## 4 Discussion
### Determination Method of \(T_{2c}\) Based on Fractal Analysis  
Before using the TC model to predict the permeability of mixtures, it is necessary to determine \(T_{2c}\). In this study, the \(T_{2c}\) was determined by an evaporation test, which is time-consuming. A simple method is thus needed to determine the \(T_{2c}\). According to previous studies, fractal dimensions can be used to determine the \(T_{2c}\) because fractal dimensions can reflect the characteristics of the pore distribution in mixtures. The fractal dimensions were determined based on the fractal theory presented in Section 2.1.2. According to Equation (10), there is a liner relationship between \(lg[1-\varphi+V(\leq r)]\) and \(\lg T_{2}\). The slope of the curve of \(lg[1-\varphi+V(\leq r)]\) versus \(lgT_{2}\) was obtained by linear regression. The linearity of the points at the beginning and end of the curves was relatively poor because of the scale dependency of the fractal dimension. The fractal dimensions calculated with different length scales are different. To achieve accurate fractal dimensions, we adopted a unified standard for the eight mixtures: maximize the length scale under the condition of the correlation coefficient reaching 0.99. Available length scales (red points in Figure 13) of K40, K60, K80, and K100 were 4.199-25.53, 3.409-19.34, 3.181-16.83, and 2.967-14.65 ms, respectively. Available length scales of M40, M60, M80, and M100 were 0.370-5.543, 0.280-3.917, 0.227-3.181, and 0.185-2.768 ms, respectively. After the slopes of the curves were determined, the fractal dimensions were determined by Equation (11). The fractal dimensions of the eight mixtures are shown in Figure 13.  
Figure 12: The predicted permeability coefficients versus the measured permeability coefficients. Red solid line indicates a slope of 1:1. The closer the black point is to the line, the smaller the difference between the predicted and measured permeability coefficient is. Figure 13: _Cont._As shown in Figure 13, the fractal dimensions of mixtures with the same kind of clay mineral decreased as the clay content increased. According to fractal theory [45, 48], the more uniform the pore distribution of the mixture is, the smaller the fractal dimension of the mixture is. This indicates that the uniformity of the pore distribution increases as the clay content increases. For mixtures with the same clay content, the fractal dimensions of M-Q mixtures were larger than those of K-Q mixtures. Figure 13: The fractal dimensions (\(D\)) of the eight mixtures: (**a**) K40; (**b**) K60; (**c**) K80; (**d**) K100; (**e**) M40; (**f**) M60; (**g**) M80; (**h**) M100. The red solid line is the fitting line of the red circular point. The Equation in the box is the Equation of the line. \(R^{2}\) indicates the correlation coefficient of the Equation. \(D\) was calculated by Equation (11). mixtures. The reason why the M-Q mixture had larger fractal dimensions is that the uniformity of the pore distribution was worse. However, the microscopic mechanism of the difference among fractal dimensions of mixtures with different mineral compositions was not clear. The uniformity of pore distribution was determined by the pore size. The free water distributed in large pore and bound water distributed in small pore. There was a relationship between fractal dimension and bound water or free water. So, the fractal dimensions could be used to predict \(T_{2c}\) in this study. The microscopic mechanism of the close relationship between fractal dimension and bound water or free water was also not clear. The two microscopic mechanisms will be explored in future work. Figure 14 shows the relationships between \(T_{2c}\) and fractal dimensions. The relationships between \(T_{2c}\) and fractal dimensions of M-Q mixtures and K-Q mixtures are different, but \(T_{2c}\) increases with an increase in the fractal dimension.