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Journal of Computational Mathematics

Hybrid high-order methods for flow simulations in extremely large discrete fracture networks
Alexandre Ern1; Florent Hédin2; Géraldine Pichot2; Nicolas Pignet3
1 Université Paris-Est, CERMICS (ENPC), 6 et 8 av. Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France and Inria, 2 rue Simone Iff, 75589 Paris, France
2 Inria, 2 rue Simone Iff, 75589 Paris, France and Université Paris-Est, CERMICS (ENPC), 6 et 8 av. Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France
3 EDF Lab Paris-Saclay, 7 boulevard Gaspard Monge, 91120 Palaiseau, France
The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 375-398.

We investigate the computational performance of hybrid high-order methods applied to flow simulations in extremely large discrete fracture networks (over one million of fractures). We study the choice of basis functions, the trade-off between increasing the polynomial order and refining the mesh, and how to take advantage of polygonal cells to reduce the number of degrees of freedom.

Published online:
DOI: 10.5802/smai-jcm.92
Classification: 65N30, 65N50, 86A05
Keywords: discrete fracture networks, flow simulations, hybrid high-order methods
Alexandre Ern 1; Florent Hédin 2; Géraldine Pichot 2; Nicolas Pignet 3

1 Université Paris-Est, CERMICS (ENPC), 6 et 8 av. Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France and Inria, 2 rue Simone Iff, 75589 Paris, France
2 Inria, 2 rue Simone Iff, 75589 Paris, France and Université Paris-Est, CERMICS (ENPC), 6 et 8 av. Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France
3 EDF Lab Paris-Saclay, 7 boulevard Gaspard Monge, 91120 Palaiseau, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     author = {Alexandre Ern and Florent H\'edin and G\'eraldine Pichot and Nicolas Pignet},
     title = {Hybrid high-order methods for flow simulations in extremely large discrete fracture networks},
     journal = {The SMAI Journal of computational mathematics},
     pages = {375--398},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Alexandre Ern; Florent Hédin; Géraldine Pichot; Nicolas Pignet. Hybrid high-order methods for flow simulations in extremely large discrete fracture networks. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 375-398. doi : 10.5802/smai-jcm.92. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.92/

[1] F. Bassi; L. Botti; A. Colombo; D. A. Di Pietro; P. Tesini On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, J. Comput. Phys., Volume 231 (2012) no. 1, pp. 45-65 | DOI | MR | Zbl

[2] Lourenço Beirão da Veiga; Franco Brezzi; Luisa Donatella Marini; Alessandro Russo Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM, Math. Model. Numer. Anal., Volume 50 (2016) no. 3, pp. 727-747 | DOI | Numdam | MR | Zbl

[3] Matías Fernando Benedetto; Stefano Berrone; Sandra Pieraccini; Stefano Scialò The virtual element method for discrete fracture network simulations, Comput. Methods Appl. Mech. Eng., Volume 280 (2014), pp. 135-156 | DOI | MR | Zbl

[4] Matías Fernando Benedetto; Stefano Berrone; Stefano Scialò A Globally Conforming Method for Solving Flow in Discrete Fracture Networks Using the Virtual Element Method, Finite Elem. Anal. Des., Volume 109 (2016) no. C, pp. 23-36 | DOI

[5] S. Berrone; S. Scialò; F. Vicini Parallel meshing, discretization, and computation of flow in massive discrete fracture networks, SIAM J. Sci. Comput., Volume 41 (2019) no. 4, p. c317-c338 | DOI | MR | Zbl

[6] Stefano Berrone; Andrea Borio; Alessandro D’Auria Refinement strategies for polygonal meshes applied to adaptive VEM discretization, Finite Elem. Anal. Des., Volume 186 (2021), 103502 | DOI | MR

[7] Stefano Berrone; Sandra Pieraccini; Stefano Scialò An optimization approach for large scale simulations of discrete fracture network flows, J. Comput. Phys., Volume 256 (2014), pp. 838-853 | DOI | MR | Zbl

[8] E. Bonnet; O. Bour; N. E. Odling; P. Davy; I. Main; P. Cowie; B. Berkowitz Scaling of fracture systems in geological media, Reviews of Geophysics, Volume 39 (2001) no. 3, pp. 347-383 | DOI

