A hybrid mortar virtual element method for discrete fracture network simulations. (English) Zbl 1351.76048
Summary: The most challenging issue in performing underground flow simulations in Discrete Fracture Networks (DFN) is to effectively tackle the geometrical difficulties of the problem. In this work we put forward a new application of the Virtual Element Method combined with the Mortar method for domain decomposition: we exploit the flexibility of the VEM in handling polygonal meshes in order to easily construct meshes conforming to the traces on each fracture, and we resort to the mortar approach in order to “weakly” impose continuity of the solution on intersecting fractures. The resulting method replaces the need for matching grids between fractures, so that the meshing process can be performed independently for each fracture. Numerical results show optimal convergence and robustness in handling very complex geometries.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74R10 | Brittle fracture |
| 76S05 | Flows in porous media; filtration; seepage |
Keywords:
discrete fracture networks; virtual element method; mortar method; fracture flows; Darcy flowsReferences:
| [1] | Fidelibus, C.; Cammarata, G.; Cravero, M., Hydraulic characterization of fractured rocks, (Abbie, M.; Bedford, J. S., Rock Mechanics: New Research (2009), Nova Science Publishers Inc.: Nova Science Publishers Inc. New York) |
| [2] | Bear, J.; Cheng, A. H.-D., Modeling Groundwater Flow and Contaminant Transport, Theory and Applications of Transport in Porous Media, vol. 23 (2010), Springer · Zbl 1195.76002 |
| [3] | Adler, P. M., Fractures and Fracture Networks (1999), Kluwer Academic: Kluwer Academic Dordrecht |
| [4] | Cacas, M. C.; Ledoux, E.; de Marsily, G.; Tillie, B.; Barbreau, A.; Durand, E.; Feuga, B.; Peaudecerf, P., Modeling fracture flow with a stochastic discrete fracture network: calibration and validation: 1. The flow model, Water Resour. Res., 26, 479-489 (1990) |
| [5] | Dershowitz, W. S.; Fidelibus, C., Derivation of equivalent pipe networks analogues for three-dimensional discrete fracture networks by the boundary element method, Water Resour. Res., 35, 2685-2691 (1999) |
| [6] | Nœtinger, B.; Jarrige, N., A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks, J. Comput. Phys., 231, 1, 23-38 (2012) · Zbl 1415.76460 |
| [7] | Nœtinger, B., A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks accounting for matrix to fracture flow, J. Comput. Phys., 283, 205-223 (2015) · Zbl 1351.76280 |
| [8] | Berrone, S.; Pieraccini, S.; Scialò, S., A PDE-constrained optimization formulation for discrete fracture network flows, SIAM J. Sci. Comput., 35, 2, B487-B510 (2013) · Zbl 1266.65188 |
| [9] | 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, A908-A935 (2013) · Zbl 1266.65187 |
| [10] | Berrone, S.; Pieraccini, S.; Scialò, S., An optimization approach for large scale simulations of discrete fracture network flows, J. Comput. Phys., 256, 838-853 (2014) · Zbl 1349.76806 |
| [11] | Formaggia, L.; Antonietti, P.; Panfili, P.; Scotti, A.; Turconi, L.; Verani, M.; Cominelli, A., Optimal techniques to simulate flow in fractured reservoir, (ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery (2014)) |
| [12] | Hyman, J. D.; Gable, C. W.; Painter, S. L.; Makedonska, N., Conforming Delaunay triangulation of stochastically generated three dimensional discrete fracture networks: a feature rejection algorithm for meshing strategy, SIAM J. Sci. Comput., 36, 4, A1871-A1894 (2014) · Zbl 1305.74082 |
| [13] | Pichot, G.; Erhel, J.; de Dreuzy, J., A mixed hybrid mortar method for solving flow in discrete fracture networks, Appl. Anal., 89, 10, 1629-1643 (2010) · Zbl 1387.65122 |
| [14] | Pichot, G.; Erhel, J.; de Dreuzy, J., A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks, SIAM J. Sci. Comput., 34, 1, B86-B105 (2012) · Zbl 1387.65121 |
| [15] | Mustapha, H.; Dimitrakopoulos, R., High-order stochastic simulation of complex spatially distributed natural phenomena, Math. Geosci., 42, 5, 457-485 (2010) · Zbl 1194.86040 |
