Quantitative Methods in Reservoir Engineering

4. Streamline Tracing and Complex Variables, 52......Page 1
6. Radial Flow Analysis, 108......Page 2
11. Effective Properties in Single and Multiphase Flows, 212......Page 3
17. Static and Dynamic Filtration, 306......Page 4
About the Author, 472......Page 5
Preface......Page 6
1 Motivating Ideas and Governing Equations......Page 10
EXAMPLES OF INCORRECT FORMULATIONS......Page 12
THOMPSON’S MAPPING......Page 0
LOGARITHMIC SOLUTIONS AND BEYOND......Page 20
FUNDAMENTAL AERODYNAMIC ANALOGIES......Page 21
Example 2-1. Single straight-line fracture in an isotropic circular reservoir containing incompressible fluid.......Page 28
Example 2-2. Line fracture in an anisotropic reservoir with incompressible liquids and compressible gases.......Page 36
Example 2-3. Effect of nonzero fracture thickness.......Page 41
Example 2-4. Flow rate boundary conditions.......Page 43
Example 2- 5. Uniform vertical velocity along the fracture.......Page 44
Example 2-6. Uniform pressure along the fracture.......Page 46
Example 2-7. More general fracture pressure distributions.......Page 47
Example 2-8. Velocity conditions for gas flows.......Page 48
Example 2-9. Determining velocity fields.......Page 49
Example 3- 1. Straight- line shale segment in uniform flow.......Page 52
Example 3- 3. Mineralized faults, anisotropy, and gas flow.......Page 58
Discussion 4-1. The classical streamfunction.......Page 61
Discussion 4-2. Streamfunction for general fluids in heterogeneous and anisotropic formations.......Page 64
Discussion 4- 3. Subtle differences between pressure and streamfunction formulations.......Page 66
Discussion 4-4. Streamline tracing in the presence of multiple wells.......Page 69
Discussion 4-5. Streamfunction expressions for distributed line sources and vortexes.......Page 72
Discussion 4-6. Streamfunction solution using complex variables techniques.......Page 74
Discussion 4-8. Generalized streamline tracing and volume flow rate computations.......Page 77
Discussion 4-9. Streamline tracing in 3D flows.......Page 79
Discussion 4-10. Tracer movement in 3D reservoirs.......Page 82
FLUID FLOW INSTABILITIES......Page 85
5 Flows in Complicated Geometries......Page 88
WHAT IS CONFORMAL MAPPING?......Page 89
Example 5-1. The classic radial flow solution.......Page 93
Example 5-2. Circular borehole with two symmetric radial fractures.......Page 95
Example 5- 3. Circular borehole with two uneven, opposite, radial fractures; or, a single radial fracture.......Page 97
Example 5-4. Circular borehole with multiple radial fractures.......Page 98
Example 5- 5. Straight shale segment at arbitrary angle.......Page 100
Example 5-6. Infinite array of straight-line shales.......Page 103
Example 5- 7. Pattern wells under aquifer drive.......Page 104
THREE- DIMENSIONAL FLOWS......Page 105
Example 5-9. Finite line source with prescribed pressure.......Page 106
Example 5- 10. Finite line source with prescribed flow rate.......Page 108
Example 5-11. Finite conductivity producing fracture having limited areal extent.......Page 109
BOREHOLE INTERACTIONS......Page 110
Example 5-13. Producing fracture near multiple wells under aquifer drive.......Page 111
Example 5- 14. Producing fractures near multiple wells in solid wall reservoirs.......Page 112
Example 5- 15. Straight- line shale segment near multiple wells in uniform flow.......Page 113
Examples 5- 16 and 5- 17. Nonproducing faults in solid wall and aquifer- driven reservoirs.......Page 114
Example 5-18. Highly curved fractures and shales.......Page 115
Example 6- 1. Steady liquids in homogeneous media.......Page 117
Example 6-2. Simple front tracking for liquids in homogeneous, isotropic media.......Page 118
Example 6- 3. Steady- state gas flows in homogeneous, isotropic media.......Page 120
TRANSIENT COMPRESSIBLE FLOWS......Page 122
Example 6-4. Numerical solution for steady flow.......Page 123
Example 6- 5. Explicit and implicit schemes for transient compressible liquids.......Page 125
Example 6-6. Transient compressible gas flows.......Page 127
FINITE DIFFERENCES: BASIC CONCEPTS......Page 131
FORMULATING STEADY FLOW PROBLEMS......Page 135
STEADY FLOW PROBLEMS: SEVEN CASE STUDIES......Page 137
WELLS AND INTERNAL BOUNDARIES......Page 145
POINT RELAXATION METHODS......Page 149
OBSERVATIONS ON RELAXATION METHODS......Page 152
ISOTROPY AND ANISOTROPY: FLUID INVASION IN CROSS- BEDDED SANDS......Page 162
8 Curvilinear Coordinates and Numerical Grid Generation......Page 169
