## 3 Results and Discussion
### Simple Regression  
Linear/nonlinear simple regression with a 95% confidence level is used to determine the correlation between the fracture toughness and index properties. According to the results of the Pearson correlation, there is a high correlation among the KIC calculated using the [PERSON] et al. [44], [PERSON] [45], and [PERSON] et al. [46] methods for the dry (Table 4) and saturated states (Table 5). Therefore, only the results of the Fowell method are used to determine the relationship between the KIC and index properties. Firstly, the correlation between the fracture toughness of the dry and saturated states with the index properties was investigated for all of the samples. The simple linear regression showed poor results. This is because limestone has a wide range of textural and petrological features. Therefore, according to the textural diversity of limestone, petrographic studies of the samples were conducted using optical microscopy and the samples were classified in five classes accordingly (Table 6). Mud supported limestones were placed in class 1A, grain supported samples were categorized as class 1B, sandy limestones were placed in class 2, coarse crystal dolomite limestones were placed in class 3A, and microcrystalline dolomite limestones were placed in class 3B. By applying this classification, more meaningful correlations were obtained between the fracture toughness and physical characteristics of the studied limestones (Table 7). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rock Code** & **Class** & **Petrography** & **Rock Code** & **Class** & **Petrography** \\ \hline R5 & 1A & Mudstone & R10 & 2 & Sandy limestone \\ R18 & 1A & Wackestone & R8 & 2 & Sandy limestone \\ R20 & 1A & Wackestone & R7 & 2 & Sandy limestone \\ R21 & 1A & Framestone & R2 & 2 & Sandy limestone \\ R19 & 1B & Floatstone & R1 & 2 & Sandy limestone \\ R4 & 1B & Packstone & R22 & 3A & Dolomite limestone \\ R23 & 1B & Packstone & R16 & 3A & Dolostone \\ R15 & 1B & Packstone & R29 & 3A & Dolomitic limestone \\ R6 & 1B & Grainstone & R17 & 3A & Dolomite limestone \\ R3 & 1B & Grainstone & R30 & 3A & Dolomite limestone \\ R13 & 1B & Grainstone & R24 & 3A & Dolostone \\ R14 & 1B & Grainstone & R25 & 3B & Dolo mudstone \\ R12 & 1B & Rudstone & R28 & 3B & Dolo mudstone \\ R9 & 2 & Sandy Limestone & R26 & 3B & Dolo mudstone \\ R11 & 2 & Sandy Limestone & R27 & 3B & Dolo mudstone \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Rock classes based on petrographic study. \begin{table}
\begin{tabular}{l c c c} \hline \hline  & & **KICW-Dy (MPa m\({}^{12}\))** & **KIC-Dy (MPa m\({}^{12}\))** \\ \hline KIC-Dy (MPa m\({}^{1/2}\)) & RMSE & 0.0108 & 0.0181 \\  & R2 (\%) & 99.91 & 99.75 \\  & Adj. R2 (\%) & 99.91 & 99.74 \\  & \(p\)-Value & 0.000 & 0.000 \\ Equation & KICW-Dy = 1.11203 KIC-Dy \(-\) & KIC-Dy = 1.1329 KIC-Dy \(-\) \\  & 0.00633 & 0.018 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Correlation among KIC determined from three different methods under a dry condition. \begin{table}
\begin{tabular}{l c c c} \hline \hline  & & **KICW-Sat (MPa m\({}^{12}\))** & **KIC-Sat (MPa m\({}^{12}\))** \\ \hline KIC-Sat (MPa m\({}^{1/2}\)) & RMSE & 0.0109 & 0.01492 \\  & R2 (\%) & 99.94 & 99.89 \\ KIC-Sat (MPa m\({}^{1/2}\)) & Adj. R2 (\%) & 99.94 & 99.89 \\  & \(p\)-Value & 0.000 & 0.000 \\  & \(p\)-Value & \(\mathrm{KICW-Sat}=1.11653\) KIC-Sat \(-\) & KIC-Sat = 1.13465 KIC-Sat \(-\) \\ Equation & & 0.00839 & 0.01444 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Correlation among KIC determined from three different methods under a saturated state. The relationship between the K\({}_{\text{IC}}\) and index properties for each class were determined under dry (Figure 10) and saturated conditions (Figure 11). The best relationship was determined for each class as well as for all of the rock samples based on simple linear/nonlinear regression analyses (Table 8). Classification of the samples according to petrographic studies showed acceptable relationships obtained from simple linear/nonlinear regression.