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1. Nodal Discretization of Two-Phase Discrete Fracture Matrix models

Author

Roland Masson, Julian Hennicker, Konstantin Brenner, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Section de mathématiques [Genève], Université de Genève (UNIGE), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Finite volume method, Discretization, Mathematical analysis, 0207 environmental engineering, Upwind scheme, 02 engineering and technology, Classification of discontinuities, 01 natural sciences, 010305 fluids & plasmas, Discontinuity (linguistics), Matrix (mathematics), Nonlinear system, 0103 physical sciences, Fracture (geology), 020701 environmental engineering, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

This chapter reviews the nodal Vertex Approximate Gradient (VAG) discretization of two-phase Darcy flows in fractured porous media for which the fracture network is represented as a manifold of co-dimension one with respect to the surrounding matrix domain. Different types of models and their discretizations are considered depending on the transmission conditions set at matrix fracture interfaces accounting for fractures acting either as drains or both as drains or barriers. Difficulties raised by nodal discretizations in heterogeneous media are investigated and solutions to solve these issues are discussed. It includes the adaptation of the porous volumes at nodal unknowns and discontinuous saturations accounting for the jumps induced by the discontinuity in space of the capillary pressure functions. A new Multi-Point upwind scheme is also introduced for the approximation of the mobilities at matrix fracture interfaces to address the issue of fluxes not defined at faces. The most accurate approach is based on the extension of the discontinuous pressure model to two-phase Darcy flows taking into account the discontinuities of both the pressures and saturations at matrix fracture interfaces. As opposed to single phase flows, It improves the accuracy even in the case of fracture acting as drains. On the other hand this approach can still exhibit a robustness issue in terms of nonlinear convergence.

Published
2021
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2. Modeling and Analysis of the Coupling in Discrete Fracture Matrix models

Author

Roland Masson, Martin J. Gander, Julian Hennicker, Section de mathématiques [Genève], Université de Genève (UNIGE), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), University of Côte d’Azur, CNRS, LJAD, Université de Genève = University of Geneva (UNIGE), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
Model order reduction, Coupling, Numerical Analysis, Asymptotic analysis, Discrete fracture, Applied Mathematics, Mathematical analysis, 010103 numerical & computational mathematics, 01 natural sciences, Fourier analysis and asymptotic analysis and functional analysis and error estimates and model order reduction and discrete fracture matrix models, Mathematics::Numerical Analysis, Reduced order, Physics::Geophysics, Computational Mathematics, Matrix (mathematics), symbols.namesake, 65N55, 65Z05, 46N20, 76S05, Fourier analysis, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; This paper deals with the derivation and analysis of reduced order elliptic PDE models on fractured domains. We use a Fourier analysis to obtain coupling conditions between subdomains, when the fracture is represented as a hypersurface embedded in the surrounded rock matrix. We compare our results to prominent examples from the literature, for diffusive models. In a second step, we present error estimates for the reduced order models in terms of the fracture width. For the proofs, we rely on a combination of Fourier analysis, asymptotic expansions and functional analysis. Finally, we study the behaviour of the error of the reduced order solutions on numerical test cases, when the fracture width tends to zero.

Published
2021
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3. The Gradient Discretisation Method for Two-phase Discrete Fracture Matrix Models in Deformable Porous Media

Author

Francesco Bonaldi, Roland Masson, Konstantin Brenner, Jérôme Droniou, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Monash University [Melbourne], COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)

Subjects
Physics, Darcy's law, Discretization, discrete fracture matrix models, Poromechanics, Linear elasticity, Mathematical analysis, Constitutive equation, poromechanics, 010103 numerical & computational mathematics, 01 natural sciences, convergence analysis, 010101 applied mathematics, two-phase Darcy flows, Matrix (mathematics), Flow (mathematics), Linear approximation, 0101 mathematics, Gradient Discretization, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [3], which covers a large class of conforming and non conforming discretizations. This framework allows a generic convergence analysis of the coupled model using a combination of discrete functional tools. Here, we describe the model together with its numerical discretisation, and we state the convergence result, whose proof will be detailed in a forthcoming paper. This is, to our knowledge, the first convergence result for this type of models taking into account two-phase flows and the nonlinear poro-mechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity [4].

Published
2020
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4. Vertex Approximate Gradient Discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media

Author

Roland Masson, Konstantin Brenner, El Houssaine Quenjel, COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), ANR-16-CE06-0009,CHARMS,Modèles de Réservoirs Quantitatifs pour les Systèmes Hydrothermaux Complexes(2016), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Imagination, Numerical Analysis, Capillary pressure, Physics and Astronomy (miscellaneous), Discretization, Capillary action, Applied Mathematics, media_common.quotation_subject, Mathematical analysis, Energy flux, 010103 numerical & computational mathematics, 01 natural sciences, Upper and lower bounds, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, 0101 mathematics, Saturation (chemistry), Porous medium, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], media_common, Mathematics
Abstract

International audience; In this article, a new nodal discretization is proposed for two-phase Darcy flows in heterogeneous porous media. The scheme combines the Vertex Approximate Gradient (VAG) scheme for the approximation of the gradient fluxes with an Hybrid Upwind (HU) approximation of the mobility terms in the saturation equation. The discretization in space incorporates naturally nodal interface degrees of freedom (d.o.f.) allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves. It is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term. Numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3D large Discrete Fracture Matrix (DFM) models.

