## 4 Discussion
To analyze how the \(f\)-parameter is related to characteristics of pore/crack space, we calculate the effective elastic moduli of carbonate rock using the five mentioned above methods varying the model parameters in a wide range. Then, among the obtained solutions for the effective bulk and shear moduli, we choose those that could be called \"similar\" and associate the \(f\)-parameter provided by a Type 2 model with the respective crack characteristics provided by a selected Type 1 model. We analyze three pairs of models (Type 1 model \(-\) Type 2 model): (1) GSA-SC \(-\) GSA-\(f\), (2) Berryman-SC \(-\) Berryman-\(f\), and (3) DEM \(-\) Berryman-\(f\). The results of our analysis demonstrate that the \(f\)-parameter can be associated with cracks in a double porosity model. Thus, an increase in the crack porosity (Figures 2 and 3) or crack density (Figure 4) is accompanied by an increase in the \(f\)-parameter values. For cracks of aspect ratio \(10^{-3}\) and higher, if we keep the crack porosity constant but increase the spherical porosity the \(f\)-parameter becomes to decrease. This is explained by diminishingthe crack contribution to the overall effect of porosity on the elastic moduli. This result is consistent with the findings of [34] showing that the \(f\)-parameter is proportional to the logarithm of the ratio of permeability to porosity. It is interesting that this effect is not observed for very thin (aspect ratio around \(10^{-4}\)) gas-saturated cracks. In this case, high values of the \(f\)-parameter are seen in the whole range of spherical porosity. These results suggest that \(f\)-models have a good chance to detect and quantify the zones of enhanced fracturing and hence permeability. Note that a good correlation between the \(f\)-parameter with crack density gives favor to \(f\)-models in characterizing the fractured zones. The \(f\)-models (Type 2 models) contain only one parameter in contrast to Type 1 models which incorporate two parameters to specify cracks. Commonly, a wide range of combinations (crack porosity, crack aspect ratio) could be inverted from measured elastic wave velocities and density, based on the rock-physics modeling. These characteristics of cracks are very difficult to estimate experimentally in order to impose bounds on the possible solutions. The use of crack density allows one to decrease the non-uniqueness in the characterization of fractured zones. Note that [PERSON]'s [57; 58] model for effective elastic properties of rock with aligned cracks also operates with a single parameter characterizing cracks, namely, crack density instead of the pair \"crack aspect ratio, crack porosity\". Besides, the inversion based on the \(f\)-models works faster compared to the inversion based on the self-consistent models due to the smaller number of unknowns and absence of iterations. This fact is significant in the case when the inversion should be performed for a large number of wells with extended depth intervals of interest. The theoretical models should be validated, ideally, on artificial samples with known geometry of inclusions. Such a validation of \(f\)-model was carried out in [29]. However, in the case of rocks, this type of validation is quite complex and may be possible with the use of digital rock approach based on data of electron microscopy or computed tomography. Besides, the parameter \(f\) in the second model has no an experimentally measurable analog. In addition, along this way, other problems arise. For example, so-called segmentation problem exists that gives a solution to the question of where the solid material ends and the void begins. This problem is not easily solved for carbonate samples due to their specific microstructure. Another problem in the digital rock approach is that the properties of intergranular material should be incorporated in the digital model but they are difficult to determine. We have a few examples of \"indirect\" validating of the effective medium models.