Numerical Calculation of Multidimensional Miscible Displacement
Authors: Peaceman D.W. , Rachford H.H. 
Published: December 2007

Abstract
A system of partial differential equations describing miscible displacement of fluids in porous media is derived. The system takes into account the influence of gravity, the spatial distribution of permeability, diffusion, and fluid viscosities and densities. A numerical procedure for approximating solutions to the differential systems has been tested for a horizontal two-dimensional geometry. In end-to-end displacements of oil with less-viscous solvents, the numerical solutions exhibited fingering qualitatively similar to that observed in laboratory models. Small random spatial variations in permeability about the mean value are sufficient to initiate fingering. Quantitative comparisons of computed results with laboratory data show good agreement.

Introduction
Miscible flooding of oil by solvent is receiving increasing consideration for field use. Unfortunately, such floods potentially present severe problems in loss of recovery through the by-passing of oil by expensive solvents. Consequently, their economic evaluation requires sound techniques for predicting recovery. The purposes of this work are to present a finite-difference method for calculating the multidimensional displacement of oil by solvent and to investigate the validity of the method by comparing results of calculations with data from displacements in laboratory models. The formulation of the method to simulate the model experiments treats a case of limited scope in the description of solvent flooding in the reservoir. The model experiments were carried out with fluids that were assumed to form an incompressible, ideal, two-component system with constant diffusivity. Establishing the validity of the method even in its present form provides a major step toward the goal of quantitatively evaluating individual solvent flooding projects. First, it demonstrates the feasibility of calculating the course of, displacements which are dominated by an inherent macroscopic instability, i.e., viscous fingering. In addition, the method provides almost the only practical means of examining the effect of the size and extent of reservoir inhomogeneities on the development and propagation of the fingers, taking into account the important influences of diffusion and gravitational segregation. It might be inferred that such studies cannot yield results of practical value because in displacements dominated by macroscopic instability the pattern of finger development and the resulting performance should depend critically on small variations in the geometric distribution of reservoir inhomogeneities. The present method offers the capability inherent in computational techniques of predicting performance reproducibly with any arbitrary distribution of reservoir properties and, as such, provides a means of evaluating the sensitivity of behavior to uncertainties in reservoir definition and, thus, assessing the reliability of prediction of performance.

THE PHYSICAL PROBLEM

DATA USED FOR QUANTITATIVE TESTS OF CALCULATIONS
In this work, a calculation for treating the miscible-displacement process is tested by comparing calculated results with experimental data. The experiments chosen for comparison have been described in detail, and the quantitative features are summarized in a later section. Briefly, oil was flooded by solvent of equal density from a thin rectangular channel in Lucite packed with uniform Ottawa sand. Under these conditions, two-dimensional geometry is considered adequate to represent the process. In as much as the present work is concerned with testing of the method by comparison with specific two-dimensional data, the system of equations presented in the succeeding section will be oriented to defining the physical system specifically.

THE DIFFERENTIAL EQUATIONS
Suppose x, y to be a Cartesian coordinate system and define h(x, y) to be the height of a point above a horizontal reference plane.
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