Induction probes radial responses

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. 

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. 

These last three equations allow us to investigate the radial responses of a two-coil induction probe in detail, in other words, to evaluate the influence of the borehole, the invasion zone and the formation as a function of the induction probe length, L, for various geo-electric parameters. 

Considering radial responses of two-coil induction probes in the range of small parameters it is natural to investigate cases when the resistivity within the intermediate zone changes as a continuous function. In accord with eq. 4.59 we have ... 

Radial Responses of Two-coil Induction Probes Displaced with Respect to the Borehole Axis... 

Until now we have considered radial and frequency responses of two-coil induction probes located on the borehole axis. However, under real conditions if special centering devices are not used induction probes are usually displaced with respect to the axis, and for this reason it is essential to investigate the influence of this displacement on the radial responses of induction probes. 

Significant technical difficulties related with measuring field components at such high frequencies, deterioration of radial response of the induction probe specially when invasion zone has intermediate resistivity [pi < p2 < pz), increase of the influence of currents in the borehole, they all essentially reduce principal possibilities to use frequencies in order to eliminate completely the effect of induced currents in the surrounding medium. 

In this chapter we will consider radial and vertical responses of a two-coil induction probe located on the borehole axis as the bed has a finite thickness. The presence of the invasion zone will also be taken into account. The main attention is paid to the analysis of the influence of such factors as ... 

This investigation will allow us to better understand the influence of a formation thickness on radial and vertical responses of a two-coil induction probe. 

This relation is useful for investigating radial responses of induction probes in the presence of an invasion zone as well as for evaluation of the influence of caverns. 

In conclusion let us make the following comment. Doll s theory does not take into account the skin effect. For this reason we can expect that both vertical and radial responses, specially the first one, derived from eq. 6.8 6.10 will be in many cases different from actual responses. Correspondingly in the next section we will consider responses of two-coil induction probes, assuming that the skin effect in the formation and in the surrounding medium is not subjected to the presence of borehole and invasion zone. 

For improving the vertical response of a two-coil induction probe it is necessary to decrease the relative contribution from the surrounding medium with respect to the signal caused by currents in the formation against which a two-coil induction probe is located. In other words, in this case the influence of more remote parts of the medium has to be reduced providing a significant signal from currents induced in that part of the medium which is located relatively close to the probe. Thus, improvement of radial and vertical responses of a two-coil induction probe is related to the development of multi-coil probes which have to satisfy opposite requirements. 

Methods of choosing probe parameters are based on the use of differential and integral responses of two-coil induction probes. The differential radial response defines a signal from a thin cylindrical shell, expressed in units of the signal, caused by currents in a uniform conducting medium. In accord with Doll s theory, described above, we have ... 

It is obvious that the radial differential response of the multi-coil induction probe does not depend on the distribution of conductivity in the radial direction if within every cylindrical layer the resistivity remains constant. 

The second radial response of the two-coil induction probe is its integral response (direct and inverse ones). The direct integral radial response defines the signal caused by currents in the cylinder referred to that from currents in a uniform medium as a function of... 

This method of superposition of radial responses of two-coil induction probes does not practically allow us to find parameters of probes which are sensitive only to certain parts of medium, similar to special probes in electrical logging as micrologs. 

This function is the geometric factor of the cylinder for a four-coil induction probe with internal focusing and parameters p and c, i.e. it is the integral radial response. Let us notice that normalization of geometric factor allows us to compare radial responses of various probes. 

Introducing a fifth compensating coil, coefficient K becomes equal to zero, and correspondingly the radial response somewhat improves at its initial part. From this consideration follows that with an increase of the length of two-coil induction probes, forming a differential four-coil probe, the cylinder radius, characterized by small values of geometric factor, increases, provided that the primary electromotive force is compensated. 

Coefficient K2 exceeds unity, and therefore the radial response approaches its asymptote slower than that of a two-coil probe, i.e. the four-coil induction probe possesses a greater depth of investigations with respect to a two-coil induction probe of the same length. 

Figure 7.5 presents the integral radial response of a four-coil probe with parameters p = 0.4 and c = 0.05. Unlike the radial response of a two-coil induction probe, in this case function G r/L) at the beginning has small but negative values, near r/L = 0.27 it changes sign and monotonically approaches unity. 

Until now it has been assumed that at all points of a medium induced currents are shifted in phase by 90°, i.e. the skin effect is absent. Now we will investigate radial responses of the probe, making use of results of the exact solutions in a medium with two cylindrical interfaces when the four-coil induction probe is located on the borehole axis. 

As was mentioned above every multi-coil induction probe can be considered as a sum of two-coil probes namely the basic induction probe and additional coil probes which provide improvement of the radial response. Electromotive forces induced in these probes can have the opposite sign to that in the basic probe as well as the same sign. However, the probes where the electromotive force has opposite sign play the most essential role and correspondingly, only they will be taken into consideration here. 

At the initial part of the radial response, G has negative values which, due to compensation of the electromotive force of the primary field, are much smaller than those of two-coil induction probes. Near point Uq when... 

In Chapter 4 we investigated an influence of displacement, tq, from the borehole axis on the radial responses of a two-coil induction probe. These results allow us to consider the dependence of radial responses of multi-coil probes on the value of displacement Tq. 

Considering the field of the magnetic dipole as well as induced currents in a uniform conducting medium it was established that with an increase of frequency the role of those parts of the medium which arc relatively close to the probe increases. For this reason the vertical response of a two-coil induction probe significantly improves, but simultaneously the influence of currents induced in the formation, with respect to those in the borehole and in the invasion zone, becomes lesser. Unlike of a two-coil induction probe the influence of the medium directly surrounding the multi-coil probe is very small, and an increase of frequency up to a certain limit does not practically change its radial response. 

In fact, the integral response, as well as the differential one, defining a signal in receiver coils due to induced currents in an arbitrary cylindrical layer with a constant resistivity, present the basic element of these calculations. However, the presence of caverns, deviation from radial distribution of resistivity because of nonuniform penetration of a borehole filtrate into a formation, its finite thickness are factors which can influence the focusing features of multi-coil induction probes. In order to eliminate the influence of these factors and to increase the depth of investigation, regardless of the geoelectric section, we will consider in this chapter another approach, based on the use of a two-coil probe and a simultaneous measurement at two or more frequencies if the quadrature component is measured. 

Consider the determination of the field of the transversal magnetic dipole in a medium with two cylindrical interfaces. Analysis of the solution permits us to investigate the influence of the resistivity and the radius of the invasion zone on the radial response of probes with transversal induction coils. 

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. 

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