## 3 Results  
### Flow Depth and Velocity for Different Check Dam Conditions  
The flow in the channel could then be classified under three stages: (i) the flow at the startup, (ii) the peak flow, and (iii) the flow after shutting down the inflow pipe valve. Figure 5 depicts the typical hydrograph of the flow that occurred during the experiment, which exhibited three stages of flow.  
The opening of the inflow valve initiated the startup stage, which continued until the maximum water depth in the channel was reached. The peak flow occurred at this stage. The inflow rate determined the extent of the startup phase. The outflow was only encountered when this phase was ended, and the water depth went beyond the check dam height. The mean velocity and water depth in the channel remained constant until the end of the second stage. The d  
Figure 4: The laboratory conducted to measure the infiltration rates through a topsoil, open to atmospheric pressure and with natural soil boundaries: (**a**) the physical model with identical vertical columns (**b**); the cumulative infiltration after 1.5 h. [21]  
Figure 5: The typical hydrograph of flow along the channel during experiments, showing three flow stages.  
determined by deducting the time of start-up and shutdown periods from 1.5 h (the overall time of the experiment). The shutting down of the inflow pipe valve initiated the third stage. The water flow through the channel was sustained for a limited period after shutting down the inflow pipe valve due to the channel longitudinal slope. The water level at this stage was gradually reduced to the height of the check dam in the downstream, causing a complete cessation of the outflow while sustaining the ponding water in the channel.  
Table 2 presents the flow characteristics estimated for different experimental cases. The table shows that the time to downstream (i.e., the time lag between upstream and downstream of the channel), \(t_{d}\) (min), decreased with the increase of inflow rate, irrespective of check dam height in the downstream. The relationship between the inflow rate and the average value of \(t_{d}\) (min) for the grassed channels with a coefficient of determination, \(R^{2}\), is shown in Figure 5(a).  
Figure 6: The experimental models developed to predict (**a**) the time to downstream \(t_{\rm d}\) (min), (**b**) the time to peak \(t_{\rm p}\) (min), and (**c**) the water level at peak flow \(y_{\rm p}\) (m).  
Figure 8: The values of cumulative infiltration, \(F\), in the grassed channel after 2 h for different check dam heights.  
Figure 7: The changes in infiltration rate \(f_{p}\) and cumulative infiltration \(F\) with time in the grassed channel for different inflow rates: (**a**) \(h_{w}=10\) cm; (**b**) \(h_{w}=15\) cm; (**c**) \(h_{w}=20\) cm.  
The infiltration rate of natural soil (2 cm/h) was considered as the final constant infiltration rate, \(f_{c}\)[11]. The analysis of \(f_{\rm P}\) was performed for the values of \(y\), \(b\), \(m\), and \(v\), and the time \(t\). The results produced an improved version of the modified Kostiakov model with a coefficient of determination \(R^{2}=0.69\), root mean square error (\(RMSE\)) = 3.1 cm/h, and values of 60 and 0.8 for the non-dimensional constants, \(K_{k}\) and \(\alpha\), respectively:  
\[f_{p}=60\ y^{1.5}\ b^{0.01}\ m^{1.1}\ v^{0.4}\ t^{-0.8}+2 \tag{6}\]  
This equation can be further simplified and generalized as  
\[f_{p}=K_{k}\ F_{\rm hp}\ t^{-\alpha}+f_{c} \tag{7}\]  
where \(F_{\rm hp}\) is a function that represents the four hydraulic parameters as  
\[F_{\rm hp}=y^{1.5}\ b^{0.01}\ m^{1.1}\ v^{0.4} \tag{8}\]