Principles of Dielectric Logging Theory

By Alex Kaufman and Jean-Marc Donadille

Principles of Dielectric Logging Theory covers the theory of dielectric logging - from first principles to development of new tools - for those involved in the design, development and interpretation of dielectric logging. Dielectric logging has recently experienced a revival in the oil and gas industry and is now considered a preferred method for estimating the saturation of reservoirs. Topics covered in this book include the use of dielectric tools in different types of fields, interpretation methods, the Maxwell phenomena, and how to develop new dielectric tools for future use. Users will find this book includes enough detail that newcomers, seasoned professionals and researchers find it useful.

• Covers both the theory and applications of dielectric logging
• Provides comprehensive guidance on how to develop dielectric tools
• Explains what physically takes place in the presence of a dielectric probe

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 28, 2021
• ISBN: 9780128222843

Alex Kaufman

Emeritus Professor A.Kaufman has 28 years’ experience of teaching at the geophysical department in Colorado School of Mines He received his PhD. in Institute of Physics of the Earth (Moscow) and degree of Doctor of Science from the Russian Academy of Science . From 1981 to 20015 he published 14 monographs by Academic Press and Elsevier, describing different geophysical methods. Most of them are translated and published in Russia and China. He also holds three patents, which found application in the surface and borehole geophysics. A. Kaufman is a honorary member of SEG.

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Index

Preface

Although dielectric logging has been used in borehole geophysics for more than 60 years, the underlying theory has not yet been described. For this reason, we, the authors, made an attempt to fill this gap. In writing this book we have relied extensively on several excellent books on the theory of dielectrics.

Chapter 1 begins with the topic that is generally the starting point of electromagnetics courses: electrostatics. From Coulomb's law and the effect induced by an electric charge, the concepts of potential, electric field, and voltage are introduced. We also detail how the volume and surface charge densities arise under the action of an electric field. The two Maxwell equations are derived for a static field starting from the Coulomb's law, as well as the corresponding field conditions at interfaces between media of different electrical properties. The Poisson equation that rules the potential is also introduced.

Chapter 2 describes the behavior of conduction currents in a stationary electric field and introduces the conductivity of a medium. At first, it may surprise to find a full chapter devoted to conduction currents and conductivity in a book about dielectric logging, as the latter is generally associated with displacement currents, and permittivity. However, when dielectric measurements are performed, we always observe conduction currents in addition to displacement currents. In the low frequency range of modern dielectric tools, conduction currents play even the dominant role.

Still considering stationary fields, Chapter 3 focuses on the dielectric effects that occur in a medium. The phenomenon of dielectric polarization is explained from a microscopic perspective as the change of the volumetric distribution of the bound charges. The main physical properties linked to the polarization of a medium are introduced: the susceptibility, and of course the dielectric permittivity. Maxwell's equations related to a static electric field in a dielectric medium are also presented, introducing the electric induction, or displacement vector. In view of the application to dielectric logging, the effective permittivity of a heterogeneous medium is also established.

Time varying fields are considered in Chapter 4. The elastic and the orientational types of polarization are first described as a modification in the volumetric distribution of different types of bound charges. Proceeding from first principles, the equation ruling the relaxation of bound particles is derived, then solved, when the electric field behaves as a step function. This allows obtaining the general expression for the permittivity as a function of time. In the frequency domain, due to the phase shift between the displacement vector and the electric field, we introduce the fundamental concept of complex-valued permittivity, whose expression is derived in the time and frequency domains.

In Chapter 5, Maxwell phenomenon is described in detail, where charges collect on interfaces between media of different electrical properties. Displacement currents are introduced, by analyzing how they appear and behave with time inside a capacitor. We then detail the dependency of the volume and surface distributions of charges with frequency, permittivity, and conductivity. Maxwell phenomenon is then described step by step, using a circuit with a capacitor as the running example. By considering the capacitor as a multilayered dielectric, we then unveil the mechanism of charge collection on the interfaces between media of different electrical properties.

Chapter 6 focuses on the determination of the effective permittivity of a heterogeneous medium. Starting with the example of a circuit with a capacitor, we establish the relation between the effective permittivity, the electric field, and the vector of displacement. A review is then made of the most popular models for predicting the complex effective permittivity of a heterogeneous medium. While most of them are generalizations to the case of the complex permittivity of the early model by Maxwell for estimating the effective conductivity of a medium, some of these models are at the core of the petrophysical interpretation of dielectric logging today.

