Two-layered medium

Example I Two-layered Medium (Invasion Zone is Absent) 

The resistivity of a borehole is relatively high (ctj (T2)- In accord with eq. 4.89  

When the probe length, L, exceeds several times the borehole radius, Oi, function Gi(a) is usually much smaller than ja - In fact, eq. 4.97 is valid when conditions 4.93 are met. In this case function ja decreases from unity to 0.3 if the probe length, L, is equal to 1 m. Correspondingly, function Gi(a) does not exceed 0.01, if ai = 0.1 m. For this reason we have  

As is seen from Table 2.2 an increase of conductivity of the bed, d2, or a frequency of more than 100 times (change of parameter a2/xcj from 0.01 to 256 for L = 1 m) leads to a decrease of the function 7 /(T2 almost of three times. For example if the frequency equals 60 kHz a change of the parameter a2/iw from 0.01 to 0.64 corresponds to a change of resistivity, p2, from 48 to 0.8 ohm-m, and the ratio Q f/ o increases almost 30 times. 

The borehole is more conductive (cti T2)- In this case we have  

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Fig. 2-21. Geoelectrochemical hodograph (metal accumulation, m, versus time, x) for a two-layer medium.

As follows from results obtained above, we have for the field on the borehole axis, provided that the formation has infinite thickness (two layered-medium with one cylindrical interface) ... 

First we will consider a two-layered medium (the invasion zone is absent). 

For a two-layered medium the following a2/[Pg.201]

Now we have to find the leading term of the expansion of the integral in cq. 4.24. Let us start from the simplest case of a two-layered medium (the invasion zone is absent) ... 

As follows from eqs. 4.57 the sum of geometric factors, as in the case of two-layered medium, is equal to unity ... 

As in the case of the two-layered medium we have instead of unity in eq. 4.95 the term which takes into account the skin effect in the bed and allows us in every given case to define the range where eq. 4.94 can be applied. 

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. 

Repeating calculations performed for a two-layered medium we again derive the same term proportional to that is ... 

The integrand of eq. 4.140 in general use has two types of singularities namely branch points and poles. First we will consider a two-layered medium when the invasion zone is absent. Analysis of zeroes of a determinant of function Ci, as well as calculations shows... 

The ratio Ui/Ai is plotted along the abscissa, where A] = 2Trh = 27r(2/c7i/iw)i/2 (/j jg the thickness of the skin layer in the borehole). The index of the curves is Ps/pj. In the case of a two-layered medium p2 = pa- For three-layered medium every set of curves has index ... 

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. 

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-... 

Making use of the approach considered in detail in section 4.3, the secondary electrical field in the borehole (two-layered medium) in the range of very small parameters can be presented as ... 

As in the case of a two-layer medium the integral from the first term at the initial part of integration along with field /7 "(A , L) do not contain odd powers of k. 

A comparison of the exact solution and asymptotic formulae shows that the field in a two-layered medium becomes practically the same as in a uniform medium with formation resistivity, if ... 

Substituting expressions for field components through potentials in a two-layered medium (eq. 10.28) into eqs. 10.75-10.76 we obtain ... 

Abstract. In this study we examined the numerical methods of solving the direct problem of electrical sormding with direct current for a layered model with complex relief contact boundaries. The solution was obtained by the method of integral equations. The system of integral equations for the solution of the direct problem of electrical soimding with direct current for a layered relief medium was estabhshed. Numerical simulation of the field for two-layered medium with various shapes of relief contact boundaries was conducted. We obtained the density of distribution of secondary sources on contact bormdaries. 

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