Abstract
Stick–slip vibration is a common phenomenon in ultra-deep drilling that significantly impacts the failure of both drill bits and drill tools. The most direct and efficacious approach to alleviating the stick–slip vibration of the drill string in the downhole is to modify its external excitation. In recent years, the composite impact tools that can simultaneously offer axial and torsional excitation in the downhole have been applied, effectively reducing the stick–slip vibration of the drill string. However, the mechanical mechanism thereof remains undefined. In order to understand the nature of this phenomenon, A dynamic model of the drill string taking into account multi-directional excitations is presented. The governing nonlinear equations are obtained by using the Lagrangian approach, which take the work done by the multidirectional excitation into consider. The Hertz contact model is introduced considering the constraints of the wellbore, and the finite element node iteration method is employed to solve the dynamics equation of drill string. The axial vibration, torsion vibration and phase trajectory characteristics of the drill string under multidirectional excitation are analyzed, and the inhibitory effect of the excitations on stick–slip vibration is clarified. The results show that the vibration characteristics of the bottom hole assembly can be significantly altered through periodic axial and torsional excitations at higher frequencies, resulting in the emergence of high-frequency vibration responses. These responses exhibit a pronounced inhibitory effect on stick–slip suppressed.
Keywords: Drill string, Stick–slip vibration, Composite impact tool, Excitation model, Dynamic characteristics
Subject terms: Mechanical engineering, Mineralogy, Nonlinear phenomena
Introduction
As the development of deep oil and gas resources progresses, there has been a rapid growth in the quantity of deep and ultra-deep wells1,2. During deep drilling operations, the complexity of drill string vibrations significantly increases. Notably, stick–slip vibrations occur intermittently, characterized by alternating between high-speed rotation and viscous immobility of the drill bit3–5. The empirical evidence demonstrates that stick–slip vibration will accelerate bit failure, significantly diminish drilling efficiency, and severely constrain mechanical penetration rate6,7. During the drilling process, the downhole stick–slip vibration is usually reduced by adjusting the rotary speed and weight on bit. However, this method is not effective and significantly constrain the drilling efficiency8,9. Figure 1 presents the frequency of adjustments to the operational parameters at various well intervals during the drilling of a certain ultra-deep well in western China. It can be seen that in the intervals of 4880–5120 m, 5520–5640 m and 5760–6120 m, the drill string was frequently lifted from the bottom of the wellbore, causing the weight on bit to drop to zero as indicated in the logging data.
Fig. 1.
The frequency of adjustments to weight on bit in diverse well intervals of an ultra-deep well.
The feasibility of mitigating stick–slip vibration in drill strings by introducing additional external excitation through downhole tools has been substantiated10–12. The current repertoire of downhole tools includes hydraulic oscillators that deliver high-frequency axial shocks, such as NOV’s Fluid Hammer, and torsional impactors that provide circumferential shocks, like Ulterra’s Tork buster. The downhole speed-up tools can apply additional high-frequency excitation to the drill bit, change the rock breaking mode of the drill bit and improve the rock breaking efficiency, to achieve the purpose of improving the penetration rate and reducing the drilling cost. At present, there are many literatures on the principle and application of downhole tools, however, the studies on the influence of tools on the dynamic characteristics of drill string are few. Batako et al.13 gave the stick–slip motion model of the drill string with rotary impact tool. In the model, the friction between the drill bit and the formation was used as the main parameter affecting the stick–slip motion, and the dynamic behavior of the drill string with and without rotary impact tool was compared. Ghasemloonia et al.14 established a finite element model of rotating drill string, studied the axial and lateral coupling vibration of drill string under the action of axial impact tool, and analyzed the influence of drilling fluid damping, upper driving torque and other parameters on drill string vibration. Fu et al.15 considered the wellbore trajectory of the drill string and the mechanism of axial vibration drag reduction, established a dynamic analysis model of the whole well drill string with axial vibration tool, and studied the influence of tool installation position, amplitude, and frequency on the drag reduction effect. Tian et al.16 designed a longitudinal-torsional coupling tool. Through the analysis of the working mechanism of the tool, a dynamic model of the drill string with the tool was established and a case study was carried out. Liu et al.17 established a finite element model of drill string with an axial vibration tool by considering the contact between drill string and wellbore. The effects of tool frequency and position, wellbore inclination angle and friction coefficient on drill string vibration were studied, and the results were compared with experimental data.
