An Analytical Model Of One-Dimensional Miscible Displacement In Porous Media
Authors: Jankovic M.S. 
Published: March 2010

Abstract
A mathematical model having an analytical solution that is valid both for stable and unstable one-dimensional first-contact miscible displacement in homogeneous and heterogeneous porous media has not been reported in the literature. This paper's main objective is to present such a definitive model. This model is described by the diffusion-convection equation for which an approximate parametric solution is found to be a one parameter family of curves.
Furthermore, simple formulae for cumulative oil recovery, solvent effluent concentration and oil cut are derived and then successfully matched with numerous published experimental data. All the matches are excellent and definitely superior to those obtained from alternative models. Many new results are obtained and some conclusions from previous investigators corroborated.

Introduction
The first contact miscible displacement process has been discussed in many excellent texts, and will not be reviewed here(3,12,14). Instead, the mathematical simulation of this process will be discussed.
With varying degrees of success(3,5,7,12), a number of different mathematical models have been used to describe one-dimensional first-contact stable miscible displacement which occurs for a favorable solvent oil mobility ratio (M &lt; 1) in homogeneous porous media. The solutions of the partial differential equations governing this displacement are either numerical or analytical in form. In the latter case, they are given in terms of the error function for a variety of initial and boundary conditions(3,5,7,12,13). However, the utility of these solutions is limited because there is often a lack of agreement between these solutions and experimental data.
Miscible displacement in heterogeneous media is much more difficult to model because it is less tractable physically, and invariably does not lend itself to a simple mathematical formulation. This formulation is further complicated in the case of unstable miscible displacement (M&gt; 1) which is accompanied by viscous fingering. No mathematical model which can accurately describe this problem and has an analytical solution has been reported. The only known model to have an analytical solution(11) (Koval's Method) is an analogy of the immiscible displacement theory of Buckley and Leverett. Indeed, Koval's formula for fractional solvent flow and oil recovery are identical to those derived for unstable immiscible displacement and commonly used in waterflood calculations(8), provided that Koval's factor K is substituted by the water-oil mobility ratio M. Thus, his solution satisfies the 1-D immiscible equation and not the diffusion-convection equation. This approach cannot be accurate because it neglects the important miscible phenomena of diffusion and dispersion. Nevertheless, Koval obtained satisfactory agreement with experimental data from miscible displacement tests in which diffusion and dispersion had little influence on the formation of viscous fingering(14). When this influence is not negligible, which was the case for part of the data given in Koval's paper, his method fails. Furthermore, it cannot be used to match time-dependent data; thus, its application is quite limited. The model presented here will be shown to be a superior alternative to Koval's model and all other known models.