This paper addresses the issue of stick-slip instabilities experienced by drag bit-equipped rotary drilling systems operating at large depths. In particular, on the basis of a simple model of the drillstring combined to an advanced bit/rock interface law, it is shown by way of a case study that a critical rotation speed exists. This critical speed separates two unstable dynamical regimes. Above it, instabilities grow on a slow timescale as compared to the resonant period of the torsional pendulum modeling the drilling apparatus. Below it, instabilities grow on a fast timescale. Accordingly, the model is more prone to nonlinear instabilities such as torsional stick-slip oscillations when operated at rotation speeds below the critical speed than it is above it.
Rotary drilling systems are known to experience instability regimes of various natures, e.g., bit bouncing or bit whirling, that are detrimental to the drilling performance and tool life. In this paper, we focus on another type of instability: stick-slip oscillations. In particular, we revisit the stability analysis of the discrete model proposed by Richard et al.  to study the self-excited torsional and axial vibrations of deep drilling systems.
Specific to the model is the definition of the drilling action as the sum of two components, one representing fragmentation and excavation of rocks by the bit cutters, the other a frictional contact process taking place on the cutter wearflats. As a direct consequence of the bit rotary motion, the cutting process, which is a function of the height of the rock ridge in front of the cutters, introduces a so-called regenerative effect in the model governing equations that are therefore of the retarded differential equation type with a discrete state-dependent delay.