Analysis of Uncertainty and Sensitivity in Tailings Dam Breach-Runout Numerical Modelling

Abstract

Tailings dam breaches (TDBs) and subsequent flows can pose significant risk to public safety, the environment, and the economy. Numerical runout models are used to simulate potential tailings flows and understand their downstream impacts. Due to the complex nature of the breach-runout processes, the mobility and downstream impacts of these types of failures are highly uncertain. We applied the first-order second-moment (FOSM) methodology to a database of 11 back-analyzed historical tailings flows to evaluate uncertainties in TDB runout modelling and conducted a sensitivity analysis to identify key factors contributing to the variability of the HEC-RAS model output, including at different locations along the runout path. The results indicate that prioritizing resources toward advancements in estimating the values of primary contributors to the sensitivity of the selected model outputs is necessary for more reliable model results. We found that the total released volume is among the top contributors to the sensitivity of modelled inundation area and maximum flow depth, while surface roughness is among the top contributors to the sensitivity of modelled maximum flow velocity and flow front arrival time. However, the primary contributors to the sensitivity of the model outputs varied depending on the case study; therefore, the selection of appropriate rheological models and consideration of site-specific conditions are crucial for accurate predictions. The study proposes and demonstrates the FOSM methodology as an approximate probabilistic approach to model-based tailings flow runout prediction, which can help improve the accuracy of risk assessments and emergency response plans.

Supplementary Information
The online version contains supplementary material available at 10.1007/s10230-024-00970-w.

Full text

Vol.:(0123456789)
Mine Water and the Environment
https://doi.org/10.1007/s10230-024-00970-w
TECHNICAL ARTICLE
Analysis ofUncertainty andSensitivity inTailings Dam Breach‑Runout
Numerical Modelling
NegarGhahramani1,2 · DanielA.M.Adria1,3· NahyanM.Rana4· MarceloLlano‑Serna5· ScottMcDougall1·
StephenG.Evans6· W.AndyTake7
Received: 25 July 2023 / Accepted: 19 January 2024
© The Author(s) 2024
Abstract
Tailings dam breaches (TDBs) and subsequent flows can pose significant risk to public safety, the environment, and the
economy. Numerical runout models are used to simulate potential tailings flows and understand their downstream impacts.
Due to the complex nature of the breach-runout processes, the mobility and downstream impacts of these types of failures
are highly uncertain. We applied the first-order second-moment (FOSM) methodology to a database of 11 back-analyzed
historical tailings flows to evaluate uncertainties in TDB runout modelling and conducted a sensitivity analysis to identify
key factors contributing to the variability of the HEC-RAS model output, including at different locations along the runout
path. The results indicate that prioritizing resources toward advancements in estimating the values of primary contributors
to the sensitivity of the selected model outputs is necessary for more reliable model results. We found that the total released
volume is among the top contributors to the sensitivity of modelled inundation area and maximum flow depth, while surface
roughness is among the top contributors to the sensitivity of modelled maximum flow velocity and flow front arrival time.
However, the primary contributors to the sensitivity of the model outputs varied depending on the case study; therefore, the
selection of appropriate rheological models and consideration of site-specific conditions are crucial for accurate predictions.
The study proposes and demonstrates the FOSM methodology as an approximate probabilistic approach to model-based
tailings flow runout prediction, which can help improve the accuracy of risk assessments and emergency response plans.
Keywords Tailings dam breach analysis (TDBA)· Numerical modelling· Runout analysis· Uncertainty analysis·
Sensitivity analysis· FOSM· HEC-RAS 2D
Introduction
Preamble
Tailings dam breaches (TDBs) and subsequent downstream
tailings flows can pose significant risk to public safety, the
environment, and the economy (Blight 2009; Ghahramani
etal. 2020; Rana etal. 2021a; Santamarina etal. 2019).
Runout models have been used to simulate the behaviour and
characteristics of potential tailings flows, including inunda-
tion area, runout distance, flow velocity, flow depth, and
arrival time (Ghahramani etal. 2022; Martin etal. 2019;
Pirulli etal. 2017). Researchers use TDB runout modelling
to understand the complex physical mechanisms and the
downstream impacts of tailings flows in diverse terrains,
whereas mine owners and industry consultants rely on the
results of TDB analyses (TDBAs) to assign consequence
* Negar Ghahramani
nghahramani@eoas.ubc.ca
1 Department ofEarth, Ocean andAtmospheric Sciences, The
University ofBritish Columbia, Vancouver, Canada
2 WSP, Lakewood, CO, USA
3 Knight Piésold, Vancouver, BC, Canada
4 Klohn Crippen Berger, Toronto, ON, Canada
5 Red Earth Engineering, Brisbane, Australia
6 Department ofEarth andEnvironmental Sciences, University
ofWaterloo, Waterloo, ON, Canada
7 Department ofCivil Engineering, Queen’s University,
Kingston, ON, Canada

