Vertex-centred discretization of multiphase compositional Darcy flows on general meshes

Eymard, Robert; Guichard, Cindy; Herbin, Raphaèle; Masson, Roland

doi:10.1007/s10596-012-9299-x

Cited by 71 publications

(83 citation statements)

References 32 publications

(47 reference statements)

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“…In the numerical tests proposed in this section, we use the Vertex Approximate Gradient scheme, denoted VAG, introduced in [17] and developed for compositional multiphase flows in porous media in [18]. Following the gradient scheme framework, we first recall the construction of the VAG scheme, then we present several numerical experiments which are focused on the simulation of oil migration in a basin with discontinuous capillary pressures.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The aim of this section is first to validate the scheme presented previously, by comparison with a classical upwind approximation of the mobility terms widely used in the oil industry. More precisely, we compare the centered approximation of the relative permeabilities corresponding to our scheme (14), with an upwind first order approximation of the relative permeability of each phase with respect to the sign of the phase Darcy flux (see [3,22], and [18] in the framework of the VAG scheme). A second objective is to assess the Hypothesis (4g) by comparison of the results obtained with k min > 0 with those obtained in the limit case k min = 0 which matches the physical model.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In the case of multiphase flows simulations, we define, between a cell and one of its vertices, a Darcy flux of a phase as described in details in [18]. The advantage of this scheme is that it allows to eliminate all values (u K ) K∈M with respect to the values (u v ) v∈V , leading to linear systems which are well suited to domain decomposition and parallel computing.…”

Section: The Vertex Approximate Gradient Schemementioning

confidence: 99%

See 2 more Smart Citations

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Eymard¹,

Guichard

Herbin

et al. 2013

Z Angew Math Mech

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The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and approximate pressure gradients thanks to monotony and compactness arguments under an assumption of non-degeneracy of the phase relative permeabilities. Strong convergence results stem from the convergence of the norms of the gradients of pressures, which demand handling the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: The Vertex Approximate Gradient Schemementioning

confidence: 99%

See 1 more Smart Citation

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Eymard¹,

Guichard

Herbin

et al. 2013

Z Angew Math Mech

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“…The VAG scheme has been introduced for the discretization of multiphase Darcy flows in [14] for immiscible flows and in [15] for compositional models. The VAG scheme basically uses nodal unknowns like the CVFE method but it also keep the cell-centred unknowns, which are eliminated at the linear algebra level without any fill-in.…”

Section: Introductionmentioning

confidence: 99%

Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Brenner

Groza

Guichard

et al. 2014

Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems

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This paper presents a finite volume discretization of two-phase Darcy flows in discrete fracture networks taking into account the mass exchange between the matrix and the fracture. We consider the asymptotic model for which the fractures are represented as interfaces of codimension one immersed in the matrix domain, leading to the so called hybrid dimensional Darcy flow model. The pressures at the interfaces between the matrix and the fracture network are continuous corresponding to a ratio between the normal permeability of the fracture and the width of the fracture assumed to be large compared with the ratio between the permeability of the matrix and the size of the domain. The discretization is an extension of the Vertex Approximate Gradient (VAG) scheme to the case of hybrid dimensional Darcy flow models. Compared with Control Volume Finite Element (CVFE) approaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rocktypes at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The convergence of the scheme is proved under the assumption that the relative permeabilities are bounded from below by a strictly positive constant. This assumption is needed in the convergence proof in order to take into account discontinuous capillary pressures in particular at the matrix fracture interfaces. The efficiency of our approach compared with CVFE discretizations is shown on two numerical examples of fracture networks in 2D and 3D.

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“…For instance to discretize the Richards equation, Celia et al (1990) consider Finite Element (FE), Narasimhan and Witherspoon (1976) apply Finite Volume (FV) with two-point flux approximation, Manzini and Ferraris (2004) use diamond cell FV, and Knabner and Schneid (2002) employ Mixed Finite Element (MFE). Several other FV methods and variants exist, including the MPFA, SUSHI and VAG schemes, see Eymard et al (2012) and references therein for details. In this work, we propose an alternative approach by applying and comparing DDFV and DG methods.…”

Section: Introductionmentioning

confidence: 99%

Comparison of DDFV and DG Methods for Flow in Anisotropic Heterogeneous Porous Media

Baron

Coudière

Sochala

2013

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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Re´sume´-Comparaison des me´thodes DDFV et DG pour des e´coulements en milieu poreux he´te´roge`ne anisotrope -Nous pre´sentons un travail pre´liminaire visant a`simuler l'injection de gaz dans les aquife`res profonds. Des e´coulements instationnaires monophasiques sont conside´re´s. Nous comparons des sche´mas volumes finis en dualite´discre`te (DDFV, Discrete Duality Finite Volume) et de Galerkin discontinu (DG, Discontinuous Galerkin) utilise´s pour discre´tiser le terme diffusif. Une formule de diffe´rentiation re´trograde d'ordre deux est utilise´e pour la discre´tisation en temps. D'une part, les sche´mas DDFV sont simples a`imple´menter, pre´servent les proprie´te´s physiques et pre´sentent souvent une super convergence en norme L 2 . D'autre part, les me´thodes DG sont flexibles, permettent des ordres de convergence arbitrairement e´leve´s et leur fondement the´orique en font un choix adapte´a`de nombreux proble`mes. La me´thode de Galerkin a`pe´nalisation inte´rieure ponde´re´e syme´trique est choisie dans ce travail. La pre´cision et la robustesse de ces deux sche´mas sont teste´es et compare´es sur des cas tests varie´s, notamment en milieu he´te´roge`ne anisotrope. Abstract -Comparison of DDFV and DG Methods for Flow in Anisotropic Heterogeneous PorousMedia -We present a preliminary work to simulate gas injection in deep aquifers. Unsteady singlephase flows are considered. We compare Discrete Duality Finite Volume (DDFV) and Discontinuous Galerkin (DG) schemes applied to discretize the diffusive term. The second-order backward differentiation formula is used for the time-stepping method. On the one hand, the DDFV methods are easy to implement, ensure a preservation of physical properties and offer superconvergence in the L 2 -norm on a regular basis. On the other hand, the DG methods are flexible, allow arbitrary order of accuracy, and their ample theoretical foundation make them a reliable choice for many computational problems. We consider here the symmetric weighted interior penalty Galerkin method. Accuracy and robustness of these two schemes are tested and compared on various test cases, especially in anisotropic heterogeneous media.

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Vertex-centred discretization of multiphase compositional Darcy flows on general meshes

Cited by 71 publications

References 32 publications

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Comparison of DDFV and DG Methods for Flow in Anisotropic Heterogeneous Porous Media

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