## 4 Discussion
### Estimation of Surface Energy  
In this work, we are able to estimate the surface energy in three different ways: (1) from mechanical tests in the SCB configuration; (2) from contact angle measurements coupled with the Owens-Wendt methodology; and (3) from atomic force microscopy providing the local adhesion force. The first method is a destructive one, and the energy needed to fracture the rock is estimated via the maximum applied load, corresponding to the energy spent to create intra and inter-granular cracks. For the second and third methods, the surface energy is inferred in a non-destructive way from surface properties through wettability attributes and small-scale adhesion pressure. From the KIC values, one can estimate the fracture surface energy \(\gamma\) using the following equation:  
\[\gamma=\frac{\text{K}_{\text{IC}}^{2}}{2\text{E}} \tag{5}\]  
with E, the Young's modulus of the tested material. Note that Equation (5) is valid for plane stress conditions: in the case of plane strain conditions, E needs to be replaced by E/(1 - v2) where v is [PERSON]'s ratio of the tested material. As we did not measure [PERSON]'s ratio in our rocks, we used Equation (5), knowing that the inferred fracture surface energy would in any case only differ by a few percent. The surface energy values are given in Table 8. The fracture toughness of Obourg Chalk is higher than the one of Ciply Chalk, but in contrast, the fracture surface energy, which depends on KIC2 (Equation (5)), is about half that of Ciply Chalk. This higher surface energy in Ciply Chalk results from the much lower Young's modulus in that rock. Following the Owens-Wendt model, the linear regression shown in Figure 14a yielded a dispersive and polar component of 22.3 and 1.1 mJ/m2 respectively, for Obourg Chalk, and 22.8 and 2.5 mJ/m2 for Ciply Chalk. Consequently, the total surface free energy is 23.4 mJ/m2 for Obourg Chalk and 25.3 mJ/m2 for Ciply Chalk. The same method \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Obourg Chalk**} & \multicolumn{3}{c}{**Ciply Chalk**} \\ \hline \multirow{3}{*}{**Surface Energy (J/m2)**} & \multirow{3}{*}{**Dry Rock**} & **Water-Saturated** & \multirow{3}{*}{**Dry Rock**} & **Water-Saturated** & \multirow{3}{*}{\(\lambda\)=\(\frac{\gamma^{(\text{sat})}}{\gamma^{(\text{sat})}}\)} \\  & & **Saturated Rock** & & & **Saturated Rock** & \\ \hline \multirow{2}{*}{\begin{tabular}{c} From contact angle measurements \\ \end{tabular} } & \multirow{2}{*}{0.0234} & not & \multirow{2}{*}{X} & \multirow{2}{*}{0.0253} & not & \multirow{2}{*}{X} \\  & & measurable & & & measurable & \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} From AFM \\ \end{tabular} } & \multirow{2}{*}{0.0207} & \multirow{2}{*}{0.0165} & \multirow{2}{*}{0.80} & \multirow{2}{*}{0.0196} & \multirow{2}{*}{0.0141} & \multirow{2}{*}{0.72} \\ \hline \multirow{2}{*}{
\begin{tabular}{c} From KIC measurements \\ \end{tabular} } & \multirow{2}{*}{0.5270} & \multirow{2}{*}{0.4150} & \multirow{2}{*}{0.79} & \multirow{2}{*}{1.01} & \multirow{2}{*}{0.84} & \multirow{2}{*}{0.83} \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Surface energy derived from the three different methods tested in this work. was applied on two single calcite crystals using only three fluids (water, diiodomethane, and formamide), yielding a dispersive and polar component of 35.2 and 63.1 mJ/m\({}^{2}\), respectively, with a total surface energy of 98.3 mJ/m\({}^{2}\), whereas the roughness obtained through laser confocal microscopy was 2766 nm on average (Table 6). Following [PERSON] et al. (2015), our AFM study is also able to provide an estimation of the rock surface energy from the adhesion force through the following relationship:  
\[\gamma=\frac{\text{F}_{\text{adh}}}{4\pi\text{R}} \tag{6}\]  
which for Obourg Chalk yields a value \(\gamma^{\text{(dry)}}\) = 20.7 mJ/m\({}^{2}\) in dry conditions, and a smaller value \(\gamma^{\text{(wet)}}\) = 16.5 mJ/m\({}^{2}\) in wet conditions.