## 6 Pros and Cons of the WPM and HBm: Conclusions
Figure 19 shows the percentage change in SG between the value calculated at \(\delta_{crit}\) and the other \(\delta\). The percentage change has the highest values at \(\delta=0^{\circ}\) when SG is calculated with HBm and at \(\delta=90^{\circ}\) (which corresponds to \(\delta^{*}=0^{\circ}\)) when SG is calculated with WPM. As the prediction with HBm seems to \"underestimate\" (compared to WPM) the mud weight at low \(R\), the use of SG which corresponds to \(\delta_{crit}\) for \(R=1.1\) results in a maximum percentage decrease equal to \(-8\%\) that occurs at \(\delta=0^{\circ}\). In the range \(\delta=15^{\circ}\)-\(75^{\circ}\), the maximum percentage decrease is \(-4\%\) (Figure 19a) The percentage decrease at high \(R\) when \(\delta=0^{\circ}\) becomes high (\(-13\%\)) and the occurrence of tensile failure (hydraulic fracturing) must be carefully investigated in order to find a properly safe mud weight window. According to [52], the lower SG for hydraulic fracturing occurs at \(\delta=0^{\circ}\). The maximum percentage decrease calculated with the WPM in the range \(\delta=15^{\circ}\)-\(45^{\circ}\) is \(-4\%\) for all the far field stress anisotropy \(R\) and pore fluid pressures \(p_{f}\). We highlight that the percentage change in Figure 19b has been truncated at \(\delta=50^{\circ}\) when \(R=1.5\), because of the anomalous drop that was discussed previously. Based on the outcomes of this paper and those reported in [52; 53; 54], the critical inclination for shales is expected to occur in the range \(\delta_{crit}=30^{\circ}\)-\(60^{\circ}\). This range of \(\delta_{crit}\) corresponds to \(\beta_{w}=40^{\circ}\)-\(60^{\circ}\) at the boundary of the borehole. Consequently, the use of \(\delta_{crit}\), for the prediction of the mud pressure in cases with other inclinations \(\delta\), results in a reduction of the number of compression tests to be carried out for determining the strength  
Figure 18: Coefficient of variation of SG at different \(R\) and \(p_{f}\). parameters, especially with the HBm criterion. The shale specimens can be tested with compression tests at three inclinations within the range \(\beta_{w}\) = 40\({}^{\circ}\)-60\({}^{\circ}\). Finally, the use of the HBm, with the assumption of the dimensional constant fixed at \(m_{\beta w}\) = 6 \(\pm\) 2 for shale rock [61], and the direct use of the uniaxial compressive strength \(\sigma_{c\beta w}\) measured (and not obtained from the data regression) with uniaxial compressive tests can avoid carrying out triaxial compressive tests which are costly and challenging. Future research developments will focus on the same sensitivity analyses of this study coupled with the interpretation of lab tests carried out on shale rocks with WPM and HBm. A further development will be the analysis of a real case of wellbore instability, by using reference mud weights based on the post-drilling experience coupled with the strength properties of shale rock obtained by lab tests. Conceptualization, C.D. ; methodology, C.D. ; software, W.L. ; formal analysis, C.D. ; investigation, C.D. and W.L. ; resources, C.D., W.L. and K.Y. ; writing--original draft preparation, C.D. ; writing--review and editing, C.D., W.L. and K.Y. ; supervision, C.D. All authors have read and agreed to the published version of the manuscript. [PERSON]'s scholarship as a visiting PhD student at Politecnico di Torino (PoliTo-CSC) was granted by \"Nationally-sponsored Graduate Program of China for High-level Universities\". Grant number: (202208340046). Not applicable. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. The authors declare no conflict of interest.