Advertisement

Computational Geosciences

, Volume 20, Issue 5, pp 997–1011 | Cite as

Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium

  • Vivette Girault
  • Kundan Kumar
  • Mary F. Wheeler
Original Paper

Abstract

We consider an iterative scheme for solving a coupled geomechanics and flow problem in a fractured poroelastic medium. The fractures are treated as possibly non-planar interfaces. Our iterative scheme is an adaptation due to the presence of fractures of a classical fixed stress-splitting scheme. We prove that the iterative scheme is a contraction in an appropriate norm. Moreover, the solution converges to the unique weak solution of the coupled problem.

Keywords

Fractured porous media flow Iterative coupling Biot model Fixed stress splitting 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alboin, C., Jaffré, J., Roberts, J.E., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media. In: Z. Chen and R. E. Ewing, editors, Fluid flow and transport in porous media: mathematical and numerical treatment, Contemporary mathematics, volume 295, pages 13–24. American Mathematical Society (2002)Google Scholar
  2. 2.
    Biot, M.A.: Consolidation settlement under a rectangular load distribution. J. Appl. Phys. 12(5), 426–430 (1941)CrossRefGoogle Scholar
  3. 3.
    Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)CrossRefGoogle Scholar
  4. 4.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer-Verlag 3rd edition (2008)Google Scholar
  5. 5.
    Bukač, M., Yotov, I., Zakerzadeh, R., Zunino, P.: Partitioning strategies for the interaction of a fluid with a poroelastic material based on Nitsche’s coupling approach. Comput. Methods Appl. Mech. Eng. 292, 138–170 (2015)CrossRefGoogle Scholar
  6. 6.
    Castonguay, S.T., Mear, M.E., Dean, R.H., Schmidt, J.H.: Predictions of the growth of multiple interacting hydraulic fractures in three dimensions. Soc. Petrol. Eng. (2013). SPE 166259-MSGoogle Scholar
  7. 7.
    Chin, L.Y., Raghaven, R., Thomas, L.K.: Fully-coupled geomechanics and fluid-flow analysis of wells with stress-dependent permeability. In: 1998 SPE International Conference and Exhibition, Beijing, China Nov, pp 2–6 (1998)Google Scholar
  8. 8.
    Chin, L.Y., Thomas, L.K., Sylte, J.E., Pierson, R.G.: Iterative coupled analysis of geomechanics and fluid flow for rock compaction in reservoir simulation. Oil and Gas Science and Technology 57(5), 485–497 (2002)CrossRefGoogle Scholar
  9. 9.
    Coussy, O.: A general theory of thermoporoelastoplasticity for saturated porous materials. Transp. Porous Media 4, 281–293 (1989)CrossRefGoogle Scholar
  10. 10.
    Dean, R.H., Schmidt, J.H.: Hydraulic-fracture predictions with a fully coupled geomechanical reservoir simulator. SPE J., 707–714 (2009)Google Scholar
  11. 11.
    Delshad, M., Hosseini, S.A., Kong, X., Tavakoli, R., Wheeler, M.F.: Modeling and simulation of carbon sequestration at Cranfield incorporating new physical models. Int. J. Greenhouse Gas Control 18, 463–473 (2013)CrossRefGoogle Scholar
  12. 12.
    Fung, L.S.K., Buchanan, L., Wan, R.G.: Coupled geomechanical-thermal simulation for deforming heavy-oil reservoirs. J. Can. Pet. Tech. 33(4) (1994)Google Scholar
  13. 13.
    Gai, X., Dean, R.H., Wheeler, M.F., Liu, R.: Coupled geomechanical and reservoir modeling on parallel computers. In: The SPE Reservoir Simulation Symposium, Houston, Texas Feb, pp 3–5 (2003)Google Scholar
  14. 14.
    Galvis, J., Sarkis, M.: Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron. trans. Numer Anal. 26, 350–384 (2007)Google Scholar
  15. 15.
