Abstract
In this paper, flow in Discrete Fracture Networks (DFN) is solved using a Mortar Mixed Hybrid Finite Element Method. To solve large linear systems derived from a nonconforming discretization of stochastic fractured networks, a Balancing Domain Decomposition is used. Tests on three stochastically generated DFN are proposed to show the ability of the iterative solver SIDNUR to solve the flow problem.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Arbogast, T., Cowsar, L.C., Wheeler, M.F., Yotov, I.: Mixed finite element methods on non-matching multiblock grids. SIAM J. Numer. Anal 37, 1295–1315 (2000)
Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. ESAIM Math. Model. Numer. Anal. 19(1), 7–32 (1985). http://eudml.org/doc/193443
Bernardi, C., Hecht, F.: Error indicators for the mortar finite element discretization of the Laplace equation. Math. Comput. 71(240), 1371–1403 (2002)
Bernardi, C., Maday, Y., Patera, A.T.: Domain decomposition by the mortar element method. In: Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters (Beaune, 1992). NATO Advanced Science Institutes Series C Mathematical and Physical Sciences, vol. 384, pp. 269–286. Kluwer Academic, Dordrecht (1993)
Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. XI, Paris, 1989–1991. Pitman Research Notes in Mathematics Series, vol. 299, pp. 13–51. Longman Sci. Tech., Harlow (1994)
Bernardi, C., Rebollo, T.C., Hecht, F., Mghazli, Z.: Mortar finite element discretization of a model coupling Darcy and stokes equations. ESAIM Math. Model. Numer. Anal. 42(3), 375–410 (2008) http://eudml.org/doc/250402
Bjørstad, P.E., Widlund, O.B.: Solving elliptic problems on regions partitioned into substructures. In: Birkhoff, G., Schoenstadt, A. (eds.) Elliptic Problem Solvers II, pp. 245–256. Academic, New York (1984)
Bonnet, E., Bour, O., Odling, N., Davy, P., Main, I., Cowie, P., Berkowitz, B.: Scaling of fracture systems in geological media. Rev. Geophys. 39(3), 347–383 (2001)
Bourgat, J.F., Glowinski, R., Le Tallec, P., Vidrascu, M.: Variational formulation and algorithm for trace operator in domain decomposition calculations. In: Chan, T., Glowinski, R., Périaux, J., Widlund, O. (eds.) Domain Decomposition Methods, pp. 3–16. SIAM, Philadelphia (1989)
Cacas, M.C., Ledoux, E., de Marsily, G., Barbeau, A., Calmels, P., Gaillard, B., Magritta, R.: Modeling fracture flow with a stochastic discrete fracture network: calibration and validation. 1. the flow model. Water Resour. Res. 26(3), 479–489 (1990)
Chen, Y., Davis, T.A., Hager, W.W., Rajamanickam, S.: Algorithm 887: Cholmod, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35(3), 22: 1–22:14 (2008). doi:10.1145/1391989.1391995. http://doi.acm.org/10.1145/1391989.1391995
Cowsar, L.C., Mandel, J., Wheeler, M.F.: Balancing domain decomposition for mixed finite elements. Math. Comput. 64(211), 989–1015 (1995). doi:10.2307/2153480
Davy, P., Bour, O., de Dreuzy, J.R., Darcel, C.: Flow in Multiscale Fractal Fracture Networks, vol. 261, pp. 31–45. Geological Society, London (2006). Special publications
Davy, P., Le Goc, R., Darcel, C., Bour, O., de Dreuzy, J.R., Munier, R.: A likely universal model of fracture scaling and its consequence for crustal hydromechanics. J. Geophys. Res. 115, B10411 (2010)
de Dreuzy, J.R., Pichot, G., Poirriez, B., Erhel, J.: Synthetic benchmark for modeling flow in 3D fractured media. Comput. Geosci. 50, 59–71 (2013). doi:10.1016/j.cageo.2012.07.025. http://hal.inria.fr/hal-00735675
Dershowitz, W.S., Einstein, H.H.: Characterizing rock joint geometry with joint system models. Rock Mech. Rock Eng. 21(1), 21–51 (1988)
Dohrmann, C.R.: A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25, 246–258 (2003)
Dryja, M., Proskurowski, W.: On preconditioners for mortar discretization of elliptic problems. Numer. Linear Algebra Appl. 10(1–2), 65–82 (2003). doi:10.1002/nla.312. http://dx.doi.org/10.1002/nla.312
Dverstop, B., Andersson, J.: Application of the discrete fracture network concept with field data: possibilities of model calibration and validation. Water Resour. Res. 25(3), 540–550 (1989)
