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Theoretical modeling and experimental study of rock-breaking depth in particle jet impact drilling process
Journal of Petroleum Science and Engineering 183 (2019) 106419
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Theoretical modeling and experimental study of rock-breaking depth in particle jet impact drilling process
T
Fushen Rena,∗∗, Tiancheng Fangb,∗, Xiaoze Chengc a
School of Mechanical Science and Engineering, Northeast Petroleum University, Daqing, Heilongjiang 163318, China School of Civil Engineering and Architecture, Northeast Petroleum University, Daqing, Heilongjiang 163318, China c The China National Petroleum Corporation, Beijing 100724, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Particle jet impact Rock-breaking depth Semi-analytical model Dynamic simulation Experimental study
Particle jet impact drilling technology is an efficient method to improve rock-breaking speed by high-speed particles and water-jet impact. The process and damage stage of rock-breaking by combined impact of highspeed water-jet and particles are very complicated, therefore, it is of great significance to study the rock-breaking depth in different damage stages for particle jet impact drilling. Therefore, laboratory test was carried out to study the effects of rock breaking depth on particle diameter, impact velocity and particle mixing ratio. Then, the semi-analytical model of rock-breaking depth under particle jet impact was established in stable damage condition, and the effect factors of rock-breaking depth were numerically analyzed. Based on SPH-FEM coupling algorithm, the dynamical simulation model of particle jet impact rock was established, and the simulation of rock-breaking process and damage evolution under particle jet impact was carried out. Effects of different particle jet parameters on rock-breaking depth were studied by comparing the semi-analytical mode calculation, dynamic simulation and experimental results. Analysis results showed that rock-breaking depth by particle jet impact increases with the increase of jet velocity and particle mixing ratio in high-pressure water-jet, and decreases with the increase of particle diameter. According to the requirements of particle jet impact drilling technology and the analysis findings, it is recommended to adopt water-jet velocity of 200–220 m/s, particle mixing ratio in water-jet of 0.20%, and particle diameter of 1.0 mm. Research results can provide theoretical basis for the parameters optimization and field application of particle jet impact drilling technology.
1. Instruction The particle jet impact drilling technology is an efficient rockbreaking method mainly based on high-frequency impact of high-speed metal particles, supplemented by high-pressure water jet and mechanical drill bit to break rock. It can be effective to overcome the drilling difficulties of hard and abrasive formations in deep and ultra-deep wells (Cui et al., 2011; Zhu et al., 2014). The rock-breaking mechanism of this technology is that high-speed particle jet impacts on rock beforehand to form an annular rock breaking zone, so the confining pressure of deep well can be released in advance. Then, the rock can be broken quickly and effectively under the grinding action of mechanical drill bit. In recent years, particle jet impact rock-breaking technology has developed rapidly and has been well applied in oil drilling, underground engineering and other fields (Wei et al., 2013; Momber, 2015). At present, the research and analysis of damage and broken depth
∗
under particle jet impact are mainly based on the combination of finite element analysis and experimental research. The advantages and disadvantages of particle impact drilling technology were discussed and the rock-breaking mechanism was pointed (Wu et al., 2008; Kuang et al., 2012). Based on fully decoupled fluid-structure coupling theory, Li et al. (2009) established the numerical analysis model and fluidstructure coupling analysis model of ultra-high pressure water jet impacting rock medium. According to experimental conditions, Anwar et al. (2013) established the finite element model of overlapping abrasive water jet (AWJ) grinding footprint. A symmetrical pre-splitting finite element model by picks without water-jet and picks assisted with water-jet were established, and the effects of different configurations on rock stress distribution and cutting force are analyzed by experiments (Liu et al., 2015). Wang et al. (2017) established the particle impact rock-breaking depth model based on cavity expansion theory, and studied the dynamic damage evolution of rock-breaking process. Based
Corresponding author. Corresponding author. E-mail addresses: [email protected] (F. Ren), [email protected] (T. Fang), [email protected] (X. Cheng).
∗∗
https://doi.org/10.1016/j.petrol.2019.106419 Received 29 March 2019; Received in revised form 22 August 2019; Accepted 23 August 2019 Available online 26 August 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
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debris.
on SPH method, Zhao et al. (2017a,b) analyzed the influence of particle jet parameters on rock-breaking volume and impact depth. Jiang et al. (2017) simulated and analyzed the rock-breaking process by using JH-2 constitutive model, and studied the influence of jet parameters on rock damage field. Wang et al. (2018) established the stress model of abrasive slurry jet impacting rock, and analyzed the stress distribution rule under abrasive slurry jet impact. The mathematical model of particle jet impact rock was established, and the stress analysis and rock-breaking area were analyzed and studied (Ren et al., 2018a,b). However, there are few researches on the rock-penetration depth in the field of particle jet coupling impact drilling at present, and also little study on rock failure under different damage state. Therefore, in order to study the penetration depth and rock failure under particle jet impact, firstly, laboratory test of particle jet impact rock was carried out. Then, the mathematical model of rock-breaking depth under particle jet impact at stable damage state is established, and also the dynamic simulation of rock-breaking process is carried out with SPH-FEM coupling method. Then, the influence of water-jet velocity, particle diameter and particle mixing ratio in high-pressure water-jet on rockbreaking depth is studied and analyzed by combining semi-analytical model calculation, dynamical simulation and experimental research. The research results would provide theoretical basis and support for the prediction of rock-breaking depth and damage velocity of particle jet impact drilling, and also the optimization of technological parameters.
2.2. Experimental materials and processes The experimental materials used in particle jet impact rock breaking experiment are shown in Fig. 2. High density granite with the size of 250 mm × 250 mm × 100 mm is used for experimental rock samples. The impact particles are made of Q235 steel spherical particles. The accelerating nozzle is of constant speed type made of cemented carbide. The experimental method and process mainly include preparation stage, rock breaking experiment stage and parameter measurement stage, as shown in Fig. 3. 2.3. Experimental samples The rock was impacted by particles with different diameters at different impact velocities, and the rock samples after crushing experiments are shown in Fig. 4. The rock-breaking depth is obtained by measuring the maximum penetration depth, and the error can be reduced by taking average value through many experiments and repeated measurements. 3. The semi-analytical model of rock-breaking depth In the process of particle jet impact drilling, the combined impact of high-speed water-jet with metal particles on rock is quite complicated. The damage and destruction of rock can be divided into three stages: initial rock-breaking stage, damage accumulation stage and stable damage stage. At the initial rock-breaking stage, rock in deep well has no initial damage, and the rock-breaking depth under particle jet impact is small at this moment; At the damage accumulation stage, with particle jet continuously impact, the cumulative damage of rock gradually increases, and the increase of rock-breaking depth obtained each time gradually increases; When at stable damage stage, the rock-breaking depth impacted by each time is basically stable. Due to the rock being basically at stable damage state in process of particle jet continuous impacting, therefore, it is of great significance to study the rockbreaking depth at stable damage state under particle jet impact for the analysis of particle jet drilling process.
2. Particle jet impact rock-breaking device and laboratory test 2.1. The particle jet impact rock-breaking device The self-developed particle jet impact rock breaking test device is shown in Fig. 1, and it mainly includes power module, particle injection module, simulated top drive module, simulated bottom-hole module, water circulation module, pressure gauge and high pressure pipeline. The system power module is mainly composed of a high-pressure mud pump with 110kw, which can provide 32 MPa and 10.8m3/h of hydraulic power. The particle injection device module mainly comprises a particle storage tank and a screw propeller, and the working principle is to uniformly inject a certain proportion of metal particles into a high-pressure pipeline through screw propeller. The simulated top drive module is used to realize the drilling process of drill pipe driven by top-motor, and can achieve the goal of constant pressure, constant speed and constant torque drilling. The simulated bottom-hole module is used to simulate the bottom-hole of rock. The water circulation module is mainly comprised of water tank and filtering device for realizing water circulation and separation of metal particles from rock
3.1. Cumulative damage and mechanical properties under particle jet impact In process of particle jet impact rock, the water-jet carries metal particles through the acceleration action of special nozzle, forming continuous water-jet and intermittent particles to impact rock at high velocity, so as to achieve the purpose of rock fragmentation. The following assumptions are made before theoretical model is established: (1) The continuous impact of high-speed water-jet on rock is regarded that rock is under constant pressure; (2) Intermittent impact of particles can be regarded as multiple cyclic impact of dynamic load; (3) The collision between particles and water jet is not considered. The expression of continuous impact pressure Pw of water-jet is:
Pw =
ρw Qvf SA
(1)
where, ρw is the density of water-jet, Q is the rated flow, vf is the impact velocity of water-jet, SA is the water-jet impact zone. Under the cyclic impact of particles on rock, apart from the damage pits on the surface, there will also be cracks and damages in rock, so the mechanical properties of rock will inevitably change. When the rock has been damaged, the cumulative damage curve equation (He et al., 2016; Jin et al., 2014) can be expressed as:
Fig. 1. Particle jet impact rock-breaking device. 2
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Fig. 2. The experimental materials.
