Numerical modeling of direct current electrical resistivity with 3D FEM based on PCG algorithm

For the direct current electrical resistivity prospecting, the requirements for the precision and the speed of inversion are higher and higher. Thus, an optimization scheme is proposed to improve the precision and the speed of FEM numerical modeling. 1D compression storage model for coefficient matrix is designed, and an index array is set up for matrix element indexing with its row number and column number. Compared with that of the variable bandwidth storage, the memory space of model greatly reduces. In order to improve the speed of numerical modeling, preconditioning conjugate gradient (PCG) algorithm is used to solve large sparse linear system in FEM. For the PCG algorithm, and the diagonal block matrix in Jacobi iteration is used as the preconditioning matrix, the inversion of which is convenient to be solved and doesn’t occupy memory spacing. In addition, the abnormal potential method is applied to solving the point source field problem and the precision near the source point is greatly improved. The practicality of the calculation scheme is verified by taking electrical resistivity prospecting for two-layer foundation as an example. Using the above calculation scheme, FEM numerical modeling is carried out to simulate advanced detection of water-bearing faults in a tunnel by means of the electrical resistivity method, and the corresponding model tests are performed. The comparison shows that the numerical modeling results are basically consistent with the experimental data, and the precision and the speed of numerical modeling greatly improve.