Dynamics, Nonlinear Instabilities, and Control of Drill-strings

Abstract

Drill strings are flexible, slender structures, which are many kilometers long. They are used to transmit the rotary motion to the drill bit in the process of drilling a borehole. Due to the flexibility of the drill string and nonlinear interactions between the drill bit and rock, these systems often experience severe vibrations, and these vibrations may cause excessive wear of the drill bit and equipment damage. The aim of this dissertation effort is to further the understanding of the underlying mechanism leading to the undesired vibratory motions of drill strings, as well as to develop a viable control strategy that is applicable for mitigation of harmful vibrations. A reduced-order drill-string model with coupled axial and torsional dynamics is constructed. Nonlinear effects associated with dry friction, loss of contact, and the state-dependent delay, which all arise from cutting mechanics are considered. For the sake of analyses, a non-dimensionalized form of the governing equations is provided. Next, in order to study the local stability of the drill-string system, a linear system associated with the state-dependent delay is derived. The stability analysis of this linearized system is carried out analytically by using the D-subdivision scheme. The obtained results are illustrated in the terms of stability crossing curves, which are presented in the plane of non-dimensional rotation speed and non-dimensional cutting depth; non-dimensional rotation speed and cutting coefficient, respectively. For the nonlinear analysis, a numerical continuation method is developed and used to follow periodic orbits of systems with friction, loss of contact, and state-dependent delay. Bifurcation diagrams are constructed to capture the possible routes from either a nominal stable operational state or a stable limit-cycle motion without stick-slip to a limit-cycle motion with stick-slip. It is shown that the system can experience subcritical Hopf bifurcations of equilibrium solutions and cyclic fold bifurcations. Furthermore, with the preceding work, an observer-based on controller design is proposed by using a continuous pole placement method for time delay systems. The effectiveness of the controller in suppressing stick-slip behavior is shown through simulations. The primary contributions of this dissertation are summarized as follows: i) analytical determination of the stable operational region; ii) revelation of the routes to torsional stick-slip vibrations; and iii) construction of a feasible control scheme to mitigate the destructive vibrations caused by complex nonlinear effects.