Fig. 3. The experimental method and process.
Fig. 4. The experimental samples.
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Fig. 5. Stress analysis of particle impact rock breaking process. Fig. 7. Variation of rock-breaking depth with the water-jet velocity. Table 1 Material attribute parameters. Material parameters
Unit
Value
Constant parameter of rock
Unit
Value
The rock density ρr The particle density ρs The water-jet density ρw The compacted strain η∗
kg/m3 kg/m3 kg/m3
2650 7850 1000
GPa MPa MPa
16.6 10 54
–
0.1
Modulus of elasticity E Shear strength G Compressive strength FC Friction coefficient μ
–
0.04
time, V is the volume of single particle, nl is the particle number that can be contained in the cross-sectional area of nozzle outlet. When metal particles impact the undamaged rock (assuming that the rock has no initial damage), the stress relationship of rock can be obtained according to Coulomb-Friction law (Wang et al., 2018; Yin et al., 1998; Zhao et al., 2017a,b):
σθ = μσr
(4)
where, μ is the sliding friction coefficient, σr is the radial stress of undamaged rock, σθ is the tangential stress. Luk and Forrestal (1987) established the penetration depth models of spherical projectiles based on dynamic cavity expansion theory. According to such theoretical model, the relationship between radial stress and impact velocity of spherical projectiles (Luk et al., 1991) is:
σr = ξFc + ζρr v 2
(5)
where, ρr is the density of rock, Fc is the compressive strength of rock, v is the particle velocity, ξ and ζ is respectively represent rock parameters related to the constitutive characteristics. The expression of ξ and ζ is as follows: ∗
(
)
2
∗2/3
∗
∗2
⎧ ζ = 3G + η 1 − 3G + 3η − 4η∗ + η Eγ γ 2E 2(1 − η ) γ ⎪ ⎪ ξ = 2(1 − lgη∗)/3 ⎨ 2/3 ⎪ γ = ⎡ 1 + G 3 − (1 − η∗) ⎤ ⎪ 2 E ⎦ ⎣ ⎩
(
)
(6)
η∗
is the volume compaction strain of rock, G is the shear where, strength of rock, E is the elastic modulus of undamaged rock. When the rock is damaged by Nth particles, the mechanical properties of rock at undamaged stage and damaged stage can be obtained according to the strain equivalence principle and cumulative damage curve equation (2):
Fig. 6. Variation of rock breaking depth with impact times and particle diameter.
Dn = α − β ln(k / n − p)
(2)
⎧ ε = σrN/ EN =σr / E σrN = σr (1 − DN ) ⎨ ⎩ EN = E (1 − DN )
where, n is the number of particle impacts per unit area, Dn is cumulative damage factors, α and β indicate parameters related to rock damage, p and k is the parameters related to water-jet pressure. Since the impact frequency of particles can reach thousands of times per minute in process of particle jet impact, such high-frequency impact can be considered as uniform impact, so the number n of particle impacts per unit area is:
n = vs t / Vnl
(7)
where, ε is the elastic strain of rock, σrN is the radial stress at damaged stage, EN is the elastic modulus at damaged stage. From equations (4) and (7), it can be concluded that the radial stress at damaged stage under impact of Nth particle is expressed as:
σrN = (ξFc + ζρr v 2)(1 − DN )
(3)
(8)
According to equations (4)–(8), it can be obtained that the radial stress at damaged stage is not only related to impact number, but also
where, vs is the incorporation rate of particles, t is the incorporation 4
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Fig. 8. The coupling process of SPH and FEM.
where, η is the ratio of particle velocity and water-jet velocity, L0 is the length of nozzle constriction. According to the force analysis, the expressions for radial resistance Fr and tangential resistance Fθ of rock (Rajput and Iqbal, 2017) can be obtained as follows:
dFr = 2πR2σr sin θdθ
(10)
dFθ = 2πR2σθ sin θdθ
(11)
where, R is the radius of particle, θ is the angle between solved point and Z axis. The sum of radial resistance and tangential resistance in vertical direction is:
dFh = πR2 (σr sin 2 θ + 2σθ sin2 θ)dθ
(12)
Integrating equation (4), the sum of resistance Fh can be obtained:
⎧ Fh = πR2∫θ (σr sin 2 θ + 2σθ sin2 θ)dθ 0 ⎨ θ = arccos((R − h p)/ R) ⎩
Fig. 9. The model of particle jet impact rock-breaking.
(13)
related to the constitutive material constants, mechanical properties, and impact velocity.
where, θ is the upper integral limit, h p is the rock-breaking depth. When at undamaged stage, the sum of resistance Fh can be obtained by the combined equation (4), equation (5) and equation (12):
3.2. Mechanical model under particle impact
Fh = πR2σr
θ
(sin 2 θ + 2μ sin2 θ)dθ
(14)
When under damaged stage, the sum of resistance Fh is:
When the certain mass m of steel particle impact rock at certain initial velocity v0 , the force acting on rock is shown in Fig. 5. σθ represents tangential stress on rock surface, and the compressive stress is positive in stress analysis, as well tensile stress is negative. After passing through the acceleration of special nozzle (Ren et al., 2018a,b), particle jet would impact rock at high velocity. At this time, the relationship between particle velocity v0 and water-jet impact velocity vf can be obtained by data regression analysis (Ren et al., 2017):
Fh = πR2σrN
∫0
θ
(sin 2 θ + 2μ sin2 θ)dθ
(15)
According to stress analysis of rock at undamaged stage and damaged state respectively, the sum of resistance in rock-breaking process is related to mechanical characteristics and particle diameter. 3.3. Establishment of rock-breaking depth model
= ⎧ v0 ηvf
η = 0.823 − 1.25e−3L0 − 2.29e−5L02 ⎨ ⎩
∫0
The relation between particle velocity change and impact resistance can be known by Newton's second law in rock-breaking process:
(9) 5
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Fig. 10. The damage cloud of particle jet impact rock-breaking.
Fig. 11. The rock failure analysis under particle jet impact.
m
dh p dv dv dv =m = mv = −Fh dt dh p dh p dt
and particle diameters. From the growth trend, it can be seen the rockbreaking depth increases with impact times, and the increase extent is smaller and smaller. After impact times are fifth, it can be considered the damage of rock has reached stable state. From the quantitative relationship, with particle diameter being fixed, the rock-breaking depth of first impact is often small, about 1/6 of particle diameter. Before the second impact occurs, mechanical properties of damaged rock have changed, so it is easier to break rock and the rock-breaking depth has increased greatly. The rock-breaking depth basically tends to be stable after the fifth time, and it is about 1/4–1/3 of particle diameter in stable state. The variation of rock-breaking depth with water-jet impact velocity is shown in Fig. 7 (the particle diameter is 1.0 mm). The rock-breaking depth gradually increases with increase of water-jet impact velocity. Therefore, higher water-jet velocity should be used as much as possible to obtain larger rock-breaking depth within allowable range of conditions.
(16)
The differential equation between rock-breaking depth and impact velocity can be obtained by simplifying equation (16):
dh p = −mv / Fh dv
(17)
Differentiating operation on equation (17), the rock-breaking depth h p at undamaged rock impacted by a single particle and the rockbreaking depth h pN after being impacted by Nth particle, can be obtained respectively:
hp =
h pN =
−mvdv
0
∫ηv
θ πR2σr 0
∫ (sin 2 θ + 2μ sin2 θ)dθ
f
0
−mvdv
f
πR2σrN ∫0 (sin 2 θ + 2μ sin2 θ)dθ
∫ηv
θ
(18)
(19)
3.4. Analysis of rock-breaking depth
4. Dynamic simulation and analysis of rock failure
Table 1 is the material properties of particle, water-jet and rock in process of particle jet impact rock. The semi-analytical model of rockbreaking depth is calculated with material parameters. Fig. 6 shows the variation of rock-breaking depth with impact times
4.1. Dynamic simulation method and coupling treatment In the process of particle jet impact rock, high pressure and large deformation will be generated under impact of high-speed water-jet. 6
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Fig. 12. The effect of particle jet impact rock-breaking.
Fig. 14. The variation of rock-breaking depth with water-jet velocity. Fig. 13. Variation of rock-breaking depth with impact times.
In dynamic simulation process of particle impacts on rock, the improved maximum principal stress model is used to describe the failure behavior (Jiang et al., 2015), and when the maximum principal stress exceeds failure criterion, the rock would be out of work immediately. When the radial stress exceeds tensile and compressive strength of rock, the rock will suffer from tensile failure and compressive fracture; when the tangential stress exceeds shear strength of rock, shear failure occurs. And then, the failed rock elements are removed. When in coupling dynamic simulation, the point-to-surface contact algorithm is used to define coupling, SPH particles are defined as slave nodes, and FEM elements of contact are defined as master surfaces. The coupling process is shown in Fig. 8.
