A mixed finite element method for Darcy flow in fractured porous media with non-matching grids
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 2, pp. 465-489.

We consider an incompressible flow problem in a N-dimensional fractured porous domain (Darcy's problem). The fracture is represented by a (N - 1)-dimensional interface, exchanging fluid with the surrounding media. In this paper we consider the lowest-order (ℝ T0, ℙ0) Raviart-Thomas mixed finite element method for the approximation of the coupled Darcy's flows in the porous media and within the fracture, with independent meshes for the respective domains. This is achieved thanks to an enrichment with discontinuous basis functions on triangles crossed by the fracture and a weak imposition of interface conditions. First, we study the stability and convergence properties of the resulting numerical scheme in the uncoupled case, when the known solution of the fracture problem provides an immersed boundary condition. We detail the implementation issues and discuss the algebraic properties of the associated linear system. Next, we focus on the coupled problem and propose an iterative porous domain/fracture domain iterative method to solve for fluid flow in both the porous media and the fracture and compare the results with those of a traditional monolithic approach. Numerical results are provided confirming convergence rates and algebraic properties predicted by the theory. In particular, we discuss preconditioning and equilibration techniques to make the condition number of the discrete problem independent of the position of the immersed interface. Finally, two and three dimensional simulations of Darcy's flow in different configurations (highly and poorly permeable fracture) are analyzed and discussed.

DOI : https://doi.org/10.1051/m2an/2011148
Classification : 76S05,  35Q86,  65L60
Mots clés : Darcy's equation, fractured porous media, mixed finite element, unfitted mesh, fictitious domain, embedded interface, extended finite element
@article{M2AN_2012__46_2_465_0,
author = {D'Angelo, Carlo and Scotti, Anna},
title = {A mixed finite element method for Darcy flow in fractured porous media with non-matching grids},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {465--489},
publisher = {EDP-Sciences},
volume = {46},
number = {2},
year = {2012},
doi = {10.1051/m2an/2011148},
zbl = {1271.76322},
language = {en},
url = {www.numdam.org/item/M2AN_2012__46_2_465_0/}
}
D’Angelo, Carlo; Scotti, Anna. A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 2, pp. 465-489. doi : 10.1051/m2an/2011148. http://www.numdam.org/item/M2AN_2012__46_2_465_0/

[1] C. Alboin, J. Jaffré, J.E. Roberts and C. Serres, Modeling fractures as interfaces for flow and transport in porous media, in Fluid flow and transport in porous media : mathematical and numerical treatment (South Hadley, MA, 2001), Contemp. Math., Amer. Math. Soc. 295 (2002) 13-24. | MR 1911534 | Zbl 1102.76331

[2] P. Angot, F. Boyer and F. Hubert, Asymptotic and numerical modelling of flows in fractured porous media. ESAIM : M2AN 43 (2009) 239-275. | EuDML 250659 | Numdam | MR 2512496 | Zbl 1171.76055

[3] T. Arbogast, L.C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295-1315 (electronic). | MR 1756426 | Zbl 1001.65126

[4] D.N. Arnold, R.S. Falk and R. Winther, Preconditioning in H(div) and applications. Math. Comp. 66 (1997) 957-984. | MR 1401938 | Zbl 0870.65112

[5] R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM : M2AN 37 (2003) 209-225. | Numdam | MR 1991197 | Zbl 1047.65099

[6] R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Eng. 198 (2009) 3352-3360. | MR 2571349 | Zbl 1230.74169

[7] I.I. Bogdanov, V.V. Mourzenko, J.-F. Thovert and P.M. Adler, Two-phase flow through fractured porous media. Phys. Rev. E 68 (2003) 026703. | MR 2011291

[8] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, Springer-Verlag, New York 15 (1991). | MR 1115205 | Zbl 0788.73002

[9] E. Burman and P. Hansbo, A unified stabilized method for Stokes' and Darcy's equations. J. Comput. Appl. Math. 198 (2007) 35-51. | MR 2250387 | Zbl 1101.76032

[10] C. D'Angelo and P. Zunino, A finite element method based on weighted interior penalties for heterogeneous incompressible flows. SIAM J. Numer. Anal. 47 (2009) 3990-4020. | MR 2576529 | Zbl pre05815164

[11] C. D'Angelo and P. Zunino, Robust numerical approximation of coupled stokes and darcy flows applied to vascular hemodynamics and biochemical transport. ESAIM : M2AN 45 (2011) 447-476. | Numdam | MR 2804646 | Zbl 1274.92010

[12] N. Frih, J.E. Roberts and A. Saada, Modeling fractures as interfaces : a model for Forchheimer fractures. Comput. Geosci. 12 (2008) 91-104. | MR 2386967 | Zbl 1138.76062

[13] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, Springer-Verlag, Berlin 5 (1986). | MR 851383 | Zbl 0585.65077

[14] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191 (2002) 5537-5552. | MR 1941489 | Zbl 1035.65125

[15] V. Martin, J. Jaffré and J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667-1691 (electronic). | MR 2142590 | Zbl 1083.76058

[16] N. Moës, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing. Internat. J. Numer. Methods Eng. 46 (1999) 131-150. | Zbl 0955.74066

[17] C.E. Powell and D. Silvester, Optimal preconditioning for Raviart-Thomas mixed formulation of second-order elliptic problems. SIAM J. Matrix Anal. Appl. 25 (2003) 718-738 (electronic). | MR 2081231 | Zbl 1073.65128

[18] A. Quarteroni and A. Valli, Numerical Aproximation of Partial Differential Equations. Springer (1994). | MR 1299729 | Zbl 1151.65339

[19] A. Reusken, Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput. Vis. Sci. 11 (2008) 293-305. | MR 2425497 | Zbl 1261.76015

[20] P. Zunino, L. Cattaneo and C.M. Colciago, An unfitted interface penalty method for the numerical approximation of contrast problems. Appl. Num. Math. 61 (2011) 1059-1076. | MR 2820966 | Zbl 1232.65152