Displacement Stability of Water Drives in Water-Wet Connate-Water-Bearing Reservoirs
Authors: Hagoort J. 
Published: December 2007

Abstract
We studied the displacement stability of the water-drive process in a water-wet connate-water-bearing reservoir, as described by the Buckley-Leverett displacement model. The reservoir bas a simple, two-dimensional canal-like geometry. We considered the viscous and capillary forces, but excluded gravitational effects. The following stability criterion holds. The displacement is unstable if the mobility ratio of the fluids behind and ahead of the shock front is greater than 1, provided the wavelength of the instabilities is smaller Man the canal width. This shock mobility ratio is always smaller than the end-point mobility ratio, which characterizes stability of the Muskat type of displacement. We therefore consider the stability of Buckley-Leverett displacement to be better than that of the Muskat type of displacement. Capillary forces determine the wavelength of the instabilities, and the wavelength that predominates is the one that contributes the maximum to the energy dissipation. This wavelength was calculated for a first-order perturbation; it is proportional to the dimensionless number giving the ratio of capillary forces to viscous forces, and is also a function of the characteristics of the porous medium. If this wavelength is much greater than the canal width, the displacement behaves as a stable one. These results were verified by experiments carried out in a well defined transparent flow model. In agreement with our stability consideration, both stable and unstable displacements were found, and the predicted wavelengths were of the same order of magnitude as those observed.

Introduction
An important factor determining the efficiency of the water-drive precess is its displacement stability. For displacements according to the Muskat model, in which oil and water flow in separate macroscopic regions, and gravitational forces are absent, Chuoke et al. determined that a necessary condition for instability is that the mobility of the displacing water be higher than that of the displaced oil. Chuoke et al. also accounted for capillary effects by defining an effective interfacial tension between the fluids in the porous medium analogous to the interfacial tension in the bulk fluids. It then appeared that capillary forces determine the wavelength of the instabilities. The above results hold for displacements in neutral-wet media in which oil and water do flow in separate regions. For displacements in water-wet connate-water-bearing media, where, in general, oil and water flow simultaneously, the Muskat model no longer holds and should be replaced by the Buckley-Leverett model, as the latter describes displacement in terms of saturation-dependent permeabilities and capillary pressures. The first attempt to tackle the stability problem for the Buckley-Leverett model was carried out by Rachford using a simplified perturbation theory and numerical calculations. However as the nonlinearity of the perturbation equations was violated and the consequences are incalculable, his results are questionable. In his numerical calculations he solved the pertinent differential equations while introducing instabilities via randomly distributed permeability variations. For relatively low end-point permeability variations. For relatively low end-point mobility ratios he found no severe instabilities. However, the number of cases he studied were too few for general conclusions to be drawn from them, in particular on the effect of capillarity. Croissant studied the problem of stability in water-wet media through physical experiments. He showed that the width of the instabilities increases with increasing capillary forces. Qualitatively he ascribed this width to concurrent imbibition at the flanks of the instability. Despite these studies, we feel that a satisfactory description of the stability of the water-drive process in water-wet media has still to be achieved. This paper presents an attempt. The problem has been paper presents an attempt. The problem has been kept as simple as possible; it has been restricted to displacements in a two-dimensional, canal-like reservoir in which gravitational forces are excluded. The approach is also simple in that we used a rather stylized picture of the displacement process. We further treated the effect of capillarity by using a simple energetic argument.
SPEJ
P. 63