A Case Study on Trim Blast Fragmentation Optimization Using the MBF Model and the MSW Blast Vibration Model at an Open Pit Mine in Canada

Abstract

A project for monitoring, analysis, and modelling trim blasting at an open pit copper/molybdenum mine in Canada in 2021 was conducted using Orica’s in-house developed proprietary tools, which includes the accelerometer near-field vibration monitoring system, the Multiple Seed Waveform near-field and far-field blast vibration model, and the multiple blasthole fragmentation model. Both models are built on the fundamentals and empirical aspects of blasting mechanics. They input full blast design parameters and site-specific information from the near-field signature hole blast vibration. The blast vibration at the highwall from trim blasts was predicted with multiple seed waveform blast vibration model. The effects of parameters from the trim blast design scenarios on fragmentation and vibration at the highwall were simulated at the mine site. Improved blast designs for finer fragmentation and controlled vibration were proposed based on the modelling. The multiple blasthole fragmentation model can simulate real blast designs in three dimensions with the actual geometry of the blasted overburden and free faces. The multiple blasthole fragmentation model can explicitly model all blast design parameters, including explosive properties, loading configuration, and delay timing of each explosive charge. The multiple seed waveform blast vibration model uses multiple seed waveforms at different distances from a blasthole and is suitable for near-field blast vibration at the highwalls. Through the application of the two models, various design scenarios can be explored, and rock fragmentation can be optimized while controlling blast vibration levels. These models have been successfully applied to several open pit mines and quarries around the world, including Canada, the United States, and Africa.

Highlights

• A case application of innovative technologies to optimizing field trim blast designs.

• New field technique to collect site input data and evaluate effectiveness of presplit blasting.

• Near-field blast vibration monitoring and modelling in parallel with fragmentation modelling.

Appendices

Appendix A: PPV Relating to Three-Dimensional Strain (Yang 2016)

Yang (2012a) related blast vibration to dynamic strain. The following analysis relates the PPV to the dynamic strain. In order to facilitate the derivation of the relationship between the PPV and the dynamic strain in blast vibration, principal wavelength and principal frequency of a blast vibration waveform are described below. Principal frequency (f_p) is defined from the zero-crossing time interval (Δt) at the peak vibration amplitude (Yang 2012a, 2012b; Yang 2015):

$$f_{p} = \frac{1}{2 \cdot \Delta t}.$$

(1)

Figure 20 illustrates the concepts of the principal frequency (f_p). For a sinusoidal wave, f_p becomes the frequency of the sinusoidal wave. The principal wavelength is

$$\lambda_{p} = 2\Delta t \cdot c,$$

(2)

where c is the p-wave velocity of the ground.

Fig. 20

Zero-crossing time interval (Δt) for principal frequency and wavelength

Figure 21 shows the particle velocity vector ($\vec{\nu }(\vec{r},t),\vec{\nu }(\vec{r} + \Delta \vec{r},t)$) and displacement ($u(t),\,\upsilon (t),\,w(t)$) produced by detonation of explosive charges at two adjacent points ($\Delta \vec{r}$ is small). At any time t, the relative displacement ($\Delta u(t),\Delta \nu (t),\Delta w(t)$) between the two points A (with a location vector $\vec{r}$) and B (with a location vector $\vec{r} + \Delta \vec{r}$ ) produces the strain E($\Delta \vec{r}$, t) (Eq. 3) at the point A (Spencer 1980):

$$E\left( {\vec{r},t} \right) = \left| {\begin{array}{*{20}c} {\frac{\partial u}{{\partial x}}} & {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} & {\frac{\partial w}{{\partial x}}} \\ {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} & {\frac{\partial v}{{\partial y}}} & {\frac{\partial v}{{\partial z}} + \frac{\partial w}{{\partial y}}} \\ {\frac{\partial u}{{\partial z}} + \frac{\partial w}{{\partial x}}} & {\frac{\partial v}{{\partial z}} + \frac{\partial w}{{\partial y}}} & {\frac{\partial w}{{\partial z}}} \\ \end{array} } \right|.$$

(3)

Fig. 21

Particle velocity and displacement produced by blast vibration at two adjacent points

The displacement gradient in (3) above at a time t is defined as

$$\frac{\partial u}{{\partial x}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{u(x + \Delta x,t) - u(x,t)}{{\Delta x}}.$$

