Comparison of Alternating-Direction and Successive Overrelaxation Techniques in Simulation of Reservoir Fluid Flow
Authors: Bjordammen J. , Coats K.H. 
Published: December 2007

Abstract
This study was performed to compare the capability and computing efficiency of successive overrelaxation (SOR) and alternating-direction (ADI) techniques in simulating pressure maintenance by water and gas injection. The calculations simulated two-phase flow and accounted for effects of capillary pressure, relative permeability, gravity and reservoir heterogeneity. The two techniques investigated were applied to the iterative, simultaneous solution of the two flow equations. Several variations of the SOR method were used: point (PSOR), point symmetric (PSSOR), line (LSOR) and line symmetric (LSSOR). The SOR methods were applied in simultaneous solution of the two partial difference equations describing the two-phase flow. Results showed that, for the oil-water simulation problems investigated here, the ADI iterative technique is superior to all variations of the SOR technique employing single relaxation factors. For all three oil-water problems the best single-value relaxation factor in the SOR technique was found to be unity. The total computing time required for simultaneous solution with ADI ranged from approximately 45 to 75 percent of that required using the best SOR technique, namely, LSOR, when the unity relaxation factor was employed in the latter technique. A significant improvement in the SOR computational requirements was obtained in the PSOR and LSOR simulation of one of the three oil-water problems-a 100 grid point two-dimensional simulation. The improved program, using combinations of relaxation factors, resulted in the reduction of LSOR computing requirements to approximately 94 percent of that required using ADI. Due to the relative complexity of the procedures involved in producing the improved SOR simulation programs, it was not considered feasible to apply these methods to the simulation of the other oil-water problems. Comparative results indicate that similar improvements in the LSOR simulation of the 300 and 625 grid point oil-water problems would still leave LSOR inferior to ADI on a computing time basis. In the simulation of a 100 grid point gas-oil cross-section, an optimized LSOR simulation using a number of relaxation factors required approximately 76 percent of the computing time that was used in the ADI simulation. The best LSOR run employing a single relaxation factor (w= 1.65) required approximately 83 percent of the ADI computing time. A satisfactory PSOR simulation of this problem could not be obtained.

Introduction
A variety of mathematical techniques are available for numerical solution of the partial differential equations governing multidimensional multiphase fluid flow in reservoirs. This work was performed to compare the capability and computing efficiency of two such techniques. The model (set of equations) employed simulates the three-dimensional, unsteady-state flow of two immiscible, incompressible phases and is applicable to pressure maintenance-type problems involving flank or pattern water injection or gas injection. The equations account for effects of gravity, capillarity, relative permeability and arbitrary reservoir geometry and heterogeneity. The model consists of two partial differential equations expressing conservation of mass of each flowing phase. The model equations were expressed in implicit finite difference form and solved simultaneously for the wetting and nonwetting phase flow potentials.