Wave Propagation in Drilling, Well

Logging and Reservoir

Applications

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Publishers at ScrivenerMartin Scrivener (martin@scrivenerpublishing.com)

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Wilson C. Chin, Ph.D., M.I.T. Stratamagnetic Soft ware, LLC,

Houston, Texas

Wave Propagation in Drilling, Well Logging

and Reservoir Applications

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Library of Congr ess Cataloging-in-Publication Data:

ISBN 978-1-118-92589-8

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

And to those who enjoy a good mystery or an exasperating puzzle, duti-fully examine our photograph. Carefully peruse the bow, the way it’s held and executed. “What eff ect does its position and velocity have on the sound created? How do waves that refl ect from the ends behave at the contact itself and how do they travel aft erwards? What changes when the environ-ment changes?”

To Jamie,

vii

Contents

Preface xxi

Acknowledgements xxiii

1 Overview and Fundamental Ideas 1

1.1 Th e Classical Wave Equation 2

1.1.1 Fundamental properties 2

1.1.2 Refl ection 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

viii Contents

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-infi nite 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

Contents ix

1.9 First-Order Partial Diff erential Equations 46

1.10 References 49

2 Kinematic Wave Th eory 50

2.1 Whitham's Th eory 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 Th e 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 fi eld 62

2.3.3 Th e energy singularity 62

2.3.4 Example 2-6. Modeling dynamically steady motions 63

2.4 High-Order Kinematic Wave Th eory 64

2.4.1 Basic assumptions 64

2.4.2 Th e general amplitude equation 65

2.4.3 Method of multiple scales 66

2.4.4 Generalized wave results 68

2.4.5 Th e low-order limit 70

2.5 Eff ect 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

x Contents

2.6 Th ree-Dimensional Kinematic Wave Th eory 76

2.6.1 Wave irrotationality 77

2.6.2 Th e 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 fl ows 87

3.2.3 Acoustic simplifi cations 87

3.2.4 Th ree-dimensional wave equation 88

3.2.5 Modal solution 88

3.2.6 Th e 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 Th e 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 fl ow stability 105

3.5.3 Stability of irrotational fl ows 106

3.6 References 106

Contents xi

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 fi eld (Structural damage;

Formation damage; Directional drilling;

Increasing rate of penetration; Improved

MWD tools and mud motors; Formation

imaging; Psychological discomfort) 117

4.1.5 Elastic line model of the drillstring 119

4.1.5.1 Early eff orts 119

4.1.5.2 Elastic line simplifi cations 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 diff erential equations 126

4.2.2.1 Damped wave equation 126

4.2.2.2 External forces and displacement sources 127

xii Contents

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

(Laboratory drillbit data; Single-tooth

impact results) 136

4.2.4.5 Modeling drillbit kinematics using

“displacement sources” (Analogies from

earthquake seismology) 139

4.2.5 Global energy balance 142

4.2.5.1 Formulation summary 142

4.2.5.2 Energy considerations (Th e drillstring;

Th e surface; Combined drillstring/

surface system) 142

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 fi xed 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 Th e general impedance mode 147

4.2.6.8 Modeling the constants alpha, beta

and gamma 149

4.2.7 Finite diff erence modeling 149

4.2.7.1 Elementary considerations 149

4.2.7.2 Transient fi nite diff erence modeling

(Th e solution methodology; Stability of

the scheme; Grid sizes, time steps, and

convergence) 151

Contents xiii

4.2.8 Complete formulation and numerical solution 156

4.2.8.1 Th e boundary value problem 156

4.2.8.2 Computational objective 157

4.2.8.3 Diff erence approximations 157

4.2.9 Modeling pipe-to-collar area changes 159

4.2.9.1 Matching conditions 160

4.2.9.2 Finite diff erence 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 (MWD-VSP; Vibration

logging of the formation) 169

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

(An inelastic impact model; Elastic

impacts, with stress eff ects) 174

4.2.11.6 Separating incident from refl ected waves

(Delay line method; Diff erential technique;