The components of the strain are related to the relative displacement at any time instance t as per the following equations Yang (2012a, b):

$$\begin{gathered} \Delta u(t) = \frac{\partial u}{{\partial x}}(x_{b} - x_{a} ) + \frac{\partial u}{{\partial y}}(y_{b} - y_{a} ) + \frac{\partial u}{{\partial z}}(z_{b} - z_{a} ) \hfill \\ \Delta v(t) = \frac{\partial v}{{\partial x}}(x_{b} - x_{a} ) + \frac{\partial v}{{\partial y}}(y_{b} - y_{a} ) + \frac{\partial v}{{\partial z}}(z_{b} - z_{a} ) \hfill \\ \Delta w(t) = \frac{\partial w}{{\partial x}}(x_{b} - x_{a} ) + \frac{\partial w}{{\partial y}}(y_{b} - y_{a} ) + \frac{\partial w}{{\partial z}}(z_{b} - z_{a} ). \hfill \\ \end{gathered}$$

(4)

The magnitude of the relative displacement between A and B can be expressed as

$$d(t) = \sqrt {\Delta u^{2} (t) + \Delta v^{2} (t) + \Delta w^{2} (t)} .$$

(5)

On the other hand,

$$d(t) = \left| {\int\limits_{0}^{t} {\left[ {\vec{v}(\vec{r} + \Delta \vec{r},\tau ) - \vec{v}(\vec{r},\tau )} \right]d\tau } } \right|,$$

(6)

$$d(t) = \left| {\int\limits_{0}^{t} {\left[ {\vec{v}(\vec{r} + \Delta \vec{r},\tau ) - \vec{v}(\vec{r},\tau )} \right]d\tau } } \right| \le \int\limits_{0}^{t} {\left| {\vec{v}(\vec{r} + \Delta \vec{r},\tau ) - \vec{v}(\vec{r},\tau )} \right|} d\tau ,$$

(7)

where $\vec{\nu }(\mathop{r}\limits^{\rightharpoonup} ,\tau )$ and $\vec{\nu }(\vec{r} + \Delta \vec{r},\tau )$ are the velocity vector of the points A and B.

We take the length (e.g. calculation grid space) of $\Delta \vec{r}$ to be less than a quarter of the principal wavelength (Yang 2012a) of the particle velocity wave and the phase difference between $\vec{\nu }(\vec{r},t)$ and $\vec{\nu }(\vec{r} + \Delta \vec{r},t$ to be less than a quarter of the principal period (2 $\Delta t$) of the particle velocity wave, as shown in Fig. 22. Consequently, at any time, the difference in particle velocity between the points A and B cannot be greater than the PPV. From the mean value theorem for integrals:

$$d(t) \le \int\limits_{0}^{t} {\left| {\vec{v}(\vec{r} + \Delta \vec{r},\tau ) - \vec{v}(\vec{r},\tau )} \right|} d\tau = \left| {\vec{v}(\vec{r} + \Delta \vec{r},\xi ) - \vec{v}(\vec{r},\xi )} \right|t \le PPV \cdot D,$$

(8)

where $0 \le \xi \le t \le D$. D is the duration of the waveforms. The equation above shows that PPV is related to the upper bound of the relative displacement d(t) between the two adjacent points A and B.

Fig. 22

A sketch of time synchronized waves at Points A and B

If the directional cosines of the vector of PPV are known (n,m,l) and if two additional points C and D are chosen such that the four points (A, B, C, D) are not co-planar and they are within distances less than a quarter of the principal wavelength, as shown in Fig. 23, the strain in Eq. (3) corresponding to the largest possible relative displacement adjacent to the point A can be estimated by solving nine independent Eq. (9) to obtain the nine displacement gradients ($\left( {\frac{\partial u}{{\partial x}},\frac{\partial u}{{\partial y}},\frac{\partial u}{{\partial z}},\frac{\partial v}{{\partial x}},\frac{\partial v}{{\partial y}},\frac{\partial v}{{\partial z}},\frac{\partial w}{{\partial x}},\frac{\partial w}{{\partial y}},\frac{\partial w}{{\partial z}}} \right)$):

