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18. Mixed and Nitsche's discretizations of Coulomb frictional contact-mechanics for mixed dimensional poromechanical models

Author

Mohamed LAAZIRI, Laurence Beaude, Franz Chouly, Roland Masson, Bureau de Recherches Géologiques et Minières (BRGM) (BRGM), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-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), Bureau de Recherches Géologiques et Minières (BRGM), ANDRA, I-Site BFC project NAANoD, and ANR-17-EURE-0002,EIPHI,Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité(2017)

Subjects
Nitsche's method, Contact mechanics, Coulomb friction, Mechanics of Materials, Mechanical Engineering, Stabilized mixed method, Poromechanics, Discrete Fracture Matrix model, Computational Mechanics, General Physics and Astronomy, [PHYS.MECA]Physics [physics]/Mechanics [physics], Computer Science Applications
Abstract

International audience; This work deals with the discretization of single-phase Darcy flows in fractured and deformable porous media, including frictional contact at the matrix-fracture interfaces. Fractures are described as a network of planar surfaces leading to so-called mixed-dimensional models. Small displacements and a linear poro-elastic behavior are considered in the matrix. One key difficulty to simulate such coupled poro-mechanical models is related to the formulation and discretization of the contact mechanical sub-problem. Our starting point is based on the mixed formulation using facewise constant Lagrange multipliers along the fractures representing normal and tangential stresses. This is a natural choice for the discretization of the contact dual cone in order to account for complex fracture networks with corners and intersections. It leads to local expressions of the contact conditions and to efficient semi-smooth nonlinear solvers. On the other hand, such a mixed formulation requires to satisfy a compatibility condition between the discrete spaces restricting the choice of the displacement space and potentially leading to sub-optimal accuracy. This motivates the investigation of two alternative formulations based either on a stabilized mixed formulation or on the Nitsche's method. These three types of formulations are first investigated theoritically in order to enhance their connections. Then, they are compared numerically in terms of accuracy and nonlinear convergence. The sensitivity to the choice of the formulation parameters is also investigated. Several 2D test cases are considered with various fracture networks using both P1 and P2 conforming Finite Element discretizations of the displacement field and an Hybrid Finite Volume discretization of the mixed-dimensional Darcy flow model.

Published
2023
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20. Verification benchmarks for single-phase flow in three-dimensional fractured porous media.

Author

Inga Berre, Wietse M. Boon, Bernd Flemisch, Alessio Fumagalli, Dennis Gläser, Eirik Keilegavlen, Anna Scotti, Ivar Stefansson, Alexandru Tatomir, Konstantin Brenner, Samuel Burbulla, Philippe R. B. Devloo, Omar Durán, Marco Favino, Julian Hennicker, I-Hsien Lee, Konstantin Lipnikov, Roland Masson, Klaus Mosthaf, Maria Giuseppina Chiara Nestola, Chuen-Fa Ni, Kirill D. Nikitin, Philipp Schädle, Daniil Svyatskiy, Ruslan M. Yanbarisov, and Patrick Zulian

Published
2020

39. 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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41. 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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43. A hybrid-dimensional compositional two-phase flow model in fractured porous media with phase transitions and Fickian diffusion

