3D DC resistivity numerical modeling by natural-infinite element coupling method

Numerical modeling of direct current (DC) resistivity models plays an essential role in the inversion of measured resistivity data and the interpretation of underground resistivity structure. In order to solve the difficulties of complex geoelectric models in the subdivision and conventional numerical algorithms in dealing with truncated boundaries, a new 3D natural-infinite element coupling method was proposed based on the natural element method and infinite element method. The coupled algorithm behaves well on filling or subdividing complex models by flexibly arranging natural nodes in the study area. The coupled infinite element helps to avoid loading truncated boundary conditions and keep the sparse stiffness matrix remain symmetric and independent of the current source. At first, we deduced the fundamental equation of the quasistatic DC field. Then, the 2D and 3D natural element methods were introduced, and the formula of the 3D shape function and its derivative were provided according to the chain derivative method of compound functions. Besides, we used the 3D multi-directional mapping infinite element to act as the boundary element, which helps to construct the 3D natural-infinite element coupling method. Finally, four numerical examples were tested and compared with the numerical result by the conventional finite element method, finite-infinite element method, and COMSOL Multiphysics software. The numerical results show that the coupled method has the advantages of having high accuracy, helping narrow down the study area, needing no update of the stiffness matrix, and facilitating the subdivision of complex models.