Preface ...................................................... xiii
Acknowledgements .............................................. xvi
List of Symbols .............................................. xvii
1 Interfacial Curvature and Contact Angle .................... 1
1.1 Interfacial Tension ........................................ 1
1.2 Young-Laplace Equation ..................................... 4
1.3 The Young Equation and Contact Angle ....................... 7
1.3.1 The Young Equation as an Energy Balance ............ 10
1.3.2 Interfacial Tension, Roughness and Wettability ..... 11
1.3.3 Capillary Rise ..................................... 14
1.3.4 Historical Interlude: Thomas Young and the
Marquis de Laplace ................................. 16
2 Porous Media and Fluid Displacement ....................... 17
2.1 Pore-Space Images ......................................... 17
2.1.1 Statistical and Process-Based Pore-Space
Reconstruction ..................................... 22
2.1.2 Definition of a Porous Medium, Representative
Volumes, Porosity and Saturation ................... 29
2.2 Pore-Scale Networks and Topological Description ........... 32
2.2.1 Transport Networks ................................. 33
2.2.2 Network Construction ............................... 35
2.2.3 Generalized Network Models ......................... 45
2.2.4 Topological Descriptors of the Pore Space .......... 50
2.3 Wettability and Displacement .............................. 56
2.3.1 Thermodynamic Description of Displacement
Processes .......................................... 56
2.3.2 Displacement Sequences ............................. 59
2.3.3 Wettability and Wettability Change ................. 60
2.3.4 Surface Roughness and Contact Angle Hysteresis ..... 65
2.3.5 Effective Contact Angle and Curvature .............. 69
3 Primary Drainage .......................................... 73
3.1 Entry Pressures and Fluid Configurations .................. 74
3.1.1 Wetting Layers ..................................... 79
3.1.2 Entry Pressures for Irregular Throats .............. 82
3.2 Macroscopic Capillary Pressure in Drainage ................ 85
3.3 Bundle of Tubes Model and the Throat Size Distribution .... 89
3.3.1 Prediction of Capillary Pressure from Images ....... 92
3.4 Invasion Percolation ...................................... 95
3.4.1 Scaling Relations in Invasion Percolation .......... 99
3.4.2 Displacement under Gravity and Gradient
Percolation ....................................... 105
3.4.3 Invasion Percolation, Normal Percolation and
Flow .............................................. 109
3.5 Final Saturation and Maximum Capillary Pressure .......... 110
4 Imbibition and Trapping .................................. 115
4.1 Layer Flow, Swelling and Snap-Off ........................ 116
4.1.1 Roof Snap-Off during Drainage ..................... 122
4.2 Piston-Like Advance and Pore Filling ..................... 126
4.2.1 Piston-Like Throat Filling ........................ 126
4.2.2 Cooperative Pore Filling .......................... 128
4.2.3 Competition between Snap-Off and Cooperative
Pore Filling ...................................... 132
4.2.4 Frequency of Different Filling Events ............. 135
4.2.5 Dynamics of Filling ............................... 141
4.2.6 Displacement as a Series of Metastable States ..... 144
4.3 Displacement Patterns in Imbibition ...................... 145
4.3.1 Percolation with Trapping ......................... 145
4.3.2 Invasion Percolation with Trapping ................ 147
4.3.3 Frontal Advance ................................... 148
4.3.4 Cluster Growth .................................... 149
4.3.5 Phase Diagrams for Capillary-Controlled
Displacement ...................................... 149
4.3.6 Infiltration or Unstable Imbibition under
Gravity ........................................... 157
4.4 Macroscopic Capillary Pressure ........................... 161
4.5 Interfacial Area ......................................... 165
4.6 Capillary Trapping and Residual Saturation ............... 168
4.6.1 Direct Imaging of Trapped Clusters and
Percolation Theory ................................ 168
4.6.2 Effect of Initial Saturation ...................... 176
5 Wettability and Displacement Paths ....................... 188
5.1 Definitions and Capillary Pressure Cycles ................ 188
5.2 Oil and Water Layers ..................................... 194
5.2.1 Pinned Water Layers and Forced Snap-Off ........... 194
5.2.2 Forced Water Injection and Oil Layer Formation .... 196
5.2.3 Recap of Displacement Processes ................... 200
5.3 Capillary Pressures and Wettability Indices .............. 201
5.3.1 Wettability Trends and Relationships between
Indices ........................................... 204
5.3.2 Displacement Statistics in Mixed-Wet Systems ...... 208
5.4 Trapping in Mixed-Wet and Oil-Wet Media .................. 211
5.4.1 Layer Connectivity as a Function of Initial
Water Saturation .................................. 213
5.4.2 Pore-Scale Observation of Trapping in Mixed-Wet
Systems ........................................... 216
6 Navier-Stokes Equations, Darcy's Law and Multiphase
Flow ..................................................... 219
6.1 Navier-Stokes Equations and Conservation of Mass ......... 219
6.1.1 Flow in a Pipe .................................... 221
6.1.2 The Washburn Equation ............................. 225
6.1.3 Flow in Wetting Layers ............................ 228
6.1.4 Reynolds Number and the Stokes Equation ........... 233
6.2 Darcy's Law and Permeability ............................. 236
6.2.1 Permeability of a Bundle of Capillary Tubes ....... 238
6.2.2 Typical Permeability Values ....................... 239
6.2.3 The Leverett J Function ........................... 243
6.2.4 Computing Flow Fields on Pore-Space Images ........ 247
6.3 The Multiphase Darcy Law and Relative Permeability ....... 254
6.3.1 Historical Interlude: Muskat, Leverett and
Buckingham ........................................ 254
