Generalized Forchheimer Flows of Isentropic Gases

Çelik, Emine; Hoang, Luan; Kieu, Thinh

doi:10.1007/s00021-016-0313-2

Cited by 10 publications

(25 citation statements)

References 23 publications

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“…We note that the global L ∞ -norm of u is bounded by (4.38) in Theorem 4.4 which is "quasi-homogeneous" in its L β x,t -norm for some β > 0. This extends previous results in [4,29]. In section 5, we establish L 2−a -estimates for the gradient of u(x, t) in Theorem 5.2.…”

Section: Introductionsupporting

confidence: 79%

“…Proof. We follow the proof of Proposition 6.2 in [4]. We use Sobolev inequality (2.15) in Lemma 2.3: Applying Hölder's inequality to the first integral on the right-hand side of (4.26) with conjugate exponentsκ α α+δ−a andκ α (κ−1)α+a−δ , we get…”

Section: )mentioning

confidence: 99%

“…This model is widely accepted as the standard form of the Forchheimer's two term-law. Based on the arguments by Muskat and Ward, we proposed in [4] the following adaptation for (1.5) − ∇p = N i=0 a i ρ α i |v| α i v, (1.8) where N ≥ 1, α 0 = 0 < α 1 < . .…”

Section: Introductionmentioning

confidence: 99%

“…, a N > 0. Then (1.9) can be rewritten as g(ρ|v|)v = −∇p + ρ g, (1.11) With this rule, we now follow the method in [4] to derive the basic PDE and its corresponding boundary condition. Multiplying both sides of (1.11) by ρ, we obtain g(|ρv|)ρv = −ρ∇p + ρ 2 g.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Section: Introductionsupporting

confidence: 79%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2018

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“…Such a simplification is commonly used in petroleum engineering. The full treatment requires more accurate models for the flows and can use the similar analysis as in [5,6].…”

Section: Modelmentioning

confidence: 99%

Fluid flows of mixed regimes in porous media

Çelik

Hoang

Ibragimov

et al. 2017

Journal of Mathematical Physics

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In porous media, there are three known regimes of fluid flows, namely, pre-Darcy, Darcy and post-Darcy. Because of their different natures, these are usually treated separately in literature. To study complex flows when all three regimes may be present in different portions of a same domain, we use a single equation of motion to unify them. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure gradient is either small or large. We estimate the pressure and its gradient for all time in terms of initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on initial and boundary data, and the structural stability for the equation are established.

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Maximum estimates for generalized Forchheimer flows in heterogeneous porous media

Çelik

Hoang

2017

Journal of Differential Equations

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This article continues the study in [3] of generalized Forchheimer flows in heterogeneous porous media. Such flows are used to account for deviations from Darcy's law. In heterogeneous media, the derived nonlinear partial differential equation for the pressure can be singular and degenerate in the spatial variables, in addition to being degenerate for large pressure gradient.Here we obtain the estimates for the L ∞ -norms of the pressure and its time derivative in terms of the initial and the time-dependent boundary data. They are established by implementing De Giorgi's iteration in the context of weighted norms with the weights specifically defined by the Forchheimer equation's coefficient functions. With these weights, we prove suitable weighted parabolic Poincaré-Sobolev inequalities and use them to facilitate the iteration. Moreover, local in time L ∞ -bounds are combined with uniform Gronwall-type energy inequalities to obtain long-time L ∞ -estimates.

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Generalized Forchheimer Flows of Isentropic Gases

Cited by 10 publications

References 23 publications

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

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

Fluid flows of mixed regimes in porous media

Maximum estimates for generalized Forchheimer flows in heterogeneous porous media

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