## 3 Methodology
### Calculating the Roughness Correction Coefficient--The Formula Method
\begin{table}
\begin{tabular}{l l l} \hline
**Author/Year** & **Formula** & **Description** \\ \hline [PERSON], 1951 [20] & \(e^{2}=\frac{a^{2}}{C},C=1+\text{m}\left(\frac{\Delta}{2}\right)^{1.5},\text{m}=17\) & a is the magnitude of the discontinuity \\ [PERSON], 1969 [21] & \(e^{2}=\frac{a^{2}}{C},C=1+\text{m}\left(\frac{\Delta}{2}\right)^{1.5},\text{m}=8.\) & \\ [PERSON], 1982 [20] & \(e^{2}=\frac{a^{2}}{C},C=1+\text{m}\left(\frac{\Delta}{2}\right)^{1.5},\text{m}=2 0.5\) & \\ [PERSON], 1978 [22] & \(e=\text{a}\left[1-0.9\text{exp}\left(-0.56\frac{a}{\sigma_{\text{E}}}\right) \right]^{\frac{1}{3}}\) & \(\sigma_{\text{E}}\) is the standard deviation of the varying aperture over the mechanical aperture. \\ [PERSON], 1981 [23] & \(e^{3}=a^{3}\frac{1-C}{1+C}\) & C = 0.25 in the study by [PERSON] et al. (1996) [24]. \\ [PERSON], 1978 [5] & \(e=\frac{a^{2}}{JRC^{2}C}\left(e^{2}=\frac{a^{2}}{C}\text{and}\ C=\frac{|RC^{ 5}}{e^{2}}\right)\) & \\ [PERSON], 1995 [25] & \(e^{2}=\frac{a^{2}}{C},C=1.1\sim 1.7\) & a = 100–500 \(\mu\)m. \\ [PERSON], 1995 [26] & \(e=\text{a}\left[1+\frac{\sigma_{\text{E}}^{2}}{a^{2}}\right]^{-\frac{1}{2}}\) & C is the contact ratio. \\ [PERSON], 1996 [24] & \(e^{3}\approx a^{3}\left[1-1.5\frac{\sigma_{\text{gen}}^{2}}{a^{3}}+\ldots \right](1-2c)\) & \\ [PERSON] et al., 1999 [27] & \(e=\text{a}\left\langle a\right\rangle/\tau^{1/3}\) & \(\sigma_{\text{apert}}\) is the standard deviation of the mean \\ [PERSON] et al., 1999 [28] & \(e^{3}=a^{3}\left[1-\frac{1.13}{1+0.191\left(2a/\sigma_{\text{gen}}\right)^{1. 33}}\right]\) & \\ [PERSON], 2001 [29] & \(e=\frac{a^{2}}{JRC^{2}C}\left(\text{u}_{\text{s}}\leq 0.75\text{u}_{\text{sp}}\right)\) & \\ [PERSON] et al., 2011 [30] & \(e=\text{a}^{1/2}JRC_{\text{mob}}\left(\text{u}_{\text{s}}\geq\text{u}_{\text{ sp}}\right)\) & \(\text{not}\) \\  & \(e^{3}=a^{3}\left[1-1.0\frac{\sigma_{\text{gen}}}{a}\right]\left(1-\frac{ \sigma_{\text{gen}}}{a}\frac{\sqrt{0.4\text{exp}}}{10}\sqrt{\text{Re}}\right)\) & \(\sigma_{\text{slope}}\) is the standard deviation of the local slope of the fracture surface. \\ [PERSON] and [PERSON], 2011 [31] & \(e=\text{a}\left[\left(1-0.034\text{m}_{\text{c}}^{-0.565}\right)^{\text{RC}_{ \text{a}}}\right]^{\frac{1}{3}}\) or & \(JRC_{a}\) is the average of the joint roughness coefficients for the upper and lower rock \\  & \(e=\text{a}\left(1-2.25\frac{\sigma_{\text{E}}}{a}\right)^{\frac{1}{3}}\) & fracture profiles. \\ [PERSON] and [PERSON], 2013 [15] & \(e=\frac{a}{1+2\zeta_{2}^{2}}\), Re \(<1\) & \(Z_{2}\) is the root mean square of the first \\  & \(e=\frac{a}{1+2\zeta_{2}^{2}}+\left(0.0006+0.004Z_{2}^{2.5}\right)\left(\text{Re }-1\right)\), Re \(\geqq 1\) & deviation of the profile. \\ [PERSON] et al., 2015 [32] & \(e^{3}=a^{3}\left[0.94-5.0\frac{\sigma_{\text{E}}^{2}}{E}\right]\) & \(\sigma_{\text{s}}\) is the standard deviation of the mechanical aperture during shear. \\ [PERSON] et al., 2019 [33] & \(e^{2}=\frac{a^{2}}{C},C=1+\text{m}\left(\frac{\Delta}{2}\right)^{0.61912}, \text{m}=0.6138\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Representative empirical relations that mathematically link the hydraulic aperture and the mechanical aperture. greater variance represents a rougher surface [19]. Since the value \(\sigma_{h}^{\Delta}\) is the statistical value within the range for the crack, it is more representative (hereinafter, \(\sigma_{h}^{\Delta}\) is referred to as \(\sigma_{\mathrm{s}}\)). [PERSON] [24], [PERSON] [30], and [PERSON] [28] also proposed corrected formulas for the cubic law, and all three of them discussed the relationship between the hydraulic gap width (\(e\), in mm units) and the mechanical gap width (\(a\), in mm units) by analyzing and studying the local cubic law, with each suggesting a formula suited to their respective studies. Of these, the final relationships determined by [PERSON] and [PERSON] only differ in the coefficient of the \((\sigma_{\mathrm{s}}^{2}/a^{2})\) term. To observe the difference, both are calculated together. The formula proposed by [PERSON] et al.