# Settlement Foundations by Exploring the Collapse of Unsaturated Soils
## 3 Validation of the Theoretical Framework
Figure 3: Variation in void ratio with vertical stress. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \hline \(\text{Ks}\) & 0.75 [m/day] \\ \(\alpha\) & 0.44 [1/m] \\ n & 1.103 \\ 1 & 0.5 \\ \(\theta_{r}\) & \(3.59\times 10^{-23}\) \\ \(\theta_{s}\) & 0.37 \\ \(\chi_{\text{f}}\) & \(4\times 10^{-10}\) [1/Pa] \\ \(\chi_{\text{p}}\) & \(1\times 10^{-4}\) [1/Pa] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of hydraulic parameters related to the studied soil. previous part. Figure 5 exhibits the model's geometry, where the soil domain was defined as having a width of 6 m and a height of 2 m.  
Figure 5a presents the mechanical boundary conditions. Vertical displacements are constrained at the lower base of the soil mass (at a depth of 2 m). Horizontal displacements are also restricted at the vertical boundaries of the soil domain (lateral boundaries). Prior to any mechanical or hydraulic loading, an initial stress state corresponding to the self-weight of the soil is established. Gravity, or weight force, is therefore represented as body force throughout the whole volume of the layer. Initially, the study assumed a stationary condition, introducing the 100 kPa load that was incrementally increased following a linear parametric function to reproduce the construction process. Figure 5b illustrates the hydraulic boundary conditions. Lateral boundaries allow no mass flux for water, meaning zero mass transfer. Considering the waterproofing inside the building, a zero-mass flux for water is applied at the boundary beneath the building. The water table, located 1.35 m below the surface, defines a pressure of 0 kPa at this depth. The initial conditions of the model are set based on the measured water content and soil suction at different depths, corresponding to the hydrostatic condition. The suction throughout the  
Figure 5: Finite element mesh and geometry. (**a**) Mechanical boundary conditions. (**b**) Mechanical boundary conditions. (**c**) Adopted mesh. soil profile changes linearly from 200 kPa at the surface to 0 kPa at the 1.35 depth. The main boundary condition of the model is the infiltration rate. We considered that precipitation occurs over a 60-day period, considering an infiltration rate of 40 mm/day. These boundary conditions can be applied as an atmospheric time-dependent boundary to the surface of the soil, as shown in Figure 5b. This can be applied using an inlet boundary condition, defining the normal inflow velocity. Figure 5c shows the adopted mesh. It was applied to the entire domain using a free triangular mesh type, including 1269 triangular elements and 681 mesh vertices. The mesh quality was entirely assessed, with a 0.72 element quality minimum and an element quality average of 0.94, producing a well-structured mesh suitable for precise numerical simulations. The element area ratio is 0.42. The minimum element size is 3.75 \(\times\) 10\({}^{-4}\) m, and the maximum element size is 0.11 m.  
In summary, the analysis consists of three phases. Initially, a vertical stress of 100 kPa is applied. Subsequently, infiltration is simulated by applying a boundary condition with a 40 mm/day infiltration rate for 60 days. Positive pressure values (\(\text{Hp}\geq 0\)), derived from the hydraulic model, representing pore pressure, were incorporated into the EBBM to calculate effective stresses. Negative pressure values (\(\text{Hp}<0\)) expressed as a positive value (\(-\text{Hp}\)) were introduced into the EBBM as suction. The wetting-induced collapses were assessed by assuming that the applied vertical load remained constant. COMSOL Multiphysics 6.1 software incorporates both the EBBM model and the Richards' equation as built-in modules. The coupling in the hydromechanical model is partial: although changes in pore pressure affect soil deformation, the resulting deformation itself has no impact on pore pressure.