Instability regimes and self-excited vibrations in deep drilling systems

Abstract

This paper analyzes the stability of the discrete model proposed by Richard et al. [1,2] to study the self-excited axial and torsional vibrations of deep drilling systems. This model, which relies on a rate-independent bit/rock interaction law, reduces to a coupled system of state-dependent delay differential equations governing the axial and angular perturbations to the stationary motion of the bit. A linear stability analysis indicates that, although the steady-state motion of the bit is always unstable, the nature of the instability depends on the nominal angular velocity Ω 0 of the drillstring imposed at the rig. On the one hand, if Ω 0 is larger than a critical velocity Ω c , the angular dynamics is responsible for the instability. However, on the timescale of the resonance period of the drillstring viewed as a torsional pendulum, the system behaves like a marginally stable one, provided that exogenous perturbations are of limited magnitude. The instability then only appears on a much larger timescale, in the form of slowly growing oscillations that ultimately lead to an undesired drilling regime such as bit-bouncing or stick-slip vibrations. On the other hand, if Ω 0 is smaller than Ω c , the instability manifests itself on the timescale of the bit motion due to a dominating unstable axial dynamics; perturbations to the steady-state motion then rapidly degenerate into stick-slip limit cycles or bit-bouncing. For typical deep drilling field conditions, the critical angular velocity Ω c is virtually independent of the axial force acting on the bit and of the bit bluntness. It can be approximated by a power law monomial, a function of known parameters of the drilling system and of the intrinsic specific energy (a quantity characterizing the energy required to drill a particular rock). This approximation holds on account that the dissipation in the drilling structure is negligible with respect to that taking place through the bit/rock interaction, as is typically the case. These findings are further illustrated on an example of deep drilling and shown to match the trends observed in the field.

A distributed parameter systems view of control problems in drilling

“…In Depouhon and Detournay [2014], a linear stability analysis of a lumped model shows the importance of the state-dependent delay t N (·). Stabilization of state-dependent state delays systems has been investigated in Bekiaris-Liberis et al [2012].…”

Systems with uncertain and variable delays in the oil industry: some examples and first solutions

“…As appears, the conditionḊ < 1 is violated on several occasions. The interested reader can refer to Depouhon and Detournay (2014); Di Meglio and Aarsnes (2015) for a comprehensive model of the drilling/cutting process at stake.…”

Passive Flutter Suppression Using a Nonlinear Tuned Vibration Absorber

“…Limit cycle oscillations (LCOs) are encountered in a number of real engineering applications including aircraft [1,2] , machine tools [3,4] , automotive disk brakes [5,6] and oil-drilling rigs [7] . LCOs often limit the performance and can also endanger the safety of operation [8] .…”