## 4 Results  
### Testing of Rock-Physics Model Selected for LBI Analysis on Experimental Data  
Before applying the constructed rock-physics model for the LBI analysis of carbonate rocks we test it on available log data on elastic wave velocities, porosity, and density. For each log depth, we construct a function having the form  
\[\psi=\left[a\frac{\left|V_{P}^{calc}-V_{P}^{\exp}\right|}{0.5\left(V_{P}^{calc} +V_{P}^{\exp}\right)}+b\frac{\left|V_{S}^{calc}-V_{S}^{\exp}\right|}{0.5\left(V _{S}^{calc}+V_{S}^{\exp}\right)}\right]100\% \tag{17}\]  
This function is a linear combination of relative differences between theoretical (_calc_) and experimental (exp) velocities \(V_{P}\) and \(V_{S}\). In view of high quality of the shear wave velocity in the used log data we put \(a=b=0.5\) in the function \(\psi\). The rock-physics model is considered applicable if we can find an aspect ratio that gives a value of function \(\psi\) not exceeding 10% (we call it acceptable value). This value usually characterizes the log data quality. Note that many solutions for aspect ratio can be found that provide an acceptable value of \(\psi\). Among the found acceptable solutions we choose only the one that satisfies the minimum of function \(\psi\) (the most probable solution) for the current log depth.  
Figure 4 shows aspect ratios inverted from log data (2371 depth points) of a well penetrating carbonate reservoir rocks (limestones) of West Siberia (Russia). Another set of aspect ratios shown in Figure 4 is derived from well logs of limestone layers within a shale formation [54]. We select only the depths where calcite volume concentration is at least 90% (17 depth points). Figure 5 demonstrates the quality of the solutions found. The horizontal axis shows the experimental velocities and the vertical axis demonstrates the relative difference between the experimental and theoretical velocities. The negative values mean that the theoretical velocity is lower than the experimental. As seen, for the both data sets, the misfit between the theoretical and experimental velocities is within 10% excepting two points for the second dataset.  
Figure 4: (**a**) Compressional and (**b**) shear wave velocities for a model of calcite matrix with randomly oriented brine-saturated voids versus decimal logarithms of porosity (poros) and void’s aspect ratio (AR). Red signs show values for West Siberian limestones. Blue signs are used for limestones from layers within a shale formation [54].  
As seen even a simplest rock-physics model can be used to explain a behavior of elastic wave velocities of carbonate rocks (limestones) at the log scale. However, the modeling with constant elastic moduli of mineral matrix may be not too successful for laboratory data. According to our experience, this model can produce overestimated compressional wave velocities. This is explained by the fact that carbonate rocks contain closed porosity. As a rule, the log data give the total porosity (from density logs) whereas in laboratory the porosity is measured by the sample saturation. As a result, the laboratory measurements of porosity characterize effective (or connected) porosity instead of the total porosity. This means that rock-physics modeling at the core scale should include the \"matrix\" elastic moduli in the list of unknown parameters considering the \"matrix\" as a part of the rock that does not participate in the fluid motion.