1 Copyright © 2005 by ASME MODELING OF STICK-SLIP IN MULTIBODY DRILLING SYSTEMS Y.A. Khulief Department of Mechanical Engineering King Fahd University of Petroleum & Minerals KFUPM Box 1767, Dhahran 31261 Saudi Arabia khulief@kfupm.edu.sa F.A. Al-Sulaiman Department of Mechanical Engineering King Fahd University of Petroleum & Minerals KFUPM Box 238, Dhahran 31261 Saudi Arabia falehas@kfupm.edu.sa S. Bashmal Department of Mechanical Engineering King Fahd University of Petroleum & Minerals KFUPM Box 5069, Dhahran 31261 Saudi Arabia sbashmal@kfupm.edu.sa ABSTRACT Drillstring vibration is one of the major causes for a deteriorated drilling performance. Field experience revealed that it is crucial to understand the complex vibrational mechanisms experienced by a drilling system in order to better control its functional operation and improve its performance. Sick-slip oscillations due to contact between the drilling bit and formation is known to excite severe torsional and axial vibrations in the drillstring. A multibody dynamic model of the drilling system including the drillpipes, drillcollars, and the rotary drive is formulated. The equation of motion of the rotating drillstring is derived using Lagrangean approach in conjunction with the finite element method. The model accounts for the gyroscopic effect, the inertia coupling, the effect of the gravitational force field, and the stick-slip interaction forces. Explicit expressions of the finite element inertia coupling and axial stiffening matrices are derived using a consistent mass formulation. Modal transformations are invoked to obtain a reduced order modal form of the dynamic equations. The developed model is integrated into a computational scheme to calculate time-response of the drillstring system in the presence of stick-slip excitations. Key words: Drillstring, Stick-Slip Vibrations, Finite Element, Modal Transformations INTRODUCTION The problem of drillstring vibrations has been recognized for many years as one of the prime causes of deterioration in drilling performance, and was subjected to some early investigations as reported in the literature [1-5]. Field observations based on downhole and surface vibration measurements have indicated that drillstrings exhibit severe vibrations. These vibrations are observed to become more severe at the bottomhole assembly (BHA). A typical drillstring configuration is shown in Fig. 1. The main reasons for drillstring vibrations are due to contact of the bit with the formation and contact of the drillstring (drill pipe, drill collars and stabilizers) with the borehole. Bent pipes and misalignment of the drillstring represent additional causes for drillstring vibrations. Such vibrations, in general, consist of axial, flexural and torsional deformations [3]. However, stick-slip is considered as the most detrimental type of torsional vibration to the service life of the drillstring and downhole equipment. Successive stick-slip oscillations induce large cyclic stresses, which can lead to fatigue problems, reduction of bit life, unexpected changes in drilling direction, and may even result in failure of the drillstring. Several dynamic formulations have been reported for investigating specific aspects of drillstring vibrational behavior. A few investigations tackled the stick-slip aspect in drilling systems. One of the major difficulties in modeling stick-slip arises from the inaccurate description of some involved parameters and downhole boundary conditions. Moreover, the analysis of the stick-slip phenomenon is numerically challenging, because the static and kinetic friction mechanisms normally result in discontinuities in the dynamic model, [6]. However, most of the reported stick-slip investigations have attributed the associated oscillations to static friction effects resulting from rock/bit interaction [7-9]. A few models of drillstring stick-slip were based on a single degree of freedom torsional pendulum [10-15], wherein a rigid body with constant mass and moment of inertia was used to model the BHA and a linear spring to model the drillstring. Most often, the friction is taken as a non-linear function and is fitted using field data [12, 16]. Leine et al. [17] addressed the combined torsional (stick- slip) and lateral (whirling) vibrations. They used a low- dimensional model to describe the stick-slip whirl interaction. The BHA was modeled as a rigid disk at the end of a massless flexible drillpipe. Although, such simple models provided some insight into this complex phenomenon, they ignored the continuum nature of the drillstring. In order to capture the multi-degree of freedom elastic behavior of the drillstring, some higher-order models were introduced. Yigit et al. [18, 19] presented dynamic models of a rotating drillstring based on the Lagrangean formulation and the assumed modes method. One model Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 24-28, 2005, Long Beach, California USA DETC2005-84225