Resistivity Modeling:

Propagation, Laterolog and Micro-Pad Analysis

by

Wilson C. Chin, Ph.D., M.I.T.

Stratamagnetic Software, LLC

Houston, Texas and Beijing, China

July 2016

Table of Contents

Preface, viii

Acknowledgements, xiii

  1. 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

1.4.1 Example 1 - Magnetic field results, 15

1.4.2 Example 2 - Electric field results, 16

1.4.3 Example 3 - Anisotropic resistivity results, 17

1.5 Closing Remarks, 25

1.6 References, 25

2. Axisymmetric Transient Models, 26

2.1 Physical Ideas, Engineering Models and Numerical Approaches, 27

2.1.1 Axisymmetric transient model – theory, 28

2.1.2 Numerical considerations, 30

2.1.2.1 Differential equation and finite difference

representation , 30

2.1.2.2 Matching conditions at horizontal bed layer

interfaces, 32

2.1.2.3 Matching conditions at radial interfaces, 33

2.1.2.4 Iterative solution by row relaxation, 34

2.1.3 Classic dipole solution, 35

2.1.4 Additional calibration models, 36

2.2 Transient Axisymmetric Coil Source Calculations, 37

2.2.1 R2D-6.for calculations (200 ´ 200 constant mesh), 38

2.2.1.1 Calculation 1 with R2D-6.for (200 ´ 200 constant mesh), 38

2.2.1.2 Calculation 2 with R2D-6.for (200 ´ 200 constant mesh), 40

2.2.1.3 Calculation 3 with R2D-6.for (200 ´ 200 constant mesh) 42

2.2.2 R2D-6.for calculations (very large 400 ´ 400 constant mesh), 43

2.2.2.1 Calculation 1 for R2D-6.for (very large 400 ´ 400 constant mesh), 43

2.2.2.2 Calculation 2 for R2D-6.for (very large 400 ´ 400 constant mesh), 46

2.2.2.3 Calculation 3 for R2D-6.for (very large 400 ´ 400 constant mesh) 48

2.2.3 R2D-7-Two-Horiz-Layer-No-Collar.for calculations (very large 400 ´ 400 constant mesh), 51

2.2.4 R2D-7-Two-Radial-Layer-Medium-No-Collar.for calculations (very large 400 ´ 400 constant mesh) , 53

2.2.5 R2D-8-GECF-MWDCollar-Larger-Mesh.for calculations (very large 400 ´ 400 constant mesh) , 55

2.2.5.1 Frequency, 400 kHz, MWD steel collar effects , 55

        1. Frequency, 2 MHz, MWD steel collar effects , 55

2.2.6 Detailed Results with R2D-6.for (200 ´200 constant mesh), 56

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 Formulations64

3.1 Laterolog Voltage Modeling and Interpretation Approach, 65

3.1.1 Direct current voltage formulation, 66

3.1.2 Finite differencing in anisotropic media, 67

3.2 Current Trajectories from Streamfunction Analysis, 68

3.2.1 Large cumulative errors along electric paths, 68

3.2.2 Streamfunction formulation derivation, 69

3.3 Voltage Calculations and Current Trajectories, 71

3.3.1 Example voltage and streamline calculations72

Run 1. Conductivities sv = 1.0, sh = 1.01, 74

Run 2. Conductivities sv = 1.01, sh = 1.0, 76

Run 3. Conductivities sv = 1, sh = 10, 78

Run 4. Conductivities sv = 10, sh = 1, 80

3.3.2 Tool design and data interpretation, 83

3.4 Current and Monitor Electrodes, 85

3.5 References, 85

4. Direct Current Models for Micro-Pad Devices, 86

4.1 Three-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

4.4.1 Physical considerations and path orientations, 90

4.4.2 Vertical well applications, 92

Run 1. Conductivities sv = 1.0, sh = 1.01, 92

Run 2. Conductivities sv = 1.01, sh = 1.0, 94

Run 3. Conductivities sv = 1, sh = 10, 96

Run 4. Conductivities sv = 10, sh = 1, 98

4.4.3 Horizontal well applications100

Run 5. Conductivities sv = 1.0, sh = 1.01, 100

Run 6. Conductivities sv = 1.01, sh = 1.0, 102

Run 7. Conductivities sv = 1, sh = 10, 104

Run 8. Conductivities sv = 10, sh = 1, 106

4.5 Closing Remarks, 108

4.6 References, 108

5. MWD/LWD Antenna Coil Modeling, 109

5.1 Axisymmetric and 3D Model Validation, 109

    1. Modeling a Center-Fed Linear Dipole Transmitter Antenna, 117

5.3 More Antenna Concepts, 127

      1. Linear dipole antennas, 127

      2. MWD/LWD applications - reconfigurable antennas, 127

      3. Fly-swatter receivers, interesting thoughts, 132

5.3.3.1 Full fly-swatter computations, 144

Example 5-1a. Horizontal transmitter coil in uniform isotropic media, 147

Example 5-2a. Vertical transmitter coil in uniform isotropic media, 151

Example 5-3a. Horizontal transmitter coil in layered isotropic media, 153

Example 5-4a. Vertical transmitter coil in layered isotropic media, 154

5.3.3.2 Half fly-swatter computations, 155

Example 5-1b. Horizontal transmitter coil in uniform isotropic media, 158

Example 5-2b. Vertical transmitter coil in uniform isotropic media, 159

Example 5-3b. Horizontal transmitter coil in layered isotropic media, 160

Example 5-4b. Vertical transmitter coil in layered isotropic media, 161

5.4 References, 162

 

