Numerical Solution Of The Multi-Dimensional Buckley-Leveret Equation By A Sampling Method
Authors: Albright Norman , Concus Paul , Proskurowski Wlodzimierz 
Published: December 2007

Abstract
A method developed earlier for solving numerically the one-dimensional Buckley-Leverett equation for two phase immiscible flow in a porous medium is extended to the case of non-uniform flow in two space dimensions. The method has the feature of tracking solution discontinuities sharply for purely hyperbolic problems, without requiring devices such as the introduction of artificial dissipation. It is found that the method is computationally efficient for solving a numerical example for the fivespot configuration of water flooding of a petroleum reservoir.

Introduction
In the study of the simultaneous flow of two incompressible, immiscible fluids through a porous medium, one is led to a first-order nonlinear partial differential equation, the Buckley-Leverett partial differential equation, the Buckley-Leverett equation. When capillary pressure effects are small or absent, this equation is hyperbolic in nature. For the purely hyperbolic case (zero capillary pressure) it is well known that, in general, solutions develop discontinuities in finite time, even for smooth initial data. These discontinuities correspond to propagating fronts between the two fluids. If a small, but non-zero, amount of capillary pressure is present then the fronts that are developed will not be perfectly sharp, but will correspond to a large change in fluid saturation over a small, but non-zero, distance.
The representation of such discontinuities, or near discontinuities, usually causes difficulty for conventional numerical methods, which are based on differencing or other discretizations, because of underlying assumptions on the smoothness of solutions. These methods often require devices such as the introduction of artificial dissipation.
The purpose of our study is the development of a numerical method that is inherently capable of tracking sharp fronts in porous flow problems. This development is based on extending to the porous flow case the Chorin-Glimm random choice method, which was formulated originally for tracking solution discontinuities for the equations of gas-dynamics. The first stages of the extension are initiated in and are continued in the present paper.

THE SATURATION EQUATION
The Buckley-Leverett saturation equation for the flow of two immiscible, incompressible liquids through a homogeneous porous medium in the absence of capillary pressure and gravitational effects, in a region free of sources or sinks, is
(1)
where v = (vx,vy) is the total velocity, and s(x;t) and f(s) are respectively the saturation and fractional flow of the wetting liquid. (For convenience, the porosity (a constant) has been absorbed into the other variables and does not appear explicitly in (1).)
Typically f(s) has the S-shape shown in Figure 1. This shape, which consists of a convex and concave part joined by an inflection point, is fundamental in determining the behavior of solutions of (1). This behavior is qualitatively more complicated than that of solutions of the hyperbolic equations occurring in gas dynamics, for which there is no inflection in the function corresponding to f(s). The presence of an inflection permits the possibility of a juxtaposition of propagating discontinuities and expansion waves. Related consequences are reported in, where it is observed that some numerical methods may not give the correct solution of the limiting purely hyperbolic problem as the capillary pressure approaches zero.
BASIC NUMERICAL PROCEDURE