Skip to main content

Numerical convergence study of iterative coupling for coupled flow and geomechanics


In this paper, we consider algorithms for modeling complex processes in porous media that include fluid and structure interactions. Numerous field applications would benefit from a better understanding and integration of porous flow and solid deformation. Important applications in environmental and petroleum engineering include carbon sequestration, surface subsidence, pore collapse, cavity generation, hydraulic fracturing, thermal fracturing, wellbore collapse, sand production, fault activation, and waste disposal, while similar issues arise in biosciences and chemical sciences as well. Here, we consider solving iteratively the coupling of flow and mechanics. We employ mixed finite element method for flow and a continuous Galerkin method for elasticity. For single-phase flow, we demonstrate the convergence and convergence rates for two widely used schemes, the undrained split and the fixed stress split. We discuss the extension of the fixed stress iterative coupling scheme to an equation of state compositional flow model coupled with elasticity and a single-phase poroelasticity model on general hexahedral grids. Computational results are presented.

This is a preview of subscription content, access via your institution.


  1. Abousleiman, Y., Cheng, A.H.D., Cui, L., Detournay, E., Roegiers, J.C.: Mandel’s problem revisited. Géotechnique 46, 187–195 (1996)

    Article  Google Scholar 

  2. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  Google Scholar 

  3. Charlez, P.A.: Rock Mechanics: Theoretical Fundamentals, vol. 1. Editions Technip, Paris (1991)

    Google Scholar 

  4. Chen, H.Y., Teufel, L.W., Lee, R.L.: Coupled fluid flow and geomechanics in reservoir study—I. Theory and governing equations, paper SPE 30752 presented at the SPE Annual Technical Conference & Exhibition, Dallas (1995)

  5. Chin, L.Y., Raghavan, R., Thomas, L.Y.: Fully-coupled geomechanics and fluid-flow analysis of wells with stress-dependent permeability, paper SPE 48857 presented at the SPE International Conference and Exhibition, Beijing (1998)

  6. Coussy, O.: Mechanics of Porous Continua. Wiley, New York (1995)

    Google Scholar 

  7. Dean, R.H., Gai, X., Stone, C.M., Minkoff, S.E.: A comparison of techniques for coupling porous flow and geomechanics. SPE J. 11, 132–140 (2006)

    Article  Google Scholar 

  8. Dean, R.H., Schmidt, J.H.: Hydraulic-fracture predictions with a fully coupled geomechanical reservoir simulator. SPE J. 14, 707–714 (2009)

    Article  Google Scholar 

  9. Gai, X.: A coupled geomechanics and reservoir flow model in parallel computers. PhD thesis, The University of Texas at Austin, Austin (2004)

  10. Geertsma, J.: The effect of fluid pressure decline on volumetric changes of porous rocks. Trans. AIME 210, 331–340 (1957)

    Google Scholar 

  11. Lawrence Livermore National Laboratory: Hypre user’s manual (2011)

  12. Kim, J., Tchelepi, H.A., Juanes, R.: Stability, accuracy, and efficiency of sequential methods for coupled flow and geomechanics. SPE J. 16(2), 249–262 (2011)

    Article  Google Scholar 

  13. Kim, J., Tchelepi, H.A., Juanes, R.: Rigorous coupling of geomechanics and multiphase flow with strong capillarity, paper SPE 141268 presented at the SPE Reservoir Simulation Symposium, The Woodlands (2011)

  14. Lange, T., Shparlinski, I.E.: Distribution of some sequences of points on elliptic curves. J. Math. Cryptol. 1, 1–11 (2007)

    Article  Google Scholar 

  15. Liu, R.: Discontinuous Galerkin for mechanics. PhD thesis, The University of Texas at Austin, Austin (2004)

  16. Mainguy, M., Longuemare, P.: Coupling fluid flow and rock mechanics: formulation of the partial coupling between reservoir and geomechanical simulators. Oil & Gas Sci. Technol. 57(4), 355–367 (2002)

    Article  Google Scholar 

  17. Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput Geosci 17, 479–496 (2013)

    Article  Google Scholar 

  18. Pan, F.: Development and application of a coupled geomechanics model for a parallel compositional reservoir simulator. PhD thesis, The University of Texas at Austin, Austin (2009)

  19. Pan, F., Sepehrnoori, K., Chin, L.Y.: A new solution procedure for a fully coupled geomechanics and compositional reservoir simulator, paper SPE 119029 presented at the SPE Reservoir Simulation Symposium, The Woodlands (2009)

  20. Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite elements methods for poroelasticity I: The continuous in time case. Comput. Geosci. 11, 131–144 (2007)

    Article  Google Scholar 

  21. Raghavan, R., Chin, L.Y.: Productivity changes in reservoirs with stress-dependent permeability, SPE Reserv. Eval. & Eng. 7(4), 308–315 (2004)

    Google Scholar 

  22. Settari, A., Walters, D.A.: Advances in coupled geomechanics and reservoir modeling with applications to reservoir compaction, paper SPE 51927 presented at the SPE Reservoir Simulation Symposium, Houston (1999)

  23. Thomas, S.G.: On some problems in the simulation of flow and transport through porous media. PhD thesis, The University of Texas at Austin, Austin (2009)

  24. Tran, D., Nghiem, L., Buchanan, L.: An overview of iterative coupling between geomechanical deformation and reservoir flow, paper SPE/PS-CIM/CHOA 97879 presented at the SPE International Thermal Operations and Heavy Oil Symposium, Calgary, Canada (2005)

  25. Wheeler, M.F., Xue, G., Yotov, I.: Accurate cell-centered discretizations for modeling multiphase flow in porous media on general hexahedral and simplicial grids. SPE J. 17(3), 779–793 (2011)

    Article  Google Scholar 

  26. Li, X., Zienkiewicz, O.C.: Multiphase flow in deforming porous media and finite element solutions. Comput. Struct. 45(2), 211–227 (1992)

    Article  Google Scholar 

Download references


A. Mikelić would like to thank the Institute for Computational Engineering and Science (ICES), UT Austin for hospitality in April 2009, 2010, 2011, and 2012. The research by M. F. Wheeler and B. Wang were partially supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences through the DOE Energy Frontier Research Center: The Center for Frontiers of Subsurface Energy Security (CFSES) under contract no. DE-SC0001114. The authors would also like to thank Dr. B. Ganis and Dr. R. Liu for their help in setting up the unstructured grid example.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Andro Mikelić.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mikelić, A., Wang, B. & Wheeler, M.F. Numerical convergence study of iterative coupling for coupled flow and geomechanics. Comput Geosci 18, 325–341 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Poroelasticity
  • Iterative coupling
  • Contraction mapping
  • Compositional flow
  • Multipoint flux mixed finite element method