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Design and optimization for bench blast based on Voronoi diagram
International Journal of Rock Mechanics & Mining Sciences 66 (2014) 30–40
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International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Design and optimization for bench blast based on Voronoi diagram Jun Liu a,n, Pinyu Sun b, Fangxue Liu c, Mingsheng Zhao d a
Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing, PR China Institute of Engineering Safety and Disaster Prevention, Hohai University, Nanjing, PR China c College of Mechanics and Materials, Hohai University, Nanjing, PR China d Guizhou Xinlian Blasting Engineering Limited Corp, Guizhou, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 15 March 2013 Received in revised form 5 September 2013 Accepted 20 November 2013 Available online 18 January 2014
A method is proposed for bench blasting in open pit mines that can automatically determine the blastholes positions, calculate the explosive charge mass, and identifies the initiation sequence in the case of the hole-by-hole initiation. An analytical solution of the blasthole and row spacing is first deduced based on the concept of the explosive charge maximization. Then for a specified blast area, the coordinates of the blastholes are determined by a cluster of reference polylines. A Voronoi diagram of the blast area is constructed by means of the reference points of the blasthole coordinates. Based on the Voronoi diagram, the explosive charge mass of a blasthole can be easily calculated by the volume formula of charge calculation. An algorithm that can identify the initiation sequence for the hole-by-hole blast is developed from the Voronoi diagram. A code for the bench blast design is developed in Cþ þ . The proposed method can automatically complete the whole bench blast design according to the blast area configuration and several parameters required by this method. The results of practical application show that the proposed method can greatly reduce the amount of the design work and validly improve the blast results. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Bench blast Voronoi diagram Hole-by-hole blast Free surface Blast design Initiation sequence
1. Introduction Bench blasting is the most common blasting technique in surface mines and quarries, and has been widely used in the fields such as civil, hydraulic, hydroelectric and transportation construction excavations. Therefore, an optimized bench blast design would be of significant benefit to the bench blast construction operators since the bench blast has been the most common rock blasting activity. By definition, bench blast is blasting in a vertical or subvertical blasthole or a row of blastholes towards a free vertical surface. The boreholes are distributed row by row. In the traditional bench blast, more than one row of blastholes can be blasted in the same round. A time delay in the detonation between the rows creates new free surfaces for each row. In general, the millisecond delay initiation of the explosive charge is widely used in bench blast. However, there exist some limitations and shortages in the millisecond blasting method such as there are not enough compensation spaces during a blast blasthole detonating, the size distribution of fragments in muck pile is non-uniform, the strongly ground vibration induced by blasting would occur because of the n Correspondence to: Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering 1 Xikang Road Nanjing, Jiangsu 210098 China. Tel.: þ 86 25 837 87172; fax: þ 86 25 837 13073. E-mail address: [email protected] (J. Liu).
1365-1609/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2013.11.012
large amount of explosive charge per delay, and tight bottom frequently appears in muckpile. Due to the varying nature of rock properties and geology as well as free surface conditions, reliable theoretic formulae are still unavailable at present and in most cases blast design is carried out by personal experience. As efforts to find more scientific and the reliable method to improve or optimize bench blast design, various research studies were initiated. These works can be divided into four types from different viewpoints: (1) the empirical formulae based on field measurement (EFFM) and simplified analytical equation (SAE); (2) numerical modeling; (3) prevention hazard of bench blast; (4) the artificial neural network (ANN) and computer-aided design (CAD) of bench blast. For the EFFM, the empirical relationships between design parameters and blast results were built by statistical analysis based on field measurements [1–3]. However, large amount of field test data are necessary in this method. Additionally, the methods have poor on-site applicability. For SAE, the simplified analytical equations between blasting conditions and blasting results were deduced based on simplification and assumptions of rock mass and explosives etc [4]. These equations can only reflect approximate relationships since some factors that influence the blasting results are neglected. In recent decades, the numerical modeling methods have been employed to predict results of bench blast such as the Finite element Method (FEM) [5–8], the Discrete Element Method (DEM)
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[9–11] and the Discontinuous Deformation Analysis (DDA) [12–15]. The blast design would be modified by the feedback from the simulated results. However, the optimum and selection of parameters of bench blast only aim at one or several factors instead of the complete blast design. Otherwise, the numerical method is difficult to apply in the routine production of bench blast since numerical modeling is a time consuming process and some parameters required by numerical modeling are difficult to determine. Therefore, the numerical method is difficult to be used to directly guide blast design, but it can well serve as a way of research. A number of researches have optimized the bench blast from the perspectives of the hazard prevention of bench blast such as control of blast induced vibration [16–20] and flyrock [21–25]. However, it is biased to optimize a bench blast design only from the angle of hazard prevention. An ideal design should be a compromise between the lowest hazards and the best blast results. Therefore, the factors, which may affect the bench blast results, should be comprehensively considered between the hazards and the results of bench blast. Artificial Neural Networks (ANN) is a type of intelligent tool to understand the complex problems. Once the network has been trained, with sufficient number of sample data sets such as the previous bench blast parameters and blast results, it can make predictions, on the basis of its previous learning, about the output related to new input data set of similar pattern [26–30]. Nevertheless, a large number of sample data sets with respect to bench blast are required to train the network. Additionally, the CAD technique has been developed to optimize bench blast design in recent decades [31]. In general, a computer-aided blast design system consists of two modules: one uses theoretical and empirical formulae and procedures to design a blast based on user supplied geological and mechanical data, while the other is an expert system that analyses some factors that influence on blast results and recommends remedial action using knowledge based rules [32–34]. The CAD method has the same limitation as that of ANN in terms of bench blast design. To overcome the limitations and the shortages of the millisecond blasting in which the blastholes are initiated row by row, the hole-by-holy initiation techniques were proposed as the emergence of Orica0 s high-precision delay detonator [35] and have been widely used in the bench blast. In the hole-by-holy initiation pattern, the blastholes in a blast area are initiated one by one, i.e., each blasthole is independently initiated according to a certain sequence of time and space. Thus, there exists a time delay between the blastholes, which can create more new free surfaces for the blastholes blasted subsequently. The advantages of the hole-by-hole blast include: the number of free surfaces for a blasthole would increase as its neighboring blastholes are gradually initiated; the boulder frequency is reduced since the rock mass is well fragmented; and the blast induced vibration is also significantly reduced. Nevertheless, the precise and accurate timing delays are required in the holy-by-hole initiation pattern and an automatic design method is also required when a large number of blastholes need to be initiated. The ShotPlus software [35] developed by Orica is a blast initiation sequencing program according to survey data detailing blasthole coordinates and identifiers, geological boundaries, pit design information. The ShotPlus allows the evaluation and optimization of blast design with Orica products. However, the ShotPlus cannot automatically implement the whole bench blast design, for example, the blasthole coordinates are input manually. In this work, a design method of hole-by-hole bench blast is developed based on the Voronoi diagram. This method can automatically determine the blastholes positions, the initiation sequence, explosive charge mass and other parameters required in
31
blasting construction. The detailed algorithm was designed and the corresponding code was also developed on the platform of VC þ þ. In the following sections, the determination of blastholes positions is first presented. Then the Voronoi diagram of blast area is introduced. Furthermore, the methods of determining explosive charge mass and the initiation sequence of the blastholes are illustrated. Finally, an example of practical application is introduced to demonstrate the performance of the proposed method.
2. Blasthole positions determination Determining the blasthole positions is the first step in the bench blast design. The distribution of blastholes was empirically determined in the bench blast design according to the blastability of rock, bench height and blasthole diameter. In this work, an adaptive scheme is proposed by which blasthole spacing (defined as the distance between two neighboring blastholes in a same row) and row spacing (defined as the distance between two neighboring rows) can be automatically calculated based on the known conditions such as the specific charge, the bench height, the blasthole depth and the blasthole diameter. Then a series of polylines, which serve to calculate the coordinates of the blasthole centers, are generated by moving the bench crest back towards to the inner blast area. Furthermore, the blasthole positions can be obtained by means of reference circles that are forwarded along the polylines at equal intervals (the blasthole spacing).
Fig. 1. The description method of the blast area configuration.
Fig. 2. The schematic of removing a segment whose direction vector is opposite to that of the corresponding segment in the previous polyline in the process of the reference polyline generation.
