Permanent fronts in two-phase flows in a porous medium

A family of exact solutions for a model of a one-dimensional horizontal flo w of two immiscible, incompressible fluids in a porous medium, including th e effects of capillary pressure, is obtained analytically by solving the go verning singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is s hown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting-o utflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analyt ically and numerically in detail in the case when the permeabilities and th e capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks-Corey model are also presented. In bo th models three global bifurcations occur. By using a geometric dynamical s ystem approach, the nonlinear stability of the permanent fronts is establis hed analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is giv en as follows. The instantaneous upstream (downstream) velocity of an arbit rary non-quasistationary front is equal to the velocity of a permanent fron t whose shape coincides up to two leading orders with the instantaneous sha pe of the non-quasistationary front at the upstream (respectively, downstre am) location. The upstream and downstream locations of the front undergo in stantaneous translations governed by modified nonsingular hyperbolic equati ons. The portion of the front in between these locations undergoes a diffus ive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic-hyperb olic domain decomposition for computing non-quasistationary fronts.