## Results and Discussion
### Effect of Inclination Angle  
To study the effect of the fracture inclination angle on the volume flow rate, the microchannel cross-sectional area was assumed to be 16 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)), and the circular and square microchannel shapes were selected. Three cases with different inclination angles (measured from a vertical position) were investigated as 0\({}^{\circ}\), 13.89\({}^{\circ}\), and 25.82\({}^{\circ}\). The 0\({}^{\circ}\) inclination angle refers to a vertical microchannel. The slanted microchannel forms an oblique cylinder, where the sides are not perpendicular to its bases. The height of the cylinder, whether it is straight vertical or oblique, should be the same (i.e., measure the perpendicular distance from the base to the top surface). Therefore, the volume of the oblique cylinder is the same as that of the straight vertical cylinder if the cross-sectional area and height are the same. In such a case, the volumes of the vertical and the slanted microchannels are constant. Figure 17 shows velocity streamlines through the cement-rock samples. Figure 16: (**a**) Percentage of increase in the volume flow rate vs. number of microchannels, and (**b**) effect of increasing number of microchannels on the volume flow rate. \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Number of Microchannels** & **Cross-Section Area (m\({}^{2}\))** & **Contact Area (m\({}^{2}\))** & **Total Volume Flow Rate (m\({}^{3}\)/s)** & **Percentage of Increase** \\ \hline
0 & 0 & 0 & 5.79 \(\times\) 10\({}^{-10}\) & / \\
1 & 2.44 \(\times\) 10\({}^{-7}\) & 1.13 \(\times\) 10\({}^{-5}\) & 6.34 \(\times\) 10\({}^{-10}\) & 9.50\% \\
2 & 4.92 \(\times\) 10\({}^{-7}\) & 2.25 \(\times\) 10\({}^{-5}\) & 6.85 \(\times\) 10\({}^{-10}\) & 18.31\% \\
3 & 7.41 \(\times\) 10\({}^{-7}\) & 3.37 \(\times\) 10\({}^{-5}\) & 7.32 \(\times\) 10\({}^{-10}\) & 26.42\% \\
4 & 9.85 \(\times\) 10\({}^{-7}\) & 4.51 \(\times\) 10\({}^{-5}\) & 7.65 \(\times\) 10\({}^{-10}\) & 32.12\% \\
5 & 1.27 \(\times\) 10\({}^{-6}\) & 5.91 \(\times\) 10\({}^{-5}\) & 7.89 \(\times\) 10\({}^{-10}\) & 36.27\% \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Effects of the number of microchannels on the volume flow rate. As the inclination angle increased, the flow direction had a tendency towards the microchannel rather than going directly upward. The main difference between a straight vertical microchannel and a slanted microchannel is the contact area of the microchannel. The slanted microchannel with a larger contact area will enable fluid to contact and access the channel more easily. The volume flow rate was calculated based on the mass flow extracted through post-processing. The cross-sectional area and contact area were also extracted. The summary of the results for circular and square shaped microchannels is shown in Tables 11 and 12, respectively. Results shown in Figure 18a indicate that the microchannel contact area increases with an increasing inclination angle. Combining the results from Figure 18b, we conclude that \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Case** & **Inclination Angle, \(\alpha\) (Degree)** & **Volume Flow Rate (m3/s)** & **Cross-Sectional Area (m2)** & **Contact Area (m2)** \\ \hline Circle-i2 & 25.82 & 6.29 \(\times\) 10\({}^{-10}\) & 3.22 \(\times\) 10\({}^{-8}\) & 4.40 \(\times\) 10\({}^{-6}\) \\ Circle-i4 & 13.89 & 6.28 \(\times\) 10\({}^{-10}\) & 3.20 \(\times\) 10\({}^{-8}\) & 4.33 \(\times\) 10\({}^{-6}\) \\ Base case & 0 & 6.25 \(\times\) 10\({}^{-10}\) & 3.18 \(\times\) 10\({}^{-8}\) & 4.31 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Volume flow rate and contact area of slanted microchannels with circular shape.