddin et al. (2002) explain that the lower asphaltene envelope could not be identified with this technique because resolubilization of asphaltene occurs gradually. 14.5.1.3

Light-scattering technique

Jamaluddin et al. (2002) concluded that the near-infrared (NIR) lightscattering technique enabled the construction of the upper and lower asphaltene deposition envelopes. This technique is implemented using a visual PVT-cell equipped with visual observation glass plates, placed on opposite sides as depicted in Figure 14-38 by Jamaluddin et al. (2002). The transmittance of an optimized laser light through the oil sample undergoing an isothermal depressurization test in the PVT-cell is measured as a function of pressure. Therefore, partial light scattering caused by gas bubbles and asphaltene precipitates reduces the light transmission through the oil sample. The bubble-point, and the upper and lower asphaltene deposition pressures can be conveniently identified as illustrated in Figure 14-39 by Jamaluddin et al. (2002). Measurements of the transmitted light using a solid detection system (SDS) based on the deviation of the transmitted light measurements from the Beer’s law predictions is illustrated in Figure 14-40 by Wang et al. (2004). P > Poap

P < Poap

NIR NIR Light transmittance

Light receiver

Light transmittance

Light receiver

Figure 14-38. Light transmittance principle (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM).

508

NIR response

Asphaltene content on filter

Light transmittance power (mW)

1.00

8.0 Poap (upper) = 37 Mpa

0.75

6.0

0.50

4.0 Pb = 29 Mpa

0.25

2.0 Pap (lower) = 26 Mpa

0.00 0

20

40

60

0.0 100

80

Pressure (MPa)

Asphaltene portion on filter from SARA (%, w/w)

Formation Damage by Organic Deposition

Figure 14-39. Light transmittance response test at 82 C using Oil B (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM).

35 Beer’s law l = 35800 exp(–11ρoil)

Oil A1

Transmitted light (mW)

30 25 20 15

onset (P ~13000 psi)

10 5

Pb

0 0

10000

5000

15000

P (psia)

Figure 14-40. Asphaltenes in Oil A are predicted to be unstable at all pressures. SDS trace (from DBR) and fit to Beer’s Law equation, in which I = the transmitted light intensity and oil = the live-oil density (g/mL) (after Wang et al., ©2004; reprinted by permission of the Society of Petroleum Engineers).

Formation Damage by Organic Deposition

14.5.1.4

509

Filtration technique

Jamaluddin et al. (2002) concluded that the filtration technique enables the construction of the upper and lower asphaltene deposition envelopes faster than the gravimetric technique but slower than the acoustic resonance and light-scattering techniques. The filtration technique utilizes a visual PVTcell similar to the light-scattering technique. The oil sample placed in the PVT-cell is mixed well using a magnetic mixer, thus preventing gravity settling of the asphaltene precipitates, during the isothermal depressurization tests. Small amounts of oil samples are taken at various pressures and the asphaltene precipitates are filtered at the same pressure and temperature conditions of the PVT-cell. Then, the saturates, aromatics, resins, and asphaltenes present in the materials filtered out of the oil are determined at various pressures by laboratory analytical procedures, referred to as the SARA analyses. Jamaluddin et al. (2002) presents the asphaltene content and resin/asphaltene ratio as a function of pressure for typical oil. Figure 14-39 by Jamaluddin et al. (2002) compares the results obtained by the filtration and light-scattering techniques. The trends of the results obtained by these techniques may differ for a number of reasons. For example, sufficiently small size precipitate particles may pass through the filter used in filtration, part of the filtered materials may resolubilize in dense oil, and increased oil density at below the bubble-point pressure conditions may change the light transmission through the oil (Jamaluddin et al., 2002). 14.5.2 Asphaltene Precipitation and Stability Prediction Methods As stated by Wang and Buckley (2001), the prediction of conditions for the OAF or OAP is essentially based on two approaches: 1. Asphaltene phase behavior (APB) and 2. Comparison of the measured and predicted refractive indices. These are described in the following. 14.5.2.1

Asphaltene phase behavior

This approach is based on investigating the APB thermodynamically with parameters estimated through the representation of the high pressure and high temperature (HTHP) experimental data using an appropriate model

510

Formation Damage by Organic Deposition

(already explained above) (Wang and Buckley, 2001). Jamaluddin et al. (2002) state, “The asphaltene precipitation envelope (APE) is defined as the pressure at which the precipitation of colloidally dispersed asphaltene is detected at a given temperature.” (Figure 14-41 by Jamaluddin et al. (2002)). Figure 14-42 by Jamaluddin et al. (2002) demonstrate that the asphaltene deposition envelope can be generated by modeling using an equation of state, such as the Soave–Redlich–Kwang (SRK) equation of state with volume correction. 14.5.2.2

Refractive index method (RIM)

This approach is based on investigating the mixture solubility by measuring the refractive index (RI) which relates to the square root of the molar volume of the precipitant causing the asphaltene flocculation, leading to the formation of asphaltene precipitate (Wang and Buckley, 2001). Buckley et al. (1998) have shown that the RI of a live-oil can be predicted using the equation given below with the PVT data:     2 1 n2 − 1 n −1 = n2 + 2 live–oillo Bo n2 + 2 stock–tank–oilsto + 752 × 10

−6 Rs

Bo

m 

xi i

(14-31)

i=1

Example reservoir conditions

Pressure

L

S-L

Upper asphaltene envelope

V-L Equilibrium S-L-V Lower asphaltene envelope

Temperature

Figure 14-41. Pressure–temperature (P–T) diagram (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM).

511

Formation Damage by Organic Deposition 100 Simulated phase envelope Measured Pb Simulated upper poan Measured upper poan Simulated lower poan Measured lower poan

Pressure (MPa)

80

60

40

20

0

0

100

200

Temperature (°C)

300

400

Figure 14-42. Measured and simulated phase envelopes for Oil A (after Jamaluddin et al., ©2002; being used with the permission of the Petroleum Society of CIM. The author thanks the Petroleum Society for the use of this material with a reminder that copyright remains with the Petroleum Society and that no other copies may be made without the express written consent of the Petroleum Society of CIM.)

Where n = RI (dimensionless), Bo is the oil formation volume factor (rb/stb), Rs is the gas solubility in oil (scf gas/stb oil), xi denotes the mole fraction of species i (dimensionless), and i denotes the molar refraction of species i (ml/mole). The Flory–Huggins regular solution theory expresses the interaction parameter proportionally to the enthalpy of mixing as (Huggins, 1941; Flory, 1942; Hirschberg et al., 1984; Wang and Buckley, 2001) √  = m /RT   a − m 2  or m = a − /RT 1/2 m (14-32) Where  is the Flory–Huggins interaction parameter (dimensionless), vm denotes the molar volume of the mixture excluding the asphaltene (ml/mole), a and m are the solubility parameters of the asphaltene and the mixture excluding the asphaltene MPa1/2  T is the absolute temperature (K), and R is the universal gas constant (8.31441 J/K-mole). m is calculated as the volume-weighted average using m =

m  i=1

i i

(14-33)

512

Formation Damage by Organic Deposition

Where m denotes the total number of species excluding the asphaltene, i is the solubility parameter MPa1/2 , and i is the volume fraction of species i in a mixture (dimensionless). Wang and Buckley (2001) estimate the molar volume of the flocculant (precipitant) p (ml/mole) from the PVT data using p ml/mole = 15890 × 105 Vg /11957Rs

(14-34)

Where Vg (rb) is the actual volume and Rs (scf/stb) is the solubility of the precipitant gas dissolved in crude oil at reservoir oil conditions. Wang et al. (2004) demonstrated that the RI measured at the onset of asphaltene flocculation RIp rendering asphaltenes unstable is related to the solubility parameter (essentially linearly) and hence the square root of the molar volume of the flocculant dissolved in oil as shown in Figure 14-43a by Wang et al. (2004) for a typical oil of 312 API and 232 g/mole average molecular weight, containing 3.7% (wt) n-C7 asphaltene. Comparison of the RI of live-oil predicted using Eq. (14-31) and the onset refractive index RIp measured at 188 F displayed in Figure 14-43b by Wang et al. (2004) indicates that they do not intersect and therefore the oil is stable in the considered pressure range. However, when mixed with a gas, the oil becomes unstable at different pressures at 70, 140, and 210 F temperatures as indicated by the intersection of the predicted onset refractive index curve with the measured refractive index curve, as shown in Figure 14-43c by Wang et al. (2004). In Figure 14-43d the plot of the onset of asphaltene flocculation pressure vs. temperature obtained from Figure 14-43c delineates the regions of stable and unstable asphaltene conditions. Civan (2006b) correlated these data as described below. Civan (2006b) states, The Arrhenius (1889) and Vogel–Tammann–Fulcher Equations (VTF) (Vogel, 1921, Tammann and Hesse, 1926, and Fulcher, 1925) have been widely facilitated for correlation of the temperature dependency of the parameters of various physical and chemical processes. As described by Civan (2004, 2005a), these are asymptotic exponential functions, expressed in a general form, as: ln f = ln fc −

E R T − Tc

(14-35)

where f represents a temperature dependent parameter (unit determined by the type of property), fc a pre-exponential coefficient (unit determined

Formation Damage by Organic Deposition

513

Figure 14-43. Effect of temperature on asphaltene onset pressure for asphaltenes destabilized by addition of lift gas. (a) Oil C onset conditions. (b) Oil C asphaltenes are predicted to be stable at all pressures. (c) Addition of lift gas 149 m3 /m3  destabilizes asphaltenes from Oil C (solid lines: RI of oil+lift gas; dashed lines: PRI at each temperature). (d) Onset pressure increases with increasing temperature for asphaltenes from Oil C + lift gas (after Wang et al., ©2004; reprinted by permission of the Society of Petroleum Engineers).

by the type of property), T and Tc the actual and critical-limit absolute temperatures (K), respectively, E the activation energy (J/kmol), and R the universal gas constant (J/kmol-K). Because Eq. (14-35) is a threeparameter empirical equation, a minimum of three data points are required for determination of its parameters. Whereas, the Arrhenius equation assumes Tc = 0. Therefore, a minimum of two data points are sufficient for estimation of the parameters of the Arrhenius equation.

514

Formation Damage by Organic Deposition 8.00 Arrhenius equation

7.95

Vogel–Tammann– Fulcher equation

InP, P in psia

7.90

7.85 Arrhenius equation InP = 8.950 – 647.1/T R 2 = 0.99

7.80

Vogel–Tammann– Fulcher equation InP = 8.281 – 95.37/(T – 360.) R 2 = 1.00

7.75

7.70

0

0.002

0.004

0.006

0.008

1/(T-Tc), T and Tc in R

Figure 14-44. Correlation of the onset pressure of asphaltene flocculation vs. temperature data of Wang et al. (2004) using the Arrhenius and Vogel–Tammann–Fulcher type equations (after Civan, 2006; reprinted by permission of the Society of Petroleum Engineers).

Figure 14-44 by Civan (2006b) presents an accurate correlation of the predicted onset pressure of asphaltene flocculation vs. temperature data of Wang et al. (2004) by means the Arrhenius and Vogel–Tammann– Fulcher (VTF) equations.

14.6 ALGEBRAIC MODEL FOR FORMATION DAMAGE BY ASPHALTENE PRECIPITATION IN SINGLE PHASE Minssieux (1997) has demonstrated that the predominant mechanisms of the asphaltene deposition can be identified by means of the Wojtanowicz et al. (1987, 1988) analytic models. Minssieux also observed that the asphaltene precipitates existing in the injected oil can pass into porous media without forming an external filter cake. The characteristics of the oils used are given in Tables 14-2 and 14-3, and the conditions and results of the coreflood experiments are given

515

Formation Damage by Organic Deposition Table 14-2 Characteristics of Stock-Tank-Oils∗

Field

Reservoir SARA ANALYSIS temperature Res/Asph Viscosity ( C) Sat Ar Resins Asph. ratio (cP 20 )

Weyburn Lagrave Hassi– Messaoud Boscan (Reference) ∗

50

401

461

80

657

228

75

4

119

705

255

33

0.15

15

37

81

85

34

5.3

14

16



API

13

19 22

29

7.7

43

1580

43

24

10

After Minssieux, ©1997 SPE; reprinted by permission of the Society of Petroleum Engineers.

Table 14-3 Characteristics of Crude and Asphaltenes∗ Crude origin % Asphaltene Average MW ( API) (weight) (vpo/toluene)

Weyburn 29

Asphaltenes analysis H/C

O/C

100

0025

%S

Tmax ( C) pyrolysis

53

6000

Lagrave 43

4

“7000–8700”

100

0010

380

416

H. Messaoud 45

015

1120 “well scales”

088

0034

080

420

8000

114

0039

670

406

Boscan (10 ) “as a reference” ∗

107

416

After Minssieux, ©1997 SPE; reprinted by permission of the Society of Petroleum Engineers.

in Tables 14-4 and 14-5 by Minssieux (1997). The analyses of typical data according to Wojtanowicz et al. (1987, 1988) formulae are given in Figures 14-45 and 14-46 by Minssieux (1997). Figure 14-45 shows the results of the analysis of the GF3 test data considering the possibility of the gradual surface deposition, single pore plugging, and in situ cake formation by pore-filling mechanisms in formation damage. As can be  seen, only the K/K0 vs. PV (pore volume) data yields a straightline plot, indicating that the damage mechanism is the gradual surface deposition. In the case of the GV5 data, Figure 14-46 indicates that the damage mechanism is the in situ cake formation by pore filling, because K0 /K vs. PV data yields a straight-line plot for this case (see Table 10-1).

516

Formation Damage by Organic Deposition Table 14-4 Conditions of Core Floods∗ Petrophysical data

Test ref.

Type of rock T( C) Crude used

ø%

GF 1

Fontainebleau sandstone

50

Weyburn

131

GF 2 GF 3

id. id.

id. id.

id. id.

136 137

KH (mD)

107 87 774

Injection rate (cm3 /hour)

50 80 10 10 20 50 80

GF 12

id.

80

H. Messaoud

8

6

10

GVM 5

50

Weyburn

247

29

10

GVM 10

Vosges sandstone id.

50

Weyburn

243

122

GVM 13 GVR 8 GVR 11

id. id. id.

80 50 80

Lagrave Weyburn Lagrave

26 226

73 152

10 5 10 10

GP 9

Palatinat sandstone

80

H. Messaoud

226

11

23

2

GP 14

id.

id.

id.

HMD 11

Res. rock from HMD id.

id.

id.

id.

id.

HMD 26 ∗

71

067

10 5 5

8

After Minssieux, ©1997 SPE; reprinted by permission of the Society of Petroleum Engineers.

14.7 PLUGGING–NONPLUGGING PATHWAYS MODEL FOR ASPHALTENE DEPOSITION IN SINGLE PHASE Ali and Islam (1997, 1998) considered only asphaltene deposition and resorted to a simplified, single-phase formation damage modeling approach according to Gruesbeck and Collins (1982a). Here, their model is presented in a manner consistent with the rest of the presentation of this chapter. Also, a few missing equations are supplied. Note that this model applies for undersaturated oils.

Table 14-5 Conditions of Core Flood Tests∗

Test ref.

Crude used

Average amount of deposits (mg/g rock)

K reduction (%) after 50 PV

Observations Model type of porous medium (pure silica)

GF 1 GF 2

Weyburn W.

0.30

Uniform

20 42.5

GF 3 GF 12

W. H.MD

0.21 0.34

Uniform Decreasing

58.5 0 (47 after 150 PV)

GVM 5

W.

1.0

Decreasing

88

GVM 10 GVM 13

W. Lagrave

0.48 –

Decreasing –

89 0

(14-55)

Using the input data given in Table 14-6, Ali and Islam (1998) obtained the results presented in Figures 14-47–14-49.

14.8 TWO-PHASE AND DUAL-POROSITY MODEL FOR SIMULTANEOUS ASPHALTENE–PARAFFIN DEPOSITION Here, the formulation of Civan’s (1995a) model is presented. 14.8.1

Considerations of the Model

The reservoir fluid system can be single-phase or multiphase depending on the prevailing reservoir conditions. Above the bubble-point pressure conditions, the oil is undersaturated and single phase. Below the bubblepoint pressure conditions, the oil is saturated and can be two-phase. Civan (1995a) developed a model that is applicable for both conditions. His model also considered the possibility of simultaneous deposition of asphaltenes and paraffins. As stated by Civan (1995a), “Although they are a mixture of different molecular weight components, the paraffins and

522

Formation Damage by Organic Deposition Table 14-6 Model Parameters∗ Experimental Parameters W0 , wt% K, mD n0 , mg/g q, mL/min

1   API r , g/mL Pore volume, cm3 Run 1, mL/min Run 2, mL/min Run 3, mL/min Run 4, mL/min

3 11.3 200 0.5, 1, 2, 3 0.35 29.29 2.71 136 0.5 1 2 3

Adjustable Parameters , cm ma /mo S ma K1  hour −1 K2  hour −1 ke  cm−1 kd  second−1 uc  cm · second−1 kp  cm−1 b cm−1  fp  Kpi , mD Knpi /Kpi

0.1 15 100 0.05 n0 20 0.008 6.3 0.00085 through 0.01 0.032 0.2 50 10 0.82 10 11.3 10

∗ Modified after Ali and Islam, ©1998 SPE; reprinted by permission of the Society of Petroleum Engineers.

asphaltenes are lumped into two groups as the paraffin, p, and the asphaltene, a, pseudo-components. The other components of the oil are grouped as the oil pseudo-component, o, which acts as a solvent. The mixture of the various gases are grouped as the gas pseudo-component, g.” Thus, Civan’s (1995a) model considers four pseudo-components: (a) paraffin, par, (b) asphaltene, asp, (c) oil, o, and (d) gas, g. The system of the fluids and the porous formation is considered in three phases as the vapor, V, liquid, L, and solid, S, following Ring et al. (1994). The solid

Formation Damage by Organic Deposition

523

Figure 14-47. Permeability reduction for injection at 0.5 mL/min rate (after Ali and Islam, ©1998 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 14-48. Permeability reduction for injection at 2 mL/min rate (after Ali and Islam, ©1998 SPE; reprinted by permission of the Society of Petroleum Engineers).

phase is considered in two parts: (1) porous matrix (unchanged), and (2) organic deposits (varying). Civan (1995a) considers that the paraffin and asphaltene transport may occur in both dissolved and precipitate forms depending on the state of saturation of the oil phase. This assumption is supported by Mansoori (1997) who points out that “   asphaltenes are partly dissolved and partly in colloidal state (in suspension) in oil peptized (or stabilized) primarily by resin molecules that are adsorbed on asphaltene surface.” The permeability impairment may occur by (a) gradual pore size reduction, and (b) pore throat plugging, obstruction, and sealing. The ratio of the plugging and nonplugging paths vary by organic deposition. Single- or two-phase fluid conditions may exist depending on whether the condition is above or below the bubble-point pressure. The various phases are assumed at thermal equilibrium within the bulk volume. Non-Newtonian fluid behavior is considered for high concentrations

524

Formation Damage by Organic Deposition

Figure 14-49. Permeability reduction for injection at 3 mL/min rate (after Ali and Islam, ©1998 SPE; reprinted by permission of the Society of Petroleum Engineers).

of organic precipitates and solutes. Non-Darcy flow behavior is assumed for flow through passages narrowing due to precipitation. 14.8.2

Porosity and Permeability Relationships

The porosity and permeability alterations are predicted based on a modified version of the plugging and nonplugging pathways concept of Gruesbeck and Collins (1982a). Relatively smooth and large diameter flowpaths are assumed to mainly undergo a surface deposition and are called nonplugging. Highly tortuous and variable diameter flowpaths are called plugging. The retainment of organic deposits in the plugging pathways occurs by jamming and pore throat blocking. Consider that po and npo denote the pore volume fractions and p and np are the fractions of the bulk volume occupied by organic deposits of the plugging and nonplugging pathways of the porous media. Thus, the instantaneous porosities in the plugging and nonplugging flow paths are given, respectively, by

p = po − p

(14-56)

np = npo − np

(14-57)

Although Gruesbeck and Collins (1982a) assume characteristic constant values, it is reasonable to consider that the fractions of the bulk volume

Formation Damage by Organic Deposition

525

containing the plugging and nonplugging pathways vary during deposition and are estimated by, because of the lack of a better theory, fp =

p

(14-58)

fnp =

np

(14-59)

The instantaneous and initial porosities of the porous medium are given, respectively, by

= p + np

(14-60)

o = po + npo

(14-61)

The total organic deposit volume fraction and the instantaneous porosity are given, respectively, by  = p + np

(14-62)

= o − 

(14-63)

The rate of deposition in the plugging pathways can be expressed by p /t = kp up p p

(14-64)

subject to the initial condition p = po 

t=0

(14-65)

for which kd = 0

when t ≥ tp

kd = 0

t < tp

(14-66) (14-67)

Here, tp is the time of initiation of the particle bridges and jamming. This is the time at which the ratio of the pore throat to particle diameter drops to below its critical value determined by the following empirical correlation (Civan, 1990, 1996a):    

 Dt Dt Dt (14-68) <  = A 1 − exp −BRep + C Dp Dp cr Dp cr

526

Formation Damage by Organic Deposition

in which the particle Reynolds number is given by

Rep = p p uDp p /  p

(14-69)

The symbol p denotes the tortuosity of the plugging paths. The rate of deposition in the nonplugging tubes can be expressed by (Civan, 1994, 1995, 1996a) np = kd unp np 2/3 np − ke np e w − cr t

(14-70)

subject to the initial condition t=0

(14-71)

ke = 0

w ≥ cr

(14-72)

ke = 0

w < cr

(14-73)

np = npo  Here,

For simplification purposes, Civan (1995a) assumed that organic deposits are sticky and, therefore, once deposited they cannot be removed anymore. Consequently, the second term in Eq. (14-70) can be dropped. Mansoori (1997) tends to support this argument. Although Leontaritis (1998) considered the possibility of erosion of deposits, it is not apparent if he actually implemented this possibility in his calculational steps. kd and ke are the surface deposition and mobilization rate constants, respectively. e is the fraction of the uncovered deposits estimated by e = exp −k

(14-74)

Where k is an empirical coefficient. cr is the minimum shear stress necessary to mobilize the surface deposits. w is the wall shear stress given by the Rabinowitsch–Mooney equation (Metzner and Reed, 1955): w = k′ 8/D n

′

(14-75)

in which the interstitial velocity, v, is related to the superficial velocity, u, as follows (Dupuit, 1863): v=

unp

(14-76)

527

Formation Damage by Organic Deposition

Where the symbol np denotes the tortuosity of the nonplugging paths, and the mean pore diameter is given by 

1/2 (14-77) D ∼ 4 2np Knp / np The permeabilities of the plugging and nonplugging pathways are given by the following empirical relationships (Civan, 1994a,c):

and

n 

Kp = Kpo exp − po − p 1 = Kpo exp −np 1 Knp = Knpo



np

npo

n 2

= Knpo

np 1−

npo

n2

(14-78)

(14-79)

Where  n1 , and n2 are empirical constants. Then, the average permeability of the porous medium is given by K = fp Kp + fnp Knp

(14-80)

The superficial flows in the plugging and nonplugging pathways are given respectively, by up =

uKp K

(14-81)

unp =

uKnp K

(14-82)

The total superficial flow is given by (Gruesbeck and Collins, 1982a) u = fp up + fnp unp

(14-83)

Considering the simultaneous deposition of paraffins and asphaltenes, p and np in Eqs (14-56) through (14-62) denote the sum of the paraffins and asphaltenes, that is, p = p par + p asp

(14-84)

np = np par + np asp

(14-85)

528

14.8.3

Formation Damage by Organic Deposition

Description of Fluid and Species Transport

The preceding treatment of the porous media impairment phenomena implies that the suspended particle and dissolved species concentrations may be different in the plugging and nonplugging pathways. Then, separate sets of balance equations are required for the plugging and nonplugging pathways. Consequently, the numerical solution would require a highly intensive computational effort. However, this problem can be conveniently circumvented by assuming that there is hydraulic interaction between these pathways (i.e., they are not isolated from each other). 1. The mass balances are considered for the following four pseudocomponents: a. Gas b. Oil c. Suspended paraffins and asphaltenes d. Dissolved paraffins and asphaltenes 2. Total thermal equilibrium energy balance is considered. 3. Non-Newtonian fluid description using the Rabinowitsch–Mooney equation is resorted. 4. The Forhheimer equation for the non-Darcy flow description is used. 5. The average flow is defined as a volume fraction weighted linear sum of the flow through the plugging and nonplugging paths according to Gruesbeck and Collins (1982a). 6. The average permeability is defined as a volume weighted linear sum of the permeabilities of the plugging and nonplugging paths according to Gruesbeck and Collins (1982a). a. In the plugging paths, a snowball deposition effect is represented by an exponential decay function. b. In the nonplugging paths, a gradual pore size reduction, represented by the power law function, is considered. 7. The precipitation of the asphaltene and paraffin is predicted, applying Chung’s (1992) thermodynamic model, discussed previously, for nonideal solutions to determine the cloud point and the quantity of the precipitates to be formed. The total mass balance of the gas component is given by



 ˙ g V + m ˙ g L = 0

SV V + SL L wg L + V uV + L uL wg L + m t x

(14-86)

Formation Damage by Organic Deposition

529

The first, second, and third terms respectively represent the accumulation, transport, and well production. Assuming that the oil component exists only in the liquid phase and does not vaporize into the gas phase, the oil component mass balance is given by    SL L woL + L uL woL + m ˙ OL = 0 t x

(14-87)

for which Ring et al. (1994) assumed woL 10. Considering that organic precipitates only exist in the liquid phase, because it is the wetting phase for these particles, the suspended paraffin and asphaltene particle mass balances are expressed by





SL L wp L + p p + ˙ p L uL L wp L + m t x    wp L 

SL DpL  p = asphaltene or paraffin = x x

(14-88)

Note that L wp L = p L = p p L

(14-89)

If the particle density, p , is assumed constant, and the suspended particle content is expressed by the volume fraction of organic substance (paraffin or asphaltene), pL , according to Eq. (14-89), then Eq. (14-88) can be simplified as  



p m ˙ pL p L   =

SL DpL  + +

SL p L + u  t x L p L t p x x p = asphaltene or paraffin

(14-90)

Note that both Ring et al. (1994) and Civan (1996a) neglected the term on the right, representing the dispersion of particles. The mass balances of the paraffin and asphaltene dissolved in oil is given by      m ˙ iL   SL L xiL +  u x + =

SL DiL L xiL  t x L L iL Mi x x i = asphaltene or paraffin

(14-91)

530

Formation Damage by Organic Deposition

where S is the saturation,  is the density, t is the time, x is distance, u is the volume flux, pL is the volume fraction of the organic precipitates in the liquid phase, wpL denotes the mass fraction, xiL is the mole fraction of organic dissolved in the oil, Mi is the molecular weight and DiL is the dispersion coefficient. p /t represents the volume rate of retention of organic deposits in porous media determined according to Eqs (14-62), (14-63), and (14-70). Assuming that the various phases are at thermal equilibrium at a temperature of TV = TL = TS = T , the total porous media energy balance is given by   SV V UV + SL L UL + par par Upar + asp asp Uasp t

 + 1 − − par − asp S US  + V uV HV + L uL HL + q˙ V + q˙ L x   P P  SV V + SL L + par par + asp asp = uV V + uL L + x x x 

T + 1 − − par − asp s  (14-92) x where U and H are the internal energy and enthalpy, respectively, q˙ V and q˙ L denote the energy loss, p is pressure, denotes the thermal conductivity, and T is temperature. Ring et al. (1994) simplified Eq. (14-92) as  

SV V UV + SL L UL + Sp p Up + 1 − s Us t   T  

+ V HV uV + L HL uL + QL + QH + QE = x x x

(14-93)

The first, second, and last terms represent the accumulation, convection, and conduction heat transfer. The QL  QH and QE terms represent the heat carried away by production at wells, heat losses into formation surrounding the reservoir, and the external heat losses. The deposition of organic precipitates in porous media reduces the flow passages causing the fluids to accelerate. Therefore, Darcy’s law

Formation Damage by Organic Deposition

531

is modified as following, considering the inertial effects and horizontal flow, according to the Forchheimer equation (Civan, 1996a): uJ = −−1 J NndJ KpJ /x 

J = V or L

(14-94)

where K is the permeability, pJ is the fluid pressure, and the non-Darcy number is given by NndJ = 1 + ReJ −1

(14-95)

in which the porous media Reynolds number is given by ReJ =

J uJ K J

(14-96)

where  is the inertial flow coefficient, and J and J denote the density and viscosity of a fluid phase J. Note that the formulations presented here are applicable for multidimensional cases encountered in the field if /x is replaced by · and a vector-tensor notation is applied. 14.8.4

Phase Transition

The source terms appearing on the right of Eqs (14-86)–(14-93) are considered a sum of the internal (rock–fluid and fluid–fluid interactions) and external (wells) sources. When the oil is supersaturated, the internal contribution to the source terms in Eq. (14-88) is determined as the excess quantity of organic content of oil above the organic solubility at saturation conditions determined by Chung’s (1992) thermodynamic model:

s s /t xpL > xpL (14-97) m ˙ pL =  xpL − xpL m ˙ pL = 0 

s xpL < xpL 

p = asphaltene or paraffin

(14-98)

s where xpL represents the mole-fraction of dissolved organic at saturation. Civan (1995a) carried out case studies similar to Ring et al. (1994) using the Sutton and Roberts data (1974). Figure 14-50 shows a comparison of the predicted and the measured permeability impairments by paraffin deposition for cases below and above bubble-point pressure. Note that, above the bubble point pressure, only the liquid phase exists and there is more severe formation damage. Whereas, below the bubble point pressure, both the liquid and the vapor phases exist and there is less severe formation damage.

