Preface xiii
Acknowledgements xvii
1 Pressure Transient Analysis and Sampling in Formation Testing 1
Pressure transient analysis challenges 1
Background development 3
1.1 Conventional Formation Testing Concepts 5
1.2 Prototypes, Tools and Systems 6
1.2.1 Enhanced Formation Dynamic Tester (EFDT®) 9
1.2.2 Basic Reservoir Characteristic Tester (BASIC-RCT(TM)) 13
1.2.3 Enhancing and enabling technologies 15
Stuck tool alleviation 16
Field facilities 17
1.3 Recent Formation Testing Developments 17
1.4 References 20
2. Spherical Source Models for Forward and Inverse Formulations 21
2.1 Basic Approaches, Interpretation Issues and Modeling Hierarchies 23
Early steady flow model 23
Simple drawdown-buildup models 23
Analytical drawdown-buildup solution 25
Phase delay analysis 26
Modeling hierarchies 28
2.2 Basic Single-Phase Flow Forward and Inverse Algorithms 36
2.2.1 Module FT-00 36
2.2.2 Module FT-01 37
2.2.3 Module FT-03 38
2.2.4 Forward model application, Module FT-00 39
2.2.5 Inverse model application, Module FT-01 41
2.2.6 Effects of dip angle 43
2.2.7 Inverse "pulse interaction" approach using FT-00 46
2.2.8 FT-03 model overcomes source-sink limitations 49
2.2.9 Module FT-04, phase delay analysis, introductory for now 52
2.2.10 Drawdown-buildup, Module FT-PTA-DDBU 55
2.2.11 Real pumping, Module FT-06 59
2.3 Advanced Forward and Inverse Algorithms 61
2.3.1 Advanced drawdown and buildup methods Basic steady model 61
Validating our method 63
2.3.2 Calibration results and transient pressure curves 65
2.3.3 Mobility and pore pressure using first drawdown data 67
2.3.3.1 Run No. 1. Flowline volume 200 cc 68
2.3.3.2 Run No. 2. Flowline volume 500 cc 69
2.3.3.3 Run No. 3. Flowline volume 1,000 cc 71
2.3.3.4 Run No. 4. Flowline volume 2,000 cc 73
2.3.4 Mobility and pore pressure from last buildup data 74
2.3.4.1 Run No. 5. Flowline volume 200 cc 74
2.3.4.2 Run No. 6. Flowline volume 500 cc 76
2.3.4.3 Run No. 7. Flowline volume 1,000 cc 77
2.3.4.4 Run No. 8. Flowline volume 2,000 cc 78
2.3.4.5 Run No. 9. Time-varying flowline volume inputs from FT-07 79
2.3.5 Phase delay and amplitude attenuation, anisotropic media with dip - detailed theory, model and numerical results 81
2.3.5.1 Basic mathematical results 82
Isotropic model 82
Anisotropic extensions 82
Vertical well limit 83
Horizontal well limit 83
Formulas for vertical and horizontal wells 83
Deviated well equations 84
Deviated well interpretation for both kh and kv 85
Two-observation-probe models 86
2.3.5.2 Numerical examples and typical results 88
Example 1. Parameter estimates 89
Example 2. Surface plots 90
Example 3. Sinusoidal excitation 91
Example 4. Rectangular wave excitation 94
Example 5. Permeability prediction at general dip angles 96
Example 6. Solution for a random input 98
2.3.5.3 Layered model formulation 99
2.3.5.4 Phase delay software interface 100
2.3.5.5 Detailed phase delay results in layered anisotropic media 103
2.3.6 Supercharging and formation invasion introduction, with review of analytical forward and inverse models 110
2.3.6.1 Development perspectives 111
2.3.6.2 Review of forward and inverse models 113
FT-00 model 113
FT-01 model 117
FT-02 model 118
FT-06 and FT-07 models 119
FT-PTA-DDBU model 122
Classic inversion model 123
Supercharge forward and inverse models 123
Multiple drawdown and buildup inverse models 129
Multiphase invasion, clean-up and contamination 133
System integration and closing remarks 138
2.3.6.3 Supercharging summaries - advanced forward and inverse models explored 139
