# Tailings Dams Failures: Updated Statistical Model for Discharge Volume and Runout

^{1}

^{2}

^{*}

## Abstract

**:**

_{F}) and the maximum distance travelled by the tailings (D

_{max}) in the event of a tailings dam failure, based on physical parameters of the dams. The dataset of historical tailings dam failures is updated from the one used by Rico et al., (Floods from tailings dam failures, Journal of Hazardous Materials, 154 (1) (2008) 79–87) for their regression model. It includes events out of the range of the dams contained in the previous dataset. A new linear regression model for the calculation of D

_{max}, which considers the potential energy associated with the released volume is proposed. A reduction in the uncertainty in the estimation of D

_{max}when large tailings dam failures are evaluated, is demonstrated. Since site conditions vary significantly it is important to directly consider the uncertainty associated with such predictions, rather than directly using these types of regression equations. Here, we formally quantify the uncertainty distribution for the conditional estimation of V

_{F}and D

_{max}, given tailings dam attributes, and advocate its use to better represent the tailings dam failure data and to characterize the risk associated with a potential failure.

## 1. Introduction

_{F}) that could potentially be released, and the distance to which the material may travel in a downstream channel, called the run-out distance (D

_{max}). 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 V

_{F}and D

_{max}using TSF parameters provides a better way to interpret the TSF failure data and to characterize the risk associated with a potential failure.

_{F}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] V

_{F}is calculated using the total impounded volume (V

_{T}) in Mm

^{3}as in Equation (1)

_{max}) is obtained using V

_{F}and the dam’s height in meters at the time of failure (H) as in Equation (2)

^{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 D

_{max}is proposed introducing the predictor (H

_{f}), which is defined as:

_{F}and D

_{max}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

_{T}), released volume (V

_{F}), and the distance traveled by the tailings (D

_{max}) 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.

_{F}and V

_{T}, and D

_{max}with H × V

_{F}(called dam factor in [10]) using the updated dataset (plots in log scale), and D

_{maX}with H

_{f}, (Equation (3)). V

_{F}and V

_{T}show a linear relationship in the log form, while for D

_{max}, there is greater dispersion with the dam factor and H

_{f}.

_{F}was estimated in the same way as in Rico et al. [10] using the predictor V

_{T}with a log-log (power) transformation and the updated data. For the estimation of D

_{max}three models were considered:

- A generalized linear model (glm) with the Gaussian family using a log link function (D
_{max}.2 in Table 2). - A model D
_{max}.3 which uses the new predictor H_{f}.

_{F}was used to fit the D

_{max}regressions.

_{F}and D

_{max}in three historical failures was compared across models using the original and the updated datasets.

## 3. Results and Discussion

#### 3.1. Released Volume of Tailings

_{F}.1 as seen in the R

^{2}, standard error and 5-fold cross validation results in Table 3. In Samarco, 58% of the tailings contained at the Fundão dam were released (P.15 in Table 1), whereas in the incident of the Gypsum tailings dam in Texas only 1.2% of the contained tailings were released (P.17 in Table 1). The Gypsum tailings dam incident (P.17) was not included in the updated regression (Equation (4)) because it was a minor release, different from the characteristics of the rest of the dataset (identified as an outlier with high influence, Table 3), and had a strong effect in the normality of the residuals.

^{2}= 0.887; standard error: 0.315.

_{F}) it is important to consider the uncertainty of prediction in terms of log(V

_{F}), and then transform the prediction intervals to real space to determine the proper uncertainty intervals for V

_{F}. Tests for the residuals from the fit provided by Equation (4) indicate that a Gaussian distribution cannot be rejected for the residuals at the 5% level (Shapiro-Wilk test p-value = 0.1161). The prediction intervals are then computed at the desired significance level for log(V

_{F}) and then transformed to real space for V

_{F}.

_{F}and 90th prediction interval for Samarco, Mt. Polley and the Gypsum TSF using Equation (4). These cases were selected based on the influence they have in the regression of V

_{F}with the updated data (Table 3). The prediction intervals are wider and the predicted mean is larger for Samarco and Mt. Polley when the original dataset is used. In the case of the Gypsum tailings dam, the observed value is not within the prediction interval in neither regression because the volume released was so small and the probability of exceeding it was very high (the same was observed even when that data point was included in the regression).

