Stick-slip vibration analysis of a coupling drilling-rock system with two delays and one delay-dependent coefficient

Drilling depths exceeding 10,000 m have recently been achieved, but the increasing depth poses challenges due to the flexible drill-string, making it more susceptible to stick-slip vibration. This vibration can lead to accidents such as excessive bit wear and tool falling into the well. Previous studies on modeling polycrystalline diamond compact (PDC) bits have focused on symmetrically distributed complete blade bits, which are not commonly used. To address this gap, this study introduces PDC bits that closely resemble the actual ones, with a structural configuration of both complete blades and part-blades symmetrically distributed. We also consider the impact of mud and rock chips on the drill bit by introducing a delay-dependent coefficient of response damping change. This study develops a stick-slip vibration model for a coupling drilling-rock system with two state-dependent delays tn1 and tn2, and one delay-dependent coefficient. Using the crossing curve method and numerical studies, we obtain different types of crossing curves reflecting the stability switching characteristics of the system in the dimensionless delay τ2τ1 space. During stick-slip vibration, τ1 mutates from 1.69 s to 2.51 s, and τ2 mutates from 1.3 s to 2.16 s, both within the stable region of the crossing curve. The critical steady state and the stable state typically cross the stable and unstable regions of the crossing curve in their respective times. Increasing the torsional damping within the range of 0.04 to 0.08 can accelerate system stabilization or delay the arrival of the critical steady state. Moreover, increasing the drive rotation speed within the range of 4 to 6 eliminates stick-slip vibration and accelerates the system towards a steady state. In conclusion, this study predicts the drill-string dynamics of a symmetrically distributed PDC bit with complete blades and part-blades. The insights gained can help minimize stick-slip vibration and provide recommendations for drilling operation parameters.