基于黏聚型裂纹本构关系的煤岩水力压裂韧性破坏模型
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摘要
线弹性断裂力学作为一种十分成功的断裂理论框架,已被广泛地应用于表征固体材料中裂纹扩展行为。对于线弹性岩石断裂力学来说,岩石一般被简化为脆性材料,相对于裂纹尺寸及试件尺寸,其裂纹尖端前断裂过程区(Fracture process zone,FPZ)范围很小可以被忽略。而另一方面,煤的破坏形式通常表现为韧性破坏,即其应力峰值后存在明显的应变软化区。对于这种韧性材料,其断裂过程区尺寸范围相对较大且会对材料的断裂行为产生很大的影响,因此线弹性断裂理论不再适用于描述煤体中裂纹扩展。而黏聚型模型(Cohesive zone model,CZM)被证明是一种有效的理论工具,能够描述韧性材料断裂过程区中的断裂行为。在该黏聚型本构模型理论中,裂纹尖端前的断裂过程区被简化为一条闭合的裂纹或闭合的裂纹面(分别对应二维及三维情况),其中断裂过程区内非线性断裂行为通过黏聚力与相对位移之间的本构关系进行表征。通过对煤进行圆盘形紧凑拉伸试验建立了不同煤阶煤(其中包括弱黏煤、气煤、肥煤、贫瘦煤及无烟煤)的黏聚型裂纹本构关系,试验结果表明,随着煤试件煤阶的升高,其初始刚度及峰值载荷逐渐升高,最大张开位移逐渐降低,试验峰后软化阶段载荷-CTOD曲线趋于线性变化且破坏形式逐渐趋于脆性破坏。采用Karihaloo多项式黏聚型本构方程对5种煤阶煤软化曲线进行拟合,得到煤体中黏聚型裂纹模型本构关系的一般形式。针对煤层松软的力学特性和韧性破坏特征,建立了基于黏聚型裂纹本构关系的煤岩水力压裂多场耦合方程组,包括多孔介质变形方程、孔隙渗流方程、裂隙渗流方程及Karihaloo多项式本构关系方程。并采用包含裂隙流水压自由度的黏聚型界面单元法进行数值模拟。结合大型真三轴水力压裂实验,验证所得煤岩水力压裂模型的正确性;根据数值模拟和物理实验结果,讨论了煤岩松软的力学特性及其裂纹尖端过程区对水力压裂的影响。
Linear elastic fracture mechanics(LEFM) has become an enormously successful theory framework in characterizing the crack propagation in a wide range of solid materials.For linear elastic rock fracture mechanics,rocks is generally simplified to brittle materials,in which the fracture process zone(FPZ),i.e.the region ahead of the crack tip where micro-cracks initiate and coalesce,is small enough to be ignored compared to the size of the crack and the size of the specimen.On the other hand,coals usually exhibit quasi-brittle failure behaviors,char-acterized by a strain softening regime after the peak stress.For the ductile material,the size of FPZ is considerably large and has strong impact on the fracture behavior,and thus the theory of LEFM is no longer suitable for characterizing the crack propagation in coals.The cohesive zone model(CZM) has proven a useful theoretical tool to describe the fracture behavior in the FPZ of ductile materials.In the theory of CZM,the FPZ in front of the real crack tip is lumped hypothetically into a discrete line or plane corresponding to two-dimensional or three-dimensional cases,respectively,and the nonlinear behavior occurrence in the FPZ is represented by a con-stitutive equation that relates the cohesive stresses to the displacement jump across this line or plane.The consti-tutive relationships of CZMs for the different rank coals,including weakly caking coals,gas coals,fat coals,meager-lean coals and anthracite,have been determined by the disk-shaped compact tension(DC(T)) tests.The results show that the initial stiffness and peak loads increase with the coal rank increase.As the coal rank increases,the critical crack separation displacement reduces,and meanwhile the shape of the postpeak softening curves tends to be linear and the failure mode becomes more brittle.To arrive at a general form of constitutive relationships of CZMs in coals,the Karihaloo's polynomial cohesion-separation law is applied to fit the softening curves of the five different rank coals.By considering the mechanical properties and the ductile failure of coals,the authors establish the multi-physics coupling equations for the hydraulic fracturing in coals,including the deformation equation of porous media,fluid flow equation in the pores,fluid flow equation in the fractures and Karihaloo's polynomial constitutive equation,and simulate it with cohesive interface elements. Combined with a large-scale hydraulic fracturing experiment,the obtained mathematical model of hydraulic fracturing in coals is verified,and the effect of coal properties and the fracture process zone on hydraulic fracturing is discussed.
