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Contents
Preface, xi
1. Motivating Ideas and Governing Equations, 1
Examples of incorrect formulations, 3
Darcy’s equations for flow in porous media, 7
Logarithmic solutions and beyond, 11
Fundamental aerodynamic analogies, 12
Problems and exercises, 18
2. Fracture Flow Analysis, 19
Example 2-1. Single straight-line fracture in an isotropic circular
reservoir containing incompressible fluid, 19
Example 2-2. Line fracture in an anisotropic reservoir with
incompressible liquids and compressible gases, 27
Example 2-3. Effect of nonzero fracture thickness, 32
Example 2-4. Flow rate boundary conditions, 34
Example 2-5. Uniform vertical velocity along the fracture, 35
Example 2-6. Uniform pressure along the fracture, 37
Example 2-7. More general fracture pressure distributions, 38
Example 2-8. Velocity conditions for gas flows, 39
Example 2-9. Determining velocity fields, 40
Problems and exercises, 41
3. Flows Past Shaly Bodies, 43
Example 3-1. Straight-line shale segment in uniform flow, 43
Example 3-2. Curved shale segment in uniform flow, 49
Example 3-3. Mineralized faults, anisotropy, and gas flow, 49
Problems and exercises, 50
4. Streamline Tracing and Complex Variables, 52
Discussion 4-1. The classical streamfunction, 52
Discussion 4-2. Streamfunction for general fluids in heterogeneous and
anisotropic formations, 55
Discussion 4-3. Subtle differences between pressure and streamfunction
formulations, 57
Discussion 4-4. Streamline tracing in the presence of multiple wells, 60
vi
Discussion 4-5. Streamfunction expressions for distributed line sources
and vortexes, 63
Discussion 4-6. Streamfunction solution using complex variables
techniques, 65
Discussion 4-7. Circle Theorem: Exact solutions to Laplace’s equation, 66
Discussion 4-8. Generalized streamline tracing and volume flow rate
computations, 68
Discussion 4-9. Streamline tracing in 3D flows, 70
Discussion 4-10. Tracer movement in 3D reservoirs, 73
Fluid flow instabilities, 76
Problems and exercises, 78
5. Flows in Complicated Geometries, 79
What is conformal mapping? 80
Using “simple‿ complex variables, 82
Example 5-1. The classic radial flow solution, 84
Example 5-2. Circular borehole with two symmetric radial fractures, 86
Example 5-3. Circular borehole with two uneven, opposite, radial
fractures; or a single radial fracture, 88
Example 5-4. Circular borehole with multiple radial fractures, 89
Example 5-5. Straight shale segment at arbitrary angle, 91
Example 5-6. Infinite array of straight-line shales, 94
Example 5-7. Pattern wells under aquifer drive, 95
Three-dimensional flows, 96
Example 5-8. Point spherical flow, 97
Example 5-9. Finite line source with prescribed pressure, 97
Example 5-10. Finite line source with prescribed flow rate, 99
Example 5-11. Finite conductivity producing fracture having
limited areal extent, 100
Example 5-12. Finite conductivity nonproducing fracture having
limited areal extent, 101
Borehole interactions, 101
Example 5-13. Producing fracture near multiple wells under aquifer
drive, 102
Example 5-14. Producing fractures near multiple wells in solid
wall reservoirs, 103
Example 5-15. Straight-line shale segment near multiple wells in
uniform flow, 104
Examples 5-16 and 5-17. Nonproducing faults in solid wall and
aquifer-driven reservoirs, 105
Example 5-18. Highly curved fractures and shales, 106
Problems and exercises, 107
6. Radial Flow Analysis, 108
Example 6-1. Steady liquids in homogeneous media, 108
Example 6-2. Simple front tracking for liquids in homogeneous,
isotropic media, 109
vii
Example 6-3. Steady-state gas flows in homogeneous, isotropic media, 111
Transient compressible flows, 113
Example 6-4. Numerical solution for steady flow, 114
Example 6-5. Explicit and implicit schemes for transient compressible
liquids, 116
Example 6-6. Transient compressible gas flows, 118
Problems and exercises, 121
7. Finite Difference Methods for Planar Flows, 122
Finite differences: basic concepts, 122
Formulating steady flow problems, 126
Steady flow problems: seven case studies, 128
Isotropy and anisotropy: fluid invasion in cross-bedded sands, 153
Problems and exercises, 158
8. Curvilinear Coordinates and Numerical Grid Generation, 160
General coordinate transformations, 162
Thompson’s mapping, 163
Some reciprocity relations, 164
Conformal mapping revisited, 165
Solution of mesh generation equations, 167
Problems and exercises, 172
9. Steady-State Reservoir Applications, 174
Governing equations, 176
