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Continuum Models of Two-Phase Flow in Porous Media

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Mathematical Modelling and Analysis of Complex Fluids and Active Media in Evolving Domains

I discuss two models of two-phase fluid flow in which undercompressive shock waves have been discovered recently. In the first part of the talk, the focus is on two-phase flow in porous media. Plane waves are modeled by the one-dimensional Buckley-Leverett equation, a scalar conservation law. The Gray-Hassanizadeh model for rate-dependent capillary pressure adds dissipation and a BBM -type dispersion, giving rise to undercompressive waves. Two-phase flow in porous media is notoriously subject to fingering instabilities, related to the classic Saffman-Taylor instability. However, a two dimensional linear stability analysis of sharp planar interfaces reveals a criterion predicting that weak Lax shocks may be stable or unstable to long-wave two-dimensional perturbations. This surprising result is related to the hyperbolic-elliptic nature of the system of linearized equations. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. In the second part of the talk, I describe a phase field model of a resident fluid being displaced by injected air in a thin tube (a microscopic pore). PDE simulations reveal the appearance of a rarefaction wave together with a faster undercompressive wave that terminates at the spherical cap tip of the injected air. Preliminary analysis and ODE simulations help to explain this structure.

This talk is part of the Isaac Newton Institute Seminar Series series.

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