Mixed Formulation of a Linearized Lubrication Fracture Model in a Poro-elastic Medium

  • Vivette GiraultEmail author
  • Mary F. Wheeler
  • Kundan Kumar
  • Gurpreet Singh
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)


We analyse and discretize a mixed formulation for a linearized lubrication fracture model in a poro-elastic medium. The displacement of the medium is expressed in primary variables while the flows in the medium and fracture are written in mixed form, with an additional unknown for the pressure in the fracture. The fracture is treated as a non-planar surface or curve according to the dimension, and the lubrication equation for the flow in the fracture is linearized. The resulting equations are discretized by finite elements adapted to primal variables for the displacement and mixed variables for the flow. Stability and a priori error estimates are derived. A fixed-stress algorithm is proposed for decoupling the computation of the displacement and flow and a numerical experiment is included.


Poro-elasticity Biot Lubrication Mixed formulation Finite-elements Fixed stress split algorithm 


  1. 1.
    Alboin C, Jaffré J, Roberts JE, Serres C (2001) Modeling fractures as interfaces for flow and transport in porous media. In: Chen Z, Ewing RE (eds) Fluid flow and transport in porous media: mathematical and numerical treatment. South Hadley, MA. (Vol 295 of Contemporary Mathematics). American Mathematical Society, Providence, RI, pp 13–24 (2002)Google Scholar
  2. 2.
    Babus̆ka I (1972/73) The finite element method with Lagrangian multipliers. Numer Math 20:179–192Google Scholar
  3. 3.
    Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164CrossRefGoogle Scholar
  4. 4.
    Bourgeat A, Mikelić A, Piatnitski A (2003) On the double porosity model of a single phase flow in random media. Asymptot Anal 34(3–4):311–332MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brenner SC, Scott LR (2008) The mathematical theory of finite element methods, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  6. 6.
    Brezzi F (1974) On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev Française Automat Inform Rech Opér Anal Numér 8(R–2):129–151MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brezzi F, Douglas J Jr, Durán R, Fortin M (1987) Mixed finite elements for second order elliptic problems in three variables. Numer Math 51(2):237–250MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brezzi F, Douglas J Jr, Marini LD (1985) Two families of mixed finite elements for second order elliptic problems. Numer Math 47(2):217–235MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ciarlet PG (1991) Basic error estimates for elliptic problems. In: Handbook of numerical analysis, Vol II. North-Holland, Amsterdam, pp 17–351Google Scholar
  10. 10.
    Dean RH, Schmidt JH (2009) Hydraulic-fracture predictions with a fully coupled geomechanical reservoir simulator. SPE J 14(04):707–714CrossRefGoogle Scholar
  11. 11.
    Fasano A, Mikelić A, Primicerio M (1998) Homogenization of flows through porous media with permeable grains. Adv Math Sci Appl 8(1):1–31MathSciNetzbMATHGoogle Scholar
  12. 12.
    Galvis J, Sarkis M (2007) Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron Trans Numer Anal 26:350–384MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ganis B, Girault V, Mear M, Singh G, Wheeler MF (2014) Modeling fractures in a poro-elastic medium. Oil Gas Sci Tech 69(4):515–528CrossRefGoogle Scholar
  14. 14.
    Girault V, Kumar K, Wheeler MF (2016) Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium. Comput Geosci 20(5):997–1011MathSciNetCrossRefGoogle Scholar
  15. 15.
    Girault V, Pencheva G, Wheeler MF, Wildey T (2011) Domain decomposition for poroelasticity and elasticity with DG jumps and mortars. Math Models Methods Appl Sci 21(1):169–213MathSciNetCrossRefGoogle Scholar
  16. 16.
    Girault V, Raviart P-A (1986) Finite element methods for Navier-Stokes equations: theory and algorithms, vol 5. Springer Series in Computational Mathematics. Springer, BerlinGoogle Scholar
  17. 17.
    Girault V, Wheeler MF, Ganis B, Mear ME (2013) A lubrication fracture model in a poro-elastic medium. ICES Report 13-32, Institute for Computational Engineering and Sciences, University of Texas at AustinGoogle Scholar
  18. 18.
    Girault V, Wheeler MF, Ganis B, Mear ME (2015) A lubrication fracture model in a poro-elastic medium. Math Models Methods Appl Sci 25(4):587–645MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ingram R, Wheeler MF, Yotov I (2010) A multipoint flux mixed finite element method on hexahedra. SIAM J Numer Anal 48(4):1281–1312MathSciNetCrossRefGoogle Scholar
  20. 20.
    Jerison DS, Kenig CE (1981) The Neumann problem on Lipschitz domains. Bull Am Math Soc (NS) 4(2):203–207MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lions J-L, Magenes E (1972) Non-homogeneous boundary value problems and applications, vol I. Springer, New YorkCrossRefGoogle Scholar
  22. 22.
    Martin V, Jaffré J, Roberts JE (2005) Modeling fractures and barriers as interfaces for flow in porous media. SIAM J Sci Comput 26(5):1667–1691MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mikelić A, Wheeler MF (2013) Convergence of iterative coupling for coupled flow and geomechanics. Comput Geosci 17(3):455–461MathSciNetCrossRefGoogle Scholar
  24. 24.
    Phillips PJ, Wheeler MF (2007) A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case. Comput Geosci 11(2):131–144MathSciNetCrossRefGoogle Scholar
  25. 25.
    Scott LR, Zhang S (1990) Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp 54(190):483–493MathSciNetCrossRefGoogle Scholar
  26. 26.
    Showalter RE (2000) Diffusion in poro-elastic media. J Math Anal Appl 251(1):310–340MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wheeler MF, Xue G, Yotov I (2014) Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity. Comput Geosci 18(1):57–75MathSciNetCrossRefGoogle Scholar
  28. 28.
    Witherspoon PA, Wang JSY, Iwai K, Gale JE (1980) Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour Res 16(6):1016–1024CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Vivette Girault
    • 1
    Email author
  • Mary F. Wheeler
    • 2
  • Kundan Kumar
    • 3
  • Gurpreet Singh
    • 2
  1. 1.Sorbonne Universités, UPMC Univ. Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Center for Subsurface Modeling, Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  3. 3.Porous Media Group, Mathematics InstituteUniversity of BergenBergenNorway

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