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Tables of Contents for Principles of Induction Logging

Chapter/Section Title

Page #

Page Count

Acknowledgments

ix

List of Symbols

xi

Introduction

1

4

Basic electromagnetic laws and Maxwell's equations

5

114

Coulomb's law

5

29

Example I: Normal component of the electric field caused by a planar charge distribution

10

5

Example II: Effect of a conductor situated within an electric field

15

19

Biot Savart law

34

18

Example I: The magnetic field of a straight wire line

44

2

Example II: The vector potential and magnetic field of the current flowing in a circular loop

46

5

Example III: The magnetic field of a grounded electrode in a uniform conducting medium

51

1

The postulate of conservation of charge and the distribution of charges in conducting media

52

15

Example I: Exponential variation

61

1

Example II: Sinusoidal variation

62

5

Faraday's law and the first Maxwell equation

67

25

Example I: The vortex electric field of a solenoid

71

2

Example II: The vortex electric field of a magnetic dipole in a free space

73

3

Example III: The inductive electric field due to the magnetic field of a current flowing in a circular loop

76

3

Example IV: Induction of a current in a thin conducting ring situated within a primary alternating field

79

10

Example V: Behavior of the electromagnetic field at the early stage and high frequencies in a conducting medium

89

3

Electromagnetic field equations

92

15

Relationships between various responses of the electromagnetic field

107

12

Electromagnetic field of the magnetic dipole in a uniform conducting medium

119

24

Methods for the solution of direct problems of induction logging

143

44

The method of separation of variables

144

2

The method of shells

146

13

Derivation of approximate boundary conditions on a shell surface

147

2

Calculation of the electromagnetic field caused by induced currents in one shell

149

2

The field in a presence of two confocal shells

151

8

The method of integral equations

159

11

Approximate methods of field calculation in induction logging

170

17

Doll's theory of induction logging

170

6

The approximate theory of induction logging, taking into account the skin effect in the external area

176

11

Electromagnetic field of a vertical magnetic dipole on the axis of a borehole

187

124

Formulation of the boundary problem

187

2

Derivation of the formula for the vertical component of the magnetic field

189

13

The quadrature component of the magnetic field at the range of very small model parameters

202

11

Radial characteristics of a two-coil induction probe at the range of small parameters

213

9

Influence of the skin effect in the formation on the radial characteristics of a two-coil induction probe

222

7

Example I: Two-layered medium

226

2

Example II: Three-layered medium

228

1

Asymptotic behavior of the magnetic field in the borehole in the range of small parameters

229

7

Behavior of the field on the borehole axis in the near and far zones

236

9

Frequency responses of the magnetic field of the vertical magnetic dipole on the borehole axis

245

4

Influence of finite dimensions of induction probe coils

249

20

Electrical field of a current ring in a medium with cylindrical interfaces

269

21

Electrical field of a single-layer coil in a medium with cylindrical interfaces

274

3

Both transmitter and receiver of the induction probe are single-layer coils

277

13

Radial responses of two-coil induction probes displaced with respect to the borehole axis

290

9

The influence of magnetic permeability and dielectric constant in induction logging

299

12

Quasistationary magnetic field of a vertical magnetic dipole in a formation with a finite thickness

311

54

Derivation of formulae for the vertical component of the magnetic field of a vertical magnetic dipole

311

8

The field of the magnetic dipole located outside the bed

315

1

The field of the magnetic dipole located within the bed

316

2

The field of the vertical magnetic dipole in the presence of a thin conducting plane

318

1

The vertical responses of the two-coil induction probe in the range of small parameters

319

1

Geometric factor of an elementary layer

319

12

Geometric factor of a layer with a finite thickness

322

9

The theory of the two-coil induction probe in beds with a finite thickness

331

21

Curves of profiling with a two-coil induction probe in a medium with two horizontal interfaces

352

13

Thick conductive bed

363

1

Thick resistive bed

364

1

Thin conductive bed

364

1

Thin resistive bed

364

1

The two-coil induction probe on the borehole axis, when the bed has a finite thickness

365

20

Doll's theory of the two-coil induction probe located on the borehole axis when a formation has a finite thickness

366

5

The theory of a two-coil induction probe, taking into account the skin effect in an external medium

371

5

Influence of the finite thickness of the formation on the magnetic field behavior

376

9

Multi-coil differential induction probes

385

78

Methods of determination of probe parameters

386

9

Physical principles of multi-coil differential probes

395

2

Radial and vertical responses of the differential probe 1.L--1.2

397

18

Radial and vertical responses of probes 6F1M, 4F1 and 4F1.1

415

22

The influence of finite height of the invasion zone on radial responses of probes 6F1M, 4F1 and 4F1.1

437

4

Three-coil differential probe

441

12

The influence of eccentricity on focusing features of multi-coil induction probes

453

3

Choice of a frequency for differential probes

456

1

Determination of the coefficient of differential probes

457

6

Induction logging based on measuring the inphase component of the secondary field or the quadrature component difference of type QHz (ω1) - ω1/ω2 QHz (ω2)

463

14

Transient induction logging

477

56

The transient field of the magnetic dipole in a uniform medium

478

18

Transient field of the vertical magnetic dipole on the borehole axis at the late stage

496

6

Apparent resistivity curves of the transient method in a medium with cylindrical interfaces

502

6

The transient responses of a vertical magnetic dipole in a formation with a finite thickness

508

20

About a nonstationary field of the electric dipole

528

5

Principles of induction logging with transversal induction coils

533

72

Electromagnetic field of the magnetic dipole in a uniform isotropic medium

533

2

Boundary problem for the horizontal magnetic dipole on the borehole axis

535

15

Magnetic field on the borehole axis in the near zone

550

8

The magnetic field on the borehole axis in the far zone

558

8

The magnetic field in a medium with two cylindrical interfaces

566

5

Cylindrical surface with transversal resistance T

571

4

The magnetic field in a medium with one horizontal interface

575

5

The magnetic field of the horizontal dipole in the formation with finite thickness

580

18

Curves of profiling with a two-coil induction probe in a medium with horizontal interfaces

598

7

The influence of anisotropy on the field of the magnetic dipole in a conducting medium

605

22

Anisotropy of a layered medium

605

3

Electromagnetic field of the magnetic dipole in a uniform anisotropic medium

608

9

Magnetic field in an anisotropic medium with two horizontal interfaces

617

10

Mathematical modeling of the response of induction logging tools in 3D geometries

627

12

References

639

2

Subject Index

641