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

• Volume: 46, Issue: 2, page 465-489
• ISSN: 0764-583X

top

Abstract

top
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.

How to cite

top

D’Angelo, Carlo, and Scotti, Anna. "A mixed finite element method for Darcy flow in fractured porous media with non-matching grids∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.2 (2011): 465-489. <http://eudml.org/doc/222154>.

@article{D2011,
abstract = {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.},
author = {D’Angelo, Carlo, Scotti, Anna},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Darcy’s equation; fractured porous media; mixed finite element; unfitted mesh; fictitious domain; embedded interface; extended finite element; Darcy's equation},
language = {eng},
month = {12},
number = {2},
pages = {465-489},
publisher = {EDP Sciences},
title = {A mixed finite element method for Darcy flow in fractured porous media with non-matching grids∗},
url = {http://eudml.org/doc/222154},
volume = {46},
year = {2011},
}

TY - JOUR
AU - D’Angelo, Carlo
AU - Scotti, Anna
TI - A mixed finite element method for Darcy flow in fractured porous media with non-matching grids∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/12//
PB - EDP Sciences
VL - 46
IS - 2
SP - 465
EP - 489
AB - 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.
LA - eng
KW - Darcy’s equation; fractured porous media; mixed finite element; unfitted mesh; fictitious domain; embedded interface; extended finite element; Darcy's equation
UR - http://eudml.org/doc/222154
ER -

References

top
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.
2. P. Angot, F. Boyer and F. Hubert, Asymptotic and numerical modelling of flows in fractured porous media. ESAIM : M2AN43 (2009) 239–275.
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).
4. D.N. Arnold, R.S. Falk and R. Winther, Preconditioning in H(div) and applications. Math. Comp.66 (1997) 957–984.
5. R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM : M2AN37 (2003) 209–225.
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.
7. I.I. Bogdanov, V.V. Mourzenko, J.-F. Thovert and P.M. Adler, Two-phase flow through fractured porous media. Phys. Rev. E68 (2003) 026703.
8. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, Springer-Verlag, New York 15 (1991).
9. E. Burman and P. Hansbo, A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math.198 (2007) 35–51.
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.
11. C. D’Angelo and P. Zunino, Robust numerical approximation of coupled stokes and darcy flows applied to vascular hemodynamics and biochemical transport. ESAIM : M2AN45 (2011) 447–476.
12. N. Frih, J.E. Roberts and A. Saada, Modeling fractures as interfaces : a model for Forchheimer fractures. Comput. Geosci.12 (2008) 91–104.
13. V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, Springer-Verlag, Berlin 5 (1986).
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.
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).
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.
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).
18. A. Quarteroni and A. Valli, Numerical Aproximation of Partial Differential Equations. Springer (1994).
19. A. Reusken, Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput. Vis. Sci.11 (2008) 293–305.
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.

top

NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.