Th ree-wave formulation; Digital analysis

methods) 179

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 – “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

xiv Contents

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 diff erence modeling 208

4.3.7.1 Pentadiagonal diff erence equations 209

4.3.7.2 Finite diff erence beam recipe 210

4.3.7.3 Additional modeling considerations

(Borehole wall contacts; Modeling steady

state oscillations; Simulating area

changes) 211

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 eff ects on bending 221

4.4.2.3 Finite diff erence modeling 222

4.4.2.4 WOB/TOB (Weight-on-bit/Torque-on-bit) 222

4.4.2.5 Applications to MWD telemetry 223

4.4.2.6 Example Fortran implementation 223

4.4.2.7 Whirling motions (Example 4-6. Machine

shaft example; Example 4-7. Generalized

whirl) 225

4.4.2.8 Causes of whirling motions 226

Contents xv

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 diff erential equations 230

4.5.4.2 Finite diff erencing 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 (Test A. Smooth

drilling and making hole; Test B. Rough

drilling with bit bounce; Model limitations

and extensions) 239

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 Diff erential equation 259

5.1.3 Area and material discontinuities 259

5.1.4 Mud acoustic formulation 261

5.1.5 Example 5-1. Idealized refl ections 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

xvi Contents

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 eff ects 269

5.3 Transient Finite Diff erencing Modeling 272

5.3.1 Basic diff erence 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

(Packaging constraints; Shock and

vibration; Mud erosion; Power requirements;

High pressure and temperature; Fluid

mechanics problems) 287

5.5.3.6 Mud pulse telemetry – the transmission

channel 289

5.5.3.7 Th e transmission channel uphole 290

5.5.3.8 Telemetry design objectives 291

5.5.3.9 Additional practical considerations 292

5.5.3.10 Th e theoretical maximum 293

5.5.3.11 Acoustic signals in the annulus 293

5.6 Recent MWD Developments 294

5.7 References 303

Contents xvii

6 Geophysical Ray Tracing 306

6.1 Classical Wave Modeling – Eikonal Methods and Ray

Tracing 307

6.1.1 Th e 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 Fermat’s 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 satisfi es least time condition 312

6.3 Fermat’s Principle Revisited Via Kinematic Wave Th eory 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 Th e 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 Th e 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 Th e seismic limit 319

6.5.1.5 Example 6-4. Simple rock formations 319

6.6 Subtle High-Order Eff ects 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

xviii Contents

6.7.3 Finite diff erence calculation of travel times 325

6.7.4 Diffi culties with simple diff erence formulation 326

6.7.4.1 Two space dimensions 326

6.7.4.2 Th ree 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 Eff ect of low-order dissipation 332

7.1.1.2 Eff ect of variable background fl ow 332

7.1.2 Waves in fi nite depth water 333

7.2 Sources of Hydrodynamic Loading 334

7.3 Instabilities Due to Heterogeneity 334

7.4 References 337

8 Borehole Electromagnetics - Diff usive and Propagation

Transients 338

8.1 Induction and Propagation Resistivity 339

8.2 Conductive Mud Eff ects 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 Eff ects – Invasion and Eccentricity 356

8.6 References 357

9 Reservoir Engineering – Steady, Diff usive 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 infi nite 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

Contents xix

10 Borehole Acoustics - New Approaches to Old Problems 367

10.1 Stoneley Waves in Permeable Wells - Background 368

10.1.1 Analytical simplifi cations 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 Eff ects 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 (Defi ning

the grid; Creating the governing equations;

Specifying the problem domain) 390

10.4 References 391

Cumulative Refrences 394

Index 410

About the Author 419

xxi

Preface

I was a troubled young man haunted by ghosts of unsolved problems. Everywhere I traveled, questions without answers plagued me. Th ey were petroleum in nature, originating from oilfi elds far and near, following engineers to home offi ces and balance sheets, lingering on fl atbeds and electronic bea-cons. Many phenomena were real and repeatable but could not be explained. And what was inexplicable could not be solved, improved or optimized.

Take drilling vibrations – many issues here. For one, almost all drillstring failures are blamed on resonance; yet, drillers could not drill at resonance (to enhance penetration rates) even if they tried. Or catastrophic lateral vibrations occurring at the neutral point – why can’t these violent events be detected at the surface even in vertical wells? And why have researchers not yet modeled stick-slip vibrations, bit-bounce and rate-of-penetration as they depend on BHA and lab-derived rock-bit interaction data in their simulation tools?

