[1] Hubbert M K, Willis D G. Mechanics of hydraulic fracturing[J]. Transactions of Society of Petroleum Engineers of AIME, 1957(210):153-168.
[2] 庄茁, 柳占立, 王涛, 等. 页岩水力压裂的关键力学问题[J]. 科学通报, 2016, 61(1):72-81. Zhuang Zhuo, Liu Zhanli, Wang Tao, et al. The key mechanical problems on hydraulic fracture in shale[J]. Chinese Science Bulletin, 2016, 61(1):72-81.
[3] 庄茁, 柳占立, 王永亮. 页岩油气高效开发中的基础理论与关键力学问题[J]. 力学季刊, 2015, 36(1):11-25. Zhuang Zhuo, Liu Zhanli, Wang Yongliang. Fundamental theory and key mechanical problems of shale oil gas effective extraction[J]. Chinese Quarterly of Mechanics, 2015, 36(1):11-25.
[4] 姚军, 孙海, 黄朝琴, 等. 页岩气藏开发中的关键力学问题[J]. 中国科学:物理学力学天文学, 2013, 43(12):1527-1547. Yao Jun, Sun Hai, Huang Zhaoqin, et al. Key mechanical problems in the development of shale gas reservoirs[J]. Science China:Physics, Mechanics& Astronomy, 2013, 43(12):1527-1547.
[5] Yuan B, Wood D A, Yu W. Stimulation and hydraulic fracturing technology in natural gas reservoirs:Theory and case studies (2012-2015)[J]. Journal of Natural Gas Science and Engineering, 2015, 26(3):1414-1421.
[6] Wu R. Some fundamental mechanics of hydarulic fracturing[D]. Atlanta:School of Civil and Environmental Engineering, Georgia Institute of Technology, 2006.
[7] Ching H. Yew, Xiaowei W. Mechanics of hydraulic fracturing[M]. 2nd. Houston:Gulf Professional Publishing, 2014.
[8] Brady B, Elbel J, Mack M, et al. Cracking rock:Progress in fracture treatment design[J]. Oilfield Review 1992, 4(4):4-17.
[9] Li Q, Xing H, Liu J, et al. A review on hydraulic fracturing of unconventional reservoir[J]. Petroleum, 2015, 1(1):8-15.
[10] Kovalyshen Y. Fluid-driven fracture in poroelastic medium[D]. Minnesota:Faculty of the Graduate School, University of Minnesota, 2010.
[11] Sone H. Mechanical properties of shale gas reservoir rocks, and its relation to the in-situ stress variation observed in shale gas reservoirs[D]. Palo Alto:Stanford University, 2012.
[12] Taleghani A D, Gonzalez M, Shojaei A. Overview of numerical models for interactions between hydraulic fractures and natural fractures:Challenges and limitations[J]. Computers and Geotechnics, 2016(71):361-368.
[13] Tang C A, Tham L G, Lee P K K, et al. Coupled analysis of flow, stress and damage (FSD) in rock failure[J]. International Journal of Rock Mechanics and Mining Sciences, 2002, 39(4):477-489.
[14] Adachi J, Siebrits E, Peirce A, et al. Computer simulation of hydraulic fractures[J]. International Journal of Rock Mechanics and Mining Sciences, 2007, 44(5):739-757.
[15] Hofmann H. Development of enhanced geothermal systems (EGS) in Northern Alberta[D]. Edmonton:University of Alberta, 2015.
[16] Min K S. Numerical modeling of hydraulic fracture propagation using thermo-hydro-mechanical analysis with brittle damage model by finite element method[D]. College Station:Department of Petroleum Engineering, Texas A&M University, 2013.
[17] Carter B J, Desroches J, Ingraffea A R, et al. Simulating fully 3D hydraulic fracturing[M]. Manhattan:John Viley and Sons, 2000:525-557.
[18] Ahn C H, Dilmore R, Wang J Y. Development of innovative and efficient hydraulic fracturing numerical simulation model and parametric studies in unconventional naturally fractured reservoirs[J]. Journal of Unconventional Oil and Gas Resources, 2014(8):25-45.
[19] Carrier B, Granet S. Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model[J]. Engineering Fracture Mechanics, 2012(79):312-328.