Mine Water and the Environment
classifications and develop emergency response plans (Cana-
dian Dam Association (CDA) 2021).
A recent benchmarking study by Ghahramani etal.
(2022), involving four numerical models commonly used
in TDBAs, indicated a high level of uncertainty in model
inputs. Some of these uncertainties were attributed to incom-
plete site-specific observational data and laboratory and in-
situ measurements, the resulting challenges associated with
selecting proper input parameters (e.g. the estimation of
released volume/hydrograph and the selection of rheological
models and their associated parameters), and the subjectiv-
ity in the model calibration process. The study highlighted
the need for additional back-analysis of historical tailings
flows to better understand and quantify the sensitivities of
output variables in modelling results, and the importance of
developing a systematic probabilistic approach for runout
analysis in TDBA practice (Ghahramani etal. 2022). The
CDA (2021) TDBA guidelines also list additional sources
of uncertainties in topographic data quality, failure modes,
and triggering factors.
High levels of uncertainty in input variables (e.g. total
released volume, rheological parameters, surface roughness,
breach parameters) can in turn lead to high uncertainty in
output variables (e.g. runout distance, inundation area, flow
velocity). The uncertainty in model outputs is quantified by
studying the distribution of possible outcomes with respect
to the uncertainty in input parameters. This type of uncer-
tainty analysis is useful when evaluating the reliability and
accuracy of model results and has been a practice for dec-
ades in various engineering activities, such as structural,
geotechnical, hydraulic, aerospace, and manufacturing pro-
cesses (e.g. Baecher and Christian 2005; Burges and Letten-
maier 1975).
Identifying the dominant controls of the uncertainty in
modelling results can help determine which inputs require
further consideration and/or higher investments in time/
budget. One method to achieve this is a sensitivity analysis,
the aim of which is to investigate how changes in input varia-
bles affect the output results (Borgonovo and Plischke 2016;
Razavi etal. 2021). Uncertainty and sensitivity analyses are
related, but have different meanings and purposes. Figure1
illustrates the concept of uncertainty and sensitivity meas-
ures and the distinction between them. Together, uncertainty
and sensitivity analyses can help the modeller understand,
enhance, and communicate the quality and reliability of the
model outcomes to support well-informed decision-making.
x1
x2
xn
Model Z=f(x1,x2,...xn)
Input
Output
Uncertainty Analysis
Sensitivity Analysis
Output Probability Distribution
z
x3
xn
x2x1
+1 Z
-1 Z
a)
b)
c)
z = Mean
Z = Standard deviation
Fig. 1 Framework illustration of numerical model uncertainty and
sensitivity analyses and distinctions between them. Box (a) displays
a numerical model with its input and output variables. Box (b) illus-
trates the results of uncertainty analysis, which provides the distri-
bution of possible outcomes with respect to the uncertainty in input
parameters while box c) illustrates the results of sensitivity analysis,
which identifies the dominant controls of the uncertainty in modelling
results

