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.