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A nonlinear dynamics model and qualitative analysis are presented to study the key effective factors for coupled axial/torsional vibrations of a drill string, which is described as a simplified, equivalent, flexible shell under axial rotation. Here, after dimensionless processing, the mathematical models are obtained accounting for the coupling of axial and torsional vibrations using the nonlinear dynamics qualitative method, in which excitation loads and boundary conditions of the drill string are simplified to a rotating, flexible shell. The analysis of dynamics responses is performed by means of the Runge-Kutta-Fehlberg method, in which the rules that govern the changing of the torsional and axial excitation are revealed, and suggestions for engineering applications are also given. The simulation analysis shows that when the drill string is in a lower-speed rotation zone, the torsional excitation is the key factor in the coupling vibration, and increasing the torsional stress of the drill string more easily leads to the coupling vibration; however, when the drill string is in a higher-speed rotating zone, the axial excitation is a key factor in the coupling vibration, and the axial stress in a particular interval more easily leads to the coupling vibration of the drill string.

The drill string consists of several drill pipes, drill collars, stabilizers, and connections (crossover sub), subjected to some heavy and complex dynamic loadings caused by different sources, such as bit and drill string interactions with the formations, torque exerted by the rotary table or top drive, buckling, and misalignment. By producing different states of stress, these loads might result in excess vibrations and lead to failure of the drilling tools. Moreover, rotation of the rotary table or top drive on the surface might be transformed into a turbulent movement in the downhole. Three forms of vibrations that have been identified for the drill string are axial, torsional, and lateral vibrations, as shown in Figure

Three forms of drill string vibration.

The coupling vibration of the drill string can lead to severe vibration, and this energy boosts the amplitude of the string vibration, increases bending and impacts with the borehole, and leads to the early fatigue of tools and the reduction of bit life. Moreover, impacts with the borehole wall tend to form an overgauge hole or produce problems with the directional control of the well and also increase the surface torque [

In accord with the geometric nonlinear characteristics of drill strings, and by adopting the combination of nonlinear dynamics and FEM (finite element method), Sampaio et al. [

Based on the specific problem of the coupled vibration of the drill string, scholars usually analyze the drill string to find the relationship between the drilling process parameters and the modal shapes and then change the drilling parameters to avoid the resonance of the drill string, which is a very practical control method; however, the key factor is to discover the relationship between the key dynamics parameters and the coupled vibrations of the drill string. Considering the complexity of the practical dynamics of the drill string system and the comprehensiveness of the quantitative analysis directly, this study employs the dimensionless method to investigate the mechanism of the coupled vibration of the drill string qualitatively. This will reveal which key factors affect the coupled vibration and how they function, which can be a basis for the quantitative analysis of the coupling vibration of the drill string.

During the process of drilling, the transverse vibration of the drill string, which is intense at the bit in the bottom of the well and attenuates quickly along the drill string, mainly contributes to the bottom hole assembly (BHA); however, the axial and torsional vibration affects the entire drill string.

In addition, several stabilizers support the lower portion of the drill string and absorb most transverse vibration energy. In other words, the transverse motion of the stabilized section mainly contributes to the transverse dynamics at the bit. The drill collars are assumed to be rigid for torsional vibrations; that is to say, the torsional deformations are assumed to take place only in the drill pipe, which can be justified since the drill collars are much stiffer than the drill pipe in torsion. Therefore, the transverse vibration of the BHA is decoupled from the upper segments of the drill string, and the entire drill string is assumed to be fixed at the top and free at the bit for the axial and torsional motion [

Based on the above analyses, ignoring the transverse vibration of the drill string, this work only analyzes the mechanism of the coupled axial and torsional vibration of the upper portion of the BHA. Here, assuming the centerline of the shell before deformation as the axis, the drill string is simplified to a flexible rotation shell instead of simply a supported beam, which will ignore most torsional vibration. The dynamics model is shown in Figure

The mechanical model of the rotation, flexible shell.

