Induction probes

Falk, J. L., Zhang, J., Chen, R., and Lau, C. E., A schedule induction probe technique for evaluating abuse potential Comparison of ethanol, nicotine and caffeine, and caffeine-midazolam interaction. Special Issue Behavioural pharmacology of alcohol. Behavioural Pharmacology 5(4-5), 513-520, 1994. 

The concept of the geometric factor for an assembly of elementary rings with centers located on the axis of the borehole plays an essential role in Doll s theoretical approach. By using such geometrical factors Doll was able to calculate the electromotive force, arising in the receiver and caused by various parts of a medium, and to investigate the vertical and radial responses of different induction probes. 

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. 

Let us notice that in most cases, the field created by currents in the coil of an induction probe is equivalent to that of a magnetic dipole. 

As was mentioned above, real coils in induction probes can very often be considered as magnetic dipoles. For this reason, let us consider the behavior of the magnetic field of a magnetic dipole in more detail. Suppose that a magnetic dipole is located at the origin of a spherical system of coordinates, as shown in Fig. 1.31a, with its moment oriented along the z-axis. Then, in accord with eq. 1.123, we have ... 

The expression for the field in eq. 1.126 is used to evaluate the primary field magnitude of an induction probe. 

This feature of the field is sometimes used in order to control the quality of an induction probe consisting of coils and wires. In the case when the magnetic field is created by one single coil, the points where the component Hz vanishes are easily calculated. 

An alternating electromagnetic field in which the current flow is tangential to interfaces between media of different conductivity so that the normal component of the electric field is zero and that charges do not arise. This happens for example when an induction probe is located on the axis of a borehole and the medium possesses cylindrical symmetry in this case, the electric field has a pure inductive character. 

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. 

Figure 2.1. Two-coil induction probe in a uniform conducting medium.

Proceeding from these equations we will investigate the behavior of the electromotive force induced in the receiver of the two coil induction probe as well as the main features of the distribution of induced currents. 

Suppose that a receiver coil is significantly smaller than the length of the induction probe. In other words, it will be assumed that all turns of the coil have the same area... 

If the area of the receiver coil of the induction probe is small with respect to its length (Fig. 2.1), one can assume that within this area the magnetic field is uniform, and it is directed perpendicular to the horizontal plane, i.e. ... 

Now let us investigate the general case with the two-coil induction probe located in a uniform conducting medium. At the point of the receiver the magnetic field in accord with eq. 2.21 is described by an equation for its complex amplitude ... 

In the case in which the field or the electromotive force is investigated in the receiver of the two-coil induction probe the distance R is replaced by the length of the probe, L, that... 

Furthermore, we will focus on the magnetic field and the electromotive force at the receiver of the two-coil induction probe. Substituting eq. 2.37 into 2.27 the magnetic field Hz can be represented as a sum of two components, namely the quadrature component which is shifted in phase by 90° with respect to the primary magnetic field, Hq, or the current in the source, and the inphase component which is shifted in phase by 0° or 180° with respect to the primary field, and we have ... 

Equations 2.47-2.48 describe the field and the electromotive force with an error not exceeding 10%, provided that the parameter p is smaller than 0.1. In this case the quadrature component of the electromotive force, containing the information about the conductivity, constitutes only 1% or less of the primary electromotive force. For this reason cancellation of the latter in the induction probe is usually performed with a high accuracy. 

By analogy the behavior of the inphase component of the magnetic field can also be explained with the use of the inphase component of currents. Here it is appropriate to notice the following. Unlike the previous case a zone of currents which gives the main contribution to the inphase component of the magnetic field is present in a confined zone, a position which essentially depends on conductivity and frequency. In particular, with a decrease of frequency is shifted far away from the induction probe and when it is located at... 

It is appropriate to notice that there will be cases when an electric field cannot be described by component only. In such cases, for example, when there is a displacement of an induction probe with respect to the well s axis, a more general approach to the solution of the boundary problem will be used. 

The minimal radius of the shell naturally coincides with the radius of nonconducting part of the induction probe. In those cases, when the argument mr is small mr 1), it is convenient to use approximate expressions for shell coefficients. For x > 0 functions h x) and Ki x) tend to x/2 and jx correspondingly and therefore instead of eq. 3.51... 

When these functions are found along the contour I, we can determine the electric field inside the borehole by making use of the computational formula 3.95. This approach has been used for the investigation of radial and vertical responses of induction probes when the formation has a finite thickness. 

Figure 3.5. Position of an elementary ring with respect to the induction probe.

Now we will find a signal at the receiver of a two-coil induction probe caused by an induced current from this ring. As was shown in Chapter 2 the current induced in the elementary unit ring is ... 

Now we will investigate the behavior of the elementary geometric factor g in detail. In accord with eq. 3.104 it is very simple to show that the geometric factor q depends on the angle under which both coils of the induction probe are seen from points of the corresponding ring, and it is equal to ... 

Now we will describe a method which under certain conditions takes correctly into account the skin effect, i.e. interaction of currents in a conducting medium. The idea of this method is very simple. Let us present all current space around the induction probe as a sum of two areas, namely ... 

The magnetic field of the vertical magnetic dipole, ho, in a horizontal layered medium is expressed in the explicit form. For example, if the induction probe is located symmetrically with respect to the formation boundary we have ... 

This method was suggested almost 35 years ago, and it was very useful in developing the interpretation of induction logging, the determination of frequencies and geoelectric parameters of a section, where focusing induction probes are effective. 

Understanding this feature of field behavior is important for the further development of the interpretation of induction logging. Moreover, some of the induction probes, currently used in practice, are based on measuring both components of the magnetic field. 

In this chapter we will derive an expression for the vertical component of the magnetic field on the axis of a borehole when the source of the primary field is a vertical magnetic dipole and the formation has an infinite thickness. Special attention will be paid to the analysis of frequency responses of quadrature and inphase components of the field, including their asymptotic behavior. The influence of various parameters of a geoelectric section will also be investigated. Such questions as the influence of finite dimensions of coils, displacement of the induction probe wdth respect to the borehole axis, the role of magnetic permeability and dielectric constant will be studied. 

All media surrounding the induction probe are uniform and isotropic. 

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