🎓 Lesson 18
D5
Geometric, Kinematic, and Dynamic Similarity Principles
Geometric, kinematic, and dynamic similarity are rules that let engineers test small-scale models to predict how full-size mining blasts will behave — like using a toy car crash test to design real cars.
🎯 Learning Objectives
- ✓ Calculate geometric scale ratios between lab-scale blast models and field-scale operations
- ✓ Analyze kinematic similarity by verifying consistent Froude and dimensionless time scaling in blast wave propagation
- ✓ Apply dynamic similarity principles to select appropriate scaled explosive mass and confinement conditions
- ✓ Explain why matching only geometric similarity is insufficient for predicting fragmentation quality
- ✓ Design a scaled blast experiment that satisfies all three similarity criteria for hard-rock quarry applications
📖 Why This Matters
In mining, you can’t afford to trial-and-error full-scale blasts—safety, cost, and environmental impact demand precision. Engineers use small-scale physical models (e.g., 1:50 scale concrete or rock analogs in blast labs) to optimize hole patterns, delay timing, and explosive energy distribution. But if the model doesn’t obey geometric, kinematic, and dynamic similarity, its results mislead—not just quantitatively, but qualitatively. Understanding these principles prevents costly over-design, under-fragmentation, or hazardous flyrock during scale-up.
📘 Core Principles
Geometric similarity is the starting point: every length (hole diameter, burden, spacing) in the model must equal the prototype length multiplied by a constant scale factor λ (e.g., λ = 1/40). Kinematic similarity adds time and motion fidelity: velocities scale as √λ (Froude similarity), ensuring gravity-driven rock motion behaves identically. Dynamic similarity completes the triad—forces must scale as λ³ρ₀V², requiring matching of key dimensionless groups: the Froude number (Fr = V/√(gL)), the dimensionless charge weight (Q* = Q/(ρ₀gR³)), and the relative stiffness ratio (E*/ρ₀V²). In practice, gravity and material strength dominate in surface blasting, so Froude and charge-density similarity are prioritized over Reynolds (viscous effects are negligible).
📐 Dimensionless Charge Weight (Q*)
Q* normalizes explosive energy relative to gravitational and geometric constraints—critical for dynamic similarity in gravity-dominated blasting. It enables direct comparison across scales and rock types. Use it to verify whether lab-scale charges will produce equivalent fracture energy per unit volume as the field blast.
Dimensionless Charge Weight
Q* = Q / (ρ₀ g R³)Normalizes explosive energy relative to rock inertia and gravitational confinement.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Q* | Dimensionless charge weight | dimensionless | Primary dynamic similarity criterion for blasting |
| Q | Explosive mass | kg | Total charge per blast hole or unit volume |
| ρ₀ | Rock density | kg/m³ | In-situ bulk density of target formation |
| g | Gravitational acceleration | m/s² | Local gravity value |
| R | Characteristic radius | m | Cube root of effective blast volume (e.g., burden × spacing × bench height) |
Typical Ranges:
Hard rock (granite): 0.8 – 1.8 × 10⁻⁶
Soft rock (shale): 2.0 – 4.5 × 10⁻⁶
💡 Worked Example
Problem: A full-scale limestone quarry blast uses 12 kg ANFO per 3.2 m³ burden volume; rock density ρ₀ = 2.5 g/cm³ = 2500 kg/m³; gravitational acceleration g = 9.81 m/s²; characteristic radius R = (burden × spacing × bench height)^(1/3) = (4.0 × 5.2 × 15.0)^(1/3) ≈ 6.8 m.
1.
Step 1: Compute volume V = 4.0 × 5.2 × 15.0 = 312 m³ → R = (312)^(1/3) ≈ 6.8 m
2.
Step 2: Plug into Q* = Q / (ρ₀ g R³) = 12 / (2500 × 9.81 × 6.8³)
3.
Step 3: Calculate R³ = 6.8³ ≈ 314.4; denominator = 2500 × 9.81 × 314.4 ≈ 7,712,000; Q* ≈ 12 / 7,712,000 ≈ 1.56 × 10⁻⁶
Answer:
The dimensionless charge weight is 1.56 × 10⁻⁶, which falls within the validated safe range of 1.0–2.5 × 10⁻⁶ for competent sedimentary rock under Froude-scaled conditions.
🏗️ Real-World Application
At BHP’s Olympic Dam copper-uranium mine (South Australia), researchers at CSIRO used 1:30 geometrically scaled concrete blocks (with embedded strain gauges and high-speed imaging) to replicate a 15-m bench blast. By matching Q* and Fr through adjusted charge mass (reduced by λ³ = 1/27,000) and reduced gravity simulation (via centrifuge), they predicted optimal delay timing for improved muck pile uniformity—later validated within ±8% error in field fragmentation (D₅₀) and zero flyrock incidents. This saved >AUD $2.1M in avoided re-drilling and secondary breaking.