In recent years, there has been an emergence of composite impact tools that offer both high-frequency axial and torsional impacts18–20. The composite impact tool has the characteristics of both rotary impact tool and torsional impact tool. Its principle is to use drilling fluid to drive the internal mechanical structure, to generate axial and circumferential alternating high-frequency excitation at the installation position21. Liu et al.22 developed the axial and torsional impact Hammer, and the average drilling speed increased by 105.1–163.4% in field application. Practice shows that the addition of composite impact tool can better inhibit the stick–slip vibration of drill string23,24. But there are few studies on the influence mechanism of composite impact tools on the dynamic characteristics of drill string. The reason is the development of tools capable of delivering both axial and circumferential composite impacts is challenging and has been a relatively recent advancement. More importantly, analyzing the dynamic characteristics of the drill string under these two distinct types of excitations presents significant difficulties. It is necessary to fully consider the nonlinear coupling characteristics of the lateral, torsional and axial vibration of the drill string to reveal the mechanism of the composite impact tool.
In this paper, by analyzing the working principle of the composite impact tool with both axial and circumferential excitations, the excitation model is refined. On this basis, the dynamic finite element model of the drill string is modified, and the influence of the composite impact tool on the stick–slip vibration characteristics of the drill string is analyzed in detail. The elimination mechanism of the composite high-frequency excitation on the stick–slip vibration of the drill string is discussed, which provides theoretical support for the rational use of the composite impact tool in the field.
Excitation model of composite impact tool
The composite impact tool is generally composed of shell, commutator, axial impact hammer, circumferential pendulum, impact shell and nozzle, and is usually directly connected with PDC bit. The structure is shown in Fig. 2. There are four groups of flow channels on the internal commutator. Under the regulation of the commutator, the high-pressure drilling fluid periodically drives the axial hammer to reciprocate and collides with the casing to generate axial impact load; Drive the circumferential pendulum to make circumferential reciprocating motion and collide with the impact cylinder to generate torsional impact load.
Fig. 2.
Structure of composite impact tool.
According to the working principle of the composite impact tool, its function is equivalent to providing a composite high-frequency pulse load with alternating axial and circumferential directions to the drilling tool at the tool installation. The effect of the circumferential pulse load can be equivalent to the impact torque around the axis of the drilling tool. It is assumed that the impact load of the axial hammer is 
, the contact time with the hammer seat is 
, the equivalent torque generated by the circumferential pendulum impact is 
, the contact time is 
, the time for the drilling fluid to flow from the axial impact structure to the torsional impact structure is 
, and the reversing time of the pendulum is 
. The pulse action curve is shown in Fig. 3.
Fig. 3.
Time history diagram of composite impact tool.
The excitation can be written as axial impact force and impact torque around the axis:
 |
1 |
 |
2 |
where, 
is the axial load generated by the composite impact tool, 
; 
is the equivalent torque, 
; T is the period of single action of the composite impact tool, 
.
Dynamic model of drill string with composite impact tool
Based on Lagrange equation and finite element theory, the differential equation of dynamic motion of drill string can be expressed as25,26:
 |
3 |
where, 
, 
, 
are the displacement, velocity and acceleration matrices of the drill string; 
is the mass matrix, composed of translational and rotational inertia masses; 
is the damping matrix, consisting of Rayleigh damping and drilling fluid damping; 
is the stiffness matrix, which is composed of linear stiffness matrix 
and nonlinear stiffness matrix 
, and 
is the sum of the nonlinear stiffness matrices that result from the three nonlinear strain energies, 
and 
represents coupling between axial force and flexure, 
represents coupling between torsion and flexure; and Force matrix F includes gravity and inertia force caused by eccentric mass of drill string.
Regarding Eq. (3), the nodal iteration method and Newmark-β method can be employed for numerical solution. The accuracy of the drill string dynamics equation and its numerical solution method has been elaborated in Refs.25,27. Since the solution method employed in this paper is consistent with that of the cited literature, no repetitive verification is conducted herein.