_{F}should be constrained by the total volume of tailings available since in all the cases presented in Table 4, the 95th percentile is more than what is physically possible (more than the total impounded are released). In this case, finding the probability associated with totally emptying the dam would be a better approach for risk estimation. We developed an online pp that has the capability of computing the probability of exceeding a value of V

_{F}specified by the user (available at https://columbiawater.shinyapps.io/ShinyappRicoRedo/). In this manner, the uncertainty around V

_{F}can be considered when estimating D

_{max}. The app also provides Q5, Q50 and Q75 of V

_{F}. From Table 1, P.10, P.13, P.28, and P.29 were nearly or totally emptied.

#### 3.2. Run-Out Distance (D_{max})

_{max}.1 (the original model by Rico et al. [10]) using the updated and original datasets, shows that the uncertainty increases when the new data points are introduced; R

^{2}is reduced and the cross validated error increases (Table 5). The Samarco failure is an influential observation in all the D

_{max}regressions except for D

_{max}.3 (Table 5). The distance traveled by the tailings reached 637 km, although more than 90% of the tailings stayed within 120 Km of the dam, the rest were transported in the Doce river all the way to the Atlantic Ocean [15]. The Bonsal TSF (P.7) is also identified as an influential observation (Table 5), this incident had a large D

_{max}(close to 1 km) compared to the released volume of tailings (the released volume of tailings was approximately 0.01 m

^{3}). The failure mode reported at P.7 was overtopping so it is likely that the tailings dam had a large proportion of water at the time of failure. P.12 also appears as an influential observation; in that case the distance traveled by the tailings was only 30 meters, which is small considering the dam height (18 m) and the released volume of tailings (0.038 Mm

^{3}).

^{2}, and 5-fold CV) and the analysis of the residual plots, the best model found was D

_{max}.3, which uses the new predictor H

_{f}and has the form:

_{max}.1) with the original and the updated datasets, and includes the results of D

_{max}.3 for three of the influential observations. The uncertainty distributions are obtained as before by considering the residuals associated with Equation (5), testing for Gaussian structure (Shapiro-Wilk test p-value = 0.388), and then computing the prediction intervals at the desired significance levels, and transforming them to real space. Figure 2 has examples of the prediction intervals obtained from D

_{max}.1 O and D

_{max}.3 U compared to the observations. From the results in Table 6 and Figure 2 is evident that the additional data used to fit the regressions of D

_{max}dramatically reduces the uncertainty bounds in the prediction of large events such as Samarco and Mt. Polley, although in smaller D

_{max}events such as Bonsal, Los Frailes or Omai, the uncertainty is similar or it increases. The probability of exceeding the observed run-out distance shown in Table 6 was calculated transforming the observation to the log space, and evaluating its location in the distribution associated to the prediction interval (t distribution). This is exemplified in Figure 3.

_{max}, rather than using the conditional mean directly is illustrated by the examples in Table 6. For the Samarco incident, the mean value of the predicted D

_{max}is 174 km using D

_{max}.3, while the predicted 5th (95th) percentile is 10 (2933) km. The D

_{max}reported from the actual failure was 637 km downstream, which based on the uncertainty distribution associated with the regression equation, has a probability of exceedance of approximately 22% using model D

_{max}.3. In this case the tailings were deposited directly in the Doce River [4], transporting the tailings all the way to the Atlantic Ocean, whereas for other TSF failures an immediate river receptor may not be there, limiting the travel distance. Consequently, if just the conditional mean of the regression equations such as those developed by Rico et al. [10] is used, then one would be rather poorly informed as to the range of potential consequences of a failure. For a probabilistic risk evaluation then, for Samarco, the concern would have been the greater than 600 km impact with a 22% chance rather than the modest 174 km indicated by the regression. It is important to highlight that the observed V

_{F}was used to fit all the D

_{maX}regressions but in reality this value might not be known prior to a failure. Therefore, the uncertainty of the estimation of V

_{F}from Equation (4) has to also be considered in the predictions of D

_{max}, which will increase the uncertainty. This is also an issue with Rico’s approach reported in Equations (1) and (2), and it is acknowledged in their paper. In the online app we developed, this is addressed and D

_{max}can be calculated taking into account the uncertainty around V

_{F}in a two-step model.

