## Results and Discussion
### Effect of Microchannel Size  
For each microchannel shape, three different sizes of microchannel cross-sectional areas, 64 pixel\({}^{2}\), 49 pixel\({}^{2}\), and 36 pixel\({}^{2}\) (equal to the physical area of 0.2273 mm\({}^{2}\), 0.1741 mm\({}^{2}\), and 0.1279 mm\({}^{2}\), respectively), were investigated. Tables 4, 6 and 7 show the volume flow rate through cement-rock models containing microchannels with a cross-sectional area of 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)), 49 pixel\({}^{2}\) (0.1741 mm\({}^{2}\)), and 36 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)), respectively. The highest volume flow rate occurs through the rock section, which is consistent with the findings from the previous section. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Section**} & \multicolumn{6}{c}{**Volume Flow Rate (m\({}^{3}\)/s)**} \\ \cline{2-7}  & **Base Case** & **Circle 64** & **Oval 64** & **Square 64** & **Rectangle 64** & **Triangle 64** \\ \hline Cement & 6.31 \(\times\) 10\({}^{-13}\) & 5.13 \(\times\) 10\({}^{-13}\) & 5.10 \(\times\) 10\({}^{-13}\) & 5.12 \(\times\) 10\({}^{-13}\) & 5.03 \(\times\) 10\({}^{-13}\) & 5.02 \(\times\) 10\({}^{-13}\) \\ Interface & 2.19 \(\times\) 10\({}^{-10}  
In order to compare the effects of the different sizes of microchannel cross-sectional areas simultaneously, three sets of data, shown in Table 8, were also plotted in Figure 14. The numbers one to five represent different shapes of microchannel (varying from circle to triangle). According to the results shown in Figure 14, for each shape of microchannel, the volume flow rate increases with the increasing microchannel cross-sectional area. This finding is consistent with the underlying physics of the process implied by [PERSON]'s law, which also states that volume flow rate is proportional to cross-sectional area. Moreover, for a microchannel with a cross-sectional area of 36 pixel\({}^{2}\) (0.1741 mm\({}^{2}\)), the volume flow rates for all shapes are close to each other. Therefore, it can be assumed that there is not much \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Section**} & \multicolumn{6}{c}{**Volume Flow Rate (m\({}^{3}\)/s)**} \\ \cline{2-7}  & **Base Case** & **Circle 49** & **Oval 49** & **Square 49** & **Rectangle 49** & **Triangle 49** \\ \hline Cement & \(6.31\times 10^{-13}\) & \(5.15\times 10^{-13}\) & \(5.11\times 10^{-13}\) & \(5.15\times 10^{-13}\) & \(5.12\times 10^{-13}\) & \(5.09\times 10^{-13}\) \\ Interface & \(2.19\times 10^{-10}\) & \(1.82\times 10^{-10}\) & \(1.81\times 10^{-10}\) & \(1.82\times 10^{-10}\) & \(1.81\times 10^{-10}\) & \(1.80\times 10^{-10}\) \\ Rock & \(3.59\times 10^{-10}\) & \(3.55\times 10^{-10}\) & \(3.53\times 10^{-10}\) & \(3.55\times 10^{-10}\) & \(3.53\times 10^{-10}\) & \(3.55\times 10^{-10}\) \\ Microchannel & / & \(9.36\times 10^{-11}\) & \(9.58\times 10^{-11}\) & \(9.40\times 10^{-11}\) & \(9.54\times 10^{-11}\) & \(9.77\times 10^{-11}\) \\ Total & \(5.79\times 10^{-10}\) & \(6.31\times 10^{-10}\) & \(6.31\times 10^{-10}\) & \(6.32\times 10^{-10}\) & \(6.31\times 10^{-10}\) & \(6.34\times 10^{-10}\) \\ \% of increase & / & \(8.98\%\) & \(8.98\%\) & \(9.15\%\) & \(8.98\%\) & \(9.50\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Volume flow rate through cement–rock sample: comparison of the base case (no microchannel) and microchannels with a 49 pixel\({}^{2}\) (0.1741 mm\({}^{2}\)) cross-sectional area.