532

Formation Damage by Organic Deposition

Figure 14-50. Comparison of the Sutton and Roberts (1974) experimental data and simulation results for permeability reduction by organic deposition below and above bubble point pressure.

14.9 SINGLE-POROSITY AND TWO-PHASE MODEL FOR ORGANIC DEPOSITION Ring et al. (1994) developed a two-phase model considering only the paraffin precipitation. They assumed that (1) oil is always saturated with the paraffin, (2) the solution is ideal, (3) paraffin deposition obeys a firstorder kinetics, (4) pores undergo an irreversible continuous plugging, and (5) permeability reduction obeys a power law: K Ko



o

m

(14-99)

Ring et al. (1994) determined that m 8 for paraffin deposition. Wang et al. (1999) and Wang and Civan (2005a,b) developed improved models considering the simultaneous deposition of asphaltenes and paraffins. These models incorporate the essential features of Civan’s (1995a) dual-porosity model for a single-porosity treatment. The formulation of the Wang and Civan (2005a,b) model and its experimental verification are described in the following. However, their formulation is presented in a manner consistent with the material presented in this book.

Formation Damage by Organic Deposition

14.9.1

533

Model Formulation∗

Wang and Civan (2005a,b) state (©2005 ASME, reprinted by permission of the American Society of Mechanical Engineers) The organic deposition during oil production by primary depletion under isothermal conditions is preferentially of the asphaltene types. Prior to the initiation of primary oil recovery, asphaltene is completely dissolved in the reservoir oil. Upon production, the reservoir oil pressure continuously declines. Asphaltene begins to separate and precipitate from the oil when the pressure decreases to the onset of asphaltene precipitation pressure Pp . Asphaltene precipitation attains a maximum value around the bubble-point pressure. When the pressure decreases in the below bubble-point region, then some asphaltene precipitates are redissolved back into oil. However, in some situations encountered in the field involving temperature variation, paraffin and asphaltene may precipitate and deposit together. For example, injection of fluids at temperatures colder than the reservoir oil temperature during hydraulic fracturing or acid stimulation processes may reduce the conditions of the reservoir oil to below the cloud point of paraffin. After the fracturing, the well begins to produce oil and the pressure in the near wellbore region declines. As a result of the simultaneous effect of temperature decrease and pressure depletion, paraffin and asphaltene may deposit in porous rock simultaneously in the near-wellbore region. For such applications, it is necessary to use a model for simultaneous deposition of paraffin and asphaltene in porous media.

Wang and Civan (2005a,b) used the real-solution theory to describe the dissolution/precipitation of paraffin in crude oil, the polymer-solution theory to describe the dissolution/precipitation of asphaltene in oil; an improved one-dimensional, three-phase, and four-pseudo-component model to represent the transport of paraffin and asphaltene precipitates; and a deposition model including the static and dynamic pore surface depositions and pore throat plugging to describe the deposition of paraffin and asphaltene. The model was developed for analysis of the laboratory core flow tests as well as for field-scale impairment by organic deposition. The capillary pressure effects between vapor and liquid phases have been neglected. The oil, gas, and solid phases were assumed at thermal equilibrium. ∗ Reprinted by permission of the Society of Petroleum Engineers from Wang et al., ©1999 SPE, SPE 50746 paper.

534

Formation Damage by Organic Deposition

The oil, gas, paraffin, and asphaltene pseduo-components are denoted by O, G, P, and A, respectively. The vapor and the liquid phases are denoted by V and L, respectively. The paraffin and asphaltene pseudocomponents exist only in the liquid phase (a suspension of the oil, and the paraffin and asphaltene particles) and as organic deposits in porous media. Considering both the free and the dissolved gases, the gas component mass balance equation is given by   SV V + SL L wGL +  · V uV + L uL wGL = 0 t

(14-100)

where represents the porosity of the porous media, Sv  v  uv are the saturation, density, and flux of the vapor phase, respectively, and SL  L  uL are the saturation, density, and flux of the liquid phase, respectively. wGL represents the mass fraction of the dissolved gas in the liquid phase and t denotes the time. The divergence operator is given by  ≡ i/x + j/y + k/z where i, j, and k denote the unit vectors in the x y, and z Cartesian directions. Considering that the oil component exists only in the liquid phase, its mass balance is given by   SL L wOL +  · L uL wOL = 0 t

(14-101)

where wOL is the mass fraction of the oil component in the liquid phase. The volume fraction is resorted to express the concentrations of the paraffin and asphaltene particles suspended in the oil because the permeability impairment in porous media can be expressed conveniently. On the other hand, mass fraction is more convenient to express the concentrations of the paraffin and asphaltene dissolved in the oil. The paraffin mass balance equation is expressed by considering that it may be partly dissolved and/or suspended as particles in the liquid phase and deposited in porous media:   SL L wP + SL P P + P P +  · uL L wP + uL P P = 0 t (14-102) where A is the volume fraction of the suspended paraffin in the liquid phase and P is the density of the paraffin. wPL represents the mass fraction of the dissolved paraffin in the liquid phase. wSPL is the mass ratio of the paraffin precipitates suspended in the liquid phase to the

Formation Damage by Organic Deposition

535

liquid phase. P is the volume fraction of the deposited paraffin in the bulk porous media. The asphaltene mass balance equation is written similarly as   SL L wA + SL A A + A A +  · uL L wA + uL A A = 0 t (14-103) in which A is the volume fraction of the suspended asphaltene in the liquid phase and A is the density of asphaltene. wAL represents the mass fraction of the dissolved asphaltene in the liquid phase. wSAL is the mass ratio of the asphaltene precipitates suspended in the liquid phase to the liquid phase. A is the volume fraction of the deposited asphaltene in the bulk porous media. The vapor- and liquid (suspension)-phase volumetric fluxes are given by Darcy’s equation, respectively, as uV = −

kRV K · PV + V gk V

(14-104)

uL = −

kRL K · PL + L gk L

(14-105)

where K is the absolute permeability tensor of the porous media, and kRV and V are the relative permeability and viscosity of the vapor phase, respectively. kRL and L are the relative permeability and viscosity of the liquid phase, respectively. The capillary pressure between the vapor and liquid phases is neglected. Thus, PV = PL = P represents the pressure of the pore fluids. The total thermal equilibrium energy balance equation is expressed as  

SV V HV + SL L HL + P P HP +A A HA t  + 1 − − P − A F HF +  · V uV HV + L uL HL   

SV V + SL L + P P + A A =· T (14-106) + 1 − − P − A F

where HV  HL  HP  HA , and HF are the enthalpies, and V  L  P  A , and F are the thermal conductivities of the vapor and liquid phases, paraffin and asphaltene, and porous media, respectively. T is the equilibrium temperature of the system.

536

Formation Damage by Organic Deposition

The saturations of the flowing vapor and liquid phases add up to 1.0: SV + SL = 10

(14-107)

The paraffin and asphaltene deposition rates are given, respectively, by

P = kdP SL P − keP P L − LcrP + ktP uL SL P t

A = kdA SL A − keA A L − LcrA + ktA uL SL A t

(14-108) (14-109)

in which the first term represents the pore surface deposition and kd is the pore surface deposition rate coefficient. The second term represents the entrainment of the surface deposits by the flowing fluid and ke is the pore surface entrainment rate coefficient. L is the interstitial velocity of the liquid (oil) phase and Lcr is the critical interstitial velocity of the liquid phase required for mobilization or entrainment of the pore surface deposits. The third term represents the pore throat plugging deposition and kt is the plugging deposition rate coefficient. The pore surface entrainment rate coefficient is assigned as when L > crL  j = P A

kej = keji  kej = 0

otherwise

j = P A

(14-110) (14-111)

keji is the value of the entrainment rate coefficient. The pore throat plugging deposition coefficient is considered based on the following criteria: ktj = ktji 1 + j P + A  ktj = 0

Dt ≤ Dtcr  j = P A

otherwise

j = P A

(14-112) (14-113)

ktji is the initial value of the plugging deposition rate constant. j represents empirically determined constants. The instantaneous porosity is given by

= i − P − A

(14-114)

and the instantaneous permeability is estimated by (Civan et al., 1989) K = fp Ki  / i 3

(14-115)

Formation Damage by Organic Deposition

537

where i and Ki are the initial porosity and permeability of the porous media, respectively. fp is a parameter representing the extent of pore connectivity. 14.9.2

Model Assisted Analysis of Laboratory Data

Wang and Civan (2005a,b) solved the model equations using an implicit finite-difference method with a block-centered grid system. They discretized the time derivative by the first-order backward finite-difference approximation and the space derivatives by means of a second-order central finite-difference approximation. They generated stable numerical solutions using uniform time increments and grid sizes. They determined the best estimates of the parameters by history matching of experimental data. The data used in the test cases and the best estimates of the model parameters are presented in Tables 14-7 and 14-8 by Wang and Civan (2005a,b). The typical case studies carried out by Wang and Civan (2005a,b) are briefly described in the following. Case 1 – Simultaneous Paraffin and Asphaltene Deposition

Sutton and Roberts (1974) first heated a Berea sandstone core saturated with a Shannon Sand crude oil to 544 C and then cooled the outlet of the core to 211 C for 2 hours without any flow. The cloud point of the oil used in their experiment was 378 C. The paraffin and asphaltene contents of the crude oil were 4.1 and 0.7 weight percents, respectively. Then, they conducted a flow experiment by injecting the Shannon Sand crude oil at a rate of 0.38 ft/day (Ring et al., 1994). The temperature of the outlet of the core was kept at 211 C. They first simulated the static pore surface deposition during 2 hours of cooling without fluid flow. Then, they simulated the damage during flow. The pore throat plugging and deposit entrainment did not take place as indicated by the estimated values of rate constants given in Table 14-7. Figure 14-51 shows that the simulated results are satisfactory. Case 2 – Simultaneous Paraffin and Asphaltene Deposition

Sutton and Roberts (1974) injected a Muddy formation crude oil into a Berea sandstone core at a rate of 0.30 ft/day (Ring et al., 1994). The outlet temperature of the core was kept at 211 C. The cloud point temperature

538

Formation Damage by Organic Deposition Table 14-7 Parameters for Cases 1 and 2∗

Experiment Experimental Conditions Tin   C Tout   C Bubble-point pressure, psia Flow rate, ft3 /day Back pressure, psia Gas pseudo-component Mg , g/mole gsc  g/cm3 Oil pseudo-component Mo , g/mole osc  g/cm3 Paraffin pseudo-component MP , g/mole wPS % Psc  g/cm3 TPM   C HP , cal/g-mole Asphaltene pseudo-component MA , g/mole wAs (%) Asc  g/cm3

1

2

54.4 21.1 2050 0.38 1900

54.4 21.1 2050 0.30 1900

16.0 0.00083

16.0 0.00083

104.11 0.72

122.51 0.75

522.4 4.1 0.83 75.7 26,000

478.7 6.1 0.98 71.7 23,600

2,500 0.7 1.1

2,500 0.1 1.1

30.5 2.5 0.25 0.405

30.5 2.5 0.25 0.314

Core Properties L, cm D, cm

o , fraction Ko , Darcy Simulation Parameters Grid spacing, x, cm Time increment, t, sec Number of blocks Deposition Parameters Dptcr , cm vcrL , cm/sec kdP = kdA , 1/sec kePi = ketAi , 1/cm ktPi = ktAi , 1/cm P = A , constant

2.54 40 12

2.54 40 12

N/A N/A 0.0241 0.0 0.0 0.0

N/A N/A 0.0182 0.0 0.0 0.0

Deposition mechanism

Surface deposition

Surface deposition

∗

Modified after Wang et al., ©1999 SPE; reprinted by permission of the Society of Petroleum Engineers, and Wang and Civan, ©2005b ASME; reprinted by permission of the American Society of Mechanical Engineers).

Table 14-8 Parameters for Case 3∗ GF1

GF3

GV10

GV5

GP9

HMD26

Core Properties Initial Permeability, Ko  mD Initial Porosity, o , % Length of Core Sample, cm Diameter of Core Sample, cm

107 13.1 6.0 2.3

77.4 13.7 6.0 2.3

18.0 24.3 6.0 2.3

29.0 24.7 6.0 2.3

1.1 22.6 6.0 2.3

0.67 7.1 6.0 2.3

Oil Properties API of Crude Oil Asphaltene Content, wt. %

29.0 5.3

29.0 5.3

29.0 5.3

29.0 5.3

43.0 0.15

43.0 0.15

Experimental Data Flow Rate, cm3 /hour Temperature,  C

50 50

10 50

10 50

10 50

10 80

8 80

Deposition Parameters Dptcr  cm crL  cm/sec kd  1/sec ke  1/cm kt  1/cm , constant fp , constant

0.00048 0.0 0.0017 0.0 0.07 35 0.91

0.0145 0.0018 0.69 0.0 0.0 1.0

0.01 0.0051 0.003 0.0 0.0 1.0

0.01 0.0128 0.012 0.0 0.0 1.0

0.000643 0.0275 0.0006 0.0 0.0 1.0

0.00016 0.0 0.0065 0.0 0.035 0.0 0.96

Surface deposition, deposit entrainment

Surface deposition, deposit entrainment

Surface deposition, deposit entrainment

Surface deposition, deposit entrainment

Deposition Mechanism

After Wang et al., ©1999 SPE, and Wang and Civan, ©2005c SPE; reprinted by permission of the Society of Petroleum Engineers.

Surface deposition, pore throat plugging

539

∗

Surface deposition, pore throat plugging

Formation Damage by Organic Deposition

TESTS

540

Formation Damage by Organic Deposition

Ratio of average permeability to initial permeability (K/Ko, fraction)

1.0 Experiment 1 data

0.8

Simulated data for Experiment 1

0.6 0.4 0.2 0.0

0

2 3 4 1 Injection pore volume of oil (PV)

5

Figure 14-51. Comparison of the simulation results with the Experiment 1 data of Sutton and Roberts (1974) (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

of the oil sample is 35 C. The paraffin and asphaltene contents of the crude oil were 6.1 and 0.1 weight percents, respectively. The pore surface depositions occurred, but the pore throat plugging and deposit entrainment did not occur as indicated by the values of the rate constants given in Table 14-7. Because the oil and core properties are very close for Cases 1 and 2, the permeability damage is also similar, as shown in Figures 14-51 and 14-52. Figure 14-52 shows the satisfactory simulated results.

Ratio of average permeability to initial permeability (K/Ko, fraction)

1.0 Experiment 2 data 0.8

Simulated data for Experiment 2

0.6 0.4 0.2 0.0

0

2 3 4 1 Injection pore volume of oil (PV)

5

Figure 14-52. Comparison of the simulation results with the Experiment 2 data of Sutton and Roberts (1974) (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

541

Formation Damage by Organic Deposition

Case 3 – Asphaltene Deposition

Ratio of average permeability to initial permeability (K/Ko, fraction)

Wang and Civan (2005a) describe that Minssieux (1997) used the dead oils obtained from the Weyburn oil (Canadian oil) containing 5.3% asphaltene and 8.5% resin by weight, and the Hassi–Messaoud crude oil (Algerian oil) containing 0.15% asphaltene. Therefore, the fluid system used in the core tests was single-phase. Because Minssieux (1997) did not report the compositions of these oils, Wang and Civan (2005a,b) approximated the compositions of the Weyburn and Hassi–Messaoud crude oils using the compositions of the dead-oils that Hirschberg et al. (1984) have provided. Wang and Civan (2005a,b) adjusted the shift parameters, eccentric factor, and critical properties of the C7+ fraction to match the present oil asphaltene contents. Minssieux (1997) used the Fontainebleau, Vosges, and Palatinat sandstones (permeability ranging from 0.67 to 107 mD and porosities ranging from 7.1 to 24.7%) and a HMD (Hassi–Messaoud) reservoir rock sample. Minssieux (1997) injected the dead-oils into the horizontal core samples (5.0 to 7.0 cm long and 2.3 cm diameter) at flow rates ranging from 8 to 10 cm3 /hour. Minssieux (1997) conducted the core tests at 50 C and 80 C temperatures by applying a 145 psia backpressure. The simulation results compared well with the measurements of the six Minssieux (1997) experiments, referred to as GF1, GF3, GV5, GV10, GP9, and HMD26 as shown in Figures 14-53–14-58 by Wang and Civan (2005a), respectively. The 1

0.9

0.8 Experiment data

0.7

Simulated results 0.6

0

10

20 30 40 50 60 Injection pore volume of oil (PV)

70

Figure 14-53. Comparison of the simulated results with the Experiment GF1 data of Minssiuex (1997) (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

542

Formation Damage by Organic Deposition

Ratio of average permeability to initial permeability (K/Ko, fraction)

1.0 Experiment data Simulated results

0.8

0.6

0.4

0.2

0

10

20 30 40 50 60 Injection pore volume of oil (PV)

70

Ratio of average permeability to initial permeability (K/Ko, fraction)

Figure 14-54. Comparison of the simulated results with the Experiment GF3 data of Minssiuex (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

1 0.9

Experiment data

0.8

Simulated results

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20 30 40 50 Injection pore volume of oil (PV)

60

Figure 14-55. Comparison of the simulated results with the Experiment GV5 data of Minssiuex (1997) (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

mechanisms involved in each cases are identified based on the estimated values of the rate constants given in Table 14-8 by Wang and Civan (2005c).

543

Formation Damage by Organic Deposition

Ratio of average permeability to initial permeability (K/Ko, fraction)

1.0 0.9

Experiment data

0.8

Simulated results

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

10 20 30 40 Injection pore volume of oil (PV)

50

Ratio of average permeability to initial permeability (K/Ko, fraction)

Figure 14-56. Comparison of the simulated results with the Experiment GV10 data of Minssiuex (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

1.0 Experiment data 0.9

Simulated results

0.8 0.7 0.6 0.5

0

10 20 30 Injection pore volume of oil (PV)

40

Figure 14-57. Comparison of the simulated results with the Experiment GP9 data of Minssiuex (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

544

Formation Damage by Organic Deposition

Ratio of average permeability to initial permeability (K/Ko, fraction)

1.0 0.9 0.8 0.7 Experiment data

0.6 0.5

Simulated results 0

100

200 300 400 500 600 Injection pore volume of oil (PV)

700

Figure 14-58. Comparison of the simulated results with the Experiment HMD26 data of Minssiuex (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

14.9.3 Simulation of Field-Scale Asphaltene Deposition by Pressure Depression in Oil Reservoirs during Primary Recovery For field-scale simulation of asphaltene deposition and its effect on oil productivity, Wang and Civan (2005c) incorporated their model presented in the previous section into the U.S. Department of Energy BOASTVHS three-dimensional and three-phase black-oil simulator (Chang et al., 1992). Then, they simulated a number of production scenarios involving vertical and horizontal wells completed in a horizontal, homogeneous, and isotropic reservoir at isothermal conditions. For illustration purposes, Wang and Civan (2005c) considered a reservoir formation located at 10,000 ft depth and having a 25 mD initial permeability and a 25% initial porosity. Initially the reservoir oil was undersaturated at 5283 psia pressure and 212 F temperature conditions. Its gravity was 19 API at the stock-tank conditions and bubble-point was 2050 psia. This oil contained 16.1% (by weight) asphaltene. According to the asphaltene precipitation chart given in Figure 14-59 by Wang and Civan (2005a,b), the onset of asphaltene precipitation pressure is 5056 psia. The precipitation amount increases by pressure depression below 5056 psia. The maximum asphaltene precipitation occurs at the bubble-point pressure. Thereafter, the precipitation amount decreases if the pressure decreases below the bubble-point. The simulation results

Formation Damage by Organic Deposition

545

Figure 14-59. Asphaltene Precipitation Curve (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

obtained by implementing four vertical wells and two horizontal wells in this reservoir are described as follows. Case 1 – Four Equally Spaced Vertical Production Wells

Wang and Civan (2005a,b) considered four equally spaced vertical production wells completed in a reservoir having 4200 ft by 4200 ft lateral dimensions and 50 ft thickness as shown in Figure 14-60. The flowing bottom pressure was assumed as 1000 psia, the initial productivity index

Figure 14-60. Reservoir with four vertical wells (Case 1) (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

546

Formation Damage by Organic Deposition

as 1.0 bbl/day-psi, and the wellbore radius as 0.5 ft for the wells completed in this reservoir. Because of the symmetrical positioning of the wells, Wang and Civan (2005c) presented the results for one well only in Figures 14-61–14-65 by taking advantage of the symmetry.

Figure 14-61. Simulated reservoir oil pressure profile at different production times for a vertical well (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

Figure 14-62. Simulated reservoir oil asphaltene precipitate concentration profile at different production times for a vertical well (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

Formation Damage by Organic Deposition

547

Figure 14-63. Simulated reservoir rock asphaltene deposit profile in reservoir at different production times for a vertical well (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

Figure 14-64. Simulated reservoir rock porosity profile at different production times for a vertical well (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

Their simulation results indicate that the vertical well began producing at a rate of 4000 bbl/day with a rapid decline to 100 bbl/day; the asphaltene deposition problem was not only limited to the near-wellbore region but it propagated the whole reservoir formation. However, the near-wellbore damage was more pronounced. Figure 14-66 shows the rapid productivity

548

Formation Damage by Organic Deposition

Figure 14-65. Simulated reservoir rock permeability profile at different production times for a vertical well (after Wang and Civan, ©2005 ASME; reprinted by permission of the American Society of Mechanical Engineers).

Figure 14-66. Simulated well productivity decline with production time for vertical and horizontal wells (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

index decline trend of the vertical well compared to that of the horizontal well discussed below. Case 2 – Two Equally Spaced Horizontal Production Wells

Wang and Civan (2005c) considered two equally spaced horizontal wells, as shown in Figure 14-67, in a reservoir of the same properties as

Formation Damage by Organic Deposition

549

Figure 14-67. Reservoir with two horizontal wells (Case 2) (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

described in Case 1. However, the lateral dimensions of the reservoir were taken as 4200 ft by 2100 ft. The flowing bottomhole pressure was assumed as 1000 psia, the initial productivity index as 17.5 bbl/day-psi, and the wellbore radius as 0.5 ft for the wells completed in this reservoir. Because of the symmetrical positioning of the wells, Wang and Civan (2005c) presented the results for one well only in Figures 14-68–14-73 by taking advantage of the symmetry. Wang and Civan (2005c) determined

Figure 14-68. Simulated reservoir oil pressure profile at different production times for a horizontal well (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

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Formation Damage by Organic Deposition

Figure 14-69. Simulated reservoir oil dissolved asphaltene concentration profile at different production times for a horizontal well (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

that the horizontal well began producing at a relatively high rate of 70,000 bbl/day. Consequently, the average reservoir pressure and the production rate decreased rapidly. Initially, the asphaltene deposited more within the wellbore region where the oil pressure became lower than the onset of asphaltene precipitation pressure (Figure 14-68). As emphasized by Wang and Civan (2005c), the maximum asphaltene precipitation and the lowest solubility conditions occur when the bubble-point is reached at various locations throughout the reservoir. Therefore, the maximum precipitation and lowest solubility conditions progress from the wellbore location to the rest of the reservoir as the pressure drops further with continued production (Figures 14-69 and 14-70). The asphaltene solubility increases and precipitation decreases in regions behind the bubble-point pressure front where the pressure is below 2050 psia bubble-point pressure. The highest asphaltene precipitation, and porosity and permeability impairment occurred at about a 100 ft distance from the wellbore (Figures 14-70–14-73). Figure 14-66 depicts that the productivity decline trend of a horizontal well is more favorable compared to that of a vertical well. The above-described simulation case studies carried out by Wang and Civan (2005a,b,c) revealed that the permeability impairment by asphaltene deposition lowered the production rate of a well rapidly towards the lowest economic production rate limit suggesting the abandonment of the

Formation Damage by Organic Deposition

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Figure 14-70. Simulated reservoir oil asphaltene precipitate concentration profile at different production times for a horizontal well (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 14-71. Simulated asphaltene deposit profile at different production times for a horizontal well (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

well in a short period of time. For example, the vertical well in Case 1 should be abandoned after 4 years if a 10 bbl/day production is considered as the economic production limit. On the other hand, the horizontal well in Case 2 should be abandoned after 3 years if a 100 bbl/day production is considered as the economic production limit for the horizontal well. Such practices will cause a substantial loss of productivity from the asphaltenic

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Formation Damage by Organic Deposition

Figure 14-72. Simulated reservoir rock porosity profile at different production times for a horizontal well (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

Figure 14-73. Simulated reservoir rock permeability profile at different production times for a horizontal well (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

oil reservoirs. Wang and Civan (2005c) state that “Although some stimulation treatments can be applied to enhance the production, not much oil can be recovered from the reservoirs because the formation has been damaged throughout the reservoir and the reservoir pressure is rather low. Once the formation has been damaged by asphaltene deposition, it is almost impossible to restore the productivity.”

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14.9.4 Preventing Asphaltene Deposition in Petroleum Reservoirs by Early Water-Injection Wang and Civan (2005c) emphasized that the maximum asphaltene precipitation and the lowest solubility conditions occur when the bubblepoint is reached and the reverse conditions are observed as the pressure declines to below or increases to above the bubble-point pressure (see Figure 14-59). Haskett and Tartera (1965) observed severe asphaltene deposition occurrence on vertical wellbores as the oil pressure approached the bubble point but deposition could be avoided by reducing the pressure sufficiently below the bubble point. However, such practice may neither be feasible to accomplish throughout a reservoir nor desirable. The same can be accomplished by maintaining a reservoir pressure above the onset of the asphaltene precipitation pressure as long as possible. Therefore, they recommend avoiding formation damage due to asphaltene deposition by applying an early water injection into such reservoirs. They have demonstrated the validity of this application by the following exercises. Case 3 – One Vertical Early Water-Injection Well and Three Vertical Production Wells

Three of the four production wells described in Case 1 were operated at a 5057 psia bottomhole flowing pressure, maintained higher than the onset of the asphaltene precipitation pressure. The remaining well was converted to a water injection well operated at 8000 psia bottomhole injection pressure. All four wells started operation at the same time. The economic limit of the production wells was set higher than Case 1 at a 100 bbl/day in the early-water injection scheme in view of the cost involved in the water injection well. The reservoir fluid pressure and the oil and water production trends are shown in Figure 14-74. Wang and Civan (2005c) determined that the early water-injection scheme yielded over three times the oil production obtained by the primary recovery described in Case 1. Case 4 – One Horizontal Early Water-Injection Well and One Horizontal Production Well

One of the two production wells described in Case 2 was operated at a 5057 psia bottomhole flowing pressure, maintained higher than the onset of the asphaltene precipitation pressure. The remaining well was

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Formation Damage by Organic Deposition

Figure 14-74. Simulated reservoir performance for Case 3 (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

converted to a water-injection well operated at a 8000 psia bottomhole injection pressure. Both wells started operation at the same time. The economic limit of the production wells was set higher than Case 2 at a 200 bbl/day in the early-water injection scheme because of the high cost involved in the water injection well. Figure 14-75 shows the reservoir fluid pressure and the oil and water production trends. Wang and Civan

Figure 14-75. Simulated reservoir performance for Case 4 (after Wang and Civan, ©2005 SPE; reprinted by permission of the Society of Petroleum Engineers).

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(2005c) determined that the early water-injection scheme produced more than twice the oil production obtained by the primary recovery described in Case 2.

Exercises 1. Based on the data given in Figure 14-45, determine the value of the empirical constant C5 in Eq. T1-1 given in Chapter 10, Table 10-1. 2. Based on the data given in Figure 14-46, determine the value of the empirical constant C6 in Eq. T1-2 given in Chapter 10, Table 10-1. 3. Apply Eq. (14-4) with the data given in Figures 14-20 and 14-21. Determine the values of various parameters of this equation from a least-squares fit to the linear plot of lnX vs. 1/T. 4. Prepare a three-dimensional plot of Eq. (14-5) showing the variation of the amount of asphaltene adsorbed on pore surface as a function of the amount of adsorbed asphaltene and concentration of asphaltene in oil. 5. Prepare a set of Langmuir isotherms based on Eq. (14-12) for a set of assumed parameter values. 6. Prepare a linear plot of the asphaltene adsorption data given in Figure 14-30 according to Eq. (14-12) and determine the values of the various parameters. 7. Consider Oil A containing 0.3 weight % asphaltenes and 0.1 weight % resins, and Oil B containing 12.5 weight % asphaltenes and 5.0 weight % resins. Which of these two different oils is likely to cause formation damage by asphaltene deposition? Explain the reason. 8. Analyze the data of Burya et al. (2001) given in Figures 14-32 and 14-33 to determine the parameters of their aggregation kinetics model described in this chapter.

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PART V Assessment of the Formation Damage Potential – Testing, Simulation, Analysis, and Interpretation

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LABORATORY EVALUATION OF FORMATION DAMAGE

Summary In this chapter, a brief review of the commonly practiced laboratory procedures and measurement techniques is presented. The evaluation of the core samples, preparation and characterization of the fluid samples, and experimental design, instrumentation, and measurement techniques are described. The laboratory procedures required for investigation of the various formation damage problems are discussed. Guidelines, program, and protocol for laboratory formation damage testing are provided. The application of these techniques and methodologies is demonstrated for determination of the formation damage potential of typical reservoir production alternatives.

15.1 INTRODUCTION Frequently, the formation damage potential of petroleum-bearing formations and methods of circumventing and remediation of formation damage are investigated by subjecting the reservoir core samples to flow at near in situ conditions in the laboratory. The scenarios planned for field applications are simulated in the laboratory under controlled conditions and the response of the core samples under these conditions are measured. The tests carried out over a range of variables yield valuable data and insight into the reaction of the core samples to fluid conditions and its effect on the alteration of the core properties. These data can be 559

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used for model-assisted analysis of the processes leading to formation damage. This exercise yields important information about the relative contributions of the various mechanisms to formation damage and help determine the values of the relevant process parameters. This information can be used to simulate the formation damage processes at the field scale. This, then, provides a valuable tool for quickly reviewing and screening the various alternative scenarios and optimizing the field applications to avoid or minimize the formation damage problems in the field. Development of meaningful laboratory testing and data interpretation techniques for assessment of the formation damage potential of petroleum-bearing formations under actual scenarios of field operations and for evaluation of techniques for restoration and stimulation of damaged formations are essential for efficient exploitation of petroleum reservoirs. Experimental systems and procedures should be designed to extract meaningful and accurate experimental data. The data should be suitable for use with the available analytical interpretation methods. This is important to develop reliable empirical correlations, verify mathematical models, identify the governing mechanisms, and determine the relevant parameters. These are then used to develop optimal strategies to mitigate the adverse processes leading to formation damage during reservoir exploitation. As expressed by Thomas et al. (1998), Laboratory testing is a critical component of the diagnostic procedure followed to characterize the damage. To properly characterize the formation damage, a complete history of the well is necessary. Every phase, from drilling to production and injection, must be evaluated. Sources of damage include drilling, cementing, perforating, completion and workover, gravel packing, production, stimulation, and injection operations. A knowledge of each source is essential. For example, oil-based drilling mud may cause emulsion or wettability changes, and cementing may result in scale formation in the immediate wellbore area from pH changes. Drilling damage in horizontal wells can be very high because of the long exposure time during drilling (mud damage and the mechanical action of the drill pipe on the formation face); thus, the well’s history may indicate several potential sources and types of damage.

For meaningful formation damage characterization, laboratory core flow tests should be conducted under certain conditions (Porter, 1989; Mungan, 1989):

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1. Samples of actual fluids and formation rocks and all potential rock– fluid interactions should be considered. 2. Laboratory tests should be designed in view of the conditions of all field operations, including drilling, completion, stimulation, and present and future oil and gas recovery strategies and techniques. 3. The ionic compositions of the brines used in laboratory tests should be the same as the formation brines and injection brines involving the field operations. 4. Cores from oil reservoir should be unextracted to preserve their native residual oil states. This is important because Mungan (1989) says that “Crude oils, especially heavy and asphaltenic crudes, provide a built-in stabilizing effect for clays and fines in the reservoir, an effect that would be removed by extraction.”

15.2 FUNDAMENTAL PROCESSES OF PRACTICAL IMPORTANCE FOR FORMATION DAMAGE IN PETROLEUM RESERVOIRS Formation damage in petroleum-bearing formation occurs by various mechanisms and/or processes, depending on the nature of the rock and fluids involved, and the in situ conditions. The commonly occurring processes involving rock–fluid and fluid–fluid interactions and their effects on formation damage by various mechanisms have been reviewed by numerous studies, including Mungan (1989), Gruesbeck and Collins (1982a), Khilar and Fogler (1983), Sharma and Yortsos (1987a,b,c), Civan (1992, 1994a,c, 1996a,b), Wojtanowicz et al. (1987, 1988), Masikevich and Bennion (1999), and Doane et al. (1999). The fundamental processes causing formation damage can be classified as following: 1. 2. 3. 4.

Physico-chemical Chemical Hydrodynamic Mechanical

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Laboratory Evaluation of Formation Damage

5. Thermal 6. Biological. Laboratory tests are designed to determine, understand, and quantify the governing processes, their parameters and dependency on the in situ and various operational conditions, and their effect on formation damage. Laboratory tests help determine the relative contributions of various mechanisms to formation damage. For convenience, the frequently encountered formation damage mechanisms can be classified into two groups (Amaefule et al., 1988; Masikevich and Bennion, 1999): (1) fluid–fluid interactions and (2) fluid–rock-particle interactions. The fluid–fluid interactions include (a) emulsion blocking, (b) inorganic deposition, and (c) organic deposition. The fluid–rock-particle interactions include (a) mobilization, migration, and deposition of in situ fine particles, (b) invasion, migration, and deposition of externally introduced fine particles, (c) alteration of particle and porous media properties by surface processes such as absorption, adsorption, wettability change, and swelling, and (d) damage by other processes, such as countercurrent imbibition, grinding and mashing of solids, and surface glazing that might occur during drilling of wells (Bennion and Thomas, 1994).

15.3 SELECTION OF RESERVOIR-COMPATIBLE FLUIDS Figure 15-1 by Masikevich and Bennion (1999) outlines the typical information, tests, and processes necessary for laboratory testing and optimal design, and selection of fluids for reservoir compatibility. Hence, Masikevich and Bennion (1999) classify the effort necessary for fluid testing and design into six steps: 1. Identification of the fluid and rock characteristics 2. Speculation of the potential formation damage mechanisms 3. Verification and quantification of the pertinent formation damage mechanisms by various tests 4. Investigation of the potential formation damage mitigation techniques 5. Development of the effective bridging systems to minimize and/or avoid fluids and fines invasion into porous media 6. Testing of candidate fluids for optimal selection.

Rock type Reservoir type Depth Temperature Pressure Porosity Permeability Pore throat size Wettability Saturations Swelling clays Mobile clays Acid gas Mobile particles Completion type History & other issues

Postulate on impairment mechanisms

Design and optimize to mitigate impairment mechanisms

Validate impairment mechanisms

Emulsion blocking

Y

Oil–Water test

Y

Best demulsifier

Precipitates

Y

Oil–Oil test Water–Water test

Y

Best wax, Scale, Alkalinity control

Migrating clays

Y

Critical velocity test

Y

Best cation

Swelling clays

Y

Clay swelling test

Y

Best cation or polymer

Phase trapping/ blocking

Y

Phase trap test

Y

Best alcohol, Oil, IFT reducers

Wettability alteration

Y

Wettability test APIRP 42

Y

Modify surfactant package

Solids invasion

Y

B A S E F L U I D

Design bridging system if required

Test whole fluid

Calcium Carbonate (acid soluble)

Threshold pressure Regain permeability

Sized salt (water soluble)

Leak off rate

Sized resin (oil soluble)

Effectiveness of stimulation

Fiber (for oil wells) Redesign is possible

Thin section petrography denotes severity of solids invasion

Laboratory Evaluation of Formation Damage

Identify characteristics of the rock–fluid system

Figure 15-1. Outline of fluid design and development strategies and the process to circumvent reservoir issues (after Masikevich and Bennion, ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

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15.4 EXPERIMENTAL SETUP FOR FORMATION DAMAGE TESTING The design of apparata for testing of reservoir core samples with fluids varies with specific objectives and applications. Typical testing systems include core holders, fluid reservoirs, pumps, flow meters, sample collectors, control systems for temperature, pressure or flow, and data acquisition systems. The degree of sophistication of the design of the core testing apparatus depends on the requirements of particular testing objectives, conditions, and expectations. High quality and specific purpose laboratory core testing facilities can be designed, constructed, and operated for various research, development, and service activities. Ready-made systems are also available in the market. Figures 15-2–15-4 by Doane et al. (1999) describe, respectively, the typical designs of a primitive system that operates at ambient laboratory temperature, and overbalanced and underbalanced core testing apparata that operate at reservoir temperature. The schematic drawing given in

Figure 15-2. Primitive drilling fluid evaluation system (after Doane et al., ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

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Reservoir fluid source High pressure N2 or air source

Flow control valve

Vacuum Mud

Injection piston

Pressure transducer

Coreholder

BPR1

Mud

Separator

Collection buffer

N2

Ruska pump

Wet test 1

BPR2

Oven To annulus pump

Wet test 2

Figure 15-3. Overbalanced reservoir condition fluid leak-off evaluation system (after Doane et al., ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum). High pressure N2 or air source

Flow control valve

Reservoir fluid source (at pres) Vacuum Mud Injection piston

Pressure transducer

BPR1

N2 Mud

Oven

Collection buffer

Constant pressure dispalcement pump (@ Pcirc < Pres)

Combined flow of mud & produced oil/gas

Coreholder

Seperator Wet test 1

BPR2 (at Pcirc) Wet test 2

To annulus pump

Figure 15-4. Underbalanced reservoir condition fluid leak-off evaluation system (after Doane et al., ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

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Laboratory Evaluation of Formation Damage

Figure 15-2 indicates that primitive core testing systems consist of a core holder, a pressure transducer controlling the pressure difference across the core, an annulus pump to apply an overburden pressure over the rubber slieve containing the core plug, a reservoir containing the testing fluid such as a drilling mud or filtrate, a displacement pump to pump the testing fluid into the core plug, and an effluent fluid collection container, such as a test tube. There is no temperature control on this system. It operates at ambient laboratory conditions. The schematic drawing given in Figure 15-3 shows the elements of a typical overbalanced core testing apparatus. This system has been designed for core testing at near in situ temperature and stress conditions, although other features are similar to that of the primitive system shown in Figure 15-2. The schematic given in Figure 15-4 shows the elements of a typical underbalanced core testing apparatus, which also operates at near in situ temperature and stress conditions. Core flood tests can be conducted in one-dimensional linear (Figures 15-2–15-4) and radial flow modes. Saleh et al. (1997) used cylindrical cores with holes and Tóth et al. (2005) used disk-shaped cores with holes for radial flow experiments. Bouhroum and Civan (1994) used a pie-shaped segment of a disk-shaped core as a representative element for radial flow experiments. Figure 15-5a–d by Saleh et al. (1997) show the schematic of typical radial flow models. Radial models better represent the effect of the converging or diverging flows in the near-wellbore formation. However, linear models are preferred for convenience in testing and preparation of core samples. The majority of the core flow tests facilitate horizontal core plugs because the application of Darcy’s law for horizontal flow does not include the gravity term and the analytical derivations used for interpretation of the experimental data is simplified. This approach provides reasonable accuracy for single-phase fluids flowing through small diameter core plugs. However, when multiphase fluid systems with significantly different properties and particulate suspensions are flown through the core plugs, an uneven distribution of fluids and/or suspended particles can occur over the cross-sectional areas of cores. This phenomenon complicates the mathematical solution of the governing equations necessary for interpretation of the experimental data. In particular, errors arise because, frequently, the transport phenomena occurring in core plugs are described as being one-dimensional along the cores. In order to alleviate this problem, it is more convenient to conduct core flow tests using vertical core plugs. Consequently, the gravity term is included in Darcy’s law

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567

Figure 15-5. Systems for horizontal wellbore studies: (a) core holder design, (b) overall productivity evaluation, (c) overall injectivity evaluation, and (d) drilling fluid evaluation (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Saleh, S. T., Rustam, R., El-Rabaa, W., and Islam, M. R., “Formation Damage Study with a Horizontal Wellbore Model”, pp. 87–99, ©1997; reprinted with permission from Elsevier Science).

based on the flow potential concept, but errors associated with uneven distribution of fluid properties over the cross-sectional area of the core ˇ nanský and Široký (1985) used a vertical plugs are avoided. Therefore, Cerˇ core holder. The dimensions of the core plugs are important parameters and should be carefully selected to extract meaningful data. Typically 1–2 in (2.54–5.08 cm) diameter and 1–4 in (2.54–10.58 cm) long cylindrical cores are used. The aspect ratio of a core plug is defined by the diameterto-length ratio. Small diameter cores introduce more boundary effects near the cylindrical surface covered by the rubber sleeve. This, in turn, introduces errors in model-assisted data interpretation and analysis when one-dimensional models are used, as frequently practiced in many applications for computational convenience and simplification purposes. On the other hand, short cores do not allow for sufficient distance for investigation of the effect of the precipitation and dissolution processes and depth of invasion (Fambrough and Newhouse, 1993; Gadiyar and Civan,

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Figure 15-5. (Continued )

1994; Doane et al., 1999). Longer cores are required for measurement of sectional or spatial porosity and permeability alteration. As described by Doane et al. (1999), a number of special purpose core holders have been designed. Figure 15-6 shows a single core for which only the pressures at inlet and outlet ports can be measured during fluid leak-off tests. This type of system is usually used with small core plugs. It only yields core response, integrated over the core length. It does not yield information about the formation damage distribution (or profile) along the core. As shown in Figure 15-7, long cores equipped with intermediate

Laboratory Evaluation of Formation Damage

569

Figure 15-5. (Continued )

pressure taps can provide information on sectional permeability alteration over the core length. Especially, core holders designed for tomographic analysis using sophisticated techniques, such as NMR, CAT-scan, and so on, may provide additional internal data. However, it is not always possible to obtain sufficiently long core plugs. In this case, several core plugs of the same diameter can be placed into a long core holder to construct a long core (20–40 cm long), and capillary contact membranes

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Laboratory Evaluation of Formation Damage

Figure 15-5. (Continued )

are placed between the core plugs to maintain capillary continuity (Doane et al., 1999). As emphasized by Doane et al. (1999), small diameter core plugs are not sufficient for testing of heterogeneous porous rocks. Therefore, full diameter core plugs have been used to alleviate this problem. But, Doane et al. (1999) warn, full diameter core plugs would not be representative when significant anisotropy exists between the horizontal and the vertical permeabilities, such as in typical carbonate formations. For the latter case, they recommend the core holder arrangement shown in Figure 15-8. In this system, the two opposing side surfaces of the core plugs are flattened

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Laboratory Evaluation of Formation Damage

Mud out

Mud in

Filter cake

Overburden pressure

Core plug Effluent filtrate Initial hydrocarbon flow direction

Regain hydrocarbon flow direction

Figure 15-6. Core holder design for drilling fluid leak-off evaluation systems (modified after Doane et al., ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum). BP

Gas Oil

Oil flow rate BP

T Core

PERF

Gas Oil

BP

Gas Brine CaCO3

T

LC

Brine

Pressure drop profile

Figure 15-7. Scale damage evaluation system allowing for measurement of sectional pressure drops (after Shaughnessy and Kline, ©1983 SPE; reprinted by permission of the Society of Petroleum Engineers).

572

Laboratory Evaluation of Formation Damage Oven

Transducer system Reservoir fluid source

Core

Extraction buffer

Constant pressure pump

To annulus pump

Gas

Mud

Oil

Water

Figure 15-8. Cross-flow fluid leak-off evaluation system for heterogeneous full diameter cores (after Doane et al., ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

by facing off and the fluid is flown over the side surface by means of a specially designed sleeve. This provides larger surfaces exposed to fluid to include the effect of the heterogeneous features of the core plugs. Obtaining and testing representative samples of fractured formations are difficult (Doane et al., 1999). Actual core samples containing natural fractures are preferred, but they are often difficult to obtain because core samples are usually poorly consolidated and may not include natural fractures (Doane et al., 1999). Then, a hydraulic fracturing apparatus can be used to prepare artificially fractured core plugs, as shown in Figure 15-9 by Doane et al. (1999). Figure 15-10 by Erna et al. (2003) is a schematic of an API cone/plate filtration cell used to perform static and dynamic filtration tests. For dynamic filtration, the cone is rotated to

573

Laboratory Evaluation of Formation Damage Confining sleeve Fractured core sample

Conventional radial collection head

Mud circulating head

Figure 15-9. Core holder design for fractured core flow evaluation systems (after Doane et al., ©1999; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

Air

Mud filling N2

Heat Mud 5° Paper or plug Back pressure Balance

Computer

Figure 15-10. Dynamic filtration cell (after Erna et al., ©2003 SPE; reprinted by permission of the Society of Petroleum Engineers).

generate the desired fluid shear effect over the surface of a core sample or filter paper. In static filtration, the cone is not rotated. For various purposes, frequently, the laboratory core tests are carried out using several core samples run either in series or in parallel. For example, the adverse effects of acidizing can be studied using sufficiently long core samples, the effect of which can be accomplished using cores in series. The effect of the permeability contrast on flooding

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Laboratory Evaluation of Formation Damage

of heterogeneous porous formations may be studied using different permeability cores run in parallel. Parallel core runs are also preferred for checking the reproducibility of the results and for testing different cores for different periods of time. For example, Figure 15-11 by Vasquez et al. (2003, 2005) describes an experimental system designed for simultaneous chemical treatment of three different core samples under near in situ pressure and temperature conditions. In this laboratory setup, the brine and mineral oil are first filtered out of their particulate matter, heated to the desired test temperature Ruska pump

Back pressure regulator

F

Filters Check valves

Polymer system

Oil

Water

Computer

Heating bath

Back pressure regulator

Pressure transducer

Nitrogen

Nitrogen

Date acquisition system

F

Cooler

Fluid catcher beaker

Sample collection beaker

Core 3

Core 2

Core 1

Pressure transducer

Back pressure regulator

Pump Fluid catcher beaker

Figure 15-11. Experimental Setup (after Vasquez et al., 2003, ©2003, Society of Petroleum Engineers Inc.).

Laboratory Evaluation of Formation Damage

575

using a coiled-tubing heating bath, and then fed into the three cores in a parallel flow mode. Several check valves are placed along the flow tubings to prevent the possibility of any backflow. An overburden pressure is applied around the rubber or silicone sleeves containing the core samples by pumping water or an appropriate fluid. Vasquez et al. (2005) recommend using sandpacks in Monel tubes for testing of chemical treatments at high temperatures, such as for the high-temperature conformance polymer systems. The silicon sleeves used under such conditions to maintain a proper overburden pressure often failed and had to be replaced frequently. The differential-pressure transducers are used to measure the pressure values at various locations throughout the experimental system. The temperature is measured using thermocouples and maintained at a prescribed value using temperature controllers. The system is equipped with a computer-aided data acquisition system. Typically, pressure difference across the cores, back pressures applied to the core samples, the various fluid temperatures, and the temperatures of the core samples are recorded as a function of time. This system has the capability of testing 1-in. diameter and 4-in. long core samples at temperatures ranging from the ambient to 350 F 177 C with about 200 psi (14 atm) overall pressure difference across the core samples. An overburden pressure of at least 500 psi (34 atm) over the injection fluid pressure is required during the core tests. The injection flow rates typically considered in the experiments are between 0.5 and 9.0 ml/min. Typically a 2% (by weight) KCl brine is used to prevent core damaging because of the interactions of the clay minerals present in the cores with the injected water. Vasquez et al. (2003, 2005) designed the experimental system shown in Figure 15-11 to enable operation in three phases: core conditioning, chemical treatment, and evaluation of the chemical treatment effectiveness. In Phase 1, the conditioning of the core samples to the reservoir fluid conditions is accomplished by flowing brine and mineral oil in the normal forward direction. To establish the reservoir fluid conditions in the test cores, first the core samples are saturated by injecting a 2% (wt) KCl brine and the absolute permeability of the cores are determined using the operation flow mode described in Figure 15-12a. Second, the water is displaced by injecting a mineral oil using the operation flow mode described in Figure 15-12b. Then, the residual oil saturation is established by injecting a 2% (wt) KCl brine using the operation mode described in Figure 15-12a. In Phase 2, Vasquez et al. (2003, 2005) applied the chemical treatment. The chemical treatment of the core samples is accomplished by injection of the treatment fluid in the reverse direction opposite to the normal flow

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Laboratory Evaluation of Formation Damage

Ruska pump

Polymer system

Water

Oil

Computer

Data acquisition system

Core 3

Core 2

Core 1

Cooler

Nitrogen

Nitrogen

F

(a)

Figure 15-12. Water Flow Diagram for: (a) Initial water saturation, water saturation to obtain residual oil saturation and evaluation of effectiveness of treatment. (b) Oil Saturation. (c) Polymer System Saturation (after Vasquez et al., 2003, ©2003, Society of Petroleum Engineers Inc.).

of the brine and oil from the reservoirs. Vasquez et al. (2003, 2005) explain that this operation mode, described in Figure 15-12c, can simulate the treatment of the production wells. In Phase 3, Vasquez et al. (2003, 2005) evaluated the effectiveness of the chemical treatment and its expected impact on the core samples. Typically, the effect of chemical treatment on the permeability of the core samples is determined by measuring the permeability by injecting a 2% (wt) KCl brine in the formal forward direction. Then, the percent change of permeability is calculated using the following formula based on the initial permeability Ki before treatment and the final permeability Kf after treatment: PCP = 1 − Kf /Ki  100

(15-1)

577

Laboratory Evaluation of Formation Damage Ruska pump

Polymer System

Water

Oil

Computer

Data acquisition system

Core 3

Core 2

Core 1

Cooler

Nitrogen

Nitrogen

F

(b)

Figure 15-12. (Continued )

15.5 RECOMMENDED PRACTICE FOR LABORATORY FORMATION DAMAGE TESTS∗ The guidelines provided by Marshall et al. (1997) for proper laboratory testing for formation damage investigations are presented in the following sections.

∗ Reproduced with permission of the Society of Petroleum Engineers from SPE 38154 paper by Marshall et al., ©1997 SPE.

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Laboratory Evaluation of Formation Damage

Ruska pump

F

Polymer system

Water

F

Data acquisition system

Core 3

Core 2

Core 1

Cooler

Nitrogen

Nitrogen

Oil

Computer

(c)

Figure 15-12. (Continued )

15.5.1

Introduction

The following procedure has been designed to provide a methodology for assessing formation damage in a variety of testing situations. Consequently, it is not rigorous in all areas and operator selection of the precise method or technique used is necessary at several points in the procedure. This procedure will however serve to minimize variability in test results if it is accurately followed and, if departures from this procedure are documented, it will help in the comparison of data derived from different tests. This procedure is not meant to provide detailed instructions on the use of the various pieces of equipment referred to. It is assumed that the reader will have a good working knowledge of the principles and practices involved in formation damage testing.

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When reporting results obtained using this procedure, details of all departures from these recommendations should be recorded in the final report along with details of methods used where more than one option is provided. 15.5.2 15.5.2.1

Core Preparation and Characterization Cutting and trimming the plugs

Plugs should be cut to give a minimum diameter of 1 in. (2.54 cm). Larger plugs, sized to fit particular core holders are preferable. The samples should have a minimum length of 1 in. (2.54 cm) and should be taken from the center of the core to minimize the impact of any coring fluid invasion. The plugging method and drill bit lubricant used during plugging will be determined by the state of preservation of the sample and the reservoir type. Cutting consolidated core. A standard core analysis rotary core plugger should be used with lubricant selected as below: a. Well-preserved core. If the core is well preserved then the appropriate fluid for the zone from which it is derived should be used; that is, formation brine or crude oil for an oil well and formation brine for a gas well (using gas as a lubricant may cause precipitation from formation brine inside the core). b. Poorly preserved core. If the core is poorly preserved and will require cleaning prior to testing, then the plugging fluid should be an inert mineral oil–fluids such as crude oil or formation brine should not be used in this case as they may cause precipitation within the core. Cutting unconsolidated core. One of the following two methods should be used depending on the state of the sample: a. Homogeneous sample. Plunge plugging, where a sharp-edged metallic cylinder is pushed into the soft rock, should be used. b. Inhomogeneous sample. If the sample contains hard nodules, cement patches, or lamination then a rotary plugger should be used with either compressed air (sample at ambient temperature) or liquid nitrogen (frozen sample) as a lubricant.

580

Laboratory Evaluation of Formation Damage

Trimming the plugs. After plugging, the samples should be trimmed (lubricant as for plugging) to give flat end faces perpendicular to the plug sides. The samples should be stored under the plugging fluid prior to cleaning or testing. The end faces of all plug samples should be cleaned of fines/rock flour generated during trimming. 15.5.2.2

Mounting and labeling the plugs

At this stage the samples should be encased in inert material leaving the end faces exposed, using PTFE tape, together with heat-shrink tubing. For poorly consolidated samples it may also be necessary to apply restraining grids to the plug end faces. The samples should be assigned a wellbore and a formation end face which are annotated on the side of the plug and not on the end faces. 15.5.2.3

Cleaning and drying

Once a cleaning and drying method has been selected, it should be identical for all samples in a particular study. The terms “Native State,” “Cleaned Sample,” and “Restored State” are defined as follows. Native state cores. Native State samples should not be cleaned and should be prepared to Swi (irreducible or connate water saturation) with the appropriate single-phase fluid using one of the methods described in the Section “Preparation to base saturation”. Non-native state cores. For core samples not in their Native State, cleaning is required. Cleaned samples may be obtained in one of the following ways: a. Samples which do not contain delicate or sensitive minerals can be cleaned in standard core analysis soxhlets using solvents or solvent mixtures and dried in a high temperature 90 C oven. Samples containing delicate or sensitive minerals such as illite or smectite should be cleaned by continuous immersion in cold static solvents and dried by critical point drying. If the equipment required is not available or if a faster technique is required, the samples can be cleaned using continuous immersion soxhlets (soxhlet extraction) and dried in a low temperature 60 C oven. Cleaned and dried samples should be stored in a desiccator prior to measurement of base parameters.

Laboratory Evaluation of Formation Damage

581

Note: Commonly used solvents for cleaning include toluene, xylene, methanol, chloroform, and acetone. There is no general agreement over which solvent is best in a given situation. b. Samples can be cleaned in core holders by flowing miscible solvents (or solvent mixtures). c. Samples can be “cleaned” by flowing alternatively light mineral oil (or crude oil if sufficient quantities are available) and formation brine until complete displacement of original fluids is achieved. The cleaning is ended by flowing oil to Swi and if necessary replacing the mineral oil with crude oil by miscible flooding at Swi . Reservoir temperature should be used. Restored State Samples. If a Restored State sample is required, this can be achieved by first of all following the procedures outlined in the Section “Nonnative state cores”, saturating the sample as described in Section 15.5.2.5, and then aging at reservoir conditions for an extended period of time (3–6 weeks). 15.5.2.4

Plug selection

Selection of duplicates. Prior to the flood tests, a sufficient number of duplicate plugs should be selected so that the entire test program can be conducted using essentially the same sample of rock. The following criteria are to be used during plug selection: a. Similar permeability (preferably within 20% as determined by Ka (or Ko for Native State samples) measurement b. Similar grain size/pore throat size distribution (determined by scanning electron microscopy (SEM), thin section, and possibly mercury injection of plug trims or carcass material) c. Similar composition/lithology (determined by X-ray diffraction analysis (XRD), SEM, thin sections of trims/carcass or computerized tomography (CT) scans of plugs) It is difficult to quantify the parameters in (b) and (c), and the comparative suitability of duplicates must be made using expert judgment. 15.5.2.5

Plug saturation

100% saturation is defined as being within 2% of base saturation.

582

Laboratory Evaluation of Formation Damage

Saturation with formation brine. Cleaned samples should initially be saturated with formation brine. This may be achieved using one of the following methods as determined by the cleaning process previously employed: a. If the samples have been cleaned using immersion methods then the dry samples are placed in a saturating vessel. Vacuum is applied and then formation brine is introduced and sufficient pressure is applied to ensure 100% formation brine saturation (as determined by weighing). b. If the samples have been cleaned by flowing solvents in a core holder ending with 100% methanol saturation (as determined by effluent composition) then the methanol can be exchanged by miscible flooding with formation brine to 100% water saturation (as determined by effluent composition). Gas phase should be eliminated by applying back pressure. Preparation to base saturation. Prior to the flood test, for both Native State and Restored State, the samples should be prepared to a defined saturation/capillary pressure. For both oil and gas wells, irreducible formation brine saturation should be used. There are various methods which can be used to achieve this saturation. These include porous plate, dynamic core holder, and ultracentrifuge. Due to the wide variety of core sample characteristics and test objectives, no one method can be recommended as being the most applicable for all situations. After preparation to base saturation, samples should be stored under the appropriate fluid and conditions prior to return permeability testing.

15.5.3 15.5.3.1

Fluid Preparation Simulated formation water

Simulated formation water (SFW) should be prepared using analytical grade inorganic salts to obtain the appropriate levels of the ions, as determined by elemental analysis, and then degassed. The SFW should be filtered to 045 .

Laboratory Evaluation of Formation Damage

15.5.3.2

583

Fluids used for initial and final permeabilities

Kerosene or inert mineral oil. Kerosene or inert mineral oil should be filtered to 045 . Formation brine. Formation brine, if available, should be filtered to 045  at reservoir temperature. Alternatively, SFW should be freshly prepared as discussed in the Section 15.5.3.1. Crude oil. It is usual to use dead crude but because it may contain a certain amount of produced water, this should be removed. The crude oil should be filtered using a 045- filter at a temperature above the wax appearance temperature. Gases. Oxygen-free nitrogen, filtered through a 045- filter, should be used. After filtration, the nitrogen should be humidified at inlet pressure conditions to prevent the sample from drying out during testing. 15.5.3.3

Wellbore fluids

Drilling fluid (whole mud). Drilling fluids to be used in return permeability testing should be as representative as possible. In the case of laboratory-prepared muds, they should contain all the components of the proposed formulation including weighting agents and contaminants and should be mixed according to standard API procedures where available. Laboratory muds should be artificially aged by hot-rolling for 16 hours at the relevant bottomhole temperature, prior to testing. These should also be passed through a mesh sieve to simulate mud conditioning where relevant (mesh size to reflect shale shaker screen sizes used in the field). In the case of field muds, the mud should be sheared on a Silverson mixer with the appropriate head (square hole emulsor screen for invert muds, 0-polymer head for water-based muds) for 5 minutes per laboratory barrel (350 ml) prior to use to ensure that the mud is in a representative state. Again, the mud should be passed through a mesh sieve to simulate mud conditioning where relevant. Drilling fluid (filtrate). Drilling fluid should be prepared as above, then filtered such that the filtrate obtained is representative of the filtrate lost to the formation through the filter cake.

