Fluid flows of mixed regimes in porous media

In porous media, there are three known regimes of fluid fl ows, namely, pre-Darcy,Darcy, and post-Darcy. Because of their different natures, these are usually treatedseparately in the literature. To study complex flows when all three regimes may bepresent in different portions of a same domain, we use a single equation of motion tounify them. Several scenarios and models are then considered for slightly compressiblefluids. A nonlinear parabolic equation for the pressure is derived, which is degeneratewhen the pressure gradient is either small or large. We estimate the pressure andits gradient for all time in terms of initial and boundary data. We also obtain theirparticular bounds for large time which depend on the asymptotic behavior of theboundary data but not on the initial one. Moreover, the continuous dependence of thesolutions on initial and boundary data and the structural stability for the equation areestablished.