## 3 Methodology and Materials
### Rock-Physics Model of Effective Elastic Properties Used for LBI Analysis  
We apply the simplest rock-physics model described above to analyze how the porosity and pore/crack opening affect the LBI of carbonate rocks. We consider a polycrystalline calcite isotropic matrix and randomly oriented ellipsoidal voids saturated with formation water. All voids have the same shape and are filled with the same fluid. The void's aspect ratio is supposed to vary in a very wide range: from 0.0001 to 1. The void's volume concentration changes from 0 to a limiting value. For spherical pores (aspect ratio is 1) the limiting porosity is 40%. The limiting porosity value for other voids is chosen such that the crack density is equal to unity. The crack density in the effective medium theory is specified by the formula \(\varepsilon=\frac{3\Phi}{4\pi\alpha}\), where \(\alpha\) is the aspect ratio and \(\phi\) is the void's volume concentration. The crystalline matrix is an isotropic polycrystal of calcite characterizing by the elastic wave velocities, compressional (\(V_{p}\)) and shear (\(V_{s}\)), equal to 6.54 and 3.35 km/s and density 2.71 g/cm\({}^{3}\)[54]. The properties of formation water are \(V_{p}\) is 1.6 km/s and density is 1.1 g/cm\({}^{3}\). To relate the model parameters with the effective elastic properties of rock, we use the classical self-consistent method of the effective medium theory [43; 44]. In the effective medium theory, the effective stiffness tensor \(\mathbf{C}^{*}\) relates the stress and strain fields averaged over the representative volume of the heterogeneous medium via the [PERSON]'s law \((\langle\sigma_{ij}\rangle=C^{*}_{ijkl}\langle\epsilon_{kl}\rangle,\ i,\ j,\ k,\ l=1,\ 2,\ 3)\). According to the self-consistent method, the formula for the effective stiffness tensor the has the form (in the tensorial form) [43; 44]  
\[\mathbf{C}^{*}=\Big{\langle}\mathbf{C}(\mathbf{x})[\mathbf{I}-\mathbf{g}( \mathbf{x})(\mathbf{C}(\mathbf{x})-\mathbf{C}^{*})]^{-1}\Big{\rangle}\Big{\langle} [\mathbf{I}-\mathbf{g}(\mathbf{x})(\mathbf{C}(\mathbf{x})-\mathbf{C}^{*})]^{- 1}\Big{\rangle}^{-1}, \tag{14}\]  
where the angular brackets indicate the volume averaging; \(\mathbf{x}\) is a point within a representative volume whose physical properties coincide with the properties of the rock; \(\mathbf{C}(\mathbf{x})\) is the stiffness tensor of a heterogeneity at the point \(\mathbf{x}\); \(\mathbf{I}\) is the unit tensor of the fourth rank having the components \(I_{ijkl}=\frac{1}{2}\Big{(}\delta_{ikl}\delta_{jl}+\delta_{il}\delta_{jk} \Big{)}\), \(i\), \(j\), \(k\), \(l=1\), \(2\), \(3\). The volume averaging in Equation (14) is performed over all components with the use of their volume concentrations. In this method the ellipsoidal shape is assumed for all heterogeneities. The tensor \(\mathbf{g}\) depends on the shape of ellipsoidal heterogeneities and the effective properties. This tensor has the form [44; 55]  
\[ \begin{array}{l}g_{kmln}=-\frac{1}{4}(\widetilde{a}_{knm}+\widetilde{a}_{mlnk }+\widetilde{a}_{knlm}+\widetilde{a}_{mnlk}),\\ \widetilde{a}_{kmln}\equiv\frac{1}{4\pi}\frac{1}{\sqrt{\,0\,}}\int\eta_{nm} \Lambda_{kl}^{-1}d\Omega,\ d\Omega\equiv\sin\theta d\theta d\varphi,\\ \Lambda_{kl}\equiv C^{*}_{kmln}n_{mn},\ n_{mn}\equiv n_{m}n_{n},\\ n_{1}=\frac{1}{a_{1}}\sin\theta\cos\varphi,\,n_{2}=\frac{1}{a_{2}}\sin\theta \sin\varphi,\,n_{3}=\frac{1}{a_{3}}\cos\theta,\end{array}  \tag{15}\]  
where \(a_{1}\), \(a_{2}\), and \(a_{3}\) are semi-axes of ellipsoids describing the shape of heterogeneities. Commonly, the shape of heterogeneities is modeled by ellipsoids of revolution with \(a_{1}=a_{2}=a\) and \(a\geq a_{3}\) (oblate spheroids). In this case the shape can be characterized by an aspect ratio that is equal to \(a_{3}/a\). As seen Equation (14) contains the term \(\mathbf{C}^{*}\) in the both left- and right-hand sides. In this case, the solution is found by iterations.