The Generation and Application of K-Orthogonal Grid Systems
Authors: Gunasekera D. , Cox J. , Lindsey P. 
Published: December 2007

Abstract
Unstructured K-orthogonal PEBI grids with permeability defined on triangles (or tetrahedra) have been used successfully by previous authors for mildly anisotropic systems. This paper presents two unstructured K-orthogonal grid systems, in which permeability is defined on cells. The first is the previously mentioned K-orthogonal PEBI grids; the other is the dual of a PEBI grid constructed by aggregating triangles (or tetrahedra), which is termed a composite tetrahedral grid. Such grids, carefully generated, enable the accurate modeling of highly anisotropic and heterogeneous systems. Good K-orthogonal grids for highly anisotropic systems can be generated by transforming the physical space into an isotropic computational space in which an orthogonal grid is generated. The steps involved in generating K-orthogonal grids and their application to reservoir simulation, namely, anisotropy scaling, point distribution, triangulation (or tetrahedralization), triangle aggregation, cell generation, transmissibility calculation, grid smoothing, well connection factors and cell renumbering for linear algebra, are described. This paper also describes how independently generated multiple domains are merged to form a single grid.
The paper presents 2D and 3D simulation results for single phase and multi-phase problems in well test and full field situations. The grids are tested under high anisotropy, high mobility ratios, complex geometries and grid orientations, in order to establish the true limitations of K-orthogonal grids. The error due to non-orthogonality is reported for each cell, suggesting regions where multi-point flux approximations may be of advantage. Relative merits of PEBI and composite tetrahedral grids are also discussed.
The grids are applicable to multi-layered, multi-phase well test and full field simulations, with full heterogeneity and anisotropy limited to a spatially varying kv/kh ratio. Technical contributions in the paper include, definition and generation of composite tetrahedral grids, the process of generating good K-orthogonal PEBI and composite tetrahedral grids, algorithms for computing volumes, transmissibilities, well connections and cell renumbering for general K-orthogonal grids.

Introduction
Flow simulations on grids based on triangles have been used by various authors inside and outside the petroleum industry. Winslow used control volumes formed around each node of a triangular grid by joining the edge midpoints to the triangle centroids for solving a 2D magnetostatic problem. This technique was applied to reservoir simulation by Forsyth, and is commonly known as the control volume finite element (CVFE) method. Cottrell et al. used control volumes formed by joining the perpendicular bisectors of triangle edges of a Delaunay triangulation for solving semiconductor device equations. Heinemann et al. applied this technique to reservoir simulation, which is known as the PEBI or the Voronoi method. Further work on the CVFE method was presented by Fung and on the PEBI method by Palagi and Gunasekerat. Both Forsyth and Fung handled heterogeneous problems by defining permeability to be constant over a triangle. Aavatsmark and Verma derived an alternative difference scheme based on the CVFE method in which permeabilities are defined as constant within control volumes. This approach handles boundaries of layers with large permeability differences better than the traditional CVFE method and, as with the traditional method, it leads to a multi-point flow stencil, hence referred to as an MPFA scheme. By contrast, the PEBI method reduces to a two point flow stencil. Heinemann et al. and Amado et al. extended the PEBI method to handle anisotropic heterogeneous systems by defining permeability to be constant within a triangle and by adjusting the angle between triangle edges and cell boundaries. This approach has two problems: firstly, handling layers of contrasting permeabilities is poor, secondly in highly anisotropic systems the angle condition between triangle edges and cell boundaries may become so severe that cells begin to overlap, as shown in Verma.
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