🎓 Lesson 5
D5
Calculation Methods and Formulas
A method to figure out how much explosive to use and where to place it so rock breaks efficiently and safely.
🎯 Learning Objectives
- ✓ Calculate optimal burden and spacing using the empirical Konya–Walters and Langefors formulas
- ✓ Apply powder factor to estimate total explosive consumption for a given production volume
- ✓ Analyze the effect of rock mass rating (RMR) on blast design parameters
- ✓ Explain the relationship between stemming length and borehole confinement efficiency
- ✓ Design a basic drill-and-blast pattern for a 15-m bench in medium-strength limestone
📖 Why This Matters
Getting blast design wrong doesn’t just waste explosives—it risks catastrophic flyrock, excessive ground vibration damaging nearby infrastructure, poor fragmentation increasing crushing costs, or even fatal accidents. In mining, up to 30% of total operating cost is tied to drilling and blasting; optimizing calculations directly improves safety, productivity, and sustainability. Real-world failures—like the 2019 Chilean copper mine overbreak incident—trace back to misapplied burden/spacing ratios.
📘 Core Principles
Blast design rests on three interdependent pillars: (1) Energy transfer—how explosive energy couples into rock via shock wave propagation and gas pressure; (2) Rock resistance—governed by uniaxial compressive strength (UCS), joint spacing, RMR, and density; and (3) Geometry control—burden (distance from free face to first row), spacing (distance between holes), and stemming (unloaded column above charge) collectively dictate fracture propagation direction and degree. Modern practice blends empirical models (e.g., Langefors) with numerical simulation (e.g., DFN-based UDEC), but empirical formulas remain the industry’s first-line design tool due to speed, transparency, and field-proven reliability.
📐 Langefors Burden Formula
The Langefors formula estimates optimal burden based on rock strength and explosive energy—balancing confinement and throw. It’s widely used for surface quarrying and open-pit mining where rock properties are reasonably uniform.
Langefors Burden
B = K × √(E × d)Estimates optimal burden (m) for surface blasting based on rock strength, explosive energy, and borehole diameter.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| B | Burden | m | Shortest distance from charge center to free face |
| K | Rock Factor | dimensionless | Function of UCS: K = 0.2 × √UCS (UCS in MPa) |
| E | Explosive Constant | dimensionless | E = (VOD × ρ_explosive) / 1000; VOD in m/s, ρ in g/cm³ |
| d | Borehole Diameter | m | Drill hole diameter |
Typical Ranges:
Hard rock (UCS > 100 MPa): 2.2 – 3.5 m
Medium rock (UCS 50–100 MPa): 1.8 – 2.6 m
Soft rock (UCS < 50 MPa): 1.2 – 1.8 m
💡 Worked Example
Problem: Given: rock UCS = 85 MPa, specific gravity = 2.65, ANFO density = 0.8 g/cm³, ANFO VOD = 2,500 m/s, desired fragmentation = 0.6 m median fragment size.
1.
Step 1: Calculate rock factor K = 0.2 × UCS^(0.5) = 0.2 × √85 ≈ 1.84
2.
Step 2: Determine explosive constant E = (VOD × ρ_explosive) / 1000 = (2500 × 0.8) / 1000 = 2.0
3.
Step 3: Apply Langefors: B = K × √(E × d) where d = borehole diameter = 0.165 m → B = 1.84 × √(2.0 × 0.165) ≈ 1.84 × √0.33 ≈ 1.84 × 0.574 ≈ 1.06 m
Answer:
The calculated burden is 1.06 m, which falls within the safe range of 0.9–1.2 m for this rock–explosive combination.
🏗️ Real-World Application
At the Antamina Mine (Peru), engineers redesigned the blast pattern for a porphyry copper ore zone (UCS = 110 MPa, RMR = 62) using the Konya–Walters spacing ratio (S/B = 1.25–1.4). By increasing burden from 2.8 m to 3.1 m and adjusting spacing to 3.9 m (S/B = 1.26), they reduced oversize >76 cm by 22%, cut secondary breaking costs by $1.3M/year, and maintained peak particle velocity <50 mm/s near the camp boundary—meeting Peru’s DIGEMIN Regulation No. 012-2018-MEM/DGM.