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 ...

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