3-D induction log modelling with integral equation method and domain decomposition pre-conditioning

Abstract

SUMMARY The deployment of electromagnetic (EM) induction tools while drilling is one of the standard routines for assisting the geosteering decision-making process. The conductivity distribution obtained through the inversion of the EM induction log can provide important information about the geological structure around the borehole. To image the 3-D geological structure in the subsurface, 3-D inversion of the EM induction log is required. Because the inversion process is mainly dependent on forward modelling, the use of a fast and accurate forward modelling tool is essential. In this paper, we present an improved version of the integral equation (IE) based modelling technique for general anisotropic media with domain decomposition pre-conditioning. The discretized IE after domain decomposition equals a fixed-point equation that is solved iteratively with either the block Gauss–Seidel or Jacobi pre-conditioning. Within each iteration, the inverse of the block matrix is computed using a Krylov subspace method instead of a direct solver. An additional reduction in computational time is obtained by using an adaptive relative residual stopping criterion in the iterative solver. Using this domain decomposition scheme, numerical experiments show computation time reductions by factors of 1.97–2.84 compared to solving the full-domain IE with a GMRES solver and a contraction IE pre-conditioner. Additionally, the reduction of memory requirement for covering a large area of the induction tool sensitivity enables acceleration with limited GPU memory. Hence, we conclude that the domain decomposition method is improving the efficiency of the IE method by reducing the computation time and memory requirement.