Industry: Mining and Natural Resources
Product: ModelRisk
Application: Mine rehabilitation planning under uncertainty

The 61% bond release nobody could pencil

A Queensland coal mine, four years post-closure, applied for $42M of progressive bond release based on rehabilitation completion of 1,150 ha of the 1,560 ha disturbed footprint. The state regulator required demonstrated landform stability above factor-of-safety 1.5 and vegetation-cover above 70% on >90% of the rehabilitated area. The deterministic engineering report said all targets were met. The Monte Carlo rerun — driven by erosion-rate variability, slope-angle settlement, and probabilistic seed-germination dynamics — said the joint probability of meeting all release criteria across the 1,150 ha was only 61%, with a 39% chance the regulator would require remedial works averaging $7.4M before release. The bond filing was restructured to phased release; the first $28M was approved on time.

The landform factor-of-safety distribution below is where the gap opens: the deterministic report quoted a single F.o.S. of 1.68, comfortably above the 1.5 gate, but the distribution carries a 14% probability of falling below it — a tail the point estimate simply cannot show.


Rehabilitation outcomes are nonlinear, slow-moving, and dominated by tail events — drought years for vegetation, intense rainfall events for landform stability. Deterministic compliance reports are exactly the wrong instrument for predicting their joint pass-rate.

Where landform stability actually fails

The capped waste-rock dump has a designed batter slope of 18° (3.1:1), within the geotechnical recommendation of 16-20°. Stability is governed by the factor of safety:

F.o.S. = (cohesion + γ × z × cos²(α) × tan(φ)) / (γ × z × sin(α) × cos(α))

with cohesion c (kPa), unit weight γ, depth z, internal friction angle φ. The deterministic calculation used central-tendency values for each input (c = 12 kPa, γ = 19 kN/m³, φ = 32°, α = 18°) and returned F.o.S. = 1.68 — comfortably above the regulator's 1.5 minimum.

The probabilistic rebuild used:

• Cohesion c: LogNormal(μ = ln 12, σ = 0.35) kPa — soil cohesion is highly variable across the cap.
• Friction angle φ: truncated Normal(32°, 2.5°, [25°, 38°]).
• Settled slope α: Triangular(17°, 19°, 22°) — settling steepens the slope over the first decade.
• Effective unit weight γ: Normal(19, 0.8) kN/m³.
• Pore-pressure ratio ru during 1-in-50-year rainfall: Beta(2, 8) × 0.4, mean 0.08, P95 0.21.

Mean F.o.S. = 1.62; P10 = 1.48; P5 = 1.45. The probability that F.o.S. falls below 1.5 = 14%. The deterministic 1.68 was right about the central case but silent on the tail (charted above).

Vegetation cover under climate variability

The revegetation target is 70% canopy and ground cover across >90% of the area, achieved by year 4. The probabilistic model treats cover as the output of a logistic growth process modulated by annual rainfall:

V(t) = K(t) / (1 + ((K(t) - V₀) / V₀) × exp(-r(t) × t))

with:

• Carrying capacity K(t): Triangular(55%, 78%, 88%) — site-specific maximum cover.
• Establishment rate r(t): Normal(2.2, 0.4) /yr (truncated to [1.0, 3.5]), with each of the four years independently exposed to drought.
• Annual rainfall: a 22% probability of drought years (<450 mm) that suppress r by 60%.
• Initial cover V₀: Beta(3, 7) × 30% + 10%.

After four years, mean cover = 73%, bounded above by the site carrying capacity: P10 = 63%, P90 = 82%. Probability of clearing the 70%-on-90%-area test = 69%.


Joint pass rate is not the product of marginals

The deterministic report treated landform stability and vegetation cover as independent compliance tests, multiplying their pass probabilities to estimate joint pass rate. That is wrong twice over: first, the events are positively correlated (both depend on rainfall, soil compaction, and contractor quality), and second, the regulator inspects them on the same site visit. The model applied a Gaussian copula with ρ = 0.55 between F.o.S. and vegetation cover, raising the joint pass rate from 59% (naive independence) to 61% — and exposing the $7.4M expected remedial cost in the 39%-fail tail.

Why Monte Carlo, not a point estimate

The deterministic report said: F.o.S. 1.68, vegetation cover 75%, "all targets met, recommend full release." The probabilistic report said: joint pass probability 61%, expected remedial cost in the 39% fail tail $7.4M, recommend phased release. Same site, opposite filing strategy — because the deterministic report describes the centre of the distribution, not the percentile the regulator's gate is set at.

Tornado: what's actually moving the joint pass rate


Settled slope angle leads — every degree of additional steepening drops F.o.S. by 0.08 and pass rate by 5 percentage points. Year-4 rainfall is second. Cohesion variability and seed-germination success round out the top four.

What the model changed

• Bond release restructured from single-application $42M to phased $28M now / $14M in year 6 contingent on year-5 monitoring outcomes — the regulator's preferred risk-sharing structure.
• Erosion-control supplementary works ($3.1M) commissioned on the two steepest batters before the next wet-season inspection, lifting F.o.S. P5 from 1.45 to 1.55.
• Native-grass overseeding programme commissioned in the 100 ha of lowest-cover areas — moved joint pass probability from 61% to 74% at a cost of $1.8M, an expected NPV of +$4.3M against the avoided remedial cost.

ModelRisk functionality used

• Gaussian copula between landform stability and vegetation cover with ρ = 0.55, raising joint pass probability from 59% (naive independence) to 61% — and exposing the right denominator for the regulator's compliance test.
• Logistic-growth model for vegetation cover with rainfall-modulated growth rate and a 22%-probability drought switch.
• Beta-distributed pore-pressure ratio for the 1-in-50-year rainfall event, propagated through the F.o.S. formula and lifting P(F.o.S. < 1.5) to 14% — undetectable in the deterministic central-case calculation.
• Joint-test reporting producing the 61% joint pass-probability figure that supported the phased bond-release strategy.
• Expected-remedial-cost calculation conditional on falling into the 39% fail tail — $7.4M, sized the contingency reserve and the year-5 monitoring scope.

A "compliant" rehabilitation, measured deterministically, can still carry a meaningful chance of failing inspection — and only a distribution can put a price on that risk.