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Consequently, the blastholes can be located in blast area by site measurement in drilling construction. 2.1. Row and blasthole spacing determination In the millisecond blasting, the burdens of blastholes in the front row are the interval from the blastholes to the bench free surface (named front row burden) and the burdens of other blastholes are the row spacing (named burden) because the blastholes are initiated row by row. However, the burdens of blastholes are altered in hole-by-hole blasting since the blastholes are initiated one by one. Under the circumstances, the blastholes distribution pattern must be reconsidered with respect to millisecond blasting. The burden of a blasthole is the minimum distance from the blasthole to free faces that are created by its blasted neighboring blastholes. In fact, the identical burden for all blastholes is preferred in hole-by-hole blasting. Therefore, the
square distribution pattern, in which the row spacing is equal to the blasthole spacing, is employed in this work. To calculate blastholes0 positions in a specified blast area, row spacing or blasthole spacing (the following is called spacing) must be first determined. To determine the spacing, a scheme of explosive charge maximization for all blastholes is adopted. The explosive charge maximization scheme means all blastholes should be charged as much explosive as possible since the aim of drilling a blasthole is to contain explosive, which can save cost if a blasthole is charged enough explosive. However, the maximum explosive charge for a blasthole is limited to ensure the rock cast in the direction of the burden instead of the stemming. The relationship between the burden and the stemming can be expressed as
ls ¼ λa
Fig. 3. The example of generating the reference polylines in a blast area.
Fig. 4. The process illustration for generating blastoreholes in a blast area.
Fig. 5. The example of the blastoreholes distribution in a blast area by the proposed method.
ð1Þ
J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 30–40
where ls is the stemming length; a is the spacing; λ is an empirical coefficient, a ratio of the stemming length to the spacing. In general, λ Z 1, to ensure the correct cast direction. Thus, the maximum explosive charge Q max for a blasthole can be calculated as follows: 1 2 Q max ¼ π d ðh ls Þρe 4
ð2Þ
where d is the blasthole diameter; h is the blasthole depth; and ρe is the charge density. On the other hand, the explosive charge mass can also be calculated by the specific charge q, the bench height H and the spacing a, i.e., Q ¼ a2 Hq
ð3Þ
substituting Eq. (1) into Eq. (2), letting Eq. (2) be equal to Eq. (3), and then solving the equation, yields the spacing a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 d πρe ðλ π d ρe þ 16HqhÞ λπ d ρe a¼ ð4Þ 8Hq 2.2. Blast area description In this work, a blast area is described by free boundaries and infinite boundaries. The infinite boundary means a separate line
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between the consecutive blast areas since the bench blast is a gradual promotion process. The free surfaces of a blast area can be described by its bench crest and bench toe. Bench crest and bench toe of a blast area can be simplified as two polylines whose vertices coordinates are obtained by site measurement. An example of blast area configuration is shown in Fig. 1, where the boundaries represent the separate lines between two consecutive blasts, and the bench crest and bench toe represent bench slope, respectively. 2.3. Reference polylines generation To locate blastholes according to the spacing determined by Eq. (4), a cluster of polyines with uniform interval are generated by moving the bench crest to the inner blast area at equal intervals of the spacing. The normal and direction vectors of segments that compose the bench crest polylines are first calculated. Then each segment of the bench crest is moved at a spacing distance along the direction of the segment0 s normal vector that denotes to the inner blast area. All the moved segments are extended to ensure that a intersection is obtained between the neighboring segments or between the segments and the blast area polygon. Moreover, the intersection points between neighboring segments or between the segments and blast area polygon are calculated. Thus, a new polyline can be obtained by connecting these intersection points in
Fig. 6. The example of the Voronoi diagram of the blast area configuration (a) for the top bench; (b) for the bottom bench; and (c) for the combined top-bottom bench.
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turn. Then let the new polyline become a new bench crest and repeat the above process. The process would be stopped when any vertex on a newly generated polyline does not lie in the blast area polygon. It should be noted that the first movement of the bench crest should be at an interval of the front row burden ðBf Þ that are designated by user. Some segments in the polyline may become shorter than the corresponding segments in the initial bench crest polyline during the process of the polyline movement. The order of the two endpoints for a segment would be altered with the segment becoming shorter and shorter. As a result, the direction vectors of the segments may be opposite to those of the corresponding segments in the initial bench crest polyline, which would create some closed domains that cannot be used to generate blastholes and lead to disorder of the algorithm (see Fig. 2). To solve this problem, an extra algorithm is developed. A segment would be removed in the current polyline if its direction vector is opposite to that of the corresponding segment in the previous bench crest polyline. At the same time, the intersect point between the two neighboring segments that are connected with the removed segment is calculated and the two endpoints of the removed segment are replaced by the intersect point (see Fig. 2). The detailed algorithm of polylines generation is shown in Fig. A.1. Fig. 3 shows an example of polylines generation in the blast area shown in Fig. 1.
detailed algorithm of blastholes generation in a blast area is shown in Fig. A.2. Figs. 4 and 5 show an example of blastholes generation in the blast area shown in Fig. 1. To conveniently cite a blasthole coordinate in the following algorithm, an integer index is assigned to each blasthole, which is composed of a row number and a column number. The blastholes whose centers lie on the same polyline have the same row number. For the blastholes with the same row number, their column numbers are assigned according to the generation order.
3. Voronoi diagram construction In this work, a method is proposed to calculate the explosive charge mass and initiation sequence of each blasthole on the basis of Voronoi diagram of the blast area, which provide convenient and efficient way for bench blast design. A Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects (subsets) in the space. These objects are usually called the sites, the generators or the seeds and to each such object one associates a corresponding Voronoi cell, namely the set of all points in the given space whose distance to the given object is not greater than their distance to the other objects. Suppose P is a set of n distinct points in the plane, i.e., P ¼ fp1 ; p2 ; U U U ; pn g, if a point q lies in a cell containing pi , the Voronoi cell T i can be expressed by T i ¼ fq : dðq; pi Þ o dðq; pj Þj pi ; pj A P; pi a pj ; 1 r i; j r ng
2.4. Blasthole positions determination After the reference polylines are generated, the blasthole positions can be determined by virtue of moving a reference circle along each polyline respectively. A margin distance ðdl Þ between a blasthole and the boundary is left to protect overbreak since a closed blast area is considered in the blasthole generation algorithm proposed in this work. A direction convention for all reference polylines is appointed, i.e., from left to right or the opposite direction. Therefore, if the starting point of a reference polyline lies on the boundary of the blast area, a point on the reference polyline that is a margin distance away from the starting point of the reference polyline is selected as the center of the first reference circle. The radius of a reference circle is selected as the spacing. Then the two intersection points of the reference circle and the reference polyline are calculated. The intersection point which is at larger distance from the starting point of the reference polyline is selected as the center of the next reference circle. The centers of the reference circle are the consequent blasthole positions. The movement of a reference circle would be stopped if a newly generated blasthole lies on the extended line of the reference polyline, i.e., the center of the circle locates outside the blast area. The above process is repeated on all polylines. The
ð5Þ
where d denotes the Euclidean distance. In this work, the central points of blastholes are regarded as a set of distinct points P according to the definition of the Voronoi diagram. Then the open source software Qhull [36] was incorporated into our code to generate the Voronoi diagram. However, in the Voronoi diagram generated by Qhull, some Voronoi cells, Table 1 The vertices and data marks of the blast area polygon. Vertex no.
X (m)
Y (m)
Data mark
1 2 3 4 5 6 7 8 9 10 11 12
29607.8984 29584.834 29555.2773 29541.761 29527.4316 29511.2109 29488.5329 29470.5329 29507.4316 29545.2773 29584.834 29620.8984
50880.6412 50860.1043 50863.9835 50869.9982 50879.7167 50901.671 50913.3507 50893.3507 50859.7167 50840.9835 50838.1043 50864.6412
1 1 1 1 1 1 1 1 1 1 1 1
Fig. 7. The example of the initiation sequence determination.
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which are adjacent to the boundaries of the blast area, are not closed domain. To make each Voronoi cell be a closed polygon, the intersection points between the Voronoi cells and the blast area polygon are calculated. In addition, the central points of the blastholes are projected to the bottom plane of the bench along the axis direction of blasthole. Then the projection points are also submitted to create a Voronoi diagram. The contour of the blast area except the bench crest is also projected to the plane of bottom bench along the vertical direction. Then combine the projected contour with the bench toe as bottom contour of the blast area. The intersection points are also calculated between the bottom Voronoi cells that are adjacent to the projected boundaries and the bottom blast area polygon. The algorithm of Voronoi diagram generation is listed in Fig. A.3. As an example, the Voronoi diagram of the blast area configuration shown in Fig. 1 is drawn in Fig. 6. Fig. 6(a) shows the Voronoi diagram of the top bench and Fig. 6(b) for the bottom bench. To more clearly illustrate the Voronoi diagram of the top and bottom bench, a three dimensional figure, which combines the Voronoi diagrams of the top and bottom bench, is also shown in Fig. 6(c).
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explosive charge mass Q i of the ith blasthole can be calculated by Q i ¼ ρr HSi q
ð7Þ
where ρr is the density of the rock mass and H is the bench height. In some cases, the explosive charge mass of a blasthole in the first row calculated by Eq. (7) may be larger than that of other blastholes because of the larger areas of the bottom Voronoi cell, which will result in a wrong throw direction because the stemming height is less than minimum burden. For example, a wrong bench toe may be measured when a muckpile of last blast is remained, which may induce a larger bottom area of Voronoi cell that is adjacent to the bench toe. To solve this problem, the stemming height is increased to its burden and the explosive charge mass is recalculated by Eq. (8). 1 2 Q 0i ¼ π d ρe ðhi bi Þ 4
ð8Þ
where Q 0i is the modified explosive charge mass; hi is blasthole depth and hi ¼ H þ H o in which H o is over-drilling depth; bi is the burden of the ith blasthole. The algorithm of explosive charge calculation is shown in Fig. A.3. 4.2. Initiation sequence identification
4. Blasting parameters determination The Voronoi diagram is also employed to determine the initiation sequence of hole-by-hole blast. A Voronoi cell is a convex polygon and a side of a Voronoi cell is defined as a potential free
4.1. Explosive charge mass In the Voronoi diagram, each Voronoi cell exactly contains a blasthole. The media within a Voronoi cell is regarded as the destruction area due to the explosives denotation from the blasthole inside the Voronoi cell because the media is much closer to the blasthole than to other blastholes by the definition of the Voronoi diagram. Therefore, the explosive charge mass of a blasthole can be calculated using the volume formula, i.e., the product of the area of the Voronoi cell surrounding the blasthole, the bench height and the specific charge. Nevertheless, to obtain more accurate explosive charge mass of a blasthole, the average area of the top and the bottom Voronoi cell area is used in the volume formula, i.e., Si ¼ ðSt þ Sb Þ=2
ð6Þ
where Si is the average value of the top and bottom Voronoi cell corresponding to the ith blasthole; St and Sb are the areas of the top and bottom Voronoi cell of the ith blasthole. Thus, the
Table 2 The input parameters required by the proposed method. Input parameters
Value
Ratio of stemming length to spacing, λ Burden of the first row, Bf (m) Borehole diameter, d (m) Bench height, H (m) Over-drilling depth, Ho (m) Charge factor, q (kg/m3) Charge density, ρe (kg/m3) Rock density, ρr (kg/m3) Delay between rows, t r (ms) Delay between holes, t h (ms) Delay inside holes, t i (ms) Distance to the separate lines, dl (m)
1.25 4 0.16 12 2.5 0.46 1250 2700 42 25 400 1
Fig.8. The blast area configuration of an applied example.
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surface of a blasthole. The number of the potential free surface for a blasthole is the number of sides of its Voronoi cell. In other words, a blasthole has the same number of the potential free surface as the neighboring Voronoi cells since the two neighboring Voronoi cells share a common side. The free surface of a blasthole will increase as its neighboring blastholes are initiated. A blasthole would be initiated if the number of its free surfaces reach or exceed 2, i.e., its two neighboring blastholes are initiated. The complete initiation sequence can be obtained by checking the free surfaces of each blasthole at each delay time step. However, there may be multiple blastholes whose free faces are equal or greater than 2 at one moment. For this case, two principles are raised: (1) a blasthole whose free faces are greater than others is first initiated and (2) a blasthole whose row number is smaller than others is first initiated. Two schemes are provided for determining the first initiation blasthole, i.e., designated by user, or generated by default. For the latter, the first generated blasthole is regarded as the first initiation blasthole. The algorithm of determining initiation sequence is shown in Fig. A.4. However, the algorithm shown in Fig. A.4 can only be used to determine the relative initiation sequence instead of the exact initiation moment. In a practical bench blast, the delay time between blastholes is realized by detonator whose delay time is assigned beforehand by the manufacturer. Thus, an algorithm must be developed to meet the correct initiation sequence identified by the algorithm shown in Fig. A.4 according to the selected detonators. In general, three types of detonators with different firing time, which include the delay between rows (t r ), the delay between blastholes (t h ) and the delay inside blastholes (ti), can completely realize the hole-by-hole initiation for the arbitrary amount of blastholes. Fig. A.5 shows the algorithm of determining initiation moment conforming to the detonators with the designated firing times. Fig. 7 is the initiation sequence of the blastholes shown in Fig. 5 obtained by the proposed algorithm.
respectively. li ¼ 4Q i =π d ρe e
2
ð9Þ
li ¼ hi 4Q i =π d ρe s
2
ð10Þ
5. Applications and discussions To verify the performance of the design and optimization method proposed in this paper, a practical hole-by-hole bench blast in a quarry open pit is selected as an application case. The blast area mainly consists of dolomite and limestone that are used as solvent in steelmaking. The emulsion explosive is used in this case. For this purpose, the blasting fragment size is limited within 1 m. The vertices coordinates of blast area polygon, which are obtained by situ measurements, are listed in Table 1. The configuration is also shown in Fig. 8. A data mark is assigned to each vertex of the blast area polygon, which is used to designate whether a segment whose starting point is the vertex lies on the bench crest or not, if yes, mark¼ 1, else mark ¼ 1. The vertices of the blast area polygon are listed in a counter clockwise order. 5.1. Design process In this example, the detailed design flow using the proposed method is introduced. The input blasting parameters, which are required by the method, are first listed in Table 2. The blastholes distribution is shown in Fig. 9, the Voronoi diagram is hown in Fig. 10, and the final initiation sequence are sequentially shown in Fig. 11. There are 111 blastholes generated by the proposed algorithm in this blast area. Three types of detonators with different firing time are applied to realize the hole-by-hole initiation. Finally, a complete list of blasting parameters for all blastholes can be given by the proposed algorithm, which is convenient for blasting construction. The parameters of part blastholes (no. 51–60) are listed in Table 3.
4.3. Other blasting parameters
5.2. Burden dispersion
After the explosive charge mass of a blasthole is determined, e s the charging length li and stemming length li for the blasthole can be easily calculated in accordance with Eqs. (9) and (10),
In general, the mechanical properties and the blastability of rock mass in a blast area are basically the same. Therefore, a smaller variation of burdens for all blastholes is preferred in bench
Fig. 9. The blastoreholes distribution of the applied example.
J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 30–40
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Fig. 10. The Voronoi diagram of the applied example (a) for the top bench; (b) for the bottom bench; and (c) for the combined top-bottom bench.
blast design to achieve ideal distribution of fragments sizes. In the hole-by-hole initiation technique, a blasthole0 s burden would dynamically change with the variety of the initiation sequence of its neighboring blastholes. To demonstrate the performance of the proposed method, a determination method of burden is proposed to illustrate the burden dispersion as shown in Fig. 12. In Fig. 12, blasthole B and blasthole C would be blasted before blasthole A.
The dashed area would disappear due to the initiation of blasthole B and blasthole C. Then the burden of blasthole A is the distance from the center of blasthole A to the the common tangent line of blasthole B and blasthole C. To illustrate dispersion of the burdens, the standard deviation of all blastholes was calculated for the example introduced in Section 5.1, the value is 0.78. Furthermore, a relative deviation of a
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Fig. 11. The final initiation sequence of the applied example.
Table 3 The list of output parameters for 10 holes. Hole no.
