Evolution of microstructure in unstable porous media flow: A relaxational approach

We study the flow of two immiscible fluids of different density and mobilit y in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a meso scopic scale and the mixing zone grows in time-an example of evolution of m icrostructure. A simple set of assumptions on the physics of this two-phase flow in a porous medium leads to a mathematically ill-posed problem-when u sed to establish a continuum free boundary problem, We propose and motivate a relaxation of this "nonconvex" constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well-posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility qu otient. (C) 1999 John Wiley & Sons, Inc.