## 2 Materials and Methods
### Application of Effective Medium Theory  
The effective medium theory gives an opportunity to relate the measured (or effective) properties with the rock composition and characteristics of microstructure. When constructing a rock physics model, first, a studied rock is replaced by a model medium that reflects the principal features of the rock's microstructure. Then, parameters describing the rock composition and microstructure should be specified. The compositional parameters include the volume concentrations and elastic moduli of components. The microstructural parameters characterize the shape, orientation, and connectivity of components. Finally, a method of the effective medium theory is to be chosen in order to relate the studied physical properties (for example, elastic) and the model parameters. The aim of RP models is to predict how the physical properties vary for different scenarios in changing rock composition and/or microstructure. For example, we can estimate how the elastic wave velocities change if we fill the pores with another fluid, or increase/decrease the porosity, or change the pore shape, or insert other minerals and organic material in the rock. For the RP modeling (RPM), we applied three methods of effective medium theory: self-consistent approximation (SCA) of [PERSON] [25], the differential effective medium (DEM) approach [26], and the Kuster-Toksoz method [27]. These methods are widely used in geophysical practice. Each of the methods relates the effective bulk and shear moduli to the rock composition and shape of ellipsoidal inclusions characterized by aspect ratios. In all of the methods, the inclusions are assumed to be ellipsoids of revolution and only one aspect ratio is required to describe their shape. In the case of voids, the aspect ratio characterizes their relative opening. The DEM and Kuster-Toksoz methods assume the existence of a host matrix. The SCA considers all heterogeneities as ellipsoidal inclusions embedded in the matrix with effective properties. The question definitely arises of how the results depend on the EMT method selected for computations. We performed a special analysis that allowed us to give an answer to this question. We compared the results obtained with the methods for two simple basic models of carbonate rock: (1) calcite matrix with quasi-isometric pores and (2) calcite matrix with cracks. The formulas of the self-consistent approximation [25], DEM [26], and Kuster-Toksoz [27] methods are presented in Appendix A. The formulas of these methods can be also found in [28]. When constructing the RP model, all possible information on rock's composition, microstructure, and measured physical properties should be taken into account. The results of thin section analysis and tomography studies provide valuable data on rock'smicrostructure. Thus, the thin sections give information on the heterogeneity's type and their mutual distribution in the rock's volume. However, the shape of heterogeneities (specifically, voids) can be more reliably estimated from the tomography data, since the thin sections provide only 2D information. For rocks with a complicated and hierarchical inner structure, the constructed RP model may have many unknown parameters. At the same time, the number of available experimental data is limited. This leads to the fact that many sets of model parameters can be found which produce values of effective properties rather close to experimental ones. In order to overcome this obstacle, a sensitivity study of the constructed RP model should be performed. Usually, this helps to simplify the model or decrease a number of unknown parameters. To construct RP models for the studied rocks, we selected the model medium such that it reflected the specific features of the rock's microstructure recognized by the thin-section analysis and tomography. Thus, we should take into account the existence of porous micritic cement confirmed by the thin section analysis.