SPH has self-adaptability and can effectively overcome calculation termination problem caused by mesh distortion when simulating large deformation and discontinuous medium mechanics problems. At the same time, FEM has high efficiency and accuracy in calculating deformation of continuous media (Zhang and Zhao, 2014; Zhang et al., 2011). Therefore, the coupling method of SPH and FEM is used to simulate the process of particle jet impact on rock. SPH is used to simulate water-jet and FEM is used to simulate rock and particles. The coupled simulation method can not only obtain relatively accurate damage characteristics, but also better simulate such problems as instantaneous large impact, large deformation and high strain rate (Lin et al., 2014; Ma et al., 2010). 7
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respectively. The boundary conditions of bottom and side adopt nonreflective boundary conditions to eliminate the effect of stress wave reflection on rock damage. 4.3. Dynamic simulation and damage evolution of rock failure under particle jet impact 4.3.1. Damage evolution of rock-breaking process under particle jet impact According to the analysis of multi-particle jet impact rock-breaking model, the damage cloud picture of rock at different impact time is shown in Fig. 10. It can be seen that with increase of impact time, the damage is gradually expanding outward from the edge of water-jet, and damage mainly occurs below particle jet impact area. The main damage area presents funnel shape. Meanwhile, the damage cracks will continue to expand with the duration of impact time, and the longitudinal damage expansion effect is more obvious than that of transverse expansion. 4.3.2. The analysis of rock failure under particle jet impact Fig. 11 shows the rock failure process under particle jet impact with velocity of 200 m/s, and shear failure and tensile failure are the main types of rock fragmentation. The impact dynamic pressure would be formed when particle impact rock and it will diffuse in form of stress waves. At the initial stage of collision, the rock fragmentation firstly represents as shear failure. With increase of impact times, tensile failure cracks germinate around the shear failure area, and these cracks expand transversely and longitudinally. The propagation of longitudinal cracks is more obvious than that of longitudinal cracks.
Fig. 15. Variation of rock-breaking depth with particle diameter.
4.4. Effect factors of rock-breaking depth 4.4.1. Effect of impact times on rock-breaking depth Based on rock-breaking model of particle jet coupling impact, the rock-breaking effect and depth under different particle numbers are analyzed (the water-jet velocity is 205 m/s, and the particle diameter is 1.0 mm), as shown in Fig. 12. According to the analysis, it can be seen that rock can be destroyed to a greater extent by combined of multiple particles and water-jet, and the destruction area is approximately rectangular hole-shaped. According to quantitative analysis of rockbreaking depth, it can be concluded that the failure depth under particle jet coupled impact is not simple superposition, and the failure depth would increase with increase of impact times. The main reason is that under the first particle impact, the rock not only produces small failure area, but also produces some obvious and hidden fissures in the lower surface layer. The existence of damages and cracks would cause the mechanical properties of rock changed. When the damaged rock area was impacts again, it is easy to break the damaged rock, and moreover, it will form deeper destruction. Therefore, the cycle impact of particles will cause the accumulation of damage inside rock and form destruction pits. Besides, the water-jet can remove rock debris, meanwhile, can aggravate crack expansion and rock damage. The depth curve of each particle impact according to the simulation results is shown in Fig. 13. From dynamic simulation and variation curve of rock-breaking depth, it can be seen that the rock-breaking depth increases gradually with increase of impact times. The rockbreaking pit formed by the first impact is shallow, and with the continuous impact of particles, the depth and width of the rock-breaking pit gradually increase. When impact times increase to five, the rockbreaking depth of per impact time is basically stable. The results of dynamic simulation and numerical calculation are basically consistent, which verifies the correctness of the model.
Fig. 16. Variation of rock-breaking depth with jet distance. Table 2 The experimental parameters. Experimental parameters
unit
value
The The The The The The
m/s % mm ° r/min s
145–220 0.15 1.0 0 30 10
water-jet velocity particle mixing ratio in water-jet particle diameter water-jet angle bit speed experimental time
4.2. Physical models The process of particle jet coupling impact rock is analyzed by using finite element method. Rock and particles are modeled by Lagrange method, and water-jet is modeled by Smooth particle hydrodynamics algorithm. The impact area is encrypted when mesh was generated. The geometric model diagram is shown in Fig. 9. The rock model presents rectangle with size of 60 × 30 mm. While the physical model is established, assuming that particles impact rock uniformly, and the collision constraint between particles is ignored in simulation study. JH-2 damage model is used for rock model in numerical calculation, and STEEL-4340 and WATER model are used for particles and water-jet
4.4.2. Effect of water-jet velocity on rock-breaking depth The particle velocity will be different under the acceleration of water-jet at different velocities. The curve graph on rock-breaking depth in stable damage state is drawn at different water-jet velocity, 8
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Fig. 17. The rock-breaking samples at different water-jet velocities.
particle diameter increasing, and the growth trend is somewhat slower. 4.4.4. Effect of jet distance on rock-breaking depth The jet distance plays an important role on rock-breaking depth. The rock-breaking depth curve with different jet distance is shown in Fig. 16. According to variation of rock-breaking depth with jet distance, it can be seen that the rock-breaking depth gradually increases with increase of jet distance at the beginning, and when jet distance exceeds the optimum distance, the rock-breaking depth begins to decrease. The main reason is that when jet distance is small, the particles are still in acceleration state, so jet distance is very small and the rock-breaking depth is low. When jet distance exceeds the optimal distance, the rockbreaking depth would decrease again due to the jet scattering and particle deceleration. The optimum distance is about 10–20 mm. 5. Analysis and comparative research According to theoretical analysis and application test, the parameters involved in process of particle jet impact rock-breaking mainly include impact velocity of water-jet, particle mixing ratio in water-jet and particle diameter, etc. The method of combining semi-analytical model calculation, dynamic simulation and experimental research to study the effect of such parameters on rock-breaking depth is of great value to development and application of particle jet impact drilling technology and selection of parameters.
Fig. 18. The curve of rock-breaking depth with the change of the water-jet velocity. Table 3 The experimental parameters. Experimental parameters
unit
value
The The The The The The
mm m/s % ° r/min s
0.8–1.4 205 0.15 8 30 30
5.1. Analysis and research methods particle diameter water-jet velocity particle mixing ratio in water-jet water-jet angle bit speed experimental time
The calculation method of rock-breaking depth can be obtained according to the following steps: Firstly, according to mathematical model and dynamic simulation, the rock-breaking depth h pN of single particle in damage stable state can be obtained by equation (19); Secondly, according to cross-sectional area of special nozzle, the maximum number nl of particles is determined; Then, particles number n and impacts frequency f at any position in a period of time can be shown as:
shown in Fig. 14. According to the effect of water-jet velocity on rock-breaking depth, with increase of water-jet velocity, the rock-breaking depth will also increase. Therefore, when particle diameter is constant, the larger water-jet velocity is, the larger particle velocity is obtained, and the larger rock-breaking depth is obtained under particle jet impact.
n = Qvs t / V
(20)
f = n/ nl
(21)
If the nozzle presents a certain angle, the coefficient ϖ should also be taken into account (Ren et al., 2018a,b); Finally, the rock-breaking depth H can be obtained under theoretical conditions within a certain time:
4.4.3. Effect of particle diameter on rock-breaking depth The steel particles with different diameters have different failure effects. The failure depth under different diameters of particle is simulated as a function of particle diameters. The rock-breaking depth curve in stable damage state is shown in Fig. 15. According to variation of rock-breaking depth with particle diameter in stable damage stage, it can be seen that the rock-breaking depth gradually increases with
H=
h pN Qvs t ϖVnl
(22)
By solving equation (22), the rock-breaking depth in a certain time 9
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Fig. 19. Experimental samples with different particle diameter.
Fig. 20. The variation of rock-breaking depth with particle diameter. Fig. 22. The relationship between the particle mixing ratio and the depth of rock breaking.
