Anisotropic Modeling with Geometric Multigrid Preconditioned Finite Element Method

Formation anisotropy in complicated geophysical environments can have a significant impact on data interpretation of electromagnetic surveys. To facilitate full 3D modeling of arbitrary anisotropy, we have adopted an h-version geometric multigrid preconditioned finite-element method based on vector basis functions. By using a structured mesh, instead of an unstructured one, our method can conveniently construct the restriction and prolongation operators for multigrid implementation, and then recursively coarsen the grid with the F-cycle coarsening scheme. The geometric multigrid method is used as a preconditioner for the biconjugate-gradient stabilized method to efficiently solve the linear system resulting from the finite-element method. Our method avoids the need of interpolation for arbitrary anisotropy modeling as in Yee’s grid-based finite-difference method, and it is also more capable of large-scale modeling with respect to the p-version geometric multigrid preconditioned finite-element method. A numerical example in geophysical well logging is included to demonstrate its numerical performance. Our h-version geometric multigrid preconditioned finite-element method is expected to help formation anisotropy characterization with electromagnetic surveys in complicated geophysical environments.

Cementation exponent as a geometric factor for the elastic properties of granular rocks

A geometric factor properly describing the microstructure of a rock is compulsory for effective medium models to accurately predict the elastic and electrical rock properties, which, in turn, are of great importance for interpreting data acquired by seismic and electromagnetic surveys, two of the most important geophysical methods for understanding the earth. Despite the applications of cementation exponent for the successful modeling of electrical rock properties, however, there has been no demonstration of cementation exponent as the geometric factor for the elastic rock properties. We have developed a workflow to model the elastic properties of clean and normal granular rocks through the combination of effective medium modeling approaches using cementation exponent as the geometric factor. Based on the dedicated modeling approaches, we find that cementation exponent can be adequately used as a geometric factor for the elastic properties of granular rocks. Further results highlight the effects of cementation exponent on the elastic and joint elastic-electrical properties of granular rocks. The results illustrate the promise of cementation exponent as a geometric link for the joint elastic-electrical modeling to better characterize the earth through integrated seismic and electromagnetic surveys.