Abstract: A transient mathematical model for modeling the transport of cuttings in the drilling mudflow during drilling horizontal and directional wells is presented. The model is based on a two-layer model that takes into account the cuttings in the fluid flow and the motionless cutting substrate. The relationships for calculating the mass exchange between the substrate and the cuttings in the flow, the friction and heat transfer between the cuttings and the fluid flow, as well as the heat transfer between the fluid and the substrate with the annulus. Options for simulating Newtonian and non-Newtonian fluid motion are implemented. It is assumed that the fluid moves in the annular channel formed by the well wall and the drill string. The presence of eccentricity in the channel is taken into account. To simulate the motion and heat transfer, a system of mass, momentum, and energy equations is solved for each component separately. The numerical implementation is based on the finite volume method, the convective terms are written in the counterflow form. The results of the test calculations demonstrating the efficiency of the model proposed are presented. The problem on the displacement of cuttings from an arbitrarily oriented well is considered.