Formation Testing:
Low Mobility Pressure Transient Analysis
Wilson C. Chin, Ph.D., MIT
Stratamagnetic Software, LLC,
Yanmin Zhou, Yongren Feng and Qiang Yu
China Oilfield Services Limited, Beijing
Table of Contents
Preface, vii
Acknowledgements, ix
1. Basic Ideas, Interpretation Issues and Modeling Hierarchies, 1
1.1 Background and Approaches, 1
1.2 Modeling Hierarchies, 5
1.3 Experimental Methods and Tool Calibration, 13
1.4 References, 24
2. Single-Phase Flow Forward and Inverse Algorithms, 25
2.1 Overview, 25
2.2 Basic Model Summaries, 27
2.2.1 Module FT-00, 28
2.2.2 Module FT-01, 30
2.2.3 Module FT-03, 30
2.2.4 Forward model application, Module FT-00, 31
2.2.5 Inverse model application, Module FT-01, 33
2.2.6 Effects of dip angle, 35
2.2.7 Inverse "pulse interaction" approach using FT-00, 37
2.2.8 Computational notes, 40
2.2.9 Source model limitations and more complete model, 41
2.2.10 Phase delay analysis, Module FT-04, 43
2.2.11 Drawdown-buildup, Module FT-PTA-DDBU, 45
2.2.12 Real pumping, Module FT-06, 48
2.2.13 Closing remarks, 50
2.2.14 References, 50
3. Advanced Drawdown and Buildup Interpretation in Low Mobility Environments, 51
3.1 Basic Steady Flow Model, 51
3.2 Transient Spherical Flow Models, 53
3.2.1 Forward or direct analysis, 53
3.2.2 Dimensionless formulation, 54
3.2.3 Exact solutions for direct problem, 55
3.2.4 Special limit solutions, 56
3.2.5 New inverse approach for mobility and pore pressure prediction, 58
3.3 Multiple-Drawdown Pressure Analysis (Patent Pending), 59
3.3.1 Background on existing models, 59
3.3.2 Extension to anisotropic, no-skin applications, 60
3.3.2.1 Method 1 - Drawdown-alone test, 61
3.3.2.2 Method 2 - Single-drawdown-single-buildup test, 62
3.3.2.3 Method 3 - Double-drawdown-single-buildup test, 62
3.4 Forward Analysis with Illustrative Calibration, 64
3.5 Mobility and Pore Pressure Using First Drawdown Data, 66
3.5.1 Run No. 1, Flowline volume 200 cc, 66
3.5.2 Run No. 2, Flowline volume 500 cc, 69
3.5.3 Run No. 3, Flowline volume 1,000 cc, 71
3.5.4 Run No. 4, Flowline volume 2,000 cc, 73
3.6 Mobility and Pore Pressure from Last Buildup Data, 74
3.6.1 Run No. 5, Flowline volume 200 cc, 74
3.6.2 Run No. 6, Flowline volume 500 cc, 76
3.6.3 Run No. 7, Flowline volume 1,000 cc, 77
3.6.4 Run No. 8, Flowline volume 2,000 cc, 78
3.6.5 Run No. 9, Time-varying flowline volume, 79
3.7 Tool Calibration in Low Mobility Applications, 81
3.7.1 Steady flow model, 81
3.7.2 Example 1, Calibration using early-time buildup data, 81
3.7.3 Example 2, Calibration using early-time buildup data, 86
3.7.4 Example 3, Example 1 using drawdown data, 89
3.7.5 Example 4, Example 2 using drawdown data, 91
3.8 Closing Remarks, 93
3.9 References, 94
4. Phase Delay and Amplitude Attenuation for Mobility Prediction in Anisotropic Media with Dip, 95
4.1 Basic Mathematical Results, 96
4.1.1 Isotropic model, 96
4.1.2 Anisotropic equations, 98
4.1.3 Vertical well solution, 99
4.1.4 Horizontal well solution, 100
4.1.5 Formulas for vertical and horizontal wells, 101
4.1.6 Deviated well equations, 101
4.1.7 Deviated well interpretation for both kh and kv, 103
4.1.8 Two-observation-probe models, 105
4.2 Numerical Examples and Typical Results, 107
4.2.1 Example 1, Parameter estimates, 108
4.2.2 Example 2, Surface plots, 109
4.2.3 Example 3, Sinusoidal excitation, 110
4.2.4 Example 4, Rectangular wave excitation, 113
4.2.5 Example 5, Permeability prediction at general dip angles, 115
4.2.6 Example 6, Solution for a random input, 117
4.3 Layered Model Formulation, 118
4.3.1 Homogeneous medium, basic mathematical ideas, 118
4.3.2 Boundary value problem for complex pressure, 120
4.3.3 Iterative numerical solution to general formulation, 120
