## 2 Simulation Methodology
### Fracture Initiation and Propagation Criteria  
Various criteria have been proposed to predict fracture initiation and propagation [39; 40; 41; 42], mainly based on linear elastic fracture mechanics (LEFM) theory and damage mechanics. The LEFM method uses characteristic parameters such as the stress intensity factor and energy release rate as criteria for fracture propagation judgement. The LEFM is unable to predict the fracture initiation and is typically used to predict the propagation of cracks that have been opened in the material. Another major method of crack prediction is based on damage mechanics, in which the microdefects inside the material are considered. By selecting appropriate damage variables, the damage evolution equation of the damage-containing materials is derived using continuum mechanics [43; 44; 45].  
In this paper, a linear Traction-Separation Law based on damage mechanics is used to judge the crack propagation behavior. It assumes that the failure of the cohesive elements is characterized by the progressive degradation of the material stiffness, which is driven by a damage process. The failure mechanism consists of two ingredients: a damage initiation criterion and a damage evolution law.  
The stress and strain follow the linear elastic relationship before the damage occurs in a cohesive element. Traction \(\mathbf{t}=\{t_{n},t_{s},t_{t}\}\), where \(t_{n}\) is the traction perpendicular to the possible fracture surface, and \(t_{s}\), \(t_{t}\) are two tangential tractions perpendicular to each other on possible fracture surfaces. \(t_{n}^{0}\), \(t_{s}^{0}\), and \(t_{t}^{0}\) represent the limit values of nominal stress, respectively. Once any component of the traction reaches the limit value, the cohesive element begins to be damaged, and the fracture occurs. This damage initiation criteria can be expressed as [46]:  
\[\max\left\{\frac{\langle t_{n}\rangle}{t_{n}^{0}},\frac{t_{s}}{t_{s}^{0}},\frac{t_ {t}}{t_{t}^{0}}\right\}=1 \tag{8}\]  
where the symbol \(\langle\ \rangle\) denotes the Macaulay bracket and signifies that a pure compressive deformation or stress state does not initiate damage.  
With the injection of fracturing fluid, the traction increases as the fracture surface separates from each other. With the increase in separation degree, due to the degradation (or softening behavior) of the stiffness of the material, the traction decreases [47] until the fracture tip is completely destroyed, and the tractive strength of the fracture surface drops to zero. The damage evolution law can be characterized by a bilinear cohesion traction-separation relationship [48], shown in Figure 2, which can be expressed as follows:  
\[t=\left\{\begin{array}{cc}K_{0}\Delta\,&0\leq\Delta\leq\Delta^{0}\\ (1-D)K_{0}\Delta,&\Delta^{0}\leq\Delta\leq\Delta^{max}\\ 0,&\Delta^{max}\leq\Delta\end{array}\right. \tag{9}\]  
where \(K_{0}\) is the initial stiffness of the cohesive element, \(\Delta^{0}\) is the critical displacement at which the fracture tip begins to damage, \(\Delta^{max}\) is the maximum displacement obtained during the loading process, and the damage index D is used to describe the damage degree of the cohesive element.