Modelling ductile failure of coals in hydraulic fracturing based on the constitutive equations of cohesive cracks

Linear elastic fracture mechanics (LEFM) has become an enormously successful theory framework in char- acterizing 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 im- pact on the fracture behavior,and thus the theory of LEFM is no longer suitable for characterizing the crack propaga- tion 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 dis- placement 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 post-peak 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 de- formation 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 hy- draulic 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.