An Approximate Model With Physically Interpretable Parameters for Representing Miscible Viscous Fingering

Abstract

Summary. An approximate viscous fingering model is derived with physically related parameters. Plausible assumptions are made physically related parameters. Plausible assumptions are made about some limits on finger fluid mixing, and the use of a fingering function is suggested. It is shown that for the horizontal linear problem, a hyperbolic partial-differential equation governs the problem, a hyperbolic partial-differential equation governs the solvent fractional flow behavior. Satisfactory agreement with classic miscible displacement experiments is demonstrated for a reasonable choice of parameters. when gravity is introduced into the equation, the same model gives adequate agreement with the rate dependency observed in a vertical displacement experiment. Recommendations are made concerning extension of the model for use in three-dimensional (3D) compositional simulations. Introduction It has long been recognized that miscible displacement with a low-viscosity fluid driving a more viscous oil will be an unstable process leading to development of viscous fingers. Although process leading to development of viscous fingers. Although laboratory experiments (see for example Perkins et al.) and perturbation theory have indicated the manner in which fingers perturbation theory have indicated the manner in which fingers will be initiated and grow, there is little direct evidence that fully developed finger growth will necessarily have reproducible stochastic properties on a macroscopic scale. Nevertheless, there remains an expectation that the fingering process should be amenable to being represented in a deterministic model whose parameters should be related to specific aspects of the physical phenomena involved. The position can be compared with the development of turbulence theory in ordinary fluid flow, although currently the equivalent analysis methods are much less developed for unstable miscible displacement in porous media. Two empirical models have been suggested by Koval and Todd and Longstaff to give a basis for computation of miscible displacement. Both models suffer from adoption of empiricism in which the principal parameters involved have little or no direct physical significance. The parameters can be fitted to simple one-dimensional (1D) laboratory experiments, for example, but translation of the same parameter values to a 3D reservoir is an uncertain undertaking. The work of Dougherty represented an attempt to set up a deterministic model with simplifying approximations and physically significant parameters, but the need to fit three different "diffusion" constants with rather large variations between experiments has not encouraged use of this model for general computation. The present paper examines a different and possibly more plausible set of simplifying assumptions than those adopted by plausible set of simplifying assumptions than those adopted by Dougherty, which results in a single hyperbolic equation in this analysis. The new model allows self-consistent approximations to be made for spatial variations of fingered fluid viscosities and densities that are necessary for calculating gravitational segregation rates. Suggestions are made for generalization of the model to 3D applications, which observe the vectorial nature of the fingering phenomenon. Because the basic physical approximations in this case phenomenon. Because the basic physical approximations in this case can be translated consistently to the 3D case, this model should be capable of giving more dependable results in field applications. Koval and Todd and Longstaff Models and Their Physical Limitations The Koval method assumes that the solvent fractional-flow behavior can be treated on a basis similar to immiscible displacement theory, with the relative permeabilities being proportional to solvent and oil volume fractions, respectively. It is assumed that there is some average effective solvent viscosity, independent of local solvent concentrations, that should be used in the mobility terms. Thus, FsK = ..............................(1) (1-C) 1 + C Koval was able to fit the effluent concentration behavior from a number of miscible displacement experiments by assuming = 0.22 and a one-fourth-power viscosity mixing law to give = + .................................(2) It is surprising that the results of this method do appear to give good agreement with measurements for a range of viscosity ratios using the fixed expression for Ms' but there is no physical justification for this success. The method provides no definition of the total effective mobility of the combined flow, which is needed for the pressure equation in a numerical simulator, nor is it clear that the effective solvent density to be used when gravitational segregation is to be computed should also be based on =0.22. The Todd and Longstaff method is based on somewhat similar assumptions, with relative permeabilities proportional to the respective component volume fractions; but in this case, effective viscosities of both solvent and oil are modified by use of the mixing rules, = ..................................(3) and .................................(4) where w is a mixing parameter and = + ............................(5) This then gives the fractional flow equation: FsTL = ..............................(6) SPERE P. 551