## 3 Theoretical Model
### Predicted Stress Paths  
Before stress path integration, the following initial conditions have been defined: the degree of saturation calculated from the value of [PERSON]'s parameter \(B\) (Equation (30)), the value of the stress ratio, \(\eta\) (in case of isotropic consolidation \(\eta_{0}=0\)), porosity, mean effective stress and back pressure, with corresponding compressibilites of soil skeleton and pore fluid. Equation (27) can be solved numerically, assuming an incremental step of the stress ratio, \(\Delta\eta\). For all cases presented in the paper the increment of \(\Delta\eta=0.001\) has been used, ensuring the stability of the solution. At the beginning of a single step increment, the mean effective stress corresponding to the current stress ratio \(\eta\) is calculated by using Equation (27). Afterwards, the stress deviator and excess pore pressure are calculated by employing Equations (24) and (26). At the end of a given step, compressibilities, porosity and degree of saturation are updated.  
Integration of Equation (27) for different values of the \(B\) parameter enables the coverage of stress space (\(p^{\prime}\),\(q\)) by corresponding stress paths. Under partial saturation conditions (see Figure 2), stress paths are located between responses characteristic of a fully saturated soil and a completely dry soil. Figures 18 and 19 show the distribution of stress paths predicted by the model for different degrees of saturation and the corresponding Skempton's \(B\) parameters for OZM50 tailings and Skarpa sand, respectively.  
Predicted changes of degree of saturation (Equation (31)) and porosity (Equation (32)) corresponding to the model simulations shown in Figures 18 and 19 were presented in Figures 20 and 21, respectively.  
Figure 17: Volumetric strains developing during isotropic unloading in OZM50 tailings.  
Figure 19: Theoretical prediction of stress paths for different saturations levels for Skarpa sand.  
Figure 18: Theoretical prediction of stress paths for different saturations levels for OZM50 tailings.  
Figure 21: Model prediction of changes in porosity corresponding to simulations presented in: (**a**) Figure 18 and (**b**) Figure 19.  
Figure 20: Model prediction of changes in degree of saturation during shearing presented in: (**a**) Figure 18 and (**b**) Figure 19.  
For both types of soil, the model predicts stress paths quite well, both qualitatively and quantitatively. To quantify the prediction accuracy of the model, the maximum deviators obtained in the laboratory tests were compared with those obtained theoretically. For this purpose, the percentage measure of error, \(\zeta\), was applied.  
\[\zeta=\left(\frac{q_{\max}^{\text{experiment}}}{q_{\max}^{\text{model}}}-1\right) \cdot 100\% \tag{39}\]  
The results presented in Figure 24 show that for all cases the error was less than 16.1% and the average error was 7.7%, which in the domain of soil mechanics is quite satisfactory, although in the majority of tests analysed the model somewhat overestimated the test results.  
Figure 23: Theoretical predictions of triaxial compression tests of partially saturated Skarpa sand vs. experimental results.  
Figure 24: Prediction errors for maximum deviators.