## Results and Discussion
### Effect of Microchannel Size
\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 36** & **Oval 36** & **Square 36** & **Rectangle 36** & **Triangle 36** \\ \hline Cement & \(6.31\times 10^{-13}\) & \(5.18\times 10^{-13}\) & \(5.17\times 10^{-13}\) & \(5.17\times 10^{-13}\) & \(5.16\times 10^{-13}\) & \(5.16\times 10^{-13}\) \\ Interface & \(2.19\times 10^{-10}\) & \(1.83\times 10^{-10}\) & \(1.83\times 10^{-10}\) & \(1.83\times 10^{-10}\) & \(1.83\times 10^{-10}\) & \(1.83\times 10^{-10}\) \\ Rock & \(3.59\times 10^{-10}\) & \(3.56\times 10^{-10}\) & \(3.55\times 10^{-10}\) & \(3.54\times 10^{-10}\) & \(3.56\times 10^{-10}\) & \(3.56\times 10^{-10}\) \\ Microchannel & / & \(9.07\times 10^{-11}\) & \(9.11\times 10^{-11}\) & \(9.14\times 10^{-11}\) & \(9.20\times 10^{-11}\) & \(9.20\times 10^{-11}\) \\ Total & \(5.79\times 10^{-10}\) & \(6.30\times 10^{-10}\) & \(6.30\times 10^{-10}\) & \(6.29\times 10^{-10}\) & \(6.31\times 10^{-10}\) & \(6.31\times 10^{-10}\) \\ \% of increase & / & \(8.81\%\) & \(8.81\%\) & \(8.64\%\) & \(8.98\%\) & \(8.98\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Volume flow rate through cement–rock sample: comparison of the base case (no microchannel) and microchannels with a 36 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)) cross-sectional area. Figure 13: Effect of the microchannel contact area on the volume flow rate: data for cases with 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)) microchannel. difference in the volume flow rate for different shapes if the microchannel cross-sectional area is lower than 0.1741 mm\({}^{2}\). For further analysis of the contact area size effect, the contact area data for 49 pixel\({}^{2}\) (0.1741 mm\({}^{2}\)) and 36 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)) cases were also extracted (Table 9). As the microchannel cross-sectional area decreases, the linear relationship between the contact area size and the volume flow rate becomes weaker. This may be because, for smaller microchannel cross-sectional areas, there is not much difference between the actual values of the volume flow rates, which causes the data to cluster on the graph. Another reason could be the error caused by not being able to control the microchannel cross-sectional area accurately. The actual cross-sectional area of the embedded microchannel depends on the meshing of the 3D model, and the meshing is automatically generated  
Figure 14: Volume flow rate through a cement–rock sample: effect of variable microchannel shape and cross-sectional area. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{**Microchannel Shape**} & \multicolumn{3}{c}{**Volume Flow Rate (m\({}^{3}\)/s)**} \\ \cline{2-4}  & **0.2273 mm\({}^{2}\) (64 Pixel\({}^{2}\))** & **0.1741 mm\({}^{2}\) (49 Pixel\({}^{2}\))** & **0.1279 mm\({}^{2}\) (36 Pixel\({}^{2}\))** \\ \hline
1. Circle & 6.32 \(\times\) 10\({}^{-10}\) & 6.31 \(\times\) 10\({}^{-10}\) & 6.30 \(\times\) 10\({}^{-10}\) \\
2. Oval & 6.33 \(\times\) 10\({}^{-10}\) & 6.31 \(\times\) 10\({}^{-10}\) & 6.30 \(\times\) 10\({}^{-10}\) \\
3. Square & 6.34 \(\times\) 10\({}^{-10}\) & 6.32 \(\times\) 10\({}^{-10}\) & 6.29 \(\times\) 10\({}^{-10}\) \\
4. Rectangle & 6.35 \(\times\) 10\({}^{-10}\) & 6.31 \(\times\) 10\({}^{-10}\) & 6.31 \(\times\) 10\({}^{-10}\) \\
5. Triangle & 6.37 \(\times\) 10\({}^{-10}\) & 6.34 \(\times\) 10\({}^{-10}\) & 6.31 \(\times\) 10\({}^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Volume flow rate through a cement–rock sample: effect of variable microchannel shape and the size of the cross-sectional area.