NEW ASPECTS OF THROUGH-CASING RESISTIVITY THEORY

A closed-form solution for a basic model of the through-casing resisti vity measurement is found, The solution reveals fundamental characteri stics of the measurement and the dependence of these characteristics o n the parameters of the casing and formation. We start our considerati on with an analysis of the current distribution in the vicinity of a s ingle electrode energizing the casing. This analysis shows that the di stribution becomes axially symmetrical at a distance of just a few rad ii from the injection point, As a consequence, the number and the phys ical location of the electrodes used to inject electric current become unimportant if the measurement electrodes are not placed too close to the sensing electrodes.We also derive st closed-form solution for the distribution of the current flowing in a homogeneous casing embedded in a homogeneous formation, This consideration results in an expressio n specifying the so-called characteristic length. The electric field p otential is analyzed as a function of the vertical and radial distance s from the current injection electrode, Explicit closed-form expressio ns are derived for the electric potential distribution at a distance t hat is greater or less than the characteristic length. A transition fr om a cylindrical to spherical pattern of the current flow occurs at a radial distance controlled by the characteristic length, This phenomen on bounds the ''current tube'' resistance. An analytical result is als o found for the casing surrounded with a homogeneous cement sheath. Th e concept of the characteristic length is related closely to the geome trical factor of the through-casing resistivity measurement, It follow s from our solution that the geometrical factor displays a significant dependence on the resistivity of the formation and the parameters of the casing. This dependence should be taken into account in the quanti tative interpretation of through-casing resistivity logs.