The dynamical model governing the 3D kinematics of a drill bit is constructed for rotary drilling applications for which the bit is guided by a push-the-bit rotary steerable system. The evolution of the bit trajectory, and thus of the borehole geometry, is a consequence of the interaction between the borehole, a geometric object, and the drilling structure, a mechanical object. In this respect, the model describing this evolution consists of the association between (i) a model of the near-bit region of the drillstring, (ii) a model of the bit/rock interaction, and (iii) kinematic relationships relating the motion of the bit into the rock to geometric variables for the borehole evolution. The mathematical formulation of these three elements yields a set of functional differential equations with secular terms accounting for a delayed influence of the borehole geometry on the bit trajectory. The parameters entering these relations account for the loads and properties of the drilling structure and for the properties of the bit and rock formation.
Three length scales are identified in the response of the directional drilling system; they correspond to short-, intermediate-, and long-range behaviors. The short-range response is associated with the dimensions of the bit, the small scale of the problem. It corresponds to fast variations of the bit orientation. On the intermediate-range, the wellbore trajectory converges to a quasi-constant curvature solution, if it is stable. On the long-range, the borehole curvature slowly varies and for appropriate set of drilling parameters the borehole converges toward a stationary helical path. Finally, the stability and rate of convergence on the intermediate and long range are investigated.