## 4. Conclusions

_{f}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 V

_{F}is presented in Equation (4), and the new model to estimate D

_{max}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.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Left: Relationship between V

_{F}and V

_{T}in ×10

^{6}m

^{3}. Center: D

_{max}in km in relation to the dam factor (H × V

_{F}). Right: D

_{max}in km in relation to H

_{f}(H × (V

_{F}/V

_{T}) × V

_{F}). All plots are in the log-log scale. Note that the D

_{max}vs. H

_{f}plot is much tighter than the D

_{max}vs the dam factor plot. CB are added points from Chambers and Bowker [15] and R are from Rico [10].

**Figure 2.**Examples of prediction intervals from D

_{max}. 1 O (Rico’s original equation), D

_{max}.3 U and the observations for D

_{max}of past failures (obs).

**Figure 3.**t distributions showing the probability of exceedance of the Samarco and Mt. Polley observed D

_{max}and the distributions obtained with model D

_{max}.3 around the predicted mean value of D

_{max}in the log space.

No | Mine | Year | H (m) | V_{T} (×10^{6} m^{3}) | D_{max} (km) | V_{F} (×10^{6} m^{3}) | Failure Type ^{a} | Source |
---|---|---|---|---|---|---|---|---|

1 | (unidentified), Southwestern USA | 1973 | 43 | 0.5 | 25 | 0.17 | SI | Rico |

2 | Aitik mine, Sweden (Boliden Ltd.) | 2000 | 15 | 15 | 5.2 | 1.8 | ER | CB |

3 | Arcturus (Zimbawe) | 1978 | 25 | 1.7–2 Mt | 0.3 | 0.0211 ^{b} | OT | Rico |

4 | Bafokeng, South Africa | 1974 | 20 | 13 ^{c} | 45 | 3 | SE | Rico |

5 | Balka Chuficheva, Russia | 1981 | 25 | 27 | 1.3 | 3.5 | SI | CB |

6 | Bellavista, Chile | 1965 | 20 | 0.45 | 0.8 | 0.07 | EQ | Rico |

7 | Bonsal, North Carolina, USA | 1985 | 6 | 0.038 | 0.8 | 0.011 | OT | CB |

8 | Cerro Negro No. (3 of 5) | 1965 | 20 | 0.5 | 5 | 0.085 | EQ | Rico |

9 | Cerro Negro No. (4 of 5) | 1985 | 40 | 2 | 8 | 0.5 | EQ | Rico |

10 | Churchrock, New Mexico, United Nuclear | 1979 | 11 | 0.37 | 110 ^{d} | 0.37 | FN | Rico/CB |

11 | Cities Service, Fort Meade, Florida | 1971 | 15 | 12.34 | 120 | 9 | SE | Rico |

12 | Deneen Mica Yancey County, North Carolina, USA | 1974 | 18 | 0.3 | 0.03 | 0.038 | SI | CB |

13 | El Cobre New Dam | 1965 | 19 | 0.35 | 12 | 0.35 | EQ | CB |

14 | El Cobre Old Dam | 1965 | 35 | 4.25 | 12 | 1.9 | EQ | Rico |

15 | Fundão-Santarem, Minas Gerais, Brazil (Samarco) | 2015 | 90 | 55 | 637 | 32 ^{e} | ST | CB |

16 | Galena Mine (1974) | 1974 | 9 | NA | 0.61 | 0.0038 | OT | Rico |

17 | Gypsum Tailings Dam (Texas, USA) | 1966 | 11 | 7 ^{f} | 0.3 | 0.085 | SE | CB |

18 | Hokkaido, Japan | 1968 | 12 | 0.3 | 0.15 | 0.09 | EQ | Rico |

19 | Imperial Metals, Mt Polley, British Columbia, Canada | 2014 | 40 | 74 | 7 | 23.6 | FN | CB |

