## 2 Materials and Methods
### Roughness Evaluation  
To study the effect of the fault surface roughness on the slip stability, the surfaces of samples were roughened either with #1200 or #60 silicon carbide grit sandpapers. Thereby, two kinds of fault surfaces with different roughness are obtained. After the surface roughening was completed, the INNO-SCAN handheld 3D scanner was used to measure the surface data to obtain contour coordinates of the fault surface topography. The instrument has a scanning accuracy of 0.025 mm. In this study, the surface roughness was quantified to distinguish the fault surfaces roughened with different sandpapers. In 1973, [PERSON] first proposed the concept of the joint roughness coefficient (JRC), which is used to describe the roughness of joint surface quantitatively. Ten typical contour lines of JRC in the range of 0-20 were proposed [58; 59]. In order to better quantify the roughness of joint surfaces, the following statistical parameters were defined based on the coordinate information of the two-dimensional contour line of the joint surface: root mean square value (_RMS_), centerline mean value (_CLA_), first derivative root mean square (\(Z_{2}\)), second derivative root mean square (\(Z_{3}\)), direction characteristic parameter (\(Z_{4}\)), and structure function (_SF_), etc. In this study, two roughness parameters, the \(Z_{2}\) proposed by [PERSON] [60] in 1962 and the \(Z_{2s}\) proposed by [PERSON] [61] in 2000, were tested to assess the surface roughness of faults. According to the test results, the most suitable surface roughness quantification parameters for this study were selected. The applied formulas were:  
\[Z_{2}=\sqrt{\frac{1}{L}\int\limits_{x=0}^{x=L}\left(\frac{\mathrm{d}y}{ \mathrm{d}x}\right)^{2}}\mathrm{d}x=\sqrt{\frac{1}{L}\sum\limits_{i=1}^{n-1} \frac{\left(y_{i+1}-y_{i}\right)^{2}}{x_{i+1}-x_{i}}} \tag{2}\]  
where \(L\) is the horizontal length of the contour line, and \(y_{i}\) is the height of the contour line when the abscissa is equal to \(x_{i}\), and  
\[Z_{2s}=\left\{ \begin{array}{c}\frac{1}{\left(N_{x}-1\right)\left(N_{y}-1 \right)}\left[\frac{1}{\left(\Delta x\right)^{2}}\sum\limits_{i=1}^{N_{x}-1} \sum\limits_{j=1}^{N_{y}-1}\frac{\left(Z_{i+1,j+1}-Z_{i,j+1}\right)^{2}+\left( Z_{i+1,j}-Z_{i,j}\right)^{2}}{2}+\right]\\ \frac{1}{\left(\Delta y\right)^{2}}\sum\limits_{j=1}^{N_{y}-1}\sum\limits_{j =1}^{N_{y}-1}\frac{\left(Z_{i+1,j+1}-Z_{i+1,j}\right)^{2}+\left(Z_{i+1}-Z_{i,j}\right)^{2}}{2}\end{array} \right]\right\}^{\frac{1}{2}} \tag{3}\]  
where \(N_{x}\) was the number of points along the x-axis; \(N_{y}\) was the number of points along the y-axis; \(\Delta x\) and \(\Delta y\) were the sampling steps along the x and y axes; and \(Z_{i,j}\) was the height at the coordinates \(\left(x_{i},y_{i}\right)\). Both methods need to take points on the fault surface and then calculate the roughness parameters according to the coordinates of the points. The selection of the spacing between adjacent points is very important, and different point spacing may lead to different results. A test was carried out to select a suitable point spacing. As shown in Figure 4, the size of the fault surface is 100 mm \(\times\) 150 mm, and the transverse direction is the shearing direction. The main step is to take three sets of contour lines along the shearing direction on the same fault surface. Each set of contour lines contains more than 950 contour lines. In each group of contour lines, the interval between adjacent contour lines is 0.1 mm. In different contour line groups, the distances between adjacent contour lines are 0.03 mm, 0.03 mm, and 0.04 mm, respectively. The point spacing on the contour lines is 0.03 mm, 0.05 mm, 0.1 mm, 0.3 mm, 0.5 mm, 0.7 mm, and 1 mm. \(Z_{2}\) and \(Z_{2s}\) are calculated according to the coordinates of the point. Figure 5a shows the standard deviations of the roughness parameters at different positions on fault surfaces.