1. Introduction
The failure of tailings storage facilities can have disastrous consequences for nearby communities, the environment, and for the mining companies, who may consequently face high financial and reputational costs. Tailings are waste resulting from mining operations and are commonly deposited as slurry behind earthen or masonry dams. We refer to this form of tailing storage facility as TSF. In 2015, the breach of the Fundão TSF at Samarco mine in Minas Gerais (jointly owned by BHP Billiton Brasil and Vale S.A.) resulted in 19 fatalities, and was declared the worst environmental disaster in Brazil’s history. The company entered an agreement with the Federal Government of Brazil and other public authorities to remediate and compensate for the impacts over a 15 years period. Jointly, BHP and Vale recognized a US
$ 2.4 billion provision for potential obligations under the agreement [
1,
2]. Twenty-one company executives were charged with qualified murder, and up until September 2017 the mine had not resumed operations. The ecosystems impacts caused by a TSF failure can last for many years depending on the nature of the tailings. Samarco is in the process of restoring 5000 streams, 16,000 hectares of Permanent Conservation Areas along the Doce River basin, and 1200 hectares in the riverbanks [
3]. It is estimated that the livelihoods of more than 1 million people were affected because of the failure [
4]. Improvements in the design, monitoring, management, and risk analysis of TSFs are needed to prevent future failures and to estimate the consequences of a breach.
The design of tailings dams has changed significantly from the 1930s to the present [
5,
6,
7]. Construction of the early TSFs was done by trial and error [
8]. During the 1960’s and 1970’s geomechanical engineering started to be used to assess the behavior of the tailings and the stability of the impoundments [
8]. Currently, various studies are required to approve a TSF design and increasingly the plans for remediation and closure of the impoundments have to be included in the feasibility phase. Breach assessments are now part of the requirements in the permitting process of a new TSF or an expansion in many countries. Different parameters need to be estimated while conducting these assessments [
9]. These include the volume of tailings (
VF) that could potentially be released, and the distance to which the material may travel in a downstream channel, called the run-out distance (
Dmax). Empirical regression equations for this purpose were developed by Rico et al. [
10] using historical TSF failure data, and are commonly used to characterize such failures (similar empirical relationships have been developed for dams holding water [
11,
12], but the lack of tailings and differences in design and construction make them inapplicable to tailings dams). However, at site conditions in the mines can vary substantially and there is considerable residual uncertainty associated with the conditional mean value estimated by these equations. In this paper, we rigorously update these regression equations using an updated data set, and characterize the uncertainty associated with the prediction. Using the uncertainty distribution for the conditional estimation of
VF and
Dmax using TSF parameters provides a better way to interpret the TSF failure data and to characterize the risk associated with a potential failure.
The calculation of
VF is of particular importance for inundation analyses. Typically, TSFs are not totally emptied in case of failure (as opposed to water dams), and only a portion of the tailings are released [
10]. In TSFs containing a large amount of water (supernatant pond), the breach would usually result in an initial flood wave followed by mobilized/liquefied tailings [
9]. Therefore, the methods developed to estimate the released volume of water or the inundation extent from a regular dam (such as the water dam break-flood analyses methods in [
13,
14]), do not apply to tailings dams. Empirical equations based on past failures, dam height, and the impounded volume of tailings, are commonly used to get a first estimate of the volume of tailings that could be released and the run-out distance. In Rico et al. [
10]
VF is calculated using the total impounded volume (
VT) in Mm
3 as in Equation (1)
and the outflow run-out distance travelled by the tailings in km (
Dmax) is obtained using
VF and the dam’s height in meters at the time of failure (
H) as in Equation (2)
Many investigators directly use such regression equations in a deterministic way to specify exposure. However, at site conditions vary significantly, and there is considerable uncertainty that needs to be quantified. This uncertainty increases as we consider TSF volumes that are near or beyond the range of the data included in the regression equation. Equation (1) predicts that approximately a third of the tailings in the impoundment (including water) will be the outflow volume. This approach may result in unrealistic estimates when liquefaction is a known risk as it does not take into account the tailings mass rheology (viscosity and yield stress) [
9]. As Rico et al. [
10] point out, some parameters contributing to the uncertainty in the predictions include sediment load, fluid behavior (depending on the type of failure), topography, the presence of obstacles stopping the flow, and the proportion of water stored in the tailings dam (linked to meteorological events or not). Therefore, it is important to account for the uncertainty in these estimates to derive a probabilistic measure of risk that also accounts for how well the regression fits in a certain range of values of the predictors.
