## 3 Description of the Simulation Model  
### Theoretical Aspects of the Method Used  
As stated in the Introduction section, slope stability analysis is important for the prediction of soil slippage, slides and deformation caused by various mechanical loads, and it is also imposed by safety concerns. For a section of the dam of the TP no. 1 Compartment B described in Section 2, such an analysis was conducted using the geomechanics module of COMSOL Multiphysics version 5 software. This software was chosen for the analysis, in consideration of its complexity, its extensive use in the literature, as presented in Section 1, and also based on the expertise of the authors' team [64; 65] in solving engineering problems using the same software.  
Within the model studied, the software used [PERSON]'s law for the computation of the pore pressure in the soil and the fluid flow through the porous medium, while the Mohr-Coulomb criterion was used to represent the soil elastic-plastic behavior during the analysis. The Mohr-Coulomb criterion is a mathematical model which describes the response of a certain material to normal and shear stresses, and it is widely used in geoengineering.  
This technique of determining the factor of safety is called the shear strength reduction (SSR) method. According to [66; 67], the SSR technique for slope stability analysis involves systematic finite element analysis to determine a stress reduction factor (SRF) that brings the slope to the verge of failure, thus obtaining the FOS value. By definition [68], the factor of safety of a slope is the \"ratio of actual soil shear strength to the minimum shear strength required to prevent failure,\" or the factor by which soil shear strength must be reduced to bring a slope to the verge of failure.  
The shear strengths of all materials in the model were progressively reduced by the SRF until collapse occurred during a multistep approach carried out automatically by the software, based on the algorithm: (1) Develop the model of the slope. Add the material properties established (strength and deformation). Compute and record the maximum deformation; (2) Increase the value of the SRF, add the reduced Mohr-Coulomb material parameters to the model. Re-compute and record the maximum deformation; (3) Repeat step 2 until the model does not converge. In other words, the strength parameters of the material are reduced until the slope fails. This critical SRF that is obtained just before the failure, is the actual FOS for the slope.  
In this method, the Mohr-Coulomb material parameters are defined as functions of the SRF. The Mohr-Coulomb yield function defining the associated plastic potential can be expressed as [55; 69]:  
\[F=Q=m\cdot\sqrt{I_{2}}+\alpha\cdot I_{1}-k \tag{1}\]  
where  
\[\alpha=\frac{\sin\phi}{3};\;k=C\cdot\cos\phi \tag{2}\]  
Figure 3: Tailings Pond no. 2.  
With this technique, the FOS affects the cohesion as well as the angle of internal friction. Cohesion describes how strongly a material will stick together. The angle of internal friction describes the frictional shear resistance of the soil. The parametrized angle of internal friction \(\phi\) and cohesion \(C\) are expressed [67,70] as:  
\[\begin{split}\phi&=\arctan\left(\frac{\tan\phi_{u}}{ \text{SRF}}\right)_{\text{(for $p<0$)}}+\arctan\left(\frac{\tan\phi_{u}}{\text{SRF}}\right)_{\text{(for $p\geq 0$)}}\\ C&=\frac{c}{\text{SRF}}\end{split} \tag{3}\]  
where \(c\) is the cohesion of the material, \(\phi_{u}\) and \(\phi_{s}\) are the angles of internal friction for unsaturated and saturated soils, and \(p\) is the pore pressure as defined by [PERSON]'s law. The dam section is modelled in 2D in order to reduce the computation time, using the plane strain approximation method and including the hydrostatic pressure and gravity effects.