Study on Muck Pile Shape of Open-Pit Bench Blasting Based on PFC

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Introduction
Current numerical simulation is mainly divided into the fnite element method based on continuum mechanics and the discrete element method based on Newton's classical mechanics [1]. Te simulation of rock blasting research mainly adopts the fnite element method, and the rock is regarded as a continuum medium [2]. Comparatively speaking, the discrete element method has excellent advantages in predicting rock motion and the blasting efect during blasting.
An et al. [3,4] proposed a hybrid fnite-discrete element method to simulate the dynamic fracture, fragmentation, and muck pile formation in blasting mining and calibrated the FDEM by simulating the process of blasting crater formation. It is found that FDEM can reasonably simulate the stress and fracture propagation process and the formation process. Te mixed fnite element and discrete element method is used to simulate the process of rock breaking and throwing and the formation of muck piles caused by blasting under diferent conditions. Various factors afecting explosive rock breaking are studied. Prasad et al. [5] studied the infuence of blasting design parameters on rock fragments by conducting tests and determining the rock fragment size and maximum fragment size using computer software. It is found that the length of blast hole plugging and charge consumption greatly infuence rock fragmentation. Hudaverdi and Akyildiz [6] proposed a new classifcation method for predicting blasting fying rock throwing distance based on feld measurement and conducted multiple regression analyses on blasting data to establish a prediction formula for blasting fying rock. Yu et al. [7] used the simulation method composed of the fnite element simulation of explosion load calculation and bonded particle method simulation of bench blasting to study the rock movement caused by blasting. Tey verifed the simulation results through theoretical analysis and feld monitoring. Yan et al. [8] established a new method based on the 3DEC program to simulate the cracking and throwing process of blasting funnel and bench blasting considering the rock fragmentation size. It is found that the discretized simulation of the bench blasting profle considering the degree of rock fragmentation is consistent with the prediction results of the traditional ballistics theory and that the maximum casting distance and muck pile height obtained by the two methods have little diference. Su et al. [9] used discrete element PFC2D software to study the relationship between blasting parameters and the shape of the muck pile. Te research results show that the throwing distance is directly proportional to the height of the bench and inversely proportional to the size of the resistance line. Te relationship between the muck pile height and blasting parameters is opposite to the throwing distance. Procházka [10] proposed a discrete hexagonal element method instead of the method to solve the persistent problem and analyzed the generation of the rockburst problem. Zhou et al. [11] compared the diference between the particle fow and other simulation methods. Tey used the particle fow method to simulate biaxial, Brazilian splitting, chamber excavation, and rock caving. Yang et al. [12] adopted the particle fow method to simulate the explosion of the cylindrical charge and the damage process of the rock mass. Te authors established the explosion particle expansion loading method and the dynamic boundary treatment method according to the discrete element mechanism and compared the blasting funnel action index with the feld test results to verify the rationality of the numerical model of cylindrical charge explosion. Ma et al. [13] established the numerical model of blasting rock breakage and throwing based on the feld experimental data and blasting theory and studied the infuence of blasting parameters on the efect of rock throwing. It is found that the single consumption of explosives and the hole distance have the most signifcant infuence on the throwing efect of rock, and the calculation formula of the limit value of the single consumption of explosives and the hole distance is obtained. Te mechanism and mechanical behavior of deep bench blasting throwing are revealed.
Tere are a lot of joints and fssures in the rock. Te rock in the region is divided into rock blocks of diferent sizes, and the interior of the rock blocks can be regarded as a continuum medium. Nevertheless, the rock mass in the whole region should be studied as a discontinuous medium due to its internal structural plane. However, the particle discrete element method cannot be limited by the size of rock deformation. It can better simulate rock media cracking, separation, cracks, and other discontinuous changes [14]. In this work, the simulation of open-pit bench blasting used particle fow software PFC5.0 and the blasting efect is optimized from two aspects of row delay and resistance line [15,16]. It has a specifc guiding signifcance for blasting engineering with similar working conditions.

Particle Flow Discrete Element Theory
2.1. Applying the Blasting Load. Te grain fow discrete element program in PFC cannot directly defne particles of explosive material, giving relevant explosion material properties to simulate the dynamic process. Te particle expansion exerts the load method for the explosive load, and the simulation results are closer to reality.
Due to the explosion site's complexity and danger, the explosion load's active form still relies on some empirical formulas. In the existing studies, the application of the explosion load usually adopts the triangular distributed load model. It can well refect the fuctuation characteristics of the outgoing load, as shown in Figure 1. Terefore, the explosion process can be simulated by applying the corresponding stress wave to the borehole wall.