Mine Water and the Environment
To quantify uncertainty, probabilistic methods such as
first-order second-moment (FOSM) (Baecher and Christian
2005; Lee and Mosalam 2005; Llano-Serna etal. 2018;
Kim etal. 2020; Nadim 2007) and Monte Carlo simula-
tions (MCS) (Kleijnen 1995; Razavi etal. 2021; Tonkin and
Doherty 2009) have become popular. The FOSM method
has been shown to be a computationally efficient tool in dif-
ferent engineering applications where more computation-
ally expensive methods, such as MCS, are not possible (e.g.
Kunstmann and Kinzelbach 2000; Kunstmann etal. 2002;
Nadim 2007; Wang and Hsu 2009). This method approxi-
mates the mean and variance of a model output variable of
interest as a function of the mean and variance of the input
factors and their correlations (Baecher and Christian 2005).
An advantage of the FOSM method is that it can provide the
uncertainties of an output variable from each input variable
separately and/or by considering all input variables together
(Kim etal. 2020). This method has been used for uncertainty
quantification in water quality modelling and groundwater
modelling (Dettinger and Wilson 1981; Kunstmann and Kin-
zelbach 2000; Kunstmann etal. 2002; Wang and Hsu 2009),
for the analysis of the probability of geotechnical failure and
potential consequences (Baecher and Christian 2005; Kim
etal. 2020; Nadim 2007), and to investigate the sensitiv-
ity of the seismic demand of a structure to potential future
earthquakes (Lee and Mosalam 2005). This track record of
success in related problems made the FOSM method a prom-
ising candidate to capture the uncertainty in TDB runout
modelling in this study.
Scope andObjectives
Various hydraulic modelling and landslide runout model-
ling tools are available for TDBAs (Canadian Dam Asso-
ciation (CDA) 2021; Ghahramani etal. 2022; McDougall
2017). The entire breach-runout process in a tailings dam
is complex and strongly dependent on site-specific condi-
tions, and the physical mechanisms of tailings flows remain
poorly understood. As such, simplifications are made at
almost every stage of the model development, from the
mathematical differential equations to the initial and bound-
ary conditions. In addition, there is uncertainty associated
with the estimation of the model inputs. As a result, there are
different sources of uncertainty associated with numerical
models (Ghahramani etal. 2022; Martin etal. 2022; Pir-
ulli etal. 2017). To the best of our knowledge, only one
recent study (Melo and Eleutério 2023) has investigated the
sensitivity of tailings dam breach inundation mapping to
rheological parameters through a probabilistic approach and
those authors highlighted the lack of research on probabilis-
tic approaches, particularly for TDBAs.
To address this gap, we used a database of 11 back-
analyzed tailings flow cases to assess the uncertainties
in TDB runout modelling using the FOSM method. The
Hydrologic Engineering Centre’s River Analysis System
(HEC-RAS) numerical model, developed as a publicly
accessible tool by the U.S. Army Corps of Engineers, was
used for the back-analysis (Adria 2022; Brunner 2020).
The main objectives of this study were to: (1) identify the
primary contributors to the sensitivity of key model out-
puts (inundation area, maximum flow velocity, maximum
flow depth, and front flow arrival time) among the selected
input variables (total released volume, yield stress, viscos-
ity, surface roughness, breach width, and breach formation
time), (2) study the variation of sensitivity estimates along
the flow runout path, and (3) investigate the applicability
of the FOSM method for probabilistic runout modelling
in prediction applications.
Methodology
The FOSM Method
The FOSM approach is a numerical probabilistic method
in which the mean and variance of the model output vari-
ables can be estimated by the first-order approximation of
a Taylor series expansion, using the mean and variance of
the input variables (Baecher and Christian 2005; Nadim
2007). If the number of uncertain input variables is n, this
method requires either evaluating
n
partial derivatives of
the performance function or performing a numerical
approximation using evaluations at
2n+1
points. We used
the latter approach in this study. For an output function
Z=f(
X
1
,X
2
,…X
n)
, in which
X1,X2,…Xn
are random varia-
bles, using the first-order approximation, the mean, μz, and
variance, σz
2, of the function Z, become:
where:
𝜇Xi
and
𝜎Xi
2 are the means and the variances of
model inputs for
i=1, 2, …,n
,
n
is the number of inputs,
and
COV (Xi,Xj)
is the covariance between input variables
XiandXj
. If it is assumed that the variables are uncorre-
lated, the second term on the right side of Eq.2 vanishes
(Baecher and Christian 2005; Nadim 2007).
Although FOSM is a linearization technique, it can be
applied to models with non-linear output functions. The
FOSM method linearizes the non-linear output function by
approximating it as a Taylor series expansion around the
mean values of the input variables. Therefore, it assumes
(1)
𝜇Z≈f(𝜇Xi)=f(𝜇X1,𝜇X2,…𝜇Xn)
(2)
𝜎
2
Z≈
n
∑
i=1
𝜎2
Xi
(𝜕f
𝜕x
i
)
2
+
∑
n
i=1
∑
n
j≠i
𝜕f
𝜕x
i
𝜕f
𝜕x
j
COV (Xi,Xj
)

