Induction logging

Figure 4-294 shows a set of resistivity logs run in a sand-shale sequence of the Gulf Coast. We have one wireline dual induction log, one MWD resistivity log, a wiper-MWD resistivity log and one gamma ray log. 

Figure 4-294. Comparison of compensated dual-resistivity logs run while drilling and in a wiper pass with the wireline induction logs. (Courtesy Anadrill [113]. ...

Figure 4-304. Resistivities and invasion determination with the Dual Induction log (DIL ) and the laterolog-8. Courtesy Schlumberger.) ( trade mark of Schlumberger.)...

Dual induction In the shallow penetration log, a higher resistivity is obtained relative to a free gas zone due to hydrate dissociation. This is a very tricky interpretation problem with Rw changing, and gas and hydrate both being insulators. The deep induction log shows high resistivity and mimics an ice-bearing reservoir. However, this log is confounded by Rw changing, and by gas and hydrate both being insulators... 

P. TYGELAND P. HUBRAL —TRANSIENT WAVES IN LAYERED MEDIA A.A. KAUFMAN AND GV. KELLER — INDUCTION LOGGING... 

In those cases, when the borehole axis is perpendicular to the boundaries between formations, the current flow path in the medium forms a circle, located in a horizontal plane and centered on the borehole axis. Correspondingly, induction logging is very sensitive to thin conductive layers, but it has difficulty in detecting relatively thin and resistive beds. 

H. Doll also introduced the differential multi-coil probes, which became very efficient logging tools and defined the path of development and application of induction logging over almost forty years. The use of these differential measurements in induction logging provides a result in which the effect of the borehole fluid, and in many cases also the invasion zone, on measurements is greatly reduced. Such devices are described in detail in this monograph. 

In almost sixty years, since the first development by H. Doll, research on various aspects of induction well-logging has been carried out around the world, and there have been some rather significant advances in theory, interpretation, probe design and equipment. Moreover, completely new modifications of induction logging have been developed and their principles are described in our monograph. As a result of the efforts of scientists and engineers in the United States, former Soviet Union and other countries, induction welllogging has become the most powerful tool for a determination of formation conductivity in uncased wells. 

Because much of the development of induction logging was done in proprietary research by logging services and oil companies, the technical articles that appeared in journals do not properly reflect the real volume of research that has been done on the method. For this reason, it is probably impossible to attribute the proper respect to everyone who has contributed in the development of induction well-logging in the western community. Among those who carried through the work started by H. Doll, arc J. H. Moran,... 

In the USSR, parallel development of theory, interpretation and equipment of induction logging, based on Doll s concepts of the geometric factor and focusing probe, was started at almost the same time. Also, during this research, new modifications of induction logging were introduced and some of them became conventional and are now used over the world. 

At the end of the 1960 s serious attention was paid to other modifications to induction logging. One of them is based on the use of transient fields, when measurements are performed in the absence of the primary magnetic field (Kaufman and Sokolov, 1972). The study of the secondary fields, caused by induced currents in a medium with either cylindrical or horizontal interfaces allowed one to describe the radial and vertical responses of the two-coil probe and find the most optimal range of time for measurements. 

Finally, theoretical investigations were performed which demonstrate that induction probes with special orientations of coils allow us to evaluate an anisotropy of formations (Kaufman and Kagansky, 1971). This study is also useful for application of induction logging in horizontal wells. 

In this monograph we describe the physical principles and theory of almost all possible modifications of induction logging. At the same time, such topics as inverse problems are out of the scope of this book. 

A need for a fundamental understanding of principles on which induction logging is... 

V. Druskin and L. Knizhnerman in 3D mathematical modeling of the response of induction logging tools in complicated models of a medium. 

We would like to note that the theory of induction logging presented in this volume can be applied not only to logging after drilling but to logging while drilling as well. 

Special attention will be paid to the set of equations which describes the quasistation-ary or quasistatic fields and provides an accurate model for induction logging, with the exception of dielectric logging where very high frequencies are used. 

The theory of induction logging is mainly based on the assumption that realistic models of a medium and field possess axial symmetry around axis of the borehole. There are however several important exceptions, such as ... 

We should note again that eq. 1.85 was developed from experiments in which direct currents were used. However, as will be shown later, the equation remains valid for quasistationary fields which are of most interest in induction logging. In the practical system of units, the magnetic field is expressed in amperes per meter. 

Making use of eq. 1.121, let us explore the influence of the radius of the loop on the magnetic field on the 2-axis. This will be useful in understanding the concept of geometrical factor widely used in the theory of induction logging. 

