## 3 Results  
By comparing the effective elastic properties obtained for two types of carbonate rock, we can associate the \(f\)-parameter values with the crack's aspect ratio and porosity. For the estimation of effective elastic moduli of the double porosity model we apply the classical self-consistent method of the effective medium theory (GSA-SC), Berryman's method (Berryman-SC), and DEM. In the case of GSA-SC and Berryman-SC, the pores, cracks, and calcite grains are embedded in effective media simultaneously, while in the case of the DEM, we insert the saturated voids in the calcite matrix sequentially: cracks at first and then the spherical pores. Regarding the calculation procedure for the self-consistent methods (GSA-SC and Berryman-SC), we use an iterative scheme, in which properties of effective media were obtained in the previous iteration step. In the initial step, we take the elastic characteristics of effective media calculated with the Hill approximation (average of the results provided by Voigt and Reuss methods). The iterative procedure is stopped if the maximum change of moduli is less than 0.01 GPa. We vary the aspect ratio of cracks \(\alpha_{\text{crack}}\) from \(10^{-4}\) to 0.1, and crack porosity \(\phi_{\text{crack}}\) from 0 to 2%, and we calculate the effective properties for all possible combinations of these parameters. Instead of \(\alpha_{\text{crack}}\) and \(\phi_{\text{crack}}\), we use the decimal logarithm of these parameters as it is more convenient for calculation and visualization. For obtaining \(f\)-model properties we use the generalized singular approximation method (GSA-\(f\)) and modification of the Berryman method (Berryman-\(f\)) with the introduced parameters \(f\) that is changing from 0 to 1. All these calculations are performed for different but fixed porosity of spherical pores \(\phi_{\text{pore}}\) varying from 2 to 20%. Then, we choose the pairs of moduli (bulk and shear) from the double porosity model (Type 1) and \(f\)-model (Type 2) whose difference is less than 0.1 GPa. This value of the threshold is chosen since, commonly, in geophysics, the bulk and shear moduli of minerals and rocks are specified with one decimal place. Figures 2 and 3 show scatter plots resulting in pairs of \(\log_{10}(\alpha_{\text{crack}})\) and \(\log_{10}(\phi_{\text{crack}})\) versus \(f\)-parameter colored depending on the values of \(f\)-parameter. The volume fraction of spherical pores is fixed for each realization. We consider the cases when the spherical porosity equals 2, 5, 15, and 20%. Besides we analyze the behavior of the \(f\)-parameter vs. crack density for fixed spherical porosity (Figure 4). The crack density is a convenient parameter combining the crack porosity and aspect ratio thereby providing a single parameter to characterize fractured zones. The formula for the crack density has the form  
\[\varepsilon= \frac{3\Phi\text{crack}}{4\pi\alpha\text{crack}}. \tag{11}\]  
In Figures 2 and 3 we observe some tendencies common for three pairs of methods: (1) GSA-SC \(-\) GSA-\(f\), (2) Berryman-SC \(-\) Berryman-\(f\), and (3) DEM \(-\) Berryman-\(f\). As follows from Figures 2 and 3 the values of the \(f\)-parameter reflecting pore connectivity increase with increasing crack porosity for a fixed aspect ratio. Assuming brine saturated voids (Figure 2), for low porosities (less than 10%) the presence of a high volume of cracks increases the \(f\)-parameter more significantly than for higher porosities. As seen in Figure 2, the maximum values of the \(f\)-parameter depend on the spherical porosity of brine-filled pores. Thus, the \(f\)-parameter reaches a maximum of 0.7 (Berryman-SC \(-\) Berryman-\(f\)) and 0.6 (DEM \(-\) Berryman-\(f\)) for 20% volume fractions of spherical pores. However, for low porosity (2%) these values are higher attaining, respectively, 0.98 and 0.97. As follows from Figure 2 the possible maximum value of \(f\)-parameter tends to decrease with increasing porosity and it is true for all pairs of methods.