Published
2020
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5. Two-Phase Darcy Flows in Fractured and Deformable Porous Media, Convergence Analysis and Iterative Coupling

Author

Jérôme Droniou, Konstantin Brenner, Francesco Bonaldi, Roland Masson, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), Monash university, University of Côte d’Azur, CNRS, LJAD, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and School of Mathematics, Monash University

Subjects
Physics, Finite volume method, Darcy's law, Discretization, Constitutive equation, Mathematical analysis, 010103 numerical & computational mathematics, 01 natural sciences, Finite element method, 010101 applied mathematics, Nonlinear system, Flow (mathematics), Linear approximation, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Summary We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. In this work, the model is derived and discretized using the gradient discretization method which covers a large class of conforming and non conforming discretizations. This framework allows a generic convergence analysis of the coupled model using a combination of discrete functional tools. The convergence of the discrete solution to a weak solution of the model is proved using a priori and compactness estimates. This is, to our knowledge, the first convergence result for this type of models taking into account two-phase flows and the nonlinear poro-mechanical coupling including the cubic nonlinear dependence of the fracture conductivity on the fracture aperture. Previous related works consider a linear approximation obtained for a single-phase flow by freezing the fracture conductivity. Numerical experiments are presented to illustrate this result using a Two-Point Flux Approximation cell centered finite volume scheme for the flow and a P2 finite element method for the mechanics. Iterative coupling algorithms are investigated to solve the coupled discrete nonlinear systems at each time step of the simulation.

Published
2020
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6. Asymptotic Analysis for the Coupling Between Subdomains in Discrete Fracture Matrix Models

Author

Julian Hennicker, Martin J. Gander, and Roland Masson

Subjects
Coupling, Physics, Elliptic operator, Asymptotic analysis, Flow (mathematics), Mathematical analysis, Layer (object-oriented design), Differential operator, Porous medium, Physics::Geophysics, Matrix (geology)
Abstract

We study the behavior of solutions of PDE models on domains containing a heterogeneous layer of aperture tending to zero. We consider general second order differential operators on the outer domains and elliptic operators inside the layer. Our study is motivated by the modeling of flow through fractured porous media, when one represents the fractures as entities of co-dimension one with respect to the surrounding rock matrix.

Published
2020
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7. Two-phase Discrete Fracture Matrix models with linear and nonlinear transmission conditions

Author

Roland Masson, Joubine Aghili, Laurent Trenty, Konstantin Brenner, Julian Hennicker, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), Université de Genève = University of Geneva (UNIGE), Agence Nationale pour la Gestion des Déchets Radioactifs (ANDRA), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and Université de Genève (UNIGE)

Subjects
Darcy's law, Continuous phase modulation, Finite volume method, MSC: 65M08 - 65N08 - 76S05, Discretization, Finite volume discretization, Mathematical analysis, Phase (waves), 010103 numerical & computational mathematics, Two-phase Darcy flow, 010502 geochemistry & geophysics, 01 natural sciences, Matrix (mathematics), Nonlinear system, Nonlinear transmission condition, Continuity equation, Modeling and Simulation, Discrete fracture network, General Earth and Planetary Sciences, Discrete fracture matrix model, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 0105 earth and related environmental sciences, Mathematics
Abstract

International audience; This work deals with two-phase Discrete Fracture Matrix models coupling the two-phase Darcy flow in the matrix domain to the two-phase Darcy flow in the network of fractures represented as co-dimension one surfaces. Two classes of such hybrid-dimensional models are investigated either based on nonlinear or linear transmission conditions at the matrix-fracture interfaces. The linear transmission conditions include the cell-centred upwind approximation of the phase mobilities classicaly used in the porous media flow community as well as a basic extension of the continuous phase pressure model accounting for fractures acting as drains. The nonlinear transmission conditions at the matrix-fracture interfaces are based on the normal flux continuity equation for each phase using additional interface phase pressure unknowns. They are compared both in terms of accuracy and numerical efficiency to a reference equi-dimensional model for which the fractures are represented as full-dimensional subdomains. The discretization focuses on Finite Volume cell-centred Two-Point Flux Approximation which is combined with a local nonlinear solver allowing to eliminate efficiently the additional matrix-fracture interfacial unknowns together with the non-linear transmission conditions. 2D numerical experiments illustrate the better accuracy provided by the nonlinear transmission conditions compared to their linear approximations with a moderate computational overhead obtained thanks to the local nonlinear elimination at the matrix-fracture interfaces. The numerical section is complemented by a comparison of the reduced models on a 3D test case using the Vertex Approximate Gradient scheme.

