Efficient 3D electromagnetic modelling in the presence of anisotropic conductive media, using integral equations

Abstract

We present a novel technique to simulate numerically the measurements performed by a borehole induction-logging tool in 3D anisotropic rock formations. The simulations are based on an integral equation formulation. Previously, such a formulation was considered impractical for solving large-scale problems due to the resulting large full matrix. To overcome this difficulty, we assume a uniform background model and make use of a uniform grid whereupon there is no need to construct explicitly all of the entries of the full Green?s function matrix. Using a uniform background model, the entries of the corresponding electric and magnetic Green?s tensors are relatively easy to calculate. In the presence of a uniform grid (not necessarily cubic), it is only necessary to calculate the first row of the resulting electric Green?s function matrix. Further, because the matrix is block Toeplitz, it can be rewritten into a block circulant form, and therefore matrix-vector multiplication can be efficiently performed with two FFTs and one inverse FFT. This strategy reduces the computation cost from O(N ´ N) to O(N ´ log2N). In addition to the substantial computer savings, the FFT technique also substantially reduces memory storage requirements because only the first row and the first column in the block Toeplitz matrix are needed to perform the computations of the remaining entries of the matrix. Numerical simulations of the measurements performed with an induction tool in dipping and anisotropic rock formations are benchmarked against accurate 3D finite-difference and 1D codes. These benchmark exercises show that the newly developed integral-equation algorithm produces accurate and efficient simulations for a variety of borehole and formation conditions.