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Dimensionally reduced flow models in fractured porous media: crossings and boundaries. (English) Zbl 1391.76747

Summary: For the simulation of fractured porous media, a common approach is the use of co-dimension one models for the fracture description. In order to simulate correctly the behavior at fracture crossings, standard models are not sufficient because they either cannot capture all important flow processes or are computationally inefficient. We propose a new concept to simulate co-dimension one fracture crossings and show its necessity and accuracy by means of an example and a comparison to a literature benchmark. From the application point of view, often the pressure is known only at a limited number of discrete points and an interpolation is used to define the boundary condition at the remaining parts of the boundary. The quality of the interpolation, especially in fracture models, influences the global solution significantly. We propose a new method to interpolate boundary conditions on boundaries that are intersected by fractures and show the advantages over standard interpolation methods.

MSC:

76S05 Flows in porous media; filtration; seepage
86-08 Computational methods for problems pertaining to geophysics
41A05 Interpolation in approximation theory
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[1] Abdelaziz, Y; Hamouine, A, A survey of the extended finite element, Comput. Struct., 86, 1141-1151, (2008)
[2] Acosta, M; Merten, C; Eigenberger, G; Class, H; Helmig, R; Thoben, B; Müller-Steinhagen, H, Modeling non-isothermal two-phase multicomponent flow in the cathode of PEM fuel cells, J. Power Sources, 159, 1123-1141, (2006)
[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, June 17-21, 2001, Mount Holyoke College, South Hadley, Massachusetts, American Mathematical Soc., vol. 295, pp. 13-25 (2002)
[4] Angot, P; Boyer, F; Hubert, F; etal., Asymptotic and numerical modelling of flows in fractured porous media, Model. Math. Anal. Numér., 23, 239-275, (2009) · Zbl 1171.76055
[5] Assteerawatt, A.: Flow and transport modelling of fractured aquifers based on a geostatistical approach. PhD thesis, Universität Stuttgart (2008)
[6] Bear, J., Tsang, C.F., Marsily, G.: Flow and contaminant transport in fractured rocks. Academic, San Diego (1993)
[7] Berkowitz, B, Characterizing flow and transport in fractured geological media: a review, Adv. Water Resour., 25, 861-884, (2002)
[8] 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, a908-a935, (2013) · Zbl 1266.65187
[9] D’Angelo, C; Scotti, A, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids, ESAIM: Math. Model. Numer. Anal., 46, 465-489, (2012) · Zbl 1271.76322
[10] Dietrich, P., Helmig, R., Sauter, M., Hötzl, H, Köngeter, J, Teutsch, G: Flow and transport in fractured porous media. Springer, Berlin (2005)
[11] Dogan, MO; Class, H; Helmig, R, Different concepts for the coupling of porous-media flow with lower-dimensional pipe flow, CMES: Comput. Model. Eng. Sci., 53, 207-234, (2009) · Zbl 1231.76308
[12] Dolbow, J.: An extended finite element method with discontinuous enrichment for applied mechanics. PhD thesis, Northwestern University (1999)
[13] Dolbow, J; Moës, N; Belytschko, T, Discontinuous enrichment in finite elements with a partition of unity method, Finite Elem. Anal. Des., 36, 235-260, (2000) · Zbl 0981.74057
[14] Erbertseder, K; Reichold, J; Helmig, R; Jenny, P; Flemisch, B, A coupled discrete/continuum model for describing cancer therapeutic transport in the lung, PLoS One, 7, e31,966, (2012)
[15] Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy’s problem in networks of fractures. MOX report 32 (2012) · Zbl 1299.76254
[16] Fumagalli, A., Scotti, A.: An efficient XFEM approximation of Darcy flows in fractured porous media. MOX report 53 (2012) · Zbl 1329.76166
[17] 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, 3523-3540, (2004) · Zbl 1068.74076
[18] Hansbo, P, Nitsche’s method for interface problems in computational mechanics, GAMM-Mitteilungen, 28, 183-206, (2005) · Zbl 1179.65147
[19] Huang, H; Long, TA; Wan, J; Brown, WP, On the use of enriched finite element method to model subsurface features in porous media flow problems, Comput. Geosci., 15, 721-736, (2011) · Zbl 1237.76193
[20] Martin, V; Jaffré, J; Roberts, JE, Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., 26, 1667-1691, (2005) · Zbl 1083.76058
[21] Matthäi, SK; Belayneh, M, Fluid flow partitioning between fractures and a permeable rock matrix, Geophys. Res. Lett., 31, 7602-6, (2004)
[22] Mohammadi, S.: Extended Finite Element Method. Wiley, New York (2008)
[23] Neumann, SP, Trends, prospects and challenges in quantifying flow and transport through fractured rocks, Hydrogeol. J., 13, 124-147, (2005)
[24] Neunhäuserer, L: Diskretisierungsansätze zur Modellierung von Strömungs- und Transportprozessen in geklüftet-porösen Medien. PhD thesis, Universität Stuttgart. http://elib.uni-stuttgart.de/opus/volltexte/2003/1477 (2003)
[25] Nordbotten, J., Celia, M., Dahle, H., Hassanizadeh, S.: Interpretation of macroscale variables in Darcy’s law. Water Resour. Res. 43(8) (2007)
[26] Swedish Nuclear Power Inspectorate (SKI): The International Hydrocoin Project-Background and Results. Organization for Economic Co-operation and Development, Paris (1987)
[27] Tsang, YW; Tsang, C, Channel model of flow through fractured media, Water Resour. Res., 23, 467-479, (1987)
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