Resistivity Modeling (eBook)

Wilson C. Chin earned his Ph.D. from M.I.T. and his M.Sc. from Caltech. He has published fifteen books describing his original research in reservoir engineering, formation testing, managed pressure drilling, wave propagation, Measurement While Drilling (MWD) and electromagnetic well logging, over one hundred papers and three dozen patents.?Mr. Chin has consulted for well known domestic and international oil and gas corporations, and during the past two decades, won five prestigious research contracts and awards in petroleum exploration and production with the United States Department of Energy.

Preface xi

Acknowledgements xvii

1 Physics, Math and Basic Ideas 1

1.1 Background, Industry Challenges and Frustrations 1

1.2 Iterative Algorithms and Solutions 2

1.3 Direct Current Focusing from Reservoir Flow Perspective 5

1.4 General Three-Dimensional Electromagnetic Model 11

1.5 Closing Remarks 25

1.6 References 25

2 Axisymmetric Transient Models 26

2.1 Physical Ideas, Engineering Models and Numerical Approaches 27

2.2 Transient Axisymmetric Coil Source Calculations 37

2.3 Effects of Frequency, from Induction, to Propagation, to Dielectric 59

2.4 Depth of Investigation 60

2.5 Closing Remarks Related to Interpretation 61

2.6 References 63

3 Steady Axisymmetric Formulations 64

3.1 Laterolog Voltage Modeling and Interpretation Approach 65

3.2 Current Trajectories from Streamfunction Analysis 68

3.3 Voltage Calculations and Current Trajectories 71

Run 1. Conductivities sigmav = 1.0, sigmah = 1.01 74

Run 2. Conductivities sigmav = 1.01, sigmah = 1.0 76

Run 3. Conductivities sigmav = 1, sigmah = 10 78

Run 4. Conductivities sigmav = 10, sigmah = 1 80

3.4 Current and Monitor Electrodes 85

3.5 References 85

4 Direct Current Models for Micro-Pad Devices 86

4.1 Th ree-Dimensional, Anisotropic, Steady Model 87

4.2 Finite Difference Approach and Subtleties 88

4.3 Row versus Column Relaxation 88

4.4 Pads Acting on Vertical and Horizontal Wells 90

Run 1. Conductivities sigmav = 1.0, sigmah = 1.01 (vertical well) 92

Run 2. Conductivities sigmav = 1.01, sigmah = 1.0 (vertical well) 94

Run 3. Conductivities sigmav = 1, sigmah = 10 (vertical well) 96

Run 4. Conductivities sigmav = 10, sigmah = 1 (vertical well) 98

Run 5. Conductivities sigmav = 1.0, sigmah = 1.01 (horizontal well) 100

Run 6. Conductivities sigmav = 1.01, sigmah = 1.0 (horizontal well) 102

Run 7. Conductivities sigmav = 1, sigmah = 10 (horizontal well) 104

Run 8. Conductivities sigmav = 10, sigmah = 1 (horizontal well) 106

4.5 Closing Remarks 108

4.6 References 108

5 Coil Antenna Modeling for MWD Applications 109

5.1 Axisymmetric and 3D Model Validation 109

5.2 Modeling a Center-Fed Linear Dipole Transmitter Antenna 117

5.3 More Antenna Concepts 127

5.4 References 162

6 What is Resistivity? 163

6.1 Resistance in Serial and Parallel Circuits, Using Classical Algebraic Approach 163

6.2 Resistance in Serial and Parallel Circuits, Using Differential Equation Approach 165

6.3 Isotropy and Anisotropy in Cross-bedded Sands 167

6.4 Tool Measurements and Geological Models 171

6.5 References 172

7 Multiphase Flow and Transient Resistivity 173

7.1 Immiscible Buckley-Leverett Linear Flows Without Capillary Pressure 176

7.2 Molecular Diffusion in Fluid Flows 183

7.3 Immiscible Radial Flows with Capillary Pressure and Prescribed Mudcake Growth 193

7.4 Immiscible Flows with Capillary Pressure and Dynamically Coupled Mudcake Growth - Theory and Numerics 208

7.5 Immiscible Flows with Capillary Pressure and Dynamically Coupled Mudcake Growth - Detailed Examples 223

7.6 Simple Example in Time Lapse Logging 234

7.7 Resistivity Distributions Variable in Space and Time 247

7.8 References 250

8 Analytical Methods for Time Lapse Well Logging Analysis 251

8.1 Experimental Model Validation 251

8.2 Characterizing Mudcake Properties 255

8.3 Porosity, Permeability, Oil Viscosity and Pore Pressure Determination 259

8.4 Examples of Time Lapse Analysis 268

8.5 References 273

Cumulative References 274

Index 276

About the Author 282