Viscous-fingering minimization in uniform three-dimensional porous media

Abstract

In this paper, we consider a radial displacement of a viscous fluid by another one of much lower viscosity through a three-dimensional uniform porous medium. It is well known that when a less viscous fluid is pumped at a constant injection rate, very complex interfacial patterns are formed. The control and eventual suppression of these instabilities are relevant to a large number of areas in science and technology. Here, we use a variational approach to search for an analytical form of an optimal flow rate so that the interface between two almost neutrally buoyant fluids grows, but the emergence of interfacial disturbances is minimized. We find a closed analytical solution for the ideal flow rate which surprisingly does not depend on either the properties of the fluids or the permeability of the porous medium.