Chapter 7 is in many ways different from the others: it presents in a historical perspective how the dielectric theory has been applied for the search of hydrocarbon. Throughout the chapter, we relate the development of dielectric logging, from the inception in the 1950s to the modern tools of today. The early probes and their magnetic dipole source response are first discussed. We then detail the petrophysical reasons that prompt the latest generation of dielectric tools to include a very high frequency, and sometimes operate at multiple frequencies. In the latest tools, the dipoles are placed in two orthogonal orientations, longitudinal and transverse to the pad axis, and operate at multiple frequencies and for different spacings, offering multiple measurements that facilitate the estimation of the permittivity and conductivity dispersion of the formation.

Hopefully, this book can be useful to at least two categories of people. Since chapters 1 to 6 cover the general theory of dielectrics, they can serve as a reference for anybody who wants to delve into this subject, either purely academically (it can be used as a textbook), or with an industrial application in mind. In addition, people from the oil and gas industry or academia can use the book to gain a deep understanding of dielectric logging.

We are indebted to all colleagues who have contributed to the area of dielectrics. We particularly wish to express our thanks and appreciation to Dr. S. Akselrod,  Dr. A. Sihvola, Dr. P. Sen, Dr. T. Habashy, Dr. G. Itskovich, Dr. Y. Dashevskiy, and Dr. L. Mossé, for their useful comments and suggestions.

Part I

Time-invariant electric field

1. The electric field of static charges 3

2. Stationary electric field 33

Chapter 1

The electric field of static charges

Abstract

This chapter covers the main notions of electrostatics: Coulomb's law, the electric field, the voltage, the potential, how the volume and surface charge densities develop under the action of an electric field, etc. The first two Maxwell equations (Gauss law for the electric field and Maxwell–Faraday law) are formally derived for a static field starting from Coulomb's law, as well as the corresponding field conditions at interfaces between media of different electrical properties. The Poisson equation that rules the potential is also introduced. Formal expressions for the electric field and its potential are developed for a number of situations: near a point source, near surface charges, and for cases of an electric dipole, a double layer, a conductor, a capacitor, and system of capacitors.

Keywords

Field; voltage; potential; charge; double layer; capacitor; electric; electrostatic; Coulomb law

1.1 Interaction of electric charges and Coulomb's law

The force of interaction between electric charges is one of the fundamental concepts of physics. The French physicist Charles de Coulomb was the first to study quantitatively this remarkable phenomenon in experiments carried out at the end of the 18th century. To present the results of these investigations, Coulomb introduced the concept of elementary charges de(q) and de(p) around some points q and p, respectively. This means that regions where elementary charges are located are much smaller than the distance between these regions, while elementary charges themselves can have an arbitrary value. Correspondingly, such volumes are called elementary one and their dimensions depend on the distance between points p and q. Coulomb's law then states that the force acting on the elementary charge located around the point p, caused by the second elementary charge located around point q, is given by very simple expression:

(1.1)

Here de(q) and de(p) are the numerical values of the elementary charges. Lqp is the vector directed from the point q to the point p, and its magnitude is the distance between these points, and ε0 is a constant (Fig. 1.1A). In the standard international system of units (SI) force is measured in newtons (N), which has dimensions of mass times acceleration, or kilogram-meter per second square (kg.m.s−2); charge is measured in coulombs (C); and distance is measured in meters (Kaufman, 1992). The constant ε0 then has the value

FIG. 1.1 (A) Interaction between elementary charges, (B) electric field of volume charges, (C) electric field of surface charges, (D) elementary voltage.

where "F is a unit called the farad." Note that the value of charge equals to 1 coulomb is many orders greater than the charge of the electron, as

The vector separation Lqp can be written as

Here is a unit vector directed along line from point q to point p, and its substitution into Eq. (1.1) gives the familiar inverse square law of force

(1.2)

In other words, the electric force of interaction between two elementary charges is directly proportional to the product of the charge strength and inversely proportional to the square of the distance between them. This law has a similar expression to Newton's law of gravitation. However, unlike mass, which is always positive, electric charge can be either positive or negative. This means that the electric force F(p) has the same direction as the unit vector when two charges have the same sign, that is the force acts to push the charges apart and repulsion is observed. When the product of charges is negative, the force F(p) points in the opposite direction from and attraction takes place.