Linear elastic fracture mechanics(LEFM) has become an enormously successful theory framework in characterizing the crack propagation in a wide range of solid materials.For linear elastic rock fracture mechanics,rocks is generally simplified to brittle materials,in which the fracture process zone(FPZ),i.e.the region ahead of the crack tip where micro-cracks initiate and coalesce,is small enough to be ignored compared to the size of the crack and the size of the specimen.On the other hand,coals usually exhibit quasi-brittle failure behaviors,char-acterized by a strain softening regime after the peak stress.For the ductile material,the size of FPZ is considerably large and has strong impact on the fracture behavior,and thus the theory of LEFM is no longer suitable for characterizing the crack propagation in coals.The cohesive zone model(CZM) has proven a useful theoretical tool to describe the fracture behavior in the FPZ of ductile materials.In the theory of CZM,the FPZ in front of the real crack tip is lumped hypothetically into a discrete line or plane corresponding to two-dimensional or three-dimensional cases,respectively,and the nonlinear behavior occurrence in the FPZ is represented by a con-stitutive equation that relates the cohesive stresses to the displacement jump across this line or plane.The consti-tutive relationships of CZMs for the different rank coals,including weakly caking coals,gas coals,fat coals,meager-lean coals and anthracite,have been determined by the disk-shaped compact tension(DC(T)) tests.The results show that the initial stiffness and peak loads increase with the coal rank increase.As the coal rank increases,the critical crack separation displacement reduces,and meanwhile the shape of the postpeak softening curves tends to be linear and the failure mode becomes more brittle.To arrive at a general form of constitutive relationships of CZMs in coals,the Karihaloo's polynomial cohesion-separation law is applied to fit the softening curves of the five different rank coals.By considering the mechanical properties and the ductile failure of coals,the authors establish the multi-physics coupling equations for the hydraulic fracturing in coals,including the deformation equation of porous media,fluid flow equation in the pores,fluid flow equation in the fractures and Karihaloo's polynomial constitutive equation,and simulate it with cohesive interface elements. Combined with a large-scale hydraulic fracturing experiment,the obtained mathematical model of hydraulic fracturing in coals is verified,and the effect of coal properties and the fracture process zone on hydraulic fracturing is discussed.
引文
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[14] CARRIER B,GRANET S.Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model[J].Engineering Fracture Mechanics,2012,79(Pt8):312-328.
[15] NGUYEN V P,LIAN H,RABCZUK T,et al.Modelling hydraulic fractures in porous media using flow cohesive interface elements[J].Engineering Geology,2017,225:68-82.
[16] SHEN W,ZHAO Y P.Combined effect of pressure and shear stress on penny-shaped fluid-driven cracks[J]. Journal of Applied Mechanics,2018,85(3):031003.
[17] SHEN W,ZHAO Y P.Quasi-static crack growth under symmetrical loads in hydraulic fracturing[J]. Journal of Applied Mechanics,2017,84(8):081009.
[18] WAGONER M P.Disk-shaped compact tension test for asphalt concrete fracture[J]. Experimental Mechanics,2005,45(3):270-277.
[19] SONG S H,WAGONER M P,PAULINO G H,et al.δ25 Crack opening displacement parameter in cohesive zone models:Experiments and simulations in asphalt concrete[J].Fatigue&Fracture of Engineering Materials&Structures,2010,31(10):850-856.
[20] KARIHALOO B L,XIAO Q Z.Asymptotic fields at the tip of a cohesive crack[J]. International Journal of Fracture,2008,150(1-2):55-74.
