## 2 Testing Conditions and Scheme
### Test Scheme  
To investigate the deformation and failure behaviors of sandstone at different burial depths and assess the impact of intermediate principal stress on sandstone failure under actual triaxial stress conditions, triaxial compression tests were conducted on sandstone under varying confining pressures. The simulated depth gradients for this test were 600 m, 1000 m, and 1400 m. Based on initial in situ stress measurement results from various countries [20], an empirical formula for calculating the axial stress of deep sandstone specimens was derived.  
\[\sigma_{lmax}=\sigma_{1}=6.7+0.0444H \tag{1}\]  
\[\sigma_{lmin}=\sigma_{2}=0.8+0.0329H \tag{2}\]  
\[\sigma_{v}=\sigma_{3}=0.027H \tag{3}\]  
The corresponding vertical stresses can be 16.2 MPa, 27 MPa, and 37.8 MPa, respectively. \(H\) represents the burial depth, with the unit being meters. According to the empirical formula, the ratio of vertical stress \(\sigma_{V}\) to the maximum stress \(\sigma_{lmax}\) and the minimum stress \(\sigma_{lmin}\) ranges from 1.2 to 2.0.  
Therefore, set \(\sigma_{2}/\sigma_{3}\) are 1.2, 1.4, 1.6, and 1.8, respectively. Each group has three test pieces. Specific values are shown in Table 1:  
Set the maximum principal stress \(\sigma_{1}\) along the Z direction, the intermediate principal stress \(\sigma_{2}\) along the Y direction, and the minimum principal stress \(\sigma_{3}\) along the X direction. The spatial stress diagram corresponding to the actual test is shown in Figure 4. The specific stress loading procedure is as follows:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Depth** & \(\sigma_{2}/\sigma_{3}\) & \(\sigma_{2}/\)**MPa** & \(\sigma_{3}/\)**MPa** \\ \hline \multirow{4}{*}{1} & \multirow{4}{*}{600 m} & 1.2 & 19.4 & \multirow{4}{*}{16.2} \\  & & 1.4 & 22.7 & \\  & & 1.6 & 25.9 & \\  & & 1.8 & 29.2 & \\ \hline \multirow{4}{*}{2} & \multirow{4}{*}{1000 m} & 1.2 & 32.4 & \multirow{4}{*}{27.0} \\  & & 1.4 & 37.8 & \\  & & 1.6 & 43.2 & \\  & & 1.8 & 48.6 & \\ \hline \multirow{4}{*}{3} & \multirow{4}{*}{1400 m} & 1.2 & 45.4 & \multirow{4}{*}{37.8} \\  & & 1.4 & 52.9 & \\ \cline{1-1}  & & 1.6 & 60.5 & \\ \cline{1-1}  & & 1.8 & 68.0 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Experimental scheme.  
The test loading system applies a 1 MPa prestress at a loading rate of 0.5 kN/s in the \(Z\) direction in load control mode to ensure that the test piece will not be displaced during loading.  
1. Following step 1, apply hydrostatic pressure simultaneously at a loading rate of 0.5 kN/s in the X, Y, and Z directions until reaching the stress value \(\sigma\)3 in the X direction, then cease loading in the X direction.
2. Continue applying hydrostatic pressure in the Y and Z directions at the loading rate of 0.5 kN/s until the stress value reaches \(\sigma_{2}\), then stop loading in the Y direction.  
After the sandstone specimen reaches the initial in situ stress state in the X and Y directions, the system switches to load-average speed-increase control mode with a loading rate of 0.5 kN/s. Axial loading and acoustic emission detection are conducted simultaneously. An acoustic emission sensor is positioned on the side, and a sufficient amount of coupling agent is applied between the sensor and the rock sample surface to ensure full coupling and enhance the stability of data acquisition.