Quadrature conductivity

X is the equivalent quadrature conductance with a weak dependence on salinity. 

This offers principally the possibihty of a determination of X Qy from the quadrature conductivity measurement ... 

The quadrature component of the conduction from the pore-throat region dominates the overall quadrature conduction for the combined circuit because it is not bypassed by a large in-phase component as in the main pore region. 

This equation can be readily integrated by quadrature, and the initial velocity of changes in the electrical conductivity of the Au/ZnO structure can be obtained in the form... 

A sequence of independent simulations, conducted at a set of points [X] spanning a path from X to Ao, then allows one to estimate the free-energy difference by numerical quadrature ... 

Duplexer Using Quadrature Hybrids The circuit devised by Umathum is shown in Fig. 17 and it works as follows While the pulse is on, the crossed diode packages Dj and Dj are heavily conducting and, because of the A/4 lines, a virtually infinite impedance appears at ports 2 and 3 of quadrature hybrid 1 (QHl) that causes QHl to funnel all the incoming power to port 4, which is connected to the (matched) probe. At the crossed diode packages >, and >2 leak voltages t/ i and each 2 V, will... 

From these expressions it is apparent that at low frequencies the quadrature component of the induced current is dominant and is directly proportional to the conductivity of the ring and to the frequency, while it does not depend on the inductance. This behavior can readily be explained as follows if we neglect the flux caused by the induced current the total flux through the ring is the same as the primary one, that is ipo- As it changes with time, we have ... 

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. 

Thus in this range the quadrature component of the magnetic field is directly proportional to the conductivity and the frequency and inversely proportional to the probe length. As will be shown later this dependence on frequency and conductivity remains valid also for a nonuniform medium. 

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. 

If the conductivity and frequency are held constant the curve shows a change in the quadrature component related to the current density Jo when the distance of the observation point from the dipole source is increased. On the other hand, the observation point can be held fixed in the medium and the conductivity or the frequency can be varied over a wide range. This permits us to explain the main features of the behavior of the quadrature component of the magnetic field following from the distribution of the quadrature component of the current density. 

The analysis of the field of a magnetic dipole in a uniform conducting medium (Chapter 2) has clearly demonstrated that with an increase of the distance from the source the quadrature component of induced currents becomes smaller with respect to that corresponding to Doll s theory. Moreover, comparison of the vertical component of the magnetic field on... 

Taking into account the magnetic field caused by currents in the borehole with conductivity [Pg.182]

Let us notice that in the case of a medium with two coaxial cylindrical interfaces one can derive a field equation, which is valid for higher frequencies and conductivities of the borehole and the invasion zone. However, in this case the field on the borehole axis in a medium with one cylindrical interface has to be known. From calculation of the field in this medium we can obtain values of the field on the borehole axis as its radius is equal to that of the invasion zone of the given model. Then, having replaced the central part of the invasion zone by a medium with the borehole resistivity, we obtain a three-layered medium, and correspondingly the quadrature component of the magnetic field is defined from equation ... 

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. 

First we will assume that the skin effect manifests itself only in a bed. Then for the quadrature component of the field and the apparent conductivity for two- and threelayered media we have (see eqs. 3.123-3.127) ... 

In a more general case, when the conductivity of the borehole is greater than that of the bed oi (T2 the influence of the skin effect often manifests itself in a lesser degree than in the previous case. With an increase of the probe length, the effect caused by the interaction of currents in the formation becomes more noticeable. It is related with two factors. First of all, the influence of the currents in the borehole is reduced since function Gi a) decreases. Second, the relative contribution of the quadrature component of the currents in removed parts of the medium (which are smaller than those in Doll s theory) increases. 

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. 

Therefore with a decrease of the probe length, L, the quadrature component of the magnetic field tends to that in a uniform medium with the conductivity of the borehole. At the same time, regardless of how small the distance is between the transmitter and the receiver, the inphase component of the secondary field coincides with that corresponding to a uniform medium with the conductivity of the formation. 

As is well known, the current density of induced currents, J,p, at any point can be presented as a sum of two components, namely the inphase and quadrature ones. The inphase and quadrature components of induced currents are shifted in phase by 180° (or 0°) and 90° with respect to the dipole current. Distribution of these components. In and QJ(j, is essentially different. The quadrature component of the current is dominant near the source and rapidly decreases with an increase of the distance from the dipole. With an increase of frequency and conductivity of the bed, dimensions of the area where the quadrature component prevails become smaller. 

In a wide range of frequencies and conductivities of borehole, invasion zone and bed, the quadrature component prevails near an induction probe, and the skin effect in the bed manifests itself in the same manner as in a uniform medium with the resistivity of the bed. 

Frequency responses of the quadrature component, Q hz, for a two-layered medium has one maximum which to some extent increases with an increase of resistivity of the borehole. The position of the maximum is mainly defined by the resistivity of the formation. For example, an increase of the borehole conductivity of more than 100 times only slightly shifts the maximum to a range of lower frequencies. In some cases when the invasion zones is relatively large we can observe two maxima. 

For illustration values of quadrature and inphase components of for various displacements e = ro/a, as the two-coil induction probe is located parallel to the borehole axis are given in Table 4.8. The ratio of conductivities = 1/16. 

Calculations based on data given in Table 4.8 demonstrate the validity of this relation. Therefore, the inphase component of the field as well as the term of the quadrature component proportional to are in this range of frequencies defined only by the conductivity of the formation and, correspondingly, do not depend on the position of the induction probe with respect to the borehole axis regardless of its length. 

From comparison of eqs. 4.252-4.254 it is seen that the quadrature component of the electromotive force is less influenced by the magnetic susceptibility than the inphase component. For this reason at the range of very small parameters, when the susceptibility is relatively small, the conductivity of a formation is defined by the quadrature component of the electromotive force while measurements of the inphase component allow us to determine the magnetic susceptibility. If parameter p = apui/2Y is not small and... 

We assume that, for most cases which are of great practical interest in induction logging, data presented in this table coincide with the magnetic field of a direct current. Therefore, by measuring the inphase component with a three-coil induction probe or with a probe of two coils, parameter s and, respectively, coefficient of the probe are defined. This enables us to calculate, the apparent conductivity by making use of quadrature component data. 

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. ... 

It is essential that this result does not depend on ratio of the bed thickness to the probe length, H/L, as well as the ratio of conductivities. In other words, with a decrease of parameters L/h the bed becomes transparent for the inphase component regardless of how the probe length is small. It means that within this range of parameters L//ii, the vertical response of the inphase component is much worse than that of the quadrature one. 

In the range of small parameters the quadrature component of the field is directly proportional to frequency and conductivity. Such behavior of the quadrature component is inherent to Doll s domain, which therefore represents the left-hand asymptote of frequency response of function Qhz L/hi). With an increase of parameter L/h the quadrature component increases, reaches a maximum and then oscillating goes to zero. Thus at the left part of the frequency response of the secondary field the quadrature component prevails while at the right part the inphase component Inh is dominant. 

Making use of relation between the quadrature component of field /ij and the apparent conductivity ... 

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... 

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