## 4 Microstructural Analysis
### Pore Scale Network Model
The throat length \(l_{t}\) is defined by subtracting the two pore lengths \(l_{i}\) and \(l_{j}\) from the throat total length \(l_{ij}\) (the Euclidean distance from pore \(i\) center to pore \(j\) center)  
\[l_{t}=l_{ij}-l_{i}-l_{j} \tag{1}\]  
The pore lengths \(l_{i}\) and \(l_{j}\) are defined by  
\[l_{i}=l_{i}^{t}\left(1-\alpha\frac{r_{t}}{r_{i}}\right) \tag{2}\]  
\[l_{j}=l_{j}^{t}\left(1-\alpha\frac{r_{t}}{r_{j}}\right) \tag{3}\]  
where \(r_{i}\), \(r_{j}\), and \(r_{t}\) are the radii of pore \(i\), pore \(j\), and throat, respectively; \(l_{i}^{t}\) and \(l_{j}^{t}\) are the Euclidean distances between the centers of pore \(i\) and pore \(j\) to the throat center; \(\alpha\) is the pore-throat segmentation coefficient to decide the pore-throat interface along the pore channels [33], which is chosen to be 0.6 in this study. In the present work, the maximum ball algorithm (see Figure 10) of [PERSON] and [PERSON] [33] is applied to extract the pore network out of the REVs of the carbonate and sandstone samples. By comparing Tables 2 and 3, it can be seen that the porosity of REV2 is closest to the porosity of the whole specimen. Therefore, the REV2 of each sample is chosen for the scrutiny of the pore network. Figure 11 shows the obtained stick-and-ball representation of the pore morphology when applying the maximum ball based extraction of pore network to the REV2 of each sample. The respective volumes of the \"pore\" (i.e., the balls) and the \"throats\" (i.e., the sticks) as well as their connectivity can be derived from the pore network model. More statistical details of the obtained pore-throat model are provided in Table 4.  
Figure 11: Pore network model of REV2 of different samples. (**a**) Carbonate rock A. (**b**) Carbonate rock B. (**c**) Carbonate rock C. (**d**) Carbonate rock D. (**e**) Carbonate rock E. (**f**) Sandstone S.  
Figure 10: Schematic of the length definition for pores and throats reproduced from [33], with permission from American Physical Society, 2022. As can be seen from Table 4, carbonate samples had a much higher coordination number compared to the sandstone sample, indicating better connectivity between the pores. Moreover, the volume of pores in the carbonate samples is significantly higher than that of the sandstone sample. Among the carbonate samples, much higher porosity (see Table 3) and more large pores (see average radius of pores in Table 4) can be observed for sample A. As shown in Figure 11, the REVs of sample B, D and E contained a large number of evenly distributed and well-connected pores compared to the other carbonate samples. However, large pores were scarcer in samples B, D and E compared to samples A and C, which could be concluded from the smaller average radius/volume of pore for the former in Table 4. The number of pores in the REV of sample C was smaller than the counterparts of the others (see number of pores in Table 4) and the pores prevailed in the right half of the REV (see Figure 11c). The large average radius/volume of pore for sample C in Table 4 indicates the prevalence of large pores. As for the sandstone sample, a low coordination number and a small average radius/volume of pores were observed, suggesting small pores and bad connectivity between the pores.