Multidimensional Displacement of a Viscous Phase by a Non-Viscous Phase
Authors: Modine Alan D. , Coats Keith H. 
Published: December 2007

Abstract
A mathematical model has been formulated for simulating three-dimensional displacement of a viscous fluid by a displacing fluid of zero viscosity. The model has been incorporated into a FORTRAN IV computer program for application in low-rate, high-permeability systems. Where applicable, the zero-viscosity program reduces computer time by a factor of 5 to 10 relative to conventional two- and three-dimensional programs.
To determine the area of applicability, a gas-oil cross-section model representation of a high-dip, high-permeability reservoir was simulated with the zero-viscosity and conventional two-dimension programs for a range of flow rates up to 80 percent programs for a range of flow rates up to 80 percent of the critical rate. In comparing the two solutions, the conventional one was assumed to be the correct one because its program is based upon a more physically realistic model than that of the physically realistic model than that of the zero-viscosity solution. The two solutions agreed at rates up to 50 percent of the critical; at 80 percent they disagreed significantly. This indicates percent they disagreed significantly. This indicates that the zero-viscosity model, which is quite simple and inexpensive to apply, can be used with accuracy at rates up to at least 50 percent of the critical. This area of applicability is important in improving computational capability, for it is at these lower rates that the conventional programs are excessively costly. At the higher rates, where the zero-viscosity solution is not accurate, the conventional programs are easy and economical to apply.
The zero-viscosity model accounts for capillary and gravitational forces, effects of viscosity and relative permeability for the displaced phase, and arbitrary reservoir heterogeneity. The program handles up to 1,800 blocks on an in-core basis.

Introduction
Computational difficulties caused by slow or metastable convergence in gas-oil calculations using conventional two-phase reservoir simulation programs have been correlatable with the effects of programs have been correlatable with the effects of low viscosity in the gas phase. In many such problems, a very small deviation in the calculated problems, a very small deviation in the calculated flow potentials causes a large deviation in the calculated gas flow due to the low viscosity. Thus, the program is trying to converge on a small variation in potential, which makes the computations difficult.
A previous method of overcoming this difficulty has been to introduce in the conventional two-phase calculations an artificial resistance to gas flow; this method causes a more significant variation in the calculated flow potential. This paper describes a new method for treatment of gas-oil problems in which a zero-viscosity gas phase is used. Both methods are based on the assumption that oil mobility is the controlling factor in the displacement and that the behavior is insensitive to gas mobility over a relatively wide range. We show that the two methods give identical results, and since the correct gas mobility is bracketed by the two methods, we may conclude that either method gives valid results for low rate displacements. The chief advantage of the zero-viscosity program is lower computing costs. This report presents a mathematical description of the zero-viscosity model and compares saturation distributions calculated for several typical problems using the zero-viscosity and conventional two-phase programs.

ZERO-VISCOSITY MODEL
The zero-viscosity model simulates the immiscible displacement of a viscous fluid by a displacing fluid of zero viscosity. The method includes the effects of capillary and gravitational forces, relative permeability and viscosity in the displaced phase, permeability and viscosity in the displaced phase, and arbitrary reservoir heterogeneity.
SPEJ
P. 325