Three-dimensional induction logging problems, Part I: An integral equation solution and model comparisons

A 3-D frequency-domain solution based on a volume integral equation approach has been implemented to simulate induction log responses. In our treatment of the problem, we assume that the electrical properties of the bedding as well as the borehole and invasion zones can exhibit transverse anisotropy. The solution process uses a Krylov subspace iteration to solve the scattering equation, which is based on the modified iterative dissipative method. Internal consistency checks and comparisons with mode matching and finite-difference solutions for vertical borehole models demonstrate the accuracy of the solution. There are no known analytical solutions for induction log responses arising from deviated boreholes intersecting horizontal bed boundaries. To simulate such responses requires the numerical solution of Maxwell's equations in three dimensions along with independent tests to validate the solution approach and its accuracy. In this paper, we compare two independent 3-D frequency-domain solutions for the problem, based on finite differences and the integral equation technique. Specific examples in the quasi-static limit are studied, including a 45degrees deviated borehole that intersects formation bed boundaries as well as cases where the bedding exhibits transverse anisotropy. All comparisons made in this paper show very good agreement and demonstrate, for the first time, verifiable induction log responses in the presence of deviated boreholes. We also show that responses arising from deviated boreholes can be significant and must be accounted for properly when interpreting induction logs.