## 3 Theoretical Model
### Soil Skeleton Compressibility Functions  
In addition to soil porosity and pore fluid compressibility, the basic parameters of the model, there are two functions of soil skeleton compressibility, i.e., the isotropic function, \(\kappa_{s}^{\eta,l}\), and the deviatoric function, \(\kappa_{s}^{\eta}\), whose coefficients should be determined in triaxial tests for soils characterised by similar state parameters.  
The function of deviatoric compressibility is determined in pure shearing tests under triaxial conditions during which mean effective stress is kept constant. For dry post-flotation copper tailings OZM50 in contractive state, volumetric strains as a function of stress ratio can be approximated by the following function (Figure 15):  
\[\varepsilon_{v}=D_{1}\exp[D_{2}(\eta-\eta_{\text{CM}})], \tag{33}\]  
where \(\eta_{\text{CM}}\) corresponds to the Coulomb-Mohr yield surface.  
The coefficients in Function (33), estimated by the least squares method for OZM50 tailings, are \(D_{1}=2.98\times 10^{-2}\) and \(D_{2}=3.11\times 10^{-3}\).  
According to Equations (8) and (33), the deviatoric compressibility function can take the following form:  
\[\kappa_{s}^{\eta,l}=\frac{\partial\varepsilon_{v}}{\partial\eta}=D_{1}D_{2} \exp[D_{2}(\eta-\eta_{\text{CM}})]. \tag{34}\]  
In turn, the isotropic compressibility function can be determined in isotropic consolidation tests. Figure 16 shows volumetric strains that developed during isotropic compression applied to a specimen of the same post-flotation tailings OZM50. In this case, they can be nicely approximated by the logarithmic function:  
\[\varepsilon_{v}=A_{1}\ln(1+A_{2}p^{\prime}), \tag{35}\]  
Figure 15: Volumetric strains versus stress ratio during pure shearing under triaxial conditions for contractive OZM50 tailings.  
whose coefficients estimated by the method of least squares are \(A_{1}=2.97\times 10^{-2}\) and \(A_{2}=6.7\times 10^{-3}\).  
The isotropic compressibility function is a derivative of the volumetric strains function, this time calculated with respect to mean effective stress:  
\[\kappa_{\rm s}^{p^{\prime},l}=\frac{\partial\varepsilon_{\rm v}}{\partial p^ {\prime}}=\frac{A_{1}A_{2}}{1+A_{2}p^{\prime}}. \tag{36}\]  
Under undrained conditions, the contractive response of sheared soil manifests itself in a reduction of mean effective stress as a result of pore pressure build-up. In the primary model, the reduction of mean effective stress which corresponds to unloading was not formally taken into account but it has quite a significant influence on its predictions. Therefore, it was necessary to differentiate between isotropic compressibility for loading and unloading and to introduce into the model the isotropic compressibility under unloading, \(\kappa_{\rm s}^{p^{\prime},\rm un}\), which should also be determined by laboratory tests.  
The results of an isotropic unloading test on contractive OZM50 tailings are shown in Figure 17. In order to approximate volumetric strains, which decrease during isotropic unloading, an exponential function was adopted:  
\[\varepsilon_{\rm v}=A_{1}^{\rm un}p^{\prime A_{2}^{\rm un}}+\text{const.} \tag{37}\]  
The coefficients estimated by the method of least squares are \(A_{1}^{\rm un}=-2.23\times 10^{-2}\) and \(A_{2}^{\rm un}=-1.92\times 10^{-1}\).  
Then the function of isotropic compressibility under unloading takes the following form:  
\[\kappa_{s}^{p^{\prime},\rm un}=\frac{\partial\varepsilon_{\rm v}}{\partial p^ {\prime}}=A_{1}^{\rm un}A_{2}^{\rm un}p^{\prime(A_{2}^{\rm un}-1)}. \tag{38}\]  
Figure 16: Volumetric strains developed during isotropic consolidation in OZM50 tailings.