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A mixed finite element method for Darcy flow in fracturedporous media with non-matching grids∗

Published online by Cambridge University Press:  19 December 2011

Carlo D’Angelo  and
Anna Scotti
Carlo D’Angelo
Affiliation:
MOX-Dip. di Matematica “F. Brioschi” Politecnico di Milano via Bonardi 9, 20133 Milano, Italy. [email protected]
Anna Scotti
Affiliation:
MOX-Dip. di Matematica “F. Brioschi” Politecnico di Milano via Bonardi 9, 20133 Milano, Italy; [email protected]
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Abstract

We consider an incompressible flow problem in a N-dimensional fracturedporous domain (Darcy’s problem). The fracture is represented by a(N − 1)-dimensional interface, exchanging fluid with the surroundingmedia. In this paper we consider the lowest-order(ℝ T0, ℙ0) Raviart-Thomas mixed finite elementmethod for the approximation of the coupled Darcy’s flows in the porous media and withinthe fracture, with independent meshes for the respective domains. This is achieved thanksto an enrichment with discontinuous basis functions on triangles crossed by the fractureand a weak imposition of interface conditions. First, we study the stability andconvergence properties of the resulting numerical scheme in the uncoupled case, when theknown solution of the fracture problem provides an immersed boundary condition. We detailthe implementation issues and discuss the algebraic properties of the associated linearsystem. Next, we focus on the coupled problem and propose an iterative porousdomain/fracture domain iterative method to solve for fluid flow in both the porous mediaand the fracture and compare the results with those of a traditional monolithic approach.Numerical results are provided confirming convergence rates and algebraic propertiespredicted by the theory. In particular, we discuss preconditioning and equilibrationtechniques to make the condition number of the discrete problem independent of theposition of the immersed interface. Finally, two and three dimensional simulations ofDarcy’s flow in different configurations (highly and poorly permeable fracture) areanalyzed and discussed.

Keywords

Darcy’s equationfractured porous mediamixed finite elementunfitted meshfictitious domainembedded interfaceextended finite element
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 465 - 489
Copyright
© EDP Sciences, SMAI 2011

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References

Alboin, C., Jaffré, J., Roberts, J.E. and Serres, C., 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. Google Scholar
Angot, P., Boyer, F. and Hubert, F., Asymptotic and numerical modelling of flows in fractured porous media. ESAIM : M2AN 43 (2009) 239–275. Google Scholar
Arbogast, T., Cowsar, L.C., Wheeler, M.F. and Yotov, I., Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295–1315 (electronic). Google Scholar
Arnold, D.N., Falk, R.S. and Winther, R., Preconditioning in H(div) and applications. Math. Comp. 66 (1997) 957–984. Google Scholar
Becker, R., Hansbo, P. and Stenberg, R., A finite element method for domain decomposition with non-matching grids. ESAIM : M2AN 37 (2003) 209–225. Google Scholar
Becker, R., Burman, E. and Hansbo, P., A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Eng. 198 (2009) 3352–3360. Google Scholar
Bogdanov, I.I., Mourzenko, V.V., Thovert, J.-F. and Adler, P.M., Two-phase flow through fractured porous media. Phys. Rev. E 68 (2003) 026703. Google ScholarPubMed
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, Springer-Verlag, New York 15 (1991).
Burman, E. and Hansbo, P., A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math. 198 (2007) 35–51. Google Scholar
D’Angelo, C. and Zunino, P., A finite element method based on weighted interior penalties for heterogeneous incompressible flows. SIAM J. Numer. Anal. 47 (2009) 3990–4020. Google Scholar
D’Angelo, C. and Zunino, P., Robust numerical approximation of coupled stokes and darcy flows applied to vascular hemodynamics and biochemical transport. ESAIM : M2AN 45 (2011) 447–476. Google Scholar
Frih, N., Roberts, J.E. and Saada, A., Modeling fractures as interfaces : a model for Forchheimer fractures. Comput. Geosci. 12 (2008) 91–104. Google Scholar
V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, Springer-Verlag, Berlin 5 (1986).
Hansbo, A. and Hansbo, P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191 (2002) 5537–5552. Google Scholar
Martin, V., Jaffré, J. and Roberts, J.E., Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667–1691 (electronic). Google Scholar
Moës, N., Dolbow, J. and Belytschko, T., A finite element method for crack growth without remeshing. Internat. J. Numer. Methods Eng. 46 (1999) 131–150. Google Scholar
Powell, C.E. and Silvester, D., Optimal preconditioning for Raviart–Thomas mixed formulation of second-order elliptic problems. SIAM J. Matrix Anal. Appl. 25 (2003) 718–738 (electronic). Google Scholar
A. Quarteroni and A. Valli, Numerical Aproximation of Partial Differential Equations. Springer (1994).
Reusken, A., Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput. Vis. Sci. 11 (2008) 293–305. Google Scholar
Zunino, P., Cattaneo, L. and Colciago, C.M., An unfitted interface penalty method for the numerical approximation of contrast problems. Appl. Num. Math. 61 (2011) 1059–1076. Google Scholar