## 2 Materials and Methods  
### Experimental Procedure  
The methodology used to achieve the objective of the present study was performed in two stages: (i) estimation and comparison of the SWRC using [PERSON] model (including the fractal, Burdine, and Mualem restrictions) and (ii) evaluation of the hydraulic conductivity function (HCF) using the van Genuchten equation via the Mualem model. In the first stage, it was necessary to use Equation (1) to adjust the \(\psi_{m}\) data previously reported in [29] to compare it with the SWRC obtained by [PERSON] model. This allowed the researchers to collate the \(\psi_{m}\) with the SWRC using [PERSON] model. Following this, using Equation (3), the adjustment of the grain-size distribution was performed using MATLAB software in the following manner:  
1. The model to be used (Mualem or Burdine) was defined. For fractal analysis, the relation s = D/E that satisfies Equation (7) was obtained;
2. The \(m\), \(n\), and \(s\) parameter values for every model to be evaluated (i.e., geometrical pore, larger pore, neutral pore, Mualem, and Burdine), were selected;
3. Diameter increments of 5.5 \(\upmu\)m, \(m\) increments of 0.001 (a conditional of the loop during each iteration through the first stage), and an error of 0.03 (which means a coefficient of determination of 0.97) were established;
4. The grain-size data for each contaminant concentration (i.e., 0 mg/kg, 25 mg/kg, and 50 mg/kg), which corresponds to experimental frequency (0 to 1) and diameter (\(\upmu\)m), was introduced;
5. For the first iteration, an initial diameter of 1 \(\upmu\)m and an initial value of 0.001 for the parameter \(m\) were used. Then, those parameters are evaluated in Equation (3) for the entire data of the grain size analysis, obtaining a proposed frequency;
6. If the error is less than or equal to 0.03 (as previously defined), the parameters \(m\) and \(n\) are obtained to evaluate Equation (1), assuming \(\psi_{m}\) as the unknown value. In other cases, a new diameter (Diameter = diameter + increment) and parameter \(m\) value (\(m=m+\) increment) are proposed, and the process is repeated from step 5.  
Once the parameters \(m\) and \(n\) were obtained, they were optimized by minimizing the sum of squared errors (\(SSE\)). The coefficient of determination (\(R^{2}\)) was obtained by applying Equation (8) as an index to evaluate the linear relationship between the frequency of the grain-size analysis and the frequency of Equation (1). Its value varies from 0 to 1 (1 is a perfect linear adjustment, while 0 means that the model represents none of the data) [40].  
\[R^{2}=1-SSE/SST=1-\left[\Sigma\big{(}Y-\hat{Y}\big{)}^{2}/\Sigma\big{(}Y-\overline {Y}\big{)}^{2}\right] \tag{8}\]  
where \(SST\) is the total sum of squares, \(Y\) is the dependent variable, \(\hat{Y}\) is the estimated dependent variable, and \(\overline{Y}\) is the mean of the dependent variable.  
In the second stage, the HCF was evaluated using the van Genuchten method since the flow of contaminants is a constant concern in soil mechanics. In this regard, the HCF should be evaluated because contaminants travel along the flow paths at different velocities; it depends not only on the saturation rate but also on the stress state of the soil.