Acted on by the force F(p), the charge located at p will, unless otherwise restrained, move with acceleration a(p) following Newton's second law of motion:

(1.3)

where m is the mass of the elementary charge and a is its acceleration. As follows from Eq. (1.1) the force acting on the charge around the point q is the opposite of the force acting on the charge around point p, that is

(1.4)

Thus, Newton's Third Law (for every action, there is equal and opposite reaction) holds for Coulomb's force. Although the mathematical sum of the forces acting on the charges is zero,

it should be remembered that these forces act at separate points: each elementary charge feels only the force caused by the other charge. Let us make one comment. Of course, the charge cannot be located at the point, as its volume would be equal to zero. Yet for simplicity, instead of "charge around point p we write or speak of charge at the point p". Next, we consider a generalization of Coulomb's law to determine the electric force caused by a collection of elementary charges distributed throughout a volume of arbitrary size (Fig. 1.1B). This generalization is based on the principle of superposition, which states that force of interaction between two charges is independent of the presence of other charges. Consider an elementary volume dV(q) surrounding the point q and containing a uniform distribution of charge, so that the total charge in this volume can be represented as the product of the charge density times the volume:

(1.5)

Here δ(q) is the volume density of charges, which, by definition, is equal to the amount of charge in a unit volume of space. Its dimension is a charge per unit volume, or coulomb's per cubic meter, written

In general, the density δ(q) can vary from point to point. Using the principle of superposition, as well as Eqs. (1.1) and (1.5), the force F(p) can be written as

(1.6)

It should be obvious that, in performing the integration over the volume V, the point q varies, while the point p (where the force is being computed) remains the same. F(p) is thus a summation of forces having different magnitudes and directions, but all applied at the same point p. The vector F(p) can be resolved into its components by introducing a system of coordinates. For instance, in Cartesian (rectangular) coordinates with

the components of the force are

(1.7)

and

Here

It is important to emphasize that Coulomb's law is fundamental, as it is valid in any medium; that is the force of interaction between two charges remains the same regardless of the medium where charges reside. Indeed, the right-hand side of Eq. (1.1) does not contain any quantity that depends on the physical properties of matter, such as conductivity, dielectric permittivity, or magnetic constant. For instance, the force of interaction between charges de(p) and de(q) in a free space does not change if we place them inside of any medium.

1.2 The electric field

Next, we introduce the concept of an electric field. Like the force of interaction F(p) the electric field E(p) is a vector attached to a point p, defined as the ratio between the electric force and the elementary charge at p:

(1.8)

As follows from Eq. (1.6) we have for the electric field caused by any distribution of charges (Smythe, 1939)

(1.9)

In the traditional SI unit the electric field is measured in volts per meter (Vm−1). Several important observations follow from Eq. (1.9):

1. Electric charges are only the sources of an electric field, which is independent on time.

2. This field, caused by charges, exists at any point p, regardless of the presence or absence of a charge at this point. One can say that the field E(p) is waiting for the moment when elementary charge will be placed at the point p and move it with some acceleration.

3. The electric field generated by a given distribution of charge does not depend on the physical properties of the medium in which the charges reside. In other words, the electric field due to these charges remains the same whether the charge exists in free space or in a nonuniform conducting and polarizable medium. This remarkable feature follows from Eq. (1.9), because only the absolute physical constant ε0 appears in this equation.

4. Coulomb's law was not derived from other equations, and in this sense it is the fundamental physical law that allows us to describe the behavior of the stationary electric field. Consequently, the basic equations of the field E(p) will be derived from Coulomb's law as well as the principle of superposition.

5. Unlike the force of interaction, the vector function E(p) is called the field, because it depends on a position of the observation point p only.

1.3 Surface density of charges

Until now we described an equation for the electric field caused by real distribution of charges, characterized by its volume density. Now suppose that a charge distribution with volume density δ(q) is confined within a layer of an arbitrary shape whose thickness h(q) is much smaller than the distance between the observation point p and point q of the volume V(q) occupied by the layer (Fig. 1.1C), that is

(1.10)

Assume, in addition, that the volume density δ(q) varies only in directions locally parallel to the layer's surface and not along its thickness. An elementary volume surrounding a point q in the layer can be written as and then the elementary charge in this volume becomes

(1.11)

We have for the electric field due to this charge

or

(1.12)

where

(1.13)

By definition, the surface density σ(q) is equal to the elementary charge per unit area dS. From Eq. (1.11) it follows that decreasing the volume density and increasing the layer thickness in a way that keeps their product constant gives the same elementary charge in a volume near the surface; therefore the field dE(p) also does not change, provided of course that condition 1.10 holds—that is, the layer does not become too thick. The reverse limiting procedure leads to the concept of a pure surface charge density. In Eq. (1.13) let the thickness h(q) decrease while the density δ(q) increases so that the surface density remains the same. As the thickness goes to zero, we arrive in the limit at a surface distribution confined to the surface element dS. The charge located at this surface element

(1.14)

is by definition equal to the charge inside the volume Note that in this limiting procedure the surface density of charge σ(q) always has a finite value, but the volume density becomes singular, as

(1.15)

In reality any volume density is always finite, because charge occupies some finite volume. Surface charge is a mathematical concept introduced to simplify calculations when the thickness of the region in which charge resides is tiny compared to other dimensions of the problem. Applying the principle of superposition, Eq. (1.12) gives for the electric field, caused by surface distribution of charge

(1.16)

Integration over a surface S (a double integral) is generally easier than integration over a volume (a triple integral). In a similar manner sometimes it is useful to introduce two other mathematical concepts: point charge and linear density of charges, which may greatly simplify calculation of the electric field.

