## 2 Numerical Simulation Calculation Model
### Model Control Equations  
#### 2.1.1 Cohesive Pore Pressure Unit  
(1) Cohesive pore pressure unit introduction  
In this paper, the cohesive unit with pore pressure is used to simulate the initiation and extension of hydraulic fractures and the laminar flow and filtration of fluid into the matrix inside the fracture. The cohesive unit has two main functions:  
1. To simulate hydraulic fracture initiation and extension based on the traction-separation criterion;
2. To describe the tangential flow of the fluid within the fracture and the normal flow perpendicular to the wall. Figure 2 shows the distribution of the 12 nodes of the 3D hole pressure cohesive cell. Nodes 1, 2, 3, and 4 form the upper surface of the cell; nodes 5, 6, 7, and 8 form the lower surface of the cell; and nodes 9, 10, 11, and 12 are the pore pressure nodes. The stress-strain relationship of the cohesive cell before damage occurs is consistent with the following linear elastic deformation:  
\[t=\left\{ \begin{array}{c}t_{n}\\ t_{s}\\ t_{t}\end{array} \right\}=k\varepsilon= \begin{bmatrix}k_{mn}&k_{ns}&k_{nt}\\ k_{ns}&k_{ss}&k_{st}\\ k_{nt}&k_{st}&k_{tt}\end{bmatrix}  \tag{1}\]  
Figure 1: Research roadmap. Figure 2: Three\(-\)dimensional hole pressure cohesive cell. where \(t\) is the stress tensor of the cohesive cell and is dimensionless; \(t_{n}\) is the normal stress of the cell, and the units are MPa; \(t_{s}\) and \(t_{t}\) are the tangential stresses of the unit, and the units are MPa; \(k\) is the stiffness matrix of the cell and is dimensionless; and \(\varepsilon\) is the unit stress and is dimensionless. When a cohesive unit is used to simulate material damage, its geometric thickness is generally defaulted to 0 before the simulation, while the concept of intrinsic thickness is introduced to replace the actual thickness of the unit to avoid singularities in the calculation, which is generally set to 1. Therefore, the displacement and strain of the cohesive unit are numerically equal. Unit damage model  
The damage mechanism of the cohesive unit follows the traction strain criterion, mainly based on the stress to which the unit is subjected as the basis for unit damage and failure. Before the damage of the unit, the displacement of the unit should be proportional to the stress. When the stress reaches the ultimate strength of the material, the unit begins to damage, the stress that can be withstood with the increase in displacement gradually decreases to zero, and then the unit fails. Figure 3 shows a schematic diagram of the damage intrinsic model in the normal direction of the cohesive unit, and the other two tangential directions have the same damage mode mechanism. When the displacement normal to the unit is smaller than the initial damage displacement of the unit \(\delta_{0}\), with the increase in the displacement, the stress normal to the unit increases, and the unit at this stage shows the characteristics of linear elasticity and the specific characteristics controlled by the unit penalty stiffness \(k_{n}\). When the normal stress of the cohesive unit reaches the strength of the material \(T_{max}\), the unit begins to damage. With the increase in the normal displacement, the stress normal to the unit gradually decreases. In this stage, the cohesive unit shows the characteristics of softening. When the normal displacement reaches the displacement of the complete destruction of the unit \(\delta_{f}\), the unit bears the stress of 0, and the unit completely fails (in the figure, \(T\) is the current strain before the damage to the stiffness of the obtained tension, Pa; \(T_{max}\) is the material tensile strength, Pa; \(T\) is the actual tensile stress, Pa; D is the damage factor; and \(\delta_{f}\), \(\delta_{m}\), and \(\delta_{0}\) are the displacement of the completely destroyed unit, the maximum displacement during the loading process, and the displacement upon initial damage, respectively, m). Initial cracking guidelines of the unit  
Maximum stress criterion  
Figure 3: T–S damage criterion for cohesive cells.