Asymptotic Regimes in Miscible Displacements in Random Porous Media
Authors: Yang Z.M. , Yortsos Y.C. 
Published: December 2007

Abstract
We study the asymptotic behavior of miscible displacements in porous media in the two limits, where a permeability-modified aspect ratio, RL = L/H(kV/kH)1/2, becomes large or small, respectively. The first limit is known as Vertical Equilibrium (VE), while the second leads to the Dykstra-Parsons (DP) problem. In either case, the problem reduces to the solution of a single integro-differential equation. It is shown that the VE and DP regimes are at the two opposite limits of the parameter RL, although the two coincide in the special case M = 1. This should clarify the relation between the two regimes, which at present is somewhat confusing (Lake, 1989). By comparison with High Resolution Simulation (HRS), we investigate the validity of these approximations. For the case of unstable displacement, we show that the evolution of transverse averages, hence of the corresponding Viscous Fingering (VF) models, depend on RL. We investigate the development of a VF model to describe viscous fingering in weakly heterogeneous porous media under VE conditions, and compare with the various existing empirical models (such as the Koval, Todd-Longstaff and Fayers models).

Introduction
Two different displacement regimes, the Vertical Equilibrium (VE) and the Dykstra-Parsons (DP) regimes, have been used to model displacements in porous media. Vertical equilibrium is a commonly made assumption to describe fluid displacement. (A more accurate terminology should be Trans- verse Equilibrium (TE), which will be followed here, since gravity is actually not necessary for its application). The parameter RL = L/H(kv/ksr)1/2 was suggested as the controlling factor for a system to approach TE, and was analytically shown to be so by Yortsos. In this regime, because of the large aspect ratio, transverse (vertical) flow is extensive. The Dykstra-Parsons (DP) approach is used to describe processes when there is no transverse flow (for example, in case of non-communicating layers). The use of either TE or DP leads to a considerable simplification due to a reduction in the number of primary variables, and facilitates the development of average models and scale-up. This reduction is of interest to coarse-gridding in reservoir simulation.
Although the two regimes have been used extensively, the range of their applicability remains unclear, and even confusing. This is particularly the case for unstable displacements and viscous fingering problems. It is one of the objectives of this paper to delineate the region of validity of these two regimes. We proceed with an asymptotic analysis of the full problem in the two limits RL&gt;&gt;1 and RL&lt;&lt; 1 and show that the TE and DP regimes are obtained in the respective limits. In both cases, the problem is reduced to the solution of a single integro differential equation. For the TE case, the limit obtained is akin to to the lubrication approximation in viscous flows. To study the validity of these asymptotic approximations, we investigate the solution of the full problem using RL as a parameter. In the full model, both pressure and concentration fields are solved. We use standard methods of High Resolution Solution (HRS) to study displacement patterns as well as the behavior of transverse averages in concentration and mobility, needed for the assessment and development of viscous fingering models. The analysis is for 2-D geometries and for the miscible problem only. Extension to other processes, including immiscible processes and gravity effects, is straightforward, but it will not be reported here.