## 2 Existing Brittleness Indices
### Mineral-Based Brittleness Indices (MBIs)
The second problem is related to the inability to use Equation (1) or its particular forms, reported in different papers, if no external data are available. Equation (1) itself does not provide a way to calculate brittleness index: external tests are necessary to obtain the factors \(a_{i}\), \(a_{i}^{{}^{\prime}}\), \(b_{i}\), and \(b_{i}^{{}^{\prime}}\), so in fact, MBIs do not provide a definition of brittleness--instead, such equations provide correlations between mineral composition and some sort of parameter that is claimed to be brittleness. Definition of brittleness is not given here--brittleness as physical property of rock can usually be defined from mechanical perspective, based on failure mechanics. Consequently, TBIs, dealing with real rock mechanical behavior under loading, provide some insight on brittleness nature. Nevertheless, Equation (1) can provide valuable data on factors controlling brittleness. While there is no strict definition of brittleness factor and mineral distribution factor, there is still an opportunity to use them as physically meaningful parameters, rather than correlation parameters obtained from regression problem solution. In fact, the assumption of brittleness of heterogeneous rock being a function of brittleness of each mineral with certain effect from geometry of grains is naturally close to the well-developed methods of predicting other mechanical properties (starting from dynamic elastic moduli) of heterogeneous media. As a result, search of parameters in Equation (1) can be performed not from the perspective of finding the best match between calculated MBI and TBI obtained from laboratory tests. Instead, a certain mathematical model of effective brittleness of mineral composition can be constructed with respect to rock specific inner geometry. If such a model can be constructed, Equation (1) can be used to obtain MBI using data on inner structure of the rock. This will be an important step in solving the problem of BI scaling. Although it is natural to consider presence of inner inhomogeneities (e.g., natural fractures) as a factor influencing on brittleness, currently there is no way to mathematically include such inhomogeneities into the brittleness prediction model. Nevertheless, presence of fractures will probably reveal in mineral distribution factor from Equation (1). The following algorithm can be proposed as the first step to solve brittleness index scaling problem:  
1. The inner structure of rock samples is studied in detail to obtain mineral composition and geometry;
2. Rock samples are subjected to loading under laboratory condition. TBIs are obtained from the stress-strain curve (the details on this determination will be given below, in the corresponding section);
3. Data on inner structure are used to construct a mathematical model of the studied rocks. This model is used to establish the relationship between model parameters, including mineral-specific brittleness factor \(a\) and mineral distribution factor \(b\), and brittleness obtained from loading tests. Note that mathematical model mentioned here is not obliged to be a certain rock physics model--any mathematical model, e.g., correlation dependencies--can be used for solving the problem of brittleness evaluation;
4. Upscaling procedure is established: it is expected that the established relationship between inner structure and composition of the sample and its brittleness remains valid for samples of other sizes, including bigger samples. Nevertheless, bigger samples can contain inhomogeneities of scales exceeding the typical size of the laboratory studied cores. As a result, it is necessary to include inhomogeneities of bigger sizes into the model to predict brittleness of the bigger samples;
5. An intermediate step can be completed to verify the model: if loading tests are performed on samples of different sizes, the assumption of the model being valid at any scale can be directly checked.