584

Laboratory Evaluation of Formation Damage

Filtrate should be obtained either by centrifuging the sample or by using a High Temperature High Pressure (HTHP) fluid loss cell. The filtrate collected is then further filtered to approximately one-third of the average pore throat diameter of the reservoir core and should be used within 16 hours of filtering. Solids-free completion fluid. Solids-free completion fluids (e.g., brine, acid) should be prepared including all additives planned for the well. They should be filtered to the appropriate specification for that formation and/or field practice. Completion fluid containing solids. This section refers mainly to Loss Control Material (LCM). These fluids should be prepared such that the particle size distribution is representative of that expected downhole. Filtering of the prepared fluid is optional, depending on the type of effects to be measured.

15.5.4 15.5.4.1

Test Procedures Wellbore fluid placement

The prepared sample for evaluation should be loaded into a core holder capable of attaining reservoir net confining pressure and temperature ratings for the matching of reservoir in situ conditions. Pressures and flow rates should be continuously logged as functions of time. The core sample should be mounted in the horizontal position for analysis. The confining stress on the sample should be gradually increased while at the same time increasing the pore pressure of the fluid in place to maintain a net confining stress ratio equivalent to the in situ reservoir stress conditions. The rate of increase of net stress on the sample should not exceed 1000 psi (68 bar) per hour. The test apparatus and sample should be heated to an equivalent reservoir temperature. During heating, the pore pressure and confining stress should be adjusted to maintain initial conditions. Monitoring of the sample temperature and applied pore and confining pressures during this process is required to determine when reservoir conditions have been attained. The sample should be allowed to stabilize at the test temperature and pressure for at least 4 hours before testing begins.

Laboratory Evaluation of Formation Damage

585

Note: In order to prevent damage in the sample due to fines mobilization during flow testing it is recommended that a separate critical velocity test be performed to determine the flow rates which can be applied without causing permeability reductions due to fines migration. Preparation of the critical velocity sample should be the same as the preparation technique used for the test samples. 15.5.4.2

Initial permeability

Formation fluid should be flowed in the production direction (from “formation to wellbore”) by injection at constant rate. Where the critical velocity is not known, the flow rate should be as low as possible yet sufficient to generate a measurable pressure drop. Where the critical velocity is known for the test material, then the flow rate should be ≤50% of the critical rate. The differential pressure across the sample should be recorded. Particular regard should be paid to anomalies caused by mobilization of fine material within the test sample. The flow should be maintained until the pressure drop has stabilized and does not vary by more than 5% for a minimum of 10 pore volumes. Fluid flow is ceased once initial permeability is established. 15.5.4.3

Drilling fluid placement

Whole mud. To simulate well conditions, drilling fluid should be flowed over the “wellbore” face of the sample. The drilling fluid should be preheated prior to placement to match the bottomhole temperature. The drilling fluid should be applied to the sample face at the same overbalance pressure as in the reservoir and should be dynamically circulated over the face of the test sample for a minimum of 4 hours. Where comparative testing of mud on the formation is required the mud flow rate will be a constant for each mud type. During circulation the drilling fluid pressure and pore pressure should be recorded to ensure the values remain stable (less than 5% variation). During dynamic drilling fluid circulation the amount of fluid invasion into the test sample should be monitored at the “formation” end of the sample. The method of monitoring should be recorded. Invasion volume as a function of time should be recorded to allow the evaluation of spurt loss as the mud cake builds up and the effectiveness of the cake to prevent filtrate invasion into the test sample (leak-off).

586

Laboratory Evaluation of Formation Damage

Static drilling fluid placement should follow the dynamic placement. During the static placement the mud pressure should be maintained without flowing fluid over the “wellbore” face of the sample. The static placement should be for a minimum of 16 hours. As in the dynamic placement, recording of invasion volume as function of time measured at the “formation” face of the sample during the static placement is required to monitor mud cake performance. Following the static placement the mud should be dynamically circulated for a minimum of 1 hour. A fluid system that requires a wash or breaker fluid will need to have a step included into the test sequence in which placement and contact with this fluid are simulated. Mud filtrate.. Ten pore volumes of the mud filtrate should be injected at the reservoir temperature through the core in the “wellbore to formation” direction at 1 ml/min and the pressure differential measured. 15.5.4.4

Completion fluid placement

Solids-free completion fluid. Ten pore volumes of the solids-free completion fluid should be injected through the core in the wellbore to formation direction at the static reservoir temperature. The fluid should be injected at a rate similar to that used when establishing initial permeability (see Section 15.5.4.2). Fluid loss control pill. Fluid loss control pills should be exposed to the wellbore face at the appropriate overbalance. This should be carried out at static reservoir temperature for a representative amount of time as determined by the operation being simulated, but a 16-hour minimum is recommended. Fluid loss over time and differential pressure during the exposure period should be recorded as well as the filter cake thickness. 15.5.4.5

Cleanup treatments

Any cleanup treatment should be flowed over the wellbore face of the sample at a pressure differential appropriate to the reservoir conditions. Fluid should be allowed to flow through the core once breakthrough is achieved and the pressure differential should be maintained until a representative amount of fluid has been lost. A shut-in period either before or after circulating the fluid may be utilized if appropriate to simulate

Laboratory Evaluation of Formation Damage

587

field practice. The amount of fluid lost into the core over time and the differential pressure should be recorded. 15.5.4.6

Production simulation

After the placement of drilling fluid (and possibly other fluids) it is important to simulate a return to production in the “formation to wellbore” direction. This can be performed under conditions of constant pressure or constant flow rate. The permeability is determined after either of these methods. a. Determination of flow rate at constant pressure (drawdown). Drawdown should be performed by decreasing the pressure at the wellbore end of the plug and maintaining the formation end pressure at pore pressure allowing flow through the plug, mud cake, and mud. The pressure drop should simulate that to be used in the reservoir. Drawdown should be continued until constant flow rate is achieved. If this is not achieved, the fact should be included in the report. Pressure and flow rate should be measured throughout this procedure. Note: Where permitted by the core holder design, flow regimes incorporating flow across the face of the core should be used. b. Determination of differential pressure at constant flow rate. This procedure differs from (a) by using a constant flow rate and measuring the corresponding differential pressures across the core. The flow rate should be representative of the flux at the wellbore face. Flow should be continued until constant pressure is achieved. The pressure required for the initiation of flow should be recorded. 15.5.4.7

Determination of return permeability

The final or return permeability measurement (or measurements) can be made at two different stages: a. Immediately after Section 15.5.4.6, and when a stable flow rate/pressure drop has been achieved, the permeability of the tested core plug can be measured. b. After preparation of the core plug in the same way as for the initial permeability (see Section 15.5.2), the permeability can be measured. To determine the return permeability, repeat the procedure used in Section 15.5.4.2 to measure the initial permeability, ensuring that identical fluids and flow rates are used.

588

15.5.5

Laboratory Evaluation of Formation Damage

Reporting

A written report should be produced using Table 15-1 as a guide to composition. The report should contain all the section headings listed in the left-hand column of Table 15-1. The content of each section cannot be specified to cover all situations but suggested elements, which should be included to facilitate interpretation and use of the results obtained, are included in the right-hand column of the table.

15.6 PROTOCOL FOR STANDARD CORE FLOOD TESTS Van der Zwaag (2004) proposed a unified test protocol for conducting standard laboratory tests for assessment of the formation damage potential of fluids involving drilling and well operations. Figure 15-13 by Van der Zwaag (2004) describes the various tests carried out under the three sections of this protocol, namely “Information”, “Simulation”, and “Analysis”. Tables 15-2a–15-4 describe the details of the information gathered and reported as a result of the formation damage studies. Core flood tests can be conducted in various ways depending on the convenience and requirements of specific applications and/or the interpretation methods available. Core floods are usually accomplished under the conditions of constant pressure difference across the core plugs or constant flow through the core plugs. This makes convenient interpretation of the experimental data by means of a mathematical model. However, maintaining constant pressure difference and constant flow conditions in actual test conditions may be difficult and may not be truly accomplished, in spite of the use of high-quality equipment. Therefore, interpretation methods that can allow for variable flow conditions are preferred. Core flood tests can be conducted using single or multiple core holders. Multiple core holders may be run in series or parallel depending on the specific reasons. Parallel core holders may be required for various reasons. Similar core plugs in parallel may be flown simultaneously and certain formation damage effects, such as filter cake thickness, porosity, or precipitate quantity, can be measured separately and over different time periods for each core plug. Tests to establish the

Laboratory Evaluation of Formation Damage

589

Table 15-1 List of Elements to be Included in Report Section Headings

Suggested Elements to Be Included

Date of issue Reporting authors Type of formation material used in tests

• e.g., core/outcrop/synthetic

Objectives of the test and background

• • • • •

Mineralogy of material used in test

• Pore throat size and method of determination

Requesting party Well formation to be investigated Origin of formation material Origin of brines/crudes/fluids Project history used

• Results of XRD, SEM, and thin section analyses

before and after treatment(s) Characteristics of core plug

• Preserved or restored; if restored, method used • • • • •

Fluids used in test

for core restoration Cutting method and lubricants used Water saturation, Swi Method of preparation to base saturation Aging time (restoration) Cleaning methods (details to include equipment, solvents, and procedures used)

• Muds: full compositional information and

preparation methods, field or laboratory source, details of additional solids/contaminants • Brines: chemical analysis • Oil: crude (live/dead), synthetic • Gas: type, humidified or not Test conditions

• Plug dimensions and orientation relative to

bedding • Surface area of plug face exposed to fluids • Core holder dimensions and rig schematic

(including diagram) • Core and fluid temperatures (e.g., if lower than

core temperature during dynamic phase) • Pore pressure • Overburden pressure • Overbalance pressure (table continued on next page)

590

Laboratory Evaluation of Formation Damage Table 15-1 (Continued)

Section Headings

Suggested Elements to Be Included • Drawdown pressure/flow rate • Volume of fluid produced during drawdown • Exposure period (including dynamic/static

periods) • Flow rate during dynamic period

(if applicable) • Shear rates at core face during dynamic period

(if applicable) Results

• • • • • • • • • • • • •

Core plug reference Depth taken (if known/applicable) Fluids used in test Test sequence (e.g., flood/drawdown/ breaker drawdown) Permeability, initial and final Pressure drop across core with time Flow rate with time Cumulative volume flowed with time Viscosity of fluids Fluid loss vs. time Temperature and pressure at which fluid collected Breakthrough pressure (if applicable) Comments on test by investigator

Deviations from recommended practice Interpretation Reproduced with permission of the Society of Petroleum Engineers from SPE 38154 paper by Marshall et al., ©1997 SPE.

effect of the permeability contrast, such as for conformance control studies, require floods using parallel core plugs with different permeabilities (Prada et al., 2000; Vasquez et al., 2003, 2005). Core plugs may be run in series to simulate the effect of formation damage over long distances. Core flood tests can be conducted in two ways: (1) interrupted core floods and (2) continuous core floods (Haggerty and Seyler, 1997). In interrupted core flood tests, the fluid injection is interrupted at certain time intervals and permeability is measured. In continuous core flood tests, the effective permeabilities are measured during the injection process.

591

Laboratory Evaluation of Formation Damage Fluids analysis

Fluids preparations

Simulation

Core analysis

Advanced analysis methods

Drilling and trimming of core plugs Labelling of core plugs Preparation of formation fluids used for core plug saturation and formation damage testing

Cleaning and drying of core plugs

Viscosity and density of formation fluids at relevant temperatures

Sample weight, air permeability, bulk volume by mercury immersion, helium porosity, grain density

Pore size distribution mineralogy, SEM, thin section

Sample weight saturated and bouyant state, bulk, pore and grain volume, bulk density, porosity, grain density

NMR relaxation, NMR imaging

Initial saturations of mobile and immobile fluid phase, sample weight

NMR relaxation, NMR imaging, cryogenic SEM

Saturation with formation brine

Establishment of base saturation

Selection and preparation of the formation damage test apparatus Mounting of the core plug Loading of the core plug sample for baseline production simulation Heating of the formation damage test apparatus Well fluids preparation

Baseline production simulation

Well fluids density, rheological parameters well fluid composition

Baseline permeability Charging of the formation damage apparatus with the well fluid Preparations for filtrate production (fluid loss, fluid leak off) measurements

Well fluid solids particle size distribution

Filtrate production sampling

Establishment of well fluid overbalance pressure and start of formation damage test Accumulated filtrate production volumes vs. Time record of pressures and temperature

Viscosity and density of produced filtrate at relevant temperature Termination of the formation damage test New well fluid for compatibility testing? Yes No Preparations for well production simulation by return flood Production simulation (return flood)

Return permeability

After test well fluid sampling

New well fluid for compatibility testing? Yes No Depressurization and cooling off of the core plug sample

Well fluids density, rheological parameters, solids particle size distribution, well fluid composition

Sample weight, visual characterisation Preparations for post exposure analysis Saturations, air permeability helium porosity

NMR relaxation, NMR Imaging, SEM, cryo SEM, thin section analysis

Fields with double lined boarders indicate procedural elements that require decisions regarding several alternatives Fields with dotted lined boarders indicate minimum requirement analysis methods for “screening” type formation damage tests Fields filled grey indicate additional analysis for “diagnostic” type studies

Figure 15-13. Flow scheme for formation damage testing (Simulation and Analysis section) (after Van der Zwaag, ©2004 SPE; reprinted by permission of the Society of Petroleum Engineers).

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Laboratory Evaluation of Formation Damage

Table 15-2a Tables for Exchange of Information from Field Team to Laboratory TABLE 1 – GENERAL INFORMATION

TABLE 2 – WELL FLUID INFORMATION

8. Type of FD-problem 1. Client Single well fluid Contact Fluid compatibility, several well 2. Field under investigation fluids Well(s) 9. Type of test fluids Formation(s) Drilling in reservoir 3. Origin of core material Cased and perforated hole Coring depth Open hole Storage time and conditions Sand screens 4. Type of core material (coring method) Conventional Gravel pack Plastic/aluminum liner Others Completion fluids Gel Cement “Sponge” Soaps Other 5. Type of coring fluid Displacement brine Base fluid Kill fluid Weight material Others Viscosifier Frac fluid Fluid loss reducers Stimulation fluids HCl Emulsifiers HF Other chemicals Others 6. Origin of fluid samples 10. Composition and properties of well Formation water fluids Oil and gas Fluid composition Well fluids Rheology 7. Type of FD-assessment API/HTHP filter loss properties Pre-analyses Filtrate composition Post- (damage) analysis Filtrate rheology TABLE 3 – AVAILABLE ROCK AND RESERVOIR FLUID INFORMATION

TABLE 4 – DOWNHOLE OPERATIONS CONDITIONS

11. Lithology Consolidated SS Poorly consolidated SS Unconsolidated SS Carbonate Chalk 12. Appr. Permeability 13. Mineralogy and Petrophysics XRD SEM

18. Temperature and pressure conditions Reservoir temperature Dynamic downhole temperature Overburden pressure Reservoir pore pressure 19. Well geometry, time aspects, and pump rate Hole diameter Length of reservoir section

Laboratory Evaluation of Formation Damage

593

Table 15-2a (Continued)

14. 15.

16.

17.

Thin sections Pore size distributions Others Sensitive clays Reservoir zone Gas zone Oil zone Water zone Transition zone Formation water composition Calcium Barium Strontium Metals Sulphate Carbonate Oil, gas, condensate composition Wax/Paraffin content Bitumen/Pyrobitumen Asphaltenes

BHA dimensions Drill pipe dimensions ROP Interruptions (e.g. Logging, WOW) Total open hole time Pump rate 20. Production data Production drawdown pressure Production rate Drainage radius Flow boundaries

Table 15-2b Tables for Exchange of Information from Laboratory to Field Team TABLE 5 – CHARACTERISTICS OF CORE MATERIAL AND RESTORATION METHODS

TABLE 6 – WELL FLUID-TO-RESERVOIR SAMPLE EXPOSURE CONDITIONS

21. Preliminary Characterisation CT-Scan of reservoir core (y/n) Well preserved Dried out, mechanically damaged Homogeneous, inhomogeneous Consolidated, unconsolidated 22. Cutting of plugs Cutting method and lubricants Cutting direction in relation to bedding planes Trimming and endface preparations 23. Preparations “Native State” “Fresh” Cleaned and restored

26. Core holder design Axial/linear flow geometry Radial flow geometry Short core D  L < 1  3 Long core D  L > 1  3 Multiple pressure ports Tandem set-up Other design features 27. Test conditions Core temperature Static well fluid temperature Well fluid temperature during circulation Confinement pressure Pore pressure (table continued on next page)

594

Laboratory Evaluation of Formation Damage Table 15-2b (Continued)

TABLE 5 – CHARACTERISTICS OF CORE MATERIAL AND RESTORATION METHODS

TABLE 6 – WELL FLUID-TO-RESERVOIR SAMPLE EXPOSURE CONDITIONS

24. Cleaning and restoration Soxhlet-solvent extraction Solvent flood (Hot flush) Solvent flood (Cold flush) Mineral oil flood (Hot flush) Mineral oil flood (Hot and viscous) 25. Restoration to Swi By flooding By centrifuge By porous plate method With mineral oil With high viscosity mineral oil With reservoir oil (dead) With reservoir oil (live) With humidified nitrogen With reservoir gas aging Aging

Well fluid pressure and overbalance Injection rate (clear fluids only) 28. Filtration conditions Dynamic filtration conditions Circulation rate Calculated shear rate Static filtration conditions Exposure time 1. dynamic 2. static as function of throughput 29. Production Simulation Constant rate Constant p (drawdown)

TABLE 7 – FLUIDS ANALYSIS

TABLE 8 – CORE ANALYSIS

30. Pore fluids Viscosity of pore fluids at p, T Density of pore fluids at p, T 31. Laboratory well fluids Fluid composition Rheology API/HTHP filter loss properties Filtrate composition Filtrate rheology Solid particle size distribution Other information from fluid supplier 32. Filtration monitoring Total filtration volume/mass Filtration volume/mass vs. Time Calc. static filtration coefficients Calc. dynamic filtration coefficients 33. Effluent analysis Composition Density/viscosity Particle size distribution

34. Conventional core analysis measurements Porosity Air permeability Brine permeability 35. Baseline and return permeability In formation damage core holder At reservoir temperature At room temperature In separate core holder After aging 36. Saturation determination Dean Stark Extraction Nuclear Magnetic Resonance (BVI/FFI) 37. Petrophysics Nuclear Magnetic Resonance T1 or T2 NMR-Imaging (M0, T1, T2) Computed Tomography Scanning Others

Laboratory Evaluation of Formation Damage

595

Table 15-2b (Continued) Particle composition analysis Bacteria Oil in Water

38. Petrographical methods Scanning Electron Microscopy (SEM) Cryogenic SEM Thin sections analysis Backscattered Electron Microscopy others 39. Mineralogy/Chemistry X-Ray Diffraction X-Ray Fluorescence Others

After Van der Zwaag, ©2004 SPE; reprinted by permission of the Society of Petroleum Engineers.

15.7 LABORATORY PROCEDURES FOR EVALUATION OF COMMON FORMATION DAMAGE PROBLEMS The laboratory procedures required for evaluation of common formation damage problems are described in this section according to Keelan and Koepf (1977). They classified the frequently encountered formation damage problems into four groups: 1. The blocking of pore channels by solids introduced by drilling, completion, workover, or injection fluids 2. Clay–water reaction that yields clay hydration and swelling, or clay particle dispersion and pore plugging by movement with produced or injected water 3. Liquid block that normally is caused by extraneous water introduced into the formation at the wellbore during drilling, coring, completion, or workover 4. The caving and subsequent flow of unconsolidated sands into the wellbore.

15.7.1

Evaluation of Liquid Block Problem

As explained in Figure 15-14 by Keelan and Koepf (1977), “liquid block reduces effective permeability to the hydrocarbon.” Before damage, the original mobile water saturation range is in 020 < Sw < 080.

596

Table 15-3 Principle Well Fluid Exposure Methods Type of Well Fluid

Filtration Regime

Core Holder Features

Scope

Not recommended for

Clear brines or filter-cake-building fluids, which are placed in the well

Hydrostatic core holder Static filtration

Semidynamic filtration

Well fluids operated under dynamic filtration conditions (e.g., drilling Dynamic fluids, fracturing fluids) filtration

Well end of the core plug is provided with a spacer ring Hydrostatic core holder Well end of the core plug is provided with a spacer ring and a flow head or stirrer.

Verification of formation damage mechanisms related to static filtration and filtrate penetration Testing brittle, ductile, or unconsolidated rock materials.

As in static filtration Tests where agitation of fluids is required for extended testing duration Extended fluid stability testing

Long-term testing Well productivity estimates for fluids outside the fluid type specifications

Well productivity estimates for fluids outside the fluid type specifications

Core holder with open core Dynamic leak-off measurements plug end (well end) Open well end is connected Well productivity estimates to a flow channel with defined geometry and a fluid circulation system.

Well fluid compatibility studies including drilling or fracturing fluids

After Van der Zwaag, ©2004 SPE; reprinted by permission of the Society of Petroleum Engineers.

Mechanically weak rock materials

Laboratory Evaluation of Formation Damage

Well fluids operated under static filtration conditions (e.g., stimulation fluids, kill pills, workover fluids)

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Laboratory Evaluation of Formation Damage

Table 15-4 Formation Damage Measurement Parameters on Different Scales Core plug scale or Centimetre (cm) scale:

– Filtration parameters. – Flow energy parameters (permeability). – Fluid phase saturations and wettability.

Intermediate scale or Millimetre (mm) scale:

– Solid particle distributions – Pore fluid saturation distribution – Filtrate distribution

Pore scale or Micrometer m scale:

– – – – –

Pore size or particle size distributions. Products of rock–fluid interaction, Products of fluid–fluid interaction, Filter cake texture, Identification of well fluid additives and their location in the core plug after exposure.

After Van der Zwaag, ©2004 SPE; reprinted by permission of the Society of Petroleum Engineers. 1.0

Relative permeability : Fraction

.9

A

.8

Irreducible water

Residual oil

.7 Kro =

.6

KO KA

.5 .4 .3

D

.2

Krw =

.1 0

KW KA

c A

0

10

20

D

30

40

50

60

c 70

B

E

Damage by clay hydration or movement

B

80

90

100

Water saturation : Percent pore space

Figure 15-14. Effect of water block on relative permeability (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

After extraneous water incompatible with the formation invades the porous media, the irreducible water saturation raises to about 34% from its original value of 0.20. Hence, the line AA′ shifts to the line DD′ .

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Laboratory Evaluation of Formation Damage

Consequently, the relative permeability of the oil at the irreducible water saturation decreases from the original 0.9 value to about 0.3, which amounts to a threefold reduction of permeability to the hydrocarbon. On the other hand, damage by clay hydration and fine particles plugging caused by incompatible extraneous fluid invasion increases the residual oil from its original 02 Sw = 08 to about 026 Sw = 074. Hence, the line BB′ shifts to the line CC′ . Consequently, the permeability to water also decreases. As a remedial action, Keelan and Koepf (1977) recommend treatments inferred by the capillary pressure equation: pc =

2  cos r

(15-2)

where pc denotes the capillary pressure necessary for water retention,  is the surface tension between water and hydrocarbon, is the contact angle between the water and hydrocarbon, and r is the pore radius. Equation (15-2) indicates that water retention can be reduced by workover schemes reducing the surface tension and/or increasing the contact angle to favor a less water-wet condition.

15.7.2 Evaluation of Drilling Mud Damage Potential and Removal Keelan and Koepf (1977) explain that drilling muds contain solid particles that form a filter cake over the wellbore wall, the filter cake restricts the mud flow into the near well bore formation, but some filtrate and fine particle invasion are unavoidable and usually occurs. The filtrate may react with the resident formation clays causing clay swelling, mobilization, and migration. The released particles and the fine particles carried into the formation by the filtrate can plug the pores and reduce the permeability of the formation. The water-based filtrates increase the irreducible water saturation and create water block and hydrocarbon permeability reduction. As depicted in Figure 15-6 by Doane et al. (1999) or Figure 15-10 by Erna et al. (2003), the face of a core sample is exposed to mud under a pressure difference across the core. As described by Keelan and Koepf (1977), test sequences can be conducted with and without the presence of mobile hydrocarbons in core plugs.

Laboratory Evaluation of Formation Damage

15.7.2.1

599

Tests in absence of mobile oil

Figure 15-15 by Keelan and Koepf (1977) delineates the test sequence without the presence of mobile hydrocarbons and shows the equations used to determine the magnitude of formation damage or remediation. Keelan and Koepf (1977) explain that “This test indicates impairment of

Low K

Same as medium K

% Mud damage K – KD × 100 = A KA

Medium K

High K

A Formation brine K

Mud-off

B

Same as medium K

Note pore volumes of filtrate required, rate of fluid loss, and pressure drop

Mud scape % Acid improvement =

KF – KA × 100 KA

Formation brine K

D

E Acidize

Damage due to 1. Mud solids 2. Clay hydration and/or movement

May require preflush, acidization and afterflush

F Formation brine K

Figure 15-15. Test sequence without the presence of mobile hydrocarbons (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

600

Laboratory Evaluation of Formation Damage

productivity by clay hydration and movement of fines into the formation during the drilling operation, and any benefit of the fines’ removal when the well flow in a reverse direction into the wellbore.” The core plug is saturated with the brine to be tested and may or may not contain irreducible, immobile oil. Hence, the water-block effect is eliminated because the water saturation is constant. During these tests, the filtrate volume or rate vs. the filtration time is measured until mud-off. If the experimental design permits, the filter cake properties, such as porosity, permeability, and thickness, and the effluent fines and liquid volumes should also be measured. The pressure difference applied to the core plug should be determined by scaling from the planned drilling over balance pressure (Keelan and Koepf, 1977). 15.7.2.2

Tests in presence of mobile oil

Figure 15-16 by Keelan and Koepf (1977) delineates the test sequence with the presence of mobile hydrocarbons and shows the equations used to determine the magnitude of formation damage or remediation. They explain that this test indicates “the water-block potential of a formation.” In this test, the water saturation varies and is calculated by measuring the effluent filtrate volume. Any permeability reduction remaining, after the production of all the injected, extraneous filtrate water, is attributed to clay hydration and/or mud-solids invasion (Keelan and Koepf, 1977). 15.7.2.3

Determination of damage potential

Figure 15-17 by Keelan and Koepf (1977) depicts the results of the evaluation tests of two muds, referred to as Muds A and B. Figure 15-17 indicates that Mud A causes more damage than Mud B. In the case of Mud A, the return permeability is only 6% of the initial permeability, while it is 54% for Mud B. Keelan and Koepf (1977) conducted evaluation tests for two different drilling mud fluids, specially prepared for stabilizing the formation to avoid formation damage during water flooding. Keelan and Koepf (1977) used fresh cores containing irreducible oil. They recommend running tests with the presence of irreducible oil because they explain that “The presence of residual oil, or associated organic compounds, sometimes protects clay surfaces, making them less sensitive to alteration when contacted by incompatible brines.” They injected coarsely filtered mud filtrates (thus containing fine particles) into core plugs and measured the permeability

Laboratory Evaluation of Formation Damage Low K

Same as medium K

Medium K

Formation brine K

601

High K

A

Ko or Kg at irreducible Sw

Same as medium K

B

% Mud damage K – KE × 100 = B KB

C Mud-off

% Acid improvement =

KG – KB × 100 KB

D Mud scape

Ko or Kg

Acidize

Ko or Kg

E

F

Damage due to 1. Mud solids 2. Clay hydration and/or movement 3. Relative permeability (water block)

G

Figure 15-16. Test sequence with the presence of mobile hydrocarbons (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

impairment. Figure 15-18 by Keelan and Koepf (1977) presents the results of injecting formation brine, filtrate, and injection brine samples into the core plugs. As can be seen, the effective permeability is 30% higher for KCl mud filtrate compared to that of lignosulfonate mud filtrate.

602

Laboratory Evaluation of Formation Damage 2.0 1.8

Brine permeability : Fraction of original

1.6 1.4 Mud-off

1.2 1.0 Formation brine (prior to mud-off)

.8

Formation brine (after mud-off)

.6

∆P = 300

∆P = 150

Mud B

.4 .2 0

∆P = 40 ∆P = 350 0

20

40

60

80

100

∆P = 500 120

140

160

Mud A 180

200

Cumulative injection : Pore volumes

Figure 15-17. Mud damage evaluation (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

15.7.2.4

Damage removal

For purposes of remediation treatment after damage, Keelan and Koepf (1977) evaluated two types of acids: (1) “a regular mud acid containing about 7% inhibited HCl and a low surface-tension agent,” and (2) a mud acid “composed of 3% HF and 12% HCl.” They explain that “Recommended use of the HF acid required a preflush with 15% HCl, injection of the HF acid, and an after flush with diesel oil containing 20% of a mutual solvent.” Based on the results presented in Figure 15-19, Keelan and Koepf (1977) summarized their interpretation of the acid treatment results as following: 1. Similar permeability reduction to each filtrate was noted in these test cores. 2. In the cores contacted with KCl filtrate, HF acid yielded 136% higher permeability to injection brine than did the regular mud acid, and

603

Laboratory Evaluation of Formation Damage 2.0 1.8

KCI filtrate damage

1.4

Lignosulphonate filtrate damage

1.2 1.0

Inj. brine *

Filtrate *

.2

Formation brine *

.4

Injection brine *

.6

Filtrate *

.8 Formation brine *

Liquid permeability : Fraction of original @residual oil

1.6

0 * Samples contain residual oil

Figure 15-18. Permeability to injection brine following exposure to KCl and lignosulfonate mud filtrate (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

resulted in a net permeability enhancement above initial. The regular mud acid was not effective, and final permeability to injection brine was no higher than when the acid wash was not used. 3. In the lignosulfonate-contacted cores, the regular mud acid and the HF acid were equally effective, and each yielded a permeability greater than the original. 4. In summary, either mud was suitable if HF acid was used for remedial treatment. If the regular mud acid was to be used, the most suitable drilling mud would be lignosulfonate.