$$\begin{gathered} n \cdot PPV \cdot D = \frac{\partial u}{{\partial x}}(x_{b} - x_{a} ) + \frac{\partial u}{{\partial y}}(y_{b} - y_{a} ) + \frac{\partial u}{{\partial z}}(z_{b} - z_{a} ) \hfill \\ m \cdot PPV \cdot D = \frac{\partial v}{{\partial x}}(x_{b} - x_{a} ) + \frac{\partial v}{{\partial y}}(y_{b} - y_{a} ) + \frac{\partial v}{{\partial z}}(z_{b} - z_{a} ) \hfill \\ l \cdot PPV \cdot D = \frac{\partial w}{{\partial x}}(x_{b} - x_{a} ) + \frac{\partial w}{{\partial y}}(y_{b} - y_{a} ) + \frac{\partial w}{{\partial z}}(z_{b} - z_{a} ) \hfill \\ n \cdot PPV \cdot D = \frac{\partial u}{{\partial x}}(x_{c} - x_{a} ) + \frac{\partial u}{{\partial y}}(y_{c} - y_{a} ) + \frac{\partial u}{{\partial z}}(z_{c} - z_{a} ) \hfill \\ m \cdot PPV \cdot D = \frac{\partial v}{{\partial x}}(x_{c} - x_{a} ) + \frac{\partial v}{{\partial y}}(y_{c} - y_{a} ) + \frac{\partial v}{{\partial z}}(z_{c} - z_{a} ) \hfill \\ l \cdot PPV \cdot D = \frac{\partial w}{{\partial x}}(x_{c} - x_{a} ) + \frac{\partial w}{{\partial y}}(y_{c} - y_{a} ) + \frac{\partial w}{{\partial z}}(z_{c} - z_{a} ) \hfill \\ n \cdot PPV \cdot D = \frac{\partial u}{{\partial x}}(x_{d} - x_{a} ) + \frac{\partial u}{{\partial y}}(y_{d} - y_{a} ) + \frac{\partial u}{{\partial z}}(z_{d} - z_{a} ) \hfill \\ m \cdot PPV \cdot D = \frac{\partial v}{{\partial x}}(x_{d} - x_{a} ) + \frac{\partial v}{{\partial y}}(y_{d} - y_{a} ) + \frac{\partial v}{{\partial z}}(z_{d} - z_{a} ) \hfill \\ l \cdot PPV \cdot D = \frac{\partial w}{{\partial x}}(x_{d} - x_{a} ) + \frac{\partial w}{{\partial y}}(y_{d} - y_{a} ) + \frac{\partial w}{{\partial z}}(z_{d} - z_{a} ). \hfill \\ \end{gathered}$$

(9)

Fig. 23

Point A and adjacent three points B, C, and D, which are not co-planar

Equation 9 demonstrate that the PPV is related to the largest possible three-dimensional dynamic strain.

Appendix B: PPV as Controlling Parameter for Rock Fragmentation (Yang 2015)

Near a blasthole, the PPV is directly related to the borehole pressure and is, therefore, easy to estimate. The blast vibration PPV and its attenuation with the charge weight scaled distance can be easily measured by signature hole blast monitoring. The PPV in rock induced from explosive detonation is used in the MBF model as the key parameter that determines the size distribution of the rock fragmentation (Yang 2015), which is calculated in the vicinity of explosive charges (Appendix C).

Nucleation in Seaman’s model (Seaman et al 1976) occurs as the addition of new voids to the existing set. The number of voids with random orientation and distribution nucleated is governed by a nucleation rate function for both ductile and brittle materials:

$$\mathop N\limits^{ \bullet } = \mathop {N_{0} }\limits^{ \bullet } \exp [\left( {P_{s} - P_{n0} } \right)/P_{1} ]\,P_{s} > P_{n0}$$

$$= 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,P_{s} \le P_{no} ,$$

(10)

where$\mathop {N_{0} }\limits^{ \bullet }$, $P_{n0}$, and $P_{1}$ are material constants. $P_{s}$ is the void pressure in the solid material, normal to the plane of the cracks, and $P{}_{n0}$ is the threshold pressure for crack nucleation. For simplicity, a constant rock breaking time (Δt) is assumed at a site, Eq. (10) can be rewritten as

$$\mathop N\limits^{ \bullet } {\Delta t} = \mathop {N_{0} }\limits^{ \bullet } {\Delta }t{ }\exp \left[ {\frac{{\left( {P_{s} {\Delta }t - P_{n0} {\Delta }t} \right)}}{{P_{1} {\Delta t}}}} \right].$$

Let $N = \mathop N\limits^{ \bullet } \Delta t$,$N_{0} = \mathop {N_{0} }\limits^{ \bullet } \Delta t$, $\frac{{P}_{s}\mathrm{\Delta t}-{P}_{n0}\mathrm{\Delta t}}{{P}_{1}\mathrm{\Delta t}}=\frac{\left(PPV-{PPV}_{0 }\right){\rho }_{r}}{{P}_{1}\mathrm{\Delta t}}= \frac{PPV-{PPV}_{0 }}{\frac{{P}_{1}\mathrm{\Delta t}}{{\rho }_{r}}}=\frac{PPV-{PPV}_{0 }}{\eta }$, $\eta =\frac{{p}_{1\Delta t}}{{\rho }_{r}},$ where ${\rho }_{r}$ is the rock mass per unit volume and Impulse Theorem is applied above, so, we have

$$N={N}_{0}\mathit{exp}\left(\frac{PPV-PP{V}_{0}}{\eta }\right),$$

where N₀ and $\eta$ are material constants, PPV (m/s) is the peak particle velocity at the calculation grid point, PPV₀ is the threshold PPV for crack nucleation. The average fragment size is inversely related to the number of fractures:

$$\overline{x} = \frac{{x_{0} }}{{N_{0} \exp \left( {\frac{{PPV - PPV_{0} }}{\eta }} \right)}},$$

(11)

where $\frac{{x_{0} }}{{N_{0} }}$ = the effective size of the initial fragment (m), ${x}_{0}$ is the average size of the in situ blocks defined by jointing, and ${N}_{0}$ is the average number of effective fractures within a block and is expected to increase with the intensity of the shock wave. PPV₀ may be related to the critical strain for rock breakage${PPV}_{0}=c{\varepsilon }_{c}$, where c is the sonic velocity of the rock. ${\varepsilon }_{c}$ may be estimated by ${\varepsilon }_{c}=\frac{UCS}{E}$.

