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

Abstract

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).

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References

  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, USA

  3. Almi, S., Maso, G.D., Toader, R.: Quasi-static crack growth in hydraulic fracture. Nonlinear Anal. Theory Methods Appl. 109, 301–318 (2014)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

  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)

    MathSciNet  MATH  Google Scholar 

  9. Bourdin, B.: Image segmentation with a finite element method. Math. Model. Numer. Anal. 33(2), 229–244 (1999)

    MathSciNet  MATH  Google 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)

  11. Bourdin, B., Francfort, G., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Bourdin, B., Francfort, G., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1–3), 1–148 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Braides, A.: Approximation of Free-Discontinuity Problems. Springer, Berlin (1998)

    Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

  18. Chambolle, A.: An approximation result for special functions with bounded variations. J. Math. Pures Appl. 83, 929–954 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, 2 edn. North-Holland, Amsterdam (1987)

  20. Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer Verlag, Berlin (2008)

    Google 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)

    MathSciNet  Google 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)

    MATH  Google 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. http://slsedp.cedram.org/slsedp-bin/fitem?id=SLSEDP_2011-2012

  25. Francfort, G., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)

    MathSciNet  MATH  Google 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)

    MathSciNet  Google 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)

    MathSciNet  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

  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)

    MathSciNet  MATH  Google 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)

  37. Lee, J.J.: Robust error analysis of coupled mixed methods for Biot’s consolidation model. J. Sci. Comput. 69(2), 610–632 (2016)

    MathSciNet  MATH  Google 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)

    MathSciNet  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  44. Liu, R.: Discontinuous galerkin finite element solution for poromechanics. Ph.D. thesis, The University of Texas at Austin (2004)

  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)

  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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

  52. Mikelić, A., Wheeler, M., Wick, T.: Phase-field modeling of pressurized fractures in a poroelastic medium. ICES Report 14-18 (2014)

  53. Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17(3), 455–462 (2012)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  56. Mikelić, A., Wheeler, M.F., Wick, T.: A quasi-static phase-field approach to pressurized fractures. Nonlinearity 28(5), 1371–1399 (2015c)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Google 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)

    MathSciNet  MATH  Google 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)

    MATH  Google 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)

    MathSciNet  Google Scholar 

  65. Sneddon, I.N., Lowengrub, M.: Crack Problems in the Classical Theory of Elasticity. SIAM Series in Applied Mathematics. Wiley, Philadelphia (1969)

    Google 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). https://doi.org/10.1177/1081286518801050

    MathSciNet  Google 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)

    MATH  Google Scholar 

  69. Wick, T.: Coupling fluid–structure interaction with phase-field fracture. J. Comput. Phys. 327, 67–96 (2016a)

    MathSciNet  MATH  Google Scholar 

  70. Wick, T.: Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity. Comput. Mech. 57(6), 1017–1035 (2016b)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Google 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)

  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 

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Correspondence to T. Wick.

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A.M. would like to thank Institute for Computational Engineering and Science (ICES), UT Austin for hospitality during his sabbatical stays. The research by M. F. Wheeler was partially supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences through DOE Energy Frontier Research Center: The Center for Frontiers of Subsurface Energy Security (CFSES) under Contract No. DE-FG02-04ER25617, MOD. 005. The work of T. Wick was supported through an ICES Postdoc fellowship, the Humboldt foundation with a Feodor-Lynen fellowship and through the JT Oden faculty research program. Currently T. Wick is supported by the DFG-SPP 1748 program.

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Mikelić, A., Wheeler, M.F. & Wick, T. Phase-field modeling through iterative splitting of hydraulic fractures in a poroelastic medium. Int J Geomath 10, 2 (2019). https://doi.org/10.1007/s13137-019-0113-y

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Keywords

  • Hydraulic fracturing
  • Phase field formulation
  • Nonlinear elliptic system
  • Computer simulations
  • Poroelasticity

Mathematics Subject Classification

  • 35Q74
  • 35J87
  • 49J45
  • 65K15
  • 74R10