Or consider the most urgent problem in MWD mud pulse telemetry. Strong signal strength is needed to overcome attenuation in deep wells, typically achieved by tightening valve clearances, which implies severe power and erosion penalties. But what about harnessing nature itself – phasing downward traveling waves from the pulser, which ultimately refl ect upwards, to superpose in-phase with upgoing waves created a split-second later? Or, in physics parlance, employ “constructive wave inter-ference” where one literally amplifi es signals for free? Problem is, if the conventional solid piston pulser model is applied, waves that initially travel in opposite directions can never interact aft er refl ection, making reinforce-ment and modeling impossible. So what now?

Fast forward to geophysical ray tracing. Almost all studies employ the three-dimensional wave equation and Fermat’s “Principle of Least Time.” Th is “must” be correct, aft er all, who can argue with Fermat? Th e Fermat, no less. However, this well known principle applies to conservative media only. When attenuation exists, least times for travel along rays do not apply. And worse, what if wave dissipation were only known empirically, say as an “imaginary

xxii Preface

frequency” function and not from diff erential equations? How do we model phase distortions due to amplitude? And what about heterogeneities?

Th en there’re Stoneley waves. Most borehole studies have us believe that numerically intensive processing is required to hunt for elusive tidbits lurking in unsuspecting waveforms – this means, naturally, service com-pany fees. Th e truth is, almost all of the physical properties identifi ed in well known studies can be summarized in a few equations derived from the “kinematic wave theory” pioneered at Caltech and M.I.T. Moreover, permeability can be accurately predicted using rather simple formulas. Inexpensively. Th e list goes on and on.

Over the years, I have collected numerous examples of wave propaga-tion problems that seemingly defi ed explanation and solution – however, applying innovative methods, we have solved all of them through analysis and logic. But rather than communicate these results in dry, abstract and esoteric scientifi c papers, I have chosen to motivate our tools and results in one comprehensive volume focusing on several key unifying themes, a book which may ultimately serve more purposes than those intended.

On this launch of John Wiley’s new Advances in Petroleum Engineering series, we are pleased to present the solutions to all of the above prob-lems and more. Originally, the publisher and I discussed the possibility of starting with a math-oriented project, one that I had endorsed. However, math books already proliferate and still another would be hopelessly lost among thousands. Th us, we decided to include only the most germane approaches, together with a concise exposition of “kinematic wave” ideas, then of “displacement sources,” and fi nally, address the insurmountable petroleum challenges cited earlier.

No other quote from classical literature is more appropriate to this eff ort than one from Sherlock Holmes, in Th e Stock-Broker’s Clerk, by Sir Arthur Conant Doyle – “I am afraid that I rather give myself away when I explain. Results without causes are much more impressive.” And to those who enjoy a good mystery or an exasperating puzzle, dutifully examine the Dedication page. Carefully peruse the bow, the way it’s held and executed. What eff ect does its position and velocity have on the sound created? How do waves that refl ect from the ends behave at the contact itself and how do they travel aft erwards? What changes when the environment changes? Th e key to our models can be found in this photograph, and in this book, one which we hope will stand on its math and engineering merits for many years.

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

Email: wilsonchin@aol.comPhone: (832) 483-6899

xxiii

Acknowledgements

Th e author gratefully acknowledges the contributions of numerous col-leagues who have added to his experiences, perspectives and insights over the years – friends and individuals who studied perplexing situa-tions through their scientifi c curiosity and who have, through their col-lective eff orts, shown how seemingly disparate phenomena share more in common than their apparent diff erences. In particular, I thank Boeing, Schlumberger, Halliburton, British Petroleum, China National Petroleum Corporation, China National Off shore Oil Corporation, GE Oil & Gas and others, for motivating many of the problems considered in this book and contributing to the scientifi c literature as a result.

Phillip Carmical, Acquisitions Editor and Publisher, has been extremely supportive of this book project and others in progress. His philosophy, to explain scientifi c principles the way they must be told, with equations and algorithms, is refreshing in an environment oft en shrouded in secrecy and commercialism. In a world increasingly dominated by fi nite element models where monotonous gridding and computer graphics substitute for physics and progress, the need for true engineering insight is now more important than ever, particularly in the race for economic superiority. Some will disagree, but mathematics will always have the fi rst word, and more oft en than not, the last. So Phil, thanks again.

