Modal Reduction Technique for Predicting the Onset of Chaotic Behavior due to Lateral Vibrations in Drillstrings

Drillstrings used for oil and gas exploration and extraction consist of a drillpipe (slender columns on the order of 3–5 km long), drill collars (DCs) (thick-walled large-diameter pipes), stabilizers (cylindrical elements with short sections and diameter near that of the borehole), and a rock-cutting tool that uses rotational energy to penetrate the soil. Several types of vibrations ensue from these motions and play a major role in added costs resulting from unforeseen events such as abandoning holes, replacing bits, and fishing severed bottom-hole assemblies (BHAs). It is thus of critical importance to understand, predict, and mitigate the severe vibrations experienced by drillstrings and BHA to optimize drilling time while lowering fuel consumption and related emissions of NOX and/or other pollutants. In this paper, we present a dynamical analysis of the behavior of drillstrings due to the violent lateral vibrations (LVs) DCs may experience as a result of rotating drillstrings. The behavior is represented by a system of two coupled nonlinear ordinary equations that are integrated numerically with a finite element analysis based on Timoshenko beam (TB) formulation combined to a modal condensation technique to reduce the computational time. Various nonlinear dynamical analysis tools, such as frequency spectrum, Poincaré maps, bifurcation diagrams, and Lyapunov exponents (LE), are used to characterizing the response. The DC section between two stabilizers is essentially modeled as a Jeffcott rotor with nonlinearity effects included. The model builds on two earlier models for the finite element formulation and the treatment of chaotic vibrations. Nonlinearity appears in the form of drillstring/borehole contact force, friction, and quadratic damping. The DC flexibility is included to allow investigation of bending modes. The analysis takes into account the length of time to steady state, number of subintervals, presence of rigid body modes, number of finite elements, and modal coordinates. Simulations results indicate that by varying operating conditions, a spectrum of behaviors from periodic to chaotic may be observed.