Forward modeling of direct-current resistivity data on unstructured grids using an adaptive mimetic finite-difference method

Abstract

An effective solver for the direct-current (DC) resistivity forward-modeling problem should be capable of accommodating arbitrary electrode layouts, adapting the mesh to topography and complex geologic structures, and adaptively refining the mesh to ensure high solution accuracy. We use the capability of the mimetic finite-difference method in naturally accommodating meshes with nonconforming elements to develop adaptive and parallel cell-based (CB) and vertex-based (VB) mimetic schemes for the forward modeling of DC resistivity data on unstructured meshes. A diffusion problem in mixed form and Poisson’s problem are solved in the CB and VB schemes, respectively. The mesh adaptivity involves an iterative h-refinement conducted by a regular subdivision of the marked elements together with a 2-irregularity condition. Goal-oriented residual- and gradient-based error estimators are used to mark the elements for refinement in the CB and VB schemes, respectively. To evaluate the accuracy and efficiency of the mimetic schemes, we use two benchmark models with analytical solutions and standard linear and quadratic finite-element solutions. The numerical results for apparent resistivity from both mimetic schemes accurately approximate the reference analytical and numerical values. Furthermore, on similar meshes, the VB mimetic scheme is found to be less demanding than the CB method in terms of computational resource requirement. We also develop a grid-based mesh generation technique based on the body-centered cubic lattice to generate high-quality initial meshes. Using benchmark examples, we validate this mesh generation method and demonstrate its versatility by discretizing a model of a thin dike where a standard Delaunay mesh generator fails to generate a mesh. Moreover, we display the effectiveness of the presented mimetic approach and the grid-based mesh generation technique using a realistic example with topography and parallelization over a large number of sources.