## 4 Rock Crushing with a Drill Bit  
Soil crushing with a drill bit can be described as a determining process in the drilling of oil wells. In this regard, great efforts have been made in the field of the mathematical modeling of this process in order to solve the following problems: determining the dynamics of the bit in order to ensure stable drilling and reduce the wear of the drilling tool, and evaluating the penetration rate and particle size distribution of the drilled rock. The models of bit dynamics as a rule include torsional and axial degrees of freedom that are subject to corresponding linear dynamic equations involving inertial, stiffness, and damping terms and describing the process of vibrations. The model of bit-rock interaction allows the coupling of the torsional and axial vibrations and the building of a unified system of nonlinear equations describing the bit oscillations. [PERSON] et al. proposed the model of bit-rock interaction [84] where the friction forces and torques in the system are connected by a time-delay term determined by the penetration of the bit blades into the rock. This approach was used in several works (e.g., see [85] and the references given there). [PERSON] et al. in [86] have supplemented this model by adding into consideration the drill string represented as a series of alternating springs and point masses. The direct numerical simulation study on the rock-breaking process is presented by [PERSON] et al. [87]. The dynamic rock breaking process under the compound impact is simulated by the finite element method using special cohesive interface elements. As a result of modeling, in particular, it is possible to determine both the rate of penetration and the distribution of the size and shape of the particles of the drilled rock. At the same time, such a direct method for modeling the interaction between a drill and soil is extremely complex and time-consuming. Therefore, the use of this approach in a complex global model of the drilling process seems hardly possible. The enormous amount of different uncontrollable factors in the process of drilling cannot be directly considered in the model. Thus, the probabilistic approaches are commonly used for modeling such processes. It is quite natural to use the approach based on the concept of a branching process for modeling particle distribution over mass or size as it is done by [PERSON] [88]. In general, the branching process describes the growth of a particle population from the probabilistic point of view [89; 90]. Several studies have been devoted to the development of rock fragmentation models, including [PERSON] [90], [PERSON] [91], and [PERSON] [92]. For the particle mass probability density, one can obtain the integro-differential equation (see [90; 91]). A power-law dependence of fragmentation intensity for the mass is commonly used (e.g., see [90]). Models for solid body crushing are presented in [93; 94; 95; 96]. It is possible to consider three types of soil in relation to its interaction with a bit. The first type (\"rocky soil\") is the soil with characteristic particle size which is comparable with the size of the bit. This type of soil will be crushed into smaller parts by the drilling bit. Soil of the second type (\"sandy soil\") consists of particles that are much smaller than the bit. This soil is not crushed but mixed during drilling. The third type is \"mixed soil\" which is a mixture of the first and the second types, which are to be modeled separately. Thus, the problem is reduced to the description of soils of the first and second types. There are models based on the concept of a branching process, both for the soil of the first and second type, but there is a fundamental difference between these soils. The first type of soil is crushed with a bit, while the particles of sandy soil are not transformed during drilling. To build a branching process model, it is necessary to set the characteristic time of the process: the time needed for the initial set of particles to be fragmented into smaller ones.