Anisotropic mesh adaptation for 3D anisotropic diffusion problems with application to fractured reservoir simulation

TL;DR

This paper develops anisotropic mesh adaptation techniques for 3D anisotropic diffusion problems, focusing on maximum principle preservation and applying these methods to fractured reservoir simulation to improve solution accuracy.

Contribution

It introduces new sufficient conditions for maximum principle preservation and explores different metric tensors for effective anisotropic mesh adaptation in 3D diffusion problems.

Findings

The fourth metric tensor improves maximum principle satisfaction.

Mesh adaptation concentrates elements in high-error regions.

Application to reservoir simulation reduces unphysical solutions.

Abstract

Anisotropic mesh adaptation is studied for linear finite element solution of 3D anisotropic diffusion problems. The M-uniform mesh approach is used, where an anisotropic adaptive mesh is generated as a uniform one in the metric specified by a tensor. In addition to mesh adaptation, preservation of the maximum principle is also studied. Some new sufficient conditions for maximum principle preservation are developed, and a mesh quality measure is defined to server as a good indicator. Four different metric tensors are investigated: one is the identity matrix, one focuses on minimizing an error bound, another one on preservation of the maximum principle, while the fourth combines both. Numerical examples show that these metric tensors serve their purposes. Particularly, the fourth leads to meshes that improve the satisfaction of the maximum principle by the finite element solution while concentrating elements in regions where the error is large. Application of the anisotropic mesh adaptation to fractured reservoir simulation in petroleum engineering is also investigated, where unphysical solutions can occur and mesh adaptation can help improving the satisfaction of the maximum principle.