6. What is Resistivity?, 163

6.1 Resistance in Serial and Parallel Circuits, Using Classical

Algebraic Approach, 163

6.1.1 Series circuits , 163

6.1.2 Parallel circuits, 164

6.1.3 Complicated circuits, 164

6.2 Resistance in Serial and Parallel Circuits, Using Differential

Equation Approach, 165

6.2.1 Cores arranged in series, 165

6.2.2 Effective conductivity and resistivity and harmonic

averaging, 166

6.2.3 Cores arranged in parallel, 166

    1. Isotropy and Anisotropy in Cross-bedded Sands, 167

      2. 6.3.1 Cross-bedded sands, 167

      6.3.2 Numerical results, 169

    2. Tool Measurements and Geological Model, s171

    3. References, 172

7. Multiphase Flow and Transient Resistivity, 173

7.1 Immiscible Buckley-Leverett Linear Flows Without Capillary Pressure, 176

7.1.1 Theory and mathematical modeling, 176

7.1.2 Example boundary value problems, 178

7.1.2.1 General initial value problem, 178

7.1.2.2 General boundary value problem for

infinite core, 179

7.1.2.3 Mudcake-dominated invasion, 180

7.1.2.4 Shock velocity, 181

7.1.2.5 Pressure solution, 182

    1. Molecular Diffusion in Fluid Flows, 183

      2. 7.2.1 Exact lineal flow solutions, 184

      7.2.2 Numerical analysis, 185

      7.2.3 Diffusion in cake-dominated flows, 186

      7.2.4 Resistivity migration, 186

      7.2.4.1 Lineal diffusion and undiffusion example, s188

      7.2.4.2 Radial diffusion and undiffusion examples, 191

    2. Immiscible Radial Flows with Capillary Pressure and Prescribed Mudcake Growth, 193

      3. 7.3.1 Governing saturation equation, 193

      7.3.2 Numerical analysis, 195

      7.3.3 Fortran implementation, 196

      7.3.4 Typical calculations, 196

      7.3.5 Mudcake-dominated flows, 202

      7.3.6 Unshocking a saturation discontinuity, 205

    3. Immiscible Flows with Capillary Pressure and Dynamically Coupled Mudcake Growth – Theory and Numerics, 208

      4. 7.4.1 Flows without mudcakes, 208

      7.4.2 Modeling mudcake coupling , 215

      7.4.3 Unchanging mudcake thickness, 217

      7.4.4 Transient mudcake growth , 219

      7.4.5 General immiscible flow model, 222

    4. Immiscible Flows with Capillary Pressure and Dynamically Coupled Mudcake Growth – Detailed Examples, 223

      5. 7.5.1 Example 1, Single probe, infinite anisotropic media, 224

      7.5.2 Example 2, Single probe, three layer medium, 227

      7.5.3 Example 3, Dual probe pumping, three layer medium, 229

      7.5.4 Example 4, Straddle packer pumping, 231

    5. Simple Example in Time Lapse Logging, 234
    6. Resistivity Distributions Variable in Space and Time, 247

      7. 7.7.1 Archie’s Law, 247

      7.7.2 Closing remarks, 249

    7. References, 250

8. Analytical Methods for Time Lapse Well Logging Analysis, 251

    1. Experimental Model Validation, 251

      2. 8.1.1 Static filtration test procedure, 251

      8.1.2 Dynamic filtration testing, 252

      8.1.4 Formation evaluation from invasion data, 253

      8.1.5 Field applications2, 54

    2. Characterizing Mudcake Properties, 255

      3. 8.2.1 Simple extrapolation of mudcake properties, 255

      8.2.2 Radial mudcake growth on cylindrical filter paper, 257

    3. Porosity, Permeability, Oil Viscosity and Pore Pressure Determination, 259

      4. 8.3.1 Simple porosity determination, 260

      8.3.2 Radial invasion without mudcake, 260

      8.3.2.1 Problem 1, 262

      8.3.2.2 Problem 2, 264

      8.3.3 Time lapse analysis using general muds, 265

      8.3.3.1 Problem 1, 266

      8.3.3.2 Problem 2, 267

    4. Examples of Time Lapse Analysis268

      5. 8.4.1 Formation permeability and hydrocarbon viscosity, 268

      8.4.2 Pore pressure, rock permeability and fluid viscosity, 271

    5. References, 273

 

Cumulative References, 274

Index, 276

About the Author, 282