Stability of Solutions to Generalized Forchheimer Equations of any Degree

ABSTRACT. The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids and study the initial boundary value problem for the resulting degenerate parabolic equation for pressure with the time-dependent flux boundary condition. We estimate L ∞ -norm for pressure and its time derivative, as well as other Lebesgue norms for its gradient and second spatial derivatives. The asymptotic estimates as time tends to infinity are emphasized. We then show that the solution (in interior L ∞ -norms) and its gradient (in interior L 2−δ -norms) depend continuously on the initial and boundary data, and coefficients of the Forchheimer polynomials. These are proved for both finite time intervals and time infinity. The De Giorgi and Ladyzhenskaya-Uraltseva iteration techniques are combined with uniform Gronwall-type estimates, specific monotonicity properties, suitable parabolic Sobolev embeddings and a new fast geometric convergence result.

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“…[3,28,32], see also section 3 of [19] for our proof for the Dirichlet boundary condition). The regularity of weak solutions is treated in [9].…”

Section: Estimates Of Solutionsmentioning

confidence: 99%

“…Second, to take advantage of available estimates in our previous work [18]. Third, to make clear our ideas and techniques without involving much more complicated technical details in case that the Degree Condition is not met (see [19]); such case will be investigated in our future work.…”

mentioning

confidence: 99%

Properties of Generalized Forchheimer Flows in Porous Media

2014

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“…Lemma (cf. , Lemma 2.7) Let

θ > 0

and let

y (t) \geq 0, h (t) > 0, f (t) \geq 0

be continuous functions on

[0, \infty)

that satisfy

y' (t) + h (t) y {false(t false)}^{θ} \leq f (t), \forall t > 0.

Then

y (t) \leq y (0) + {[E n v true(\frac{f false(t false)}{h false(t false)} true)]}^{\frac{1}{θ}}, for all t \geq 0.

\int_{0}^{\infty} h (t) d t = \infty

then

\underset{t \to \infty}{\lim \sup} y (t) \leq \underset{t \to \infty}{\lim \sup} {[\frac{f false(t false)}{h false(t false)}]}^{\frac{1}{θ}} .

Lemma (Discrete Gronwall's inequality) Let

θ > 0, Δ t > 0

. Assume

{f_{n}}_{<...>}

…”

Section: Notations and Auxiliary Resultsunclassified

“…Lemma 2.6 is the discrete version of Lemma 2.5. It can be proved by following the ideas of the Lemma 2.7in . The proof is given in the Appendix.…”

Section: Notations and Auxiliary Resultsmentioning

confidence: 99%

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

Kieu

2017

Numerical Methods Partial

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In this article, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long-time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all time is established in a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequalitytype is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates. K E Y W O R D SDarcy-Forchheimer flow, error analysis, Galerkin FEM, nonlinear degenerate parabolic equations, numerical analysis

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“…A general Forchheimer equation, which is studied in has the form

g (| u |) u = - \nabla p,

where

g (s) \geq 0

is a function defined on

[0, \infty)

. When

g (s) = α, α + β s, α + β s + γ s^{2}, α + γ_{m} s^{m - 1},

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.…”

Section: Mathematical Preliminaries and Auxiliariesmentioning

confidence: 99%

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

Kieu

2015

Numerical Methods Partial

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In this paper, we consider the generalized Forchheimer flows for slightly compressible fluids in porous media. Using Muskat's and Ward's general form of Forchheimer equations, we describe the flow of a single-phase fluid in R d ,d ≥ 2 by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.

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Stability of Solutions to Generalized Forchheimer Equations of any Degree

Cited by 29 publications

References 15 publications

Properties of Generalized Forchheimer Flows in Porous Media

Properties of Generalized Forchheimer Flows in Porous Media

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

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

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