Predicting the extent of blast-induced damage in rock masses

Predicting the extent of blast-induced damage in rock masses

International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53 Contents lists available at SciVerse ScienceDirect International Journal o...

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International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53

Contents lists available at SciVerse ScienceDirect

International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms

Predicting the extent of blast-induced damage in rock masses Fernando Garcı´a Bastante a,n, Leandro Alejano a, Jose Gonza´lez-Cao b a b

Universidad de Vigo, Departamento de Ingenierı´a de los Recursos Naturales y Medio Ambiente, Campus Lagoas-Marcosende, 36310 Vigo, Spain ´tica Aplicada II, Campus Lagoas-Marcosende, 36310 Vigo, Spain Universidad de Vigo, Departamento de Matema

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 September 2011 Received in revised form 20 April 2012 Accepted 27 July 2012 Available online 22 August 2012

We describe a new model for predicting the extent of blast-induced damage (BID) in rock masses. The model is based on Langefors’ theory of rock blasting and the hypothesis that the maximum burden parameter defined by Langefors is a good indicator of BID. We have incorporated the effect of decreased internal energy of the gases as they expand to reach the walls of the borehole and then fitted the model to the available experimental data. Our model involves three easily obtainable parameters, namely, the energy loaded in the borehole, the coupling factor and the rock constant, and the mean gas isentropic expansion factor, which depends on the type of explosive used and on borehole and cartridge diameters. For this factor, we used approximate values independent of the type of explosive. The model indicates that there is a positive relationship between the internal energy of the explosive gases, once they have expanded inside the borehole, and the volume of damaged rock. This relationship is a function of the rock constant, a fundamental rock mass property that defines the maximum extent of damage under predefined charge conditions. The proposed model provides quite acceptable estimates of the fitted data, considering that just a single parameter is incorporated in the statistical procedure: the relationship between maximum burden and BID. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Blast-induced damage Langefors’ theory Blasting Burden

1. Introduction Growing interest in the topic addressed in this paper over the past few decades has led to several lines of research, such as identifying rock mass failure mechanisms and understanding their importance; estimating the extent of blast-induced damage (BID) in remaining rock mass from data collected in situ; developing predictive models to estimate the extent of BID; and estimating how the geomechanical parameters of the damaged rock mass are affected. In relation to this last point, Saiang [1] pointed out that ‘‘much of the effort in blast damage quantification has been largely focused on defining the depth or extent of the damage, and less on assessing its inherent properties, such as its strength and stiffness’’. Over the past decade, researchers have also begun using numerical models to analyse how BID affects the stability of the excavated rock mass [2–4]. Raina et al. [5] wrote an excellent review of the most important milestones and references for these different research lines. Since this study focuses on models for estimating BID, we need to clarify the meaning of the term. In referring to damage to rock mass caused by blasting, a distinction must be drawn between excavation beyond the desired limits and damage entailing a significant or noticeable decrease in the mechanical properties of the rock mass due to the

creation of new discontinuities or the enlargement of existing discontinuities on scales ranging from metres to micrometres [6]. Oriard [7] defines damage to include ‘‘not only the breaking and rupturing of rock beyond the desired limits of excavation but also an unwanted loosening, dislocation and disturbance of the rock mass’’; Singh [8] uses the term to refer to changes in the properties of the rock mass; and Scoble et al. [9] uses it to describe a decrease in integrity and quality. In this paper, we take BID to mean the rock mass area—as measured from the borehole axis—that undergoes a measurable increase in fracturing as a result of the blast, resulting, in turn, in a measurable decrease in rock strength properties. Under the assumption that this area is represented both by the lengthening of radial cracks around the borehole axis and by increased fracturing in core samples taken perpendicular to the axis, an analysis of possible models for predicting BID can be carried out using a relatively small amount of experimental data, considering the magnitude of the problem posed. This article is structured as follows. First, we review and critique the main models for predicting BID. We then describe how we developed the proposed model. Finally we describe and analyse the results obtained in the light of existing empirical evidence.

2. Models for estimating BID n

Corresponding author. E-mail address: [email protected] (F. Garcı´a Bastante).

1365-1609/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2012.07.023

Due to the large number of parameters on which the extent of damage to remaining rock mass may theoretically depend,

F. Garcı´a Bastante et al. / International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53

the complexity of the mechanisms that interact in the breakage process and possible BID variability in what is, apparently, a single blast scenario, BID has generally been estimated using rather simplistic models that exclude variables as important as the presence of water, blast sequence and geometry, confinement of the explosive and rock structure. Many of these models were developed for a technical (rather than scientific) purpose, namely, as blast design tools. It is no surprise, therefore, that they approach a complex issue in simplistic terms. These models can be divided into two categories: those which use peak particle velocity (PPV) as an indicator of the extent of damage, and those which use other approaches. 2.1. PPV-based models BID has traditionally been identified by defining a PPV threshold above which rock fracturing begins [10,11]. This approach is justified for two reasons: (1) the simplest mechanical model—a plane wave in an elastic medium—indicates that the stress in a medium is proportional to the particle velocity and that the material breaks if this velocity exceeds a critical value; and (2) it is possible to make a quasi-functional approximation between the PPV, the explosive charge (Q) and the distance from the blast to the measurement point (Ds) by means of a statistical analysis of blast data (PPV, Q and Ds) that can be recorded with relative ease using a seismograph. Although this criterion has been adopted as an engineering tool that, when used properly, makes it possible to establish damage-control guidelines for blast design, it lacks scientific motivation. Vibrations are usually measured at a considerable distance from blastholes, whereas the effect on PPV of measurements taken close to the blasthole is the circumstance of interest when one wishes to estimate the extent of damage to rock mass. Holmberg and Persson [12,13] therefore extended the model (in what follows, the H–P model), beginning with a widely used vibration propagation model for rock mass [14] in which PPV increases in scale as a function of charge and distance ð1Þ

where K, a and b are parameters determined using statistical regression techniques and Ds is the distance—usually radial— from the point in question to the charge. Eq. (1) can be expressed in terms of charge concentration (l) and charge length (x) as b PPV ¼ KðlxÞa D s

