A Control Volume Scheme for Flexible Grids in Reservoir Simulation

Abstract

Abstract The commonly used Cartesian and hybrid local grid refinements have the disadvantage that the base grid is always Cartesian, which makes it difficult to align along reservoir heterogeneities. Also, the locally orthogonal Voronoi and k-orthogonal perpendicular bisector (PEBI) grids are only useful for isotropic or limited anisotropic permeability distributions. The finite-difference approach proposed in this paper overcomes limitations of existing flexible gridding schemes in modeling full, anisotropic and asymmetric permeability tensors and permeability heterogeneity. The new scheme assumes uniform properties inside control-volumes and it can be used for control volumes formed around vertices of triangles in two-dimensions and tetrahedra in three-dimensions. It can thus be used with Voronoi grids in two-dimensions, and median (i.e. CVFE) and boundary adapting grids (BAG) in two- and three-dimensions. Applications of Voronoi grids in three-dimensions are limited due to geometrical considerations. The proposed method can also be used with grids constructed to align along major reservoir heterogeneities, wells and streamlines. Several applications of this method for two- and three-dimensional problems are presented. Introduction Petroleum reservoir simulation involves the numerical solution of mass and/or energy conservation equations in discretized form over a grid. A grid is called flexible (or unstructured) when it is made up of polygons (polyhedra in three-dimensions) whose shape and size can vary from place to place in the reservoir. This paper addresses two important aspects of flexible grids;grid construction, andnumerical procedure to solve fluid flow equations on these grids. Solution of fluid and heat flow equations on flexible grids have been discussed extensively in the literature (e.g. Heinrich 1987, Heinemann 1988, Palagi 1992 and Prakash 1987). Most of the existing numerical schemes assume either an isotropic permeability tensor or a symmetric and anisotropic permeability tensor. Existing methods do not provide sufficient flexibility to align grids with (a) reservoir bed boundaries, (b) beds with varying permeability tensors, (c) faults and (d) horizontal/inclined wells. The new method proposed in this work provides greater flexibility than existing approaches. PREVIOUS WORK In this section, some of the most important previous works related to the topic of this paper are described. Local Grid Refinement (LGR). Local grid refinement involves using fine grid inside coarse base grid. This is done only in selected regions, e.g., near-well regions, in regions of wide saturation variation, in highly heterogeneous regions, etc. This reduces computation time compared to uniform fine grids. The two commonly used methods are Cartesian local grid refinement (CLGR) (Ciment and Sweet 1973 and Nacul 1991) as shown in Figure 1 and hybrid local grid refinement (HLGR) as shown in Figure 2 (Pedrosa and Aziz, 1985). Both the Cartesian and hybrid LGR require a base grid which is Cartesian. Hence it becomes difficult to align the grid with varying facies horizontal wellbores etc. Also, additional discretization errors are introduced at the boundaries of coarse and refined regions. Control Volume Finite Element (CVFE) Method. The control volume finite element (CVFE) method uses triangular mesh in two-dimensions (Figure 3) and tetrahedral mesh in three-dimensions. Control volumes are formed around gridnodes by joining the midpoint of the triangles' edges with a point a inside the triangle. Different locations of point a give rise to different forms of the flow term between grid nodes. When a is at the barycenter, the resulting grid is called CVFE type grid (Heinrich, 1987, calls this the median grid. Forsyth (1989) applied the CVFE method to thermal reservoir simulation problems. However, all his numerical examples were for homogeneous reservoirs. Fung et al. (1991) also studied the CVFE method and implemented it in a general purpose thermal simulator. They converted the algebraic approximation of the conservation equation to a form similar to those in conventional finite-difference simulators, thus making its implementation in existing reservoir simulators easy. P. 215^