Properties of Generalized Forchheimer Flows in Porous Media

Hoang, Luan; Kieu, Thinh; Phan, Tuoc

doi:10.1007/s10958-014-2045-2

Cited by 18 publications

(45 citation statements)

References 25 publications

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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: 95%

See 1 more Smart Citation

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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“…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: 95%

“…Let x ∈ R d , 0 < T < ∞, t ∈ (0, T ] be the spatial and time variable. A general Forchheimer equation, which is studied in [12,13,15,17] has the form g(|u|)u = −∇p, (2.1)Numerical Methods for Partial Differential Equations…”

mentioning

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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“…It is used to unify the models (1.2), (1.3), (1.4), and as a framework for interpretation of different experimental or field data. It is analyzed numerically in [8,21,27], theoretically in [2,[12][13][14][15]18] for single-phase flows, and also in [16,17] for two-phase flows. For compressible fluids, especially gases, the dependence of coefficients a i 's on the density ρ is essential.…”

Section: Introductionmentioning

confidence: 99%

Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media

2018

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This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source terms and boundary fluxes are integrated into the model. It covers isentropic gas flows, ideal gases and slightly compressible fluids. We derive a doubly nonlinear parabolic equation for the so-called pseudopressure, and study the corresponding initial boundary value problem. The maximum estimates of the solution are established by using suitable trace theorem and adapting appropriately the Moser's iteration. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications.

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“…These equations are analyzed numerically in [7][8][9], theoretically in [4,6,[10][11][12][13] for single phase flows, and also in [14,15] for two-phase flows.…”

Section: Introductionmentioning

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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Properties of Generalized Forchheimer Flows in Porous Media

Cited by 18 publications

References 25 publications

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

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

Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

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