15.7.3

Evaluation of Hydraulic Fracturing Fluids

As explained by Keelan and Koepf (1977) fracturing fluids cause formation damage by water-block, solids invasion associated with fluid leak-off, and clay hydration in the near-fracture formation. Therefore, it is important to use compatible fluids and fluid-loss additives. Hence, they

604

Laboratory Evaluation of Formation Damage 2.2 1.8

KCI filtrate damage & recovery by acidization

Lignosulphonate filtrate damage & recovery by acidization

1.4 1.2

1.0

Sample identification

Injection brine @ sor

Formation brine @ sor

Acidize

Filtrate

Formation brine @ sor

0

Inj. brine @ sor

.2

Formation brine @ sor

.4

Acidize

.6

Filtrate

.8

Formation brine @ sor

Liquid permeability : Fraction of original @residual oil (sor)

1.6

Regular acid Hydrofloric (HF)

Figure 15-19. Permeability improvement by HF and HCl acid treatment following exposure to KCl and lignosulfonate mud filtrate (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

recommend performing tests on core samples extracted from the reservoir formation, in which fractures will be created. In these tests, the spurt loss, fluid-loss coefficient, effect of additives, acid solubility of formation, and fines release with the acid reaction are typically determined (Keelan and Koepf, 1977).

15.7.4

Evaluation of Workover and Injection Fluids

These tests indicate the incompatibility of clays with the extraneously introduced water, including filtered formation brine and filtered mud filtrate without the presence of mobile hydrocarbons (Keelan and

Laboratory Evaluation of Formation Damage

605

Koepf, 1977). Such tests can also be used to evaluate the effectiveness of clay stabilizers added to workover and injection fluids (Keelan and Koepf, 1977). Keelan and Koepf (1977) state that “Use of filtered workover fluids removes plugging solids and results in evaluation of damage resulting from clay swelling and/or clay-particle movement.” The rock–water system is considered compatible when the formation permeability does not decrease by fluid injection. Keelan and Koepf (1977) state that “The clays damage productivity either by swelling in place or by release from their anchor point and subsequent movement to block pore channels. The inclusion of certain ions in workover and injection fluids often offers a relatively inexpensive and effective stabilization of the clays and prevention of productivity impairment.” Figure 15-20 by Keelan and Koepf (1977) presents the test sequence and the equations necessary for determining formation damage for evaluation of the compatibility of the injection and workover fluids with the formation clays. Figure 15-21 by Keelan and Koepf (1977) shows the results of injecting brines with and without KCl and CaCl2 addition. Injecting a brine, rather than the formation brine, into a core sample A reduced the permeability to 50% of its formation brine permeability. Injecting a brine containing 100 ppm KCl into a core sample B doubled its formation brine permeability. However, injecting a brine containing 100 ppm CaCl2 reduced the permeability to 50% of its formation brine permeability. Figure 15-22 by Keelan and Koepf (1977) shows that consecutively decreasing concentrations of KCl and CaCl2 in the injected brines yields permeabilities above the initial formation brine permeability. Keelan and Koepf (1977) concluded that KCl treatment is favorable even though the data appear unusual. Keelan and Koepf (1977) recommend the water–oil relative permeability measurements as a practical approach to damage assessment in core plugs. Keelan and Koepf (1977) express that the fluids compatible with the core material should typically yield the relative permeability curves similar to those shown between the AA′ and BB′ lines in Figure 15-14. However, Keelan and Koepf (1977) explain that, when a filtered injection brine is injected into a core containing irreducible oil, a specific value of the water relative permeability, denoted by Point E in Figure 15-14, is obtained. This particular value represents the water relative permeability at the injection-wellbore formation face, whereas, the water relative permeability at a sufficiently long distance from the well bore is represented by Point B.

4

Formation brine K

Formation brine K

Formation brine K

Formation brine K

A

B 5% CaCl2

5% KCl

10,000 ppm brine

5% CaCl2

Low ppm

9

A

5% KCl

10,000 ppm brine

B 5% KCl

5,000 ppm brine 2,500 ppm brine

8

10,000 ppm brine

5% CaCl2

5,000 ppm brine

5,000 ppm brine

Damage factor % K – KB × 100 = A KA

2,500 ppm brine B Low ppm

Damage due to 1. Clay hydration and/or movement

2,500 ppm brine Low ppm

Figure 15-20. Test sequence for evaluation of injection and workover fluids using filtered fluids in the absence of mobile hydrocarbons (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

Laboratory Evaluation of Formation Damage

Formation brine K

A

7

6

5

Formation brine K

3

Formation brine K

2

Formation brine K

1

High K

Medium K

Formation brine K

Low K

606

Sample number

607

Laboratory Evaluation of Formation Damage 2.0

1.6

Sample A

Sample

Sample

B

C

1.4 1.2 1.0 .8

0

100 ppm CaCl2 *

Formation brine

100 ppm KCl *

.2

Formation brine

.4

Injection brine

.6 Formation brine

Liquid permeability : Fraction of original

1.8

* Mixed with injection brine

Figure 15-21. Permeability to injection brine with and without KCl and CaCl2 addition (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

15.7.5 Evaluation of Workover Damage and Remedial Chemicals Figure 15-23 by Keelan and Koepf (1977) describes the testing schemes and equations necessary for determining the damage and evaluation of remedial chemical treatment of water block in the presence of mobile hydrocarbons. Keelan and Koepf (1977) facilitate chemicals reducing surface tension to remove the water forming the water-block. 15.7.6 Determination of Critical Interstitial Fluid Velocity and pH for Hydrodynamic Detachment of Fines in Porous Media Flow Rate Sensitivity Test The drag force acting upon a fine particle attached to the pore surface is proportional to the interstitial velocity and viscosity of the fluid and

608

Laboratory Evaluation of Formation Damage 2.0

Sample D Sample E

Test sequence

1.8

ppm KCl ppm CaCl2

1.4 1.2

1.0

Injection water

20 ppm *

50 ppm *

.2

100 ppm *

.4

1,000 ppm *

.6

10,000 ppm *

.8

Formation brine

Liquid permeability : Fraction of original

1.6

0

* Mixed with injection brine

Figure 15-22. Permeability with reduced concentrations of KCl and CaCl2 solutions (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

the surface area of the particle, as discussed in Chapter 8. As the fluid velocity is increased gradually, a critical interstitial fluid velocity necessary for detachment of fine particles from the pore surface can be reached. Amaefule et al. (1987) state that “The critical velocity is dependent on the ionic strength and pH of the carrier fluid, interfacial tension, pore geometry and morphology, and the wettability of the rock and fine particles.” Then, the particles are hydrodynamically removed from the pore surface and entrained by the fluid flowing through porous media. Fine particles migrating downstream with the fluid may encounter and plug narrow pore throats by a jamming process. This causes the pressure difference across the core to increase and the permeability to decrease. Therefore, from a practical point of view, the critical interstitial velocity is characterized as the interstitial velocity at which permeability reduction

609

Laboratory Evaluation of Formation Damage Low K

Same tests as medium K

Medium K

Formation brine K

K gas @ irreducible S w

Inject workover fluid

K gas after workover vs. gas volumes

Inject remedial chemicals

K gas after treatment vs. gas volumes

High K

A

B

C

D

Same tests as medium K

% Damage ratio after workover K – KD × 100 = B KB

% Damage ratio after treatment K – KF × 100 = B KB

% K gas return after treatment =

E

F

KF × 100 KB

Damage due to 1. Clay hydration and/or movement 2. Solids in workover and remedial solution 3. Relative permeability (water block)

Figure 15-23. Test sequence for evaluation of workover fluid damage and remedial chemicals with the presence of mobile hydro carbons (after Keelan and Koepf, ©1977 SPE; reprinted by permission of the Society of Petroleum Engineers).

and pressure differential increase begin as the fluid velocity is increased gradually from a sufficiently low value (Gruesbeck and Collins, 1982a; Gabriel and Inamdar, 1983; Egbogah, 1984; Amaefule et al., 1987, 1988; Miranda and Underdown, 1993).

610

Laboratory Evaluation of Formation Damage

The theory of the critical velocity determination is based on Forchheimer’s (1914) equation, given below, which describes flow through porous media for conditions ranging from laminar to inertial flow:  dp  − + g sin = u + u2 dx K 

(15-3)

where p

 and u denote the fluid pressure, density, viscosity, and volumetric flux, respectively; x denotes the distance in the flow direction; K and denote the permeability and inertial flow coefficient of the porous media, respectively, g is the gravitational acceleration, and is the angle of inclination. The interstitial velocity, v is given by the DuPuit (1863) equation, accounting for the effect of the irreducible fluid saturations, as v = u



   1 − Swc − Sro  = q A1 − Swc − Sro 

(15-4)

where Swc and Sro denote the connate water and residual oil saturations.  denotes the porosity,  denotes the tortuosity, A denotes the crosssectional area of porous media normal to flow, and q is the volumetric flow rate. Considering horizontal flow, and constant fluid and core properties averaged over the core length, Eq. (15-3) can be integrated over the core length for applications to laboratory core tests. In view of Eq. (15-4), the resultant expression can be given in terms of the interstitial fluid velocity as p L = +  K

  2 1 − Swc − Sro  L  2

(15-5)

or in terms of the flow rate as p L L = + 2 q q KA A

(15-6)

At sufficiently low fluid velocities, the fine particles remain attached to the pore surface and, therefore, there is no formation damage by fines migration and p/ remains constant as depicted schematically in Figure 15-24a by Amaefule et al. (1988). However, when the fluid velocity is gradually increased, first a critical velocity at which particle

611

Laboratory Evaluation of Formation Damage

K True critical velocity

∆P U

(a)

(b)

U

U

∆P U Effect of inertia (c)

U

Figure 15-24. Method for laboratory determination of critical velocity for particle mobilization (after Amaefule et al., ©1988; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

detachment by hydrodynamic forces begins, is reached, and then the value of p/ or p/q continuously increases (Figure 15-24a) and permeability continuously decreases (Figure 15-24b) by fines migration and deposition in porous media. As emphasized by Amaefule et al. (1987, 1988), the increase in p/ or p/q at high flow rates may be due to both fines migration and inertial flow effects (Figure 15-24c). For decoupling these two effects, Amaefule et al. (1987) propose a subsequent velocity reducing test. When the flow rate is reduced gradually, the p/q value should reach its original value measured during the velocity increasing test, if the critical velocity has not been reached during the previous velocity increasing test. However, if the critical velocity has been exceeded during the velocity increasing tests, permeability impairment by fines mobilization and deposition would have occurred. Therefore, a subsequent velocity decreasing test will yield a value of p/q different than the previous value measured during the velocity increasing test. Figure 15-25 by Amaefule et al. (1987) presents a schematic illustration of their approach to decoupling the fines migration and inertial effects using KCl-saturated brine (55 kppm KCl) and kerosene oil at different saturation levels: (1) 100% water, (2) saturated with oil in the presence of irreducible water, and (3) 100% oil. They determined that the critical velocity is zero for oil flowing through cores at irreducible water saturation. Figure 15-26,

612

Laboratory Evaluation of Formation Damage

Pressure difference/flow rate, ∆p/q

Pressure difference/flow rate, ∆p/q

Permeability alteration due to fines

True critical flow rate

(a)

No permeability alteration

Apparent critical flow rate

(b)

Flow rate, q

Flow rate, q

Normalized permeability K /Kinitial, % initial

Figure 15-25. Effect of flow regime: (a) permeability alteration by mobilized particles during Darcy flow, and (b) apparent permeability alteration by mobilized particles during non-Darcy flow (after Amaefule et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers).

100

Critical velocity for

Critical velocity for KCl = 0.1624 cm

(NaCL /CaCl2) = 0.0289 cm s

90

A = 11.36 cm2 ø = 18.1 %

A = 11.36 cm2 ø = 17.8 %

s

80

70

0

0.02

0.01

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.2

Interstitial velocity, v (cm/s)

Figure 15-26. Normalized permeability vs. interstitial fluid velocity for two brines (after Amaefule et al., ©1988; reprinted by permission of the Canadian Institute of Mining, Metallurgy and Petroleum).

Laboratory Evaluation of Formation Damage

613

based on experiments conducted with Berea core samples by Amaefule et al. (1988), shows that critical velocity is higher for KCl brines than NaCl/CaCl2 brines. Amaefule et al. (1987) verified their proposed approach by a series of tests: they injected a standard operating (SOP) brine containing 50 kppm NaCl and 5 kppm CaCl2 into a field core sample (2.54 cm diameter and 12.2% porosity) and measured p/q. During the increasing flow cycle, they determined the critical velocity to be 0.0674 cm/s based on their plot of data shown in Figure 15-27. They also measured the effluent brine pH during the flow tests, as it may provide some evidence of the physico-chemical interactions of the aqueous solution with the formation (Amaefule et al., 1987; Millan-Arcia and Civan, 1992). As shown in Figure 15-27, during the flow test, with increasing SOP brine velocity, the pH first decreased from the initial value of 8.45, reached a minimum value of 6.74, and then increased to a value of 7.5. As can be seen from Figure 15-27, the minimum pH coincides with the critical interstitial velocity. This finding is particularly significant. Therefore, Keelan and Amaefule (1993) recommend monitoring injection and produced water pH as an integral part of critical interstitial velocity determination. Amaefule et al. (1987) explain the change of pH to attain a minimum at the critical velocity due to the increase of the K + ion in the aqueous phase by exchange of the K + cations of the formation minerals with the Na+ and Ca2+ cations of the aqueous phase. Following the velocity increasing cycle, Amaefule et al. (1987) conducted a velocity reducing cycle until a velocity of 0.027 cm/s was reached. As can be seen in Figure 15-27b, the permeability first reduced to 66% of its initial value and then to 55% after an additional one pore volume of the SOP brine injection at the same rate of 0.027 cm/s, indicating the dependency of the permeability reduction to the brine throughput as a result of simultaneous entrainment, migration, and redeposition of fine particles in porous media. When Amaefule et al. (1987) reversed the flow, they observed a rebound of permeability to 90% of the initial permeability because of dislodging of particles from clogged pores, but the permeability decreased to 87% of initial after a one pore volume brine injection. Amaefule et al. (1987) explain this affect due to the fines entrainment and then migration and redeposition phenomena. They observed a similar effect on the effluent brine pH variation.

100

8.0

4.5

98

7.9

96

Permeability ratio K/Kinitial, % initial

4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9

(a)

7.8 Kwc = 70.54 md Porosity = 12.2%

94

0

1

2

3

4

5

6

7

8

Nominal flow rate, q (cc/min)

9

92

(b)

7.7 7.6

90

7.5

88

K /KI

7.4

86 7.3

84

7.2

82

pH

7.1

80

7.0

78 76

6.9

74

6.8

72

6.7

70 0.00

10

Critical velocity = .0674 cm/s

0.05

0.10

0.15

0.20

0.25

6.6 0.30

Interstitial velocity, v (cm/s)

Figure 15-27. Injection of SOP brine into a field core at 3000 psi pressure: (a) pressure drop per flow rate vs. flow rate and (b) instantaneous to initial permeability ratio and pH vs. interstitial fluid velocity (after Amaefule et al., ©1987 SPE; reprinted by permission of the Society of Petroleum Engineers).

Laboratory Evaluation of Formation Damage

Pressure difference/flow rate, ∆p /q

4.3

pH

4.4

614

4.6

615

Laboratory Evaluation of Formation Damage

15.7.7

Scaling from Laboratory to Bottomhole

Miranda and Underdown (1993) developed a method for scaling laboratory data to the bottomhole dimensions based on the schematic given in Figure 15-28. For this purpose, the interstitial fluid velocities expressed in terms of the parameters of the core plug and perforated wellbore are equated as shown below, modified here, considering the effect of the irreducible fluid saturations Swc and Sro , and tortuosity :     q q c = = (15-7) A1 − Swc − Sro  core A1 − Swc − Sro  wellbore The cross-sectional area A of the core plug is known. They expressed the total inflow area of the perforated wellbore interval by 

A = Es Ns Li 072rh + 03r r 2 + h′2 (15-8)

where q denotes the flow rate,  is porosity, Es denotes the shot efficiency in percent, Ns is the perforation density expressed as the number of shots per foot, Li is the interval length in ft, r is the radius of perforations in

Casing

Cylinder portion

Cone portion

Cement

Figure 15-28. Perforation tunnel model for scaling laboratory core tests (after Miranda and Underdown, ©1993 SPE; reprinted by permission of the Society of Petroleum Engineers).

616

Laboratory Evaluation of Formation Damage

ft, and h and h′ denote the lengths of the cylindrical and conical portions of the perforation tunnel, respectively. The factors 0.7 and 0.3 represent the fractions of flow entering the cylindrical and conical sections of the perforation tunnel, determined as inferred by the studies of Deo et al. (1987).

15.8 EVALUATION OF THE RESERVOIR FORMATION DAMAGE POTENTIAL BY LABORATORY TESTING – A CASE STUDY Haggerty and Seyler (1997) conducted an extensive laboratory investigation of formation damage by mud cleanout acids (MCA) and injection waters in Aux Vases sandstone reservoirs. A brief description of their studies conducted for evaluation of the reservoir formation damage potential is presented in this section. 15.8.1

Formation Evaluation

Seyler (1998) conducted extensive analyses of the core samples obtained from Aux Vases reservoirs. Seyler (1998) examined over 150 thin sections. The petrographical analyses conducted included the following: 1. Petrographical analysis (PA). Standard optical microscopy using thin sections was carried out. For this purpose, they stained the thin sections with potassium ferricyanide and alizarine red to distinguish and detect the carbonate phases. Petrographical characteristics and attributes such as grain composition and size, cementing agents, porosity types, and reservoir quality were determined. 2. X-ray diffraction analysis (XRD). The XRD analyses determined the type and semiquantitative composition of the minerals present in the samples. 3. Scanning electron microscopy (SEM) with energy dispersive X-ray microbeam (EDX) analysis. This is referred to as the SEM/EDX analysis. For this purpose, the samples were sputter-coated with gold and palladium. The SEM analyses identified the pore-lining minerals, and the EDX analysis determined the elemental composition. The analyses of the individual core samples are presented in Table 15-5 by Haggerty and Seyler (1997). They concluded that Aux Vases formation

617

Laboratory Evaluation of Formation Damage

Table 15-5 Aux Vases Samples: Bulk Weight Percentage of Clay Minerals; Absolute Percentage of other Minerals Depth ft

Perm. mD

Porosity %

I %

I/S %

C %

BC %

Q %

Kf %

Pf %

Cc %

D %

845 816 849 795 880 807 825 823 778 837 678 829 684 24 29 34

13 17 15 12 00 26 27 26 15 12 08 17 12 00 00 00

25 32 37 43 19 26 37 33 19 24 15 28 18 00 00 00

64 86 56 81 32 85 68 55 142 76 218 84 168 912 936 718

00 00 00 00 20 00 00 00 00 00 00 00 00 00 00 156

692 642 850 864 862 721 779 914 827 618 788 727 803 845 906

00 30 11 06 04 89 23 00 02 205 06 02 50 05 04

29 74 39 40 49 61 90 44 71 43 64 22 31 37 39

185 167 39 29 39 34 37 19 38 33 42 152 19 15 00

00 00 01 00 03 00 00 00 00 00 00 00 00 00 00

808 565 854 727 763 635

00 02 06 02 04 00

82 82 26 78 63 25

60 309 67 166 125 275

00 00 00 00 00 00

Farrar 1 McCreery, Dale Consolidated Field – API 1205523456 3,190.5 3,191.5 3,192.7 3,194.0 3,195.7 3,197.9 3,199.4 3,201.0 3,203.3 3,205.8 3,207.1 3,208.4 3,209.9 3,212.7 3,216.7 3,219.2

490 815 1060 1160 558 568 419 355 279 158 18 25 −01 NA NA NA

204 234 256 236 198 248 243 214 186 214 146 131 07 NA NA NA

32 28 25 34 26 31 21 31 21 24 31 17 52 27 15 46

14 15 11 25 15 17 15 23 21 21 42 19 47 37 20 44

07 53 07 5 06 43 09 68 07 48 08 56 06 42 09 63 04 46 05 51 07 8 05 41 19 118 00 64 00 35 01 91

Gallagher Drilling Company 2 Mack, Zeigler Field – API 1205523750 2,605.5 2,606.5 2,608.5 2,610.5 2,611.5 2,612.5 2,614.5 2,617.5 2,618.5 2,620.5 2,621.5 2,623.7 2,623.8 2,625.5 2,629

249 2160 490 640 1240 1520 356 895 478 478 560 NA −01 −01 −01

144 251 221 221 241 255 236 245 248 235 238 NA 88 170 110

55 37 29 24 17 42 23 05 12 20 22 24 25 25 10

27 27 12 14 08 19 12 03 06 21 07 21 19 25 11

11 94 22 87 19 6 22 6 19 43 33 94 37 71 15 23 43 62 60 101 72 101 52 97 53 97 49 99 30 51

Budmark 2 Morgan Coal, Energy Field – API 1219923465 2,387.6 2,388.4 2,390.1 2,392.7 2,394.7 2,395.2

1840 2460 690 850 690 43

213 217 236 233 206 136

16 12 14 08 10 16

09 11 07 03 05 09

24 19 25 17 30 41

5 42 47 27 45 66

(table continued on next page)

618

Laboratory Evaluation of Formation Damage Table 15-5 (Continued)

Depth ft

Perm. mD

Porosity %

I %

I/S %

C %

BC %

Q %

Kf %

Pf %

Cc %

D %

937 907 883 806 676

20 33 13 12 12

10 23 05 08 08

05 05 81 84 285

02 03 02 02 01

25 26 32 29

07 13 16 11

129 101 60 47

05 04 03 05

Superior Oil Company 1 Price, Boyd Field – API 1208101972 2,129.0 2,131.0 2,133.0 2,134.0 2,135.0

00 17 1180 810 00

73 177 151 114 70

12 08 05 37 04

09 06 03 26 03

05 15 07 25 11

26 29 15 88 18

Superior Oil Company 7 Sanders, Boyd Field – API 1208101950 2,141.0 2,144.0 2,151.0 2,155.0

420 3620 1400 580

216 242 215 195

11 04 14 06

08 03 10 05

07 07 16 13

26 15 41 24

809 842 849 883

Perm.= permeability, D = dolomite, Cc = calcite, C = chlorite, I = illite, I/S = illite/smectite, BC = bulk content, Kf = K-feldspar (microcline or orthoclase), Pf = plagioclase feldspar, Q = quartz. After Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey.

core samples contained 65–90% quartz, 3–15% feldspar, 0–15% calcite, and 2–7% clay minerals. Haggerty and Seyler (1997) determined that calcite is the primary porefilling mineral. They described the observed three types of pore-filling calcite as “In relative order of abundance, they are (1) patchy cement filling intergranular porosity (Figure 15-29c); (2) framework grains such as marine fossil fragments, ooids, and pelloids; and (3) minute, late-stage euhedral crystals (Figure 15-29a) on diagenetic clay minerals that coat framework grains and line pores.” Haggerty and Seyler (1997) describe the pore-lining minerals to be, “in descending order of abundance, dominantly diagenetic clay minerals, calcite, partially dissolved feldspars, solid hydrocarbons, anatase, barium-rich celestite, and traces of dolomite.” Haggerty and Seyler (1997) observed that “Pores in Aux Vases sandstone reservoirs are lined with, and may be bridged by, diagenetic clay minerals that consist of an intimately intergrown mixture of mixed-layered illite/smectite, chlorite, and illite (Figure 15-30). Although clay minerals constitute only 2–7% of the bulk mineral content, SEM analysis indicates that clay minerals coat more than 95% of pore surfaces. Therefore, an understanding of the composition and response of these diagenetic clay minerals to injected fluids is of utmost importance when selecting drilling muds and stimulation methods.” “Chlorite

Laboratory Evaluation of Formation Damage

619

(a)

Figure 15-29. SEMs comparing acid-soaked with untreated wafers sliced from 1-in.-diameter core plugs from two wells in Energy Field: (a) An untreated, criticalpoint-dried sample (2405.2-ft depth, Budmark 3 Burr Oak) shows a late-stage, minute calcite crystal with sharp euhedral edges (arrow) on top of diagnetic clay minerals; its position indicates that it precipitated after the clay minerals. This type of calcite is the first to be affected by exposure to MCA or 15% HCl; (b) A minute calcite crystal (arrow) was etched on this sample (2393.5-ft depth, Budmark 2 Morgan Coal) after 30 minutes of soaking in MCA. Compare the etched faces with the straight euhedral crystal faces shown in Figure 15-29a; clay minerals do not appear to have been affected; (c) An untreated sample (2393.5-ft depth, Budmark 2 Morgan Coal) shows patchy calcite cement filling pores (arrows). This type of calcite is much more extensive than the minute late-stage crystals shown in Figure 15-29a. Dissolution of this type of calcite would improve porosity and permeability of reservoirs; (d) Partial dissolution of pore-filling calcite cement (arrow) after 30 minutes of soaking the sample (depth 2393.5-ft, Budmark 2 Morgan Coal) in MCA. Compare this example with the untreated calcite cement shown in Figure 15-29c. The edges of the calcite are etched and contrast sharply with the flat, smooth cleavage surfaces in the untreated sample. Note that the clay minerals appear to have been unaffected (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

identified by XRD and SEM/EDX analyses in Aux Vases samples is typically not iron-rich, but contains approximately equal amounts of iron and magnesium.” “Reservoirs containing chlorite rich in iron may be more susceptible to formation damage than those containing other varieties of chlorite because they may form insoluble iron oxides or iron hydroxides.”

620

Laboratory Evaluation of Formation Damage

(b)

(c)

Figure 15-29. (Continued )

Haggerty and Seyler (1997) report that the mixed-layered illite/ montmorillinite (smectite) varieties are the only water-sensitive, expandable clay minerals they found in the Aux Vases core samples.

Laboratory Evaluation of Formation Damage

621

(d)

Figure 15-29. (Continued )

15.8.2 15.8.2.1

Experimental Studies Focus and design

The primary objective of the studies by Haggerty and Seyler (1997) is the experimental investigation of formation damage by (1) mud cleanout acids and (2) injection waters in Aux Vases sandstone reservoirs. Haggerty and Seyler (1997) describe that Development of sandstone reservoirs in the Illinois Basin typically includes these steps: 1. Drilling with freshwater mud 2. Perforating the potential reservoir zone, if casing is used, or open-hole completions with casing cemented above the producing zone 3. Preflushing with 15% HCl or MCA to remove drilling mud 4. Cleaning out perforations or the wellbore with MCA 5. Initial swabbing to retrieve stimulation fluids and induce oil flow toward the wellbore 6. Hydraulic fracturing using a freshwater gelled pad and sand propant 7. Final swabbing during the production test.

622

Laboratory Evaluation of Formation Damage

Figure 15-30. SEM shows a close-up of typical intergrown diagnetic clay minerals coating grains and lining pores in Aux Vases reservoirs. This sample was critical-pointdried to preserve the morphology of hydrated clay minerals composed of intimately intergrown mixed-layered illite/smectite, chlorite, and illite. Fresh cores with preserved fluids needed for critical point drying were available from only two wells. This sample (2405-ft depth, Budmark 3 Burr Oak, Energy Field) was chosen because the well is near the Budmark 2 Morgan Coal well sampled for this study (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

Table 15-6 (adapted from Piot and Perthuis, 1989) clearly indicates that exposure of Aux Vases formation to acid solution can activate some reactions with the resident minerals. In an effort to simulate the field practice in the laboratory investigations, Haggerty and Seyler (1997) state that The experiments focused on five objectives: 1. Determine how MCA containing 15% HCl with additives, typically used during completion and/or stimulation, affects pore-lining minerals and the permeability of Aux Vases reservoir rocks by conducting dynamic, constant rate injection coreflood experiments 2. Investigate how 15% HCl and MCA affects crude oil from Aux Vases reservoirs by conducting compatibility experiments 3. Examine how exposure to various salinity brines affects permeability in samples of Aux Vases reservoirs by conducting coreflood experiments

Table 15-6 Reactions of Aux Vases Sandstone Minerals with Well Fluids Solubility in 15% HCI

Chemical Composition

Released

Insoluble

SiO2

None

Low

KAlSi3 O8 

K+1 Al+3 Si+4

Na-feldspar

Low Low to medium Low to medium

Insoluble

None

Illite

High

Low

Mixed-layered

High

Low Low Low to medium High High Insoluble Insoluble

NaAlSi3 O8  Fe MgKx Al2 Si4−x Alx  O10 OH2 Fe MgKx Al2 Si4−x Alx  O10 OH2 1 /2 Ca Na•7Al Mg Fe4− Si Al8 O20 OH4 •nH2 O Mg Fe5 Al FeAl Si3 O10  OH8 CaCO3 Ca Mg FeCO3 TiO2 Ba SrSO4

Mg+2 Fe+2 Al+3 Si+4 Ca+2 CO− 3 Mg+2 Fe+2 Ca+2 CO− 3 None None

Med Insoluble

C,OH,H,S,N Fe2 S

C+4 S None

Quartz K-feldspar

Illite/smectite Chlorite Calcite Fe-dolomite Anatase Barite-celestite Solid H-carbon Pyrite

High Low to high Low Low Low Low to medium Low

K+1 Al+3 Si+4 Ca+2 Na+1 Fe+2 Mg+2 Al+3 Si+4

+4

623

Adapted from Piot and Perthuis (1989). After Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey.

