Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
A Mixed Finite Element Method for Modeling the Fluid Exchange Between Microcirculation and Tissue Interstitium Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media

verfasst von : Bernd Flemisch, Alessio Fumagalli, Anna Scotti

Erschienen in: Advances in Discretization Methods

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This paper presents a review of the available mathematical models and corresponding non-conforming numerical approximations which describe single-phase fluid flow in a fractured porous medium. One focus is on the geometrical difficulties that may arise in realistic simulations such as intersecting and immersed fractures. Another important aspect is the choice of the approximation spaces for the discrete problem: in mixed formulations, both the Darcy velocity and the pressure are considered as unknowns, while in classical primal formulations, a richer space for the pressure is considered and the Darcy velocity is computed a posteriori. In both cases, the extended finite element method is used, which allows for a complete geometrical decoupling among the fractures and rock matrix grids. The fracture geometries can thus be independent of the underlying grid thanks to suitable enrichments of the spaces that are able to represent possible jumps of the solution across the fractures. Finally, due to the dimensional reduction, a better approximation of the resulting boundary conditions for the fractures is addressed.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel On a PDE-Constrained Optimization Approach for Flow Simulations in Fractured Media
Nächstes Kapitel Modeling of Fracture in Polycrystalline Materials
Literatur
1.
Abdelaziz, Y., Hamouine, A.: A survey of the extended finite element. Comput. Struct. 86 (11–12), 1141–1151 (2008)CrossRef
2.
Alboin, C., Jaffré, J., Roberts, J.E., Wang, X., Serres, C.: Domain decomposition for some transmission problems in flow in porous media. In: Numerical Treatment of Multiphase Flows in Porous Media (Beijing, 1999). Lecture Notes in Physics, vol. 552, pp. 22–34. Springer, Berlin (2000)
3.
Alboin, C., Jaffré, J., Roberts, J.E., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media. In: Fluid Flow and Transport in Porous Media, Mathematical and Numerical Treatment: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Fluid Flow and Transport in Porous Media, Mathematical and Numerical Treatment, Mount Holyoke College, South Hadley, Massachusetts, 17–21 June 2001, vol. 295, pp. 13–25. American Mathematical Society, Providence (2002)
4.
Alboin, C., Jaffré, J., Roberts, J.E., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media. In: Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment (South Hadley, MA, 2001). Contemporary Mathematics, vol. 295, pp. 13–24. American Mathematical Society, Providence (2002)
5.
Amir, L., Kern, M., Martin, V., Roberts, J.E.: Décomposition de domaine et préconditionnement pour un modèle 3D en milieu poreux fracturé. In: Proceeding of JANO 8, 8th Conference on Numerical Analysis and Optimization (2005)
6.
7.
Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. M2AN Math. Model. Numer. Anal. 43 (2), 239–275 (2009)
8.
Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Verzotti, N.: Mimetic finite difference approximation of flows in fractured porous media. Technical Report, Politecnico di Milano (2015)MATH
9.
Bear, J.: Dynamics of Fluids in Porous Media. American Elsevier, New York (1972)MATH
10.
Benedetto, M.F., Berrone, S., Pieraccini, S., Scialò, S.: The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Eng. 280 (0), 135–156 (2014)MathSciNetCrossRef
11.
Berrone, S., Pieraccini, S., Scialò, S.: On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. 35 (2), 908–935 (2013)MathSciNetCrossRefMATH
12.
Berrone, S., Pieraccini, S., Scialò, S.: An optimization approach for large scale simulations of discrete fracture network flows. J. Comput. Phys. 256 (0), 838–853 (2014)MathSciNetCrossRefMATH
13.
Bonn, W.M., Nordbotten, J.M.: Robust discretization of flow in fractured porous media. arXiv:1601.06977 [math.NA] (2016)
14.
Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2007)CrossRefMATH
15.
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Computational Mathematics, vol. 15. Springer, Berlin (1991)
16.
Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (4), 328–341 (2012)MathSciNetMATH
17.
Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104 (7), 472–501 (2015)MathSciNetCrossRef
18.
D’Angelo, C., Scotti, A.: A Mixed Finite Element Method for Darcy Flow in Fractured Porous Media with Non-Matching Grids. Technical Report, MOX, Mathematical Department, Politecnico di Milano, 2010MATH
19.
D’Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. Math. Model. Numer. Anal. 46 (02), 465–489 (2012)CrossRefMATH
20.
Dassi, F., Perotto, S., Formaggia, L., Ruffo, P.: Efficient geometric reconstruction of complex geological structures. Math. Comput. Simul. 46 (02), 465–489 (2014)MathSciNet
21.
Del Pra, M., Fumagalli, A., Scotti, A.: Well posedness of fully coupled fracture/bulk Darcy flow with XFEM. Technical Report 25/2015, Politecnico di Milano (2015). Submitted to: SIAM Journal on Numerical Analysis
22.