UCS is the dynamic uniaxial compressive strength. In the future, with more measurement, UCS may be modelled as a monotonically increasing function of PPV to account for the strain rate effect on rock breakage. The higher the PPV, the higher the strain rate and the higher the rock breakage critical strain (${\varepsilon }_{c}$), and, therefore, the higher the UCS should be.

During blasting, rock mass near blastholes can have variable deformability, therefore, various failure modes at different points can coexist at a time instance, such as shear failure, tensile cracking, or compressive crushing. Explicitly to model each mode may not be practical. UCS is one of the most important mechanical properties of rocks and one of the easiest to measure among other properties. It is most frequently used strength index. In addition, from UCS, many other rock strength indexes can be estimated such as shear strength, tensile strength, and Young’s modulus (Goodman 1989). Therefore, MBF model chooses dynamic UCS of the rock at a site as an input to simplify the rock fragmentation modelling. Inaccuracies of using UCS in the model could be contained in the calibration parameters after the model is calibrated at a site.

η is related to rock brittleness: for more brittle rock η is smaller; more ductile rock, η is larger. On the other hand, for the same rock, the higher the strain rate, the more micro cracks could participate in the rock fracture process and finer rock fragments could be generated. Therefore, η tends to become small.

For simplicity, the MBF model uses the Rosin Rammler (${R\left(x\right)=1- {e}^{-\left(\frac{x}{\overline{x} }\right)}}^{n}$) function to describe the size distribution of the rock fragments at a point. R(x) is the cumulative volumetric fraction for fragment sizes smaller than x. The mean fragment size $\overline{x }$ is calculated from Eq. (11). Although the Rosin Rammler function has some deficiencies (Sanchidrián and Ouchterlony 2017), it is simple with only two parameters to fit the measurement data. Future development of the MBF model is considered incorporating more advanced functions.

In the present modelling, $\delta =\frac{{x}_{0}}{{N}_{0}}$, $\eta$ and n are hard to measure. For example, ${N}_{0}$ is an increasing function with the intensity of the stress/strain generated from explosive charge detonations. When the stress/strain is low, only few pre-existing joints divide rock into sub-blocks. However, when the stress/strain is high, more pre-existing joints can be activated generating more sub-blocks. Only three parameters: $\delta =\frac{{x}_{0}}{{N}_{0}}$, η, and n are used as calibration parameters since the blast design parameters are all explicitly input to the MBF model.

Appendix C: A Complete Scaling Law to Model VoD, Nonlinearity, and Scale Up (Yang 2018)

A near-field signature hole PPV versus scaled distance is established from the site characterization using the near-field signature hole vibration monitoring for each site. Each charge is treated as assemble of charge elements with length equal to its diameter. For each charge element, a complete (dual) charge weight scaled distance law is established in the model—two scaling laws joined at the scaled distance sd_c = 0.5 m/kg^0.5, where it is assumed that the blast vibration can be safely recorded as the nearest distance (Fig. 24). The borehole wall PPV (${u}_{b})$ and the corresponding scaled distance ${sd}_{b}$ are obtained. ${u}_{b}$ is obtained from the borehole pressure that is determined from the explosive property, blasthole diameter, and loading coupling and the rock Hugoniot parameters. The ${sd}_{b}$ is calculated by assuming the contributing charge segment has the length of the diameter of the charge and the distance is the blasthole radius.

Fig. 24

Two scaling laws joined at the joining point on a logarithmic scale of the PPV axis

The complete (dual) charge weight scaling law for a charge element is intended to model the contributions to dynamic pressure or peak particle velocity at the vicinity of explosive charges from their detonation. The properties of each explosive charge and nonlinearity of rock in the complete PPV charge weight scaling law are modelled explicitly. At the close vicinity of a borehole wall, only charge elements of which the shock wave arrival time within the contributing time window contribute to the pressure rather than the whole charge. The effects from multiple charges are obtained using a variable time window associate with each charge element and a non-linear charge weight integration. Both the best-fit (average) and the 97.5% upper bound regressions for the charge weight scaling law are used as the base for Monte Carlo modelling of the geological random effects causing the variation of PPV (Yang 2018).