Drillstring Dynamics 

This research is built on a novel model recently developed at the University of Minnesota to analyze the selfexcited vibrations that occur when drilling with PDC bits. The discrete model takes into consideration the axial and the torsional modes of vibration and the coupling between these two modes through rateindependent bitrock interaction laws that account for the cutting and the frictional process. The cutting forces are proportional to the depth of cut, a variable that depends on a previous axial position of the bit at an a priori unknown time. The frictional process mobilized at the wearflats/rock interface can vanish when the bit moves upward. The present research deals with (i) adding new elements in the model, namely a more realistic bit geometry and a supplementary degree of freedom in the description of the drillstring, (ii) optimization of the computational algorithm, (iii) a linear stability analysis of the trivial stationary motion, and (iv) a comprehensive analysis of the system response. A linear stability analysis reveals that the steadystate solution of such a system is unstable, and that selfexcited oscillations develop. Numerical simulations show that these oscillations may sometimes degenerate into stickslip vibrations. It was found that the key quantity governing the mean response of the system is a parameter κ, which is related to the occurrence of losses of frictional contact at the wearflats/rock interface. Indeed, the mean rate of penetration and the mean torque depend on this quantity, as well as on the parameters characterizing the system. The study focused on determining the minimum set of parameters influencing κ and to map the dependence of κ on those parameters. It is shown, for example, that the velocity weakening torque law is a consequence of the diminution of the axial vibrations with the prescribed angular velocity. Moreover, it is accompanied with a decrease of the drilling efficiency, since more energy in average is dissipated by frictional contact. 
Maps of stability showing regions where the bit is bouncing in the parametric space of the weightonbit and the bit bluntness were computed as well as the dependence of these regions upon the other parameters of the system. Furthermore, it appears that there exists a limit rate of penetration beyond which the axial vibrations experienced by the bit are too strong, leading to bit bouncing. We also quantified how fast the amplitudes of the torsional vibrations are growing, by computing the first time of stick. A systematic parametric study reveals the existence of unexpected “antiresonance zones”, characterized by low amplitudes of torsional oscillations, and identical tendencies as observed in the field; an increase of the weightonbit can trigger stickslip vibrations, while an increase of the angular velocity can inhibit these oscillations. 