6.3.2 Assumptions Inherent in the Multiphase Darcy
Law ............................................... 255
6.4 Capillary Number and Pore-Scale Dynamics ................. 257
6.4.1 Macroscopic Flow Patterns for Imbibition .......... 257
6.4.2 Capillary Number and the Perturbative Effect of
Flow Rate ......................................... 263
6.4.3 Layer Conductance and Viscous Effects ............. 265
6.4.4 Correlation Lengths for Percolation-Like
Displacement ...................................... 270
6.4.5 Correlation Length and Residual Saturation ........ 273
6.4.6 Mobilization of Trapped Ganglia ................... 275
6.4.7 Ganglion Dynamics, Connectivity and Flow
Regimes ........................................... 279
6.4.8 Viscous and Capillary Forces as an Energy
Balance ........................................... 285
6.4.9 Direct Computation of Multiphase Flow ............. 288
6.5 Extensions to the Multiphase Darcy Law ................... 298
6.5.1 Infiltration and Phase Field Models ............... 298
6.5.2 Accounting for Non-Equilibrium Effects ............ 300
6.5.3 Averaged Equations from Energy, Momentum and
Entropy Balance ................................... 301
6.5.4 Consideration of Trapped Phases and Other
Approaches ........................................ 303
6.6 Flow Regimes ............................................. 304
6.6.1 Dimensionless Numbers ............................. 305
6.6.2 Viscous Fingering and DLA ......................... 306
6.6.3 Summary of Regime Diagrams ........................ 309
7 Relative Permeability .................................... 315
7.1 Water-Wet Media .......................................... 315
7.1.1 Primary Drainage .................................. 315
7.1.2 Waterflooding ..................................... 318
7.1.3 Predictions of Relative Permeability .............. 319
7.1.4 Relative Permeabilities for Different Rock
Types ............................................. 322
7.1.5 Effect of Initial Saturation ...................... 325
7.2 Effect of Wettability .................................... 326
7.2.1 Oil-Wet Media ..................................... 327
7.2.2 Cross-Over Saturation and Waterflood Recovery ..... 329
7.2.3 Mixed-Wet Media ................................... 330
7.2.4 Hysteresis in the Water Relative Permeability ..... 333
7.2.5 Hysteresis in the Oil Relative Permeability ....... 336
7.2.6 Features of Relative Permeability in Mixed-Wet
Rocks ............................................. 338
7.2.7 Guidelines for Assessing Wettability .............. 340
7.3 Effect of Capillary Number and Viscosity Ratio ........... 343
7.3.1 Relative Permeability in the Viscous Limit ........ 344
7.3.2 Viscosity Ratio and Viscous Coupling .............. 346
7.4 Empirical Models ......................................... 347
7.4.1 Relative Permeability of a Bundle of Tubes ........ 348
7.4.2 Functional Forms for Relative Permeability and
Capillary Pressure ................................ 350
7.4.3 Empirical Models in Hydrology ..................... 352
8 Three-Phase Flow ......................................... 354
8.1 Spreading ................................................ 354
8.2 Contact Angles and the Bartell-Osterhof Equation ......... 358
8.2.1 Wetting and Spreading in Three-Phase Flow ......... 360
8.2.2 Why Ducks Don't Get Wet ........................... 361
8.2.3 Wettability States in Three-Phase Flow ............ 363
8.3 Oil Layers ............................................... 364
8.3.1 Mixed-Wettability and Layer Stability ............. 365
8.3.2 Three Phases in Capillary/Gravity Equilibrium ..... 366
8.3.3 Layer Conductance and Relative Permeability ....... 368
8.4 Displacement Processes ................................... 373
8.4.1 Fluid Configurations .............................. 373
8.4.2 Configuration Changes and Layer StabiHty .......... 375
8.4.3 Multiple Displacement ............................. 375
8.4.4 Displacement Paths ................................ 378
8.5 Three-Phase Relative Permeability ........................ 381
8.5.1 Pore Occupancy and Saturation Dependence .......... 386
8.5.2 Predictions of Three-Phase Relative
Permeability ...................................... 389
8.5.3 Trapping in Three-Phase Flow ...................... 391
8.5.4 Direct Imaging of Trapped Phases in Three-Phase
Flow .............................................. 397
8.5.5 Empirical Models in Three-Phase Flow .............. 397
9 Solutions to Equations for Multiphase Flow ............... 402
9.1 Conservation Equations for Multiphase Flow ............... 402
9.1.1 Equations in One Dimension and the Fractional
Flow .............................................. 405
9.1.2 Waterflooding and Spontaneous Imbibition .......... 405
9.1.3 Exemplar Relative Permeabilities and Capillary
Pressures ......................................... 407
9.1.4 Boundary Conditions for One-Dimensional Flow
Problems .......................................... 411
9.2 Buckley-Leverett Analysis for Two-Phase Flow ............. 411
9.2.1 Dimensionless Variables and Wavespeeds ............ 413
9.2.2 Shocks ............................................ 414
9.2.3 Constructing a Solution for Saturation ............ 415
9.2.4 Recovery Calculations ............................. 418
9.2.5 Example Recovery Curves ........................... 418
9.3 Analysis of Imbibition ................................... 422
9.3.1 Capillary Dispersion and Fractional Flow .......... 422
9.3.2 Example Solutions ................................. 427
9.3.3 Experimental Analysis of Spontaneous Imbibition ... 430
9.4 Recovery, Imbibition and the Trillion-Barrel Question .... 432
Appendix. Exercises ........................................... 437
References .................................................... 447
Index ......................................................... 475
The plate section can be found between pages 140 and 141
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