Published
2019
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8. Hybrid Finite Volume discretization of two-phase Discrete Fracture Matrix models with nonlinear interface solver

Author

Julian Hennicker, Laurent Trenty, Konstantin Brenner, Joubine Aghili, Roland Masson, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Université de Genève (UNIGE), Agence Nationale pour la Gestion des Déchets Radioactifs (ANDRA), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), and Université de Genève = University of Geneva (UNIGE)

Subjects
Materials science, Discrete fracture, Interface (Java), Mathematical analysis, Phase (waves), Finite volume discretization, 010103 numerical & computational mathematics, Solver, 16. Peace & justice, 01 natural sciences, 010305 fluids & plasmas, Nonlinear system, Matrix (mathematics), 0103 physical sciences, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2018
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9. Nearwell local space and time refinement in reservoir simulation

Author

Arthur Moncorgé, Walid Kheriji, Roland Masson, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), Centre scientifique et Technique Jean Feger (CSTJF), TOTAL FINA ELF, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and TOTAL-Scientific and Technical Center Jean Féger (CSTJF)

Subjects
General Computer Science, Discretization, local time stepping, Porous media, MathematicsofComputing_NUMERICALANALYSIS, 010103 numerical & computational mathematics, 01 natural sciences, Domain (mathematical analysis), Theoretical Computer Science, Applied mathematics, nearwell refinement, 0101 mathematics, Mathematics, Numerical Analysis, Darcy's law, Spacetime, Applied Mathematics, Multiphase flow, Mathematical analysis, Domain decomposition methods, Darcy flow, optimized Schwarz methods, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010101 applied mathematics, domain decomposition, Reservoir simulation, Flow (mathematics), Modeling and Simulation, two phase Darcy flow, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In reservoir simulations, nearwell regions usually require finer space and time scales compared with the remaining of the reservoir domain. We present a domain decomposition algorithm for a two phase Darcy flow model coupling nearwell regions locally refined in space and time with a coarser reservoir discretization. The algorithm is based on an optimized Schwarz method using a full overlap at the coarse level. The main advantage of this approach is to apply to fully implicit discretizations of general multiphase flow models and to allow a simple optimization of the interface conditions based on a single phase flow equation.

Published
2015
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10. Gradient discretization of hybrid dimensional Darcy flows in fractured porous media

Author

Cindy Guichard, Mayya Groza, Konstantin Brenner, Roland Masson, Gilles Lebeau, COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Numerical Analysis, Geophysics and Ecology (ANGE), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement - Direction Centre-Est (Cerema Direction Centre-Est), Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema), and Université Nice Sophia Antipolis (... - 2019) (UNS)

Subjects
Finite volume method, Discretization, Applied Mathematics, Numerical analysis, Mathematical analysis, Isotropy, Geometry, 010103 numerical & computational mathematics, 01 natural sciences, law.invention, 010101 applied mathematics, Computational Mathematics, Flow (mathematics), law, Cartesian coordinate system, Hexahedron, [MATH]Mathematics [math], 0101 mathematics, Porous medium, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; This article deals with the discretization of hybrid dimensional model of Darcy flow in fractured porous media. These models couple the flow in the fractures represented as the surfaces of codimension one with the flow in the surrounding matrix. The convergence analysis is carried out in the framework of Gradient schemes which accounts for a large family of conforming and nonconforming dis-cretizations. The Vertex Approximate Gradient (VAG) scheme and the Hybrid Finite Volume (HFV) scheme are applied to such models and are shown to verify the Gradient scheme framework. Our theoretical results are confirmed by a few numerical experiments performed both on tetrahedral and hexahedral meshes in heterogeneous isotropic and anisotropic media.

Published
2015
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11. Numerical analysis of a two-phase flow discrete fracture matrix model

Author

Julian Hennicker, Jérôme Droniou, Roland Masson, School of Mathematical Sciences [Clayton], Monash University [Clayton], COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), TOTAL S.A., TOTAL FINA ELF, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
Series (mathematics), Discretization, Applied Mathematics, Numerical analysis, Mathematical analysis, 010103 numerical & computational mathematics, Codimension, Classification of discontinuities, 01 natural sciences, Gradient discretisation method, Physics::Geophysics, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 010101 applied mathematics, Computational Mathematics, Matrix (mathematics), Convergence analysis, Convergence (routing), Two-phase flow, Discrete fracture matrix model, 0101 mathematics, Two phase Darcy flow, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Hybrid-dimensional model
Abstract

International audience; We present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since pressure discontinuities at the matrix-fracture interfaces are permitted. Additionally, a layer of damaged rock at the matrix-fracture interfaces is accounted for. The numerical analysis is carried out in the general framework of the Gradient Discretisation Method. Compactness techniques are used to establish convergence results for a wide range of possible numerical schemes; the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis. A series of numerical experiments conclude the paper, with a study of the influence of the damaged layer on the numerical solution.

