Industry: Mining and Natural Resources
Product: ModelRisk
Application: Geological modeling

Turning a kriged copper block model into a grade-tonnage distribution

A pre-feasibility deck for a 280 Mt porphyry-copper resource at 0.62% Cu reported a single contained-metal number: 1.74 Mt of copper. The deterministic resource estimate hid the question that mattered to the project's investment committee: what is the probability that contained metal falls below the 1.40 Mt floor below which the concentrator IRR drops under the 12% hurdle? The probabilistic kriging rerun showed that probability at 13% — a tail the single-point ore-reserve report had no way to expose. Aggregating 200 conditional realisations over the 7-million-block model turned the single 1.74 Mt point into the distribution below: mean 1.71 Mt, P10 1.37 Mt, P90 2.08 Mt, with the deterministic estimate sitting almost exactly at the mean and the economic floor biting into the lower tail.


The deposit, a Tier-2 porphyry in the central Andes, had 142,000 m of diamond-drill core on a 60 m × 60 m × 15 m block grid. The geomodel team used ModelRisk to build a sequential-Gaussian-simulation (SGS) wrapper around the kriged block model and propagate grade, tonnage, density and recovery uncertainty into a single grade-tonnage distribution.

Why kriging variance is not resource uncertainty

The deterministic kriged block model returned a mean grade of 0.62% Cu with a kriging variance of 0.0034. Reading kriging variance as "the uncertainty in the resource" is a textbook mistake: kriging variance measures only the conditional estimation error at a point, not the joint variability of the deposit. Two adjacent blocks each estimated at 0.6% Cu ± 0.06 with a correlation of 0.7 do not behave like two independent estimates — they move together.

The probabilistic rebuild replaced ordinary kriging with 200 conditional SGS realisations, each respecting:

• Grade variogram: spherical model, range 220 m horizontal / 35 m vertical, sill σ² = 0.18 (log-grade).
• Grade distribution per block: LogNormal in raw % Cu space, declustered-cell mean 0.62%, σ_log = 0.42.
• Density: Normal(μ = 2.71, σ = 0.04) t/m³ — modest uncertainty but it matters at the 280 Mt scale.
• Boundary indicator: Bernoulli at the mineralised-domain edge, calibrated against 38 cross-domain holes — probability of a block belonging to the ore domain rather than waste rises smoothly from 0.05 outside the wireframe to 0.95 inside.

What the grade-tonnage curve actually showed

Aggregating 200 realisations over the 7-million-block model gave a contained-copper distribution with mean 1.71 Mt, median 1.69 Mt, P10 = 1.37 Mt, P90 = 2.08 Mt. The deterministic 1.74 Mt fell almost exactly at the mean — but the 1.40 Mt economic floor sat just above P10, giving a 13% probability of falling short.

The grade-tonnage curve traced the trade-off at every cut-off: at 0.40% Cu the project mines 251 Mt (1.74 Mt contained, mean); at 0.55% cut-off it mines 178 Mt (1.39 Mt). The point of the curve is that the cut-off decision is itself probabilistic — at the 0.55% cut-off, P10 contained metal is 1.12 Mt, below floor.


Why Monte Carlo, not a point estimate

The kriged number was 1.74 Mt Cu, IRR 14.6%, payback 5.3 years. The probabilistic model preserved that near the mean but added the things the point estimate could not: a 13% probability of breaching the 1.40 Mt economic floor, an IRR P10 of 9.8% (below hurdle) and a payback P90 of 7.1 years. The investment committee's question was not "what is the expected resource?" — it was "how often do we lose?" — and that requires a distribution.

Tornado on contained metal

Sensitivity analysis on contained-metal P10 ranked the inputs:


Grade-variogram sill (the σ² that controls deposit-scale grade swing) dominated — a +/-20% perturbation moved P10 by 0.21 Mt. Boundary-domain probability around the wireframe edges contributed another 0.11 Mt. Bulk density and recovery were minor.

What the model changed

• 32 additional infill holes commissioned in the SE quadrant where SGS realisations diverged most, dropping σ_log from 0.42 to 0.31 and P(< floor) from 13% to 5%.
• Cut-off grade lowered from 0.55% to 0.45% Cu — the grade-tonnage P10 envelope showed this protected contained metal against the high-σ tail without materially eroding head grade.
• Concentrator design throughput sized to the probabilistic mean (1.71 Mt Cu), not an optimistic point estimate — saved $48M in oversized milling capacity.

ModelRisk functionality used

• Sequential Gaussian simulation on a 7-million-block model, producing 200 conditional realisations of grade and density that preserve the spherical variogram structure.
• Bernoulli boundary indicator with smooth probability ramp at the mineralised-domain edge, replacing the hard wireframe and reducing inside-out volume bias by 4.2%.
• Cut-off-grade sweep from 0.30% to 0.70% Cu, computing P10/P50/P90 grade-tonnage curves and exposing the cut-off at which the P10 falls under the economic floor.
• Tornado on P10 contained metal ranking variogram sill, domain probability, density, recovery, with the +/-20% input perturbation translated into Mt-Cu impact.
• Cross-realisation correlation preserved across blocks via the same SGS path, so the aggregate distribution reflects deposit-scale joint variability — not the false-precision sum of independent block errors.

In probabilistic geological modelling, the resource is not a number — it is a curve, and the only question worth asking is where on that curve the project's break-even falls.