A simple method for solving the Bussian equation for electrical conduction in rocks

Abstract

One of the most general and effective models for calculating the complex electrical conductivity and relative dielectric permittivity of rocks saturated with pore fluids is that of Bussian. Unlike most models, it is non-linear and cannot be solved algebraically. Consequently, researchers use reiterating numerical routines to obtain a solution of the equation, and then only for the real part of the solution. Here we present a different approach to the solution that uses conformal mapping in the complex plane, and implements it within Maple^TM. The method is simple and elegant in that it requires, for example, only 3 lines of code in Maple^TM 11 and little programming experience. The approach has been shown to be as precise as using the classical reiterating bisection method for real data implemented in C++ on an ordinary desktop computer to within a probability over 1 in 10⁹. However, the conformal mapping approach is 52 times as fast. We show once more that the Bussian equation breaks down for low fluid conductivities, but recommend it (with the modified Archie's law) for use with rocks saturated with high salinity fluids when the matrix is conductive.