Wave Propagation in Drilling, Well Logging and Reservoir Applications

by

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

Stratamagnetic Software, LLC

Houston, Texas and Beijing, China

 

Table of Contents

Preface, xiii

Dedication , xv

Acknowledgements, xvi

1. Overview and Fundamental Ideas, 1

1.1 The Classical Wave Equation, 2

1.1.1 Fundamental properties, 2

1.1.2 Reflection properties, 5

1.1.2.1 Example 1-1. Rigid end termination, 5

1.1.2.2 Example 1-2. Stress-free end, 6

1.1.2.3 Note on acoustics, 6

1.2 Fundamental Representation, 7

1.2.1 Taylor series, 7

1.2.2 Fourier series, 7

1.3 Separation of Variables and Eigenfunction Expansions, 8

1.3.1 Example 1-3. String with pinned ends and general initial

conditions, 9

1.3.2 Example 1-4. String with distributed forces, 10

1.3.3 Example 1-5. Alternative boundary conditions, 11

1.3.4 Example 1-6. Mixed boundary conditions, 11

1.3.5 Example 1-7. Problems without initial conditions, 13

1.3.5.1 Example 1-7a. Naive approach, 13

1.3.5.2 Example 1-7b. Correct approach, 14

1.3.5.3 Example 1-7c. Faster approach, 14

1.3.6 Example 1-8. Dissipative wave solution, 14

1.4 Standing Versus Propagating Waves, 16

1.4.1 Standing waves, 16

1.4.2 Propagating waves, 16

1.4.3 Combined standing and propagating waves, 17

1.4.4 Characterizing propagating waves, 17

1.5 Laplace Transforms, 20

1.5.1 Wave equation derivation, 20

1.5.2 Example 1-9. String falling under its own weight, 21

1.5.3 Example 1-10. Semi-infinite string with a general end support, 22

1.5.3.1 Example 1-10a. Rectangular pulse, 25

1.5.3.2 Example 1-10b. Impulse response, 25

1.5.3.3 Example 1-10c. Incident sinusoidal wavetrain, 26

1.6 Fourier Transforms, 26

1.6.1 Example 1-11. Propagation of an initially static disturbance, 27

1.6.2 Example 1-12. Directional properties, special wave , 28

1.7 External Forces Versus Boundary Conditions, 30

1.7.1 Single point force, 30

1.7.2 Properties of point loads, 32

1.7.2.1 Example 1-13. Boundary conditions versus forces, 32

1.7.2.2 Couples or dipoles, 33

1.7.2.3 Multiple forces and higher order moments, 36

1.7.2.4 Symmetries and anti-symmetries, 36

1.7.2.5 Impulse response, 36

1.7.2.6 On the subtle meaning of impulse, 39

1.7.2.7 Example 1-14. Incorrect use of impulse response, 39

1.7.2.8 Additional models, 39

1.7.2.9 Other delta function properties, 40

1.8 Point Force and Dipole Wave Excitation, 42

1.8.1 Example 1-15. Finite string excited by a time-varying concentrated point force, 42

1.8.2 Example 1-16. Finite string excited by a time-varying point dipole (i.e., a force couple), 44

1.8.3 Example 1-17. Splitting of an applied initial disturbance, 45

1.9 First-Order Partial Differential Equations, 46

1.10 References, 49

2. Kinematic Wave Theory, 50

2.1 Whitham's Theory in Nondissipative Media, 51

2.1.1 Uniform media, 52

2.1.2 Example 2-1. Transverse beam vibrations, 52

2.1.3 Example 2-2. Simple longitudinal oscillations, 52

2.1.4 Example 2-3. Asymptotic stationary phase expansion, 53

2.1.5 Simple consequences of KWT, 54

2.1.6 Nonuniform media, 56

2.1.7 Example 2-4. Numerical integration, 56

2.1.8 Ease of use is important to practical engineering, 57

2.2 Simple Attenuation Modeling, 57

2.2.1 The Q-model, 57

2.2.2 Relating Q to amplitude in space, 58

2.2.3 Relating Q to standing wave decay, 59

2.2.4 Kinematic wave generalization, 59

2.3 KWT in Homogeneous Dissipative Media, 60

2.3.1 Example 2-5. General initial value problem in uniform media, 61

2.3.2 Singularities of the kinematic field, 62

2.3.3 The energy singularity, 62

2.3.4 Example 2-6. Modeling dynamically steady motions, 63

2.4 High-Order Kinematic Wave Theory, 64

2.4.1 Basic assumptions, 64

2.4.2 The general amplitude equation, 65

2.4.3 Method of multiple scales, 66

2.4.4 Generalized wave results, 68

2.4.5 The low-order limit, 70

2.5 Effect of Low-Order Nonuniformities, 70

