An integral transform solution to the fundamental problem in resistivity logging

Abstract

This paper develops the analysis necessary for the computation of the response of a resistivity tool within a well bore as it traverses a thin‐invaded bed. The solution for the potential induced by a steady current source (point current electrode) is formulated in terms of both Fourier cosine and Fourier sine transforms with arbitrary coefficients. A suitable matching of the necessary boundary conditions results in a system of singular integral equations. An iterative solution (Neumann series) is obtained for these transform coefficients which, in turn, are used to determine the potential at an arbitrary point of measurement. The theory is applied to some typical focused resistivity tools, and the results are found to be in close agreement with similar results obtained via a resistor network (analog) solution.