Bifurcation and chaos in drillstring dynamics

G.H.M. van der Heijden

A model, consisting of a system of ordinary differential equations, is discussed, describing the lateral vibrations of a stabilised drillstring as is used in drilling oil wells. It turns out that the nonlinearity introduced by the bearing clearance (the play between stabiliser and borehole wall) gives rise to complicated dynamics. For clearances less than (approximately) the mass eccentricity of the drillstring synchronous forward whirl dominates asymptotic motion. For clearances larger than the mass eccentricity, and relatively high rotary speeds, whirl is no longer possible and one observes (in co-rotating co-ordinates) several types of periodic solutions, each of them undergoing a series of period-doubling bifurcations ending up in chaotic motion described by a strange attractor with unusual dimensional properties. Coulomb friction between stabiliser and wall causes the forward whirl to become unstable at certain driving frequencies, resulting in nonsynchronous self-excited oscillations of large amplitude. There are several possibilities, then, for a transition from forward to backward drillstring motion which may induce strongly fluctuating bending moments.

Chaos, Solitons & Fractals 3, 219-247 (1993)