Instability in Water Flooding Oil from Water -Wet Porous Media Containing Connate Water
Authors: Rachford H.H. 
Published: December 2007

Abstract
This work presents a first-order analysis of the instability underlying viscous fingering in adverse viscosity-ratio water floods. It extends previous analyses of frontal instabilities, which were carried out with equations for parallel plate models, by including effects of the saturation transition zone observed behind the front in water floods in water wet systems. This zone tends to insulate incipient fingers from the high-mobility water; thus conditions for the onset of fingering differ from those in the parallel plate theory. Finite-difference solutions of the two-dimensional equations of displacement in porous media exhibited the predicted stability characteristics in six hypothetical field- and laboratory-scale floods in rectangular reservoirs. In contrast to results with parallel plate systems, this paper concludes that for water-wet reservoirs, laboratory models scaled by the usual criteria are also correctly scaled for frontal instability. Further, fingering in the systems studied can occur in any saturation range behind the front, and may occur at an intermediate saturation even though stability obtains both at the saturation corresponding to the Buckley-Leverett front and near residual oil saturation. Other points of contrast are that the likelihood of occurrence of fingering may not increase as flow rate or viscosity difference increases, but may be sensitive to changes in the relative permeability and capillary pressure functions.

Introduction
The recovery of oil by water flooding frequently involves displacing the oil by water of a lower viscosity. Displacement of a fluid by a less viscous one may lead to gross channeling or fingering like that observed in solvent floods, in which it severely lowers recovery efficiency. In addition to adverse effects on recovery, it has been suggested that the unstable movement causing fingering may interfere with interpretation of scaled model studies of proposed water floods, since the instability in the model might not be faithfully scaled to that in the reservoir prototype. In view of the serious implications of this possible breakdown of widely used scaled model techniques, it is the purpose of this paper to examine the question further. Instabilities in the solutions of systems of differential equations imply a loss of smooth dependence on initial and boundary conditions. Thus, the possibility exists that in using models whose scaling is based on the differential system there may arise size- and rate-dependent factors which are not properly scaled. This possibility was examined in detail by Chouke et al., who analyzed the instability of frontal advance in a related problem, water-oil displacement in parallel plate models, in which a moving interface separates two regions of constant, unequal mobilities. First-order perturbation theory predicts the existence of a critical wave length for the growth of perturbations:
and a wave length of maximum instability of .
The interpretation is that wave lengths in a perturbation which are longer than will grow. Thus, if the width of a two-dimensional channel is greater than, fingers will grow, and the spacing of the fingers which grow at the maximum rate will be approximately. It is important that the higher the velocity and/or the difference in flow resistance, the lower is, and thus the greater the number of fingers that can grow in a given model. In applying these conclusions to porous media, a pseudo-interfacial tension, was assumed for the invasion front. Since this would not necessarily be equal to the liquid-liquid interfacial tension, an unknown constant was substituted for in the foregoing expression for.
JPT
P. 133ˆ