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 3D geological structure in the subsurface, 3D inversion of the EM induction log is required. Because the inversion process is mainly dependent on forward modelling, the use of fast and accurate forward modelling 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 preconditioning. The discretised IE after domain decomposition equals a fixed-point equation that is solved iteratively with either the block Gauss-Seidel or Jacobi preconditioning. 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. Numerical experiments show a maximum reduction in computational time of 35 per cent compared to solving the full-domain IE with a conventional GMRES solver. 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.

翻译：电磁（EM）感应随钻测井技术是辅助地质导向决策过程的标准常规方法之一。通过反演电磁感应测井数据获得的电导率分布，可为井眼周围地质结构提供重要信息。为对地下三维地质结构进行成像，需采用电磁感应测井的三维反演技术。由于反演过程主要依赖于正演模拟，采用快速且精确的正演模拟方法至关重要。本文提出了一种改进型积分方程（IE）建模技术，该技术适用于一般各向异性介质，并引入了区域分解预条件。经区域分解后的离散积分方程等价于一个不动点方程，可通过块高斯-赛德尔或块雅可比预条件进行迭代求解。在每次迭代中，利用Krylov子空间方法而非直接求解器计算块矩阵的逆矩阵。通过在迭代求解器中采用自适应相对残差终止准则，进一步降低了计算时间。数值实验表明，与采用传统GMRES求解器求解全区域积分方程相比，计算时间最多可减少35%。此外，由于覆盖感应测井仪大范围灵敏区域所需的内存需求降低，可在有限GPU内存下实现加速。因此，我们认为区域分解方法通过减少计算时间和内存需求，有效提高了积分方程方法的效率。