Steady areal flow: generalized log r solution, 177
Streamline tracing in curvilinear coordinates, 181
Calculated steady flow examples, 183
Example 9-1. Well in Houston, 184
Example 9-2. Well in Dallas, 189
Example 9-3. Well in center of Texas, 190
Example 9-4. Fracture across Texas, 192
Example 9-5. Isothermal and adiabatic gas flows, 194
Mesh generation: several remarks, 197
Problems and exercises, 201
10. Transient Compressible Flows: Numerical Well Test
Simulation, 202
Example 10-1. Transient pressure drawdown, 203
Example 10-2. Transient pressure buildup, 207
Problems and exercises, 211
11. Effective Properties in Single and Multiphase Flows, 212
Example 11-1. Constant density liquid in steady linear flow, 212
Example 11-2. Lineal multiphase flow in two serial cores, 215
Example 11-3. Effective properties in steady cylindrical flows, 219
Example 11-4. Steady, single-phase, heterogeneous flows, 219
Example 11-5. Time scale for compressible transients, 219
Problems and exercises, 221
viii
12. Modeling Stochastic Heterogeneities, 222
Observations on existing models, 222
A mathematical strategy, 224
Example 12-1. Contractional fractures, 226
Problems and exercises, 228
13. Real and Artificial Viscosity, 229
Real viscosity and shockwaves, 229
Artificial viscosity and fictitious jumps, 232
Problems and exercises, 234
14. Borehole Flow Invasion, Lost Circulation, and Time
Lapse Logging, 235
Borehole invasion modeling, 235
Example 14-1. Thin lossy muds (that is, water), 236
Example 14-2. Time-dependent pressure differentials, 237
Example 14-3. Invasion with mudcake effects, 237
Time lapse logging, 238
Lost circulation, 243
Problems and exercises, 244
15. Horizontal, Deviated, and Modern Multilateral Well
Analysis, 245
Fundamental issues and problems, 246
Governing equations and numerical formulation, 252
Example calculations, 266
Example 15-1. Convergence acceleration, two deviated horizontal
gas wells in a channel sand, 267
Example 15-2. Dual-lateral horizontal completion in a fractured,
dipping, heterogeneous, layered formation, 270
Example 15-3. Stratigraphic grids, drilling dome-shaped structures, 273
Example 15-4. Simulating-while-drilling horizontal gas wells
through a dome-shaped reservoir, 275
Example 15-5. Modeling wellbore storage effects and compressible
borehole flow transients, 281
Problems and exercises, 287
16. Fluid Mechanics of Invasion, 288
Qualitative ideas on formation invasion, 290
Background literature, 294
Darcy reservoir flow equations, 297
Moving fronts and interfaces, 303
Problems and exercises, 305
17. Static and Dynamic Filtration, 306
Simple flows without mudcake, 306
Flows with moving boundaries, 312
Coupled dynamical problems: mudcake and formation interaction, 316
Dynamic filtration and borehole flow rheology, 325
ix
Concentric power law flows without pipe rotation, 334
Concentric power law flows with pipe rotation, 336
Formation invasion at equilibrium mudcake thickness, 337
Dynamic filtration in eccentric boreholes, 338
Problems and exercises, 340
18. Formation Tester Applications, 341
Problems and exercises, 351
19. Analytical Methods for Time Lapse Well Logging Analysis, 352
Experimental model validation, 352
Characterizing mudcake properties, 356
Porosity, permeability, oil viscosity, and pore pressure determination, 360
Examples of time lapse analysis, 367
Problems and exercises, 372
20. Complex Invasion Problems: Numerical Modeling, 373
Finite difference modeling, 373
Example 20-1. Lineal liquid displacement without mudcake, 381
Example 20-2. Cylindrical radial liquid displacement without cake, 386
Example 20-3. Spherical radial liquid displacement without cake, 389
Example 20-4. Lineal liquid displacement without mudcake, including
compressible flow transients, 391
Example 20-5. Von Neumann stability of implicit time schemes, 393
Example 20-6. Gas displacement by liquid in lineal core without
mudcake, including compressible flow transients, 395
Example 20-7. Simultaneous mudcake buildup and displacement front
motion for incompressible liquid flows, 399
Problems and exercises, 407
21. Forward and Inverse Multiphase Flow Modeling, 408
Immiscible Buckley-Leverett lineal flows without capillary pressure, 409
Molecular diffusion in fluid flows , 416
Immiscible radial flows with capillary pressure and prescribed mudcake
growth, 424
Immiscible flows with capillary pressure and dynamically coupled
mudcake growth, 438
Problems and exercises, 452
Cumulative References, 453
Index, 462
About the Author, 472
