Grid-Orientation Effects and the Use of Orthogonal Curvilinear Coordinates in Reservoir Simulation
Authors: Robertson G.E. 
Published: December 2007

Robertson, G.E., Woo, P.T., Members SPE-AIME, Chevron Oil Field P.T., Members SPE-AIME, Chevron Oil Field Research Co., La Habra, Calif.

Abstract
Grid orientation effects in reservoir simulation recently have received considerable attention. Several authors have proposed methods to reduce or to eliminate these effects. However, the proposed methods require reprogramming of simulators based on standard techniques. The reprogramming effort can be considerable if the numerical model is highly complex, such as in steamflood simulation. An orthogonal curvilinear coordinate system that essentially eliminates the problem of grid orientation was investigated. With no reprogramming, this computing grid can be used readily in existing simulators. Such grids were used to study pattern steamfloods and pattern waterfloods. The results are compared in detail with those obtained by using conventional Cargesian grids. Grid orientation is shown to have a more pronounced effect on the saturation fronts than oil recovery, whether in a steamflood or a waterflood. It is concluded that orthogonal curvilinear grids can be used easily to estimate pattern flood performance without modification of the solution method.

Introduction
In a simulation study of unfavorable mobility displacement, the areal orientation of the grid system, with respect to the location of the injectors and producers, can affect the calculated results. This is referred to as the "grid-orientation" effect. Fig. 1 illustrates two Cartesian grid orientations for one-eighth of a repeated five-spot pattern. The "parallel" orientation has a direct-flow pattern. The "parallel" orientation has a direct-flow path along the grid lines between the injector and path along the grid lines between the injector and producer, whereas the path is indirect in the producer, whereas the path is indirect in the "diagonal" system. Using a repeated five-spot pattern, Todd et al., Holloway et al., and Yanosik and McCracken investigated grid-orientation effects in gas or waterfloods. Coats et al. reported grid-orientation effects in steamfloods. In all cases, the investigators found that grid orientation significantly affected the movement of saturation fronts and the breakthrough time. Oil recovery was affected to a lesser extent. The effect appears to increase as the mobility ratio becomes more unfavorable and as the transition zone across the saturation front shortens. Todd et al. presented the two-point mobility approximation to alleviate the grid-orientation effect. Holloway et al. introduced a transmissibility modification scheme to reduce this effect further. While these two schemes can be introduced readily into simulators using explicit relative permeability, it is difficult to implement the schemes in simulators using implicit or semi-implicit relative permeability. Yanosik and McCracken observed that the grid-orientation effect probably was caused by the lack of flow from a grid block to its diagonal neighbors and introduced a nine-point finite-difference formula to account for this flow.
SPEJ
p. 13