K+1 Al+3 Si+4

Laboratory Evaluation of Formation Damage

Exposure to Fluids

Mineral

624

Laboratory Evaluation of Formation Damage

4. Investigate the effects of long-term contact of 15% HCl and MCA with pore-lining minerals in reservoir samples by conducting static soak experiments 5. Compare XRD analyses of the bulk mineralogy and SEM/EDX analyses of pore-lining minerals with flood results to identify minerals that would be most affected by fluids commonly used during drilling, completion, and stimulation of Aux Vases reservoirs.

Therefore, Haggerty and Seyler (1997) carried out five sets of bench experiments, with the specific objectives described in Table 15-7. The direct contact experiments were conducted to determine the effect of the acids on the physical properties of crude oil. In the coreflood tests, they continuously injected excessive amounts (25–50 pore volumes) of fluid during coreflood experiments. Haggerty and Seyler (1997) state that their coreflood experiments most closely represent the completely flushed reservoir zones and, under these conditions, the precipitates cannot deposit and cause formation damage in porous media within the time scale of the convective flow. The acid soak experiments served the purpose of observing the long-term effects of reactions in unflushed and incompletely flushed zones. 15.8.2.2

Description and preparation of materials

Core Plugs. Table 15-8 shows the sources and available data of the Aux Vases reservoir core samples used by Haggerty and Seyler (1997). Onein.-diameter (2.54-cm) core plugs were extracted out of 4-in.-diameter whole cores with maximum possible lengths permitted by drilling. In the coreflood experiments, they used core plugs of 1 in. (2.54 cm) to 2.5 in. (6.35 cm) length. In the MCA and HCl soak experiments, they used 1-in.-diameter (2.54-cm) and 0.25-in.-thick (0.635-cm) core plug wafers. Fluids. Halliburton, Inc., provided the MCA solution containing “15% HCl in a proprietary formulation of surfactants, suspending agents, antisludge agents, clay mineral stabilizers, iron-sequestering agents, and corrosion inhibitors,” as stated by Haggerty and Seyler (1997). The characteristics of the waters used by Haggerty and Seyler (1997) are described in Table 15-9. The Aux Vases formation brine was obtained from the Budmark No. 3 Morgan Coal lease in Energy Field, filtered, and then used in coreflood tests with a 13.7% total dissolved solids (TDS) content.

Table 15-7 Experimental Overview Field, Well

Depth (ft)

To Determine

Direct contact: crude oil and acids

Crude oils

Boyd, Baldridge B5 Bizot

2,170

Compatibility of 15% HCl vs. MCA

Dale, Farrar 2 McCullum Community Energy, Budmark 2 Morgan Coal Zeigler, Gallagher Drilling 1 Alex

3,158–3,176 2,385–2,395 2,615–2,630

MCA

Energy, Budmark 2 Morgan Coal

2,392.1

MCA 15% HCl

Zeigler, Gallagher Drilling 2 Mack Energy, Budmark 2 Morgan Coal

2,627 2,393.5

MCA

Dale Cons, Farrar 1 McCreery

3,198.7

MCA

Energy, Budmark 2 Morgan Coal

2,391.1

Waters, various salinities

Boyd, Superior Oil 9 Sanders

2,163

Energy, Budmark 2 Morgan Coal Energy, Budmark 2 Morgan Coal Zeigler, Gallagher Drilling 2 Mack Dale Cons., Farrar 1 McCreery Energy, Budmark 2 Morgan Coal

2,390 2,388 2,611 3,200.6 2,388.3

15% HCl

Energy, Budmark 2 Morgan Coal

2,392.8

MCA

Energy, Budmark 2 Morgan Coal

2,393

Coreflood: continuous injection at a constant rate

Coreflood: interrupted injection, constant rate

Core waterflood: interrupted injection, constant rate

Acid soak

After Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey.

Effects on permeability and pore-lining minerals

Effects of interrupting injection and soaking sample in MCA; simulates potential damage after injection and before swabbing Sensitivity of rock to injected water of varying salinities; note permeability changes

Long-term reaction of reservoir rock to MCA and 15% HCl

625

Fluids

Laboratory Evaluation of Formation Damage

Type of Experiment

626

Laboratory Evaluation of Formation Damage

Table 15-8 Samples Used for the Experiments and Methods Used to Describe Them Field, Well, and Well ID

Thin Section

Depth ft

SEM/EDX

Energy 2 Morgan Coal 1219923465

2,393.5 2,388.3 2,391.1 2,390 2,388 2,393 2,392.8

Yes Yes Yes No No Yes Yes

No Yes No Yes No No Yes

No Yes No Yes No No Yes

Dale 1 McCreery 1205523456

3,198.7 3,200.6

Yes No

No No

Yes No

Zeigler 2 Mack 1205523750

2,611 2,627

Yes Yes

Yes Yes

Yes Yes

Boyd 9 Sanders 1208102628

2,163

No

No

No

After Haggerty and Geological Survey.

Seyler,

1997;

reprinted

by

XRD

permission

of

the

Illinois

State

A fresh water mixture containing 1.2% TDS and its mixtures with the formation brine at various proportions, as described in Table 15-9, were synthetically prepared and used in the coreflood tests. Haggerty and Seyler (1997) measured the resistivities of these mixtures and then estimated their TDS in ppm units using the TDS–resistivity correlation developed by Demir (1995): TDS =

678609 1022T R12853 w

(15-9)

where Rw denotes the resistivity of water in -m and T denotes the water temperature in  F. 15.8.2.3

Equipment

Haggerty and Seyler (1997) performed their coreflood tests using a TEMCO™ integrated coreflood apparatus. This system is described

Ionic composition meq/l Na+

Ba2+

Fe3+

Mg2+

Cl

341

1883

00

04

173

2394

16

14

01

00

13

Type

Ca

Formation brine Supply water

2+

−

Rw

TDS

HCO3

pH

-m

%

10

17

61

0063

137

135

15

53

−

0433

12

95%(1)–5%∗

0064

135

90%(1)–10%

0068

124

75%(1)–25%

0076

105

50%(1)–50%

0124

∗

282

2−

SO4

95% (by volume) of formation brine and 5% (by volume) of the supply water. meq/l = mole wt/charge per liter. After Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey.

595

Laboratory Evaluation of Formation Damage

Table 15-9 Characteristics of Water Mixtures

627

628

Laboratory Evaluation of Formation Damage

Figure 15-31. Temco integrated coreflood apparatus (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

schematically in Figure 15-31. A conventional Hassler-type coreholder is placed in an oven for temperature control. The core is placed inside a rubber sleeve between metal end pieces. Spacers are attached to adjust for different core lengths. The space between the coreholder and rubber sleeve is filled with a pressurized hydraulic fluid. A confining pressure pump and gauge system applies pressure to the rubber sleeve containing the core sample. The confining pressure applied over the core plugs prevents the bypassing of the injected fluids around core plugs and the mixing of the injected and hydraulic fluids. The coreholder is connected to oil and water reservoirs and the oil and water recycling cylinders. The coreholder is equipped with an inlet pressure transducer and gauge

Laboratory Evaluation of Formation Damage

629

and a back pressure regulator and gauge. The other auxiliary equipment includes a nitrogen gas cylinder, rotameters, pressure valves, and a wet test meter (Oil and Gas Section – ISGS, 1993). 15.8.2.4

Coreflood tests

Haggerty and Seyler (1997) report that the core plugs were cleaned in a CO2 /solvent core cleaner and vacuum-dried. Then the porosity and permeability to nitrogen gas were measured. The baseline liquid permeability was measured by injecting brine at the rate of 15 cm3 /min continuously into the core plugs. They determined the 15 cm3 /min rate by scaling the typical reservoir fluid velocity of 14 ft/day and 68.6 bbl/day in a 10-feet-thick pay zone, using the scaling coefficient of unity LVw = 10 for 65% oil recovery at water breakthrough (Kyte and Rapoport, 1958; Delclaud, 1991). This corresponds to 0.296 cm/min or 1.5 cc/min flow through 1-in.-diameter core plugs. They measured the pressure difference between the inlet and the outlet of the core plugs and calculated the effective liquid permeability of the core plugs using Darcy’s law. Haggerty and Seyler (1997) estimated the overburden pressure of the Aux Vases formations located at the depths of 2100–3200 ft to be in the range of 2100–3200 psi by assuming a gradient of 1.0 psi/ft for typical sedimentary basins (Levorsen, 1967). Assuming the reservoir fluid is normally pressured and using a hydrostatic pore pressure gradient of 0.45 psi/ft, they estimated the pore fluid pressure to be 1200 psi. The bottomhole temperatures of wells in the Aux Vases formations vary from 75 to 98 F 24 –36 C. However, they performed the coreflood tests at 1000 psig (6895 kPa) confining pressure and 75 F 24 C temperature. They assumed that the effects of the differences between the test and the field conditions are negligible based on the arguments by Amyx et al. (1960) and Eickmeier and Ramey (1970). They conducted the flow tests at constant injection rates. The pressure difference across the core plugs typically varied between 10 and 50 psi (69.8–345 kPa) for injection at a 14 ft/day rate. A 50–75 psi (345–517 kPa) back pressure was sufficient to maintain single phase and avoid CO2 gas bubbles. As stated by Haggerty and Seyler (1997), tests conducted using core plugs have certain inherent limitations: 1. Their small size represents a very small percentage of the total reservoir; therefore the entire range of effects that introduced fluid may have on reservoir behavior cannot be fully determined, and

630

Laboratory Evaluation of Formation Damage

2. Each experiment represents a discrete phase of the drilling, completion, or stimulation process.

However, coreflood tests conducted at near in situ conditions can help explain the reactions between pore-lining minerals and extraneous fluids introduced by drilling, completion, and stimulation operations (Haggerty and Seyler, 1997). Then, the cumulative effects of the rock– fluid interactions on formation damage can be determined by simulation or other means. 15.8.2.5

Experimental results

Haggerty and Seyler (1997) conducted a number of tests with 15% HCl and 15% HCl-MCA to determine the effect of the clay stabilizing agents present in the MCA provided by Halliburton, Inc. Calcite dissolution with the MCA. The HCl in the MCA dissolves calcite by the reaction: CaCO3 + 2HCl −→ ←− CO2g + CaCl2 + H2 O

(15-10)

The produced carbon dioxide CO2  gas dissolves in the aqueous phase at elevated pressures, but separates as the effluent solution comes out of the core. Haggerty and Seyler (1997) conducted four types of tests. These tests and their results are summarized as follows. Coreflood tests with MCA. They conducted interrupted and continuous acid corefloods using a 15% HCl-MCA on samples described in Tables 15-6 and 15-7. They injected ten pore volumes of the acid solution into the core plugs for complete flushing of the cores to simulate the total flushing of the near-wellbore formation during acid stimulation. The primary objective of the interrupted corefloods was to investigate the effects of the MCA solution on the formation without the presence of other reservoir fluids (i.e., oil and brine). Therefore, Haggerty and Seyler (1997) injected MCA into a dry core sample. The acid injection was interrupted at certain time intervals and the permeability of the core was measured. Figures 15-32 and 15-33 show the measured permeabilities of two different core plugs. The acid dissolved the calcite cement and permeability increased. Although some

Laboratory Evaluation of Formation Damage

631

Figure 15-32. An interrupted, constant flow rate, acid coreflood exposed a core plug (3198.7-ft depth, Farrar 1 McCreery, Dale Cons. Field) to MCA for 74 hours. Permeability increased with MCA–rock contact time as calcite cement dissolved, dislodging some fine grains that were flushed out of the core sample (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

fine particles may have been unleashed by calcite dissolution, damage by fine particles migration and deposition was not observed because the fine particles were flushed out of the core plugs by an excessive amount of acid injection (ten pore volumes). In fact, after 24 hours of exposure, Haggerty and Seyler (1997) detected fine-grained sand, siltsized grains of nonclay minerals, and diagenetic clay particles in the effluent. The continuous acid corefloods were conducted to simulate the flushing of the near wellbore formation in the presence of formation fluids. For this purpose, the cores were restored to their reservoir conditions by a series of displacement processes. First, the cores were saturated with brine, the brine was displaced by oil up to irreducible water saturation, and the cores were allowed to establish oil–water equilibrium by soaking them in oil for 48 hours. The oil present in the cores, prepared in this way, was displaced with brine and then MCA was injected to displace the brine and effective permeabilities were measured during continuous acid injection. As indicated in Figure 15-34 by Haggerty and Seyler (1997), the permeability first decreased rapidly and then increased continuously.

632

Laboratory Evaluation of Formation Damage

Figure 15-33. Permeability changes during a 30-hour, interrupted, constant flow rate 15 cm3 /min coreflood test using a 1-in.-diameter core plug (2392.1-ft depth, Budmark 2 Morgan Coal, Energy Field). Permeability increased with MCA–rock sample contact (no flow) and flow times. The increase is more pronounced in the McCreery core plug (Figure 15-32) because of the dissolution of large amounts of calcite cement aligned along crossbedding laminae; the Budmark 2 Morgan Coal sample did not have as much calcite cement (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

Haggerty and Seyler (1997) attribute the initial decreasing of permeability to carbon dioxide CO2  gas production. Figure 15-35a and b are the photomicrograph and SEM photomicrograph, respectively, of a thin section taken from a core plug exposed to MCA for 75 hours. These photomicrographs clearly show that calcite cement was dissolved and the pores were enlarged. Calcite dissolution in MCA soak tests. The MCA soak tests were conducted statically (without flow) at ambient room conditions to observe the effect of acid on pore-lining minerals. These tests determine the effect of the acid remaining in locations far away from the wellbore after acid stimulation. Figures 15-29a and 15-36 by Haggerty and Seyler (1997) show that untreated core samples are loosely cemented and friable sandstone with grains coated with diagenetic clay minerals. Figure 15-29c shows pores filled with patchy calcite cement in an untreated sample. Figures 15-29c

Laboratory Evaluation of Formation Damage

633

Figure 15-34. Permeability of a sample (2627.5-ft depth, Gallagher Drilling Co. 2 Mack, Zeigler Field) varied significantly during a 4.5-hour, continuous, constant rate coreflood test using MCA (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

and d show the pores of 30 minute acid-treated cores. Comparison of Figure 15-29a to Figure 15-29b and Figure 15-29c to Figure 15-29d indicate that some crystals and patches of calcite cement were partially dissolved after 30 minutes, but the diagenetic clay minerals were not affected. Haggerty and Seyler (1997) attribute this to the function of clay-stabilizing additives present in the MCA. Coreflood tests with 15% HCl without additives. In order to determine the effect of acid treatment without the clay stabilizers, Haggerty and Seyler (1997) conducted a number of tests using an aqueous solution containing 15% HCl only. As shown in Figure 15-37, the permeability of the core plug increased. They attributed the permeability increase to calcite dissolution and concluded that the aluminum- or magnesium-rich chlorites in the Aux Vases formations do not produce iron gels that could cause permeability reduction. The effluent ion analyses presented in Table 15-10 by Haggerty and Seyler (1997) clearly indicate leaching, dissolution, and disintegration of the pore-lining diagenetic clay minerals and the pore-filling cements by the 15% HCl solution coreflood tests. The SEMs shown in

634

Laboratory Evaluation of Formation Damage

(a)

Figure 15-35. (a) Photomicrograph of a thin section made from the core plug after 74 hours of exposure to MCA shows that coarse-grained laminae, filled with calcite cement prior to the coreflood, developed a channelized pore system due to total dissolution of calcite cement along the laminae. As a result, large oversize pores formed (arrow); (b) SEM photomicrograph of the same sample after coreflood also shows enlarged pores (arrow) and diagenetic clay minerals coating siliciclastic framework grains. The framework grains have bimodal size distribution and are either very fine grained 100 m or medium grained at 250 m or greater (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

Figures 15-38a–c by Haggerty and Seyler indicate the deposition of various precipitates in treatment with 15% HCl without the stabilizing agents, as a result of the reaction of the acid with the pore minerals and brine. Figure 15-38d shows the remaining quartz minerals stripped off the clay mineral coatings by the 15% HCl acid solution. Soak tests with 15% HCl without additives. Haggerty and Seyler (1997) observed significantly more mineral precipitation, as shown in Figures 15-38a–d, with samples soaked in 15% HCl compared to those soaked in MCA.

Laboratory Evaluation of Formation Damage

635

(b)

Figure 15-35. (Continued )

Coreflood tests for water sensitivity. Haggerty and Seyler (1997) injected different salinity brines into Aux Vases core plugs and conducted two types of water sensitivity tests: (1) determination of the critical salt concentration and (2) permeability impairment and restoration. In the first type test, permeability of the core plugs were monitored while injecting brines with slowly reduced salinities. For this purpose, they began injecting first the formation brine TDS = 120 000 ppm, and then brines were progressively diluted with deionized water. Figure 15-39 by Haggerty and Seyler (1997) show that Aux Vases formation is sensitive to waters having salinities below the salinity of the formation water and the water sensitivity is more pronounced at lower salinities. In the second type test, the formation, fresh water, and formation brines progressively diluted by fresh water were injected in separate tests until equilibrium and the permeability variations were measured. Figure 15-40 by Haggerty and Seyler (1997) shows that permeability decreased in all cases, even though halite precipitation from formation brine and dissolution in freshwater later caused temporary increase of permeability. They could not restore the permeability by injecting a brine of higher salinity into a core, exposed to fresh water, possibly because

636

Laboratory Evaluation of Formation Damage

Figure 15-36. SEM of a critical-point-dried sample from a typical Aux Vases reservoir (2400-ft depth, Budmark 3 Burr Oak, Energy Field) shows pore-lining and porebridging diagnetic clay minerals (closely intergrown mixed-layered illite/smectite, chlorite, and illite) and no quartz overgrowths (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

Permeability (mD)

80

60

40

20

0

0.0

0.5

1.0

1.5

2.0

Hours

Figure 15-37. Permeability of a core plug (2393.5-ft depth, Budmark 2 Morgan Coal, Energy Field) changed as the HCl–rock contact time increased during a continuous, constant rate acid flood test using 15% HCI (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

637

Laboratory Evaluation of Formation Damage

Table 15-10 Composition of Effluent from HCl Coreflow: Energy Field Sample, 2392.6-ft Depth, Budmark 2 Morgan Coal Well Effluent sampled at 1 hour 2 hours 2 days

1 hour 2 hours 2 days

mg/l Al

As

B

Ba

Be

265 250 564 Cd

02 01 01 Co

026 011 01 Cr

218 068 083 Cu

0004 0001 0002 Fe

004 046 001 K

040 046 034 La

1 hour 2 hours 2 days

9 7 6 Na

052 019 028 Ni

1 hour 2 hours 2 days

895 168 154 Sr

1 hour 2 hours 2 days

171 163 235

131 574 494 Ti 010 013 011

81 111 151 Li 019 047 012 Li

194 106 921 Mg 926 177 485 Mg

109 409 158 Mn

Ca 8250 1550 3130

Mo

154 409 749 Mn

408 212 154 Mo 178 157 192 Zr

008 014 004 Tl

0010 0009 0022 V

01 01 01 Zn

01 01 01

002 070 020

164 174 080

007 007 010

After Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey.

the clay minerals permanently swelled and/or migrated to and plugged the pore throats. Haggerty and Seyler (1997) concluded that permeability impairment by water flooding of Aux Vases reservoirs can only be avoided by injecting waters as saline as the formation brines. Oil–acid compatibility. Haggerty and Seyler (1997) investigated the compatibility of oil with 15% HCl without additives and with 15% HCl-MCA containing proprietary additives. They mention that the typical additives used in MCA solutions include demulsifying and/or antisludging agents.

638

Laboratory Evaluation of Formation Damage

(a)

Figure 15-38. SEMs illustrate the effects of soaking Aux Vases core plug wafers (2392.4-ft depth, Budmark 2 Morgan Coal, Energy Field) in a 15% HCl solution without any of the additives found in MCA. The most common precipitates in these samples are shown in Figures 15-38a–c. (a) Gypsum crystals precipitated because of the reaction of pore minerals and formation brine with 15% HCl. The EDX analysis shows that S and Ca are the predominant detectable elements in these crystals; (b) Blebs that do not display any crystalline structure and appear to be an amorphous or gel-like substance containing Fe and Cl, as identified by EDX analysis (The EDX unit used in this study detects elements with atomic numbers of 6 or greater.); (c) In another amorphous material, EDX analysis has identified Ca, Cl, Al, Si, and Fe. All these precipitates are attributed to the reaction of pore minerals and formation brine with 15% HCl because they have not been observed in any untreated samples; (d) The smooth quartz grains without their original clay mineral coatings (Figure 15-36) indicates widespread removal, disintegration, or dissolution of diagnetic clay minerals — the effects of longterm soaking in 15% HCl (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

As depicted in Figure 15-41 by Haggerty and Seyler (1997), 15% HCl without any additives formed an emulsified sludge with the oil and, therefore, it has the potential of reducing the fluid mobility in the reservoir, whereas the MCA used in their tests did not form any sludge, indicating that the antisludging agents were effective for preventing the emulsification process.

Laboratory Evaluation of Formation Damage

639

(b)

(c)

Figure 15-38. (Continued )

Summary of results. Haggerty and Seyler (1997) summarized the findings of their experimental investigations in Table 15-11. Presentation of results in concise forms, similar to Table 15-11, provides a convenient

640

Laboratory Evaluation of Formation Damage

(d)

Figure 15-38. (Continued )

Figure 15-39. Various mixtures of produced brine and deionized water were injected into a core plug (2388-ft depth, Budmark 2 Morgan Coal, Energy Field). Decreasing salinity corresponded with decreasing permeability, which suggests that injection of fluids less saline than formation brine can cause formation damage (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

Laboratory Evaluation of Formation Damage

641

Figure 15-40. Changes in permeability of core plugs tested with water of varying salinities. The most pronounced decrease occurred in high permeability samples from Energy and Boyd Fields. An early but brief increase in permeability in a low permeability sample from Energy Field was probably due to dissolution of halite precipitated when brine evaporated during air drying (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey).

means of translating the results of extensive and lengthy work to practical recipes for the benefit of the field operators and decision-makers.

15.8.2.6

Conclusions

Haggerty and Seyler (1997) determined that typical Aux Vases reservoir formation is “a poorly cemented, soft, friable, fine-grained sandstone with pores lined with diagenetic clay minerals (Figure 15-36). The diagenetic clay mineral suite in Aux Vases reservoirs is a closely intergrown mixture of mixed-layered illite/smetite, chlorite, and illite. No kaolinite was found in the Aux Vases reservoir rocks sampled.” They recommended that injection waters should be as saline as the formation brines and a properly formulated mud cleanout acid should be used to reduce formation damage.

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Laboratory Evaluation of Formation Damage

Figure 15-41. (a) Emulsified sludge formed immediately upon contact of Aux Vases oil with 15% HCl; (b) When MCA was added to Aux Vases oil, sludge did not form, presumably because of the surfactant in the MCA (after Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey). Table 15-11 Summary of Experimental Results Experiment

Figures

Change in Permeability

Crude oil/15% HCl

15-41

(Decrease)∗

Crude oil/MCA Continuous corefloods with MCA Continuous coreflood with HCl Discontinous coreflood with MCA HCl soak MCA soak Water sensitivity Gradual increase in freshwater Boyd (2163-ft depth) Energy (2388-ft depth) Energy (2190-ft depth) Dale (3200.6-ft depth) Zeigler (2611-ft depth)

15-41 15-33 and 15-34 15-37 15-32 15-38a–d 15-29b and 15-29d

(No change)∗ Increase 0–60% Increase up to 30% Increase up to 600% Could not measure Could not measure

15-39 15-40 15-40 15-40 15-40 15-40

Decrease 15–56% −600% to −760% −745% to −936% +346% to −235% +53% to −537% 0 to −584%

∗

Inferred from experiment results, not directly measured. After Haggerty and Seyler, 1997; reprinted by permission of the Illinois State Geological Survey.

Laboratory Evaluation of Formation Damage

643

Exercises 1. How does liquid block affect the flow capability of fluids in porous media? 2. How differently does a 15% HCl containing mud acid function from a 15% HCl containing aqueous acid solution? 3. Explain the purpose of oil–acid compatibility tests. 4. How does the Forchheimer equation help distinguish the true critical interstitial fluid velocity for particle mobilization in porous media? 5. Consider the experimental data given in Table 15-12, obtained in a laboratory test by flowing a 1.5 centipoise viscosity oil through a 5.0-cm-long and 2.5-cm-diameter sandstone core plug of 95 mD permeability and 25% porosity at the ambient temperature and do the following: a) Prepare a plot of pressure difference across the core plug divided by interstitial velocity vs. the interstitial velocity. b) Plot permeability against the interstitial velocity, Darcy velocity, and flow rate. c) Determine the critical interstitial velocity required for fine particle mobilization. 6. Referring to the study by Haggerty and Seyler (1997), do the following: a) Describe the properties of the core samples used in a concise and systematic manner. b) Describe the primary tests conducted and their objectives and significant findings in a concise and systematic manner.

Table 15-12 Experimental flow test data Pressure difference (atm)

25 50 120 140 190 230 140 85

Fluid flown (ml)

08 16 58 63 120 105 76 100

Time interval (seconds)

52 47 56 42 45 27 30 54

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Laboratory Evaluation of Formation Damage

c) Analyze the results reported in Figure 15-37 by means of the straight-line plotting methods of Wojtanowicz et al. (1987, 1988). d) Explain the significance of the results reported in Figure 15-39. e) Explain the significance of the results reported in Figure 15-40.

C

H

A

P

T

E

R

16

FORMATION DAMAGE SIMULATOR DEVELOPMENT∗ Summary An overview of the formation damage models pointing out their common bases and special features has been presented in previous chapters. In this chapter, the methodology for development of a formation damage simulator and applications for typical cases are presented. The equations forming the various models are classified into several groups: algebraic equations, ordinary differential equations, partial differential equations, initial and boundary conditions, and constraints. The numerical calculation schemes are developed for computer programming.

16.1 INTRODUCTION In spite of many experimental studies of the formation damage of oil- and gas-bearing formations, there have been only a few reported attempts to mathematically model the relevant processes and develop formation damage simulators. The use of these models in actual reservoir formation damage analysis and management has been rather limited because of the difficulties in understanding and implementing these models, as well as due to the limitations in the applicability of these models. Most present formation damage models consider a single fluid phase and the dominant formation damage mechanisms are assumed to be the mobilization, ∗ Parts of this chapter have been reprinted with permission of the U.S. Department of Energy and the Society of Petroleum Engineers from Civan (1994a,c).

645

646

Formation Damage Simulator Development

migration, and retention of fine particles in porous matrix. Although these models have been validated using experimental data obtained from reservoir core samples under controlled laboratory conditions, their applicability is rather limited in the field conditions. Most formation damage cases encountered in actual reservoirs are associated with multiphase flow and other factors which are not considered in the present singlephase formation damage models. In addition, determination of the model parameters has not been well addressed. Formation damage refers to permeability impairment by alteration of porous media due to rock–fluid and fluid–fluid interactions in geological porous formations. The phenomena leading to formation damage is a rather complicated process involving mechanical, physical, thermal, biological, and chemical factors. A formation damage model is a mathematical expression of the permeability impairment due to the alteration of the porous media pore structure and surface characteristics. This is an equation of permeability of formation undergoing an alteration. This must be a dynamic model, which is coupled with a porous media fluid flow model to predict the mutual effects of formation damage and flow conditions in oil and gas reservoirs. Therefore, although the main emphasis and objective are to develop a formation damage model, we must also address the modeling of fluid flow in porous media. Thus, the basic constituents of the overall modeling effort involve (1) porous media realization, (2) formation damage model, (3) fluid and species transport model, (4) numerical solution, (5) parameter estimation, and (6) model validation and application. In reality, porous matrix and the fluids contained within pore volume display a discrete structure. For convenience, however, a continuum approach using average properties defined over representative elemental porous media volume is preferred (Civan, 2002f). Model development begins with the realization of porous media. A conceptual view of the various levels of analysis and modeling needs to be developed in representing the processes occurring in a real system (the reservoir). The steps from microscopic to macroscopic scales are required for integration from local to global representation. Although it would be more rigorous to proceed through these steps, we often resort to a continuum modeling approach using the average properties over representative elemental porous media for simplification purposes. The loss of information on the process details is then compensated by empirical formulations. Empiricism cannot be avoided because of the irregular structure of geological porous media and disposition of various fluid phases and particulate matter. Porous media is considered in two parts: (1) the

Formation Damage Simulator Development

647

flowing phase, denoted by the subscript f, consisting of a suspension of fine particles flowing through and (2) the stationary phase, denoted by the subscript s, consisting of the porous matrix and the particles retained.

16.2 DESCRIPTION OF FUNDAMENTAL MODEL EQUATIONS The basic model equations are the mathematical expressions for the following (Civan, 1994a,c): 1. Mass balance a. Fluid phases (Gas/oil/water) b. Species – Solid (indigenous/external, water-wet/oil-wet/intermediately wet, swelling/nonswelling) – Ionic (anions/cations) – Molecular – Associates 2. Momentum balance – Fluid phases (Gas/oil/water) (Forchheimer/Darcy) 3. Porosity–Permeability–Texture relationship (equation of permeability variation) 4. Particle transport efficiency factor 5. Swelling rate a. Formation b. Particle 6. Pore throat plugging criteria and rate 7. Particle mobilization rate 8. Internal and external filter cake formation 9. Plug-type deposition rate 10. Pore surface deposition rate 11. External fluid infiltration rate a. Oil-based b. Water-based c. Emulsion 12. Effective porosity 13. Interface particle and species exchange rates 14. Critical salt concentration (salinity shock)

648

15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Formation Damage Simulator Development

Critical velocity (velocity shock) Critical temperature (temperature shock) Wettability index (water-wet, oil-wet, intermediately wet) Phase equilibrium conditions Non-Darcy coefficient correlation Dynamic pore size distribution Dynamic pore throat size distribution Chemical equilibrium Particle size growth Other phenomena.

