## 3 Results
### Young's Modulus and Brinell Hardness  
Figure 7 presents the stress-strain curves of all the unconfined compression tests performed on the cylindrical samples; it can be observed that the initial loading curves all  
Figure 6: Comparison of selected indentation load–displacement curves under different confining pressures: (**a–e**) \(\sigma_{1}=\sigma_{2}=0\)–20 MPa, respectively; (**f**) relation between the confining pressure and displacement.  
consist of a curved segment, which is mainly due to the microdefects (e.g., microfractures and micropores) in the shale gradually becoming closer under the action of loading, thus making the sample denser [34]. In the subsequent calculation of mechanical parameters, this part is not taken into account, and only the linear segment afterwards is considered. Based on the obtained results in Table 2, it can be inferred that the three samples exhibit a high degree of homogeneity. The average Young's modulus (\(E\)) for the samples is determined to be 40.04 GPa, while the average unconfined compressive strength (\(UCS\)) measures at 101.79 MPa. Additionally, Poisson's ratio (\(\
u\)) is calculated to be 0.25.  
The relationship between Young's modulus and confining pressure is demonstrated in Figure 8a. Without any confining pressure, the average Young's modulus for the shale sample was calculated as 39.50 GPa using the Oliver-Pharr method. Comparing this value with the macroscopic Young's modulus obtained from unconfined compression tests (i.e., 40.04 GPa), a minimal difference of merely 1.35% is observed. These findings suggest that the mechanical parameters derived from microindentation testing, conducted at a maximum load of 100 N, aptly represent the overall macroscopic mechanical characteristics of shale. With the escalation of confining pressure, the measured Young's modulus demonstrates an upward trend and exhibits a positive correlation. When subjected to a confining pressure of 20 MPa, the measured Young's modulus reaches 46.20 GPa, showing a notable increase of 16.96% compared to the modulus under no confining pressure. These findings highlight the substantial impact of confining pressure on altering the mechanical properties within shale reservoirs, particularly in deeper formations characterized by higher confining pressures. Thus, it can be inferred that the amplification of rock's Young's modulus due to confining pressure will be more pronounced in these deeper reservoirs.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shale Samples** & \(UCS\) (MPa) & \(E\) (GPa) & \(\
u\) \\ \hline
1\# sample & 101.38 & 40.03 & 0.25 \\
2\# sample & 99.38 & 39.98 & 0.25 \\
3\# sample & 104.60 & 40.10 & 0.25 \\ Average values & 101.79 & 40.04 & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mechanical parameters of the shale samples under unconfined compression tests.  
Figure 7: Stress–strain curves of the unconfined compression tests.  
The relationship between Brinell hardness and confining pressure is presented in Figure 8b. It becomes evident that the behavior of Brinell hardness follows a similar trend to that of Young's modulus, with an increase observed as the confining pressure rises. Without any confining pressure, the Brinell hardness measures at 56.08, while under a confining pressure of 20 MPa, it elevates to 81.58, signifying a growth of 45.47%. Overall, based on the experimental results, it can be inferred that in the actual in situ reservoir environment, the application of confining pressure enhances the strength and hardness of shale, resulting in a reduced depth for proppant embedding. This phenomenon proves beneficial for shale gas production and long-term stability.