2.5.1 Detailed formal analysis, 71

2.5.2 Wave energy and momentum, 71

2.5.3 Example 2-7. String with variable properties, 73

2.5.4 Computational solution, 73

2.5.5 Dynamically steady problems, 74

2.5.6 Waves in nonuniform moving media, 75

2.5.7 Average Lagrangian formalism, 75

2.5.8 Example 2-8. Wave action conservation, 75

2.6 Three-Dimensional Kinematic Wave Theory, 76

2.6.1 Wave irrotationality, 77

2.6.2 The ray equation, 78

2.6.3 Frequency variation, 78

2.6.4 Energy variation, 79

2.6.5 Ray topology, 79

2.6.6 Example 2-9. Acoustics application, 79

2.7 References, 80

3. Examples from Classical Mechanics, 82

3.1 Example 3-1. Lateral Vibration of Simple Beams, 82

3.1.1 Example 3-1a. Hinged ends, 84

3.1.2 Example 3-1b. Clamped end, other end free, 84

3.2 Example 3-2. Acoustic Waves in Waveguides, 85

3.2.1 Simple waveguides, 85

3.2.2 Simple hydraulic flows, 87

3.2.3 Acoustic simplifications, 87

3.2.4 Three-dimensional wave equation, 88

3.2.5 Modal solution, 88

3.2.6 The dispersion relation, 90

3.2.7 Physical interpretation, 90

3.2.8 MWD notes, 91

3.2.9 Phase and group velocity, 91

3.2.10 The velocity potential, 93

3.2.11 Modeling MWD sources, 94

3.3 Example 3-3. Gravity-Capillary Waves in Deep Water, 96

3.3.1 Governing Laplace equation, 96

3.3.2 Boundary conditions, kinematic and dynamic, 97

3.3.3 Problem solution, 98

3.3.4 Energy considerations, 99

3.4 Example 3-4. Fluid-Solid Interaction - Waves on Elastic

Membranes, 100

3.4.1 Governing Rayleigh equation, 101

3.4.2 Boundary conditions for potential, 102

3.4.3 Eigenvalue bounds, 103

3.5 Example 3-5. Problems in Hydrodynamic Stability, 104

3.5.1 Neutral stability diagrams, 104

3.5.2 Borehole flow stability, 105

3.5.3 Stability of irrotational flows, 106

3.6 References, 106

4. Drillstring Vibrations: Classic Ideas and Modern Approaches, 109

4.1 Typical Downhole Vibration Environment, 110

4.1.1 What is wave motion?, 110

4.1.2 Drillstring vibration modes, axial, torsional and lateral, 111

4.1.2.1 Axial vibrations, 111

4.1.2.2 Transverse vibrations, 112

4.1.2.3 Torsional vibrations, 113

4.1.2.4 Whirling vibrations, 113

4.1.2.5 Coupled axial, torsional and lateral vibrations, 113

4.1.2.6 Transient and dynamically steady oscillations, 114

4.1.2.7 Understanding the environment, 114

4.1.3 Long-standing vibrations issues, 115

4.1.3.1 Example 4-1. Case of the missing waves, 115

4.1.3.2 Example 4-2. Looking for resonance in all the wrong places, 116

4.1.3.3 Example 4-3. Drillstrings that don't drill, 116

4.1.3.4 Example 4-4. Modeling coupled vibrations, 116

4.1.3.5 Example 4-5. Energy transfer mechanisms, 116

4.1.4 Practical applications, 117

4.1.4.1 Anecdotal stories, 117

4.1.4.2 Applications to the field, 117

(Structural damage; Formation damage; Directional drilling; Increasing rate of penetration; Improved MWD tools and mud motors; Formation imaging; Psychological discomfort)

4.1.5 Elastic line model of the drillstring, 119

4.1.5.1 Early efforts, 119

4.1.5.2 Elastic line simplifications, 120

4.1.5.3 Historical precedents, 120

4.1.5.4 Our focus, 121

4.1.6 Objectives and discussion plan, 122

4.2 Axial Vibrations, 123

4.2.1 Pioneering axial vibration studies, 124

4.2.2 Governing differential equations, 126

4.2.2.1 Damped wave equation, 126

4.2.2.2 External forces and displacement sources, 127

4.2.2.3 Dynamic and static solutions, 128

4.2.2.4 Free-fall as a special solution, 128

4.2.2.5 More on AC/DC interactions, 129

4.2.3 Conventional separation of AC/DC solutions, 129

4.2.3.1 Sign conventions, 130

4.2.3.2 Static weight on bit, 131

4.2.4 Boundary conditions - old and new ideas, 132

4.2.4.1 Surface boundary conditions, 132

4.2.4.2 Conventional bit boundary conditions, 133

4.2.4.3 Modeling rock-bit interactions, 134

4.2.4.4 Empirical notes on rock-bit interaction, 136

(Laboratory drillbit data; Single-tooth impact results)

