## 6 Pros and Cons of the WPM and HBm: Conclusions  
The results presented in Sections 4 and 5 evidenced that the SG predicted by the HBm is lower than SG predicted by the WPM at least in the range \(\delta=0^{\circ}\)-40\({}^{\circ}\). The highest gap is \(-9\%\) and occurs when the far field stress anisotropy is \(R=1.1\). On the other hand, the SG predicted by the WPM is substantially lower than SG predicted by the HBm in the range \(\delta=40^{\circ}\)-90\({}^{\circ}\), when \(R>1.1\). Furthermore, SG calculated with WPM has a sudden drop, in the range \(\delta=50^{\circ}\)-90\({}^{\circ}\), which appears anomalous. The highest gap is \(-37\%\) and occurs at \(\delta=90^{\circ}\) when the far field stress anisotropy is \(R=1.5\). The inclination \(\delta=90^{\circ}\) seems uncommon, as discussed in Section 2.2. However, this case becomes \(\delta^{\star}=0^{\circ}\) (rotated reference system) and can often occur in real fields. We calculated the coefficient of variation of SG [CV = (Standard Deviation/Mean) \(\times\) 100]. Figure 18 shows the comparison between the coefficient of variation of SG calculated with the two criteria. The coefficient of variation related to HBm is CV \(\leq\) 3% when \(R=1.1\) and CV \(\leq\) 5% at higher values of \(R\). The coefficient of variation related to WPM is CV \(\leq\) 4% when \(R=1.1\) and shows an increasing extent of variability with \(R\). This result confirms that the SG prediction with WPM should be avoided in the range \(\delta=60^{\circ}\)-90\({}^{\circ}\) when \(R>1.1\). This analysis suggests that a conservative estimate of SG can be obtained with WPM at low far field stress anisotropy and with HBm at high far field stress anisotropy. This conclusion needs further investigations, always based on lab measurements of shale prop  
Figure 17: Numerical simulations of rock plasticity with HB by using the \(\sigma_{c}\delta_{\rm pu}\) shown in Figure 16. \(R=1.5\) with \(p_{f}=11\) MPa. Legend of colors: light red indicates the elastic state; dark green indicates at yield; light green \({}^{\prime\prime}\)at yield in past\({}^{\prime\prime}\), i.e., during the numerical convergence. The dotted blue line indicates the inclination of the weakness planes. (**a**) \(\delta=45^{\circ}\), (**b**) \(\delta=0^{\circ}\), (**c**) \(\delta^{\star}=0^{\circ}\), (**d**) \(\delta^{\star}=0^{\circ}\) calculated with the mud pressure predicted by the WPM (Figure 15d). erties. In fact, the HBm function fitted the experimental data of the Tournemire shale quite well, even though the minimum strength is predicted at \(\beta_{w}=40^{\circ}\) and the experimental one is at \(\beta_{w}=60^{\circ}\). On the other hand, the WPM function fitted the experimental data rather poorly, even though the predicted and experimental minimum strengths occur at the same \(\beta_{w}=60^{\circ}\). Figure 12 indicates how many inclinations of the weakness planes must be tested in uniaxial/triaxial compression with the two criteria. The WPM predicts the majority of mud pressure peaks in correspondence of \(\beta_{w}=50^{\circ}\)-\(60^{\circ}\) and therefore needs a lower number of lab tests compared to HBm. Nevertheless, WPM provides anomalous results in some conditions. On the other hand, the WPM is implemented in the majority of the commercial software of geomechanics. The prediction of SG with WPM and HBm, in a real field, must also take into account the uncertainties in the estimation of the variables that affect the mud pressure: inclination of the weak planes; magnitude of the far field stresses and pore fluid pressure. All these issues suggest that the limit mud weight could be higher than that predicted at a specific \(\delta\). The higher value of SG, which corresponds to \(\delta_{crit}\), could be considered the reference SG at least with small \(\delta\) and low \(R\) for HBm and with large \(\delta\) and high \(R\) for WPM. The results of the analyses reported in Section 4 indicated that \(\delta_{crit}=30^{\circ}\) for WPM and \(\delta_{crit}=60^{\circ}\) for HBm.