New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry

Traditional resistivity tools are designed to function in vertical wells. In horizontal well environments, the interpretation of resistivity logs becomes much more difficult because of the nature of 3-D effects such as highly deviated bed boundaries and invasion. The ability to model these 3-D effects numerically can greatly facilitate the understanding of tool response in different formation geometries. Three‐dimensional modeling of induction tools requires solving Maxwell’s equations in a discrete setting, either finite element or finite difference. The solutions of resulting discretized equations are computationally expensive, typically on the order of 30 to 60 minutes per log point on a workstation. This is unacceptable if the 3-D modeling code is to be used in interpreting induction logs. In this paper we propose a new approach for solutions to Maxwell’s equations. The new method is based on the spectral Lanczos decomposition method (SLDM) with Krylov subspaces generated from the inverse powers of th...