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 the Maxwell operator. This new approach significantly speeds up the convergence of standard SLDM for the solution of Maxwell's equations while retaining the advantages of standard SLDM such as the ability of solving for multiple frequencies and eliminate completely spurious modes. The cost of evaluating powers of the matrix inverse of the stiffness operator is effectively equivalent to the cost of solving a scalar Poisson's equation. This is achieved by a decomposition of the stiffness operator into the curl-free and divergence-free projections. The solution of the projections can be computed by discrete Fourier transforms (DFT) and preconditioned conjugate gradient iterations. The convergence rate of the new method improves as frequency decreases, which makes it more attractive for low-frequency applications. We apply the new solution technique to model induction logging in geophysical prospecting applications, giving rise to two orders of magnitude convergence improvement over the standard Krylov subspace approach and more than an order of magnitude speed-up in terms of overall execution time. This makes it feasible to routinely use 3-D modeling for model-based interpretation, a breakthrough in induction logging and interpretation.