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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.

A three-dimensional finite-difference ͑FD͒ method is used to simulate sonic wave propagation in a borehole with an inhomogeneous solid formation. The second-order FD scheme solves the first-order elastic wave equations with central differencing in both space and time via staggered grids. Liao's boundary condition is used to reduce artificial reflections from the finite computational domain. In the staggered grids, sources have to be implemented at the discrete center in order to maintain the appropriate symmetry in an axisymmetric borehole environment. The FD scheme is validated for multipole sources in three special media: ͑i͒ a homogeneous medium; ͑ii͒ a homogeneous formation with a fluid-filled borehole; and ͑iii͒ a horizontally layered formation. The staircase approximation of a circular borehole introduces little error in dipole wave fields, although it causes a noticeable phase velocity error in the monopole Stoneley wave. This error has been drastically reduced by using a material averaging scheme and finer grids. Numerical examples show that the FD scheme can accurately model 3-D elastic wave propagation in complex borehole environments.

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We present a systematic study of the problem of reconstructing the scattering matrix from experimental data. Insufficient data lead to ambiguities; a general technique for finding ambiguities i s given. The cases of isospinless spin-0-spin-4 scattering (two amplitudes) and pion-nucleon scattering (four amplitudes) a r e treated in detail. The ambiguities present in both cases when the R and A parameters a r e unknown have a profound effect on partial-wave analysis, leading to confusion and misidentification of resonances. It is shown that only eight measurements a r e needed to reconstruct the pion-nucleon amplitudes, rather than nine a s averred by Bilenkii and Ryndin.