The aim of the work is a mathematical modeling of the rock crushing during drilling and removal of the drilling cuttings (sludge) to the surface by drilling fluid. The process of rock destruction is described using the mathematical theory of fragmentation. The distribution of sludge particles in size and mass depends on such factors as the properties of the drilled rock, the rate of penetration, the type of bit, and the output power. After the formation of sludge, the process of its removal to the surface is modeled. The drilling fluid together with the rock particles is considered as a heterogeneous multiphase medium in which the carrier phase – the drilling fluid – is a non-Newtonian fluid. The flow of such a medium is described using a mixture model in the framework of the multi-fluid approach. This results in a system of nonlinear partial differential equations, for which a new closure relation is derived. To solve the system, the SIMPLE algorithm is used. As a result, the flow properties are studied with the inclusion of particles of various sizes. In particular, for particles of small size due to the action of plastic stresses in a non-Newtonian drilling fluid, an equilibrium mode arises in which the particles move with the drilling fluid without slipping. This is the fastest mode of delivery of sludge to the surface. The specific dimensions of such particles depend on the parameters of the drilling process. In particular, the appropriate size range can be adjusted by changing the parameters of the drilling fluid.
