Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media

Coclite, Giuseppe Maria; Mishra, Siddhartha; Risebro, Nils Henrik; Weber, Franziska

doi:10.1007/s10596-014-9410-6

Cited by 17 publications

(28 citation statements)

References 21 publications

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“…2. An interesting study of a two-phase Brinkman type of model is found in [9]. A two-phase formulation based on Darcy's equations of the following form is studied with s = s w…”

Section: (Communicated By Paola Goatin)mentioning

confidence: 99%

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Compressible and viscous two-phase flow in porous media based on mixture theory formulation

Qiao¹,

Wen²,

Evje³

2019

Networks &Amp; Heterogeneous Media

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The purpose of this work is to carry out investigations of a generalized two-phase model for porous media flow. The momentum balance equations account for fluid-rock resistance forces as well as fluid-fluid drag force effects, in addition, to internal viscosity through a Brinkmann type viscous term. We carry out detailed investigations of a one-dimensional version of the general model. Various a priori estimates are derived that give rise to an existence result. More precisely, we rely on the energy method and use compressibility in combination with the structure of the viscous term to obtain H 1 -estimates as well upper and lower uniform bounds of mass variables. These a priori estimates imply existence of solutions in a suitable functional space for a global time T > 0. We also derive discrete schemes both for the incompressible and compressible case to explore the role of the viscosity term (Brinkmann type) as well as the incompressible versus the compressible case. We demonstrate similarities and differences between a formulation that is based, respectively, on interstitial velocity and Darcy velocity in the viscous term. The investigations may suggest that interstitial velocity seems more natural to use in the formulation of momentum balance than Darcy velocity.

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“…2. An interesting study of a two-phase Brinkman type of model is found in [9]. A two-phase formulation based on Darcy's equations of the following form is studied with s = s w…”

Section: (Communicated By Paola Goatin)mentioning

confidence: 99%

“…In [9] a notion of weak solutions to the Brinkman model (1.13) is introduced and convergence of an iterative approximation as well as full numerical scheme is demonstrated. Numerical experiments show that the numerical approximation is quite sensitive to the choice of µ and creates oscillatory behavior.…”

Section: (Communicated By Paola Goatin)mentioning

confidence: 99%

Compressible and viscous two-phase flow in porous media based on mixture theory formulation

Qiao¹,

Wen²,

Evje³

2019

Networks &Amp; Heterogeneous Media

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“…The second step in deriving the BVE-model is applying formal asymptotic analysis, with respect to γ, to the dimensionless BTP-model (61). The existence of a weak solution (S γ i , p γ ,U γ ,W γ ) for model (61) with appropriate initial and boundary conditions is established in [9]. We assume in this section that each component in (S γ i , p γ ,U γ ,W γ ) is smooth and can be written in terms of the asymptotic expansions…”

Section: Asymptotic Analysismentioning

confidence: 99%

On Darcy- and Brinkman-type models for two-phase flow in asymptotically flat domains

Armiti-Juber

Rohde

2018

Comput Geosci

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We study two-phase flow for Darcy and Brinkman regimes. To reduce the computational complexity for flow in vertical equilibrium various simplified models have been suggested. Examples are dimensional reduction by vertical integration, the multiscale model approach in [Guo et al., 2014] or the asymptotic approach in [Yortsos, 1995]. The latter approach uses a geometrical scaling. We show the efficiency of the approach in asymptotically flat domains for Darcy regimes. Moreover, we prove that it is vastly equivalent to the multiscale model approach.We apply then asymptotic analysis to the two-phase flow model in Brinkman regimes. The limit model is a single nonlocal evolution law with a pseudo-parabolic extension. Its computational efficiency is demonstrated using numerical examples. Finally, we show that the new limit model exhibits overshoot behaviour as it has been observed for dynamical capillarity laws [Hassanizadeh and Gray, 1993].spatially three-dimensional settings [20]. Therefore it is an important issue to perform model reductions if possible, in particular for any kinds of limit regimes.

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“…For relevant results on the analysis and the numerical approximation of a twophase flow model in porous media we refer the reader to [5]. Related work on the mathematical analysis of mechanical models of Hele-Shaw-type have been presented by Perthame et al [14,15].…”

Section: 24mentioning

confidence: 99%

Analysis and Simulations on a Model for the Evolution of Tumors Under the Influence of Nutrient and Drug Application

Trivisa¹,

Weber²

2018

SIAM J. Numer. Anal.

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We investigate the evolution of tumor growth relying on a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the role of nutrient and drug application in the progression of tumors. We construct an explicit convergent numerical scheme to approximate solutions of the nonlinear system of partial differential equations. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The same numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug both shrinkage and growth of tumors can occur. The role of inhomogeneous boundary conditions, vascularization and anisotropies in the development of tumor shape irregularities are discussed.

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Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media

Cited by 17 publications

References 21 publications

Compressible and viscous two-phase flow in porous media based on mixture theory formulation

Compressible and viscous two-phase flow in porous media based on mixture theory formulation

On Darcy- and Brinkman-type models for two-phase flow in asymptotically flat domains

Analysis and Simulations on a Model for the Evolution of Tumors Under the Influence of Nutrient and Drug Application

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