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A reduced model for Darcy’s problem in networks offractures∗

Published online by Cambridge University Press:  02 July 2014

Luca Formaggia ,
Alessio Fumagalli ,
Anna Scotti  and
Paolo Ruffo
Luca Formaggia
Affiliation:
MOX - Dipartimento di Matematica “F. Brioschi” – Politecnico di Milano - via Bonardi 9, 20133 Milan, Italy.. [email protected]; [email protected]
Alessio Fumagalli
Affiliation:
MOX - Dipartimento di Matematica “F. Brioschi” – Politecnico di Milano - via Bonardi 9, 20133 Milan, Italy.. [email protected]; [email protected] IFP Energies nouvelles – 1 and 4, avenue de Bois-Prau, 92852 Rueil-Malmaison Cedex, France. ; [email protected]
Anna Scotti
Affiliation:
MOX - Dipartimento di Matematica “F. Brioschi” – Politecnico di Milano - via Bonardi 9, 20133 Milan, Italy.. [email protected]; [email protected]
Paolo Ruffo
Affiliation:
ENI Spa – Exploration and Production Division – 5° Palazzo Uffici, Room 4046 E, GEBA Dept. - via Emilia 1, San Donato Milanese, 20097 (MI), Italy.; [email protected]
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Abstract

Subsurface flows are influenced by the presence of faults and large fractures which actas preferential paths or barriers for the flow. In literature models were proposed tohandle fractures in a porous medium as objects of codimension 1. In this work we considerthe case of a network of intersecting fractures, with the aim of deriving physicallyconsistent and effective interface conditions to impose at the intersection betweenfractures. This new model accounts for the angle between fractures at the intersectionsand allows for jumps of pressure across intersections. This fact permits to describe theflow when fractures are characterized by different properties more accurately with respectto other models that impose pressure continuity. The main mathematical properties of themodel, derived in the two-dimensional setting, are analyzed. As concerns the numericaldiscretization we allow the grids of the fractures to be independent, thus in generalnon-matching at the intersection, by means of the extended finite element method(XFEM). This increases the flexibility of the method in the case of complexgeometries characterized by a high number of fractures.

Keywords

Reduced modelsfractured porous mediaXFEM
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1089 - 1116
Copyright
© EDP Sciences, SMAI 2014

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References

R.T. Adams, Sobolev Spaces, vol. 65. Pure and Applied Mathematics. Academic Press (1975). Google Scholar
P.M. Adler and J.-F. Thovert, Fractures and fracture networks. Springer (1999). Google Scholar
P.M. Adler, J.-F. Thovert and V.V. Mourzenko, Fractured porous media. Oxford University Press (2012). Google Scholar
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), vol. 295. Contemp. Math.. Amer. Math. Soc. Providence, RI (2002) 13–24. Google Scholar
C. Alboin, J. Jaffré, J.E. Roberts, X. Wang and C. Serres. Domain decomposition for some transmission problems in flow in porous media, vol. 552. Lect. Notes Phys. Springer, Berlin (2000) 22–34. CrossRefGoogle Scholar
L. Amir, M. Kern, V. Martin and J.E. Roberts, Décomposition de domaine et préconditionnement pour un modèle 3D en milieu poreux fracturé, in Proc. of JANO 8, 8th Conf. Numer. Anal. Optim. (2005). 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
J. Bear, C.-F. Tsang and G. de Marsily, Flow and contaminant transport in fractured rock. Academic Press, San Diego (1993). CrossRefGoogle 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. Engrg. 198 (2009) 3352–3360. Google Scholar
Berkowitz, B., Characterizing flow and transport in fractured geological media: A review. Adv. Water Resources 25 (2002) 861–884. Google Scholar
Berrone, S., Pieraccini, S. and Scialò, S., On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. 35 (2013) 908–935. Google Scholar
Berrone, S., Pieraccini, S. and Scialò, S., A PDE-constrained optimization formulation for discrete fracture network flows. SIAM J. Sci. Comput. 35 (2013). Google Scholar
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Springer (2010). Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15. Comput. Math. Springer Verlag, Berlin (1991). Google Scholar
D’Angelo, C. and Scotti, A., A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: M2AN 46 (2012) 465–489. Google Scholar
A. Ern and J.L. Guermond, Theory and practice of finite elements. Appl. Math. Sci. Springer (2004). Google Scholar
B. Faybishenko, P.A. Witherspoon and S.M. Benson, Dynamics of fluids in fractured rock, vol. 122. Geophysical monographs. American geophysical union (2000). CrossRefGoogle Scholar
A. Fumagalli, Numerical Modelling of Flows in Fractured Porous Media by the XFEM Method. Ph.D. thesis, Politecnico di Milano (2012). Google Scholar
A. Fumagalli and A. Scotti, A numerical method for two-phase ow in fractured porous media with non-matching grids, in vol. 62 of Adv. Water Resources (2013) 454–464. CrossRefGoogle Scholar
A. Fumagalli and A. Scotti, A reduced model for flow and transport in fractured porous media with non-matching grids, Numer. Math. Advanced Applications 2011. Edited by A. Cangiani, R.L. Davidchack, E. Georgoulis, A.N. Gorban, J. Levesley and M.V. Tretyakov. Springer Berlin, Heidelberg (2013) 499–507. Google Scholar
B. Gong, G. Qin, C. Douglas and S. Yuan, Detailed modeling of the complex fracture network of shale gas reservoirs. SPE Reservoir Evaluation and Engrg. (2011). Google Scholar
Hansbo, A. and Hansbo, P., A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Engrg. 193 (2004) 3523–3540. Google Scholar
M. Hussein and D. Roussos, Discretizing two-dimensional complex fractured fields for incompressible two-phase flow. Int. J. Numer. Methods Fluids (2009). Google Scholar
J. Jaffré, V. Martin and J.E. Roberts, Generalized cell-centered finite volume methods for flow in porous media with faults, in Finite volumes for complex applications III (Porquerolles, 2002). Hermes Sci. Publ., Paris (2002) 343–350. Google Scholar
Jaffré, J., Mnejja, M. and Roberts, J.E., A discrete fracture model for two-phase flow with matrix-fracture interaction. Procedia Comput. Sci. 4 (2011) 967–973. Google Scholar
Karimi-Fard, M., Durlofsky, L.J. and Aziz, K., An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators. SPE J. 9 (2004) 227–236. 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. Google Scholar
Mustapha, H., A new approach to simulating flow in discrete fracture networks with an optimized mesh. SIAM J. Sci. Comput. 29 (2007) 1439–1459. Google Scholar
A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, vol. 23. Springer Ser. Comput. Math. Springer-Verlag, Berlin (1994). Google Scholar
M. Sahimi, Flow and transport in porous media and fractured rock. Wiley-VCH, Weinheim (2011). Google Scholar