Table 4 The experimental parameters. Experimental parameters
unit
value
The The The The The The
% m/s mm ° r/min s
0.12–0.20 205 1.0 8 30 30
particle mixing ratio in water-jet water-jet velocity particle diameter water-jet angle bit speed experimental time
under the theoretical state can be obtained, and the analysis method is also applicable to calculation of the rock-breaking depth in dynamical simulation process. 5.2. Effect of water-jet velocity on rock-breaking depth In order to analyze the effect of water-jet velocity on rock-breaking depth, particle jet impact rock-breaking experiment was carried out under experimental parameters shown in Table 2. The rock-breaking
Fig. 21. Rock-breaking samples with different particle proportions. 10
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Therefore, it is recommended to use particle mixing ratio of 0.20% in engineering application.
samples are shown in Fig. 17. The experimental results of rock-breaking depth at different waterjet velocities are compared and analyzed with theoretical calculation and simulation results, as shown in Fig. 18. According to comparative analysis, when water-jet velocity is low, the rock-breaking depth in a certain time is small. When water-jet velocity exceeds 180 m/s, the rock-breaking depth would increase rapidly. And water-jet velocity is greater than 200 m/s, the rock-breaking depth exceeds 15 mm in 10s, which is equivalent to drilling speed close to 6 m/h, while the drilling speed of hard formation is about 2 m/h under the conventional drilling (Wu et al., 2008). At present, the rockbreaking speed is about 3 times faster than that under conventional drilling, which can meet the rock-breaking technological requirements of hard formation. The higher water-jet velocity is, the greater rockbreaking depth is, however, the higher requirements for drilling pipelines and equipment are. Overall consideration, it is recommended to select the water-jet velocity of 200 m/s∼220 m/s under the engineering conditions.
6. Conclusion (1) The mathematical model of rock-breaking depth under particle jet impact is established, it is concluded that the rock-breaking depth of stable damage stage is studied, and it can be applied to analysis of the whole rock-breaking process. Then, the variation of rockbreaking depth with impact times, water-jet velocity and particle diameter is analyzed. (2) Based on SPH-FEM coupling algorithm and explicit finite element method, the dynamic simulation of particle jet impact rock is carried out, and the effect of different impact times, water-jet velocity and particle diameter on rock-breaking depth is analyzed. Analysis results show that the change range of rock-breaking depth is basically gentle when impact times reach five times, and rock-breaking depth of single particle gradually increases with increase of waterjet velocity and particle diameter; (3) The effect of water-jet velocity, particle diameter and particle mixing ratio in water-jet on rock-breaking depth is studied by the verification method combining semi-analytical solution, dynamic simulation and experimental study, and the parameter optimization scheme is proposed for engineering application. The analysis results show that rock-breaking depth would increase with increase of water-jet velocity and particle mixing ratio in water-jet, and decrease with increase of particle diameter. In drilling engineering application, it is recommended to use water-jet velocity of 200–220 m/s, steel particles with particle diameter of 1.0 mm and particle mixing ratio in water-jet of 0.20%.
5.3. Effect of particle diameter on rock-breaking depth In process of particle jet impact rock-breaking, the carrying capacity of high-speed water-jet to particles with different diameters is different, so the rock-breaking depth within a certain time is also different. In order to study the effect of particle diameters on rock-breaking depth, an experimental study was carried out according to relevant parameters in Table 3. The rock-breaking samples are shown in Fig. 19. According to experimental samples and rock-breaking depth under different particle diameters, the depth gradually decreases with increase of diameter, which is different from the rock-breaking depth of single particle in stable damage state. The main reason is that with increase of diameter, impact frequency decreases in several multiple, so the rock-breaking depth decreases with increase of particle diameter. The trend and depth of theoretical calculation and experimental results are basically consistent, and the average error is less than 10%. Analysis of variation curves shows that the smaller particle diameter is, the greater rock-breaking depth is, however, the smaller impact kinetic energy of particles is, so too small particles will cause relatively low kinetic energy. Therefore, when water-jet velocity and particle mixing ratio in water-jet are constant, the rock-breaking depth and impact kinetic energy should be considered comprehensively, and it is suggested to select particle diameter of 1.0 mm in application testing (see Fig. 20).
Conflicts of interest All authors declare that there was no any conflict of interest. Acknowledgements The study was funded by National Natural Science Foundation of China (11972113), the Post-doctoral Research Start-up Fund Project of Heilongjiang Province (LBH-Q15018), and the Guiding Innovation Fund Project of Northeast Petroleum University (2019YDL-07). Appendix A. Supplementary data
5.4. Effect of particle mixing ratio in water-jet on rock-breaking depth
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.petrol.2019.106419.
The particle impact frequency per unit area of rock is changed by controlling particle mixing ratio in high-pressure water-jet. Therefore, particle jet impact rock experiment was carried out using the experimental parameters shown in Table 4, and the rock-breaking samples are shown in Fig. 21. According to experimental data and theoretical calculation, the effect of particle mixing ratio on rock-breaking depth is analyzed, as shown in Fig. 22. From the comparative analysis, it can be seen that with increase of particle mixing ratio in water-jet, the rock-breaking depth shows an upward trend. The main reason is that with increase of particle mixing ratio, particle impact frequency per unit area of rock also increases, thus realizing increase of rock-breaking depth. The average error between experimental result and theoretical calculation is about 3%–5%, which mainly results in energy loss due to the interaction of particles, so experimental results are generally lower than theoretical calculation. According to comparative analysis, when particle mixing ratio in water-jet reaches 0.20%, the rock-breaking depth in experiment for 30s can reach 52 mm (equivalent to the drilling speed of 6.2 m/h), which already can meet the rock-breaking technology requirements.
References Anwar, S., Axinte, D.A., Becker, A.A., 2013. Finite element modelling of abrasive waterjet milled footprints. Wear 303 (1–2), 426–436. http://doi.org/10.1016/j.jmatprotec. 2012.09.006. Cui, M., Zhai, Y.H., Guo, D.J., 2011. Experimental study of rock breaking effect of steel particles. J. Hydrodyn. Ser. B 23 (2), 241–246. http://doi.org/10.1016/s10016058(10)60109-6. He, C., Jin, J.F., Zhou, X.J., Cheng, Y., Chang, X.X., Yuang, W., 2016. Damage constitutive model of rock subjected to coupled static loadings and cyclic impacts. Nonferrous Metals Sci. Eng. 7 (4), 114–120. Jiang, H.X., Du, C.L., Liu, S.Y., Gao, K.D., 2015. Numerical analysis on damage field of rock fragmentation with water jet. J Cent. South Univ. (Science and Technology) 46 (1), 287–294. https://doi.org/10.11817/j.issn.1672−7207.2015.01.039. Jiang, H.X., Liu, Z.H., Gao, K.D., 2017. Numerical simulation on rock fragmentation by discontinuous water-jet using coupled sph/fea method. Powder Technol. 312, 248–259. http://doi.org/10.1016/j.powtec.2017.02.047. Jin, J.F., Li, X.B., Qiu, C., Tao, W., Zhou, X.J., 2014. Evolution model for damage accumulation of rock under cyclic impact loadings and effect of static loads on damage Evolution. Chin. J. Rock Mech. Eng. 33 (8), 1662–1671. http://doi.org/10.13722/j. cnki.jrme.2014.08.017. Kuang, Y.C., Zhu, Z.P., Jiang, H.J., Wu, K.S., 2012. The experimental study and numerical simulation of single-particle impacting rock. Acta Pet. Sin. 17 (6), 779–788. http:// doi.org/10.7623/syxb201206019.
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doi.org/10.7623/syxb201802012. Wang, F.X., Wang, R.H., Zhou, W.D., Chen, G.C., 2017. Numerical simulation and experimental verification of the rock damage field under particle water jet impacting. Int. J. Impact Eng. 102 (Complete), 169–179. http://doi.org/10.1016/j.ijimpeng. 2016.12.019. Wang, F.C., Zhou, D.P., Xu, Q.W., Qiang, C.H., Guo, C.W., 2018. Mathematical model of rock stress under abrasive slurry jet impact based on contact mechanics. Int. J. Rock Mech. Min. Sci. 107, 1–8. http://doi.org/10.1016/j.ijrmms.2018.04.010. Wei, L., Tie, Y., Zhichao, Z., Lianqun, L., 2013. Rock response mechanism and rock breaking test analysis for impact of high frequency vibration drilling tool. Petrol. Drill. Tech. 41 (6), 25–28. http://doi.org/10.3969/j.issn.1001-0890.2013.06.005. Wu, K.S., Gu, J.F., Kuang, Y.C., Li, M., Zhai, Z.M., 2008. Comment on particle impact drilling technology. J. Southwest Petrol. Univ. (Science&Technology Edition) 30 (2), 142–146. Yin, F., Qian, Q., Wang, M., Yan, S., Yan, D., 1998. Penetration depth of projectile oblique into target. Explos. Shock Waves 18 (1), 69–76. Zhang, Q.B., Zhao, J., 2014. A review of dynamic experimental techniques and mechanical behaviour of rock materials. Rock Mech. Rock Eng. 47 (4), 1411–1478. http://doi.org/10.1007/s00603-013-0463-y. Zhang, Z.C., Qiang, H.F., Gao, W.R., 2011. A new SPH-FEM coupling method and its application in impact dynamics. Explos. Shock Waves 31 (3), 243–249. http://doi. org/10.1631/jzus.A1000257. Zhao, J., Zhang, G.C., Xu, Y.J., Zhou, Y., Wang, R.Y., Xing, X.Y., Li, J.B., 2017a. SPHbased numerical simulation and experimental study on rock breaking by particle impact. Explos. Shock Waves 37 (3), 479–486. http://doi.org/10.11883/10011455(2017)_03-0479-08. Zhao, J., Zhang, G.C., Xu, Y.J., Wang, R.H., Zhou, W.D., Han, L.X., Zhou, Y., 2017b. Mechanism and effect of jet parameters on particle waterjet rock breaking. Powder Technol. 313, 231–244. http://doi.org/10.1016/j.powtec.2017.03.026. Zhu, X.H., Tang, L.P., Tong, H., 2014. Effects of high-frequency torsional impacts on rock drilling. Rock Mech. Rock Eng. 47 (4), 1345–1354. http://doi.org/10.1007/s00603013-0461-0.