[20] Chen Z, Bunger A P, Zhang X, et al. Cohesive zone finite elementbased modeling of hydraulic fractures[J]. Acta Mechanica Solida Sinica, 2009, 22(5):443-452.
[21] Therrien R, Sudicky E A. Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media[J]. Journal of Contaminant Hydrology, 1996, 23(1-2):1-44.
[22] Devloo P R B, Fernandes P D, Gomes S M, et al. A finite element model for three dimensional hydraulic fracturing[J]. Mathematics and Computers in Simulation, 2006, 73(1-4):142-155.
[23] Hunsweck M J, Shen Y, Lew A J. A finite element approach to the simulation of hydraulic fractures with lag[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2013, 37(9):993-1015.
[24] Ouyang S, Carey G F, Yew C H. An adaptive finite element scheme for hydraulic fracturing with proppant transport[J]. International Journal for Numerical Methods in Fluids, 1997, 24(7):645-670.
[25] Wang S Y, Sun L, Au A S K, et al. 2D-numerical analysis of hydraulic fracturing in heterogeneous geo-materials[J]. Construction and Building Materials, 2009, 23(6):2196-2206.
[26] Chen Z. Finite element modelling of viscosity-dominated hydraulic fractures[J]. Journal of Petroleum Science and Engineering, 2012(88):136-144.
[27] Wangen M. Finite element modeling of hydraulic fracturing in 3D[J]. Computational Geosciences, 2013, 17(4):647-659.
[28] Taleghani A. Analysis of hydraulic fracture propagation in fractured reservoirs:An improved model for the interaction between induced and natural fractures[D]. Austin:Department of Petroleum and Geosystems Engineering, University of Texas at Austin, 2009.
[29] Taleghani A, Olson J E. Numerical modeling of multistranded-hydraulic-fracture propagation:Accounting for the interaction between induced and natural fractures[J]. Society of Petroleum Engineers, 2011(16):575-581.
[30] Taleghani A D, Olson J E. Analysis of multistranded hydraulic fracture propagation:An improved model for the interaction between induced and natural fractures[C]//Annual Technical Conference and Exhibition. New Orleans, Louisiana:Society of Petroleum Engineers, 2009.
[31] Lecampion B. An extended finite element method for hydraulic fracture problems[J]. Communications in Numerical Methods in Engineering, 2009, 25(2):121-133.
[32] Mohammadnejad T, Khoei A R. An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model[J]. Finite Elements in Analysis and Design, 2013(73):77-95.
[33] Gordeliy E, Peirce A. Coupling schemes for modeling hydraulic fracture propagation using the XFEM[J]. Computer Methods in Applied Mechanics and Engineering, 2013(253):305-322.
[34] Salimzadeh S, Khalili N. A three-phase XFEM model for hydraulic fracturing with cohesive crack propagation[J]. Computers and Geotechnics, 2015(69):82-92.
[35] Wang H. Numerical modeling of non-planar hydraulic fracture propagation in brittle and ductile rocks using XFEM with cohesive zone method[J]. Journal of Petroleum Science and Engineering, 2015(135):127-140.
[36] Hossain M M, Rahman M K. Numerical simulation of complex fracture growth during tight reservoir stimulation by hydraulic fracturing[J]. Journal of Petroleum Science and Engineering, 2008, 60(2):86-104.
[37] Pecher R. Boundary element simulation of petroleum reservoirs with hydraulically fractured wells[D]. Galgary:Department of Chemical and Petroleum Engineering, University of Calgary, 1999.
[38] Lecampion B, Detournay E. An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag[J]. Computer Methods in Applied Mechanics and Engineering, 2007, 196(49):4863-4880.
[39] Sheibani F. Solving three-dimensional problems in natural and hydraulic fracture development:Insight from displacement discontinuity modeling[D]. Austin:Department of Civil, Architectural and Environmental Engineering, University of Texas at Austin, 2013.
[40] Behnia M, Goshtasbi K, Fatehi Marji M, et al. On the crack propagation modeling of hydraulic fracturing by a hybridized displacement discontinuity/boundary collocation method[J]. Journal of Mining and Environment, 2011, 2(1):1-16.
[41] Al-Busaidi A, Hazzard J F, Young R P. Distinct element modeling of hydraulically fractured Lac du Bonnet granite[J]. Journal of Geophysical Research:Solid Earth, 2005, 110(B06302).
