Two-phase Flow in Porous Media: Sharp Fronts and the Saffman-Taylor Instability

Start: 03/28/2012 - 1:15pm
End  : 03/28/2012 - 2:15pm

Applied Math Seminar

Michael Shearer (NCSU)


Plane waves for two phase flow in a porous medium are modeled by the one-dimensional Buckley-Leverett equation, a  scalar conservation law.  In the first part of the talk, we study traveling wave solutions of the equation modified by the Gray-Hassanizadeh model for rate-dependent capillary pressure. The modification adds a BBM-type dispersion to the classic equation, giving rise to undercompressive waves.  

In the second part of the talk, we analyze stability of sharp planar interfaces  (corresponding to Lax shocks)  to two-dimensional perturbations, which involves a  system of partial differential equations. The Saffman-Taylor analysis predicts instability of planar fronts, but their calculation lacks the dependence on saturations in the Buckley-Leverett equation. Interestingly, the  dispersion relation  we derive leads to the conclusion that some interfaces are long-wave stable  and  some  are not. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion,  verify   the stability predictions at the hyperbolic level.
This is joint work with Kim Spayd and Zhengzheng Hu.


RN 103