Robust iterative schemes for non-linear poromechanics

  • Manuel Borregales
  • Florin A. Radu
  • Kundan Kumar
  • Jan M. Nordbotten
Original Paper
  • 1 Downloads

Abstract

We consider a non-linear extension of Biot’s model for poromechanics, wherein both the fluid flow and mechanical deformation are allowed to be non-linear. Specifically, we study the case when the volumetric stress and the fluid density are non-linear functions satisfying certain assumptions. We perform an implicit discretization in time (backward Euler) and propose two iterative schemes for solving the non-linear problems appearing within each time step: a splitting algorithm extending the undrained split and fixed stress methods to non-linear problems, and a monolithic L-scheme. The convergence of both schemes are shown rigorously. Illustrative numerical examples are presented to confirm the applicability of the schemes and validate the theoretical results.

Keywords

Biot’s model L-schemes MFEM Convergence analysis Fixed-stress method Coupled problems Poromechanics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abousleiman, Y., Cheng, A.H.D., Cui, L., Detournay, E., Roegiers, J.C.: Mandel’s problem revisited. Géotechnique 46(2), 187–195 (1996)CrossRefGoogle Scholar
  2. 2.
    Adler, J.H., Gaspar, F.J., Hu, X., Rodrigo, C., Zikatanov, L.T.: Robust block preconditioners for Biot’s model. arXiv:1705.08842 (2017)
  3. 3.
    Almani, T., Kumar, K., Dogru, A.H., Singh, G., Wheeler, M.F.: Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics. Comput. Methods. Appl. Mech. Eng. 311, 180–207 (2016)CrossRefGoogle Scholar
  4. 4.
    Armero, F.: Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions. Comput. Methods. Appl. Mech. Eng. 171(3), 205–241 (1999)CrossRefGoogle Scholar
  5. 5.
    Armero, F., Simo, J.C.: A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems. Int. J. Numer. Meth. Eng. 35(4), 737–766 (1992)CrossRefGoogle Scholar
  6. 6.
    Bangerth, W., Kanschat, G., Heister, T.: Deal. II Differential equations analysis library (2014)Google Scholar
  7. 7.
    Bause, M., Radu, F.A., Kocher, U.: Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods. Appl. Mech. Eng. 320(15), 745–768 (2017)CrossRefGoogle Scholar
  8. 8.
    Biot, M.A.: Consolidation settlement under a rectangular load distribution. J. Appl. Phys. 12(5), 426–430 (1941)CrossRefGoogle Scholar
  9. 9.
    Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)CrossRefGoogle Scholar
  10. 10.
    Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182–185 (1955)CrossRefGoogle Scholar
  11. 11.
    Both, J.W., Borregales, M., Nordbotten, J.M., Kumar, K., Radu, F.A.: Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017)CrossRefGoogle Scholar
  12. 12.
    Both, J.W., Kumar, K., Nordbotten, J.M., Radu, F.A.: Iterative methods for coupled flow and geomechanics in unsaturated porous media. In: Proceedings of the Sixth Biot conference on poromechanics, Paris (2017).  https://doi.org/10.1061/9780784480779.050
  13. 13.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, Springer Ser. Comput. Math, vol. 15. Springer, New York (2012)Google Scholar
  14. 14.
    Castelletto, N., White, J.A., Ferronato, M.: Scalable algorithms for three-field mixed finite element coupled poromechanics. J. Comput. Phys. 327, 894–918 (2016)CrossRefGoogle Scholar
  15. 15.
    Castelletto, N., White, J.A., Tchelepi, H.A.: Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics. Int. J. Numer. Anal. Meth. Geomech. 39(14), 1593–1618 (2015)CrossRefGoogle Scholar
  16. 16.
    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 Gas Sci. Technol. 57(5), 485–497 (2002)CrossRefGoogle Scholar
  17. 17.