[9] Houman Borouchaki; Paul-Louis George Quality mesh generation, C. R. Acad. Sci., Paris, Sér. II, Fasc. b, Méc., Volume 328 (2000) no. 6, pp. 505-518 | DOI | Zbl

[10] Houman Borouchaki; Patrick Laug; Paul-Louis George Parametric surface meshing using a combined advancing-front generalized Delaunay approach, Int. J. Numer. Methods Eng., Volume 49 (2000) no. 1-2, pp. 233-259 | DOI | MR | Zbl

[11] Erik Burman; Alexandre Ern An unfitted hybrid high-order method for elliptic interface problems, SIAM J. Numer. Anal., Volume 56 (2018) no. 3, pp. 1525-1546 | DOI | MR | Zbl

[12] M. C. Cacas; E. Ledoux; G. de Marsily; B. Tillie; A. Barbreau; E. Durand; B. Feuga; P. Peaudecerf Modeling fracture flow with a stochastic discrete fracture network: calibration and validation: 1. The flow model, Water Resources Research, Volume 26 (1990) no. 3, pp. 479-489 | DOI

[13] Florent Chave; Daniele A. Di Pietro; Luca Formaggia A hybrid high-order method for passive transport in fractured porous media, GEM. Int. J. Geomath., Volume 10 (2019) no. 1, 12, 34 pages | DOI | MR | Zbl

[14] M. Cicuttin; D. A. Di Pietro; A. Ern Implementation of discontinuous skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming, J. Comput. Appl. Math., Volume 344 (2018), pp. 852-874 | DOI | MR | Zbl

[15] Mattheo Cicuttin; Alexandre Ern; Nicolas Pignet Hybrid high-order methods. A primer with application to solid mechanics, SpringerBriefs in Mathematics, Springer, 2021 | DOI

[16] Bernardo Cockburn; Daniele A. Di Pietro; Alexandre Ern Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM, Math. Model. Numer. Anal., Volume 50 (2016) no. 3, pp. 635-650 | DOI | Numdam | MR | Zbl

[17] Philippe Davy; Romain Le Goc; Caroline Darcel A model of fracture nucleation, growth and arrest, and consequences for fracture density and scaling, J. Geophys. Res. Solid Earth, Volume 118 (2013) no. 4, pp. 1393-1407 | DOI

[18] Philippe Davy; Romain Le Goc; Caroline Darcel; Olivier Bour; Jean-Raynald de Dreuzy; R. Munier A likely universal model of fracture scaling and its consequence for crustal hydromechanics, J. Geophys. Res. Solid Earth, Volume 115 (2010) no. B10 | DOI

[19] Jean-Raynald de Dreuzy; Géraldine Pichot; Baptiste Poirriez; Jocelyne Erhel Synthetic Benchmark for Modeling Flow in 3D Fractured Media, Comput. Geosci., Volume 50 (2013), pp. 59-71 | DOI

[20] Daniele A. Di Pietro; Jérôme Droniou The hybrid high-order method for polytopal meshes. Design, analysis, and applications, 19, Springer, 2020, xxxi+525 pages | DOI

[21] Daniele A. Di Pietro; Alexandre Ern A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Eng., Volume 283 (2015), pp. 1-21 | DOI | MR | Zbl

[22] Daniele A. Di Pietro; Alexandre Ern; Simon Lemaire An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., Volume 14 (2014) no. 4, pp. 461-472 | DOI | MR | Zbl

[23] D. Doolaeghe; P. Davy; J. D. Hyman; C. Darcel Graph-based flow modeling approach adapted to multiscale discrete-fracture-network models, Phys. Rev. E, Volume 102 (2020) no. 5, 053312 | DOI | MR

[24] Jocelyne Erhel; Jean-Raynald de Dreuzy; Baptiste Poirriez Flow simulation in three-dimensional discrete fracture networks, SIAM J. Sci. Comput., Volume 31 (2009) no. 4, pp. 2688-2705 | DOI | MR | Zbl

[25] Luca Formaggia; Alessio Fumagalli; Anna Scotti; Paolo Ruffo A reduced model for Darcy’s problem in networks of fractures, ESAIM, Math. Model. Numer. Anal., Volume 48 (2014) no. 4, pp. 1089-1116 | DOI | MR | Zbl

