Comparative analysis of computational schemes for FEM modeling of 3D time-domain geoelectromagnetic fields excited by a horizontal grounded-wire source

Abstract

Two mathematical formulations are considered for calculating the time-domain electromagnetic fields excited by a horizontal grounded-wire source in a 3D geological medium. The first of them is formulated for a vector potential (A-formulation), the second one is formulated for the electric field strength (E-formulation). Both formulations are also considered when using the primary-secondary field approach (PSFA), where the primary field is calculated in a horizontally layered medium. The calculation of 3D non-stationary fields is performed using the vector finite element method (FEM). For time discretization, both two-point and three-point implicit schemes are considered. As the initial field, we used the stationary field of the direct current, the corresponding mathematical formulations are also presented using the PSFA. To verify computational schemes, a comparison with the results of other authors is given. For the problems of marine electrical exploration, we present a comparison of two formulations, analyze the effectiveness of PSFA and the use of a three-point implicit scheme instead of a two-point one. The results of computational experiments have shown that when using both formulations, the use of PSFA and the three-point implicit scheme “with backward overstepping” (instead of the two-point implicit scheme) can reduce computational costs by more than an order. In this case, both formulations give solutions that are close to each other in accuracy and computational costs. However, the implementation of the E-formulation using the PSFA requires certain thoroughness and in some cases can lead to a sharp deterioration in the numerical solution accuracy at late times. This is due to the fact that when using the vector FEM, the numerical solution can lead to a nonzero constant field, which is the gradient of some scalar function.