Supercharge math model development 139
Conventional zero supercharge model 141
Supercharge extension 142
2.3.6.4 Drawdown only applications 144
Example DD-1. High overbalance 144
Example DD-2. High overbalance 150
Example DD-3. High overbalance 154
Example DD-4. Qualitative pressure trends 158
Example DD-5. Qualitative pressure trends 161
Example DD-6. "Drawdown-only" data with multiple inverse scenarios for 1 md/cp application 163
Example DD-7. "Drawdown-only" data with multiple inverse scenarios for 0.1 md/cp application 168
2.3.6.5 Drawdown - buildup applications 173
Example DDBU-1. Drawdown-buildup, high overbalance 173
Example DDBU-2. Drawdown-buildup, high overbalance 177
Example DDBU-3. Drawdown-buildup, high overbalance 180
Example DDBU-4. Drawdown-buildup, 1 md/cp calculations 184
Example DDBU-5. Drawdown-buildup, 0.1md/cp calculations 188
2.3.7 Advanced multiple drawdown - buildup (or, "MDDBU") forward and inverse models 193
2.3.7.1 Software description 193
2.3.7.2 Validation of PTA-App-11 inverse model 200
2.3.8 Multiphase flow with inertial effects -Applications to borehole invasion, supercharging, clean-up and contamination analysis 217
2.3.8.1 Mudcake dynamics 217
2.3.8.2 Multiphase modeling in boreholes 220
2.3.8.3 Pressure and concentration displays 222
Example 1. Single probe, infinite anisotropic media 223
Example 2. Single probe, three layer medium 228
Example 3. Dual probe pumping, three layer medium 230
Example 4. Straddle packer pumping 231
Example 5. Formation fluid viscosity imaging 233
Example 6. Contamination modeling 234
Example 7. Multi-rate pumping simulation 234
2.4 References 236
3 Practical Applications Examples 237
3.1 Non-constant Flow Rate Effects 238
3.1.1 Constant flow rate, idealized pumping, inverse method 239
3.1.2 Slow ramp up/down flow rate 245
3.1.3 Impulsive start/stop flow rate 250
Closing remarks 255
3.2 Supercharging - Effects of Nonuniform Initial Pressure 256
Conventional zero supercharge model 256
Supercharge "Fast Forward" solver 258
3.3 Dual Probe Anisotropy Inverse Analysis 264
3.4 Multiprobe "DOI," Inverse and Barrier Analysis 273
3.5 Rapid Batch Analysis for History Matching 281
3.6 Supercharge, Contamination Depth and Mudcake Growth in "Large Boreholes" - Lineal Flow 289
Mudcake growth and filtrate invasion 289
Time-dependent pressure distributions 292
3.7 Supercharge, Contamination Depth and Mudcake Growth in Slimholes or "Clogged Wells" - Radial Flow 292
3.8 References 294
4 Supercharge, Pressure Change, Fluid Invasion and Mudcake Growth 295
Conventional zero supercharge model 295
Supercharge model 296
Relevance to formation tester job planning 298
Refined models for supercharge invasion 299
4.1 Governing equations and moving interface modeling 300
Single-phase flow pressure equations 300
Problem formulation 303
Eulerian versus Lagrangian description 303
Constant density versus compressible flow 304
Steady versus unsteady flow 305
Incorrect use of Darcy's law 305
Moving fronts and interfaces 306
Use of effective properties 308
4.2 Static and dynamic filtration 310
4.2.1 Simple flows without mudcake 310
Homogeneous liquid in a uniform linear core 311
Homogeneous liquid in a uniform radial flow 313
Homogeneous liquid in uniform spherical domain 314
Gas flow in a uniform linear core 315
Flow from a plane fracture 317
4.2.2 Flows with moving boundaries 318
Lineal mudcake buildup on filter paper 318
Plug flow of two liquids in linear core without cake 321
4.3 Coupled Dynamical Problems: Mudcake and Formation Interaction 323