Erhel, J., Guyomarc’h, F.: An augmented conjugate gradient method for solving consecutive symmetric positive definite systems. SIAM J. Matrix Anal. Appl. 21(4), 1279–1299 (2000)
Erhel, J., de Dreuzy, J.R., Poirriez, B.: Flow simulations in three-dimensional discrete fracture networks. SIAM J. Sci. Comput. 31(4), 2688–2705 (2009). doi:10.1137/080729244
Frank, J., Vuik, C.: On the construction of deflation-based preconditioners. SIAM J. Sci. Comput. 23(2), 442–462 (2001). doi:10.1137/S1064827500373231. http://dx.doi.org/10.1137/S1064827500373231
Frey, P.J., George, P.L.: Maillages: applications aux éléments finis. Hermès sciences publ. DL1999 (53-Mayenne), Paris (1999). http://opac.inria.fr/record=b1094298
Gander, M.J., Tu, X.: On the origins of iterative substructuring methods. In: Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2014) (same volume)
Glowinski, R., Wheeler, M.F.: Domain decomposition and mixed finite element methods for elliptic problems. In: Glowinski, R., Golub, G.H., Meurant, G.A., Periaux, J. (eds.) Domain Decomposition Methods for Partial Differential Equations, pp. 144–172. SIAM, Philadelphia (1988)
Kim, H.H., Dryja, M., Widlund, O.B.: A BDDC method for mortar discretizations using a transformation of basis. SIAM J. Numer. Anal. 47(1), 136–157 (2008)
Le Tallec, P.: Domain decomposition methods in computational mechanics. In: Oden, J.T. (ed.) Computational Mechanics Advances, vol. 1, No. 2, pp. 121–220. North-Holland, Amsterdam (1994)
Le Tallec, P., De Roeck, Y., Vidrascu, M.: Domain decomposition methods for large linearly elliptic three-dimensional problems. J. Comput. Appl. Math. 34, 93–117 (1991)
Mandel, J.: Balancing domain decomposition. Commun. Appl. Numer. Methods 9, 233–241 (1993)
Mandel, J., Brezina, M.: Balancing domain decomposition: theory and computations in two and three dimensions. Technical Report UCD/CCM 2, Center for Computational Mathematics, University of Colorado at Denver (1993)
Meurant, G.: Computer Solution of Large Linear Systems. Elsevier Science B.V., Amsterdam (1999)
Pencheva, G., Yotov, I.: Balancing domain decomposition for mortar mixed finite element methods. Numer. Linear Algebra Appl. 10(1–2), 159–180 (2003). doi:10.1002/nla.316. http://dx.doi.org/10.1002/nla.316
Pichot, G., Erhel, J., de Dreuzy, J.R.: A mixed hybrid mortar method for solving flow in discrete fracture networks. Appl. Anal. 89(10), 1629–1643 (2010)
Pichot, G., Erhel, J., de Dreuzy, J.R.: A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks. SIAM J. Sci. Comput. 34(1), B86–B105 (20 pp.) (2012)
Poirriez, B.: Etude et mise en oeuvre d’une méthode de sous-domaines pour la modélisation de l’écoulement dans des réseaux de fractures en 3d. Ph.D. thesis, University of Rennes 1 (2011)
Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999)
Tang, J.M., Nabben, R., Vuik, C., Erlangga, Y.A.: Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods. J. Sci. Comput. 39(3), 340–370 (2009). doi:10.1007/s10915-009-9272-6. http://dx.doi.org/10.1007/s10915-009-9272-6
Vohralik, M., Maryska, J., Severyn, O.: Mixed and nonconforming finite element methods on a system of polygons. Appl. Numer. Math. 57, 176–193 (2007)
Wheeler, M.F., Yotov, I.: A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43(3), 1021–1042 (2005). doi:10.1137/S0036142903431687. http://dx.doi.org/10.1137/S0036142903431687
Acknowledgements
This work was supported by the French National Research Agency, with the ANR-07-CIS7 project MICAS, and by INRIA with the ARC-INRIA GEOFRAC project.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Pichot, G., Poirriez, B., Erhel, J., de Dreuzy, JR. (2014). A Mortar BDD Method for Solving Flow in Stochastic Discrete Fracture Networks. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-05789-7_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05788-0
Online ISBN: 978-3-319-05789-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)