4.2.4.5 Modeling drillbit kinematics using "displacement

sources", 139

(Analogies from earthquake seismology)

4.2.5 Global energy balance, 142

4.2.5.1 Formulation summary, 142

4.2.5.2 Energy considerations, 142

(The drillstring; The surface; Combined drillstring

/surface system)

4.2.5.3 Detailed bit motions, 144

4.2.6 Simple solution for rate-of-penetration, 145

4.2.6.1 Field motivation, 145

4.2.6.2 Simple analytical solution, 146

4.2.6.3 Classic fixed end, 146

4.2.6.4 Classic free end, 146

4.2.6.5 Other possibilities, 147

4.2.6.6 Simple derivative model, 147

4.2.6.7 The general impedance mode, 147

4.2.6.8 Modeling the constants alpha, beta and gamma, 149

4.2.7 Finite difference modeling, 149

4.2.7.1 Elementary considerations, 149

4.2.7.2 Transient finite difference modeling, 151

(The solution methodology; Stability of the scheme;

Grid sizes, time steps, and convergence)

4.2.8 Complete formulation and numerical solution, 156

4.2.8.1 The boundary value problem, 156

4.2.8.2 Computational objective, 157

4.2.8.3 Difference approximations, 157

4.2.9 Modeling pipe-to-collar area changes, 159

4.2.9.1 Matching conditions, 160

4.2.9.2 Finite difference model, 160

4.2.9.3 Generalized formulation, 161

4.2.9.4 Alternative boundary conditions, 161

4.2.10 Example Fortran implementation, 162

4.2.10.1 Code fragment, 162

4.2.10.2 Modeling dynamically steady problems, 165

4.2.10.3 Jarring issues and stuck pipe problems, 167

4.2.11 Drillstring and formation imaging, 168

4.2.11.1 Drillstring imaging, 169

4.2.11.2 Seeing ahead of the bit: MWD-VSP and vibration

logging, 169

(MWD-VSP; Vibration logging of the formation)

4.2.11.3 Notes on rock-bit interaction, 171

4.2.11.4 Basic mathematical approach, 173

4.2.11.5 More rock-bit interaction models, 174

(An inelastic impact model; Elastic impacts, with stress

effects)

4.2.11.6 Separating incident from reflected waves, 179

(Delay line method; Differential technique; Three-

wave formulation; Digital analysis methods)

4.3 Lateral Bending Vibrations, 184

4.3.1 Why explain this drilling paradox?, 184

4.3.2 Lateral vibrations in deepwater operations, 185

4.3.2.1 Marine risers, 185

4.3.2.2 Bending vibrations in directional control, 186

4.3.2.3 Plan for remainder of chapter, 186

4.3.3 A downhole paradox Ė "Case of the vanishing waves", 186

4.3.3.1 Physical features observed at failure, 187

4.3.3.2 Field evidence widely available, 187

4.3.3.3 Wave trapping, a simple analogy, 189

4.3.3.4 Extension to general systems, 190

4.3.4 Why drillstrings fail at the neutral point, 191

4.3.4.1 Beam equation analysis, 192

4.3.4.2 Kinematic wave modeling, 193

4.3.4.3 Bending amplitude distribution in space, 199

4.3.4.4 Designing safe drill collars, 202

4.3.4.5 Viscous dissipation, 203

4.3.5 Surface detection of downhole bending disturbances, 203

4.3.5.1 Detecting lateral vibrations, 203

4.3.5.2 Nonlinear axial equation, 204

4.3.5.3 Detecting lateral vibrations from the surface, 205

4.3.6 Linear boundary value problem formulation, 206

4.3.6.1 General linear equation, 206

4.3.6.2 Auxiliary conditions, 207

4.3.7 Finite difference modeling, 208

4.3.7.1 Pentadiagonal difference equations, 209

4.3.7.2 Finite difference beam recipe, 210

4.3.7.3 Additional modeling considerations, 211

(Borehole wall contacts; Modeling steady state

oscillations; Simulating area changes)