Coordinate X (m)
Coordinate Y (m)
Hole depth (m)
Over-drilling (m)
Explosive charge (kg)
Charging depth (m)
Burden (m)
Stemming (m)
Initiation time (ms)
51 52 53 54 55 56 57 58 59 60
29608.3 29604.0 29599.7 29595.4 29591.1 29586.7 29581.0 29575.3 29569.6 29563.8
50875.6 50871.8 50868.0 50864.1 50860.3 50856.5 50856.6 50857.3 50858.1 50858.8
14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5
2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
186.343 183.051 184.902 221.584 221.584 221.584 206.134 189.655 172.215 159.080
7.41437 7.28336 7.35700 8.81654 8.81654 8.81654 8.20181 7.54611 6.85220 6.32960
5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346
7.08563 7.21664 7.14300 5.68346 5.68346 5.68346 6.29819 6.95389 7.64780 8.17040
1433 1475 1492 1534 534 576 618 660 702 744
burden from the mean of all burdens is also calculated according to Eq. (11).
si ¼ ðBi BÞ=B
ð11Þ
where si is the relative deviation from the mean for the ith blasthole; Bi is the burden of the ith blasthole; B is the mean of all burdens. The relative deviation of each blasthole for the example introduced in Section 5.1 is shown in Fig. 13. Fig. 13 shows that the relative deviations of the blastholes except for the front row blastholes and the blastholes adjacent to the boundary are almost same as the mean value. For the blastholes in the front row, the relative deviations are larger because their burdens are designated by user. As for the blastholes adjacent to the boundary, the larger relative deviations are mainly due to the complex geometrical shape of the blast area. 5.3. Ratio of spacing to burden
Fig. 12. The proposed method of burden determination.
For design purposes, it is common to consider the spacing in terms of the burden on a blasthole in the form of a ratio. For most situations the a/B (where B is burden) ratio0 s value will be from 1 to 2; although in certain instances where conditions are favorable, larger ratios have been used with satisfactory results [37]. In this work, blastholes fired independent one another will require a spacing ratio between 1 and 1.5, a 1.41 value being the ideal geometric balance for breakage of massive material [37]. For the example introduced in Section 5.1, the mean of burdens is used to calculate the ratio of spacing to burden since each blasthole has
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Fig. 13. The relative deviation of each blastorehole for the applied example.
different burden. The consequent ratio is 1.38 for this example, which approached to the ideal value 1.41 [37]. 5.4. Other issues For the example introduced above, the bulk ratio, which is a ratio of the mass of the fragments whose maximum sizes exceed 1 m to the amount mass of blasting muckpile, is approximately 1%. The sizes of the most fragments range from 10 cm to 90 cm and the distribution of the fragments sizes is basically reasonable. Meanwhile, the blast-induced vibration was also recorded in the form of velocity waveforms at the site that is 500 m away from the blast area. The peak particle velocity (PPV) of the waveform is also compared with that of a previous blast with the same scale at the approximately same measurement site. The result shows that the PPV reduces by 43% compared to that of the previous blast. The method proposed in this work can directly be applicable to inclined blastholes since the Voronoi diagram is constructed by projecting the coordinates of the blastholes on the top bench to the bottom bench along the axis directions of the blastholes and constructing the bottom Voronoi diagram according to the projected points. This operation is suitable to both vertical and inclined blastholes. In addition, the method can also be applicable to decked charges after a slight modification, i.e., the charge weight for each blasthole can be divided into multi-charge according to the request of user. After a slight modification, the proposed method can also be used to the other blasting technique such as the tunnel digging blasting.
6. Conclusions A method of bench blast design was proposed based on the Voronoi diagram. An algorithm, which can automatically calculate the coordinates of the blastholes, was developed. Then a Voronoi diagram of a blast area was constructed by taking the coordinates of blastholes as the reference points. Moreover, a cell in a Voronoi diagram is regarded as the destruction area induced by the blasthole inside the Voronoi cell. Thus, the explosive charge mass of a blasthole can be easily calculated according to the volume formula. Based on the Voronoi diagram, the initiation sequence of the hole-by-hole initiation can also be obtained by identifying the number of the free surfaces of a blasthole. By comparison with the existing methods, the features of the proposed method include the following: (1) The row and blasthole spacing can be determined by an analytical solution based on the concept of the explosive charge mass maximization.
(2) The blasthole location can be automatically determined according to the blast area configuration and the other parameters required by the algorithm. (3) The Voronoi diagram of blast area facilitates calculating the explosive charge mass and determining the initiation sequence in the case of hole-by-hole initiation. (4) A complete blast design can be automatically implemented by the proposed method and the design results are conveniently used in the blasting construction since the detailed information for each blasthole can be listed in the form of a table. The practical application results show that the burdens of all the blastholes approach to the uniform distribution; the ratio of spacing to burden is approximately the same as the empirical value; the blast induced vibration is validly reduced; the distribution of the fragments sizes is basically reasonable.
Acknowledgments This work was supported by the grants from the National Natural Science Foundation of China (No. 51174076) and the National Basic Research Program of China (973 Project) (No. 2007CB714104). These grants are gratefully acknowledged.
Appendix A. supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ijrmms.2013.11.012. References [1] Adhikari GR. Burden calculation for partially changed blast design conditions. Int J Rock Mech Min Sci 1999;36:253–6. [2] Adhikari GR. Selection of blasthole diameter for a given bench height at surface mines. Int J Rock Mech Min Sci 1999;36:843–7. [3] Yang HS, Rai P. Characterization of fragment size vis-à-vis delay timing in quarry blasts. Powder Technol 2011;211:120–6. [4] Yang RL, Kavetsky A, Mckenzie CK. A two-dimensional kinematic model for predicting muckpile shape in bench blasting. Int J Min Geol Eng 1989;7:209–26. [5] Liqing L, Katsabanis PD. A numerical study of the effects of accurate timing on rock fragmentation. Int J Rock Mech Min Sci 1997;34:817–35. [6] Jia Z, Chen G, Huang S. Computer simulation of open pit bench blasting in jointed rock mass. Int J Rock Mech Min Sci 1998;35:476. [7] Cho SH, Kaneko K. Rock fragmentation control in blasting. Mater Trans 2004;45:1722–30. [8] Zhu ZM. Numerical prediction of crater blasting and bench blasting. Int J Rock Mech Min Sci 2009;46:1088–96. [9] Preece DS. A numerical study of bench blast row delay timing and its influence on percent-cast. In: Proceedings of the eighth international conference of the international association for computer method and advances in geomechanics (IACMAG). West Virginia; 22–28 May 1994. p. 1–10.
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[10] Preece DS, Tidman JP, Chung SH. Expanded rock blast modeling capabilities of DMC-BLAST, including buffer blasting. In: Proceedings of the 13th annual symposium on explosives and blasing research. Las Vegas; 2–5 February 1997. p. 125–34. [11] Preece DS, Chung SH. Development and application of a 3-D rock blast computer modeling capability using discrete elements-DMCBLAST-3D. In: Proceedings of 27th Annual Conference on Explosive and Blasting Technique (I). Orlando; 28-31 January 2001. p. 11-8. [12] Mortazavi A, Katsabanis PD. Modelling burden size and strata dip effects on the surface blasting process. Int J Rock Mech Min Sci 2001;38:481–98. [13] Zhu CY, Dai C, Jiang QH. Numerical simulation of bench blasting by the discontinuous deformation analysis method (in Chinese). Chin J Rock Mech Eng 2002;21:2461–4. [14] Jing L, Stephansson O. Implicit discrete element method for block systemsdiscontinuous deformation analysis (DDA). In: Jing L, Stephansson O, editors. Dev Geotech Eng. Amsterdam: Elsevier; 2007. p. 317–64. [15] Ning YJ, Yang J, An XM, Ma GW. Modelling rock fracturing and blast-induced rock mass failure via advanced discretisation within the discontinuous deformation analysis framework. Comp Geotech 2011;38:40–9. [16] Afeni TB, Osasan SK. Assessment of noise and ground vibration induced during blasting operations in an open pit mine-A case study on Ewekoro limestone quarry, Nigeria. Min Sci Tech 2009;19:0420–4. [17] Amnieh HB, Mozdianfard MR, Siamaki A. Predicting of blasting vibrations in Sarcheshmeh copper mine by neural network. Safety Sci 2010;48:319–25. [18] Khandelwal M, Kankar PK, Harsha SP. Evaluation and prediction of blast induced ground vibration using support vector machine. Min Sci Tech 2010;20:0064–70. [19] Shi XZ, Chen SHR. Delay time optimization in blasting operations for mitigating the vibration-effects on final pit walls0 stability. Soil Dyn Earthq Eng 2011;31:1154–8. [20] Dehghani H, Ataee PM. Development of a model to predict peak particle velocity in a blasting operation. Int J Rock Mech Min Sci 2011;48:51–8. [21] Bajpayee TS, Rehak TR, Mowrey GL, Ingram DK. Blasting injuries in surface mining with emphasis on flyrock and blast area security. J Safety Res 2004;35:47–57.
