Analysis of generalized Forchheimer flows of compressible fluids in porous media

Abstract

This work is focused on the analysis of non-linear flows of slightly compressible fluids in porous media not adequately described by Darcy's law. We study a class of generalized nonlinear momentum equations which covers all three well-known Forchheimer equations, the so-called two-term, power, and three-term laws. The non-linear Forchheimer equation is inverted to a non-linear Darcy equation with implicit permeability tensor depending on the pressure gradient. This results in a degenerate parabolic equation for the pressure. Two classes of boundary conditions are considered, given pressure and given total flux. In both cases they are allowed to be unbounded in time. The uniqueness, Lyapunov and asymptotic stabilities, and other long-time dynamical features of the corresponding initial boundary value problems are analyzed. The results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters. Some numerical simulations are also included.

Properties of Generalized Forchheimer Flows in Porous Media

“…Let g 1 (s) = g(s, a (1) ) and g 2 (s) = g(s, a (2) ) be two functions of class FP(N, α), where a (1) and a (2) belong to D. For k = 1, 2, let p k = p k (x, t; a (k) ) be the solution of (3.1) and (3.2) with K = K(ξ, a) and the same boundary flux ψ.…”

“…Generalized Forchheimer equations, studied in [1,17], are of the form: where α, β, γ, m, γ m are empirical constants, we have Darcy's law, Forchheimer's two-term, three-term and power laws, respectively. In this paper, the function g in (2.1) is a generalized polynomial with non-negative coefficients, that is, g(s) = a 0 s α 0 + a 1 s α 1 + .…”

“…Even so, most of these papers consider incompressible fluids only. In our previous works [1,[17][18][19], we proposed and studied generalized Forchheimer equations for slightly compressible fluids in porous media. Such mathematical generalization is appropriate and useful.…”

Fluid flows of mixed regimes in porous media

“…In order to obtain this, we first establish in Lemma 5.1 the perturbed monotonicity for our degenerate PDE. It is then proved in Theorem 5.2 that the difference P (x, t) between the two solutions which correspond to two different coefficient vectors a (1) and a (2) is estimated in terms of their initial difference P (x, 0) and | a (1) − a (2) |, see (5.14). Moreover, when time goes to infinity, this difference can be controlled by | a (1) − a (2) | only, see (5.15).…”

“…[25]. The post-Darcy regime has been attracted attention recently with the (nonlinear) Forchheimer models, see [1,5,6,12,[15][16][17][18]24] and references therein. In contrast, the (nonlinear) pre-Darcy regime is virtually ignored.…”