A Nine-Point, Finite-Difference Reservoir Simulator for Realistic Prediction of Adverse Mobility Ratio Displacements
Authors: Yanosik J.L. , McCracken T.A. 
Published: December 2007

Abstract
Reservoir simulators based on five-point difference techniques do not predict the correct recovery performance for unfavorable mobility-ratio, piston-type performance for unfavorable mobility-ratio, piston-type displacements. For a developed five-spot pattern, the predicted performance depends on the grid orientation predicted performance depends on the grid orientation (parallel or diagonal) used. This paper discusses the development and testing of a nine-point, finite-difference reservoir simulator. Developed five-spot-pattern flood predictions are presented for piston-type displacements predictions are presented for piston-type displacements with mobility ratios ranging from 0.5 to 50-0. We show that the predicted fronts are realistic and that very little or no difference exists between the results of parallel and diagonal grids. The maximum difference in the recovery curves is less than 1.5 %.
The nine-point-difference method is extended to any grid network composed of rectangular elements. Results for two example problems - a linear flood and a direct line-drive flood - indicate the extension is correct.
The techniques discussed here can be applied directly in the development of any reservoir simulator. We anticipate that the greatest utility will be in the development of simulators for the improved oil-recovery processes that involve unfavorable mobility ratio processes that involve unfavorable mobility ratio displacements. Examples are miscible flooding, micellar/ polymer flooding (water displacing polymer), and direct polymer flooding (water displacing polymer), and direct steam drive.

Introduction
Miscible displacement oil-recovery methods often are characterizedby a large viscosity ratio between the oil and its miscible fluid andby a very low immobile oil saturation behind the displacement front.
These conditions represent an unfavorable mobility-ratio, piston-type displacement. They differ from a conventional piston-type displacement. They differ from a conventional gas drive, where a substantial mobile oil saturation remains behind the displacement front.
Reservoir simulators based on five-point, finitedifference techniques do not predict the correct performance for unfavorable mobility-ratio, piston-type performance for unfavorable mobility-ratio, piston-type displacements. Results of an areal simulation for a developed five-spot flood depend on the grid orientation (diagonal or parallel, Fig. 1). Grid orientation significantly influences the predicted recovery performance and displacement front positions. performance and displacement front positions. A nine-point, finite-difference reservoir simulator is described. Predictions of piston-type displacements in a developed five-spot pattern are presented for mobility ratios ranging from 0.5 to 50. We show that the predicted fronts are realistic and that very little or no predicted fronts are realistic and that very little or no difference exists between the results of parallel and diagonal grid orientations.
A formulation of the nine-point, finite-difference technique applicable to any rectangular grid network is presented. Results for two example two-dimensional presented. Results for two example two-dimensional problems, a linear flood, and a direct line-drive flood problems, a linear flood, and a direct line-drive flood indicate that the formulation is correct for nonsquare grid networks.

Background
Grid-orientation effects for five-point reservoir simulators were demonstrated by Todd et al. They studied two developed five-spot grid systems - a diagonal grid and a parallel grid. These grid systems are shown in Fig. 1. parallel grid. These grid systems are shown in Fig. 1. The diagonal grid represents a quarter of a five-spot pattern, with grid lines at 45 degrees to a line connecting the pattern, with grid lines at 45 degrees to a line connecting the injector and producer. The parallel grid represents one-half of a five-spot pattern, with grid lines either parallel or perpendicular to the lines connecting the parallel or perpendicular to the lines connecting the injector-producer pads.
SPEJ
P. 253