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 solv ed iterativ ely 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 iterativ e solv er. 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.

Inversion of electromagnetic induction log while drilling provides structural information around borehole for geosteering decision process. Imaging a complex three-dimensional (3D) anisotropic formation is very challenging due to the significant cost of the forward solver. To overcome this challenge, we have developed an inversion algorithm using an adjoint Integral Equation Method for logging while drilling scenarios in anisotropic media with several inexpensive approaches in this study. We use a quasi-Newton method with L-BFGS algorithm and implement a matrix-free adjoint Integral Equation method for efficient update of inverted model parameters. In addition, we propose the use of limited number of iterations for solving the adjoint equation as an approximation to the gradient for a faster computation time. To regularize the inverted models, we adopt the use of multiplicative regularization and weighted L2 total variation. Additionally, we impose priori information in our formulations, such as 2D parameterization and the range of conductivity values. We show a numerical example of logging while drilling inversion on a 2.5D faulted anisotropic formation. Numerical experiments show that our inversion workflow shows a good structural agreement of the inverted model within the sensitive range of the tool.

Real-time 3D imaging of the induction log is essential for improving the decision-making process in geosteering. To fulfill this need, we investigated various strategies for reducing the computational cost for 3D modelling of induction logs using the integral equation (IE) method, including the use of iterative Krylov solver, convolution with FFT algorithm, contraction IE formulation, computation acceleration with GPUs, and domain decomposition. We present two cases example to demonstrate the implementation of IE with these strategies. In the first case, we show that the application of domain decomposition allows one to only discretize the inhomogeneous domain and save the computation cost in the case of isolated domains. We present a logging while drilling scenario on a complex model for the second case. Our implementation of the efficient IE on GPUs enables significant acceleration and allows the computation of 3D forward modelling within less than two minutes for each local 3D simulation domain with approximately two million grid cells on a laptop. The implementation of domain decomposition formulation shows a different arrangement of solving IE by decomposing the domain.