Nonlinear stochastic drill-string vibrations

Lateral vibrations of drill-strings used in oil well operations are considered. A finite elements based discretization procedure leads to a nonlinear dynamic system which is used to represent the drill-string inertia and stuffiness characteristics, as well as the elasticity of the wall of the well. Due to the erratic pattern and the uncertainty of the forces at the drill-bit, a stochastic dynamics approach is adopted in investigating the problem. The method of statistical linearization is used, and expressions for determining an equivalent linear system to model the drill-string dynamics are derived. Further, a Monte Carlo simulation of the system dynamics is conducted by means of an Auto Regressive Moving Average (ARMA) digital filter and by integrating the equations of motion using the Newmark scheme. Numerical results pertaining to data obtained by measurement while drilling (MWD) tools are presented. It is hoped that this study will enhance the interest in using stochastic dynamics techniques in drilling system analysis and design, as they can capture quite appropriately the inherent uncertainty of the bit forces and, potentially, of other sources.