Theory Of Equivalent Artificial Earth Networks And Their Application In Electric Well Logging
Authors: Barlai Zoltan 
Published: December 2007

Abstract
An electric network is suggested for calibration of electric well logging equipment. The correspondence between the network and the three-dimensional field of measurement is a unique one achieved by the theoretical design. As a benefit, accuracy of electric well logging can be improved in ultradeep boreholes.

Introduction
For a better control of the accuracy and reliability of well logging equipment, the comparison of several repeatedly recorded well logs is generally preferred. Several curves are recorded for the same borehole by using the same sonde with two or more different but calibrated sets of the surface instrument, and vice versa. Then the accuracy and reliability of the well logging equipment is characterized by the goodness of the fit of those repeatedly recorded logs.
However, this kind of control cannot be considered as a strict quantitative proof of the accuracy. This is so because perfectly accurate surface instrument and borehole sonde cannot perfectly accurate surface instrument and borehole sonde cannot be provided for such a comparison. In addition, subsurface formations comprising the field of measurement are never exactly known.
From the preceding remarks it can be concluded that the design and the setting up of accurate control tools, connected to the input terminals of the well logging cable cores and lowered into the borehole, are of utmost importance.

RESISTANCE MATRIX AND EQUIVALENT RESISTANCE NETWORK OF SUBSURFACE FORMATIONS
For the control of electric well logging equipment, the application of equivalent resistance networks is suggested. They are applicable when the frequency of the electromagnetic field is below 1,000 cycle and electric connection between the sonde and the subsurface formations is achieved by electrodes.
A criteria is suggested for making the resistance network as being equivalent to the three-dimensional field of measurement. The criteria suggested may be stated as follows: The electric potentials generated at the terminals of the network by feed potentials generated at the terminals of the network by feed currents ought to be the same as those created by the same currents at the electrodes of the well logging down-hole sonde.
The electrode potentials generated by the feed currents at the electrodes in the three-dimensional field can be described by means of an equivalent resistance matrix [R]. The general form of matrix [R], in case of a sonde composed of n electrodes, is shown in the Appendix in Eq. A-1.
The matrix presented by Eq. A-1 is a symmetric square matrix of order n. The number of the independent matrix elements, w, is determined by the following formula.
(1)
The matrix elements of Eq. A-1 in the principal diagonal, R ii, are eventually grounding resistances of the electrodes, and are expressed by Eq. 2.
(2)
where rii = the grounding radius of the electrode with a serial number i, and R a,ii = the apparent grounding resistivity of monoelectrode (resistivity logging) sonde represented by electrode i. The matrix elements R ik outside the principal diagonal are transfer resistances between the electrodes i and k and described by Eq. 3.
(3)
where L ik is the distance between the electrodes i and k, and R a, ik is the apparent resistivity measured by a normal resistivity logging) sonde composed of electrodes i and k.