Li, G.S., Liao, H.L., Huang, Z.W., Shen, Z.H., 2009. Rock damage mechanisms under ultrahigh pressure water jet impact. J. Mech. Eng. 45 (10), 284–293. http://doi.org/10. 3901/JME.2009.10.284. Lin, X.D., Lu, Y.Y., Tang, J.R., Ao, X., Zhang, L.M., 2014. Numerical simulation of rock breaking by abrasive water jet based on SPH-FEM coupling algorithm. J. Vib. Shock 33 (18), 170–176. http://doi.org/10.13465/j.cnki.jvs.2014.18.028. Liu, X., Liu, S., Ji, H., 2015. Mechanism of rock breaking by pick assisted with water jet of different modes. J. Mech. Sci. Technol. 29 (12), 5359–5368. http://doi.org/10.1007/ s12206-015-1137-3. Luk, V.K., Forrestal, M.J., 1987. Penetration into semi-infinite reinforced-concrete targets with spherical and ogival nose projectiles. Int. J. Impact Eng. 6 (4), 291–301. http:// doi.org/10.1016/0734-743X(87)90096. Luk, V.K., Forrestal, M.J., Amos, D.E., 1991. Dynamic spherical cavity expansion of strainhardening materials. J. Appl. Mech. 58 (1), 1–6. http://doi.org/10.1115/1.2897150. Ma, G.W., Wang, X.J., Li, Q.M., 2010. Modeling strain rate effect of heterogeneous materials using sph method. Rock Mech. Rock Eng. 43 (6), 763–776. http://doi.org/10. 1007/s00603-010-0089-2. Momber, A., 2015. Fracture toughness effects in geomaterial solid particle erosion. Rock Mech. Rock Eng. 48 (4), 1573–1588. http://doi.org/10.1007/s00603-014-0658-x. Rajput, A., Iqbal, M.A., 2017. Ballistic performance of plain, reinforced and pre-stressed concrete slabs under normal impact by an ogival-nosed projectile. Int. J. Impact Eng. 110, 57–71. https://doi.org/10.1016/j.ijimpeng.2017.03.008 https://search. crossref.org/?q=Ballistic+performance+of+plain%2C+reinforced+and+prestressed+concrete+slabs+under+normal+impact+by+an+ogival-nosed +projectile. Ren, F.S., Cheng, X.Z., Li, Y., Wang, B.J., Zhu, A.H., Zhao, L., 2017. Mathematical modeling and experimental analysis of coupled particles and jet rock breaking. Eng. Mech. 34 (2), 249–256. http://doi.org/10.6052/j.issn.1000-4750.2015.08.0660. Ren, F.S., Fang, T.C., Cheng, X.Z., Chang, Y.L., 2018a. Rock-breaking stress analysis and rock-breaking region under particle-waterjet impact. Acta Pet. Sin. 39 (09), 1070–1080. http://doi.org/10.7623/syxb201809011. Ren, F.S., Fang, T.C., Cheng, X.Z., 2018b. Parameter optimization and experimental study of particle jet impact rock breaking nozzle. China Petrol. Mach. 46 (02), 5–11. http://
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Theoretical modeling and experimental study of rock-breaking depth in particle jet impact drilling process
T
Fushen Rena,∗∗, Tiancheng Fangb,∗, Xiaoze Chengc a
School of Mechanical Science and Engineering, Northeast Petroleum University, Daqing, Heilongjiang 163318, China School of Civil Engineering and Architecture, Northeast Petroleum University, Daqing, Heilongjiang 163318, China c The China National Petroleum Corporation, Beijing 100724, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Particle jet impact Rock-breaking depth Semi-analytical model Dynamic simulation Experimental study
Particle jet impact drilling technology is an efficient method to improve rock-breaking speed by high-speed particles and water-jet impact. The process and damage stage of rock-breaking by combined impact of highspeed water-jet and particles are very complicated, therefore, it is of great significance to study the rock-breaking depth in different damage stages for particle jet impact drilling. Therefore, laboratory test was carried out to study the effects of rock breaking depth on particle diameter, impact velocity and particle mixing ratio. Then, the semi-analytical model of rock-breaking depth under particle jet impact was established in stable damage condition, and the effect factors of rock-breaking depth were numerically analyzed. Based on SPH-FEM coupling algorithm, the dynamical simulation model of particle jet impact rock was established, and the simulation of rock-breaking process and damage evolution under particle jet impact was carried out. Effects of different particle jet parameters on rock-breaking depth were studied by comparing the semi-analytical mode calculation, dynamic simulation and experimental results. Analysis results showed that rock-breaking depth by particle jet impact increases with the increase of jet velocity and particle mixing ratio in high-pressure water-jet, and decreases with the increase of particle diameter. According to the requirements of particle jet impact drilling technology and the analysis findings, it is recommended to adopt water-jet velocity of 200–220 m/s, particle mixing ratio in water-jet of 0.20%, and particle diameter of 1.0 mm. Research results can provide theoretical basis for the parameters optimization and field application of particle jet impact drilling technology.
1. Instruction The particle jet impact drilling technology is an efficient rockbreaking method mainly based on high-frequency impact of high-speed metal particles, supplemented by high-pressure water jet and mechanical drill bit to break rock. It can be effective to overcome the drilling difficulties of hard and abrasive formations in deep and ultra-deep wells (Cui et al., 2011; Zhu et al., 2014). The rock-breaking mechanism of this technology is that high-speed particle jet impacts on rock beforehand to form an annular rock breaking zone, so the confining pressure of deep well can be released in advance. Then, the rock can be broken quickly and effectively under the grinding action of mechanical drill bit. In recent years, particle jet impact rock-breaking technology has developed rapidly and has been well applied in oil drilling, underground engineering and other fields (Wei et al., 2013; Momber, 2015). At present, the research and analysis of damage and broken depth
∗
under particle jet impact are mainly based on the combination of finite element analysis and experimental research. The advantages and disadvantages of particle impact drilling technology were discussed and the rock-breaking mechanism was pointed (Wu et al., 2008; Kuang et al., 2012). Based on fully decoupled fluid-structure coupling theory, Li et al. (2009) established the numerical analysis model and fluidstructure coupling analysis model of ultra-high pressure water jet impacting rock medium. According to experimental conditions, Anwar et al. (2013) established the finite element model of overlapping abrasive water jet (AWJ) grinding footprint. A symmetrical pre-splitting finite element model by picks without water-jet and picks assisted with water-jet were established, and the effects of different configurations on rock stress distribution and cutting force are analyzed by experiments (Liu et al., 2015). Wang et al. (2017) established the particle impact rock-breaking depth model based on cavity expansion theory, and studied the dynamic damage evolution of rock-breaking process. Based
Corresponding author. Corresponding author. E-mail addresses: [email protected] (F. Ren), [email protected] (T. Fang), [email protected] (X. Cheng).
∗∗
https://doi.org/10.1016/j.petrol.2019.106419 Received 29 March 2019; Received in revised form 22 August 2019; Accepted 23 August 2019 Available online 26 August 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
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debris.
on SPH method, Zhao et al. (2017a,b) analyzed the influence of particle jet parameters on rock-breaking volume and impact depth. Jiang et al. (2017) simulated and analyzed the rock-breaking process by using JH-2 constitutive model, and studied the influence of jet parameters on rock damage field. Wang et al. (2018) established the stress model of abrasive slurry jet impacting rock, and analyzed the stress distribution rule under abrasive slurry jet impact. The mathematical model of particle jet impact rock was established, and the stress analysis and rock-breaking area were analyzed and studied (Ren et al., 2018a,b). However, there are few researches on the rock-penetration depth in the field of particle jet coupling impact drilling at present, and also little study on rock failure under different damage state. Therefore, in order to study the penetration depth and rock failure under particle jet impact, firstly, laboratory test of particle jet impact rock was carried out. Then, the mathematical model of rock-breaking depth under particle jet impact at stable damage state is established, and also the dynamic simulation of rock-breaking process is carried out with SPH-FEM coupling method. Then, the influence of water-jet velocity, particle diameter and particle mixing ratio in high-pressure water-jet on rockbreaking depth is studied and analyzed by combining semi-analytical model calculation, dynamical simulation and experimental research. The research results would provide theoretical basis and support for the prediction of rock-breaking depth and damage velocity of particle jet impact drilling, and also the optimization of technological parameters.
