## 2 Materials and Methods
### Determination of Microchannel Shapes  
Cement paste is a heterogeneous material [20; 21; 22]. Even if it is under uniform loading conditions, unlike the homogeneous isotropic materials, the stress distribution inside the cement column will not be uniform [21]. Anisotropic stress distribution would result in the diversity and the complexity of microcracks inside the material. When different microcracks form a microchannel, the actual shape, size, and length of the microchannel cannot be determined. Due to technical and instrumental limitations, the realistic structure of microchannels is still unclear. In this case, it is suitable to first assume a pre-determined shape for a microchannel. Theoretically, a complex shape can be considered as a combination of simple shapes. Therefore, it is better to first have a comprehensive understanding of the effects of basic shapes of microchannels before studying the complex shapes.  
Two kinds of fracture models are commonly used. One considers fractures that have a polygonal shape, and the other considers those that have a circular or oval shape [23]. A triangle is the polygon that has the least number of sides. A circle is a special polygon with countless sides. The square and rectangle can be considered as similar shapes but with different ratios of length and width. The same situation occurs in the case of a circle and an oval. Therefore, in the current study, we have considered circular, oval, square, rectangular, and triangular shapes of microchannels (Figure 4). The next step is to determine the appropriate sizes for those shapes of microchannels.  
Since the binarized CT image of the cement-rock sample is a cross-sectional view, we actually use the term \"size\" of the artificial microchannel to indicate the cross-sectional area of the microchannel. The maximum cross-sectional area of the microchannel is 64 pixel\({}^{2}\), which is equal to a physical size of 0.2273 mm\({}^{2}\). This is because we assume that the artificial microchannel is limited to the interface region and, therefore, it should not touch the boundaries of the interface. Figure 4 shows various shapes and parameters of the microchannels with a cross-sectional area of 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)). The height and width of all shapes are rounded to the nearest integers because the minimum unit in the graphical modification software is 1 pixel. It is anticipated that a microchannel with a larger cross-sectional will have a greater effect on the results than that of the one with a smaller cross-sectional area. Therefore, in order to minimize errors (due to the size effect of the cross-sectional area), the artificial microchannel with a cross-sectional area of 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)) were used to investigate the effect of the microchannel shape. To investigate the effect of the size of the microchannel cross-sectional area, two more  
Figure 3: The binarized CT images. (**a–c**): rock phase, cement phase, interface between cement and rock.  
microchannel cross-sectional areas were used, which were 49 pixel\({}^{2}\) (0.1741 mm\({}^{2}\)) and 36 pixel\({}^{2}\) (0.1279 mm\({}^{2}\)).  
The microchannels were embedded into the binarized CT images of the cement-rock interface using Macromedia Firework 8 software. To investigate the effect of the microchannel shape, we have used only one single microchannel in each case. Figure 5 shows the modified CT images for different shapes of simulated microchannels with a cross-sectional area of 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)). For example, a black circular shape with an area of 64 pixel\({}^{2}\) (0.2273 mm\({}^{2}\)) was drawn on the right-center of the white interface region, which is the case shown in Figure 5a. A total of 93 images with such modifications were prepared and named in order. For other shapes, similar steps were followed.