## 2 Existing Brittleness Indices
### Mineral-Based Brittleness Indices (MBIs)  
Lithological influence on brittleness of a heterogeneous rock serves as a basis to provide certain ways to compute brittleness index based on rock composition. Mineral weight faction was accepted by many researchers [17; 18; 19; 20; 21; 22] as a basic parameter to evaluate brittleness. These studies reveal effects of different minerals on brittleness: e.g., presence of quartz has positive effect rock brittleness, while increase in clay fraction results into decrease of brittleness of the rock. A number of correlations have been proposed for different formations--[PERSON] et al., 2019 [4] highlight 7 ways to calculate mineral-based brittleness indices suitable for Barnett, Neuquen Basin, Haynesville, Wolfcamp, and other shales. A general form of equation for calculating brittleness index from mineral composition was proposed by [PERSON] et al., 2010 [23]  
\[MBI=\sum_{i}a_{i}b_{i}M_{i}\bigg{/}\sum_{i}a_{i}{}^{\prime}b_{i}{}^{\prime} M_{i}. \tag{1}\]  
Figure 1: Classification of brittleness indices. Here, \(M_{i}\) stands for the weight fraction of the \(i\)-th mineral; \(a_{i}\), \(a_{i}^{{}^{\prime}}\), \(b_{i}\), and \(b_{i}^{{}^{\prime}}\) are empirically driven coefficients. The first pair, \(a_{i}\) and \(a_{i}^{{}^{\prime}}\), represent mineral specific brittleness factors: these parameters show the positive and negative effects of the particular mineral on overall brittleness respectively. The second pair, \(b_{i}\) and \(b_{i}^{{}^{\prime}}\), are associated with the specifics of mineral distribution in the homogeneous rock: these parameters are affected by the geometrical properties of the inner structure of the rock. Equation (1) makes it possible to take into account the effects of various minerals on brittleness of the whole composition with regard to mineral-specific effects. The majority of MBIs obtained for different fields and formations can be set using the form of Equation (1) with site-specific values of \(a_{i}\), \(a_{i}^{{}^{\prime}}\), \(b_{i}\), and \(b_{i}^{{}^{\prime}}\) factors typical for the studied rock. There are not many examples of Equation (1) being incapable of predicting brittleness of heterogeneous rocks: a notable exception has been reported by [PERSON] et al., 2016 [PERSON] et al., 2016 [PERSON] et al., 2016, who suggested introducing porosity into consideration. There are two major problems with usage of MBIs. First of all, mineral brittleness factors \(a\) and mineral distribution factors \(b\) are correlation parameters. These parameters can differ from field to field a lot, so it is complicated to set a general correlation. Moreover, there is no way to measure these factors distinctively: only brittleness of the whole rock can be obtained from various kinds of tests. A certain regression problem can be postulated: consider the set of tests providing evaluations of brittleness for a number of rock samples with known mineral compositions. In fact, only weight fractions of all minerals can be obtained from independent tests, so the regression problem of establishing the proper MBI correlation for the given set of data can be deduced to finding the numbers of \(a_{i}\), \(a_{i}\), \(b_{i}\), and \(b_{i}^{{}^{\prime}}\) factors in Equation (1), satisfying the mineral weight fractions and providing the MBIs corresponding to real brittleness of the studied rock samples. This regression problem is usually ill-posed: depending on the number of studied minerals and number of rock samples in the studied collection, there can be either too few or too many experimental results for obtaining the single stable solution. In the first case, there will be not enough data to obtain a single solution for the inverse problem; in the second case, the system can become overdetermined and therefore inconsistent. It is necessary to apply certain methods to obtain the proper correlation parameters for the studied set of data.