Scientific Engine Reference
BlastCAD’s backend analytics engine implements a deterministic set of industry-standard empirical models for blast performance prediction. This page documents the mathematical basis, constants, and calibration guidance for each model.
Table of contents
- 1. Fragmentation — Kuz-Ram Model
- 2. Ground Vibration — Scaled Distance Model
- 3. Livingston Crater Theory
- 4. Volume Computation — Voronoi Tessellation
- 5. Explosive Energy
- 6. Detonation Timing — Scatter Analysis
- 7. Flyrock Estimation
- References
1. Fragmentation — Kuz-Ram Model
The Kuz-Ram model is the combination of three separate contributions, forming the most widely validated fragmentation prediction framework in the mining industry.
1.1 Mean Fragment Size (Kuznetsov, 1973 / Cunningham, 1983)
\[x_{mean} = A \times \left(\frac{V_0}{Q}\right)^{0.8} \times \left(Q \times \frac{RWS}{115}\right)^{\frac{1}{6}}\]| Parameter | Description |
|---|---|
| $A$ | Rock Factor (5–13+) |
| $V_0$ | Volume of rock broken per hole (m³) |
| $Q$ | Charge mass per hole (kg) |
| $RWS$ | Relative Weight Strength (ANFO = 100) |
1.2 Uniformity Index (Cunningham, 1983)
\[n = \left(2.2 - 14\frac{B}{d}\right) \times \left(1 - \frac{W}{B}\right)^{0.5} \times \left(\frac{1+S/B}{2}\right)^{0.1} \times \left(\frac{L_c}{H}\right)^{0.1}\]1.3 Rosin-Rammler Distribution (Rosin & Rammler, 1933)
\[P(x) = \left[1 - e^{-\left(\frac{x}{x_c}\right)^n}\right] \times 100\] \[x_c = \frac{x_{mean}}{0.693^{1/n}}\]Output KPIs: P20, P50 (median), P80, oversize fraction.
See Fragmentation Analysis for the full parameter guide and tuning instructions.
2. Ground Vibration — Scaled Distance Model
2.1 Peak Particle Velocity (Attewell & Farmer, 1976)
\[PPV = K \times \left(\frac{R}{\sqrt{MIC}}\right)^{-\alpha}\]| Parameter | Default | Description |
|---|---|---|
| $K$ | 1140 | Site transmission constant |
| $\alpha$ | 1.6 | Attenuation exponent |
| $R$ | User input (m) | Distance to receptor |
| $MIC$ | Computed (kg) | Maximum Instantaneous Charge (8 ms window) |
2.2 Minimum Safe Distance
\[R_{min} = \sqrt{MIC} \times \left(\frac{K}{V_{limit}}\right)^{1/\alpha}\]Where $V_{limit}$ is the regulatory PPV limit in mm/s.
Applicable standard: AS 2187.2-2006 (Australian standard for storage and use of explosives).
3. Livingston Crater Theory
3.1 Critical Radius
\[R_c = C_b \times Q^{1/3}\]3.2 Breakage Radius
\[R_b = C_b \times \Delta^{1/3} \times Q^{1/3}\]| Parameter | Default | Description |
|---|---|---|
| $C_b$ | 0.84 | Strain energy factor (hard rock) |
| $\Delta$ | 0.6 | Depth ratio ($B / R_c$) |
| $Q$ | Computed (kg) | Charge mass |
BlastCAD uses $R_b$ as the recommended burden. Holes where the design burden exceeds $R_b$ are flagged.
4. Volume Computation — Voronoi Tessellation
Per-hole breakage volume is computed using a Voronoi tessellation of all hole collar positions projected onto the ring plane:
- Project all collar coordinates into the 2D ring plane (U, V coordinates).
- Compute the Voronoi diagram — each hole’s cell defines its area of influence $A_i$.
- Multiply by the ring burden: $V_{0,i} = A_i \times B$
For single-hole rings or rings where Voronoi is degenerate, the fallback method averages the burden area across the ring circumference.
5. Explosive Energy
Energy per hole:
\[E_{hole} = \sum_{segments} m_{seg} \times e_{exp}\]Where $e_{exp}$ is the specific energy of the explosive in MJ/kg, derived from RWS:
\[e_{exp} \approx e_{ANFO} \times \frac{RWS}{100} = 3.85 \times \frac{RWS}{100} \quad \text{(MJ/kg)}\]Powder Factor:
\[PF = \frac{\sum m_{holes}}{\sum V_{holes} \times \rho_{rock}} \quad \text{(kg/t)}\]Default rock density: 2700 kg/m³ (adjustable in Project Settings).
6. Detonation Timing — Scatter Analysis
For each detonator with nominal delay $t_0$ and scatter percentage $\sigma$:
\[t_{actual} \in \left[t_0 - t_0 \frac{\sigma}{100},\; t_0 + t_0 \frac{\sigma}{100}\right]\]The worst-case MIC is computed by placing all holes at the boundary of their scatter window such that the maximum simultaneous charge is maximised within any 8 ms window.
7. Flyrock Estimation
\[D_{fly} = \frac{V_0^2 \sin(2\theta_{worst})}{g}\]Worst-case ejection angle $\theta_{worst} = 45°$ gives $\sin(90°) = 1$, maximizing range.
Ejection velocity $V_0$ is estimated empirically from the powder factor and stemming column length. The model is conservative — actual flyrock is typically less than the predicted maximum.
References
- Kuznetsov, A. (1973). The mean diameter of the fragments formed by blasting rock. Soviet Mining Science, 9(2), 144–148.
- Cunningham, C.V.B. (1983). The Kuz-Ram model for prediction of fragmentation from blasting. First Int. Symposium on Rock Fragmentation by Blasting.
- Rosin, P. & Rammler, E. (1933). The laws governing the fineness of powdered coal. Journal of the Institute of Fuel.
- Attewell, P.B. & Farmer, I.W. (1976). Principles of Engineering Geology. Chapman & Hall.
- Livingston, C.W. (1956). Fundamentals of rock failure. Quarterly of the Colorado School of Mines, 51(3).
- Langefors, U. & Kihlström, B. (1963). The Modern Technique of Rock Blasting. Wiley.
- AS 2187.2-2006. Explosives — Storage and use — Part 2: Use of explosives. Standards Australia.