Use of Irregular Grid in Reservoir Simulation

Abstract

Abstract In this paper three different schemes for the finite-difference approximation of ( ) terms with irregular spacings of the grid points are investigated. Several methods of selecting the grid spacing are considered, and numerical results are presented for some carefully selected examples. presented for some carefully selected examples. Results indicate that one of the most commonly used finite-difference schemes may unnecessarily result in large discretization errors; the other two schemes discussed result in greater accuracy and are no more difficult to use. Introduction Finite-difference methods are used widely for solving partial-differential equations encountered in petroleum reservoir simulation. In many practical problems it is necessary to refine the grid in certain problems it is necessary to refine the grid in certain parts of the reservoir, thus resulting in an irregular parts of the reservoir, thus resulting in an irregular grid spacing for the system. For example, the local refinement of grid is usually necessary in the study of the flow behavior near a well, i.e., the case of coning problems. On the other hand, it is usually advantageous to make the grid locally coarse over aquifers and large gas caps. In spite of the importance of irregular grids in reservoir simulation and other engineering problems, little is known about the effect of various schemes on the over-all discretization error. Most literature on the subject of irregular grid spacings is concerned with the treatment of irregular boundaries. Here, the properties of several difference approximations to properties of several difference approximations to the second-order operator Au = [ ] are investigated. Operator A is encountered in all diffusion-type equations, such as multiphase flow in porous media and heat conduction with temperature-dependent conductivity. For simplicity, we limit our study to one space variable in this study. All results reported here can be extended readily to any number of space variables. FINITE-DIFFERENCE APPROXIMATIONS Various finite-difference approximations to be considered may be expressed as where U is the exact solution to the differential equation at Point i and the approximate solution at i is denoted by u . The finite-difference approximation is obtained by replacing the differential operator A in the equations by the difference operator L. The local discretization error is denoted by e(Ui). Two different problems arise when we try to construct the operator L. One results from the unequal spacing of grid points. The second results from the fact that for K not equal constant, the values of K usually are available only at grid points, and some operators require values K points, and some operators require values K between grid points, which must be approximated. These two problems can be studied essentially independently; we will deal in this paper primarily with the spatial distribution problem. For this purpose we will assume mean weighting of K as purpose we will assume mean weighting of K as follows: SCHEME 1 - "BLOCK CENTERED GRID" In the case of the one-dimensional problem, we divide the system in grid blocks of arbitrary length Deltax as shown in Fig. IA. The grid points at which the differential equation is to be approximated are then placed at the center of each block, i.e., Delta i+1/2 = (xi+1 - xi) = 1/2 (Delta xi + Delta xi+1). We now write the difference operator as (1) SPEJ P. 103