A model of fracture for elliptic problems with flux and solution jumps

Philippe Angot

doi:10.1016/S1631-073X(03)00300-5

Mathematical Problems in Mechanics/Numerical Analysis

A model of fracture for elliptic problems with flux and solution jumps
[Un modèle de fracture pour des problèmes elliptiques avec sauts de flux et de solution]

Présenté par : Philippe G. Ciarlet

Philippe Angot ¹

¹ Université de la Méditerranée Aix-Marseille II, CMI–LATP, UMR CNRS 6632, 39, rue F. Joliot Curie, 13453 Marseille cedex 13, France

Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 425-430.

Abstract (VO)
Résumé

A new model of fracture for elliptic problems combining flux and solution jumps as immersed boundary conditions is proposed and proved to be well-posed. An application of this model to the flow in fractured porous media is also proposed including the cases of “impermeable fracture” and “fully permeable fracture” satisfying the so-called “cubic law”, as well as intermediate cases. A finite volume scheme on general polygonal meshes is built to solve such problems. Since no unknown is required at the fracture interface, the scheme is as cheap as standard schemes for the same problems without fault. The convergence of the scheme can be proved to the weak solution of the problem. With weak regularity assumptions, we also establish for the discrete H¹₀ and L² norms some error estimates in $𝒪 (h)$ , where h is the maximum diameter of the control volumes of the mesh.

Pour des problèmes elliptiques, un nouveau modèle de fracture combinant des sauts de la solution et du flux comme conditions aux limites immergées est proposé et on montre qu'il est bien posé. On propose également une application de ce modèle à l'écoulement dans des milieux poreux fissurés incluant les cas de « fracture imperméable » et de « fracture totalement perméable » satisfaisant la « loi cubique », ainsi que des cas intermédiaires. Un schéma en volumes finis est construit pour résoudre de tels problèmes sur des maillages polygonaux généraux. Comme aucune inconnue n'est nécessaire sur l'interface de fracture, ce schéma est aussi économique que des schémas standards pour résoudre les mêmes problèmes sans faille. On peut prouver la convergence de ce schéma vers la solution faible du problème. De plus, avec des hypothèses faibles de régularité, on établit pour la norme discrète H¹₀ et pour la norme L² des estimations d'erreur en $𝒪 (h)$ , où h désigne le pas du maillage, i.e. le diamètre maximum des volumes finis du maillage.

Reçu le : 2003-05-22
Accepté le : 2003-06-10
Publié le : 2003-09-11

DOI : 10.1016/S1631-073X(03)00300-5

Affiliations des auteurs :

Philippe Angot ¹

¹ Université de la Méditerranée Aix-Marseille II, CMI–LATP, UMR CNRS 6632, 39, rue F. Joliot Curie, 13453 Marseille cedex 13, France

@article{CRMATH_2003__337_6_425_0,
     author = {Philippe Angot},
     title = {A model of fracture for elliptic problems with flux and solution jumps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {425--430},
     publisher = {Elsevier},
     volume = {337},
     number = {6},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00300-5},
     language = {en},
}

TY  - JOUR
AU  - Philippe Angot
TI  - A model of fracture for elliptic problems with flux and solution jumps
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 425
EP  - 430
VL  - 337
IS  - 6
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00300-5
LA  - en
ID  - CRMATH_2003__337_6_425_0
ER  -

%0 Journal Article
%A Philippe Angot
%T A model of fracture for elliptic problems with flux and solution jumps
%J Comptes Rendus. Mathématique
%D 2003
%P 425-430
%V 337
%N 6
%I Elsevier
%R 10.1016/S1631-073X(03)00300-5
%G en
%F CRMATH_2003__337_6_425_0

Philippe Angot. A model of fracture for elliptic problems with flux and solution jumps. Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 425-430. doi : 10.1016/S1631-073X(03)00300-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00300-5/

Bibliographie
Cité par

[1] Ph. Angot, Mathematical and numerical modelling for a fictitious domain method with jump and penalized immersed boundary conditions, Preprint, Thèse HDR, Univ. Méditerranée Aix-Marseille II, sept. 1998

[2] Ph. Angot Finite volume methods for non smooth solution of diffusion models; application to imperfect contact problems, Proc. 4th Int. Conf. NMA'98, Sofia (Bulgarie) (O.P. Iliev; M.S. Kaschiev; S.D. Margenov; Bl.H. Sendov; P.S. Vassilevski, eds.), World Sci. Publications (1999), pp. 621-629