Published
2018
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12. Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Author

Konstantin Brenner, Roland Masson, Laurent Jeannin, Mayya Groza, Jeanne Pellerin, COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), ENGIE E&P International [La Défense], Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin] (WIAS), Forschungsverbund Berlin e.V. (FVB) (FVB), Supported by ENGIE EP, Storing and GeoPor ANR project, ANR-13-JS01-0007,GEOPOR,Approche géométrique pour les écoulements en milieux poreux: théorie et numérique(2013), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and Laboratoire Jean Alexandre Dieudonné (JAD)

Subjects
Capillary pressure, Mathematical optimization, Discretization, 010103 numerical & computational mathematics, 01 natural sciences, Fluid dynamics, Discrete fracture network, 0101 mathematics, Computers in Earth Sciences, [MATH]Mathematics [math], Mathematics, Darcy's law, Mathematical analysis, Two-phase Darcy flow, MSC 65M08 and 76S05, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Computational Theory and Mathematics, Heterogeneous porous media, Two-phase flow, Finite Volume Discterization, Energy source, Porous medium
Abstract

International audience; Fully implicit time-space discretizations applied to the two-phase Darcy flow problem lead to the systems of nonlinear equations, which are traditionally solved by some variant of Newton's method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton's method is not invariant with respect to a nonlinear change of variable. In this regard the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two phase water-gas Darcy flow in fractured porous media for which the fractures are modeled as interfaces of co-dimension one. The problem is discretized using an extension of Vertex Approximate Gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom.

Published
2017
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13. Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation

Author

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

Subjects
Nonlinear system, Compact space, Applied Mathematics, Mathematical analysis, Computational Mechanics, Richards equation, Two-phase flow, Saturation (chemistry), Porous medium, Finite element method, Pressure gradient, Mathematics
Abstract

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.

Published
2013
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14. Vertex-centred discretization of multiphase compositional Darcy flows on general meshes

Author

Cindy Guichard, Robert Eymard, Roland Masson, Raphaèle Herbin, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Laboratoire Jean Alexandre Dieudonné (LJAD), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), and Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS)

Subjects
Discretization, Geometry, 010103 numerical & computational mathematics, 01 natural sciences, Stencil, law.invention, Finite element, law, Heterogeneous anisotropic media, Polygon mesh, Cartesian coordinate system, 0101 mathematics, Computers in Earth Sciences, Mathematics, Finite volume method, Hydrogeology, Mathematical analysis, Finite element method, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Permeability (earth sciences), Computational Theory and Mathematics, General meshes, Compositional multiphase Darcy flow, Finite volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; This paper concerns the discretization on general 3D meshes of multiphase compositional Darcy flows in heterogeneous anisotropic porous media. Extending Coats' formulation to an arbitrary number of phases, the model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium, and dynamically manages phase appearance and disappearance. The spatial discretization of the multiphase compositional Darcy flows is based on a generalization of the Vertex Approximate Gradient scheme (VAG), already introduced for single phase diffusive problems. It leads to an unconditionally coercive scheme for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes, and the number of unknowns on tetrahedral meshes is considerably reduced, compared with usual cell-centred approaches. The efficiency of our approach is exhibited on the nearwell injection of miscible CO2 in a saline aquifer taking into account the vaporization of H2O in the gas phase as well as the precipitation of salt.

Published
2012
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15. A relaxation method for two-phase flow models with hydrodynamic closure law

Author

Frédéric Coquel, Quang Huy Tran, Roland Masson, Michaël Baudin, and Christophe Berthon

Subjects
Conservation law, Partial differential equation, Applied Mathematics, Numerical analysis, Degenerate energy levels, Mathematical analysis, Computational Mathematics, Nonlinear system, symbols.namesake, Riemann problem, Simple (abstract algebra), Law, symbols, Mathematics, Numerical stability
Abstract

This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided.

Published
2004
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16. Domain decomposition and splitting methods for Mortar mixed finite element approximations to parabolic equations

Author

Roland Masson, Stéphanie Gaiffe, and Roland Glowinski

Subjects
Computational Mathematics, Partial differential equation, Discretization, Applied Mathematics, Numerical analysis, Mathematical analysis, Projection method, Domain decomposition methods, Parabolic partial differential equation, Finite element method, Mortar methods, Mathematics
Abstract

We introduce in this article a new domain decomposition algorithm for parabolic problems that combines Mortar Mixed Finite Element methods for the space discretization with operator splitting schemes for the time discretization. The main advantage of this method is to be fully parallel. The algorithm is proven to be unconditionally stable and a convergence result in $$\mathcal{O}$$ (Δt/h 1/2) is presented.