Equations of set (A) are valid for constant fields, but there will be additional effects to consider for alternating fields. In other words, when time-varying electromagnetic fields are considered, there is another source for the magnetic field in addition to the conduction currents. However, for the so-called quasistationary field, the influence of the second source of the magnetic field (displacement currents) is negligible as this is the case in induction logging, and equations of set (A) can be applied. 

Such a relationship between charge density and electric field at any point of a medium is essential to the definition of the quasistationary behavior which is responsible for many effects in induction logging. 

In induction logging, the receiver of an induction probe measures the field caused by the currents induced in the surrounding medium. As will be shown later, these currents in most cases flow along circles. This is why it is appropriate to explore some features of the electromagnetic field caused by a sole current ring. 

It is important to note that in induction logging the quadrature component is usually measured. 

Thus at low frequencies the quadrature component is directly proportional to the primary field, frequency and conductivity. It is important to state that this behavior stands when more complicated conductors are considered. It is also appropriate to notice that eq. 1.239 for the quadrature component is very basic in the theory of induction logging as developed by H. Doll. 

There must, however, be some sources of the magnetic field which maintain the primary field when the source is switched off. These sources are induced surface currents which are situated close to the source of the primary field if this one is located outside the conductor. If the source is located within the conductor, as in induction logging, induced currents initially exist near the source only. 

We will now investigate the relationship between frequency domain and time domain responses. In most cases considered in this section a transient electromagnetic field is excited by a step function current in the source. Moreover the theory of the transient induction logging described in this monograph will be developed for this type of excitation. For this reason the relationship between frequency response and transient response corresponding to this single type of excitation will be our principal concern. The information we need is obtained through use of the Fourier transform which takes the well... 

Simplicity of this model allows us to investigate not only frequency responses of the magnetic field, measured by a receiver, but also a distribution of currents in a conducting medium. Such will help us to understand deeper physical principles of the induction logging as well as some approximate methods of calculation of fields which are widely used for interpretation. 

In conventional induction logging instead of amplitude and phase, quantities such as magnitudes of quadrature and sometimes inphase components are measured. 

In conclusion, let us present eq. 2.54 in the explicit form, since some advantages of measuring the amplitude and the phase in the induction logging will be considered in detail. In accord with this equation we have ... 

METHODS FOR THE SOLUTION OF DIRECT PROBLEMS OF INDUCTION LOGGING... 

The main task of the theory of induction logging is to determine the dependence of the quasistationary electromagnetic field, measured by a probe receiver, on the resistivity of a medium. Our investigations will naturally be based on Maxwell s equations. As was shown in Chapter 1 the problem of field determination can be formulated in the following way. All space can be represented as a sum of areas with constant parameters fii and a, where fii is the magnetic permeability and is the conductivity of area Di. Within every area Di electric and magnetic fields satisfy Helmholtz equations ... 

It is clear that we have formulated a boundary problem for harmonic fields which are used in conventional induction logging. The transition to a nonstationary field can be easily done applying Fourier integral. 

Thus a solution of the direct problem of induction logging consists of determination of a function which satisfies the Helmholtz equation inside every area and the boundary conditions ... 

In this chapter we will describe basic methods which have been mainly used in solving boundary problems of induction logging. 

Both cases, when either cylindrical or horizontal interfaces are present, are of great practical interest in developing the theory of induction logging. 

In fact, during the last 30 years the use of integral equations has allowed us to move significantly forward in the theory and interpretation of induction logging. This is the main reason why we will describe here only this numerical method. At the same time it is reasonable to point out that both methods have been used, provided that a model of the medium and a field have cylindrical symmetry with the common axis. Until now this restriction has not permitted us to investigate a field behavior in the case when the boundaries between a formation and a surrounding medium are not perpendicular to the borehole axis. 

Figure 3.4. Models in induction logging with axial symmetry.

Approximate Methods of Field Calculation in Induction Logging... 

In this section we will describe two methods of field calculation which have played an essential role in developing such aspects of induction logging as ... 

In 1949 Henry Doll developed an approximate theory of induction logging. The basis of this theory is the assumption that for a sufficiently resistive medium and at relatively low frequencies one can neglect interaction of induced currents. For this reason the phase of these currents is 90°, regardless of the distance from the transmitter coil, and the measured signal is a sum of elementary signals created by currents in various parts of the medium, which depend on the conductivity of the corresponding part of the medium only. 

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