A generalization of the Darcy–Forchheimer equation involving an implicit, pressure-dependent relation between the drag force and the velocity

Bulíček, Miroslav; Málek, Josef; Žabenský, Josef

doi:10.1016/j.jmaa.2014.11.053

Cited by 10 publications

(6 citation statements)

References 30 publications

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“…This shows that V = {0}, i.e., V contains only the zero function on (0, 1). Trivially then, the zero function is the unique global minimizer of the energy E in (12) and M E is a singleton. Note that in this case the energy is only trivially convex.…”

Section: Case λ 1 > λmentioning

confidence: 99%

“…In Section 3.1 we will complete the strong formulation in Problem 2.1, which is somewhat still formal since it lacks a law at the transition zone; indeed, we will introduce a multi-valued weak setting so as to be able to treat the transition zone without imposing any given law on it. We refer the reader to [12] for a multi-valued monotone operator approach for pressure-dependent permeabilities; note that the operator therein is continuous in velocity.…”

Section: Velocity-pressure Constitutive Lawmentioning

confidence: 99%

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Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies

Fumagalli

Patacchini

2022

ESAIM: M2AN

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Depending on the physical and geometrical properties of a given porous medium, fluid flow can behave differently, going from a slow Darcian regime to more complicated Brinkman or even Forchheimer regimes for high velocity. The main problem is to determine where in the medium one regime is more adequate than others. In order to determine the low-speed and high-speed regions, this work proposes an adaptive strategy which is based on selecting the appropriate constitutive law linking velocity and pressure according to a threshold criterion on the magnitude of the fluid velocity itself. Both theoretical and numerical aspects are considered and investigated, showing the potentiality of the proposed approach. From the analytical viewpoint, we show existence of weak solutions to such model under reasonable hypotheses on the constitutive laws. To this end, we use a variational approach identifying solutions with minimizers of an underlying energy functional. From the numerical viewpoint, we propose a one-dimensional algorithm which tracks the transition zone between the low- and high-speed regions. By running numerical experiments using this algorithm, we illustrate some interesting behaviors of our adaptive model on academic cases and on small networks of intersecting fractures.

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Section: Case λ 1 > λmentioning

confidence: 99%

Section: Velocity-pressure Constitutive Lawmentioning

confidence: 99%

Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies

Fumagalli

Patacchini

2022

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…This shows that V = {0}, i.e., V contains only the zero function on (0, 1). Trivially then, the zero function is the unique global minimizer of the energy E in (19) and M E is a singleton. Note that in this case the energy is only trivially convex.…”

Section: Case λ 1 > λmentioning

confidence: 99%

“…We will complete in Section 3.1 the strong formulation (6), which is somewhat formal since it lacks an interface constitutive relation; indeed, we will introduce a multi-valued weak setting so as to be able to treat the interface without imposing any given law on it. We refer the reader to [19] for a multi-valued monotone operator approach for pressure-dependent permeabilities; note that the operator therein is continuous in velocity.…”

Section: Velocity-pressure Constitutive Lawmentioning

confidence: 99%

Model adaptation in a discrete fracture network: existence of solutions and numerical strategies

Fumagalli,

Patacchini

2021

Preprint

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Fractures are normally present in the underground and are, for some physical processes, of paramount importance. Their accurate description is fundamental to obtain reliable numerical outcomes useful, e.g., for energy management. Depending on the physical and geometrical properties of the fractures, fluid flow can behave differently, going from a slow Darcian regime to more complicated Brinkman or even Forchheimer regimes for high velocity. The main problem is to determine where in the fractures one regime is more adequate than others. In order to determine these low-speed and high-speed regions, this work proposes an adaptive strategy which is based on selecting the appropriate constitutive law linking velocity and pressure according to a threshold criterion on the magnitude of the fluid velocity itself. Both theoretical and numerical aspects are considered and investigated, showing the potentiality of the proposed approach. From the analytical viewpoint, we show existence of weak solutions to such model under reasonable hypotheses on the constitutive laws. To this end, we use a variational approach identifying solutions with minimizers of an underlying energy functional. From the numerical viewpoint, we propose a one-dimensional algorithm which tracks the interface between the low-and high-speed regions. By running numerical experiments using this algorithm, we illustrate some interesting behaviors of our adaptive model on a single fracture and small networks of intersecting fractures.

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“…For the Forchheimer equations and other related models of fluid flows in porous media that differ from the ubiquitous Darcy's law, the interested reader is referred to the books [3,21,24]. Regarding their mathematical analysis in the case without rotation, see [4,5,11,13,19,[22][23][24] for incompressible fluids, see [2,[6][7][8][9][14][15][16][17] for compressible fluids, and references therein. For more information about fluid flows in rotating porous media, see [25] and, also, our previous mathematical study [10].…”

Section: Introductionmentioning

confidence: 99%

Studying a doubly nonlinear model of slightly compressible Forchheimer flows in rotating porous media

Çelik¹,

Hoang²,

Kieu³

2021

Preprint

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We study the generalized Forchheimer flows of slightly compressible fluids in rotating porous media. In the problem's model, the varying density in the Coriolis force is fully accounted for without any simplifications. It results in a doubly nonlinear parabolic equation for the density. We derive a priori estimates for the solutions in terms of the initial, boundary data and physical parameters, emphasizing on the case of unbounded data. Weighted Poincaré-Sobolev inequalities suitable to the equation's nonlinearity, adapted Moser's iteration and maximum principle are used and combined to obtain different types of estimates.

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A generalization of the Darcy–Forchheimer equation involving an implicit, pressure-dependent relation between the drag force and the velocity

Cited by 10 publications

References 30 publications

Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies

Model adaptation for non-linear elliptic equations in mixed form: existence of solutions and numerical strategies

Model adaptation in a discrete fracture network: existence of solutions and numerical strategies

Studying a doubly nonlinear model of slightly compressible Forchheimer flows in rotating porous media

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