Numerical Solutions of the Equations for One-Dimensional Multi-Phase Flow in Porous Media
Authors: Gottfried Byron S. , Guilinger W.H. , Snyder R.W. 
Published: December 2007

Abstract
Two numerical methods are presented for solving the equations for one-dimensional, multiphase flow in porous media. The case of variable physical properties is included in the formulation, although gravity and capillarity are ignored. Both methods are analyzed mathematically, resulting in upper and lower bounds for the ratio of time step to mesh spacing.
The methods are applied to two- and three-phase waterflooding problems in laboratory-size cores, and resulting saturation and pressure distributions and production histories are presented graphically. Results of the two-phase flow problem are in agreement with the predictions of the Buckley-Leverett theory. Several three-phase flow problems are presented which consider variations in the water injection rate and changes in the initial oil- and water-saturation distributions. The results are different physically from the two-phase case; however, it is shown that the Buckley-Leverett theory can accurately predict fluid interface velocities and displacing-fluid frontal saturations for three-phase flow, providing the correct assumptions are made.
The above solutions are used as a basis for evaluating the numerical methods with respect to machine time requirements and allowable time step for a fixed mesh spacing.

Introduction
Considerable progress has been made in recent years in obtaining numerical solutions of the equations for two-phase flow in porous media. Douglas, Blair and Wagner2 and McEwen11 present different methods for solving the one-dimensional case for incompressible fluids with capillarity (the former using finite differences, the latter with an approach based upon characteristics). Fayers and Sheldon4 and Hovanesian and Fayers8 have extended these studies to include the effects of gravity. West, Garvin and Sheldon,14 in a pioneer paper, treat linear and radial systems with both capillarity and gravity and they also include the effects of compressibility. Douglas, Peaceman and Rachford3 consider two-dimensional, two-phase, incompressible flow with gravity and capillarity and Blair and Peaceman1 have extended this method to allow for compressible fluids. No one, however, has examined the case of three-phase flow, even for the relatively simple case of one-dimensional flow of incompressible fluids in the absence of gravity and capillarity.
In obtaining a numerical technique for simulating forward in situ combustion laboratory experiments, Gottfried5 has developed a method for solving the one-dimensional, compressible flow equations with any number of flowing phases. Gravity and capillarity are not included in the formulation. The method has been used successfully, however, for two- and three-phase problems in a variable-temperature field with sources and sinks.
This paper examines the algorithm of Gottfried more critically. Two numerical methods are presented for solving the one-dimensional, multi-phase flow equations with variable physical properties. Both methods are analyzed mathematically, and are used to simulate two- and three-phase waterflooding problems. The numerical solutions are then taken as a basis for comparing the utility of the methods.

Problem Statement
Consider a one-dimensional system in which capillarity, gravity and molecular diffusion are negligible. If n immiscible phases are present, n 2, the equation describing the flow of the ith phase is:12Equation 1
where all terms can vary with x and t.