Published
2002
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17. [Untitled]

Author

Pascal Joly and Roland Masson

Subjects
Sobolev space, Elliptic operator, Operator (computer programming), Trace (linear algebra), Wavelet, Preconditioner, Applied Mathematics, Diagonal, Mathematical analysis, Galerkin method, Computer Science::Numerical Analysis, Mathematics::Numerical Analysis, Mathematics
Abstract

The diagonal preconditioning in wavelet basis enables one to obtain an optimal preconditioner for Galerkin discretizations of elliptic operators in Sobolev norms of both positive and negative smoothness. We develop these techniques in order to solve efficiently the bi-Laplacian or the bidimensional Stokes problem in ψ–ω formulation using a diagonal preconditioning in wavelet basis for the H−1/2(∂Ω) boundary operator that relates the trace of ∂nψ to the trace of ω.

Published
2000
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18. Wavelet Methods for Second-Order Elliptic Problems, Preconditioning, and Adaptivity

Author

Roland Masson and Albert Cohen

Subjects
Computational Mathematics, Elliptic operator, Constant coefficients, Operator (computer programming), Wavelet, Applied Mathematics, Multiresolution analysis, Mathematical analysis, MathematicsofComputing_NUMERICALANALYSIS, Domain decomposition methods, Boundary value problem, Laplace operator, Mathematics
Abstract

Wavelet methods allow us to combine high-order accuracy, efficient preconditioning techniques, and adaptive approximations in order to solve efficiently elliptic operator equations. Many difficulties remain, in particular, related to the adaptation of wavelet decompositions to bounded domains with prescribed boundary conditions, leading to possibly high constants in the ${\cal O}(1)$ preconditioning. In this paper we consider the framework of conforming domain decomposition to generate our wavelet bases and second-order operators. We emphasize the choice of the wavelets near the boundary of the tensor product reference domain in order to optimize the efficiency of the diagonal preconditioning of elliptic operators. In order to improve the constants obtained by such diagonal preconditionings, we propose to take into account interactions between the scales through the computation of a sparse approximate inverse (SPAI) on a set of nonzero entries obtained from the compression of the operator itself in the wavelet basis. The efficiency of these methods is illustrated by solving elliptic second-order problems with variable or constant coefficients and homogeneous boundary conditions on a uniform discretization. Finally, we propose a coupling of the iterative solver with an adaptive space refinement technique. On the Laplacian model problem, our experiments show that this algorithm generates an optimal nonlinear approximation of the solution.

Published
1999
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19. Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Author

Mayya Groza, Konstantin Brenner, Cindy Guichard, and Roland Masson

Subjects
Vertex (graph theory), Capillary pressure, Matrix (mathematics), Darcy's law, Discretization, Mathematical analysis, Convergence (routing), Fracture (geology), Porous medium, Physics::Geophysics, Mathematics
Abstract

This paper presents the Vertex Approximate Gradient (VAG) discretization of a two-phase Darcy flow in discrete fracture networks (DFN) 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 with continuous pressures at the matrix fracture interface. 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 but cover the case of discontinuous capillary pressures. The efficiency of our approach compared with CVFE discretizations is shown on a 3D fracture network with very low matrix permeability.

Published
2014
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20. BIORTHOGONAL SPLINE WAVELETS ON THE INTERVAL FOR THE RESOLUTION OF BOUNDARY PROBLEMS

Author

Roland Masson

Subjects
Hermite spline, Applied Mathematics, Perfect spline, Mathematical analysis, Biorthogonal polynomial, Mathematics::Numerical Analysis, Spline (mathematics), Smoothing spline, Computer Science::Graphics, Modeling and Simulation, Biorthogonal system, Spline interpolation, Thin plate spline, Mathematics
Abstract

We construct biorthogonal spline wavelets on the interval for different spline orders. We then discuss the accuracy of the construction and test the wavelet transform. Building the corresponding wavelet basis for [Formula: see text] boundary conditions we use it for the resolution of elliptic boundary problems both with pure Galerkin and Petrov-Galerkin schemes. The results are compared from stability and convergence points of view. The main conclusion is that using the biorthogonal counterpart of spline functions as trial spaces for low order splines provides much more advantages than using spline functions themselves as trial spaces for high order splines.

