Mathematical Treatment Of The Miscible Displacement Problem
Authors: Oguztoreli M. , Ali S.M. Farouq 
Published: March 2010

Abstract
Miscible displacement or solvent flooding is the basis for a wide range of enhanced oil recovery methods, both thermal and non-thermal. From a mathematical standpoint, it is a difficult problem, and an improper treatment of the miscible phenomena in the simulations of various enhanced recovery methods can produce erroneous results. The purpose of this paper is to point out the difficulties of the problem, in particular for the two-dimensional case, and then offer recommendations for the solution schemes reviewed in this work.
For a two-dimensional geometry, the miscible displacement problem involves the solution of the combined convective diffusion and Darcy-continuity equations. The velocity field is implicitly dependent on the solvent concentration, while an apparent diffusion coefficient tensor includes the combined effects of molecular diffusion and directional velocity disperision. The resulting system of nonlinear equations can be solved by a number of methods, none of which are entirely satisfactory. Numerical dispersion and stability are of concern.
Two distinct approaches are used to solve the two dimensional convective-diffusion equation, for the general case, viz. fluids of different viscosities and densities. The first scheme involves a discretization of the convective term that results in a numerical dispersion term, which is included in the finite difference analog to eliminate a major portion of the truncation error. The second scheme utilizes a truncation error cancellation procedure, with weighted spatial averages. A third scheme based on the operator compact implicit method is reviewed for the linear problem. Direct solution is used in all schemes considered, in comparison to ADIP proposed by the previous investigators.
It is shown that the degree of convergence of the finitie differencesolution to the exact solution of the differential system is governed by the characteristic ratio of convection to diffusion rates (the Peclet number), and the time step size.
The paper presents a series of computational examples, showing the importance of the problem, and gives practical guidelines concerning the solution of this important problem, which enters into many recovery simulations.

Introduction
The convective-diffusion transport phenomenon describes the miscible displacement of oil by a solvent. Velocity dispersion enhances the diffusion process in both longitudinal and transverse directions of the gross fluid movement. The dispersion mechanism is intrinsic of the flow field and subsequently dependent on the local velocity direction. Dispersion is modelled by Fick's law with an apparent diffusion coefficient tensor of rank two to describe the inherent directional dependence.
Analytical solutions are not available for the general miscible displacement problem due to reservoir heterogeneity and the nonlinearity of the governing equations. Finite difference approximations applied to the two-dimensional problem gives the standard five-point difference formulas(ll). These formulas suffer at high Peclet numbers in their inability to track sharp concentration fronts on a coarse mesh. Upstream differences for the convective term introduce a first-order numerical dispersion error which may be several orders of magnitude larger than the physical dispersion term. Central-differences reduce the numerical dispersion at the expense of second order errors which generate nonphysical oscillations or overshoot.