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Computers & Geosciences 32 (2006) 352–359 www.elsevier.com/locate/cageo
Monte Carlo simulation as a tool to predict blasting fragmentation based on the Kuz–Ram model$ Mario A. Morina,, Francesco Ficarazzob a
Department of Mining Engineering, University of British Columbia, Vancouver, BC, Canada b Dipartimento di Georisorse e Territorio, Politecnico di Torino, Italy
Received 26 November 2004; received in revised form 19 June 2005; accepted 20 June 2005
Abstract Rock fragmentation is considered the most important aspect of production blasting because of its direct effects on the costs of drilling and blasting and on the economics of the subsequent operations of loading, hauling and crushing. Over the past three decades, significant progress has been made in the development of new technologies for blasting applications. These technologies include increasingly sophisticated computer models for blast design and blast performance prediction. Rock fragmentation depends on many variables such as rock mass properties, site geology, in situ fracturing and blasting parameters and as such has no complete theoretical solution for its prediction. However, empirical models for the estimation of size distribution of rock fragments have been developed. In this study, a blast fragmentation Monte Carlo-based simulator, based on the Kuz–Ram fragmentation model, has been developed to predict the entire fragmentation size distribution, taking into account intact and joints rock properties, the type and properties of explosives and the drilling pattern. Results produced by this simulator were quite favorable when compared with real fragmentation data obtained from a blast quarry. It is anticipated that the use of Monte Carlo simulation will increase our understanding of the effects of rock mass and explosive properties on the rock fragmentation by blasting, as well as increase our confidence in these empirical models. This understanding will translate into improvements in blasting operations, its corresponding costs and the overall economics of open pit mines and rock quarries. r 2005 Elsevier Ltd. All rights reserved. Keywords: Monte Carlo simulation; Blasting; Fragmentation prediction; Kuz–Ram model
1. Introduction to rock fragmentation
$ Code available from server at http://www.iamg.org/CGEditor/index.htm. Corresponding author at: Room 517-6350, Mining Engineering, University of British Columbia, Stores Road-UBC, Vancouver, Canada V6T 1Z4. Tel.: +1 6048275089; fax: +1 6048225599. E-mail address:
[email protected] (M.A. Morin).
Some 30 years ago, Mackenzie (1966, 1967) presented his now classic conceptual curves showing the cost dependence of the different mining unit operations on the degree of fragmentation at the Quebec-Cartier iron ore open pit mine, namely:
Drilling Blasting Loading
0098-3004/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2005.06.022
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Hauling Crushing Mackenzie’s objective was to determine the cost curves based on the mean fragmentation size. MacKenzie showed that loading, hauling and crushing costs decreased with increasing rock fragmentation while drilling and blasting costs increased with increasing rock fragmentation. As shown in Fig. 1, the summation
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of costs resulted in an inverted cost curve with a minimum fragmentation cost and thus optimum fragmentation size. This concept of optimum fragmentation is critical for optimizing a drilling and blasting program that minimizes the entire cost for a mining operation (Hustrulid, 1999; Kanchibotla, 2001; La Rosa, 2001; JKTech Pty, 2004). Mechanical crushing and grinding are particularly expensive operations at a mine and considerable cost and throughput benefits can be obtained by breaking the rock using explosives effectively instead (Eloranta, 1997; Simangunsong, et al., 2003; Katsabanis, et al., 2004). Being able to predict the fragmentation of a rock mass broken up by blasting offers significant advantages in providing muck with a desirable size distribution and specifications for aggregate purposes or mill feed. It is possible to design a blast to provide riprap (an erosion protection measure) to line up dams and channels that meet engineered specifications. Knowing the size distribution for a particular blast and rock mass conditions, the contractor can adapt the blasting if possible or take into account the material that will not meet the required specifications into the project bid. Such knowledge can also be used for sizing and selecting crushers and conveyor systems. An undersize material-handling system will be a bottleneck; an oversize system will be wasteful and underutilized.
