Experimental Study on Axial Impact Mitigating Stick-Slip Vibration with a PDC Bit

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Introduction
Drilling investment accounts for the main part of the cost of oil and gas development.Safe and efficient drilling can greatly reduce the cost and improve the economic benefits of oil field.Stick-slip vibration may cause early damage of drilling tools, low rate of penetration (ROP), and high nonproductive time.Particularly, when a PDC bit is used to drill hard strata, stick-slip vibration becomes more intense [1].
e elimination of stick-slip vibration is helpful to improve ROP and reduce underground safety hazards.erefore, stick-slip vibration and its suppression methods have been an important topic of research in the drilling industry [2].
In the process of rotary drilling, the discontinuous crushing of the rock is one of the main reasons leading to the stick-slip vibration in the downhole.erefore, the key to reduce the stick-slip vibration of the drillstring is to suppress the vibration in the rock-breaking process of the bit [3,4].To better understand and ultimately eliminate stick-slip vibration, many drilling service providers have developed downhole vibration measurement instruments, evaluation systems, and vibration-reduction drilling tools.e drilling assembly design, constant torque tool, reactive-torque-free bit, and axial impact drilling technology have been proposed, which can make the bit move with more stability and continuously cut rocks, thus reducing the bit's viscosity and slip-off phenomenon [5,6].
ese techniques are effective in reducing the stick-slip vibration of the bit and have been proved by a considerable number of field tests.In particular, axial impactors have an obvious effect in restraining the stick-slip vibration of PDC bits, which fully verifies that the impact load plays a key role in restraining stick-slip and other harmful vibrations of the bit [7].
In the research of stick-slip vibration, some researchers simplified the bit, drill collar, and drill pipe into a torsional pendulum model, assuming the boundary at the bit as a velocity-dependent friction boundary, and applied extra force to reduce stick-slip vibration [8,9].However, other researchers think that cutting tooth rock-breaking behavior is the reason for stick-slip.
erefore, through the establishment of a single cutter/rock contact model, stick-slip and its mitigation methods can be studied.As shown in the literature [10], the numerical simulation method reveals that the torsional impact force can reduce the time of cutting and rock breaking, thus reducing the possibility of stick slip.
Some researchers have tried to study the law of stick-slip vibration through laboratory experiments.But some of the devices they built did not take into account the axial flexibility [11][12][13], and some did not truly reproduce the process of PDC bit drilling [14][15][16][17][18][19].To better understand the mechanism and features of mitigating stick slip by axial impact, this article uses similarity theory and a two-degreesof-freedom mass-spring-damper model to develop a laboratory experiment apparatus and investigate stick-slip mitigation by axial impact and how parameters such as bit back-rake angle affect the effects of vibration suppression.

Experimental Model Design
2.1.Mathematical Model of the Drillstring System.In the process of rock breaking by a PDC bit, a PDC cutter has both axial and torsional motions.e coupling of torsional and axial motions occurs at the cutting teeth.
erefore, the simulation device of a rotary drilling system should consider both axial and torsional dynamics.In this paper, a twodegrees-of-freedom (DOF) mass-spring-damper model and similarity theory are used to design the experimental parameters [20][21][22][23].
e simplified model of the drill string system is shown in Figure 1.Torsional stiffness of the drill pipe is simplified as a torsion spring K T , and axial stiffness of the drill pipe is simplified as a compression/tension spring K a .
e rotational damping of drilling fluid is simplified to a linear damping C T , while the axial damping of drilling fluid is ignored.e bottom hole assembly (BHA) is simplified to a stiff rotary table with an inertia of I b and a mass of M b .e top drive rotary speed, rotary displacement, torque, and axial displacement are represented by Ω 0 , Φ 0 , T 0 , and U 0 , respectively.
e rotary speed, rotary displacement, torque, and axial displacement of the bit and BHA are denoted by Ω, Φ, T, and U, respectively.WOB and torque are the axial component W and torsional component T b of the bit cutting force, respectively.e system axial dynamics is as follows: and the torsional dynamics is expressed as e interaction between the bit and the formation consists of cutting forces and friction forces, which produce reactive force of W b and reactive torque T b in the axial and torsional directions, respectively.erefore, we have where T c and T f represent the cutting and frictional components of torque and W c and W f refer to the cutting and frictional components of WOB.