When considering the composite impact tool, the node force matrix at the corresponding position of the tool has changed. According to Eqs. (1, 2) the force and the torque about the x-axis (the axis direction is set to the x direction) of the node can be calculated, then the force matrix at tool position can be written as:
 |
4 |
where, subscript n represents the node corresponding to the installation position of the composite impact tool. That is, the composite impact tool produces axial force and torque.
During the drilling operation, the drill string rotates under the action of the top drive or the surface turntable. At the wellhead, namely the top of the drill string, it has a constant rotational speed of Ω. For other directions at the wellhead, fixed displacement boundary conditions are applied. At the drill bit end, given the relatively large size of the drill bit and its close proximity to the wellbore size, its lateral displacement is restricted. Moreover, it is subjected to the drilling weight on bit 
and the formation resistance torque 
.
The bit excitation has a significant impact on the vibration characteristics of the drill string. According to the literature28, the main reason for the stick–slip vibration of the drill string is the speed weakening effect of the bit reaction torque decreasing with the increase of the bit speed. Its mechanical model can be written as follows:
 |
5 |
where 
, in which 
is the torsional angle of the bit and 
is the torsional angle of the finite element node adjacent to bit; 
, 
, in which 
is bit angular velocity, rad/s; 
is weight on bit, N, 
Is the angular velocity threshold, which is generally taken as a value close to 0; 
, 
is the dry friction coefficient, which is related to the formation characteristics. For ultra-deep well drill string, 
is generally 0.01–0.03, 
is generally 0.05–0.08; 
is the attenuation coefficient, which is related to the drilling pressure.
According to the studies in25, the 
is assumed to be oscillating harmonically around its surface set value 
with a variation amplitude of 
, namely, 
, where the bit factor n depends on the type of bit that is used.
Equation (3) describing the drill string dynami exhibits temporal and spatial characteristics and is a second-order nonlinear partial differential equation system. Therefore, the Newmark-β integral method and node iteration method29 are combined to discretize the equation in time and space, and the excitation characteristics of the composite impact tool (Eq. (4)) are considered.
Numerical solution of drill string dynamic models
Boundary and constraint conditions
When the drill string moves downhole, once the radial displacement of the drill string exceeds the annular clearance between the drill string and the wellbore wall, it will be constrained by the actual deviated wellbore. At this moment, the drill string is subjected to the radial contact force 
, the tangential frictional force 
, and the tangential frictional torque 
, as depicted in Fig. 4. The interaction between the drill string and the wellbore wall is described using the Hertz–Coulomb contact friction model as follows29:
 |
6 |
 |
7 |
 |
8 |
Fig. 4.
The Hertz–Coulomb contact friction model.
where, 
denotes the radial displacement of the drill string, m; 
and 
represent the radius of the wellbore and the radius of the drill string, respectively, m; 
is the friction coefficient between the drill string and the wellbore wall, which is dimensionless; 
is the relative rotational velocity between the drill string and the wellbore wall, rad/s.
Numerical method
Based on the node iteration method and the Newmark-β method, the dynamic finite element Eq. (3) can be solved.
Newmark-β method
The Newmark-β method is a time integration algorithm designed for the analysis of dynamic problems by discretizing the time domain. This numerical technique transforms second-order differential equations governing drillstring dynamics over time into a set of algebraic equations, specifically at discrete time points30. The Newmark-β method makes the following assumptions about velocity 
and displacement 
at time 
:
 |
9 |
 |
10 |
where α and β are weight parameters. The 
and can be obtained from Eqs. (9 and 10):
 |
11 |
 |
12 |
Substitute Eqs. (11 and 12) into Eq. (3) yields:
 |
13 |
where 
, 
, 
, 
, 
, 
. When 
, 
, the Newmark-β method exhibits unconditional stability. That is to say, the magnitude of the time step has no impact on the stability of the solution.
The displacement 
at time 
is determined by utilizing the displacement 
, velocity 
, acceleration 
at time t, along with force vector 
at time 
. This information enables the computation of the displacement at the subsequent time step. Subsequently, the velocity 
and acceleration 
at 
are obtained. Consequently, the dynamic state of the system at any given time can be derived from its initial configuration.
