Linear Stability of Filtration Flow of a Gas and Two Immiscible Liquids with Interfaces

The stability of the vertical flow that occurs when gas displaces oil from a reservoir is investigated. It is assumed that the oil and gas areas are separated by a layer saturated with water. This method of oil displacement, called water-alternating-gas injection, improves the oil recovery process. We consider the linear stability of two boundaries that are flat at the initial moment, separating, respectively, the areas of gas and water, as well as water and oil. The instability of the interfaces can result in gas and water fingers penetrating into the oil-saturated area and causing residual oil. Two cases of perturbation evolution are considered. In the first case, only the gas–water interface is perturbed at the initial moment, and in the second case, small perturbations of the same amplitude are present on both surfaces. It is shown that the interaction of perturbations at interfaces depends on the thickness of the water-saturated layer, perturbation wavelength, oil viscosity, pressure gradient and formation thickness. Calculations show that perturbations at the oil–water boundary grow much slower than perturbations at the gas–water boundary. It was found that, with other parameters fixed, there is a critical (or threshold) value of the thickness of the water-saturated layer, above which the development of perturbations at the gas–water boundary does not affect the development of perturbations at the water–oil boundary.