Determination of the Model Scale Parameters.
In this section, dimensional analysis is used to derive the model scale parameters.Equations ( 1) and ( 2) adopt the MLT dimension system, and their dimension matrixes are presented in Tables 1 and 2, respectively.It is seen from Table 1 that the rank of the dimension matrix of the axial dynamics is 3, which indicates that there are three independent variables among six physical quantities in equation (1).erefore, 3 (6 − 3 � 3) similarity criteria exist.As for Table 2, the rank of the dimension matrix of equation ( 2) is two, which means only two variables are independent among seven physical quantities.Moreover, given the dimensionless parameter Φ, 4 (7 − 2 − 1 � 4) similarity criteria are required.Since the mathematic models have already been presented, the integral analogy approach can be used for derivation of the similarity criteria.
According to integral analogy approach, equations ( 4) and ( 5) can be obtained by using the 2nd term in equation ( 1) and 3rd term in equation ( 2) to divide the other terms in respective equations: e ratios of physical quantities are then used to replace the corresponding derivatives, and seven invariants are obtained: and C t represent ratios of corresponding physical quantities, we can conclude that the criterion for similarity between the model and actual case should be  3.

Experimental Apparatus.
e schematic diagram of the laboratory model is illustrated in Figure 2. e driving motor drives the torsion bar spring to rotate, and the torsion bar passes torque to the rotary inertia plate through the torque drive mechanism and finally to the drill bit.e torsional spring simulates the torsional elasticity of the drillstring, the rotary plate simulates the rotational inertia of the BHA, and the mass block simulates the mass of the drill string.e drillstring drives the motor to simulate the rotary table, and the speed can be adjusted through the frequency converter.
e speed adjustment range is 0 ∼ 500 revolutions per minute (RPM).
e servo motor drives the rotor of the impactor to rotate and produces impact force, which can be adjusted through the frequency converter, and the impact force is transmitted to the bit through the drive shaft.e servo motor drives the lift up and down to simulate the rise and fall of the hook, and the spring simulates the axial elasticity of the drill string.
e lower part of the lift is connected with the concentrated mass block through the disc spring, beam, and column, and the concentrated mass block simulates the mass of the drill string.e permanent magnet DC motor is connected to the end of the torsional spring to simulate the drill string rotation damping.e torsion natural frequency is ω t � 0.97 Hz, and the axial natural frequency is ω v � 6.5 Hz.Table 1: Dimensions of the axial dynamics of the drillstring system.

Physical quantity Dimension
Table 2: Dimensions of the torsional dynamics of the drillstring system.

Analysis of Torsional Vibration Mechanism.
In this section, the drilling experiment under ordinary condition (no axial impact) is carried out to demonstrate that the apparatus can reproduce stick-slip vibration and study features of stick-slip vibration.e rotation speed, WOB, and torque measured while drilling are shown in Figure 3. e average values of WOB and torque are 4.7 kN and 130 N m, respectively, and the ratio of torque to WOB is 27.7 N m/ kN.For the convenience of description, the time is divided into three stages according to the amplitude of rotation speed fluctuation: stable period (0-5 S), transitional period (6-13 s), and stick-slip period (14-19 s). e amplitude of rotation speed fluctuation in the three stages is 24 RPM, 62 RPM, and 120 RPM, respectively.e amplitude of WOB fluctuation is 0.78 kN, 0.86 kN, and 0.99 kN, respectively, and the amplitude of torque fluctuation is 93 N m, 96 N m, and 124 N m, respectively.e spectrum analysis of rotation speed, WOB, and torque is shown in Figure 4. Stable period: main frequency of rotation speed is 0.48 Hz and second-order frequency is 0.97 Hz; the main frequency of WOB is close to the amplitude of the second-order and third-order frequency.e main frequency and second-order frequency of torque are 0.488 Hz and 0.244 Hz, and there is no significant peak near the natural frequency ω t of torque.Transitional period: the main frequencies of rotation speed, bit pressure, and torque are all ω t , and the amplitude of the second-order frequency (0.488 Hz) in the WOB spectrum is close to that of the main frequency.Stick-slip period: the main frequency of each spectrum is ω t , and the amplitude of the frequency is significantly different from that of the second-order frequency.
e main frequency of torque is close to the natural frequency ω t of the system, which resonates and ultimately leads to stick-slip vibration.
e spectrum analysis shows that the main frequency of stick-slip vibration is the first-order torsional natural frequency of BHA and is not related to bit-rock interaction, which is consistent with the conclusions obtained from field measurements.Bit-rock interactions produce torques of different frequencies, but only first-order resonance can be excited.