Next proceeding from Coulomb's law and principle of superposition we begin to study the two main features of the electric field in any medium and first introduce the concept of voltage.

1.4 Voltage of the electric field

Consider an elementary displacement dl (Fig. 1.1D), where the field E(p) is constant at its points. By definition, the dot product

(1.17)

is called the voltage of the field E(p) along the elementary path dl. It is clear that the voltage can be positive, negative, or zero depending on the angle between the field and path.

The voltage between two points a and b, caused by the electric field along an arbitrary path L (Fig. 1.2A), is defined to be

(1.18)

FIG. 1.2 (A) Voltage along arbitrary path, (B) voltage along radius-vector, (C) voltage along radius-vector and arc, (D) voltage along an arbitrary path, (E) voltage along closed path. (F) elementary contour around surface charge.

Here (E, dl) is the angle between the electric field and the displacement vector dl at each point of the path. By definition of the work of a force, the product E ⋅ dl is actually the work performed by the electric field in moving a unit positive charge along the path dl. This product has dimension of work per unit charge, which defines the unit volt. The integral in Eq. (1.18) represents the work in carrying a unit positive charge between two end points a,b of some path L. The familiar device called a voltmeter measures this work and, in accordance to Eq. (1.18), certainly is a very clever instrument that, in principle, divides a path into many elements, calculates elementary work along each, and performs the integration.

1.4.1 Independence of voltage on a path of integration

Starting from Coulomb's law and principle of charge conservation it will be proven a remarkable feature of the voltage caused by a charge—namely, that the voltage between two points is independent of the path taken between these points:

(1.19)

Here Li are different paths with the same end points. This feature is not obvious from the definition of voltage. In fact, as the path changes its length and shape change, as well as the direction and magnitude of the electric field along it. Thus, the elementary dot products E ⋅ dl vary considerably between different paths, but it turns out that their sum remains the same for all possible paths. Moreover, applying the principle of superposition we will conclude that this property is valid for any distribution of charges, and it holds in any medium with arbitrary electrical parameters. It should be stressed that this path independence of the voltage holds for the stationary (time-invariant) electric field and in general does not apply to time-varying fields. At the beginning we assume that the source of the field is a single elementary charge de. Its electric field is

Consider first the simple case when the path of integration is located along the radius vector Lqp passing through points q, a, and b (Fig. 1.2B). Then the voltage is easily calculated as a simple integral

because

Integration gives

(1.20)

Next consider the case when points a and b lie at the end of different radius vectors Lqa and Lqb, and a path of integration L1 is shown in Fig. 1.2C. Assume that point b is further away from the point charge than a. This path consists of two pieces. The first piece is a circular arc ab′ that extends from the tip of Lqa until it intersects Lqb. The second piece proceeds along the radius vector Lqb until it reaches b. For this path the voltage is

The integral along the arc ab′vanishes because the dot product dl⋅Lqp is zero at each point along the path. Thus, the voltage between points a and b is again equal to

(1.21)

as . In other words, the voltage depends only on the radial distances between the two points and the charge. Consider now a slightly more complicated path ab1b2b that first proceeds outward along the radius vector Lqato the point b1, then swings along a circular arc from b1 to b2 on the radius Lqb, and then proceeds outward along this vector to point b. The integral along the circular arc is zero, as before, and we have

(1.22)

because points b1 and b2 are located at the same radial distance from the charge.

We can now generalize this result to an arbitrary path between the points a and b (Fig. 1.2D). In fact, any element of this path can be represented as a sum

Here dl1 is a displacement along the radius vector Lqp, and dl2 is a circular arc at the same radius. Thus, we have for the elementary voltage

(1.23)

Using this approach, we represent an arbitrary path as a sum of circular arcs and displacements along radius vectors. All integrals along arcs are zero, while the sum of integrals along the radius vectors is an alternating sum that collapses to two terms:

We have established the first fundamental feature of a stationary field that obeys Coulomb's law: the voltage between two points is independent of the path of integration (Eq. 1.19). We assumed that the field is caused by a single point charge, but as was pointed out, applying the principle of superposition, this result remains valid for arbitrary distribution of