## 2 Materials and Methods
### The Estimation of the Infiltration Rates  
The infiltration flow rate was estimated using a mass-balance method. This method requires regular measurements of inflow and outflow rates. The measurements were conducted every 5 min for a period of 1.5 h. The equation of the mass-balance method is as follows:  
\[Q_{f}=(Q_{in}-Q_{out})-\frac{\Delta S}{\Delta t} \tag{1}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Instruments** & **Application** & **Function** \\ \hline Upstream tank & \begin{tabular}{l} Placed under the inflow pipe; some stones \\ were placed in the bottom of it \\ \end{tabular} & \begin{tabular}{l} Regulate inflow and reduce \\ turbulence \\ \end{tabular} \\ Electronic flowmeter & \begin{tabular}{l} Attached to the inflow pipe \\ \end{tabular} & \begin{tabular}{l} Measure, regulate, and control the \\ inflow rate \\ \end{tabular} \\ Staff pages & \begin{tabular}{l} Fixed at three points along the channel as \\ point 1: 5 m from upstream, point 2: middle \\ of the channel, point 3: 5 m from \\ downstream \\ \end{tabular} & \begin{tabular}{l} Measure water level \\ \end{tabular} \\ Current meter & \begin{tabular}{l} Movable among the three water level \\ measurement points along the channel \\ \end{tabular} & \begin{tabular}{l} Measure the mean flow velocities \\ \end{tabular} \\ Downstream tank & \
\begin{tabular}{l} Placed after the check dam \\ \end{tabular} &
\begin{tabular}{l} Measure the outflow using the \\ volumetric method \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The instruments used in the experiments for the measurement of water movements. Figure 3: The instruments used in the experiments: (**a**) upstream inflow tank; (**b**) electronic flowmeter; (**c**) staff gage; (**d**) current meter; (**e**) downstream tank with valve. where \(Q_{f}=\) infiltration flow rate (m\({}^{3}\)/s), \(Q_{in}=\) inflow rate (m\({}^{3}\)/s), \(Q_{out}=\) outflow rate (m\({}^{3}\)/s), and \(\frac{\Delta S}{\Delta t}=\) change in channel storage (m\({}^{3}\)/s) over a time interval, \(\Delta t\). The infiltration flow rate in a channel can be estimated using the following equation [29]:  
\[Q_{f}=\frac{f}{360,000}\;PL \tag{2}\]  
where \(f=\) infiltration rate (cm/h), \(P=\) wetted perimeter (m), \(L=\) channel length (m), and the 360,000 is the unit normalization factor (the unit of \(Q_{f}\) is m\({}^{3}\)/s, whereas \(f\) is cm/h). Equation (2) can be rewritten as follows to calculate the infiltration rate in a grassed channel:  
\[f=\frac{360,000\;Q_{f}}{PL} \tag{3}\]  
The modified Kostiakov model [27] for estimation of infiltration rate was adapted in the present study for the development of an improved model for the estimation of infiltration rate of a grassed channel with check dams. The infiltration capacity in unlined irrigation and drainage channels can be estimated using a modified Kostiakov model:  
\[f_{p}=K_{k}\;t^{-\alpha}+f_{c} \tag{4}\]  
where \(f_{p}=\) infiltration rate (cm/h), \(t=\) time (h), \(f_{c}=\) final infiltration rate (cm/h), and \(K_{k}\) and \(\alpha\) are non-dimensional constants. The parameters \(K_{k}\) and \(\alpha\) are usually obtained from experimental data and they do not have any physical meaning [28; 35; 36]. The experimental tests conducted by [37] revealed that \(K_{k}\) can have a value ranging from 0.225 to 1.1, and \(\alpha\) can be between 0.458 and 0.669. However, the experimental tests conducted by [38] on a wide range of soil types revealed that the values of the parameters can have a much wider range, 7.76 to 333.92 for \(K_{k}\) and 0 to 0.77 for \(\alpha\). The estimation accuracy of the modified Kostiakov model can be significantly improved by adding flow velocity and channel cross-section parameters (i.e., mean velocity \(v\), side slope \(m\), base width \(b\), and water level \(y\)), as follows [21; 39]:  
\[f_{p}=K_{k}\;y^{C1}\;b^{C2}\;m^{C3}\;y^{C4}\;t^{-\alpha}+f_{c} \tag{5}\]  
where C1, C2, C3, and C4 are the constants for the introduced hydraulic parameters, \(y\), \(b\), \(m\), and \(v\), respectively.