    Girault, V., Pencheva, G.V., Wheeler, M.F., Wildey, T.M.: Domain decomposition for poroelasticity and elasticity with DG jumps and mortars. Math. Models Methods Appl. Sci. 21(1), 169–213 (2011)CrossRefGoogle Scholar
  16. 16.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986)CrossRefGoogle Scholar
  17. 17.
    Girault, V., Wheeler, M.F., Ganis, B., Mear, M.E.: A lubrication fracture model in a poro-elastic medium. Math. Models Methods Appl. Sci. 25(4), 587–645 (2015)CrossRefGoogle Scholar
  18. 18.
    Kim, J., Tchelepi, H.A., Juanes, R.: Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits. Comput. Methods Appl. Mech. Engrg. 200(23-24), 2094–2116 (2011)CrossRefGoogle Scholar
  19. 19.
    Kim, J., Tchelepi, H.A., Juanes, R.: Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods Appl. Mech. Engrg. 200(13-16), 1591–1606 (2011)CrossRefGoogle Scholar
  20. 20.
    Lewis, R.W., Schrefler, B.A.: the Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2nd edition. Wiley, New York (1998)Google Scholar
  21. 21.
    Lions, J. -L., Magenes, E.: non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer-Verlag, New York (1972)CrossRefGoogle Scholar
  22. 22.
    Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005)CrossRefGoogle Scholar
  23. 23.
    Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and mechanics. Comput. Geosci. 17(3), 455–461 (2013)CrossRefGoogle Scholar
  24. 24.
    Mikelić, A., Wheeler, M.F., Wick, T.: A Phase-Field Approach to the Fluid-Filled Fracture Surrounded by a Poro-Elastic Medium ICES Report 13-15, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin Texas (2013)Google Scholar
  25. 25.
    Mikelić, A., Wheeler, M.F., Wick, T.: A quasi-static phase-field approach to the fluid-filled fracture. Nonlinearity 28(5), 1371–1399 (2015)CrossRefGoogle Scholar
  26. 26.
    Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case. Comput. Geosci. 11(2), 131–144 (2007)CrossRefGoogle Scholar
  27. 27.
    Pop, I.S., Radu, F., Knabner, P.: Mixed finite elements for the Richards’ equation. Linearization procedure. J. Comput. Appl. Math. 168(1-2), 365–373 (2004)CrossRefGoogle Scholar
  28. 28.
    Radu, F.A., Nordbotten, J.M., Pop, I.S., Kumar, K.: A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015)CrossRefGoogle Scholar
  29. 29.
    Settari, A., Mourits, F.M.: Coupling of geomechanics and reservoir simulation models. In: Siriwardane and Zema, editors, Comp. Methods and Advances in Geomech., pages 2151–2158, Balkema, Rotterdam (1994)Google Scholar
  30. 30.
    Showalter, R.E.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251(1), 310–340 (2000)CrossRefGoogle Scholar
  31. 31.
    Small, J.C., Booker, J.R., Davis, E.H.: Elasto-plastic consolidation of soil. Int. J. Solids Struct. 12(6), 431–448 (1976)CrossRefGoogle Scholar
  32. 32.
    von Terzaghi, K.: Theoretical Soil Mechanics. Wiley, New York (1943)CrossRefGoogle Scholar
  33. 33.
    Wick, T., Singh, G., Wheeler, M.F.: Pressurized fracture propagation using a phase-field approach coupled to a reservoir simulator. In: SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas. Society of Petroleum Engineers, SPE 168597-MS submitted (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Vivette Girault
    • 1
  • Kundan Kumar
    • 2
  • Mary F. Wheeler
    • 3
  1. 1.Sorbonne Universités, UPMC Univ. Paris 06CNRS, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Realfagbygget, Mathematics Institute, Allegaten 41University of BergenBergenNorway
  3. 3.Center for Subsurface Modeling, The Institute for Computational Engineering Sciences (ICES)The University of Texas at AustinAustinUSA

Personalised recommendations