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A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium. (English) Zbl 1317.74028

##### MSC:
 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74S05 Finite element methods applied to problems in solid mechanics 76S05 Flows in porous media; filtration; seepage 35Q86 PDEs in connection with geophysics 74A45 Theories of fracture and damage
deal.ii
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##### References:
 [1] W. Bangerth, T. Heister, G. Kanschat, et al., Differential Equations Analysis Library, 2012. [2] M. A. Biot, Consolidation settlement under a rectangular load distribution, J. Appl. Phys., 12 (1941), pp. 426–430. · JFM 67.0837.02 [3] M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), pp. 155–164. · JFM 67.0837.01 [4] M. A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 25 (1955), pp. 182–185. · Zbl 0067.23603 [5] M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. R. Hughes, and C. M. Landis, A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Engrg., 217 (2012), pp. 77–95. · Zbl 1253.74089 [6] B. Bourdin, C. Chukwudozie, and K. Yoshioka, A variational approach to the numerical simulation of hydraulic fracturing, SPE Journal, Conference Paper 159154-MS, 2012. [7] B. Bourdin, G. A. Francfort, and J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, 48 (2000), pp. 797–826. · Zbl 0995.74057 [8] B. Bourdin, G. A. Francfort, and J.-J. Marigo, The variational approach to fracture, J. Elasticity, 91 (2008), pp. 1–148. · Zbl 1176.74018 [9] S. Burke, C. Ortner, and E. Süli, An adaptive finite element approximation of a generalized Ambrosio-Tortorelli functional, Math. Models Methods Appl. Sci., 23 (2013), pp. 1663–1697. · Zbl 1266.74044 [10] B. Carrier and S. Granet, Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model, Eng. Fract. Mech., 79 (2012), pp. 312–328. [11] H.-Y. Chen, L. W. Teufel, and R. L. Lee, Coupled fluid flow and geomechanics in reservoir study I. Theory and governing equations, SPE Journal, Conference Paper 30752, 1995. [12] Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. · Zbl 1092.76001 [13] O. Coussy, Mechanics of Porous Continua, Wiley, New York, 1995. · Zbl 0838.73001 [14] O. Coussy, Poromechanics, Wiley, New York, 2004. [15] T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 140–158. · Zbl 0884.65021 [16] R. H. Dean and J. H. Schmidt, Hydraulic fracture predictions with a fully coupled geomechanical reservoir simulator, SPE Journal, Conference Paper 116470-MS, 2008. [17] J. Desroches, E. Detournay, B. Lenoach, P. Papanastasiou, J. R. A. Pearson, M. Thiercelin, and A. Cheng, The crack tip region in hydraulic fracturing, Proc. R. Soc. Lond., 447 (1994), pp. 39–48. · Zbl 0813.73051 [18] E. Detournay and A. Peirce, On the moving boundary conditions for a hydraulic fracture, Internat. J. Engrg. Sci., 84 (2014), pp. 147–155. [19] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), pp. 1319–1342. · Zbl 0966.74060 [20] X. Gai, A Coupled Geomechanics and Reservoir Flow Model on Parallel Computers, Ph.D. thesis, The University of Texas at Austin, Austin, TX, 2004. [21] B. Ganis, V. Girault, M. Mear, G. Singh, and M. F. Wheeler, Modeling fractures in a poro-elastic medium, Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles, 69 (2014), pp. 515–528. [22] B. Ganis, M. E. Mear, A. Sakhaee-Pour, M. F. Wheeler, and T. Wick, Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method, Comput. Geosci., 18 (2014), pp. 613–624. · Zbl 1392.76078 [23] D. I. Garagash and E. Detournay, The tip region of a fluid-driven fracture in an elastic medium, ASME J. Appl. Mech., 67 (2000), pp. 183–192. · Zbl 1110.74448 [24] A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond., 221 (1921), pp. 163–198. [25] P. Gupta and C. A. Duarte, Simulation of non-planar three-dimensional hydraulic fracture propagation, Int. J. Numer. Anal. Meth. Geomech., 38 (2014), pp. 