## 3 Results and Discussion
### The Relationship between Compressive Strength and Capillary Porosity  
Based on the analysis in Figures 6-10, the capillary porosities of the cement stone increased and the compressive strengths decreased with the increase in the _w/c_ ratio at the same hydration time. The compressive strength and the capillary porosity of the cement stone were negatively correlated. To present the relationship between compressive strength and porosity of the cement stone, three widely accepted functions, the Balshin function, Ryshkewitch function, and Schiller function [45; 46] were chosen and described as follows:  
\[y=a(1-x/100)^{b} \tag{1}\]  
Ryshkewitch function,  
\[y=ae^{-bx} \tag{2}\]  
Schiller function,  
\[y=aln(b/x) \tag{3}\]  
One linear function,  
\[y=ax+b \tag{4}\]  
where, \(y\) is the compressive strength of the cement stone, MPa; \(x\) is the capillary porosity of the cement stone, in percentage; \(a\) and \(b\) are fitting parameters. In this study, the Balshin function, Ryshkewitch function, and Schiller function were used to fit the relationship between compressive strength and porosity, and the results are shown in Figure 11. According to the analysis in Figure 11, the correlation coefficients (R) between the measured compressive strength and the predicted results of Balshin function, Ryshkewitch function, and Schiller function were no less than 0.970 (i.e., R\({}^{2}\geq\) 0.94) (Figure 11a-c), which indicated that the correlation between compressive strength and the porosity of cement paste is high. A linear function was used to fit the relationship between compressive strength and porosity. The result is plotted in Figure 11d, and the correlation coefficient was 0.975 (i.e., R\({}^{2}\geq\) 0.95). Furthermore, the accuracy of the model function in predicting compressive strength was estimated based on statistical analysis. The predicted results were compared with the measured results (Table 4). Except for individual abnormal data points, the internally studentized residuals of the prediction model were all within the range of [\(-\)2, 2]. The individual abnormal data points within the range of (\(-\infty\), \(-\)2) \(\cup\) (2, \(\infty\)) of the internal studentized residuals were judged as abnormal points at the 95% confidence level, which was ignored in the regression straight line fitting. Hence, the occurrence of abnormal points will not affect the fitting results. It can also be found from Table 3 that there were no abnormal points in the fitting results of the one linear function, which indicates that \(y=-2.38x+82.76\) was the most representative function. It proved that the relationship between the compressive strength and porosity could be established by the linear function. Combining the results in Figure 10 with Section 3.2, the  
Figure 10: Pore structure of cement stone at 28 days under different _w/c_ ratios using micro-CT (**a**) 0.40; (**b**) 0.44; (**c**) 0.50.  
capillary porosity is the main factor in determining the compressive strength of the cement stone, which is consistent with the result of [PERSON] et al. [16; 45; 46]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Measured Results** & **Predicted Results** & **Residual** &
\begin{tabular}{c} **Internally** \\ **Studentized Residuals** \\ \end{tabular} \\ \hline \multirow{8}{*}{Balshin function} & 49.30 & 54.07 & \(-\)4.77 & 2.093 \\  & 47.27 & 40.60 & 6.67 & \(-\)1.714 \\  & 37.62 & 36.34 & 1.28 & \(-\)0.209 \\  & 30.15 & 33.15 & \(-\)3.00 & 0.992 \\  & 34.08 & 32.44 & 1.64 & \(-\)0.423 \\  & 30.72 & 28.38 & 2.34 & \(-\)0.744 \\  & 24.90 & 26.63 & \(-\)1.73 & 0.432 \\  & 23.91 & 22.50 & 1.41 & \(-\)0.624 \\  & 19.07 & 19.13 & \(-\)0.06 & \(-\)0.274 \\  & 11.40 & 12.83 & \(-\)1.43 & \(-\)0.033 \\  & 8.63 & 10.35 & \(-\)1.72 & \(-\)0.013 \\  & 5.74 & 9.32 & \(-\)3.58 & 0.518 \\  & 49.30 & 54.64 & \(-\)5.34 & 2.233 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical analysis results of prediction functions of compressive strength.