## 3 Results and Discussion
### Simple Regression
Fracture toughness is generally inversely related to porosity and water absorption percentage, and is directly related to the ultrasonic wave velocity, density, and point load index. It is clear from Table 8 that by applying petrographic classification to the data, better correlations between fracture toughness and index properties were obtained. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & & **n** & **V\({}_{\text{S}}\)** & **V\({}_{\text{PDy}}\)** & **V\({}_{\text{PSat}}\)** & **\(\rho\)Sat** & **\(\rho\)d** & **WA** & **RN** & **IS\({}_{\text{50 Dry}}\)** & **IS\({}_{\text{50 Sat}}\)** \\ \hline \multirow{3}{*}{All Data} & K\({}_{\text{ICFDry}}\) & 46.92 & 29.27 & 44.22 & 41.34 & 5.40 & 35.16 & 48.86 & 0.96 & 52.27 & 68.56 \\ \cline{2-10}  & K\({}_{\text{ICFDst}}\) & 58.68 & 44.09 & 73.27 & 71.23 & 28.30 & 41.22 & 57.91 & 1.82 & 29.48 & 58.98 \\ \hline \multirow{3}{*}{Class 1A} & K\({}_{\text{ICFDry}}\) & 94.28 & 65.43 & 51.56 & 97.81 & 83.07 & 95.45 & 95.84 & 23.65 & 27.52 & 2.36 \\ \cline{2-10}  & K\({}_{\text{ICFDSat}}\) & 1.66 & 4.12 & 11.58 & 5.02 & 34.64 & 3.49 & 3.43 & 2.18 & 23.83 & 1.64 \\ \hline \multirow{3}{*}{Class 1B} & K\({}_{\text{ICFDry}}\) & 46.24 & 0.62 & 18.49 & 44.49 & 35.26 & 48.58 & 47.47 & 54.02 & 87.42 & 72.93 \\ \cline{2-10}  & K\({}_{\text{ICFDy}}\) & 73.79 & 8.65 & 55.35 & 67.90 & 43.97 & 73.10 & 73.96 & 34.50 & 76.39 & 74.30 \\ \hline \multirow{3}{*}{Class 2} & K\({}_{\text{ICFDry}}\) & 83.36 & 64.00 & 85.38 & 79.39 & 0.25 & 75.86 & 83.17 & 35.52 & 98.41 & 91.78 \\ \cline{2-10}  & K\({}_{\text{ICFDy}}\) & 69.89 & 87.24 & 85.19 & 80.28 & 0.64 & 80.10 & 68.56 & 15.76 & 65.77 & 82.81 \\ \hline \multirow{3}{*}{Class 3A} & K\({}_{\text{ICFDry}}\) & 30.41 & 16.66 & 30.05 & 14.33 & 8.70 & 55.93 & 35.42 & 0.00 & 63.35 & 45.03 \\ \cline{2-10}  & K\({}_{\text{ICFDy}}\) & 56.08 & 24.81 & 51.65 & 42.39 & 26.22 & 69.65 & 58.76 & 0.95 & 49.00 & 23.58 \\ \hline \multirow{3}{*}{Class 3B} & K\({}_{\text{ICFDry}}\) & 68.80 & 27.68 & 9.00 & 38.35 & 69.34 & 44.88 & 68.53 & 41.50 & 0.12 & 25.05 \\ \cline{2-10}  & K\({}_{\text{ICFDy}}\) & 95.84 & 30.80 & 40.67 & 67.06 & 29.94 & 67.12 & 95.06 & 84.82 & 6.23 & 64.02 \\ \hline \end{tabular}
\end{table} 
Table 7: Linear correlation between the index properties and dry and saturated fracture toughness for all of the limestones and individual classes. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Dependent Variable** & **Independent Variables** & **Predictive Model** & **R\({}^{2}\) (\%)** & _p-Value_ \\ \hline \multirow{3}{*}{All Data} & K\({}_{\text{ICF-Dry}}\) & I\({}_{\text{SS0-Sat}}\) & y = 0.891 \(\times\)\(@^{0.596}\) & 69.5 & 0.000 \\  & K\({}_{\text{ICF-Sat}}\) & V\({}_{\text{P-Sat}}\) & y = 0.008 \(\times\)\(@^{2.758}\) & 75.2 & 0.000 \\  & & V\({}_{\text{P-Sat}}\) & y = \(-0.087\times\)\(@^{}\)\(+\)\(1.62\) & 97.9 & 0.011 \\  & n & y = 0.044\(\times\)\(@^{}\)\(+\)\(0.994\) & 94.3 & 0.029 \\ \multirow{3}{*}{Class 1A} & K\({}_{\text{ICF-Dry}}\) & \(\rho\)Sat & y = 1.4586 \(\times\)\(@^{}\)\(-\)\(0.032\) & 93.1 & 0.035 \\  & pd & y = \(-3.229\)ln(x) + 4.257 & 95.5 & 0.023 \\  & W.A. & y = 0.999 \(\times\)\(@^{0.107\times}\) & 96.0 & 0.020 \\  & I\({}_{\text{SS0-Sat}}\) & y = 0.616 ln(x) + 0.871 & 73.1 & 0.003 \\  & I\({}_{\text{SS0-Dry}}\) & y = 0.756 \(\times\)\(@^{0.902}\) & 88.4 & 0.000 \\  & V\({}_{\text{P-Sat}}\) & y = 0.003 \(\times\)\(@^{0.324}\) & 80.3 & 0.003 \\  & n & y = 1.567 \(\times\)\(@^{}\)\(@^{}\)\(@^{}\)\(@^{}\)\(@^{}\)\(@^{}\)\(@^{}\)\(@^{}\)\(@^{}\)\(@^{}\)\(@text{}\)\(@^{}\