Mathematical Model of an Unstable Miscible Displacement
Authors: Dougherty E.L. 
Published: December 2007

DOUGHERTY, E.L., CALIFORNIA RESEARCH CORP., LA HABRA, CALIF. JUNIOR MEMBER AIME

Abstract
A phenomenological theory for a one-dimensional unstable miscible displacement similar in type to the Buckley-Leverett model but including the effects of mixing is proposed An equation giving the fractional flow of pure solvent through an oil phase containing dissolved solvent is derived including the effects of gravity and nonhomogeneities. Numerical integration of the resulting pair of nonlinear hyperbolic partial differential equations gives volumes of dissolved and undissolved solvent as functions of space and time. Parameters in the model which characterize mixing due to dispersion were determined from experimental data; the parameters correlated with viscosity ratio. The results indicate that the rate of dispersive mixing is proportional to volumetric flow rate Incorporated into our equations were Koval's heterogeneity factor H, which characterizes mixing due to channeling. This allowed satisfactory predictions of displacement behavior observed in horizontal floods in nonhomogeneous cores; but in most cases the values of H which we used were considerably larger than those used by Koval.

Introduction
It has been shown that for favorable viscosity ratios, the diffusion equation with convection satisfactorily describes the behavior of a miscible displacement in a porous media. Numerous attempts to develop a satisfactory mathematical description for the case of unfavorable viscosity ratio have been reported, but they have met with only partial success. These attempts, which are reviewed by Koval, have taken two approaches:development of a flow model akin to the Buckley-Leverett approach for immiscible displacement neglecting the effects of mixing andsimultaneous application of Darcy's Law for flow and the diffusion equation with a convection term for mass transfer.
We set out to construct a mathematical model of Type 1 including, though, the effects of mixing. Based upon a set of hypotheses, we derived a pair of non-linear hyperbolic partial differential equations which describe in a one-dimensional system the combined effects of flow and mixing. To work with these equations it was necessary to develop a fractional flow formula. The combined system was integrated numerically using the method of characteristics.In our equations the mixing process is characterized by four parameters. Three of these, labeled beta, p and p, account for dispersive type mixing. The fourth is the heterogeneity factor H, proposed by Koval to account for channeling due to nonhomogeneities in the porous media. Values of the parameters which would cause agreement between theory and experiment were determined by trial-and-error for miscible floods conducted in horizontal cores of both homogeneous and heterogeneous materials. Calculations were also performed for vertical floods in homogeneous cores.The purpose of this paper isto present the details of the mathematical analysis,to present the results of the calculations,to consider what light the results shed on the mixing process in an unstable miscible displacement, andto provide a firmer foundation for the correlation technique developed by Koval for predicting the behavior of unstable miscible floods.

STATEMENT OF PROBLEM
The problem is to construct a mathematical model which describes in one dimension the observed behavior in an unstable miscible displacement. The approach is phenomenological in that the equations are based on assumptions which violate certain physical precepts known to apply to the displacement phenomenon. However, the assumptions do allow us to account mathematically for the more essential phenomena. The work is of value if we can quantitatively predict experimental results which heretofore could not be predicted, even though the synthetic nature of the model belies complete explanation of the observations.We assume that the system is comprised of two contiguous flowing phases. One phase, which we call free solvent, has the properties of pure solvent. We call the fraction of the pore volume occupied by this phase the solvent saturation, designated s.
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