## 3 Methodology
### The Roughness Correction Coefficient--The Numerical Method  
To calculate the roughness correction coefficient for each set of roughness data, a fracture flow model was developed using a flow field simulation. Normal temperature and pressure conditions were set in the model, the fluid material was water, the permeability model applied involved the cubic law and a steady state, and the formula for the roughness correction coefficient was consistent with Equation (8). The pore size was calculated using the corresponding imported slit width data. To simplify the numerical calculations, the intercepted data on the fractured surfaces and their gap widths were rotated by the same angle in the opposite direction according to the angle of rotation of the intercept box to ensure that the limit of the data range was parallel with the axis of the coordinates. The water flow was determined in the x direction, and the initial head was set to 0. The data for each group, including data on the fractured surfaces and the corresponding gap widths, were imported into the established model to calculate the roughness correction coefficient. Figure 4 is a schematic diagram of the model flow and the flow field. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Author/Year** & **Fieldwork Symbols** & **Formula** & **Description** \\ \hline [PERSON], 1969 [21] & Eq-L & \(\frac{e}{\mathbf{a}}=\mathbf{C}\) & \(C=1+\mathrm{m}\left(\frac{\Delta}{2\Delta}\right)^{1.5}\), _and_\(\mathrm{m}=8.8\). \\ [PERSON] et al., 1996 [24] & Eq-Zi & \(\frac{e}{\mathbf{a}}=\sqrt[]{\mathbf{1}-\mathbf{1}.5}\;\sigma_{\mathrm{s}}^{ 2}/a^{2}\) & \(\sigma_{\mathrm{s}}\) equals 0, and the ratio of e/a \\ [PERSON] et al., 1999 [28] & Eq-M & \(\frac{e}{\mathbf{a}}=\sqrt[]{\mathbf{1}-\mathbf{1}\mathbf{1}\mathbf{1}\mathbf{ 1}\mathbf{0}\mathbf{1}\mathbf{0}\mathbf{1}\left(2\left(\mathbf{a}/\sigma_{ \mathrm{s}}\right)\right)^{1.08}}\) & \(\sigma_{\mathrm{s}0}\) is the standard deviation of the initial aperture value. \\ [PERSON] et al., 2011 [30] & Eq-X & \(\frac{e}{\mathbf{a}}=\sqrt[]{\mathbf{1}-\mathbf{1}.0\sigma_{\mathrm{s}}^{2}/a^ {2}}\) & \(\sigma_{\mathrm{s}0}\) is the standard deviation of the initial aperture value. \\ [PERSON] et al., 2019 [19] & Eq-Zh & \(\frac{e}{\mathbf{a}}=\mathbf{C}\) & \(C=1+\mathrm{m}\left(\frac{\Delta}{2\Delta}\right)^{0.61912}\), and \(\mathrm{m}=0.6138\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Formulas for the relationship between fractured width and roughness. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Formula**} & \multicolumn{8}{c}{**Truncated Frame Size (Percent)**} \\  & **10\%** & **20\%** & **30\%** & **40\%** & **50\%** & **60\%** & **70\%** & **80\%** & **90\%** & **100\%** \\ \hline Eq-L & 1.0816 & 1.0582 & 1.0118 & 1.0099 & 1.0089 & 1.0078 & 1.0073 & 1.0069 & 1.0067 & 1.0066 \\ Eq-Zh & 2.5210 & 2.9724 & 1.6342 & 1.7491 & 1.8106 & 1.7956 & 1.7666 & 1.7706 & 1.8230 & 1.9185 \\ Eq-X & 1.0112 & 1.0255 & 1.3840 & 1.5634 & 1.4650 & 1.4364 & 1.4604 & 1.4696 & 1.4296 & 1.3916 \\ Eq-Zi & 1.0169 & 1.0392 & 1.8956 & 2.4361 & 2.0307 & 1.9503 & 2.0221 & 2.0467 & 1.9156 & 1.8015 \\ Eq-M & 1.0198 & 1.0418 & 1.3862 & 1.5077 & 1.4563 & 1.4375 & 1.4527 & 1.4589 & 1.4351 & 1.4108 \\ Num & 1.0916 & 1.1460 & 1.4409 & 2.2475 & 2.0752 & 1.9182 & 2.0088 & 2.1018 & 2.0138 & 1.7916 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of calculating roughness correction coefficient (S4). Figure 5: The curves depicting the variations in the roughness correction coefficient with respect to size were obtained using different formulas (**(a)**: Eq-M; **(b)**: Eq-Zi; **(c)**: Eq-X; **(d)**: Eq-Zh; **(e)**: Eq-L; **(f)**: Num). The roughness correction coefficients for all the samples showed minimal variation when using Eq-L. With the exception of a pronounced change observed in the curve when using Eq-Zh, the roughness correction coefficient for shear-cut samples S1, S2, and S3 remains relatively stable when using different formulas with respect to size.