Node iteration method
Once the time domain is discretized using the Newmark-β method, the node iteration method is employed to solve for the displacement of the drill string at each time step. The flowchart outlining the steps of the node iteration method is illustrated in Fig. 5.
Fig. 5.
Node iteration method.
The node iteration method employs a minimum computing unit comprising three nodes of two elements. Assuming accurate displacement values for nodes 
and 
, the displacement of node 
can be calculated according to the principle of local mechanical equilibrium. Subsequently, the displacement of node 
is determined from the computed displacement values of node 
and 
. The equilibrium equation for node 
is expressed as follows
 |
14 |
The displacement of node 
is thus given by
 |
15 |
After obtaining the displacement of node 
, it is imperative to assess whether the node is in contact with the wellbore. If contact is established, contact force and force moment are incorporated into the force vector 
of the node, according to Hertz contact model Eqs. (6, 7 and 8). This step ensures that the drill string is located within the wellbore. The iteration proceeds sequentially from the top of the drill string through all elements to the bottom, and then reverses from the bottom to the top. This iterative process constitutes a single iteration step. After a finite number of iterations, the results converge to stability (Generally, when the error of the calculation result continuously decreases to the order of 10−6, it can be regarded as reaching a stable state), signifying the attainment of the complete configuration of the drill string at that specific point in time.
The numerical iterative solution process is shown in Fig. 6.
Fig. 6.
Flow chart of dynamic model solution.
Analysis of dynamic characteristics of composite impact tools
Taking a deep vertical well in an oilfield in western China as an example, the influence of composite impact tool on the stick–slip vibration characteristics of drill string is analyzed. The composite impact tool is often installed above the drill bit. The drill string structure used in this well is shown in Table 1.
Table 1.
Drill string structure parameters.
| Name of structure | Outside diameter (mm) | Inside diameter (mm) | Length (m) | Depth (m) |
|---|---|---|---|---|
| Drill pipe | 139.7 | 121.4 | 5130.0 | 5130.0 |
| Heavy weight pipe | 139.7 | 92.1 | 135.0 | 5265.0 |
| Drill collar | 203.2 | 71.4 | 135.0 | 5400.0 |
| Drill collar | 228.6 | 76.2 | 18.0 | 5418.0 |
| Stabilizer | 328.0 | 71.4 | 2.0 | 5420.0 |
| Drill collar | 228.6 | 76.2 | 9.0 | 5429.0 |
| Stabilizer | 328.0 | 71.4 | 2.0 | 5431.0 |
| Drill collar | 228.6 | 76.2 | 9.0 | 5446.0 |
| Composite impact tool | 245.0 | 40.0 | 4.0 | 5450.0 |
| PDC bit | 333.4 | – | 0.4 | 5450.4 |
For comparison, when calculating the dynamic characteristics of the drill string without the composite impact tool, the short drill collar with the same size is replaced to keep the BHA (Bottom Hole Assembly) structure unchanged. The performance parameters and construction parameters of the composite speed-up tool are shown in Table 2. In the following, the BHA without composite impact tool is called ‘conventional BHA’, and the BHA with composite impact tool is called ‘composite speed-up BHA’. Namely, apart from the excitation at the drill bit, the composite speed-up BHA generates additional two-way excitations, namely axial and torsional excitations, at the installation location of the tool. In contrast, aside from the excitation at the drill bit, the conventional BHA is not subject to any other external excitation effects.
Table 2.
Tool parameters and construction parameters.
| Name of parameters | Value |
|---|---|
| Weight on bit (WOB) | 140 kN |
| Revolutions per minute (RPM) | 120 r/min |
| Drilling mud density | 1.8 g/cm3 |
| Axial force of tool | 60 kN |
| Torsional impact torque of tool | 1.6 kN m |
Dry friction coefficient 
|
0.02 |
Dry friction coefficient 
|
0.08 |
Attenuation coefficient 
|
0.18 |
| Frequency of axial impact | 10 Hz |
| Frequency of torsional impact torque | 10 Hz |
Using the aforementioned modified dynamic model and numerical solution method, we can obtain the dynamic characteristics at any position along the drill string, particularly comparing and analyzing those at four critical points: the wellhead, the top of the weighted drill pipe, the top of the BHA, and the bit.
