A transmission‐line model for electrical logging through casing

Abstract

Electrical logging in cased holes is a subject of strong current interest. In this paper we develop the conventional theory of physical responses of electrical logging through casing where the formation resistivity changes vertically and where the conductance of the casing varies. The theory is developed for a four‐electrode sonde with one current electrode, and three receiver electrodes that measure current leakage into the formation. All the theoretical developments in the paper are derived using approximate transmission‐line theory. An expression for the array coefficient is first derived and then used to obtain an expression for the apparent conductivity of the formation. Then a relation is established between the array coefficient, geometrical factor and conductance of the casing. This step led to a method of measuring casing conductance. Knowing casing conductance, the response of the array to vertical changes in formation conductivity is measurable. It was established, that when the array length is less than the thickness of the formation layer, the measured apparent conductivity is equivalent to the true conductivity of the formation, unlike the case of normal and induction logs. It was also established that measurement of apparent conductivity can be made accurately inside a casing of finite length when it essentially exceeds the length of the electrode array. The results contained in this paper show that electrical logging through casing is possible in wells with conventional casing strings that penetrate layered sedimentary formations.