Mechanism of Axial Impact Mitigating Stick Slip.
In this section, an experimental study on the suppression of torsional vibration by axial impact is carried out, and the results are shown in Figure 5.In this experiment, the amplitude of impact force was 5 kN and the frequency was 200 Hz.Under the impact condition, the average WOB is 4.28 kN and the average torque is 85 N m. e rotation speed, torque, and WOB fluctuation amplitude are 36 RPM, 70 N m, and 7.9 kN, respectively.
Compared with Figure 3, the fluctuation rate of WOB under the axial impact increased by 597%, but the average WOB and average torque decreased by 8.9% and 35%, respectively, and the fluctuation amplitude of rotation speed and torque decreased by 70% and 44%, respectively.Spectrum analysis of rotation speed, WOB, and torque is shown in Figure 6. e main frequencies of rotation speed, WOB, and torque are 0.97 Hz, 200 Hz, and 2.44 Hz, respectively.e main frequency of torque is far greater than the natural frequency of the system ω t , and resonance cannot occur, so the rotation speed is relatively stable.

Effects of Back-Rake Angle and Impact Frequency.
e back-rake angle of PDC teeth has an important effect on the torsional stability of drillstring.e influence of impact load amplitude and PDC back-rake angle on rotation speed is shown in Figure 7. e back-rake angle of PDC teeth determines the cutting force characteristics of the drill bit and also affects the torsional stability of the drillstring.With the impact amplitude unchanged, the drillstring fluctuation first weakens and then increases as the backrake angle increases, and there is an optimal back-rake angle with the weakest fluctuation.When impact force is zero, the PDC back-rake angle with the minimum rotation speed fluctuation amplitude is 17 °.When the amplitude of impact force increases from 1000 N to 5000 N, the optimal back-rake angle gradually decreases to 12 °.When the backrake angle is greater than 5 °, the fluctuation of rotation speed decreases as impact force increases.With the increase of the back-rake angle, the impact amplitude decreases.When the back-rake angle is greater than 15 °, the impact force will even cause the rotation speed to fluctuate more severely.
e influence of impact frequency on rotation speed is shown in Figure 8, when the back-rake angle and impact amplitude remain unchanged.e back-rake angle of the PDC bit is 10 °, and the impact amplitude is 5 KN.When the impact frequency is lower than 100 Hz, the fluctuation of rotational speed decreases linearly with the increase of the impact frequency.When the impact frequency is greater than 100 Hz, the rotation speed fluctuation is constant at 24 RPM.

Conclusions
Based on the 2 DOFs drillstring dynamics and similarity theory, this paper designed and built an experimental apparatus to study stick-slip mitigation by axial impact.e main findings of this paper are as follows: (1) In conventional drilling, the main frequency of torque is consistent with the torsion natural frequency of the  system in the period of stability and transition, resonance is generated accordingly, and the fluctuation of rotation speed increases gradually.(2) Under the impact condition, the average WOB, average torque, and torque fluctuation decreased significantly.
(3) e impact force makes the main frequency of torque far higher than the first-order natural frequency of drillstring, thus eliminating resonance and inhibiting stick slip.(4) When the amplitude and frequency of impact force are constant, there is a back-rake angle with the weakest fluctuation.(5) When the impact frequency is lower than 60 Hz, the higher the impact frequency is, the weaker the rotation speed fluctuation will be.When the impact frequency exceeds 100 Hz, the fluctuation amplitude of rotation speed remains constant.
, and C Ω0 � 1. C Tb /C W � C a is the geometry ratio of the model, namely, the ratio of the drilling footage of the model to the actual drilling footage C U , and Ca stands for the ratio of the diameter of the model bit to that of the actual bit.Table 3 demonstrates that the drill bit diameter a � 0.108 m, and the drill bit radius defined in this paper equals 0.028 m. us, C U � C a � 0.028/0.108� 0.26.In this paper, M b � 2000 kg and C Ka � C Mb � 2000/11000 � 0.18; therefore, C W � C Ka C U � 0.0468 and C Ib � C CT � C KT � C Tb � C W C a � 0.012.Model scale parameters are shown in Table

1 :
Figure 1: Schematic diagram of the 2 DOFs model of the drilling string.

Figure 6 :
Figure 6: Frequency spectrum of WOB, rotation speed, and torque under axial impact.

Table 3 :
Model scale parameters for the experimental model.