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<ul><li><p>Flow and Transportin Porous Mediaand Fractured Rock</p><p>Muhammad Sahimi</p><p>From Classical Methods to Modern ApproachesSecond, Revised and Enlarged Edition</p><p>le-texDateianlage9783527636716.jpg</p></li><li><p>Muhammad SahimiFlow and Transport in Porous Mediaand Fractured Rock</p></li><li><p>Related Titles</p><p>Clark, M. M.</p><p>Transport Modeling for Environmental Engineers and Scientists</p><p>2009</p><p>ISBN 978-0-470-26072-2</p><p>Guyer, R. A., Johnson, P. A.</p><p>Nonlinear Mesoscopic ElasticityThe Complex Behaviour of Granular Media including Rocks and Soil</p><p>2009ISBN 978-3-527-40703-3</p><p>Pinder, G. F., Gray, W. G.</p><p>Essentials of Multiphase Flow in Porous Media</p><p>2008</p><p>ISBN 978-0-470-31762-4</p><p>Klages, R., Radons, G., Sokolov, I. M. (eds.)</p><p>Anomalous TransportFoundations and Applications</p><p>2008ISBN 978-3-527-40722-4</p></li><li><p>Muhammad Sahimi</p><p>Flow and Transport in Porous Mediaand Fractured Rock</p><p>From Classical Methods to Modern Approaches</p><p>Second, Revised and Enlarged Edition</p><p>WILEY-VCH Verlag GmbH &amp; Co. KGaA</p></li><li><p>The Author</p><p>Prof. Muhammad SahimiUniversity of Southern CaliforniaDept. of Chemical Engineeringmoe@usc.edu</p><p>All books published by Wiley-VCH are carefullyproduced. Nevertheless, authors, editors, andpublisher do not warrant the informationcontained in these books, including this book, tobe free of errors. Readers are advised to keep inmind that statements, data, illustrations,procedural details or other items mayinadvertently be inaccurate.</p><p>Library of Congress Card No.: applied for</p><p>British Library Cataloguing-in-Publication Data:A catalogue record for this book is availablefrom the British Library.</p><p>Bibliographic information published by theDeutsche NationalbibliothekThe Deutsche Nationalbibliothek lists thispublication in the Deutsche Nationalbibliografie;detailed bibliographic data are available on theInternet at http://dnb.d-nb.de.</p><p> 2011 WILEY-VCH Verlag GmbH &amp; Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany</p><p>All rights reserved (including those of translationinto other languages). No part of this book maybe reproduced in any form by photoprinting,microfilm, or any other means nor transmittedor translated into a machine language withoutwritten permission from the publishers. Regis-tered names, trademarks, etc. used in this book,even when not specifically marked as such, arenot to be considered unprotected by law.</p><p>Typesetting le-tex publishing services GmbH,LeipzigPrinting and Binding Strauss GmbH,MrlenbachCover Design Adam Design, Weinheim</p><p>Printed in the Federal Republic of GermanyPrinted on acid-free paper</p><p>ISBN Print 978-3-527-40485-8</p><p>ISBN oBook 978-3-527-63669-3ISBN ePDF 978-3-527-63671-6ISBN ePub 978-3-527-63670-9ISBN Mobi 978-3-527-63672-3</p></li><li><p>V</p><p>Dedicated to the memory of my parentsHabibollah Sahimi (19161997) and Fatemeh Fakour Rashid (19282006)and toMahnoush, Ali and Niloofar</p></li><li><p>VII</p><p>Contents</p><p>Preface to the Second Edition XIX</p><p>Preface to the First Edition XXIII</p><p>1 Continuum versus Discrete Models 11.1 A Hierarchy of Heterogeneities and Length Scales 21.2 Long-Range Correlations and Connectivity 31.3 Continuum versus Discrete Models 5</p><p>2 The Equations of Change 92.1 The Mass Conservation Equation 92.2 The Momentum Equation 102.3 The Diffusion and Convective-Diffusion Equations 112.4 Fluid Flow in Porous Media 12</p><p>3 Characterization of Pore Space Connectivity: Percolation Theory 153.1 Network Model of a Porous Medium 153.2 Percolation Theory 183.2.1 Bond and Site Percolation 193.2.2 Computer Simulation and Counting the Clusters 223.2.3 Bicontinuous Porous Materials 233.3 Connectivity and Clustering Properties 233.4 Flow and Transport Properties 243.5 The