A closed-form solution for a basic model of the through-casing resistivity measurement is found. The solution reveals fundamental characteristics of the measurement and the dependence of these characteristics on the parameters of the casing and formation. We start our consideration with an analysis of the current distribution in the vicinity of a single electrode energizing the casing. This analysis shows that the distribution becomes axially symmetrical at a distance of just a few radii from the injection point. As a consequence, the number and the physical 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 a closed-form solution for the distribution of the current flowing in a homogeneous casing embedded in a homogeneous formation. This consideration results in an expression specifying the so-called characteristic length. The electric field potential is analyzed as a function of the vertical and radial distances from the current injection electrode. Explicit closed-form expressions are derived for the electric potential distribution at a distance that is greater or less than the characteristic length. A transition from a cylindrical to spherical pattern of the current flow occurs at a radial distance controlled by the characteristic length. This phenomenon bounds the "current tube" resistance. An analytical result is also found for the casing surrounded with a homogeneous cement sheath. The concept of the characteristic length is related closely to the geometrical factor of the through-casing resistivity measurement. It follows 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 quantitative interpretation of through-casing resistivity logs.

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