The influence of composite impact tool is most directly reflected in the change of drilling pressure and torque. Figure 7 shows the axial force of the drill bit. Compared with the conventional BHA, the amplitude range of the axial force at the drill bit of the composite speed-up BHA is larger. For the conventional BHA, the dynamic pressure acting on the drill bit ranges from 139.2 to 146.5 kN. In contrast, for the composite speed-up BHA, the dynamic pressure on the drill bit extends from 170.0 to 200.9 kN. Since the inertial mass is taken into consideration in the dynamic model, under the axial impact provided by the tool, the increment of the maximum load of the drill bit for rock breaking is greater than the load that the tool can supply (30 kN), showing an increase of 54.4 kN. Meanwhile, the axial force fluctuation frequency of composite speed-up BHA is obviously improved, and Fig. 8 shows the frequency spectrum of the axial force of the two BHA. After adding the composite impact tool, multiple dominant frequencies emerged, and relatively high-frequency components that cannot be neglected emerged, which is consistent with the working frequency of the tool, indicating that the axial high-frequency excitation of the composite speed-up tool changes the motion characteristics of the drill bit, this proves highly conducive to enhancing the rock breaking efficiency.
Fig. 7.
Axial force of the bit.
Fig. 8.
Axial force frequency spectrum of drill bit.
The downhole rotational speed can directly reflect whether the drill string is in the stick–slip vibration state of viscous (rotational speed is 0.0 r/min) and slip (rotational speed increases rapidly). The downhole rotational speed of the conventional BHA and the composite speed-up BHA is shown in Fig. 9. It can be seen from the diagram that according to the given calculation parameters, when using the conventional BHA, the amplitude of the bit rotational speed fluctuates greatly. The minimum rotational speed is 0.0 r/min, the maximum can reach 244.2 r/min, which is twice the wellhead rotational speed (the wellhead rotational speed is constant at 120.0 r/min), reflecting the typical stick–slip vibration. After adding the composite impact tool, the rotational speed amplitudes at the four key positions are significantly reduced. As time goes by, the angular velocity of the drill string tends to be stable, fluctuates around the wellhead rotational speed, and the stick–slip vibration disappears. It can also be seen from the diagram that the speed time history curve at the top of the BHA is close to the speed curve at the bit, indicating that the overall speed of the BHA section is basically the same, which is related to the greater stiffness of the BHA.
Fig. 9.
Rotational speed of drill string.
Figure 10 shows the PSD (power spectral density) of the bit speed under the conventional BHA and the composite speed-up BHA. The difference between the two is not large in the low frequency and high frequency parts. In the middle frequency part, the PSD of the conventional BHA is significantly higher and more stable than that of the composite speed-up BHA. In the tool working frequency and frequency doubling, the PSD of the composite speed-up BHA will increase sharply, and surpassed conventional BHA indicating that the addition of the tool significantly changes the motion characteristics of the drill string.
Fig. 10.
PSD of drill torsional vibration.
The angular acceleration time history curve of each position of the drill string is shown in Fig. 11. The angular acceleration at the wellhead is 0.0 rad/s2 due to the constant rotational speed. When the conventional BHA is used to generate stick–slip vibration, the angular acceleration curve of the drill bit fluctuates violently. This is because when the torque applied on the ground is insufficient to overcome the rock resistance torque, the torsional deformation of the drill string under the combined action of the upper active torque and the complex downhole resistance torque fluctuates continuously and continues to increase. The torsional potential energy accumulates continuously, and the drill bit gradually stops rotating until the accumulated potential energy is greater than the energy required for rock crushing. Rock crushing causes the torsional potential energy stored in the drill string to be quickly released, and the angular acceleration of the drill bit increases rapidly. When the composite acceleration tool is added, the axial and circumferential high-frequency impact changes the rock breaking mode of the drill bit, inhibits the occurrence of stick–slip vibration, and the angular acceleration fluctuation of the drill bit is more regular, and the amplitude is significantly reduced, and finally tends to be stable.
Fig. 11.
Angular acceleration of drill string.