[21] KARIHALOO B L,XIAO Q Z. Asymptotic fields ahead of mixed mode frictional cohesive cracks[J]. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f1/4r Angewandte Mathematik und Mechanik,2010,90(9):710-720.
[2] NORDGREN R P.Propagation of a vertical hydraulic fracture[J].Society of Petroleum Engineers Journal,1972,12(4):306-314.
[3] PERKINS T K,KERN L R.Widths of hydraulic fractures[J].Journal of Petroleum Engineering,1961,13(9):937-949.
[4] GEERTSMA J,DE KLERK F. A rapid method of predicting width and extent of hydraulically induced fractures[J]. Journal of Petroleum Technology,1969,21(12):1571-1581.
[5] KHRISTIANOVIC S A,ZHELTOV Y P.Formation of vertical fractures by means of highly viscous liquid[A]. Proceedings of the fourth world petroleum congress[C].Rome,1955:579-86.
[6] ZHANG X,DETOURNAY E,JEFFREY R.Propagation of a pennyshaped hydraulic fracture parallel to the free-surface of an elastic half-space[J]. International Journal of Fracture,2002,115(2):125-158.
[7] SETTARI A,CLEARY M P.Development and testing of a pseudothree-dimensional model of hydraulic fracture geometry[J]. SPE Production Engineering,1986,1(6):449-466.
[8] CARTER B J,DESROCHES J,INGRAFFEA A R,et al.Simulating fully 3D hydraulic fracturing[J]. Modeling in Geomechanics,2000,200:525-557.
[9] 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.
[10]杨健锋,梁卫国,陈跃都,等.不同水损伤程度下泥岩断裂力学特性试验研究[J].岩石力学与工程学报,2017,36(10):95-104.YANG Jianfeng,LIANG Weiguo,CHEN Yuedu,et al. Experiment research on the fracturing characteristics of mudstone with different degrees of water damage[J].Chinese Journal of Rock Mechanics and Engineering,2017,36(10):95-104.
[11] YANG J,LIAN H,LIANG W,et al. Experimental investigation of the effects of supercritical carbon dioxide on fracture toughness of bituminous coals[J]. International Journal of Rock Mechanics and Mining Sciences,2018,107:233-242.
[12] HILLERBORG A,M MODER,PETERSSON P E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[J]. Cement&Concrete Research,1976,6(6):773-781.
[13] XU X,NEEDLEMAN A.Numerical simulation of fast crack growth in brittle solids[J].Journal of the Mechanics and Physics of Solids,1994,42(9):1397-1434.
[14] CARRIER B,GRANET S.Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model[J].Engineering Fracture Mechanics,2012,79(Pt8):312-328.
[15] NGUYEN V P,LIAN H,RABCZUK T,et al.Modelling hydraulic fractures in porous media using flow cohesive interface elements[J].Engineering Geology,2017,225:68-82.
[16] SHEN W,ZHAO Y P.Combined effect of pressure and shear stress on penny-shaped fluid-driven cracks[J]. Journal of Applied Mechanics,2018,85(3):031003.
[17] SHEN W,ZHAO Y P.Quasi-static crack growth under symmetrical loads in hydraulic fracturing[J]. Journal of Applied Mechanics,2017,84(8):081009.
[18] WAGONER M P.Disk-shaped compact tension test for asphalt concrete fracture[J]. Experimental Mechanics,2005,45(3):270-277.
[19] SONG S H,WAGONER M P,PAULINO G H,et al.δ25 Crack opening displacement parameter in cohesive zone models:Experiments and simulations in asphalt concrete[J].Fatigue&Fracture of Engineering Materials&Structures,2010,31(10):850-856.
[20] KARIHALOO B L,XIAO Q Z.Asymptotic fields at the tip of a cohesive crack[J]. International Journal of Fracture,2008,150(1-2):55-74.
[21] KARIHALOO B L,XIAO Q Z. Asymptotic fields ahead of mixed mode frictional cohesive cracks[J]. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f1/4r Angewandte Mathematik und Mechanik,2010,90(9):710-720.