🎓 Lesson 5
D3
Uncertainty Quantification Fundamentals
Uncertainty quantification is the process of measuring how much we don’t know—and how that lack of knowledge affects our predictions—when simulating mining or blasting operations.
🎯 Learning Objectives
- ✓ Explain the difference between aleatory and epistemic uncertainty using blast design examples
- ✓ Calculate output uncertainty (e.g., fragmentation size distribution) using Monte Carlo sampling given input parameter distributions
- ✓ Analyze sensitivity of blast outcomes to rock mass properties using Sobol indices
- ✓ Apply polynomial chaos expansion to accelerate uncertainty propagation in fragment size prediction models
📖 Why This Matters
In open-pit mines, a 15% error in rock strength estimation can cause overbreak by 2–3 m—costing $2M/year in rehandling and delaying production schedules. Digital twins promise real-time optimization, but without quantifying uncertainty, they mislead more than they guide. UQ transforms simulation from 'what-if' guesswork into auditable, defensible engineering judgment—critical for regulatory compliance (e.g., MSHA Part 46), blast permit approvals, and ESG reporting on slope stability risks.
📘 Core Principles
UQ begins with uncertainty identification: sources include geotechnical assay variability (e.g., RQD ±12%), drill deviation (±0.5°), and explosive energy variation (±8% per ANFO batch). Next, uncertainty characterization assigns probability distributions—lognormal for rock strength, uniform for burden tolerance—based on field data and expert elicitation. Propagation then maps input uncertainties through computational models (e.g., PFC2D fragmentation simulators) to quantify output uncertainty. Finally, sensitivity analysis identifies which inputs dominate output variance—guiding where to invest in better data collection (e.g., targeted core logging vs. additional seismic surveys).
📐 First-Order Uncertainty Propagation (Linear Approximation)
Used when input uncertainties are small and model responses are approximately linear; provides fast analytical estimates of output standard deviation without full Monte Carlo simulation.
Error Propagation Law (First-Order)
(σ_f/f)² ≈ Σ (∂ln f/∂x_i)² · (σ_{x_i}/x_i)²Estimates relative standard deviation of model output f given independent, small uncertainties in inputs x_i.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| σ_f | Standard deviation of output | same as f | Quantifies spread of predicted outcome (e.g., mm for D₅₀) |
| f | Model output | depends on context | Quantity being predicted (e.g., fragment size, throw distance, vibration PPV) |
| x_i | Input parameter i | varies | Uncertain input (e.g., burden in meters, UCS in MPa) |
| σ_{x_i} | Standard deviation of input i | same as x_i | Measure of input variability from assays, surveys, or lab tests |
Typical Ranges:
Hard rock bench blasting: 3–8% relative uncertainty for geometric parameters
Rock mass strength (UCS): 15–30% coefficient of variation in heterogeneous deposits
💡 Worked Example
Problem: A blast design model predicts fragment size D₅₀ (mm) as D₅₀ = k × (B / S)^α × ρ^β, where B = burden (m), S = spacing (m), ρ = rock density (g/cm³), k = 1200, α = 0.8, β = −0.3. Given: B = 4.2 ± 0.15 m (3.6% CV), S = 5.6 ± 0.2 m (3.6% CV), ρ = 2.65 ± 0.08 g/cm³ (3.0% CV). Calculate relative uncertainty in D₅₀.
1.
Step 1: Apply logarithmic differentiation: ln(D₅₀) = ln(k) + α·ln(B) − α·ln(S) + β·ln(ρ)
2.
Step 2: Square relative uncertainties: (σ_D₅₀/D₅₀)² ≈ α²·(σ_B/B)² + α²·(σ_S/S)² + β²·(σ_ρ/ρ)²
3.
Step 3: Plug values: (σ_D₅₀/D₅₀)² = (0.8)²×(0.036)² + (0.8)²×(0.036)² + (−0.3)²×(0.030)² = 0.00207 + 0.00207 + 0.00027 = 0.00441 → σ_D₅₀/D₅₀ ≈ √0.00441 ≈ 0.0664 = 6.6%
Answer:
The relative uncertainty in D₅₀ is 6.6%, meaning predicted D₅₀ = 42 mm has a 95% confidence interval of ~39–45 mm—well within acceptable range for crusher feed control (±10%).
🏗️ Real-World Application
At Newmont’s Boddington Mine (WA), UQ was embedded in their digital twin to forecast muck pile fragmentation variability. Input uncertainties included GSI (35–52, triangular), UCS (58–112 MPa, lognormal), and explosive velocity (4200–4800 m/s, uniform). Using 10,000 Latin Hypercube samples in RS2 slope + FragSim coupling, they found D₅₀ exceeded 60 mm in 12% of realizations—triggering automatic redesign of burden and delay timing. This reduced crusher liner wear by 22% and eliminated unplanned secondary breaking events for 14 consecutive months.