[22] Kecojevic V, Radomsky M. Flyrock phenomena and area security in blastingrelated accidents. Safety Sci 2005;43:739–50. [23] Monjezi M, Bahrami A, Varjani AY. Simultaneous prediction of fragmentation and flyrock in blasting operation using artificial neural networks. Int J Rock Mech Min Sci 2010;47:476–80. [24] Rezaei M, Monjezi M, Varjani AY. Development of a fuzzy model to predict flyrock in surface mining. Safety Sci 2011;49:298–305. [25] Stojadinovic S, Pantovic R, Zikic M. Prediction of flyrock trajectories for forensic applications using ballistic flight equations. Int J Rock Mech Min Sci 2011;48:1086–94. [26] Singh TN, Singh V. An intelligent approach to prediction and control ground vibration in mines. Geotech Geol Eng 2005;23:249–62. [27] Iphar M, Yavuz M. Hakan Ak. Prediction of ground vibrations resulting from the blasting operations in an open pit mine by adaptive neuro-fuzzy inference system. Environ Geol 2008;56:97–107. [28] Kulatilake PHSW Qiong W, Hudaverdi T, Kuzu C. Mean particle size prediction in rock blast fragmentation using neural networks. Eng. Geol 2010;114:298–311. [29] Monjezi M, Rezaei M, Yazdian A. Prediction of backbreak in open-pit blasting using fuzzy set theory. Expert Syst Appl 2010;37:2637–43. [30] Verma AK, Singh TN. Intelligent systems for ground vibration measurement: a comparative study. Eng Comput 2011;27:225–33. [31] Qu SJ, Hao SH, Chen GP, Li BH, Bian GZ. The BLAST-CODE model-A computeraided bench blast design and simulation system. Int J Blast Frag 2002;6:85–103. [32] Khandelwal M. Application of an expert system for the assessment of blast vibration. Geotech Geol Eng 2012;30:205–17. [33] Smith ML, Hautala, RL. A coupled expert system approach to optimal blast design. In: Proceedings of the second international symposium on mine planning and equipment selection. Calgary; 7–9 November 1990. p. 481–7. [34] Cheimanoff N, Adda M, Duchene M, Puntous R. Expertir: an expert system for rock blasting planning in open-pit mines. In: Proceedings of the 14th world mining congress. Beijing; 14–18 May 1990. p. 799–804. [35] 〈www.oricaminingservices.com/〉. [36] 〈www.qhull.org/〉. [37] Kennedy BA. Surface mining. Maryland: Port City; 1990.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Design and optimization for bench blast based on Voronoi diagram Jun Liu a,n, Pinyu Sun b, Fangxue Liu c, Mingsheng Zhao d a
Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing, PR China Institute of Engineering Safety and Disaster Prevention, Hohai University, Nanjing, PR China c College of Mechanics and Materials, Hohai University, Nanjing, PR China d Guizhou Xinlian Blasting Engineering Limited Corp, Guizhou, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 15 March 2013 Received in revised form 5 September 2013 Accepted 20 November 2013 Available online 18 January 2014
A method is proposed for bench blasting in open pit mines that can automatically determine the blastholes positions, calculate the explosive charge mass, and identifies the initiation sequence in the case of the hole-by-hole initiation. An analytical solution of the blasthole and row spacing is first deduced based on the concept of the explosive charge maximization. Then for a specified blast area, the coordinates of the blastholes are determined by a cluster of reference polylines. A Voronoi diagram of the blast area is constructed by means of the reference points of the blasthole coordinates. Based on the Voronoi diagram, the explosive charge mass of a blasthole can be easily calculated by the volume formula of charge calculation. An algorithm that can identify the initiation sequence for the hole-by-hole blast is developed from the Voronoi diagram. A code for the bench blast design is developed in Cþ þ . The proposed method can automatically complete the whole bench blast design according to the blast area configuration and several parameters required by this method. The results of practical application show that the proposed method can greatly reduce the amount of the design work and validly improve the blast results. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Bench blast Voronoi diagram Hole-by-hole blast Free surface Blast design Initiation sequence
1. Introduction Bench blasting is the most common blasting technique in surface mines and quarries, and has been widely used in the fields such as civil, hydraulic, hydroelectric and transportation construction excavations. Therefore, an optimized bench blast design would be of significant benefit to the bench blast construction operators since the bench blast has been the most common rock blasting activity. By definition, bench blast is blasting in a vertical or subvertical blasthole or a row of blastholes towards a free vertical surface. The boreholes are distributed row by row. In the traditional bench blast, more than one row of blastholes can be blasted in the same round. A time delay in the detonation between the rows creates new free surfaces for each row. In general, the millisecond delay initiation of the explosive charge is widely used in bench blast. However, there exist some limitations and shortages in the millisecond blasting method such as there are not enough compensation spaces during a blast blasthole detonating, the size distribution of fragments in muck pile is non-uniform, the strongly ground vibration induced by blasting would occur because of the n Correspondence to: Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering 1 Xikang Road Nanjing, Jiangsu 210098 China. Tel.: þ 86 25 837 87172; fax: þ 86 25 837 13073. E-mail address: [email protected] (J. Liu).
1365-1609/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2013.11.012
large amount of explosive charge per delay, and tight bottom frequently appears in muckpile. Due to the varying nature of rock properties and geology as well as free surface conditions, reliable theoretic formulae are still unavailable at present and in most cases blast design is carried out by personal experience. As efforts to find more scientific and the reliable method to improve or optimize bench blast design, various research studies were initiated. These works can be divided into four types from different viewpoints: (1) the empirical formulae based on field measurement (EFFM) and simplified analytical equation (SAE); (2) numerical modeling; (3) prevention hazard of bench blast; (4) the artificial neural network (ANN) and computer-aided design (CAD) of bench blast. For the EFFM, the empirical relationships between design parameters and blast results were built by statistical analysis based on field measurements [1–3]. However, large amount of field test data are necessary in this method. Additionally, the methods have poor on-site applicability. For SAE, the simplified analytical equations between blasting conditions and blasting results were deduced based on simplification and assumptions of rock mass and explosives etc [4]. These equations can only reflect approximate relationships since some factors that influence the blasting results are neglected. In recent decades, the numerical modeling methods have been employed to predict results of bench blast such as the Finite element Method (FEM) [5–8], the Discrete Element Method (DEM)
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[9–11] and the Discontinuous Deformation Analysis (DDA) [12–15]. The blast design would be modified by the feedback from the simulated results. However, the optimum and selection of parameters of bench blast only aim at one or several factors instead of the complete blast design. Otherwise, the numerical method is difficult to apply in the routine production of bench blast since numerical modeling is a time consuming process and some parameters required by numerical modeling are difficult to determine. Therefore, the numerical method is difficult to be used to directly guide blast design, but it can well serve as a way of research. A number of researches have optimized the bench blast from the perspectives of the hazard prevention of bench blast such as control of blast induced vibration [16–20] and flyrock [21–25]. However, it is biased to optimize a bench blast design only from the angle of hazard prevention. An ideal design should be a compromise between the lowest hazards and the best blast results. Therefore, the factors, which may affect the bench blast results, should be comprehensively considered between the hazards and the results of bench blast. Artificial Neural Networks (ANN) is a type of intelligent tool to understand the complex problems. Once the network has been trained, with sufficient number of sample data sets such as the previous bench blast parameters and blast results, it can make predictions, on the basis of its previous learning, about the output related to new input data set of similar pattern [26–30]. Nevertheless, a large number of sample data sets with respect to bench blast are required to train the network. Additionally, the CAD technique has been developed to optimize bench blast design in recent decades [31]. In general, a computer-aided blast design system consists of two modules: one uses theoretical and empirical formulae and procedures to design a blast based on user supplied geological and mechanical data, while the other is an expert system that analyses some factors that influence on blast results and recommends remedial action using knowledge based rules [32–34]. The CAD method has the same limitation as that of ANN in terms of bench blast design. To overcome the limitations and the shortages of the millisecond blasting in which the blastholes are initiated row by row, the hole-by-holy initiation techniques were proposed as the emergence of Orica0 s high-precision delay detonator [35] and have been widely used in the bench blast. In the hole-by-holy initiation pattern, the blastholes in a blast area are initiated one by one, i.e., each blasthole is independently initiated according to a certain sequence of time and space. Thus, there exists a time delay between the blastholes, which can create more new free surfaces for the blastholes blasted subsequently. The advantages of the hole-by-hole blast include: the number of free surfaces for a blasthole would increase as its neighboring blastholes are gradually initiated; the boulder frequency is reduced since the rock mass is well fragmented; and the blast induced vibration is also significantly reduced. Nevertheless, the precise and accurate timing delays are required in the holy-by-hole initiation pattern and an automatic design method is also required when a large number of blastholes need to be initiated. The ShotPlus software [35] developed by Orica is a blast initiation sequencing program according to survey data detailing blasthole coordinates and identifiers, geological boundaries, pit design information. The ShotPlus allows the evaluation and optimization of blast design with Orica products. However, the ShotPlus cannot automatically implement the whole bench blast design, for example, the blasthole coordinates are input manually. In this work, a design method of hole-by-hole bench blast is developed based on the Voronoi diagram. This method can automatically determine the blastholes positions, the initiation sequence, explosive charge mass and other parameters required in
31
blasting construction. The detailed algorithm was designed and the corresponding code was also developed on the platform of VC þ þ. In the following sections, the determination of blastholes positions is first presented. Then the Voronoi diagram of blast area is introduced. Furthermore, the methods of determining explosive charge mass and the initiation sequence of the blastholes are illustrated. Finally, an example of practical application is introduced to demonstrate the performance of the proposed method.
2. Blasthole positions determination Determining the blasthole positions is the first step in the bench blast design. The distribution of blastholes was empirically determined in the bench blast design according to the blastability of rock, bench height and blasthole diameter. In this work, an adaptive scheme is proposed by which blasthole spacing (defined as the distance between two neighboring blastholes in a same row) and row spacing (defined as the distance between two neighboring rows) can be automatically calculated based on the known conditions such as the specific charge, the bench height, the blasthole depth and the blasthole diameter. Then a series of polylines, which serve to calculate the coordinates of the blasthole centers, are generated by moving the bench crest back towards to the inner blast area. Furthermore, the blasthole positions can be obtained by means of reference circles that are forwarded along the polylines at equal intervals (the blasthole spacing).
Fig. 1. The description method of the blast area configuration.
Fig. 2. The schematic of removing a segment whose direction vector is opposite to that of the corresponding segment in the previous polyline in the process of the reference polyline generation.
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Consequently, the blastholes can be located in blast area by site measurement in drilling construction. 2.1. Row and blasthole spacing determination In the millisecond blasting, the burdens of blastholes in the front row are the interval from the blastholes to the bench free surface (named front row burden) and the burdens of other blastholes are the row spacing (named burden) because the blastholes are initiated row by row. However, the burdens of blastholes are altered in hole-by-hole blasting since the blastholes are initiated one by one. Under the circumstances, the blastholes distribution pattern must be reconsidered with respect to millisecond blasting. The burden of a blasthole is the minimum distance from the blasthole to free faces that are created by its blasted neighboring blastholes. In fact, the identical burden for all blastholes is preferred in hole-by-hole blasting. Therefore, the
square distribution pattern, in which the row spacing is equal to the blasthole spacing, is employed in this work. To calculate blastholes0 positions in a specified blast area, row spacing or blasthole spacing (the following is called spacing) must be first determined. To determine the spacing, a scheme of explosive charge maximization for all blastholes is adopted. The explosive charge maximization scheme means all blastholes should be charged as much explosive as possible since the aim of drilling a blasthole is to contain explosive, which can save cost if a blasthole is charged enough explosive. However, the maximum explosive charge for a blasthole is limited to ensure the rock cast in the direction of the burden instead of the stemming. The relationship between the burden and the stemming can be expressed as
ls ¼ λa
Fig. 3. The example of generating the reference polylines in a blast area.
Fig. 4. The process illustration for generating blastoreholes in a blast area.
Fig. 5. The example of the blastoreholes distribution in a blast area by the proposed method.
ð1Þ
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where ls is the stemming length; a is the spacing; λ is an empirical coefficient, a ratio of the stemming length to the spacing. In general, λ Z 1, to ensure the correct cast direction. Thus, the maximum explosive charge Q max for a blasthole can be calculated as follows: 1 2 Q max ¼ π d ðh ls Þρe 4
ð2Þ
where d is the blasthole diameter; h is the blasthole depth; and ρe is the charge density. On the other hand, the explosive charge mass can also be calculated by the specific charge q, the bench height H and the spacing a, i.e., Q ¼ a2 Hq
ð3Þ
substituting Eq. (1) into Eq. (2), letting Eq. (2) be equal to Eq. (3), and then solving the equation, yields the spacing a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 d πρe ðλ π d ρe þ 16HqhÞ λπ d ρe a¼ ð4Þ 8Hq 2.2. Blast area description In this work, a blast area is described by free boundaries and infinite boundaries. The infinite boundary means a separate line
33
between the consecutive blast areas since the bench blast is a gradual promotion process. The free surfaces of a blast area can be described by its bench crest and bench toe. Bench crest and bench toe of a blast area can be simplified as two polylines whose vertices coordinates are obtained by site measurement. An example of blast area configuration is shown in Fig. 1, where the boundaries represent the separate lines between two consecutive blasts, and the bench crest and bench toe represent bench slope, respectively. 2.3. Reference polylines generation To locate blastholes according to the spacing determined by Eq. (4), a cluster of polyines with uniform interval are generated by moving the bench crest to the inner blast area at equal intervals of the spacing. The normal and direction vectors of segments that compose the bench crest polylines are first calculated. Then each segment of the bench crest is moved at a spacing distance along the direction of the segment0 s normal vector that denotes to the inner blast area. All the moved segments are extended to ensure that a intersection is obtained between the neighboring segments or between the segments and the blast area polygon. Moreover, the intersection points between neighboring segments or between the segments and blast area polygon are calculated. Thus, a new polyline can be obtained by connecting these intersection points in
Fig. 6. The example of the Voronoi diagram of the blast area configuration (a) for the top bench; (b) for the bottom bench; and (c) for the combined top-bottom bench.
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turn. Then let the new polyline become a new bench crest and repeat the above process. The process would be stopped when any vertex on a newly generated polyline does not lie in the blast area polygon. It should be noted that the first movement of the bench crest should be at an interval of the front row burden ðBf Þ that are designated by user. Some segments in the polyline may become shorter than the corresponding segments in the initial bench crest polyline during the process of the polyline movement. The order of the two endpoints for a segment would be altered with the segment becoming shorter and shorter. As a result, the direction vectors of the segments may be opposite to those of the corresponding segments in the initial bench crest polyline, which would create some closed domains that cannot be used to generate blastholes and lead to disorder of the algorithm (see Fig. 2). To solve this problem, an extra algorithm is developed. A segment would be removed in the current polyline if its direction vector is opposite to that of the corresponding segment in the previous bench crest polyline. At the same time, the intersect point between the two neighboring segments that are connected with the removed segment is calculated and the two endpoints of the removed segment are replaced by the intersect point (see Fig. 2). The detailed algorithm of polylines generation is shown in Fig. A.1. Fig. 3 shows an example of polylines generation in the blast area shown in Fig. 1.
detailed algorithm of blastholes generation in a blast area is shown in Fig. A.2. Figs. 4 and 5 show an example of blastholes generation in the blast area shown in Fig. 1. To conveniently cite a blasthole coordinate in the following algorithm, an integer index is assigned to each blasthole, which is composed of a row number and a column number. The blastholes whose centers lie on the same polyline have the same row number. For the blastholes with the same row number, their column numbers are assigned according to the generation order.
3. Voronoi diagram construction In this work, a method is proposed to calculate the explosive charge mass and initiation sequence of each blasthole on the basis of Voronoi diagram of the blast area, which provide convenient and efficient way for bench blast design. A Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects (subsets) in the space. These objects are usually called the sites, the generators or the seeds and to each such object one associates a corresponding Voronoi cell, namely the set of all points in the given space whose distance to the given object is not greater than their distance to the other objects. Suppose P is a set of n distinct points in the plane, i.e., P ¼ fp1 ; p2 ; U U U ; pn g, if a point q lies in a cell containing pi , the Voronoi cell T i can be expressed by T i ¼ fq : dðq; pi Þ o dðq; pj Þj pi ; pj A P; pi a pj ; 1 r i; j r ng
2.4. Blasthole positions determination After the reference polylines are generated, the blasthole positions can be determined by virtue of moving a reference circle along each polyline respectively. A margin distance ðdl Þ between a blasthole and the boundary is left to protect overbreak since a closed blast area is considered in the blasthole generation algorithm proposed in this work. A direction convention for all reference polylines is appointed, i.e., from left to right or the opposite direction. Therefore, if the starting point of a reference polyline lies on the boundary of the blast area, a point on the reference polyline that is a margin distance away from the starting point of the reference polyline is selected as the center of the first reference circle. The radius of a reference circle is selected as the spacing. Then the two intersection points of the reference circle and the reference polyline are calculated. The intersection point which is at larger distance from the starting point of the reference polyline is selected as the center of the next reference circle. The centers of the reference circle are the consequent blasthole positions. The movement of a reference circle would be stopped if a newly generated blasthole lies on the extended line of the reference polyline, i.e., the center of the circle locates outside the blast area. The above process is repeated on all polylines. The
ð5Þ
where d denotes the Euclidean distance. In this work, the central points of blastholes are regarded as a set of distinct points P according to the definition of the Voronoi diagram. Then the open source software Qhull [36] was incorporated into our code to generate the Voronoi diagram. However, in the Voronoi diagram generated by Qhull, some Voronoi cells, Table 1 The vertices and data marks of the blast area polygon. Vertex no.
X (m)
Y (m)
Data mark
1 2 3 4 5 6 7 8 9 10 11 12
29607.8984 29584.834 29555.2773 29541.761 29527.4316 29511.2109 29488.5329 29470.5329 29507.4316 29545.2773 29584.834 29620.8984
50880.6412 50860.1043 50863.9835 50869.9982 50879.7167 50901.671 50913.3507 50893.3507 50859.7167 50840.9835 50838.1043 50864.6412
1 1 1 1 1 1 1 1 1 1 1 1
Fig. 7. The example of the initiation sequence determination.
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which are adjacent to the boundaries of the blast area, are not closed domain. To make each Voronoi cell be a closed polygon, the intersection points between the Voronoi cells and the blast area polygon are calculated. In addition, the central points of the blastholes are projected to the bottom plane of the bench along the axis direction of blasthole. Then the projection points are also submitted to create a Voronoi diagram. The contour of the blast area except the bench crest is also projected to the plane of bottom bench along the vertical direction. Then combine the projected contour with the bench toe as bottom contour of the blast area. The intersection points are also calculated between the bottom Voronoi cells that are adjacent to the projected boundaries and the bottom blast area polygon. The algorithm of Voronoi diagram generation is listed in Fig. A.3. As an example, the Voronoi diagram of the blast area configuration shown in Fig. 1 is drawn in Fig. 6. Fig. 6(a) shows the Voronoi diagram of the top bench and Fig. 6(b) for the bottom bench. To more clearly illustrate the Voronoi diagram of the top and bottom bench, a three dimensional figure, which combines the Voronoi diagrams of the top and bottom bench, is also shown in Fig. 6(c).
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explosive charge mass Q i of the ith blasthole can be calculated by Q i ¼ ρr HSi q
ð7Þ
where ρr is the density of the rock mass and H is the bench height. In some cases, the explosive charge mass of a blasthole in the first row calculated by Eq. (7) may be larger than that of other blastholes because of the larger areas of the bottom Voronoi cell, which will result in a wrong throw direction because the stemming height is less than minimum burden. For example, a wrong bench toe may be measured when a muckpile of last blast is remained, which may induce a larger bottom area of Voronoi cell that is adjacent to the bench toe. To solve this problem, the stemming height is increased to its burden and the explosive charge mass is recalculated by Eq. (8). 1 2 Q 0i ¼ π d ρe ðhi bi Þ 4
ð8Þ
where Q 0i is the modified explosive charge mass; hi is blasthole depth and hi ¼ H þ H o in which H o is over-drilling depth; bi is the burden of the ith blasthole. The algorithm of explosive charge calculation is shown in Fig. A.3. 4.2. Initiation sequence identification
4. Blasting parameters determination The Voronoi diagram is also employed to determine the initiation sequence of hole-by-hole blast. A Voronoi cell is a convex polygon and a side of a Voronoi cell is defined as a potential free
4.1. Explosive charge mass In the Voronoi diagram, each Voronoi cell exactly contains a blasthole. The media within a Voronoi cell is regarded as the destruction area due to the explosives denotation from the blasthole inside the Voronoi cell because the media is much closer to the blasthole than to other blastholes by the definition of the Voronoi diagram. Therefore, the explosive charge mass of a blasthole can be calculated using the volume formula, i.e., the product of the area of the Voronoi cell surrounding the blasthole, the bench height and the specific charge. Nevertheless, to obtain more accurate explosive charge mass of a blasthole, the average area of the top and the bottom Voronoi cell area is used in the volume formula, i.e., Si ¼ ðSt þ Sb Þ=2
ð6Þ
where Si is the average value of the top and bottom Voronoi cell corresponding to the ith blasthole; St and Sb are the areas of the top and bottom Voronoi cell of the ith blasthole. Thus, the
Table 2 The input parameters required by the proposed method. Input parameters
Value
Ratio of stemming length to spacing, λ Burden of the first row, Bf (m) Borehole diameter, d (m) Bench height, H (m) Over-drilling depth, Ho (m) Charge factor, q (kg/m3) Charge density, ρe (kg/m3) Rock density, ρr (kg/m3) Delay between rows, t r (ms) Delay between holes, t h (ms) Delay inside holes, t i (ms) Distance to the separate lines, dl (m)
1.25 4 0.16 12 2.5 0.46 1250 2700 42 25 400 1
Fig.8. The blast area configuration of an applied example.
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surface of a blasthole. The number of the potential free surface for a blasthole is the number of sides of its Voronoi cell. In other words, a blasthole has the same number of the potential free surface as the neighboring Voronoi cells since the two neighboring Voronoi cells share a common side. The free surface of a blasthole will increase as its neighboring blastholes are initiated. A blasthole would be initiated if the number of its free surfaces reach or exceed 2, i.e., its two neighboring blastholes are initiated. The complete initiation sequence can be obtained by checking the free surfaces of each blasthole at each delay time step. However, there may be multiple blastholes whose free faces are equal or greater than 2 at one moment. For this case, two principles are raised: (1) a blasthole whose free faces are greater than others is first initiated and (2) a blasthole whose row number is smaller than others is first initiated. Two schemes are provided for determining the first initiation blasthole, i.e., designated by user, or generated by default. For the latter, the first generated blasthole is regarded as the first initiation blasthole. The algorithm of determining initiation sequence is shown in Fig. A.4. However, the algorithm shown in Fig. A.4 can only be used to determine the relative initiation sequence instead of the exact initiation moment. In a practical bench blast, the delay time between blastholes is realized by detonator whose delay time is assigned beforehand by the manufacturer. Thus, an algorithm must be developed to meet the correct initiation sequence identified by the algorithm shown in Fig. A.4 according to the selected detonators. In general, three types of detonators with different firing time, which include the delay between rows (t r ), the delay between blastholes (t h ) and the delay inside blastholes (ti), can completely realize the hole-by-hole initiation for the arbitrary amount of blastholes. Fig. A.5 shows the algorithm of determining initiation moment conforming to the detonators with the designated firing times. Fig. 7 is the initiation sequence of the blastholes shown in Fig. 5 obtained by the proposed algorithm.
respectively. li ¼ 4Q i =π d ρe e
2
ð9Þ
li ¼ hi 4Q i =π d ρe s
2
ð10Þ
5. Applications and discussions To verify the performance of the design and optimization method proposed in this paper, a practical hole-by-hole bench blast in a quarry open pit is selected as an application case. The blast area mainly consists of dolomite and limestone that are used as solvent in steelmaking. The emulsion explosive is used in this case. For this purpose, the blasting fragment size is limited within 1 m. The vertices coordinates of blast area polygon, which are obtained by situ measurements, are listed in Table 1. The configuration is also shown in Fig. 8. A data mark is assigned to each vertex of the blast area polygon, which is used to designate whether a segment whose starting point is the vertex lies on the bench crest or not, if yes, mark¼ 1, else mark ¼ 1. The vertices of the blast area polygon are listed in a counter clockwise order. 5.1. Design process In this example, the detailed design flow using the proposed method is introduced. The input blasting parameters, which are required by the method, are first listed in Table 2. The blastholes distribution is shown in Fig. 9, the Voronoi diagram is hown in Fig. 10, and the final initiation sequence are sequentially shown in Fig. 11. There are 111 blastholes generated by the proposed algorithm in this blast area. Three types of detonators with different firing time are applied to realize the hole-by-hole initiation. Finally, a complete list of blasting parameters for all blastholes can be given by the proposed algorithm, which is convenient for blasting construction. The parameters of part blastholes (no. 51–60) are listed in Table 3.
4.3. Other blasting parameters
5.2. Burden dispersion
After the explosive charge mass of a blasthole is determined, e s the charging length li and stemming length li for the blasthole can be easily calculated in accordance with Eqs. (9) and (10),
In general, the mechanical properties and the blastability of rock mass in a blast area are basically the same. Therefore, a smaller variation of burdens for all blastholes is preferred in bench
Fig. 9. The blastoreholes distribution of the applied example.
J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 30–40
37
Fig. 10. The Voronoi diagram of the applied example (a) for the top bench; (b) for the bottom bench; and (c) for the combined top-bottom bench.
blast design to achieve ideal distribution of fragments sizes. In the hole-by-hole initiation technique, a blasthole0 s burden would dynamically change with the variety of the initiation sequence of its neighboring blastholes. To demonstrate the performance of the proposed method, a determination method of burden is proposed to illustrate the burden dispersion as shown in Fig. 12. In Fig. 12, blasthole B and blasthole C would be blasted before blasthole A.
The dashed area would disappear due to the initiation of blasthole B and blasthole C. Then the burden of blasthole A is the distance from the center of blasthole A to the the common tangent line of blasthole B and blasthole C. To illustrate dispersion of the burdens, the standard deviation of all blastholes was calculated for the example introduced in Section 5.1, the value is 0.78. Furthermore, a relative deviation of a
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Fig. 11. The final initiation sequence of the applied example.
Table 3 The list of output parameters for 10 holes. Hole no.
Coordinate X (m)
Coordinate Y (m)
Hole depth (m)
Over-drilling (m)
Explosive charge (kg)
Charging depth (m)
Burden (m)
Stemming (m)
Initiation time (ms)
51 52 53 54 55 56 57 58 59 60
29608.3 29604.0 29599.7 29595.4 29591.1 29586.7 29581.0 29575.3 29569.6 29563.8
50875.6 50871.8 50868.0 50864.1 50860.3 50856.5 50856.6 50857.3 50858.1 50858.8
14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5 14.5
2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
186.343 183.051 184.902 221.584 221.584 221.584 206.134 189.655 172.215 159.080
7.41437 7.28336 7.35700 8.81654 8.81654 8.81654 8.20181 7.54611 6.85220 6.32960
5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346 5.68346
7.08563 7.21664 7.14300 5.68346 5.68346 5.68346 6.29819 6.95389 7.64780 8.17040
1433 1475 1492 1534 534 576 618 660 702 744
burden from the mean of all burdens is also calculated according to Eq. (11).
si ¼ ðBi BÞ=B
ð11Þ
where si is the relative deviation from the mean for the ith blasthole; Bi is the burden of the ith blasthole; B is the mean of all burdens. The relative deviation of each blasthole for the example introduced in Section 5.1 is shown in Fig. 13. Fig. 13 shows that the relative deviations of the blastholes except for the front row blastholes and the blastholes adjacent to the boundary are almost same as the mean value. For the blastholes in the front row, the relative deviations are larger because their burdens are designated by user. As for the blastholes adjacent to the boundary, the larger relative deviations are mainly due to the complex geometrical shape of the blast area. 5.3. Ratio of spacing to burden
Fig. 12. The proposed method of burden determination.
For design purposes, it is common to consider the spacing in terms of the burden on a blasthole in the form of a ratio. For most situations the a/B (where B is burden) ratio0 s value will be from 1 to 2; although in certain instances where conditions are favorable, larger ratios have been used with satisfactory results [37]. In this work, blastholes fired independent one another will require a spacing ratio between 1 and 1.5, a 1.41 value being the ideal geometric balance for breakage of massive material [37]. For the example introduced in Section 5.1, the mean of burdens is used to calculate the ratio of spacing to burden since each blasthole has
J. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 30–40
39
Fig. 13. The relative deviation of each blastorehole for the applied example.
different burden. The consequent ratio is 1.38 for this example, which approached to the ideal value 1.41 [37]. 5.4. Other issues For the example introduced above, the bulk ratio, which is a ratio of the mass of the fragments whose maximum sizes exceed 1 m to the amount mass of blasting muckpile, is approximately 1%. The sizes of the most fragments range from 10 cm to 90 cm and the distribution of the fragments sizes is basically reasonable. Meanwhile, the blast-induced vibration was also recorded in the form of velocity waveforms at the site that is 500 m away from the blast area. The peak particle velocity (PPV) of the waveform is also compared with that of a previous blast with the same scale at the approximately same measurement site. The result shows that the PPV reduces by 43% compared to that of the previous blast. The method proposed in this work can directly be applicable to inclined blastholes since the Voronoi diagram is constructed by projecting the coordinates of the blastholes on the top bench to the bottom bench along the axis directions of the blastholes and constructing the bottom Voronoi diagram according to the projected points. This operation is suitable to both vertical and inclined blastholes. In addition, the method can also be applicable to decked charges after a slight modification, i.e., the charge weight for each blasthole can be divided into multi-charge according to the request of user. After a slight modification, the proposed method can also be used to the other blasting technique such as the tunnel digging blasting.
6. Conclusions A method of bench blast design was proposed based on the Voronoi diagram. An algorithm, which can automatically calculate the coordinates of the blastholes, was developed. Then a Voronoi diagram of a blast area was constructed by taking the coordinates of blastholes as the reference points. Moreover, a cell in a Voronoi diagram is regarded as the destruction area induced by the blasthole inside the Voronoi cell. Thus, the explosive charge mass of a blasthole can be easily calculated according to the volume formula. Based on the Voronoi diagram, the initiation sequence of the hole-by-hole initiation can also be obtained by identifying the number of the free surfaces of a blasthole. By comparison with the existing methods, the features of the proposed method include the following: (1) The row and blasthole spacing can be determined by an analytical solution based on the concept of the explosive charge mass maximization.
(2) The blasthole location can be automatically determined according to the blast area configuration and the other parameters required by the algorithm. (3) The Voronoi diagram of blast area facilitates calculating the explosive charge mass and determining the initiation sequence in the case of hole-by-hole initiation. (4) A complete blast design can be automatically implemented by the proposed method and the design results are conveniently used in the blasting construction since the detailed information for each blasthole can be listed in the form of a table. The practical application results show that the burdens of all the blastholes approach to the uniform distribution; the ratio of spacing to burden is approximately the same as the empirical value; the blast induced vibration is validly reduced; the distribution of the fragments sizes is basically reasonable.
Acknowledgments This work was supported by the grants from the National Natural Science Foundation of China (No. 51174076) and the National Basic Research Program of China (973 Project) (No. 2007CB714104). These grants are gratefully acknowledged.
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