Three dimensional finite-element numerical modeling of geophysical electromagnetic problems using tetrahedral unstructured grids

Finite-element solutions to the three-dimensional geophysical electromagnetic forward modeling problem in the frequency domain are presented. The method is firstly examined for the solution to the E-field Helmholtz equation. Edge-element basis functions are used for the electric field. An alternative method is also used which is based on decomposing the electric field into vector and scalar potentials in the Helmholtz equation and in the equation of conservation of charge. Edge element and nodal element basis functions are used respectively for the vector and scalar potentials. This decomposition is performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the inter-element boundaries, therefore finding an efficient solution to the problem. The computational domain is subdivided into unstructured tetrahedral elements. The system of equations is discretized using the Galerkin variant of the weighted residuals method, with the approximated vector and scalar potentials as the unknowns of a sparse linear system. Both iterative and direct solvers are used for the solutions to the E-field and decomposed systems. A generalized minimum residual solver with an incomplete LU preconditioner is used to iteratively solve the system. The direct solver, MUMPS, is used to provide the direct solution to the system of equations. The forward modeling methods are validated using a number of examples. The fields generated by small dipoles on the surface of a homogeneous half-space are compared against their corresponding analytic solutions. The next example provides a comparison with the results from an integral equation method for a long grounded wire source on a model with a conductive block buried in a less conductive half-space. The decomposed method is also verified for a large conductivity contrast model where a magnetic dipole transmitter-receiver pair moves over a graphite cube immersed in brine. Solutions from the numerical approach are in good agreement with the data from physical scale modeling of this scenario. Another example verifies the solution for a resistive disk model buried in marine conductive sediments. For all examples, convergence of the solution that used potentials was significantly quicker than that using the electric field. The inductive and galvanic components of the electromagnetic response are also investigated for the above examples. Furthermore a detailed investigation of these effects are presented for models with varying conductivity contrasts. The characteristics of the solutions in terms of implicitly and explicitly enforcing the Coulomb gauge condition are investigated for the decomposed system. Numerical computations show that for the above-mentioned grounded wire example the electromagnetic response is significantly affected by the contribution from the galvanic part. By contrast for the example where a magnetic dipole excites a graphite cube immersed in brine solutions, the inductive scenario dominantly contributes to the model’s electromagnetic response.