4.3.4 Successive line over relaxation procedure, 121
4.3.5 Advantages of the scheme, 122
4.3.6 Extensions to multiple layers, 122
4.3.7 Extensions to complete formation heterogeneity, 123
4.4 Phase Delay Software Interface, 123
4.4.1 Output file notes, 126
4.4.2 Special user features, 126
4.5 Detailed Phase Delay Results in Layered Anisotropic Media, 127
4.6 Typical Experimental Results, 134
4.7 Closing Remarks - Extensions and Additional Applications, 138
4.8 References, 139
5. Four Permeability Prediction Methods, 140
5.1 Steady-State Drawdown Example, 142
5.2 Early-Time, Low-Mobility Drawdown-Buildup, 144
5.3 Early-Time, Low-Mobility Drawdown Approach, 147
5.4 Phase Delay, Non-Ideal Rectangular Flow Excitation, 148
6. Multiphase Flow with Inertial Effects, 151
6.1 Physical Problem Description, 152
6.1.1 The physical problem, 152
6.1.2 Job planning considerations, 154
6.1.3 Modeling challenges, 155
6.1.4 Simulation objectives, 156
6.1.5 Modeling overview, 157
6.2 Immiscible Flow Formulation, 159
6.2.1 Finite difference solution, 160
6.2.2 Formation tester application, 161
6.2.3 Mudcake growth and formation coupling at sandface, 163
6.2.4 Pumpout model for single-probe pad nozzles, 165
6.2.5 Dual probe and packer surface logic, 166
6.3 Miscible Flow Formulation, 168
6.4 Inertial Effects with Forchheimer Corrections, 169
6.4.1 Governing differential equations, 169
6.4.2 Pumpout boundary condition, 171
6.4.3 Boundary value problem summary, 172
6.5 References, 173
7. Multiphase Flow - Miscible Mixing Clean-Up Examples, 175
7.1 Overview Capabilities, 175
7.1.1 Example 1, Single probe, infinite anisotropic media, 176
7.1.2 Example 2, Single probe, three layer medium, 181
7.1.3 Example 3, Dual probe pumping, three layer medium, 183
7.1.4 Example 4, Straddle packer pumping, 185
7.1.5 Example 5, Formation fluid viscosity imaging, 187
7.1.6 Example 6, Contamination modeling, 188
7.1.7 Example 7, Multi-rate pumping simulation, 189
7.2 Source Code and User Interface Improvements, 191
7.2.1 User data input panel, 191
7.2.2 Source code engine changes, 193
7.2.3 Output color graphics, 195
7.3 Detailed Applications, 200
7.3.1 Run No. 1, Clean-up, single-probe, uniform medium, 200
7.3.2 Run No. 2, Clean-up, dual-probe, uniform medium, 209
7.3.3 Run No. 3, Clean-up, elongated pad, uniform medium, 213
7.3.4 Run No. 4, A minimal invasion example, 218
7.3.5 Run No. 5, A single-phase fluid, constant viscosity example, 222
7.3.6 Run No. 6, A low-permeability "supercharging" example, 224
7.3.7 Run No. 7, A tree-layer simulation, 226
8. Time-Varying Flowline Volume, 229
8.1 Transient Anisotropic Formulation for Ellipsoidal Source, 230
8.1.1 Formulation for liquids and gases, 230
8.1.2 Similarity transform, 232
8.1.3 Transient flow numerical modeling, 233
8.1.4 Finite difference equation, 234
8.1.5 Boundary condition - flowline storage with and without skin effects, 235
8.1.6 Detailed time integration scheme, 236
8.1.7 Observation probe response, 237
8.2 FT-06 Software Interface and Example Calculations, 238
8.3 Time-Varying Flowline Volume Model, 244
8.3.1 Example 1, Software calibration, 245
8.3.2 Example 2, Simple interpretation using numerical pressure data, 252
8.3.3 Example 3, Simple interpretation using numerical pressure data, 255
8.3.4 Example 4, Simple interpretation using low permeability data, 257
8.3.5 Example 5, Simple interpretation using numerical pressure data, 258
8.3.6 Example 6, Simple interpretation using numerical pressure data, 262
8.3.7 Example 7, Enhancing phase delay detection in very lowpermeability environments, 264
9. Closing Remarks, 270
Cumulative References, 281
Lead Author Publications, 287
Index, 302
About the Authors, 307