1

1Overview and Fundamental Ideas

What is wave propagation? Commonplace examples include obvious yetmathematically complicated events such as the movement of water waves onbeaches, the vibrations of guitar strings, the sonic boom underneath a high-speedairplane and children singing in hallways. Wave propagation, associated withvibration theory, hyperbolic equations and asymptotic WKB analysis, has beenstudied mathematically for three hundred years. Its literature is vast and subtle.Its developers include the greatest minds in physics. Libraries of books havebeen written on esoteric implications. The subject matter encompasses manydisciplines: optics (light), acoustics, structural vibrations, aeroelasticity,electromagnetism, hydrodynamic stability, laminar flow transition, underwatersound, philharmonic hall design, plasma stability, earthquake seismology,geophysics, and spiral galactic instabilities. How, then, does the student ofpetroleum engineering (whose work ultimately embraces practical drillingvibrations, well logging and Measurement-While-Drilling, a.k.a. “MWD,”among other evolving technologies) build an adequate technical foundationquickly without an overwhelming amount of study? How can he appreciate thephysics underlying important engineering phenomena without digesting reamsof finite element results and hoping for self-evident generalities?

Goals of the book. Despite the seriousness of the material and themathematics presented in this volume, the present book is intended to be anintroductory textbook, but in a sense not usually taken in formal courses. Wewill provide the student with a flavor for the formal constructive techniquesusually taught in advanced courses, so that he acquires enough familiarity withthe jargon to read and understand professional papers and books.

2 Wave Propagation

This is accomplished by offering a synopsis of the conventionalmathematics taught in partial differential equations courses, covering standardsolution methods, plus their strengths and weaknesses. We introduce pertinentanalysis areas to give students resources to pursue additional study. We will becorrect but not rigorous in a formal sense. Our objective is not the teaching ofclassical mathematics per se. Thorough understanding can only be achievedthrough intensive diligent exercise. We only introduce these methods, but in amanner that demonstrates their powerful capabilities or their severe limitations.

This objective allows us to communicate “the big picture,” so that thereader can at once grasp an appreciation of what has been made possible bygenerations of eminent scholars. But since it turns out that many of the wavepropagation issues encountered in modern petroleum engineering do not drawdirectly upon these techniques, this overview suffices insofar as providing thebasic foundation needed for more specialized analysis. The physical problemsparticular to our rapidly changing industry are reviewed for each of the titledapplications, and mathematical and numerical models are developed to handlethe specialized circumstances these specific applications demand.

By taking this approach, the requirements for new analysis methods andthe mathematical issues associated with their implementation appear naturally,and without intimidating theorems, proofs and corollaries. Modern subjects suchas finite difference modeling, multiple-scaling, stationary phase, monopole anddipole source properties, Fermat’s principle of least time, kinematic wave theoryand group velocity are introduced and developed naturally ... and in a simple andintuitive way. This approach to introducing specialized subjects in the simplestmanner has posed the greatest challenge. But this philosophy reflects the trialsand tribulations of the author’s own learning process, quickly taking advantageof the opportunities the Oil Patch offered after a false start in aerospaceengineering. For the reader willing to endure the formalities, the rewards areenticing: innovative ways to study borehole electromagnetics, new approachesto MWD telemetry, out-of-box ways to extract permeability from Stoneley wavemotions in borehole acoustics. Their extensions will become apparent. Theauthor believes that new lines of research will be defined that will lead toimproved efficiencies needed in modern exploration and production.

1.1 The Classical Wave Equation1.1.1 Fundamental properties.

Many students learn wave propagation by way of the undamped classical“wave equation” (e.g., see Hildebrand, 1948, or Tychonov and Samarski, 1964)

2u/ t2 - c2 2u/ x2 = 0 (1.1)

for u(x,t), where x is the propagation direction and t denotes time. We maythink of Equation 1.1 as describing the transverse vibrations of a string.

Basic Ideas and Mathematical Methods 3

If so, u(x,t) is the displacement from equilibrium, and c is the disturbancespeed (c2 = T/ l, where T is tension and l is lineal mass density).* Examplesinclude waves on violin and guitar strings. The domain of x can be finite,semi-infinite or infinite, depending on the application. But time is always zeroor positive, and future events must never affect present and past motions. This“principle of causality” governs equations of evolution, particularly “hyperbolicequations” such as Equation 1.1. Not all equations are causal. The “ellipticequations” used in reservoir flow simulation, which may look something like

2p/ x2 + 2p/ y2 = 0 (with a “+” instead of a “-“ sign) deal with “domains ofinfluence and dependence” such that every point affects and is influenced byevery other point (Chin, 1993). By contrast, “parabolic equations,” taking formssimilar to T/ t - 2T/ x2 = 0 ( > 0), deal with diffusion in space and time.