2.2. Experimental materials and processes The experimental materials used in particle jet impact rock breaking experiment are shown in Fig. 2. High density granite with the size of 250 mm × 250 mm × 100 mm is used for experimental rock samples. The impact particles are made of Q235 steel spherical particles. The accelerating nozzle is of constant speed type made of cemented carbide. The experimental method and process mainly include preparation stage, rock breaking experiment stage and parameter measurement stage, as shown in Fig. 3. 2.3. Experimental samples The rock was impacted by particles with different diameters at different impact velocities, and the rock samples after crushing experiments are shown in Fig. 4. The rock-breaking depth is obtained by measuring the maximum penetration depth, and the error can be reduced by taking average value through many experiments and repeated measurements. 3. The semi-analytical model of rock-breaking depth In the process of particle jet impact drilling, the combined impact of high-speed water-jet with metal particles on rock is quite complicated. The damage and destruction of rock can be divided into three stages: initial rock-breaking stage, damage accumulation stage and stable damage stage. At the initial rock-breaking stage, rock in deep well has no initial damage, and the rock-breaking depth under particle jet impact is small at this moment; At the damage accumulation stage, with particle jet continuously impact, the cumulative damage of rock gradually increases, and the increase of rock-breaking depth obtained each time gradually increases; When at stable damage stage, the rock-breaking depth impacted by each time is basically stable. Due to the rock being basically at stable damage state in process of particle jet continuous impacting, therefore, it is of great significance to study the rockbreaking depth at stable damage state under particle jet impact for the analysis of particle jet drilling process.
2. Particle jet impact rock-breaking device and laboratory test 2.1. The particle jet impact rock-breaking device The self-developed particle jet impact rock breaking test device is shown in Fig. 1, and it mainly includes power module, particle injection module, simulated top drive module, simulated bottom-hole module, water circulation module, pressure gauge and high pressure pipeline. The system power module is mainly composed of a high-pressure mud pump with 110kw, which can provide 32 MPa and 10.8m3/h of hydraulic power. The particle injection device module mainly comprises a particle storage tank and a screw propeller, and the working principle is to uniformly inject a certain proportion of metal particles into a high-pressure pipeline through screw propeller. The simulated top drive module is used to realize the drilling process of drill pipe driven by top-motor, and can achieve the goal of constant pressure, constant speed and constant torque drilling. The simulated bottom-hole module is used to simulate the bottom-hole of rock. The water circulation module is mainly comprised of water tank and filtering device for realizing water circulation and separation of metal particles from rock
3.1. Cumulative damage and mechanical properties under particle jet impact In process of particle jet impact rock, the water-jet carries metal particles through the acceleration action of special nozzle, forming continuous water-jet and intermittent particles to impact rock at high velocity, so as to achieve the purpose of rock fragmentation. The following assumptions are made before theoretical model is established: (1) The continuous impact of high-speed water-jet on rock is regarded that rock is under constant pressure; (2) Intermittent impact of particles can be regarded as multiple cyclic impact of dynamic load; (3) The collision between particles and water jet is not considered. The expression of continuous impact pressure Pw of water-jet is:
Pw =
ρw Qvf SA
(1)
where, ρw is the density of water-jet, Q is the rated flow, vf is the impact velocity of water-jet, SA is the water-jet impact zone. Under the cyclic impact of particles on rock, apart from the damage pits on the surface, there will also be cracks and damages in rock, so the mechanical properties of rock will inevitably change. When the rock has been damaged, the cumulative damage curve equation (He et al., 2016; Jin et al., 2014) can be expressed as:
Fig. 1. Particle jet impact rock-breaking device. 2
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Fig. 2. The experimental materials.
Fig. 3. The experimental method and process.
Fig. 4. The experimental samples.
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Fig. 5. Stress analysis of particle impact rock breaking process. Fig. 7. Variation of rock-breaking depth with the water-jet velocity. Table 1 Material attribute parameters. Material parameters
Unit
Value
Constant parameter of rock
Unit
Value
The rock density ρr The particle density ρs The water-jet density ρw The compacted strain η∗
kg/m3 kg/m3 kg/m3
2650 7850 1000
GPa MPa MPa
16.6 10 54
–
0.1
Modulus of elasticity E Shear strength G Compressive strength FC Friction coefficient μ
–
0.04
time, V is the volume of single particle, nl is the particle number that can be contained in the cross-sectional area of nozzle outlet. When metal particles impact the undamaged rock (assuming that the rock has no initial damage), the stress relationship of rock can be obtained according to Coulomb-Friction law (Wang et al., 2018; Yin et al., 1998; Zhao et al., 2017a,b):
σθ = μσr
(4)
where, μ is the sliding friction coefficient, σr is the radial stress of undamaged rock, σθ is the tangential stress. Luk and Forrestal (1987) established the penetration depth models of spherical projectiles based on dynamic cavity expansion theory. According to such theoretical model, the relationship between radial stress and impact velocity of spherical projectiles (Luk et al., 1991) is:
σr = ξFc + ζρr v 2
(5)
where, ρr is the density of rock, Fc is the compressive strength of rock, v is the particle velocity, ξ and ζ is respectively represent rock parameters related to the constitutive characteristics. The expression of ξ and ζ is as follows: ∗
(
)
2
∗2/3
∗
∗2
⎧ ζ = 3G + η 1 − 3G + 3η − 4η∗ + η Eγ γ 2E 2(1 − η ) γ ⎪ ⎪ ξ = 2(1 − lgη∗)/3 ⎨ 2/3 ⎪ γ = ⎡ 1 + G 3 − (1 − η∗) ⎤ ⎪ 2 E ⎦ ⎣ ⎩
(
)
(6)
η∗
is the volume compaction strain of rock, G is the shear where, strength of rock, E is the elastic modulus of undamaged rock. When the rock is damaged by Nth particles, the mechanical properties of rock at undamaged stage and damaged stage can be obtained according to the strain equivalence principle and cumulative damage curve equation (2):
Fig. 6. Variation of rock breaking depth with impact times and particle diameter.
Dn = α − β ln(k / n − p)
(2)
⎧ ε = σrN/ EN =σr / E σrN = σr (1 − DN ) ⎨ ⎩ EN = E (1 − DN )
where, n is the number of particle impacts per unit area, Dn is cumulative damage factors, α and β indicate parameters related to rock damage, p and k is the parameters related to water-jet pressure. Since the impact frequency of particles can reach thousands of times per minute in process of particle jet impact, such high-frequency impact can be considered as uniform impact, so the number n of particle impacts per unit area is:
n = vs t / Vnl
(7)
where, ε is the elastic strain of rock, σrN is the radial stress at damaged stage, EN is the elastic modulus at damaged stage. From equations (4) and (7), it can be concluded that the radial stress at damaged stage under impact of Nth particle is expressed as:
σrN = (ξFc + ζρr v 2)(1 − DN )
(3)
(8)
According to equations (4)–(8), it can be obtained that the radial stress at damaged stage is not only related to impact number, but also
where, vs is the incorporation rate of particles, t is the incorporation 4
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Fig. 8. The coupling process of SPH and FEM.
where, η is the ratio of particle velocity and water-jet velocity, L0 is the length of nozzle constriction. According to the force analysis, the expressions for radial resistance Fr and tangential resistance Fθ of rock (Rajput and Iqbal, 2017) can be obtained as follows:
dFr = 2πR2σr sin θdθ
(10)
dFθ = 2πR2σθ sin θdθ
(11)
where, R is the radius of particle, θ is the angle between solved point and Z axis. The sum of radial resistance and tangential resistance in vertical direction is:
dFh = πR2 (σr sin 2 θ + 2σθ sin2 θ)dθ
(12)
Integrating equation (4), the sum of resistance Fh can be obtained:
⎧ Fh = πR2∫θ (σr sin 2 θ + 2σθ sin2 θ)dθ 0 ⎨ θ = arccos((R − h p)/ R) ⎩
Fig. 9. The model of particle jet impact rock-breaking.
(13)
related to the constitutive material constants, mechanical properties, and impact velocity.
where, θ is the upper integral limit, h p is the rock-breaking depth. When at undamaged stage, the sum of resistance Fh can be obtained by the combined equation (4), equation (5) and equation (12):
3.2. Mechanical model under particle impact
Fh = πR2σr
θ
(sin 2 θ + 2μ sin2 θ)dθ
(14)
When under damaged stage, the sum of resistance Fh is:
When the certain mass m of steel particle impact rock at certain initial velocity v0 , the force acting on rock is shown in Fig. 5. σθ represents tangential stress on rock surface, and the compressive stress is positive in stress analysis, as well tensile stress is negative. After passing through the acceleration of special nozzle (Ren et al., 2018a,b), particle jet would impact rock at high velocity. At this time, the relationship between particle velocity v0 and water-jet impact velocity vf can be obtained by data regression analysis (Ren et al., 2017):
Fh = πR2σrN
∫0
θ
(sin 2 θ + 2μ sin2 θ)dθ
(15)
According to stress analysis of rock at undamaged stage and damaged state respectively, the sum of resistance in rock-breaking process is related to mechanical characteristics and particle diameter. 3.3. Establishment of rock-breaking depth model
= ⎧ v0 ηvf
η = 0.823 − 1.25e−3L0 − 2.29e−5L02 ⎨ ⎩
∫0
The relation between particle velocity change and impact resistance can be known by Newton's second law in rock-breaking process:
(9) 5
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Fig. 10. The damage cloud of particle jet impact rock-breaking.
Fig. 11. The rock failure analysis under particle jet impact.
m
dh p dv dv dv =m = mv = −Fh dt dh p dh p dt
and particle diameters. From the growth trend, it can be seen the rockbreaking depth increases with impact times, and the increase extent is smaller and smaller. After impact times are fifth, it can be considered the damage of rock has reached stable state. From the quantitative relationship, with particle diameter being fixed, the rock-breaking depth of first impact is often small, about 1/6 of particle diameter. Before the second impact occurs, mechanical properties of damaged rock have changed, so it is easier to break rock and the rock-breaking depth has increased greatly. The rock-breaking depth basically tends to be stable after the fifth time, and it is about 1/4–1/3 of particle diameter in stable state. The variation of rock-breaking depth with water-jet impact velocity is shown in Fig. 7 (the particle diameter is 1.0 mm). The rock-breaking depth gradually increases with increase of water-jet impact velocity. Therefore, higher water-jet velocity should be used as much as possible to obtain larger rock-breaking depth within allowable range of conditions.
(16)
The differential equation between rock-breaking depth and impact velocity can be obtained by simplifying equation (16):
dh p = −mv / Fh dv
(17)
Differentiating operation on equation (17), the rock-breaking depth h p at undamaged rock impacted by a single particle and the rockbreaking depth h pN after being impacted by Nth particle, can be obtained respectively:
hp =
h pN =
−mvdv
0
∫ηv
θ πR2σr 0
∫ (sin 2 θ + 2μ sin2 θ)dθ
f
0
−mvdv
f
πR2σrN ∫0 (sin 2 θ + 2μ sin2 θ)dθ
∫ηv
θ
(18)
(19)
3.4. Analysis of rock-breaking depth
4. Dynamic simulation and analysis of rock failure
Table 1 is the material properties of particle, water-jet and rock in process of particle jet impact rock. The semi-analytical model of rockbreaking depth is calculated with material parameters. Fig. 6 shows the variation of rock-breaking depth with impact times
4.1. Dynamic simulation method and coupling treatment In the process of particle jet impact rock, high pressure and large deformation will be generated under impact of high-speed water-jet. 6
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Fig. 12. The effect of particle jet impact rock-breaking.
Fig. 14. The variation of rock-breaking depth with water-jet velocity. Fig. 13. Variation of rock-breaking depth with impact times.
In dynamic simulation process of particle impacts on rock, the improved maximum principal stress model is used to describe the failure behavior (Jiang et al., 2015), and when the maximum principal stress exceeds failure criterion, the rock would be out of work immediately. When the radial stress exceeds tensile and compressive strength of rock, the rock will suffer from tensile failure and compressive fracture; when the tangential stress exceeds shear strength of rock, shear failure occurs. And then, the failed rock elements are removed. When in coupling dynamic simulation, the point-to-surface contact algorithm is used to define coupling, SPH particles are defined as slave nodes, and FEM elements of contact are defined as master surfaces. The coupling process is shown in Fig. 8.
SPH has self-adaptability and can effectively overcome calculation termination problem caused by mesh distortion when simulating large deformation and discontinuous medium mechanics problems. At the same time, FEM has high efficiency and accuracy in calculating deformation of continuous media (Zhang and Zhao, 2014; Zhang et al., 2011). Therefore, the coupling method of SPH and FEM is used to simulate the process of particle jet impact on rock. SPH is used to simulate water-jet and FEM is used to simulate rock and particles. The coupled simulation method can not only obtain relatively accurate damage characteristics, but also better simulate such problems as instantaneous large impact, large deformation and high strain rate (Lin et al., 2014; Ma et al., 2010). 7
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respectively. The boundary conditions of bottom and side adopt nonreflective boundary conditions to eliminate the effect of stress wave reflection on rock damage. 4.3. Dynamic simulation and damage evolution of rock failure under particle jet impact 4.3.1. Damage evolution of rock-breaking process under particle jet impact According to the analysis of multi-particle jet impact rock-breaking model, the damage cloud picture of rock at different impact time is shown in Fig. 10. It can be seen that with increase of impact time, the damage is gradually expanding outward from the edge of water-jet, and damage mainly occurs below particle jet impact area. The main damage area presents funnel shape. Meanwhile, the damage cracks will continue to expand with the duration of impact time, and the longitudinal damage expansion effect is more obvious than that of transverse expansion. 4.3.2. The analysis of rock failure under particle jet impact Fig. 11 shows the rock failure process under particle jet impact with velocity of 200 m/s, and shear failure and tensile failure are the main types of rock fragmentation. The impact dynamic pressure would be formed when particle impact rock and it will diffuse in form of stress waves. At the initial stage of collision, the rock fragmentation firstly represents as shear failure. With increase of impact times, tensile failure cracks germinate around the shear failure area, and these cracks expand transversely and longitudinally. The propagation of longitudinal cracks is more obvious than that of longitudinal cracks.
Fig. 15. Variation of rock-breaking depth with particle diameter.
4.4. Effect factors of rock-breaking depth 4.4.1. Effect of impact times on rock-breaking depth Based on rock-breaking model of particle jet coupling impact, the rock-breaking effect and depth under different particle numbers are analyzed (the water-jet velocity is 205 m/s, and the particle diameter is 1.0 mm), as shown in Fig. 12. According to the analysis, it can be seen that rock can be destroyed to a greater extent by combined of multiple particles and water-jet, and the destruction area is approximately rectangular hole-shaped. According to quantitative analysis of rockbreaking depth, it can be concluded that the failure depth under particle jet coupled impact is not simple superposition, and the failure depth would increase with increase of impact times. The main reason is that under the first particle impact, the rock not only produces small failure area, but also produces some obvious and hidden fissures in the lower surface layer. The existence of damages and cracks would cause the mechanical properties of rock changed. When the damaged rock area was impacts again, it is easy to break the damaged rock, and moreover, it will form deeper destruction. Therefore, the cycle impact of particles will cause the accumulation of damage inside rock and form destruction pits. Besides, the water-jet can remove rock debris, meanwhile, can aggravate crack expansion and rock damage. The depth curve of each particle impact according to the simulation results is shown in Fig. 13. From dynamic simulation and variation curve of rock-breaking depth, it can be seen that the rock-breaking depth increases gradually with increase of impact times. The rockbreaking pit formed by the first impact is shallow, and with the continuous impact of particles, the depth and width of the rock-breaking pit gradually increase. When impact times increase to five, the rockbreaking depth of per impact time is basically stable. The results of dynamic simulation and numerical calculation are basically consistent, which verifies the correctness of the model.
