Three-dimensional numerical modeling of direct current resistivity methods in mixed space-wavenumber domain

At present, in order to obtain highly accurate solution for complex (geo-electric) conditions, it is necessary to divide the model precisely in the numerical modeling of DC resistivity method, which results in greatly increases the amount of computation and memory. To solve this problem, we propose a 3D numerical simulation method in a mixed space-wavenumber domain. The method transforms the partial differential equation of DC abnormal potential into many independent 1D differential equations with different wavenumbers by performing a 2D Fourier transform along the horizontal direction. It decomposes a large-scale 3D numerical modeling problem into many 1D numerical modeling problems, which greatly reduces the computation and memory requirements of modeling, and each 1D differential equation is independent, so it has high parallelism. The vertical direction is reserved as the spatial domain, which is convenient for the subdivision, taking into account the calculation accuracy and efficiency. A finite-element algorithm combined with a chasing method is used to solve the 1D differential equations with different wavenumbers, and a contraction operator based on electromagnetic integral equation method is used to the iteration of the algorithm for the first time. Taking full advantage of the rapidity of Fourier transform, the high efficiency of solving 1D differential equations and the high parallelism of 1D problems with different wavenumbers, it can greatly reduce the amount of calculation and storage, thus greatly improves the efficiency of the algorithm. The accuracy and convergence of the algorithm are verified by using the model of a sphere in half space. Based on the model with five cubes in half space, the efficiency of the algorithm is verified by comparing with the academic Gimli package based on the finite element method. In addition, a parallel algorithm based on OpenMP is used to verify the parallelism of the proposed algorithm. The examples results show that, the 3D numerical modeling method of DC resistivity in the mixed space-wavenumber domain has the characteristics of high efficiency, high precision and highly parallel. This method provides a new way for 3D numerical simulation of DC resistivity method.