## Results and Discussion  
### Effect of Microchannel Shape  
To investigate the effects of the different microchannel shapes on the volume flow rate, the cross-sectional area of the microchannel was fixed to be 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)). The effects of five different microchannel shapes (i.e., circular, oval, square, rectangular, and triangular shapes) on the volume flow rate were determined and then compared to the  
Figure 11: Positions of selected velocity measuring points (**a**) and corresponding velocity profiles under different meshes (**b**). \begin{table}
\begin{tabular}{l c} \hline \hline
**Case** & **Rock Permeability (m\({}^{2}\))** \\ \hline k1 (krock \(<\) kinterface) & 6.908461 \(\times\) 10\({}^{-17}\) \\ k2 (krock \(=\) kinterface) & 4.9346 \(\times\) 10\({}^{-16}\) \\ k3 (krock \(>\) kinterface) & 1.70737 \(\times\) 10\({}^{-15}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rock permeability values used for three cases of simulation studies. base case. In these analyses, the volume of all microchannels with different shapes was assumed to be the same. This was achieved by keeping the heights and the cross-sectional areas of all microchannel models the same. The graphical results of the velocity streamlines for these cases are shown in Figure 12. The velocity streamlines show the direction of fluid flow through the cement-rock sample. The highest velocities were always observed in the microchannels, indicating that, compared with the base case, microchannels provided more flowable paths for fluid to travel from the inlet to the outlet. The mass flow through the cement, rock, interface and microchannel regions was extracted through post-processing. The volume flow rate of water through each section could be calculated by using Equation (1). \[\dot{V}=\frac{\dot{m}}{\rho} \tag{2}\]  
where \(\dot{V}\) is volume flow rate (m\({}^{3}\)/s), \(\dot{m}\) is mass flow rate (kg/s), and \(\rho\) is fluid density (kg/m\({}^{3}\)). Since the water density at a temperature of 25 \({}^{\circ}\)C is approximately 997 kg/m\({}^{3}\)[32; 33], the calculation results based on this \(\rho\) value are shown in Table 4 for different microchannel shapes. According to the results shown in Table 4, the percentage of the increase in volume flow rate varies from 9.15% to 10.02%. The highest volume flow rate occurs in the rock section, which indicates that the shortest flow path exists in rock. Based on [PERSON]'s law, the volume flow rate is inversely proportional to the distance between the inlet and the outlet if other parameter changes are negligible. Compared with other paths from the inlet to the outlet, the rock region is the first section that fluid flows through, which results in the shortest flow path from the inlet to the outlet. Even though [PERSON]'s equation cannot be directly applied to calculate permeability, the qualitative analyses indicate that the cement  
Figure 12: Effects of the presence of microchannels with different shapes on velocity streamlines through a cement–rock sample. (**a–f**): circular, oval, square, rectangular, triangular, base case. rock samples with a microchannel have higher overall effective permeability values than the base case. In addition, the actual values of the contact areas for microchannels of a 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)) cross-sectional area were extracted and summarized in Table 5. The contact area (i.e., the lateral surface area of the microchannel) was calculated by multiplying the perimeter of the cross-sectional shape of the microchannel by the height of the model. The microchannel has the same cross-sectional area but different shapes will have different perimeter values. Since the height of the model is constant, different perimeter values will lead to different contact areas. The isoperimetric inequality in mathematics implies that, among geometric shapes with equal cross-sectional area, the circle has the smallest perimeter length.