We consider the mixed formulation for Darcy’s flow in fractured media. We give a well-posedness result that does not rely on the imposition of pressure in part of the boundary of the fracture network, thus including a fully immersed fracture network. We present and analyze a mimetic finite difference formulation for the problem, providing convergence results and numerical tests.
Mots-clés : Flow in porous media, fracture networks, mimetic finite difference
@article{M2AN_2018__52_2_595_0,
author = {Formaggia, Luca and Scotti, Anna and Sottocasa, Federica},
title = {Analysis of a mimetic finite difference approximation of flows in fractured porous media},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {595--630},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2017028},
mrnumber = {3834437},
zbl = {1404.65228},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017028/}
}
TY - JOUR AU - Formaggia, Luca AU - Scotti, Anna AU - Sottocasa, Federica TI - Analysis of a mimetic finite difference approximation of flows in fractured porous media JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 595 EP - 630 VL - 52 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017028/ DO - 10.1051/m2an/2017028 LA - en ID - M2AN_2018__52_2_595_0 ER -
%0 Journal Article %A Formaggia, Luca %A Scotti, Anna %A Sottocasa, Federica %T Analysis of a mimetic finite difference approximation of flows in fractured porous media %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 595-630 %V 52 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017028/ %R 10.1051/m2an/2017028 %G en %F M2AN_2018__52_2_595_0
Formaggia, Luca; Scotti, Anna; Sottocasa, Federica. Analysis of a mimetic finite difference approximation of flows in fractured porous media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 595-630. doi : 10.1051/m2an/2017028. http://www.numdam.org/articles/10.1051/m2an/2017028/
[1] , and , Fractured Porous Media. Oxford University Press (2013). | MR | Zbl
[2] , , , and , Domain Decomposition for Some Transmission Problems in Flow in Porous Media. Vol. 552 of Lecture Notes in Physics. Springer, Berlin (2000) 22–34. | DOI | MR | Zbl
[3] , and , Mimetic finite differences for flow in fractures from microseismic data, in SPE Reservoir Simulation Symposium, 23–25 February, Houston, Texas, USA. Society of Petroleum Engineers (2015).
[4] , and . Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: M2AN 43 (2009) 239–275. | DOI | Numdam | MR | Zbl
[5] , and , A mimetic discretization of elliptic obstacle problems. Math. Comp. 82 (2013) 1379–1400. | DOI | MR | Zbl
[6] , and , Mimetic discretizations of elliptic control problems. J. Sci. Comput. 56 (2013) 14–27. | DOI | MR | Zbl
[7] , , and , Mimetic finite differences for nonlinear and control problems. Math. Model. Methods Appl. Sci. 24 (2014) 1457–1493. | DOI | MR | Zbl
[8] , and , Mimetic finite difference approximation of quasilinear elliptic problems. Calcolo 52 (2015) 45–67. | DOI | MR | Zbl
[9] , , and , Discontinuous Galerkin approximation of flows in fractured porous media. Technical Report 22/2016, MOX, Politecnico di Milano (2016).
[10] , , , and , Mimetic finite difference approximation of flows in fractured porous media. ESAIM: M2AN 50 (2016) 809–832. | DOI | Numdam | MR | Zbl
[11] , and , Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21 (1990) 823–836. | DOI | MR | Zbl
[12] , , and , The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Eng. 280 (2014) 135–156. | DOI | MR | Zbl
[13] , and , A globally conforming method for solving flow in discrete fracture networks using the virtual element method. Finite Elem. Anal. Des. 109 (2016) 23–36. | DOI
[14] , Characterizing flow and transport in fractured geological media: a review. Adv. Water Res. 25 (2002) 861–884. | DOI
[15] , and , Mixed Finite Element Methods and Applications. Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). | MR | Zbl
[16] and , Robust Discretization of Flow in Fractured Porous Media. Preprint arXiv: (2016). | arXiv | MR
[17] and The Mathematical Theory of Finite Element Methods. Springer, Berlin, Heidelberg (1994). | DOI | MR | Zbl
[18] , , and , Gradient discretization of hybrid dimensional Darcy flows in fractured porous media with discontinuous pressures at the matrix fracture interfaces. IMA J. Numer. Anal. 37 (2017) 1551–1585 | MR | Zbl
[19] , and , Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM: J. Numer. Anal. 43 (2006) 1872–1896. | MR | Zbl
[20] , , and , A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Eng. 196 (2007) 3682–3692. | DOI | MR | Zbl
[21] and , A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: M2AN 46 (2012) 465–489. | DOI | Numdam | Zbl
[22] ,, and , Efficient geometric reconstruction of complex geological structures. Math. Comput. Simul. 106 (2014) 163–184. | DOI | MR | Zbl
[23] , and , Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113 (2009) 325–356. | DOI | MR | Zbl
[24] , and The Mimetic Finite Difference Method for Elliptic Problems. Vol. 11 of MS&A. Modeling, Simulation and Applications. Springer, Cham (2014). | MR | Zbl
[25] , , and , Model reduction and discretization using hybrid finite volumes for flow in porousmedia containing faults. Comput. Geosci. 20 (2016) 317–339. | DOI | MR | Zbl
[26] , , and , A reduced model for Darcy’s problem in networks of fractures. ESAIM: M2AN 48 (2014) 1089–1116. | DOI | Numdam | MR | Zbl
[27] and , Dual virtual element method for discrete fractures networks. SIAM: J. Sci. Comput. 40 (2018) B228–B258. | MR | Zbl
[28] and A reduced model for flow and transport in fractured porous media with non-matching grids, in Proc. of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications. Springer-Verlag (2012). | MR | Zbl
[29] and , A numerical method for two-phase flow in fractured porous media with non-matching grids. Adv. Water Res. 62 (2013) 454–464. Computational Methods in Geologic CO2 Sequestration. | DOI
[30] and , An efficient XFEM approximation of Darcy flows in arbitrarily fractured porous media. Oil & Gas Science and Technology–Revue d’IFP Energies nouvelles 69 (2014) 555–564. | DOI
[31] , , and , On the use of enriched finite element method to model subsurface features in porous media flow problems. Comput. Geosci. 15 (2011) 721–736. | DOI | MR | Zbl
[32] , and , A discrete fracture model for two-phase flow with matrix-fracture interaction. Procedia Comput. Sci. 4 (2011) 967–973 | DOI
[33] , , , et al. An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9 (2004) 227–236. | DOI
[34] , and , Practical gridding algorithms for discrete fracture modeling workflows, in 12th European Conference on the Mathematics of Oil Recovery (2010). | DOI
[35] , , and , Modeling fractures and barriers as interfaces for flow in porous media. SIAM: J. Sci. Comput. 26 (2005) 1667–1691. | MR | Zbl
[36] , , and , Dimensionally reduced flow models in fractured porous media: crossings and boundaries. Comput. Geosci. 19 (2015) 1219–1230. | DOI | MR | Zbl
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