## 2 Materials and Methods
### Application of Effective Medium Theory
The calcite grains and pores visible with computed tomography (\"visible\" pores) are placed in this cement. This RP model contains an unknown parameter, namely, the porosity of the micrite matrix. In some works [29], the microporosity is suggested to determine the difference between the measured porosity and porosity provided by the computed tomography. In our modeling, we also followed this line since the analysis of density of the studied samples suggested that the closed porosity was quite small (around 1%). To calculate the effective elastic moduli of the micrite matrix, we applied the SCA method. To estimate the elastic moduli of the micritic matrix with calcite grains and \"visible\" pores, we used the Kuster-Toksoz method, since this method assumes the existence of a host matrix. We did not distinguish in our RP model between \"visible\" pores of different size, since the porosity was too small to do that. The difference in size of inclusions is taken into account implicitly in hierarchical RP modeling according to the principle \"from small heterogeneities to large heterogeneities\" [30]. Thus, previous studies performed for carbonate rocks showed that the \"size effect\" produces a 3% difference in bulk and shear moduli at a total porosity of 20% when vuggy porosity varies from 1 to 10% [31]. The RPM in our study was used to solve both the forward and inverse problems. The forward problem assumes the calculation of the effective elastic moduli, and then elastic wave velocities from the known composition of samples and microstructural parameters provided by different sets of experimental data we have. The theoretical values of longitudinal and transverse wave velocities (\(Vp\) and \(Vs\)) are compared with the respective experimental values. This comparison gives a possibility to answer the question of whether the RP model can be further used for the predictions mentioned above. In the case of unacceptable misfit between the theoretical and experimental velocities, the RP model should be modified. Thus, smaller voids can exist in rocks that are not recognizable with the used resolution of tomography (for example, microcracks). In this case, an inverse RP-based problem should be solved to estimate the parameters of the hidden voids. Voids filled with a contrasting substance compared to the host matrix properties (for example, gas or liquid) have a significant effect on the overall elastic moduli. If the voids have nonisomeric shape (microcracks), even their small volume concentration (less than 1%) can lead to a significant drop in the elastic moduli of rocks (see Section 3.2). Microcracks are thin enough to be detected with tomography. Thus, we should not exclude them from consideration, specifically in the case of packstones. The occurrence of microcracks in packstones was confirmed by previous lithological studies in this area [4; 5]. It should be noted that many factors except voids can affect the elastic wave velocities. These factors include the state of microfabrics (intact or stressed grains), grain boundaries, and many others. To take into account the overall effect of these factors we can consider that the elastic moduli of the host matrix are also unknown values to be inverted from experimental data. However, for isotropic rocks, only two experimental velocities are available, namely, \(Vp\) and \(Vs\). This means that, even assuming only one set of invisible voids, we should invert four unknowns from these measurements. In this case, the domain of possible solutions of the inverse problem can be quite wide. As shown below, even for the known elastic moduli of the host matrix, this domain is not unique. Therefore, considering a model with hidden voids, we assumed that the elastic moduli of the host matrix are known and equal to those of the calcite polycrystal. The difference between the experimental velocities and velocities obtained with RPM, which includes only the voids provided by tomography, was attributed to invisible voids. We also considered another type of RP model.