4.3.8 Example Fortran implementation, 212

4.3.9 Nonlinear interaction between axial and lateral bending vibrations, 215

4.4 Torsional and Whirling Vibrations, 216

4.4.1 Torsional wave equation, 216

4.4.2 Stick-slip oscillations, 219

4.4.2.1 Energy considerations, 220

4.4.2.2 Static torque effects on bending, 221

4.4.2.3 Finite difference modeling, 222

4.4.2.4 WOB/TOB (Weight-on-bit/Torque-on-bit), 4424

4.4.2.5 Applications to MWD telemetry, 223

4.4.2.6 Example Fortran implementation, 223

4.4.2.7 Whirling motions, 225

(Example 4-6. Machine shaft example;

Example 4-7. Generalized whirl)

4.4.2.8 Causes of whirling motions, 226

4.5 Coupled Axial, Torsional and Lateral Vibrations, 227

4.5.1 Importance to PDC bit dynamic, 227

4.5.2 Coupled axial, torsional and bending vibrations, 228

4.5.2.1 Example 4-8. Simple desktop experiment, 228

4.5.3 Notes on the coupled model, 229

4.5.4 Coupled axial, torsional and bending vibrations, 229

4.5.4.1 Partial differential equations, 230

4.5.4.2 Finite differencing the coupled bending equations, 231

4.5.4.3 Computational recipe, 233

4.5.4.4 Modes of coupling, 233

4.5.4.5 Numerical considerations, 234

4.5.4.6 General Fortran implementation, 235

4.5.4.7 Example calculations: bit-bounce, stick-slip,

rate-of-penetration and drillstring precession, 239

(Test A. Smooth drilling and making hole;

Test B. Rough drilling with bit bounce; Model

limitations and extensions)

4.5.4.8 Precessional instabilities, 244

4.5.4.9 Comments on Dunayevsky model, 244

4.5.4.10 Direct simulation of bit precession, 246

4.5.4.11 Drillstring vibrations in horizontal wells, 247

4.6 References, 248

5. Mud Acoustics in Modern Drilling, 257

5.1 Governing Lagrangian Equations, 258

5.1.1 Hydraulic versus acoustic motion, 258

5.1.2 Differential equation, 259

5.1.3 Area and material discontinuities, 259

5.1.4 Mud acoustic formulation, 261

5.1.5 Example 5-1. Idealized reflections and transmissions, 261

5.1.6 Example 5-2. Classical water hammer, 263

5.1.7 Example 5-3. Acoustic pipe resonances, 263

5.1.7.1 Closed-closed ends, 264

5.1.7.2 Open-open ends, 264

5.1.7.3 Closed-open ends, 264

5.1.8 Example 5-4. Passage through area obstructions, 265

5.1.9 Example 5-5. Transmission through contrasting media, 266

5.2 Governing Eulerian Equations, 267

5.2.1 Steady and unsteady hydraulic limits, 268

5.2.2 Separating hydraulic and acoustic effects, 269

5.3 Transient Finite Differencing Modeling, 272

5.3.1 Basic difference model, 272

5.3.2 Modeling area discontinuities, 273

5.3.2.1 Axial vibrations, 273

5.3.2.2 Mud acoustics, 274

5.4 Swab-Surge Modeling, 275

5.4.1 Wave physics of swab-surge, 275

5.4.2 Designing a swab-surge simulator, 277

5.5 MWD Mud Pulse Telemetry, 278

5.5.1 Basic MWD system components, 278

5.5.2 Candidate transmission technologies - with brief survey of early work, 279

5.5.3 Mud pulse telemetry - the acoustic source, 281

5.5.3.1 Positive pressure poppet valves, 281

5.5.3.2 Negative pressure valves, 283

5.5.3.3 Mud siren sources, 285

5.5.3.4 Signal generation at the source, 286

5.5.3.5 Mechanical design considerations, 287

(Packaging constraints; Shock and vibration; Mud

erosion; Power requirements; High pressure and

temperature; Fluid mechanics problems)