Dolbow, J.: An extended finite element method with discontinuous enrichment for applied mechanics. Ph.D. thesis, Northwestern University (1999)
23.
Dolbow, J., Moës, N., Belytschko, T.: Discontinuous enrichment in finite elements with a partition of unity method. Finite Elem. Anal. Des. 36 (3–4), 235–260 (2000)CrossRefMATH
24.
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer, New York (2004)CrossRefMATH
25.
Faille, I., Flauraud, E., Nataf, F., Pégaz-Fiornet, S., Schneider, F., Willien, F.: A new fault model in geological basin modelling. Application of finite volume scheme and domain decomposition methods. In: Finite Volumes for Complex Applications, III (Porquerolles, 2002), pp. 529–536. Hermes Science Publishing, Paris (2002)
26.
Faille, I., Fumagalli, A., Jaffré, J., Roberts, J.E.: A double-layer reduced model for fault flow on slipping domains with hybrid finite volume scheme. SIAM: J. Sci. Comput. (2015, in preparation). hal-01162048
27.
Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy’s problem in networks of fractures. ESAIM: Math. Model. Numer. Anal. 48, 1089–1116 (2014)MathSciNetCrossRefMATH
28.
Fries, T.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75 (5), 503–532 (2008)MathSciNetCrossRefMATH
29.
Frih, N., Martin, V., Roberts, J.E., Saâda, A.: Modeling fractures as interfaces with nonmatching grids. Comput. Geosci. 16 (4), 1043–1060 (2012)CrossRef
30.
Fumagalli, A.: Numerical Modelling of Flows in Fractured Porous Media by the XFEM Method. Ph.D. thesis, Politecnico di Milano (2012)
31.
Fumagalli, A., Scotti, A.: Numerical modelling of multiphase subsurface flow in the presence of fractures. Commun. Appl. Ind. Math. 3 (1) (2011)
32.
Fumagalli, A., Scotti, A.: A reduced model for flow and transport in fractured porous media with non-matching grids. In: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications. Springer, Berlin (2012)
33.
Fumagalli, A., Scotti, A.: An efficient XFEM approximation of Darcy flows in fractured porous media. MOX Report 53 (2012)
34.
Fumagalli, A., Scotti, A.: A numerical method for two-phase flow in fractured porous media with non-matching grids. Adv. Water Resour. 62 (Part C(0)), 454–464 (2013). Computational Methods in Geologic CO2 Sequestration
35.
Fumagalli, A., Scotti, A.: An efficient XFEM approximation of Darcy flow in arbitrarly fractured porous media. Oil Gas Sci. Technol. - Rev. d’IFP Energies Nouv. 69 (4), 555–564 (2014)MathSciNetCrossRef
36.
Hackbusch, W., Sauter, S.A.: Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math. 75 (4), 447–472 (1997)MathSciNetCrossRefMATH
37.
Hanowski, K., Sander, O.: Simulation of deformation and flow in fractured, poroelastic materials (2016). Preprint, arXiv:1606.05765
38.
39.
Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191 (47–48), 5537–5552 (2002)MathSciNetCrossRefMATH
40.
Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193 (33), 3523–3540 (2004)MathSciNetCrossRefMATH
41.
Huang, H., Long, T.A., Wan, J., Brown, W.P.: On the use of enriched finite element method to model subsurface features in porous media flow problems. Comput. Geosci. 15 (4), 721–736 (2011)MathSciNetCrossRefMATH
42.
Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (5), 1667–1691 (2005)MathSciNetCrossRefMATH
43.
Massing, A., Larson, M.G., Logg, A.: Efficient implementation of finite element methods on non-matching and overlapping meshes in 3d. arXiv preprint arXiv:1210.7076 (2012)
44.
45.
Olshanskii M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47 (5), 3339–3358 (2009)MathSciNetCrossRefMATH
46.
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)
47.
Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems. Lect. Notes Math. 606, 292–315 (1977)MathSciNetCrossRefMATH
48.
Remij, E., Remmers, J., Huyghe, J., Smeulders, D.: The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials. Comput. Methods Appl. Mech. Eng. 286, 296–312 (2015)MathSciNetCrossRef
49.
Roberts, J.E., Thomas, J.-M.: Mixed and Hybrid Methods. Handbook of Numerical Analysis, vol. II, pp. 523–639. North-Holland, Amsterdam (1991)
50.
Schwenck, N.: An XFEM-based model for fluid flow in fractured porous media. Ph.D. thesis, Department of Hydromechanics and Modelling of Hydrosystems, University of Stuttgart (2015)
51.
Schwenck, N., Flemisch, B., Helmig, R., Wohlmuth, B.: Dimensionally reduced flow models in fractured porous media: crossings and boundaries. Comput. Geosci. 19 (6), 1219–1230 (2015)MathSciNetCrossRef
52.
Tunc, X., Faille, I., Gallouët, T., Cacas, M.C., Havé, P.: A model for conductive faults with non-matching grids. Comput. Geosci. 16, 277–296 (2012)CrossRefMATH
53.
Zimmerman, R.W., Kumar, S., Bodvarsson, G.S.: Lubrication theory analysis of the permeability of rough-walled fractures. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 28 (4), 325–331 (1991)CrossRef
Metadaten
Titel
A Review of the XFEM-Based Approximation of Flow in Fractured Porous Media
verfasst von
Bernd Flemisch
Alessio Fumagalli
Anna Scotti
Copyright-Jahr
2016
Buch
Advances in Discretization Methods

Print ISBN: 978-3-319-41245-0

Electronic ISBN: 978-3-319-41246-7

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-41246-7_3

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Zur Marktübersicht
    Bildnachweise