## 4 Sensitivity Analyses: Results and Discussion
### Influence of the Far Field Stresses and Pore Fluid Pressure on Mud Weight Prediction
The range \(\delta=75^{\circ}\)-\(90^{\circ}\) exhibits a lightly higher SG calculated with Hbm (\(+2\%\)). On the other hand, the percentage decrease in Hbm reduces when \(R=1.5\). In this case, the percentage increase in Hbm is very high (\(+29\)-\(40\%\)) from \(\delta=40^{\circ}\) to \(\delta=90^{\circ}\). The reason for this high mismatch appears to be the remarkable decrease in WPM mud weight when \(R>1.1\) (Figure 10c,d). Finally, the figure shows that the change in trend (higher/lower SG) predicted by WPM and Hbm is independent of the pore fluid pressure magnitude. We performed a quantitative comparison between the predictions of WPM and Hbm, by varying the far field stress anisotropy. Figure 11a,b show the comparison of the mud weight predicted by the two criteria at different \(R\) with constant pore fluid pressure. The figures indicate that the change in trend (higher/lower SG) occurs when \(R>1.1\). Furthermore, Hbm predicts a definitely higher mud weight when \(R>1.5\) for all the inclinations \(\delta\) of the weak planes. This outcome indicates that WPM provides questionable mud weights at high \(R\), in the range \(\delta=50^{\circ}\)-\(90^{\circ}\). Figure 11c shows the percentage change in SG with the variation of the far field stresses \(\Delta R\) (at constant pore fluid pressure \(p_{f}=11\) MPa). The change in \(R\) from 1.1 to 1.5 corresponds to \(\Delta R=+36\%\) and the change in \(R\) from 1.1 to 1.25 corresponds to \(\Delta R=+14\%\). The figure indicates that the variation of SG calculated with Hbm is more sensitive to the increase in far field stress anisotropy, especially from \(\delta=30^{\circ}\) to \(\delta=90^{\circ}\). The change in SG calculated with Hbm is overall higher, except for a few inclinations of the weakness planes with \(\Delta R=+14\%\), and shows a maximum at \(\delta=45^{\circ}\). On the other hand, the change in SG calculated with WPM after the peak exhibits a substantial decrease from \(\delta=30^{\circ}\) to \(\delta=90^{\circ}\). This prediction seems to be anomalous, particularly close to \(\delta=90^{\circ}\). Figure 11d shows the comparison between the SG of the weakness planes and rock matrix when \(R=1.5\). In general, the SG of the rock matrix is expected to be lower than the SG of the weak planes. Nevertheless, the figure clearly shows that the SG of the rock matrix is higher than the SG of the weakness planes in the range \(\delta=75^{\circ}\)-\(90^{\circ}\). On the other hand, the peaks calculated with Hbm (Figure 11a-c) are higher than the peaks of the rock matrix in the range \(\delta=45^{\circ}\)-\(90^{\circ}\). The change in trend (higher/lower SG) of the predictions with WPM and Hbm occurs with increasing \(R\). This change depends on the interpolating functions of the strength. WPM provides anomalous results when \(R>1.1\). These outcomes are related to the constrain of the minimum strength at \(\beta_{w}=45^{\circ}+\phi_{w}^{\prime}/2\) and the fast growth of the interpolating function. Figures 10 and 11 indicate that the maximum mud weight occurs at a specific inclination \(\delta=\delta_{crit}\) of the weakness planes, irrespective of \(R\) and \(p_{f}\). The maximum SG predicted by WPM always occurs at \(\delta_{crit}=45^{\circ}-\phi_{w}^{\prime}/2\), as found by [52] and in this paper (\(\delta_{crit}=30^{\circ}\)). On the other hand, the maximum SG predicted by Hbm occurs at \(\delta_{crit}=50^{\circ}\) with \(\beta_{w}=40^{\circ}\), which corresponds to the minimum strength of the interpolating function. The analysis of the location of the mud weight peaks at different \(\delta\) evidenced that the correspondent angles \(\beta_{w}\) have different distribution. Figure 12 shows the frequency distribution of the angles \(\beta_{w}\) at which the peaks of the mud weight occur.