## 3 Results and Discussion
### Permeability vs. Porosities  
Figure 10 demonstrates the distribution of absolute permeability of sub-volumes on total and connected porosities. In this figure, round and square symbols indicate the permeability of sub-volumes before and after their dissolution for heterogeneous (Figure 10a-c) and naturally fractured (Figure 10d-f) samples, respectively. To establish relationships between permeability and porosity, our data were fitted using the equations \(k=a\phi^{n}\), \(k=a(\phi-\phi_{c})^{n}\), and \(k=a\phi_{con}^{n}\) (Figure 10, solid lines). The values of \(\phi_{c}\) for heterogeneous and naturally fractured samples have been taken from Table 4.  
Figure 10: Permeability vs. total (**a**,**d**), percolation threshold (**b**,**e**), and connected (**c**,**f**) porosities.  
Figure 9: Connected and total porosities of sub-volumes extracted from heterogeneous (**a**) and naturally fractured (**b**) samples before and after rock dissolution.  
Figure 10 indicates that, due to rock dissolution, not only did porosity and permeability increase, but the coefficients \(a\) and \(n\) also changed. For example, for a heterogeneous sample, before its dissolution, if the relationship between permeability and total porosity (Figure 10a) is described by the equation \(k=1.02\times 10^{7}\phi^{8.85}\), then after dissolution this equation takes the form of \(k=1.10\ \times 10^{5}\phi^{6.27}\). A similar trend is observed for fractured samples and equations \(k=a(\phi-\phi_{c})^{n}\) and \(k=a\phi_{con}^{n}\) (Figure 10b-f). Therefore, this means that due to rock dissolution, the properties of the samples changed. In other words, the change due to rock dissolution must be considered when modeling fluid flow at larger scales. We also note that the values of the power exponent \(n\) obtained in this study are significantly larger than the exponent in the K-C equation, in which \(n=3\). High values of \(n\) for the carbonate samples considered may be due to the natural rock samples having a more complex pore structure compared to the ideal porous materials, for which the K-C equation works well. Other studies have also shown higher values of \(n\)[11; 13; 15]. For example, [PERSON] obtained \(n=13\) for the dissolution of a cylindrical limestone sample during the injection of CO\({}_{2}\)[13]. [PERSON] and [PERSON] fitted permeability and porosity data during the dissolution of a limestone sample by CO\({}_{2}\) with the equation \(k\sim\phi^{n}\), for which \(n\approx 4.9\)[15].  
Figure 10 demonstrates that there is a large scattering of the permeability distribution of sub-volumes extracted from the naturally fractured sample (Figure 10d-f) compared to the distributions for heterogeneous samples (Figure 10a-c), indicating that the relationship between permeability and porosity is weak in the case of naturally fractured samples. Consequently, for such samples, the permeability should be evaluated considering other properties of the rock. However, the scattering of data in the permeability distribution for naturally fractured sample decreased after rock dissolution, which can be seen from the improvement in the correlation coefficient \(R^{2}\) (Figure 10d-f). This means that as the rock dissolves, the relationship between permeability and porosity becomes stronger. The correlation coefficients of all types of equations used for heterogeneous samples range from 0.64 to 0.78 (Figure 10a-c), while for a fractured sample this coefficient has significantly low values of 0.14-0.36 (Figure 10d-f).  
The sub-volumes extracted from naturally fractured samples have higher permeabilities (Figure 10d-f) compared to heterogeneous samples (Figure 10a-c) at relatively low porosities. Figure 10 also demonstrates that, for heterogeneous samples, the equations \(k=a(\phi-\phi_{c})^{n}\) and \(k=a\phi_{con}^{n}\) can better describe the sharp decrease in absolute permeability at low porosities (Figure 10a,c,e, solid lines) compared to the equation \(k=a\phi^{n}\).