## 5 Stick-Slip Vibrations  
The purpose of the model introduced in this paper is to study the dynamic behavior of the drillstring in an operational mode. This model combines the mechanical response of drillstring to torsional vibrations, and the bit-rock interaction. Based on [PERSON] (1998), the drillstring is divided into three main parts: (1) top rotating system, (2) drill pipes, and (3) BHA. Each element is represented in the form of a disk of mass (m) and inertia (J); the disks are connected to each other by springs with a torsional stiffness (k\({}_{t}\)) and a viscous damping coefficient (c\({}_{t}\)). Figure 3 shows the drillstring as a torsion pendulum driven by the top drive. The interaction of the drilling mud with the drillstring can be simplified in the form of viscous damping c\({}_{t}\) and c\({}_{tb}\) along the drill pipes and the BHA, respectively. It is assumed that the top drive rotates with an initial velocity \(\theta_{r}\) and provides a torque Tm (with neglecting the damping and frictional torques at the top drive). According to the fundamental principle of the rotational dynamics  
\[\frac{d\overset{\rightarrow}{L}}{dt}=\overset{\rightarrow}{M} \tag{2}\]  
\(L\) is angular momentum and \(M\) is dynamic moment of external forces  
\[\overset{\rightarrow}{L}=\text{J}\,\frac{d\theta}{dt}\overset{\rightarrow}{e} \tag{3}\]  
\(\overset{\rightarrow}{e}\) is unit vector. Substituting (3) in (2), Equation (4) is obtained. \[\text{J}\ddot{\theta}=M \tag{4}\]  
Taking into account the viscous forces and the rigidity of elements, the equation of motion becomes  
\[\text{J}_{i}\ddot{\theta_{i}}=M-C_{i+1}(\dot{\theta_{i}}-\dot{\theta_{i+1}})- k_{i+1}(\theta_{i}-\theta_{i+1})+C_{i}(\dot{\theta}_{i-1}-\dot{\theta_{i}})+k_{i}( \theta_{i-1}-\theta_{i}) \tag{5}\]  
\(C\) is viscous damping, \(k\) is torsional stiffness, \(i\) is current element, \(i+1\) is next element. By applying Equation (5), equations of motion for three elements are given in (6)  
\[\left\{ \begin{array}{cc}J_{r}\ddot{\theta}_{r}+C_{t}(\dot{\theta}_{r}-\dot{ \theta}_{p})+k_{t}(\theta_{r}-\theta_{p})=T_{m}\\ J_{p}\ddot{\theta}_{p}-C_{t}(\dot{\theta}_{r}-\dot{\theta}_{p})-k_{t}(\theta_{ r}-\theta_{p})+c_{tb}(\theta_{p}-\dot{\theta}_{b})&+k_{tb}(\theta_{p}-\theta_{ b})=0\\ J_{b}\ddot{\theta}_{b}-C_{tb}(\dot{\theta}_{p}-\dot{\theta}_{b})-k_{tb}( \theta_{p}-\theta_{b})=-T_{b}(\dot{\theta}_{b})\end{array} \right. \tag{6}\]  
\(T_{b}(\dot{\theta}_{b})=T_{ab}(\dot{\theta}_{b})+T_{fb}(\dot{\theta}_{b})\) is the torque on the bit, \(T_{ab}\approx c_{b}\cdot\dot{\theta}_{b}\) approximates the influence of the drilling mud on the bit, \(T_{fb}(\dot{\theta}_{b})\) is the non-linear friction torque due to bit-rock contact interaction, it is given in (7). \[T_{fb}(\dot{\theta}_{b})=Wob\cdot R_{b}\cdot\mu_{b}(\dot{\theta}_{b})\cdot sign (\dot{\theta}_{b}) \tag{7}\]  
Figure 5: Measured vibration using BlackBox in Well-1. _Wob_ is the weight on the bit, \(R_{b}\) is the bit radius, \(sign(\hat{\theta}_{b})\) represents the rotational direction; and \(\mu_{b}(\hat{\theta}_{b})\) is the bit dry friction, which follows the Karnopp friction model, and it is given in (8). \[\mu_{b}(\hat{\theta}_{b})=\mu_{cb}+(\mu_{sb}-\mu_{cb})e^{-\frac{\gamma_{b}}{ \gamma_{f}}|\hat{\theta}_{b}|} \tag{8}\]  
\(\mu_{sb}\) is the static friction coefficient, \(\mu_{cb}\) is the Coulomb friction coefficient, \(\gamma_{b}\) and \(\
u_{f}\) are constants verifying the following assumptions: \(0<\gamma_{b}<1\) and \(\
u_{f}>0\), to simplify calculation, it is assumed that \(\lambda=\frac{\gamma_{b}}{\gamma_{f}}\). The sign function is defined in (9). \[sign(\hat{\theta}_{b})=\ \frac{\hat{\theta}_{b}}{\left|\hat{\theta}_{b}\right|} \ \text{if}\ \hat{\theta}_{b}\
eq\ 0,\ sign(\hat{\theta}_{b})\in[-1,1]\ \text{if}\ \hat{\theta}_{b}=0 \tag{9}\]  
In the next section, simulation results will be presented with several scenarios, including input torque variation, change in WOB and bit radius variation. The simulations were carried out using the MATLAB/Simulink R2012a (7.14.0.739).