Additional information about TSF failures since Rico et al. [
10] published the above-mentioned equations is available. In this paper we update the original data used by Rico from 22 complete cases (including height, storage volume in m
3, released volume in m
3, and distance traveled) to 29 complete cases with data compiled in Chambers and Bowker [
15]. We compare the results of the original linear regressions done by Rico et al. [
10] with the results using the updated dataset. A new model for the calculation of
Dmax is proposed introducing the predictor (
Hf), which is defined as:
This variable was introduced to consider that the potential energy associated with the release volume, may be better related to the fractional volume released as opposed to the total volume of the TSF.
For each of the models we consider, and for the final model we recommend, we consider the uncertainty analysis of prediction. We compare the predicted intervals and observed values of VF and Dmax of three TSF failures across the models that were evaluated to see how well the prediction intervals fit the observed data. The indicated probability of exceedance of the observation as per each model was also assessed.
2. Materials and Methods
The height at time of failure (
H), TSF capacity (
VT), released volume (
VF), and the distance traveled by the tailings (
Dmax) are inputs in Rico’s equations (Equation (1) and Equation (2)). The data used in this analysis is a combination of the cases used by Rico and others compiled by Chambers and Bowker [
15], including failures post 2008. Seven of the 29 incidents used by Rico et al. [
10] do not have complete information or the information for volume is included in million tons, which cannot be used in the analysis without density data. These cases are in red letters in
Table 1. It is important to note that the original data did not include releases as large as the ones experienced in Samarco and Mt Polley (cases 15 and 19 in
Table 1), and this becomes relevant for future estimations of the potential risk of larger TSFs. The data on reported failures have variations in different sources; some of these variations are included as footnotes in
Table 1.
Figure 1 shows the relationships between
VF and
VT, and
Dmax with
H × VF (called dam factor in [
10]) using the updated dataset (plots in log scale), and
DmaX with
Hf, (Equation (3)).
VF and
VT show a linear relationship in the log form, while for
Dmax, there is greater dispersion with the dam factor and
Hf.
VF was estimated in the same way as in Rico et al. [
10] using the predictor
VT with a log-log (power) transformation and the updated data. For the estimation of
Dmax three models were considered:
The observed value of VF was used to fit the Dmax regressions.
The goodness of fit of each model was analyzed with residual plots, outlier identification, analysis of influential observations using Cook’s distance, and computing the root mean square error(RMSE) using a 5-fold cross validation (CV). The prediction intervals and probability of occurrence of VF and Dmax in three historical failures was compared across models using the original and the updated datasets.
4. Conclusions
The empirical equations developed by Rico et al. [
10] to estimate the volume of tailings released in a tailings dam failure and the run-out distance of the tailings were reviewed. An updated dataset provided information on dam failures that happened after the Rico et al. [
10] paper was published and includes cases of dams with larger storage capacity and height than the points in the original dataset. The introduction of the new data points in the regression reduces the uncertainty of the prediction of large failure incidents such as the one occurred in Samarco.
An improved model to estimate the run-out distance is proposed. The model uses the predictor
Hf that considers the potential energy associated with the released volume as opposed to the whole tailings impoundment volume. The model proposed has a better linear fit than the original model when using the updated dataset. The updated model to calculate
VF is presented in Equation (4), and the new model to estimate
Dmax in Equation (5). We recommend using the app we provide which contains the equations (available at
https://columbiawater.shinyapps.io/ShinyappRicoRedo/). Since we recommend using the uncertainty distribution for each “prediction” it is easiest for the user to use our web app. As data on other failures becomes available, it can be brought into the app and the model can then be automatically updated.
This paper emphasizes that these are empirical regression equations with significant uncertainty about the mean. Some investigators directly use such regression equations in a deterministic way to specify exposure. However, at site conditions vary significantly (rheology, water content, failure type, etc.), and even with the log-log regressions presented here, there is considerable uncertainty that needs to be quantified. It is important to account for the uncertainty in these estimates to derive a probabilistic measure of risk that also accounts for how well the regression fits in a certain range of values of the predictors.