20 | Itabirito (Brazil) | 1986 | 30 | NA | 12 | 0.1 | ST | Rico |

21 | La Patagua New Dam (Chile) | 1965 | 15 | NA | 5 | 0.035 | EQ | Rico |

22 | Los Frailes, near Seville, Spain (Boliden Ltd.) | 1998 | 27 | 15 | 41 | 6.8 ^{g} | FN | CB |

23 | Los Maquis No. 3 | 1965 | 15 | 0.043 | 5 | 0.021 | EQ | Rico |

24 | Merriespruit, South Africa (Harmony)-No. 4A Tailings Complex | 1994 | 31 | 7.04 | 4 ^{h} | 0.6 ^{h} | OT | CB |

25 | Mochikoshi No. 1, Japan (1 of 2) | 1978 | 28 | 0.48 | 8 | 0.08 | EQ | Rico |

26 | Mochikoshi No. 2 (Japan) | 1978 | 19 | NA | 0.15 | 0.003 | EQ | Rico |

27 | Olinghouse, Nevada, USA | 1985 | 5 | 0.12 | 1.5 | 0.025 | SE | Rico |

28 | Omai Mine, No. 1, 2, Guyana (Cambior) | 1995 | 44 | 5.25 | 80 | 4.2 | ER | Rico |

29 | Prestavel Mine-Stava, North Italy, 2, 3 (Prealpi Mineraria) | 1985 | 29.5 | 0.3 | 8 ^{i} | 0.2 | SI | Rico |

30 | Sgurigrad, Bulgaria | 1996 | 45 | 1.52 | 6 | 0.22 | SI | Rico |

31 | Stancil, Maryland, USA | 1989 | 9 | 0.074 | 0.1 | 0.038 | SI | Rico |

32 | Taoshi, Linfen City, Shanxi province, China (Tahsan Mining Co.) | 2008 | 50.7 | 0.29 | 2.5 | 0.19 | U | CB |

33 | Tapo Canyon (USA) | 1994 | 24 | NA | 0.18 | NA | EQ | Rico |

34 | Tyrone, New Mexico (Phelps Dodge) | 1980 | 66 | 2.5 | 8 | 2 | SI | Rico |

35 | Veta de Agua (Chile) | 1985 | 24 | 0.7 | 5 | 0.28 | EQ | Rico |

^{a}SI = Slope instability, EQ = Earthquake, OT = Overtopping, ER = Erosion, FN = Foundation, SE = Seepage, U = Undefined.

^{b}CB report 0.039 × 10

^{6}m

^{3}V

_{F}.

^{c}In Rico 13 × 10

^{6}m

^{3}as [16] and [17]; in CB 17 × 10

^{6}m

^{3}.

^{d}In CB 110 km as in [18] and [17], in Rico 96.5−112.6 km.

^{e}43 × 10

^{6}m

^{3}in [19].

^{f}Rico in tones.

^{g}In Rico 4.6 × 10

^{6}m

^{3}; in [20] 10

^{6}m

^{3}, in [21] 5.5 × 10

^{6}m

^{3}of tailings and 1.9 × 10

^{6}m

^{3}of acid water; in [17] 5.5 Mm

^{3}.

^{h}In Rico 2.5 Mt V

_{F}and 2 km in D

_{max}as in [22]; in [5] 0.6 × 10

^{6}m

^{3}.

^{i}In Rico 4.2 km D

_{max}; in [13] 8 km.

Name | Model |
---|---|

V_{F}.1 | log(V_{F})~log(V_{T}) |

D_{max}.1 | log(D_{max})~log(H × V_{F}) |

D_{max}.2 | D_{max}~log(H × V_{F}), glm, Gaussian, log link |

D_{max}.3 | log(D_{max})~log(H_{f}) |

**Table 3.**Results for V

_{F}.1 with the original Rico dataset (O) and updated (U) datasets (including P.17).