Published
1996
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21. Algebraic Domain Decomposition Methods for Highly Heterogeneous Problems

Author

Mikolaj Szydlarski, Hua Xiang, Frédéric Nataf, Tao Zhao, Roland Masson, Pascal Havé, IFP Energies nouvelles (IFPEN), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Algorithms and parallel tools for integrated numerical simulations (ALPINES), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Global parallel and distributed computing (GRAND-LARGE), Laboratoire de Recherche en Informatique (LRI), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Laboratoire d'Informatique Fondamentale de Lille (LIFL), Université de Lille, Sciences et Technologies-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lille, Sciences Humaines et Sociales-Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Sciences et Technologies-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lille, Sciences Humaines et Sociales-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Wuhan], Huazhong University of Science and Technology [Wuhan] (HUST)-Wuhan University [China], Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Paris-Sud - Paris 11 (UP11)-Laboratoire d'Informatique Fondamentale de Lille (LIFL), Université de Lille, Sciences et Technologies-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lille, Sciences Humaines et Sociales-Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Sciences et Technologies-Université de Lille, Sciences Humaines et Sociales-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Recherche en Informatique (LRI), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec, and Nataf, Frédéric

Subjects
Sequence, Applied Mathematics, Linear system, Mathematical analysis, Domain decomposition methods, 010103 numerical & computational mathematics, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Solver, 01 natural sciences, Matrix similarity, 010101 applied mathematics, Computational Mathematics, Matrix (mathematics), Graph (abstract data type), Applied mathematics, Adjacency list, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; We consider the solving of linear systems arising from porous media flow simulations with high heterogeneities. Using a Newton algorithm to handle the non-linearity leads to the solving of a sequence of linear systems with different but similar matrices and right hand sides. The parallel solver is a Schwarz domain decomposition method. The unknowns are partitioned with a criterion based on the entries of the input matrix. This leads to substantial gains compared to a partition based only on the adjacency graph of the matrix. From the information generated during the solving of the first linear system, it is possible to build a coarse space for a two-level domain decomposition algorithm that leads to an acceleration of the convergence of the subsequent linear systems. We compare two coarse spaces: a classical approach and a new one adapted to parallel implementation.

Published
2013

22. Vertex-centred Discretization of Multiphase Compositional Darcy Flows on General Meshes

Author

Cindy Guichard, Roland Masson, Raphaèle Herbin, Robert Eymard, Pierre Samier, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical Analysis, Geophysics and Ecology (ANGE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Analyse, Topologie, Probabilités (LATP), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (JAD), Centre scientifique et Technique Jean Feger (CSTJF), TOTAL FINA ELF, Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), TOTAL CSTJF, Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
Darcy's law, Discretization, Multiphase flow, Mathematical analysis, 010103 numerical & computational mathematics, 02 engineering and technology, Volume mesh, 01 natural sciences, Mathematics::Numerical Analysis, law.invention, Vertex (geometry), Permeability (earth sciences), 020401 chemical engineering, law, Polygon mesh, Cartesian coordinate system, 0204 chemical engineering, 0101 mathematics, Geomorphology, Geology, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; This paper introduces a vertex centred discretization on general 3D meshes of multiphase Darcy flows in heterogeneous anisotropic porous media. The model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium. The conservative spatial discretization of the Darcy fluxes is based on the Vertex Approximate Gradient scheme (VAG) which is unconditionally coercive for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes. On tetrahedral meshes, the number of unknowns is considerably reduced, by typically a factor five, compared with usual cell-centred MultiPoint Fluxes Approximations, which is a key asset for multiphase flow simulations on unstructured meshes. An adaptive choice of the pore volume at the vertices ensures the accuracy of the discretization even for coarse meshes on highly heterogeneous media. This approach can easily be implemented on existing reservoir simulators using a graph of transmissibilities for the computation of the fluxes. The efficiency of our approach is exhibited on several two phase and three phase Darcy flow examples. In particular it includes the nearwell injection of miscible CO2 in a saline aquifer taking into account the precipitation of salt.

Published
2012
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23. Vertex centred Discretization of Two-Phase Darcy flows on General Meshes

Author

Raphaèle Herbin, Cindy Guichard, Robert Eymard, Roland Masson, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Laboratoire d'Analyse, Topologie, Probabilités (LATP), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), and École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
diffusion fluxes, Discretization, two phase Darcy flows, Anisotropic porous medium, Geometry, 010103 numerical & computational mathematics, 01 natural sciences, Physics::Fluid Dynamics, QA1-939, Polygon mesh, 0101 mathematics, Mathematics, heterogeneous anisotropic media, general meshes, T57-57.97, Applied mathematics. Quantitative methods, Darcy's law, Finite volume method, Mathematical analysis, Vertex (geometry), 010101 applied mathematics, Elliptic curve, Finite volume, Hyperbolic partial differential equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

This paper concerns the discretization of multiphase Darcy flows, in the case of heterogeneous anisotropic porous media and general 3D meshes used in practice to represent reservoir and basin geometries. An unconditionally coercive and symmetric vertex centred approach is introduced in this paper. This scheme extends the Vertex Approximate Gradient scheme (VAG), already introduced for single phase diffusive problems in [9], to multiphase Darcy flows. The convergence of the VAG scheme is proved for a simplified two-phase Darcy flow model, coupling an elliptic equation for the pressure and a linear hyperbolic equation for the saturation. The ability for the VAG scheme to efficiently deal with highly heterogeneous media and complex meshes is exhibited on immiscible and miscible two phase Darcy flow models. Cet article porte sur la discrétisation des flux de Darcy polyphasiques au sein de milieux poreux hétérogènes et anisotropes, dans des maillages tridimensionnels généraux utilisés dans le contexte de la simulation de réservoir ou de bassin. Un schéma avec inconnues aux sommets [9], qui a l’avantage d’être inconditionnellement coercif et symétrique, est généralisé au cas des écoulements de Darcy polyphasiques. La convergence du schéma est démontrée sur un modèle diphasique simplifié, couplant une équation elliptique pour la pression à une équation hyperbolique linéaire pour la saturation. On illustre ensuite la capacité du schéma à prendre en compte efficacement les fortes hétérogénéités et les maillages complexes sur des exemples d’écoulements diphasiques immiscibles et miscibles.

