## 4 Conclusions  
This paper analyzed the main characteristics of 408 cubic sub-volumes extracted from heterogeneous and naturally fractured carbonate samples before and after injecting HCl solutions. An analysis of X-ray microcomputed tomography images of the sample slices showed the presence of natural fractures and vugs in sample #2, which contributed to the formation of secondary wormholes. The results also showed that the largest increases in permeability occurred in sub-volumes with poor pore connections before the injection of HCl solutions. It has been shown that the absolute permeability \(k\) of sub-volumes from heterogeneous samples correlates well with porosities both before and after the injection of HCl solutions. However, the best correlation was observed between \(k\) and the factors \(\phi r_{p}^{2}\) and \(\phi_{cont}r_{pc}^{2}\), with correlation coefficients \(R^{2}\approx 0.65-0.79\). This indicates that the mean pore radius, in combination with porosity, significantly influences the permeability of the porous medium. This is advantageous as the mean pore radius and porosity can be determined relatively easily. As the results indicate, the presence of natural fractures noticeably affected the relationship between \(k\) and other parameters of the porous medium, resulting in a low \(R^{2}\) value  
Figure 12: Permeability vs. \(\phi r_{p}^{2}\) (**a,c**) and \(\phi_{cont}r_{pc}^{2}\) (**b,d**). of approximately 0.15. However, an improvement in the correlation can be observed after rock dissolution, with an \(R^{2}\) value of approximately 0.35. The results also highlight that for the considered samples, the relationship between \(k\) and the other parameters changes shape after rock dissolution; i.e., the coefficient \(a\) and the power exponent \(n\) have different values before and after rock dissolution. The relationship between \(k\), \(\tau\), and specific surface area \(S\) is described by power laws \(k\sim\tau^{-\alpha}\) and \(k\sim S^{-\beta}\), where \(\alpha\) ranges from 8.5 to 40.1 and \(\beta\) ranges from 5.1 to 17.1, significantly exceeding the values in the Kozeny-Carman equation (\(\alpha=\beta=2\)). The results demonstrate that the relationship between connected and total porosity is described well by the parabolic equation \(\phi_{con}=a\phi^{2}+b\phi+c\) both before and after rock dissolution, giving high correlation coefficients \(R^{2}\approx 0.96-0.99\). The percolation threshold porosity values found in this article align well with those reported in [31], and are approximately three times higher than the values for sandstones. It has been shown that, as a result of rock dissolution, the connectivity between pores increases for heterogeneous samples, as evident from the decrease in the percolation threshold porosity. The results suggest that if the dissolution process has similar effects on pore connectivity in different rock types, this may provide clues about the degree of transferability of the results. This aspect deserves further research. Conceptualization, B.K.A. and B.I. ; methodology, B.K.A. ; software, D.A.B. and K.S.U. ; validation, Z.K.A., G.I.I. and K.S.U. ; formal analysis, D.A.B. ; investigation, Z.K.A. ; resources, D.A.B. ; data curation, B.K.A. ; writing--original draft preparation, Z.K.A. and B.I. ; writing--review and editing, Z.K.A. ; visualization, K.S.U. ; supervision, B.K.A. ; project administration, G.I.I. ; funding acquisition, B.K.A. All authors have read and agreed to the published version of the manuscript. This research was funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant no. AP09058419). The authors also thank Sathayev University for supporting Z.K.A. [PERSON]'s dissertation under the Ph.D. program. Not applicable.