🎓 Lesson 15
D5
Sensitivity Index Calculation Lab
Sensitivity Index tells you how much a small change in a blasting parameter—like hole spacing or charge weight—will affect the overall cost of breaking rock.
🎯 Learning Objectives
- ✓ Calculate Sensitivity Index for burden, spacing, and powder factor using partial differentiation of empirical cost models
- ✓ Analyze which blast design parameter exerts the greatest influence on total unit cost in a given geotechnical context
- ✓ Explain the economic implications of high vs. low Sensitivity Index values when selecting between alternative drill-and-blast configurations
- ✓ Apply SI results to redesign a blast pattern that reduces unit cost by ≥8% without compromising fragmentation quality
📖 Why This Matters
In open-pit mining, 25–40% of total production cost stems from drilling and blasting—but not all design choices impact cost equally. The Sensitivity Index helps engineers cut costs intelligently: instead of blindly reducing powder factor (risking poor fragmentation), you identify *which* parameter—e.g., burden or delay timing—delivers the highest ROI per engineering hour spent optimizing it. This turns cost modeling from descriptive analytics into prescriptive decision-making.
📘 Core Principles
Sensitivity analysis in blasting economics rests on two pillars: (1) a validated production cost model—typically including drill hours, explosives cost, loading/fuel, and secondary breakage penalties—and (2) differentiable relationships linking blast geometry (burden B, spacing S, hole depth H) and explosive energy (powder factor PF) to outcomes like fragmentation (P80), muckpile shape, and diggability. The Sensitivity Index is formally defined as SIₓ = (∂C/∂x) × (x₀/C₀), where x₀ and C₀ are baseline values. For non-linear models (e.g., Kuz-Ram fragmentation coupled to shovel productivity), SI is computed numerically via finite-difference approximation around operational baselines. Crucially, SI is *context-dependent*: it shifts with rock strength, equipment fleet, and haul distance—making site-specific calibration essential.
📐 Key Calculation
The Sensitivity Index for any design variable x is calculated using finite-difference approximation to avoid complex symbolic differentiation of integrated cost models. This approach is industry-standard for field-deployable analysis.
Finite-Difference Sensitivity Index
SIₓ ≈ [(C(x₊δ) − C(x₋δ)) / (2δx₀)] × (x₀ / C₀)Numerical approximation of normalized partial derivative for variable x in a production cost model C.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| SIₓ | Sensitivity Index for variable x | dimensionless | Quantifies percent change in total unit cost per percent change in variable x |
| C(x₊δ) | Unit cost at +δ% perturbation of x | $/ton | Cost evaluated after increasing x by small percentage δ (typically 3–5%) |
| C(x₋δ) | Unit cost at −δ% perturbation of x | $/ton | Cost evaluated after decreasing x by same δ |
| x₀ | Baseline value of variable x | m, kg/ton, ms, etc. | Original design value used in current operation |
| C₀ | Baseline unit cost | $/ton | Total delivered cost per tonne under baseline design |
Typical Ranges:
Burden in hard granite: 0.6 – 1.1
Powder factor in weathered limestone: 0.3 – 0.7
Delay timing (ms) in multi-row blasts: 0.1 – 0.4
💡 Worked Example
Problem: A copper mine’s baseline blast design uses burden B₀ = 3.2 m, with total unit cost C₀ = $1.82/ton. A ±5% perturbation yields C₊ = $1.91/ton at B = 3.36 m and C₋ = $1.75/ton at B = 3.04 m. Calculate SI for burden.
1.
Step 1: Compute cost change ΔC = (C₊ − C₋) / 2 = (1.91 − 1.75) / 2 = $0.08/ton
2.
Step 2: Compute variable change ΔB = 0.05 × B₀ = 0.05 × 3.2 = 0.16 m
3.
Step 3: Apply formula SI_B = (ΔC / ΔB) × (B₀ / C₀) = (0.08 / 0.16) × (3.2 / 1.82) = 0.5 × 1.758 ≈ 0.879
Answer:
The result is SI_B = 0.88, which falls within the moderate-sensitivity range of 0.7–1.2—indicating burden adjustments offer meaningful but not dominant cost leverage.
🏗️ Real-World Application
At Newmont’s Boddington Mine (Western Australia), engineers used SI analysis to resolve conflicting stakeholder priorities: drilling crews advocated wider spacing to reduce rig moves, while crushing plant managers demanded finer fragmentation. SI calculation revealed spacing had SI = 1.32 (high sensitivity), while stemming length had SI = 0.21 (low). Redesigning spacing from 4.2 m to 3.9 m—while adding 0.3 m of engineered stemming—reduced total unit cost by 11.4% and improved P80 by 18%, validating SI as a cross-functional alignment tool.