Modeling of torsional oscillation of drillstring dynamics

  • Qilong XueEmail author
  • Henry Leung
  • Leilei Huang
  • Rui Zhang
  • Baolin Liu
  • Jin Wang
  • Lixin Li
Original Paper


Drillstring vibration is an important cause of precocious failure of drilling, and torsional vibration is a particular important phenomenon. Stick/slip is a severe type of torsional drillstring oscillation that affects the efficiency of the drilling process and can cause bit damage as well as drillstring failure. In an automatic vertical drilling system (VDS), frictional force between the pads and borehole wall will reduce the drill bit instantaneous rotational speed. The pads of the implementing agencies in VDS constantly pushed against the borehole wall, making bottom hole by a cycle of nonlinear damping force, which lead to chaotic and disorder motion to the bottom drilling tool. Here, the stick–slip vibration is modeled to understand the large amplitude torsional oscillation of the drillstring. Based on the model, the causes for torsional vibrations in VDS and torsional vibrations with and without stick–slip will be evaluated. We show that pushing the borehole wall will cause the drill bit torsional vibration more serious. The results contribute to the better understanding of the dynamics of the push-the-bit VDS. The results obtained can also be applied to the same type of rotary steerable system.


Torsional oscillation Stick–slip Automatic vertical drilling system Push the bit Drillstring dynamics 

List of symbols

\(J_t \)

The top drive equivalent moment of inertia (\(\hbox {kg}\,\hbox {m}^{2}\))

\(\theta _t \)

The rotation angle at the top of the drillstring (\(\hbox {rad}\))


Gearbox gear ratio


Equivalent rotational damping of the motor rotor and gear reduction system (\(\hbox {N}\, \hbox {m}\,\hbox {s/rad}\))


The electromagnetic torque produced by the rotor (\(\hbox {N}\, \hbox {m}\))


The torque acting on the top of the drillstring to the top drive (\(\hbox {N}\, \hbox {m}\))

\(\Omega _0\)

The rotational speed of motor (\(\hbox {rad/s}\))


The spring stiffness of top drive (\(\hbox {N}\,\hbox {m/rad}\))


The viscous damping coefficient of top drive (\(\hbox {N}\, \hbox {m}\, \)s/rad)

\(\psi \)

The connection rotation angle of equivalent damper and equivalent torsion spring (\(\hbox {rad}\))


The spring stiffness of drillstring (\(\hbox {N}\,\hbox {m/rad}\))


The viscous damping coefficient of drillstring (\(\hbox {N}\, \hbox {m}\, \hbox {s/rad}\))


The drillstring moment of inertia (\(\hbox {kg}\, \hbox {m}^{2}\))


The combined mass of the drillstring (kg)


The weight on the bit (N)


The torque on the bit because of cutting rock (\(\hbox {N}\, \hbox {m}\))


The torque on the bit because of one of the pads’ push the bit (\(\hbox {N}\, \hbox {m}\))


The torque acting on the bit, \(T_{b}=T_{d}+T_{f}\) (\(\hbox {N}\, \hbox {m}\))

\(\psi _b\)

The rotation angle at the top of the drillstring (rad)

\(\varphi =\psi _t -\psi _b\)

The drillstring rotation angle difference between top and bottom (rad)


The radius of the drill bit (mm)

\(\delta _c \)

The cutting depth of one revolution of the drill bit (mm)

\(\phi \)

The bit torsional displacement (rad)


The average rate of penetration (m/s)


The difference between the total weight and the hookload (N)

\(\bar{{\omega }}_b\)

The average rotational speed of drill bit (rad/s)


The force because of the drill bit cutting action (N)

\(F_c \)

The contact force of the drill bit (N)

\(\xi \)

The coefficient of cutter inclination

\(\varepsilon \)

The rock specific strength (\(\hbox {Mpa/mm}^{2}\))

\(\alpha \)

The phase angle of the cone in the bottom of the well (rad)


The wellbore radius (mm)


The pressure of the drillstring acting on the roller through the bearing (N)


The positive pressure acting on the roller cone (N)


The driving force of the drillstring on the cone (N)


The drill bit torque caused by the wave bottom hole (\(\hbox {N}\, \hbox {m}\))


The push force of a single pad pushing against the borehole wall i= 1, 2, 3, ... (N)

\(\Delta p\)

The internal and external pressure difference (Mpa)


The piston radius of the actuator (mm)

\(\rho _w\)

The drilling fluid density (\(\hbox {g/cm}^{3}\))

\(Q_n \)

The nozzle flow (1/s)

\(C_{ne} \)

The nozzle flow coefficient

\(d_{ne} \)

The nozzle equivalent diameter (mm)

\(\theta _S\)

The initial phase angle of pads pushing (rad)

\(\theta _E\)

The termination phase angle of pads pushing (rad)

\(\omega _t\)

The rotation angular speed of bottom tool (rad/s)


The relative speed of movement of friction model (m/s)



The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the Natural Science Foundation of China (51704264) and the National Key R&D Program of China (2016YFE0202200).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Engineering and TechnologyChina University of Geosciences (Beijing)BeijingChina
  2. 2.Key Laboratory of Deep Geodrilling TechnologyMinistry of Land and ResourcesBeijingChina
  3. 3.Sinopec International Petroleum Service CorporationBeijingChina
  4. 4.Department of Electrical and Computer EngineeringUniversity of CalgaryCalgaryCanada
  5. 5.College of Petroleum EngineeringChina University of PetroleumQingdaoChina
  6. 6.Chinese Academy of Geological SciencesBeijingChina

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