Sample-Spanning Cluster and Its Backbone 253.6 Universal Properties 273.7 The Significance of Power Laws 283.8 Dependence of Network Properties on Length Scale 283.9 Finite-Size Effects 303.10 Random Networks and Continuum Models 313.11 Differences between Network and Continuum Models 333.12 Porous Materials with Low Percolation Thresholds 353.13 Network Models with Correlations 353.14 A Glance at History 36</p></li><li><p>VIII Contents</p><p>4 Characterization of the Morphology of Porous Media 394.1 Porosity 414.2 Fluid Saturation 434.3 Specific Surface Area 444.4 The Tortuosity Factor 444.5 Correlations in Porosity and Pore Sizes 454.6 Surface Energy and Surface Tension 474.7 Laplace Pressure and the YoungLaplace Equation 484.8 Contact Angles and Wetting: The YoungDupr Equation 494.9 The Washburn Equation and Capillary Pressure 504.10 Measurement of Capillary Pressure 534.11 Pore Size Distribution 544.12 Mercury Porosimetry 554.12.1 Pore Size Distribution 594.12.2 Pore Length Distribution 604.12.3 Pore Number Distribution 604.12.4 Pore Surface Distribution 604.12.5 Particle Size Distribution 604.12.6 Pore Network Models 614.12.7 Percolation Models 694.13 Sorption in Porous Media 764.13.1 Classifying Adsorption Isotherms and Hysteresis Loops 774.13.2 Mechanisms of Adsorption 784.13.2.1 Adsorption in Micropores 784.13.2.2 Adsorption in Mesopores: The Kelvin Equation 784.13.3 Adsorption Isotherms 814.13.3.1 The Langmuir Isotherm 814.13.3.2 The BrunauerEmmettTeller (BET) Isotherm 824.13.3.3 The FrenkelHalseyHill Isotherm 834.13.4 Distributions of Pore Size, Surface, and Volume 834.13.5 Pore Network Models 854.13.6 Percolation Models 864.14 Pore Size Distribution from Small-Angle Scattering Data 874.15 Pore Size Distribution from Nuclear Magnetic Resonance 884.16 Determination of the Connectivity of Porous Media 914.17 Fractal Properties of Porous Media 964.17.1 Adsorption Methods 964.17.2 Chord-Length Measurements 994.17.2.1 Chord-Length Measurements on Fracture Surfaces 994.17.2.2 Chord-Length Measurements on Thin Sections 1024.17.3 The Correlation Function Method 1034.17.4 Small-Angle Scattering 1064.17.5 Porosity and Pore Size Distribution of Fractal Porous Media 108</p></li><li><p>Contents IX</p><p>5 Characterization of Field-Scale Porous Media:Geostatistical Concepts and Self-Affine Distributions 109</p><p>5.1 Estimators of a Population of Data 1115.2 Heterogeneity of a Field-Scale Porous Medium 1135.2.1 The DykstraParsons Heterogeneity Index 1145.2.2 The Lorenz Heterogeneity Index 1155.2.3 The Index of Variation 1165.2.4 The GelharAxness Heterogeneity Index 1175.2.5 The Koval Heterogeneity Index 1175.3 Correlation Functions 1175.3.1 Autocovariance 1185.3.2 Autocorrelation 1185.3.3 Semivariance and Semivariogram 1195.4 Models of Semivariogram 1215.4.1 The Exponential Model 1215.4.2 The Spherical Model 1215.4.3 The Gaussian Model 1215.4.4 The Periodic Model 1225.5 Infinite Correlation Length: Self-Affine Distributions 1225.5.1 The Spectral Density Method 1275.5.2 Successive Random Additions 1295.5.3 The Wavelet Decomposition Method 1295.5.4 The Maximum Entropy Method 1315.6 Interpolating the Data: Kriging 1325.6.1 Biased Kriging 1345.6.2 Unbiased Kriging 1355.6.3 Kriging with Constraints for Nonadditive Properties 1365.6.4 Universal Kriging 1375.6.5 Co-Kriging 1375.7 Conditional Simulation 1385.7.1 Sequential Gaussian Simulation 1385.7.2 Random Residual Additions 1395.7.3 Sequential Indicator Simulation 1405.7.4 Optimization-Reconstruction Methods 141</p><p>6 Characterization of Fractures, Fracture Networks,and Fractured Porous Media 143</p><p>6.1 Surveys and Data Acquisition 1446.2 Characterization of Surface Morphology of Fractures 1466.2.1 Self-Similar Structures 1466.2.2 The Correlation Functions 1486.2.3 Rough Self-Affine Surfaces 1486.2.4 Measurement of Surface Roughness 1496.3 Generation of a Rough Surface: Fractional Brownian Motion 1516.4 The