Preface ........................................................ xi
Acknowledgements ............................................. xiii
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. l, 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 Mobihty 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 Remarb ............................................ 93
3.9 References ................................................ 94
4 Phase Delay and Amplitude Attenuation for Mobility
Prediction in Anisotropic Media with Dip (Patent
Pending) .................................................. 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 and К ....... 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 Three-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 - FlowUne 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 FlowHne 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 Low Permeability Environments ................ 264
9 Closing Remarks .......................................... 270
References ............................................... 281
Index ......................................................... 287
About the Authors ............................................. 293
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