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Abstract

Aiming at the problem of stick-slip vibration caused by sudden drilling resistance torque during drilling in coal seam with gangue, the drilling tool dynamic model with two degree of freedom was established based on the interaction between the bit cutting teeth and the coal seam with gangue. So, the motion differential equation of torsional vibration of drilling tool was derived, and the torsional vibration response of drilling tool was analyzed. Taking the drilling tool with drilling depth of 300 m as an example, the response laws of angular displacement, angular velocity, resistance torque, driving torque, relative motion phase trajectory and torsional vibration of the drill bit were discussed. The results show that the drilling tool has obvious stick-slip vibration under the action of sudden drilling resistance in the process of drilling in coal seam with gangue. The angular velocity of the drill bit moves alternately between the viscous stage and the sliding stage. A stable limit cycle will appear in the phase trajectory curve of the drill bit.

Keywords

Drilling tool stick-slip vibration response law coal seam with gangue phase trajectory curve

Introduction

In the process of drilling in the coal seam with gangue, the drilling resistance is complex and changeable, which causes the stick-slip vibration of drilling tool including drill pipe and drill bit during the drilling in the coal seam with gangue. The occurrence of stick-slip vibration aggravates the wear of the drilling tool, reduces the drilling efficiency, and even causes the production accidents of loosening and falling off of the drill pipes and drill bit.

Lubinski et al. deduced the critical pressure of drill string buckling and the contact force between the bent drill string and the hole wall.^1–3 Bailey and Finnie discussed the boundary conditions of the longitudinal and torsional vibration of drill pipe to obtain its natural frequency and vibration characteristics.^4–5 Aarrestad et al. studied the coupled vibration of the torsional and longitudinal vibration of the drilling tool, and proposed that the torsional vibration was related to the speed change of the drill bit.⁶ Besaisow et al. analyzed the causes of drilling tool vibration and resonance through the experimental data.⁷

Some scholars explored the characteristics of the stick-slip vibration by observing and measuring the variation law of bit speed. Belokobyl'Skii et al. established a torsional vibration model. When the angular velocity of the drill bit was less than that of the turntable, the drill pipe would distort and deform. In serious cases, the angular velocity of the drill bit changed periodically, which was the earliest study on the stick-slip vibration.⁸ N.Challamel et al. established a two degree of freedom drill string model, and demonstrated that the speed change of drill bit tended to the stick-slip vibration through simulation.⁹ Bashmal et al. proposed that the stick-slip vibration was a special form of torsional vibration. The rotational speed of the bit changed periodically.¹⁰ Richard T et al. established a mechanical model about the friction between the drill bit and rock, and found that there was a delay between the drill bit and the drilling head in the cutting process, which proved the existence of stick-slip vibration¹¹。.

Yigit A S et al. established the drill string dynamics model to analyze the influence law of different working conditions parameters on the stick-slip vibration.¹² Mou Haiwei and Zhang Xiaodong put forward that the friction between drill bit and rock layer was the main cause of stick-slip vibration.^13–14 Zhu Xiaohua et al. established a torsional oscillation model considering the friction between drill bit and rock stratum，and proposed that the continuous accumulation and release of the energy from drill pipe was the main cause of the stick-slip vibration of drilling tool.^15–16

To sum up, international experts and scholars have done more research on the stick-slip vibration of the vertical drill string in the field of oil drilling, but there is a lack of research on the stick-slip vibration during the near horizontal drilling construction of the coal seam water prevention and the gas drainage. Therefore, this paper will study and analyze the stick-slip vibration characteristics of drilling tool during the near horizontal drilling in the coal seam with gangue.

Stick-slip vibration model of drilling tool

During drilling in the coal and rock, the drilling characteristics of drilling tool are affected by drill bit type, drill pipe performance, rock stratum properties and other factors. Under the condition of retaining the main mechanical characteristics of drilling tool during drilling in the coal seam with gangue, the following assumptions are made.

Ignore the length of the drill bit which is far less than that of the drill pipe.

Consider the drill pipe as a spring with torsional stiffness.

Ignore the lateral vibration and axial vibration of the drilling tool, and only consider the torsional vibration of the drilling tool.

The drill hole and drilling tool are horizontal and concentric.

The simplified model of drilling tool is shown in Figure 1.

Figure 1.

Simplified model of drilling tool.

Assuming that the outer and inner diameter of the drill pipe are D and d respectively, the length of the drill pipe is L, the mass per unit length is m, the density of the drill pipe is ρ, the displacement of the drill pipe cross section from the drilling head is x, and the kinetic energy of the drill pipe is

\frac{1}{2} J \dot{θ_{B}^{2}} = \frac{1}{2} \int_{0}^{L} \frac{m}{8} (D^{2} + d^{2}) {(\frac{x {\dot{θ}}_{B}}{L})}^{2} d x

(1)

The equivalent moment of inertia of the system is

J = \frac{π ρ}{96} L (D^{4} - d^{4})

(2)

Where,

θ_{B}

is the angular displacement of drill bit.

{\dot{θ}}_{B}

is the angular speed of drill bit.

The torsion stiffness of the drill pipe is

K = \frac{π G}{32 L} (D^{4} - d^{4})

(3)

Where, G is the shear modulus of the drill pipe.

The force between drill bit and coal rock is mainly composed of the Coulomb friction between drill bit and coal rock and the cutting reaction force between coal rock and drill bit. When the drill bit encounters the coal gangue, the cutting strength of drill bit increases with the increase of the coal rock cohesion. So, the resistance torque between the drill bit and the coal rock is¹⁷

T_{B} = \frac{C_{0} \cdot b_{0} \cdot R_{B} \sqrt{R^{2} - {(R - h / 3 \cos α)}^{2}} \cos φ \cos (θ + α)}{\cos (β + α + θ + φ) \cos β}

+ μ b R_{B} σ_{S} [\frac{0.6}{\cos α} + \frac{1}{3} (△ L_{f} - \frac{0.6}{\cos α})] - \frac{2 μ F R_{B} sgn ({\dot{θ}}_{B})}{3}

(4)

Where,

μ

is the friction coefficient. R_B is the radius of the drill bit. C₀ is the cohesion of coal rock. F is the drill pressure. b is the width of drilling coal rock. b₀ is the crack length. h is the drilling depth per revolution.

α

is the tooth front angle of drill bit.

θ

is the frictional angle.

β

is the angle formed by the direction of cutter feeding and the shear plane of coal rock.

φ

is the internal friction angle of coal rock.

Δ L_{f}

is the wear length of composite cutting teeth.

σ_{s}

is the yield strength of coal rock under uniaxial compression. R is the composite cutting teeth radius of the drill bit.

The symbol function $sgn ({\dot{θ}}_{B})$ is

sgn ({\dot{θ}}_{B}) = {\begin{matrix} 1 {\dot{θ}}_{B} > 0 \\ - 1 {\dot{θ}}_{B} < 0 \end{matrix}

(5)

It is assumed that the initial time of the drill bit is at the junction of the transition from the viscous state to the sliding state. When the system is in the viscous phase, the bit remains stationary. Namely

{\dot{θ}}_{B} = 0

(6)

In the viscous state, the restoring force of the drill pipe is equal to the torque applied by the drill pipe to the drill bit. So

K_{D} ω t = \frac{d V}{d θ_{B}}

(7)

Where, V is the work done by the torsional elastic restoring moment of drill pipe.

V = \frac{1}{2} K (θ_{B} - ω t)^{2}

(8)

Where,

ω

is the speed of drilling head. t is the viscous time.

The recovery torque of the drill pipe in the viscous stage is less than or equal to the resistance torque on the drill bit. So

| \frac{d V}{d θ_{B}} | \leq T_{B}

(9)

With the continuous energy input from the drilling head, the recovery torque of the drill pipe in the viscous stage increases continuously. Once the recovery torque becomes greater than the resistance torque received by the drill bit, the coal gangue is broken instantly and the drill bit accelerates to rotate. Then, the drill bit enters the sliding stage.

The motion equation of the bit in the sliding stage is expressed by the Euler-Lagrange equation. So

\frac{d}{d t} (\frac{\partial E}{\partial {\dot{θ}}_{B}}) = \frac{\partial E}{\partial θ_{B}} + Q

(10)

Where, E is the kinetic energy of the drill bit.

E = \frac{1}{2} J \dot{θ_{B}^{2}}

. Q is the force acting on the drill bit which can be written as

Q = - \frac{d V}{d θ} + T_{B} + c {\dot{θ}}_{B}

(11)

Where, c is the damping coefficient.