2. The Kuz–Ram fragmentation model The Kuz–Ram model is an empirical fragmentation model (Cunningham, 1983, 1987; Lizotte, 1990) based on the Kuznetsov and Rosin–Rammler equations as well as an algorithm, developed by Cunningham, which derives the coefficient of uniformity in the Rosin–Rammler equation from blasting parameters. The model predicts fragmentation from blasting in terms of mass percent passing a given mesh size:
Smaller fragmentation occurs with higher explosive
energy input, weaker rock types and smaller blasthole diameters. More regular fragmentation sizing results from the uniform distribution of explosives in the rock mass, smaller burdens and greater spacing/burden ratios.
The problem areas in any specifically predictive blasting model are typically:
definition of relevant rock mass properties, selection of appropriate explosives performance Fig. 1. Effect of degree of fragmentation on individual unit operation costs and overall cost (after Mackenzie, 1967).
indices,
determination of actual blast fragmentation.
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This work is meant to particularly address the issue of variation and uncertainty in rock mass properties through the use of Monte Carlo simulation. Because rock mass conditions vary across a blast, the resulting fragmentation should vary as well. 2.1. The Kuznetsov equation Kuznetsov (1973) formulated a semi-empirical equation based on field investigations and a review of previous published data that related the mean fragment size to the mass of explosive, the volume blasted and the rock strength. The Kuznetsov equation, given below, relates the mean fragment size and the applied blast energy per unit volume of rock (i.e. referred to as the powder factor) as a function of rock type. 0:8 V0 1=6 Xm ¼ A QT , (1) QT where Xm is the the mean fragment size (cm), A the rock factor, V0 the rock volume (m3) broken per blasthole ¼ burden spacing bench height, QT the mass (kg) of TNT containing the energy equivalent of the explosive charge in each blasthole. Cunningham (1983) shows how the basic equation can be modified to treat various types of explosives relative to the performance of ANFO (ammonium nitrate—fuel oil, the most common bulk mining explosive) mixtures with the use of the following equation: SANFO , (2) 115 where Qe is the mass of explosive being used (kg), SANFO the relative weight strength of the explosive relative to ANFO. The equation can also be stated as a function of the powder factor or specific charge K (kg of explosives/m3 of rock) using
QT ¼ Qe
V0 1 ¼ . Qe K
Mineral processors are generally familiar with this equation, expressed as n
R ¼ eðX =X c Þ ,
(5)
where R is the mass fraction larger than size X, X the diameter of fragment (cm), Xc the characteristic size (cm), n the Rosin–Rammler exponent, and e the base of natural logarithms, 2.7183. The characteristic size, Xc, is approximately the 36.8% size retainment point on the size distribution function. The Rosin–Rammler exponent, n, is known as the uniformity coefficient. A wide variety of size distributions can be modeled with the Rosin–Rammler equation by simply changing the value of n to fit the curve. Cunningham (1987) notes that the uniformity coefficient n usually varies between 0.8 and 1.5. Since the Kuznetsov formula gives the screen size Xm for which 50% of the material would pass, substituting X ¼ X m and R ¼ 0.5 (see Fig. 2) into Eq. (6) one finds that Xc ¼
Xm ð0:693Þ1=n
.
(6)
Given that the Kuznetsov equation accounts for explosive strength and rock mass characteristics, and that the mean size is related to the characteristic size of the Rosin–Rammler distribution, the only unknown left is the uniformity coefficient. Cunningham established the applicable uniformity coefficient through several investigations, taking into consideration the impact of such factors as: blast geometry, hole diameter, burden, spacing, hole lengths and drilling accuracy. The exponent n for the Rosin–Rammler equation is estimated as follows: B 1 þ S=B 0:5 W L n ¼ 2:2 14 , (7) 1 D 2 B H
(3)
Eqs. (2) and (3) can be rewritten to calculate the mean fragmentation size Xm for a given powder factor as 115 19=30 . (4) X m ¼ AðK 0:8 ÞQe1=6 S ANFO Cunningham (1983, 1987) and later Lilly (1986) provide methodologies for evaluating the rock factor A based on the geomechanical properties of the rock mass to be blasted, typically in the range of 8–12. 2.2. The Rosin–Rammler equation The Rosin–Rammler equation is used to characterize the partial-size distribution of a material for use in a variety of applications (Rosin and Rammler, 1933).
