Modeling of borehole resistivity measurements using infinite electrical grids

Abstract

An infinite electrical resistive grid generated by a finite‐difference approximation to Poisson’s equation in cylindrical coordinates with no azimuthal variation has been used to model borehole resistivity measurements. The medium surrounding the borehole is infinite in extent and its resistivity varies only in the radial direction. As a result, the branch resistance values of the infinite grid also vary only in the radial direction. A continued fraction of Laurent operators determines the node voltages along the borehole’s surface. Use of an operational calculus based upon the finite Fourier transformation converts the continued fraction into a readily computed form. This yields new formulas for apparent resistivity, given various resistivity measuring arrays and a resistivity profile in the radial direction. In contrast to classical methods, the need to match boundary conditions at the interface of a cylindrically layered earth is eliminated, so that the procedure is just as easily applied to a continuous resistivity profile as it is to a stepped profile. Moreover, the method is computationally fast by virtue of the fast Fourier transform algorithm and the avoidance of boundary‐condition matching. The speed and versatility of this model make it useful for studying the effects of complex invasion on the response of resistivity tools.