Semi-analytical solutions for the problem of the electric potential set in a borehole with a highly conductive casing

Highly conductive thin casings pose a great challenge in the numerical simulation of well-logging instruments. Witty asymptotic models may replace the presence of casings by impedance transmission conditions in those numerical simulations. The accuracy of such numerical schemes can be tested against benchmark solutions computed semi-analytically in simple geometrical configurations. This paper provides a general approach to construct those benchmark solutions for three different models: one reference model that indeed considers the presence of the casing; one asymptotic model that avoids computations in the casing domain; and one asymptotic model that reduces the presence of the casing to an interface. Our technique uses a Fourier representation of the solutions, where special care has been taken in the analytical integration of singularities to avoid numerical instabilities.