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  • Flow and Transportin Porous Mediaand Fractured Rock

    Muhammad Sahimi

    From Classical Methods to Modern ApproachesSecond, Revised and Enlarged Edition


  • Muhammad SahimiFlow and Transport in Porous Mediaand Fractured Rock

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  • Muhammad Sahimi

    Flow and Transport in Porous Mediaand Fractured Rock

    From Classical Methods to Modern Approaches

    Second, Revised and Enlarged Edition

    WILEY-VCH Verlag GmbH & Co. KGaA

  • The Author

    Prof. Muhammad SahimiUniversity of Southern CaliforniaDept. of Chemical Engineeringmoe@usc.edu

    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.

    Library of Congress Card No.: applied for

    British Library Cataloguing-in-Publication Data:A catalogue record for this book is availablefrom the British Library.

    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.

    2011 WILEY-VCH Verlag GmbH & Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany

    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.

    Typesetting le-tex publishing services GmbH,LeipzigPrinting and Binding Strauss GmbH,MrlenbachCover Design Adam Design, Weinheim

    Printed in the Federal Republic of GermanyPrinted on acid-free paper

    ISBN Print 978-3-527-40485-8

    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

  • V

    Dedicated to the memory of my parentsHabibollah Sahimi (19161997) and Fatemeh Fakour Rashid (19282006)and toMahnoush, Ali and Niloofar

  • VII


    Preface to the Second Edition XIX

    Preface to the First Edition XXIII

    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

    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

    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

  • VIII Contents

    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

  • Contents IX

    5 Characterization of Field-Scale Porous Media:Geostatistical Concepts and Self-Affine Distributions 109

    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

    6 Characterization of Fractures, Fracture Networks,and Fractured Porous Media 143

    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

  • X Contents

    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

    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

  • Contents XI

    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

    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 R


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