Abstract：Calculations of hydraulic fracturing are very essential for the exploration of shale gas. In this paper，the shale is approximated by a homogeneous porous medium. The seepage process in rock mass and the fluid motion in cracks are described by the dynamic Darcy-Biot and Poiseuille flows respectively. The crack propagation is tracked with the phase-field method and the simulation method for the dynamic hydraulic model is proposed. The method is implemented under an explicit finite element scheme. The accuracy of the method to calculate the dynamic displacement and pressure fields is illustrated by comparing the output with the one-dimensional analytical solution. A series of numerical examples indicate that the proposed method can not only simulate the typical characteristics of dynamic hydraulic fracturing，can also calculate the crack intersection，bifurcation and three-dimensional propagation.
[1] 张有天. 岩石水力学与工程[M]. 北京：中国水利水电出版社，2005：112–114.(ZHANG Youtian. Rock hydraulics and engineering[M]. Beijing：China Water Power Press，2005：112–114.(in Chinese))
[2] TERZAGHI K. Principles of soil mechanics，IV—Settlement and consolidation of clay[J]. Engineering News-Record，1925，95(3)：874–878.
[3] BIOT M A. General theory of three-dimensional consolidation[J]. Journal of Applied Physics，1941，12(2)：155–164.
[4] BIOT M A. Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena[J]. Journal of Applied Physics，1954，25(11)：1 385–1 391.
[5] GARAGASH D I，DETOURNAY E，ADACHI J I. Multiscale tip asymptotics in hydraulic fracture with leak-off[J]. Journal of Fluid Mechanics，2011，669：260–297.
[6] ZHANG H B，LIU J S，ELSWORTH D. How sorption-induced matrix deformation affects gas flow in coal seams：a new FE model[J]. International Journal of Rock Mechanics and Mining Sciences，2008，45(8)：1 226–1 236.
[7] ADLER P M，THOVERT J F，MOURZENKO V V. Fractured porous media[M]. Oxford：Oxford University Press，2012：66–81.
[8] BOONE T J，INGRAFFEA A R. A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1990，14(1)：27–47.
[9] SCHREFLER B A，SECCHI S，SIMONI L. On adaptive refinement techniques in multi-field problems including cohesive fracture[J]. Computer Methods in Applied Mechanics and Engineering，2006，195(4)：444–461.
[10] SECCHI S，SCHREFLER B A. A method for 3-D hydraulic fracturing simulation[J]. International Journal of Fracture，2012，178(1/2)：245–258.
[11] RÉTHORÉ J，BORST R D，ABELLAN M A. A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks[J]. Computational Mechanics，2008，42(2)：227–238.
[12] IRZAL F，REMMERS J J C，HUYGHE J M，et al. A large deformation formulation for fluid flow in a progressively fracturing porous material[J]. Computer Methods in Applied Mechanics and Engineering，2013，256：29–37.
[13] GRASSL P，FAHY C，GALLIPOLI D，et al. On a 2D hydro-mechanical lattice approach for modelling hydraulic fracture[J]. Journal of the Mechanics and Physics of Solids，2015，75：104–118.
[14] 严成增，郑 宏，孙冠华，等. 模拟水压致裂的二维FDEM-flow方法[J]. 岩石力学与工程学报，2015，34(1)：67–75.(YAN Chengzeng，ZHENG Hong，SUN Guanhua，et al. A 2D FDEM-flow method for simulating hydraulic fracturing[J]. Chinese Journal of Rock Mechanics and Engineering，2015，34(1)：67–75.(in Chinese))
[15] MIKELIC A，WHEELER M F，WICK T. Phase-field modeling of a fluid-driven fracture in a poroelastic medium[J]. Computational Geosciences，2015，19(6)：1 171–1 195.
[16] 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]. Journal of the Mechanics and Physics of Solids，2015，82：186–217.
[17] VERRUIJT A. An introduction to soil dynamics[M]. Dordrecht：Springer Science and Business Media，2009：91–92.
[18] LEWIS R W，SCHREFLER B A. The finite element method in the static and dynamic deformation and consolidation of porous media[M]. New York：John Wiley，1998：453–459.
[19] FRANCFORT G A，MARIGO J J. Revisiting brittle fracture as an energy minimization problem[J]. Journal of the Mechanics and Physics of Solids，1998，46(8)：1 319–1 342.
[20] MIEHE C，WELSCHINGER F，HOFACKER M. Thermodynamically consistent phase-field models of fracture：Variational principles and multi-field FE implementations[J]. International Journal for Numerical Methods in Engineering，2010，83(10)：1 273–1 311.
[21] BORDEN M J，VERHOOSEL C V，SCOTT M A，et al.. A phase-field description of dynamic brittle fracture[J]. Computer Methods in Applied Mechanics and Engineering，2012，217：77–95.
[22] 刘国威，李庆斌，左 正. 相场断裂模型分步算法在ABAQUS中的实现[J]. 岩石力学与工程学报，2016，35(5)：1 019–1 030.(LIU Guowei，LI Qingbin，ZUO Zheng. Implementation of a staggered algorithm for a phase field model in ABAQUS[J]. Chinese Journal of Rock Mechanics and Engineering，2016，35(5)：1 019–1 030.(in Chinese))
[23] LIU G W，LI Q B，MSEKH M A，et al. Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field model[J]. Computational Materials Science，2016，121：35–47.
[24] HAMANN T，QIU G，GRABE J. Application of a coupled Eulerian–Lagrangian approach on pile installation problems under partially drained conditions[J]. Computers and Geotechnics，2015，63：279–290.
[25] ABAQUS. ABAQUS theory manual and user?s manual[K]. Providence，RI，USA：Dassault Systemes Simulia Corp.，2012.
[26] SCHANZ I M，CHENG A D. Transient wave propagation in a one-dimensional poroelastic column[J]. Acta Mechanica，2000，145(1–4)：1–18.
[27] CHUKWUDOZIE C，BOURDIN B，YOSHIOKA K. A variational approach to the modeling and numerical simulation of hydraulic fracturing under in-situ stresses[C]// Thirty-eighth Workshop on Geothermal Reservoir Engineering. Stanford：Stanford University Press，2013：11–13.
[28] 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)：5 811–5 835.
[29] DETOURNAY E，GARAGASH D I. The near-tip region of a fluid-driven fracture propagating in a permeable elastic solid[J]. Journal of Fluid Mechanics，2003，494：1–32.
[30] MIKELIC A，WHEELER M F，WICK T. A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium[J]. Multiscale Modeling and Simulation，2015，13(1)： 367–398.
[31] WHEELER M F，WICK T，WOLLNER W. An augmented-Lagrangian method for the phase-field approach for pressurized fractures[J]. Computer Methods in Applied Mechanics and Engineering，2014，271：69–85.
2017, Vol. 36 
