## 2 Materials and Methods
### Hydraulic Model  
In this study, the Van Genuchten model [31] was used to describe the hydraulic behavior of the soil, under the following Equation (1):  
\[\theta=\theta_{r}+\frac{\left(\theta_{s}-\theta_{r}\right)}{\left[1+\left( \alpha h\right)^{n}\right]^{m}} \tag{1}\]  
\[m=1-\frac{1}{n}\]  
In the Plaxis 2D software, the variables are defined using different names and symbols, which are presented in Equation (2):  
\[S(\Psi)=S_{res}+(S_{sat}-S_{res})g_{a}\big{[}1+(g_{a}|\Psi|)^{g_{a}}\big{]}^{g_{ c}} \tag{2}\]  
\[g_{c}=\frac{1-g_{n}}{g_{n}}\]  
where (\(\Psi\)) represents the degree of saturation, \(S_{res}\) represents the residual saturation, \(S_{sat}\) represents saturation under fully saturated conditions, \(\Psi\) represents the matric suction, \(g_{a}\) is a fitting parameter related to air entry into the soil, and \(g_{n}\) is a fitting parameter related to water extraction from the soil once the air entry value is exceeded [31; 49]. It is also worth noting that, regarding the parameters of the original van Genuchten equation and its adaptation within Plaxis 2D, the following equivalences are established:  
\[S_{res}=\theta_{res}\]  
\[S_{sat}=\theta_{sat}\]  
\[g_{a}=\alpha\]  
\[g_{n}=n\]  
\[g_{c}=-m\] \begin{table}
\begin{tabular}{c c c} \hline
**Properties** & **SP-SM** & **SW-SM** \\ \hline Saturated unit weight [kN/m\({}^{3}\)] & 18.9 & 18 \\ Unsaturated unit weight [kN/m\({}^{3}\)] & 15.7 & 16 \\ Deformation modulus [kN/m\({}^{2}\)] & 30,000 & 15,000 \\ Poisson’s ratio & 0.3 & 0.35 \\ Friction angle [\({}^{\circ}\)] & 38 & 36 \\ Cohesion [kN/m\({}^{2}\)] & 5 & 10 \\ Sand percentage & 93.5 & 87.8 \\ Fines percentage & 6.5 & 11.9 \\ Gravel percentage & 0 & 0.3 \\ Initial degree of saturation & 70 & 70 \\ \hline \end{tabular}
\end{table} 
Table 1: Soil properties. Additionally, Equation (3) represents the effective saturation:  
\[S_{eff}=\frac{S_{r}(\psi)-S_{res}}{S_{sat}-S_{res}} \tag{3}\]  
where \(S_{eff}\) is the effective saturation, \(S_{r}(\psi)\) is the degree of saturation as a function of suction, \(S_{res}\) is the residual saturation, and \(S_{sat}\) is the total saturation. Using the effective saturation and total water pressure, the effective water pressure can be determined with Equation (4):  
\[p_{active}=S_{eff}\cdot p_{w} \tag{4}\]  
where \(p_{active}\) is the effective water pressure. With the effective water pressure, the effective stress of the soil can be calculated using Equation (5), as shown below:  
\[\sigma^{\prime}=\sigma-p_{active} \tag{5}\]  
where \(\sigma^{\prime}\) is the effective stress and \(\sigma\) is the total stress. To analyze the behavior of flow and strength in partially saturated soils, Plaxis 2D describes the evolution of water pressure as a function of suction level, represented by the following differential Equation (6):  
\[\frac{dS(p_{w})}{d_{p_{w}}}=(S_{sat}-S_{res})\bigg{[}\frac{1-g_{n}}{g_{n}} \bigg{]}\bigg{[}g_{n}\bigg{(}\frac{g_{a}}{\gamma_{w}}\bigg{)}^{g_{n}}\cdot p_{ w}^{(g_{n}-1)}\bigg{]}\bigg{[}1+\bigg{(}g_{a}\cdot\frac{p_{w}}{\gamma_{w}} \bigg{)}^{g_{n}}\bigg{]}^{(\frac{1-2g_{n}}{g_{n}})} \tag{6}\]  
Finally, in the context of partially saturated soils, soil permeability changes with water content, as suction affects the amount of water in the pores, influencing the soil's ability to allow water flow. The following formula represents this modification:  
\[k_{rel}(S_{r})=max\left[S_{eff}^{gl}\Bigg{(}1-\Big{[}1-S_{eff}\frac{g_{n}}{g^{ -1}}\Big{]}^{(\frac{g_{n}}{g_{n}-1})}\Bigg{)}^{2},10^{-4}\right] \tag{7}\]  
Regarding the suction curves, these were obtained from previous research conducted by various authors. The suction curve for SW-SM soil was taken from the work of [PERSON]. [PERSON] et al. [33], while the suction curve for SP-SM soil was derived from the research of [PERSON] et al. [32]. Therefore, the parameters necessary to construct these curves were also derived from the data and models proposed in these studies, ensuring the accuracy and representativeness of the suction properties for each soil type in this study.