The simulation input file requires the following information (Gruesbeck and Collins, 1982a,b; Amaefule et al., 1988; Baghdikian et al., 1989; Civan et al., 1989; Chang and Civan, 1992; Civan, 1994a,c): 1. Phenomena considered: • Dissolution/precipitation of mineral salts • Mobilization/retention of particles • Cation exchange between rock and fluids • Liquid absorption by porous matrix • Particle size variation • Crystallization processes 2. Injection fluid conditions: • Rate- or pressure-specified • Particle-free or particle-containing 3. Rock properties: • Core length • Core diameter • Initial porosity • Initial permeability • Dispersion coefficient in porous media • Diffusion coefficient for liquid in porous matrix 4. Properties of the fluid and suspended particles: • Viscosity of liquid • Density of porous matrix material • Density of mineral salts • Density of clay particles • Density of externally injected particles • Critical velocity for particle mobilization • Critical salt concentration

Formation Damage Simulator Development

649

5. Stoichiometric coefficients for chemical reactions in liquid media • Aqueous Phase 6. Conditions of the problem: • Initial conditions in porous media – Concentration of ionic, molecular, and particulate species • Boundary conditions a. Injection end – Constant pressure or flux – Species concentration or flux b. Outlet end – Constant pressure – Species flux 7. Model parameters: • Delete the parameters of the mechanisms neglected for a specific problem. • Assign the measured values for the parameters that are directly measurable by laboratory procedures. • Identify the parameters for which the best estimates will be obtained by history matching. 8. Laboratory core flow test data that will be used for history matching: • Input–Output pressure differential or input volume flux vs. pore volume injected. • Effluent pH and species concentrations vs. pore volume injected. 9. Output that can be requested: • Best estimates of the unknown parameters • Predicted vs. measured data • Simulation of pressure; various species concentrations in the flowing fluid and the pore surface; porosity and permeability as functions of pore volume injected or time. Many examples of the types of mathematical expressions required for various constituents of formation damage models have been provided throughout this book.

16.3 NUMERICAL SOLUTION OF FORMATION DAMAGE MODELS Depending on the level of sophistication of the considerations, theoretical approaches, mathematical formulations, and due applications, formation

650

Formation Damage Simulator Development

damage models may be formed from algebraic equations, and ordinary and partial differential equations, or a combination of such equations. Numerical solutions are sought under certain conditions, defined by specific applications. The conditions of solution can be grouped into two classes: (1) initial conditions, defining the state of the system prior to any or further formation damage, and (2) boundary conditions, expressing the interactions of the reservoir system with its surrounding during formation damage. Typically, boundary conditions are required at the surfaces of the system through which fluids enter or leave, such as the injection and production wells or ports, or those that undergo surface processes such as exchange or reaction processes. Algebraic formation damage models relate the process variables by algebraic expressions. These are either empirical correlation and/or obtained by analytical solution of differential equation models for certain simplified cases. Numerical solution methods for linear and nonlinear algebraic equations are well developed. Ordinary differential equation models describe the variation of processes in a single variable, such as either time or one space variable. However, as demonstrated in the following sections, in some special cases, special mathematical techniques can be used to transform multivariable partial differential equations into single-variable ordinary differential equations. Amongst these special techniques are the methods of combination of variables and separation of variables, and the method of characteristics. The numerical solution methods for ordinary differential equations are well developed. Partial differential equation models involve the variation of processes in two or more independent variables. There are many numerical methods available for solution of partial differential equations, such as the finite difference method (Thomas, 1982), finite element method (Burnett, 1987), finite analytic method (Civan, 1995b), and the method of weighted sums (the quadrature and cubature methods) (Civan, 1994b,d,e, 1995b, 1996c, 1998a; Malik and Civan, 1995; Escobar et al., 1997). In general, implementation of numerical methods for solution of partial differential equations is a challenging task. Although numerical simulators can be developed from scratch as demonstrated by the examples given in the following sections, we can save a lot of time and effort by taking advantage of ready-made software available from various sources. For this purpose, the spreadsheet programs are particularly convenient and popular. Various software for solving algebraic, ordinary and partial differential equations are available.

Formation Damage Simulator Development

651

Commercially available reservoir simulators can be manipulated to consider formation damage, such as by paraffin deposition as demonstrated by Wang and Civan (2005a,b,c). In the following sections, several representative examples are presented for instructional purposes. They are intended to provide some insight into the numerical solution process. Interested readers can resort to many excellent references available in the literature for details and sophisticated methods. For most applications, however, the information presented in this chapter is sufficient and a good start for those interested in specializing in the development of formation damage simulators.

16.4 ORDINARY DIFFERENTIAL EQUATIONS In this section, several examples are given to illustrate the numerical solution of ordinary differential equation models. Specifically, the simplified formation damage and filtration models, developed in previous chapters, are solved.

16.4.1 Example 16-1: Wojtanowicz et al. Fines Migration Model a. Derive a numerical solution for the following modified Wojtanowicz et al. (1987, 1988) fines migration model: d AL c +  = q cin − c  t > 0 dt

(16-1)

d = kd c − ke  t > 0 dt

(16-2)

 = o −

  t>0

(16-3)

subject to c = 0  = 0  = o  t = 0

(16-4)

652

Formation Damage Simulator Development

b. Plot c and  vs. t using the following data until  = 0. The following data are given: A = 1 0 cm2 , L = 1 0 cm, o = 0.20, q = 0 5 cm3 / min, cin = 0.85 gr/cm3 , = 1.0 g/cm3 , kd = 0 7 min−1 , ke = 0.2 min−1 . Expanding Eq. 16-1 and then substituting Eqs (16-2) and (16-3) and rearranging yields   q c cin − c − 1 − kd c − ke  dc AL (16-5) =  dt o − A simultaneous solution of Eqs (16-2) and (16-5) as a function of time, subject to the initial conditions given by Eq. (16-4), can be readily obtained using an appropriate method, such as by the Runge–Kutta– Fehlberg four (five) method available in many ordinary differential equation solving software. Then, the porosity variation is calculated by Eq. (16-3). A typical numerical solution is presented in Figure 16-1. ˇ nanský and Široký Fines 16.4.2 Example 16-2: Cerˇ Migration Model The numerical solution is carried out for cr′ = 0. Here, the numerical ˇ nanský and Široký (1985) is described. solution approach presented by Cerˇ Define the dimensionless time and distance, respectively, by

Figure 16-1. Particle concentration and porosity vs. time.

Formation Damage Simulator Development

653

T = t/L/u

(16-6)

X = x/L

(16-7)

Thus, invoking Eqs (16-6) and (16-7), their formulation, described in Chapter 10, can be summarized as following:  + =0 T X

(16-8)

= kd L o −  − ke′ L /K T

(16-9)

The conditions of solution are given by the following initial and boundary conditions, respectively:

= o 

 = o 

 = in 

0 ≤ x ≤ L x = 0

t=0

t>0

(16-10) (16-11)

Equations (16-8) and (16-9) are a system of hyperbolic partial differential equations, which can be transformed into a system of ordinary differential equations by means of the method of characteristics as d = −f   dX

(16-12)

d = f   dT

(16-13)

f   = kd L o −  − ke′ L /K

(16-14)

in which

The characteristics are given by dT = 0  or T = constant dX

(16-15)

dX = 0 or X = constant dT

(16-16)

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Formation Damage Simulator Development

The conditions of solution for Eqs (16-8) and (16-9) are

= o   = o  0 ≤ X ≤ 1  T = 0

(16-17)

 = in  X = 0 T ≥ 0

(16-18)

Applying the condition given by Eq. (16-17), Eq. (16-12) becomes d = −kd Lo dX

(16-19)

for which the analytic solution at T = 0 and considering the boundary condition given by Eq. (16-18) is given by  = in exp −kd Lo X

(16-20)

The system of ordinary differential equations given by Eqs (16-12) and (16-13) are solved by means of the fourth-order Runge–Kutta method, subject to the conditions given by Eqs (16-17) and (16-18) along the characteristic represented by Eq. (16-16). Figure 16-2 shows the dimensionless effluent particles concentration as a function of the filtrate volume per unit area. Figures 16-3 and 16-4 show, respectively, typical suspended particle concentration and the particles retained in porous media as a function of distance along the porous media at different times. 16.4.3 Example 16-3: Civan’s Incompressive Cake Filtration without Fines Invasion Model The equations of Civan’s (1998, 1999) incompressive cake filtration model are given in Chapter 12. As described in Chapter 12, the ordinary differential equations of this model have been solved by the Runge– Kutta–Fehlberg four (five) numerical scheme (Fehlberg, 1969), subject to prescribed initial conditions. 16.4.4 Example 16-4: Civan’s Compressive Cake Filtration Including Fines Invasion Model The equations of Civan’s (1998b,c, 1999b,c) compressive cake filtration including fines invasion model are given in Chapter 12. As described

655

Formation Damage Simulator Development 1.0 5 10 15 C /C0

20

0.5

50

H = 100 mm 0

0

10

20

30 (Q/A) [m]

40

50

Figure 16-2. Experimental and simulated dimensionless concentrations vs. filtrate volume per unit cross-sectional area for the POP 1 material, using particle den3

3

3

3

sity p = 2825 kg/m  cin = 01 kg/m  in = 01 kg/m /2825 kg/m  = 354 × 3 ˇ nanský, A., 10−5 m3 /m  H = 5, 10, 15, 20, 50, and 100 mm, and u = 05 cm/s (Cerˇ and Široký, R., 1985; reprinted by permission of the AIChE, ©1985 AIChE, all rights ˇ nanský and Široký, 1982, reprinted by permission). reserved; and after Cerˇ

in Chapter 12, the ordinary differential equations of this model have been solved by the Runge–Kutta–Fehlberg four (five) numerical scheme (Fehlberg, 1969), subject to prescribed initial conditions.

16.5 PARTIAL DIFFERENTIAL EQUATIONS In this section, the application of the finite-difference method for solution of partial differential type models is illustrated by several examples. 16.5.1

Finite-Difference Method

The method of finite differences is one of many methods available for numerical solution of partial differential equations. Because of its

656

Formation Damage Simulator Development 1.0

in 0m 48 0

C

t=

40

0.5

0 32

0

24

16 0 80

2

0

0

0.5 X

1.0

Figure 16-3. Simulated dimensionless concentration vs. dimensionless distance at different times for the FINET-PES 1 material, using particle density 3

3

3

3

p = 2825 kg/m  cin = 01 kg/m  in = 01 kg/m /2825 kg/m  = 354 × 3 ˇ nanský, A., and Široký, R., 1985; 10−5 m3 /m  H = 100 mm, and u = 05 cm/s (Cerˇ reprinted by permission of the AIChE, ©1985 AIChE, all rights reserved; and after ˇ nanský and Široký, 1982, reprinted by permission). Cerˇ

simplicity and convenience, the method of finite differences is the most frequently used numerical method for solution of differential equations. This method provides algebraic approximations to derivatives so that differential equations can be transformed into a set of algebraic equations, which can be solved by appropriate numerical procedures. Although the finite difference approximations can be derived by various methods, a simple method based on the power series approach is presented here to avoid complicated mathematical derivation. Interested readers may resort to many excellent textbooks and literature available on the finitedifference method. Nevertheless, the information provided in this chapter is sufficient for many applications and for the purpose of this book. Most transport phenomenological models involve first- and second-order derivatives. Therefore, the following derivation is limited to the development of the first- and second-order derivative formulae. However, the higher-order derivative formulae can be readily derived by the same approach presented in this chapter.

657

Formation Damage Simulator Development 240

t=

160

0

24

16 0

80

in 0m 48 0 40

120

0 32

ερp [kg/m3]

200

80

40 2 0

0

0.2

0.4

0.6

0.8

1.0

X

Figure 16-4. Simulated mass of particles retained per unit volume of porous material vs. dimensionless distance at different times for the FINET-PES 1 material, using particle 3

3

3

3

density p = 2825 kg/m  cin = 01 kg/m  in = 01 kg/m /2825 kg/m  = 354× 3 ˇ nanský, A., and Široký, R., 1985; 10−5 m3 /m  H = 100 mm, and u = 05 cm/s (Cerˇ reprinted by permission of the AIChE, ©1985 AIChE, all rights reserved; and after ˇ nanský, A., and Široký, R., 1982, reprinted by permission). Cerˇ

16.5.1.1

First-order derivatives

In general, a function can be approximated by a power series as f x =

  i=0

ai xi = a0 + a1 x + a2 x2 + · · ·

(16-21)

in which a0  a1  a2 are some fitting coefficients. To determine the fitting coefficients, consider any set of three discrete function values fi−1  fi , and fi+1 located at the sample points xi−1  xi , and xi+1 , respectively, as shown in Figure 16-5. More points could be considered for better accuracy. Higher-order accurate finite-difference formulae can be derived easily using the quadrature method as described by Civan (1994b,d,e). Case 1 – First-order Accurate Finite-difference Formulae

With three points, we can write the following three linear approximations at i − 1 i, and i + 1: fi−1 = a0 + a1 xi−1

(16-22)

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Formation Damage Simulator Development

Figure 16-5. Sample points considered for the finite-difference method.

fi = a0 + a1 xi

(16-23)

fi+1 = a0 + a1 xi+1

(16-24)

If the middle point is considered as a reference point, then the locations of the three equally spaced points are given by xi−1 = −x xi = 0 xi+1 = x

(16-25)

Thus, substituting Eq. (16-24) into Eqs (16-21)–(16-23), and then solving the resultant two algebraic equations simultaneously yields the following expressions for the fitting coefficients of the quadratic expression: a0 = fi a1 =

fi − fi−1 x

(16-26) (16-27)

On the other hand, the derivative of Eq. (16-21) for linear approximation is given by df = a1 dx

(16-28)

Thus, the following forward difference formula for the first-order derivative is obtained by substituting Eq. (16-26) into Eq. (16-28): dfi−1 dfi fi − fi−1 = = dx dx x

(16-29)

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Formation Damage Simulator Development

Similarly, using Eqs (16-23) and (16-24), the following backward difference formula for the first-order derivative is obtained: dfi+1 dfi fi+1 − fi = = dx dx x

(16-30)

Case 2 – Second-order Accurate Finite-difference Formulae

With three points, we can write the following three quadratic approximations at i–1 i, and i + 1: 2 fi−1 = a0 + a1 xi−1 + a2 xi−1

(16-31)

fi = a0 + a1 xi + a2 xi2

(16-32)

2 fi+1 = a0 + a1 xi+1 + a2 xi+1

(16-33)

If the middle point is considered as a reference point, then the locations of the three equally spaced points are given by xi−1 = −x xi = 0 xi+1 = x

(16-34)

Thus, substituting Eq. (16-34) into Eqs (16-31)–(16-33), and then solving the resultant three algebraic equations simultaneously yields the following expressions for the fitting coefficients of the quadratic expression: a0 = fi a1 = a2 =

fi+1 − fi−1 2x

fi−1 − 2fi + fi+1 2 x2

(16-35) (16-36) (16-37)

On the other hand, the derivative of Eq. (16-21) for quadratic approximation is given by df = a1 + 2a2 x dx

(16-38)

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Formation Damage Simulator Development

Thus, the following forward difference formula for the first-order derivative is obtained by substituting Eqs (16-36) and (16-37) into Eq. (16-38) for a1  a2 at x = xi−1 = −x: dfi−1 −3fi−1 + 4fi − fi+1 = dx 2x

(16-39)

The central difference formula for the first-order derivative is obtained as, by substituting Eqs (16-36) and (16-37) for a1  a2 into Eq. (16-38) at x = xi = 0: dfi fi+1 − fi−1 = dx 2x

(16-40)

The backward difference formula for the first-order derivative is obtained as, by substituting Eqs (16-36) and (16-37) for a1  a2 into Eq. (16-38) at x = xi+1 = x: dfi+1 fi−1 − 4fi + 3fi+1 = dx 2x 16.5.1.2

(16-41)

Second-order derivatives (second-order accurate)

A similar procedure can be applied to derive the second (and higher)order derivative approximations. However, the formulae presented in this section are second-order accurate. Thus, consider a power series expansion as f ′ x =

 df  = bi xi = b0 + b1 x + b2 x2 + · · · dx i=0

(16-42)

Expressions similar to Eqs (16-35)–(16-37) are obtained for the fitting coefficients, given by b0 = fi′ b1 = b2 =

′ ′ fi+1 − fi−1 2x

′ ′ fi−1 − 2fi′ + fi+1

2 x2

(16-43) (16-44) (16-45)

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Formation Damage Simulator Development

The derivative of the quadratic equation is obtained from Eq. (16-42) as f ′′ = b1 + 2b2 x

(16-46)

Thus, the forward difference formula for the second-order derivative is obtained as, by substituting Eqs (16-44) and (16-45) for b1  b2 at x = xi−1 = −x into Eq. (16-46): ′ ′ + 4fi′ − fi+1 d2 fi−1 −3fi−1 = dx2 2x

(16-47)

The central difference formula for the second-order derivative is obtained as, by substituting Eqs (16-44) and (16-45) for b1  b2 at x = xi = 0 into Eq. (16-46): ′ ′ − fi−1 d2 fi fi+1 = dx2 2x

(16-48)

The backward difference formula for the second-order derivative is obtained as, by substituting Eqs (16-44) and (16-45) for b1  b2 into Eq. (16-46) for x = xi+1 = x: ′ ′ − 4fi′ + 3fi+1 d2 fi+1 fi−1 = dx2 2x

(16-49)

However, only the central second-order derivative formula is used in our models. Thus, substituting the first-order forward and backward difference formulae given by Eqs (16-39) and (16-41) into Eq. (16-48), the central second-order difference formula for the second-order derivative is obtained as ′ ′ − fi−1 d2 fi fi+1 = = dx2 2x

=

fi−1 − 4fi + 3fi+1 −3fi−1 + 4fi − fi−1 − 2x 2x 2x fi−1 − 2fi + fi+1 x2

(16-50)

Note: The same result can be obtained by applying Eqs (16-29) and (16-30) at the midpoints as, because the slope remains the same along a straight-line, dfi−1/2 fi − fi−1 = dx x

(16-51)

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Formation Damage Simulator Development

dfi+1/2 fi+1 − fi = dx x

(16-52)

and then substituting Eqs (16-51) and (16-52) into the following expression: ′ ′ − fi−1/2 d2 fi fi+1/2 = dx2 x

(16-53)

16.5.2 Example: Civan and Engler Mud Filtrate Invasion Model The normalized equations of the Civan and Engler (1994) model (see Chapter 18) are given by the transport equation: 

2 c c c − = 2 r r t

(16-54)

subject to the initial condition c = 0

0 ≤ r ≤ 1

t=0

(16-55)

and the inlet and outlet boundary conditions uc −

1 c D = ucin  Pe r c = 0 r

r = 1

r = 0 t>0

t>0

(16-56) (16-57)

For numerical solution purposes, the time–space solution domain is discretized as shown in Figure 16-6, by separating the time and space into a number of equally spaced discrete points. r and t denote the grid point spacing and time increment, respectively. Accurate solution with a uniform grid requires sufficiently small r and t, and therefore a high level of computational effort. This approach is selected here for simplicity. Computationally efficient schemes can be developed by varying the grid size in space and time. This is beyond the scope of this book. The grid system depicted in Figure 16-6 is implemented here by central finite-difference formulae, derived in the previous section. The spatial

663

Formation Damage Simulator Development Fictitious block

i =0

t, Temporal domain

Time–space computational molecule

Fictitious block

Reservoir

1

i –1

i +1

i

N

N +1

Steady-state (Final time: t→∞)

n+1 n 1 ∇

∇

r

j=0 i=0

t

Initial-state (Initial time: t = 0) N 1 i –1 i i +1 r, Spatial domain

N+1

Figure 16-6. Discretization of the time–space computational domain and the grid system.

grid points are denoted by the subscript indices i = 0 1 2     N N +1. Because the spatial grid points were placed in the center of the grid blocks, the points identified by i = 0 and i = N +1 are outside the inlet and outlet boundaries and, therefore, called fictitious points. The points designated by i = 1 2 N are the real points, called interior points. The discrete times are denoted by the superscript indices n = 0 1 2     . n = 0 denotes the initial condition, at which time the concentrations at various discrete spatial points are prescribed by Eq. (16-55). The central difference formulae necessary to develop a numerical solution are given as follows: ci ci+1 − ci−1 = r 2r

(16-58)

2 ci ci+1 − 2ci + ci−1 = r r2

(16-59)

664

Formation Damage Simulator Development

ci−1/2 ci − ci−1 = r r

(16-60)

ci+1/2 ci+1 − ci = r r

(16-61)

cn+1 cn+1 − cn = t t

(16-62)

The concentration values at the inlet and outlet boundaries are estimated by the following arithmetic averages: ci−1/2 =

ci−1 + ci 2

(16-63)

ci+1/2 =

ci + ci+1 2

(16-64)

Applying the Crank–Nicolson formulation (see Thomas, 1982), the central difference discretization of Eq. (16-54) in time and space yields  n n n − 2cin + ci−1 cn − ci−1 1 n ci+1 n i+1 i −  i r 2  2 2r n+1 n+1 n+1 n+1 n+1  + ci−1 n+1 ci+1 − 2ci n+1 ci+1 − ci−1 + i − i r 2  2r =

cin+1 − cin  i = 1 2     N and n = 1 2      t

(16-65)

The initial condition given by Eq. (16-55) can be expressed as ci = 0  i = 1 2     N n = 0

(16-66)

The inlet boundary condition given by Eq. (16-56) can be discretized as: ui=1/2 ci=1/2 −

Di=1/2 ci=1 − ci=0 = ucin  i = 1/2 Pe r

(16-67)

in which ci=1/2 =

1 c + ci=1  2 i=0

(16-68)

665

Formation Damage Simulator Development

Thus, substituting Eq. (16-68) into (16-67), the fictitious point value is determined as    Bc1 A (16-69) c0 = 1 −  where

D u  A= + 2 Per

D u  B= − 2 Per

(16-70) (16-71)

The outlet boundary condition given by Eq. (16-57) is discretized as cN +1 − cN =0 r

(16-72)

from which the fictitious point value is obtained as (16-73)

cN +1 = cN

For convenience in numerical solution, first we rearrange Eq. (16-65) as ci−1 + Bin+1 ci + Cin+1 ci+1 = Din  i = 1 2     N and An+1 i n = 1 2     

(16-74)

in which Ai =

i 2

+

i 2 r

 r   1 i Bi = −2 +  r2  t Ci =

i 2

 r

+

i 2 r



n n Din = − Ani ci−1 + Bin cin + Cin ci+1

(16-75) (16-76) (16-77) (16-78)

Next, we incorporate the fictitious point values near the inlet and outlet boundaries into Eq. (16-74). For this purpose, applying Eq. (16-74) at the

666

Formation Damage Simulator Development

inlet and outlet grid blocks by substituting i = 1 and i = N , respectively, yields n+1 An+1 + B1n+1 c1n+1 + C1n+1 c2n+1 = D1n  i = 1 1 c0

(16-79)

n+1 n+1 n+1 n+1 n+1 n An+1 N cN −1 + BN cN + CN cN +1 = DN  i = N

(16-80)

Substituting Eqs (16-69) and (16-73) into Eqs (16-79) and (16-80), respectively, yields n

n+1

   1  i = 1 B1 c1n+1 + C1n+1 c2n+1 = D n+1

 n+1 n+1  = DNn  i = N An+1 N cN −1 + BN cN

in which

B  A A1  ˜  B 1 = B1 − 1  D 1 = D1 −   A A   B N = B N + CN

(16-81) (16-82)

(16-83) (16-84)

Equation (16-81) for i = 1, Eq. (16-74) for i = 2 3     N –1 N , and Eq. (16-82) for i = N can be compiled in a form of a tri-diagonal coefficient linear matrix equation as ⎡  B1 ⎢A ⎢ 2 ⎢ ⎢ ⎢   ⎢ ⎣

C1 B2



Ai

  C2     B i Ci 

0       AN

⎡ ⎤n  ⎡ ⎤ 1 D ⎤n+1 cn+1 1 ⎢ ⎥ 0 ⎢ cn+1 ⎥ ⎢ D2 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎥ ⎥ ⎢

⎥ ⎢

⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥=⎢ D ⎥ ⎥ ⎢ ⎥ ⎢

   ⎥ ⎢

⎥ ⎢ i ⎥ ⎥ ⎦ ⎢ n+1 ⎥ ⎢

⎥ ⎢ ⎥ ⎣ ⎦ cN −1  ⎣D ⎦  n+1 BN N −1 cN DN

(16-85)

and solved by Thomas algorithm (see Thomas, 1982). Figure 16-7 shows the typical concentration profiles calculated by Civan and Engler (1994) at different times using the parameter values a = 0 08 m3 /h b = 1 67×10−5 h−1  f = 51 7 g = 1 25 h = 0 5 m rw = 0 05 m, and re = 10 m.

667

Formation Damage Simulator Development 0.30

Mud concentration, c, kg/m3

0.25

0.20

t= 0.15

10

0h

r

t=

50

0.10

t= 0.05

0.00 0.05

t= t=

5

10

1

1.05

2.05

7.05 4.05 6.05 3.05 5.05 Radial distance from wellbore, r, meters

8.05

9.05

Figure 16-7. Mud filtrate concentration vs. radial distance from wellbore at different times (reprinted from Journal of Petroleum Science and Engineering, Vol. 11, Civan, F., and Engler, T., “Drilling Mud Filtrate Invasion – Improved Model and Solution,” pp. 183–193, ©1994; reprinted with permission from Elsevier Science).

Note that in this presentation, the dimensionless quantities were defined as follows: c = C/Co  r = R − rw  re − rw   t = T To

(16-86) (16-87) (16-88)

where R denotes the radial distance, T is the time, To is a characteristic time, Co and C denote the initial and instantaneous filtrate concentrations, rw and re are the wellbore and outlet boundary radii, and Eq. (16-87) is different than the dimensionless distance  r = R rw (16-89) used by Civan and Engler (1994). Eq. (16-87) is preferred because it maps the radial distance over a unit-size domain.

