## 2 Numerical Simulation Calculation Model
### Benchmarking Example  
Hydraulic fractures usually produce complex fracture morphologies when expanding in shale reservoirs, while weak surfaces such as shale laminae and natural fractures present in the shale oil reservoirs can have an important impact on the hydraulic fracture expansion behavior and the final fracture distribution characteristics during fracture modification. In this paper, the expansion process of hydraulic fractures in a reservoir is simulated by using the method of globally embedded cohesive units. The modeling process requires prespecifying the distribution location and morphological characteristics of the laminar joints, so the corresponding preprocessing script files are written in the Python language, and different types of laminar joints are flexibly arranged by the idea of parametric modeling. The computational model is shown in Figure 6. The model size is 20 \(\times\) 20 m, and the injection point is located at the center of the model (i.e., (0,0)). Ten laminar surfaces are initially set, and each laminar surface is composed of several microlaminates with a length distribution of 2-4 m. The laminar joints are spaced at 1-2 m, and there are 37 laminar joints in the whole model. The laminated joints have no initial opening, the angle with the x-axis is 30\({}^{\circ}\), the x-direction is the minimum horizontal principal stress direction, and the y-direction is the vertical principal stress direction. Figure 5: Cohesive seepage node at the merged intersection location. The grid division is shown in Figure 7. The computation area is divided by a quadrilateral grid, the cell type is CPE4P, the cell size is approximately 0.2 m, and there are 59,612 nodes and 35,449 cells in the model, including 11,883 quadrilateral block cells and 23,566 cohesive cells. The cohesive cell type is COH2D4P. The outer boundary of the model is subjected to fixed displacement and impermeable boundary conditions, where the upper and lower boundaries constrain the displacement in the y-direction and the left and right boundaries constrain the displacement in the x-direction. To analyze the effects of different parameters on the hydraulic fracture extension behavior, sensitivity analysis of each parameter is needed. Therefore, a set of parameters needs to be selected first as the benchmark case, and the subsequent sensitivity analysis is  
Figure 6: Distribution of laminated joints and model forces. Figure 7: Finite element meshing. carried out by changing the corresponding parameters on this basis. The base parameters used in the benchmark case are shown in Table 1.  
In the benchmark case, the injection displacement is 0.002 m\({}^{2}\)/s, the injection duration is 20 s, the stress boundary condition is the effective stress, the difference between the overlying rock pressure and the minimum horizontal principal stress is 2 MPa, and the values in Table 1 are used for other parameters. Figure 8 shows the fracture pattern at the end of injection. The blue dashed line represents the initial laminar fracture, the green dashed line represents the expansion path of the hydraulic fracture in the reservoir matrix, and the red dashed line indicates the activated laminar fracture. After the hydraulic fracture starts to crack from the injection point, it expands along the direction perpendicular to the minimum horizontal principal stress. After encountering \begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameters** & **Numerical Value** \\ \hline
**Rocks** & \\ \hline Young’s modulus (GPa) & 14 \\ \hline Poisson’s ratio & 0.2 \\ \hline Density (kg/m\({}^{3}\)) & 2600 \\ \hline Tensile strength (MPa) & 6 \\ \hline Permeability coefficient (m/s) & \(1\times 10^{-7}\) \\ \hline Porosity & 0.1 \\ \hline
**Layer theory** & \\ \hline Tensile strength (MPa) & 2 \\ \hline Critical damage displacement (m) & 0.001 \\ \hline
**Fracturing fluids** & \\ \hline Injection rate (m\({}^{2}\)/s) & 0.002 \\ \hline Viscosity (Pa-s) & 0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model base input parameters.