Fig. 16. Variation of rock-breaking depth with jet distance. Table 2 The experimental parameters. Experimental parameters
unit
value
The The The The The The
m/s % mm ° r/min s
145–220 0.15 1.0 0 30 10
water-jet velocity particle mixing ratio in water-jet particle diameter water-jet angle bit speed experimental time
4.2. Physical models The process of particle jet coupling impact rock is analyzed by using finite element method. Rock and particles are modeled by Lagrange method, and water-jet is modeled by Smooth particle hydrodynamics algorithm. The impact area is encrypted when mesh was generated. The geometric model diagram is shown in Fig. 9. The rock model presents rectangle with size of 60 × 30 mm. While the physical model is established, assuming that particles impact rock uniformly, and the collision constraint between particles is ignored in simulation study. JH-2 damage model is used for rock model in numerical calculation, and STEEL-4340 and WATER model are used for particles and water-jet
4.4.2. Effect of water-jet velocity on rock-breaking depth The particle velocity will be different under the acceleration of water-jet at different velocities. The curve graph on rock-breaking depth in stable damage state is drawn at different water-jet velocity, 8
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Fig. 17. The rock-breaking samples at different water-jet velocities.
particle diameter increasing, and the growth trend is somewhat slower. 4.4.4. Effect of jet distance on rock-breaking depth The jet distance plays an important role on rock-breaking depth. The rock-breaking depth curve with different jet distance is shown in Fig. 16. According to variation of rock-breaking depth with jet distance, it can be seen that the rock-breaking depth gradually increases with increase of jet distance at the beginning, and when jet distance exceeds the optimum distance, the rock-breaking depth begins to decrease. The main reason is that when jet distance is small, the particles are still in acceleration state, so jet distance is very small and the rock-breaking depth is low. When jet distance exceeds the optimal distance, the rockbreaking depth would decrease again due to the jet scattering and particle deceleration. The optimum distance is about 10–20 mm. 5. Analysis and comparative research According to theoretical analysis and application test, the parameters involved in process of particle jet impact rock-breaking mainly include impact velocity of water-jet, particle mixing ratio in water-jet and particle diameter, etc. The method of combining semi-analytical model calculation, dynamic simulation and experimental research to study the effect of such parameters on rock-breaking depth is of great value to development and application of particle jet impact drilling technology and selection of parameters.
Fig. 18. The curve of rock-breaking depth with the change of the water-jet velocity. Table 3 The experimental parameters. Experimental parameters
unit
value
The The The The The The
mm m/s % ° r/min s
0.8–1.4 205 0.15 8 30 30
5.1. Analysis and research methods particle diameter water-jet velocity particle mixing ratio in water-jet water-jet angle bit speed experimental time
The calculation method of rock-breaking depth can be obtained according to the following steps: Firstly, according to mathematical model and dynamic simulation, the rock-breaking depth h pN of single particle in damage stable state can be obtained by equation (19); Secondly, according to cross-sectional area of special nozzle, the maximum number nl of particles is determined; Then, particles number n and impacts frequency f at any position in a period of time can be shown as:
shown in Fig. 14. According to the effect of water-jet velocity on rock-breaking depth, with increase of water-jet velocity, the rock-breaking depth will also increase. Therefore, when particle diameter is constant, the larger water-jet velocity is, the larger particle velocity is obtained, and the larger rock-breaking depth is obtained under particle jet impact.
n = Qvs t / V
(20)
f = n/ nl
(21)
If the nozzle presents a certain angle, the coefficient ϖ should also be taken into account (Ren et al., 2018a,b); Finally, the rock-breaking depth H can be obtained under theoretical conditions within a certain time:
4.4.3. Effect of particle diameter on rock-breaking depth The steel particles with different diameters have different failure effects. The failure depth under different diameters of particle is simulated as a function of particle diameters. The rock-breaking depth curve in stable damage state is shown in Fig. 15. According to variation of rock-breaking depth with particle diameter in stable damage stage, it can be seen that the rock-breaking depth gradually increases with
H=
h pN Qvs t ϖVnl
(22)
By solving equation (22), the rock-breaking depth in a certain time 9
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Fig. 19. Experimental samples with different particle diameter.
Fig. 20. The variation of rock-breaking depth with particle diameter. Fig. 22. The relationship between the particle mixing ratio and the depth of rock breaking.
Table 4 The experimental parameters. Experimental parameters
unit
value
The The The The The The
% m/s mm ° r/min s
0.12–0.20 205 1.0 8 30 30
particle mixing ratio in water-jet water-jet velocity particle diameter water-jet angle bit speed experimental time
under the theoretical state can be obtained, and the analysis method is also applicable to calculation of the rock-breaking depth in dynamical simulation process. 5.2. Effect of water-jet velocity on rock-breaking depth In order to analyze the effect of water-jet velocity on rock-breaking depth, particle jet impact rock-breaking experiment was carried out under experimental parameters shown in Table 2. The rock-breaking
Fig. 21. Rock-breaking samples with different particle proportions. 10
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Therefore, it is recommended to use particle mixing ratio of 0.20% in engineering application.
samples are shown in Fig. 17. The experimental results of rock-breaking depth at different waterjet velocities are compared and analyzed with theoretical calculation and simulation results, as shown in Fig. 18. According to comparative analysis, when water-jet velocity is low, the rock-breaking depth in a certain time is small. When water-jet velocity exceeds 180 m/s, the rock-breaking depth would increase rapidly. And water-jet velocity is greater than 200 m/s, the rock-breaking depth exceeds 15 mm in 10s, which is equivalent to drilling speed close to 6 m/h, while the drilling speed of hard formation is about 2 m/h under the conventional drilling (Wu et al., 2008). At present, the rockbreaking speed is about 3 times faster than that under conventional drilling, which can meet the rock-breaking technological requirements of hard formation. The higher water-jet velocity is, the greater rockbreaking depth is, however, the higher requirements for drilling pipelines and equipment are. Overall consideration, it is recommended to select the water-jet velocity of 200 m/s∼220 m/s under the engineering conditions.
6. Conclusion (1) The mathematical model of rock-breaking depth under particle jet impact is established, it is concluded that the rock-breaking depth of stable damage stage is studied, and it can be applied to analysis of the whole rock-breaking process. Then, the variation of rockbreaking depth with impact times, water-jet velocity and particle diameter is analyzed. (2) Based on SPH-FEM coupling algorithm and explicit finite element method, the dynamic simulation of particle jet impact rock is carried out, and the effect of different impact times, water-jet velocity and particle diameter on rock-breaking depth is analyzed. Analysis results show that the change range of rock-breaking depth is basically gentle when impact times reach five times, and rock-breaking depth of single particle gradually increases with increase of waterjet velocity and particle diameter; (3) The effect of water-jet velocity, particle diameter and particle mixing ratio in water-jet on rock-breaking depth is studied by the verification method combining semi-analytical solution, dynamic simulation and experimental study, and the parameter optimization scheme is proposed for engineering application. The analysis results show that rock-breaking depth would increase with increase of water-jet velocity and particle mixing ratio in water-jet, and decrease with increase of particle diameter. In drilling engineering application, it is recommended to use water-jet velocity of 200–220 m/s, steel particles with particle diameter of 1.0 mm and particle mixing ratio in water-jet of 0.20%.
5.3. Effect of particle diameter on rock-breaking depth In process of particle jet impact rock-breaking, the carrying capacity of high-speed water-jet to particles with different diameters is different, so the rock-breaking depth within a certain time is also different. In order to study the effect of particle diameters on rock-breaking depth, an experimental study was carried out according to relevant parameters in Table 3. The rock-breaking samples are shown in Fig. 19. According to experimental samples and rock-breaking depth under different particle diameters, the depth gradually decreases with increase of diameter, which is different from the rock-breaking depth of single particle in stable damage state. The main reason is that with increase of diameter, impact frequency decreases in several multiple, so the rock-breaking depth decreases with increase of particle diameter. The trend and depth of theoretical calculation and experimental results are basically consistent, and the average error is less than 10%. Analysis of variation curves shows that the smaller particle diameter is, the greater rock-breaking depth is, however, the smaller impact kinetic energy of particles is, so too small particles will cause relatively low kinetic energy. Therefore, when water-jet velocity and particle mixing ratio in water-jet are constant, the rock-breaking depth and impact kinetic energy should be considered comprehensively, and it is suggested to select particle diameter of 1.0 mm in application testing (see Fig. 20).
Conflicts of interest All authors declare that there was no any conflict of interest. Acknowledgements The study was funded by National Natural Science Foundation of China (11972113), the Post-doctoral Research Start-up Fund Project of Heilongjiang Province (LBH-Q15018), and the Guiding Innovation Fund Project of Northeast Petroleum University (2019YDL-07). Appendix A. Supplementary data
5.4. Effect of particle mixing ratio in water-jet on rock-breaking depth
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.petrol.2019.106419.
The particle impact frequency per unit area of rock is changed by controlling particle mixing ratio in high-pressure water-jet. Therefore, particle jet impact rock experiment was carried out using the experimental parameters shown in Table 4, and the rock-breaking samples are shown in Fig. 21. According to experimental data and theoretical calculation, the effect of particle mixing ratio on rock-breaking depth is analyzed, as shown in Fig. 22. From the comparative analysis, it can be seen that with increase of particle mixing ratio in water-jet, the rock-breaking depth shows an upward trend. The main reason is that with increase of particle mixing ratio, particle impact frequency per unit area of rock also increases, thus realizing increase of rock-breaking depth. The average error between experimental result and theoretical calculation is about 3%–5%, which mainly results in energy loss due to the interaction of particles, so experimental results are generally lower than theoretical calculation. According to comparative analysis, when particle mixing ratio in water-jet reaches 0.20%, the rock-breaking depth in experiment for 30s can reach 52 mm (equivalent to the drilling speed of 6.2 m/h), which already can meet the rock-breaking technology requirements.
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