Modeling Reservoir Geometry With Irregular Grids

Abstract

Abstract The widely used Cartesian coordinate grid has some disadvantages in the description of boundaries, faults and discontinuities. In addition, a five-point scheme can cause significant grid orientation effects. A nine-point scheme reduces this effect but makes the treatment of boundaries and heterogeneities more difficult. Orthogonal curvilinear coordinate systems have improved the modeling of reservoir shape and flow geometry. Mathematically they are based on a coordinate transformation and discretization of the transformed equations in the usual manner. Due to the lack of orthogonality additional mixed derivative terms are introduced which are difficult to discretize and therefore are usually neglected. However, because grid lines have to be coordinate lines, strictly orthogonal systems are not flexible enough to describe reservoirs of complicated shape. This paper describes a practical method for using irregular or locally irregular grids in reservoir simulation with the advantages of flexible approximation of reservoir geometry, simple treatment of boundary conditions and reduced grid orientation effects. Finite difference equations are set up by the so-called balance method. This method uses an integral formulation of the reservoir model equations equivalent to the commonly used differential equations. Integrating over grid blocks results in "balance equations" for each block. This can be done for various types of networks formed by triangles, convex quadrangles, polar meshes, curvilinear or locally refined grids. HEINRICHS1 has proved consistence for this discretization scheme. The mesh can be refined locally. Well locations can be selected as mesh points to ensure that they are situated in the center of grid blocks. For triangular grids, the more isotropic distribution of grid points diminishes the orientation effect significantly. Numerical examples are presented comparing the proposed difference scheme with a nine-point Cartesian scheme. The performance of the method is illustrated by symmetry elements and complex simulation problems.