[3] Ph. Angot; Th. Gallouët; R. Herbin Convergence of finite volume methods on general meshes for non smooth solution of elliptic problems with cracks (R. Vilsmeier; F. Benkhaldoun; D. Hänel, eds.), Finite Volumes for Complex Applications II, Hermès, 1999, pp. 215-222

[4] Ph. Angot, A model of fracture for elliptic problems with flux and solution jumps, Preprint L.A.T.P., UMR CNRS 6632 http://www.cmi.univ-mrs.fr, May 2003

[5] Ph. Angot, Th. Gallouët, R. Herbin, in preparation

[6] Flow and Contaminant Transport in Fractured Rock (J. Bear; C.-F. Tsang; G. de Marsily, eds.), Academic Press, San Diego, 1993

[7] R. Eymard; Th. Gallouët; R. Herbin Finite volume methods (P.G. Ciarlet; J.-L. Lions, eds.), Handbook of Numerical Analysis, Vol. VII, North-Holland, 2000, pp. 713-1020

[8] I. Faille; E. Flauraud; F. Nataf; S. Pégaz-Fiornet; F. Schneider; F. Willien A new fault model in geological basin modelling. Application of finite volume scheme and domain decomposition methods (R. Herbin; D. Kröner, eds.), Finite Volumes for Complex Applications III, Hermes Penton Sci. (HPS), 2002, pp. 543-550

[9] Dynamics of Fluids in Fractured Rock (B. Faybishenko; P.A. Witherspoon; S.M. Benson, eds.), Geophysical Monograph Series, 122, American Geophysical Union, Washington, DC, 2000

[10] J. Jaffré; V. Martin; J.E. Roberts Generalized cell-centered finite volume methods for flow in porous media with faults (R. Herbin; D. Kröner, eds.), Finite Volumes for Complex Applications III, Hermes Penton Sci. (HPS), 2002, pp. 357-364

