Influence induced currents

In Fig. 4 we present the final induced decision tree, as well as the partition of the (xi,X4) plane defined by its leaves, together with a projection of all the available (x, y) pairs on the same plane. These two decision variables are clearly influencing the current performance of the refinery unit, and the decision tree leaves perform a reasonable partition of the plane. To achieve better performance, we must look for operating zones that will result in obtaining mostly y = 3 values. Terminal nodes 2... 

A specific and peculiar additional possibility was suggested " from the observation that an ordered array of Ti02 nanocoils forms in particular conditions of synthesis by anodic oxidation.The interesting aspect is that in such nanostructures, photo-induced current generates a local magnetic field which influences the reactivity properties of catalytic... 

Remember 13.3 The Ohmic impedance is a complex quantity that is influenced by geometry-induced current and potential distributions. 

As was pointed out, the remarkable simplicity of Doll s theory is related to the fact that interaction of induced currents is neglected. In order to take into account this effect and improve the quality of an interpretation of logging data, a new approach, also an approximate one. was suggested (Kaufman, 1962). This method allows us relatively quickly to evaluate the field, subjected to an influence of the skin effect in a formation, when there are both cylindrical and horizontal interfaces. Much later, this rather complicated problem was solved by V. Dimitriev, L. Tabarovsky, V. Zakharov using the method of integral equations. 

It is clear that in those cases, where the influence of induced currents is negligible or completely absent, the phase is equal to zero. In other words, the magnetic field,, is equal to the primary one which varies synchronously with the transmitter current. 

It is obvious that parameter p characterizes a distance, expressed in units of the skin depth, and this fact vividly demonstrates that an influence of induced currents in a surrounding medium is not defined by the value of frequency or resistivity and the distance between the dipole and an observation point, but it depends on the ratio between this distance and the skin depth only, that is the parameter p. 

In this connection it is appropriate to emphasize that the distribution of the quadrature component of induced currents near the probe is not practieally subjected to the skin effect and only distant currents are subjected to the influence of their interaction. At the same time, the inphase component of induced currents regardless of the distance from the... 

Application of Doll s theory permits us to derive simple expressions for the quadrature component of the magnetic field in a medium with horizontal and cylindrical interfaces, and in many cases this theory allows us to evaluate with sufficient accuracy the influence of currents, induced in the borehole and in the invasion zone as well as in other parts of the medium. It is appropriate to consider formulae based on this theory as asymptotical ones, which are valid for large values of the skin depth with respect to such parameters as ... 

It is obvious that only the first term contains an inphase component of the magnetic field, which coincides with that in a uniform medium with conductivity of the formation, (72-In other words, within a range of relatively small parameters L//i induced currents in the borehole do not influence the inphase component. Similar results are obtained when an invasion zone is present. In accord with eq. 3.123 we have ... 

Again the inphase component of the magnetic field in the borehole is not practically subjected to the influence of induced currents in the borehole and in the invasion zone, and it coincides with the inphase component in a uniform medium with the formation conductivity, (73. In this approximation induced currents in the borehole and in the invasion zone contribute to the quadrature component of the field. This consideration clearly shows that the inphase component of the magnetic field has a different sensitivity to geoelectric parameters of a medium than the quadrature component, and therefore they are characterized by different depths of investigation. It is clear that the analysis of the current distribution in a uniform medium, performed in Chapter 2, is in complete agreement with these results. 

There is one common feature in all these cases, namely the strong influence of the primary electric field. In other words, in the limit we can neglect the interaction of induced currents and consider that the current density at every point of a medium is defined by the primary electric field only. In accord with the results described in Chapter 3 (Doll s... 

With an increase of resistivity, p2, (P2 > Ps) the contribution of induced currents within the invasion zone in a measured signal decreases, and correspondingly the influence of the skin effect in the bed becomes stronger. 

Calculations demonstrate that the influence of frequency and conductivity of a formation on the magnitude of the ratio Q S/So is practically the same as in the previous case. At the range of small values of parameter (T2/xa the relative contribution of currents induced in the bed constitutes about 80% while for a value of 02lMXi = 0.64 the contribution of the formation is equal to 70% but the ratio Q S/So essentially increases. For this reason with an increase of the frequency the depth of investigation of a two-layered medium by a two-coil induction probe does not change until the signal from the formation is greater or at least comparable with that caused by induced currents in the borehole. Also the natural limitation of a further increase of frequency is related with a nonunique interpretation, inasmuch as the spectrum of the quadrature component has a maximum. 

As has been shown above, the range of frequencies and conductivities, when this equation can be applied, becomes smaller with an increase of the radius of the invasion zone, U2. The influence of the frequency on the ratio < SjSo and the distribution character of induced currents essentially depends on the resistivity and the radius of the invasion zone. The deeper the penetration of the borehole filtrate and the smaller the resistivity the smaller the influence of the skin effect in the bed on the value of Q jSo- However, the relative contribution of currents in the bed also decreases. 