1397–1430. [26] K. M. D. Hals and I. Berre, Interaction between injection points during hydraulic fracturing, Water Resour. Res., 48 (2012), doi: 10.1029/2012WR012265. [27] T. Heister, M. F. Wheeler, and T. Wick, A Primal-Dual Active Set Method and Predictor-Corrector Mesh Adaptivity for Computing Fracture Propagation Using a Phase-Field Approach, Tech. Report, ICES-Preprint 14-27, The University of Texas, Austin, TX, 2014; available online from \burlhttps://www.ices.utexas.edu/research/reports/. [28] M. Hofacker, C. Miehe, and F. Welschinger, Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83 (2013), pp. 1273–1311. · Zbl 1202.74014 [29] F. Irzal, J. J. C. Remmers, J. M. Huyghe, and R. de Borst, A large deformation formulation for fluid flow in a progressively fracturing porous material, Comput. Methods Appl. Mech. Engrg., 256 (2013), pp. 29–37. · Zbl 1352.76113 [30] A. Katiyar, J. T. Foster, H. Ouchi, and M. M. Sharma, A peridynamic formulation of pressure driven convective fluid transport in porous media, J. Comput. Phys., 261 (2014), pp. 209–229. · Zbl 1349.76819 [31] Y. Kovalyshen, Fluid-Driven Fracture in Poroelastic Medium, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 2010. [32] O. A. Ladyzhenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, Transl. Math. Monogr. 23, AMS, Providence, RI, 1968. [33] B. Lecampion and E. Detournay, An implicit algorithm for the propagation of hydraulic fracture with a fluid lag, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 4863–4880. · Zbl 1173.74383 [34] A. Mikelić, B. Wang, and M. F. Wheeler, Numerical convergence study of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 18 (2014), pp. 325–341. · Zbl 1386.76115 [35] A. Mikelić and M. F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17 (2013), pp. 455–461. · Zbl 1392.35235 [36] A. Mikelić, M. F. Wheeler, and T. Wick, A Phase-Field Approach to the Fluid Filled Fracture Surrounded by a Poroelastic Medium, ICES-Preprint 13-15, The University of Texas, Austin, TX, 2013. [37] A. Mikelić, M. F. Wheeler, and T. Wick, A Quasi-static Phase-Field Approach to the Fluid Filled Fracture, ICES-Preprint 13-22, The University of Texas, Austin, TX, 2013, submitted. [38] A. Mikelić, M. F. Wheeler, and T. Wick, Phase-Field Modeling of Pressurized Fractures in a Poroelastic Medium, ICES-Preprint 14-18, The University of Texas, Austin, TX, 2014, submitted. [39] S. L. Mitchell, R. Kuske, and A. P. Peirce, An asymptotic framework for finite hydraulic fractures including leak-off, SIAM J. Appl. Math., 67 (2007), pp. 364–386. · Zbl 1110.74047 [40] B. A. Schrefler, St. Secchi, and L. Simoni, On adaptive refinement techniques in multi-field problems including cohesive fracture, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 444–461. · Zbl 1193.74158 [41] A. Settari and D. A. Walters, Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction, SPE J., 6 (2001), pp. 334–342. [42] A. Z. Szeri, Fluid Film Lubrication, 2nd ed., Cambridge University Press, Cambridge, UK, 2011. · Zbl 1001.76001 [43] M. Wangen, Finite element modeling of hydraulic fracturing on a reservoir scale in 2d, J. Petrol. Sci. Eng., 77 (2011), pp. 274–285. [44] M. Wangen, Finite element modeling of hydraulic fracturing in $$3$$D, Comput. Geosci., 17 (2013), pp. 647–659. · Zbl 1387.74116 [45] M. F. Wheeler, T. Wick, and W. Wollner, An augmented-Lagrangian method for the phase-field approach for pressurized fractures, Comput. Methods Appl. Mech. Engrg., 271 (2014), pp. 69–85. · Zbl 1296.65170 [46] T. Wick, Solving monolithic fluid-structure interaction problems in arbitrary Lagrangian Eulerian coordinates with the deal.ii library, Arch. Numer. Software, 1 (2013), pp. 1–19. [47] T. Wick, G. Singh, and M. F. Wheeler, Pressurized fracture propagation using a phase-field approach coupled to a reservoir simulator, SPE 168597-MS, SPE Proc., 2013.
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