×

zbMATH — the first resource for mathematics

Phase field modeling of fracture in multi-physics problems. III: Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. (English) Zbl 1425.74423
Summary: The prediction of fluid- and moisture-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developed cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media. It provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy-Biot-type bulk response of hydro-poro-elasticity. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. A modular concept is outlined for linking of the diffusive crack modeling with the hydro-poro-elastic response of the porous bulk material. This includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length. This proposed modular model structure is exploited in the numerical implementation by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the hydro-poro-elastic bulk response. We demonstrate the performance of the phase field formulation of fracture for a spectrum of model problems of hydraulic fracture. A slight modification of the framework allows the simulation of drying-caused crack patterns in partially saturated capillar-porous media.
For Part I and II see [the first author et al., ibid. 294, 449–485 (2015; Zbl 1423.74838); the first author et al., ibid. 294, 486–522 (2015; Zbl 1423.74837)].

MSC:
74R10 Brittle fracture
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
Software:
FEAPpv
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boone, T. J.; Ingraffea, A. R., A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media, Int. J. Numer. Anal. Methods Geomech., 14, 27-47, (1990)
[2] Rubin, A. M., Propagation of magma-filled cracks, Annu. Rev. Earth Planet. Sci., 23, 287-336, (1995)
[3] Zhang, X.; Detournay, E.; Jeffrey, R., Propagation of a penny-shaped hydraulic fracture parallel to the free-surface of an elastic half space, Int. J. Fract., 115, 126-158, (2002)
[4] Adachi, J.; Siebrits, E.; Peirce, A.; Desroches, J., Computer simulation of hydraulic fractures, Int. J. Rock Mech. Min. Sci., 44, 739-757, (2007)
[5] Bažant, Z. P.; Salviato, M.; Chau, V. T.; Viswanathan, H.; Zubelewicz, A., Why fracking works, J. Appl. Mech., 81, 1-10, (2014)
[6] Simoni, L.; Schrefler, B., Multi field simulation of fracture, (Advances in Applied Mechanics, (2014), Elsevier), 367-519
[7] Miehe, C.; Schänzel, L.; Ulmer, H., Phase field modeling of fracture in multi-physics problems. part I. balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids, Comput. Methods Appl. Mech. Engrg., 294, 449-485, (2015) · Zbl 1423.74838
[8] 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)
[9] Terzaghi, K., Erdbaumechanik auf bodenphysikalischer grundlage, (1925), F. Deuticke · JFM 51.0655.07
[10] Biot, M., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164, (1941) · JFM 67.0837.01
[11] Biot, M., Theory of finite deformations of pourous solids, Indiana Univ. Math. J., 21, 597-620, (1972) · Zbl 0218.76090
[12] Rice, J.; Clearly, M., Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents, Rev. Geophys. Space Phys., 14, 227-241, (1976)
[13] Bowen, R. M., Theory of mixtures, (Eringen, A. C., Continuum Physics, vol. III, (1976), Academic Press New York)
[14] Bedford, A.; Drumheller, D., Theories of immiscible and structured mixtures, Internat. J. Engrg. Sci., 21, 863-960, (1983) · Zbl 0534.76105
[15] Truesdell, C., Rational thermodynamics, (1984), Springer New York · Zbl 0598.73002
[16] Coussy, O., Mechanics of porous continua, (1995), John Wiley & Sons, Chichester
[17] de Boer, R., Theory of porous media, (2000), Springer Berlin · Zbl 0961.74021
[18] Ehlers, W., Foundations of multiphasic and porous materials, (Ehlers, W.; Bluhm, J., Porous Media: Theory, Experiments and Numerical Applications, (2002), Springer-Verlag Berlin), 3-86 · Zbl 1062.76050
[19] Detournay, E.; Cheng, A. H.-D., Fundamentals of poroelasticity, (Fairhurst, C., Comprehensive Rock Engineering: Principles, Practice and Projects, Vol. II, Analysis and Design Method, (1993), Pergamon Press), 113-171
[20] Coussy, O.; Dormieux, L.; Detournay, E., From mixture theory to biot’s approach for porous media, Int. J. Solids Struct., 35, 4619-4635, (1998) · Zbl 0932.74014
[21] Adler, P. M.; Thovert, J.-F.; Mourzenko, V. V., Fractured porous media, (2013), Oxford University Press Croydon · Zbl 1266.74002
[22] Schrefler, B.; Secchi, S.; Simoni, L., On adaptive refinement techniques in multi-field problems including cohesive fracture, Comput. Methods Appl. Mech. Engrg., 195, 444-461, (2006) · Zbl 1193.74158