[42] Zhang F, Damjanac B, Huang H. Coupled discrete element modeling of fluid injection into dense granular media[J]. Journal of Geophysical Research:Solid Earth, 2013, 118(6):2703-2722.
[43] Nikolic M, Ibrahimbegovic A, Miscevic P. Discrete element model for the analysis of fluid-saturated fractured poro-plastic medium based on sharp crack representation with embedded strong discontinuities[J]. Computer Methods in Applied Mechanics and Engineering, 2016(298):407-427.
[44] Deng S, Li H, Ma G, et al. Simulation of shale-proppant interaction in hydraulic fracturing by the discrete element method[J]. International Journal of Rock Mechanics and Mining Sciences, 2014(70):219-228.
[45] Shimizu H, Murata S, Ishida T. The distinct element analysis for hydraulic fracturing in hard rock considering fluid viscosity and particle size distribution[J]. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(5):712-727.
[46] Ingrid T. Micro-mechanical aspects of hydraulic fracture propagation and proppant flow and transport for stimulation of enhanced geothermal systems a discrete element study[D]. Golden:Department of Civil and Environmental Engineering, Colorado School of Mines, 2007.
[47] Zeeb C, Konietzky H. Simulating the hydraulic stimulation of multiple fractures in an anisotropic stress field applying the discrete element method[J]. Energy Procedia, 2015, 76:264-272.
[48] McClure M W, Horne R N. Discrete fracture network modeling of hydraulic stimulation:coupling flow and geomechanics[M]. Berlin:Springer Science & Business Media, 2013.
[49] Damjanac B, Cundall P. Application of distinct element methods to simulation of hydraulic fracturing in naturally fractured reservoirs[J]. Computers and Geotechnics, 2016(71):283-294.
[50] Fu P, Johnson S M, Hao Y, et al. Fully coupled geomechanics and discrete flow network modeling of hydraulic fracturing for geothermal applications[C]//Thirty-Sixth Workshop on Geothermal Reservoir Engineering, 2011. Palo Alto:Stanford University, 2011.
[51] Fu P, Johnson S M, Carrigan C R. An explicitly coupled hydro-geomechanical model for simulating hydraulic fracturing in arbitrary discrete fracture networks[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2013, 37(14):2278-2300.
[52] Yaghoubi A, Zoback M D. Hydraulic fracturing modeling using a discrete fracture network in the barnett shale[C]//American Geophysical Union, Fall Meeting 2012. Palo Alto:Stanford University, 2012.
[53] Zhang X, Jeffrey R G, Thiercelin M. Mechanics of fluid-driven fracture growth in naturally fractured reservoirs with simple network geometries[J]. Journal of Geophysical Research:Solid Earth, 2009, 114(B12):465-484.
[54] Hu M, Wang Y, Rutqvist J. Development of a discontinuous approach for modeling fluid flow in heterogeneous media using the numerical manifold method[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2015, 39(17):1932-1952.
[55] Wu Z, Wong L N Y. Extension of numerical manifold method for coupled fluid flow and fracturing problems[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2014, 38(18):1990-2008.
[56] Zhang G, Li X, Li H. Simulation of hydraulic fracture utilizing numerical manifold method[J]. Science China Technological Sciences, 2015, 58(9):1542-1557.
[57] Mikelić A, Wheeler M F, Wick T. A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium[J]. Multiscale Modeling & Simulation, 2015, 13(1):367-398.
[58] 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[J]. Computer Methods in Applied Mechanics and Engineering, 2016(304):619-655.
[59] Mikelić A, Wheeler M F, Wick T. Phase-field modeling of a fluiddriven fracture in a poroelastic medium[J]. Computational Geosciences, 2015, 19(6):1171-1195.
[60] Wick T, Singh G, Wheeler M F. Pressurized-fracture propagation using a phase-field approach coupled to a reservoir simulator[C]//SPE Hydraulic Fracturing Technology Conference. Woodlands:Society of Petroleum Engineers, 2014.
[61] Papanastasiou P. Hydraulic fracture closure in a pressure-sensitive elastoplastic medium[J]. International Journal of Fracture, 2000, 103(2):149-161.
[62] Papanastasiou P C. A coupled elastoplastic hydraulic fracturing model[J]. International Journal of Rock Mechanics and Mining Sciences, 1997, 34(3):240.e1-240.e15.