In Equation 1.1, / t and / x represent partial derivatives, or derivativesof functions of several variables relative to the particular independent variableshown, with all others held fixed. We may also express partial derivatives usingsubscript notation,

utt - c2 uxx = 0 (1.2)

Its general solution, obtained two hundred years ago by D’Alembert, is

u(x,t) = f(x+ct) + g(x-ct) (1.3)

and will find significance later, for instance, in MWD signal processing andecho cancellation. To understand why Equation 1.3 holds, we apply thederivative rule from calculus, stating that the x-derivative of h{p(x)} is{h'(p)}{dp(x)/dx} where primes denote differentiation with respect to p. Sinceux = f ' + g', uxx = f " + g", u t = c f ' - cg' and u tt = c2f " + c2g", substitution inEquation 1.2 proves Equation 1.3. This equation contains much information. Ifwe consider the u(x,t) = f(x+ct) contribution, we observe that u(x,t) must beconstant if x+ct is constant. But “x+ct = constant” is just the straight line inFigure 1.1. Since time must increase, the argument x+ct must represent left-going waves. Similarly, u(x,t) = g(x-ct) is constant along trajectories with x-ct =constant. Thus, g(x-ct) represents right-going waves. Depending upon theapplication, wave solutions may be both up-going and down-going. Equation 1.3states that solutions of Equation 1.1 can be constructed from general families ofleft and right-going waves. Lines for which x+ct and x-ct are constant areknown as “characteristics” or “rays,” and their coordinates represent the naturalor canonical variables describing the wave propagation.

*Two densities, the lineal mass density l and the mass density per unit volume, are used in thisbook. The first is appropriate to vibrating string problems satisfying l

2u/ t2 - T 2u/ x2 = 0 wherethe tension T has units of force. The second applies to drillstring vibrations and borehole acousticsproblems satisfying 2u/ t2 - E 2u/ x2 = 0 where E, either Young’s modulus or the bulk modulus,has units of force per unit area.

4 Wave Propagation

Figure 1.1. The characteristic plane.

In linear theory, characteristics within each wave family are parallel. The“method of characteristics,” used analytically and numerically, solves problemsby tracing values of the dependent-variable along rays. We will use this methodin Chapter 2 to obtain closed form solutions for use in kinematic wave theory.From Figure 1.1, the slope c in the characteristic x-t plane is the wave or “soundspeed,” e.g., the line x-ct = constant has the slope c, which is obviously thepropagation velocity with which disturbances travel. In general, a disturbancepropagates with speed c to both the left and the right. Equation 1.1 states that cis the only available wave speed. In many problems, other speeds are possible.In three-dimensional waveguides, additional speeds are possible. Waves withdifferent lengths (or, equivalently, frequencies) may travel at different speeds:an initially confined wave group may disperse and lose its identity.

“Wave dispersion” is also possible in one-dimensional problems, e.g.,bending waves on beams. We deal with the subject of dispersion later, but it isimportant to recognize that Equation 1.1 does not describe dispersion at all.And since, as discussed, disturbances satisfying Equation 1.1 propagate withshape and amplitude both intact, the classical equation does not describedissipative, attenuative or nonconservative effects, terms often usedinterchangeably. It applies to “conservative” or undamped wave motions only.Also, Equation 1.1 does not contain variable coefficients: the motion occurs inuniform or homogeneous media, in contrast to nonuniform or heterogeneousmedia. Heterogeneous media may contain spatial and temporalinhomogeneities, that is, variable coefficients in x and t. Such general waves areeasily modeled using kinematic wave theory. In Equation 1.3, “+” appears inone function, while “-“ appears in the other. Different conventions are used indifferent applications; u(x,t) = f(t+x/c) + g(t-x/c) may appear in one context, butu(x,t) = g(-t - x/c) - f(-t + x/c) may be more convenient in others. Sometimes f