Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository

Amaziane, Brahim; Antont︠s︡ev, S. N.; Pankratov, Leonid; Piatnitski, Andrey

doi:10.1137/100790215

Cited by 42 publications

(48 citation statements)

References 29 publications

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“…In this section, basing on the energy equality, we establish a priori estimates for solution to (2.24)-(2.28). Following the ideas of Galusinski and Saad [29] (see also [9,28]), we obtain the following result:…”

Section: Uniform Estimates For the Solutions To Problem (224)-(228)mentioning

confidence: 56%

“…Using these estimates and applying Lemma from , we obtain the following compactness result. Proposition For any x 0 ∈Ω∖ A n , under our standing assumptions the families

{({}, s_{m, x_{0}}^{ε})}_{ε > 0}, {({}, θ_{m, x_{0}}^{ε})}_{ε > 0}

are compact sets in the space L q ( Y ×]0, T [) for all q ∈[1, ∞ [, where

θ_{m, x_{0}}^{ε} \overset{def}{=} ϱ_{g} ((), p_{m, x_{0}}^{ε} + G_{g} ((), s_{m, x_{0}}^{ε})) ((), 1 - s_{m, x_{0}}^{ε})

.…”

Section: Dilation Operator and Convergence Results For The Dilated Fumentioning

confidence: 95%

“…In this section, basing on the energy equality , we establish a priori estimates for solution to –. Following the ideas of Galusinski and Saad (see also ), we obtain the following result: Lemma Let

(〈〉, p_{g}^{ε}, p_{w}^{ε})

be a solution to problem . Then,

\frac{d}{d t} \int_{normalΩ} {normalΦ}^{ε} (x) E^{ε} (x, t) dx + \int_{normalΩ} K^{ε} (x) ({}, λ_{w} (S^{ε}) \nabla p_{w}^{ε} \cdot ((), \nabla p_{w}^{ε} - \overset{g}{true}) + λ_{g} (S^{ε}) \nabla p_{g}^{ε} \cdot ((), \nabla p_{g}^{ε} - ϱ_{g} ((), p_{g}^{ε}) \overset{g}{true})) dx = 0

in the sense of distributions.…”

Section: Uniform Estimates For the Solutions To Problem (224)–(228)mentioning

confidence: 90%

“…However, the situation is quite different for immiscible compressible two‐phase flow in porous media, where, only recently, few results have been obtained. In a previous paper , we gave a homogenization result for immiscible compressible two‐phase, such as water–gas, flow in porous media. The problem is formulated in terms of a coupled system of diffusion–convection equations in a domain with periodic microstructure with rapidly oscillating porosity function and absolute permeability tensor.…”

Section: Introductionmentioning

confidence: 99%

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Homogenization of a model for water-gas flow through double-porosity media

Amaziane

Pankratov

2015

Math. Meth. Appl. Sci.

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This paper presents a study of immiscible compressible two-phase, such as water and gas, flow through double porosity media. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the standard Darcy-Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, that is, where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium becomes discontinuous. Consequently, the model involves highly oscillatory characteristics. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. We obtain the convergence of the solutions, and a macroscopic model of the problem is constructed using the notion of two-scale convergence combined with the dilatation technique.

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Section: Uniform Estimates For the Solutions To Problem (224)-(228)mentioning

confidence: 56%

“…Using these estimates and applying Lemma from , we obtain the following compactness result. Proposition For any x 0 ∈Ω∖ A n , under our standing assumptions the families

{({}, s_{m, x_{0}}^{ε})}_{ε > 0}, {({}, θ_{m, x_{0}}^{ε})}_{ε > 0}

are compact sets in the space L q ( Y ×]0, T [) for all q ∈[1, ∞ [, where

θ_{m, x_{0}}^{ε} \overset{def}{=} ϱ_{g} ((), p_{m, x_{0}}^{ε} + G_{g} ((), s_{m, x_{0}}^{ε})) ((), 1 - s_{m, x_{0}}^{ε})

.…”

Section: Dilation Operator and Convergence Results For The Dilated Fumentioning

confidence: 95%

(〈〉, p_{g}^{ε}, p_{w}^{ε})

be a solution to problem . Then,

\frac{d}{d t} \int_{normalΩ} {normalΦ}^{ε} (x) E^{ε} (x, t) dx + \int_{normalΩ} K^{ε} (x) ({}, λ_{w} (S^{ε}) \nabla p_{w}^{ε} \cdot ((), \nabla p_{w}^{ε} - \overset{g}{true}) + λ_{g} (S^{ε}) \nabla p_{g}^{ε} \cdot ((), \nabla p_{g}^{ε} - ϱ_{g} ((), p_{g}^{ε}) \overset{g}{true})) dx = 0

in the sense of distributions.…”

Section: Uniform Estimates For the Solutions To Problem (224)–(228)mentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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Homogenization of a model for water-gas flow through double-porosity media

Amaziane

Pankratov

2015

Math. Meth. Appl. Sci.

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“…Recently, in [7], an existence result has been shown in the case where the densities of each phase depend on the corresponding pressure. This approach is also used in [8,9] to treat a homogenization problem of immiscible compressible water-gas flow in porous media. For miscible and compressible flow, we refer to [10,11] for more details.…”

Section: Introductionmentioning

confidence: 99%

Two-Component Two-Compressible Flow in a Porous Medium

Caro

Saad

2011

Acta Appl Math

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We prove existence of solutions of a two-compressible (liquid and gas) phase flow model in porous media with two components (water and hydrogen). This model is obtained by writing the mass conservation for each component in each phase. We suppose that the mass exchange between dissolved hydrogen and hydrogen in the gas phase is supposed finite. This mass exchange is modeled by a source term on each mass conservation equations.

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Mathematical analysis of the two‐phase two‐component fluid flow in porous media by an artificial persistent variables approach

Vrbaški,

Žgaljić Keko

2024

Math Methods in App Sciences

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This paper deals with the existence of weak solutions of the system that describes the two‐phase two‐component fluid flow in porous media. Both two‐phase and possible one‐phase flow regions are taken into account. Our research is based on a global pressure, an artificial variable that allows us to partially decouple the original equations. As a second primary unknown for the system, we choose the gas pseudo‐pressure, a persistent variable which coincides with the gas pressure in the two‐phase regions while it does not have physical meaning in one‐phase flow regions, when only the liquid phase is present. This allows us to introduce an another persistent variable that is an artificial variable in one‐phase flow regions and a physical variable in two‐phase flow regions—the capillary pseudopressure. We rewrite the system's equations in a fully equivalent form in terms of the global pressure and the gas‐pseudo pressure. In order to prove the existence of weak solutions of obtained system, we also use the capillary pseudo‐pressure. By using it, we can decouple obtained equations on the discrete level. This allows us to derive the existence result for weak solutions in more tractable way.

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Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository

Cited by 42 publications

References 29 publications

Homogenization of a model for water-gas flow through double-porosity media

Homogenization of a model for water-gas flow through double-porosity media

Two-Component Two-Compressible Flow in a Porous Medium

Mathematical analysis of the two‐phase two‐component fluid flow in porous media by an artificial persistent variables approach

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