    Coussy, O.: A general theory of thermoporoelastoplasticity for saturated porous materials. Trans. Por. Med. 4(3), 281–293 (1989)CrossRefGoogle Scholar
  18. 18.
    Coussy, O.: Mechanics of Porous Continua. Wiley, New York (1995)Google Scholar
  19. 19.
    Coussy, O.: Mechanics of Porous Continua. Wiley, New York (2004)Google Scholar
  20. 20.
    Detournay, E., Cheng, A.H.D.: Fundamentals of Poroelasticity, vol. 2, chap. 5. Pergamon Press, Oxford (1993)Google Scholar
  21. 21.
    Doster, F., Nordbotten, J.M.: Full pressure coupling for geo-mechanical multi-phase multi-component flow simulations, paper SPE 173232 presented at the SPE Reservoir Simulation Symposium Houston (2015)Google Scholar
  22. 22.
    Fung, L.S.K., Buchanan, L., Wan, R.G.: Coupled geomechanical-thermal simulation for deforming heavy-oil reservoirs. J. Can. Pet. Technol. 33(04) (1994)Google Scholar
  23. 23.
    Gai, X., Dean, R.H., Wheeler, M.F., Liu, R.: Coupled geomechanical and reservoir modeling on parallel computers, paper SPE 79700 presented at the SPE Reservoir Simulation Symposium Houston (2003)Google Scholar
  24. 24.
    Gai, X., Wheeler, M.F.: Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity. Numer. Methods. Partial. Diff. Equations 23(4), 785–797 (2007)CrossRefGoogle Scholar
  25. 25.
    Girault, V., Kumar, K., Wheeler, M.F.: Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium. Comput. Geosci. 20(5), 997–1011 (2016)CrossRefGoogle Scholar
  26. 26.
    Haga, J.B., Osnes, H., Langtangen, H.P.: Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media. Int. J. Numer. Anal. Meth. Geomech. 35(13), 1466–1482 (2011)Google Scholar
  27. 27.
    Jeannin, L., Mainguy, M., Masson, R., Vidal-Gilbert, S.: Accelerating the convergence of coupled geomechanical-reservoir simulations. Int. J. Numer. Anal. Meth. Geomech. 31(10), 1163–1181 (2007)CrossRefGoogle Scholar
  28. 28.
    Jha, B., Juanes, R.: A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotech. 2(3), 139–153 (2007)CrossRefGoogle Scholar
  29. 29.
    Kim, J., Tchelepi, H., Juanes, R.: Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits. Comput. Methods. Appl. Mech. Eng. 200(23–24), 2094–2116 (2011)CrossRefGoogle Scholar
  30. 30.
    Kim, J., Tchelepi, H., Juanes, R.: Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods. Appl. Mech. Eng. 200(13–16), 1591–1606 (2011)CrossRefGoogle Scholar
  31. 31.
    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)CrossRefGoogle Scholar
  32. 32.
    Kumar, K., Pop, I., Radu, F.: Convergence analysis of mixed numerical schemes for reactive flow in a porous medium. SIAM J. on Numer. Anal. 51(4), 2283–2308 (2013)CrossRefGoogle Scholar
  33. 33.
    Lee, S., Mikelic, A., Wheeler, M.F., Wick, T.: Phase-field modeling of proppant-filled fractures in a poroelastic medium. Comput. Methods. Appl. Mech. Eng. 312, 509–541 (2016)CrossRefGoogle Scholar
  34. 34.
    Lewis, R.W., Schrefler, B.A.: The finite element method in the static and dynamic deformation and consolidation of porous media, 2nd edn. Wiley, Chichester (1998)Google Scholar
  35. 35.
    Lewis, R.W., Sukirman, Y.: Finite element modelling of three-phase flow in deforming saturated oil reservoirs. Int. J. Numer. Anal. Meth. Geomech. 17(8), 577–598 (1993)CrossRefGoogle Scholar
  36. 36.
    List, F., Radu, F.A.: A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016)CrossRefGoogle Scholar
  37. 37.
    Mandel, J.: Consolidation Des Sols (Étude mathématique). Gé,otechnique 3(7), 287–299 (1953)Google Scholar
  38. 38.
    Mikelić, A., Wang, B., Wheeler, M.F.: Numerical convergence study of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18(3-4), 325–341 (2014)CrossRefGoogle Scholar
  39. 39.
