Structural stability of generalized Forchheimer equations for compressible fluids in porous media

Abstract

We study the generalized Forchheimer equations for slightly compressible fluids in porous media. The structural stability is established with respect to either the boundary data or the coefficients of the Forchheimer polynomials. An inhomogeneous Poincare-Sobolev inequality related to the non-linearity of the equation is used to study the asymptotic behavior of the solutions. Moreover, we prove a perturbed monotonicity property of the vector field associated with the resulting non-Darcy equation, where the correction is linear in the coefficients of the Forchheimer polynomials.

Properties of Generalized Forchheimer Flows in Porous Media

“…As shown in [17], all constants d j , c j , C j and C appearing in estimates in the previous sections when a varies among the vectors a (1) , a (2) , a (1) ∨ a (2) and a (1) ∧ a (2) , can be made dependent ofχ(D), but independent of a. We still denote them by d j , c j , C j and C, respectively, in this section.…”

“…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 + .…”

“…The function H(ξ) can be compared with ξ and K(ξ) by 17) where c 1 , c 2 > 0 depend on χ( a). Next, we recall important monotonicity properties.…”

On the well-posedness of generalized Darcy–Forchheimer equation

“…To see this, we multiply equation (2) by ϕ ∈ X and integrate over . We then apply the conditions in equation (3) to the Green's formula (14):…”

Fluid flows of mixed regimes in porous media

“…These can be rewritten in terms of ψ(x, t) by using a specific extension. For instance, the harmonic extension is utilized in [11] with the use of norm relations in [19].…”