Author

Roland Masson, Joubine Aghili, Jean-Raynald de Dreuzy, Laurent Trenty, Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Géosciences Rennes (GR), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire des Sciences de l'Univers de Rennes (OSUR)-Centre National de la Recherche Scientifique (CNRS), Observatoire des Sciences de l'Univers de Rennes (OSUR), Agence Nationale pour la Gestion des Déchets Radioactifs (ANDRA), 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), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Laboratoire Jean Alexandre Dieudonné (JAD), ANDRA, Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UNIV-RENNES), 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)-Université Côte d'Azur (UCA), COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA), Université de Rennes (UR)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire des Sciences de l'Univers de Rennes (OSUR), Université de Rennes (UR)-Institut national des sciences de l'Univers (INSU - CNRS)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut national des sciences de l'Univers (INSU - CNRS)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut national des sciences de l'Univers (INSU - CNRS)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), 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
Materials science, Physics and Astronomy (miscellaneous), two-phase Darcy flow, Upwind scheme, 010103 numerical & computational mathematics, 010502 geochemistry & geophysics, 01 natural sciences, Physics::Geophysics, finite volume scheme, Discrete Fracture Models, 0101 mathematics, Diffusion (business), 0105 earth and related environmental sciences, Numerical Analysis, Darcy's law, Applied Mathematics, Mechanics, Backward Euler method, Fick's laws of diffusion, Fickian diffusion, Computer Science Applications, phase transitions, Computational Mathematics, Nonlinear system, two-point flux approximation, Modeling and Simulation, Fracture (geology), Two-phase flow, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; This paper presents an extension of Discrete Fracture Matrix (DFM) models to com-positional two-phase Darcy flow accounting for phase transitions and Fickian diffusion. The hybrid-dimensional model is based on nonlinear transmission conditions at matrix fracture (mf) interfaces designed to be consistent with the physical processes. They account in particular for the saturation jump induced by the different rock types, for the Fickian diffusion in the fracture width, as well as for the thermodynamical equilibrium formulated by complementary constraints. The model is validated by numerical comparison with a reference equi-dimensional model using a TPFA approximation in space and a fully implicit Euler time integration. It is also compared with the usual approach based on an harmonic averaging of the transmissivities at mf interfaces combined with a two-point upwinding of the mobilities jumping over the mf interfaces. Our approach is shown to provide basically the same accuracy than the equi-dimensional model as opposed to the classical harmonic averaging approach which is shown to exhibit physical inconsistency. It is then applied to simulate the desaturation by suction at the interface between a fractured Callovo-Oxfordian argilite storage rock and a ventilation tunnel with data set provided by Andra.

Published
2021
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44. Total-velocity-based finite volume discretization of two-phase Darcy flow in highly heterogeneous media with discontinuous capillary pressure

Author

Jérôme Droniou, 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 (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), School of Mathematical Sciences [Clayton], Monash University [Clayton], 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), 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 ANR-16-CE06-0009,CHARMS,Modèles de Réservoirs Quantitatifs pour les Systèmes Hydrothermaux Complexes(2016)

Subjects
Finite volume method, Darcy's law, Discretization, Applied Mathematics, General Mathematics, Weak solution, Degrees of freedom (physics and chemistry), Upwind scheme, 010103 numerical & computational mathematics, Mechanics, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Flow (mathematics), 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

This work proposes a finite volume scheme for two-phase Darcy flow in heterogeneous porous media with different rock types. The fully implicit discretization is based on cell-centered, as well as face-centered degrees of freedom in order to capture accurately the nonlinear transmission conditions at different rock type interfaces. These conditions play a major role in the flow dynamics. The scheme is formulated with natural physical unknowns, and the notion of global pressure is only introduced to analyze its stability and convergence. It combines a two-point flux approximation of the gradient normal fluxes with a Hybrid Upwinding approximation of the transport terms. The convergence of the scheme to a weak solution is established taking into account the discontinuous capillary pressure at different rock type interfaces and the degeneracy of the phase mobilities. Numerical experiments show the additional robustness of the proposed discretization compared with the classical Phase Potential Upwinding approach.

Published
2020
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45. A robust VAG scheme for a two-phase flow problem in heterogeneous porous media

Author

Roland Masson, El Houssaine Quenjel, Konstantin Brenner, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), 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
Imagination, Darcy's law, Chemical substance, heterogeneous capillary pressure, Discretization, media_common.quotation_subject, 65M22, two-phase Darcy flow, 010103 numerical & computational mathematics, VAG scheme MSC : 35K65, 6T99, 01 natural sciences, 65M08, Vertex (geometry), 010101 applied mathematics, Nonlinear system, finite volume scheme, Applied mathematics, Two-phase flow, 0101 mathematics, Porous medium, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], media_common, Mathematics
Abstract

International audience; A positive Vertex Approximate Gradient (VAG) scheme is proposed to discretize the total velocity formulation of two-phase Darcy flow problems in heterogeneous porous media. The discretization is based on the physical variables and allows for multiple rock types with highly contrasted petrophysical and hydrodynamical properties. The numerical experiment shows that, compared to the Phase Potential Upwind (PPU) version of VAG scheme, this new discretization is more robust and efficient in terms of nonlinear convergence

