Small- and Waiting-Time Behavior of a Darcy Flow Model with a Dynamic Pressure Saturation Relation

We address the small-time evolution of interfaces (fronts) for the pseudoparabolic generalization \[ {\partial u\over \partial t} = {\partial\over \partial x} \left( u^\alpha {\partial u\over \partial x} + u^\beta {\partial^2 u \over \partial x \partial t} \right) \] of the porous-medium equation, identifying regimes in which the local behavior remains fixed for some finite time and others in which it changes instantaneously. A number of phenomena beyond those exhibited by the porous-medium equation are elucidated, including retreating fronts and novel types of local behavior. Related results for the important limit case \[ {\partial u\over \partial t} ={\partial\over \partial x}\left(u^\beta {\partial^2 u \over \partial x \partial t} \right) \] are also described.

[1]  Josephus Hulshof,et al.  Infiltration in porous media with dynamic capillary pressure: travelling waves , 2000, European Journal of Applied Mathematics.

[2]  G. I. Barenblatt,et al.  A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow , 1993 .

[3]  John R. King,et al.  Asymptotic results for the Stefan problem with kinetic undercooling , 2000 .

[4]  Michael A. Celia,et al.  Dynamic Effect in the Capillary Pressure–Saturation Relationship and its Impacts on Unsaturated Flow , 2002 .

[5]  D. Aronson.  The porous medium equation , 1986 .

[6]  William G. Gray,et al.  Thermodynamic basis of capillary pressure in porous media , 1993 .

[7]  John R. King,et al.  The selection of Saffman-Taylor fingers by kinetic undercooling , 2003 .

[8]  W. Kath,et al.  Waiting‐Time Behavior in a Nonlinear Diffusion Equation , 1982 .

[9]  Alain Bourgeat,et al.  Effective two-phase flow through highly heterogeneous porous media: Capillary nonequilibrium effects , 1998 .

[10]  C. M. Elliott,et al.  On the Cahn-Hilliard equation with degenerate mobility , 1996 .

[11]  J. Vázquez,et al.  Persistence of corners in free boundaries in Hele-Shaw flow , 1995, European Journal of Applied Mathematics.

[12]  G. I. Barenblatt,et al.  A mathematical model of turbulent heat and mass transfer in stably stratified shear flow , 1993, Journal of Fluid Mechanics.

[13]  C. M. Elliott,et al.  The viscous Cahn-Hilliard equation , 2002 .

[14]  J. M. Oliver,et al.  Thin-film modelling of poroviscous free surface flows , 2005, European Journal of Applied Mathematics.

[15]  R. E. Grundy.  Local Similarity Solutions for the Initial-value Problem in Non-linear Diffusion , 1983 .

[16]  G. I. Barenblatt,et al.  Mathematical model of the non equilibrium water oil displacement in porous strata , 1997 .

[17]  Luis A. Caffarelli,et al.  How an Initially Stationary Interface Begins to Move in Porous Medium Flow , 1983 .

[18]  Padrón Victor,et al.  SOBOLEV REGULARIZATION OF A NONLINEAR ILL-POSED PARABOLIC PROBLEM AS A MODEL FOR AGGREGATING POPULATIONS , 1998 .

[19]  John R. King,et al.  Analysis of a Darcy Flow Model with a Dynamic Pressure Saturation Relation , 1998, SIAM J. Appl. Math..

[20]  Michael A. Celia,et al.  Dynamic Effect in the Capillary Pressure–Saturation Relationship and its Impacts on Unsaturated Flow , 2002 .

[21]  A. Friedman,et al.  Regularity of the Free Boundary for the One-Dimensional Flow of Gas in a Porous Medium , 1979 .

[22]  John Ockendon,et al.  “Waiting-time” Solutions of a Nonlinear Diffusion Equation , 1982 .

[23]  R. Blossey,et al.  A thin-film equation for viscoelastic liquids of Jeffreys type , 2005, The European physical journal. E, Soft matter.

[24]  Two-dimensional Stokes and Hele-Shaw flows with free surfaces , 1999, European Journal of Applied Mathematics.