Assuming that the drill bit rotates clockwise at high speed after breaking the viscous state, the motion equation of the drill bit in the sliding stage can be obtained by combining (10) and (11). So

J θ_{B}^{. .} + c {\dot{θ}}_{B} + K (θ_{B} - ω t) + T_{B} = 0

(12)

The critical equilibrium position of the corresponding drill bit is

θ_{B} = ω t

(13)

The characteristic roots of the homogeneous equation corresponding to Equation (12) are a pair of conjugated complex roots, and the angular displacement response of the drill bit in the sliding stage formed by equation (12) is

θ_{B} = A e^{- ξ ω_{n} t} \sin (\sqrt{1 - ξ^{2}} ω_{n} t + ψ) + ω t - \frac{T_{B}}{K}

(14)

The angular velocity response of the drill bit in sliding stage is

{\dot{θ}}_{B} = A ω_{n}^{2} e^{- ξ ω_{n} t} [\sqrt{1 - ξ^{2}} \cos (\sqrt{1 - ξ^{2}} ω_{n} t + ψ)] - A ω_{n}^{2} e^{- ξ ω_{n} t} [ξ \sin (\sqrt{1 - ξ^{2}} ω_{n} t + ψ)] + ω

(15)

ω_{n} = \sqrt{\frac{K}{J}}

(16)

ξ = \frac{c}{2 J} / \sqrt{\frac{K}{J}}

(17)

Where,

ω_{n}

is the inherent frequency.

ξ

is the damping ratio. A is the amplitude.

ψ

is the initial phase which is determined by the initial conditions.

The expression for the amplitude A can be written as

A = - \sqrt{{(θ_{B 0} + \frac{T_{B}}{K})}^{2} + {[\frac{φ - {\dot{θ}}_{B 0} + ξ ω_{n} (θ_{B 0} + \frac{T_{B}}{K})}{\sqrt{1 - ξ^{2}} ω_{n}}]}^{2}}

(18)

The expression for the initial phase

ψ

can be written as

ψ = - \arctan \frac{\sqrt{1 - ξ^{2}} ω_{n} (θ_{B 0} + \frac{T_{B}}{K})}{φ - {\dot{θ}}_{B 0} + ξ ω_{n} (θ_{B 0} + \frac{T_{B}}{K})}

(19)

Where,

θ_{B 0}

is the initial angular displacement in the sliding phase.

{\dot{θ}}_{B 0}

is the initial angular velocity in the sliding phase.

The motion equation (12) can be written in the form of an equation of state. Let

{\dot{θ}}_{B} = y

(20)

So, the differential equation of phase trajectory can be derived.

\frac{d y}{d θ_{B}} = [c {\dot{θ}}_{B} + K (θ_{B} - ω t) + T_{B}] \div (- J)

(21)

The equation of isoclinic line family is derived from the above equation. So

[c {\dot{θ}}_{B} + K (θ_{B} - ω t) + T_{B}] + a J y = 0

(22)

The absolute angular displacement and absolute angular velocity response of the drill pipe can be obtained, and the phase trajectory curve can be drawn in the phase plane area.

Stick-slip vibration response characteristics of drilling tool

Taking the length of drilling tool as 300 m as an example, the stick-slip vibration characteristics of drilling tool when drilling in coal seam with gangue is studied. Relevant parameters during drilling construction are shown in Table 1.

Table 1.

Related parameters of drilling tool.

Parameter name	Value	Parameter name	Value
$D$ (mm)	73	$R_{B}$ （mm）	47
$ρ$ (kg/m³)	7850	$φ$	$30^{0}$
$G$ (Pa)	$8 \times 10^{8}$	$β$	$45^{0}$
$μ_{K}$	0.5	$C_{0}$	1.5
$ξ$	0.03	$σ_{S}$	6.5
F (kN)	20	b	9.1
L (m)	300	$b_{0}$	0
d (mm)	54.6	$α$	$15^{0}$
$ω$ (rad/s)	15.8	$△ L_{f}$	0.8
$μ_{S}$	0.8	R (mm)	6.5

The model solving process is shown in Figure 2.

Figure 2.

Model solving flow chart.

The vibration response of the drilling tool after the solution is shown in Figure 3.

Figure 3.

Absolute angular displacement response curve.

