## 4 Numerical Results and Discussion
### Complexity of Hydraulic Fracture Network  
A complex fracture network is the key to obtaining considerable economic benefits. The formation of a complex network can enlarge the stimulated reservoir volume and cause the fluid in the matrix to permeate along the shortest distance, as well as reduce the driving pressure difference required for the oil and gas flow [50] Therefore, a quantitative assessment of the complexity of the fracture network is required. The total length of the main hydraulic fracture and the effective area of the fracture path in the model were extracted. Here, the effective area of the fracture path is defined as the total area of the closed area surrounded by the equivalent polyline of the hydraulic fracture, which can reflect the area influenced by different fracture shapes on a reservoir to a certain extent. In general, the effective area of reticulated fractures is larger than that of strip-type fractures, and the effective area of zigzag extended fractures is larger than that of unidirectional extended fractures. The main fracture length and the effective area of a fracture path under different in situ stresses are summarized in Table 3 and Figure 17. With the increase in horizontal in situ stress difference, the length of the main fracture first increased and then declined. The maximum occurred when the horizontal in situ stress difference was 6 MPa, where the in situ stress ratio was 1.12, and the longest main fracture was 53.1 m. This indicates that an excessive horizontal in situ stress difference will lead to a single shape and direction of a hydraulic fracture, which is not conducive to the complexity of a fracture network. In general, the effective area of the fracture path decreased with the increase in the horizontal in situ stress difference. Its value was higher when the horizontal in situ stress difference was at a low level (0-2 MPa). This may be attributed to the secondary fracture with a large angle to the direction of the main fracture. The result shows that the key to increasing the complexity of the hydraulic fracture network is to form secondary fractures with a larger angle to the main fracture to enlarge the network morphology. \begin{table}
\begin{tabular}{c c c c} \hline \(\sigma_{H}\)–\(\sigma_{h}\)(MPa) & \(\sigma_{H}\)/\(\sigma_{h}\) & \ \begin{tabular}{c} **Length of Main** \\ **Fractures (m)** \\ \end{tabular}  & \begin{tabular}{c} **Effective Area of** \\ **Fracture Path (m\({}^{2}\))** \\ \end{tabular}  \\ \hline
0 & 1.0 & 37.3 & 123.7 \\
1 & 1.02 & 43.1 & 126.4 \\
2 & 1.04 & 40.5 & 122.3 \\
3 & 1.06 & 45.4 & 92.6 \\
4 & 1.08 & 47.6 & 86.7 \\
5 & 1.10 & 48.7 & 62.4 \\
6 & 1.12 & 53.1 & 90.3 \\
7 & 1.14 & 48.4 & 47.2 \\
8 & 1 Figures 18 and 19 show the length of the main fracture and maximum aperture under different tensile strengths of rock and cementing strengths of natural fractures, respectively. The fracture parameters are listed in Table 4. It can be seen that with the increase in rock tensile strength, the length of the main fracture decreased, in general. When the tensile strength of rock increased from 3 to 9 MPa, the length of the main fracture decreased from 40.4 to 35.1 m. This is because fracturing rock needs more energy when the tensile strength is higher, and it is difficult for a fracture to propagate in rock. Thus, excessive rock tensile strength is not conducive to stimulation treatments. It is better to select reservoirs with appropriate rock tensile strength for fracturing operations when conducting a fracturing feasibility study, for better economic benefits. In addition, it was found that the maximum aperture of the hydraulic fracture increased as the length of the crack decreased. This may because a longer length results in additional leak-off and reduces the fluid pressure in fractures [15], which makes the fracture more narrow. Figure 17: Fracture length under different in situ horizontal stresses. Figure 18: Fracture length under different tensile strengths of rock.