## 2 Laboratory Tests on Nonlinear Flow of Fractures  
### Sample Preparation  
In this study, fractures in granite were taken to investigate nonlinear flow characteristics; the samples were cored with a diameter of 50 mm and length of 70 mm, due to the limitation of the apparatus.  
To obtain a fractured granite sample, as shown in Figure 1a, firstly, an intact sample of the size suitable for the equipment was prepared; the sample was then placed on the mold, and it was ensured that the burins on the top and bottom were in contact with the sample. Then, by applying the vertical loading after fixing the sample at the central position of the mold via the screws on both sides, it was preloaded. Next, the screws were released to allow the sample to be in an unconfined state. Finally, vertical loading continued to be applied until the sample was split into two halves, as shown in Figure 1b. The planarity after splitting was different because of the different internal mineral distribution, which resulted in different roughnesses of the fracture surface. The fractured sample is shown in Figure 2, and the three samples are named G1, G2, and G3, respectively.  
Figure 1: (a) Preparation of specimens. (b) Two halves of the specimen.  
Through three-dimensional scanning of these fracture surfaces, a large number of point sets of three-dimensional coordinates can be obtained. Based on these point sets, a visual image of the fracture surface can be obtained (Figure 3), and the joint roughness coefficient (JRC) of the fracture surface can be calculated according to the following formulas [59; 60]:  
\[Z_{2}=\left[\frac{1}{\left(n-1\right)\left(\Delta x\right)^{2}}\sum_{i=1}^{n-1 }\left(Z_{i+1}-Z_{i}\right)^{2}\right]^{1/2} \tag{1}\]  
\[\text{JRC}=32.69+32.98\times\text{lg}Z_{2} \tag{2}\]  
where \(Z_{2}\) is the root mean square slope of the profiles based on the extracted data, \(Z_{i}\) represents the coordinates of the fracture surface profile, n is the number of data points, and \(\Delta x\) is the interval between the data points. Specifically, the mean JRC values of Samples G1, G2, G3 were 13.01, 13.50, and 13.97, respectively.  
Figure 3: _Cont._  
Figure 2: Photographs of fractured sample.  
The test schemes are shown in Table 1. Each sample was used to conduct two tests: unfilled and filled with quartz sand for the consideration of the real state of fractures in practical engineering [61; 62]. For the filled cases, the diameters of the quartz sand were 50 mesh, 100 mesh, and 200 mesh, respectively. In order to ensure that the same filling quality was used under each operating condition, 0.3 g of each filling material was weighed accurately. Then, the filling material was evenly placed on the fracture surface by hand or using a small brush. The sample was weighed pre- and post- filling to ensure that no filling material was lost during the process and that the filling amount was the same under each condition. Care was taken to ensure that the fillings were laid evenly on the fracture surface. The sample was wrapped with adhesive tape to prevent the leakage of quartz sand during the placement process of the samples, as shown in Figure 4. During the experiments, permeable stones were placed at both ends of the sample to prevent the quartz sand from being washed away.  
Figure 3: Three dimensional scanning of fracture surfaces: (**a**) G1 (JRC = 13.01), (**b**) G2 (JRC = 13.50), (**c**) G3 (JRC = 13.97).