Arash Andrea Roknian; Anna Scotti; Alessio Fumagalli Free convection in fractured porous media: A numerical study, Advances in Water Resources, Volume 202 (2025), p. 104988 | DOI:10.1016/j.advwatres.2025.104988
Nathan Shauer; José B. Villegas S.; Bernd Flemisch; Philippe R. B. Devloo Fluid flow analysis of discrete fracture networks using a locally conservative stable mixed finite element method, Computational Geosciences, Volume 29 (2025) no. 2, p. 24 (Id/No 13) | DOI:10.1007/s10596-025-10358-6 | Zbl:8038729
Xinyu Wu; Hui Guo; Ziyao Xu; Yang Yang A reinterpreted discrete fracture model for Darcy–Forchheimer flow in fractured porous media, Advances in Water Resources, Volume 179 (2023), p. 104504 | DOI:10.1016/j.advwatres.2023.104504
Xin Zhang; Q. Jane Wang A Unified Analogy-Based Computation Methodology From Elasticity to Electromagnetic-Chemical-Thermal Fields and a Concept of Multifield Sensing, ASME Open Journal of Engineering, Volume 1 (2022) | DOI:10.1115/1.4053910
Noé Brice Nkoumbou Kaptchouang; Lionel Gélébart Multiscale coupling of FFT-based simulations with the LDC approach, Computer Methods in Applied Mechanics and Engineering, Volume 394 (2022), p. 29 (Id/No 114921) | DOI:10.1016/j.cma.2022.114921 | Zbl:1507.65279
Mohamed Kara; Salim Mesbahi; Philippe Angot The fictitious domain method with sharp interface for elasticity systems with general jump embedded boundary conditions, Advances in Applied Mathematics and Mechanics, Volume 13 (2021) no. 1, pp. 119-139 | DOI:10.4208/aamm.oa-2019-0119 | Zbl:1488.65569
Hui Guo; Wenjing Feng; Ziyao Xu; Yang Yang Conservative numerical methods for the reinterpreted discrete fracture model on non-conforming meshes and their applications in contaminant transportation in fractured porous media, Advances in Water Resources, Volume 153 (2021), p. 103951 | DOI:10.1016/j.advwatres.2021.103951
Alessio Fumagalli; Anna Scotti A mathematical model for thermal single-phase flow and reactive transport in fractured porous media, Journal of Computational Physics, Volume 434 (2021), p. 26 (Id/No 110205) | DOI:10.1016/j.jcp.2021.110205 | Zbl:7508521
Ziyao Xu; Yang Yang The hybrid dimensional representation of permeability tensor: a reinterpretation of the discrete fracture model and its extension on nonconforming meshes, Journal of Computational Physics, Volume 415 (2020), p. 28 (Id/No 109523) | DOI:10.1016/j.jcp.2020.109523 | Zbl:1440.76089
Lars H. Odsæter; Trond Kvamsdal; Mats G. Larson A simple embedded discrete fracture-matrix model for a coupled flow and transport problem in porous media, Computer Methods in Applied Mechanics and Engineering, Volume 343 (2019), pp. 572-601 | DOI:10.1016/j.cma.2018.09.003 | Zbl:1440.76147
Alessio Fumagalli; Eirik Keilegavlen; A. Anciaux-Sedrakian; Q. H. Tran Dual Virtual Element Methods for Discrete Fracture Matrix models, Oil Gas Science and Technology – Revue d’IFP Energies nouvelles, Volume 74 (2019), p. 41 | DOI:10.2516/ogst/2019008
Philippe Angot Well-posed Stokes/Brinkman and Stokes/Darcy coupling revisited with new jump interface conditions, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 52 (2018) no. 5, pp. 1875-1911 | DOI:10.1051/m2an/2017060 | Zbl:1414.35161
Alessio Fumagalli; Isabelle Faille A double-layer reduced model for fault flow on slipping domains with an hybrid finite volume scheme, Journal of Scientific Computing, Volume 77 (2018) no. 2, pp. 885-910 | DOI:10.1007/s10915-018-0740-8 | Zbl:1404.76256
Philippe Angot; Benoît Goyeau; J. Alberto Ochoa-Tapia Asymptotic modeling of transport phenomena at the interface between a fluid and a porous layer: Jump conditions, Physical Review E, Volume 95 (2017) no. 6 | DOI:10.1103/physreve.95.063302
Bernd Flemisch; Alessio Fumagalli; Anna Scotti A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media, Advances in Discretization Methods, Volume 12 (2016), p. 47 | DOI:10.1007/978-3-319-41246-7_3
Michel Belliard; M. Chandesris; J. Dumas; Y. Gorsse; D. Jamet; C. Josserand; B. Mathieu An analysis and an affordable regularization technique for the spurious force oscillations in the context of direct-forcing immersed boundary methods, Computers Mathematics with Applications, Volume 71 (2016) no. 5, pp. 1089-1113 | DOI:10.1016/j.camwa.2016.01.024 | Zbl:1443.65155
Xavier Tunc; Isabelle Faille; Thierry Gallouët; Marie Christine Cacas; Pascal Havé A model for conductive faults with non-matching grids, Computational Geosciences, Volume 16 (2012) no. 2, pp. 277-296 | DOI:10.1007/s10596-011-9267-x | Zbl:1348.76167
M. Belliard; I. Ramière Fictitious domain methods for two-phase flow energy balance computations in nuclear components, International Journal for Numerical Methods in Fluids, Volume 68 (2012) no. 8, pp. 939-957 | DOI:10.1002/fld.2587 | Zbl:1427.76118
Philippe Angot On the well-posed coupling between free fluid and porous viscous flows, Applied Mathematics Letters, Volume 24 (2011) no. 6, pp. 803-810 | DOI:10.1016/j.aml.2010.07.008 | Zbl:1402.76122
Philippe Angot A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 348 (2010) no. 11-12, pp. 697-702 | DOI:10.1016/j.crma.2010.04.022 | Zbl:1194.35317
Philippe Angot; Franck Boyer; Florence Hubert Asymptotic and numerical modelling of flows in fractured porous media, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 43 (2009) no. 2, pp. 239-275 | DOI:10.1051/m2an/2008052 | Zbl:1171.76055
Isabelle Ramière; Philippe Angot; Michel Belliard A general fictitious domain method with immersed jumps and multilevel nested structured meshes, Journal of Computational Physics, Volume 225 (2007) no. 2, pp. 1347-1387 | DOI:10.1016/j.jcp.2007.01.026 | Zbl:1122.65115
Isabelle Ramiere; Philippe Angot; Michel Belliard, 17th AIAA Computational Fluid Dynamics Conference (2005) | DOI:10.2514/6.2005-4709
Philippe Angot A unified fictitious domain model for general embedded boundary conditions, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 341 (2005) no. 11, pp. 683-688 | DOI:10.1016/j.crma.2005.09.046 | Zbl:1082.65589
Philippe Angot A model of fracture for elliptic problems with flux and solution jumps., Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 337 (2003) no. 6, pp. 425-430 | DOI:10.1016/s1631-073x(03)00300-5 | Zbl:1113.76455

Cité par 25 documents. Sources : Crossref, zbMATH

^☆ An extended version of this paper with some more details can be found in Angot (Preprint L.A.T.P.).

Commentaires - Politique