Convergence of the Homogenization Process for a Double-Porosity Model of Immiscible Two-Phase Flow

Bourgeat, Alain; Luckhaus, Stephan; Mikelić, Andro

doi:10.1137/s0036141094276457

Cited by 125 publications

(144 citation statements)

References 15 publications

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“…In fact, in [9] a different law is derived by starting with another ansatz; our analysis recovers the nonlinear function of [9]. For rigorous homogenization results of nonlinear equations we refer the reader to [7] for a double porosity model, to [6] for a stochastic setting, and to [4,17] for models of capillary hysteresis. All these results concern the case of nondegenerate coefficients.…”

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confidence: 93%

Homogenization of Degenerate Two-Phase Flow Equations with Oil Trapping

Schweizer¹

2008

SIAM J. Math. Anal.

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Abstract. We consider the one-dimensional degenerate two-phase flow equations as a model for water drive in oil recovery. The effect of oil trapping is observed in strongly heterogeneous materials with large variations in the permeabilities and in the capillary pressure curves. In such materials, a vanishing oil saturation may appear at interior interfaces and inhibit the oil recovery. We introduce a free boundary problem that separates a critical region with locally vanishing permeabilities from a strictly parabolic region and we give a rigorous derivation of the effective conservation law.

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confidence: 93%

Homogenization of Degenerate Two-Phase Flow Equations with Oil Trapping

Schweizer¹

2008

SIAM J. Math. Anal.

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“…To this end, we use the localization (dilation, unfolding) operator T ε , and its properties, see e.g. in [4], and [5]. Taking into account the lower-semicontinuity of the norm with respect to the weak convergence, we have for all t ∈ (0, T ) lim inf…”

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confidence: 99%

Homogenization Limit of a Model System for Interaction of Flow, Chemical Reactions, and Mechanics in Cell Tissues

Jäger

Mikelić

Neuss‐Radu³

2011

SIAM J. Math. Anal.

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Abstract. In this article we obtain rigorously the homogenization limit for a fluid-structurereactive flow system. It consists of cell tissue and intercellular liquid, transporting solutes. The cell tissue is supposed linearly elastic and deforming with a viscous non-stationary flow. The elastic moduli of the tissue change with cumulative concentration value. In the limit, when the scale parameter goes to zero, we obtain the quasi-static Biot system, coupled with the upscaled reactive flow. Effective Biot's coefficients depend on the reactant concentration. Additionally to the weak two-scale convergence results, we prove convergence of the elastic and viscous energies. This then implies a strong two-scale convergence result for the fluid-structure variables. Next we establish the regularity of the solutions for the upscaled equations. In our knowledge, it is the only known study of the regularity of solutions to the quasi-static Biot system. The regularity is used to prove the uniqueness for the upscaled model.

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“…For the unfolding operator defined in the fully periodic context (classical homogenization), we refer to [8,11] and to [36] for an application of this technique relevant to the present context . Below, we list some simple but useful propositions.…”

Section: The Boundary Unfolding Operatormentioning

confidence: 99%

“…This technique has been systematically introduced by [11] (see also [8] for similar ideas) and has been applied for homogenization problems, in particular for periodic homogenization. Relevant references to this work are [32] and [12].…”

Section: Introductionmentioning

confidence: 99%

Rigorous upscaling of rough boundaries for reactive flows

Kumar

Helvoort²,

Pop

2014

Z Angew Math Mech

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We consider a mathematical model for reactive flow in a channel having a rough (periodically oscillating) boundary with both period and amplitude ε. The ions are being transported by the convection and diffusion processes. These ions can react at the rough boundaries and get attached to form the crystal (precipitation) and become immobile. The reverse process of dissolution is also possible. The model involves non‐linear and multi‐valued rates and is posed in a fixed geometry with rough boundaries. We provide a rigorous justification for the upscaling process in which we define an upscaled problem defined in a simpler domain with flat boundaries. To this aim, we use periodic unfolding techniques combined with translation estimates. Numerical experiments confirm the theoretical predictions and illustrate a practical application of this upscaling process.

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Convergence of the Homogenization Process for a Double-Porosity Model of Immiscible Two-Phase Flow

Cited by 125 publications

References 15 publications

Homogenization of Degenerate Two-Phase Flow Equations with Oil Trapping

Homogenization of Degenerate Two-Phase Flow Equations with Oil Trapping

Homogenization Limit of a Model System for Interaction of Flow, Chemical Reactions, and Mechanics in Cell Tissues

Rigorous upscaling of rough boundaries for reactive flows

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