Simulation of Flow in Fractured Poroelastic Media: A Comparison of Different Discretization Approaches

  • I. Ambartsumyan
  • E. Khattatov
  • I. YotovEmail author
  • P. Zunino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


We study two finite element computational models for solving coupled problems involving flow in a fracture and flow in poroelastic media. The Brinkman equation is used in the fracture, while the Biot system of poroelasticity is employed in the surrounding media. Appropriate equilibrium and kinematic conditions are imposed on the interfaces. We focus on the approximation of the interface conditions, which in this context feature the interaction of different variables, such as velocities, displacements, stresses and pressures. The aim of this study is to compare the Lagrange multiplier and the Nitsche’s methods applied to enforce these non standard interface conditions.


Interface Condition Lagrange Multiplier Method Saddle Point Problem Poroelastic Medium Brinkman Equation 
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The first three authors have been partially supported by the NSF grants DMS 1115856 and DMS 1418947. The third and fourth authors have been partially supported by the DOE grant DE-FG02-04ER25618. The authors thank Martina Bukac and Rana Zakerzadeh for their contribution to the development of the software used in this paper.


  1. 1.
    Badia, S., Quaini, A., Quarteroni, A.: Coupling biot and navier-stokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys. 228(21), 7986–8014 (2009). zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Beavers, G., Joseph, D.: Boundary conditions at a naturally impermeable wall. J. Fluid. Mech 30, 197–207 (1967)CrossRefGoogle Scholar
  3. 3.
    Biot, M.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)zbMATHCrossRefGoogle Scholar
  4. 4.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013). zbMATHGoogle Scholar
  5. 5.
    Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the stokes equations. In: Hackbusch, W. (ed.) Efficient solutions of elliptic systems (Kiel, 1984). Notes on Numerical Fluid Mechanics, vol. 10, pp. 11–19. Vieweg, Braunschweig (1984)CrossRefGoogle Scholar
  6. 6.
    Bukac, M., Yotov, I., Zakerzadeh, R., Zunino, P.: Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Comput. Methods Appl. Mech. Eng. 292, 138–170 (2015). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burman, E., Fernández, M.A.: Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility. Comput. Methods Appl. Mech. Eng. 198, 766–784 (2009)zbMATHCrossRefGoogle Scholar
  8. 8.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)zbMATHGoogle Scholar
  9. 9.
    Girault, V., Vassilev, D., Yotov, I.: Mortar multiscale finite element methods for Stokes-Darcy flows. Numer. Math. 127(1), 93–165 (2014). zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hansbo, P.: Nitsche’s method for interface problems in computational mechanics. GAMM-Mitt. 28(2), 183–206 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Lesinigo, M., D’Angelo, C., Quarteroni, A.: A multiscale darcy-brinkman model for fluid flow in fractured porous media. Numer. Math. 117(4), 717–752 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Pitkäranta, J.: Boundary subspaces for the finite element method with Lagrange multipliers. Numer. Math. 33(3), 273–289 (1979). zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Saffman, P.: On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50, 93–101 (1971)zbMATHGoogle Scholar
  14. 14.
    Showalter, R.E.: Poroelastic filtration coupled to Stokes flow. In: Control theory of partial differential equations, Lectures Notes Pure Applied Mathematics, vol. 242, pp. 229–241. Chapman & Hall/CRC, Boca Raton (2005).

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • I. Ambartsumyan
    • 1
  • E. Khattatov
    • 1
  • I. Yotov
    • 1
    Email author
  • P. Zunino
    • 2
  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Department of Mechanical Engineering and Materials ScienceUniversity of PittsburghPittsburghUSA

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