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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 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 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 conpressible 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 Exanple 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 12. Modeling Stochastic Heterogeneities, 222 Observations on existing m6dels, 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 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 analys;, 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

Library of Congress subject headings for this publication: Oil reservoir engineering