[63] Shojaei A, Dahi Taleghani A, Li G. A continuum damage failure model for hydraulic fracturing of porous rocks[J]. International Journal of Plasticity, 2014(59):199-212.
[64] Biot M A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. low frequency range[J]. The Journal of the Acoustical Society of America, 1956(28):168-178.
[65] Biot M A. Theory of propagation of elastic waves in a fluid-saturated porous solid. Ⅱ. higher frequency range[J]. The Journal of the Acoustical Society of America, 1956(28):179-191.
[66] Batchelor G K. An introduction to fluid dynamics[M]. Cambridge:Cambridge University Press, 1967.
[67] Garagash D I. Propagation of a plane-strain hydraulic fracture with a fluid lag:Early-time solution[J]. International Journal of Solids and Structures, 2006, 43(18):5811-5835.
[68] Yao Y. Linear elastic and cohesive fracture analysis to model hydraulic fracture in brittle and ductile rocks[J]. Rock Mechanics and Rock Engineering, 2012, 45(3):375-387.
[69] ABAQUS. Abaqus documentation version 6.14[EB/OL].[2016-07-10]. http://129.97.46.200:2080/v6.14/index.html.
[70] Howard G C, Fast C R. Optimum fluid characteristics for fracture extension[C]//Drilling and Production Practice. New York:American Petroleum Institute, 1957.
[71] Sarris E, Papanastasiou P. Modeling of hydraulic fracturing in a poroelastic cohesive formation[J]. International Journal of Geomechanics, 2012, 12(2):160-167.
[72] Peirce A, Detournay E. An implicit level set method for modeling hydraulically driven fractures[J]. Computer Methods in Applied Mechanics and Engineering, 2008, 197(33):2858-2885.
[73] Vandamme L M, Roegiers J C. Poroelasticity in hydraulic fracturing simulators[J]. Journal of Petroleum technology, 1990, 42(09):1199-1203.
[74] Valkó P P, Economides M J. Fluid-leakoff delineation in high-permeability fracturing[J]. Society of Petroleum Engineers, 1999, 14(2):110-116.
[75] Carter R D. Derivation of the general equation for estimating the extent of the fractured area[J]. Drilling and Production Practices, 1957. 261-270.
[76] Moes N, Dolbow J, Belytschko. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering, 1999, 46(1):131-150.
[77] 王涛, 高岳, 柳占立, 等. 基于扩展有限元法的水力压裂大物模试验的数值模拟[J]. 清华大学学报(自然科学版), 2014, 54(10):1304-1309. Wang T, Gao Y, Liu Z L, et al. Numerical simulations of hydraulic fracturing in large objects using an extended finite element method[J]. Journal of Tsinghua University:Science and Technology, 2014, 54(10):1304-1309.
[78] Gupta P, Duarte C A. Simulation of non-planar three-dimensional hydraulic fracture propagation[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2014, 38(13):1397-1430.
[79] Duarte C A, Oden J T. An h-p adaptive method using clouds[J]. Computer Methods in Applied Mechanics and Enginering, 1996, 139(1):237-262.
[80] 王杰, 李世海, 张青波. 基于单元破裂的岩石裂纹扩展模拟方法[J]. 力学学报, 2015, 47(1):105-118. Wang J, Li S, Zhang Q. Simulation of crack propagation of rock based on splitting elements[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(1):105-118.
[81] 魏怀鹏, 易大可, 李世海, 等. 基于连续介质模型的离散元方法中弹簧性质研究[J]. 岩土力学与工程学报,2006, 25(6):1159-1169. Wei H P, Yi D K, Li S H, et al. Study on spring properties of continuum-based discrete element method[J]. Chinese Journal of Rock Mechanics and Engineering, 2006, 25(6):1159-1169.
[82] 刘洋,李世海,刘晓宇. 基于连续介质离散元的双重介质渗流应力耦合模型[J]. 岩石力学与工程学报, 2011, 30(5):951-959. Liu Yang, Li Shihai, Liu Xiaoyu. Coupled fluid flow and stress computation model of dual media based on continuum-medium distinct element method[J]. Chinese Journal of Rock Mechanics and Engineering, 2011, 30(5):951-959.