GENERAL COORDINATE TRANSFORMATIONS......Page 171
SOME RECIPROCITY RELATIONS......Page 173
CONFORMAL MAPPING REVISITED......Page 174
SOLUTION OF MESH GENERATION EQUATIONS......Page 176
9 Steady-State Reservoir Applications......Page 183
GOVERNING EQUATIONS......Page 185
STEADY AREAL FLOW: GENERALIZED LOG R SOLUTION......Page 186
STREAMLINE TRACING IN CURVILINEAR COORDINATES......Page 190
CALCULATED STEADY FLOW EXAMPLES......Page 192
Example 9-1. Well in Houston.......Page 193
Example 9-2. Well in Dallas.......Page 198
Example 9-3. Well in center of Texas.......Page 199
Example 9-4. Fracture across Texas.......Page 201
Example 9-5. Isothermal and adiabatic gas flows.......Page 203
MESH GENERATION: SEVERAL REMARKS......Page 206
10 Transient Compressible Flows: Numerical Well Test Simulation......Page 211
Example 10- 1. Transient pressure drawdown.......Page 212
Example 10-2. Transient pressure buildup.......Page 216
Example 11- 1. Constant density liquid in steady linear flow.......Page 221
Example 11-2. Lineal multiphase flow in two serial cores.......Page 224
Example 11-5. Time scale for compressible transients.......Page 228
OBSERVATIONS ON EXISTING MODELS......Page 231
A MATHEMATICAL STRATEGY......Page 233
Example 12-1. Contractional fractures.......Page 235
REAL VISCOSITY AND SHOCKWAVES......Page 238
ARTIFICIAL VISCOSITY AND FICTITIOUS JUMPS......Page 241
BOREHOLE INVASION MODELING......Page 244
Example 14-1. Thin lossy muds (that is, water).......Page 245
Example 14-3. Invasion with mudcake effects.......Page 246
TIME LAPSE LOGGING......Page 247
LOST CIRCULATION......Page 252
15 Horizontal, Deviated, and Modern Multilateral Well Analysis......Page 254
FUNDAMENTAL ISSUES AND PROBLEMS......Page 255
GOVERNING EQUATIONS AND NUMERICAL FORMULATION......Page 261
EXAMPLE CALCULATIONS......Page 275
Example 15- 1. Convergence acceleration, two deviated horizontal gas wells in a channel sand.......Page 276
Example 15- 2. Dual- lateral horizontal completion in a fractured, dipping, heterogeneous, layered formation.......Page 279
Example 15-3. Stratigraphic grids, drilling dome-shaped structures.......Page 282
Example 15- 4. Simulating- while- drilling horizontal gas wells through a dome- shaped reservoir.......Page 284
Example 15- 5. Modeling wellbore storage effects and compressible borehole flow transients.......Page 290
16 Fluid Mechanics of Invasion......Page 297
QUALITATIVE IDEAS ON FORMATION INVASION......Page 299
BACKGROUND LITERATURE......Page 303
DARCY RESERVOIR FLOW EQUATIONS......Page 306
MOVING FRONTS AND INTERFACES......Page 312
SIMPLE FLOWS WITHOUT MUDCAKE......Page 315
FLOWS WITH MOVING BOUNDARIES......Page 321
COUPLED DYNAMICAL PROBLEMS: MUDCAKE AND FORMATION INTERACTION......Page 325
DYNAMIC FILTRATION AND BOREHOLE FLOW RHEOLOGY......Page 334
CONCENTRIC POWER LAW FLOWS WITHOUT PIPE ROTATION......Page 343
CONCENTRIC POWER LAW FLOWS WITH DRILLPIPE ROTATION......Page 345
FORMATION INVASION AT EQUILIBRIUM MUDCAKE THICKNESS......Page 346
DYNAMIC FILTRATION IN ECCENTRIC BOREHOLES......Page 347
18 Formation Tester Applications......Page 350
EXPERIMENTAL MODEL VALIDATION......Page 361
CHARACTERIZING MUDCAKE PROPERTIES......Page 365
POROSITY, PERMEABILITY, OIL VISCOSITY, AND PORE PRESSURE DETERMINATION......Page 369
EXAMPLES OF TIME LAPSE ANALYSIS......Page 376
FINITE DIFFERENCE MODELING......Page 382
Example 20- 1. Lineal liquid displacement without mudcake.......Page 390
Example 20- 2. Cylindrical radial liquid displacement without cake.......Page 395
Example 20- 3. Spherical radial liquid displacement without cake.......Page 398
Example 20- 4. Lineal liquid displacement without mudcake, including compressible flow transients.......Page 400
Example 20-5. Von Neumann stability of implicit time schemes.......Page 402
Example 20-6. Gas displacement by liquid in lineal core without mudcake, including compressible flow transients.......Page 404
Example 20- 7. Simultaneous mudcake buildup and displacement front motion for incompressible liquid flows.......Page 408
21 Forward and Inverse Multiphase Flow Modeling......Page 417
IMMISCIBLE BUCKLEY- LEVERETT LINEAL FLOWS WITHOUT CAPILLARY PRESSURE......Page 418
MOLECULAR DIFFUSION IN FLUID FLOWS......Page 425
IMMISCIBLE RADIAL FLOWS WITH CAPILLARY PRESSURE AND PRESCRIBED MUDCAKE GROWTH......Page 433
IMMISCIBLE FLOWS WITH CAPILLARY PRESSURE AND DYNAMICALLY COUPLED MUDCAKE GROWTH......Page 447
Cumulative References......Page 462
Index......Page 471
About the Author......Page 481