## 2 Test Materials and Program
### Sample Preparation  
The rock specimen was collected from a grout-reinforced service slope in Shaanxi Province. The rock is grayish in appearance, and the core was extracted from a nearby section of the intact rock mass. The mineral content of the sandstone sample was analyzed using a DX-2700 BH X-ray diffractometer (Dalian Instrument Co., Ltd., Dalian, China), and the results are presented in Table 1. According to the standard of the International Society for Rock Mechanics (ISRM) [21], the rock specimen was processed into a cylindrical specimen with a diameter of about 50 mm and a height of about 100 mm (L/D ratio of 2:1), and the wave velocity was measured. This L/D ratio was selected based on several considerations: it is commonly used in rock mechanics research and is considered effective in mitigating boundary effects caused by improper specimen dimensions. This ratio ensures a more uniform stress distribution within the specimen during testing, thereby enhancing the reliability and repeatability of the results. The selection of the 2:1 L/D ratio aligns with practices outlined in the ISRM Suggested Methods for Rock Characterization, Testing, and Monitoring [22] and ASTM D4543-19 Standard Practice for Preparing Rock Core Test Specimens [23], which, while not specifying an exact L/D ratio, support the rationale for using this dimension ratio in standard rock mechanics testing. Samples exhibiting significant discrepancies were discarded using an NM-4A Ultrasonic Pulse Velocity Tester (Cangzhou Lvely Group Co., Ltd., Cangzhou, China) to minimize the impact of sandstone variability on the test results. The samples were subsequently placed in an oven and dried at 105 \({}^{\circ}\)C for 48 h using a DZF-type vacuum drying oven (Zhengzhou Keda Machinery and Instrument Equipment Co., Ltd., Zhengzhou, China). After determining the dry mass, the samples were placed in a ZK-270-type vacuum saturation apparatus (Zhengzhou Soil Instrument Factory, Zhengzhou, China) for saturation. The saturated mass was obtained after weighing the samples following 24 h of saturation. The dry density (\(\rho_{d}\)), saturated density (\(\rho_{sa}\)), saturated moisture content (\(\omega_{st}\)), and porosity (\(n_{0}\)) of the rock specimens were calculated using Equations (1)-(4). Specifically, these parameters were computed as follows. The results are presented in Table 2.  
\[\rho_{d}=\frac{m_{s}}{A\cdot H} \tag{1}\]  
\[\rho_{sa}=\frac{m_{sa}}{A\cdot H} \tag{2}\]  
\[\omega_{st}=\frac{m_{sa}-m_{s}}{m_{s}}\times 100\% \tag{3}\]The preparation process of the fractured sandstone used in this experiment is depicted in Figure 1. A single fracture was created using a waterjet, as shown in Figure 1a. The fracture had a width of 3 mm and a length of 21 mm. To simulate the typical fracture distribution in nature, fracture inclination angles (\(\theta\), the angle between the fracture and the horizontal plane) of 15\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), and 75\({}^{\circ}\) were selected. Next, based on the formula proposed in [24], a theoretical fractal model was used to simulate the roughness of the joint profile. The fractal dimension of the joint was directly estimated from two statistical parameters: the average base length (\(l\)) and the average height (\(h\)) of the joint roughness. The mathematical expressions for the fractal dimension (\(D\)) and the joint roughness coefficient (\(JRC\)) are given by Equation (5) and Equation (6), respectively:  
\[D= log4/log\{2[1+cos(arctan(2h/l))]\} \tag{5}\] \[JRC= 85.2671(D-1)^{0.5679} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Quantiz** & **Calcite** & **Albite** & **Kaolin** & **Chlorite** \\ \hline
45.2 & 6.8 & 32.1 & 7.5 & 8.4 \\ \hline \end{tabular}
\end{table} 
Table 1: Mineral composition of sandstone (%). Figure 1: Flow chart for preparation of slurry-fractured sandstone.