    Mikelić, A., Wheeler, M.F.: Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system. J. Math. Phys. 53(12), 123702 (2012)CrossRefGoogle Scholar
  40. 40.
    Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18(3-4), 325–341 (2013)CrossRefGoogle Scholar
  41. 41.
    Nordbotten, J.M.: Stable cell-centered finite volume discretization for Biot equations. SIAM J. Numer. Anal. 54(2), 942–968 (2016)CrossRefGoogle Scholar
  42. 42.
    Pettersen, O.: Coupled flow and rock mechanics simulation optimizing the coupling term for faster and accurate computation. nt. J. Numer. Anal. Model. 9(3), 628–643 (2012)Google Scholar
  43. 43.
    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
  44. 44.
    Pop, I., 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
  45. 45.
    Prevost, J.H.: One-way versus two-way coupling in reservoir-geomechanical models. In: Proceedings of the Fifth Biot conference on Poromechanics, Vienna (2013).  https://doi.org/10.1061/9780784412992.061
  46. 46.
    Radu, F.A., Kumar, K., Nordbotten, J.M., Pop, I.S.: A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities. IMA J. Numer. Anal. (2017).  https://doi.org/10.1093/imanum/drx032
  47. 47.
    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
  48. 48.
    Radu, F.A., Pop, I.S.: Newton method for reactive solute transport with equilibrium sorption in porous media. J. Comput. Appl. Math. 234(7), 2118–2127 (2010)CrossRefGoogle Scholar
  49. 49.
    Radu, F.A., Wang, W.: Convergence analysis for a mixed finite element scheme for flow in strictly unsaturated porous media. Nonlinear Anal. Real World Appl. 15, 266–275 (2014)CrossRefGoogle Scholar
  50. 50.
    Rodrigo, C., Gaspar, F., Hu, X., Zikatanov, L.: Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Methods. Appl. Mech. Eng. 298, 183–204 (2016)CrossRefGoogle Scholar
  51. 51.
    Settari, A., Mourits, F.M.: Coupling of geomechanics and reservoir simulations models. Comput. Methods and Advances in Geomechanics (1994)Google Scholar
  52. 52.
    Settari, A., Mourits, F.M.: A coupled reservoir and geomechanical simulation system. SPE J (1998)Google Scholar
  53. 53.
    Settari, A., Walters, D.A.: Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. SPE J (2001)Google Scholar
  54. 54.
    Showalter, R.E.: Diffusion in poro-elastic media. J. Math Anal. Appl. 251(1), 310–340 (2000)CrossRefGoogle Scholar
  55. 55.
    Temam, R.M., Miranville, A.M.: Mathematical Modeling in Continuum Mechanics. Cambridge (2005)Google Scholar
  56. 56.
    Thomas, J.: Sur l’ numerique des methodes d’elements finis hybrides et mixtes. Univ. Pierre et Marie Curie thèse (1977)Google Scholar
  57. 57.
    Wan, J., Durlofsky, L., Hughes, T., Aziz, K.: Stabilized finite element methods for coupled geomechanics—reservoir flow simulations, paper SPE 79694 presented at the SPE Reservoir Simulation Symposium Houston (2003)Google Scholar
  58. 58.
    White, D., Ganis, B., Liu, R., Wheeler, M.F.: A Near-Wellbore Study with a Drucker-Prager Plasticity Model Coupled with a Parallel Compositional Reservoir Simulator, Paper SPE-182627-MS Presented at the SPE Reservoir Simulation Conference, Texas (2017)Google Scholar
  59. 59.
    White, J.A., Castelletto, N., Tchelepi, H.A.: Block-partitioned solvers for coupled poromechanics: a unified framework. Comput. Methods. Appl. Mech. Eng. 303, 55–74 (2016)CrossRefGoogle Scholar
  60. 60.
    Zienkiewicz, O.C., Paul, D.K., Chan, A.H.C.: Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems. Int. J. Numer. Meth. Eng. 26(5), 1039–1055 (1988)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.Department of Civil and Environmental EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations