## 2 Mathematical Framework
### Geomechanical Governing Equations and Model Parameterization
The plastic multiplier rate is defined as:  
\[\hat{\gamma}=\frac{\lambda^{*}}{\rho_{\mu}\tau^{*}\frac{\partial p_{c}}{ \partial p}}\left(\frac{p_{c}}{Rp_{c,0}\text{exp}\left(\frac{-\rho_{\kappa} \epsilon_{c}^{p}}{\lambda^{*}(\rho_{\kappa}-1)}\right)}\right)^{\rho_{\mu}( \rho_{\kappa}-1)/\rho_{\kappa}}, \tag{6}\]  
where \(\rho_{\mu}\) is the ratio between the modified compression and creep index, \(\tau^{*}\) is a reference time value, \(\epsilon_{p}^{p}\) is the volumetric plastic strain, and \(p_{c}\) is set equal to a lumped scalar representation of the stress state in the plane of the first and second stress invariants \((p,q)\):  
\[p_{c}=p+\frac{c}{\tan\varphi}+\frac{q^{2}}{M^{2}\left(p+\frac{c}{\tan\varphi} \right)}. \tag{7}\]  
In this last equation, \(c\) is the material cohesion, \(\varphi\) the friction angle, and \(M\) the slope of the Critical State Line in the \((p,q)\) plane:  
\[M=3\sqrt{\left(\frac{1-K_{0}}{1+2K_{0}}\right)^{2}+\frac{(1-K_{0})(1-2\
u)( \rho_{\kappa}-1)}{(1+2K_{0})(1-2\
u)\rho_{\kappa}-(1-K_{0})(1+\
u)}}, \tag{8}\]  
with \(K_{0}\) the ratio between the vertical and horizontal stress in normal consolidation state, also known as confinement factor. Finally, in Equation (6) \(R\) is the geotechnical initial overconsolidation ratio defined as \(p_{c,r,0}/p_{c,0}\), where \(p_{c,0}\) is a representation of the stress state at initial conditions and \(p_{c,r,0}\) is a parameter related to the plastic strain developed by the material before loading. For details on the numerical implementation of Equations (4)-(8) in the forward model, the reader is referred to Reference [3]. The VN visco-elasto-plastic model requires a significant number of independent material parameters, which may vary according to the selected implementation. In particular, for the formulation presented herein we need: \(\
u\), \(\lambda^{*}\), \(\rho_{\kappa}\), \(\rho_{\mu}\), \(\tau^{*}\), \(R\), \(c\), \(\varphi\), and \(K_{0}\). The present analysis focuses on the parameters that mostly characterize the VN constitutive law with respect to more traditional ones, specifically, (1) \(\lambda^{*}\), (2) \(\rho_{\kappa}\), (3) \(\rho_{\mu}\), and (4) \(R\). Parameter \(\lambda^{*}\) represents the slope of the normal consolidation profile of volumetric strain vs axial stress in a logarithmic plot, while \(\rho_{\kappa}\) allows to obtain the slope of the same profile in unloading/reloading conditions. The parameter \(\rho_{\mu}\) depends on the slope of the volumetric strain vs. time profile, in natural logarithmic scale. Finally, the ratio \(R\) provides the over-consolidation degree of the porous rock and describes the size of the implicit yield surface defined by the VN model in the plane of the first two stress invariants \((p,q)\). This set of parameters is also selected because of the availability of literature estimates and the independence on the employed unity of measure. All the remaining parameters have been fixed to standard values generally accepted for a wide range of materials (\(\
u=0.3\), \(\tau^{*}=1\) day, \(c=1\) MPa, \(\varphi=30^{\circ}\), and \(K_{0}=\
u/(1-\
u)=0.43\)).