To facilitate the calculation of the drill string strain, select the arbitrary point displacement of the drill string as follows [

Since the drill string cannot be thin, and the tangential displacements

Based on the nonlinear shell theory [

Analogous to the case of plane stress, the stress in thickness of the drill string is neglected [

Based on (

The modals of axial and torsional vibration are given as [

Using the Galerkin method to disperse (

To facilitate the analysis, replace the coefficients in (

Substituting (

Here, the differential operators are employed as in the following form:

Taking into account the principal parameter resonance and 2 : 1 internal resonance, the resonance relationship can be expressed as follows:

Taking into account the vibration response caused by the axial and torsional excitation in the case of the above resonance, the dimensionless axial and torsional excitation in the frequency domain can be expanded into the following form, including the two resonance frequencies [

Substituting (

For

For

Substituting (

The change rules for the amplitude and phase angles in the motion equation are found in (

In the present section, the numerical simulation results obtained from using the proposed model are discussed for the upper portion of the BHA. The Runge-Kutta-Fehlberg method with adaptive steps is employed to perform the simulations, aiming at obtaining the dynamic response of the coupled axial and torsional vibration in the case of resonance. The geometric properties of the upper segment are the length ^{3}, and the damping coefficients,

The Runge-Kutta-Fehlberg method is employed to analyze the average of (

Bifurcations of the system under torsional excitation.

Bifurcation diagram of the axial modal

Bifurcation diagram of the torsional modal

From the bifurcation, along with the change in torsional excitation

By only increasing the torsional excitation until

Period responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

When

Period responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

When

Period responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

By increasing the value of

The almost-periodic responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

It is well-known that the torsional vibration of the drill string is mainly generated from the rotational speed change of the bit when breaking rock intermittently. The excitation frequency is related to the rotation speed, stiffness of the drill string, and characteristics of the rock. The higher the rock hardness, the bigger the torsional stress and the greater the torsional amplitude. We know from (

Based on the above simulation analysis, we also found that torsional excitation will affect both torsional and axial vibration simultaneously. Torsional and axial vibration are similar qualitatively and in the nature of the vibration and are synchronous in form, proving that energy can transfer between the torsional and the axial vibration modal.

The above analysis accounts for the impact of torsional excitation on the coupling of the axial and torsional vibrations. Here, the impact of axial excitation will be investigated.

The bifurcation diagram (Figure

Bifurcation of the resonance frequency-multiplier.

Bifurcation diagram of the axial modal

Bifurcation diagram of the torsional modal

Increasing axial excitation until

Period responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

When

Period responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

When

Period responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

When

Period-doubled responses of the system (

Phase diagram of the axial modal

Oscillogram of the axial modal

Phase diagram of the torsional modal

Oscillogram of the torsional modal

Three-dimensional phase diagram

Poincare section

As can be seen from the above qualitative analysis of the axial excitation of the coupling vibration, we find the same phenomenon as in the torsional excitation analysis for the coupling vibration, namely, that axial excitation affects both torsional and axial vibration simultaneously and is similar qualitatively and also synchronous in form.

Along with the increase in the axial excitation, the system coupling vibration turns from period-doubling to period-three, to chaotic, and, finally, to period-doubling. So, the axial excitation parameter leads to a coupling vibration in some special zone. In the process of practical drilling, the axial jump mainly contributes to the axial excitation. The higher the rotation speed, the higher the excitation frequency. The larger the WOB becomes, the larger both the axial amplitude and the axial stress will be. We know from (

In this paper, we modeled and analyzed drill-string vibrations by focusing on the coupled axial/torsional vibrations by means of nonlinear dynamics and qualitative analysis, aiming at revealing the key, effective factors influencing the coupled vibrations. Here, the drill string was described as a simplified, equivalent, flexible shell under axial rotation, in which the excitation loads and boundary conditions of the drill string were simplified. After dimensionless processing, we built the dynamics motion equation and the average equation.

We found that the low-frequency amplitude expression of the torsional excitation and the high-frequency amplitude expression of the axial excitation in the average equation are the factors that determine the coupling vibration. Based on this procedure, numerical simulations were carried out with adaptive steps using the Runge-Kutta-Fehlberg method to discover the response of the system vibrations. Further, we found that a change in the torsional or the axial excitation affects both the torsional and the axial vibration simultaneously and that it is similar qualitatively and synchronous in form.

The results of the simulation analysis show that when the drill string is in a lower-speed rotation zone, the torsional excitation mainly contributes to the coupling vibration, increasing the torsional stress of the drill string that more easily leads to the coupling vibration. When in a higher-speed rotating zone, the axial excitation mainly contributes to the coupling vibration; so, in a particular interval, it is more likely to cause the coupling vibration of the drill string.

Consider

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge financial support by Natural Science Foundation of China, Project 11372071, and Postdoctoral Fund of China, Project 2013M541339.