Published
2012
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24. Hybrid finite volume discretization of linear elasticity models on general meshes

Author

Robert Eymard, Simon Lemaire, Roland Masson, Daniele Antonio Di Pietro, IFP Energies nouvelles (IFPEN), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), J. Fořt, J. Fürst, J. Halama, R. Herbin, F. Hubert, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), F. Hubert Eds., Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Darcy's law, Discretization, Linear elasticity, Mathematical analysis, Geometry, Hybrid finite volumes, 010103 numerical & computational mathematics, Linear interpolation, 01 natural sciences, 010101 applied mathematics, Compatibility (mechanics), Displacement field, Piecewise, General meshes, Polygon mesh, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, ComputingMethodologies_COMPUTERGRAPHICS
Abstract

The original publication is available at www.springerlink.com; International audience; This paper presents a new discretization scheme for linear elasticity models using only one degree of freedom per face corresponding to the normal component of the displacement. The scheme is based on a piecewise constant gradient construction and a discrete variational formulation for the displacement field. The tangential components of the displacement field are eliminated using a second order linear interpolation. Our main motivation is the coupling of geomechanical models and porous media flows arising in reservoir or CO2 storage simulations. Our scheme guarantees by construction the compatibility condition between the displacement discretization and the usual cell-centered finite volume discretization of the Darcy flow model. In addition it applies on general meshes, possibly non conforming such as Corner Point Geometries commonly used in reservoir and CO2 storage simulations.

Published
2011
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25. Grid Orientation Effect and MultiPoint Flux Approximation

Author

Robert Eymard, Roland Masson, Cindy Guichard, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical Analysis, Geophysics and Ecology (ANGE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS), and Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Diffusion equation, 020209 energy, Computation, Mathematical analysis, Flux, Geometry, 02 engineering and technology, Finite Volume Methods, 01 natural sciences, Stencil, Grid Orientation Effect, 010101 applied mathematics, MSC2010: 76S05, 65M08, 76E06, Five-point stencil, 0202 electrical engineering, electronic engineering, information engineering, Grid orientation, 0101 mathematics, Convection–diffusion equation, Porous medium, Two Phase Porous Media Flow, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; Some cases of nonlinear coupling between a diffusion equation, related to the computation of a pressure field within a porous medium, and a convection equation, related to the conservation of a species, lead to the apparition of the so-called grid orientation effect. We propose in this paper a new procedure to eliminate this Grid Orientation Effect, only based on the modification of the stencil of the discrete version of the convection equation. Numerical results show the efficiency and the accuracy of the method.

Published
2011
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26. Convergence of the Finite Volume MPFA O Scheme for Heterogeneous Anisotropic Diffusion Problems on General Meshes

Author

Roland Masson and Leo Agelas

Subjects
Grid pattern, Finite volume method, Anisotropic diffusion, Scheme (mathematics), Convergence (routing), Mathematical analysis, Polygon mesh, General Medicine, Diffusion (business), Anisotropy, Mathematics
Abstract

This Note proves the convergence of the finite volume MultiPoint Flux Approximation (MPFA) O scheme for anisotropic and heterogeneous diffusion problems. Its main originality is that our framework and proof deal with general polygonal and polyhedral meshes as well as with L ∞ diffusion coefficients, which is essential in practical applications. To cite this article: L. Agelas, R. Masson, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Published
2008
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27. Existence and Uniqueness of a Weak Solution to a Stratigraphic Model

Author

Véronique Gervais, Roland Masson, Robert Eymard, and Thierry Gallouët

Subjects
Lithology, Advection, Weak solution, Physics::Medical Physics, Mathematical analysis, Geometry, Physics::Geophysics, Boundary value problem, Uniqueness, Diffusion (business), Foreland basin, Physics::Atmospheric and Oceanic Physics, Linear equation, Mathematics
Abstract

In this paper, we study a multi-lithology diffusion model used to simulate the evolution through time of a sedimentary basin composed of several lithologies such as sand or shale. It is a simplified model for which the surficial flux in lithology i is taken proportional to the slope and to a lithology fraction c i s in lithology i at the top of the basin with a unitary diffusion coefficient. Thus, the sediment thickness variable satisfies a linear parabolic problem and decouples from the other unknowns. The remaining equations couple, for each lithology, a first order linear equation for the surface concentration c i s with a linear advection equation for the basin concentration, for which c i s appears as an input boundary condition at the top of the basin in case of sedimentation. The existence and uniqueness of a weak solution in L ∞ is proved for this problem.