Published
2020
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46. 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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47. 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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48. Domain decomposition methods to model heat exchanges between a well and a rock mass

Author

Floriane Louvet, Laurent Jeannin, Aden Vuddamalay, 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), and Storengy France

Subjects
geography, Hydrogeology, geography.geographical_feature_category, Petroleum engineering, Domain decomposition methods, 010103 numerical & computational mathematics, 01 natural sciences, Computer Science Applications, Physics::Geophysics, Underground gas storage, Computational Mathematics, Computational Theory and Mathematics, Heat transfer, 0101 mathematics, Computers in Earth Sciences, Rock mass classification, Geology, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Water well
Abstract

International audience; This paper focuses on the modelling of heat transfer between fluid flowing in a well and the surrounding rock mass and proposes an efficient domain decomposition approach to solve this problem. The importance of such a method is illustrated by the study of the modelling of underground gas storage in salt caverns.

Published
2020
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49. Le démantèlement des installations nucléaires

Author

Jérôme Gosset, Sophie Remont, Roland Masson, Jérôme Gosset, Sophie Remont, and Roland Masson

Subjects
Nuclear bringing down
Abstract

A l'aube du XXIe siècle, la première génération d'installations nucléaires arrive en fin de vie. Un nombre grandissant d'entre elles sont mises à l'arrêt aussi bien en France que dans les principaux pays industrialisés. Quel est le devenir de ces centrales, usines et laboratoires? Quelles solutions existent pour les démanteler? Que faire alors des déchets? Est-il possible de conserver ces installations en l'état de façon sûre? Quel est l'impact radiologique des diverses voies envisageables? Qui supporte le coût de gestion de ces installations à l'arrêt? A qui en incombe d'ailleurs la responsabilité? Comment se mettent en place en France les solutions à ce problème complexe? Voici là quelques-unes des questions auxquelles ce livre apporte des éléments de réponse. Grâce à une étude comparative des situations en France, en Allemagne et aux Etats-Unis, il met à la portée de tous les éléments de réflexions indispensables à un large débat.

Published
2022

50. Implementation of a Vertex-Centered Method Inside an Industrial Reservoir Simulator: Practical Issues and Comprehensive Comparison With Corner-Point Grids and Perpendicular-Bisector-Grid Models on a Field Case

Author

Pierre Samier and Roland Masson

Subjects
Vertex (computer graphics), Field (physics), Computer science, Energy Engineering and Power Technology, 010103 numerical & computational mathematics, Geotechnical Engineering and Engineering Geology, Grid, 01 natural sciences, Computational science, Unstructured grid, 010101 applied mathematics, Perpendicular, Point (geometry), 0101 mathematics, Simulation
Abstract

Summary Corner-point grids (CPGs) and pillar-based unstructured grids do not provide an effective work flow for translating Earth models into simulation models. Such a work flow requires grids that allow an accurate representation of the near-well flow, preserve geological accuracy, and offer flexible resolution control. Hence, a 3D unstructured approach is required. Significant work has been performed for generating unstructured grids, and modeling hydraulic-fracture flow for gas-shale simulation has given a new impulse for unstructured gridding. Recent methods such as vertex-approximate gradient (VAG) or more-mature ones such as multipoint flux approximation (MPFA) provide a numerical scheme dependent on multipoint stencil more physical than two-point flux-approximation (TPFA) methods. This paper presents the implementation of VAG and MPFA schemes inside a next-generation reservoir simulator starting from a source code calculating multipoint flux nonneighbor connections (NNCs) for any polygonal-shaped control volume. The unstructured-scheme approach has been developed as an in-house extension to a next-generation multicompany collaborative reservoir simulator that is designed for handling unstructured grids. The main issues addressed are the introduction of vertices unknowns among the usual cell-center variables, the assignment of vertices properties in the reservoir-simulator model, and the link with the well model. Incidentally, the definition of an exchange format to describe the unstructured geometry (vertices, edges, faces, control volumes) on a large reservoir-simulation model is proposed. Three simulation examples are presented, and we compare results, accuracy, and performance of multipoint-scheme methods such as VAG and MPFA on unstructured grids. Results are compared with TPFA methods on refined structured CPGs and TPFA methods on unstructured Voronoi grids. The two first test cases are academic models, and the third one is a field model.

Published
2017
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