Boundary Conditions.
Tere are two commonly used methods to simulate the boundary of the infnite medium rock mass: one is the viscous boundary and the other is the transmission boundary method. Te transmission boundary method will cause the particle velocity in the boundary range to be almost zero, and the stress wave will not propagate to the boundary, which directly leads to the sudden drop of the peak value of the stress wave near the boundary. Nevertheless, this difers from the feld's stress wave propagation law. Terefore, the viscous boundary method is closer to reality, and the simulation of an infnite medium uses the viscous boundary.
In the particle fow method, the boundary force is applied to the boundary particles near the wall to ofset the transmitted stress wave and prevent the refected stress wave from accumulating stress inside the model. Te relationship between the boundary force and the particle movement velocity is as follows: (1) R is the radius of suspended particles, ρ is the rock density, C is the stress wave velocity, and u is the particle velocity.
If incident vibration wave U exists at the boundary, the contact force between particles can be written as follows:  Shock and Vibration Blasting stress waves propagates in the medium, resulting in a dispersion efect and attenuation of stress wave intensity. Terefore, the above formula must be modifed to obtain more accurate results:

Parameter Calibration of the Particle Flow Discrete Element Rock Mass Model
Te test mining area is located on the Zaozhuang fault's north wall, and the mine's lithology is mainly medium-thick limestone. Te representative rock blocks with relatively intact rocks and no obvious cracks on the surface are selected from the limestone stones in the mining area for core drilling and sampling. Te standard rock sample with a diameter of 50 mm and a height of 100 mm was selected for the rock uniaxial compression tests, and the mechanical parameter statistics of the rock sample are shown in Table 1 [17].
Te microparameters of the model were calibrated based on the macroscopic mechanical parameters obtained from the uniaxial compression experiment. Te particle discrete element model of microparameters mainly includes the model outline of parameters and the model of contact between particles. Among them, the contact parameter between particles is the main infuencing parameter afecting the intensity of the model.
To match the parameters of the subsequent bench blasting model, the uniaxial compression test model and the bench blasting test model adopted the same particle radius range and generation mode. A plane 2D particle discrete element uniaxial compression model is established with an aspect ratio of 2 : 1, a width of 50 mm, and a height of 100 mm [18,19]. To prevent particles from escaping from the wall during loading, the stifness of the loaded wall should be much higher than particle stifness, usually set to 10∼15 times the wall stifness [20].
Te simulated curve is close to the test curve, and the error is within an acceptable range. According to the calibration law, certain microparameters can be fne-tuned to make their ftting degree higher because it has little infuence on the determined curve. Te stress-strain curves obtained by calibration and those obtained by the laboratory test are shown in Figure 2. It can be seen from the fgure that the calibrated curves obtained by adjusting each microparameters are consistent with those obtained by the uniaxial compression test. Although the slope of the simulated curve is slightly smaller than that of the test curve, the error is only 3.65%, which is within the acceptable range [21]. Te peak stress of the simulated curve is slightly larger than that of the experimental value, with an error of 1.48%. Te ftting degree of peak stress is higher than that of elastic modulus. Terefore, the obtained microparameters can be used to establish the bench model. Te microparameters of the model are shown in Table 2.

Numerical Simulation
Te particle fow discrete element model was established according to the size of the explosion zone, bench height, and blasting parameters. Te model adopted the parallel bonding model with a length of 60 m and a height of 22 m, as shown in Figure 3. Te bench height is 11 m, there are four rows of holes in total, the hole distance is 4.4 m, and the depth of the holes is 13 m. Te length of the blocking section is 3 m. Te blast hole used a cylindrical charge. According to the Starfeld superposition principle [22], the superposition of the spherical charge is equivalent to the cylindrical charge. Moreover, the total length of the spherical charge pack after superposition should be consistent with that of the columnar charge [23]. Te bursting particle expansion loading method is adopted, and the bench blasting simulation is shown in

Morphology Comparison of the Muck Pile
Te height of the muck pile formed after step blasting and the throwing distance of the rock will directly afect the efciency of rock cleaning work after blasting [24]. After the blasting test, the parameters of the muck pile formed after the step blasting test were recorded in time to ensure the site's safety. Te throwing distance of rocks was determined by a UAV's aerial shot of the blasting area and erected scales  After the test, the height of the muck pile measured was 5.53 m and the throwing distance was 42.87 m. According to Figures 5 and 6 on-site, signifcant chunks appeared in the front of the muck pile, among which the maximum chunk size was 1.32 × 1.2 m. Analysis of the rock movement process found that the large blocks are mainly generated from the rock mass between the blocked section of the step blast holes and the empty surface of the bench. Te rock breakage between the behind row blast holes is relatively sufcient, and the bulk rate is low. Te reason is that, after the detonation of the front charge, the stress wave and explosive gas also afect the rock between the behind row blast holes and the rock mass, resulting in cracks in the rock mass. Te rocks between the front and behind blast holes will collide with each other to further improve the crushing efect, so the rock breakage is relatively sufcient [25].
Te numerical simulation of the blasting heap, as shown in Figure 7, shows that the front of a large piece of rock is more. Tese chunks come from between the blast holes of the frst row of the model, and the bench-free face, especially the hole-plugging section, positions chunks of rock particles to produce more. Simulation of large pieces is greater than that of the actual rock, and the degree of rock simulation error is bigger. Te height of the muck pile by numerical simulation is 5.86 m, and the throwing distance is 39.74 m. Te simulated height is slightly larger than the actual height, with an error of 5.97%, and the throwing distance is marginally smaller than the real, with an error of 7.88%. Te error value is within the acceptable range, indicating that the PFC is feasible and accurate to simulate the form of the muck pile and throwing distance of step blasting [26].
Te error between the simulated rock size and the actual situation is signifcant, and the shape of the blast pile is relatively close to the actual situation. Te reason is that the step model is two-dimensional. Te model of detonating considered the blasting delay between rows and did not consider the feld test existing between the blasting delay between holes. Te blasting delay between rows greatly infuences the throwing distance of the muck pile, and the blasting delay between holes greatly infuences rock fragmentation.

Analysis of Influencing Factors of the Open-Pit Bench Blasting Effect
A new bench blasting model is established based on the model parameters verifed by feld tests. Te parameters of deep bench blasting are as follows: the bench height is 5.5 m, the blast hole diameter is 90 mm, the blast hole depth is 6.5 m, the blasting delay between four rows is 30 ms, the spacing of the blast hole row is 2 m, and explosive consumption per unit volume is 0.53 kg/m 3 [27]. Te left and lower boundaries of the model are nonrefective, simulating the propagation of infnite media blasting.

Analysis of the Shape of the Muck Pile with Diferent
Resistance Lines. For the bench blasting simulation, the blasting parameters, such as the charge amount of the blast hole, the blast hole distance, and the extra depth of the blast hole, were kept unchanged. Te size of the resistance lines of the frst row of blast holes near the free face of the bench was changed, set as 1.0 m, 1.25 m, 1.5 m, 1.75 m, and 2.0 m, respectively. Te blasting efect of the diferent resistance lines was analyzed for bench blasting. Figure 8 shows the shape of the muck pile when the resistance line is 1.0 m, 1.5 m, 1.5 m, 1.75 m, and 2 m. Te statistical situation of the explosive heap height and rock throwing distance is shown in Table 3. It can be seen from Figure 8 that the larger the resistance line, the larger the chunk of rock in the front of the muck pile, and the larger the chunk size, it means that the rock is not broken enough. Large rock blocks appear in the middle and rear of the muck pile, and the main reason is that the row spacing between the behind row blast holes is signifcant, resulting in insufcient rock fragmentation. Te rock blasting area was cut into fragments by longitudinal and transverse cracks, and there are only a few radial tensile cracks in the rock mass to the left of the blast holes of the last row. It indicates that the energy generated by blasting acts mainly on the rock mass on the right side of the blast holes, the rock movement is also along the direction of the minimum resistance line, and the millisecond delay blasting has a particularly protective efect on the retained rock mass.
Te height of the muck pile increases gradually with an increase in the resistance line. Moreover, the throwing distance decreases with an increase in the resistance line. Te  T h r o w n d is t a n c e 4 2 .8 7 m Shock and Vibration 5 needs to be broken and thrown. Te energy generated by the explosion with the same charge is constant, so the blasting energy allocated to unit rock decreases accordingly, resulting in insufcient throwing kinetic energy of rock, reduced throwing distance, and increased muck pile height. As shown in Table 3, when the resistance line increases by 0.25 m, the pile height increases by 6.03%, 11.70%, and 5.38%, respectively. It can be seen from the analysis that the change rate of the height of the muck pile and the throwing distance are increasing-decreasing-increasing and that there is a strong correspondence between the two. Analysis of the relationship between the resistance line w and the rock throwing distance y is carried out, and curve ftting of data points obtained the relationship between the resistance line and the rock throwing distance. Te ftting curve is drawn in Figure 9. Te correlation coefcient R 2 of the ftting formula is 0.93. When the resistance line is between 1.0 and 2.0 m, the relationship between the throwing distance and the resistance line is shown in formula (4). According to the relationship curve, the relationship between the resistance line and the rock throwing distance is nonlinear and the throwing distance of the rock decreases with an increase in the resistance line. When the resistance line is 1.0 m, the maximum throwing distance of the rock is 14.86 m: y � −4.63w 2 + 9.17w + 10.09. (4) Te relationship between the resistance line W and the height of the muck pile z was analyzed. Curve ftting was performed on data to obtain the relationship between the resistance line and the height of the muck pile, and the ftting curve is plotted in Figure 10. Te correlation coefcient R 2 of the ftting formula is 0.99. When the resistance line was between 1.0 m and 2.0 m, the relationship between the height of the muck pile and the resistance line is shown in formula (5). It can be seen from the relationship curve that the resistance line and the height of the muck pile are nonlinear and that the height of the muck pile increases with an increase in the resistance line. When the resistance line is 2 m, the maximum height of the pile is 3.87 m: Te ratio of the height of the muck pile H to the height of step h is an important index to evaluate the form of the muck pile. Presently, the bench blast casting of sound efects and signifcant economic beneft is used in engineering blasting at home and abroad, and the value range of H/h adopted by most of them is between 0.6 and 0.9. When the resistance lines are 1.0 m, 1.0 m, 1.5 m, 1.5 m, 1.75 m, and 2 m, the H/h ratios are 0.52, 0.54, 0.61, 0.64, and 0.70, respectively. When the resistance line increases by 0.25 m, the blasting muck pile's height increases by 6.03%, 11.70%, 5.38%, and 9.94%, respectively, and the throwing distance decreases by 5.92%, 6.29, 2.67%, and 24.86%, respectively. When the resistance

Analysis of the Infuence of Diferent Row Delays on the
Shape of the Blasting Pile. Te reasonableness of the millisecond interval used in blasting will seriously afect rock's blasting quality and vibration efect. Terefore, determining the reasonable blasting delay between rows for blasting design is signifcant. Rock properties, hole distance, row spacing, site requirements, and other infuencing factors should be considered to determine the delay interval. If the blasting delay between rows is too long, the blasting efect is equivalent to that of the single-hole blasting funnel playing its role one by one, and sometimes it will even afect the blasting network. If the blasting delay between rows is too short, and the new free surface has not been formed when the behind row blast holes are detonated, the millisecond blasting efect cannot be achieved. Terefore, the reasonable blasting delay between rows should be when the previous Shock and Vibration 7 row of blast holes forms a free face for the next row of blast holes. Te resistance line of the model was fxed at 1.75 m, the hole distance was 2 m, and the other bench parameters and blasting parameters were consistent with the original model. Te blasting delay interval between four rows of blast holes was kept the same, and the blasting delay time between blast holes was changed. Respectively, the delay interval between 5 ms, 10 ms, 15 ms, 20 ms, 25 ms, and 30 ms was used to simulate the model. As shown in Figure 11, the infuence of diferent delay intervals on the muck pile parameters is explored by analyzing the muck pile parameters after blasting. When marking the height and throwing distance of the muck pile, the infuence of the occasional large rock on the height and throwing distance of the muck pile is not considered because the occasional large rock will signifcantly impact the original size of the muck pile. Te size marking of the muck pile is mainly based on the height and throwing distance of the overall shape of the muck pile. Table 4 shows the data statistics of the height of the muck pile and rock throwing distance under diferent delay intervals. Ten, according to the statistical data in Table 4, the point diagram of the size of the muck pile is drawn, as shown in Figure 12. By analyzing the data in the fgure, it can be seen that when the resistance line of the front row blast hole is 1.75 m, the height of the muck pile formed is between 2.8 m and 3.9 m and the throwing distance is between 11.9 m and 13.3 m. Te overall shape of the muck pile increases with the delay interval and gradually fattens out. Te following conclusions can be drawn by analyzing the size data of the muck pile under diferent delay intervals.
When the delay interval changes between 5 ms and 10 ms, the muck pile's height decreases with the delay interval's increase, which decreases by 26.03%. Te reason mainly is the delay interval being too small. After the behind row charge detonated, the front of the rock movement distance was not enough, failing to create a free face for the back rock and impeding the back row of the rock movement. Te loose degree of rock between the front and behind blast holes is poor due to excessive extrusion, which leads to increased blasting backlash and the height of the muck pile. When the delay interval is 10 ms, the rock movement in the front row creates a sure free face and rock movement space for the rock movement in the behind row, the hindrance between rocks weakens, and the height of the muck pile decreases.
When the time interval is from 10 ms to 20 ms, the height of the muck pile increases, and the main reason for this phenomenon is the delay interval becoming larger. Te movement of the front row rock for the behind row rock movement creates enough free face and space, but the retarding efect on rock movement in the behind row decreases gradually. Meanwhile, jostling each other between rock-broken degrees is abate because the rock mass is bigger, the loose degree is high, and the high of the muck pile begins to become bigger.
When the delay interval changes between 20 ms and 30 ms, the height of the muck pile does not change significantly and the maximum change range is only 2.42%. It can be approximated that the delay interval does not afect the height of the muck pile. As shown in Figure 12, when the delay interval reaches 30 ms, the rate of large rock mass in the muck pile increases signifcantly. Tis is because when the delay interval is too large, the collision between the front and back rocks is weakened, and the crushing efect of the rocks is reduced. So this is not conducive to shovel loading and transporting rocks after blasting. Terefore, under the model condition, a delay time of 30 ms is too large to be used.
Te rock throwing distance before the delay time increases to 20 ms has been increasing, but the throwing distance decreased after 20 ms. Analysis of the throwing distance's cause decreased, mainly when the delay interval was too big. Te front row of rocks stops motion when the throw is complete, and the front row of the rocks hampers the behind row of rock movement to here; thus, throwing distance begins to decrease.
Considering the infuence of the delay interval on the height of the muck pile and the throwing distance, when the ratio of the height of the muck pile H to the height of the bench h is controlled between 0.6 and 0.9, the height of the muck pile and the throwing distance of rock should be minimized as much as possible, to reduce the space occupied by the muck pile as much as possible. Te loading mechanism has a higher efciency at shoveling rock. When the delay interval is 25 ms, and the pile height H to bench height h is between 0.6 and 0.9, the blasting muck pile's height and rock throwing distance are relatively small. Terefore, the optimal blasting delay between rows for the bench model is 25 ms.

Conclusion
By comparing the diference between the simulated result and the result in the muck pile height, the model's rationality is verifed and the error value is in an acceptable range. It is feasible to simulate the open-pit bench blasting process with PFC. At the same time, a model is established to optimize the blasting efect by adjusting the resistance line and the delay between the rows.
(1) Adjusting the resistance line size found that the muck pile height is proportional to the resistance line, and the rock throwing distance is inversely proportional to the resistance line. Te best resistance line is 1.75 m, according to the efect of the muck pile. (2) Analyzing the relationship between the resistance line w, the rock throwing distance y, and the blasting muck pile's height z found that the relationship between the resistance line and the throwing distance and the height of the muck pile is nonlinear. Te ftting relationship between the resistance line, the rock throwing distance, and the muck pile height was obtained by curve ftting with the data points. (3) Changing the blasting delay between rows found that when the delay interval changes between 5 and 30 ms, the muck pile height decreases frst, then increases, and fnally becomes stable. When the delay interval increases to 20 ms, the throwing distance of the rock increases continuously; after 20 ms, the throwing distance decreases. Considering the efect of the delay interval on the height and throwing distance of the muck pile, the optimal blasting delay between rows in the bench blasting model is 25 ms.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.