On the Linear Stability Problem for a Three-Layer Displacement in a Porous Media

Abstract

In this paper, we study the secondary oil recovery process. The oil from a porous reservoir at low pressure is pushed by a forerunner, less viscous fluid (a polymer solute). Then the well-known Saffman-Taylor instability appears. Some authors tried to minimize this instability by using a succession of intermediate liquids with constant viscosities - the multi-layer model. The surface tensions on the interfaces between liquid layers are a stabilizing factor. In some previous papers, we proved some contradictions of this multi-layer model. However, we considered that the corresponding stability problem has a solution. This model's first step (and the mathematical basis) is the three-layer model, with a single intermediate liquid. We prove that the linear stability problem for the three-layer model has no solution (in general) - the growth rates of perturbations may not exist. On the contrary, an intermediate liquid with a suitable variable viscosity can almost suppress the Saffman-Taylor instability, even if the surface tensions are missing [17].