Use of Irregular Grid in Reservoir Simulation

Abstract

ABSTRACT In order to numerically solve the system of nonlinear partial differential equations which describes multiphase flow in porous media, finite volume discretization in space and finite-difference in time are commonly used in reservoir simulation1. This requires the calculation of flow at the boundaries between adjacent blocks using the pressure information at a specific point within each block. Consequently, the knowledge of both, block boundaries and grid point locations are required. Therefore, the distribution of grid points in space and the location of block boundaries with respect to grid points influence the accuracy of finite-difference approximations. In this paper, we investigate the accuracy of six different kinds of gridding systems, which are derived from the classical point-distributed and the block-centered methods. Our goal is to obtain maximum accuracy for a given number of blocks. To achieve this goal, two approaches are studied: Given a set of grid points, decide where to place the block boundaries.Given a set of grid blocks, decide where to locate the grid points. The point-distributed gridding system is known to minimize the truncation error in the discretization of the pressure derivative (flow terms), whereas the block-centered gridding system is known to minimize the truncation error in the discretization of the volume integral (accumulation terms). When a uniform grid spacing is used both methods minimize the truncation error in the same way, with the truncation error being of second order for the midpoint transmissibility and first order for the upstream transmissibility. However, in many practical problems it is necessary to refine the grid in certain regions of the reservoir, thus resulting in a irregular grid for the system. A truncation error analysis for an irregular system using upstream transmissibility is performed. It shows that both methods are locally inconsistent, i.e., both have error of order zero. However because of the way the block boundaries are selected in the "point-distributed" methods, these methods give more accurate results when compared with other methods. Six different gridding systems are tested on two examples. The first example studied is a special single phase flow problem where a nonlinear partial differential equation has a known exact solution. This solution is used for comparison with the numerical results obtained using the six different kinds of gridding systems studied. In this case the Euclidean norm of errors is evaluated as a function of time and it shows that for the methods where the grid block boundaries are located half-way between two adjacent grid points, the truncation error tends to decrease with time. The behavior is opposite for the methods where the grid points are located in the center of the grid blocks. The second example compares the results for a two phase problem. It shows the discrepancy in the results when block-centered method is used. In some cases the results obtained from a coarse uniform grid are more accurate than the results obtained from a finer block-centered irregular grid.