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Evolution of microstructure in unstable porous media flow: A relaxational approach. (English) Zbl 0929.76136
Summary: We study the flow of two immiscible fluids of different 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 flow in 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 that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient.

MSC:
76S05 Flows in porous media; filtration; seepage
35R25 Ill-posed problems for PDEs
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