## 3 Interpretation of Laboratory Tests on Tournemire Shale
### Determination of the Strength Properties with WPM and HBm  
[PERSON] et al. (2015) [40] carried out uniaxial, triaxial and Brazilian tests on Tournemire shale specimens at different inclinations of \(\beta_{w}\) of the weakness planes. The strength parameters, necessary for the calculation of the mud pressures, are obtained with the regression of the data of these lab tests with WPM and HBm. We reduced the tensile strength of the Brazilian tests by a factor equal to 0.8, according to [60]. The tensile strength ranges from \(\sigma_{\ell\beta w}=-3.8\) MPa at \(\beta_{w}=90^{\circ}\) to \(\sigma_{\ell\beta w}=-4.6\) MPa at \(\beta_{w}=0^{\circ}\). The regressions carried out with HBm at different \(\beta_{w}\) were set up by using the complete data set (uniaxial, triaxial and Brazilian tests). The linear regression of the data [\(\sigma_{3}\), \((\sigma_{1}-\sigma_{3})^{2}\)] enabled the calculation of the instantaneous uniaxial compressive strengths \(\sigma_{c\beta w}\) and the instantaneous constants \(m_{\beta w}\). Table 1 reports the rock properties obtained through this process. Figure 6a shows that the calculated uniaxial compressive strength matches well with the results of the lab tests. The figure indicates that the lower uniaxial compressive strength measured in lab tests (\(\sigma_{c\beta w}=15.27\) MPa) occurs at \(\beta_{w}=60^{\circ}\). The minimum uniaxial compressive strength obtained from the regression (\(\sigma_{c\beta w}=17.40\) MPa) occurs at \(\beta_{w}=45^{\circ}\). Figure 6a also shows the interpolating function of the regression data of \(\sigma_{c\beta w}\). The function is a second-order polynomial (parabola), which is used for the calculation of the mud pressure with Equation (6). The minimum of the parabola occurs close to \(\beta_{w}=40^{\circ}\). The effect of this mismatch on the mud pressure will be discussed in the next section. Figure 6c shows the percentage increase in uniaxial compressive strength \(\sigma_{c\beta w}\) compared to the minimum at \(\beta_{w}=40^{\circ}\). The maximum percentage increase occurs at \(\beta_{w}=90^{\circ}\) and \(\beta_{w}=0^{\circ}\), as expected. The instantaneous constants \(m_{\beta w}\) (Figure 6b and Table 1) change with \(\beta_{w}\) from \(m_{\beta w}=3.90\) to \(m_{\beta w}=7.16\), in agreement with the value of \(m=6\pm 2\), suggested by [61], for shale. Figure 6b also shows the interpolating function of the regression data of \(m_{\beta w}\), which is a second-order polynomial (parabola). It is worth to note that the instantaneous dimensionless constant \(m_{\beta w}\) does not exhibit a well-defined trend in several data regressions [52] and an average value must be assumed. The average value for our dataset is \(m_{\beta w}=5.16\). Figure 6d shows the percentage increase/decrease calculated with \(m_{\beta w}=5.16\) and the function of \(m_{\beta w}\). The figure indicates that the variation is \(\cong\pm 30\%\) and in the range \(\beta_{w}=0^{\circ}\)-\(55^{\circ}\)\(m_{\beta w}=5.16\) is higher than the \(m_{\beta w}\) of the function. We investigated the mud pressure sensitivity to the instantaneous constants \(m_{\beta w}\) in Section 4. Figure 7a shows the comparison between the experimental results and the linear regression data obtained with WPM. The figure shows two regressions that were obtained with the complete experimental dataset (CD) and with the experimental data set without the Brazilian tests (NB). The figure indicates that the best matching of the experimental data is obtained with the regression without the Brazilian tests. The value of the uniaxial \begin{table}
\begin{tabular}{c c c} \hline \(\beta_{w}\) (\({}^{\circ}\)) & \(\sigma_{c\beta w}\) (MPa) & \(m_{\beta w}\) (-) \\ \hline
0 & 25.97 & 3.90 \\
30 & 20.09 & 4.69 \\
45 & 17.40 & 4.39 \\
60 & 18.12 & 5.60 \\
90 & 29.60 & 7.16 \\ \hline \end{tabular}
\end{table} 
Table 1: Strength parameters of the shale obtained with the HBm regression of lab data.