A high-dimensional model to study the self-excited oscillations of rotary drilling systems

Abstract

This paper analyzes the self-excited axial and torsional vibrations of rotary drilling systems with a model that combines a multi-degrees of freedom representation of the drilling structure with a rate-independent bit-rock interface law. The state-dependent delay introduced by the bit-rock interaction is captured by a bit trajectory function, which leads to the formulation of a nonlinear system of coupled partial differential equation (PDE) and ordinary differential equations (ODEs). This paper describes an algorithm that utilizes the Chebyshev spectral method to convert the system of PDE–ODEs into a system of coupled ODEs. The use of Chebyshev polynomials rather than Fourier series leads to a computationally more efficient algorithm, in view of the non-periodic nature of the bit trajectory function. By linearizing the system of coupled PDE–ODEs, this algorithm is further extended to assess numerically the stability of the high-dimensional model. The predictions of the linear stability analysis are validated with published results as well as with time-domain simulations. It is shown, as anticipated by Liu et al. (2014), that the stable region in the space of the operating parameters shrinks with increasing spatial resolution of the high-dimensional model, suggesting that the effects of damping on the stability of drilling system is generally negligible in the range of practical drilling parameters. Extensive parametric studies have been carried out using the high-dimensional model to analyze the influence of various factors on the torsional stick–slip oscillations. The bit wearflats are shown to play an important role in delaying or even suppressing the occurrence of torsional stick–slip oscillations. Finally, the axial vibrations propagating along the drillstring are revealed in the form of traveling waves, whereas the torsional vibrations take the form of standing waves, whose frequency is the same as the torsional resonance of the drillstring excited by the bit-rock interaction.