ð2Þ

Holmberg and Persson [12,13] assumed that the experimental model in Eq. (1), in its differential form, is derived from the following: dq ¼ Ds b=a dQ with q ¼ ðPPV=KÞ1=a Z a1 a b=a b=a D dðPPVÞ ¼ K al dx D dx s s the integral form of which gives the H–P model  Z a b=a dx PPV ¼ K l D s

velocity is obtained from the ‘energy contribution’ dq (called ‘vibration intensity’ by Holmberg and Persson) of each differential of the charge. Eq. (4), which provides a simple and feasible methodology for rock mass damage control, has been widely used in recent decades with various different combinations of rock masses and explosives [15]. For hard rock masses (primarily granite and gneiss), Persson et al. [6] established K ¼700, a ¼0.7, b ¼1.5, and critical PPV as between 0.7 and 1 m/s, corresponding to stress levels of 7–10 MPa, in accordance with the aforementioned twodimensional elastic model, with a P-wave propagation velocity of 5000 m/s in the rock mass (Cp) and a Young’s modulus of 50 GPa. Fig. 1 shows BID according to this model for a charge height of 4 m along an axis perpendicular to the centre of gravity of the charge. The model derived from the scaling law depicted in Eq. (1) was criticised by Blair and Minchinton [16], who wrote the following about Eq. (4): ‘‘the Holmberg–Persson model does not even supply a first-order approximation to either the near-field or far-field radiation from a blasthole’’. Their argument was based on analytical and numerical solutions for the radiation pattern of a charge in an elastic medium that differs from the model obtained with Eq. (4). Moreover, this radiation pattern will depend on assumptions made about both the model of material behaviour ˜ o et al. [17], for example, obtained a radiation pattern for a (Trivin fracturing medium) and the model of pressure transmitted to the ground. Other Blair and Minchinton objections to the H–P model include the fact that the model fails to take into account the vectorial nature of vibration velocity and the fact that stress is only proportional to the vibration velocity in the plane simplification of the model. Other possible objections include the following: (1) because Eq. (1) is an approximation to reality, the dispersion of the residuals obtained after statistically fitting the model is usually high, which means that results must be interpreted probabilistically; (2) it seems rather extreme to assume that the values of K, a and b obtained from far-field data are valid in the vicinity of the blasthole, that is, to extrapolate Eq. (1) far beyond the range of the experimental data; and (3) K, a and b should possibly be obtained from data recorded inside the rock mass—making the datacollection process more difficult and expensive—rather than by the usual practice of recording data on the surface. Nevertheless, Ouchterlony et al. [18] declared that the fact ‘‘that the PPV approach has been a working engineer’s tool despite all this may depend on the fact that it gives a consistent and sufficiently accurate way of estimating the relative load of damage effect incurred by the different charges types.’’

2.5

ð3Þ

H-P (0.7 m/s) H-P (1 m/s) Langefors SNRA

2

ð4Þ

where Ds is now the distance between each differential of the charge (dx) and the point of measurement and is therefore a function of charge length (x). This expression is compatible with Eqs. (1) and (2), since, for Ds bx, this distance is approximately equal to the radial distance. Note that the differential form of an expression such as Eq. (1), obtained in the experimental and integral form cannot be definitively determined except by making some sort of assumption—as in Eqs. (3) and (4)—according to which maximum vibration

BID (m)

b PPV ¼ KQ a D s

45

1.5 1 Abbreviations: BID, blast-induced damage. H-P, Holmberg-Persson. SNRA, Swedish National Road Administration.

0.5 0 0

0.2

0.4

0.6

0.8 1 l (kg/m)

1.2

1.4

Fig. 1. Estimation of BID from charge concentration.

1.6

1.8

F. Garcı´a Bastante et al. / International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53

Lu and Hustrulid [19] also criticised Eq. (4), citing a mathematical error in the derivation of the model. According to those authors, the differential form of Eq. (1) is as follows: b dðPPVÞ ¼ KðldxÞa D s

ð5Þ

This is indeed equivalent to assuming that Eq. (1) establishes a relationship between differential and non-absolute values. However, this is not the case in practice; Eq. (1) refers to measured values of PPV and Q, as expressed in Eq. (2). Iverson et al. [20] proposed directly using Eq. (1) to determine PPV in the near field, with Ds as the mean distance from the point of measurement to the charge (the National Institute for Occupational Safety and Health (NIOSH)-Modified Holmberg–Persson Approach) rather than the radial distance. However, Iverson et al. [21], noting that this approach underestimates the PPV for distances of less than 1 m, argued that the distance should be calculated instead using a weighted approach, with the inverse of the distance raised to the power b used to weight the distance for each element of the charge. These authors concluded: ‘‘This weighting approach results in a design curve that compares closely with the Holmberg–Persson integration design curve with assurance that a math error is not introduced.’’ In practice, the similarity between the two models will depend on the values of a and b used in Eq. (1). Fig. 2 shows the models described for estimating near-field PPV and the model represented by Eq. (1), all based on the Persson et al. [6] parameters. The charge concentration is 1 kg/ m, the charge length is 4 m and the PPV is calculated on an axis perpendicular to the charge centre of gravity. As the figure shows, when charge length is included in Eq. (1), PPV decreases in the vicinity of the blasthole with respect to the values obtained by direct application of Eq. (1); this is reasonable, at least in qualitative terms. Lu and Hustrulid [19] and Iverson et al. [21] also mentioned other models for estimating PPV, namely, the Colorado School of Mines (CSM) approach, the hydrodynamic approach and the Lu–Hustrulid approach. However, these models are more complex and require more parameters than the H–P model, so their practical usefulness is not clear. In keeping with the approach described so far, BID for a given rock mass can be estimated from the charge concentration (l); references to this approach can be found in [22] and in the

10000

Swedish National Road Administration (SNRA) table [23], whose values are reproduced in Fig. 1. The SNRA table estimates BID for borehole diameters between 45 mm and 51 mm; it provides information on the type and calibre of each explosive and also shows charge concentration (expressed as a charge concentration equivalent to Dynamex, an ammonia gelatin dynamite). The SNRA table indirectly reflects the influence of the coupling factor (f), which is the relationship between charge calibre and borehole diameter. In the table, values of f tend to increase in line with l. According to Ouchterlony et al. [18], the SNRA table partly ¨ originated in the research of Sjoberg [24,25], who measured BID by mapping freshly created fractures in coring holes associated with blasting for road tunnel construction in Gothenburg and established a linear relationship between BID and charge concentration. The table also draws on the H–P model equation (Fig. 1). Because ¨ Sjoberg took measurements with just two types of explosives, whereas the SNRA table includes about ten different explosives, Ouchterlony infers that the H–P model played an important role in the construction of the table, which has been used as a reference for BID estimation in Sweden for the past two decades. The SNRA table was subsequently revised with the publication of Anl¨aggningsAMA 98 [26]. No significant changes were introduced with regard to damage estimation and the revised version is essentially a simplified version of the original with the names and calibres of the explosives excluded.

2.2. Other predictive models From 1991 to 1996, the Swedish Rock Engineering Research ˚ Foundation (SveBeFo) conducted blast-damage research at Vanga dimension-stone quarry in southern Sweden that, among other goals, was aimed at improving the SNRA table [27]. The results of this research can be found in Olsson and Bergqvist [28–30], Ouchterlony [31] and Olsson and Ouchterlony [32]. To measure damage to the rock mass after each blast—initially with a single hole and later with three or four holes in a row—the researchers extracted blocks measuring about 2 m3 from the remaining rock, sawed them along a plane perpendicular to the borehole axis and used dye-penetrant inspection to carry out fracture mapping.

peak particle velocity (mm/s)

46

1000

Abbreviations: H-P, Holmberg-Persson; NIOSH, National Institute for Occupational Safety and Health

H-P MODEL MODEL Eq. (1) NIOSH-Average NIOSH-Weighting

0.1

100 1.0 Distance (m) Fig. 2. Models of PPV attenuation in the vicinity of the blasthole.

