Drill-string model for coupled lateral-torsional vibrations with stochastic non-proportional damping

Abstract

In this dissertation, drill-string dynamics are analyzed. The drill-string consists of a rotor-like structure that drills rock formations until the oil reservoir is reached. The structure in question is extremely slender and prone to different non-linear phenomena. A numerical model is developed, where the Finite Element Method is used to discretize a continuous drill-string geometry, which leads to a system of non-linear differential equations. Geometric nonlinearities includes the static effects of relevant axial forces of the problem. The model considers a continuous unbalance force approach not commonly used in the literature and lateral-torsional generalized impact forces. In the sequence, distinct damping models are presented for the drill-string lateral dynamics. Three damping ratio relations are explored, originating different proportional damping matrices. Later, uncertainties are introduced in the damping matrices with the random matrix theory, which adds global non-parametric uncertainties to the damping term. With this stochastic model, it is – to some extent – acknowledged that the complete nature of the dissipation forces in the process might be unknown and that a nonlinear dynamic with fluid-structure interaction may present non-proportional damping. A model order reduction technique is used, based on the most relevant modes, and numerical simulations are conducted in order to obtain the time-domain response in different drilling configurations. With these, maps detailing possible regimes are presented. The Monte Carlo Method is applied for numerical simulations of the stochastic models. Maps containing probabilities of events are then calculated. Finally, the impact of the proportional damping hypothesis is qualitatively evaluated.