A 3-D FINITE-ELEMENT ALGORITHM FOR DC RESISTIVITY MODELLING USING THE SHIFTED INCOMPLETE CHOLESKY CONJUGATE GRADIENT METHOD

Abstract

An accurate and efficient 3-D finite-element (FE) forward algorithm for DC resistivity modelling is developed. First, the total potential is decomposed into the primary potential caused by the source current and the secondary potential caused by changes in the electrical conductivity. Then, the boundary value problem for the secondary field and its equivalent variational problem are presented, where the finite-element method is used. This removes the singularity caused by the primary potential, resulting in an accurate 3-D resistivity model. Secondly, I introduce the row-indexed sparse storage mode to store the coefficient matrix and the shifted incomplete Cholesky conjugate gradient (SICCG) iterative method to solve the large linear system derived from the 3-D FE calculation. The SICCG method converges very quickly and requires much less computer storage, while the traditional ICCG or modified ICCG (MICCG) method usually fails for an irregular grid. The SICCG method is more efficient than the direct method, i.e. the elimination solver with the banded Cholesky factorization. Also, it has an advantage over the symmetric successive overrelaxation preconditioning conjugate gradient method. Numerical examples of a three-layered model with high conductivity contrast and a vertical contact show that the results from the secondary potential FE method agree well with analytic solutions. With the same grid nodes, much higher accuracy from the solution of the secondary potential than those solving the total potential can be achieved. Also a 3-D cubic body is simulated, and the dipole-dipole apparent resistivities agree well with the results from other methods. By defining the analytical solution of a vertical contact as the primary potential, a more complicated model with several 3-D inhomogeneities near the vertical contact is simulated. The presented method also obtains good results for this model.