## 3 Methodology
### The Roughness Correction Coefficient--The Numerical Method
In contrast, the roughness correction coefficient for tension samples S4 and S5 exhibits significant variations with size when using the other formulas, except for a minor change observed in the curve when using Eq-Zh. Eventually, the coefficient gradually stabilizes. The suggested coefficient being less than 1 results in the opposite trend in the variation in the curve when calculating with Eq-Zh compared to the other formulas. \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 & 1.0598 & 1.0376 & 1.0360 & 1.0291 & 1.0231 & 1.0189 & 1.0141 & 1.0132 & 1.0129 & 1.0126 \\ Eq-Zh & 1.7806 & 2.3314 & 2.9868 & 3.4119 & 3.8425 & 4.2180 & 4.0435 & 4.3534 & 4.6486 & 4.8899 \\ Eq-X & 2.3916 & 1.3681 & 1.3673 & 1.3575 & 1.3238 & 1.3018 & 1.2979 & 1.3043 & 1.3203 & 1.3562 \\ Eq-Zi & 9.8563 & 1.7369 & 1.7371 & 1.7080 & 1.6216 & 1.5691 & 1.5587 & 1.5741 & 1.6139 & 1.7067 \\ Eq-M & 1.7474 & 1.3946 & 1.3936 & 1.3873 & 1.3626 & 1.3453 & 1.3425 & 1.3475 & 1.3597 & 1.3859 \\ Num & 5.0726 & 1.8825 & 2.4237 & 2.1008 & 2.0180 & 1.8159 & 1.6998 & 1.7720 & 1.8208 & 1.9584 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of calculating roughness correction coefficient (S5). From the results of the calculations in Table 3 and Figure 5, we determine the following:  
1. The roughness correction coefficient obtained using the Eq-Zh calculations exhibits a significant overestimation, deviating greatly from the actual values, particularly for S2, attaining a maximum value of 30. On the other hand, the roughness correction coefficient determined using Eq-L is excessively uniform, limiting its ability to capture the variations in the roughness levels across fissure surfaces. Consequently, we will not discuss or analyze the results obtained using Eq-Zh and Eq-L further. 2. The calculations using the different types of formulas indicate that the roughness correction factor changes minimally when varying the fracture size for shear fractures (S1, S2, S3), whereas the roughness correction factor varies with fluctuations in the fracture size for tensile fractures (S4, S5), but the extent of the variation differs. For instance, the variation when using Eq-Zi is relatively significant, aligning with our actual understanding of the fracture roughness. 3. Irrespective of the type of rock, the roughness correction factors vary across varying sizes, predominantly ranging from 1.0 to 3.0, with the exception of a minimal number of outlier data points. Across these differences, S5 exhibits the highest degree of variation in the roughness correction factor compared to the others. 4. The results of the calculations using the Eq-X and Eq-M formulas exhibit the greatest stability across various types of rock. 5. The roughness correction coefficient calculated using these numerical methods varies significantly with sample size, with fewer fluctuations observed in the compressive-shear rock samples compared to the tensile rock samples. For the different calculation methods used, all of the samples except for S5 exhibit a noticeable trend of the roughness correction coefficient gradually stabilizing as the size increases, indicating that scale has an effect on roughness. In comparison, compressive-shear fractures tend to stabilize more readily in terms of their roughness, while larger sizes are required for tensile fractures to stabilize.