## 2 Materials and Methods
### Strength Criteria  
Sedimentary rocks are porous materials and can exhibit a significant strength decrease with the increase in the pore water pressure. Consequently, the pore water pressure must be taken into account in the mechanical analysis. The concept of effective stress was originally introduced by [PERSON] [44] for soils. According to [45], the theory of poroelasticity demonstrates that the deformation of the porous material is proportional to the effective stress:  
\[\sigma^{\prime}_{ij}=\sigma_{ij}-\alpha p_{f}\delta_{ij}, \tag{1}\]  
where: \(\sigma^{\prime}_{ij}\) and \(\sigma_{ij}\) are the effective and total stress tensors, respectively; \(\alpha\) is the Biot coefficient, function of the bulk moduli of the rock skeleton, rock grains and fluid; \(p_{f}\) is the pore pressure in the porous medium; \(\delta_{ij}\) is the Kronecker function. Equation (1) assumes that the rock is linearly elastic. This assumption is not directly applicable to rocks subject to failure processes [46]. The [PERSON]'s principle of effective stress for soils (\(\alpha=1\)) seems to be the most appropriate definition to be used in rock failure analysis [46; 47]. All the subsequent analyses are carried out with the effective stresses. The deviatoric stress (\(\sigma_{1}-\sigma_{3}\)) is reported in terms of total stresses, because it is always an effective stress: \([\sigma_{1}-\sigma_{3}]=[(\sigma_{1}-p_{f})-(\sigma_{3}-p_{f})]=[\sigma^{ \prime}_{1}-\sigma^{\prime}_{3}]\). Rock failure is generally approached in terms of isotropic behavior; however, the actual behavior of many rocks is anisotropic. Shales are frequently affected by parallel weakness planes, with reduced strength compared to intact material. The analysis of failure in such rocks needs a strength criterion which accounts for the strength anisotropy. Figure 1: Flowchart of the study. The most widely used criterion is WPM, that considers a set of parallel discontinuities on which failure occurs in the form of slip. WPM written in terms of principal stresses (\(\sigma_{1}\) and \(\sigma_{3}\)) becomes  
\[\left(\sigma_{1}-\sigma_{3}\right)_{slip}=\frac{2(c_{w}^{\prime}+\sigma_{3}^{ \prime}\tan\phi_{w}^{\prime})}{\left(1-\frac{\tan\phi_{w}^{\prime}}{\tan\beta_ {w}}\right)\sin 2\beta_{w}}\] (2a) where \[\beta_{w}\], \[c_{w}^{\prime}\] and \[\phi_{w}^{\prime}\] are the inclination, the cohesion and the friction angle of the weakness planes, respectively. Figure 2 shows the plot of Equation (2a): the minimum strength occurs at \[\beta_{w}=45^{\circ}+\phi_{w}^{\prime}/2\] and for values of \[\beta_{w}\], close to \[90^{\circ}\] and in the range \[0^{\circ}\] - Figure 1:0, slip on the plane of weakness cannot occur. Within these ranges, shear failure occurs through the rock material in a direction not controlled by the plane of weakness [47]. The plateau of constant strength represents the resistance of the rock in the ranges of Figure 1:1 that are not predicted by Equation (2a). The plateau of constant strength can be described by the Mohr-Coulomb criterion (M-C):  
Figure 1:2  
where \(c^{\prime}\) is the cohesion, \(\phi^{\prime}\) is the friction angle, \(\sigma_{c}\) is the uniaxial compressive strength of the intact rock and \(N_{\phi}\) is the slope of the strength envelope. WPM needs the calculation of the strength with Equations (2a) and (2b) for a given \(\beta_{w}\). The equation that predicts the lowest strength must be considered for failure. WPM is a simple criterion that does not take into account the contemporary development of slip and failure in the intact rock, observed in laboratory tests [48]. Furthermore, the plateau of constant strength is not always observed in the experimental tests. The use of WPM needs the determination of the cohesion \(c_{w}^{\prime}\) and the friction angle \(\phi_{w}^{\prime}\) of the weakness planes and the cohesion \(c^{\prime}\) and the friction angle \(\phi^{\prime}\) of the rock matrix (plateau strength).