Stochastic modeling and simulation of the coupled axialtorsional vibration of a drill-string

Abstract

This dissertation investigates the coupled axial-torsional vibration of drill-strings in the presence of lithology uncertainties. This work has three main objectives: (i) to perform a deterministic numerical investigation through a continuous axialtorsional coupled model, considering the geometric nonlinearities and the bit-rock interaction; (ii) to propose a novel stochastic model to describe uncertainties on bitrock interaction by using Itˆo stochastic differential equations, and; (iii) to analyze the influence of the main parameters on stochastic response by comparing the statistics obtained from the results. The continuous model is discretized by means of the finite element method, considering the geometric nonlinearities and a bit-rock interaction model that is qualitatively validated with experimental results. The deterministic results showed that the geometric nonlinearities did not affect the response and, thus, are ignored in the subsequent analyses. The model is reduced using the modal basis and the results are analyzed. Then, a stochastic model is constructed to describe lithology heterogeneity. This model considers two stochastic processes: OrnsteinUhlenbeck process and a novel coupled process. The novel process is capable of describing better the physics when severe stick-slip happens. The statistics of the response show that the heterogeneity on rock formation induces severer vibrations at the bit. Also, the differences between the processes are small when bit does not stick, but the novel process causes worse vibrations when bit sticks.