Generalized Forchheimer flows of compressible fluids in heterogeneous porous media

The first part of this dissertation studies the generalized Forchheimer flows of slightly compressible fluids in heterogeneous porous media. Such flows are used to account for deviations from the ubiquitous Darcy's law. In heterogeneous media, the media's porosity and coefficients of the Forchheimer equation are functions of the spatial variables. In describing the dynamics of fluid flows, we derive a nonlinear partial differential equation for the pressure.
This parabolic equation is degenerate in its gradient and can be both singular and degenerate in the spatial variables.
Suitable weighted Lebesgue norms for the pressure, its gradient and time derivative are estimated. The weights used for these norms are specifically defined by the porosity function and the Forchheimer equation's coefficient functions. The continuous dependence on the initial and boundary data is established for the pressure and its gradient with respect to those corresponding norms. Asymptotic estimates are derived even for unbounded boundary data as time tends to infinity. We then go further to obtain the estimates for the $L\infty$-norms of the pressure and its time derivative in terms of the initial and the time-dependent boundary data. They are established by implementing De Giorgi's iteration in the context of the above weighted norms. Suitable weighted parabolic Poincar'e-Sobolev inequalities are proved and then used to facilitate the iteration. Moreover, local in time $L\infty$-bounds are combined with uniform Gronwall-type energy inequalities to obtain long-time $L\infty$-estimates.

While the first part is about slightly compressible fluids, the second part of this dissertation is focused on generalized Forchheimer flows for a larger class of compressible fluids. In particular, we study the isentropic gas flows in porous media. By using Muskat's and Ward's general form of the Forchheimer equations, we describe the fluid dynamics by a doubly nonlinear parabolic equation for the appropriately defined pseudo-pressure. The volumetric flux boundary condition is converted to a time-dependent Robin-type boundary condition for this pseudo-pressure. We study the corresponding initial boundary value problem, and estimate the $L\infty$ and $W1,2-a$ (with ) norms for the solution on the entire domain in terms of the initial and boundary data. It is carried out by using a suitable trace theorem and an appropriate modification of Moser's iteration.