A Weighted Nine-Point Finite-Difference Scheme For Eliminating The Grid Orientation Effect In Numerical Reservoir Simulation
Authors: Ko Stephen C.M. , Au Anthony D.K. 
Published: December 2007

Abstract
Previous attempts to solve the grid orientation problem in numerical modeling of pattern flood problem in numerical modeling of pattern flood performance by finite-difference techniques have met performance by finite-difference techniques have met with only limited success. In fact, none of the previously reported finite-difference methods were able previously reported finite-difference methods were able to solve this long-standing problem completely, especially for difficult examples which involved simulation of adverse mobility ratio piston-like displacements with small transition zones.
This paper describes a weighted nine-point finite-difference scheme for square grid networks which provides numerical solutions independent of grid orientation. The numerical solution obtained from the weighted nine-point scheme is a linear combination of two five-point finite-difference solutions with grid coordinates rotated at 45 degrees to each other. The proposed scheme has been successfully applied to unfavorable mobility ratio piston-like displacement problems with short transition zones which the previous finite-difference methods have failed to solve adequately.
Numerical experiments for shock mobility ratio ranging from 1 to 50 showed that the weighting factors used in the proposed method are a function of shock mobility ratio only, and are independent of grid block size and treatment of interblock mobility. Both the single-point upstream mobility method and the two-point upstream mobility method were implemented in the weighted nine-point simulator. As expected, the two-point upstream mobility method reduces numerical dispersion of the flood fronts as a result of lower spatial truncation error. It is also demonstrated that proper allocation of injection and production in the rotated grid network eliminates the production in the rotated grid network eliminates the "horn-like" displacement front reported in a previous investigation using a nine-point difference schemes.

Introduction
The grid orientation problem in numerical modeling of pattern flood performance has been a long-standing problem since it was first recognized in 1972 by Todd et all while developing improved methods for black oil simulations. They proposed a second-order two-point upstream mobility method which was adequate only for problems with about unit shock mobility ratio (Ms). Later in 1974, Coats et al reported severe grid orientation effects in steam flood simulations. They found that the numerical solutions for the two grid orientations did not converge as the grids were refined.
In 1975, Holloway et al presented a diagonal flow modification technique in conjunction with two point upstream mobility to reduce grid orientation effects. However, their results showed only marginal improvements over that achieved by the two-point upstream mobility method alone.
The use of curvilinear grid in an attempt to eliminate the grid orientation effect was investigated by Robertson and Woo. This method is difficult to extend to multi-irregular areal flood patterns. Furthermore, the convergence of the numerical solution was not substantiated in their work.
The application of a nine-point finite-difference approximation was first published in the petroleum reservoir simulation literature by Yanosik and McCracken. The method showed some improvements over the previous five-point finite-difference methods for reducing grid orientation effects in adverse mobility ratio (Ms less than 20) piston-like displacement problems. However, unusual "horn-like" or "bullet-like" displacement fronts were observed for more adverse mobility ratios (Ms greater than 20) and smaller saturation transition zones.
Recently some successes of eliminating the grid orientation effects were claimed by the use of variational and finite element methods.