Zifeng Li

Petroleum Engineering Institute, Yanshan University, Qinhuangdao, Hebei, 066004, China

ABSTRACT This paper analyses the physical models in “Kapitaniak M, Hamaneh VV, Chˊavez JP, Nandakumar K,Wiercigroch M.Unveiling complexity of drill-string vibrations: experiments and modelling.International Journal of Mechanical Sciences 2015;101-102:324-327”.The results are that the physical models described in the original paper are clearly incorrect.For the physical model used to explain the axial vibration on a drill string,using a lumped mass-axial spring with damping to model a long slender drill string is not advisable.For the physical model used to explain the torsional vibration on a drill string,using a torsional spring-pendulum with damping to model a long slender drill string is not advisable.The drill string should be constrained in the well bore.As an example,appropriate physical and mathematical models of drill string axial and torsional vibrations are recommended.

Keywords:Drill string Vibration Axial Torsion Boundary

1.Introduction

Physical models are the most important in the mechanical analysis.Hundreds papers have been published on axial vibration,torsional vibration, coupled vibration and buckling.Recently [1],published a paper which investigates complex drill-string dynamics,including stability,on a novel experimental rig.The work in the present paper analyses the correctness and integrity of the physical models,statements and equations presented in that paper,and finally, appropriate physical and mathematical models of drill string axial and torsional vibration are recommended.

2.Original physical models

Fig.1 shows the physical models used to describe drill string torsional vibration[Fig.10 in Ref.[1]].BHA means the bottom hole assembly.Fig.2 shows the helical buckling deformations occurring along the drill string[Fig.15 in Ref.[1]].For more details,please see Ref.[1].

3.Comments

(1) In Fig.1, the left-hand side shows a torsional springpendulum with damping; however, the figure caption says that it is a lumped mass model.The reason for this contradictory can be found in the previous papers published by Nandakumar K and Wiercigroch M(2013a,2013b),in which the authors built a modified physical model(Fig.4)from the original axial-torsional vibrations(Fig.3).Fig.1 is a simplified version of Fig.4.

In Fig.4, the lumped mass M is used to study the axial vibrations,and torsional inertial I is used to study the torsional vibrations.“The lumped mass M′′in Fig.3 is changed to“m”in Fig.4; “lumped torsional inertial I′′in Fig.3 is changed to“I” in Fig.4; “the drill-string model with a lumped mass approximating the BHA(m,I)”is listed in the figure caption,only emphasizing on the lumped mass, but the lumped torsional inertia is missing.From Figs.4-1, they removed“m”from the figure,without changing the“lumped mass”to“lumped torsional inertial in the figure caption”.

Fig.1.Left panel depicts a lump mass model, whereas the right panel shows a continuous equivalent modelled by using FEM in Ref.[1].

Fig.2.Development of helical deformation of flexible shaft during stick-slip motion for pre-buckled configuration.(a) Initial straight configuration, (b) pre-buckled configuration, (c) shape of the flexible shaft during stick phase in Ref.[1].

(2) The left-hand side in Fig.1 shows a torsional springpendulum with damping, nevertheless, the right-hand side in Fig.1 is a continuous rod without damping.The mass distribution is changed as well as the damping.They are far from equivalent.

Fig.3.The original physical model of the axial-torsional vibrations.The model has two degrees-of-freedom (translation U, and rotation Φ of the bit).The rotary table at the top is driven at constant angular velocity Ω0 and a hook load H0 is applied to top of drill-pipes.Reactions from rock,W,and there active torque,T,are acting on the BHA in Refs.[10,11].

Fig.4.Schematic of the drill-string model with a lumped mass approximating the BHA(m,I), and a flexible cylinder modelling the stiffness (Ka,Kt) and damping (Ca,Ct) in Refs.[10,11].

(3) Using a torsional spring-pendulum with damping to model the long slender drill string to examine the drill string torsional vibrations is not recommended.Because the characteristics of the torsional spring-pendulum with damping are significantly different from that of the long slender drill string in a well bore filled with fluid,the results are nearly useless.The torsion vibration is a problem of vibration wave propagation along the drill string under bottom excitation,the solution of which gives the angle,the stresses,etc., at any time and any place; however the lumped torsional inertial approach can only solve the issues of particle angle.They are two different problems.The majority of researchers investigating this phenomenon establish a mathematical model directly on the long slender drill string in the well bore filled with fluid.Their work shows that the differential equation to describe this phenomenon is a onedimensional torsional wave equation with damping [2-9].

Fig.5.The drill pipe in Fig.2 is shown to be outside of the well bore.

(4) Using a lumped mass-axial spring with damping to model the long slender drill string to examine the drill string axial vibration is not recommended too.Because the characteristics of the lumped mass-axial spring with damping are significantly different from that of a long slender drill string in a well bore filled with fluid,the results are nearly useless.The axial vibration is a problem of vibration wave propagation along the drill string under bottom excitation, the solution of which gives the displacement, the stresses, etc., at any time and any place;however the lumped-mass approach solve only the issues of particle displacement.They are two different problems.The majority of researchers investigating this phenomenon establish a mathematical model directly on the long slender drill string in the well bore filled with fluid.Their work shows that the differential equation to describe this phenomenon is a one-dimensional axial wave equation with damping [2-9].

(5) The drill string should be constrained in the well bore.If we draw a diagram of the well bore wall, we will find that the drill pipe shown in Fig.2b and c is outside of the well bore,as shown in Fig.5.

(6) A typical drill string includes an 8 1/2′′bit, an 8′′drill collar,and a 5′′drill pipe.The ratio of the bit diameter/drill pipe diameter of the typical drill string is not as large as shown in Fig.2.

4.Physical and mathematical models for drill string axial and torsional vibrations

For the recommend physical and mathematical models for drill string axial and torsional vibrations, please see Refs.[2-7].

5.Conclusions

(1) It is not recommended to use a torsional spring pendulum with damping to model the torsional vibration of a long slender drill string.

(2) It is also not recommended to use a lumped mass-axial spring with damping to model the axial vibration of a long slender drill string.

(3) The drill string should be constrained in the well bore.

(4) Finally,physical and mathematical models of drill string axial and torsional vibration are introduced.

Declaration of competing interests

The authors declare that they have no conflict of interests.

Acknowledgments

The paper is supported by the National Natural Science Foundation of China (Grant No.51674220).