Preface to the Second Edition .............................. XIX
Preface to the First Edition ............................. XXIII
1 Continuum versus Discrete Models ............................. 1
1.1 A Hierarchy of Heterogeneities and Length Scales ........ 2
1.2 Long-Range Correlations and Connectivity ................ 3
1.3 Continuum versus Discrete Models ........................ 5
2 The Equations of Change ...................................... 9
2.1 The Mass Conservation Equation .......................... 9
2.2 The Momentum Equation .................................. 10
2.3 The Diffusion and Convective-Diffusion Equations ....... 11
2.4 Fluid Flow in Porous Media ............................. 12
3 Characterization of Pore Space Connectivity: Percolation
Theory ...................................................... 15
3.1 Network Model of a Porous Medium ....................... 15
3.2 Percolation Theory ..................................... 18
3.3 Connectivity and Clustering Properties ................. 23
3.4 Flow and Transport Properties .......................... 24
3.5 The Sample-Spanning Cluster and Its Backbone ........... 25
3.6 Universal Properties ................................... 27
3.7 The Significance of Power Laws ......................... 28
3.8 Dependence of Network Properties on Length Scale ....... 28
3.9 Finite-Size Effects .................................... 30
3.10 Random Networks and Continuum Models ................... 31
3.11 Differences between Network and Continuum Models ....... 33
3.12 Porous Materials with Low Percolation Thresholds ....... 35
3.13 Network Models with Correlations ....................... 35
3.14 A Glance at History .................................... 36
4 Characterization of the Morphology of Porous Media .......... 39
4.1 Porosity ............................................... 41
4.2 Fluid Saturation ....................................... 43
4.3 Specific Surface Area .................................. 44
4.4 The Tortuosity Factor .................................. 44
4.5 Correlations in Porosity and Pore Sizes ................ 45
4.6 Surface Energy and Surface Tension ..................... 47
4.7 Laplace Pressure and the Young-Laplace Equation ........ 48
4.8 Contact Angles and Wetting: The Young-Dupre Equation ... 49
4.9 The Washburn Equation and Capillary Pressure ........... 50
4.10 Measurement of Capillary Pressure ...................... 53
4.11 Pore Size Distribution ................................. 54
4.12 Mercury Porosimetry .................................... 55
4.13 Sorption in Porous Media ............................... 76
4.14 Pore Size Distribution from Small-Angle Scattering
Data ................................................... 87
4.15 Pore Size Distribution from Nuclear Magnetic
Resonance .............................................. 88
4.16 Determination of the Connectivity of Porous Media ...... 91
4.17 Fractal Properties of Porous Media ..................... 96
5 Characterization of Field-Scale Porous Media:
Geostatistical Concepts and Self-Affine Distributions ...... 209
5.1 Estimators of a Population of Data .................... 111
5.2 Heterogeneity of a Field-Scale Porous Medium .......... 113
5.3 Correlation Functions ................................. 117
5.4 Models of Semivariogram ............................... 121
5.5 Infinite Correlation Length: Self-Affine
Distributions ......................................... 122
5.6 Interpolating the Data: Kriging ....................... 132
5.7 Conditional Simulation ................................ 138
6 Characterization of Fractures, Fracture Networks, and
Fractured Porous Media ..................................... 143
6.1 Surveys and Data Acquisition .......................... 144
6.2 Characterization of Surface Morphology of Fractures ... 146
6.3 Generation of a Rough Surface: Fractional Brownian
Motion ................................................ 151
6.4 The Correlation Function for a Rough Surface .......... 152
6.5 Characterization of a Single Fracture ................. 152
6.6 Characterization of Fracture Networks ................. 156
6.7 Characterization of Fractured Porous Media ............ 170
7 Models of Porous Media ..................................... 279
7.1 Models of Porous Media ................................ 279
7.2 Continuum Models ...................................... 285
7.3 Models Based on Diagenesis of Porous Media ............ 299
7.4 Reconstruction of Porous Media ........................ 202
7.5 Models of Field-Scale Porous Media .................... 205
8 Models of Fractures and Fractured Porous Media ............. 213
8.1 Models of a Single Fracture ........................... 213
8.2 Models of Fracture Networks ........................... 215
8.3 Reconstruction Methods ................................ 229
8.4 Synthetic Fractal Models .............................. 232
8.5 Mechanical Models of Fracture Networks ................ 234
8.6 Percolation Properties of Fractures ................... 241
8.7 Models of Fractured Porous Media ...................... 247
9 Single-Phase Flow and Transport in Porous Media:
The Continuum Approach ..................................... 253
9.1 Derivation of Darcy's Law: Ensemble Averaging ......... 253
9.2 Measurement of Permeability ........................... 256
9.3 Exact Results ......................................... 257
9.4 Effective-Medium and Mean-Field Approximations ........ 265
9.5 Cluster Expansion ..................................... 269
9.6 Rigorous Bounds ....................................... 271
9.7 Empirical Correlations ................................ 273
9.8 Packings of Nonspherical Particles .................... 274
9.9 Numerical Simulation .................................. 275
9.10 Relation between Permeability and Electrical
Conductivity .......................................... 291
9.11 Relation between Permeability and Nuclear Magnetic
Resonance ............................................. 292
9.12 Dynamic Permeability .................................. 295
9.13 Non-Darcy Flow ........................................ 297
10 Single-Phase Flow and Transport in Porous Media:
The Pore Network Approach .................................. 299
10.1 The Pore Network Models ............................... 301
