We present a possible framework for the numerical simulation of flow in fractured porous media that couples mimetic finite differences for the porous matrix with a finite volume scheme for the flow in the fractures. The resulting method is theoretically analyzed in the case of a single fracture. Moreover, several numerical experiments show the capability of the method to deal also with complicated networks of fractures. Thanks to the implementation of rather general coupling conditions, it encompasses both “conductive fractures”, i.e., fractures with high permeability and “sealed fractures”, i.e., fractures with low permeability which act as a flow barrier.
DOI: 10.1051/m2an/2015087
Mots-clés : Mimetic finite differences, flow in fractured porous media
@article{M2AN_2016__50_3_809_0, author = {Antonietti, Paola F. and Formaggia, Luca and Scotti, Anna and Verani, Marco and Verzott, Nicola}, title = {Mimetic finite difference approximation of flows in fractured porous media}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {809--832}, publisher = {EDP-Sciences}, volume = {50}, number = {3}, year = {2016}, doi = {10.1051/m2an/2015087}, mrnumber = {3507274}, zbl = {1381.76231}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015087/} }
TY - JOUR AU - Antonietti, Paola F. AU - Formaggia, Luca AU - Scotti, Anna AU - Verani, Marco AU - Verzott, Nicola TI - Mimetic finite difference approximation of flows in fractured porous media JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 809 EP - 832 VL - 50 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015087/ DO - 10.1051/m2an/2015087 LA - en ID - M2AN_2016__50_3_809_0 ER -
%0 Journal Article %A Antonietti, Paola F. %A Formaggia, Luca %A Scotti, Anna %A Verani, Marco %A Verzott, Nicola %T Mimetic finite difference approximation of flows in fractured porous media %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 809-832 %V 50 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015087/ %R 10.1051/m2an/2015087 %G en %F M2AN_2016__50_3_809_0
Antonietti, Paola F.; Formaggia, Luca; Scotti, Anna; Verani, Marco; Verzott, Nicola. Mimetic finite difference approximation of flows in fractured porous media. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Volume 50 (2016) no. 3, pp. 809-832. doi : 10.1051/m2an/2015087. http://www.numdam.org/articles/10.1051/m2an/2015087/
P.M. Adler, J.-F. Thovert and V.V. Mourzenko, Fractured Porous Media. Oxford University Press (2013). | MR | Zbl
O. Al-Hinai, S. Srinivasan and M.F. Wheeler, 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).
C. Alboin, J. Jaffré, J.E. Roberts, X. Wang and C. Serres, Domain decomposition for some transmission problems in flow in porous media. In vol. 552 of Lecture Notes Phys. Springer, Berlin (2000) 22–34. | MR | Zbl
Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: M2AN 43 (2009) 239–275. | DOI | Numdam | MR | Zbl
, and ,Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems. SIAM J. Numer. Anal. 51 (2013) 654–675. | DOI | MR | Zbl
, , and ,A mimetic discretization of elliptic obstacle problems. Math. Comput. 82 (2013) 1379–1400. | DOI | MR | Zbl
, , and ,Mimetic discretizations of elliptic control problems. J. Sci. Comput. 56 (2013) 14–27. | DOI | MR | Zbl
, and ,Mimetic finite differences for nonlinear and control problems. Math. Models Methods Appl. Sci. 24 (2014) 1457–1493. | DOI | MR | Zbl
, , and ,Mimetic finite difference method for shape optimization problems. Lect. Notes Comput. Sci. Eng. 103 (2015) 125–132. | MR | Zbl
, and ,Mimetic finite difference approximation of quasilinear elliptic problems. Calcolo 52 (2015) 45–67. | DOI | MR | Zbl
, and ,J. Bear, C.-F. Tsang and G. de Marsily, Flow and contaminant transport in fractured rock. Academic Press, San Diego (1993).
Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. | DOI | MR | Zbl
, and ,Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl
, , , , and ,Numerical results for mimetic discretization of Reissner-Mindlin plate problems. Calcolo 50 (2013) 209–237. | DOI | MR | Zbl
, and ,L. Beirao da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. Springer (2014). | MR | Zbl
The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Engrg. 280 (2014) 135–156. | DOI | MR | Zbl
, , and ,A globally conforming method for solving flow in discrete fracture networks using the virtual element method. Finite Elements Anal. Design 109 (2016) 23–36. | DOI
, and ,Numerical solution of saddle point problems. Acta numerica 14 (2005) 1–137. | DOI | MR | Zbl
, and ,On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. 35 (2013) A908–A935. | DOI | MR | Zbl
, and ,A PDE-constrained optimization formulation for discrete fracture network flows. SIAM J. Sci. Comput. 35 (2013) B487–B510. | DOI | MR | Zbl
, and ,An optimization approach for large scale simulations of discrete fracture network flows. J. Comput. Phys. 256 (2014) 838–853. | DOI | MR | Zbl
, and ,A parallel solver for large scale DFN flow simulations. SIAM J. Sci. Comput. 37 (2015) C285–C306. | DOI | MR | Zbl
, , and ,K. Brenner, M. Groza, C. Guichard, G. Lebeau and R. Masson, Gradient discretization of hybrid dimensional Darcy flows in fractured porous media. In Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, edited by J. Fuhrmann, M. Ohlberger and C. Rohde. Springer (2014) 527–535. | MR
K. Brenner, M. Groza, C. Guichard, G. Lebeau and R. Masson, Gradient discretization of hybrid dimensional Darcy flows in fractured porous media. Technical Report. HAL archives-ouvertes hal-01097704 (2014). | MR
K Brenner, J Hennicker, R Masson and P Samier, Gradient discretization of hybrid dimensional Darcy flows in fractured porous media with discontinuous pressures at the matrix fracture interfaces. In MAMERN VI-2015 HAL-01147495 (2015). | MR
Innovative mimetic discretizations for electromagnetic problems. J. Comput. Appl. Math. 234 (2010) 1980–1987. | DOI | MR | Zbl
and ,Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. | DOI | MR | Zbl
, and ,A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005) 1533–1551. | DOI | MR | Zbl
, and ,Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16 (2006) 275–297. | DOI | MR | Zbl
, and ,A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3682–3692. | DOI | MR | Zbl
, , and ,Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. | DOI | Numdam | MR | Zbl
, and ,Mimetic scalar products of discrete differential forms. J. Comput. Phys. 257 (2014) 1228–1259. | DOI | MR | Zbl
, and ,B. da Veiga Lourenco, K. Lipnikov and G. Manzini, The mimetic finite difference method for elliptic problems. Vol. 11 of MS&A. Model. Simul. Appl. Springer, Cham (2014). | MR | Zbl
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
and ,Finite volume schemes for diffusion equations: Introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24 (2014) 1575–1619. | DOI | MR | Zbl
,Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. 23 (2013) 2395–2432. | DOI | MR | Zbl
, , and ,J. Droniou, R. Eymard and R. Herbin, Gradient schemes: generic tools for the numerical analysis of diffusion equations. To appear in Special issue – Polyhedral discretization for PDE: ESAIM: M2AN 50 (2016) Doi:. | DOI | Numdam | MR
G.D. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces. In Partial differential equations and calculus of variations. Vol. 1357 of Lect. Notes Math. Springer, Berlin (1988) 142–155. | MR | Zbl
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. In Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis, VII. Edited by P.G. Ciarlet and J.-L. Lions. North Holland (2000) 713–1020. | MR | Zbl
A numerical method for two-phase flow in fractured porous media with non-matching grids. Computational Methods in Geologic CO2 Sequestration. Adv. Water Res. 62 (2013) 454–464. | DOI
and ,G. Guennebaud, Benoît Jacob, et al., Eigen v3. http://eigen.tuxfamily.org (2010).
Comparison of cell-and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture–matrix system. Adv. Water Res. 32 (2009) 1740–1755. | DOI
, , , and ,A discrete fracture model for two-phase flow with matrix-fracture interaction. Proc. Comput. Sci. 4 (2011) 967–973. | DOI
, and ,An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9 (2004) 227–236. | DOI
, , , et al.,A multilevel multiscale mimetic method for two-phase flows in porous media. J. Comput. Phys. 227 (2008) 6727–6753. | DOI | MR | Zbl
, and ,Mimetic finite difference method. Physics-compatible numerical methods. J. Comput. Phys. 257 (2014) 1163–1227. | DOI | MR | Zbl
, and ,B.T. Mallison, M.H. Hui and W. Narr, Practical gridding algorithms for discrete fracture modeling workflows. In 12th European Conference on the Mathematics of Oil Recovery (2010).
Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667–1691. | DOI | MR | Zbl
, and ,A Gabriel-Delaunay triangulation of 2D complex fractured media for multiphase flow simulations. Comput. Geosci. 18 (2014) 989–1008. | DOI | Zbl
,J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods. In Vol. 2 of Handbook of Numerical Analysis. Finite Element Methods (Part I). Elsevier (1991) 523–639. | MR | Zbl
The CGAL Project, CGAL User and Reference Manual. CGAL Editorial Board, 4.6 edition (2015).
Cited by Sources: