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Numerical simulation of fracking in shale rocks: current state and future approaches. (English) Zbl 1364.76221
Summary: Extracting gas from shale rocks is one of the current engineering challenges but offers the prospect of cheap gas. Part of the development of an effective engineering solution for shale gas extraction in the future will be the availability of reliable and efficient methods of modelling the development of a fracture system, and the use of these models to guide operators in locating, drilling and pressurising wells. Numerous research papers have been dedicated to this problem, but the information is still incomplete, since a number of simplifications have been adopted such as the assumption of shale as an isotropic material. Recent works on shale characterisation have proved this assumption to be wrong. The anisotropy of shale depends significantly on the scale at which the problem is tackled (nano, micro or macroscale), suggesting that a multiscale model would be appropriate. Moreover, propagation of hydraulic fractures in such a complex medium can be difficult to model with current numerical discretisation methods. The crack propagation may not be unique, and crack branching can occur during the fracture extension. A number of natural fractures could exist in a shale deposit, so we are dealing with several cracks propagating at once over a considerable range of length scales. For all these reasons, the modelling of the fracking problem deserves considerable attention. The objective of this work is to present an overview of the hydraulic fracture of shale, introducing the most recent investigations concerning the anisotropy of shale rocks, then presenting some of the possible numerical methods that could be used to model the real fracking problem.

MSC:
 76S05 Flows in porous media; filtration; seepage 76Mxx Basic methods in fluid mechanics 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
Software:
 [1] Abdollahi, A; Arias, I, Phase-field simulation of anisotropic crack propagation in ferroelectric single crystals: effect of microstructure on the fracture process, Model Simul Mater Sci Eng, 19, 074010, (2011) [2] Abou-Chakra Guéry, A; Cormery, F; Shao, JF; Kondo, D, A multiscale modeling of damage and time-dependent behavior of cohesive rocks, Int J Numer Anal Methods Geomech, 33, 567-589, (2009) · Zbl 1273.74234 [3] Adachi, J; Siebrits, E; Peirce, A; Desroches, J, Computer simulation of hydraulic fractures, Int J Rock Mech Mining Sci, 44, 739-757, (2007) [4] Adachi, JI; Detournay, E, Plane strain propagation of a hydraulic fracture in a permeable rock, Eng Fract Mech, 75, 4666-4694, (2008) [5] Agwai AG, Madenci E (2010) Predicting crack initiation and propagation using XFEM, CZM and peridynamics: a comparative study. In: Electronic components and technology conference (ECTC) · Zbl 1042.74002 [6] Albuquerque, EL; Sollero, P; Fedelinski, P, Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems, Comput Struct, 81, 1703-1713, (2003) [7] Aliabadi, MH; Saleh, AL, Fracture mechanics analysis of cracking in plain and reinforced concrete using the boundary element method, Eng Fract Mech, 69, 267-280, (2002) [8] Ambati, M; Gerasimov, T; Lorenzis, L, A review on phase-field models of brittle fracture and a new fast hybrid formulation, Comput Mech, 55, 383-405, (2015) · Zbl 1398.74270 [9] Amor, H; Marigo, JJ; Maurini, C, Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments, J Mech Phys Solids, 57, 1209-1229, (2009) · Zbl 1426.74257 [10] Aplin, AC; Macquaker, JHS, Mudstone diversity: origin and implications for source, seal, and reservoir properties in petroleum systems, AAPG Bull, 95, 2031-2059, (2011) [11] Aragón, AM; Soghrati, S; Geubelle, PH, Effect of in-plane deformation on the cohesive failure of heterogeneous adhesives, J Mech Phys Solids, 61, 1600-1611, (2013) [12] Asadpoure, A; Mohammadi, S, Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method, Int J Numer Methods Eng, 69, 2150-2172, (2007) · Zbl 1194.74358 [13] Asta, M; Beckermann, C; Karma, A; Kurz, W; Napolitano, R; Plapp, M; Purdy, G; Rappaz, M; Trivedi, R, Solidification microstructures and solid-state parallels: recent developments, future directions, Acta Mater, 57, 941-971, (2009) [14] Atluri, S; Zhu, T, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput Mech, 22, 117-127, (1998) · Zbl 0932.76067 [15] Babuška, I; Melenk, JM, The partition of unity method, Int J Numer Methods Eng, 4, 607-632, (1997) · Zbl 0951.65128 [16] Babuška I, Banerjee U, Osborn J (2003) Meshless and generalized finite element methods: a survey of some major results. In: Griebel M, Schweitzer MA (eds) Lecture notes in computational science and engineering: meshfree methods for partial equations, vol 26. Springer, Berlin, pp 1-20 [17] Barenblatt, GI, The mathematical theory of equilibrium cracks in brittle fracture, Adv Appl Mech, 7, 55-129, (1962) [18] Barla, M; Beer, G, Special issue on advances in modeling rock engineering problems, Int J Geomech, 12, 617-617, (2012) [19] Barsoum, RS, Further application of quadratic isoparametric finite elements to linear fracture mechanics of plate bending and general shells, Int J Fract, 11, 167-169, (1975) [20] Bebendorf M (2008) Hierarchical matrices. Springer, Berlin · Zbl 1151.65090 [21] Béchet, E; Minnebot, H; Möes, N; Burgardt, B, Improved implementation and robustness study of the X-FEM for stress analysis around cracks, Int J Numer Methods Eng, 64, 1033-1056, (2005) · Zbl 1122.74499 [22] Belytschko, T; Gu, L; Lu, Y, Fracture and crack growth by element free Galerkin methods, Model Simul Mater Sci Eng, 2, 519-534, (1994) [23] Belytschko, T; Krongauz, Y; Organ, D; Fleming, M; Krysl, P, Meshless methods: an overview and recent developments, Comput Methods Appl Mech Eng, 139, 3-47, (1996) · Zbl 0891.73075 [24] Belytschko, T; Lu, Y; Gu, L, Element-free Galerkin methods, Int J Numer Methods Eng, 37, 229-256, (1994) · Zbl 0796.73077 [25] Belytschko, T; Lu, Y; Gu, L, Crack-propagation by element-free Galerkin methods, Eng Fract Mech, 51, 295-315, (1995) [26] Belytschko, T; Lu, Y; Gu, L; Tabbara, M, Element-free Galerkin methods for static and dynamic fracture, Int J Numer Methods Eng, 32, 2547-2570, (1995) · Zbl 0918.73268 [27] Belytschko, T; Ventura, G; Xu, J, New methods for discontinuity and crack modelling in EFG, Lecture notes in computational science and engineering: meshfree methods for partial differential equations, 26, 37-50, (2003) · Zbl 1036.74049 [28] Benedetti, I; Aliabadi, MH, A fast hierarchical dual boundary element method for three-dimensional elastodynamic crack problems, Int J Numer Methods Eng, 84, 1038-1067, (2010) · Zbl 1202.74192 [29] Biner, S; Hu, SY, Simulation of damage evolution in composites: a phase-field model, Acta Mater, 57, 2088-2097, (2009) · Zbl 1400.74099 [30] Biani N (2004) Discrete Element Methods. In: Encyclopedia of computational mechanics. John Wiley & Sons, Ltd [31] Bjørlykke, K, Clay mineral diagenesis in sedimentary basinsa key to the prediction of rock properties. examples from the north sea basin, Clay Miner, 33, 15-34, (1998) [32] Bjørlykke, K, Principal aspects of compaction and fluid flow in mudstones, Geol Soc Lond Spec Publ, 158, 73-78, (1999) [33] Bobet, A; Fakhimi, A; Johnson, S; Morris, J; Tonon, F; Yeung, MR, Numerical models in discontinuous media: review of advances for rock mechanics applications, J Geotech Geoenviron Eng, 135, 1547-1561, (2009) [34] Bobko, C; Ulm, FJ, The nano-mechanical morphology of shale, Mech Mater, 40, 318-337, (2008) [35] Bohacs, KM; Lazar, OR; Demko, TM, Parasequence types in shelfal mudstone strataquantitative observations of lithofacies and stacking patterns, and conceptual link to modern depositional regimes, Geology, 42, 131-134, (2014) [36] Bonet, J; Peraire, J, An alternating digital tree (ADT) algorithm for 3D geometric searching and intersection problems, Int J Numer Methods Eng, 31, 1-17, (1991) · Zbl 0825.73958 [37] Bordas, S; Rabczuk, T; Zi, G, Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment, Eng Fract Mech, 75, 943-960, (2008) [38] Borden, MJ; Verhoosel, CV; Scott, MA; Hughes, TJ; Landis, CM, A phase-field description of dynamic brittle fracture, Comput Methods Appl Mech Eng, 217, 77-95, (2012) · Zbl 1253.74089 [39] Bourdin, B; Francfort, GA; Marigo, JJ, Numerical experiments in revisited brittle fracture, J Mech Phys Solids, 48, 797-826, (2000) · Zbl 0995.74057 [40] Boyer, C; Kieschnick, J; Suarez-Rivera, R; Lewis, RE; Waters, G, Producing gas from its source, Oilfield Rev, 18, 36-49, (2006) [41] Brebbia, CA; Domínguez, J, Boundary element methods for potential problems, Appl Math Model, 1, 372-378, (1977) · Zbl 0373.31007 [42] Breitenfeld, M; Geubelle, P; Weckner, O; Silling, S, Non-ordinary state-based peridynamic analysis of stationary crack problems, Comput Methods Appl Mech Eng, 272, 233-250, (2014) · Zbl 1296.74099 [43] Budiansky B, Wu TT (1962) Theoretical prediction of plastic strains of polycrystals. Harvard University, Cambridge [44] Cai, Y; Zhuang, X; Augarde, C, A new partition of unity finite element free from the linear dependence problem and possessing the delta property, Comput Methods Appl Mech Eng, 199, 1036-1043, (2010) · Zbl 1227.74065 [45] Campilho, RDSG; Banea, MD; Chaves, FJP; Silva, LFM, Extended finite element method for fracture characterization of adhesive joints in pure mode I, Comput Mater Sci, 50, 1543-1549, (2011) [46] Carter, E; Fast, GHC (ed.), Optimum fluid characteristics for fracture extension, 261-270, (1957), Washington [47] Chen, CS; Pan, E; Amadei, B, Fracture mechanics analysis of cracked discs of anisotropic rock using the boundary element method, Int J Rock Mech Mining Sci, 35, 195-218, (1998) [48] Chessa, J; Wang, H; Belytschko, T, On the construction of blending elements for local partition of unity enriched finite elements, Int J Numer Methods Eng, 57, 1015-1038, (2003) · Zbl 1035.65122 [49] Cho, N; Martin, CD; Sego, DC, A clumped particle model for rock, Int J Rock Mech Mining Sci, 44, 997-1010, (2007) [50] Clayton, J; Knap, J, A geometrically nonlinear phase field theory of brittle fracture, Int J Fract, 189, 139-148, (2014) [51] Combescure, A; Gravouil, A; Grégoire, D; Rethoré, J, X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation, Comput Methods Appl Mech Eng, 197, 309-318, (2008) · Zbl 1169.74593 [52] Cruse, TA; Rizzo, FJ, A direct formulation and numerical solution of the general transient elastodynamic problem I, J Math Anal Appl, 22, 244-259, (1968) · Zbl 0167.16301 [53] Cundall, PA; Hart, RD, Numerical modelling of discontinua, Eng Comput, 9, 101-113, (1992) [54] Dal Maso, G; Toader, R, A model for the quasistatic growth of brittle fracture. existence and approximation results, Arch Ration Mech Anal, 162, 101-135, (2002) · Zbl 1042.74002 [55] Davies, RJ; Almond, S; Ward, RS; Jackson, RB; Adams, C; Worrall, F; Herringshaw, LG; Gluyas, JG; Whitehead, MA, Oil and gas wells and their integrity: implications for shale and unconventional resource exploitation, Marine Pet Geol, 56, 239-254, (2014) [56] Day-Stirrat, RJ; Aplin, AC; Środoń, J; Pluijm, BA, Diagenetic reorientation of phyllosilicate minerals in paleogene mudstones of the podhale basin, southern Poland, Clays Clay Miner, 56, 100-111, (2008) [57] Day-Stirrat, RJ; Dutton, SP; Milliken, KL; Loucks, RG; Aplin, AC; Hillier, S; Pluijm, BA, Fabric anisotropy induced by primary depositional variations in the silt: Clay ratio in two fine-grained slope Fan complexes: Texas gulf coast and northern north sea, Sediment Geol, 226, 42-53, (2010) [58] Dayal, K; Bhattacharya, K, A real-space non-local phase-field model of ferroelectric domain patterns in complex geometries, Acta Mater, 55, 1907-1917, (2007) [59] Borst, R, Fracture in quasi-brittle materials: a review of continuum damage-based approaches, Eng Fract Mech, 69, 95-112, (2002) [60] Denda, M; Araki, Y; Yong, YK, Time-harmonic BEM for 2-D piezoelectricity applied to eigenvalue problems, Int J Solids Struct, 26, 7241-7265, (2004) · Zbl 1124.74327 [61] Denzer, R; Barth, FJ; Steinmann, P, Studies in elastic fracture mechanics based on the material force method, Int J Numer Methods Eng, 58, 1817-1835, (2003) · Zbl 1032.74702 [62] Detournay, E, Propagation regimes of fluid-driven fractures in impermeable rocks, Int J Geomech, 4, 35-45, (2004) [63] Droser, ML; Bottjer, DJ, A semiquantitative field classification of ichnofabric: research method paper, J Sediment Res, 56, 558-559, (1986) [64] Duflot, M, A meshless method with enriched weight functions for three-dimensional crack propagation, Int J Numer Methods Eng, 65, 1970-2006, (2006) · Zbl 1114.74064 [65] Duflot, M, A study of the representation of cracks with level sets, Int J Numer Methods Eng, 70, 1261-1302, (2007) · Zbl 1194.74516 [66] Dugdale, DS, Yielding of steel sheets containing slits, J Mech Phys Solids, 8, 100-104, (1960) [67] Elices, M; Guinea, GV; Gómez, J; Planas, J, The cohesive zone model: advantages, limitations and challenges, Eng Fract Mech, 69, 137-163, (2002) [68] Eliyahu, M; Emmanuel, S; Day-Stirrat, RJ; Macaulay, CI, Mechanical properties of organic matter in shales mapped at the nanometer scale, Marine Pet Geol, 59, 294-304, (2015) [69] Erdogan, F; Sih, G, On the crack extension in plates under plane loading and transverse shear, J Fluids Eng, 85, 519-525, (1963) [70] Eshelby, JD, The force on an elastic singularity, Philos Trans R Soc Lond Math Phys Sci Ser A, 244, 87-112, (1951) · Zbl 0043.44102 [71] Eshelby, JD, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc R Soc Lond Math Phys Sci Ser A, 241, 376-396, (1957) · Zbl 0079.39606 [72] Eshelby, JD, The elastic field outside an ellipsoidal inclusion, Proc R Soc Lond Math Phys Sci Ser A, 252, 561-569, (1959) · Zbl 0092.42001 [73] Eshelby JD (1970) Energy relations and the energy momentum tensor in continuum mechanics, chapter in elastic behavior of solids. McGraw-Hill, New York [74] Fagerström, M; Larsson, R, Theory and numerics for finite deformation fracture modelling using strong discontinuities, Int J Numer Methods Eng, 66, 911-948, (2006) · Zbl 1110.74815 [75] Feng, H; Wnuk, MP, Cohesive models for quasistatic cracking in inelastic solids, Int J Fract, 48, 115-138, (1991) [76] Feng, YT; Han, K; Owen, DRJ, Combined three-dimensional lattice Boltzmann method and discrete element method for modelling fluid-particle interactions with experimental assessment, Int J Numer Methods Eng, 81, 229-245, (2010) · Zbl 1183.76843 [77] Fernández-Méndez, S; Huerta, A, Imposing essential boundary conditions in mesh-free methods, Comput Methods Appl Mech Eng, 193, 1257-1275, (2004) · Zbl 1060.74665 [78] Fineberg, J; Gross, SP; Marder, M; Swinney, HL, Instability in dynamic fracture, Phys Rev Lett, 67, 457, (1991) [79] Fleming, M; Chu, YA; Moran, B; Belytschko, T; Lu, YY; Gu, L, Enriched element-free Galerkin methods for crack tip fields, Int J Numer Methods Eng, 40, 1483-1504, (1997) [80] Francfort, GA; Marigo, JJ, Revisiting brittle fracture as an energy minimization problem, J Mech Phys Solids, 46, 1319-1342, (1998) · Zbl 0966.74060 [81] Frederix K, Van Barel M (2008) Solving a large dense linear system by adaptive cross approximation. TW Reports · Zbl 1196.65064 [82] Freund LB (1998) Dynamic fracture mechanics. Cambridge University Press, Cambridge · Zbl 0712.73072 [83] Frey, J; Chambon, R; Dascalu, C, A two-scale poromechanical model for cohesive rocks, Acta Geotech, 8, 107-124, (2013) [84] Fries, TP, A corrected XFEM approximation without problems in blending elements, Int J Numer Methods Eng, 75, 503-532, (2008) · Zbl 1195.74173 [85] Fries TP, Matthies HG (2004) Classification and overview of meshfree methods. Technical Report 2003-3, Technical University Braunschweig, Brunswick, Germany [86] Gale JF, Holder J (2010) Natural fractures in some US shales and their importance for gas production. In: Geological Society, London, Petroleum Geology conference series, vol 7. Geological Society of London, pp 1131-1140 [87] Gale, JF; Reed, RM; Holder, J, Natural fractures in the barnett shale and their importance for hydraulic fracture treatments, AAPG Bull, 91, 603-622, (2007) [88] Gao, H, A theory of local limiting speed in dynamic fracture, J Mech Phys Solids, 44, 1453-1474, (1996) [89] Gao, X; Kang, X; Wang, H, Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material, Theor Appl Fract Mech, 51, 73-85, (2009) [90] García-Sánchez, F; Rojas-Díaz, R; Sáez, A; Zhang, C, Fracture of magnetoelectroelastic composite materials using boundary element method (BEM), Theor Appl Fract Mech, 47, 192-204, (2007) [91] García-Sánchez, F; Sáez, A; Domínguez, J, Traction boundary elements for cracks in anisotropic solids, Eng Anal Bound Elem, 28, 667-676, (2004) · Zbl 1130.74462 [92] García-Sánchez, F; Sáez, A; Domínguez, J, Anisotropic and piezoelectric materials fracture analysis by BEM, Comput Struct, 83, 804-820, (2005) [93] García-Sánchez, F; Sáez, A; Domínguez, J, Two-dimensional time-harmonic BEM for cracked anisotropic solids, Eng Anal Bound Elem, 30, 88-99, (2006) · Zbl 1195.74230 [94] García-Sánchez, F; Zhang, C, A comparative study of three BEM for transient dynamic crack analysis of 2-D anisotropic solids, Comput Mech, 40, 753-769, (2007) · Zbl 1191.74053 [95] Ghajari, M; Iannucci, L; Curtis, P, A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media, Comput Methods Appl Mech Eng, 276, 431-452, (2014) · Zbl 1423.74882 [96] Ghosh, S; Lee, K; Moorthy, S, Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method, Int J Solids Struct, 32, 27-62, (1995) · Zbl 0865.73060 [97] Glorioso JC, Rattia A (2012) Unconventional reservoirs: basic petrophysical concepts for shale gas. In: SPE/EAGE European unconventional resources conference and exhibition-from potential to production [98] Gravouil, A; Möes, N; Belytschko, T, Non-planar 3D crack growth by the extended finite element and level sets—part II: level set update, Int J Numer Methods Eng, 53, 2569-2586, (2002) · Zbl 1169.74621 [99] Grytsenko, T; Galybin, A, Numerical analysis of multi-crack large-scale plane problems with adaptive cross approximation and hierarchical matrices, Eng Anal Bound Elem, 34, 501-510, (2010) · Zbl 1244.74218 [100] Guiggiani, M; Krishnasamy, G; Rudolphi, TJ; Rizzo, FJ, A general algorithm for the numerical solution of hypersingular boundary integral equations, J Appl Mech, 59, 604-614, (1992) · Zbl 0765.73072 [101] Gumerov, NA; Duraiswami, R, Fast multipole methods on graphics processors, J Comput Phys, 227, 8290-8313, (2008) · Zbl 1147.65012 [102] Guo, XH; Tin-Loi, F; Li, H, Determination of quasibrittle fracture law for cohesive crack models, Cem Concret Res, 29, 1055-1059, (1999) [103] Gürses, E; Miehe, C, A computational framework of three-dimensional configurational-force-driven brittle crack propagation, Comput Methods Appl Mech Eng, 198, 1413-1428, (2009) · Zbl 1227.74070 [104] Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Springer, New York · Zbl 0951.74003 [105] Gurtin, ME; Podio-Guidugli, P, Configurational forces and the basic laws for crack propagation, J Mech Phys Solids, 44, 905-927, (1996) · Zbl 1054.74508 [106] Ha, YD; Bobaru, F, Studies of dynamic crack propagation and crack branching with peridynamics, Int J Fract, 162, 229-244, (2010) · Zbl 1425.74416 [107] Hackbusch, W, A sparse matrix arithmetic based on $$\cal {H}$$-matrices. part I: introduction to $$\cal {H}$$-matrices, Computing, 62, 89-108, (1999) · Zbl 0927.65063 [108] Hamada, S, GPU-accelerated indirect boundary element method for voxel model analyses with fast multipole method, Comput Phys Commun, 182, 1162-1168, (2011) [109] Hattori G (2013) Study of static and dynamic damage identification in multifield materials using artificial intelligence, BEM and X-FEM. PhD thesis, University of Seville [110] Hattori, G; Rojas-Díaz, R; Sáez, A; Sukumar, N; García-Sánchez, F, New anisotropic crack-tip enrichment functions for the extended finite element method, Comput Mech, 50, 591-601, (2012) · Zbl 1312.74029 [111] Hattori G, Sáez A, Trevelyan J, García-Sánchez F (2014) Enriched BEM for fracture in anisotropic materials. In: Mallardo V, Aliabadi MH (eds) Advances in boundary element & meshless methods XV, EC Ltd, UK, pp 309-314 · Zbl 1352.74284 [112] Hattori G, Serpa AL (2016) Influence of the main contact parameters in finite element analysis of elastic bodies in contact. Key Eng Mater. Wear and Contact Mechanics II:214-227 · Zbl 0955.74066 [113] Hatzor, YH; Benary, R, The stability of a laminated voussoir beam: back analysis of a historic roof collapse using DDA, Int J Rock Mech Mining Sci, 35, 165-181, (1998) [114] Hatzor, YH; Wainshtein, I; Mazor, DB, Stability of shallow karstic caverns in blocky rock masses, Int J Rock Mech Mining Sci, 47, 1289-1303, (2010) [115] Heintz, P, On the numerical modelling of quasi-static crack growth in linear elastic fracture mechanics, Int J Numer Methods Eng, 65, 174-189, (2006) · Zbl 1111.74043 [116] Heintz, P; Larsson, F; Hansbo, P; Runesson, K, Adaptive strategies and error control for computing material forces in fracture mechanics, Int J Numer Methods Eng, 60, 1287-1299, (2004) · Zbl 1060.74627 [117] Henry, H, Study of the branching instability using a phase field model of inplane crack propagation, Europhys Lett, 83, 16004, (2008) [118] Henry, H; Adda-Bedia, M, Fractographic aspects of crack branching instability using a phase-field model, Phys Rev E, 88, 060401, (2013) [119] Henshell, RD; Shaw, KG, Crack tip finite elements are unnecessary, Int J Numer Methods Eng, 9, 495-507, (1975) · Zbl 0306.73064 [120] Ho, NC; Peacor, DR; Pluijm, BA, Preferred orientation of phyllosilicates in gulf coast mudstones and relation to the smectite-illite transition, Clays Clay Miner, 47, 495-504, (1999) [121] Hower, J; Eslinger, EV; Hower, ME; Perry, EA, Mechanism of burial metamorphism of argillaceous sediment: 1. mineralogical and chemical evidence, Geol Soc Am Bull, 87, 725-737, (1976) [122] Hu, W; Ha, YD; Bobaru, F, Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites, Comput Methods Appl Mech Eng, 217, 247-261, (2012) · Zbl 1253.74008 [123] Hussain, M; Pu, S; Underwood, J, Strain energy release rate for a crack under combined mode I and mode II, Fract Anal ASTM-STP, 560, 2-28, (1974) [124] I.C.G. Inc. (2013) UDEC (universal distinct element code) [125] Irwin, GR, Analysis of stresses and strains near the end of a crack traversing a plate, J Appl Mech, 24, 361-364, (1957) [126] Jarvie, DM; Hill, RJ; Ruble, TE; Pollastro, RM, Unconventional shale-gas systems: the Mississippian barnett shale of north-central Texas as one model for thermogenic shale-gas assessment, AAPG Bull, 91, 475-499, (2007) [127] Jing, L, A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering, Int J Rock Mech Mining Sci, 40, 283-353, (2003) [128] Jing, L; Stephansson, O, Discrete element methods for granular materials, No. 85, 399-444, (2007), Amsterdam [129] Kaczmarczyk, Ł; Nezhad, MM; Pearce, C, Three-dimensional brittle fracture: configurational-force-driven crack propagation, Int J Numer Methods Eng, 97, 531-550, (2014) · Zbl 1352.74284 [130] Karamnejad, A; Nguyen, VP; Sluys, LJ, A multi-scale rate dependent crack model for quasi-brittle heterogeneous materials, Eng Fract Mech, 104, 96-113, (2013) [131] Katzav, E; Adda-Bedia, M; Arias, R, Theory of dynamic crack branching in brittle materials, Int J Fract, 143, 245-271, (2007) · Zbl 1197.74111 [132] Ke, CC; Chen, CS; Ku, CY; Chen, CH, Modeling crack propagation path of anisotropic rocks using boundary element method, Int J Numer Anal Methods Geomech, 33, 1227-1253, (2009) · Zbl 1273.74250 [133] Ke, CC; Chen, CS; Tu, CH, Determination of fracture toughness of anisotropic rocks by boundary element method, Rock Mech Rock Eng, 41, 509-538, (2008) [134] Keller, MA; Isaacs, CM, An evaluation of temperature scales for silica diagenesis in diatomaceous sequences including a new approach based on the miocene monterey formation, California, Geo-Marine Lett, 5, 31-35, (1985) [135] Kienzler R, Herrmann G (2000) Mechanics in material space with applications to defect and fracture mechanics. Springer, New York · Zbl 0954.74001 [136] King, GE, Thirty years of gas shale fracturing: what have we learned? in SPE annual technical conference and exhibition, Soc Pet Eng, 2, 900-949, (2010) [137] Kolk, K; Weber, W; Kuhn, G, Investigation of 3D crack propagation problems via fast BEM formulations, Comput Mech, 37, 32-40, (2005) · Zbl 1158.74513 [138] Kozicki, J; Donzé, FV, Yade-open DEM: an open-source software using a discrete element method to simulate granular material, Eng Comput, 26, 786-805, (2009) · Zbl 1257.74189 [139] Kraaijeveld, F; Huyghe, JM; Remmers, JJC; Borst, R, Two-dimensional mode I crack propagation in saturated ionized porous media using partition of unity finite elements, J Appl Mech, 80, 020907, (2013) [140] Kröner, E, Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls, Zeitschrift für Physik, 151, 504-518, (1958) [141] Kuhn, C; Müller, R, A continuum phase field model for fracture, Eng Fract Mech, 77, 3625-3634, (2010) · Zbl 1003.83012 [142] Laborde, P; Pommier, J; Renard, Y; Salaün, M, High-order extended finite element method for cracked domains, Int J Numer Methods Eng, 64, 354-381, (2005) · Zbl 1181.74136 [143] Larsson, R; Fagerström, M, A framework for fracture modelling based on the material forces concept with XFEM kinematics, Int J Numer Methods Eng, 62, 1763-1788, (2005) · Zbl 1121.74316 [144] Larter, S, Some pragmatic perspectives in source rock geochemistry, Marine Pet Geol, 5, 194-204, (1988) [145] Lekhnitskii SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day, San Francisco · Zbl 0119.19004 [146] Lisjak, A; Grasselli, G, A review of discrete modeling techniques for fracturing processes in discontinuous rock masses, J Rock Mech Geotech Eng, 6, 301-314, (2014) [147] Liu G (2003) Meshfree methods: moving beyond the finite element method. CRC Press, Florida [148] Liu, P; Islam, M, A nonlinear cohesive model for mixed-mode delamination of composite laminates, Compos Struct, 106, 47-56, (2013) [149] Liu YJ (2009) Fast multipole boundary element method: theory and applications in engineering. Cambridge University Press, Cambridge [150] Liu, YJ; Mukherjee, S; Nishimura, N; Schanz, M; Ye, W; Sutradhar, A; Pan, E; Dumont, NA; Frangi, A; Sáez, A, Recent advances and emerging applications of the boundary element method, Appl Mech Rev, 64, 030802, (2011) [151] Liu, ZL; Menouillard, T; Belytschko, T, An XFEM/spectral element method for dynamic crack propagation, Int J Fract, 169, 183-198, (2011) · Zbl 1283.74096 [152] Lorentz, E; Cuvilliez, S; Kazymyrenko, K, Modelling large crack propagation: from gradient damage to cohesive zone models, Int J Fract, 178, 85-95, (2012) [153] Love AEH (1944) A treatise on the mathematical theory of elasticity. C. J. Clay and Sons [154] Macek, RW; Silling, SA, Peridynamics via finite element analysis, Finit Elem Anal Des, 43, 1169-1178, (2007) [155] MacLaughlin, MM; Doolin, DM, Review of validation of the discontinuous deformation analysis (DDA) method, Int J Numer Anal Methods Geomech, 30, 271-305, (2006) · Zbl 1140.74574 [156] Macquaker JHS, Howell JK (1999) Small-scale ($$<$$ 5.0 m) vertical heterogeneity in mudstones: implications for high-resolution stratigraphy in siliciclastic mudstone successions. J Geol Soc 156(1):105-112 [157] Macquaker, JHS; Taylor, KG; Gawthorpe, RL, High-resolution facies analyses of mudstones: implications for paleoenvironmental and sequence stratigraphic interpretations of offshore ancient mud-dominated successions, J Sediment Res, 77, 324-339, (2007) [158] Maerten, F, Adaptive cross-approximation applied to the solution of system of equations and post-processing for 3D elastostatic problems using the boundary element method, Eng Anal Bound Elem, 34, 483-491, (2010) · Zbl 1244.74172 [159] Mahabadi, OK; Lisjak, A; Munjiza, A; Grasselli, G, Y-geo: new combined finite-discrete element numerical code for geomechanical applications, Int J Geomech, 12, 676-688, (2012) [160] Mathias, SA; Fallah, AS; Louca, LA, An approximate solution for toughness-dominated near-surface hydraulic fractures, Int J Fract, 168, 93-100, (2011) · Zbl 1283.74065 [161] Mathias, SA; Reeuwijk, M, Hydraulic fracture propagation with 3-D leak-off, Transp Porous Media, 80, 499-518, (2009) [162] Maugin, GA, Material force: concepts and applications, Appl Mech Rev, 23, 213-245, (1995) [163] Maugin, GA; Trimarco, C, Pseudomomentum and material forces in nonlinear elasticity: variational formulations and applications to brittle fracture, Acta Mech, 94, 1-28, (1992) · Zbl 0780.73014 [164] Mehmani, A; Prodanović, M; Javadpour, F, Multiscale, multiphysics network modeling of shale matrix gas flows, Transp Porous Media, 99, 377-390, (2013) [165] Meidani, H; Desbiolles, JL; Jacot, A; Rappaz, M, Three-dimensional phase-field simulation of micropore formation during solidification: morphological analysis and pinching effect, Acta Mater, 60, 2518-2527, (2012) [166] Melenk, J; Babuška, I, The partition of unity finite element method: basic theory and applications, Comput Methods Appl Mech Eng, 139, 289-314, (1996) · Zbl 0881.65099 [167] Menouillard, T; Song, JH; Duan, Q; Belytschko, T, Time dependent crack tip enrichment for dynamic crack propagation, Int J Fract, 162, 33-49, (2010) · Zbl 1425.74442 [168] Meschke, G; Dumstorff, P, Energy-based modeling of cohesive and cohesionless cracks via X-FEM, Comput Methods Appl Mech Eng, 196, 2338-2357, (2007) · Zbl 1173.74384 [169] Mi, Y; Aliabadi, MH, Three-dimensional crack growth simulation using BEM, Comput Struct, 52, 871-878, (1994) · Zbl 0900.73900 [170] Miehe, C; Gürses, E, A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment, Int J Numer Methods Eng, 72, 127-155, (2007) · Zbl 1194.74444 [171] Miehe, C; Gürses, E; Birkle, M, A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization, Int J Fract, 145, 245-259, (2007) · Zbl 1198.74008 [172] 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 Eng, 199, 2765-2778, (2010) · Zbl 1231.74022 [173] 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) · Zbl 1202.74014 [174] Miller, RE; Tadmor, EB, The quasicontinuum method: overview, applications and current directions, J Comput Aided Mater Des, 9, 203-239, (2002) [175] Möes, N; Belytschko, T, Extended finite element method for cohesive crack growth, Eng Fract Mech, 69, 813-833, (2002) [176] Möes, N; Dolbow, J; Belytschko, T, A finite element method for crack growth without remeshing, Int J Numer Methods Eng, 46, 131-150, (1999) · Zbl 0955.74066 [177] Mohaghegh, SD, Reservoir modeling of shale formations, J Nat Gas Sci Eng, 12, 22-33, (2013) [178] Mohammadi S (2008) Extended finite element method: for fracture analysis of structures. Wiley, New York · Zbl 1132.74001 [179] Moonen, P; Carmeliet, J; Sluys, L, A continuous-discontinuous approach to simulate fracture processes in quasi-brittle materials, Philos Mag, 88, 3281-3298, (2008) [180] Moonen, P; Sluys, L; Carmeliet, J, A continuous-discontinuous approach to simulate physical degradation processes in porous media, Int J Numer Methods Eng, 84, 1009-1037, (2010) · Zbl 1202.74158 [181] Motamedi, D; Milani, AS; Komeili, M; Bureau, MN; Thibault, F; Trudel-Boucher, D, A stochastic XFEM model to study delamination in PPS/Glass UD composites: effect of uncertain fracture properties, Appl Compos Mater, 21, 341-358, (2014) [182] Motamedi, D; Mohammadi, S, Dynamic analysis of fixed cracks in composites by the extended finite element method, Eng Fract Mech, 77, 3373-3393, (2010) [183] Motamedi, D; Mohammadi, S, Fracture analysis of composites by time independent moving-crack orthotropic XFEM, Int J Mech Sci, 54, 20-37, (2012) [184] Mousavi, S; Grinspun, E; Sukumar, N, Higher-order extended finite elements with harmonic enrichment functions for complex crack problems, Int J Numer Methods Eng, 86, 560-574, (2011) · Zbl 1216.74027 [185] Mueller, R; Kolling, S; Gross, D, On configurational forces in the context of the finite element method, Int J Numer Methods Eng, 53, 1557-1574, (2002) · Zbl 1114.74489 [186] Mueller, R; Maugin, GA, On material forces and finite element discretizations, Comput Mech, 29, 52-60, (2002) · Zbl 1053.74048 [187] Munjiza A (2004) The combined finite-discrete element method. Wiley, New York · Zbl 1194.74452 [188] Murdoch, LC; Germanovich, LN, Analysis of a deformable fracture in permeable material, Int J Numer Anal Methods Geomech, 30, 529-561, (2006) · Zbl 1140.74542 [189] Näser, B; Kaliske, M; Dal, H; Netzker, C, Fracture mechanical behaviour of visco-elastic materials: application to the so-called Dwell-effect, ZAMM J Appl Math Mech, 89, 677-686, (2009) · Zbl 1168.74010 [190] Näser, B; Kaliske, M; Müller, R, Material forces for inelastic models at large strains: application to fracture mechanics, Comput Mech, 40, 1005-1013, (2007) · Zbl 1160.74003 [191] Nguyen, V; Rabczuk, T; Bordas, S; Duflot, M, Meshless methods: a review and computer implementation aspects, Math Comput Simul, 79, 763-813, (2008) · Zbl 1152.74055 [192] Nishimura, N; Yoshida, K-I; Kobayashi, S, A fast multipole boundary integral equation method for crack problems in 3D, Eng Anal Bound Elem, 23, 97-105, (1999) · Zbl 0953.74074 [193] Nobile, L; Carloni, C, Fracture analysis for orthotropic cracked plates, Compos Struct, 68, 285-293, (2005) [194] Okiongbo, KS; Aplin, AC; Larter, SR, Changes in type II kerogen density as a function of maturity: evidence from the kimmeridge Clay formation, Energy Fuels, 19, 2495-2499, (2005) [195] Oliveira, HL; Leonel, ED, Cohesive crack growth modelling based on an alternative nonlinear BEM formulation, Eng Fract Mech, 111, 86-97, (2013) [196] Oliveira, HL; Leonel, ED, An alternative BEM formulation, based on dipoles of stresses and tangent operator technique, applied to cohesive crack growth modelling, Eng Anal Bound Elem, 41, 74-82, (2014) · Zbl 1297.74151 [197] Organ, D; Fleming, M; Terry, T; Belytschko, T, Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Comput Mech, 18, 225-235, (1996) · Zbl 0864.73076 [198] Ortega, JA; Ulm, FJ; Abousleiman, Y, The effect of the nanogranular nature of shale on their poroelastic behavior, Acta Geotech, 2, 155-182, (2007) [199] Ortega, JA; Ulm, FJ; Abousleiman, Y, The nanogranular acoustic signature of shale, Geophysics, 74, d65-d84, (2009) [200] Ortega, JA; Ulm, FJ; Abousleiman, Y, The effect of particle shape and grain-scale properties of shale: a micromechanics approach, Int J Numer Anal Methods Geomech, 34, 1124-1156, (2010) · Zbl 1273.74295 [201] Oterkus, E; Madenci, E; Weckner, O; Silling, S; Bogert, P; Tessler, A, Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot, Compos Struct, 94, 839-850, (2012) [202] Özenç, K; Kaliske, M; Lin, G; Bhashyam, G, Evaluation of energy contributions in elasto-plastic fracture: a review of the configurational force approach, Eng Fract Mech, 115, 137-153, (2014) [203] Papanastasiou, P, The effective fracture toughness in hydraulic fracturing, Int J Fract, 96, 127-147, (1999) [204] Passey QR, Bohacs K, Esch WL, Klimentidis R, Sinha S (2010) From oil-prone source rock to gas-producing shale reservoir-geologic and petrophysical characterization of unconventional shale-gas reservoirs. SPE, Beijing 8 [205] Peirce, A; Detournay, E, An implicit level set method for modeling hydraulically driven fractures, Comput Methods Appl Mech Eng, 197, 2858-2885, (2008) · Zbl 1194.74534 [206] Peltonen, C; Marcussen, Ø; Bjørlykke, K; Jahren, J, Clay mineral diagenesis and quartz cementation in mudstones: the effects of smectite to illite reaction on rock properties, Marine Pet Geol, 26, 887-898, (2009) [207] Pepper AS, Corvi PJ (1995) Simple kinetic models of petroleum formation. Part III: modelling an open system. Marine Pet Geol 12(4):417-452 [208] Perkins, E; Williams, JR, A fast contact detection algorithm insensitive to object sizes, Eng Comput, 18, 48-62, (2001) · Zbl 1015.74079 [209] Pichler, B; Dormieux, L, Cohesive zone size of microcracks in brittle materials, Eur J Mech A/Solids, 26, 956-968, (2007) · Zbl 1122.74044 [210] Pichler, B; Dormieux, L, Instability during cohesive zone growth, Eng Fract Mech, 76, 1729-1749, (2009) [211] Pilipenko, D; Fleck, M; Emmerich, H, On numerical aspects of phase field fracture modelling, Eur Phys J Plus, 126, 1-16, (2011) [212] Pine, RJ; Owen, DRJ; Coggan, JS; Rance, JM, A new discrete fracture modelling approach for rock masses, Geotechnique, 57, 757-766, (2007) [213] Planas, J; Elices, M, Asymptotic analysis of a cohesive crack: 1. theoretical background, Int J Fract, 55, 153-177, (1992) [214] Portela, A; Aliabadi, MH; Rooke, DP, Dual boundary element incremental analysis of crack propagation, Comput Struct, 46, 237-247, (1993) · Zbl 0825.73888 [215] Potyondy, DO; Cundall, PA, A bonded-particle model for rock, Int J Rock Mech Mining Sci, 41, 1329-1364, (2004) [216] Prechtel, M; Ronda, PL; Janisch, R; Hartmaier, A; Leugering, G; Steinmann, P; Stingl, M, Simulation of fracture in heterogeneous elastic materials with cohesive zone models, Int J Fract, 168, 15-29, (2011) · Zbl 1283.74073 [217] Provatas N, Elder K (2011) Phase-field methods in materials science and engineering. Wiley, New York [218] R.S. Ltd. (2004) ELFEN 2D/3D numerical modelling package [219] Rabczuk, T; Bordas, S; Zi, G, On three-dimensional modelling of crack growth using partition of unity methods, Comput Struct, 88, 1391-1411, (2010) [220] Rabczuk, T; Song, JH; Belytschko, T, Simulations of instability in dynamic fracture by the cracking particles method, Eng Fract Mech, 76, 730-741, (2009) [221] Rahm, D, Regulating hydraulic fracturing in shale gas plays: the case of Texas, Energy Policy, 39, 2974-2981, (2011) [222] Ravi-Chandar, K; Knauss, WG, An experimental investigation into dynamic fracture: II. microstructural aspects, Int J Fract, 26, 65-80, (1984) [223] Ravi-Chandar, K; Knauss, WG, An experimental investigation into dynamic fracture: III. on steady-state crack propagation and crack branching, Int J Fract, 26, 141-154, (1984) [224] Rice, JR, A path independent integral and the approximate analysis of strain concentration by notches and cracks, J Appl Mech, 33, 379-386, (1968) [225] Richardson, CL; Hegemann, J; Sifakis, E; Hellrung, J; Teran, JM, An XFEM method for modeling geometrically elaborate crack propagation in brittle materials, Int J Numer Methods Eng, 88, 1042-1065, (2011) · Zbl 1242.74156 [226] Rojas-Díaz, R; García-Sánchez, F; Sáez, A, Analysis of cracked magnetoelectroelastic composites under time-harmonic loading, Int J Solids Struct, 47, 71-80, (2010) · Zbl 1193.74038 [227] Rojas-Díaz, R; Sáez, A; García-Sánchez, F; Zhang, C, Time-harmonic greens functions for anisotropic magnetoelectroelasticity, Int J Solids Struct, 45, 144-158, (2008) · Zbl 1167.74407 [228] Rokhlin, V, Rapid solution of integral equations of classical potential theory, J Comput Phys, 60, 187-207, (1985) · Zbl 0629.65122 [229] Runesson, K; Larsson, F; Steinmann, P, On energetic changes due to configurational motion of standard continua, Int J Solids Struct, 46, 1464-1475, (2009) · Zbl 1236.74017 [230] Saad, Y; Schultz, MH, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J Sci Stat Comput, 7, 856-869, (1986) · Zbl 0599.65018 [231] Sáez, A; Gallego, R; Domínguez, J, Hypersingular quarter-point boundary elements for crack problems, Int J Numer Methods Eng, 38, 1681-1701, (1995) · Zbl 0831.73077 [232] Sáez, A; García-Sánchez, F; Domínguez, J, Hypersingular BEM for dynamic fracture in 2-D piezoelectric solids, Comput Methods Appl Mech Eng, 196, 235-246, (2006) · Zbl 1120.74846 [233] Saleh, AL; Aliabadi, MH, Crack growth analysis in concrete using boundary element method, Eng Fract Mech, 51, 533-545, (1995) [234] Saleh, AL; Aliabadi, MH, Boundary element analysis of the pullout behaviour of an anchor bolt embedded in concrete, Mech Cohes Frict Mater, 1, 235-249, (1996) [235] Saleh, AL; Aliabadi, MH, Crack growth analysis in reinforced concrete using BEM, J Eng Mech, 124, 949-958, (1998) [236] Samimi, M; Dommelen, J; Geers, M, A three-dimensional self-adaptive cohesive zone model for interfacial delamination, Comput Methods Appl Mech Eng, 200, 3540-3553, (2011) · Zbl 1239.74087 [237] Samimi, M; Dommelen, J; Geers, MGD, An enriched cohesive zone model for delamination in brittle interfaces, Int J Numer Methods Eng, 80, 609-630, (2009) · Zbl 1176.74195 [238] Samimi, M; Dommelen, JAW; Kolluri, M; Hoefnagels, JPM; Geers, MGD, Simulation of interlaminar damage in mixed-mode bending tests using large deformation self-adaptive cohesive zones, Eng Fract Mech, 109, 387-402, (2013) [239] Saouma, VE; Ayari, ML; Leavell, DA, Mixed mode crack propagation in homogeneous anisotropic solids, Eng Fract Mech, 27, 171-184, (1987) [240] Scherer, M; Denzer, R; Steinmann, P, On a solution strategy for energy-based mesh optimization in finite hyperelastostatics, Comput Methods Appl Mech Eng, 197, 609-622, (2008) · Zbl 1169.74666 [241] Schieber, J; Southard, JB; Schimmelmann, A, Lenticular shale fabrics resulting from intermittent erosion of water-rich mudsinterpreting the rock record in the light of recent flume experiments, J Sediment Res, 80, 119-128, (2010) [242] Schlüter, A; Willenbücher, A; Kuhn, C; Müller, R, Phase field approximation of dynamic brittle fracture, Comput Mech, 54, 1141-1161, (2014) · Zbl 1311.74106 [243] Sethian, J, Fast marching methods, SIAM Rev, 41, 199-235, (1999) · Zbl 0926.65106 [244] Sfantos, GK; Aliabadi, MH, A boundary cohesive grain element formulation for modelling intergranular microfracture in polycrystalline brittle materials, Int J Numer Methods Eng, 69, 1590-1626, (2007) · Zbl 1194.74503 [245] Shenoy, V; Miller, R; Tadmor, E; Rodney, D; Phillips, R; Ortiz, M, An adaptive finite element approach to atomic-scale mechanicsthe quasicontinuum method, J Mech Phys Solids, 47, 611-642, (1999) · Zbl 0982.74071 [246] Shi G, Goodman RE (1988) Discontinuous deformation analysis—a new method for computing stress, strain and sliding of block systems. In: The 29th US symposium on rock mechanics (USRMS). American Rock Mechanics Association · Zbl 0831.73077 [247] Sih GC (1991) Mechanics of fracture initiation and propagation. Springer, Berlin [248] Silling, SA, Reformulation of elasticity theory for discontinuities and long-range forces, J Mech Phys Solids, 48, 175-209, (2000) · Zbl 0970.74030 [249] Silling, SA; Askari, E, A meshfree method based on the peridynamic model of solid mechanics, Comput Struct, 83, 1526-1535, (2005) [250] Silling, SA; Epton, M; Weckner, O; Xu, J; Askari, E, Peridynamic states and constitutive modeling, J Elast, 88, 151-184, (2007) · Zbl 1120.74003 [251] Simpson, R; Trevelyan, J, A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics, Comput Methods Appl Mech Eng, 200, 1-10, (2011) · Zbl 1225.74117 [252] Soeder, DJ; Sharma, S; Pekney, N; Hopkinson, L; Dilmore, R; Kutchko, B; Stewart, B; Carter, K; Hakala, A; Capo, R, An approach for assessing engineering risk from shale gas wells in the united states, Int J Coal Geol, 126, 4-19, (2014) [253] Spatschek, R; Brener, E; Karma, A, Phase field modeling of crack propagation, Philos Mag, 91, 75-95, (2011) [254] Środoń, J, Nature of mixed-layer clays and mechanisms of their formation and alteration, Annu Rev Earth Planet Sci, 27, 19-53, (1999) [255] Steinmann, P; Ackermann, D; Barth, FJ, Application of material forces to hyperelastostatic fracture mechanics. II. computational setting, Int J Solids Struct, 38, 5509-5526, (2001) · Zbl 1066.74508 [256] Steinmann P, Maugin GA (eds) (2005) Mechanics of material forces. Springer, Berlin · Zbl 1087.74006 [257] Stroh, AN, A theory of the fracture of metals, Adv Phys, 6, 418-465, (1957) [258] Stumpf, H; Le, KC, Variational principles of nonlinear fracture mechanics, Acta Mech, 83, 25-37, (1990) · Zbl 0726.73099 [259] Tada, R; Siever, R, Pressure solution during diagenesis, Annu Rev Earth Planet Sci, 17, 89, (1989) [260] Takezawa, A; Kitamura, M, Phase field method to optimize dielectric devices for electromagnetic wave propagation, J Comput Phys, 257, 216-240, (2014) · Zbl 1349.82123 [261] Távara, L; Mantič, V; Salvadori, A; Gray, LJ; París, F, Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids, Comput Mech, 51, 535-551, (2013) · Zbl 1312.74058 [262] Tay, TE; Sun, XS; Tan, VBC, Recent efforts toward modeling interactions of matrix cracks and delaminations: an integrated XFEM-CE approach, Adv Compos Mater, 23, 391-408, (2014) [263] Tayloer, LM; Preece, DS, Simulation of blasting induced rock motion using spherical element models, Eng Comput, 9, 243-252, (1992) [264] Thyberg, B; Jahren, J; Winje, T; Bjørlykke, K; Faleide, JI; Marcussen, Ø, Quartz cementation in late cretaceous mudstones, northern north sea: changes in rock properties due to dissolution of smectite and precipitation of micro-quartz crystals, Marine Pet Geol, 27, 1752-1764, (2010) [265] Tillberg, J; Larsson, F; Runesson, K, On the role of material dissipation for the crack-driving force, Int J Plast, 26, 992-1012, (2010) · Zbl 1426.74278 [266] Travasso, RDM; Castro, M; Oliveira, JCRE, The phase-field model in tumor growth, Philos Mag, 91, 183-206, (2011) [267] Tsesarsky, M; Hatzor, YH, Tunnel roof deflection in blocky rock masses as a function of joint spacing and friction-a parametric study using discontinuous deformation analysis (DDA), Tunn Undergr Space Technol, 21, 29-45, (2006) [268] Ulm, FJ; Abousleiman, Y, The nanogranular nature of shale, Acta Geotech, 1, 77-88, (2006) [269] Ulm FJ, Constantinides G, Delafargue A, Abousleiman Y, Ewy R, Duranti L, McCarty DK (2005) Material invariant poromechanics properties of shales. Poromechanics III. biot centennial (1905-2005). AA Balkema Publishers, London, pp. 637-644 · Zbl 1426.74257 [270] Ulm, FJ; Vandamme, M; Bobko, C; Alberto Ortega, J; Tai, K; Ortiz, C, Statistical indentation techniques for hydrated nanocomposites: concrete, bone, and shale, J Am Ceram Soc, 90, 2677-2692, (2007) [271] Vallejo, LE, Shear stresses and the hydraulic fracturing of Earth dam soils, Soils Found, 33, 14-27, (1993) [272] Meer, FP; Dávila, CG, Cohesive modeling of transverse cracking in laminates under in-plane loading with a single layer of elements per ply, Int J Solids Struct, 50, 3308-3318, (2013) [273] Vaughn A, Pursell D (2010) Frac attack: risks, hype, and financial reality of hydraulic fracturing in the shale plays. Reservoir Research Partners and TudorPickering Holt and Co, Houston [274] Vernerey, FJ; Kabiri, M, Adaptive concurrent multiscale model for fracture and crack propagation in heterogeneous media, Comput Methods Appl Mech Eng, 276, 566-588, (2014) · Zbl 1423.74930 [275] Vignollet, J; May, S; Borst, R; Verhoosel, CV, Phase-field models for brittle and cohesive fracture, Meccanica, 49, 1-15, (2014) [276] Voigt W (1928) Lehrbuch der Kristallphysik. B.G. Teubner, Leipzig · JFM 54.0929.03 [277] Wang, CY; Achenbach, JD, 3-D time-harmonic elastodynamic green’s functions for anisotropic solids, Philos Trans R Soc Lond Math Phys Sci Ser A, 449, 441-458, (1995) · Zbl 0852.73011 [278] Wang, Y; Wang, Q; Wang, G; Huang, Y; Wang, S, An adaptive dual-information FMBEM for 3D elasticity and its GPU implementation, Eng Anal Bound Elem, 37, 236-249, (2013) · Zbl 1351.74146 [279] Warpinski, NR; Teufel, LW, Influence of geologic discontinuities on hydraulic fracture propagation (includes associated papers 17011 and 17074), J Pet Technol, 39, 209-220, (1987) [280] Warren, TL; Silling, SA; Askari, A; Weckner, O; Epton, MA; Xu, J, A non-ordinary state-based peridynamic method to model solid material deformation and fracture, Int J Solids Struct, 46, 1186-1195, (2009) · Zbl 1236.74012 [281] Williams, LA; Crerar, DA, Silica diagenesis, II. general mechanisms, J Sediment Res, 55, 312-321, (1985) [282] Wünsche, M; García-Sánchez, F; Sáez, A, Analysis of anisotropic Kirchhoff plates using a novel hypersingular BEM, Comput Mech, 49, 629-641, (2012) · Zbl 1398.74214 [283] Wünsche, M; Zhang, C; Kuna, M; Hirose, S; Sladek, J; Sladek, V, A hypersingular time-domain BEM for 2D dynamic crack analysis in anisotropic solids, Int J Numer Methods Eng, 78, 127-150, (2009) · Zbl 1183.74343 [284] Xiao, QZ; Karihaloo, BL, Asymptotic fields at frictionless and frictional cohesive crack tips in quasibrittle materials, J Mech Mater Struct, 1, 881-910, (2006) [285] Xu, D; Liu, Z; Liu, X; Zeng, Q; Zhuang, Z, Modeling of dynamic crack branching by enhanced extended finite element method, Comput Mech, 54, 1-14, (2014) · Zbl 1398.74422 [286] Yang, B; Ravi-Chandar, K, A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks, Int J Fract, 93, 115-144, (1998) [287] Yang, Y; Aplin, AC, Definition and practical application of mudstone porosity-effective stress relationships, Pet Geosci, 10, 153-162, (2004) [288] Yao, Y, Linear elastic and cohesive fracture analysis to model hydraulic fracture in brittle and ductile rocks, Rock Mech Rock Eng, 45, 375-387, (2012) [289] Yoffe EH (1951) LXXV. The moving griffith crack. Philos Mag 42(330):739-750 · Zbl 0043.23504 [290] Yoshida, K-I; Nishimura, N; Kobayashi, S, Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D, Int J Numer Methods Eng, 50, 525-547, (2001) · Zbl 1004.74078 [291] Zamani, A; Gracie, R; Reza Eslami, M, Cohesive and non-cohesive fracture by higher-order enrichment of XFEM, Int J Numer Methods Eng, 90, 452-483, (2012) · Zbl 1242.74177 [292] Zeng, T; Shao, JF; Xu, W, Multiscale modeling of cohesive geomaterials with a polycrystalline approach, Mech Mater, 69, 132-145, (2014) [293] Zeng, X; Li, S, A multiscale cohesive zone model and simulations of fractures, Comput Methods Appl Mech Eng, 199, 547-556, (2010) · Zbl 1227.74054 [294] Zhang, C, A 2D hypersingular time-domain traction BEM for transient elastodynamic crack analysis, Wave Motion, 35, 17-40, (2002) · Zbl 1120.74849 [295] Zhang, GQ; Chen, M, Dynamic fracture propagation in hydraulic re-fracturing, J Pet Sci Eng, 70, 266-272, (2010) [296] Zhang, X; Jeffrey, RG; Detournay, E, Propagation of a hydraulic fracture parallel to a free surface, Int J Numer Anal Methods Geomech, 29, 1317-1340, (2005) · Zbl 1140.74544 [297] Zhang, Z; Huang, K, A simple J-integral governed bilinear constitutive relation for simulating fracture propagation in quasi-brittle material, Int J Rock Mech Mining Sci, 48, 294-304, (2011) [298] Zhong, H; Ooi, ET; Song, C; Ding, T; Lin, G; Li, H, Experimental and numerical study of the dependency of interface fracture in concrete-rock specimens on mode mixity, Eng Fract Mech, 124, 287-309, (2014) [299] Zhou, Q; Liu, HH; Bodvarsson, GS; Oldenburg, CM, Flow and transport in unsaturated fractured rock: effects of multiscale heterogeneity of hydrogeologic properties, J Contam Hydrol, 60, 1-30, (2003) [300] Zhuang, X; Augarde, C; Bordas, S, Accurate fracture modelling using meshless methods and level sets: formulation and 2D modelling, Int J Numer Methods Eng, 86, 249-268, (2011) · Zbl 1235.74346 [301] Zhuang, X; Augarde, C; Mathiesen, K, Fracture modelling using a meshless method and level sets: framework and 3D modelling, Int J Numer Methods Eng, 92, 969-998, (2012) · Zbl 1352.74312 [302] Zi, G; Belytschko, T, New crack-tip elements for XFEM and applications to cohesive cracks, Int J Numer Methods Eng, 57, 2221-2240, (2003) · Zbl 1062.74633 [303] Zi, G; Rabczuk, T; Wall, W, Extended meshfree methods without branch enrichment for cohesive cracks, Comput Mech, 40, 367-382, (2007) · Zbl 1162.74053