10.2 Exact Formulation and Perturbation Expansion .......... 303
10.3 Anomalous Diffusion and Effective-Medium
Approximation ......................................... 326
10.4 Archie's Law and the Effective-Medium Approximation ... 322
10.5 Renormalization Group Methods ......................... 324
10.6 Renormalized Effective-Medium Approximation ........... 329
10.7 The Bethe Lattice Model ............................... 332
10.8 Critical Path Analysis ................................ 333
10.9 Random Walk Method .................................... 337
10.10 Non-Darcy Flow ....................................... 338
11 Dispersion in Flow through Porous Media .................... 342
11.1 The Phenomenon of Dispersion .......................... 342
11.2 Mechanisms of Dispersion Processes .................... 342
11.3 The Convective-Diffusion Equation ..................... 343
11.4 The Dispersivity Tensor ............................... 345
11.5 Measurement of the Dispersion Coefficients ............ 346
11.6 Dispersion in Systems with Simple Geometry ............ 354
11.7 Classification of Dispersion Regimes in Porous
Media ................................................. 359
11.8 Continuum Models of Dispersion in Porous Media ........ 361
11.9 Fluid-Mechanical Models ............................... 363
11.10 Pore Network Models .................................. 367
11.11 Long-Time Tails: Dead-End Pores versus Disorder ...... 370
11.12 Dispersion in Short Porous Media ..................... 372
11.13 Dispersion in Porous Media with Percolation
Disorder ............................................. 374
11.14 Dispersion in Field-Scale Porous Media ............... 382
11.15 Numerical Simulation ................................. 403
11.16 Dispersion in Unconsolidated Porous Media ............ 410
11.17 Dispersion in Stratified Porous Media ................ 412
12 Single-Phase Flow and Transport in Fractures and
Fractured Porous Media ..................................... 415
12.1 Experimental Aspects of Flow in a Fracture ............ 416
12.2 Flow in a Single Fracture ............................. 418
12.3 Conduction in a Fracture .............................. 429
12.4 Dispersion in a Fracture .............................. 435
12.5 Flow and Conduction in Fracture Networks .............. 441
12.6 Dispersion in Fracture Networks ....................... 447
12.7 Flow and Transport in Fractured Porous Media .......... 450
13 Miscible Displacements ..................................... 467
13.1 Factors Affecting the Efficiency of Miscible
Displacements ......................................... 469
13.2 The Phenomenon of Fingering ........................... 473
13.3 Factors Affecting Fingering ........................... 476
13.4 Gravity Segregation ................................... 480
13.5 Models of Miscible Displacements in Hele-Shaw Cells ... 481
13.6 Averaged Continuum Models of Miscible Displacements ... 487
13.7 Numerical Simulation .................................. 492
13.8 Stability Analysis .................................... 495
13.9 Stochastic Models ..................................... 500
13.10 Pore Network Models .................................. 509
13.11 Crossover from Fractal to Compact Displacement ....... 511
13.12 Miscible Displacements in Large-Scale Porous Media ... 522
13.13 Miscible Displacements in Fractures .................. 514
13.14 Main Considerations in Miscible Displacements ........ 515
14 Immiscible Displacements and Multiphase Flows:
Experimental Aspects and Continuum Modeling ................ 529
14.1 Wettability and Contact Angles ........................ 519
14.2 Core Preparation and Wettability Considerations ....... 521
14.3 Measurement of Contact Angle and Wettability .......... 524
14.4 The Effect of Surface Roughness on Contact Angle ...... 527
14.5 Dependence of Dynamic Contact Angle and Capillary
Pressure on Capillary Number .......................... 527
14.6 Fluids on Rough Self-Affine Surfaces: Hypodiffusion
and Hyperdiffusion .................................... 529
14.7 Effect of Wettability on Capillary Pressure ........... 531
14.8 Immiscible Displacement Processes ..................... 535
14.9 Mobilization of Blobs: Choke-Off and Pinch-Off ........ 540
14.10 Relative Permeability ................................ 543
14.11 Measurement of Relative Permeabilities ............... 544
14.12 Effect of Wettability on Relative Permeability ....... 552
14.13 Models of Multiphase Flow and Displacement ........... 553
14.14 Fractional Flows and the Buckley-Leverett Equation ... 554
14.15 The Hilfer Formulation: Questioning the Macroscopic
Capillary Pressure ................................... 556
14.16 Two-Phase Flow in Unconsolidated Porous Media ........ 557
14.20 Two-Phase Flow in Fractured Porous Media ............. 572
15 Immiscible Displacements and Multiphase Flows: Network
Models ..................................................... 575
15.1 Pore Network Models of Capillary-Controlled Two-
Phase Flow ............................................ 575
15.2 Simulating the Flow of Thin Wetting Films ............. 585
15.3 Displacements with Two Invaders and Two Defenders ..... 588
15.4 Random Percolation with Trapping ...................... 593
15.5 Crossover from Fractal to Compact Displacement ........ 593
15.6 Pinning of a Fluid Interface .......................... 596
15.7 Finite-Size Effects and Devil's Staircase ............. 598
15.8 Displacement under the Influence of Gravity:
Gradient Percolation .................................. 599
15.9 Computation of Relative Permeabilities ................ 601
15.10 Models of Immiscible Displacements with Finite
Capillary Numbers .................................... 608
15.11 Phase Diagram for Displacement Processes ............. 623
15.12 Dispersion in Two-Phase Flow in Porous Media ......... 624
15.13 Models of Two-Phase Flow in Unconsolidated Porous
Media ................................................ 617
15.14 Three-Phase Flow ..................................... 620
15.15 Two-Phase Flow in Fractures and Fractured Porous
Media ................................................ 631
References ................................................. 633
Index ...................................................... 701
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