Probabilistic Stability Analysis of Stacks: How to Translate Safety Factor into Risk Language

In mining operations with large stockpiles, especially in environments with limited space, rugged topography, and heterogeneous materials, the relevant question is no longer "Does the safety factor meet the requirements?and it becameWhat is the probability of failure associated with this structure, and is this level of risk acceptable given the consequences involved?.

This paradigm shift is not just technical, it's business-related. It redefines how the company prioritizes investments in drainage, monitoring, reinforcement, and closure strategies. Probabilistic studies of tailings piles consistently show that two scenarios with the same average safety factor can present very different orders of magnitude of probability of failure, depending on the variability of the parameters, the quality of the data, and how uncertainties are handled.

This article discusses how to structure probabilistic stability analyses for fuel piles, how to interpret safety factor distributions, failure probabilities, and reliability indices, and, most importantly, how to translate these results into understandable and actionable risk language for senior management, regulators, and other stakeholders.

Why is the deterministic safety factor insufficient in batteries?

The deterministic safety factor, calculated using a single set of parameters for strength, specific weight, and neutral pressures, will continue to be a central design indicator; however, in isolation, it does not provide the risk insight required today by boards of directors, investors, and regulatory bodies. In high piles, some limitations are especially relevant:

• First, the heterogeneity of the material.Given the mixture of blocks, soils, fines, oxides, reprocessed materials, and partially saturated zones, adopting a single cohesion-friction angle pair as representative of the entire rock mass is unrealistic. In many cases, the critical failure surface develops in specific zones associated with facies changes or low-resistance interfaces that are not captured by average parameters.
• Secondly, hydrological and hydrogeotechnical uncertainty is high.On sloping or hillside slopes, the interaction between extreme rainfall, recharge, surface runoff, preferential flow, and internal drainage is complex in space and time. A "representative" water level adopted in the design may be systematically displaced from reality, especially during seasonal transition periods or extreme events, altering the equilibrium condition in a way not captured by deterministic calculations.
• Third, there is a false numerical precision.When reporting “FS = 1.52”, it implicitly communicates a degree of accuracy that does not exist. In practice, this value is the result of assumptions about parameters, model, geometries, and boundary conditions, each with its own uncertainty intervals. Structures with the same nominal FS can, in fact, operate with very different levels of reliability.

In short, the safety factor is necessary, but not sufficient. It is a point in a problem that is, by nature, distributed. Probabilistic analysis comes in precisely to reveal the structure of this distribution.

Decision structures: from fixed parameters to random variables

The central logic of probabilistic analysis is to explicitly address uncertainty. What was previously a single number becomes modeled as a random variable, with distribution, dispersion, and correlations. In waste rock piles, this applies to parameters such as shear strength, effective cohesion, angle of friction, natural and saturated specific weight, position and variability of the water line, operational overloads, geometries, construction sequences, and even the presence of weak layers or reprocessing zones.

These variables are represented by probability density functions, for example normal, lognormal or beta distributions, defined based on tests, back-analyses, monitoring data and expert judgment. From there, a performance function is defined, usually of the type G = R minus S, where R represents the "resistance" and S the "demand" and where failure occurs when G is less than or equal to zero, equivalent to FS less than or equal to 1.

To propagate uncertainties to the safety factor, methods such as Monte Carlo simulation with thousands or tens of thousands of parameter samples are used, as well as first- and second-order reliability methods that allow capturing non-linear responses with lower computational cost. In cases of significant spatial variability, it is possible to employ random fields and an approach of "random fields"which recognize that the material is not only variable, it is variable in space."

The result is a probability distribution of the safety factor or the G performance index. In this way, the question “What is FS?and then the question is askedWhat is the FS distribution, what is its mean, what is its standard deviation, what is the probability of it being below 1.0, and how does this compare to the reliability objectives defined for the structure?.

Probability of failure and reliability index

The crucial step in discussing risk is converting the FS distribution into a probability of failure. This probability, usually represented by Pf, is the area of the FS distribution to the left of 1. In practical terms, it is the fraction of possible scenarios in which the system fails, given the set of uncertainties considered.

This probability can be expressed on an annual basis, for example Pf equals 10-⁴ per year, over a useful life horizon, for example, a cumulative Pf of approximately 10-³ in 20 years, or in portfolio terms, for example, “in a hypothetical universe of 10,000 batteries with this level of reliability, approximately one failure is expected over the lifetime of the product”.

Associated with Pf, the reliability index β is defined, which measures, in standard deviation units, the distance between the average state of the system and the fault surface. The larger β, the smaller Pf. For corporate management, the mathematical detail is less relevant than the interpretation patterns that are built upon these numbers. Some points are central.

Two projects with the same average safety factor (FS) can have completely different β and Pf values if, for example, one is based on poorly investigated parameters with high dispersion, and the other on a dense and consistent database. Investments in geotechnical investigation, advanced testing, and monitoring can increase β without necessarily altering the average FS, only reducing uncertainty. In many cases, a structure with a "modest" FS, but with well-controlled uncertainties, is more probabilistically safer than another with a nominally high FS, but supported by optimistic assumptions.

This logic opens the door to more refined decisions. Instead of simply seeking "higher FS" (financial safety margins), the company begins to evaluate where each real invested generates the greatest reduction in Pf (performing phases), whether in physical reinforcements, drainage, monitoring, or knowledge.

Acceptable risk levels: connecting probability, consequence, and governance.

Establishing acceptable risk levels for piles is essentially a governance decision. Engineering provides quantification, scenarios, and... trade-offsHowever, defining what is acceptable depends on risk appetite, regulatory context, social expectations, and ESG commitments.

In practice, acceptability results from the combination of probability of failure, consequence, and controllability. The same probability may be considered acceptable in a structure with predominantly operational consequences, but unacceptable in a structure whose failure could affect communities, strategic water resources, or critical infrastructure.

