Thinh Kieu

Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids

Abstract The generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior L ∞ {L^{\infty}} -estimates for the pressure, its gradient and time derivative, and the interior L 2 {L^{2}} -estimates for its Hessian. The De Giorgi and Ladyzhenskaya–Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity.

A family of steady two-phase generalized Forchheimer flows and their linear stability analysis

We model multi-dimensional two-phase flows of incompressible fluids in porous media using generalized Forchheimer equations and the capillary pressure. First, we find a family of steady state solutions whose saturation and pressure are radially symmetric, and velocities are rotation-invariant. Their properties are investigated based on relations between the capillary pressure, each phase’s relative permeability and Forchheimer polynomial. Second, we analyze the linear stability of those steady states. The linearized system is derived and reduced to a parabolic equation for the saturation. This equation has a special structure depending on the steady states which we exploit to prove two new forms of the lemma of growth of Landis-type in both bounded and unbounded domains. Using these lemmas, qualitative properties of the solution of the linearized equation are studied in details. In bounded domains, we show that the solution decays exponentially in time. In unbounded domains, in addition to their stability...

Symplectic-mixed finite element approximation of linear acoustic wave equations

We apply mixed finite element approximations to the first-order form of the acoustic wave equation. The semidiscrete method exactly conserves the system energy. A fully discrete method employing the symplectic Euler time method in time exactly conserves a positive-definite pertubed energy functional that is equivalent to the actual energy under a CFL condition. In addition to proving optimal-order

An expanded mixed finite element method for generalized Forchheimer flows in porous media

L^\infty (L^2)

Stability of Solutions to Generalized Forchheimer Equations of any Degree

L∞(L2) estimates, we also develop a bootstrap technique that allows us to derive stability and error bounds for the time derivatives and divergence of the vector variable beyond the standard under some additional regularity assumptions.

Properties of Generalized Forchheimer Flows in Porous Media

We consider generalized Forchheimer flows of either isentropic gases or slightly compressible fluids 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

Analysis of expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

L^\infty