[26] André Fourno; Tri-Dat Ngo; Benoit Noetinger; Christian La Borderie FraC: A new conforming mesh method for discrete fracture networks, J. Comput. Phys., Volume 376 (2019), pp. 713-732 | DOI | MR | Zbl

[27] Alessio Fumagalli; Eirik Keilegavlen Dual virtual element method for discrete fractures networks, SIAM J. Sci. Comput., Volume 40 (2018) no. 1, p. b228-b258 | DOI | MR | Zbl

[28] Alessio Fumagalli; Eirik Keilegavlen; Stefano Scialò Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations, J. Comput. Phys., Volume 376 (2019), pp. 694-712 | DOI | MR | Zbl

[29] Gaël Guennebaud; Benoît Jacob et al. Eigen v3, http://eigen.tuxfamily.org, 2010

[30] Florent Hédin; Géraldine Pichot; Alexandre Ern HHO-DFN: a Hybrid High Order (HHO) method for the simulation of flow in large tridimensional Discrete Fracture Networks, Numerical Mathematics and Advanced Applications ENUMATH 2019, Springer (2019)

[31] Jeffrey D. Hyman; Carl W. Gable; Scott L. Painter; Nataliia Makedonska Conforming Delaunay triangulation of stochastically generated three dimensional discrete fracture networks: a feature rejection algorithm for meshing strategy, SIAM J. Sci. Comput., Volume 36 (2014) no. 4, p. a1871-a1894 | DOI | MR | Zbl

[32] Jeffrey D. Hyman; Satish Karra; Nataliia Makedonska; Carl W. Gable; Scott L. Painter; Hari S. Viswanathan dfnWorks: A discrete fracture network framework for modeling subsurface flow and transport, Comput. Geosci., Volume 84 (2015), pp. 10-19 | DOI

[33] Intel MKL, https://software.intel.com/en-us/mkl, 2019

[34] S. Karra; D. O’Malley; J. D. Hyman; H. S. Viswanathan; G. Srinivasan Modeling flow and transport in fracture networks using graphs, Phys. Rev. E, Volume 97 (2018) no. 3, 033304 | DOI

[35] Patrick Laug; Géraldine Pichot Mesh Generation and Flow Simulation in Large Tridimensional Fracture Networks, MASCOT2018 - 15th Meeting on Applied Scientific Computing and Tools (IMACS Series in Computational and Applied Mathematics), Volume 22 (2019)

[36] J. Maillot; P. Davy; R. Le Goc; C. Darcel; J.-R. de Dreuzy Connectivity, permeability, and channeling in randomly distributed and kinematically defined discrete fracture network models, Water Resources Research, Volume 52 (2016) no. 11, pp. 8526-8545 | DOI

[37] Vincent Martin; Jérôme Jaffré; Jean E. Roberts Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., Volume 26 (2005) no. 5, pp. 1667-1691 | DOI | MR | Zbl

[38] Jiří Maryška; Otto Severýn; Martin Vohralík Numerical simulation of fracture flow with a mixed-hybrid FEM stochastic discrete fracture network model, Comput. Geosci., Volume 8 (2004) no. 3, pp. 217-234 | DOI | MR | Zbl

[39] Ani Miraçi; Jan Papež; Martin Vohralík A multilevel algebraic error estimator and the corresponding iterative solver with p-robust behavior, SIAM J. Numer. Anal., Volume 58 (2020) no. 5, pp. 2856-2884 | DOI | MR | Zbl

[40] Tri-Dat Ngo; André Fourno; Benoit Noetinger Modeling of transport processes through large-scale discrete fracture networks using conforming meshes and open-source software, Journal of Hydrology, Volume 554 (2017), pp. 66-79 | DOI

[41] G. Pichot; J. Erhel; J.-R. de Dreuzy A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks, SIAM J. Sci. Comput., Volume 34 (2012) no. 1, p. b86-b105 | DOI | MR | Zbl

[42] Martin Vohralík A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, SIAM J. Numer. Anal., Volume 45 (2007) no. 4, pp. 1570-1599 | DOI | MR | Zbl

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