It means that induced currents concentrate near the source, and the secondary field is almost equal by magnitude to the primary field but it has the opposite sign. However, at the far zone unlike the near one the influence of the formation resistivity remains regardless of the frequency. 

With a further increase of frequency the influence of induced currents in the bed becomes smaller, and the frequency responses in a three-layered medium almost coincide with those for a two-layered medium when the resistivity outside the borehole is equal to that of the invasion zone, P2-... 

In Chapter 7 we will consider in detail multi-coil induction probes which essentially allow a decrease on influence on induced currents in the borehole as well as in the invasion zone. At that time questions related to magnetic permeability will not be investigated more. For this reason let us here briefly demonstrate that multi-coil probes can be applied in order to reduce the influence of induced currents in a conductive and magnetic medium of the borehole (Fig. 4.46). As the first example we will consider a three-coil induction probe consisting of one generator and two measuring coils (Fig. 4.46a). The distance between the later is significantly less than that between the transmitter and receiver coils. Moments of receiver coils are chosen in such a way that the electromotive force in a free space, is equal to zero. 

As a second example of a probe which simultaneously measures conductivity and magnetic permeability and reduces the influence of induced currents in the borehole, we will consider a four-coil symmetrical probe (Fig. 4.46b). In accord with eq. 4.262 electromotive force induced in receivers of this probe, when it is located in a uniformly conductive... 

Comparing responses of quadrature and inphase components we can see that in the range of small parameters induced currents in the surrounding medium have an influence on the inphase component, In/i , which is much stronger than that on the quadrature component, Qhz. In the limit, as parameter L/h tends to zero, the inphase component of the magnetic field approaches to that of a uniform medium with the conductivity of surrounding medium, a2. ... 

Therefore, Doll s theory is in fact the theory of very small parameters, which characterize the linear dimensions of a model, expressed in units of the skin depth. For example, with a decrease of the probe length parameter L/hi decreases also. From a physical point of view this means that the influence of induced currents near the dipole, which are shifted in phase by 90° and do not interact with each other, increases. 

As is seen from Figs. 5.23-5.31, all curves of the apparent conductivity at the left-hand part, i.e. within the range of small parameters, are parallel to the axis of abscissa that corresponds to Doll s domain but with an increase of CT2/cti the influence of the skin effect manifests earlier. This behavior is in complete agreement with our understanding of the distribution of quadrature component of induced currents in a conducting medium. In fact, with an increase of the distance from the dipole this component becomes smaller than that according to Doll s theory, and since with an increase of conductivity of the surrounding medium the role of this part of the medium also increases, deviation between results of calculation by exact and approximate solutions also increases. Practically this... 

Due to the internal skin effect with an increase of frequency induced currents concentrate more near the source, and correspondingly the influence of the surrounding medium decreases. It is vividly seen from curves of the apparent conductivity as the bed thickness exceeds the probe length (Figs. 5.26-5.31). Therefore, we can select such high frequency for which the influence of induced currents in the surrounding medium will already be practically negligible. 

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 a similar manner as before the different character of the influence of induced currents in the borehole and in the formation on the quadrature and the inphase components will be emphasized. 

As was demonstrated above, function GI2 characterizes a signal caused by induced currents in an infinitely long cylinder. At the same time it is useful to observe the influence of different parts of this cylinder. With this purpose curves of function Gj2 are presented in Fig. 6.4. They demonstrate the influence of relatively thin cylinders with various position with respect to the induction probe. If the probe length exceeds the cylinder radius, the part of the borehole which directly surrounds the probe provides the main contribution to the signal caused by induced currents in the borehole. 

Therefore, in both cases induced currents in the borehole as well as in the invasion zone do not have an influence on the inphase component of the secondary field, and it coincides with that corresponding to a horizontally layered medium. 

In this section we will consider again the influence of induced currents on the behavior of the quadrature component of the magnetic field on the borehole axis when the formation has a finite thickness. However, unlike the previous sections we will proceed from the results of calculations based on a solution of integral equations with respect to tangential components of the field. This method of the solution of the value of the boundary problem has been described in detail in Chapter 3. This analysis is mainly based on numerical modeling performed by L. Tabarovsky and V. Dimitriev. 

This analysis of the focusing features of the probe l.L-1.2 with p = 0.4 and c = 0.05 in media with cylindrical interfaces allows us to establish the range of frequencies as well as parameters of borehole and invasion zone when induced currents within them do not have an influence on the measured signal. If the resistivity of the invasion zone exceeds that of the formation, p2 > Pa, and as/ay < 4, the apparent conductivity practically coincides... 

Now we will briefly discuss vertical responses of probes with external focusing coils. It is obvious that in the case when the whole probe is located against the formation the influence of currents induced in the surrounding medium is stronger than that for the basic two-coil probe (Fig. 7.23a). 

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. 

Any multi-coil induction probe, regardless of the amount of transmitter and receiver coils, by no means performs focusing of the field as it takes place, for example, in optics. In effect, every induction probe, except a two-coil one, is a differential system measuring such a difference of signals in receivers that the influence of induced currents in the borehole and in the invasion zone is significantly reduced. 

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. 

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