Potential theory of electrical well logging



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This paper describes some of the electrode arrangements used for measuring electrical resistivity in a bore hole. For a single electrode sonde, the electrode surface is treated as a general ellipsoid. A solution to the potential distribution in a homogeneous material is given; the potential of the electrode surface and a surface integral for the current flowing from any part of the surface is included. The general ellipsoid is reduced to a prolate spheroid, a circular disk, and a sphere. For each case, a sensitivity equation is derived such that apparent resistivity is expressed as a calibration constant times potential divided by current. The electrode surface may be divided into parts such that one part is a measuring electrode, the other parts are guards. A prolate spheroid with this division of surface is the guard ring electrode and is treated in detail. Contact resistance is defined as to be measured in ohm meters[squared] and the effects on single electrode sondes are considered. A volume reslstivity-contact resistivity cell is described and brief conclusions of experiments on contact resistivity are given. Multiple electrode sondes are treated in two groups. Two, three, and four electrode sondes are considered together because each has a current source, a current sink, and two potential electrodes. One or two of the electrodes are located remotely for a three or two electrode sonde. Additional electrodes may be added for special purposes. The five electrode limestone sonde and the seven electrode laterolog is considered. A sensitivity equation is derived for each type. The basic potential theory required to compute response curves for multiple electrode sondes is the solution of potential distribution about a point current source under given conditions. Five conditions with plane and cylindrical interfaces are treated here. The results are given In figures for easy reference. Numerical computation from the equations is very long and is most often done on modern electronic computers. In the summary, section V, Is a review of sensitivity equations for the various sondes. Notes on substituting the calculated potentials to get apparent resistivity for the particular condition are given.