[23] Huang, N. C.; Russell, S. G., Hydraulic fracturing of a saturated porous medium—I: general theory, Theor. Appl. Fract. Mech., 4, 201-213, (1985)
[24] Huang, N. C.; Russell, S. G., Hydraulic fracturing of a saturated porous medium—II: special cases, Theor. Appl. Fract. Mech., 4, 215-222, (1985)
[25] Savitski, A. A.; Detournay, E., Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions, Int. J. Solids Struct., 39, 6311-6337, (2002) · Zbl 1032.74640
[26] Garagash, D. I.; Detournay, E., Plane-strain propagation of a fluid-driven fracture: small toughness solution, J. Appl. Mech., 72, 916-926, (2005) · Zbl 1111.74411
[27] Secchi, S.; Schrefler, B., A method for 3-d hydraulic fracturing simulation, Int. J. Fract., 178, 245-258, (2012)
[28] Secchi, S.; Schrefler, B., Hydraulic fracturing and its peculiarities, Asia Pac. J. Comput. Eng., 1, 1-21, (2014)
[29] Larsson, R.; Runesson, K.; Sture, S., Embedded localization band in undrained soil based on regularized strong discontinuity. theory and FE-analyisis, Int. J. Solids Struct., 33, 3081-3101, (1996) · Zbl 0919.73279
[30] Steinmann, P., Formulation and computation of geometrically non-linear gradient damage, Internat. J. Numer. Methods Engrg., 46, 757-779, (1999) · Zbl 0978.74006
[31] Callari, C.; Armero, F., Finite element methods for the analysis of strong discontinuities in coupled poro-plastic media, Comput. Methods Appl. Mech. Engrg., 191, 4371-4400, (2002) · Zbl 1124.74324
[32] Callari, C.; Armero, F.; Abati, A., Strong discontinuities in partially saturated poroplastic solids, Comput. Methods Appl. Mech. Engrg., 199, 1513-1535, (2010) · Zbl 1231.74105
[33] de Borst, R.; Réthoré, J.; Abellan, M.-A., A numerical approach for arbitrary cracks in a fluid-saturated medium, Arch. Appl. Mech., 75, 595-606, (2006) · Zbl 1168.74447
[34] Irzal, F.; Remmers, J. J.C.; Huyghe, J. M.; de Borst, R., A large deformation formulation for fluid flow in a progressively fracturing porous material, Comput. Methods Appl. Mech. Engrg., 256, 29-37, (2013) · Zbl 1352.76113
[35] Kraaijeveld, F.; Huyghe, J. M.; Remmers, J. J.C.; de Borst, R., Two-dimensional mode I crack propagation in saturated ionized porous media using partition of unity finite elements, J. Appl. Mech., 80, 1-12, (2013)
[36] Réthoré, J.; de Borst, R.; Abellan, M., A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks, Comput. Mech., 42, 227-238, (2008) · Zbl 1154.76053
[37] Mohammadnejad, T.; Khoei, A., Hydro-mechanical modeling of cohesive propagation in multiphase porous media using the extended finite element method, Int. J. Numer. Anal. Methods Geomech., 37, 1247-1279, (2013)
[38] Gordeliy, E.; Detournay, E., A fixed grid algorithm for simulating the propagation of a shallow hydraulic fracture with a fluid lag, Int. J. Numer. Anal. Methods Geomech., 35, 602-629, (2011) · Zbl 1274.74433
[39] Grassl, P.; Fahy, C.; Gallipoli, D.; Wheeler, S. J., On a 2d hydro-mechanical lattice approach for modelling hydraulic fracture, J. Mech. Phys. Solids, 75, 104-118, (2015) · Zbl 1349.74033
[40] Francfort, G. A.; Marigo, J. J., Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46, 1319-1342, (1998) · Zbl 0966.74060
[41] Bourdin, B.; Francfort, G.; Marigo, J.-J., The variational approach to fracture, (2008), Springer
[42] Hakim, V.; Karma, A., Laws of crack motion and phase-field models of fracture, J. Mech. Phys. Solids, 57, 342-368, (2009) · Zbl 1421.74089
[43] Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83, 1273-1311, (2010) · Zbl 1202.74014
[44] 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. Methods Appl. Mech. Engrg., 217-220, 77-95, (2012) · Zbl 1253.74089
[45] Verhoosel, C. V.; de Borst, R., A phase-field model for cohesive fracture, Internat. J. Numer. Methods Engrg., 96, 43-62, (2013) · Zbl 1352.74029
[46] Chukwudozie, C.; Bourdin, B.; Yoshioka, K., A variational approach to the modeling and numerical simulation of hydraulic fracturing under in-situ stresses, (Proceedings, Thirty-Eighth Workshop on Geothermal Reservoir Engineering, (2013), Stanford University Stanford, California)
[47] Mikelic, A.; Wheeler, M. F.; Wick, T., A quasistatic phase field approach to fluid filled fractures, technical report ICES report 13-22, the institute for computational engineering and science, (2013), The University of Texas at Austin
[48] Mikelic, A.; Wheeler, M.; Wick, T., A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium, SIAM Multiscale Modeling Simul., 13, 367-398, (2015) · Zbl 1317.74028
[49] Wheeler, M. F.; Wick, T.; Wollner, N., An augmented-Lagrangian method for the phase-field approach for pressurized fractures, Comput. Methods Appl. Mech. Engrg., 271, 69-85, (2014) · Zbl 1296.65170
[50] Whitaker, S., Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying, Adv. Heat Transfer, 13, 119-203, (1977)
[51] Kowalski, S.; Rybicki, A., Drying stress formation by inhomogeneous moisture and temperature distribution, Transp. Porous Media, 24, 139-156, (1996)
[52] Kowalski, S. J., Thermomechanics of drying processes, (2003), Springer Berlin · Zbl 1038.74002
[53] Richards, L. A., Capillary conduction of liquids through porous media, J. Appl. Phys., 1, 318-333, (1931) · Zbl 0003.28403
[54] Bear, J., Dynamics of fluids in porous media, (1972), Dover Publications New York · Zbl 1191.76001
[55] Ambrosio, L.; Tortorelli, V. M., Approximation of functionals depending on jumps by elliptic functionals via \(\gamma\)-convergence, Comm. Pure Appl. Math., 43, 999-1036, (1990) · Zbl 0722.49020
[56] Borden, M. J.; Hughes, T. J.R.; Landis, C. M.; Verhoosel, C. V., A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework, Comput. Methods Appl. Mech. Engrg., 273, 100-118, (2014) · Zbl 1296.74098
[57] Miehe, C.; Hofacker, M.; Welschinger, F., A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits, Comput. Methods Appl. Mech. Engrg., 199, 2765-2778, (2010) · Zbl 1231.74022
[58] Miehe, C., A multi-field incremental variational framework for gradient-extended standard dissipative solids, J. Mech. Phys. Solids, 59, 898-923, (2011) · Zbl 1270.74022
[59] Terzaghi, K., Theoretical soil mechanics, (1943), Wiley New York
[60] Frémond, M.; Nedjar, B., Damage, gradient of damage and principle of virtual power, Int. J. Solids Struct., 33, 1083-1103, (1996) · Zbl 0910.73051
[61] Frémond, M., Non-smooth thermomechanics, (2002), Springer Verlag · Zbl 0990.80001
[62] Pham, K.; Amor, H.; Marigo, J.; Maurini, C., Gradient damage models and their use to approximate brittle fracture, Int. J. Damage Mech., 20, 618-652, (2011)
[63] de Boer, R.; Ehlers, W., The development of the concept of effective stresses, Acta Mech., 83, 77-92, (1990) · Zbl 0724.73195
[64] Babuška, I., Error-bounds for finite element method, Numer. Math., 16, 322-333, (1971) · Zbl 0214.42001
[65] Babuška, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179-192, (1973) · Zbl 0258.65108
[66] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag · Zbl 0788.73002
[67] Zienkiewicz, O. C.; Taylor, R.; Zhu, J. Z., The finite element method: its basis and fundamentals, (2005), Elsevier
[68] Sandhu, R.; Wilson, E., Finite-element analysis of seepage in elastic media, J. Eng. Mech. Div., 95, 641-652, (1969)
[69] Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 337-344, (1984) · Zbl 0593.76039
[70] Papastavrou, A.; Steinmann, P.; Stein, E., Enhanced finite element formulation for geometrically linear fluid saturated porous media, Mech. Cohesive-Frictonal Mater., 2, 185-203, (1997)
[71] Zhou, X. X.; Chow, Y. K.; Leung, C. F., Hybrid and enhanced finite element methods for problems of soil consolidation, Internat. J. Numer. Methods Engrg., 69, 221-249, (2007) · Zbl 1194.74494
[72] 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)
[73] Roels, S.; Moonen, P.; de Proft, K.; Carmeliet, J., A coupled discrete-continuum approach to simulate moisture effects on damage processes in porous materials, Comput. Methods Appl. Mech. Engrg., 195, 7139-7153, (2006) · Zbl 1331.76118
[74] Grasberger, S.; Meschke, G., Drying shrinkage, creep and cracking of concrete: from coupled material modelling to multifield structural analyses, (de Borst, R.; Mang, H. A.; Bićanić, N.; Meschke, G., Computational Modelling of Concrete Structures, (2003), Balkema, Rotterdam), 433-442
[75] Lewis, R. W.; Strada, M.; Comini, G., Drying-induced stresses in porous bodies, Internat. J. Numer. Methods Engrg., 11, 1175-1184, (1977)
[76] Scherer, G. W., Theory of drying, J. Amer. Ceram. Soc., 73, 3-14, (1990)
[77] Peron, H.; Laloui, L.; Hu, L.-B.; Hueckel, T., Formation of drying crack patterns in soils: a deterministic approach, Acta Geotech., 8, 215-221, (2013)
[78] Bourdin, B.; Marigo, J.-J.; Maurini, C.; Sicsic, P., Morphogenesis and propagation of complex cracks induced by thermal shocks, Phys. Rev. Lett., 112, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.