## 2 Existing Brittleness Indices
### Log-Based Brittleness Indices (LBs)
The following failure mechanics equation was used to evaluate critical energy release rate [27]  
\[G_{\rm C}=K_{IC}^{2}/E^{\prime}, \tag{7}\]  
where \(K_{IC}\) is fracture toughness, and \(E^{\prime}\) is equal to [PERSON]'s modulus \(E\) for plane stress state and \(E^{\prime}=E/(1-v^{2})\) for plane strain (this difference emerges from [PERSON]'s law form for the cases of one zero principal stress--plane stress state and one zero principal strain--plane strain state). Note that dynamic elastic moduli were used in Equation (7) to predict critical energy release rate. Plane strain state was considered by [PERSON] et al., 2014 [21] to analyze hydraulic fracture propagation. Fracture toughness was obtained from existing correlations, so well logs were used to estimate profiles of critical energy release rate. Consequently, Equation (4) was modified to incorporate fracture toughness and critical energy release rate. The following equations were proposed for brittleness evaluation [21]  
\[LBI_{6}=\frac{LBI_{3,norm}+G_{C,norm}}{2},\ \ \ LBI_{7}=\frac{LBI_{3,norm}+K_{IC,norm}}{2 },\ \ \ LBI_{8}=\frac{LBI_{3,norm}+E_{dyn,norm}}{2}\ \, \tag{8}\]  
where versions of \(LBI_{3}\), critical energy release rate, and fracture toughness are normalized as [21]  
\[LBI_{3,norm}=\frac{LBI_{3-LBI_{\min}}}{LBI_{3\max}-LBI_{\min}},\ \ \ G_{\rm C,norm}=\frac{G_{\rm Cmax}-G_{C}}{G_{\rm Cmax}-G_{\min}},\ \ \ K_{IC,norm}=\frac{K_{IC,\max}-K_{IC}}{K_{IC,\max}-K_{IC}}\ . \tag{9}\]An analogue of Equation (1) can be constructed to cover the LBIs based on elastic moduli (Equations (4)-(8)): their general form can be written as  
\[LBI=\sum_{i}c_{i}E_{i}\bigg{/}\sum_{i}c_{i}^{\prime}E_{i}, \tag{10}\]  
where \(c_{i}\) is weight factor specific for \(i\)-th elastic modulus \(E_{i}\) obtained from well logs data interpretation. Dynamic moduli are used in majority of Equations (4)-(8): e.g., fracture toughness \(K_{JC}\) used in Equation (7) was, in fact, obtained from a linear correlation between fracture toughness and dynamic Young's modulus typical for Barnett shale studied by [PERSON] et al., 2014 [21]. Acquisition of weight factors remains the task of regression problem in a way similar to the one discussed above: laboratory studies on core material remain the main source of data for finding the proper correlation coefficients. Respectively, all problems discussed in the previous section remain actual, including the problem of difference between scales of laboratory tests and well logs data. The variety of methods to calculate brittleness leaves several questions open. First of all, how does one choose the proper correlation for any formation? The majority of studies mentioned here were devoted to Barnett shale, but properties of other formations can differ a lot [16]. Secondly, it is important to know how to obtain empirical coefficients used in correlations, and, more importantly, are the results of brittleness evaluation performed by varied methods sensitive to the choice of used correlation and empirical coefficients? [PERSON] et al., 2019 [4] have studied these problems for four different formations: Marcellus, Chattanooga, Bakken, and Niobrara formations. Different correlations for evaluation of brittleness index were compared for these formations. Detailed study had been performed on data obtained from well logs--we would like to highlight the results obtained by [PERSON] et al., 2019 [4]. Figure 2 represents the summary of these results: brittleness indices obtained using different correlations for the same data are shown as profiles along the wells. Certain empirical parameters and correlations were obtained with respect to the formation lithology. Figure 2: Brittleness evaluation from different correlations for four formations. Modified after [PERSON] et al., 2019 [4]. Symbols represent predicted brittleness using the correlations proposed by authors mentioned in the legends with respect to equations in the current paper. Several important points can be seen from the figure. First of all, brittleness indices vary a lot.