10.0

BID DATA (m)

F. Garcı´a Bastante et al. / International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Table 1

Olsson and Berqvist (1993) Olsson and Berqvist (1996) Sjöberg(1980) Mean values

Source

Equationn

Chapman–Jouguet theory of detonation

pCJ ¼ rD2 =ð1þ gÞ

Abbreviations: BID, blast-induced damage.

FðgÞ ¼

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 l (kgdx/m) xroot square (f)

BIDðmÞ ¼ 1:9xlðkgdx=mÞ

ð6Þ

Note the good correlation between Olsson and Bergqvist’s results in the coordinate system and, once a coupling factor function was introduced, between Olsson and Bergqvist’s results ¨ and the Sjoberg model. This suggests, as we shall see below, that this variable should be included in predictive damage estimation models. Olsson and Bergqvist [29] fitted their average results—the data from the seven three- or four-hole blasts—by means of regression, according to the following model: R0 c ¼ kxpbh ph ¼ 0:125xrxD2 xf

2 g1

ð9Þ ð10Þ

gg ð11Þ

ðg þ 1Þg þ 1

ph ¼ pe f

ð12Þ

2g1

 1:5ð Þ pr f ¼ pd 2r

D 0:25 c

Liu and Katsabanis [35]

Fig. 3. Experimental measurements of BID versus l  Of.

Fig. 3 shows the maximum crack lengths obtained by Olsson and Bergqvist, 1993 [28] taken from [33] and Olsson and Bergqvist, 1996 [29]. The figure shows the results for 12 singlehole blasts, corresponding to the 1993 study. For the 1996 study, the figure shows the mean values for several blasts and, in five of those cases, the highest values obtained, as well as two values corresponding to three-hole blasts. The horizontal axis represents the product of the charge concentration, expressed in terms of Dynamex-equivalent kg/m (kgdx/m) multiplied by the root of the coupling factor. The figure ¨ also shows Sjoberg’s linear predictive model, whereby BID can be estimated for low charge densities according to the following expression:

ð8Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 for ideal gases g ¼ 1 þ 2e pe ¼ FðgÞrD2

Isentropic flow equations

0

47

Grady and Kipp [36]

U s ¼ C þ kvr

ð14Þ

P d ¼ 1:5rD2 ðrr =rÞ0:28 ðc=DÞ0:26

ð15Þ

N ¼ pf

Theory of elasticity

ð13Þ

e_ y ¼ vr ¼



rr ce_

2=3 ð16Þ

6K IC

vr r

ð17Þ

pr

ð18Þ

rr c

n pCJ is the detonation pressure, g is the isentropic expansion factor of the gaseous products in the Chapman–Jouguet state of detonation; e (J/kg) is the heat of reaction of the explosive; pe is the explosion pressure; c is the elastic P-wave velocity in rock; pr is the shock amplitude at distance r; Pd is the initial dynamic pressure; F is the borehole diameter; rr is the rock density; e is the strain rate; ey is the circumferential strain rate; KIC is the fracture toughness; N is the number of cracks generated along the circumference in a cylindrical hole subjected to e; vr is the radial particle velocity; Us is the shock wave velocity in the rock; C and k are experimentally determined parameters; C is approximately equal to the sound velocity of rock c.

Table 2 Values of F(g) according to different models. Explosive Eq. (9) Gurit Emulet 20 Kimulux Detonex 80

Cooper [38]

Kamlet and Jacobs [39]

Johansson and Persson [40]

Kamlet and Short [41]

0.213 0.120 0.205 0.140

0.121 –

0.123 –

0.124 –

0.140 0.118 0.141 0.116

0.119 0.116

0.120 0.116

0.122 0.119

ð7Þ

where R’c is the BID minus the borehole radius, D is the detonation velocity of the explosive, r is the density of the explosive and g1 is the average isentropic expansion factor of the gases from their initial density to their final expanded state in the borehole. Ph is an estimate of the pressure in the gases as they expand to completely fill the borehole. The parameters k, b and g1 were estimated as 1.52, 0.68 and 1.68, respectively, in the regression model. Ouchterlony [31] exhaustively analysed Olsson and Bergqvist’s results and built a more complex model—intended to be more dimensionally consistent [34]—using relationships from different theories and models, the most important of which are listed in Table 1. Requiring that the number of cracks be at least 2, Ouchterlony obtained, from Eqs. (16)–(18), the expression for the minimum dynamic pressure required to initiate cracks in the walls of the borehole and, using this expression together with Eqs. (16)–(18) and (13) and setting F at twice the BID value, Ouchterlony obtained the critical pressure expression for the propagation of BID-length cracks. Finally, using log-normal regression models, Ouchterlony compared different models for estimating Ph,

concluding that the following model best fitted the data 0:25   2BID ðD=cÞ

f

pffiffiffiffiffiffiffiffi!2=3 BID ¼ 0:566 ph K IC

ð19Þ

Ph is obtained from Eq. (12), g1 is 1.1, as determined from regression along with the other two constants appearing in the right-hand term of Eq. (19), and g is obtained from Eq. (9). Ouchterlony thus described an elegant solution to the estimation of BID using Eq. (19), which, according to the hypothesis in Eq. (13), is a representation of the crack propagation criterion proposed by Griffith [37]. Nevertheless, Eq. (19) poses important drawbacks, some of which were identified by Ouchterlony in his research. These drawbacks are described below. Because Eq. (9) assumes ideal gas behaviour it cannot be used to estimate the value of g for solid explosives. For solid explosives g depends on its density (r). This means that the value of the function of g, F(g), which is used in Eq. (10) to estimate the explosion pressure differs considerably from the values obtained in the literature, some of which [38–41] are listed in Table 2.

F. Garcı´a Bastante et al. / International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53

48

The values for Cooper were calculated using the following relationship between explosive density (r) and density in the detonation state (rCJ)

rcj ¼ 1:386r0:96

ð20Þ

This relationship was obtained from experimentally derived data [38] for a range of densities covering the entire spectrum of industrial explosives. Applying the condition of tangency between the isentropic and Hugoniot equations of the products in the Chapman–Jouguet state, from the theory of detonation we obtain

g

r rCJ r

ð21Þ

From Eqs. (20) and (21), it follows that

g

1 1:386r0:04 1

ð22Þ

Note, in Table 2, the great discrepancy between the value of F(g) as determined by Eq. (9) and the values obtained for the other models for low-detonation-velocity explosives. Estimating g1 is difficult because fit depends not only on the composition of the explosive gases but also on the thermodynamic state considered and, therefore, on the degree of expansion or coupling factor. The polytropic approximation defined by Eq. (12), with a single general value of g1, is therefore rough. Persson et al. [6] suggested a value of 1.5 and Bauer [42] suggested a value of 1.3. Because this issue has received little attention, we estimated the value of g1 using the simple equation of state proposed by Cook [43] PaðvÞ ¼ nRT,

aðvÞ ¼ vaðvÞ

ð23Þ

where P and v are pressure and volume, respectively, a(v) is covolume (tabulated by Cook [44] for densities of upto 0.7 g/cc) and n, T and R are the number of moles of gas, temperature and the universal gas constant, respectively. If the pressure in the borehole is approximated using the isotherm, and given that, according to Eq. (12), what interests us is a polytropic model comprising the explosion state (E), from Eq. (23) we obtain  g1 1 vaðvÞ v Fa ¼ ¼ ¼ Fvg ð24Þ ½vaðvÞE vE Hence, for every pair (Fa, Fv) on a log–log scale, the exponent in question is determined by the slope of Eq. (24). Because expansion in the borehole in contour blasting involves densities considerably lower than those tabulated by Cook, we used the expression proposed by Hustrulid [45] to estimate the value of a(v)

aðvÞ ¼ 1:1e0:473=v

ð25Þ

In Fig. 4, the model represented by Eq. (24) is plotted for a state (E) corresponding to an initial density of 1.0 g/cc. It can be

6 5

LN (Fa)

4

GELATIN WATERGEL PETN ANFO ANFO-Al EMULSION-Al EMULSION

Gamma = 2.20

Gamma = 1.95 Cook

3 Gamma = 1.70

2 1 0

Gamma = 1.35 3

5 V/VE

10

Fig. 4. Estimation of the isentropic expansion factor.

15

20

observed that g1 is dependent on the final state, Fv or v/vE, and that its value decreases with the degree of expansion. Fig. 4 shows that taking an isentropic exponent that is independent of the degree of expansion implies that rough estimates of gas pressures will be obtained. The above results can be compared with results obtained with codes that use more sophisticated equations of state. Lo´pez Sa´nchez [46] calculated the ratio of useful work (w) for different expansion states and explosives using the W-Detcom code [47] which incorporates several equation-of-state models. Having considered data for various explosives, including those most commonly used in blasting, we arbitrarily selected the results obtained with the Becker–Kistiakowsky–Wilson equation of state with S-parameterisation [48], assuming ideal detonation and complete reaction. Given that gas energy (e) in state (E) is approximately equal to constant-volume explosion heat (Qv), we can approximate ideal gas behaviour between each two states of expansion (i, j) corresponding to the specific volumes (vi, vj) Q v wi  ei and

and

ei ej ¼ wj wi ¼ Dji

pi vi gij  cte

between i,j

with

ei ¼

pi vi

gij 1 ð26Þ

Developing the above expressions, we can fit the data this time in terms of energy. If Fi,j denotes the relationship vj/vi, then ! F i,j ei ð27Þ Ln ¼ gij LnðF i,j Þ ei Dji With the value gij for every Fi,j, we can calculate the value of g1 for every Fv. These results were superimposed over those of Cook in Fig. 4. With the exception of the emulsion, with a fairly constant value of g1 throughout the expansion process of around 2.2 (indicating rapid energy release), the other explosives behave as outlined above. Thus, the values of g1 defined first by Holmberg and Persson and later by Bauer are consistent with those obtained from Cook’s isotherm and could be considered as lower bounds. The values obtained from Lo´pez Sa´nchez are logically higher, given that the slope of the isentropic curve is greater than the corresponding isothermal curve. It therefore seems that if we must use a single value of g1 that is independent of explosive type and expansion, it would need to be considerably larger than the value of 1.1 obtained by Ouchterlony [31]. Returning to the origins of Eq. (19), using Eq. (13)—obtained from Liu and Katsabanis [35]—is apparently very appropriate for the attenuation of a wave along its path. If we accept the twodimensional approximation (in which stress is proportional to particle velocity) in the far field, the exponent of attenuation with distance, 1.5(D/c)0.25, is of the same order as those proposed by various authors for estimating peak particle velocity attenuation in the far field, namely 1.45 [6], 1.46 [49] and 1.57 [50]. These exponents are derived from the analysis of a wide range of data and experiences, so naturally their value is purely statistical; in other words, they do not reflect local circumstances such as the type of rock through which a particular wave is transmitted. The term in Eq. (19) that reflects the ratio between velocities could take this factor into account. In their paper, Liu and Katsabanis (hereinafter L–K) began by using the equation for the isentropic expansion of gases and the Hugoniot equation of the rock—Eq. (14)—to calculate dynamic pressure on the rock mass. Then, using regression, they related the dynamic pressure to a set of parameters that roughly represented the detonation pressure and the ratio between explosive impedance and rock mass impedance Eq. (15). Fig. 5

F. Garcı´a Bastante et al. / International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53

1

1000

PD/PCJ

0.8 0.7 0.6 0.5

100

10 0.6

0.8

1 1.2 2/(1+z)

1.4

1.6

1

1.8

10 R/Ro

Fig. 5. Relationship between dynamic pressure and Chapman–Jouguet detonation pressure as a function of the impedance ratio (obtained from the Liu and Katsabanis, 1993 model).

shows the relationship between the initial dynamic pressure on the walls of the borehole (PD) and the detonation pressure (PCJ) as a function of the ratio between the explosive/rock impedance (z)   PD 2 ð28Þ ¼f 1 þz P CJ The figure was obtained by applying the L–K model to the data provided in their article. This relationship, clear and important in this dynamic model with coupled charges, was rejected by Ouchterlony in applying Eq. (13) after a regression analysis of various models. This can perhaps be explained by the fact that the energy of the explosive shock wave contributes only marginally to rock breakage by providing the basic conditions for the process [51]. To estimate the stress wave attenuation as a function of distance, Liu and Katsabanis used the relationship between particle velocity in the blasthole walls (vr)—when the blasthole expands to a radius r and the wavefront achieves a distance R—and the particle velocity at the wavefront (vR) as represented by vR ¼ ðr=RÞvr

Limestone/Magnafrac 1000 Granite/ANFO Limestone/ANFO Oil Shale/ANFO

peak particle velocity (m/s)

0.9

0.4 0.4

49

ð29Þ

Using numerical methods, the authors used the Hugoniot equation of the rock to calculate the values of vr and r during the gas expansion process. They analysed various rock/explosive combinations and used regression to obtain, for each combination, the parameter for the exponent of Eq. (13), with values ranging from 1.1 to 1.8. By analysing all of the cases, again using regression, they finally reached the general form of Eq. (13). In a subsequent paper, Liu and Tidman [52] analysed Liu and Katsabanis’s work and proposed using the cube root (rather than the fourth root) of the velocity ratio, after checking the theoretical model by measuring stress at different distances from the blasthole during experimental blasts in granodiorite. The experimental data showed considerable dispersion. Note that the model does not introduce a non-elastic component for attenuation of the wave as it moved through the rock mass. Particle velocity according to Eq. (29) decreases along a slope that changes in value as the distance from the blasthole to the wavefront increases in response to changes in pressure as the walls of the hole expand. Fig. 6 shows, on a log–log scale, particle velocity attenuation for some of the cases presented by Liu and Katsabanis. As can be observed, the change in particle velocity with distance can be represented by a straight line on a log–log scale; this is probably due to the regression and the small relative weight of the quadratic component of the particle velocity, which is used to determine Pr according to Eqs. (14) and (18).

Fig. 6. Model of PPV attenuation in the vicinity of the blasthole for different materials (Liu and Katsabanis, 1993).

It therefore appears that it was quite by chance that Liu and Katsabanis obtained exponents in the same range as the values obtained experimentally in the far field for the particle-velocity damping coefficient, given that that effect in the L–K model is determined by the gas expansion process in the blasthole. As a result, including the function appearing in the exponent of Eq. (13) in Ouchterlony’s model does not seem sufficiently justified on theoretical grounds. In short, Ouchterlony’s model depends on many parameters, some of which, like detonation velocity dependence on the calibre of the cartridge, are difficult to determine accurately. Other parameters, including fracture toughness, can only be determined by means of specifically designed tests. This gives rise to the need to specify the methodology employed because, as Iqbal and Mohanty [53] pointed out, results may vary depending on which test recommended by the International Society for Rock Mechanics (ISRM) is used. Moreover, the model is based on rough estimates of both borehole pressure and wave attenuation, uses too many variables for the supporting data and, unlike the H–P model, is difficult to apply in engineering. The NIOSH has also developed BID prediction models [54,55]. Drawing on the work of Ash [56], the NIOSH proposed the following two predictors of the extent of overbreak (BIDN) sffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffi ! 2BIDN Ph 2:65 ¼ 25 ð30Þ f P eANFO rr 2BIDN

f

!

sffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2:65 ¼ 25 RBS

rr

ð31Þ

Ph is estimated from Cook’s isotherm, PeANFO is the pressure of the ammonium nitrate/fuel oil (ANFO) explosion and RBS is the relative bulk strength of the explosive with respect to ANFO. Eq. (31) is only valid for coupled charges. Both [54,55] recommend using the first equation to estimate BIDN. Hustrulid and Iverson [55] used Eq. (30) to evaluate drifting data from the Kiruna mine during a drift-driving programme conducted by the LKAB Group in the early 1990s, involving measurement of overbreak from several blasts in both mineral and waste rock. The authors started from the assumption that overbreak is largely independent of the perimeter charge if buffer row design is correct. In comparing the overbreak measurements with results obtained by applying in Eqs. (30) and (31) to the charge in the buffer holes, Hustrulid and Iverson found that the pressure-based Eq. (30) was a good predictor of overbreak, unlike the energy-based Eq. (31), which underestimated overbreak. The authors calculated overbreak as about 22 times the borehole

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diameter—measured from the buffer row—for both mineral and waste rock. For the pressure-based equation, overbreak was 19 and 23 times the borehole diameter for mineral and waste rock, respectively; for the energy-based equation, these figures fell to 10 and 13, respectively. Entering the values of D and c, as proposed by Hustrulid and Iverson, in Ouchterlony’s Eq. (19), the damage zone is approximately proportional to the product of (Ph)(F)1.5, which means that this model is not compatible with Eq. (30)—recommended by NIOSH—provided, of course, that we accept that the relationship between the extent of overbreak (BIDN) and BID is approximately linear.

3. Estimating BID using Langefors’ theory We describe a simple and highly practical new model whose results are quite consistent with the data in Fig. 1. The starting point for this model is Langefors’ theory of rock blasting [51]. According to this theory, the charge concentration required to tear the burden (V) by blasting a row of holes containing elongated charges is given by lf  0:9cV 2 =s with c  0:07=V þ 1:2c0 þ 0:004V

ð32Þ

where s is the relative weight strength of the explosive with respect to Langefors’ standard dynamite, and where c0 is determined by the minimum specific energy consumption needed to break the bottom of a borehole with a concentrated charge. A factor of 1.2 is introduced as a margin to account for c0 variations in the rock mass and the product of c ¼1.2xc0 is the rock constant. According to Langefors, c0 is usually between 0.28 kg/m3 and 0.35 kg/m3 for most rocks, with variation ranging from 0.2 kg/m3 to 0.8 kg/m3 approximately. The other rock factor terms in V represent Rittinger’s law and the component required to heave the rock mass enough to completely detach the burden. If we assume that the damage zone can be represented as a function of the maximum burden that can be broken with charge concentration lf in a rock mass whose rock constant is c0, we can estimate this function by simply solving for V and comparing the results with experimental measurements. Fig. 1 includes the V obtained with Eq. (32), taking the c0 value given by Langefors for hard Swedish rock (0.35 kg/m3) and the Dynamex explosive used by the SNRA (s¼0.92) as the reference explosive. The similarity between the results of the Langefors-based and the SNRA data is evident. Hence, this model based on the energy loaded in the borehole and another parameter dependent on the rock mass seems to be a good starting point for estimating BID. This strongly correlated data, as inferred from Fig. 1, suggests that the parameter V, as defined by Eq. (32), might be a good indicator of the extent of damage—although it could well also indicate that the development of the SNRA table was influenced by Langefors’ theory. Nonetheless, the empirical value of the table—which has seen more than 15 years of practical use without undergoing significant changes—is by no means negligible. Since the criterion proposed for determining BID is defined by the relationship between the energy loaded in the borehole and the minimum energy required to cause breakage, it seems logical to include the coupling factor, given its influence on variations in the internal energy of the expanding gases. According to Eq. (27), the relationship between gas energy from state vE to state vF in which the gases fill the borehole completely is determined by  1gF eF vF ¼ ð33Þ eE vE where gF is the isentropic expansion factor between state vE and state vF, which depends on both the explosive type and

expansion. As discussed above, in accordance with Cook’s isotherm a minimum value between 1.50 and 1.35 appears to be appropriate for expansions of 10–20 times the initial volume. Introducing the influence of the coupling factor f in the equation for V, we obtain lf f

2ðgF 1Þ

 0:9c0 V2 =s

ð34Þ

Thus, the terms in V have been eliminated from the rock factor, which simplifies subsequent calculations and removes the (inconvenient) possibility of obtaining negative values. The surface created inside the rock mass is very small compared to the surface created in the breakage face and, moreover, no rock detachment takes place there, so it seems legitimate to eliminate both terms. Note that strictly speaking, V is no longer the burden defined by Langefors [51] according to Eq. (32).

4. Model development and analysis Development of the proposed model was difficult due to the lack of experimental data and the wide range of methodologies used to collect whatever data is available. As mentioned above, Sweden’s SveBeFo is the institution that has conducted the most experimental research in this area. All references cited so far deal with hard rocks common in Sweden (granite and gneiss). At least on this point, uniformity is assured as Langefors’ c0 can be considered representative of such materials. The available information sources are the SNRA table data and SveBeFo’s research results. The SNRA table must be rejected for its apparent lack of direct experimental evidence. As for the work by Olsson and Bergqvist, a distinction must be drawn between the 12 results from the first study (1993)—corresponding to single-hole blasts—and the 7 results from the second study (1996)—corresponding to mean values for three-hole blasts. A more recent data source is an article by the Swedish researchers Kilebrant et al. [57], who, for six tests on hard granite gneiss, measured BID in core samples using various observation techniques (core mapping and ultrasonic velocity, porosity and density measurements). Given the large number of parameters eliminated from the models that may have a significant influence on the extent of damage and the small amount of heterogeneous data available, it seems reasonable to ensure that the model does not introduce any bias with respect to the available information. Fig. 7 shows the relationship between the BID and burden as calculated using Eq. (34). Note that we drew on [33] for Olsson and Bergqvist’s (1993) values for explosive characteristics and damage extent; likewise, we drew on [31] for Olsson and Bergqvist’s (1996) values, but taking the maximum and not the average value for the extent of damage. The density of the detonating cord was taken from Ouchterlony [31]: 1250 kg/m3 (as compared to the 1050 kg/m3 cord used in the Olsson and Bergqvist (1993) study). The value of gF was set at 1.3. The figure also shows the distribution of the residuals with respect to the expansion factor. The residuals were defined as the difference between the experimental data and the linear regression trend. Interestingly, for small expansion factors of less than 10, the figure shows a clear bias that is consistent with the observations made in the previous section. The lower the expansion factor, the higher the coefficient g. Fig. 8 recreates the same graph with the value of gF set at 1.7. It can be observed that, as gF increases, the residuals become more concentrated for small expansion factors of less than 10 and, moreover, that the residuals shift to the right for large expansion factors of above 20; note, however, that the latter occurs with a

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Fig. 7. Relationship between experimental BID and V for gF ¼1.3.

51

Fig. 9. Relationship between experimental BID and BID as given by the model for a variable gF.

1 BID DATA(m) = BID MODEL(m) 2 0.9 R = 0.96 (Kilebrant et al., 2010)

BID DATA(m)= 1,01 BID MODEL (m)

BID DATA (m)

0.8 R2 = 0.77 (Olsson and Bergqvist, 1993) BID DATA(m) = 0,97 BID MODEL(m)

0.7 R2 = 0.70 (Olsson and Bergqvist, 1996) 0.6 0.5 0.4 0.3

Abbreviations: BID, blast-induced damage.

0.2 Fig. 8. Relationship between experimental BID and V for gF ¼1.7.

very small number of experimental points. Although there are only two data points in the expansion range of 10–20 times the original volume, it seems reasonable to assume that, in this range, the appropriate value of gF falls somewhere between the values given above. It needs to be borne in mind that the value of gF depends not only on the degree of expansion but also on other factors such as explosive type and density. For the analysed data set, values of gF ranging from 1.3 to 1.7 were taken, but we just as easily could have considered the range 1.4–1.8: both the regression trend (0.72 V) and the standard deviation of the residuals (s ¼13 cm) are very stable within this range of variation of gF. We therefore made a simple visual fit. Since gF actually follows a path that varies with the degree of expansion, this path should theoretically be divided into sections that represent this variability. For example, Fig. 9 shows the results obtained for gF set at 1.7 for expansion factors of upto 8.5; beyond that, gF is set at 1.1. These values were chosen in order to represent a gF of 1.65 to 1.55 for expansions of 10–20 times the original volume. These values are more consistent with the gF values obtained using the data from [46]. In this figure, the trend of the model (0.74 V) changes slightly: the residuals are more concentrated and the value of s is slightly lower (12 cm). The figure also depicts the residual distribution, with the residuals tending to increase in line with BID values and dispersion seeming to decrease with the expansion factor. To be able to carry out a truly meaningful statistical analysis that would support these conclusions, more data are needed in the areas with high values of BID, charge density and expansion factor. Since the data come from non-homogeneous tests, the partial response of the model for each data set is shown in Fig. 10. From the figure—in which the range of the horizontal axis has been limited to increase clarity—we can infer that, despite the nonhomogeneity of the data sources, the results can fortunately be considered to broadly come from the same population group. The absence of a high degree of bias is the relevant aspect of the figure.

0.1 0.1

0.2

0.3

0.4 0.5 0.6 0.7 0.8 BID MODEL (m) = 0,74V (m)

0.9

1

Fig. 10. Relationship between experimental BID and BID as given by the model for a variable gF (disaggregated data).

This suggests—bearing in mind the limited nature of our conclusions due to the number of observations—that the extent of damage can be conveniently estimated on the basis of Kilebrant’s core sampling, a far less onerous approach than measurement of the extent of radial cracks in pre-sawn blocks of rock. Finally, if the extent of BID is taken as 75% of the value of V as defined by Eq. (34), we obtain qffiffiffiffiffiffiffiffiffiffiffiffi ðg 1Þ BID  0:8f F lf s=c0 ð35Þ

5. Final remarks If the proposed model is applied using the original data from the SNRA table, there is a high degree of linear correlation between the BID and the V found with Eq. (34), as well as low bias (3 cm). However, the trend is 0.98 V instead of 0.74 V, so the model represented by Eq. (35) predicts, on average, BID values that are 75% of the value of the points in Fig. 1. This is due to the fact that most of the experimental data result in smaller damage zones than those predicted by the SNRA table. Moreover, the value of the trend cannot be generalised, strictly speaking, because we did not work with experimental data for different types of rock mass. Note also that, when applied to coupled charges, the H–P model and Eq. (34) generally do not coincide despite the proximity of the results, as can be observed in Fig. 1 for the charge length considered (4 m). For a given charge concentration, the H–P model is highly dependent on charge length, unlike Eq. (34). This is due to the fact that Holmberg and Persson assumed that the entire length of the charge contributes to particle velocity in the vicinity of the blasthole.

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52

In the model’s derivation of gF, any of an infinite number of possible paths could have been chosen, some with smoother transitions between expansion states, or even different paths for each type of explosive. Many more experimental data are required in order to perform a rigorous analysis. From the previous section, we can conclude that there is not much difference in the derived model: when the data are analysed globally, BIDE(0.72–0.74)V for the range of gF values considered. For example, if we take gF values of 1.90 and 1.35 (rather than the 1.7 and 1.1 used), BID is equal to 0.73 V, there is very little change in the standard deviation of the residuals but about 6 cm of bias appears in the model. It is therefore evident that the estimated values of gF, which reflect results that are acceptable overall for the analysed data set, are not necessarily the best estimates for each particular case. Other considerations could have been taken into account. For example, we can infer from the theory of Langefors or of Persson et al. [58] that, for a given charge density and high coupling factors, there is no significant variation in the breakage power of the charge. This may well indicate that, in such cases, the extent of the damage is of the same order as that of the corresponding coupled charge. Eq. (35) predicts an extent of damage proportional to f (gF  1). There is also the issue of the quasi-ideal behaviour of emulsions, with their rapid delivery of gas energy, indicating that the appropriate value of gF, according to Fig. 4, should be around 2. Indeed, open questions remain that can only be clarified with the standardized collection and analysis of new experimental data, including the effect of anisotropy of the rock mass on the BID. If the rock mass is homogeneous and isotropic, the implicit model assumption that the rock constant is a rock mass property that defines the BID seems reasonable. Otherwise, the spatial distribution of the BID is highly variable being influenced by the structural and stress anisotropy. Finally, note that Eq. (35) is compatible with the NIOSH models. Taking into account that sE672 RBS/r and lf ¼f2r&F2/4, from Eq. (35) we obtain   2BID g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 36f F RBS=co ð36Þ

the maximum burden defined by Langefors’ theory is a good indicator parameter for predicting BID. The incorporation of the effect of the drop in internal gas energy during the expansion process is consistent with the notion that the shock wave acts primarily as an initiating cause of damage, where as the energy of the gas bubble ultimately defines the extent of the damage. Also consistent with this notion are the parameters used in the model: the explosive energy, the coupling factor, the rock constant and the gas isentropic expansion factor. The fact that this model is easy to interpret would indicate that a positive relationship exists between the internal energy of explosive gases once expanded inside the borehole and the volume of damaged rock, with this relationship dependent on the rock mass as characterised by the rock constant. We also discussed other models for estimating BID, highlighting the crudeness of their formulations. In contrast, the proposed model provides quite acceptable estimates of the fitted data, considering that just one parameter is incorporated into the statistical procedure: the relationship between maximum burden and BID. The model yields what seems to be an acceptable range of BID values within the range of variation of the rock constant. Nevertheless, it cannot objectively be viewed as more or less approximate than the other models, given the scarcity of homogeneous experimental data. The lack of this kind of data is undoubtedly the Achilles’ heel of all BID prediction models.

Acknowledgements The authors thank the Spanish Ministry of Science and Technology for financial support awarded under contract BIA200909673 for the research project titled ‘‘Studies of Underground Excavations in Rock Masses’’, which funded the research described in this paper. Kelly Dickeson and Ailish M.J. Maher provided assistance with English usage in a version of this manuscript.

f

For Swedish granite, ANFO and coupled charges, the extent of BID is around 30 F, more than twice the overbreak predicted by the NIOSH energy-based equation. This result is more consistent with the magnitude of the Kiruna mine drifting data, considering that the extent of BID has to be greater than the extent of overbreak, which was approximately 22 F. Working under the aforementioned conditions and taking into account the range of rock-constant values, BID values would be in the range of 20–40 times the borehole diameter, i.e. similar to the burden value used in practice. If the borehole is loaded with an explosive whose calibre is equal to one third the value of F, then the value of gF is 1.68 and the extent of damage drops to 3–6 times the borehole diameter—in other words, about half the typical spacing between pre-splitting boreholes. According to Langefors, lf should be around 90 F2 (expressed in terms of kg/m of Langefors’ standard dynamite) for charges in contour blasting. The coupling factor to obtain this charge concentration (r E1400 kg/m3, RBSE2) would be 0.28. The extent of damage under these conditions (gF ¼1.63) would be 4–8 times the borehole diameter.

6. Conclusions The proposed model for estimating the extent of BID in rock masses has a solid theoretical foundation—Langefors’ theory of rock blasting—and makes an apparently reasonable assumption:

References [1] Saiang D. Damaged rock zone around excavation boundaries and its interac˚ Sweden: Lulea˚ University of tion with shotcrete. Licentiate thesis. Lulea, Technology; 2004. [2] Saiang D, Nordlund E. Numerical study of the mechanical behaviour of the damaged rockmass around a deep underground excavation. In: Proceedings ˚ 9–11 June 2008. of the 5th international conference on mass mining. Lulea; p. 803–13. [3] Saiang D, Nordlund E. Numerical analyses of the influence of blast-induced rock around shallow tunnels in brittle rock. Rock Mech Rock Eng 2009;42: 421–48. [4] Saiang D. Stability analysis of the blast-induced damage zone by continuum and coupled continuum-discontinuum methods. Eng Geol 2010;116:1–11. [5] Raina AK, Chakraborty AK, Ramulu M, Jethwa JL. Rock mass damage from underground blasting, a literature review, and lab- and full scale tests to estimate crack depth by ultrasonic method. Int J Blast Frag 2000;4:103–25. [6] Persson PA, Holmberg R, Lee J. Rock blasting and explosives engineering. CRC Press; 1993. [7] Oriard LL. Blasting effects and their control. In: Hustrulid WA, editor. Underground mining methods handbook. New York: SME/AIME; 1982. p. 1590–603. [8] Singh SP. Mining industry and blast damage. J Mines Met Fuels 1992:465–72. [9] Scoble MJ, Lizotte YC, Paventi M, Mohanty BB. Measurement of blast damage. Min Eng 1997:103–8. [10] Langefors U, Kihlstrom B, Westerberg H. Ground vibrations in blasting. Water Power 1958(parts I–III):335–8 390–95; 421–24. [11] Edwards AT, Northwood TD. Experimental studies of the effects of blasting on structures. The Engineer 1960;210:538–46. [12] Holmberg R, Persson PA. The swedish approach to contour blasting. In: Proceedings of the 4th conference on explosives and blasting technique. New Orleans; 1–3 February 1978. p. 113–27. [13] Holmberg R, Persson PA. Design of tunnel perimeter blasthole patterns to prevent rock damage. In: Tunneling 79: Proceedings of the second international symposium. London; 12–16 March 1979. p. 280–83.

F. Garcı´a Bastante et al. / International Journal of Rock Mechanics & Mining Sciences 56 (2012) 44–53

[14] Duvall WI, Johnson ChF, Meyer AVC, Devine JF. Vibrations from instantaneous and millisecond-delayed quarry blasts. US Department of the Interior, Bureau of Mines. Report RI 6151. US; 1963. [15] Persson PA. The relationship between strain energy, rock damage, fragmentation, and throw in rock blasting. Int J Blast Frag 1997;1:99–110. [16] Blair DP, Minchinton A. On the damage zone surrounding a single blasthole. Int J Blast Frag 1997;1:59–72. ˜ o LF, Mohanty B, Munjiza A. Seismic radiation patterns from cylindrical [17] Trivin explosive charges by analytical and combined finite-discrete element methods. In: Proceedings of the 9th international symposium on rock fragmentation by blasting. Granada; 13–17 September 2009. p. 415–25. [18] Ouchterlony F, Olsson M, Bergqvist I. Towards new Swedish recommendations for cautious perimeter blasting. Int J Blast Frag 2002;6:235–61. [19] Lu W, Hustrulid W. The Lu–Hustrulid approach for calculating the peak particle velocity caused by blasting explosives and blasting technique. In: Proceedings of the EFEE 2nd world conference. Prague; 10–12 September 2003. p. 291–300. [20] Iverson SR, Kerkering JC, Hustrulid W. Application of the NIOSH-modified Holmberg–Persson approach to perimeter blast design. In: Proceedings of the 34th annual conference on explosives and blasting technique. New Orleans; 27–30 January 2008. 2:p. 1–33. [21] Iverson SR, Hustrulid W, Johnson JC, Tesarik D, Akbarzadeh Y. The extent of blast damage from a fully coupled explosive charge. In: Proceedings of the 9th international symposium on rock fragmentation by blasting. Granada; 13–17 September 2009. p. 459–68. [22] Gustafsson R. Te´cnica sueca de voladuras. Sweden: Spi Nora; 1977. [23] SNRA. Cautious blasting, careful blasting and scaling, rock engineering ¨ directions for the construction of Ringen and Yttre Tvarleden. Swedish National Road Administration. Project directions ANV 00031:1; 1st rev, 1995-09-15. Stockholm; 1995 [in Swedish]. ¨ [24] Sjoberg C, Larsson B, Lindstrom M, Palmqvist K. A blasting method for controlled crack extension and safety underground. ASF project no. 77/224. Nitro Consult, Gothenburg; 1977. ¨ [25] Sjoberg C. Cracking zones around slender borehole charges. In: Proceedings of annual discussion meeting BK-79. Swedish rock construction committee. Stockholm; 1979. p. 53–98. ¨ [26] AnlaggningsAMA-98. General materials and works description for construc¨ tion work, section CBC: Bergschakt. Svensk Byggtjaanst: Stockholm; 1999 [in Swedish]. [27] Saiang D. Blast-induced damage: a summary of SVEBEFO investigations. Lulea˚ University of Technology. Technical report. Department of Civil and Environmental Engineering Division of Rock Mechanics; 2008. [28] Olsson M, Bergqvist I. Crack growth in rock during cautions blasting. SveBeFo report no. 3, Swedish Rock Engineering Research, Stockholm; 1993. [29] Olsson M, Bergqvist I Crack growth during multiple hole blasting. SveBeFo report no.18, Swedish Rock Engineering Research, Stockholm; 1996. [30] Olsson M, Bergqvist I. Crack propagation in rock from multiple hole blastingsummary of work during the period 1993–96. SveBeFo report no. 32, Swedish Rock Engineering Research, Stockholm; 1997. [31] Ouchterlony F. Prediction of crack lengths in rock after cautious blasting with zero interhole delay. Int J Blast Frag 1997;1:417–44. [32] Olsson M, Ouchterlony F. New formula for blast-induced damage in the remaning rock. SveBeFo report no. 65, Swedish Rock Engineering Research, Stockholm; 2003. [33] Hustrulid W. Blasting principles for open pit mining. volume 2: theoretical foundations. Rotterdam, Netherlands: AA Balkema; 1999. [34] Buckingham E. On physically similar systems; illustrations of the use of dimensional equations. Phys Rev 1914;4:345–76. [35] Liu Q, Katsabanis PD. A theoretical approach to the stress waves around a borehole and their effect on rock crushing. In: Proceedings of the fourth international symposium on rock fragmentation by blasting. Vienna; 5–8 July 1993. p. 9–16.

53

[36] Grady DE, Kipp ME. Dynamic rock fragmentation. In: Atkinson BK, editor. Fracture mechanics of rock. London: Academic Press; 1987. p. 429–75. [37] Griffith AA. The phenomena of rupture and flow in solids. Philos Trans R Soc 1921;A221:163–98. [38] Cooper P. Explosives engineering. NY, USA: Wiley-VCH; 1996. [39] Kamlet MJ, Jacobs SJ. Chemistry of detonations I. A simple method for calculating detonation properties of C–H–N–O explosives. J Chem Phys 1968;48:23–35. [40] Johansson CH, Persson PA. Density and pressure in the Chapman–Jouguet plane as functions of initial density of explosives. Nature 1966;212:1230–1. [41] Kamlet MJ, Short JM. The chemistry of detonations. VI. A ‘Rule for Gamma’ as a criterion for choice among conflicting detonation pressure measurements. Combust Flame 1980;38:221–30. [42] Bauer A. Wall control blasting in open pits. In: Baumgartner P, editor. Rock breaking and mechanical excavation, 30. Quebec, Montreal: Harpell’s Press; 1982. p. 3–10. [43] Cook MA. An equation of state for gases at extremely high pressures and temperatures from the hydrodynamic theory of detonations. J Chem Phys 1947;15:518–24. [44] Cook MA. The science of high explosives. NY, USA: Reinhold Pub Corp; 1958 American Chemical Society Monograph Series No. 139. [45] Hustrulid WA. Practical, yet technically sound, design procedure for pre-split blasts. In: Proceedings of the 33rd annual ISEE conference on explosives and blasting technique. Nashville; 28–31 January 2007. p. 26. [46] Lo´pez Sa´nchez LM. Evaluacio´n de la energı´a de los explosivos mediante modelos termodina´micos de detonacio´n. PhD thesis. Madrid: UPM. ETSI Minas; 2003. [47] Sanchidria´n JA, Lo´pez LM. Calculation of the explosives useful work-comparison with cylinder test data. In: Proceedings of the second world conference on explosives and blasting technique. Prague; 10–12 September 2003. p. 357–61. [48] Hobbs ML, Baer MR. Calibrating the BKW-EOS with a large product species database and measured CJ properties. In: Proceedings of the 10th international symposium on detonation, Boston; 12–16 July 1993. p. 409–18. [49] Dowding CH. Construction vibrations. New Jersey, US: Prentice Hall; 2000. [50] Hendron AJ, Oriard LL. Specifications for controlled blasting in civil engineering projects. In: Proceedings of the North American rapid excavation and tunneling conference. Chicago; 5–7 June 1972. 2: p. 1585–609. [51] Langefors U, Kihlstrom B. The modern technique of rock blasting. NY, US: Wiley; 1967. [52] Liu Q, Tidman JP. Estimation of the dynamic pressure around a fully loaded borehole. Canmet/MRL technical report 95-014; 1995. [53] Iqbal MJ, Mohanty B. Experimental calibration of ISRM suggested fracture toughness measurement techniques in selected brittle rocks. Rock Mech Rock Eng 2007;40(5):453–75. [54] Hustrulid WA, Johnson JC. A gas pressure-based drift round blast design methodology. In: Proceedings of the 5th international conference on mass ˚ Sweden; 9–11 June 2008. p. 657–69. mining. Lulea, [55] Hustrulid WA, Iverson SR. Evaluation of Kiruna mine drifting data using the NIOSH design approach. In: Proceedings of the 9th international symposium on rock fragmentation by blasting. Granada; 13–17 September 2009. p. 497–506. [56] Ash RL. The mechanics of rock breakage (Parts I–IV). Pit and Quarry 1963;56:98–100 (2); 118–23 (3); 126–31 (4); 109–11, 114–18 (5). ˚ [57] Kilebrant M, Norrgard T, Jern M. The size of the damage zone in relation to the linear charge concentration. In: Proceedings of the 9th international symposium on rock fragmentation by blasting. Granada; 13–17 September 2009. p. 449–57. ¨ [58] Persson PA, Ladegaard-Pedersen A, Kihlstrom B. The influence of borehole diameter on the rock blasting capacity of an extended explosive charge. Int J Rock Mech Min Sci 1969;6:277–84.