Shock and Vibration

Volume 2016, Article ID 9168747, 7 pages

http://dx.doi.org/10.1155/2016/9168747

## Stick-Slip Analysis of a Drill String Subjected to Deterministic Excitation and Stochastic Excitation

Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St John’s, NL, Canada A1B 3X5

Received 13 January 2016; Accepted 27 April 2016

Academic Editor: Evgeny Petrov

Copyright © 2016 Hongyuan Qiu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using a finite element model, this paper investigates the torsional vibration of a drill string under combined deterministic excitation and random excitation. The random excitation is caused by the random friction coefficients between the drill bit and the bottom of the hole and assumed as white noise. Simulation shows that the responses under random excitation become random too, and the probabilistic distribution of the responses at each discretized time instant is obtained. The two points, entering and leaving the stick stage, are examined with special attention. The results indicate that the two points become random under random excitation, and the distributions are not normal even when the excitation is assumed as Gaussian white noise.

#### 1. Introduction

In oil and gas industry, wells are drilled for either exploration or production purposes. The drill string, a very long and slender structure, experiences various vibrations during drilling operations, which can have significant detrimental effects on the drilling system [1]. Those detrimental effects include decrease of rate of penetration (ROP), interference with measurement while drilling (MWD) tools, and causing fatigue of the drilling components. In general, three vibration modes exist in the drill string, namely, axial, torsional, and lateral. Among the three vibrations, torsional vibration has attracted significant research effort partially due to its severe negative effects on drilling efficiency and life of drill string components. In the open literature, the torsional vibration/dynamics of a drill string has been modeled in several ways [2–4]. A torsional pendulum with one or two degrees of freedom is often used [5–7] for modeling stick-slip, one special type of torsional vibration. These models are good to look at the dynamics qualitatively; however, they may not be adequate to provide more quantitative insight into the problems under investigation. Thus, some other researchers modeled the system with the Finite Element Method (FEM) [8–10]. In modeling the torsional vibration, an important factor is the excitation to the system. Some researchers accounted for the friction between the drill string and the wellbore; others focused on the resistant torque on the bit [1, 2, 5, 6, 11–13]. Existing research has revealed that the excitation from the bit-rock interaction is especially complicated [1, 6, 13], involving both friction and cutting mechanisms. Under certain conditions, the friction mechanism may cause stick-slip motion of the bit which has particularly negative effect on the drilling system. Some researchers examined the stick-slip motion and the complicated dynamics in torsion [1, 2, 5, 11–13]. Undoubtedly, these research works are helpful to understand the complex dynamics of the drill string in rotation; however, limitations exist. First, both field test and theoretical analysis have indicated that the friction mechanism between two surfaces is very complicated; the friction coefficient, in reality, is related to various different factors, such as the profile of the surface, the materials, and the lubrication conditions. The value of friction coefficient is always highly scattered. Second, the downhole condition is highly unpredictable due to the many uncertain factors in the wellbore. These factors determine that the drill string vibration and dynamics cannot be well understood with deterministic theory; rather methods from random vibration and/or stochastic dynamics would be much more powerful tools. As a matter of fact, [14] realized this point already in as early as 1950s and proposed a probabilistic model. However, research work along this direction has progressed very little since then, probably due to the conceptual complexity of random vibration. Among the very few researchers working on random vibrations of drill string, [15] investigated lateral vibration with a nonlinear random model. In the work of Chevallier, the excitation for a tricone bit and the excitation for a PDC bit were modeled as a Kanai-Tajimi process and as band limited white noise, respectively. The nonlinearity was handled with a stochastic linearization technique. In recent years, [16–19] also investigated drill string dynamics with probabilistic models. The authors’ focus was placed on the bit-rock interaction and the drilling fluid effect on dynamics. In some other papers, [20, 21] also studied the uncertainties in the weight-on-hook.

In general, there is a lack of work on torsional vibration of drill strings from the random perspective. In view of this fact, this paper, which is based on the work of [22], focuses on the random torsional vibration of a drill string. FEM is used to build the dynamic model. The central difference method is then used to find the solution, and Monte Carlo (MC) simulation is carried out to obtain the statistics of the responses. The paper is organized as follows. In Section 2, a dynamic model is developed. Following that, the solution strategies used in both deterministic and random cases are presented in Section 3. Simulation results are presented and analyzed in Section 4. Finally, conclusions are drawn in Section 5.

#### 2. Formulation

##### 2.1. Dynamic Model

The drill string investigated in this paper is schematically shown in Figure 1. For convenience of mathematical derivation, it is assumed that the top is clamped, while the ground rotates with a constant speed. This does not change the nature of problem in terms of the relative motion. The drill string, including drill pipes and drill collars, is discretized into finite elements using linear Euler-Bernoulli beam theory. If only the rotation is considered, the element stiffness matrix and mass matrix are given bywhere is the shear modulus of the drill string material, is the drill string density, is the polar moment of inertia of drill string cross section, and is the element length.