Evolution of Microstructure in Porous

We study the ow of two immiscible uids of diierent density and mobility 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 mesoscopic scale and the mixing zone grows in time | an example of evolution of microstructure. A simple set of assumptions on the physics of this two{phase ow in a porous medium leads to a mathematically ill{posed problem | when used 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 which predicts shape and evolution of the mixing proole as a function of the density diierence and mobility quotient. 1 Introduction We are interested in the ow of two immiscible uids of diierent density and mobility in a porous medium. If the more dense phase lies above the less dense one or if the more mobile phase displaces the other, the interface is observed to be unstable (we will focus on the density driven instability here). The two phases start to mix on a mesoscopic scale and the mixing zone grows in time (see for instance 16, p.261{p.267] and and 9]) | an example of evolution of microstructure. The two{phase ow between closely spaced parallel sheets of glass (the Hele{Shaw cell) can to some extend be considered an experimental simulation of a two{dimensional two{phase ow in a porous medium 13] | without simulating the inherent \stochasticity" of a porous medium on the microscopic scale. It seems natural to model the ow of two immiscible uids in a Hele{Shaw cell by a two{dimensional free boundary problem for the interface 13]. It seems more naive to model the ow in a porous medium by the same free boundary problem as it does not attempt to describe the observed mixing on a mesoscopic scale in an eecient way. Nevertheless, this model will be our starting point.