A Volume-Surface Integral Approach For Direct Current Resistivity Problems With Topography

We have developed an accurate volume-surface integral formula for 3D direct current (DC) resistivity forward modeling with heterogeneous conductivities and arbitrary homogeneous topography. First, a volume-surface integral formula is derived from its elliptic boundary value problem in terms of an artificial analytical function defined over the full space. That leads to a volume integral accounting for underground anomalous regions and a surface integral over the surface topography. Then, tetrahedral grids are used to discretize the volume anomalous bodies and triangular grids are adopted to approximate the complicated surface topography. The use of unstructured grids enables our volume-surface integral formula to deal with realistic earth models with complex geometries and conductivity distributions. Furthermore, linear shape functions are assumed in the tetrahedral and triangular elements to obtain the final system of linear equations. In the final system matrix, singularity-free analytical expressions are developed for entries arising from volume integrals over tetrahedral elements and Gaussian quadrature formulas are used to calculate surface integrals over triangular elements. To guarantee the accuracy of the final numerical solutions, direct solvers are used. At the end, three synthetic models are used to verify our newly developed volume-surface integral formula by comparison with published analytical solutions and finite-element solutions. Due to its high accuracy, solutions of our volume-surface integral approach can act as an efficient benchmark tool for other numerical solutions for complicated DC models with arbitrary homogeneous topographies.