## 2 Materials and Methods
### [PERSON]'s Method  
[PERSON]'s method [34] is another realization of the self-consistent effective medium methods [44; 45; 46; 47]. Based on the results of [PERSON] and [PERSON] [48], [PERSON] obtained expressions for calculating the effective moduli (Equation (6)) of an isotropic matrix with randomly oriented ellipsoidal isotropic inclusions. Note that in this method no host medium is assumed. The pieces of mineral material, pores, cracks, and other impurities are considered as ellipsoidal inclusions. The formulas for calculating the effective elastic moduli have the form:  
\[\sum_{i=1}^{\mathrm{N}}v_{i}(K_{i}-K_{\mathrm{SC}}^{*})P^{*i}=0,\ \sum_{i=1}^{ \mathrm{N}}v_{i}(\mu_{i}-\mathrm{u}_{\mathrm{SC}}^{*})Q^{*i}=0, \tag{6}\]  
where \(K_{\mathrm{SC}}^{*}\) and \(\mathrm{u}_{\mathrm{SC}}^{*}\) are effective bulk and shear moduli; \(K_{i}\) and \(\mu_{i}\) are the inclusion's bulk and shear moduli, \(v_{i}\) are the volume fractions of inclusions, \(N\) is the number of inclusion types. The coefficients \(P^{*i}\) and \(Q^{*i}\) depend on the inclusion's shape and properties of the effective medium. Equations of the \(P^{*i}\) and \(Q^{*i}\) for the general case of ellipsoidal inclusions are rather bulky and are not shown here. They can be found in [48]. Equation (6) contain unknown effective elastic moduli in the both left- and right-hand sides. Such equations are solved by iterations.  
Note that Equation (6) can be written in a form more convenient for computations:\[K^{*}_{\text{SC-}f}=\sum_{i=1}^{N}v_{i}K_{i}P^{ei}/\sum_{i=1}^{N}v_{i}P^{ei},\ \mu^{*}_{\text{SC-}f}=\sum_{i=1}^{N}v_{i}\mu_{i}Q^{ei}/\sum_{i=1}^{N}v_{i}Q^{ei}, \tag{7}\]  
Equations (6) and (7) are the results of a series of long-wavelength scatterings experiments. During the experiments, a small sphere of the true composite material (heterogeneous material) is embedded in a medium whose properties are changed in a controlled manner, achieving zero scattering from the composite sphere. If we change the sphere properties to comparison body properties and consider comparison body properties as provided by Equation (3) with \(f\)-parameter, we derive an alternative version of [PERSON]'s formulas accounting for pore connectivity. Note that in this case there is no need for an iteration scheme as for Equation (6) or (7).  
\[K^{*}_{\text{SC-}f}=\sum_{i=1}^{N}v_{i}K_{i}P^{Ci}/\sum_{i=1}^{N}v_{i}P^{Ci},\ \mu^{*}_{\text{SC-}f}=\sum_{i=1}^{N}v_{i}\mu_{i}Q^{Ci}/\sum_{i=1}^{N}v_{i}Q^{Ci}, \tag{8}\]  
where \(P^{Ci}\) and \(Q^{Ci}\) are evaluated similarly to those as in Equations (6) and \(K^{*}=K^{C}\), \(\mu^{*}=\mu^{C}\). The comparison body moduli \(K^{C}\) and \(\mu^{C}\) are found by analogy with Equation (3):  
\[K^{C}=(1-f)K^{M}+fK^{fl},\ \mu^{C}=(1-f)\mu^{M}+f\mu^{fl}, \tag{9}\]  
where \(K^{M},\ \mu^{M}\) are the bulk and shear moduli of the mineral matrix (the stiffest component); \(K^{fl},\ \mu^{fl}\) are the bulk and shear moduli of the fluid filling the voids (the softest component); \(f\) is an empirical parameter, reflecting connectivity of the saturated voids.  
As in the previous section, for our modeling \(N=3\) for Type 1 model and \(N=2\) for Type 2 model. Inclusions are pieces of mineral matrix, pores, and cracks in Type 1 model. For Type 2 model the inclusions are pieces of mineral matrix and pores. The pieces of mineral matrix are assumed to be spherical.