When t = 0 s, the drilling tool has just entered the sliding stage, the drilling head rotates at a uniform speed, and the angular displacement of the drilling head increases linearly. As the drilling head transmits the energy to the drill bit through the drill pipe, the angular displacement of the drill bit shows an upward trend with time. However, when the angular displacement of the drill bit reaches the value corresponding to point A in Figure 3, the angular displacement of the drill bit will not rise and remain unchanged. The angular displacement of the drill bit is shown as the horizontal line AB in the figure. It indicates that the driving torque received by the drill bit is less than the resistance torque from the coal gangue, and the drilling tool enters the viscous state.

With the continuous input of the drilling head energy, the driving torque transmitted from the drill pipe to the drill bit continues to increase. When the driving torque of the drill bit is greater than the resistance torque from the coal gangue, the coal gangue is broken instantaneously. With the instantaneous reduction of the resistance torque, the drill bit starts to rotate faster and the stick-slip vibration occurs. When the bit is drilling in the coal seam with gangue, the viscous stage and the sliding stage occur periodically alternately.

The coordinate of point A in Figure 3 is (0.272, 2.995). T = 0.272 s is the critical point of the drill bit from sliding state to viscous state. At this moment, the drilling head rotates at a uniform speed. The angular displacement of the drilling head is 4.298 rad, and the drill bit lags behind the drilling head by 1.289 rad. The driving force of the drill bit is less than the resistance torque from the coal gangue and remains in a viscous static state. The drill bit angular displacement does not increase after increasing from the initial point to point A, and the drill bit enters the viscous stage. The time of the viscous stage of the drilling tool is 0.081 s. So the period of the stick-slip vibration of the drilling tool with a length of 300 m can be expressed as

T = t_{1} + t_{2} = 0.272 + 0.081 = 0.353 s

(23)

Where, t₁ is the slide phase duration. t₂ is the viscous phase duration.

The absolute angular velocity response curve of the drilling tool is shown in Figure 4. Driven by the drilling head, the drill bit makes a periodic “sliding-viscous” movement. The movement in sliding stage is shown as the stages a and b, while the movement in viscous stage is shown as the stage c in Figure 4 respectively.

Figure 4.

Absolute angular velocity response curve.

In stage a, the drill bit starts to rotate and the movement of drilling tool enters the sliding stage. The energy of the drill pipe accumulated in the viscous stage is released. The drill bit begins to rotate faster. The absolute angular velocity of the drill bit increases gradually to exceed the angular velocity of the drilling head until it reaches the maximum when the driving torque is equal to the resistance torque.

In stage b, with the increase of resistance torque from the coal gangue, the driving torque of the drill bit is less than the resistance torque. The drill bit begins to decelerate until the angular velocity of the drill bit decreases to 0 rad/s. So, the movement of drill bit enters the viscous stage.

In stage c, the drill bit is stuck. The driving torque of the drill bit is balanced with the resistance torque. The drill bit remains stationary. With the continuous rotation of the drilling head, the drill pipe continuously accumulates energy and transmits the energy to the drill bit. When the driving torque generated by the drill bit is greater than the maximum resistance torque from the coal gangue. The coal gangue is cut and broken. The drill bit starts to slide again.

By making the difference between the angular displacement of the drilling head and the drill bit, the relative angular displacement of the drill tool is obtained as shown in Figure 5.

Figure 5.

Relative angular displacement response curve.

When the movement of drill bit enters the sliding stage, the angular velocity of the drill bit is lower than that of the drilling head. So, the relative angular displacement between the drill bit and the drilling head increases for a period of time. However, when the angular velocity of the drill bit exceeds the speed of the drilling head, the angular displacement between the drill bit and the drilling head begins to decrease gradually. When the angular velocity of the drill bit reaches the maximum, the relative angular displacement between the drill bit and the drilling head becomes minimal. Then, with the gradual decrease of the angular velocity of drill bit, the relative distance starts to increase again. When the movement of the drill bit enters the viscous stage, the drill bit stops rotating while the drilling head continues to rotate at the original constant speed. So, the relative angular displacement between the drilling head and the drill bit is the linear function of the time during the viscous stage which is shown as the inclined line AB in the figure.

The driving torque of the drill bit is equal to the product of the torsional stiffness of the drill pipe and the absolute value of the relative angular displacement. As shown in Figure 6, the driving torque of the drill bit presents a simple harmonic curve in the sliding stage and an oblique straight line in the viscous stage.

