## 3 Results and Discussion
### Infiltration and Infiltration Rate
Taking the results of the C1 soil column as an example, the change process of the infiltration rate can be roughly divided into three stages. The first stage occurs within two hours of the initial stage of infiltration. The infiltration capacity of the soil under the initial conditions is greater than the rain intensity, causing all water to infiltrate the upper layer of soil column, and the infiltration rate rapidly decreases over time. At this time, the infiltration rate is equal to the rain intensity, and this stage is defined as the rainfall intensity control stage. The time point of water drainage at the bottom of the soil column is used as the endpoint of the second stage, and the duration of the second stage is 1-8 h. At this stage, the saturation range of the upper soil layer gradually increases, the infiltration capacity of the soil gradually decreases once the soil is near saturation, and the instantaneous infiltration rate decreases over time. This stage is defined as the soil infiltration capacity control stage, and in this stage, the infiltration rate of the soil column mainly depends on the properties and initial conditions of the soil. After the time point of water drainage at the bottom of the soil column, the third stage starts, and the soil column reaches the saturation state. At this time, the infiltration rate almost no longer changes, and the infiltration rate is equal to the saturation permeability, which is basically stable at 5.04 \(\times\) 10\({}^{-5}\) cm/s, so the permeability can be calculated in the stable infiltration stage. The infiltration rate of the sample with a dry density of \(\rho_{d}\) = 1.39 g/cm\({}^{3}\) is basically stable at 1.9 \(\times\) 10\({}^{-5}\) cm/s, and the saturated permeability is \(k_{s}\) = 1.9 \(\times\) 10\({}^{-5}\) cm/s. Similar to the conclusions of [PERSON], the rainfall infiltration process can be divided into three stages: rainfall intensity control, unsaturated infiltration and saturated infiltration [42]. The average infiltration rate is defined as the infiltration rate per unit area and per unit time. The change in the infiltration rate of the soil is divided into two stages. The infiltration rate is higher at the initial stage of infiltration, and the infiltration rate gradually stabilizes in the later stage of infiltration, which is numerically referred to as the stable infiltration rate. The infiltration rate in the stable stage is an important parameter for describing the  
Figure 10: Instantaneous infiltration rate. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Soil Column No.** & **Kostiakov Model** & **R\({}^{2}\)** & **Philip Model** & **R\({}^{2}\)** \\ \hline Instantaneous & C1 & I(t) = 0.00495t\({}^{-0.50578}\) & 0.9545 & I(t) = 6 \(\times\) 10\({}^{-4}\) + 0.0036t\({}^{-0.5}\) & 0.8721 \\ Infiltration rate & C2 & I(t) = 1.08 \(\times\) 10\({}^{-4}\)t\({}^{-0.0596}\) & 0.9576 & I(t) = 5 \(\times\) 10\({}^{-4}\) + 0.00068t\({}^{-0.5}\) & 0.9623 \\  & C3 & I(t) = 1.70 \(\times\) 10\({}^{-4}\)t\({}^{-0.1532}\) & 0.9831 & I(t) = 4.5 \(\times\) 10\({}^{-4}\) + 0.00040t\({}^{-0.5}\) & 0.9473 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fitting parameters of the infiltration rate. infiltration characteristics of soil. The average infiltration rate of the soil surface at a given time is:  
\[i_{ave}=\frac{Q}{\text{A}_{\text{t}}} \tag{7}\]  
where \(i_{ave}\) is the average infiltration rate, \(Q\) is the cumulative infiltration, \(A\) is the cross-sectional area of the soil column, and \(t\) is the duration. Figure 11 shows the average infiltration rate curve of the soil column. The average infiltration rate can also be divided into three stages. The first stage is a gentle section, the second stage is a slowly decreasing section, and the third stage is a stable section, which is consistent with the three stages of the instantaneous infiltration rate.