Global estimates for generalized Forchheimer flows of slightly compressible fluids

Hoang, Luan; Kieu, Thinh

doi:10.1007/s11854-018-0064-5

Cited by 15 publications

(20 citation statements)

References 22 publications

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“…These equations are analyzed numerically in [7,19,27], theoretically in [2,[10][11][12][13]16] for single-phase flows, and also in [14,15] for two-phase flows. Our previous analysis [2,[10][11][12][13]16] was focused on a simplified model for slightly compressible fluids. Though such a minor simplification is commonly used in reservoir engineering, the mathematical rigor is compromised.…”

Section: Introductionmentioning

confidence: 99%

Generalized Forchheimer Flows of Isentropic Gases

Çelik

Hoang

Kieu

2017

J. Math. Fluid Mech.

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We consider generalized Forchheimer flows of either isentropic gases or slightly compressible fluids in porous media. By using Muskat's and Ward's general form of the Forchheimer equations, we describe the fluid dynamics by a doubly nonlinear parabolic equation for the appropriately defined pseudo-pressure. The volumetric flux boundary condition is converted to a time-dependent Robin-type boundary condition for this pseudo-pressure. We study the corresponding initial boundary value problem, and estimate the L ∞ and W 1,2−a (with 0 < a < 1) norms for the solution on the entire domain in terms of the initial and boundary data. It is carried out by using a suitable trace theorem and an appropriate modification of Moser's iteration.

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Section: Introductionmentioning

confidence: 99%

Generalized Forchheimer Flows of Isentropic Gases

Çelik

Hoang

Kieu

2017

J. Math. Fluid Mech.

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“…It is worth mentioning that our results are applicable to all commonly used Forchheimer's laws. Finally, we remark that in case of homogeneous porous media, estimates for p and its time derivative pave the way for obtaining L ∞ -estimates for the gradient, as well as strong continuous dependence and structural stability, see [13][14][15]. However, it is not known whether such results still hold true for heterogeneous media in the current study.…”

Section: Introductionmentioning

confidence: 68%

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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“…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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Global estimates for generalized Forchheimer flows of slightly compressible fluids

Cited by 15 publications

References 22 publications

Generalized Forchheimer Flows of Isentropic Gases

Generalized Forchheimer Flows of Isentropic Gases

Maximum estimates for generalized Forchheimer flows in heterogeneous porous media

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

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