Published
2004
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28. Stabilized wavelet approximations of the Stokes problem

Author

Roland Masson and Claudio Canuto

Subjects
Computational Mathematics, Spline (mathematics), Algebra and Number Theory, Partial differential equation, Wavelet, Discretization, Applied Mathematics, Norm (mathematics), Biorthogonal system, Mathematical analysis, Residual, Differential operator, Mathematics
Abstract

We propose a new consistent, residual-based stabilization of the Stokes problem. The stabilizing term involves a pseudo-differential operator, defined via a wavelet expansion of the test pressures. This yields control on the full L 2 L^2 -norm of the resulting approximate pressure independently of any discretization parameter. The method is particularly well suited for being applied within an adaptive discretization strategy. We detail the realization of the stabilizing term through biorthogonal spline wavelets, and we provide some numerical results.

Published
2001

29. A Chebyshev collocation algorithm for 2-D non-boussinesq convection

Author

P Perrot, P. Le Quéré, and Roland Masson

Subjects
Convection, Discretization, Physics and Astronomy (miscellaneous), Iterative method, Perfect gas, Physics::Fluid Dynamics, symbols.namesake, Collocation method, Fluid dynamics, Applied mathematics, Orthogonal collocation, Chebyshev collocation, Chebyshev equation, Mathematics, Numerical Analysis, Natural convection, Applied Mathematics, Mathematical analysis, Chebyshev iteration, Rayleigh number, Computer Science Applications, Computational Mathematics, Helmholtz free energy, Modeling and Simulation, symbols, Algorithm
Abstract

A Chebyshev collocation algorithm is developed to integrate the time-dependent Navier-Stokes equations for natural convection flow with large temperature differences. The working fluid is assumed to be a perfect gas and its thermophysical properties vary with temperature according to Sutherland laws. The governing equations do not allow for acoustic waves. The generalized Helmholtz and Uzawa operators which arise from time discretization are solved iteratively and the performances of several types of preconditioners and iterative schemes are examined. The algorithm is validated by computing almost Boussinesq flows and by comparing with previous results obtained with a finite difference algorithm. We investigate the effects of the temperature difference and of total mass contained within the cavity on the transition to unsteadiness in a cavity of aspect ratio 8. It is shown that these parameters have, indeed, a significant effect on the value of Rayleigh number at which unsteadiness is triggered. We also discuss the nature of the time-periodic solution which is obtained for Ra slightly supercritical values.

Published
1992
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30. Nonlinear Preconditioning: how to use a Nonlinear Schwarz Method to Precondition Newton's Method

Author

Martin J. Gander, Victorita Dolean, Roland Masson, Walid Kheriji, Felix Kwok, University of Strathclyde [Glasgow], Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Section de mathématiques [Genève], Université de Genève (UNIGE), and Dolean, Victorita

Subjects
Iterative method, 65N22, 010103 numerical & computational mathematics, 01 natural sciences, Newton's method in optimization, Mathematics::Numerical Analysis, symbols.namesake, Additive Schwarz method, FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Schwarz alternating method, QA, Newton's method, 65F10, Mathematics, Preconditioner, Applied Mathematics, Mathematical analysis, Domain decomposition methods, Numerical Analysis (math.NA), [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Computer Science::Numerical Analysis, Local convergence, 010101 applied mathematics, Computational Mathematics, Precondi-tioning Newton's Method AMS subject classifications 65M55, Computer Science::Mathematical Software, symbols, Non-Linear Preconditioning, Two-Level Non-Linear Schwarz Methods, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration, and thus converges much faster. We show in this paper that also for non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two level non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem. 1. Introduction. Non-linear partial differential equations are usually solved after discretization by Newton's method or variants thereof. While Newton's method converges well from an initial guess close to the solution, its convergence behaviour can be erratic and the method can lose all its effectiveness if the initial guess is too far from the solution. Instead of using Newton, one can use a domain decomposition iteration , applied directly to the non-linear partial differential equations, and one obtains then much smaller subdomain problems, which are often easier to solve by Newton's method than the global problem. The first analysis of an extension of the classical alternating Schwarz method to non-linear monotone problems can be found in [24], where a convergence proof is given at the continuous level for a minimization formulation of the problem. A two-level parallel additive Schwarz method for non-linear problems was proposed and analyzed in [12], where the authors prove that the non-linear iteration converges locally at the same rate as the linear iteration applied to the linearized equations about the fixed point, and also a global convergence result is given in the case of a minimization formulation under certain conditions. In [25], the classical alternating Schwarz method is studied at the continuous level, when applied

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