Existence of weak solutions for a nonlocal pseudo-parabolic model for Brinkman two-phase flow in asymptotically flat porous media

Armiti-Juber, Alaa; Rohde, Christian

doi:10.1016/j.jmaa.2019.04.049

Cited by 5 publications

(6 citation statements)

References 25 publications

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“…For this, we distinguish between two cases for thin domains with geometrical ratio

γ = H false/ L ≪ 1

: Case 1: Porous media domains with width

H = scriptO false(1 false)

and length L ≫ 1 have the parameters

β_{1} = scriptO false(γ^{2} false)

and

β_{2} = scriptO false(1 false)

. In the limit, this will lead to a solution S with reduced regularity in the x ‐direction, as the estimates on ∂ x S γ in Lemma 1 and ∂ tx S γ in Lemma 3 will vanish and the well‐posedness proof of the BVE model in Armiti‐Juber and Rohde 23 will not be valid. Case 2: Porous media domains with width

H = scriptO false(γ^{2} false)

and length

L = scriptO false(1 false)

. Thus, the parameter

β_{2} = scriptO false(γ^{- 2} false)

and the higher‐order term ∂ tzz S γ will dominate.…”

Section: Convergence Analysismentioning

confidence: 99%

“…These are essential for the convergence analysis as γ tends to zero in the next section. Note that the existence of weak solutions for the BTP-model is proved in [5], while for the BVEmodel is proved in [2].…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…We do this by proving that the reduced model is the analytical limit of the full BTP model in domains with vanishing width–length ratio γ . For the well‐posedness of the models we refer to Coclite et al 22 and Armiti‐Juber and Rohde, 23 where the existence of weak solutions for the BTP model and the BVE model, respectively, is proved. Other models with a similar structure to the BTP and BVE models have been developed to describe the two‐phase flow including dynamic effects and/or hysteresis in the phase‐pressure difference.…”

Section: Introductionmentioning

confidence: 99%

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On the limit of a two‐phase flow problem in thin porous media domains of Brinkman type

Armiti-Juber

2021

Math Methods in App Sciences

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We study the process of two‐phase flow in thin porous media domains of Brinkman type. This is generally described by a model of coupled, mixed‐type differential equations of fluids' saturation and pressure. To reduce the model complexity, different approaches that utilize the thin geometry of the domain have been suggested. We focus on a reduced model that is formulated as a single nonlocal evolution equation of saturation. It is derived by applying standard asymptotic analysis to the dimensionless coupled model; however, a rigid mathematical derivation is still lacking. In this paper, we prove that the reduced model is the analytical limit of the coupled two‐phase flow model as the geometrical parameter of domain's width–length ratio tends to zero. Precisely, we prove the convergence of weak solutions for the coupled model to a weak solution for the reduced model as the geometrical parameter approaches zero.

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“…For this, we distinguish between two cases for thin domains with geometrical ratio

γ = H false/ L ≪ 1

: Case 1: Porous media domains with width

H = scriptO false(1 false)

and length L ≫ 1 have the parameters

β_{1} = scriptO false(γ^{2} false)

and

β_{2} = scriptO false(1 false)

H = scriptO false(γ^{2} false)

and length

L = scriptO false(1 false)

. Thus, the parameter

β_{2} = scriptO false(γ^{- 2} false)

and the higher‐order term ∂ tzz S γ will dominate.…”

Section: Convergence Analysismentioning

confidence: 99%

Section: A Priori Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the limit of a two‐phase flow problem in thin porous media domains of Brinkman type

Armiti-Juber

2021

Math Methods in App Sciences

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“…The class of PPEs becomes a subject of an extensive study in the last decade. The analysis of global existence and blow-up phenomenon for solutions of PPEs with different types of nonlinearity was given in previous research [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], while some long-time behavior results were established in other works [26,30].…”

Section: Introductionmentioning

confidence: 99%

Final value problem governed by a class of time‐space fractional pseudo‐parabolic equations with weak nonlinearities

Dinh Ke,

Bao Ngoc,

Huy Tuan

2024

Math Methods in App Sciences

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We study the final value problem involving a class of semilinear fractional pseudo‐parabolic equations, where the nonlinearity probably takes values in fractional Sobolev spaces. By establishing some estimates for resolvent operators and employing the embedding related to Hilbert scales and fractional Sobolev spaces, we are able to obtain the existence and uniqueness result to the mentioned problem. In addition, the behavior of solutions at initial time is analyzed with respect to the final data. It will be shown that various cases of the nonlinearity functions meet our setting, including functions with gradient term.

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“…A theoretical or numerical analysis of the case of a general network of fractures is still missing. For analysis of two‐phase flow reduced models we refer the reader to References 23,25,28 and specifically to Reference 29 for Richards' equation. These existing models will be denoted as “local” in the following, meaning that the constitutive laws are local in both space and geometrical representation, the latter implying that the flux on any given subdomain (or boundary) is proportional to pressure gradients (or jumps) on the same subdomain.…”

Section: Introductionmentioning

confidence: 99%

Modeling and discretization of flow in porous media with thin, full‐tensor permeability inclusions

Starnoni

Berre

Keilegavlen

et al. 2021

Numerical Meth Engineering

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When modeling fluid flow in fractured reservoirs, it is common to represent the fractures as lower‐dimensional inclusions embedded in the host medium. Existing discretizations of flow in porous media with thin inclusions assume that the principal directions of the inclusion permeability tensor are aligned with the inclusion orientation. While this modeling assumption works well with tensile fractures, it may fail in the context of faults, where the damage zone surrounding the main slip surface may introduce anisotropy that is not aligned with the main fault orientation. In this article, we introduce a generalized dimensional reduced model which preserves full‐tensor permeability effects also in the out‐of‐plane direction of the inclusion. The governing equations of flow for the lower‐dimensional objects are obtained through vertical averaging. We present a framework for discretization of the resulting mixed‐dimensional problem, aimed at easy adaptation of existing simulation tools. We give numerical examples that show the failure of existing formulations when applied to anisotropic faulted porous media, and go on to show the convergence of our method in both two‐dimensional and three‐dimensional.

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Existence of weak solutions for a nonlocal pseudo-parabolic model for Brinkman two-phase flow in asymptotically flat porous media

Cited by 5 publications

References 25 publications

On the limit of a two‐phase flow problem in thin porous media domains of Brinkman type

On the limit of a two‐phase flow problem in thin porous media domains of Brinkman type

Final value problem governed by a class of time‐space fractional pseudo‐parabolic equations with weak nonlinearities

Modeling and discretization of flow in porous media with thin, full‐tensor permeability inclusions

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