A reduced model for Darcy’s problem in networks of fractures

Luca Formaggia; Alessio Fumagalli; Anna Scotti; Paolo Ruffo

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 4, page 1089-1116
  • ISSN: 0764-583X

Abstract

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

How to cite

top

Formaggia, Luca, et al. "A reduced model for Darcy’s problem in networks of fractures." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.4 (2014): 1089-1116. <http://eudml.org/doc/273287>.

@article{Formaggia2014,
abstract = {Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures.},
author = {Formaggia, Luca, Fumagalli, Alessio, Scotti, Anna, Ruffo, Paolo},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {reduced models; fractured porous media; XFEM},
language = {eng},
number = {4},
pages = {1089-1116},
publisher = {EDP-Sciences},
title = {A reduced model for Darcy’s problem in networks of fractures},
url = {http://eudml.org/doc/273287},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Formaggia, Luca
AU - Fumagalli, Alessio
AU - Scotti, Anna
AU - Ruffo, Paolo
TI - A reduced model for Darcy’s problem in networks of fractures
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 4
SP - 1089
EP - 1116
AB - Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures.
LA - eng
KW - reduced models; fractured porous media; XFEM
UR - http://eudml.org/doc/273287
ER -

References

top
  1. [1] R.T. Adams, Sobolev Spaces, vol. 65. Pure and Applied Mathematics. Academic Press (1975). Zbl0314.46030
  2. [2] P.M. Adler and J.-F. Thovert, Fractures and fracture networks. Springer (1999). 
  3. [3] P.M. Adler, J.-F. Thovert and V.V. Mourzenko, Fractured porous media. Oxford University Press (2012). Zbl1266.74002MR3237557
  4. [4] 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. Zbl1102.76331MR1911534
  5. [5] 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. Zbl1010.76050MR1876007
  6. [6] 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). 
  7. [7] P. Angot, F. Boyer and F. Hubert, Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: M2AN 43 (2009) 239–275. Zbl1171.76055MR2512496
  8. [8] J. Bear, C.-F. Tsang and G. de Marsily, Flow and contaminant transport in fractured rock. Academic Press, San Diego (1993). 
  9. [9] 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. Engrg.198 (2009) 3352–3360. Zbl1230.74169MR2571349
  10. [10] B. Berkowitz, Characterizing flow and transport in fractured geological media: A review. Adv. Water Resources25 (2002) 861–884. 
  11. [11] S. Berrone, S. Pieraccini and S. Scialò, On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput.35 (2013) 908–935. Zbl1266.65187MR3038026
  12. [12] S. Berrone, S. Pieraccini and S. Scialò, A PDE-constrained optimization formulation for discrete fracture network flows. SIAM J. Sci. Comput. 35 (2013). Zbl1266.65188MR3038028
  13. [13] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Springer (2010). Zbl1220.46002MR2759829
  14. [14] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15. Comput. Math. Springer Verlag, Berlin (1991). Zbl0788.73002MR1115205
  15. [15] C. D’Angelo and A. Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: M2AN 46 (2012) 465–489. Zbl1271.76322
  16. [16] A. Ern and J.L. Guermond, Theory and practice of finite elements. Appl. Math. Sci. Springer (2004). Zbl1059.65103MR2050138
  17. [17] B. Faybishenko, P.A. Witherspoon and S.M. Benson, Dynamics of fluids in fractured rock, vol. 122. Geophysical monographs. American geophysical union (2000). 
  18. [18] A. Fumagalli, Numerical Modelling of Flows in Fractured Porous Media by the XFEM Method. Ph.D. thesis, Politecnico di Milano (2012). 
  19. [19] 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. Zbl1273.76398
  20. [20] 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. Zbl1273.76398
  21. [21] 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). 
  22. [22] A. Hansbo and P. Hansbo, A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Engrg.193 (2004) 3523–3540. Zbl1068.74076MR2075053
  23. [23] M. Hussein and D. Roussos, Discretizing two-dimensional complex fractured fields for incompressible two-phase flow. Int. J. Numer. Methods Fluids (2009). Zbl05862329
  24. [24] 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. Zbl1177.76231MR2007435
  25. [25] J. Jaffré, M. Mnejja and J.E. Roberts, A discrete fracture model for two-phase flow with matrix-fracture interaction. Procedia Comput. Sci.4 (2011) 967–973. 
  26. [26] M. Karimi-Fard, L.J. Durlofsky and K. Aziz, An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators. SPE J.9 (2004) 227–236. 
  27. [27] 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. Zbl1083.76058MR2142590
  28. [28] H. Mustapha, A new approach to simulating flow in discrete fracture networks with an optimized mesh. SIAM J. Sci. Comput.29 (2007) 1439–1459. Zbl1251.76056MR2341795
  29. [29] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, vol. 23. Springer Ser. Comput. Math. Springer-Verlag, Berlin (1994). Zbl0803.65088MR1299729
  30. [30] M. Sahimi, Flow and transport in porous media and fractured rock. Wiley-VCH, Weinheim (2011). Zbl1219.76002

NotesEmbed ?

top

You must be logged in to post comments.