668

Formation Damage Simulator Development

Exercises 1) Consider the Wojtanowicz et al. (1987, 1988) model for fines migration described by Eqs (16-2), (16-3), and (16-5), subject to the initial conditions given by Eq. (16-4). a. Obtain a numerical solution of this model until the porosity reduces to zero. For this purpose, apply a ready-made computer program, such as based on the Runge–Kutta numerical scheme. b. Present a plot of the results similar to Figure 16-1 using the parameter values given in the book. Compare your results with those given in Figure 16-1. c. Repeat (a) using a different set of parameter values of your choice. 2) Consider the one-dimensional diffusion problem given by Eqs (2-2)– (2-5). Assume a set of reasonable values for the parameters. Approximate the infinite distance from the pore surface by a sufficiently large value. a. Transform the equations of the model into dimensionless forms using an appropriate set of dimensionless variables similar to Eqs (16-72)–(16-74). b. Convert the differential equations of the model to a set of algebraic equations by means of the finite-difference approximations. c. Compose matrix equations of the resultant algebraic equations. d. Solve the resulting matrix equation by means of a ready-made computer program. e. Present a plot of typical results at various times until the steady state. f. Present details, computer program, and typical numerical output. ˇ nanský and Široký (1985) fines migration model 3) Consider the Cerˇ given in the linear coordinate for core plugs in this chapter. a) Express their equations in the radial coordinates by considering the region around an injection well, extending over rw ≤ r ≤ re , where rw and re denote the radii of the wellbore and near-wellbore-region external boundary, respectively. b) Express the radial flow and particle deposition equations in the dimensionless time T and dimensionless radial distance R, defined as follows: T = t/to R=

r 2 − rw2 re2 − rw2

(16-90) (16-91)

Formation Damage Simulator Development

669

where t and r denote the time and radial distance from the well centerline. to is a characteristic time scale, assumed constant. c) Express the radial flow and particle deposition equations in the dimensionless time T and dimensionless radial distance R, defined as follows: T = t/to

(16-92)

R = r/rw 2 − 1

(16-93)

C

H

A

P

T

E

R

17

MODEL-ASSISTED ANALYSIS AND INTERPRETATION OF LABORATORY AND FIELD TESTS

Summary The methodology for the model-assisted analysis and interpretation of laboratory and field data using formation damage simulators is presented. The optimal strategies are developed for the identification of the governing formation damage mechanisms and their relative contributions and importance, estimation and correlation of the model parameters, model calibration by history matching, validation and improvement, and sensitivity and simulation studies. The typical laboratory and field data are reviewed. The model parameters are classified into the groups of directly measurable parameters and the parameters that can be determined by the history matching method. The impact of the errors associated with measurements and data on the calculations and predictions is delineated. An objective function is formulated in terms of the directly measurable formation damage indicators such as injectivity or productivity loss, permeability impairment, and effluent fluid conditions, including fines concentration, pH, and species content. Applications to laboratory and field systems are illustrated with several examples. 670

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

671

17.1 INTRODUCTION J. Willard Gibbs stated that “The purpose of a theory is to find that viewpoint from which experimental observations appear to fit the pattern” (Duda, 1990). Toward this end, we carry out theoretical analysis and modeling, and experiments under controlled conditions in an effort to predict the behavior of the interactions of the reservoir rock and fluid systems. However, model predictions usually involve uncertainties because of the approximations in systems description and uncertainties in the parameters and measurements. Luckert (1994) draws attention to the fact that “the models often contain only differential values while the experimental values are integrals.” Thus, for direct comparison with experimental data, the models must be transformed into integral forms either by analytical or numerical solution methods. As explained by Frenklach and Miller (1985), the predictive equations of natural phenomena, frequently called mathematical models, are usually derived in the form of differential and/or integral equations, and often solutions can only be obtained by numerical methods. Frenklach and Miller (1985) stress that the dynamic model-building process has to deal with several important issues: (1) adequacy, (2) statistical reliability of the proposed model, and (3) determination of its parameters. Frenklach and Miller (1985) describe the usual approach taken to determine the model parameters as an iterative adjustment of the parameter values until the numerical solution of the model, called the model response or prediction, fits the experimental data. They add that frequently the adjustment of the parameter values is guided by the sensitivity analysis based on the partial derivatives of the predictions of the model with respect to its parameters. Frenklach and Miller (1985) draw attention to several problems associated with this approach: (1) in the statistical sense, the sensitivity is physically meaningful only if the model is adequate, (2) sensitivity varies during a dynamic process and, therefore, point estimates of sensitivities in the parameter space are not adequate, and (3) correlating sensitivities independently of each other complicates the interpretation of the sensitivities. Frenklach and Miller (1985) circumvent these problems by incorporating the parameter estimation, adequacy test, and sensitivity analysis tasks into mathematical modeling. Their parameter estimation approach is based on developing an analytically or numerically determined functional relationship between the response and parameters of the model. For this purpose, the individual responses of the dynamic model for different prescribed

672

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

values of the parameters are obtained by means of the numerical solution of the model. Then, a functional relationship between the model responses and the prescribed parameter values is developed by applying a statistical analysis. They recommend the application of the experimental design techniques to improve the efficiency of this method. The objectives of the model-assisted analyses and interpretations include the identification of the governing formation damage mechanisms and their relative contributions and importance; estimation and correlation of the model parameters; model calibration via history matching; model verification and improvement; and sensitivity and simulation studies (Civan, 1996a). Direct measurement of all the model parameters is usually not feasible when the model involves many parameters. Therefore, many researchers (Civan et al., 1989; Ohen and Civan, 1990, 1993; Millan-Arcia and Civan, 1992; Chang and Civan, 1991, 1992, 1997; and Liu and Civan, 1995, 1996; Civan, 1994a, 1996a; Willhite et al., 1991; Vitthal et al., 1988; Civan and Nguyen, 2005) have resorted to indirect methods of inferring the values of such parameters by history matching of some experimental data. Although others (Gruesbeck and Collins, 1982a; Amaefule et al., 1988; Sharma and Yortsos, 1987a,b,c; Khilar and Fogler, 1987; Civan, 1998b) offer some analytical expressions and/or direct measurement methods, these apply only to extremely simplified models having only a few model parameters. For complicated models, history matching appears the best choice in lack of a better method. However, some parameters may be measured and the remainder can be estimated by an optimal history matching method to minimize an objective function expressing the weighted sum of the squares of the deviations between the directly measured and the model-predicted formation damage indicators such as pressure loss, permeability impairment, and effluent conditions (Civan, 1996). For this purpose, simulated annealing is appealing as a practical optimization method (Szücs and Civan, 1996; Szücs et al., 2006) because it does not require any derivative evaluations and it leads to global minimum without being trapped in one of the local minimal. However, the achievability of the uniqueness of the estimated parameter values depends on and increases by the amount of the measured data. Ucan et al. (1997) have demonstrated that uniqueness can be achieved if both the external and the internal core fluid data are used simultaneously. Typical internal data include the sectional pressure difference and fluid saturations along the core plug. Typical external data include the pressures at the core inlet and outlet, and the effluent solution properties.

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

673

Ready-made simulators can be used to determine and review the formation damage potential, under various production scenarios, to seek for optimal strategies to avoid and/or minimize formation damage in petroleum reservoirs. Simulators provide convenient, rapid, and economic means of investigating the formation damage associated with various prospects and applications, including acid treatment of wells, petroleum recovery processes, and geochemical alteration of reservoir formations, as demonstrated in the following sections.

17.2 MEASUREMENT ERROR∗ Errors associated with the measurements and input data can affect the accuracy of prediction and simulation of the formation damage processes. Measurements are uncertain numbers that are random and independent variables (Reilly, 1992). As stated by Cook (1980): Error is the uncertainty in a measured quantity. An opposite expression is accuracy which is the reliability of the measurement. Precision, on the other hand, is the repeatability or reproducibility of a measurement. Hence, the measurements can be precise but not accurate – meaning that there is a systematic error.

There are three main sources of errors that affect the accuracy of measurements (Civan, 1989). The first is the human errors resulting from improper handling of instruments and incorrect readings of indicators such as the thermometer, manometer, clock, pressure gauge, etc. The second source of error is the systematic errors in the instruments themselves. Errors involving various elements of instruments can accumulate and lead to pronounced errors in the value of the measurements. The third source of errors is the statistical errors, which are not predictable. Fluctuations in ambient pressure and temperature, and in electric power supply are examples of statistical errors. Statistical and human errors can be referred conveniently to as random errors. Human errors can be minimized by using the instruments carefully and maintaining them in good condition, but errors in instruments are systematic and often are undetected. ∗ Reproduced by permission of the Society of Petroleum Engineers from Civan, ©1989, SPE, Paper 19073.

674

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

Statistical errors are unavoidable. Thus, repeated measurements of variables should be taken to obtain statistically good results (although this will not change the systematic errors).

17.2.1

Random Error

In the following, first the relations for the error estimate referring only to the random error are discussed. The values of variables, xi , measured at i = 1 2     n, repeated tests differ somewhat from each other. Thus, an arithmetic mean value of the measured values should be used, defined by x=

n 1 x n i=1 i

(17-1)

However, the mean value alone does not indicate the extent of reproducibility of the measurements. Therefore, an estimate of the confidence limits should also be given. In this respect, one of the following forms of random error estimates can be used, assuming the random errors are normally distributed: a. Standard deviation

in which

 1/2 x = x2 − x2

x2 =

n 1 x2 n i=1 i

(17-2)

(17-3)

b. Average random error Ax = 07979x

(17-4)

Px = 06745x

(17-5)

c. Probable error

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

675

Hence, the measured value is reported in terms of its mean value and the random error or confidence limits as x ± E, where E can be represented by one of the random error expressions (i.e., x  Ax , or px ). See Mickley et al. (1957), Spiegel (1961), and Schenck, Jr (1961) for details. Composite random error in a calculated value obtained by a series of calculational steps can be estimated analytically based on the accuracy of various quantities involving these calculation steps. For this purpose, the upper bounds of the overall random error can be determined by applying the rules given in the next section. 17.2.2

Systematic Error

As stated by Ku (1969), “systematic error is a fixed deviation that is inherent in each and every measurement.” Hence, the measurements can be corrected for the systematic error if the magnitude and direction of the systematic error are known. Complex devices make it difficult to predict their accuracy. Leaks, and variation of temperature and pressure also influence the accuracy. Large volume flow-type tests suffer from sudden variation of species composition due to their larger residence times. All of these are sources of errors. The mechanical design parameters and dimensions of experimental systems immensely effect the accuracy of measurements. Careful analysis of their design and innovative improvements to increase their accuracy are vital for measurement with better accuracy. 17.2.3 Error Analysis – Propagation, Impact, Estimation Although the phenomenological descriptions of processes are generally accomplished in terms of differential equations called mathematical models, solutions can be obtained analytically for simplified cases and numerically for complicated cases. Accuracy of model predictions is dependent on various factors, including (1) the adequacy of the model, (2) the accuracy of the input data, and (3) the accuracy of the solution technique. Various sources of uncertainties affect the reliability of the predictions of models, as described in Figure 17-1 by Bu and Damsleth (1996). Experimental measurements taken under controlled test conditions to determine the input–output (or cause and affect or the parity

676

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

Figure 17-1. Sources of errors and uncertainty associated with mathematical modeling (after Bu and Damsleth, ©1996 SPE; reprinted by permission of the Society of Petroleum Engineers).

relationship) response of systems (such as core plugs undergoing a flow test) also involve uncertainties. In general, solutions of models, called model predictions, and the response of the test systems under prescribed conditions can be represented numerically or analytically by functional relationships, mathematically expressed as: f = fx1  x2      xn 

(17-6)

in which f is a system response and x1  x2  x3     denote the various input variables and parameters. Uncertainties involved in actual calculations (predictions) or measurements (experimental testing) lead to estimated or approximate results, the accuracy of which depend on the errors involved. Therefore, the actual values are the sum of the estimates and the errors. Thus, if f˜  x˜ 1  x˜ 2      x˜ n indicate the estimated values of the function and its variables, and f˜  ˜x1  ˜x2      ˜xn represent the errors or uncertainties associated with these quantities, the following equations, expressing the actual quantities as a sum of the estimated values and the errors associated with them, can be written: x1 = x˜ 1 ±  x˜ 1

(17-7)

x2 = x˜ 2 ± ˜x2

(17-8)

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

677

and so on until xn = x˜ n ± ˜xn

(17-9)

f = f˜ ± f˜

(17-10)

The estimation of the propagation and impact of errors is usually based on a Taylor series expansion (Chapra and Canale, 1998): f x1  x2  · · ·  xn  =f ˜x1  x˜ 2  · · ·  x˜ n   f ˜x1  x˜ 2  · · ·  x˜ n  1 x1 − x˜ 1  + 1! x1 f ˜x1  x˜ 2  · · ·  x˜ n  +··· + x2 − x˜ 2  x2 + higher order terms

 (17-11)

Neglecting the higher-order terms for relatively small errors, Eq. (17-11) can be written in a compact form as: f  f˜ +

n  i=1

xi − x˜ i 

f˜ xi

(17-12)

Then, applying Eqs (17-7)–(17-10) into Eq. (17-12), the error or the uncertainty in the function value can be estimated by (Chapra and Canale, 1998):    f˜  n    f˜ = ˜xi  i  (17-13)  xi  i=1 or by a norm as (Reilly, 1992) ⎡ f˜ = ⎣

n  i=1



f˜ ˜xi i xi

2 ⎤1/2 ⎦

(17-14)

The uncertainty associated with summation and/or subtraction of numbers, defined by Eqs (17-7) through (17-9), is the square root of the squares of the uncertainties in these numbers (Reilly, 1992). Thus, if f = ±x1 ± x2 ± · · · ± xn

(17-15)

678

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

then 1/2  f˜ = x1 2 + x2 2 + · · · + xn 2

(17-16)

f = x1 ×÷ x2 ×÷ · · · ×÷ xn

(17-17)

The relative uncertainty in a multiplication or division of numbers is the square root of the sum of the squares of the relative uncertainties in these numbers (Reilly, 1992). Thus, if

then f˜ = f˜



˜x1 x˜ 1

2

+



˜x2 x˜ 2

2

+ ··· +



˜xn x˜ n

2 1/2

(17-18)

and ⎧ ⎨

⎫  2 1/2 ⎬ n   ˜ x i f = f˜ ± f˜ = f˜ 1 ± ⎭ ⎩ x ˜ i i=1

(17-19)

For example, given a function like

f x = e−2x

2

(17-20)

the error in the function value as a result of using an erroneous measured value of x = 05 ± 01 can be estimated by applying Eq. (17-13) as:   2  f˜ = ˜x −4˜xe−2˜x  (17-21)

Thus, substituting x˜ = 05 and ˜x = 01 into Eqs (17-20) and (17-21) yields f˜ = 06 and f˜ = 01. Therefore, the calculated value is expressed according to Eq. (17-10) as: f = 06 ± 01

(17-22)

As another example, consider f = f x y =

x y

(17-23)

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

679

Thus 1 f = x y

f −x = 2 y y

and

(17-24)

Eq. (17-13) can be applied as:    f˜    ˜ f = ˜x   + ˜y  x 

   f˜       y 

(17-25)

Thus, Eqs (17-23)–(17-25) lead to the following relative error expression: f˜ ˜x ˜y + = ˜x ˜y f˜

(17-26)

A similar result is obtained for f = fx y = xy. It can be shown for f = fx y = x + y that f˜ ˜x + ˜y = ˜f ˜x + ˜y

(17-27)

For f = fx = xn , we can derive f˜ ˜x =n ˜x f˜

(17-28)

For f = fx = ln x, we obtain f˜ ˜x = ˜f ˜x ln x˜ 

(17-29)

Applying Eq. (17-14) for Eq. (17-23) results in f˜ = f˜



˜x x˜

2

+

A similar result for f = xy is obtained.



˜y y˜

2 1/2

(17-30)

680

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

Bu and Damsleth (1996) consider Darcy’s law as an example K=

L q A p

(17-31)

q p

(17-32)

Let b≡

Thus, they express relative error in the calculated K value as a function of the measurements involving errors as [apply Eq. (17-14)]: ⎡ 2

2 2 ⎤1/2  2

˜ ˜ ˜ K ⎣  A ˜ b˜ ⎦ L + = + + ˜ ˜ ˜  ˜ K L A b˜

(17-33)

17.2.4 Sensitivity Analysis – Stability and Conditionality Sensitivity analysis is an important tool for systematic evaluation of mathematical models (Lehr et al., 1994). Sensitivity analysis can be used for various purposes, including model validation, evaluating model behavior, estimating model uncertainties, decision-making using uncertain models, and determining potential areas of research (Lehr et al., 1994). Sensitivity analysis provides information about the effect of the errors and/or variations in the variables and/or parameters and models on the predicted behavior. Sensitivity of a model to changes in its input data determines the condition of the model (Chapra and Canale, 1998). The sensitivity of a system’s outcome or response to changes in a variable is defined by the partial derivative (Lehr et al., 1994): Sx =

f˜ ≡ f˜ ′ x

(17-34)

Relative sensitivity (Lehr et al., 1994) or the condition number (Chapra and Canale, 1998) is defined as the ratio of the relative change or error in the function to the relative change or error in the variable or parameter value. Thus, for a single parameter function, the relative sensitivity can

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

681

be expressed by means of Eq. (17-12) as (Lehr et al., 1994; Chapra and Canale, 1998): x =

x˜ f˜ ′ x˜ Sx f˜ /f˜ = = ˜x/˜x f˜ f˜

(17-35)

Thus, the condition number or relative sensitivity can be used as a criteria to evaluate the effect of an uncertainty in the x variable on the condition of a system as (Chapra and Canale, 1998): ⎧ < 1 effect in the function is attenuated ⎪ ⎪ ⎪ ⎨= 1 effect in the function is same as the variation (17-36) x  ⎪ in the variable ⎪ ⎪ ⎩ > 1 effect in the function is amplified

Given the differential equations of a model, the sensitivity equations can be formulated for determining the sensitivity trajectory. The following example by Lehr et al. (1994) illustrates the process. Consider a mathematical model given by an ordinary differential equation as: df x t = g f x t dt

(17-37)

A differentiation of the gf x t function with respect to the variable (or parameters) x leads to: g f g dg = + dx f x x

(17-38)

Substituting Eqs (17-34) and (17-37) into Eq. (17-38) and rearranging yield the following sensitivity trajectory equation:   dS g g d df ≡ x = Sx + (17-39) dt dx dt f x One of the practical applications of the sensitivity analysis is to determine the critical parameters which strongly effect the predictions of models (Lehr et al., 1994). Lehr et al. (1994) studied the sensitivity of an oil spill evaporation model. Figures 17-2 and 17-3 by Lehr et al. (1994) depict the sensitivity of the fractional oil evaporation, f , from an oil spill with respect to the initial bubble point, TB , and the rate of bubble point variation by the fraction of oil evaporated (the slope of the evaporation curve),

682

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

Sensitivity TG ~ (STG ≡ ∂f/ ∂TG)

0.0

–0.2

–0.4

–0.6

–0.8

–1.0 300

350

400

450

500

550

600

Parameter TB

Figure 17-2. Dependence of the sensitivity with respect to TG on the initial bubble point TB (modified after Lehr, W., Calhoun, D., Jones, R., Lewandowski, A., and Overstreet, R., “Model Sensitivity Analysis in Environmental Emergency Management: A Case Study in Oil Spill Modeling,” Proceedings of the 1994 Winter Simulation Conference, J. D. Tew, S. Manivannan, D. A. Sadowski, and A. F. Seila (eds.), pp. 1198–1205, ©1994 IEEE; reprinted by permission).

TG ≡ 2 TB / t f , respectively. Examination of Figures 17-2 and 17-3 reveals that the initial bubble point TB is the critical parameter, influencing the sensitivity of the oil spill evaporation model. Figure 17-2 clearly indicates that there is a strong correlation between the sensitivity with respect to the slope of the evaporation curve and the bubble point. Whereas, Figure 17-3 shows that the sensitivity with respect to the slope of the evaporation curve cannot be correlated with the slope of the evaporation curve.

17.3 MODEL VALIDATION, REFINEMENT, AND PARAMETER ESTIMATION As stated by Civan (1994a), confidence in the model cannot be established without validating it by experimental data. However, the microscopic phenomena are too complex to study each detail individually. Thus, a practical method is to test the system for various conditions to generate its input–output response data. Then, determine the model parameters such that model predictions match the actual measurements within an

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683

Sensitivity TG ~ (STG ≡ ∂f/ ∂TG)

0.0

–0.2

–0.4

–0.6

–0.8

–1.0 100

200 300 400 500 600 700 800 900 1000

Parameter TG

Figure 17-3. Dependence of the sensitivity with respect to TG on TG (after Lehr, W., Calhoun, D., Jones, R., Lewandowski, A., and Overstreet, R., “Model Sensitivity Analysis in Environmental Emergency Management: A Case Study in Oil Spill Modeling,” Proceedings of the 1994 Winter Simulation Conference, J. D. Tew, S. Manivannan, D. A. Sadowski, and A. F. Seila (eds.), pp. 1198–1205, ©1994 IEEE; reprinted by permission).

acceptable tolerance. However, some parameters may be directly measurable. A general block diagram for parameter identification and model development and verification is given in Figure 17-4 (Civan, 1994a). 17.3.1

Experimental System

The experimental system is a reservoir core sample subjected to fluid flow. The input variables are injection flow rate or pressure differential and its particles concentration, temperature, pressure, pH, etc. The output variables are the measured pressure differential, pH, and species concentration of the effluent. Frequently, data filtering and smoothing are required to remove noise from the data as indicated in Figure 17-4. However, some important information may be lost in the process. Millan-Arcia and Civan (1992) reported that frequent breakage of particle bridges at the pore throat may cause temporary permeability improvements which are real and not just a noise. Baghdikian et al. (1989) reported that accumulation and flushing of particle floccules can cause an oscillatory behavior during permeability damage.

684 Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

Figure 17-4. Steps for formation damage process identification and model development (after Civan, 1994; reprinted by permission of the U.S. Department of Energy).

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

17.3.2

685

Parity Equations

An integration of the model equations over the length of core yields the equations of a macroscopic model called the “parity equations.” However, for a complicated model of rock–fluid-particle interactions in geological porous formations it is impractical to carry out such an integration analytically. Hence, an appropriate numerical method, such as described in Chapter 16, is facilitated to generate the model response (pressure differential across the core or sectional pressure differentials, and the effluent conditions) for a range of input conditions (i.e., the conditions of the influent, confining stress, temperature, pressure, pH, etc.). 17.3.3

Parameter Estimation with Linearized Models

Luckert (1994) points out that estimating parameters using linearized model equations obtained by transformation is subject to uncertainties and errors because of the errors introduced by numerical transformation of the experimental data. Especially, numerical differentiation is prone to larger errors than numerical integration. Luckert (1994) explains this problem on the determination of the parameters K and q of the following filtration model:  q d2 t dt (17-40) =K 2 dQ dQ where t and Q denote the filtration time and the filtrate volume, respectively. This equation can be linearized by taking a logarithm as  2    dt dt log (17-41) = log K + q log 2 dQ dQ Thus, a straightline plot of Eq. (17-41) using a least-squares fit provides the values of log K and q as the intercept and slope of this line, respectively. Luckert (1994) points out, however, that this approach leads to highly uncertain results because numerical differentiation of the experimental data involves some errors, second differentiation involves more errors than the first derivative, and numerical calculation of logarithms of the second numerical derivatives introduce further errors. Therefore, Luckert (1994) recommends linearization only for preliminary parameter estimation, when the linearization requires numerical processing of

686

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

experimental data for differentiation. Luckert (1994) states that “From a statistical point of view, experimental values should not be transformed in order that the error distribution remains unchanged.” In Chapter 12, detailed examples of constructing diagnostic charts for determining the parameters of the incompressive cake filtration model by Civan (1998b) have been presented. It has been demonstrated that the model parameters can be determined from the slopes and intercepts of the straight-line plots of the experimental data according to the linearized forms of the various equations, describing the linear and radial filtration processes. Using the parameter values determined this way, Civan (1998b) has shown that the model predictions compared well with the measured filtrate volumes and cake thicknesses. The advantage of this type of direct method is the uniqueness of the parameter values as described in Chapter 12. As described in Chapter 10, Wojtanowicz et al. (1987, 1988) also used linearized diagnostic equations given in Table 10-1 for determining the parameters of their single-phase fines migration models. 17.3.4

History Matching for Parameter Identification

The model equations contain various parameters dealing with the rate equations. They are determined by a procedure similar to history matching commonly used in reservoir simulation. In this method, an objective function JX is defined as J X = Ym − Yc T W Ym − Yc  =

n 

wi ym − yc 2i

(17-42)

i=1

where X denotes the vector of the parameters to be determined by history matching, Ym is the measured values, Yc is the calculated values, W is the weighting matrix, W = V −1  V is the variance–covariance matrix of the measurement error, and n is the number of data points. Szücs and Civan (1996) and Szücs et al. (2006) facilitated an alternative formulation of the objective function, called the p-norm, which lessens the effect of the outliers in the measured data points. Then, a suitable method is used for minimizing the objective function to obtain the best estimates of the unknown model parameters (Ohen and Civan, 1990, 1993). Frequently used optimization methods are (1) trial-and-error [tedious and time-consuming]; (2) LevenbergMarquardt (Marquardt, 1963) method [requires derivative evaluation,

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687

computationally intensive]; (3) simulated annealing [algebraic and practical (Szücs and Civan, 1996; Ucan et al., 1997)]. Note that the number of measurements should be equal or greater than the number of unknown parameters. When there is less number of measurements than the unknown parameters, additional data can be generated by interpolation between the existing data points. However, for meaningful estimates of the model parameters the range of the data points should cover a sufficiently long test period to reflect the effect of the governing formation damage mechanisms. The above-described method has difficulties. First, it may require a lot of effort to converge on the best estimates of the parameter values. Second, there is no guarantee concerning the uniqueness of the parameter values determined with nonlinear models. However, some parameters can be eliminated for less important mechanisms for a given formation and fluid system. The remaining parameters are determined by a history matching procedure. In this method, the best estimates of the unknown parameters are determined in such a way that the model predictions match the measurements obtained by laboratory testing of cores within a reasonable accuracy (Civan, 1994a). Detailed examples of the history matching method by Ohen and Civan (1990, 1993) and Civan and Nguyen (2005) have been presented in Chapter 10 and here, respectively. 17.3.5

Sensitivity Analysis

Once the model is developed and verified with experimental data, studies of the sensitivity of the model predictions with respect to the assumptions, considerations, and parameters of the model can be conducted. Consequently, the factors having negligible effects can be determined and the model can be simplified accordingly. Models simplified this way are preferred for routine, specific applications. For example, Figure 17-5 by Ziauddin et al. (1999) shows the effect of including tertiary reactions on the predicted alumina and silica concentrations during gel precipitation by sandstone acidizing at different temperatures. Figure 17-6 by Gadiyar and Civan (1994) shows the effect of the acidizing reaction rate constants on permeability alteration. The solid line represents the best fit of the experimental data using the best estimates of the model parameters obtained by history matching. When the parameter values were perturbed in a random manner, a significantly different trend, represented by the dashed line, was obtained.

688

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

Figure 17-5. Sensitivity analysis for the tertiary reaction at different temperatures: (a) 22 C and (b) 66 C (after Ziauddin et al., ©1999; reprinted by permission of the Society of Petroleum Engineers).

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

689

Normalized permeability (K /K0)

1.0 Simulated (set 1) Experimental Simulated (set 2)

0.8 0.6 0.4 0.2 0.0

q = 8.6 cm3/min L = 30.5 cm 0

200

400

600

800

1000

Cumulative volume (cm3)

Figure 17-6. Dependence of the sensitivity with respect to the model parameters (after Gadiyar and Civan, ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers).

17.4 FORMATION DAMAGE POTENTIAL OF STIMULATION AND PRODUCTION TECHNIQUES Reservoir exploitation processes frequently cause pressure, temperature, and concentration changes, and rock–fluid and fluid–fluid interactions, which often adversely affect the performance of these processes. Prior to any reservoir exploitation applications, extensive laboratory, field, and simulation studies should be conducted for assessment of the formation brine and mineral chemistry and the formation damage potential of the reservoir. Consequently, optimal strategies can be designed to effectively mitigate the adverse effects and improve the oil and gas recovery. Typical examples of such detailed studies have been presented in a series of reports by Demir (1995), Haggerty and Seyler (1997), and Seyler (1998) for characterization of the brine and mineral compositions and the investigation of the formation damage potential in the Mississippian Aux Vases and Cypress formations in the Illinois Basin. Demir (1995) used the chemical data on the formation brines and minerals to interpret the geology, determine the properties (porosity, permeability, water saturation), and estimate the formation damage potential of these reservoirs.

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Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

17.4.1

Formation Minerals and Brines

The stratigraphic dispositions of the Aux Vases and Cypress formations are shown in Figure 17-7 by Leetaru (1990) given by Demir (1995). 17.4.1.1

Formation brines

Demir (1995) reports that brine samples were gathered from the oilproducing wells in the Aux Vases and Cypress formations. Also, the samples of the brines of the Cypress and Waltersburg formations, which are used for water flooding in the Aux Vases and Cypress reservoirs, were collected from the separation tanks. Prior to sample collection, chemical treatments, such as acidizing and corrosion inhibitor applications in the wells, were ceased usually for 24 hr, but at least for 4 hr. The brine

Figure 17-7. Generalized upper Valmeyeran and Chesterian geologic column (Mississippian system) of southern Illinois (from Leetaru, 1990; after Demir, 1995; reprinted by permission of the Illinois State Geological Survey).

Model-Assisted Analysis and Interpretation of Laboratory and Field Tests

691

samples were collected using the USGS method, described in Figure 17-8, and isolated from exposure to the atmosphere to avoid oxidation and degassing. The samples were identified by the well API numbers and field names, and the pH, Eh resistivity, total dissolved solids (TDS), and laboratory chemical analysis of these samples were determined. The results are summarized in Table 17-1 by Demir (1995). The methods used to analyze the formation brine samples are described in Table 17-2 according to ISGS (1993).

Figure 17-8. Schematic of the setup for collecting formation fluids in the field (after Demir, 1995; reprinted by permission of the Illinois State Geological Survey).

Table 17-1 Well Identification and Chemical Composition of Formation Waters for Selected Locations

692

Sample no.

API no.

Field name

Strat. Unit

EOR-B115 EOR-B92 EOR-B17 EOR-B70 EOR-B51 EOR-B93 EOR-B52 EOR-B107 EOR-B101 EOR-B99 EOR-B60 EOR-B9 EOR-B36 EOR-B35 EOR-B73 EOR-B41 EOR-B83 EOR-B113 EOR-B22 EOR-B23 EOR-B59

120552391900 121913194800 120270314300 121932941500 121913099200 121910790600 121913114900 120552347200 120552272100 121652490500 121992347700 120810016700 120292340200 120290127300 121932575700 120472399500 120252742400 Injc. Water 121450227300 Injec. Water 120552387700

AKIN BARNHILL BARTELSO CARMI NORTH CLAY CITY C. CLAY CITY C. CLAY CITY C. DALE CONS. DALE CONS. ELDORADO W. ENERGY KING MATTOON MATTOON NEW HARMONY C. PARKERSBURGH SAILOR SPRING C. STORMS C. TAMAROA TAMAROA ZEIGLER

Cyp. Aux V. Cyp. Aux. V. Aux V. Aux V. Cyp. Aux. V. Cyp. Aux V. Aux V. Aux V. Cyp. Aux V. Cyp. L. Cyp. Aux V. Wallersbrg Cyp. Cyp. Aux V.

Average depth Resis. (ft.) (ohm-m) Eh (mV) pH 2825 3360 967 3236 3234 3102 2940 3172 2959 2864 2364 2744 1820 1808 2611 2804 2909 1159 NA 2629

00601 00639 01382 00676 00782 00692 00670 00615 00621 00746 00632 00621 00692 01252 00707 00723 00613 01406 00877 00877 00610

−186 −234 −318 −150 −335 −287 −285 −134 −270 −213 −102 −287 −276 −314 −156 −284 −144 −59 −178 −73 −111

After Demir, 1995; reprinted by permission of the Illinois State Geological Survey.

644 689 701 592 724 656 724 534 624 656 663 681 685 736 592 674 613 722 647 668 624

Anions and Major Cations (mg/L) TDS (mg/l)

Cl

Br

145333 87000 160 132780 78000 110 49667 29000 84 125460 74000 180 100768 59000 130 116926 68000 140 124254 72000 140 137444 84000 160 140537 83000 150 108598 65000 180 132040 79000 180 141529 85000 180 117069 69000 140 56100 33000 71 116603 67000 190 116429 71000 140 146456 85000 44320 25000 64 87529 52000 140 85236 51276+ 137329 82000 190

I

SO4 NO3 CO3 HCO3

350 270 062 1300 1300 008 560 25 010 700 304 060 690