## 2 Materials and Methods
### Determination of Microchannel Inclination Angles  
A simulated microchannel with an inclination angle can be obtained by changing the location of the microchannel in each cross-sectional CT image. A graphical description of the process of generating a slanted microchannel is shown in Figure 7. Firstly, a black circular shaped microchannel was embedded into the CT image at a location shown in Figure 7a. This was considered the first CT image. Then, this black circular microchannel was moved downward one pixel and was saved as the second CT image. By repeating this step until the microchannel reached the position shown in Figure 7c, all the CT image modifications were finished. Since these CT images are the cross-sectional view of the interface region, by combining all the images in order, a three-dimensional model can be formed, which is shown in Figure 7d. The top of the cylinder corresponds to the first image and the bottom corresponds to the last image. The total moving distance of the microchannel is the distance between the location of the first CT image and that of the last CT image. In 3D view, the length of the red line in Figure 7d is the total moving distance of the artificial microchannel. Different inclination angles can be achieved by changing the value of the total moving distance. The microchannel shape was fixed as a circle and a square, respectively, and the microchannel cross-sectional area was fixed as 16 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)) for each microchannel shape. The circular and square shapes were chosen to minimize errors in the cross-sectional  
Figure 6: Modified CT images for different quantities of simulated microchannel with square shape and cross-sectional area of 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)). (**a-e**): microchannel quantity = 1, 2, 3, 4, 5. Figure 7: Graphical description of forming a slanted microchannel in the interface region. (**a–d**): first CT image, intermediate CT image, last CT image, 3D view of slanted microchannel. area of the microchannel. The cross-sectional area was chosen to be 16 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)) because this area provided the longest moving distance for the microchannel compared with the larger one. Additionally, if the microchannel cross-sectional area were too small, there would not be much difference in the final results, which is undesirable for data analysis. Therefore, after determining the shape and cross-sectional area, the next step was to determine the inclination angle of the microchannel. In this study, we assumed that a microchannel was only located at the interface region and thus the moving distance of the microchannel was limited. In the cross-sectional view of the CT image, the maximum vertical movement for a 16 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)) microchannel was 45 pixels (2.68 mm). Since the height of the 3D model was 93 pixels (5.54 mm), the maximum inclination angle was around 25.82 degrees. In order to make a uniform reduction in the inclination angle, the intermediate inclination angle was chosen to be around 13.89 degrees and the smallest inclination angle was 0 degrees, which represented the base case containing a vertical microchannel without any inclination. The different moving distances of microchannels to form the desired inclination angles are shown in Figure 8.  
When embedding inclined channels in micro-CT images of rock-cement samples, the procedure for constructing each case was slightly different. In order to generate a microchannel with a 25.82 degree inclination angle, the required moving distance of the microchannel was 45 pixels. Each image had a thickness of 1 pixel and the required height of the 3D model was 93 pixels. In this case, the microchannel was moved downward one pixel for every two images, which formed a total of 90 images. The remaining 3 images were the same as the last image. In this way, a total of 93 images were prepared and named in order.