## 3 Results
The results shown in Figure 2 suggest that this is more pronounced for sufficiently open cracks (aspect ratio is greater than \(10^{-3}\)). The reason for a decrease in the \(f\)-parameter with spherical porosity (for constant crack porosity) is that the relative volume of cracks in the total porosity (the ratio of the crack porosity to the total porosity) decreases. This result is in line with the proportionality of the \(f\)-parameter to the permeability-to-porosity ratio reported in the work [34] since the permeability is greatly affected by connected cracks. Figure 2: Cont. Figure 3: \(Cont.\)  
Figure 2: Crack aspect ratio and crack porosity vs. \(f\)-parameter as a color for fixed spherical porosity values. Voids are brine saturated. Calculations are performed for three pairs of methods: (**a**) GSA-\(f\), (**b**) [PERSON]-\(SC\)—[PERSON]-\(f\), and (**c**) DEM—[PERSON]-\(f\). Each color point is related to a solution for aspect ratio and crack porosity provided by Type 1 model. The color is controlled by the \(f\)-parameter value provided by Type 2 model. Figure 4: _Cont._  
Figure 4: The \(f\)-parameter vs. crack density for fixed spherical porosity values. Effective properties are calculated with [PERSON]’s methods, voids are (**a**) brine and (**b**) gas saturated; (**c**) with DEM and [PERSON]-\(f\) methods, voids are gas saturated. The accepted difference between the effective moduli provided by Type 1 and Type 2 models in cases (**a**-**c**) is chosen as 0.1 GPa. (**d**) Effective properties are calculated with [PERSON]’s methods, voids are gas-saturated; the accepted difference between the effective moduli provided by Type 1 and Type 2 models is chosen as 1%. The meaning of numbers in parentheses is (type of saturation, accepted difference between the effective moduli provided by Type 1 and Type 2 models). In Figure 2, a slight increase in the _f_-parameter is observed as the crack aspect ratio (or crack relative opening) decreases at the fixed crack porosity. This means that for fixed crack length, more cracks of lower aspect ratio should exist in rock to keep the same crack porosity as for more open cracks. In turn, the increase in crack number enhances the probability of cracks being connected. Thus, this result seems to be logical. Similar dependences for gas-saturated voids are shown in Figure 3. Again, we see a tendency for an increase in the _f_-parameter with crack porosity. However, now solutions for very thin cracks (aspect ratio around \(10^{-4}\)) can be obtained. For these thin cracks, we observe another behavior of the _f_-parameter with porosity. Namely, we can see the high values of connectivity parameter in the whole range of spherical porosity without a tendency to decrease with porosity. As the cracks become more open the effect of a decrease in this parameter with the spherical porosity (as is observed for brine-saturated pores) starts to manifest itself. If we compare the results of three pairs of methods, we notice that the results of the first two pairs GSA-SC \(-\) GSA-_f_, and Berryman-SC \(-\) Berryman-_f_ are almost identical for brine-saturated voids. It follows from that GSA and Berryman methods give the same effective moduli for spherical inclusions. In the case of gas-saturated voids, i.e., the media with a greater difference in elastic properties between matrix and fluids, the results on the scatter plots are slightly different (Figure 3). Figure 4 demonstrates the dependence of the _f_-parameter on the decimal logarithm of crack density. Figure 4a,b shows the results for pairs Berryman-SC \(-\) Berryman-_f_ for brine and gas-saturated models, respectively. Figure 4c exemplifies similar dependences for pair DEM \(-\) Berryman-_f_ for gas-saturated case. The chosen measure of closeness in the elastic moduli for all of these cases is 0.1 GPa. A pronounced positive correlation of the _f_-parameter with the crack density is observed.