Data | R^{2} | p-Value | Standard Error | 5-fold CV, 100 reps | Outliers | Leverage | Cook’s Distance |
---|---|---|---|---|---|---|---|

O | 0.87 | 1.209 × 10^{−9} | 0.288 | 3.3 | P.11, P.12 | P.9 | |

U | 0.815 | 1.285 × 10^{−11} | 0.402 | 11.5 | P.17 | P.19, P.15 | P.17 |

**Table 4.**Predicted and observed V

_{F}values in ×10

^{6}m

^{3}for select cases (using all the points in the original Rico dataset (O) and updated (U) data sets except P.17 in U).

Data | Points | Median Prediction * | Q5 | Q95 | Observed | Probability V_{F} > Observed | Probability V_{F} > V_{T} |
---|---|---|---|---|---|---|---|

O | Samarco (P.15) | 19.8 | 5.2 | 75.7 | 32 | 0.27 | 0.1 |

Mt Polley (P.19) | 26.8 | 6.8 | 104.8 | 23.6 | 0.56 | 0.1 | |

Gympsum (P.17) | 2.42 | 0.71 | 8.2 | 0.09 | 0.99 | 0.07 | |

U | Samarco (P.15) | 15.2 | 4.1 | 57.3 | 32 | 0.17 | 0.05 |

Mt Polley (P.19) | 20.3 | 5.3 | 76.9 | 23.6 | 0.42 | 0.05 | |

Gypsum (P.17) | 2.13 | 0.6 | 7.6 | 0.09 | 0.99 | 0.08 |

Model | Data * | R^{2} | p-Value | 5-Fold CV, 100 reps | Significant Outliers | Leverage | Cook’s Distance |
---|---|---|---|---|---|---|---|

D_{max}.1 | U | 0.44 | 5.335 × 10^{−5} | 249.5 | P.12 | P.7, P.15 | P.19, P.12 |

D_{max}.1 | O | 0.55 | 5.39 × 10^{−6} | 54.5 | P.12 | P.12, P.28 | |

D_{max}.2 | U | NA | NA | 284.9 | P.11, P.15, P.19 | P.15, P.7 | P.15 |

D_{max}.3 | U | 0.53 | 4.415 × 10^{−6} | 230.3 | P.12 | P.7, P.15 | P.7, P.12, P.19 |

**Table 6.**Predicted values (in km) using all the data points for training the models (using the observed V

_{F}).

Model | Points | Median Prediction * | Q5 | Q95 | Observed | Probability D_{max} > Observed |
---|---|---|---|---|---|---|

D_{max}.1 O | Samarco (P.15) | 294 | 19 | 4595 | 637 | 0.3 |

Mt Polley (P.19) | 141 | 10 | 2012 | 7 | 0.96 | |

Bonsal (P.7) | 0.25 | 0.02 | 3.25 | 8 | 0.014 | |

D_{max}.1 U | Samarco (P.15) | 141 | 6 | 3130 | 637 | 0.21 |

Mt Polley (P.19) | 74 | 3.6 | 1525 | 7 | 0.9 | |

Bonsal (P.7) | 0.28 | 0.01 | 6 | 0.8 | 0.23 | |

D_{max}.3 U | Samarco (P.15) | 174 | 10 | 2933 | 637 | 0.22 |

Mt Polley (P.19) | 68 | 4 | 1054 | 7 | 0.92 | |

Bonsal (P.7) | 0.3 | 0.02 | 5 | 0.8 | 0.28 |

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**MDPI and ACS Style**

Concha Larrauri, P.; Lall, U.
Tailings Dams Failures: Updated Statistical Model for Discharge Volume and Runout. *Environments* **2018**, *5*, 28.
https://doi.org/10.3390/environments5020028

**AMA Style**

Concha Larrauri P, Lall U.
Tailings Dams Failures: Updated Statistical Model for Discharge Volume and Runout. *Environments*. 2018; 5(2):28.
https://doi.org/10.3390/environments5020028

**Chicago/Turabian Style**

Concha Larrauri, Paulina, and Upmanu Lall.
2018. "Tailings Dams Failures: Updated Statistical Model for Discharge Volume and Runout" *Environments* 5, no. 2: 28.
https://doi.org/10.3390/environments5020028