For piles with potentially severe consequences, it is consistent to work with target failure probabilities on the order of 10.-⁵ to 10-⁴ per year, depending on the benchmark adopted, always connecting these values to robust monitoring plans, emergency preparedness, and system redundancy. For structures with less severe consequences, slightly higher probabilities may be accepted, provided that the decision is documented, communicated, and reviewed periodically in light of new data.

Without this structured discussion, the company tends to operate under a regime of adjectives, using terms like "safe," "adequate," or "conservative" that are not comparable between assets nor do they support consistent investment prioritization decisions. Probabilistic analysis is the bridge between qualitative discourse and quantitative risk governance.

From technical analysis to risk language: communicating to decide.

The incorporation of probabilistic analysis into risk governance is only achieved when the results can be communicated in a clear, honest, and decision-oriented manner. This requires translating technical concepts into risk language.

A good practice is to start with the practical implication, not the formula. Instead of opening with “Pf equals 5 times 10-⁴ per yearIt is more effective to say thatUnder current conditions, the probability of this pile breaking over 50 years is on the order of X percent.", then explaining how that number was obtained.

Another critical dimension is the adoption of probability ranges associated with well-defined qualitative categories. The company can, for example, standardize that events with an annual Pf between 10-⁴ and 10-³ be classified as “unlikely”, between 10-3 and 10-² as “rare”, between 10-2 and 10-¹ as “possible"Always with clear, consistent, and documented definitions. This prevents different areas from interpreting the same word in divergent ways."

Visualization also matters. FS distribution curves, highlighting the region below 1, diagrams that decompose the relative contribution of uncertainties such as water level, resistance, and geometry, risk matrices combining probability and consequence, are tools that help anchor the discussion. The goal is not to impress, but to allow a director, manager, or regulator to quickly understand where the risk lies, what drives it, and what actions are possible.

Finally, communication needs to connect risk to decision. Each probabilistic outcome should be accompanied by a clear agenda of implications, such as planned volume reduction (height), reinforcement of critical slopes, redesign of drainage systems, expansion of monitoring in sensitive areas, or revision of closure strategies.

Lessons from probabilistic studies in stacks

The experience gained from dealing with high-stakes situations allows us to draw general lessons that are useful for risk management:

• In the first place, Water tends to be the main risk driver.In several studies, relatively modest reductions in uncertainty regarding waterline positions and pore pressures, obtained through more efficient drainage, targeted instrumentation campaigns, and more consistent hydrogeological modeling, generate significant decreases in the probability of failure. In many cases, the gains in Pf (probability of failure) per investment associated with knowledge of the water level exceed those obtained by isolated structural reinforcements.
• Secondly, constructive variability is crucial.Differences in procedures between shifts, mining faces, equipment suppliers, and seasonal moisture conditions generate significant internal heterogeneity in the stockpile, with zones of lower relative density, higher fines content, or poorer drainage. Ignoring this variability and working only with average parameters may underestimate the probability of forming critical surfaces on intermediate slopes or berms.
• Thirdly, Extreme rainfall events can no longer be treated as "distant exceptions."Updated historical data and evidence of climate change indicate that the frequency and intensity of extreme events are changing. Rainfall scenarios previously classified as very remote should be reassessed, both in hydrological and probabilistic terms, with direct impacts on drainage system design and the definition of operational triggers during critical periods.

Finally, sophisticated stability models, by themselves, do not resolve data gaps. The robustness of probabilistic analysis is contingent upon the quality of the geotechnical investigation, the understanding of the water regime, and real-time monitoring. In other words, the software is leverage, but the foundation remains technical data.

Integrating probabilistic analysis into the risk management system

To generate real value, probabilistic analyses cannot be confined to a report appendix. They need to be connected to the company's risk management system processes, indicators, and decisions.

This involves defining clear corporate guidelines for target failure probability levels by structure class, based on consequence and criticality; incorporating Pf and β into corporate risk matrices, alongside consequence and detection capability metrics; establishing quantitative triggers connecting monitoring data, such as water levels, deformations, and pore pressures, to the probabilistic model, so that certain combinations trigger pre-planned actions, such as reduced operation, intensive inspections, or reinforcement interventions; and periodically reviewing probabilistic analyses in light of new field data, operational changes, structure reforms, or changes in the regulatory context.

When this happens, the safety factor ceases to be a static number and becomes a living variable, reassessed as the operation evolves. The result is pile management that is more aligned with international best practices in reliability and risk.

How VinQ supports the journey from “FS” to quantified risk.

Migrating from a reliability-focused culture to a risk-oriented culture requires combining geotechnical depth, mastery of reliability methods, and the ability to translate quantitative concepts into business decisions.

VinQ operates precisely at this intersection. In simpler terms, this means structuring probabilistic stability models aligned with the specific characteristics of the operation, with consistent integration between laboratory tests, retrospective analyses, numerical modeling, and near real-time monitoring data; quantifying failure probabilities and reliability indices compatible with the criticality of the structure, exposure to third parties, and current regulatory frameworks; identifying the main risk drivers and guiding the use of capital, showing where interventions in drainage, reinforcement, further investigation, or monitoring actually reduce Pf; supporting the definition of internal risk acceptability criteria consistent with the ESG strategy and international benchmarks; and developing technical and executive materials that allow for transparent communication of results to management, regulators, investors, and communities.

By treating the safety factor as part of a broader architecture of reliability, probability of failure, and governance, VinQ helps its clients move away from the reactive discourse of “we meet the minimum FSand move towards a proactive and sustainable position:We understand, quantify, and manage the risk of our stockpiles based on data, clear criteria, and decisions aligned with the complete asset lifecycle..