## 2 Materials and Methods
### Application of Effective Medium Theory
This RP model explains the discrepancy in experimental and theoretical velocities obtained with RPM, which incorporates only tomography-visible voids, by various factors changing the elastic moduli of the host matrix. Specifically, one of these factors is the effect of the grain boundaries. In the vicinity of the grain boundaries, the grain material can differ from those of the perfect crystal. Below, we call a substance near the grain faces \"intergranular material\". The elastic moduli of this material are included in the list of unknown parameters instead of the characteristics of hidden voids. In this case, the unknown elastic moduli of intergranular material were inverted in two steps. First, we considered that the quasi-isometric voids visible with computed tomography were placed in an isotropic host matrix with unknown elastic moduli. These moduli were inverted from the experimental velocities \(Vp\) and \(Vs\). Then, we assumed that the host matrix was also a heterogeneous medium composed of particles of calcite polycrystals surrounded by a soft material with unknown properties. This soft material plays the role of intergranular material. The volume concentration of the soft material should be rather small. We assumed that it did not exceed 1-2%. For the inverse problem solution, we applied the method of \(N\)-dimensional meshes. According to this method, first, for each parameter, a range of its possible variation is specified. Then, the whole range of variation in each parameter is divided into intervals. As a result, we obtained a \(N\)-dimensional mesh with nodes at the intersection of the interval's ends. The forward problem was solved for each node. Then, the sets of parameters providing an \"acceptable\" discrepancy (a discrepancy comparable to experimental error) between the theoretical and experimental values of elastic wave velocities were chosen as possible solutions to the inverse problem. Finally, the most probable solution was selected among the possible solutions. The most probable solution is the solution that provides the minimum value of the chosen measure of discrepancy in both the velocities \(Vp\) and \(Vs\). In this work, the discrepancy measure was taken in the form:  
\[Discrepancy\ measure=\sqrt{0.6[Diff(Vp)]^{2}+0.4[Diff(Vs)]^{2}}\cdot 100\% \tag{1}\]  
where \(Diff(Vp)\) and \(Diff(Vs)\) are the relative differences between the theoretical and experimental values of \(Vp\) and \(Vs\). velocities, respectively. In the RP model with microcracks, \(N=3\) (mineral matrix, pores, and microcracks). Instead of values of microcrack porosity and aspect ratio, we considered their decimal logarithms. Decimal logarithms were used due to the strong nonlinearity of the dependence of elastic wave velocities on the aspect ratio of cracks and their porosity in the domain of their small values. As a result, the two-dimensional region of possible change in the decimal logarithms of the desired parameters was covered with a mesh. In the mesh nodes, the forward problem of determining the velocities of elastic waves was solved. When solving the two-step inverse problem for a model with unknown properties of the host matrix, at each step, \(N=2\). For the RP model with microcracks, we assumed that a solution was acceptable if the relative difference between the theoretical and experimental velocities did not exceed the threshold value of 5% (a value comparable to the velocity measurement error). If this condition was not met for the specified ranges of parameter values, then the threshold was increased by 1%. Moreover, this increase was carried out sequentially--first, the threshold for the S-wave was increased by 1%. If solutions were not determined again, then the threshold was increased by 1% for the P-wave as well. We observed from the modeling carried out for the three samples that, in the \"worst\" case, the threshold value for velocity did not exceed 7% for P-wave and 7.5% for S-wave.