## 5 Prediction Model of Slope Flow Value and Test Time  
### Slope Flow Value and Time Prediction Model of Different Gradation Index  
Taking the gradation index (\(r\)) as an example, the relationship between the slope flow value and the test time was analyzed. From Section 3.2, its characteristics were: (1) Slope flow value decreased with the increase in gradation index; (2) the growth trend of the curve was that the slope flow value rose rapidly in the first few hours of the test, and the slope flow value tended to be stable with the increase in the test time.  
According to the variation trend of the relationship between the slope flow value and the test time, it conformed to the variation law of the exponential function BoxLucas1 model [23]. Therefore, the model was used to describe the variation of the slope flow value (\(U\)) with time (\(T\)). The curve model is as follows:  
\[U_{r}=a[1-exp(-bT)] \tag{6}\]  
In the formula: \(U_{r}\) is the slope flow value, mm; \(T\) is time, h; \(a\), \(b\) are material parameters.  
The fitting results of different gradation indexes are shown in Table 1, and the correlation coefficient R\({}_{2}\) of each fitting curve is greater than 0.9.  
It can be seen from Table 7 that the variation range of \(b\) value was small and there was no obvious rule. Therefore, the \(b\) value can be replaced by the mean value \(b\) in the calculation. The linear regression of the gradation index and the value \(a\) shows that \(a\) had a linear relationship with \(r\) when the bitumen aggregate ratio and the filler content were constant, as shown in Figure 8.  
After fitting, it was found that the function relationship between \(a\) value and gradation index (\(r\)) was as follows:  
\[a=\alpha+\beta r \tag{7}\]  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Grading Index (\(r\))**} & \multicolumn{2}{c}{**Fitting Parameters**} & \multirow{2}{*}{**Error Analysis R\({}^{2}\)**} \\ \cline{2-3}  & & \(a\) & \(b\) \\ \hline
0.33 & 1.35 & 0.26 & 0.968 \\
0.36 & 0.99 & 0.19 & 0.973 \\
0.39 & 0.77 & 0.28 & 0.989 \\
0.42 & 0.50 & 0.32 & 0.990 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Different gradation index fitting results.  
Figure 8: Fitting parameter \(a\) and gradation index \(r\) relationship curve. The red line is the fitting curve.  
In the formula: \(\alpha\) and \(\beta\) are fitting parameters.  
By substituting Formula (7) into (6), we obtain:  
\[U_{r}=(\alpha+\beta r)[1-exp(-bT)] \tag{8}\]  
Formula (8) is the BoxLucas1 correction model, and the calculation results were in good agreement with the experimental data. This formula can be used as a correction model for the prediction model of slope flow value under different gradation indexes. The model had three parameters, and the calculated results were \(\alpha=4.365\), \(\beta=-0.923\), \(b=0.2625\).  
According to the calculation parameters, and the calculation model of Equation (8), the prediction model of slope flow value with different gradation index was obtained and fitted with the test data, as shown in Figure 9. It was found that the calculation results of this formula were in good agreement with the experimental data, and the variation law of slope flow value and test time can be seen. With the increase in test time, the slope flow value first increased rapidly and then, the upward trend slowed down. As the test continued, the slope flow value remained stable.