Mine Water and the Environment
that the output can be locally approximated as a linear
function near the mean values of the input variables (Lee
and Mosalam 2005).
Output Variables
The output variables of TDB runout modelling represent the
simulated characteristics of the tailings flow downstream
of the breach. For a FOSM analysis, four main outputs are
studied: inundation area, maximum flow velocity, maximum
flow depth, and flow front arrival time. The output values of
the last three variables are measured at 50% of the observed
Zone 1 runout distance, which is defined as “the extent of
the main solid tailings deposit, which is characterized by
remotely visible or field-confirmed sedimentation, above
typical bankfull elevations if extending into downstream
river channels” (Ghahramani etal. 2020).
Input Variable Statistics
In this study, the following six input variables were selected:
total released volume, yield stress, viscosity, surface rough-
ness, breach width (considering a trapezoidal breach shape),
and breach formation time. Detailed definitions of breach
geometry and breach formation time are provided in Wahl
(1998) and Froehlich (2008). The conventions are adopted
from water-retaining dam breach practice, as they were
found to be generally suitable for tailings dam breaches by
Adria etal. (2023a). Breach formation time is only used
for erosional breach case studies that typically involve over-
topping or piping/seepage with a relatively large volume of
supernatant pond and a long breach duration. In reality, some
of these inputs might be correlated (e.g. a wider breach can
release more tailings and higher yield stress values are typi-
cally associated with higher viscosity values). However, in
the HEC-RAS numerical model, the six selected inputs are
formulated independently and are manually assigned, and
therefore are not correlated. In other words, adjusting the
breach width value in HEC-RAS does not affect the value
for the outflow volume, or adjusting the yield stress value
does not affect the viscosity value. Input variables that can
be correlated to other inputs in the HEC-RAS model, such
as solid concentration, were not considered in this study.
For example, as part of the quadratic rheology within the
numerical model, yield stress and viscosity are both a func-
tion of solid concentration. The best-fit (calibrated) input
values that are estimated in the case study back analyses
(described in the “Tailings flow back-analyses” section) are
set to be the mean of the model inputs (
𝜇Xi)
and their output
results are represented by the mean values (
𝜇Z)
of the model
outputs. Other programs or modelling tools used in TDBAs
may treat input values differently than HEC-RAS.
There are two ways to estimate the variance,
𝜎2
Z
, of the
output function Z: i) if the function
f(
X
1
,X
2
,…X
n)
is tractable,
the function can be differentiated to give a closed-form
expression for the variance of
f(
X
1
,X
2
,…X
n)
or ii) more com-
monly, it is not possible to differentiate the function directly;
therefore, the partial derivatives must be obtained through
numerical approximation approaches (Baecher and Christian
2005). In this application, since the form of function Z is
unknown, the second approach is used to approximate the
partial derivatives with the central differences method. To
find the partial derivative for each best-fit input variable, the
best-fit input value is increased and decreased by a small
increment (± 10% was used in this study), while the rest of
the variables are kept constant. The differences between the
resulting output values are then calculated and divided by
the differences between the increased and decreased input
values. This can be represented mathematically as follows:
where
𝜀i
is ± 10% of the best-fit value for the particular input.
To compute Eq.2, an estimate of the variance of the
model inputs is also needed. To achieve this, the standard
deviation and the mean values of selected variables were
estimated statistically using data from available databases.
For the total released volume and breach width, data from 41
TDB cases and 36 TDB cases, respectively, were collected
from Rana etal. (2021b). Since the total released volume is
a portion of the total impoundment volume, the ratio of the
total released volume to the total impoundment volume was
used to estimate of the mean and standard deviation. For the
breach width statistics, the top breach width data from Rana
etal. (2021b) were used, and for the FOSM analysis, the side
slopes were kept constant. For the surface roughness, 74 data
points from Chow (1959) were used. For the breach for-
mation time statistics, 27 water retaining dam failures were
compiled from Wahl (1998) and Wahl (2014). Ghahramani
etal. (2022) and Adria (2022) showed that the numerical
rheological parameter values do not necessarily correspond
with the measured rheological parameter values. However,
in the absence of sufficient calibrated yield stress and viscos-
ity data from the numerical back-analysis of historical cases,
a tailings rheology database from Martin etal. (2022) was
used as a first order approximation for estimating the mean
and standard deviation of rheological parameters. Using the
rheology database, the yield stress and viscosity data were
classified with respect to the volumetric solid concentra-
tion ranges and their means and standard deviations were
estimated for each range. Then, the coefficients of varia-
tion (CoV) of input variables were calculated as the ratio
of their standard deviation to their mean values. Tables1,
(3)
𝛿f(𝜇
Xi
)
𝛿x
i
≈1
2𝜀
i{
f
(
𝜇X1,𝜇X2,…,𝜇Xi+𝜀i,…,𝜇Xn
)
−f
(
𝜇X1,𝜇X2,…,𝜇Xi−𝜀i,…,𝜇Xn
)}

