Generalized Forchheimer flows in heterogeneous porous media

Abstract

We study the generalized Forchheimer flows of slightly compressible fluids in heterogeneous porous media. The media's porosity and coefficients of the Forchheimer equation are functions of the spatial variables. The partial differential equation for the pressure 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 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.

Maximum estimates for generalized Forchheimer flows in heterogeneous porous media

“…[19,22] for incompressible fluids, [1,[10][11][12]15] for slightly compressible fluids, and [4] for isentropic gases. The problem of Forchheimer flows in heterogeneous media, which is encountered frequently in real life applications, was started in [3]. The current article is a continuation of [3] and is focused on the L ∞ -estimates rather than L 2 .…”

“…The problem of Forchheimer flows in heterogeneous media, which is encountered frequently in real life applications, was started in [3]. The current article is a continuation of [3] and is focused on the L ∞ -estimates rather than L 2 . Below, we follow [3] in presenting the model and deriving the key partial differential equation (PDE).…”

“…The current article is a continuation of [3] and is focused on the L ∞ -estimates rather than L 2 . Below, we follow [3] in presenting the model and deriving the key partial differential equation (PDE).…”

“…∂p ∂t = ∇ · (K(x, |∇p|)∇p) on U × (0, ∞), p = ψ on Γ × (0, ∞), p(x, 0) = p 0 (x) on U, (1.6) where p 0 (x) are ψ(x, t) are given initial and boundary data. (See [3] for more details.) Here afterward, the function g(x, s) in (1.2) is fixed, hence so is K(x, ξ).…”

“…Although φ(x) belongs to (0, 1) in applications, we only assume φ(x) > 0 in this paper. As noted in [3], the PDE in (1.6) is degenerate in ∇p as |∇p| → ∞, and can be both singular and degenerate in x. For such a nonlinear PDE, finer analysis is needed to deal with different types of degeneracy and singularity.…”