Correlation Function for a Rough Surface 152</p></li><li><p>X Contents</p><p>6.5 Characterization of a Single Fracture 1526.5.1 Aperture 1536.5.2 Contact Area 1546.5.3 Surface Height 1556.5.4 Surface Roughness 1556.6 Characterization of Fracture Networks 1566.6.1 Fractures and Power-Law Scaling 1576.6.2 Distribution of Fractures Length 1596.6.3 Distribution of Fractures Displacement 1606.6.4 Distribution of Fractures Apertures 1616.6.5 Distribution of Fractures Orientation 1636.6.6 Density of Fractures 1636.6.7 Connectivity of Fracture Networks 1646.6.8 Self-Similar Structure of Fracture Networks 1676.6.9 Interdimensional Relations 1696.7 Characterization of Fractured Porous Media 1706.7.1 Analysis of Well Logs 1716.7.2 Seismic Attributes 1716.7.3 Fracture Distribution 1746.7.4 Fracture Density from Well Log Data 175</p><p>7 Models of Porous Media 1797.1 Models of Porous Media 1797.1.1 One-Dimensional Models 1807.1.2 Spatially-Periodic Models 1817.1.3 Bethe Lattice Models 1837.1.4 Pore Network Models 1847.2 Continuum Models 1857.2.1 Packing of Spheres 1867.2.2 Particle Distribution and Correlation Functions 1887.2.3 The n-Particle Probability Density 1927.2.4 Distribution of Equal-Size Particles 1937.2.4.1 Fully-Penetrable Spheres 1947.2.4.2 Fully-Impenetrable Spheres 1957.2.4.3 Interpenetrable Spheres 1967.2.5 Distribution of Polydispersed Spheres 1967.2.5.1 Fully-Penetrable Spheres 1977.2.5.2 Fully-Impenetrable Spheres 1987.2.6 Simulation of Packings of Spheres 1987.3 Models Based on Diagenesis of Porous Media 1997.4 Reconstruction of Porous Media 2017.5 Models of Field-Scale Porous Media 2057.5.1 Random Hydraulic Conductivity Models 2067.5.2 Fractal Models 2067.5.3 Multifractal Models 207</p></li><li><p>Contents XI</p><p>7.5.4 Reconstruction Methods 2087.5.4.1 The Genetic Algorithm for Reconstruction 2097.5.4.2 Reconstruction Based on Flow and Seismic Data 211</p><p>8 Models of Fractures and Fractured Porous Media 2138.1 Models of a Single Fracture 2138.2 Models of Fracture Networks 2158.2.1 Excluded Area and Volume 2168.2.2 Two-Dimensional Models 2178.2.3 Three-Dimensional Models 2208.2.4 Fracture Networks of Convex Polygons 2228.2.5 The Dual Permeability Model 2278.3 Reconstruction Methods 2298.4 Synthetic Fractal Models 2328.5 Mechanical Models of Fracture Networks 2348.6 Percolation Properties of Fractures 2418.6.1 A Single Fracture 2418.6.2 Fracture Networks 2438.7 Models of Fractured Porous Media 2478.7.1 The Double-Porosity and Double-Permeability Models 2488.7.2 Discrete Models of Fractured Porous Media 250</p><p>9 Single-Phase Flow and Transport in Porous Media:The Continuum Approach 253</p><p>9.1 Derivation of Darcys Law: Ensemble Averaging 2539.2 Measurement of Permeability 2569.3 Exact Results 2579.3.1 Fluid Flow 2579.3.2 Transport 2629.4 Effective-Medium and Mean-Field Approximations 2659.4.1 Fluid Flow 2669.4.2 Transport 2679.5 Cluster Expansion 2699.5.1 Fluid Flow 2699.5.2 Transport 2719.6 Rigorous Bounds 2719.6.1 Fluid Flow 2719.6.2 Transport 2739.7 Empirical Correlations 2739.8 Packings of Nonspherical Particles 2749.9 Numerical Simulation 2759.9.1 Random Walk Methods 2769.9.2 Lattice-Gas and Lattice-Boltzmann Methods 2849.9.2.1 Lattice-Gas Method 2849.9.2.2 Lattice-Boltzmann Method 2879.10 Relation between Permeability and Electrical Conductivity 291</p></li><li><p>XII Contents</p><p>9.11 Relation between Permeability and Nuclear Magnetic Resonance 2929.12 Dynamic Permeability 2959.13 Non-Darcy Flow 297</p><p>10 Single-Phase Flow and Transport in Porous Media:The Pore Network Approach 299</p><p>10.1 The Pore Network Models 30110.2 Exact Formulation and Perturbation Expansion 30310.2.1 Green Function Formulation and Perturbation Expansion 30410.2.2 Self-Consistent Approximation 30510.2.3 Random Walks and Self-Consistent Approximation 30610.2.4 Relation with Continuous-Time Random Walks 30710.2.5 Effective-Medium Approximation 30810.2.6 