One-dimensional two-phase generalized Forchheimer flows of incompressible fluids

We derive a nonlinear system of parabolic equations to describe the one-dimensional two-phase generalized Forchheimer flows of incompressible, immiscible fluids in porous media, with the presence of capillary forces. Under relevant constraints on relative permeabilities and capillary pressure, non-constant steady state solutions are found and classified into sixteen types according to their monotonicity and asymptotic behavior. For a steady state whose saturation can never attain either value 0 or 1, we prove that it is stable with respect to a certain weight. This weight is a function comprised of the steady state, relative permeabilities and capillary pressure. The proof is based on specific properties of the steady state, weighted maximum principle and Bernstein's estimate.