## 3 Methodology and Materials  
### Rock-Physics Models and Methods of Effective Medium Theory  
In order to analyze the effect of rock's inner structure on its brittleness a rock-physics model of the effective elastic properties should be constructed. The rock-physics model of effective physical properties is a triad consisting of the following elements: (1) model medium, (2) parameters of the model, and (3) equations relating the model parameters with the macroscopic (or effective) physical properties. The model medium should mimic general specific features of the rock microstructure. Usually, in the model medium all inclusions are represented by ellipsoids (commonly, ellipsoids of revolution). The ellipsoid shape is characterized by the aspect ratio that is a ratio of semi-axis normal to the crack's plane to the semi-axis in the crack's plane. The microstructure parameters include the aspect ratio of inclusions and parameters characterizing inclusions' orientation and mutual distribution in the rock volume. The effective medium theory (EMT) provides the equations relating the macroscopic physical properties of rock with parameters characterizing the rock composition and microstructure. The problem of effective physical properties determination is related to the so-called'many-body problem' and, in the general case, can be solved only approximately. This leads to the fact that there are many methods for calculation of the effective physical properties. Note that the effective physical properties that can be considered within the EMT approach include both elastic and transport properties. The latter are the thermal and electrical conductivity, as well as dielectric and hydraulic permeability [43; 44]. When selecting a specific EMT method, the rock microstructure should be taken into account. For example, in the case of two-component rock of'matrix-inclusion' type with completely isolated inclusions the upper Hashin-Shtrikman (HS) bound [1; 45] of effective medium theory provides the best results if the matrix is the stiffest component. In the opposite case (stiff components in a soft matrix) the lower HS bound is preferable. Commonly, in sedimentary rocks the voids (pores and cracks) are connected. In this case, the rock-physics methods that take into account the connectivity should be applied. Among these methods are the self-consistent (SC) method [43; 44; 46], differential effective medium or DEM [47], \(f\)-model of generalized singular approximation method [13]. The SC and DEM methods take into account the void's connectivity implicitly whereas the \(f\)-model incorporates a special parameter characterizing a degree of pore/crack connectivity. Strictly speaking, before applying a method of effective medium theory it should be tested on a composite material containing heterogeneities of known physical properties, shape, and distribution in the composite volume. The testing for HS bounds and \(f\)-model is presented in [48]. However, it is very difficult to produce such a test for SC and DEM since no explicit geometrical structure can be put in correspondence to some mental procedures accompanied the derivation of formulas of these methods. Thus, in the SC method, it is assumed that each heterogeneity can be represented by ellipsoid. Each ellipsoid is placed in a matrix having already effective properties. In other words, each inclusion is placed in a conglomerate made of other inclusions. In the DEM the inclusions are added in the matrix by portions controlled by a differential equation. Meanwhile, numerical tests show that composites having the same number of cracks with the same contact area, but differing in the number and length of the contacts, exhibit different elastic properties [49]. Besides, in real rocks, the shape of mineral grains, pores, cracks, and other inclusions is not strictly ellipsoidal. However, this is the only shape that can obtain an analytical solution for effective physical properties. The voids in rocks often exhibit different size and relative opening (characterized by aspect ratio).