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Numerical simulation and experimental verification of the rock damage field under particle water jet impacting
International Journal of Impact Engineering 102 (2017) 169179
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Numerical simulation and experimental verification of the rock damage field under particle water jet impacting TagedPD1X XFangxiang WangD2X X*, D3X XRuihe WangD4X X, D5X XWeidong ZhouD6X X, D7X XGuichun ChenD8X X TagedPSchool of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong 266580, China
TAGEDPA R T I C L E
I N F O
Article History: Received 15 July 2016 Revised 11 December 2016 Accepted 30 December 2016 Available online 3 January 2017 TagedPKeywords: Particle impact drilling technology Particle water jet Rock breaking by impact SPH-FEM coupling Rock damage
TAGEDPA B S T R A C T
By impacting and breaking rock with a particle water jet, particle impact drilling technology can accelerate the rate of penetration. To study the dynamic process, damage evolvements, and failure mechanisms of rock under the impact of a particle water jet, the smoothed particle hydrodynamics method (SPH) was coupled with the finite element method (FEM) to combine their advantages. Based on a stochastic algorithm, steel particles built by FEM were dispersed among water built by SPH to construct a new model of a particle water jet that can actually reflect the large-diameter and low-concentration features of the solid phase in a particle water jet. To verify the accuracy of the numerical simulation results, indoor experiments of a particle water jet impacting rock were carried out. The simulation results were demonstrated by experimental studies, which indicated that the SPH-FEM coupled method was able to simulate the rock-breaking process under a particle water jet. The simulation results revealed that rock fragmentation was achieved due to the impact dynamic load and rock softening caused by the stress wave and that the scope of the stress wave had a distinct partial effect. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction TagedPWith the continuous exploration and usage of oil and gas resources, shallow oil and gas reservoirs are gradually being exhausted, and stabilizing production is becoming more difficult. An increasing number of countries have shifted their focus to deep oil and gas resources. The number of deep wells and ultra-deep wells has continuously increased in recent years. In China, 73% of the remaining oil and gas resources are in formations below 5000 m. The number of deep wells completed by Sinopec exceeding 5000 m increased from 22 in 2006 to 190 in 2013, reaching 1052 in total. As the drilling depth has increased, the strength and hardness of the rock have increased, and the drillability has become poorer. Combined with other factors, such as the formation of strong abrasions, conventional drilling technology will be confronted with the technical bottlenecks of a low rate of penetration (ROP), long drilling cycles and high costs [1,2]. Accelerating the ROP is an important way to guarantee the integral exploration benefits of oil and gas resources. TagedPWithout changing the drilling facilities and processes using particle impacting drilling (PID) technology, the steel particles of high strength and abrasiveness were injected into the drilling fluid in an appropriate ratio through the particle injection system that bridged the drilling pump and drilling pipe and were carried to the PID bit by the drilling * Corresponding author. E-mail address: [email protected] (F. Wang). http://dx.doi.org/10.1016/j.ijimpeng.2016.12.019 0734-743X/© 2017 Elsevier Ltd. All rights reserved.
TagedP uid. The particles were ejected from nozzles on the drilling bit at a fl high velocity and formed a particle water jet. The particles impacted formation at a high frequency and broke the rock, reducing the rockbreaking threshold pressure and rock destruction specific energy and accelerating the ROP in deep hard formations and strong abrasive formations [3,4]. The oilfield practices in Utah, Texas and Sichuan Basin showed that PID technology could accelerate ROP by 2»4 times [57]. TagedPThe particle water jet had a solid-liquid two-phase flow. The solid phase was particles with a diameter of 0.0005 m»0.0015 m and its volume fraction was 1%»3%. The solid phase of the particle water jet had a large-diameter and low-concentration. Investigating the dynamic response and fracture propagation of rock under impact from the particle water jet was essential to reveal the rock-breaking mechanism of PID technology. Since PID technology was put forward, researchers have mainly studied rock-breaking laws under the impact of single, double and four particles. Thomas [8] and Gordon and Greg [9] calculated the contact press of a single particle impacting rock and analysed the fragmentation and initial fractures under compression generated by the impact of the particle. Xu et al. [10], Kuang et al. [11] and Wang et al. [12] conducted studies on the rock-breaking process of a single particle and studied the rock-breaking mechanism in view of the hydraulic energy and contact stress. However, studies of a single particle cannot reflect the interactions between particles. Thus, Wu et al. [13] and Zhao et al. [14] studied the rock-breaking mechanism under the impact of double and four particles by numerical simulations and optimized the parameters of rock breaking under the
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TagedPimpact of particles. Still, there are deficiencies in these studies. The nonlinear dynamic finite element method (FEM) was applied to simulate the rock-breaking process by the above-mentioned studies. Although it has a relatively high computational efficiency, this method encounters serious problems of mesh warp when dealing with extraordinary mesh deformation due to its intrinsic limits, resulting in simulation instability or even failure [1517]. Additionally, studying the rock-breaking mechanism under continuous particle impacting is quite different compared to that of single, double and four particles. In addition, past research on rock-breaking mechanisms neglected the influence of hydraulic factors. Thus, the extant research findings cannot be fully referenced to study the rock-breaking mechanism and guide onsite application of particle impact drilling. TagedPThe smooth particle hydrodynamics method (SPH) is a mesh-free algorithm that uses a series of particles to discretize the computational domain [18]. SPH has advantages in dealing with large mesh deformation problems and discrete medium dynamics and has been used in numerical simulation of rock breaking under a water jet and abrasive water jet. Bui et al. [19] built a model of a water jet and rock with SPH and studied the impact of the water jet on dry soil and saturated soil using numerical simulations. Liu et al. [20] built a model of a water jet with SPH to study the rock breaking of a conical cutter with the assistance of front and rear water jets. Wang et al. [21] built a model of an abrasive water jet with SPH. The water and abrasives in the model were constructed based on SPH particles with the same diameter, which is different from the large-diameter and low-concentration features of the solid phase in a particle water jet. However, the study method is worth discussing. TagedPCompared with FEM, the computation efficiency of SPH is relatively low [2223]. To study the damage evolution and failure process of rock impacted simultaneously by particles and a water jet, in this paper, a coupled method combining the advantages of SPH and FEM was used to simulate the three-dimensioned nonlinear impact dynamics of hard rock breaking. The influences of the stress wave on rock damage were quantitatively analysed. To reflect the largediameter and low-concentration features of the solid phase in a particle water jet, the water in the model was constructed by SPH and the steel particles were constructed by FEM. The results of the numerical simulation were compared with the results of experiments based on the standards of the dimensionless rock-breaking depth and the dimensionless rock-breaking volume, which could evaluate the rock-breaking efficiency under the impact of a particle water jet, and the validity of the numerical simulation results was verified. This study can help to improve the development and application of PID technology. 2. Basic theory of coupled SPH and FEM
TagedPwhere V is the support domain including x and x0 . Wðx¡x0 ; hÞ is the kernel function and h is the smoothing length defining the range of support domain. TagedPThe kernel function in this paper is the cubic spline function recommended by Monaghan and Lattanzio [24]: 8 2 3 > 0R<1 < 1:5¡R C 0:5R 3 ð2Þ WðR; hÞ D a£ =6 1 R<2 ; ð2¡RÞ > : 0 R2 where R D jx¡x0 j=h. For three-dimensional space problem in this paper, a D 3=ð2ph3 Þ. TagedPThe interspace of particles changes greatly at each step time during the calculation process of a large deformation problem of fluid. Proper smoothing length h should be recomputed. The derivative of the smoothing length h with respect to time t is taken in the following continuity equation: dðhðtÞÞ D hðtÞ r n: dt
ð3Þ
where n is the velocity of the particle. TagedP(2) Particle approximation TagedPThe continuous integral form of field variables is transformed into the discrete form by summing the contributions of particles in the range of the support domain by particle approximation (see Fig. 1). The value of an arbitrary function at particle i can be estimated by the weighted average of the values of the function at other neighbouring particles in the support domain of the kernel function W(R, h). TagedPThe particle approximation for the function f(x) and its derivative at particle i are given as follows: 〈fðxi Þ〉 D
N X mj jD1
rj
fðxj ÞWij ;
ð4Þ
where Wij D Wðxi ¡xj ; hÞ D Wðjxi ¡xj j; hÞ. 〈 r ¢ fðxi Þ〉 D
N X mj jD1
rj
fðxj Þ ¢ r i Wij :
x ¡x @W
x @W
where r i Wij D irij j @rijij D rijij @rijij . mj and rj are the mass and density of particle j, respectively. N is the number of particles in the range of support domain. rij is the distance between particle i and j. 2.2. SPH formulation of the governing equation for fluid flow TagedPBy discretizing the NavierStokes equation with Eqs. (4) and (5), the SPH formulation of the governing equation for fluid flow can be derived [25].
2.1. Fundamental function of SPH TagedPSPH interpolates arbitrary macroscopic variables, such as density, velocity, and energy, using a set of values of disordered points based on interpolation theory. Kernel approximation and particle approximation are pivotal steps in SPH. The partial differential equation with field variables is transformed into an integral equation by kernel approximation. The continuous form of the integral equation is transformed into the discrete form of an ordinary differential equation by particle approximation. The transformations not only assure the reliability of the equation but also make the equation convenient to solve. TagedP(1) Kernel approximation TagedPPresume an arbitrary field function f(x) that can be expressed, as follows, by kernel approximation: Z ð1Þ 〈fðxÞ〉 D fðx0 ÞWðx¡x0 ; hÞdx0 ; V
ð5Þ
Fig. 1. Particle approximation.
F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179
TagedPEquation of continuity: dri D dt
N X jD1
Table 1 Constitutive model parameters of water.
b @Wij ; b
mj vij ¢
171
ð6Þ
@xi
TagedPConservation equation of the moment: 0 1 ab ab N X s s dvai @Wij j D mj @ i 2 C 2 A b ; dt r r @x i j jD1
ð7Þ
r0 g/cm3
C m/s
S1
S2
S3
a
g0
1
1480
2.56
¡1.986
0.2286
1.397
0.49
rTagedP ock. ③The rock was assumed to be isotropic media, and the effect of pores on rock breaking was ignored.
i
TagedPConservation equation of mass: ! N pj dei 1X p m ab ab b @Wij D mj 2i C 2 vij b C i ɛi ɛj ; 2 jD1 dt 2ri ri rj @x
3.1. Model of a particle water jet ð8Þ
i
b
b
b
where vij D vi ¡vj , a and b are coordinate directions. s ab is total stress tensor. m is fluid viscosity. ɛab is shear strain. p is fluid pressure. 2.3. Coupling algorithm TagedPThe basic idea of coupling SPH with FEM was to address the large deformation region from SPH and the small deformation region from FEM. A fixed coupling algorithm was used herein, in which the computational domains of SPH and FEM were determined at the initial time [26]. SPH and FEM were coupled via the contact algorithm based on a penalty function. The contact force F(xi) imposed on SPH particles and the surface of a finite element was exerted by a virtual spring between them (Fig. 2). The detection of penetration was conducted at each step time. The contact force was imposed when there was penetration, whereas nothing was done without it. The contact force was calculated as follows [27]: N X mi mj
Wðrij Þn¡1 ¢ r xi Wðrij Þ: WðhÞn
TagedPWater in a particle water jet related to large deformation was modelled by SPH and was assumed to be a perfectly plastic material. The Mie-Grueisen equation of a state was used for water: r0 C m 1 C 1¡ g20 m¡ 2a m2 i C ðg 0 C amÞEa: ð10Þ PD h 2 3 1¡ðS1 ¡1Þm¡S2 mmC 1 ¡S3 m 2 ðm C 1Þ
where r0 is water density. C is the intercept of shock wave velocity vs and particle velocity vp. S1, S2, S3 is the slope coefficient of vs-vp curve. g 0 is Grueisen constant. a is the first order volume correction. E is internal energy per unit volume. TagedPTable 1 shows the material model parameters of water. TagedPThe steel particle was a large-diameter rigid body and was meshed by the Lagrangian finite element method. Table 2 shows the material model parameters of the particle. E and y are the Elastic module and Poisson's coefficient of a particle, respectively. TagedPThe model was established with the following steps to actually reflect the large-diameter and low-concentration features of the solid phase in a particle water jet.
ð9Þ
TagedPStep 1: A three-dimensional jet flow beam is established and meshed by FEM in terms of the size of the steel particle.
where K is a contact stiffness penalty parameter.N is the number of nodes in support domain. h is the average smoothing length. Other parameters are same as above. TagedPParticles i and j penetrated the surface of the finite element, and a contact force was exerted. There was nothing to be done with the other particles.
TagedPStep 2: According to the volume fraction of particles, the number of 3 particles n is calculated by the formula n D 6cV0 =ðpF Þ, where c is the volume fraction of particles, V0 is the total volume of a particle water jet as determined by the geometric model, and F is the diameter of the particles.
Fðxi Þ D
r rj jD1 i
Kn
3. Computation of the model TagedPModel assumptions: ①Steel particles were dispersed randomly in water. ②No deformation or rupture of particles occurred during the collision of steel particles and collision between particles and
TagedPStep 3: The number of random elements is equal to the number of particles. The centre of the elements that are selected randomly by the randomized algorithm is the origin of coordinates of the finite element model of the particle at a later time. The other elements in the jet flow beam are then deleted. TagedPStep 4: The finite element model of spherical is built particles according to the elements selected to at the central coordinates. The material model parameters of the steel particles are assigned in the finite element model. There are now only steel particles in the jet flow beam. TagedPStep 5: The jet flow beam is remeshed, and the finite element mesh of water is obtained. The material model parameters of the water are assigned in the model. TagedPStep 6: The finite elements of the water jet beam are transformed into SPH particles. The volume, mass and stress of SPH particles are equal to the corresponding physical quantities of the transformed finite elements. The jet flow beam consists of SPH particles of water
Table 2 Model parameters of a particle.
Fig. 2. Diagram of coupled SPH with FEM.
r kg/m3
E GPa
y
7800
203
0.3
172
F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179
TagedPwhere DɛP is the equivalent plastic strain increment. DmP is the equivalent volumetric strain increment. T D T=fC , T is the tensile strength. D1 and D2 are damage constants of the material. TagedPUnder the background of drilling in hard rock in China's Sichuan Basin, to coincide with the rock-breaking experiment of a particle water jet, the material model parameters utilized were set as shown in Table 3. 3.3. Geometric model and its mesh generation TagedP3.3.1. The geometric model of the rock and its mesh generation TagedPThe model of a particle water jet impacting rock is shown in Fig. 3. The size of the geometric model for rock was a cuboid of size 60 mm £ 60 mm £ 40 mm. To reflect the infinitely large rock in practical engineering, the displacement of the rock in the X, Y and Z directions was limited, and the surrounding and bottom surfaces were set as the non-reflection boundary. TagedPThe mesh of the region of impact of the particle water jet on rock was densified by a hexahedral mesh of high accuracy. The size of the mesh influences the accuracy of the results and computational cost. To obtain an appropriate size, meshes of different sizes were utilized to discretize the rock model. An enlarged drawing of a single mesh is shown in Fig. 3, where l, w and h are the length, width and height, respectively. Under the conditions that the velocity of the jet was 100 m/s, the volume fraction of particles was 1.5%, diameter of particles was 0.0005 m, and standoff was 6 mm, dimensionless rockbreaking depth H/d and dimensionless rock-breaking volume V/ (pd3/4) corresponding to the mesh of different sizes were computed. H is the rock-breaking depth, d is the jet nozzle diameter (3 mm), and V is the rock-breaking volume. The results are shown in Table 4. TagedPTable 4 shows that the size of the mesh has a significant influence on the accuracy of the results. As the size of the mesh decreases and the number of meshes increases, the dimensionless rock-breaking depth and volume obtained by the simulation increase accordingly, and the simulation results better match the experimental results (the experimental results are shown in Table 5), which indicates a higher accuracy. When the size of the mesh was reduced to less than 0.31 mm £ 0.31 mm £ 0.502 mm, the simulation results changed little as the size continuously decreased. Thus, the size of 0.31 mm £ 0.31 mm £ 0.502 mm not only satisfied the accuracy requirements but also led to a relatively high computational efficiency. In the later simulation, the rock model was discretized by a mesh of the size above.
Fig. 3. Model of a particle water jet impacting rock.
TagedPand finite elements of steel particles, composing the whole model of a particle water jet (shown as Fig. 3).
3.2. Constitutive model of rock TagedPRock breaking under particle water jet impacting produced a large transient dynamic load, a large strain, high strain rates and high pressure. The HolmquistJohnsonConcrete (H-J-C) constitutive model, which satisfied the above working conditions, was used for rock [11,28]. The equivalent yield strength is a function of the pressure, strain rate and damage. Pressure is a function of the volumetric strain, and damage accumulation is a function of the plastic volume strain, equivalent plastic strain and pressure. The normalized equivalent stress for strength is as follows: s D Að1¡DÞ C BPN ½1¡C ln_ɛ ð11Þ where s D s =fC , s is the actual equivalent stress, and fC is the uniaxial compressive strength. D is the rock damage (0 D 1). P is the equivalent pressure, and P D P=fC ._ɛ is the equivalent strain rate.ɛ_ D ɛ_ =_ɛ 0 , where ɛ_ is the actual equivalent strain rate and ɛ_ 0 is the reference strain rate.ɛ_ 0 D 1:0s¡1 , and A, B, C, N is material constants. TagedPThe rock damage D was accumulated based on the equivalent plastic strain and plastic volumetric strain. It is expressed as follows: DD
X DɛP C Dm P D1 ðP C T ÞD2
:
TagedP3.3.2. The geometric model of a particle water jet and its mesh generation TagedPThe geometric model of a particle water jet was a cylinder that was 3 mm in diameter. The length of the jet beam was the product of the duration of the jet and jet velocity. Researchers at the University of Leeds in England argued that the action time of a jet impact is
ð12Þ
Table 3. Material model parameters of the rock model. Rock density Shear modulus Compressive strength r g/cm3 G GPa fC GPa
A
2.60
0.93 1.6
20.8
0.061
Material constants B
C
Damage constants
N
K1 GPa K2 GPa K3 GPa D1
0.08 0.79 85
¡171
208
0.04
D2 1
Maximum tensile hydrostatic pressure pr GPa
Crushing volumetric pressure pc GPa
Crushing volumetric strain mc
Extreme hydrostatic pressure p1 GPa
Extreme volumetric strain m1
Equivalent maximum strength Smax
Damage factor ɛfmix
0.004
0.016
0.001
0.80
0.1
7.0
0.01
F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179 Table 4. Effect of the size of the mesh on the results. l mm
w mm
H mm
Total number of grids
H/d
V/(pd3/4)
0.43 0.39 0.35 0.31 0.27 0.23
0.43 0.39 0.35 0.31 0.27 0.23
0.562 0.542 0.522 0.502 0.482 0.462
1,385,763 1,746,763 2,251,935 2,984,938 4,098,150 5,892,028
1.69 2.17 2.78 3.01 3.05 3.08
3.87 4.97 5.93 6.89 6.98 7.05
TagedPso short that the broken hole forms in milliseconds to microseconds [29]. However, indoor experiments usually measure time in seconds. If the particle water jet beam was continuous in the numerical simulation, the length of the jet beam would be several hundred meters, and the computational requirements would be tremendous. Thus, before carrying out the numerical simulation, an appropriate duration of the jet needs to be determined so that the simulation results could match the indoor experiment results and the computation requirement would not be so great. TagedPAccording to the previous research [30], geometric models of different jet velocities were built within 600 ms. The diameter of the particles was 0.0005 m. The number of particles was calculated according to a particle volume fraction of 1.5%. The particles were meshed with a hexahedral mesh. The standoff was 6 mm. The variations of the dimensionless rock-breaking depth and rock-breaking volume with the duration of the jet were revealed by the simulation results, as shown in Fig. 4. TagedPAs shown in Fig. 4, rock breaking mainly occurs in the initial 100»200 ms. When the duration of the jet reaches 300 ms, the rate of the increase of the dimensionless rock-breaking depth and
173
TagedPvolume decreases greatly over time. Therefore, when the duration of the jet exceeds 300 ms, extension of the jet time will not influence the depth and volume of the broken rock hole. TagedPSimilar conclusions could be drawn via the indoor experiments of a particle water jet impacting rock. The experimental procedure is described in Section 4. Fig. 5 shows the variations of the dimensionless rock-breaking depth and volume of the jet obtained from experiments under different jet durations and jet velocities. Corresponding to the numerical simulation, the diameter of the particles is 0.0005 m, volume fraction of the particle is 1.5% and standoff is 6 mm. TagedPFig. 5 shows that the dimensionless rock-breaking depth and volume exhibit no significant increase over the duration of the jet within 2 s»10 s and the increase rate declines gradually. When the increase rate declines to a certain extent, the extension of the duration of the jet will not influence the depth or volume of the broken rock hole. TagedPTable 5 compares the dimensionless rock-breaking depth and volume of the simulation results at 300 ms with the results of the indoor experiment at 10 s. Table 5 shows that the numerical simulation results at 300 ms are very close to the experiment results at 10 s. The largest rate of change of the dimensionless rock-breaking depth under different jet velocities is only 9.80%, and the largest rate of change of the dimensionless rock-breaking volume is only 8.99%. TagedPIn conclusion, it is appropriate to set the duration of the jet to be 300 ms in the simulation, and it is believable and feasible to compare the results of the simulation at 300 ms with the results of the experiment at 10 s to verify the validity of the simulation results. While building the geometric model of a particle water jet, the jet time was set at 300 ms, and the length of the jet was calculated
Table 5. Comparison of the results of the simulation and experiment.
Jet velocity m/s
H/d Numerical simulation results (300ms)
Experimental results (10 s)
Rate of change %
V/(pd3/4) Numerical simulation results (300ms)
Experimental results (10 s)
Rate of change %
100 120 150 170 200
3.01 3.68 6.02 8.03 13.22
3.21 3.94 5.54 7.36 12.04
6.23 6.60 8.66 9.10 9.80
6.89 10.20 15.34 19.45 31.30
7.30 11.02 14.12 18.03 34.39
5.61 7.44 8.64 7.88 8.99
Fig. 4. Variations of the dimensionless rock-breaking depth and volume with the duration of the jet.
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F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179
Fig. 5. Variations of the dimensionless rock-breaking depth and volume with the duration of the jet.
TagedPaccording to the jet velocity. Fig. 3 shows the geometric model of a particle water jet impacting rock when the jet velocity is 100 m/s, the diameter of the particles is 0.0005 m and the volume fraction of the particle is 1.5%. Under these conditions, the length of the jet beam is 30 mm. 4. Experiment TagedPTo verify the accuracy of the numerical simulation, indoor experiments of a particle water jet impacting rock were carried out at the High Pressure Water Jet Laboratory of the China University of Petroleum (East China). 4.1. Experimental facilities and materials TagedPThe experimental facilities of rock breaking under the impact of a particle water jet mainly consisted of a high-pressure water supply system (high-pressure pump whose rated pressure was 50 MPa and maximum discharge was 530 L/min), particle injection system (the rated pressure was 50 MPa and the volume was 1200 L), particle recovery unit (the recovery rate of the particles was more than 96%) and PID bit. Particle water jet nozzles with a diameter of 3 mm were installed on the PID bit. Fig. 6 shows the main experimental facilities. TagedPThe experimental materials were steel particles and rock cores. Particles with a Rockwell hardness of 40»51 were used in the experiments. To correspond with the numerical simulation, the diameters of the particles were 0.0005 m, 0.008 m, 0.001 m, 0.0012 m and 0.0014 m. Particles with different diameters are shown in Fig. 7. The particle volume fractions were adjusted by the particle injection system to be 1.0%, 1.5%, 2.0%, 2.5% and 3.0%. The rock cores taken from the Xujiahe formation in Sichuan basin were mainly
TagedPgranite, as shown in Fig. 8. The lithologic parameters were tested before the experiment according the parameters in Table 3. 4.2. Experimental method TagedPThe experimental system in this paper is shown as Fig. 9. During the experiment, particles with high hardnesses and good abrasion resistances were injected into the drilling mud at an appropriate proportion by the particle injection system connected between the high-pressure pump and high-pressure fluid transportation pipeline, forming a uniformly dispersed particle slurry. After rock breaking, the particles and rock debris were carried to the particle recovery unit by the drilling mud, and the particles were separated, recovered and recycled by the particle recovery unit. TagedPThe experimental procedure was as follows: ①Connect devices according to Fig. 9, and turn on the pump. Change the velocity of the particle water jet by regulating the pressure of the pump. ②Adjust the particle volume fraction by the particle injection system. ③Turn on the fill-up valve to inject the particles. The particles that are brought to the jet nozzle are accelerated in the jet nozzle, impact the rock core through the fixed nozzle, and a broken hole is formed in the core. ④Run the experiment for 10 s. After 10 s, stop the pump and measure and record the rock-breaking depth and rock-breaking volume. To diminish the effect of occasional factors on the experiment results, repeat the experiment three times under the same parameters and take the average.
Fig. 7. Steel particles with different diameters.
Fig. 6. Experimental facilities.
Fig. 8. Rock cores.
F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179
175
Fig. 9. Experimental system of rock impacted by particles.
5. Results and discussions TagedPFig. 10 shows an image of rock breaking under a particle water jet impacting some point. The conditions of the numerical simulations were as follows: the jet velocity was 100 m/s, diameter of the particles was 0.0005 m, volume fraction of the particles was 1.5%, standoff was 6 mm and duration of the jet was 300 ms. As shown in Fig. 10, the SPH particles of water splashed around in the shape of an inverted umbrella after impacting the rock. The steel particles bounced back after impacting the rock. Some of them collided with the follow-up particles during bounces, and these particles may have impacted the rock again or been bumped away from rock surface. It is observed that the simulation method of SPH-FEM could reflect the actual physical conditions of rock breaking by a particle water jet.
rTagedP ock hole, and compared with the profile acquired from the simulation (seen Fig. 11). The experimental conditions were the same as those in the numerical simulation. The jet velocity was 100 m/s. The diameter of the particles was 0.0005 m, the volume fraction of the particles was 1.5%, and standoff was 6 mm. The results indicated that the shapes of the broken rock holes were consistent. Particles played a major role in the forming process of the broken rock hole. The particles impacted the bottom of the broken rock hole at a perpendicular angle, gradually deepening the broken rock hole. The water eroded the wall, enlarging the diameter of the broken rock hole. The particles collided with each other and bounced back constantly and erratically in the broken rock hole, impacting the wall of the broken rock hole irregularly, resulting in a relatively irregular shape. However, generally, the broken rock hole still had the shape of a horn mouth.
5.1. Verification of simulation results
TagedP5.1.1. Comparison of the shape of broken holes TagedPRock with broken holes formed under the impact of a particle water jet was split to obtain the shape of the profile of the broken
Fig. 10. The process of rock breaking by a particle water jet (partially enlarged).
Fig. 11. Comparison of the shapes of broken rock holes.
TagedP5.1.2. Comparison of the size of broken holes TagedPWe compare the experiment values of the dimensionless rockbreaking depth H/d and dimensionless rock-breaking volume V/ (pd3/4) with the calculation results under the same conditions. The comparison is shown in Figs. 12 and 13. TagedPFig. 12 shows the effect of the particle volume fraction on the rock-breaking efficiency under different jet velocities. It can be seen from Fig. 12 that the dimensionless rock-breaking depth and volume both increased first and then decreased with an increase in the particle volume fraction. Thus, there was an optimal volume fraction. Combined with the rock-breaking process under a particle water jet from numerical simulations, it was considered that the chance that particles collided with each other was low when the volume fraction of the particle was small. Although sparse particles in a jet meant that there was a small chance that the particles collided with each other, the number of particles that impacted rock in a unit of time was also small, and the rock-breaking efficiency was not good. As the volume fraction of the particles increased, the number of particles in the volume of the jet increased and the chance of particles colliding with each other increased. However, the collision was not as violent. The number of particles impacting rock in a unit of time increased, and the dimensionless rock-breaking depth and volume increased gradually. However, when the volume fraction reached more than 2.5%, too many particles existed in the jet, resulting in a much more violent collision between the particles. The impact kinetic energy transformed into inertial energy and dissipated quickly. The energy utilization efficiency decreased, and the rockbreaking efficiency decreased gradually. TagedPFig. 13 shows the effect of the particle diameter on the rockbreaking efficiency under different jet velocities. As the diameter of particle increased, the dimensionless rock-breaking depth and volume both increased first and then decreased, indicating that there was an optimal diameter of particles. For particles of the same
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F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179
Fig. 12. Effect of the particle volume fraction on the rock-breaking efficiency.
Fig. 13. Effect of particle diameter on the rock-breaking efficiency.
TagedPmaterial, particles with a smaller diameter have a relative smaller kinetic energy at the same velocity due to their lighter weight compared with larger particles. When impacting a rock, a lower energy of smaller particles will be delivered to the rock, resulting in a poor efficiency. On the contrary, particles with a larger diameter have a higher kinetic energy and better behaviour. However, when the diameter of particles increases to a certain degree, it is not easy for the jet to carry the particles due to their excessive weight, causing a poor acceleration process. These particles will not reach a high impact velocity. Thus, when the diameter of particles exceeds 0.0012 m, the dimensionless rock-breaking depth and volume decreased, and the rock-breaking efficiency became worse. TagedPIt can be seen from Figs. 12 and 13 that the effect of the volume fraction and diameter of particles on the rock-breaking efficiency obtained from the results of the simulation both matched well with the experimental results and corresponded with the laws in reference [31], indicating that the SPH-FEM method in this paper is correct and feasible and that the model of a particle water jet impacting a rock could be used in the analysis of rock-breaking mechanism. 5.2. Rock-breaking mechanism and process TagedPAs seen from Table 5, Figs. 12 and 13, when the jet velocity was 100 m/s, the particle diameter was 0.0005 m, particle volume fraction was 1.5%, dimensionless rock-breaking depth and volume obtained from the numerical simulation best matched the
TagedPexperimental results. Therefore, in the numerical simulation of the rock-breaking process and mechanism, the jet velocity was set to 100 m/s, diameter of particles was set to 0.0005 m and volume fraction of particles was set to 1.5%. TagedPFig. 14 shows the continuous changing process of the distribution of rock damage D under the impact of a particle water jet. Fig. 15 shows the evolution process of the damage field contour. The greyscale of the damage field contour was 0.1»1.0. Fig. 16 shows the time variation of the damage and maximum principal stress of three representative elements, A, B, and C. Shown as the blue element in Fig. 17, element A was located at the intersection of the top surface of the rock and centre line of the jet flow. Element B was located at the top surface of the rock and was 1.55d from Element A. Element C was 1.5d underneath Element A. TagedPThe particle water jet impacted the top surface of the rock at 60 ms (shown in Fig. 14(a)). The powerful impact pressure generated by the impact compressed and crushed the rock instantaneously (shown as a time variation of the maximum principal stress of element A in Fig. 16), resulting in the initial cracks. The particles continued to impact the rock after crushing the rock surface, causing the radial motion and stretch of the rock mass adjacent to the broken rock hole. A tension stress zone was then formed. A shear stress action face between the tension stress zone and its exterior rock mass took shape due to the radial stretch. Circular cracks were formed under the comprehensive effect of the tension stress and shear stress. Element B is a typical element with that type of failure
F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179
177
Fig. 16. Variations of the element maximum principal stress and damage value with time.
Fig. 17. Distribution of elements (partially enlarged). Fig. 14. Distribution of rock damage D (partially enlarged).
TagedPmode. Rock fragments were formed by the propagation and penetration of cracks under the auxiliary effect of hydraulic pressure, and a new free surface of rock appeared. The downward movement of particles was resisted by the surrounding rock mass until the velocity of particles declined to zero. The kinetic energy of the particles was then transformed into the energy of rock during the process. The rock in the state of compression beneath the contact area between the particles and rock surface released compression energy upward. The particles gained kinetic energy and transformed from the deformation energy of the rock and bounced off, and the rock stretched during the energy releasing process. That is, unloading resulted in the tensile failure of the rock (shown as the evolution process of element C in Fig. 16). The water jet constantly washed away the rock cuttings and eroded the wall of the broken hole. The rock experienced iterations of compression, unloading and recompression under the impact dynamic load of the particle swarm, forming a macroscopic broken hole. TagedPIt can be seen from Figs. 14 and 15 that the depth of the broken rock hole changed little, and that the diameter increased to a certain degree from (c) to (d) because in the initial stage, the particles bounced off the rock surface rapidly after impacting the rock due to the relatively shallow broken rock hole and had little effect on the subsequent particles. However, with increasing of depth, the particles entered into a relatively cramped space, resulting in collision with the rebounded particles and collsion of the rebounded particles with subsequent particles. The rebounded particles cannot leave the broken rock hole immediately and stack in the bottom, hindering subsequent particles from directly impacting the bottom rock. Along with the relatively large energy loss in this process, the depth of the broken rock hole changed little. However, the particles impacted the wall of broken rock hole irregularly in the collision process, Fig. 15. Evolution of the rock damage field contour (partially enlarged).
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TagedPconspicuously enlarging the diameter. The shape of the broken rock hole profile was not quiet regular due to the irregular impact of particles. TagedPThe dynamic response of the high-velocity particle water jet impacting the rock mainly presented as a wave response. A portion of the kinetic energy of particles propagated in the rock in the form of a spherical stress wave. Cracks formed on the wavefront of the stress wave under tensile stress. The energy of the impact stress wave attenuated rapidly away from the impact point. Although the microcracks in the rock resulting from the attenuated wave energy did not lead to rock failure, the microcracks increased the degree of damage of the rock around the broken rock hole (shown as element B in Fig. 16 whose damage value is 0.73), resulting in the changes of the mechanic and physical properties of the rock, such as the strength and porosity, and improving the drillability of the rock. TagedPThe analyses indicated that the damage from a particle water jet on a rock was mainly caused by the impact dynamic load and damage softening effect of the stress wave, and the damage field evolved as follows: initial cracks were formed under the impact pressure; rock fragments were formed by the propagation and penetration of cracks under the comprehensive effect of tension stress and shear stress; unloading resulted in the tensile failure of rock; propagation of the stress wave increased the range of rock damage field and decreased the crushing strength of rock; and finally, a broken rock hole formed. 5.3. Analysis of rock damage TagedPTake 7 displays elements in the Z direction located on the same plane as the element C. They are D, E, F, G, H, I and J (shown as Fig. 17), which were 0.41d, 0.83d, 1.24d, 1.65d, 2.07d, 2.48d and 2.89d away from jet centre, respectively. As shown in Fig. 18, the damage values of elements D, E, and F reached 1 during particle water jet impact. These elements were damaged completely and formed a portion of the broken rock hole. The damage values of elements G, H, and I at some distance from the jet centre declined, and the values were smaller than 1, indicating no failure. The damage value of the farthest element J was 0, indicating that element J was not damaged at all. The above results indicate that the stress wave had a distinct partial effect that was caused by the attenuation of wave energy during propagation. It can be seen from the analysis that element F was located around the wall of the broken rock hole, implying that the distance from element F to the jet centre was the radius of the broken rock hole. Suppose that the area damaged by a particle water jet has a cylindrical shape extending from the centre of the broken rock hole (shown as Figs. 14 and 15). Element I is located around the exterior wall of the cylindrical damage field, and the distance from element I to the jet centre is the radius of the
Fig. 19. Diameter of the broken hole obtained from the experiment.
TagedPdamage field. Thus the damage zone of a particle water jet can be measured qualitatively. Under the research condition of this paper, the diameter of the broken rock hole is approximately 2.48 times of diameter of the jet. Fig. 19 shows that the diameter of the broken rock hole obtained from experiment is 2.67 times of the jet diameter under the same conditions as the numerical simulation. The diameters of the simulation and experiment matched well with each other. The diameter of the damage zone was approximately 4.96 times of diameter of the jet. TagedP 6. Conclusion TagedP(1) The calculation results were verified by indoor experiments, which indicated that the SPH-FEM coupling simulation method can reflect the actual physical process of rock breaking by a particle water jet. The established numerical model can be used to analyse of the damage evolution and failure mechanism of rocks under impact from a particle water jet. TagedP(2) The damage of a particle water jet to a rock was mainly caused by the impact dynamic load and damage softening effect of the stress wave. Affected by the impact dynamic load and damage softening effect, the damage field evolved as follows: initial cracks were formed under the impact pressure; rock fragments were formed by the propagation and penetration of cracks under the comprehensive effect of tension stress and shear stress; unloading resulted in the tensile failure of rock; propagation of the stress wave increased the range of the rock damage field and decreased the crushing strength of the rock, and finally, a broken rock hole formed. TagedP(3) The effect of the stress wave had a distinct partial effect. Suppose that the area damaged by a particle water jet has a cylindrical shape extending from the centre of the broken hole of the rock. The diameter of the pit is 2.48 times greater than that of the jet diameter, and the diameter of damage zone is 4.96 times greater than that of the jet diameter under the research conditions of this paper. Acknowledgements TagedPThis work is supported by the National Science and Technology Major Project of China (No. 2011ZX05060-001), Science and Technology Major Project of CNPC (No. 2015F-1801) and Fundamental Research Funds for the Central Universities (No. 14CX06084A). References
Fig. 18. Variation of the element damage value with time.
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TagedP[19] Bui HH, Sako K, Fukagawa R. Numerical simulation of soil-water interaction using smoothed particle hydrodynamics (SPH) method. J Terramech 2007;44 (5):339–46. TagedP[20] Liu SY, Liu ZH, Cui XX, et al. Rock breaking of conical cutter with assistance of front and rear water jet. Tunnel Underground Space Technol 2014;42(5):78–86. TagedP[21] Wang JM, Gao N, Gong WJ. Abrasive water-jet machining simulation by coupling smoothed particle hydrodynamics/finite element method. Chin J Mech Eng 2010;23(5):568–73. TagedP[22] Hu DA, Han X, Xiao YH, et al. Research developments of smoothed particle hydrodynamics method and its coupling with finite element method. Chin J Theor Appl Mech 2013;45(5):639–52. TagedP[23] Bui HH, Fukagawa R, Sako K, et al. Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic-plastic soil constitutive model. Int J Numer Anal Methods Geomech 2008;32(12):1537–70. TagedP[24] Monaghan JJ, Lattanzio JC. A refined particle method for astrophysical problems. Astron Astrophys 1985;149(149):135–43. TagedP[25] Liu MB. Smoothed particle method. Singapore: World Scientific Publishing Co. Pte. Ltd.; 2003. p. 39–40. TagedP[26] Vuyst TD, Vignjevic R, Campbell JC. Coupling between meshless and finite element methods. Int J Impact Eng 2005;31(8):1054–64. TagedP[27] Vignjevic R, Vuyst TD, Campbell JC. A frictionless contact algorithm for meshless methods. Comput Model Eng Sci 2006;13(1):35–48. TagedP[28] Holmquist TJ, Templeton DW, Bishnoi KD. Constitutive modeling of aluminum nitride for large strain, high-strain rate, and high-pressure applications. Int J Impact Eng 2001;25(3):211–31. TagedP[29] Xu XH, Yu J. Rock crushing science. Beijing: China Coal Industry Publishing House; 1984. p. 257–8. TagedP[30] Kiyotoshi S, Yuichi Y, Akihisa K. Numerical simulation of water jet excavation of rock using smoothed particle hydrodynamics. In: 7th Asian Rock Mechanics Symposium; 2012. TagedP[31] Cui M, Zhai YH, Ji GD. Experimental study of rock breaking effect of steel particles. J Hydrodyn 2011;23(2):241–6.
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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Numerical simulation and experimental verification of the rock damage field under particle water jet impacting TagedPD1X XFangxiang WangD2X X*, D3X XRuihe WangD4X X, D5X XWeidong ZhouD6X X, D7X XGuichun ChenD8X X TagedPSchool of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong 266580, China
TAGEDPA R T I C L E
I N F O
Article History: Received 15 July 2016 Revised 11 December 2016 Accepted 30 December 2016 Available online 3 January 2017 TagedPKeywords: Particle impact drilling technology Particle water jet Rock breaking by impact SPH-FEM coupling Rock damage
TAGEDPA B S T R A C T
By impacting and breaking rock with a particle water jet, particle impact drilling technology can accelerate the rate of penetration. To study the dynamic process, damage evolvements, and failure mechanisms of rock under the impact of a particle water jet, the smoothed particle hydrodynamics method (SPH) was coupled with the finite element method (FEM) to combine their advantages. Based on a stochastic algorithm, steel particles built by FEM were dispersed among water built by SPH to construct a new model of a particle water jet that can actually reflect the large-diameter and low-concentration features of the solid phase in a particle water jet. To verify the accuracy of the numerical simulation results, indoor experiments of a particle water jet impacting rock were carried out. The simulation results were demonstrated by experimental studies, which indicated that the SPH-FEM coupled method was able to simulate the rock-breaking process under a particle water jet. The simulation results revealed that rock fragmentation was achieved due to the impact dynamic load and rock softening caused by the stress wave and that the scope of the stress wave had a distinct partial effect. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction TagedPWith the continuous exploration and usage of oil and gas resources, shallow oil and gas reservoirs are gradually being exhausted, and stabilizing production is becoming more difficult. An increasing number of countries have shifted their focus to deep oil and gas resources. The number of deep wells and ultra-deep wells has continuously increased in recent years. In China, 73% of the remaining oil and gas resources are in formations below 5000 m. The number of deep wells completed by Sinopec exceeding 5000 m increased from 22 in 2006 to 190 in 2013, reaching 1052 in total. As the drilling depth has increased, the strength and hardness of the rock have increased, and the drillability has become poorer. Combined with other factors, such as the formation of strong abrasions, conventional drilling technology will be confronted with the technical bottlenecks of a low rate of penetration (ROP), long drilling cycles and high costs [1,2]. Accelerating the ROP is an important way to guarantee the integral exploration benefits of oil and gas resources. TagedPWithout changing the drilling facilities and processes using particle impacting drilling (PID) technology, the steel particles of high strength and abrasiveness were injected into the drilling fluid in an appropriate ratio through the particle injection system that bridged the drilling pump and drilling pipe and were carried to the PID bit by the drilling * Corresponding author. E-mail address: [email protected] (F. Wang). http://dx.doi.org/10.1016/j.ijimpeng.2016.12.019 0734-743X/© 2017 Elsevier Ltd. All rights reserved.
TagedP uid. The particles were ejected from nozzles on the drilling bit at a fl high velocity and formed a particle water jet. The particles impacted formation at a high frequency and broke the rock, reducing the rockbreaking threshold pressure and rock destruction specific energy and accelerating the ROP in deep hard formations and strong abrasive formations [3,4]. The oilfield practices in Utah, Texas and Sichuan Basin showed that PID technology could accelerate ROP by 2»4 times [57]. TagedPThe particle water jet had a solid-liquid two-phase flow. The solid phase was particles with a diameter of 0.0005 m»0.0015 m and its volume fraction was 1%»3%. The solid phase of the particle water jet had a large-diameter and low-concentration. Investigating the dynamic response and fracture propagation of rock under impact from the particle water jet was essential to reveal the rock-breaking mechanism of PID technology. Since PID technology was put forward, researchers have mainly studied rock-breaking laws under the impact of single, double and four particles. Thomas [8] and Gordon and Greg [9] calculated the contact press of a single particle impacting rock and analysed the fragmentation and initial fractures under compression generated by the impact of the particle. Xu et al. [10], Kuang et al. [11] and Wang et al. [12] conducted studies on the rock-breaking process of a single particle and studied the rock-breaking mechanism in view of the hydraulic energy and contact stress. However, studies of a single particle cannot reflect the interactions between particles. Thus, Wu et al. [13] and Zhao et al. [14] studied the rock-breaking mechanism under the impact of double and four particles by numerical simulations and optimized the parameters of rock breaking under the
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TagedPimpact of particles. Still, there are deficiencies in these studies. The nonlinear dynamic finite element method (FEM) was applied to simulate the rock-breaking process by the above-mentioned studies. Although it has a relatively high computational efficiency, this method encounters serious problems of mesh warp when dealing with extraordinary mesh deformation due to its intrinsic limits, resulting in simulation instability or even failure [1517]. Additionally, studying the rock-breaking mechanism under continuous particle impacting is quite different compared to that of single, double and four particles. In addition, past research on rock-breaking mechanisms neglected the influence of hydraulic factors. Thus, the extant research findings cannot be fully referenced to study the rock-breaking mechanism and guide onsite application of particle impact drilling. TagedPThe smooth particle hydrodynamics method (SPH) is a mesh-free algorithm that uses a series of particles to discretize the computational domain [18]. SPH has advantages in dealing with large mesh deformation problems and discrete medium dynamics and has been used in numerical simulation of rock breaking under a water jet and abrasive water jet. Bui et al. [19] built a model of a water jet and rock with SPH and studied the impact of the water jet on dry soil and saturated soil using numerical simulations. Liu et al. [20] built a model of a water jet with SPH to study the rock breaking of a conical cutter with the assistance of front and rear water jets. Wang et al. [21] built a model of an abrasive water jet with SPH. The water and abrasives in the model were constructed based on SPH particles with the same diameter, which is different from the large-diameter and low-concentration features of the solid phase in a particle water jet. However, the study method is worth discussing. TagedPCompared with FEM, the computation efficiency of SPH is relatively low [2223]. To study the damage evolution and failure process of rock impacted simultaneously by particles and a water jet, in this paper, a coupled method combining the advantages of SPH and FEM was used to simulate the three-dimensioned nonlinear impact dynamics of hard rock breaking. The influences of the stress wave on rock damage were quantitatively analysed. To reflect the largediameter and low-concentration features of the solid phase in a particle water jet, the water in the model was constructed by SPH and the steel particles were constructed by FEM. The results of the numerical simulation were compared with the results of experiments based on the standards of the dimensionless rock-breaking depth and the dimensionless rock-breaking volume, which could evaluate the rock-breaking efficiency under the impact of a particle water jet, and the validity of the numerical simulation results was verified. This study can help to improve the development and application of PID technology. 2. Basic theory of coupled SPH and FEM
TagedPwhere V is the support domain including x and x0 . Wðx¡x0 ; hÞ is the kernel function and h is the smoothing length defining the range of support domain. TagedPThe kernel function in this paper is the cubic spline function recommended by Monaghan and Lattanzio [24]: 8 2 3 > 0R<1 < 1:5¡R C 0:5R 3 ð2Þ WðR; hÞ D a£ =6 1 R<2 ; ð2¡RÞ > : 0 R2 where R D jx¡x0 j=h. For three-dimensional space problem in this paper, a D 3=ð2ph3 Þ. TagedPThe interspace of particles changes greatly at each step time during the calculation process of a large deformation problem of fluid. Proper smoothing length h should be recomputed. The derivative of the smoothing length h with respect to time t is taken in the following continuity equation: dðhðtÞÞ D hðtÞ r n: dt
ð3Þ
where n is the velocity of the particle. TagedP(2) Particle approximation TagedPThe continuous integral form of field variables is transformed into the discrete form by summing the contributions of particles in the range of the support domain by particle approximation (see Fig. 1). The value of an arbitrary function at particle i can be estimated by the weighted average of the values of the function at other neighbouring particles in the support domain of the kernel function W(R, h). TagedPThe particle approximation for the function f(x) and its derivative at particle i are given as follows: 〈fðxi Þ〉 D
N X mj jD1
rj
fðxj ÞWij ;
ð4Þ
where Wij D Wðxi ¡xj ; hÞ D Wðjxi ¡xj j; hÞ. 〈 r ¢ fðxi Þ〉 D
N X mj jD1
rj
fðxj Þ ¢ r i Wij :
x ¡x @W
x @W
where r i Wij D irij j @rijij D rijij @rijij . mj and rj are the mass and density of particle j, respectively. N is the number of particles in the range of support domain. rij is the distance between particle i and j. 2.2. SPH formulation of the governing equation for fluid flow TagedPBy discretizing the NavierStokes equation with Eqs. (4) and (5), the SPH formulation of the governing equation for fluid flow can be derived [25].
2.1. Fundamental function of SPH TagedPSPH interpolates arbitrary macroscopic variables, such as density, velocity, and energy, using a set of values of disordered points based on interpolation theory. Kernel approximation and particle approximation are pivotal steps in SPH. The partial differential equation with field variables is transformed into an integral equation by kernel approximation. The continuous form of the integral equation is transformed into the discrete form of an ordinary differential equation by particle approximation. The transformations not only assure the reliability of the equation but also make the equation convenient to solve. TagedP(1) Kernel approximation TagedPPresume an arbitrary field function f(x) that can be expressed, as follows, by kernel approximation: Z ð1Þ 〈fðxÞ〉 D fðx0 ÞWðx¡x0 ; hÞdx0 ; V
ð5Þ
Fig. 1. Particle approximation.
F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179
TagedPEquation of continuity: dri D dt
N X jD1
Table 1 Constitutive model parameters of water.
b @Wij ; b
mj vij ¢
171
ð6Þ
@xi
TagedPConservation equation of the moment: 0 1 ab ab N X s s dvai @Wij j D mj @ i 2 C 2 A b ; dt r r @x i j jD1
ð7Þ
r0 g/cm3
C m/s
S1
S2
S3
a
g0
1
1480
2.56
¡1.986
0.2286
1.397
0.49
rTagedP ock. ③The rock was assumed to be isotropic media, and the effect of pores on rock breaking was ignored.
i
TagedPConservation equation of mass: ! N pj dei 1X p m ab ab b @Wij D mj 2i C 2 vij b C i ɛi ɛj ; 2 jD1 dt 2ri ri rj @x
3.1. Model of a particle water jet ð8Þ
i
b
b
b
where vij D vi ¡vj , a and b are coordinate directions. s ab is total stress tensor. m is fluid viscosity. ɛab is shear strain. p is fluid pressure. 2.3. Coupling algorithm TagedPThe basic idea of coupling SPH with FEM was to address the large deformation region from SPH and the small deformation region from FEM. A fixed coupling algorithm was used herein, in which the computational domains of SPH and FEM were determined at the initial time [26]. SPH and FEM were coupled via the contact algorithm based on a penalty function. The contact force F(xi) imposed on SPH particles and the surface of a finite element was exerted by a virtual spring between them (Fig. 2). The detection of penetration was conducted at each step time. The contact force was imposed when there was penetration, whereas nothing was done without it. The contact force was calculated as follows [27]: N X mi mj
Wðrij Þn¡1 ¢ r xi Wðrij Þ: WðhÞn
TagedPWater in a particle water jet related to large deformation was modelled by SPH and was assumed to be a perfectly plastic material. The Mie-Grueisen equation of a state was used for water: r0 C m 1 C 1¡ g20 m¡ 2a m2 i C ðg 0 C amÞEa: ð10Þ PD h 2 3 1¡ðS1 ¡1Þm¡S2 mmC 1 ¡S3 m 2 ðm C 1Þ
where r0 is water density. C is the intercept of shock wave velocity vs and particle velocity vp. S1, S2, S3 is the slope coefficient of vs-vp curve. g 0 is Grueisen constant. a is the first order volume correction. E is internal energy per unit volume. TagedPTable 1 shows the material model parameters of water. TagedPThe steel particle was a large-diameter rigid body and was meshed by the Lagrangian finite element method. Table 2 shows the material model parameters of the particle. E and y are the Elastic module and Poisson's coefficient of a particle, respectively. TagedPThe model was established with the following steps to actually reflect the large-diameter and low-concentration features of the solid phase in a particle water jet.
ð9Þ
TagedPStep 1: A three-dimensional jet flow beam is established and meshed by FEM in terms of the size of the steel particle.
where K is a contact stiffness penalty parameter.N is the number of nodes in support domain. h is the average smoothing length. Other parameters are same as above. TagedPParticles i and j penetrated the surface of the finite element, and a contact force was exerted. There was nothing to be done with the other particles.
TagedPStep 2: According to the volume fraction of particles, the number of 3 particles n is calculated by the formula n D 6cV0 =ðpF Þ, where c is the volume fraction of particles, V0 is the total volume of a particle water jet as determined by the geometric model, and F is the diameter of the particles.
Fðxi Þ D
r rj jD1 i
Kn
3. Computation of the model TagedPModel assumptions: ①Steel particles were dispersed randomly in water. ②No deformation or rupture of particles occurred during the collision of steel particles and collision between particles and
TagedPStep 3: The number of random elements is equal to the number of particles. The centre of the elements that are selected randomly by the randomized algorithm is the origin of coordinates of the finite element model of the particle at a later time. The other elements in the jet flow beam are then deleted. TagedPStep 4: The finite element model of spherical is built particles according to the elements selected to at the central coordinates. The material model parameters of the steel particles are assigned in the finite element model. There are now only steel particles in the jet flow beam. TagedPStep 5: The jet flow beam is remeshed, and the finite element mesh of water is obtained. The material model parameters of the water are assigned in the model. TagedPStep 6: The finite elements of the water jet beam are transformed into SPH particles. The volume, mass and stress of SPH particles are equal to the corresponding physical quantities of the transformed finite elements. The jet flow beam consists of SPH particles of water
Table 2 Model parameters of a particle.
Fig. 2. Diagram of coupled SPH with FEM.
r kg/m3
E GPa
y
7800
203
0.3
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TagedPwhere DɛP is the equivalent plastic strain increment. DmP is the equivalent volumetric strain increment. T D T=fC , T is the tensile strength. D1 and D2 are damage constants of the material. TagedPUnder the background of drilling in hard rock in China's Sichuan Basin, to coincide with the rock-breaking experiment of a particle water jet, the material model parameters utilized were set as shown in Table 3. 3.3. Geometric model and its mesh generation TagedP3.3.1. The geometric model of the rock and its mesh generation TagedPThe model of a particle water jet impacting rock is shown in Fig. 3. The size of the geometric model for rock was a cuboid of size 60 mm £ 60 mm £ 40 mm. To reflect the infinitely large rock in practical engineering, the displacement of the rock in the X, Y and Z directions was limited, and the surrounding and bottom surfaces were set as the non-reflection boundary. TagedPThe mesh of the region of impact of the particle water jet on rock was densified by a hexahedral mesh of high accuracy. The size of the mesh influences the accuracy of the results and computational cost. To obtain an appropriate size, meshes of different sizes were utilized to discretize the rock model. An enlarged drawing of a single mesh is shown in Fig. 3, where l, w and h are the length, width and height, respectively. Under the conditions that the velocity of the jet was 100 m/s, the volume fraction of particles was 1.5%, diameter of particles was 0.0005 m, and standoff was 6 mm, dimensionless rockbreaking depth H/d and dimensionless rock-breaking volume V/ (pd3/4) corresponding to the mesh of different sizes were computed. H is the rock-breaking depth, d is the jet nozzle diameter (3 mm), and V is the rock-breaking volume. The results are shown in Table 4. TagedPTable 4 shows that the size of the mesh has a significant influence on the accuracy of the results. As the size of the mesh decreases and the number of meshes increases, the dimensionless rock-breaking depth and volume obtained by the simulation increase accordingly, and the simulation results better match the experimental results (the experimental results are shown in Table 5), which indicates a higher accuracy. When the size of the mesh was reduced to less than 0.31 mm £ 0.31 mm £ 0.502 mm, the simulation results changed little as the size continuously decreased. Thus, the size of 0.31 mm £ 0.31 mm £ 0.502 mm not only satisfied the accuracy requirements but also led to a relatively high computational efficiency. In the later simulation, the rock model was discretized by a mesh of the size above.
Fig. 3. Model of a particle water jet impacting rock.
TagedPand finite elements of steel particles, composing the whole model of a particle water jet (shown as Fig. 3).
3.2. Constitutive model of rock TagedPRock breaking under particle water jet impacting produced a large transient dynamic load, a large strain, high strain rates and high pressure. The HolmquistJohnsonConcrete (H-J-C) constitutive model, which satisfied the above working conditions, was used for rock [11,28]. The equivalent yield strength is a function of the pressure, strain rate and damage. Pressure is a function of the volumetric strain, and damage accumulation is a function of the plastic volume strain, equivalent plastic strain and pressure. The normalized equivalent stress for strength is as follows: s D Að1¡DÞ C BPN ½1¡C ln_ɛ ð11Þ where s D s =fC , s is the actual equivalent stress, and fC is the uniaxial compressive strength. D is the rock damage (0 D 1). P is the equivalent pressure, and P D P=fC ._ɛ is the equivalent strain rate.ɛ_ D ɛ_ =_ɛ 0 , where ɛ_ is the actual equivalent strain rate and ɛ_ 0 is the reference strain rate.ɛ_ 0 D 1:0s¡1 , and A, B, C, N is material constants. TagedPThe rock damage D was accumulated based on the equivalent plastic strain and plastic volumetric strain. It is expressed as follows: DD
X DɛP C Dm P D1 ðP C T ÞD2
:
TagedP3.3.2. The geometric model of a particle water jet and its mesh generation TagedPThe geometric model of a particle water jet was a cylinder that was 3 mm in diameter. The length of the jet beam was the product of the duration of the jet and jet velocity. Researchers at the University of Leeds in England argued that the action time of a jet impact is
ð12Þ
Table 3. Material model parameters of the rock model. Rock density Shear modulus Compressive strength r g/cm3 G GPa fC GPa
A
2.60
0.93 1.6
20.8
0.061
Material constants B
C
Damage constants
N
K1 GPa K2 GPa K3 GPa D1
0.08 0.79 85
¡171
208
0.04
D2 1
Maximum tensile hydrostatic pressure pr GPa
Crushing volumetric pressure pc GPa
Crushing volumetric strain mc
Extreme hydrostatic pressure p1 GPa
Extreme volumetric strain m1
Equivalent maximum strength Smax
Damage factor ɛfmix
0.004
0.016
0.001
0.80
0.1
7.0
0.01
F. Wang et al. / International Journal of Impact Engineering 102 (2017) 169179 Table 4. Effect of the size of the mesh on the results. l mm
w mm
H mm
Total number of grids
H/d
V/(pd3/4)
0.43 0.39 0.35 0.31 0.27 0.23
0.43 0.39 0.35 0.31 0.27 0.23
0.562 0.542 0.522 0.502 0.482 0.462
1,385,763 1,746,763 2,251,935 2,984,938 4,098,150 5,892,028
1.69 2.17 2.78 3.01 3.05 3.08
3.87 4.97 5.93 6.89 6.98 7.05
TagedPso short that the broken hole forms in milliseconds to microseconds [29]. However, indoor experiments usually measure time in seconds. If the particle water jet beam was continuous in the numerical simulation, the length of the jet beam would be several hundred meters, and the computational requirements would be tremendous. Thus, before carrying out the numerical simulation, an appropriate duration of the jet needs to be determined so that the simulation results could match the indoor experiment results and the computation requirement would not be so great. TagedPAccording to the previous research [30], geometric models of different jet velocities were built within 600 ms. The diameter of the particles was 0.0005 m. The number of particles was calculated according to a particle volume fraction of 1.5%. The particles were meshed with a hexahedral mesh. The standoff was 6 mm. The variations of the dimensionless rock-breaking depth and rock-breaking volume with the duration of the jet were revealed by the simulation results, as shown in Fig. 4. TagedPAs shown in Fig. 4, rock breaking mainly occurs in the initial 100»200 ms. When the duration of the jet reaches 300 ms, the rate of the increase of the dimensionless rock-breaking depth and
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TagedPvolume decreases greatly over time. Therefore, when the duration of the jet exceeds 300 ms, extension of the jet time will not influence the depth and volume of the broken rock hole. TagedPSimilar conclusions could be drawn via the indoor experiments of a particle water jet impacting rock. The experimental procedure is described in Section 4. Fig. 5 shows the variations of the dimensionless rock-breaking depth and volume of the jet obtained from experiments under different jet durations and jet velocities. Corresponding to the numerical simulation, the diameter of the particles is 0.0005 m, volume fraction of the particle is 1.5% and standoff is 6 mm. TagedPFig. 5 shows that the dimensionless rock-breaking depth and volume exhibit no significant increase over the duration of the jet within 2 s»10 s and the increase rate declines gradually. When the increase rate declines to a certain extent, the extension of the duration of the jet will not influence the depth or volume of the broken rock hole. TagedPTable 5 compares the dimensionless rock-breaking depth and volume of the simulation results at 300 ms with the results of the indoor experiment at 10 s. Table 5 shows that the numerical simulation results at 300 ms are very close to the experiment results at 10 s. The largest rate of change of the dimensionless rock-breaking depth under different jet velocities is only 9.80%, and the largest rate of change of the dimensionless rock-breaking volume is only 8.99%. TagedPIn conclusion, it is appropriate to set the duration of the jet to be 300 ms in the simulation, and it is believable and feasible to compare the results of the simulation at 300 ms with the results of the experiment at 10 s to verify the validity of the simulation results. While building the geometric model of a particle water jet, the jet time was set at 300 ms, and the length of the jet was calculated
Table 5. Comparison of the results of the simulation and experiment.
Jet velocity m/s
H/d Numerical simulation results (300ms)
Experimental results (10 s)
Rate of change %
V/(pd3/4) Numerical simulation results (300ms)
Experimental results (10 s)
Rate of change %
100 120 150 170 200
3.01 3.68 6.02 8.03 13.22
3.21 3.94 5.54 7.36 12.04
6.23 6.60 8.66 9.10 9.80
6.89 10.20 15.34 19.45 31.30
7.30 11.02 14.12 18.03 34.39
5.61 7.44 8.64 7.88 8.99
Fig. 4. Variations of the dimensionless rock-breaking depth and volume with the duration of the jet.
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Fig. 5. Variations of the dimensionless rock-breaking depth and volume with the duration of the jet.
TagedPaccording to the jet velocity. Fig. 3 shows the geometric model of a particle water jet impacting rock when the jet velocity is 100 m/s, the diameter of the particles is 0.0005 m and the volume fraction of the particle is 1.5%. Under these conditions, the length of the jet beam is 30 mm. 4. Experiment TagedPTo verify the accuracy of the numerical simulation, indoor experiments of a particle water jet impacting rock were carried out at the High Pressure Water Jet Laboratory of the China University of Petroleum (East China). 4.1. Experimental facilities and materials TagedPThe experimental facilities of rock breaking under the impact of a particle water jet mainly consisted of a high-pressure water supply system (high-pressure pump whose rated pressure was 50 MPa and maximum discharge was 530 L/min), particle injection system (the rated pressure was 50 MPa and the volume was 1200 L), particle recovery unit (the recovery rate of the particles was more than 96%) and PID bit. Particle water jet nozzles with a diameter of 3 mm were installed on the PID bit. Fig. 6 shows the main experimental facilities. TagedPThe experimental materials were steel particles and rock cores. Particles with a Rockwell hardness of 40»51 were used in the experiments. To correspond with the numerical simulation, the diameters of the particles were 0.0005 m, 0.008 m, 0.001 m, 0.0012 m and 0.0014 m. Particles with different diameters are shown in Fig. 7. The particle volume fractions were adjusted by the particle injection system to be 1.0%, 1.5%, 2.0%, 2.5% and 3.0%. The rock cores taken from the Xujiahe formation in Sichuan basin were mainly
TagedPgranite, as shown in Fig. 8. The lithologic parameters were tested before the experiment according the parameters in Table 3. 4.2. Experimental method TagedPThe experimental system in this paper is shown as Fig. 9. During the experiment, particles with high hardnesses and good abrasion resistances were injected into the drilling mud at an appropriate proportion by the particle injection system connected between the high-pressure pump and high-pressure fluid transportation pipeline, forming a uniformly dispersed particle slurry. After rock breaking, the particles and rock debris were carried to the particle recovery unit by the drilling mud, and the particles were separated, recovered and recycled by the particle recovery unit. TagedPThe experimental procedure was as follows: ①Connect devices according to Fig. 9, and turn on the pump. Change the velocity of the particle water jet by regulating the pressure of the pump. ②Adjust the particle volume fraction by the particle injection system. ③Turn on the fill-up valve to inject the particles. The particles that are brought to the jet nozzle are accelerated in the jet nozzle, impact the rock core through the fixed nozzle, and a broken hole is formed in the core. ④Run the experiment for 10 s. After 10 s, stop the pump and measure and record the rock-breaking depth and rock-breaking volume. To diminish the effect of occasional factors on the experiment results, repeat the experiment three times under the same parameters and take the average.
Fig. 7. Steel particles with different diameters.
Fig. 6. Experimental facilities.
Fig. 8. Rock cores.
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Fig. 9. Experimental system of rock impacted by particles.
5. Results and discussions TagedPFig. 10 shows an image of rock breaking under a particle water jet impacting some point. The conditions of the numerical simulations were as follows: the jet velocity was 100 m/s, diameter of the particles was 0.0005 m, volume fraction of the particles was 1.5%, standoff was 6 mm and duration of the jet was 300 ms. As shown in Fig. 10, the SPH particles of water splashed around in the shape of an inverted umbrella after impacting the rock. The steel particles bounced back after impacting the rock. Some of them collided with the follow-up particles during bounces, and these particles may have impacted the rock again or been bumped away from rock surface. It is observed that the simulation method of SPH-FEM could reflect the actual physical conditions of rock breaking by a particle water jet.
rTagedP ock hole, and compared with the profile acquired from the simulation (seen Fig. 11). The experimental conditions were the same as those in the numerical simulation. The jet velocity was 100 m/s. The diameter of the particles was 0.0005 m, the volume fraction of the particles was 1.5%, and standoff was 6 mm. The results indicated that the shapes of the broken rock holes were consistent. Particles played a major role in the forming process of the broken rock hole. The particles impacted the bottom of the broken rock hole at a perpendicular angle, gradually deepening the broken rock hole. The water eroded the wall, enlarging the diameter of the broken rock hole. The particles collided with each other and bounced back constantly and erratically in the broken rock hole, impacting the wall of the broken rock hole irregularly, resulting in a relatively irregular shape. However, generally, the broken rock hole still had the shape of a horn mouth.
5.1. Verification of simulation results
TagedP5.1.1. Comparison of the shape of broken holes TagedPRock with broken holes formed under the impact of a particle water jet was split to obtain the shape of the profile of the broken
Fig. 10. The process of rock breaking by a particle water jet (partially enlarged).
Fig. 11. Comparison of the shapes of broken rock holes.
TagedP5.1.2. Comparison of the size of broken holes TagedPWe compare the experiment values of the dimensionless rockbreaking depth H/d and dimensionless rock-breaking volume V/ (pd3/4) with the calculation results under the same conditions. The comparison is shown in Figs. 12 and 13. TagedPFig. 12 shows the effect of the particle volume fraction on the rock-breaking efficiency under different jet velocities. It can be seen from Fig. 12 that the dimensionless rock-breaking depth and volume both increased first and then decreased with an increase in the particle volume fraction. Thus, there was an optimal volume fraction. Combined with the rock-breaking process under a particle water jet from numerical simulations, it was considered that the chance that particles collided with each other was low when the volume fraction of the particle was small. Although sparse particles in a jet meant that there was a small chance that the particles collided with each other, the number of particles that impacted rock in a unit of time was also small, and the rock-breaking efficiency was not good. As the volume fraction of the particles increased, the number of particles in the volume of the jet increased and the chance of particles colliding with each other increased. However, the collision was not as violent. The number of particles impacting rock in a unit of time increased, and the dimensionless rock-breaking depth and volume increased gradually. However, when the volume fraction reached more than 2.5%, too many particles existed in the jet, resulting in a much more violent collision between the particles. The impact kinetic energy transformed into inertial energy and dissipated quickly. The energy utilization efficiency decreased, and the rockbreaking efficiency decreased gradually. TagedPFig. 13 shows the effect of the particle diameter on the rockbreaking efficiency under different jet velocities. As the diameter of particle increased, the dimensionless rock-breaking depth and volume both increased first and then decreased, indicating that there was an optimal diameter of particles. For particles of the same
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Fig. 12. Effect of the particle volume fraction on the rock-breaking efficiency.
Fig. 13. Effect of particle diameter on the rock-breaking efficiency.
TagedPmaterial, particles with a smaller diameter have a relative smaller kinetic energy at the same velocity due to their lighter weight compared with larger particles. When impacting a rock, a lower energy of smaller particles will be delivered to the rock, resulting in a poor efficiency. On the contrary, particles with a larger diameter have a higher kinetic energy and better behaviour. However, when the diameter of particles increases to a certain degree, it is not easy for the jet to carry the particles due to their excessive weight, causing a poor acceleration process. These particles will not reach a high impact velocity. Thus, when the diameter of particles exceeds 0.0012 m, the dimensionless rock-breaking depth and volume decreased, and the rock-breaking efficiency became worse. TagedPIt can be seen from Figs. 12 and 13 that the effect of the volume fraction and diameter of particles on the rock-breaking efficiency obtained from the results of the simulation both matched well with the experimental results and corresponded with the laws in reference [31], indicating that the SPH-FEM method in this paper is correct and feasible and that the model of a particle water jet impacting a rock could be used in the analysis of rock-breaking mechanism. 5.2. Rock-breaking mechanism and process TagedPAs seen from Table 5, Figs. 12 and 13, when the jet velocity was 100 m/s, the particle diameter was 0.0005 m, particle volume fraction was 1.5%, dimensionless rock-breaking depth and volume obtained from the numerical simulation best matched the
TagedPexperimental results. Therefore, in the numerical simulation of the rock-breaking process and mechanism, the jet velocity was set to 100 m/s, diameter of particles was set to 0.0005 m and volume fraction of particles was set to 1.5%. TagedPFig. 14 shows the continuous changing process of the distribution of rock damage D under the impact of a particle water jet. Fig. 15 shows the evolution process of the damage field contour. The greyscale of the damage field contour was 0.1»1.0. Fig. 16 shows the time variation of the damage and maximum principal stress of three representative elements, A, B, and C. Shown as the blue element in Fig. 17, element A was located at the intersection of the top surface of the rock and centre line of the jet flow. Element B was located at the top surface of the rock and was 1.55d from Element A. Element C was 1.5d underneath Element A. TagedPThe particle water jet impacted the top surface of the rock at 60 ms (shown in Fig. 14(a)). The powerful impact pressure generated by the impact compressed and crushed the rock instantaneously (shown as a time variation of the maximum principal stress of element A in Fig. 16), resulting in the initial cracks. The particles continued to impact the rock after crushing the rock surface, causing the radial motion and stretch of the rock mass adjacent to the broken rock hole. A tension stress zone was then formed. A shear stress action face between the tension stress zone and its exterior rock mass took shape due to the radial stretch. Circular cracks were formed under the comprehensive effect of the tension stress and shear stress. Element B is a typical element with that type of failure
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Fig. 16. Variations of the element maximum principal stress and damage value with time.
Fig. 17. Distribution of elements (partially enlarged). Fig. 14. Distribution of rock damage D (partially enlarged).
TagedPmode. Rock fragments were formed by the propagation and penetration of cracks under the auxiliary effect of hydraulic pressure, and a new free surface of rock appeared. The downward movement of particles was resisted by the surrounding rock mass until the velocity of particles declined to zero. The kinetic energy of the particles was then transformed into the energy of rock during the process. The rock in the state of compression beneath the contact area between the particles and rock surface released compression energy upward. The particles gained kinetic energy and transformed from the deformation energy of the rock and bounced off, and the rock stretched during the energy releasing process. That is, unloading resulted in the tensile failure of the rock (shown as the evolution process of element C in Fig. 16). The water jet constantly washed away the rock cuttings and eroded the wall of the broken hole. The rock experienced iterations of compression, unloading and recompression under the impact dynamic load of the particle swarm, forming a macroscopic broken hole. TagedPIt can be seen from Figs. 14 and 15 that the depth of the broken rock hole changed little, and that the diameter increased to a certain degree from (c) to (d) because in the initial stage, the particles bounced off the rock surface rapidly after impacting the rock due to the relatively shallow broken rock hole and had little effect on the subsequent particles. However, with increasing of depth, the particles entered into a relatively cramped space, resulting in collision with the rebounded particles and collsion of the rebounded particles with subsequent particles. The rebounded particles cannot leave the broken rock hole immediately and stack in the bottom, hindering subsequent particles from directly impacting the bottom rock. Along with the relatively large energy loss in this process, the depth of the broken rock hole changed little. However, the particles impacted the wall of broken rock hole irregularly in the collision process, Fig. 15. Evolution of the rock damage field contour (partially enlarged).
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TagedPconspicuously enlarging the diameter. The shape of the broken rock hole profile was not quiet regular due to the irregular impact of particles. TagedPThe dynamic response of the high-velocity particle water jet impacting the rock mainly presented as a wave response. A portion of the kinetic energy of particles propagated in the rock in the form of a spherical stress wave. Cracks formed on the wavefront of the stress wave under tensile stress. The energy of the impact stress wave attenuated rapidly away from the impact point. Although the microcracks in the rock resulting from the attenuated wave energy did not lead to rock failure, the microcracks increased the degree of damage of the rock around the broken rock hole (shown as element B in Fig. 16 whose damage value is 0.73), resulting in the changes of the mechanic and physical properties of the rock, such as the strength and porosity, and improving the drillability of the rock. TagedPThe analyses indicated that the damage from a particle water jet on a rock was mainly caused by the impact dynamic load and damage softening effect of the stress wave, and the damage field evolved as follows: initial cracks were formed under the impact pressure; rock fragments were formed by the propagation and penetration of cracks under the comprehensive effect of tension stress and shear stress; unloading resulted in the tensile failure of rock; propagation of the stress wave increased the range of rock damage field and decreased the crushing strength of rock; and finally, a broken rock hole formed. 5.3. Analysis of rock damage TagedPTake 7 displays elements in the Z direction located on the same plane as the element C. They are D, E, F, G, H, I and J (shown as Fig. 17), which were 0.41d, 0.83d, 1.24d, 1.65d, 2.07d, 2.48d and 2.89d away from jet centre, respectively. As shown in Fig. 18, the damage values of elements D, E, and F reached 1 during particle water jet impact. These elements were damaged completely and formed a portion of the broken rock hole. The damage values of elements G, H, and I at some distance from the jet centre declined, and the values were smaller than 1, indicating no failure. The damage value of the farthest element J was 0, indicating that element J was not damaged at all. The above results indicate that the stress wave had a distinct partial effect that was caused by the attenuation of wave energy during propagation. It can be seen from the analysis that element F was located around the wall of the broken rock hole, implying that the distance from element F to the jet centre was the radius of the broken rock hole. Suppose that the area damaged by a particle water jet has a cylindrical shape extending from the centre of the broken rock hole (shown as Figs. 14 and 15). Element I is located around the exterior wall of the cylindrical damage field, and the distance from element I to the jet centre is the radius of the
Fig. 19. Diameter of the broken hole obtained from the experiment.
TagedPdamage field. Thus the damage zone of a particle water jet can be measured qualitatively. Under the research condition of this paper, the diameter of the broken rock hole is approximately 2.48 times of diameter of the jet. Fig. 19 shows that the diameter of the broken rock hole obtained from experiment is 2.67 times of the jet diameter under the same conditions as the numerical simulation. The diameters of the simulation and experiment matched well with each other. The diameter of the damage zone was approximately 4.96 times of diameter of the jet. TagedP 6. Conclusion TagedP(1) The calculation results were verified by indoor experiments, which indicated that the SPH-FEM coupling simulation method can reflect the actual physical process of rock breaking by a particle water jet. The established numerical model can be used to analyse of the damage evolution and failure mechanism of rocks under impact from a particle water jet. TagedP(2) The damage of a particle water jet to a rock was mainly caused by the impact dynamic load and damage softening effect of the stress wave. Affected by the impact dynamic load and damage softening effect, the damage field evolved as follows: initial cracks were formed under the impact pressure; rock fragments were formed by the propagation and penetration of cracks under the comprehensive effect of tension stress and shear stress; unloading resulted in the tensile failure of rock; propagation of the stress wave increased the range of the rock damage field and decreased the crushing strength of the rock, and finally, a broken rock hole formed. TagedP(3) The effect of the stress wave had a distinct partial effect. Suppose that the area damaged by a particle water jet has a cylindrical shape extending from the centre of the broken hole of the rock. The diameter of the pit is 2.48 times greater than that of the jet diameter, and the diameter of damage zone is 4.96 times greater than that of the jet diameter under the research conditions of this paper. Acknowledgements TagedPThis work is supported by the National Science and Technology Major Project of China (No. 2011ZX05060-001), Science and Technology Major Project of CNPC (No. 2015F-1801) and Fundamental Research Funds for the Central Universities (No. 14CX06084A). References
Fig. 18. Variation of the element damage value with time.
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