Acute Boundary Aligned Unstructured Grid Generation And Consistent Flux Approximations

Abstract

Summary Grids based on Voronoi diagrams, comprise the dual of Delaunay triangulations (DTs), and remain predominant in reservoir simulation. Voronoi-based grids are locally orthogonal, i.e., PErpendicular BIsectional(PEBI), and permit consistent two-point flux approximation if the permeability field is isotropic, or if the grid is generated to be K-orthogonal. In addition, Voronoi grids can be made to honor classical key constraints, minimizing discretization error, thereby aligning associated control volumes with solid walls, well-trajectories, and geological features: layers, shale barriers, fractures, faults, and pinch-outs. Typically, in reservoirs formed by deposition, the directional trend in the horizontal plane is not very distinct, whereas across the layers, rock properties may jump by orders of magnitude. The PEBI property associated with DTs is of major significance, and can only be exploited provided the circumcenter is used as the approximation point. This requires that the grids generated be boundary aligned, and comprised of entirely acute simplexes. The control-volume centroid is commonly used as the approximation point, because a geological feature honored acute triangulation cannot be guaranteed in the general case; especially in the presence of complex geometries and geological constraints. Development of a geological feature-based acute DT technique is presented. A boundary-aligned grid generation method is augmented with a mesh reconstruction technique, which can ensure circumcenter containment of a DT. To honor the geological feature idea of protection-circle is used. In the mesh reconstruction technique, each mesh point is optimized iteratively, using a local-advancing front method incorporating the length of opposite edges of the set of simplexes sharing it. The methods presented generate boundary-aligned acute DT, where previously proposed methods fail to ensure the acute DT property. Details of the method will be presented, together with results for a number of test cases that verify consistency of the two-point flux on the resulting boundary-aligned acute Voronoi grids.