## 3 Methodology
### Results of Calculating Directionality  
Because of the square frame applied to the data when using the formulas, we cannot judge directionality. [PERSON] (2019) proposed using an elliptic diagram for permeability to derive a tensor of the roughness correction coefficient. Analyzing the elliptic diagram provides a reliable estimate of the tensor. In addition, this method has been combined with other previous approaches to determine the directionality of the permeability coefficient in studies on seepage in fracture networks. To determine whether fractured rock masses can be represented by equivalent continuous media, [PERSON] et al. (2015) drew ellipses to assess the permeability coefficient. Similarly, [PERSON] et al. (2016) utilized equivalent elliptical curves for the permeability coefficient, drawn based on measurements of fractured openings and their intercept lengths, to assess whether the requisite conditions for using an equivalent continuum were satisfied. After taking osmotic measurements, [PERSON] (1962) and other researchers (2014) drew ellipsoids to determine the direction of penetration, represented by the maximum direction of the primary axis of the spheroid. [PERSON] (2016) also noted that the difference between the long and short axes of the elliptic curve and its smoothness are critical to determining directionality. This study was inspired by this previous research. Here, the roughness correction coefficient value was obtained using the numerical method, and elliptical graphs were analyzed for research and evaluation purposes. We surmised that being able to obtain a well-defined elliptical curve would suggest that the correction coefficient could be represented as a tensor. The results obtained by calculating S1-S5 using numerical methods were fitted using ellipses, as shown in Figure 6.  
The long and short axes of the ellipse indicate the maximum and minimum values of the roughness correction coefficient, respectively, and directionality is judged according to the size difference between the angle between the two axes and the horizontal axis [25]. The statistics measured show the angles between the short axis and the horizontal axis in Figure 6, and the results are shown in Table 8. In the table, we can see the following:  
1. The angle between the long axis, the short axis, and the horizontal line varies in each figure, resulting in an anisotropic roughness correction coefficient. 2. The roughness correction coefficient varies with direction for different types of rocks. 3. The variation in the roughness correction coefficient is indicative of the variation in the roughness of the rock samples. Additionally, the roughness will differ in different directions within the same sample, leading to variations in the flow velocity of fluid in different directions. The rotation angle can be expressed as a matrix for each angle considering the angle of rotation in the horizontal direction. It should be noted that \(\theta\) represents the rotation angle, and its corresponding matrix R is [9]:  
\[\text{R}= \begin{pmatrix}\text{cos}\theta&\text{sin}\theta\\ -\text{sin}\theta&\text{cos}\theta\end{pmatrix} .\]  
The rotation matrix from the ellipse method for permeability can be applied to solving the roughness correction coefficient [10; 12]. In the relation formula,  
\[\text{C}_{\text{an}}=\text{R}\,\text{C}_{\text{dig}}\text{R}^{\text{T}}\]  
Figure 6: Polar plots represent the elliptical fitting of the roughness correction coefficients for each sample. The results of elliptical fitting for the roughness correction coefficients for samples S1, S2, S3, S4, and S5 are (**a**–**e**), respectively. The roughness correction coefficient varies with direction, irrespective of whether it is a tensile or compressive-shear rock sample.