Mine Water and the Environment
2, 3 present the input variables with their estimated CoV
values. A greater CoV indicates greater dispersion around
the mean value. Finally, the standard deviation of a model
input can be estimated for each case study individually, by
multiplying the best-fit value (mean) of the model input and
estimated CoVs.
Sensitivity andUncertainty Estimates ofModel
Outputs
A type of sensitivity analysis can be carried out using
differentiation-based methods (Borgonovo and Plischke
2016). The FOSM methodology enables estimation of the
gradient of the output variables with respect to the input
variables, due to small local changes in model inputs.
Therefore, the partial derivatives in Eq.2 become natu-
ral sensitivity estimates. Since the partial derivative of
each input has different units from one another, we use
the equation provided by Borgonovo and Plischke (2016),
in which the sensitivity measures (Di) are normalized
and can be ranked (Eq.4). The result of this sensitivity
analysis can be used to identify the primary contributors
to the uncertainty in model outputs. The uncertainty meas-
ure (CoVZ) of each output is estimated as the ratio of the
standard deviation to the mean value, which is called the
coefficient of variation of the output variable (Eq.5).
where
Xi
is the input variable and n is the number of inputs.
This fraction quantifies how the resulting output value
changes with a particular input variable relative to the total
change in the output variable.
where
𝜎Z
and
𝜇Z
are the standard deviation and mean of
model output, respectively. The estimated standard devia-
tion and mean of each output are obtained from the FOSM
results.
In this study, the sensitivity analysis was divided into
two parts. For the first part, the 50% runout distance of
Zone 1 was selected for all local model sensitivity esti-
mates as a consistent relative location to compare all
events for maximum flow velocity, maximum flow depth,
and flow front arrival time. For the second part, the varia-
tion of sensitivity estimates was investigated at 10%, 25%,
50%, 75%, and 90% of the Zone 1 runout distance.
(4)
D
i=
�
�
�
𝜕f
(𝜇X)
𝜕Xi
dXi�
�
�
�
�
�
�
∑
n
j=1
𝜕f(𝜇X)
𝜕Xj
dXj
�
�
�
�
(5)
CoV
Z=
𝜎
Z
𝜇
Z
Table 1 Coefficient of variation (CoV) values for the selected input variables
a The ratio of the total released volume to the total impoundment volume
b The breach formation time is normalized by the breach height similar to the mean erosion rate in Walder and O’Connor (1997)
Input parameter Mean Standard deviation CoV Sample size References
Total released volumea (%) 36 25 0.7 41 Rana etal. (2021b)
Roughness (s/m1/3) 0.06 0.05 0.8 74 Chow (1959)
Yield stress (Pa) 36–297 74–127 0.4–2.1 See Table2Martin etal. (2022)
Viscosity (Pas) 0.27–1.6 0.1–2.4 0.4–1.5 See Table3Martin etal. (2022)
Breach width (m) 218 269 1.2 36 Rana etal. (2021b)
Breach formation timeb (m/h) 31 29 0.9 27 Wahl 1998 and Wahl 2014
Table 2 Coefficient of variation (CoV) values for the yield stress with
respect to volumetric solid concentration (Cv)
Data extracted from the database in Martin etal. (2022)
Parameter Cv range (%) Sample size CoV
Yield stress 10–19.9 21 0.9
20–29.9 73 1.6
30–39.9 121 2.1
40–49.9 59 1.6
50–59.9 49 1.5
60–69.9 10 0.4
Table 3 Coefficient of variation (CoV) values for the viscosity with
respect to volumetric solid concentration (Cv)
Data extracted from the database in Martin etal. (2022)
Parameter Cv range (%) Sample size CoV
Viscosity 10–19.9 7 0.6
20–29.9 31 0.9
30–39.9 59 1.5
40–49.9 17 1.2
50–59.9 5 0.4
60–69.9 – –

Mine Water and the Environment
Tailings Flow Back‑Analysis
HEC‑RAS 2D
HEC-RAS is an open-access software package that was
originally developed by the U.S. Army Corps of Engineers
(USACE) for water resource engineering and open-channel
hydraulic analysis (Adria 2022; Brunner 2020; Gibson etal.
2021, 2022). HEC-RAS 2D is a depth-integrated two-dimen-
sional model that uses the finite volume numerical method.
It is capable of dam breach-runout modelling, erosion and
sediment transport simulations, and water quality analyses
(Brunner 2020). Version 6.1 of HEC-RAS 2D (the most
current version at the time of this work) was used in this
work due to its popular application in dam breach-runout
modelling and flood risk management studies, and its dem-
onstrated capability of modelling both Newtonian and non-
Newtonian flow types (Adria 2022; Brunner 2020; