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Kuz-Ram Fragmentation Analysis Essay

The main goal of rock blasting is the fragmentation of the rock mass. Prediction of the size distribution of the fragmented rock from the rock mass characteristics, the blast design parameters (both in terms of the geometry and of the initiation sequence) and the explosive properties is a challenge that has been undertaken for decades, and is currently available to the blasting engineer in the form of formulae that relate the parameters of a given size distribution function to the rock properties and the blast design parameters.

One of the most relevant fragmentation by blasting formulae is the Kuznetsov (1973) one:

$$x_{m} = Aq^{ - 0.8} Q^{1/6}$$

(1)

where xm is the mean fragment size in cm, A is a rock strength factor in the range 7–13, q is the powder factor (or specific charge, or charge concentration—explosive mass per unit rock volume) in kg/m3 and Q is the charge per hole in kg. These three factors comply essentially with the expected behavior of rock fragmentation by blasting. Essentially, the Kuznetsov formula means that (1) the harder the rock, the bigger the fragments; (2) the higher the specific amount (powder factor) of explosive, the smaller the fragments; and (3) the larger the scale (the charge per hole is used as scale factor, and larger charges per hole indicate larger drill patterns), the larger the fragments. In Eq. 1, the amounts of explosive in the powder factor and the charge per hole are given in TNT equivalent mass. Such equivalent mass can be calculated multiplying the mass of explosive by its relative strength with respect to TNT; if such strength is θ, Eq. 1 can be written as (Kuznetsov 1973, Eq. 12):

$$x_{m} = Aq^{ - 0.8} Q^{1/6} \theta^{ - 19/30}$$

(2)

in which q and Q refer to actual mass of explosive of relative strength θ. Equation 1 or its equivalent Eq. 2 is written originally for the mean size xm of a Rosin–Rammler (RR) distribution (Rosin and Rammler 1933; Weibull 1939, 1951) that was assumed to accurately describe the fragmented rock. The Rosin–Rammler, or Weibull, cumulative distribution function is:

$$P\left( x \right) = 1 - \exp \left[ { - \left( {\frac{x}{{x_{\text{c}} }}} \right)^{n} } \right] = 1 - \exp \left[ { - \ln 2\left( {\frac{x}{{x_{50} }}} \right)^{n} } \right]$$

(3)

where xc is the characteristic size (the size for which the passing fraction is 1 − 1/e, or 63.2%) and n is a shape factor, usually quoted as uniformity index; the expression on the right is written with the median size x50 instead of xc. The use of the RR distribution for rock fragmented by blasting had been positively assessed by Baron and Sirotyuk (1967) and Koshelev et al. (1971), which is used by Kuznetsov (1973).

The Soviet literature of the time does not make it entirely clear whether Eqs. 1 or 2 refer to the mean or the characteristic size (which value is close to the mean if the shape index of the distribution is not much different than one). There has been some controversy (Spathis 2004, 2009, 2012, 2016; Ouchterlony 2016a, b) about the actual size that the Kuznetsov formula was addressing. For all practical purposes, it has been calibrated, tailored and used over the years to estimate the median size x50 (Cunningham 1987, 2005; Rollins and Wang 1990; Raina et al. 2002, 2009; Liu 2006; Cáceres Saavedra et al. 2006; Borquez 2006; Rodger and Gricius 2006; Vanbrabant and Espinosa Escobar 2006; Mitrovic et al. 2009; Engin 2009; Gheibie et al. 2009a, b; Bekkers 2009; McKenzie 2012; Sellers et al. 2012; Faramarzi et al. 2015; Jahani and Taji 2015; Singh et al. 2015; Adebola et al. 2016).

Kuznetsov derived his formula based on blasting tests in limestone specimens reported by Koshelev et al. (1971). These consisted of eleven small- to mid-scale shots in irregular limestone blocks, with RDX charges of 0.5–500 g. Kuznetsov then assessed the formula with some data from large-scale tests by Marchenko (1965), 6 blasts in limestone, 6 blasts in a medium-hard rock and 14 blasts in hard and very hard rock (the exact rock type is not reported, nor is the explosive used). A values recommended by Kuznetsov were 7, 10 and 13 for medium-to-hard, hard fissured and hard massive rocks, respectively. Kuznetsov mentions that the mean deviation of experimental data from the theoretical (i.e., predicted) data is ±15%; a detailed analysis of the data and the predictions by Eqs. 1 or 2 gives a mean error of 9%, with minimum and maximum errors of −42 and 55%, respectively (Ouchterlony 2016a). This matter will be re-assessed in this work.

About ten years after its publication, the Kuznetsov formula was popularized by Cunningham (1983), who wrote it so as to use the relative weight strength with respect to ANFO, the standard explosive in civil applications, instead of TNT, in what became the popular Kuz-Ram model. In his text, Cunningham uses the term ‘mean fragment size,’ but his mathematical definition of it implies the median; he in fact uses the symbol x50. In the Kuz-Ram papers that followed (Cunningham 1987, 2005), the term mean is also used but again contradicted by figures and equations that imply x50:

$$x_{50} = Aq^{ - 0.8} Q^{1/6} \left( {\frac{\text{RWS}}{115}} \right)^{ - 19/30}$$

(4)

Here RWS is the relative weight strength (heat of explosion, or energy in general, ratio with respect to ANFO, in percent; 115 is the relative weight strength of TNT). Cunningham (1987) adapted a blastability index proposed by Lilly (1986) to replace Kuznetsov’s numerical rock factor:

$$A = 0.06 \cdot \left( {{\text{RMD}} + {\text{JF}} + {\text{RDI}} + {\text{HF}}} \right)$$

(5)

The form of the rock mass description term (RMD) has had some changes over the years; in its final form (Cunningham 2005), it is:
  • RMD = 10 (powdery/friable), JF (if vertical joints) or 50 (massive);

  • JF (joint factor) = JPS (joint plane spacing) + JPA (joint plane angle);

  • JPS = 10 (average joint spacing SJ < 0.1 m), 20 (0.1 ≤ SJ < 0.3 m), 80 (0.3 m ≤ SJ < 0.95·(B·S)1/2, B and S being burden and spacing), 50 (SJ > (B·S)1/2). Cunningham (2005) incorporates a joint condition correction factor that multiplies the joint plane spacing, with value 1, 1.5 and 2 for tight, relaxed and gouge-filled joints, respectively;

  • JPA = 20 (dip out of face), 30 (strike perpendicular to face) or 40 (dip into face). Cunningham does not give a JPA value for horizontal planes but Lilly (1986) assigns them JPA = 10.

The rock density influence (RDI) is:
ρ being density (kg/m3). Finally, the hardness factor (HF) is:
  • HF = E/3 if E < 50, or

  • HF = σc/5 if E > 50.

σc and E being uniaxial compressive strength (MPa) and Young’s modulus (GPa).

The form of RMD implies that JF may enter twice in Eq. 5, directly and through RMD. The direct term in Eq. 5 Cunningham (1987) is in all probability a printing error and Cunningham (2005) removed it in later Kuz-Ram model updates, so that:

$$A = 0.06 \cdot \left( {{\text{RMD}} + {\text{RDI}} + {\text{HF}}} \right)$$

(6)

Cunningham (2005) also incorporated a delay-dependent factor in the central size formula, based on Bergmann et al. (1974) data:

$$\begin{aligned} A_{t} & = 0.66\left( {\frac{{\Delta T}}{{T_{\hbox{max} } }}} \right)^{3} \,-\, 0.13\left( {\frac{{\Delta T}}{{T_{\hbox{max} } }}} \right)^{2} \,-\, 1.58\left( {\frac{{\Delta T}}{{T_{\hbox{max} } }}} \right) + 2.1, \quad \frac{{\Delta T}}{{T_{\hbox{max} } }} < 1 \\ A_{t} & = 0.9 + 0.1\left( {\frac{{\Delta T}}{{T_{\hbox{max} } }} - 1} \right), \quad \frac{{\Delta T}}{{T_{\hbox{max} } }} > 1 \\ \end{aligned}$$

(7)

where \(T_{\hbox{max} } = 15.6B/c_{P}\); ΔT is in-row delay (ms), B is burden (m) and cP is P-wave velocity (m/ms). Note that At is not continuous at ΔT/Tmax = 1, but replacing the constant term 2.1 by 2.05 would make it so.
As previously stated, the various factors (e.g., the rock description pre-factor 0.06) were tailored by Cunningham to fit the median. The final form of the Kuznetsov–Cunningham formula should be (Cunningham 2005):

$$x_{50} = AA_{t} q^{ - 0.8} Q^{1/6} \left( {\frac{\text{RWS}}{115}} \right)^{ - 19/30} C\left( A \right)$$

(8)

The factor C(A) is included in order to correct the predicted median size, to be determined experimentally from data in a given site; according to Cunningham, it would normally be within the range 0.5 < C(A) < 2.0. This suggests a prediction error expected of up to about 100%. This is larger than Kuznetsov’s (1973) error bounds, but while Kuznetsov’s A values lie in the range 7–13, Cunningham (1987) covers the much wider range 0.8–22.

Besides adapting Kuznetsov’s central size formula, Cunningham (1983) followed up the conclusion of the Soviet researchers that the fragmented rock could be described by the RR function and formulated an equation for the uniformity or shape index (n in Eq. 3); in its final form, after several corrections and refinements (Cunningham 1987, 2005), the shape index for the RR distribution of rock fragments is:

$$n = n_{s} \left( {2 - 0.03\frac{B}{d}} \right)^{0.5} \left( {\frac{1 + S/B}{2}} \right)^{0.5} \left( {1 - \frac{W}{B}} \right)\left( {\frac{{l_{\text{c}} }}{H}} \right)^{0.3} \left( {\frac{A}{6}} \right)^{0.3} C(n)$$

(9)

where W is drilling deviation, lc is charge length, H is bench height and ns is a factor that accounts for the delay precision:

$$n_{s} = 0.206 + \left( {1 - R_{s} /4} \right)^{0.8} ,\quad R_{s} = 6s_{t} /\Delta T$$

(10)

where st is the standard deviation of the initiation system. The factor C(n) is, again, a variable correction to match experimental data if available; no expected range is given to it.1

The Kuznetsov–Cunningham formula is physically sound, as previously discussed. Similar forms may be obtained applying asteroid collision principles (Holsapple and Schmidt 1987; Housen and Holsapple 1990) as shown by Ouchterlony (2009b). However, the experimental data supporting the median size expression are, as mentioned above, extremely limited. Furthermore, no experimental data supporting the shape index formula seem to have been reported. The initial shape index dependence on geometry appears to originate in 2D blast modeling by Lownds (1983) (Cunninghan 1983, p. 441).2

The assessment of the Kuz-Ram model by, e.g., the JKMRC (Kanchibotla et al. 1999; Thornton et al. 2001; Brunton et al. 2001) and other publications (e.g., Ford 1997; Morin and Ficarazzo 2006; Gheibie et al. 2009a, b; Hafsaoui and Talhi 2009; Strelec et al. 2011; Tosun et al. 2014) seldom include hard experimental data and are often obscured by the lack of real knowledge of the resulting muckpile fragmentation, which hinders a reliable error assessment. A generally accepted weakness of the Kuz-Ram model is that it normally predicts too few fines in a muck pile. This led to model extensions involving one RR function for the coarse material and another for the fines, the crush zone model (CZM, Kanchibotla et al. 1999; Thornton et al. 2001; Brunton et al. 2001) and the two-component model (TCM, Djordjevic 1999) from the JKMRC. Of these, the CZM has become the one more used. The CZM rests on the assumption that the fines come from a volume around the borehole in which the rock may fail under compression (it is ‘crushed’); the radius of the crush zone is:

$$r_{\text{c}} = r_{\text{b}} \sqrt {\frac{{P_{\text{b}} }}{{\sigma_{\text{c}} }}}$$

(11)

where rb is the borehole radius, Pb is the borehole pressure and σc is the uniaxial compressive strength. The crushed volume is a fraction Fc of the volume excavated by each borehole, BHS:

$$F_{\text{c}} = \frac{{\pi \left( {r_{\text{c}}^{2} - r_{\text{b}}^{2} } \right)\left( {H - l_{\text{s}} } \right)}}{\text{BHS}}$$

(12)

where ls is the stemming length. The crushed zone maximum size of fragments is assumed to be 1 mm. The CZM uses the Kuz-Ram model3 for the coarse part (above 50% passing for competent rock, σc > 50 MPa, and 90% for soft rock, σc < 10 MPa), below which a second RR function is used, defined so as to include (for the competent rock case) the (x50, 0.5) and (1 mm, Fc) points; for the soft rock case, the grafting point is (x90, 0.9).4 Such a ‘fines’ RR function is defined by x50 and a uniformity index nf:

$$n_{f} = \frac{{\ln \left[ { - \frac{{\ln \left( {1 - F_{c} } \right)}}{\ln 2}} \right]}}{{\ln \left[ {1/x_{50} \left( {\text{mm}} \right)} \right]}}$$

(13)

An analogous expression can be obtained for the soft rock case.

Both the Kuz-Ram model and its CZM extension have been assessed with the data set of 169 blasts that is described in Sect. 3. Figure 1 shows the logarithmic errors of the size prediction:

$$e_{L} = { \ln }\left( {\frac{{x^{*} }}{x}} \right)$$

(14)

where x* and x are predicted and data sizes, respectively.5

The error of the Kuz-Ram and crush zone predictions in about half of the cases (the interquartile range) lies within an approximate range [−0.6, 0.4], roughly equivalent to relative errors −75 to +50% in nearly the whole percentage passing range. The prediction is noticeably negative-biased (the sizes predicted are smaller than the data) in most of the range except the upper end. Other conclusions from Fig. 1 are that the largest bias occurs in the central zone, which suggests that (1) the median size formula has a limited predictive capability in terms of accuracy, with a systematic error of some −30%; the precision (i.e., the random error around the central value) is also limited, with the interquartile range of about [−75, +50%]; this prediction error is larger than the assessment by Ouchterlony (2016a) on the original Kuznetsov’s formula (−42 to 55% maximum error), though the number of blasts used in the present case more than quadruples the data set used by Kuznetsov; (2) the prediction in the extremes of the range analyzed is somewhat better than in the central range which means that also the uniformity (the overall slope in the size/passing plot in log–log) of the distribution is generally overestimated.

The data from the seminal small-scale (bench height of about 1 m) blasting tests by Otterness et al. (1991) underscore the above. Plotting some percentile sizes as a function of the powder factor from a selection of 10 blasts of nearly identical geometry gives the results shown in Fig. 2. Power laws fit each percentile size quite well (the determination coefficients are given in the figure; both the pre-factor and the exponent are significant in all the fits, and the maximum p values are 0.002 for the pre-factor and 0.003 for the exponent). Ouchterlony et al. (2016) show that such pattern requires that (1) the RR exponent would need to be variable at different percentiles (which would require a different function—e.g., a piecewise RR with variable exponent), and (2) its value should depend on the specific charge; only if the exponent of the power law \(x_{P} = {\text{const}}/q^{\kappa }\) is constant in some percentile interval, then the RR exponent might not depend on q, but it would still need to vary at different percentiles. This fundamental discrepancy with the experimental evidence lies at the core that the RR function with a single n value is a poor description of the fragmentation data, and a severe hindrance for its use in state-of-the-art fragmentation prediction models.

The xP versus specific charge convergent lines pattern (which we call ‘fan’ plots) is a general characteristic of fragmentation by blasting. In fact, most of the groups of data that form the basis for this work show the same behavior as the data in Fig. 2; power laws fit the percentile size data to the powder factor quite well, and the exponent is a function of P. In well-controlled model-scale blasts, this behavior is extremely well developed; the consequences of that are discussed in Ouchterlony et al. (2016).

The facts above are not described by any known credible theory of blast fragmentation that favors a specific fragment size distribution. On the contrary, they speak in favor of developing prediction equations for blast fragmentation that do not depend on any specific size distribution function. Size and passing data have a clear physical meaning, whereas shape, or uniformity, indices are just a reduced interpretation of the data through some particular size distribution. A median, or a 20 percentile size, even a 63.2 percentile like xc, have their full meaning without any additional statement; however, a shape index has no real meaning unless it is tied to a particular distribution—a RR, a Swebrec, a lognormal, a maximum-size-transformed RR one, etc.

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