Effective-Medium Approximation and Percolation Disorder 31010.2.7 Steady-State Transport and Percolation Threshold 31110.2.8 Extensions of the Effective-Medium Approximation 31210.2.9 Effective-Medium Approximation for Anisotropic Media 31210.2.10 Continuum Models and Effective-Medium Approximation for</p><p>Site-Disordered Networks 31410.2.11 Accuracy of the Effective-Medium Approximation 31410.2.12 Effective-Medium Approximation for the Effective Permeability 31510.3 Anomalous Diffusion and Effective-Medium Approximation 31610.3.1 Scaling Theory of Anomalous Diffusion 31710.3.2 Experimental Test of Anomalous Diffusion 31910.3.3 The Governing Equation for Anomalous Diffusion 32010.4 Archies Law and the Effective-Medium Approximation 32110.5 Renormalization Group Methods 32410.6 Renormalized Effective-Medium Approximation 32910.7 The Bethe Lattice Model 33110.8 Critical Path Analysis 33310.9 Random Walk Method 33710.10 Non-Darcy Flow 338</p><p>11 Dispersion in Flow through Porous Media 34111.1 The Phenomenon of Dispersion 34111.2 Mechanisms of Dispersion Processes 34211.3 The Convective-Diffusion Equation 34311.4 The Dispersivity Tensor 34511.5 Measurement of the Dispersion Coefficients 34611.5.1 Longitudinal Dispersion Coefficient 34611.5.1.1 Concentration Measurements 34611.5.1.2 Resistivity Measurements 34811.5.1.3 The Acoustic Method 34911.5.2 Transverse Dispersion Coefficient 35011.5.3 Nuclear Magnetic Resonance Method 35111.6 Dispersion in Systems with Simple Geometry 354</p></li><li><p>Contents XIII</p><p>11.6.1 Dispersion in a Capillary Tube: The TaylorAris Theory 35611.6.2 Dispersion in Spatially-Periodic Models of Porous Media 35811.7 Classification of Dispersion Regimes in Porous Media 35911.8 Continuum Models of Dispersion in Porous Media 36111.8.1 The Volume-Averaging Method 36111.8.2 The Ensemble-Averaging Method 36211.9 Fluid-Mechanical Models 36311.10 Pore Network Models 36711.10.1 First-Passage Time and Random Walk Simulation 36711.10.2 Probability Propagation Algorithm 36811.10.3 Deterministic Models 37011.11 Long-Time Tails: Dead-End Pores versus Disorder 37011.12 Dispersion in Short Porous Media 37211.13 Dispersion in Porous Media with Percolation Disorder 37411.13.1 Theoretical Developments 37411.13.2 Experimental Measurements 38011.14 Dispersion in Field-Scale Porous Media 38211.14.1 Large-Scale Volume Averaging 38411.14.2 Ensemble Averaging 38511.14.3 Stochastic Spectral Method 38511.14.4 Continuous-Time Random Walk Approach 38811.14.4.1 Relation between the Transition Rates and the Waiting-Time</p><p>Distribution 39211.14.4.2 Continuum Limit of the CTRW 39311.14.4.3 Application to Laboratory Experiments 39511.14.4.4 Application to Field-Scale Experiments 39611.14.5 Fractional Convective-Diffusion Equation 39811.14.6 The Critical Path Analysis 40011.15 Numerical Simulation 40311.15.1 Lattice-Boltzmann Method 40411.15.2 Particle-Tracking Method 40511.15.3 Fractal Models 40611.15.4 Long-Range Correlated Percolation Model 40811.16 Dispersion in Unconsolidated Porous Media 41011.17 Dispersion in Stratified Porous Media 412</p><p>12 Single-Phase Flow and Transport in Fracturesand Fractured Porous Media 415</p><p>12.1 Experimental Aspects of Flow in a Fracture 41612.2 Flow in a Single Fracture 41812.2.1 The Reynolds Approximation 42012.2.2 Perturbation Expansion 42112.2.3 Effective-Medium Approximation 42112.2.4 Asymptotic Expression 42312.2.5 Effect of the Contact Areas 424</p></li><li><p>XIV Contents</p><p>12.2.6 Numerical Simulation 42412.2.6.1 Mapping onto Equivalent Pore Networks 42512.2.6.2 Numerical Simulation of the Reynolds Equation 42612.2.6.3 Numerical Simulations with a Three-Dimensional Fracture 42612.2.6.4 Lattice-Gas and Lattice-Boltzmann Simulations 42712.3 Conduction in a Fracture 42912.3.1 The Reynolds Approximation 43012.3.2 Perturbation Expansion 4301...</p></li></ul>

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