Question 3 of 6: Limestone Diameter Up-Sizing via Cratering Curve

Question 3: Limestone Diameter Up-Sizing via Cratering Curve (21 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Pattern dimensions, loading and timing for the 165 mm redesign

Given.

No rock UCS/E is given for this limestone (unlike Q2's iron ore), so this design uses Figure 2's cratering data (Livingston crater theory) rather than Kuznetsov/Lilly – the cratering curve directly calibrates burden-to-charge scaling from field tests in the same rock, without needing a separately-estimated rock factor.

Find. A 165 mm pattern (burden, spacing, collar) that preserves or improves the current design's fragmentation/flyrock performance.

Approach. Scale burden, spacing and collar geometrically with hole diameter (preserves the powder factor and the pattern's dimensionless ratios, per the standard similarity argument). Cross-check the result against Figure 2 by computing the scaled depth of charge (collar length divided by the cube root of the charge mass) for both the current and proposed design and reading the corresponding scaled crater volume – this validates that the new pattern does not creep toward the curve's steep, under-confined (flyrock-prone) left-hand side. Finally check worst-case flyrock throw (Lundborg) against the 300 m stand-off.

1. Linear scaling of B, S, T. $$\frac{De_2}{De_1}=\frac{165}{102}=1.618$$ $$B_2=3.0(1.618)=\boxed{4.85\ \text{m}},\quad S_2=4.0(1.618)=6.47\ \text{m},\quad T_2=2.5(1.618)=4.04\ \text{m}$$
2. Cratering-curve cross-check. Treat burden/collar and full-column charge mass as the crater-theory "depth of charge" and "charge weight" (Q = (H−T)×Area× ρ_e): $$Q_1=(12-2.5)(0.008171)(1250)=97.0\ \text{kg},\quad sd_1=\frac{T_1}{Q_1^{1/3}}=\frac{2.5}{4.60}=0.54$$ $$Q_2=(12-4.04)(0.02138)(1250)=213\ \text{kg},\quad sd_2=\frac{T_2}{Q_2^{1/3}}=\frac{4.04}{5.97}=\boxed{0.68}$$

3. Interpret the cross-check. Reading Figure 2: at sd₁=0.54 the scaled volume is ≈0.48 m³/kg (current design sits on the rising, less-confined side of the curve, well short of the sd≈0.87 peak); at sd₂=0.68 the scaled volume rises to ≈0.68 m³/kg – the linearly-scaled 165 mm design moves CLOSER to the curve's optimum (more confined, more efficient breakage per unit charge) while still sitting short of the peak, so it is not entering the over-confined/venting side either. The scaling is therefore validated: it should give comparable-or-better fragmentation efficiency than the proven current design, not worse.
4. Flyrock check (Lundborg). $$L_{max}(\text{ft})=260\,D(\text{in})^{2/3}$$ $$L_1=260(4.02)^{2/3}=657\ \text{ft}=200\ \text{m},\qquad L_2=260(6.50)^{2/3}=905\ \text{ft}=\boxed{276\ \text{m}}$$ Both stay under the 300 m stand-off; the margin narrows from ≈100 m to ≈24 m, so stemming quality on the 165 mm design must not be trimmed below the scaled 4.04 m value.

Timing: keep the same relative sequencing that proved successful at 102 mm – diagonal echelon initiation toward the quarry's open face, 17–25 ms hole-to-hole and 25–42 ms row-to-row delays (Q1(j), Q2(b)) – but note the per-hole charge nearly doubles (97 kg→213 kg), so vibration/airblast at any nearby structure should be re-checked against the site's own attenuation law before the first production shot at 165 mm.

(b) Parameters controlling maximum flyrock range

Beyond hole diameter (which sets the Lundborg ceiling used above), the governing parameters are: stemming length and quality (T/De ratio – inadequate/poor stemming is the single most common flyrock cause, allowing early gas venting at the collar); burden (an under-burdened hole, from drilling error or hole deviation, directs excess energy toward the free face as projectile velocity rather than displacement/breakage); powder factor / explosive energy per unit rock volume (excess energy beyond what is needed to break and gently displace the rock converts to fragment kinetic energy); structural discontinuities intersecting the charge or stemming column (joints, voids, mud seams venting gas along an unplanned path to the face); timing/delay scatter (an out-of-sequence detonation firing into an unrelieved or wrongly-confined face); and face condition (an irregular, previously-damaged, or steeply undercut free face reduces the effective burden locally and increases throw in that direction). For this specific case (300 m stand-off, diameter increasing to 165 mm), stemming length and quality is the parameter with the greatest leverage the operator directly controls without sacrificing the productivity gain from the larger hole.