Local lateral stability analysis of deep drilling processes

Rotary drilling systems often suffer from severe self-excited lateral vibrations, especially bit whirl. A recent study establishes a delayed measure differential inclusion (DMDI) model to describe the lateral dynamics of rotary drilling processes. It shows that the combination of bit-rock interaction and drill-string dynamics, even without the eccentricity of the drilling system or the effects of drilling fluid, can independently trigger lateral drilling instability. This paper further investigates the mechanisms of such instability from a mathematical perspective. The ideal drilling process (without any lateral vibrations) is shown to correspond to an equilibrium point of the DMDI system, and the local dynamics around this equilibrium is governed by a linear delay differential equation (DDE). The stability of this DDE determines the lateral stability of the drilling process. The proposed DDE enables the identification of key factors affecting lateral drilling stability. For the case of instability, a method is developed for the analytical prediction of the whirling mode (i.e., forward or backward whirl tendency) of symmetric straight-bladed bits with prescribed blade geometries as they diverge from the equilibrium.