5.5.3.6 Mud pulse telemetry - the transmission channel, 289

5.5.3.7 The transmission channel uphole, 290

5.5.3.8 Telemetry design objectives, 291

5.5.3.9 Additional practical considerations, 292

5.5.3.10 The theoretical maximum, 293

5.5.3.11 Acoustic signals in the annulus, 293

5.6 Recent MWD Developments, 294

5.7 References, 303

6. Geophysical Ray Tracing, 306

6.1 Classical Wave Modeling - Eikonal Methods and Ray Tracing, 307

6.1.1 The plane wave, 307

6.1.2 High frequency limit, 307

6.1.3 Eikonal equation in nonuniform media, 308

6.1.4 Continuing the series, 308

6.1.5 Integrating the eikonal equation, 308

6.1.6 Summary of ray tracing results, 310

6.2 Fermatís Principal of Least Time (via Calculus of Variations), 310

6.2.1 Travel time along a ray, 310

6.2.2 Calculus of variations, 311

6.2.3 Eikonal solution satisfies least time condition, 312

6.3 Fermatís Principle Revisited Via Kinematic Wave Theory, 312

6.4 Modeling Wave Dissipation, 313

6.4.1 Example 6-1. A simple model, 314

6.4.2 Example 6-2. Another case history, 314

6.4.3 Example 6-3. Motivating damped wave study, 314

6.4.4 The quality factor Q, 315

6.4.5 A simple example, 315

6.5 Ray Tracing Over Large Space-Time Scales, 317

6.5.1 High-order modulation equations, 317

6.5.1.1 The low-order limit, 318

6.5.1.2 Extended eikonal equations, 318

6.5.1.3 Extended eikonal equations in homogeneous medium, 318

6.5.1.4 The seismic limit, 319

6.5.1.5 Example 6-4. Simple rock formations, 319

6.6 Subtle High-Order Effects, 320

6.6.1 A low-order nonlinear wave equation, 320

6.6.2 Singularities in the low-order model, 321

6.6.3 Existence of the singularity, 321

6.6.4 Entropy conditions, 322

6.7 Travel-Time Modeling, 324

6.7.1 Applications to crosswell tomography, 324

6.7.2 Applications to surface seismics, 325

6.7.3 Finite difference calculation of travel times, 325

6.7.4 Difficulties with simple difference formulation, 326

6.7.4.1 Two space dimensions, 326

6.7.4.2 Three space dimensions, 326

6.7.4.3 Analysis of the problem, 327

6.8 References, 329

7. Wave and Current Interaction in the Ocean, 331

7.1 Wave Kinematics and Energy Summary, 331

7.1.1 Damped waves in deep water, 332

7.1.1.1 Effect of low-order dissipation, 332

7.1.1.2 Effect of variable background flow, 332

7.1.2 Waves in finite depth water, 333

7.2 Sources of Hydrodynamic Loading, 334

7.3 Instabilities Due to Heterogeneity, 334

7.4 References, 337

8. Borehole Electromagnetics - Diffusive and Propagation Transients, 338

8.1 Induction and Propagation Resistivity, 339

8.2 Conductive Mud Effects in Wireline and MWD Logging, 344

8.3 Longitudinal Magnetic Fields, 346

8.4 Apparent Anisotropic Resistivities for Electromagnetic Logging Tools in Horizontal Wells, 349

8.5 Borehole Effects - Invasion and Eccentricity, 356

8.6 References, 357

9. Reservoir Engineering - Steady, Diffusive and Propagation Models, 358

9.1 Buckley-Leverett Multiphase Flow, 358

9.1.1 Example boundary value problems, 361

9.1.2 General initial value problem, 361

9.1.3 General boundary value problem for infinite core, 362

9.1.4 Variable q(t) rate, 362

9.1.5 Mudcake dominated invasion, 363

9.1.6 Shock velocity, 363

9.1.7 Pressure solution, 364

9.2 References, 366

10. Borehole Acoustics - New Approaches to Old Problems, 367

10.1 Stoneley Waves in Permeable Wells - Background, 368

10.1.1 Analytical simplifications and new "lumped" parameters, 369

10.1.2 Properties of Stoneley waves from KWT analysis, 370

10.1.2.1 Dissipation due to permeability, 370

10.2.2.2 Phase velocity and attenuation decrement, 370

10.1.2.3 Relative magnitudes, phase and group velocities, 371

10.1.2.4 Amplitude and group velocity dependence, 372

10.2 Stoneley Wave Kinematics and Dynamics, 372

10.2.1 Energy redistribution within wave packets, 372

10.2.2 Dynamically steady Stoneley waves in heterogeneous media, 375

10.2.3 Permeability prediction from energy considerations, 376

10.2.4 Permeability prediction from phase considerations, 378

10.2.5 Example permeability predictions, 378

10.3 Effects of Borehole Eccentricity, 384

10.3.1 Industry formulations, solutions and approaches, 384

10.3.2 Successes in eccentricity modeling, 385

10.3.3 Applications to borehole geophysics, 388

10.3.3.1 General displacement approach, 389

10.3.3.2 Numerical solution strategy, 390

(Defining the grid; Creating the governing equations;

Specifying the problem domain)

10.4 References, 391

Cumulative Refrences, 394

Index, 410

About the Author, 419