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Downhole tools failures and drilling inefficiency are often attributed to drillstring vibrations during the drilling process, which can be categorized as axial, torsional and lateral vibrations. These vibrations become more pronounced in the context of deep and hard rock drilling. Many studies have been carried out to model the axial and torsional vibrations phenomena, but only few researches address the lateral vibrations of the bit. The interaction between the torsional and lateral vibrations has been analyzed by Mihajlovic and Leine. R et al. [1-2]. However, the foremost stick-slip vibrations in both models result from an empirical friction formulation rather than a physical description of the rock bit interaction. The root cause of stick-slip torsional vibration in drillstring has been studied by T. Richard et al. [3], which is a coupled axial-torsional dynamic system with state dependent delay. A. Depouhon and Besselink et al. have successively analyzed the stability of this retarded system accounting for the effects of depth and properties of drillstring, respectively [4-5].
In the current work, the whirl motions of the bit are modelled by the following assumptions: 1)the bit mass imbalance produces a centrifugal force $\vec{F}^e(t,\dot{\phi})$ that pushes the bit out of the center of the borehole, and 2) the rock-bit contact force $\vec{F}^w$ prevents the bit lateral motions and is defined $\vec{F}^w=\vec{f}(T,\phi,\textbf{x})$, where $T(u_t,\phi_t)$, the torque on bit depends on the history of the bit axial and torsional motions $u_t$ and $\phi_t$, respectively. Here $\phi$ is the rotary angle and $\textbf{x}$ is the whirl position. It is assumed that the whirl motion cannot influence the axial and torsional vibrations of the bit. Therefore, a semi-coupled axial-torsional-lateral dynamic system of the drill bit is developed with state dependent delay. In the model, $\textbf{x}(t)=\{\vec{x}(t),\vec{y}(t)\}$ is a two-dimensional time dependent vector describing the lateral displacement of the bit, which was restricted by the boundary of the borehole well with $F^c$. The stability of the delayed differential equation is studied by determining the location of the right-most poles of the associated linearized system. The stability of drillstring dynamic system with state-dependent delay is investigated in the desired angular velocity $\omega_0$ and weight on bit $W_0$. The coupled axial-torsional state-dependent delay model will support the development of drilling tests performed in the field and instrumented with down-hole measurements. These full-scale tests will also provide input for model assessment and further development.

[1] N. Mihajlovic, N. Wouw, P. Rosielle, H. Nijmeijer, ‘Interaction between torsional and lateral vibrations in flexible rotor systems with discontinuous friction’, Nonlinear Dyn (2007) 50:679–699, DOI 10.1007/s11071-006-9172-3
[2] R. I. Leine, D. H. van Campen, W. J. G. Keultjes. ‘Stick-slip Whirl Interaction in Drillstring Dynamics’. Journal of Vibration and Acoustics, Vol. 124 Õ 209, DOI: 10.1115/1.1452745
[3] Richard, T., Germay, C., Detournay, E., 2007. A simplified model to explore the root cause of stick–slip vibrations in drilling systems with drag bits. J. Sound Vib. 305, 432–456. doi:10.1016/j.jsv.2007.04.015
[4] Alexandre Depouhon, Emmanuel Detournay. ‘Instability regimes and self-excited vibrations in deep drilling systems’. Journal of Sound and Vibration 333(7):2019–2039 · March 2014, DOI: 10.1016/j.jsv.2013.10.005
[5] B. Besselink, N. van de Wouw, H. Nijmeijer. ‘A Semi-Analytical Study of Stick-Slip Oscillations in Drilling Systems’. Journal of Computational and Nonlinear Dynamics. Vol. 6 / 021006-1, DOI: 10.1115/1.4002386