Volumes & Stockpiles

Surveyors measure how much has been mined and stockpiled. Learn end-area and prismoidal (Simpson) formulas, stockpile geometry with angle of repose, surface-to-surface (DTM) volumes, bank vs loose volumes and swell, and month-end production reconciliation.

Contractors are paid by the cubic metre, mills are fed from stockpiles, and reserves are depleted by what is mined — so volume surveys directly affect money and reporting. Modern pickups use drones and laser scanners, but the underlying formulas remain the same.

An analogy

Computing volume from cross-sections is like slicing a loaf of bread, measuring each slice's area and adding them up.

Cross-section methods

Sections at regular intervals along a pit or road; the end-area method averages adjacent sections (slightly overestimates for curved shapes); Simpson's prismoidal formula is more accurate with a middle section.

Stockpiles

Loose material forms cones or windrows at the angle of repose (≈ 35–40° for crushed rock). Volumes are converted to tonnes using measured loose density, which varies with moisture and compaction.

DTM volumes

Digital terrain models (triangulated surfaces) from before and after surveys are subtracted to give cut and fill volumes — the standard method in mining software with drone or scanner data.

Reconciliation

Monthly surveyed volumes (bcm) × density compared with truck counts, belt scales and mill tonnes reveal density errors, dilution and model bias. Discrepancies feed back to resource models.

Formulas

• End-area: V = L (A₁ + A₂)/2
• Prismoidal (Simpson): V = L/6 (A₁ + 4A_m + A₂)
• Conical stockpile: V = π r² h / 3, r = h / tan φ_r
• Tonnes: t = V × loose density

Worked example: End-area vs prismoidal

Given: A₁ = 120 m², A_m = 150 m², A₂ = 170 m²; L = 40 m

Find: Volume both ways.

• End-area = 40 × 145 = 5800 m³
• Prismoidal = 40/6 × (120 + 600 + 170) = 5933 m³

Answer: 5800 vs 5933 m³ (≈ 2 % difference)

Worked example: Conical stockpile

Given: Height 12 m, repose 37°; Loose density 1.6 t/m³

Find: Tonnes.

• r = 12/tan 37° = 15.92 m
• V = π × 15.92² × 12/3 = 3187 m³
• t = 3187 × 1.6 = 5099 t

Answer: ≈ 5100 t

Common mistakes

• Wrong: Using bank density for stockpiles. — Right: Use measured loose density.
• Wrong: Wide section spacing on irregular shapes. — Right: Space sections closely or use DTM volumes.

Applications

• Contractor payments: Monthly bcm surveys of excavation.
• Stockpile inventory: Drone surveys of ROM and product piles.
• Reconciliation: Survey vs truck vs mill comparisons.

Key takeaways

• End-area: L(A₁ + A₂)/2.
• Prismoidal: L/6(A₁ + 4A_m + A₂).
• Cone: πr²h/3 with repose angle.
• Reconcile surveys with production data.

Going further

• TIN surfaces: Triangulated irregular networks for volumes.
• Density measurement: Nuclear gauges and water-replacement tests.
• Automated stockpile scanning: Fixed LiDAR on stackers and sheds.