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

✍ Scribed by Felix Otto

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 268 KB
• Volume: 52
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199907)52:7<873::aid-cpa5>3.0.co;2-t

✦ Synopsis

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 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 that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient.