Phase-field modeling through iterative splitting of hydraulic fractures in a poroelastic medium

  • A. Mikelić
  • M. F. Wheeler
  • T. WickEmail author
Original Paper
Part of the following topical collections:
  1. Numerical methods for processes in fractured porous media


We study the propagation of hydraulic fractures using the fixed stress splitting method. The phase field approach is applied and we study the mechanics step involving displacement and phase field unknowns, with a given pressure. We present a detailed derivation of an incremental formulation of the phase field model for a hydraulic fracture in a poroelastic medium. The mathematical model represents a linear elasticity system with fading elastic moduli as the crack grows that is coupled with an elliptic variational inequality for the phase field variable. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. We establish existence of a minimizer of an energy functional of an incremental problem and convergence of a finite dimensional approximation. Moreover, we prove that the fracture remains small in the third direction in comparison to the first two principal directions. Computational results of benchmark problems are provided that demonstrate the effectiveness of this approach in treating fracture propagation. Another novelty is the treatment of the mechanics equation with mixed boundary conditions of Dirichlet and Neumann types. We finally notice that the corresponding pressure step was studied by the authors in Mikelić et al. (SIAM Multiscale Model Simul 13(1):367–398, 2015a).


Hydraulic fracturing Phase field formulation Nonlinear elliptic system Computer simulations Poroelasticity 

Mathematics Subject Classification

35Q74 35J87 49J45 65K15 74R10 



  1. Adachi, J., Siebrits, E., Peirce, A., Desroches, J.: Computer simulation of hydraulic fractures. Int. J. Rock Mech. Min. Sci. 44, 739–757 (2007)Google Scholar
  2. Almani, T., Lee, S., Wheeler, M., Wick, T.: Multirate coupling forflow and geomechanics applied to hydraulic fracturing using anadaptive phase-field technique (2017). SPE RSC 182610-MS, Feb. 2017, Montgomery, Texas, USAGoogle Scholar
  3. Almi, S., Maso, G.D., Toader, R.: Quasi-static crack growth in hydraulic fracture. Nonlinear Anal. Theory Methods Appl. 109, 301–318 (2014)MathSciNetzbMATHGoogle Scholar
  4. Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Pelteret, J.P., Turcksin, B., Wells, D.: The deal.II library, version 8.5. J. Numer. Math. 25(3), 137–146 (2017)MathSciNetzbMATHGoogle Scholar
  5. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)MathSciNetzbMATHGoogle Scholar
  6. Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J.R., Landis, C.M.: A phase-field description of dynamic brittle fracture. Comput. Meth. Appl. Mech. Eng. 217, 77–95 (2012)MathSciNetzbMATHGoogle Scholar
  7. de Borst, R., Rethoré, J., Abellan, M.: A numerical approach for arbitrary cracks in a fluid-saturated porous medium. Arch. Appl. Mech. 595–606 (2006)Google Scholar
  8. Both, J., Borregales, M., Nordbotten, J., Kumar, K., Radu, F.: Robust fixed stress splitting for biots equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017)MathSciNetzbMATHGoogle Scholar
  9. Bourdin, B.: Image segmentation with a finite element method. Math. Model. Numer. Anal. 33(2), 229–244 (1999)MathSciNetzbMATHGoogle Scholar
  10. Bourdin, B., Chukwudozie, C., Yoshioka, K.: A variational approach to the numerical simulation of hydraulic fracturing. In: SPE Journal, Conference Paper 159154-MS (2012)Google Scholar
  11. Bourdin, B., Francfort, G., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000)MathSciNetzbMATHGoogle Scholar
  12. Bourdin, B., Francfort, G., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1–3), 1–148 (2008)MathSciNetzbMATHGoogle Scholar
  13. Braides, A.: Approximation of Free-Discontinuity Problems. Springer, Berlin (1998)zbMATHGoogle Scholar
  14. Burke, S., Ortner, C., Süli, E.: An adaptive finite element approximation of a variational model of brittle fracture. SIAM J. Numer. Anal. 48(3), 980–1012 (2010)MathSciNetzbMATHGoogle Scholar
  15. Cajuhi, T., Sanavia, L., De Lorenzis, L.: Phase-field modeling of fracture in variably saturated porous media. Comput. Mech. 61(3), 299–318 (2018)MathSciNetzbMATHGoogle Scholar
  16. Castelletto, N., White, J.A., Tchelepi, H.A.: Accuracy and convergence properties of the fixedstress iterative solution of twoway coupled poromechanics. Int. J. Numer. Anal. Methods Geomech. 39(14), 1593–1618 (2015)Google Scholar
  17. Castonguay, S., Mear, M., Dean, R., Schmidt, J.: Predictions of the growth of multiple interacting hydraulic fractures in three dimensions. SPE-166259-MS pp. 1–12 (2013)Google Scholar
  18. Chambolle, A.: An approximation result for special functions with bounded variations. J. Math. Pures Appl. 83, 929–954 (2004)MathSciNetzbMATHGoogle Scholar
  19. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, 2 edn. North-Holland, Amsterdam (1987)Google Scholar
  20. Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer Verlag, Berlin (2008)zbMATHGoogle Scholar
  21. Dean, R., Schmidt, J.: Hydraulic-fracture predictions with a fully coupled reservoir simulator. SPE J. 14(4), 707–714 (2014)Google Scholar
  22. Engwer, C., Schumacher, L.: A phase field approach to pressurized fractures using discontinuous Galerkin methods. Math. Comput. Simul. 137, 266–285 (2017)MathSciNetGoogle Scholar
  23. Ferronato, M., Castelletto, N., Gambolati, G.: A fully coupled 3-d mixed finite element model of Biot consolidation. J. Comput. Phys. 229(12), 4813–4830 (2010)zbMATHGoogle Scholar
  24. Francfort, G.: Un résumé de la théorie variationnelle de la rupture (2011). Séminaire Laurent Schwartz – EDP et applications, Institut des hautes études scientifiques, 2011–2012, Exposé no. XXII, 1-11.
  25. Francfort, G., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)MathSciNetzbMATHGoogle Scholar
  26. Ganis, B., Girault, V., Mear, M., Singh, G., Wheeler, M.F.: Modeling fractures in a poro-elastic medium. Oil Gas Sci. Technol. 4, 515–528 (2014)Google Scholar
  27. Gaspar, F.J., Rodrigo, C.: On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics. Comput. Methods Appl. Mech. Eng. 326, 526–540 (2017)MathSciNetGoogle Scholar
  28. Gerasimov, T., Lorenzis, L.D.: A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput. Methods Appl. Mech. Eng. 312, 276–303 (2016)MathSciNetGoogle Scholar
  29. 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(04), 587–645 (2015)MathSciNetzbMATHGoogle Scholar
  30. Gupta, P., Duarte, C.: Simulation of non-planar three-dimensional hydraulic fracture propagation. Int. J. Numer. Anal. Meth. Geomech. 38, 1397–1430 (2014)Google Scholar
  31. Heider, Y., Markert, B.: A phase-field modeling approach of hydraulic fracture in saturated porous media. Mech. Res. Commun. 80, 38–46 (2017)Google Scholar
  32. Heister, T., Wheeler, M.F., Wick, T.: A primal-dual active set method and predictor–corrector mesh adaptivity for computing fracture propagation using a phase-field approach. Comput. Meth. Appl. Mech. Eng. 290, 466–495 (2015)MathSciNetzbMATHGoogle Scholar
  33. Hong, Q., Kraus, J.: Parameter-robust stability of classical three-field formulation of Biot’s consolidation model. Electron. Trans. Numer. Anal. 48, 202–226 (2018)MathSciNetzbMATHGoogle Scholar
  34. Hwang, J., Sharma, M.: A 3-dimensional fracture propagation model for long-term water injection. In: 47th US Rock Mechanics/Geomechanics Symposium (2013)Google Scholar
  35. Irzal, F., Remmers, J.J., Huyghe, J.M., de Borst, R.: A large deformation formulation for fluid flow in a progressively fracturing porous material. Comput. Methods Appl. Mech. Eng. 256, 29–37 (2013)MathSciNetzbMATHGoogle Scholar
  36. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. In: Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (2000)Google Scholar
  37. Lee, J.J.: Robust error analysis of coupled mixed methods for Biot’s consolidation model. J. Sci. Comput. 69(2), 610–632 (2016)MathSciNetzbMATHGoogle Scholar
  38. Lee, S., Mikelić, 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 (2016a)MathSciNetGoogle Scholar
  39. Lee, S., Wheeler, M.F., Wick, T.: Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model. Comput. Methods Appl. Mech. Eng. 305, 111–132 (2016b)MathSciNetzbMATHGoogle Scholar
  40. Lee, S., Wheeler, M.F., Wick, T.: Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches. J. Comput. Appl. Math. 314, 40–60 (2017a)MathSciNetzbMATHGoogle Scholar
  41. Lee, S., Wheeler, M.F., Wick, T., Srinivasan, S.: Initialization of phase-field fracture propagation in porous media using probability maps of fracture networks. Mech. Res. Commun. 80, 16–23 (2017b)Google Scholar
  42. Lee, J.J., Mardal, K.A., Winther, R.: Parameter-robust discretization and preconditioning of Biot’s consolidation model. SIAM J. Sci. Comput. 39(1), A1–A24 (2017c)MathSciNetzbMATHGoogle Scholar
  43. Lee, S., Mikelić, A., Wheeler, M.F., Wick, T.: Phase-field modeling of two phase fluid filled fractures in a poroelastic medium (2018). SIAM Multiscale Model Simul. 16(4), 1542–1580 (2018)MathSciNetzbMATHGoogle Scholar
  44. Liu, R.: Discontinuous galerkin finite element solution for poromechanics. Ph.D. thesis, The University of Texas at Austin (2004)Google Scholar
  45. Markert, B., Heider, Y.: Recent Trends in Computational Engineering—CE2014: Optimization, Uncertainty, Parallel Algorithms, Coupled and Complex Problems, chap. Coupled Multi-Field Continuum Methods for Porous Media Fracture, pp. 167–180. Springer, Cham (2015)Google Scholar
  46. McClure, M.W., Kang, C.A.: A three-dimensional reservoir, wellbore, and hydraulic fracturing simulator that is compositional and thermal, tracks proppant and water solute transport, includes non-darcy and non-newtonian flow, and handles fracture. SPE-182593-MS (2017)Google Scholar
  47. Miehe, C., Mauthe, S.: Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. Comput. Methods Appl. Mech. Eng. 304, 619–655 (2016)MathSciNetzbMATHGoogle Scholar
  48. Miehe, C., Mauthe, S., Teichtmeister, S.: Minimization principles for the coupled problem of Darcy-Biot-type fluid transport in porous media linked to phase field modeling of fracture. J. Mech. Phys. Solids 82, 186–217 (2015)MathSciNetGoogle Scholar
  49. Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field fe implementations. Int. J. Numer. Methods Eng. 83, 1273–1311 (2010)MathSciNetzbMATHGoogle Scholar
  50. Mikelić, A., Wang, B., Wheeler, M.F.: Numerical convergence study of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18(3), 325–341 (2014)MathSciNetzbMATHGoogle Scholar
  51. Mikelić, A., Wheeler, M., Wick, T.: A phase-field approach to the fluid filled fracture surrounded by a poroelastic medium. ICES Report 13-15 (2013)Google Scholar
  52. Mikelić, A., Wheeler, M., Wick, T.: Phase-field modeling of pressurized fractures in a poroelastic medium. ICES Report 14-18 (2014)Google Scholar
  53. Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17(3), 455–462 (2012)MathSciNetzbMATHGoogle Scholar
  54. Mikelić, A., Wheeler, M.F., Wick, T.: A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium. SIAM Multiscale Model. Simul. 13(1), 367–398 (2015a)MathSciNetzbMATHGoogle Scholar
  55. Mikelić, A., Wheeler, M.F., Wick, T.: Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput. Geosci. 19(6), 1171–1195 (2015b)MathSciNetzbMATHGoogle Scholar
  56. Mikelić, A., Wheeler, M.F., Wick, T.: A quasi-static phase-field approach to pressurized fractures. Nonlinearity 28(5), 1371–1399 (2015c)MathSciNetzbMATHGoogle Scholar
  57. Murad, M.A., Loula, A.F.: Improved accuracy in finite element analysis of Biot’s consolidation problem. Comput. Methods Appl. Mech. Eng. 95(3), 359–382 (1992)MathSciNetzbMATHGoogle Scholar
  58. Murad, M.A., Loula, A.F.D.: On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37(4), 645–667 (1994)MathSciNetzbMATHGoogle Scholar
  59. Nguyen, T., Yvonnet, J., Zhu, Q.Z., Bornert, M., Chateau, C.: A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography. Comput. Methods Appl. Mech. Eng. 312, 567–595 (2016)MathSciNetGoogle Scholar
  60. Philips, P., Wheeler, M.: A coupling of mixed and galerkin finite element methods for poro-elasticity. Comput. Geosci. 12(4), 417–435 (2003)Google Scholar
  61. Rodrigo, C., Gaspar, F., Hu, X., Zikatanov, L.: Stability and monotonicity for some discretizations of the Biots consolidation model. Comput. Methods Appl. Mech. Eng. 298, 183–204 (2016)MathSciNetzbMATHGoogle Scholar
  62. Santillan, D., Juanes, R., Cueto-Felgueroso, L.: Phase field model of fluid-driven fracture in elastic media: immersed-fracture formulation and validation with analytical solutions. J. Geophys. Res. Solid Earth 122, 2565–2589 (2017)Google Scholar
  63. Schrefler, B.A., Secchi, S., Simoni, L.: On adaptive refinement techniques in multi-field problems including cohesive fracture. Comput. Meth. Appl. Mech. Eng. 195, 444–461 (2006)zbMATHGoogle Scholar
  64. Sneddon, I.N.: The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc. R. Soc. Lond. A 187, 229–260 (1946)MathSciNetGoogle Scholar
  65. Sneddon, I.N., Lowengrub, M.: Crack Problems in the Classical Theory of Elasticity. SIAM Series in Applied Mathematics. Wiley, Philadelphia (1969)zbMATHGoogle Scholar
  66. Tolstoy, I.: Acoustic, Elasticity, and Thermodynamics of Porous Media. Twenty-One Papers by M.A. Biot. Acoustical Society of America, New York (1992)Google Scholar
  67. van Duijn, C.J., Mikelić, A., Wick, T.: A monolithic phase-field model of a fluid-driven fracture in a nonlinear poroelastic medium. Math. Mech. Solids (2018). MathSciNetGoogle Scholar
  68. Wheeler, M., Wick, T., Wollner, W.: An augmented-Lagangrian method for the phase-field approach for pressurized fractures. Comput. Meth. Appl. Mech. Eng. 271, 69–85 (2014)zbMATHGoogle Scholar
  69. Wick, T.: Coupling fluid–structure interaction with phase-field fracture. J. Comput. Phys. 327, 67–96 (2016a)MathSciNetzbMATHGoogle Scholar
  70. Wick, T.: Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity. Comput. Mech. 57(6), 1017–1035 (2016b)MathSciNetzbMATHGoogle Scholar
  71. Wick, T.: An error-oriented Newton/inexact augmented Lagrangian approach for fully monolithic phase-field fracture propagation. SIAM J. Sci. Comput. 39(4), B589–B617 (2017)MathSciNetzbMATHGoogle Scholar
  72. Wick, T.: Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation. Comput. Methods Appl. Mech. Eng. 325, 577–611 (2017)MathSciNetGoogle Scholar
  73. Wick, T., Lee, S., Wheeler, M.: 3D phase-field for pressurizedfracture propagation in heterogeneous media. In: ECCOMAS and IACMCoupled Problems Proceedings, May 2015 at San Servolo, Venice, Italy (2015)Google Scholar
  74. Wick, T., Singh, G., Wheeler, M.: Fluid-filled fracture propagation using a phase-field approach and coupling to a reservoir simulator. SPE J. 21(03), 981–999 (2016)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille JordanVilleurbanne CedexFrance
  2. 2.Center for Subsurface Modeling, The Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  3. 3.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany

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