Figure 6.

Drill bit driving torque response curve.

The change of resistance torque between the drill bit and the coal rock is shown in Figure 7. The resistance torque has piecewise characteristics, which is caused by the different resistance from the coal rock in the two stages of viscosity and sliding. When the drill bit is drilling, the resistance torque is composed of the sliding friction torque and the reverse cutting torque. When the drill bit is drilling in the coal seam, it is considered that the resistance torque is constant. It shows a horizontal line in the figure, and the resistance torque is 937 N^.m . When the drill bit encounters the coal gangue, the resistance torque increases to 974 N^.m. The movement of the drill bit enters the viscous state, and the resistance torque image in the viscous stage is an inclined straight line. As the drilling head drives the drill pipe to rotate, the drill pipe continuously accumulates energy. When the driving torque transmitted to the drill bit increases to the resistance torque 1313 N^.m, the coal gangue is cut and broken. The drill bit begins to enter the sliding stage.

Figure 7.

Resistance torque response curve of drill bit.

The relative moving phase trajectory curve of the drill bit obtained after the solution of formula (22) is shown in Figure 8. It can be seen from the figure that the drill bit enters the sliding stage from the viscous stage and begins to move from the initial point a. In the process of moving from point a to point b, the drill bit accelerates to rotate. But the angular velocity of the drill bit is lower than that of the drilling head. The relative angular displacement becomes larger, while the relative angular velocity becomes smaller. At the point b, the angular velocity of the drill bit is equal to that of the drilling head. So, the relative angular displacement reaches the maximum. As the angular velocity of the drill bit continues to increase, it is greater than that of the drilling head. The relative angular displacement becomes smaller, while the relative angular velocity becomes larger. Under the action of the resistance torque, the angular acceleration of the drill bit decreases gradually until the angular velocity of the drill bit reaches the maximum at point c. At the same time, the relative angular velocity between the drill bit and the drilling head reaches the maximum and the relative angular displacement continues to decrease. In the process of moving from point c to point d, the angular velocity of the drill bit starts to decreases gradually until it is equal to that of the drilling head. The relative angular velocity reaches 0 rad/s and the relative angular displacement becomes minimal. As the angular velocity of the drill bit continues to decrease gradually, the angular velocity of the drilling head is constant and the relative angular velocity becomes larger. At the point e, the angular velocity of the drill bit decreases to 0 rad/s, the movement of drill bit enters the viscous stage from the sliding stage.

Figure 8.

Relative motion phase trajectory curve.

Conclusions

A drilling tool dynamic model with two degree of freedom was established based on the interaction between the bit cutting teeth and the coal seam with gangue. After the motion differential equation of torsional vibration of drilling tool was derived and solved, the torsional vibration response of drilling tool was analyzed. The main conclusions are drawn as follows:

The driving torque of the drill bit is equal to the product of the torsional stiffness of the drill pipe and the relative angular displacement. And the resistance torque between the drill bit and the coal rock has segmented characteristic, which is caused by the difference of resistance torque from the coal rock.

During drilling in the coal seam with gangue, the drilling tool is prone to stick-slip vibration under the action of the resistance torque from the coal gangue. The motion parameters of the drill bit changes periodically.

When the stick-slip vibration of the drilling tool occurs, the phase trajectory curve at the drill bit will show a stable limit cycle. When the drilling tool does not have stick-slip vibration, the drilling head drives the drill bit to rotate through the drill pipe. The phase trajectory curve will appear in the shape of spiral ring.

Above all, the stick-slip vibration phenomenon of the drilling tool during drilling in the coal seam with gangue was explained. Future research is recommended to analyze the influence law of relevant parameters on the stick-slip vibration deeply which can provide the theoretical basis for improving the drilling efficiency.

Footnotes

Acknowledgements

This work was supported by the key research projects of colleges and universities in Henan Province (grant number 22A440013), and the mechanical engineering discipline of henan polytechnic university.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iD

Xiaoming Han

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Stick-slip vibration characteristics of drilling tool during drilling in coal seam with gangue

Abstract

Keywords

Introduction

Stick-slip vibration model of drilling tool

Stick-slip vibration response characteristics of drilling tool

Conclusions

Footnotes

Acknowledgements

Declaration of conflicting interests

ORCID iD

References