Simultaneous mudcake buildup and filtrate invasion in a linear core (liquid flows) 323
Simultaneous mudcake buildup and filtrate invasion in a radial geometry (liquid flows) 327
Hole plugging and stuck pipe 330
Fluid compressibility 331
Formation invasion at equilibrium mudcake thickness 335
4.4 Inverse Models in Time Lapse Logging 336
Experimental model validation 336
Static filtration test procedure 337
Dynamic filtration testing 337
Measurement of mudcake properties 338
Formation evaluation from invasion data 338
Field applications 339
Characterizing mudcake properties 340
Simple extrapolation of mudcake properties 341
Radial mudcake growth on cylindrical filter paper 342
4.5 Porosity, Permeability, Oil Viscosity and Pore Pressure Determination 345
Simple porosity determination 345
Radial invasion without mudcake 346
Problem 1 348
Problem 2 350
Time lapse analysis using general muds 351
Problem 1 352
Problem 2 353
4.6 Examples of Time Lapse Analysis 354
Formation permeability and hydrocarbon viscosity 355
Pore pressure, rock permeability and fluid viscosity 357
4.7 References 360
5 Numerical Supercharge, Pressure, Displacement and Multiphase Flow Models 363
5.1 Finite Difference Solutions 364
Basic formulas 364
Model constant density flow analysis 366
Transient compressible flow modeling 369
Numerical stability 371
Convergence 371
Multiple physical time and space scales 372
Example 5-1. Lineal liquid displacement without mudcake 373
Example 5-2. Cylindrical radial liquid displacement without cake 380
Example 5-3. Spherical radial liquid displacement without cake 383
Example 5-4. Lineal liquid displacement without mudcake, including compressible flow transients 385
Example 5-5. Von Neumann stability of implicit time schemes 388
Example 5-6. Gas displacement by liquid in lineal core without mudcake, including compressible flow transients 390
Incompressible problem 391
Transient, compressible problem 392
Example 5-7. Simultaneous mudcake buildup and displacement front motion for incompressible liquid flows 396
Matching conditions at displacement front 399
Matching conditions at the cake-to-rock interface 399
Coding modifications 400
Modeling formation heterogeneities 403
Mudcake compaction and compressibility 404
Modeling borehole activity 405
5.2 Forward and Inverse Multiphase Flow Modeling 405
Problem hierarchies 406
5.2.1 Immiscible Buckley-Leverett lineal flows without capillary pressure 407
Example boundary value problems 409
General initial value problem 410
General boundary value problem for infinite core 411
Variable q(t) 411
Mudcake-dominated invasion 412
Shock velocity 412
Pressure solution 414
5.2.2 Molecular diffusion in fluid flows 415
Exact lineal flow solutions 416
Numerical analysis 417
Diffusion in cake-dominated flows 419
Resistivity migration 419
Lineal diffusion and "un-diffusion" examples 420
Radial diffusion and "un-diffusion" examples 423
5.2.3 Immiscible radial flows with capillary pressure and prescribed mudcake growth 425
Governing saturation equation 426
Numerical analysis 427
Fortran implementation 429
Typical calculations 429
Mudcake dominated flows 435
"Un-shocking" a saturation discontinuity 438
5.2.4 Immiscible flows with capillary pressure and dynamically coupled mudcake growth 441
Flows without mudcakes 441
Modeling mudcake coupling 450
Unchanging mudcake thickness 451
Transient mudcake growth 453
General immiscible flow model 457
5.3 Closing Remarks 458
5.4 References 464
Cumulative References 467
Index 481
About the Authors 498