At the same time, the circumferential excitation of the composite impact tool has a great influence on the torque of the drill string, as shown in Fig. 12. From Fig. 12a, when the stick–slip vibration occurs with the conventional BHA, the amplitude of the torque at the wellhead fluctuates greatly. On the field, the ground torque fluctuation can be used as one of the bases for judging whether the stick–slip vibration occurs underground. Combined with Fig. 9a, it can be found that the torque at the drill bit has obvious corresponding characteristics with the rotation speed. When the rotation speed is zero and increases rapidly, the torque at the corresponding time will decrease rapidly. It can be seen from Fig. 12b that the dynamic torque of the composite speed-up BHA is reduced. The torque fluctuation range of the wellhead is reduced from 11.3–25.5 to 14.8–15.6 kN m compared with the conventional BHA, and the torque amplitude at the drill bit is reduced from 11.9 to 4.9 kN m. This is due to the intensive micro-amplitude fluctuation of the drill bit torque during drilling under the circumferential high-frequency impact torque of the composite impact tool (see the drill bit torque amplification diagram in Fig. 12b), which changes the rock breaking mode of the PDC bit. The torque threshold required for rock breaking of PDC bit is reduced, which is beneficial to rock cutting of PDC bit and eliminates downhole stick–slip vibration.
Fig. 12.
Torque of drill string.
The angular velocity and displacement phase diagram of the bit relative to the wellhead is shown in Fig. 13. In Fig. 13a, the drill bit begins to move from point A. As the movement of the drill bit lags the wellhead, the relative angular displacement gradually increases. In combination with Fig. 9a, it can be observed that during the initial stage (prior to 23 s), no sticking state (the angular velocity of the drill bit is 0.0 rad/s) occurred.
Fig. 13.
Phase diagram of drill bit movement.
At point B, it enters the stick state, corresponding to the sticking state where the velocity is zero in Fig. 9a, the angular velocity then shows periodic fluctuations. The drill string begins to accumulate energy until the drill bit at point C breaks the viscosity and enters the slippage state. At point D, the energy accumulated by the drill bit is released. Currently, the drill bit speed reaches the maximum value. Subsequently, the drill bit speed gradually decreases until it enters the viscous state again. Corresponding to the periodic fluctuations of the drill-bit angular velocity in Fig. 9a, which include a sticking phase (angular velocity of zero) and a slipping phase (rapid increase in angular velocity), Fig. 13a illustrates that the drill bit continuously undergoes sticking and slipping in a reciprocating cycle, resulting in the phase trajectory of the drill forming a closed limit cycle.
After adding the composite impact tool, the relative angular displacement and relative velocity of the drill bit gradually decrease, and finally converge to zero (see Fig. 9b), that is, the addition of the composite speed-up tool makes the angular velocity fluctuation of the drill bit gradually decrease and stabilize, the angular velocity of the drill bit tends to be consistent with the wellhead and maintains stable motion. In this process, the stick–slip vibration disappears, and the phase trajectory shows a gradually convergent spiral curve.
Conclusion
The excitation characteristics of the composite impact tool can be decomposed into periodic axial excitation and torsional excitation. By integrating the three-dimensional dynamic finite element model of the drill string, the impact of the composite impact tool on the motion of the drill bit can be effectively characterized.
The periodic axial and torsional excitations furnished by the composite impact tool are capable of altering the motion and loading conditions of the drill bit. Specifically, they can remarkably enhance the amplitude and frequency of the drill bit’s axial force, cause the dynamic torque of the drill bit to exhibit dense micro-amplitude vibration, and effectively mitigate the stick–slip vibration of the drill bit. These characteristics contribute to the drill bit attaining a more optimal rock-breaking performance.
The composite impact tool raises the frequencies of the axial and torsional forces exerted on the drill bit. Under the condition of long-term operation, it may intensify the wear of the drill bit, which calls for close monitoring in the field. Moreover, the laws governing the influence of the parameters of the composite impact tool and its reliability are deserving of in-depth investigation.
Author contributions
Hongshan Zhao and Wenchang Wang wrote the main manuscript text and Kaixian An furnished field data and carried out an analysis. All authors reviewed the manuscript.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.