Curvilinear Coordinate Systems for Reservoir Simulation

This paper presents a set of curvilinear coordinate transformations that lead to no mixed derivative terms in transformed flow equations. The transformations are created by examining the transformed flow equations and by showing that the mixed derivative terms are zero if the transformation satisfies a system of differential equations that depend on the geometry and rock property distribution within the reservoir. Numerical examples with a black-oil simulator are presented to show the increased accuracy resulting from the use of the curvilinear coordinate system and the importance of eliminating the mixed derivative terms. Introduction The accurate and efficient simulation of fluid flow in a reservoir is highly dependent on the choice of the mesh upon which discrete flow equations are to be solved. In many situations, a simulation conducted on a curvilinear coordinate system is advantageous. Many authors have reported on the advantages of solving reservoir simulation problems with a curvilinear coordinate system. problems with a curvilinear coordinate system. These advantages include the elimination of grid-orientation effects, improved modeling of reservoir geometries, and reductions in CPU time. Solving a problem on a curvilinear coordinate grid system is equivalent to transforming the reservoir into a rectangle and then solving the transformed problem on the rectangle. The transformed reservoir flow equations, however, will generally contain mixed derivative terms. In turn, these terms can alter the structure of the matrix that results from discretizing the flow equations. If the structure of the coefficient matrix is altered, the efficient solvers designed for use in reservoir simulation cannot be used without major modifications. One method for eliminating the mixed derivative terms is to use an orthogonal coordinate system. This system, however, does not eliminate the mixed derivative terms for an anisotropic permeability distribution. Several authors claim that the mixed derivative terms are small and, therefore, may be neglected. This is not the case, however, with large anisotropies. To maintain the inherent advantage of the curvilinear coordinate system, we found it necessary to minimize the effect of the mixed derivatives. This paper presents a class of curvilinear coordinate transformations that lead to no mixed derivative terms. The transformations maintain the advantages of a curvilinear coordinate grid system while avoiding the use of mixed derivative terms. The transformations are generated by examining the transformed flow equations and by showing that the mixed derivative terms are zero if the transformations satisfy a system of differential equations. These equations are based on the geometry and rock property distribution within the reservoir. The resulting system of differential equations is solved by a finite-element method (FEM) developed by Aziz and Leventhal. We begin with the derivation of the curvilinear coordinate transformation and how the mixed derivative terms are eliminated. The FEM is outlined briefly, followed by a description of the inverse transformation used to construct the curvilinear grid. Numerical examples with a black-oil simulator are presented to show the increased accuracy resulting from the use of the curvilinear coordinate system, the importance of accurately representing the reservoir geometry, and the importance of eliminating the mixed derivative terms. Mathematical Formulation The scope of this study is limited to two-dimensional (2D) reservoir flow problems. The transformations developed are applicable to complex transient and multiphase problems. However, it is sufficient to consider the model for the flow equations given in Eq. 1. .......(1) (See Appendix for more details.) It is assumed that the reservoir is banded by four curves, as shown in Fig. 1. The top and bottom curves represented by f1 and f2 are functions of x, while the sides, g1 and g2, are functions of y. The coordinates of the four corner points, c1, c2, c3, and c4, are (x1, y1), (x2, y2), (x3, y3), and (x4, y4), respectively. SPEJ p. 893