Fig. 2. Typical fragmentation curve showing per cent retained as function of screen opening.
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where B is the blasting burden (m), S the blasthole spacing (m), D the blasthole diameter (mm), W the standard deviation of drilling accuracy (m), L the total charge length (m), and H the bench height (m). The above parameters are illustrated in Fig. 3. It is normally desired to have uniform fragmentation; so high values of n are preferred. Experience by Cunningham (1987) has suggested that the normal range of n for blasting fragmentation in reasonably competent ground is from 0.75 to 1.5, with the average being around 1.0. More competent rocks have higher values. The modified Kuznetsov Eq. (4), the Rosin–Rammler Eq. (5) and the estimate of the Rosin–Rammler exponent forms the basic of the Kuz–Ram formulation for blast fragmentation prediction model. The Kuz–Ram model can be applied in a variety of ways depending on the design objective. If it is possible to vary the blast design to achieve a constant mean fragmentation size (Xm), or the powder factor (K) can be held constant, thus predicting the resulting size distribution.
3. Monte Carlo-based simulation and fragmentation prediction First coined by Metropolis and Ulam (1949), Monte Carlo-based simulation methods have gained the status of a full-fledged numerical method capable of addressing complex problems. Monte Carlo simulation can be loosely described as a simulation method where the simulation results are based on a model where the input values are selected at random from representative statistical distribution functions that describe those inputs. The simulation is repeated n-times and the results themselves now described a statistical distribu-
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tion (Sobol et al., 1994; Fishman, 1996). Monte Carlo simulation methods are primarily used in situations where there is uncertainty in our inputs and where the calculated uncertainty of results accurately reflects the uncertainty of the input data. It is generally recognized that natural materials like rock tend to show considerable variety of properties. Rock strength, fracture spacing and orientation within a given rock mass can and do vary. Drilling itself can introduce variability with deviations in drill hole spacing, burden and alignment. The end result of such a variation is that the resulting fragmentation size predicted by the Kuz–Ram model will also show variability. This observation is particularly important if blasting is meant to achieve a specific purpose other that breaking up the rock mass. For example, the width of conveyor systems is typically dimensioned using the typical fragment size to be moved. If rock oversize is encountered more frequently than expected, the conveyor system will not perform as expected. Another example involves the sizing of the throat for a rock crusher. If the blasted material is coarser than expected, the crusher will be undersized, if the material is finer than expected, the crusher will be underutilized. One last example involves the production of riprap, a rock-based erosion control measure used in dam and water channel construction. A contractor wishing to blast riprap that must meet design specifications might find that the rock mass is incapable of producing such material, or that crushing and screening might be required after blasting. The extra handling will be costly and should be factored in the contractual bid. Monte Carlo-based simulation using the Kuz–Ram model can provide insight into all of these problems and help the engineer create a suitable blast design to meet a required goal.
4. Blasting principles for pits and quarries Blasting is as much an art as it is a science. Although much work has been done in the area of modeling fracture development and stress wave propagation in an isotropic/anisotropic medium, the natural variability of the rock mass often precludes such complex approaches to blast design and therefore considerable blasting design is based on relatively simple rules-of-thumb. Blast design must consider the following parameters (Ash, 1968; Jimeno et al., 1995; Hustrulid, 1999):
Fig. 3. Blasthole layout terminology.
Blasthole diameter D (m) Bench height H (m) Blasthole burden B (m) Blasthole spacing S (m)
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Blasthole subdrilling J (m) Blasthole stemming length T (m) Blasthole deviation and alignment Blasthole pattern (staggered or rectangular) Rock mass properties and discontinuities Explosive properties The most critical and important dimension in blasting is that of the burden B as it represents the rock mass to be fragmented by the explosive column. Its actual value will depend on a combination of variables including the rock characteristics, the explosive used, etc. A convenient guide used for estimating the burden is the KB ratio (KB is burden/diameter). Experience shows that when K B ¼ 30 (typical range 20–40), the blaster can usually expect satisfactory results for average field condition. The distance between adjacent blastholes, measured perpendicular to the burden, is defined as the spacing S. Ideal energy balancing between explosive charges is usually accomplished when the spacing dimension is nearly equal to double that of the burden (K S ¼ 2) when the charges are initiated simultaneously. For most conditions, the required subdrilling (J) should be 0.3 times the burden dimension yielding KJ or the subdrill to burden ratio (Ash, 1963). Stemming is the portion of blasthole that has been packed with inert material above the charge so as to confine and retain the gases produced by the explosion, thus improving the fragmentation process. Field experience shows that a KT (stemming/burden ratio) of 0.7 is a reasonable starting point. It has been found that fragmentation is significantly affected by local geological conditions. The strike and dip direction of joint sets and their frequency in the rock mass are of great importance, since the stress waves produced by the detonation of the explosive charges will be reflected at the joint surfaces. Rock mass discontinuities that are perpendicular to the blasthole axis have little effect on fragmentation. However, if they are parallel to the axis of the borehole, energy is wasted in excessive crushing in the area close to the borehole while little work is done away from the blasthole. The direction of the blast with respect to the structural conditions is of paramount importance in practice.
of future enhancements and integration into other applications. The Monte Carlo simulation model requires the following seven parameters that have a range of possible values:
Unconfined compressive strength of the intact rock Elastic modulus of the intact rock Joint’s dip Joint’s dip direction Joint’s spacing Drilling accuracy Bench’s face dip direction To describe the behaviors, triangular distributions were utilized for all parameters except the joint spacing, where both triangular and exponential distributions were used. The UCS parameter will be modeled to serve as a general example of the approach taken. A similar approach can be applied to all remaining parameters. The unconfined compressive strength value (UCS) is typically determined by an unconfined compression test where a cylindrical core sample is loaded axially to failure, with no confinement (lateral support). Conceptually, the peak value of the axial stress is taken as the unconfined compressive strength of the sample. In view of the variability of rock properties, when adequate samples are available, repeated testing may be warranted to determine average values. Unfortunately, we rarely have a unique value but a range. Fig. 4 shows a histogram for a set of nine unconfined compressive strength results for the same rock material. The histogram shows an approximate triangular distribution with the following characteristics:
Most Likely ¼ 405 with 0.238 probability Lower Bound ¼ 375 with 0.095 probability Upper Bound ¼ 455 with 0.047 probability
5. Development of the Monte Carlo simulator using Microsoft’s Visual Basic.Net Microsoft’s Visual Basic.Net offers a strong software platform for developing a Monte Carlo-based simulator. Although, there exists other dedicated commercial applications for performing Monte Carlo-based simulation, Visual Basic offers the capability to create a stand-alone program as well as the possibility
Fig. 4. UCS frequency histogram.
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A linear relationship is assumed between lower bound and most likely value as well as upper bound and most likely value. Also, the cumulative frequency (from zero to 1) can be calculated. Using a random-number generator with a range set between zero and 1, it is possible to work backwards and calculate UCS values for substitution into the Monte Carlo simulator and the analysis of one scenario. To guarantee the maximum randomness of this ‘‘scenario’’ and the independence between the parameters, seven values were generated from the seven triangular distributions, producing seven different random numbers. The Kuz–Ram model produces three parameters for defining the degree of fragmentation and the relative percentage for each size:
Oversize as the upper bound of the fragmentation; its
percentage will represent the coarser part after blasting of the fragmentation curve. Optimum-size as the most likely size; it is the size through which will pass value to the mineral processing chain (loading, crushing, hauling) Undersize as the lower bound size and it is the finer part of the fragmentation curve.
6. Application of simulator to a case study The simulation application has been designed to answer two types of problems regarding the application of the Kuz–Ram model:
Calculate
the powder factor required to obtain a certain mean fragment size Calculate fragment size distribution while the powder factor is held constant
As described above, the first approach is referred to as the ‘‘constant mean fragmentation’’ and the second one as the ‘‘constant powder factor’’. To validate the model, the simulator was compared to data obtained from a quarry, the Costiolo open-pit in
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Italy. Costiolo is an open pit mine located near Bergamo (Milano). The Ghisalberti Lime S.p.a. Company has a concession to mine Triassic limestone for producing hydrated lime. The data consist of five specific gravity tests, 19 elastic modulus and UCS tests, as well as discontinuity analysis and the blast design itself. The actual fragmentation obtained from the blast design was measured using screen (sieve) analysis. On average, the specific gravity value is estimated to be 2.68 (g/cm3). The discontinuities data were obtained from scanlines surveyed on the rock mass. 6.1. Constant powder factor analysis In this approach, the fragment size distribution and the mean fragment after blasting is predicted using a constant and known powder factor (i.e. calculated from a defined drilling pattern (burden, spacing) and mass of explosive). Under the ‘‘constant powder factor’’ analysis, the simulator uses the powder factor as an input. The results of the simulation are compared with those observed in the field to verify that the simulator predictions are essentially congruent and physically correct. Table 1 lists the triangular distribution information used by the simulator. Note that drilling accuracy was assumed to be very high. The face dip direction parameter was measured from a map. The drilling pattern used in the actual blast was entered in the simulator. Within the simulator, a drop box named ‘‘explosive properties’’ contains the relative weight strength and the density (kg/m3) of the most common explosives used in the industry and can be used to select an explosive other than ANFO. Explosive strength values and densities have been extracted from the commercial literature. If no information is available about the explosive strength, that parameter can be calculated relative to ANFO if the heat of explosion is known. The simulator offers an ‘‘initial blast design’’ as another helper tool to aid the user in selecting the appropriate blasthole burden or spacing. The number of simulations was set to 100,000 and the results of each
Table 1 Triangular distribution data for case study Parameter
UCS Elastic modulus Discontinuity dip Discontinuity dip direction Discontinuity spacing
Units
MPa GPa degree degree cm
Lower bound
Most likely
Upper bound
Frequency
Value
Frequency
Value
Frequency
Value
1 1 3 4 5
121.34 72.95 55 160 40
3 3 19 14 21
141.22 81.22 75 240 200
1 1 9 6 1
208.16 91.79 85 340 400
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simulation stored into a temporary array for later statistical analysis. The resulting population tends to be normally distributed, a consequence of the Central Limit Theorem. The results indicate a mean of 7.12 and a standard deviation of 0.738. It is of course possible to specify the range of values to either the 68%, 95% and 99% confidence limits. The simulator then calculates the mean fragment size using Eq. (4). The mass of explosive Qe is set to 50.8 kg/ hole and the explosive column diameter D is 70 mm. The bench height H is 15 m while the loaded length L above the bench toe is 11 m. The powder factor K is 0.242 kg/ m3. For the case study, the mean fragment size Xm is 42 cm with a standard deviation of 4.35 cm. Using Eq. (7), the simulator calculates n, the Rosin–Rammler exponent, as 1.273 using a burden B of 3.5 m, a blasthole spacing S of 4 m, a drill hole diameter of 90 mm and a drilling accuracy W of 0.01 m. The characteristic size Xc is the size through which 63.2% of the particles pass and is simply a scale factor. In this case study, although n is a unique value, Xm is a distribution and therefore Xc will also be a distribution. Using Eq. (5) and increasing the X value from 0 to 300 cm, it is possible to calculate all cumulative distributions. After 100,000 simulations, the simulator has essentially described all possible scenarios and defined the worst case scenarios and the expected or ‘mean case’. The mean case takes into account all mean Kuz–Ram values like the mean fragment size Xm, the mean characteristic size Xc, the mean blastability of the rock mass and the mean uniformity indexes. The Monte Carlo simulation also defines upper and lower percent ‘‘passing a given size’’ boundaries. The area between these two boundaries represents all cumulative fragments distributions that could come out by blasting with this particular design pattern and layout and provides the blast designer with a possible range of variation. The simulator calculated the mean fragment size as 42 cm and the characteristic size as 56 cm as well as the percentage between the undersize and the oversize curves. Table 2 summarizes the predicted percent passing versus the measured field passing. As shown in Fig. 5, a comparison between the simulated distribution curve and the actual curve, measured in the field, shows considerable similarity, indicating that the Kuz–Ram simulator, under a ‘‘constant powder factor’’ process, can predict fairly well the fragmentation size distribution of a blast layout. The real mean fragment size is 45 cm while the predicted value is 42 cm. 6.2. Constant mean fragmentation analysis The simulator can also be used to design a blast to achieve a given fragmentation distribution. Using the
Table 2 Constant powder factor—real and simulated percent passing
Oversize Mean Undersize
Size (cm)
Real passing (%)
Simulated passing (%)
Error (%)
100 42 15
12.40 50 17.20
13 48 15
0.6 2 2.2
Fig. 5. Constant powder factor—final cumulative distributions.
Table 3 Constant mean fragmentation—specified percent passing curve
Oversize Mean Undersize
Size (cm)
Fixed passing (%)
100 45 15
15 50 15
rock mass data previously defined and taking into account the explosive properties, the simulator can execute, in reverse, the constant powder factor process. Both the mean fragment size and the percent passing are defined. It is possible to also define the undersize or oversize percent passing, depending on the cumulative distribution that is desirable after blasting. In fact, by fixing two points, it is possible to define the entire cumulative distribution. In this application, a constant mean and undersize passing will be specified (listed in Table 3) and the simulator will be searching for a suitable drilling pattern, including spacing S and burden B values. Several burden–spacing combinations are capable of meeting the required fragmentation distribution. It is upto the blasting engineer to make the final selection based on experience. Table 4 compares the values used for the actual blast with those predicted by the simulator. Overall, the results are quite comparable.
ARTICLE IN PRESS M.A. Morin, F. Ficarazzo / Computers & Geosciences 32 (2006) 352–359 Table 4 Constant mean fragmentation—real and simulated blasting patterns results
Burden B (m) Spacing S (m)
Actual pattern
Simulated pattern
3.5 4.0
3.5 4.2
7. Conclusions The purpose of this work was to develop a Monte Carlo-based simulation program that would provide a reasonable estimate of blasting fragmentation for surface mine and quarry blast design. The initial work done to date suggests that this approach is applicable. The Monte Carlo simulator can be used to target an optimized feed size distribution for a mill or minimize fines production. By using the simulator, the blasting can be adapted to achieve consistent fragmentation designs for different rock masses or to optimize the fragmentation to maximize the throughput and performance of crushing and mill circuits. One of the drawbacks of simulation is that field data are required. It can be difficult to fit certain parameters to triangular distributions when there is not enough data. Collecting this data can also be costly and time consuming. However, with a little effort, much of this information can be gathered and accumulated during regular drilling and blasting operations. One benefit of this simulator is that the triangular distribution approach is quite amenable to the ‘‘expert’’ opinion where the blasting engineer defines the ‘‘worst-case scenario’’ or a suitable range of parameters. At this stage of development, it appears that the simulator program does its job reasonably well, based on a real case study. However, further work is required to fully validate this approach and to verify its long-term usefulness and applicability. Over time, a more sophisticated fragmentation model could be defined and integrated within this simulator.
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