| [16] | Berrone, S.; Canuto, C.; Pieraccini, S.; Scialò, S., Uncertainty quantification in discrete fracture network models: stochastic fracture transmissivity, Comput. Math. Appl., 70, 603-623 (2015) · Zbl 1443.76210 |
| [17] | Huang, Z.; Yan, X.; Yao, J., A two-phase flow simulation of discrete-fractured media using mimetic finite difference method, Commun. Comput. Phys., 16, 3, 799-816 (2014) · Zbl 1373.76172 |
| [18] | Mustapha, H.; Mustapha, K., A new approach to simulating flow in discrete fracture networks with an optimized mesh, SIAM J. Sci. Comput., 29, 4, 1439-1459 (2007) · Zbl 1251.76056 |
| [19] | Berrone, S.; Pieraccini, S.; Scialò, S.; Vicini, F., A parallel solver for large scale DFN simulations, SIAM J. Sci. Comput., 37, 3, C285-C306 (2015) · Zbl 1320.65167 |
| [20] | Karimi-Fard, M.; Durlofsky, L. J., Unstructured adaptive mesh refinement for flow in heterogeneous porous media, (ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery (2014)) |
| [21] | Jaffré, J.; Roberts, J. E., Modeling flow in porous media with fractures, discrete fracture models with matrix-fracture exchange, Numer. Anal. Appl., 5, 2, 162-167 (2012) · Zbl 1299.76256 |
| [22] | Benedetto, M. F.; Berrone, S.; Pieraccini, S.; Scialò, S., The virtual element method for discrete fracture network simulations, Comput. Methods Appl. Mech. Eng., 280, 135-156 (2014) · Zbl 1423.74863 |
| [23] | Benedetto, M. F.; Berrone, S.; Scialò, S., A globally conforming method for solving flow in discrete fracture networks using the virtual element method, Finite Elem. Anal. Des., 109, 23-36 (2016) |
| [24] | Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 1, 199-214 (2013) · Zbl 1416.65433 |
| [25] | Bernardi, C.; Maday, Y.; Patera, A. T., A new nonconforming approach to domain decomposition: the mortar element method, (Nonlinear Partial Differential Equations and Their Applications, vol. XI. Nonlinear Partial Differential Equations and Their Applications, vol. XI, Collège de France Seminar, Paris, 1989-1991. Nonlinear Partial Differential Equations and Their Applications, vol. XI. Nonlinear Partial Differential Equations and Their Applications, vol. XI, Collège de France Seminar, Paris, 1989-1991, Pitman Res. Notes Math. Ser., vol. 299 (1994), Longman Sci. Tech.: Longman Sci. Tech. Harlow), 13-51 · Zbl 0797.65094 |
| [26] | Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci., 24, 08, 1541-1573 (2014) · Zbl 1291.65336 |
| [27] | Beirão da Veiga, L.; Manzini, G., A virtual element method with arbitrary regularity, IMA J. Numer. Anal., 34, 2, 759-781 (2014) · Zbl 1293.65146 |
| [28] | Ahmad, B.; Alsaedi, A.; Brezzi, F.; Marini, L. D.; Russo, A., Equivalent projectors for virtual element methods, Comput. Math. Appl., 66, 3, 376-391 (2013) · Zbl 1347.65172 |
| [29] | Cangiani, A.; Manzini, G.; Russo, A.; Sukumar, N., Hourglass stabilization and the virtual element method, Int. J. Numer. Methods Eng., 102, 3-4, 404-436 (2015) · Zbl 1352.65475 |
| [30] | Ayuso de Dios, B.; Lipnikov, K.; Manzini, G., The nonconforming virtual element method · Zbl 1343.65140 |
| [31] | Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., Virtual element methods for general second order elliptic problems on polygonal meshes · Zbl 1332.65162 |
| [32] | Lipnikov, K.; Manzini, G.; Shashkov, M., Mimetic finite difference method, J. Comput. Phys., 257, 1163-1227 (2014) · Zbl 1352.65420 |
| [33] | Beirão da Veiga, L.; Lipnikov, K.; Manzini, G., The Mimetic Finite Difference Method for Elliptic Problems, Model. Simul. Appl., vol. 11 (2014), Springer · Zbl 1286.65141 |
| [34] | Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Revue française d’automatique, informatique, recherche opérationelle, Anal. Numér., 8, 2, 129-151 (1974) · Zbl 0338.90047 |
| [35] | Belgacem, F. B., The mortar finite element method with Lagrange multipliers, Numer. Math., 84, 2, 173-197 (1999) · Zbl 0944.65114 |
| [36] | Raviart, P.; Thomas, J., Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comput., 31, 138, 391-413 (1977) · Zbl 0364.65082 |
| [37] | Wohlmuth, B., Discretization Methods and Iterative Solvers Based on Domain Decomposition (2001), Springer · Zbl 0966.65097 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
