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

Kieu, Thinh

doi:10.1002/num.21984

Cited by 14 publications

(20 citation statements)

References 33 publications

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“…For the vector functions s 1 , s 2 , there is a positive constant

C_{5} (Ω, d, g)

such that

(K (| s_{1} |) s_{1} - K (| s_{2} |) s_{2}, s_{1} - s_{2}) \geq C_{5} ω ‖ s_{1} - s_{2} ‖_{0, β}^{2},

where

ω = {false(1 + \max {‖ s_{1} ‖_{0, β}; ‖ s_{2} ‖_{0, β}} false)}^{- a} .

Lemma (cf. , Lemma 2.4) For all vector

y, y' \in ℝ^{d}

. There exists a positive constant C 6 depending on polynomial g, the spatial dimension d and domain Ω such that

true| K (| y' |) y' - K (| y |) y true| \leq C_{6} | y' - y | .

Definition Given f(t) defined on an interval

I \subset ℝ

.…”

Section: Notations and Auxiliary Resultsmentioning

confidence: 99%

“…Lemma 2.3 (cf. [8], Lemma 2.4) For all vector y, y ∈ R d . There exists a positive constant C 6 depending on polynomial g, the spatial dimension d and domain such that |K(|y |)y − K(|y|)y| ≤ C 6 |y − y|.…”

Section: )mentioning

confidence: 99%

“…Combining (1.3) with the suggestive form (1.4) for the dependence on ρ and v , we propose the following equation (see e.g. ).

- \nabla p = \sum_{i = 0}^{N} a_{i} ρ^{α_{i}} | v |^{α_{i}} v,

where

N \geq 1, α_{0} = 0 < α_{1} < \dots < α_{N}

are real numbers, the coefficients

a_{0}, \dots, a_{N}

are positive.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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“…For the vector functions s 1 , s 2 , there is a positive constant

C_{5} (Ω, d, g)

such that

(K (| s_{1} |) s_{1} - K (| s_{2} |) s_{2}, s_{1} - s_{2}) \geq C_{5} ω ‖ s_{1} - s_{2} ‖_{0, β}^{2},

where

ω = {false(1 + \max {‖ s_{1} ‖_{0, β}; ‖ s_{2} ‖_{0, β}} false)}^{- a} .

Lemma (cf. , Lemma 2.4) For all vector

y, y' \in ℝ^{d}

. There exists a positive constant C 6 depending on polynomial g, the spatial dimension d and domain Ω such that

true| K (| y' |) y' - K (| y |) y true| \leq C_{6} | y' - y | .

Definition Given f(t) defined on an interval

I \subset ℝ

.…”

Section: Notations and Auxiliary Resultsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…Combining (1.3) with the suggestive form (1.4) for the dependence on ρ and v , we propose the following equation (see e.g. ).

- \nabla p = \sum_{i = 0}^{N} a_{i} ρ^{α_{i}} | v |^{α_{i}} v,

where

N \geq 1, α_{0} = 0 < α_{1} < \dots < α_{N}

are real numbers, the coefficients

a_{0}, \dots, a_{N}

are positive.…”

Section: Introductionmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

Kieu

2017

Numerical Methods Partial

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show abstract

“…In the continuity equation (1.6), γ is a pseudo-compressibility parameter [10] that accounts for slight change of the density of the fluid phase in the dissolution process. As stated in [14], in some conditions of γ = 0, the linear system may be singular and cannot be solved by the linear solver, and as a result, one suggested solution is to set γ to be a very small positive number to ensure an invertible coefficient matrix.…”

Section: Introduction Matrix Acidization Technique Plays An Importanmentioning

confidence: 99%

Mixed finite element-based fully conservative methods for simulating wormhole propagation

Kou

Sun

2016

Computer Methods in Applied Mechanics and Engineering

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. MIXED FINITE ELEMENT-BASED FULLY CONSERVATIVE METHODS FOR SIMULATING WORMHOLE PROPAGATION *JISHENG KOU † , SHUYU SUN ‡ , AND YUANQING WU § Abstract. Wormhole propagation during reactive dissolution of carbonates plays a very important role in the product enhancement of oil and gas reservoir. Because of high velocity and nonuniform porosity, the Darcy-Forchheimer model is applicable for this problem instead of conventional Darcy framework. We develop a mixed finite element scheme for numerical simulation of this problem, in which mixed finite element methods are used not only for the Darcy-Forchheimer flow equations but also for the solute transport equation by introducing an auxiliary flux variable to guarantee full mass conservation. In theoretical analysis aspects, based on the cut-off operator of solute concentration, we construct an analytical function to control and handle the change of porosity with time; we treat the auxiliary flux variable as a function of velocity and establish its properties; we employ the coupled analysis approach to deal with the fully coupling relation of multivariables. From this, the stability analysis and a priori error estimates for velocity, pressure, concentration and porosity are established in different norms. Numerical results are also given to verify theoretical analysis and effectiveness of the proposed scheme.

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Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

Kieu

2017

Math Methods in App Sciences

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We 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 the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in L 1 .0, T; L q . //, q 2, time derivative in L 1 .0, T; L 2 . // and gradient in L 1 .0, T; W 1,2 a . //, with a 2 .0, 1/ are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Above, the positive constants a, b, c, d are obtained from experiments. The generalized Forchheimer equation of (1.1) and (1.2) were proposed in [8-10] of the form rp D N X iD0 a i jvj˛i v. (1.3)These equations are analyzed numerically in [11][12][13], theoretically in [8,10,[14][15][16][17] for single phase flows, and also in [18,19] for two phase flows.

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

Cited by 14 publications

References 33 publications

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

Mixed finite element-based fully conservative methods for simulating wormhole propagation

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

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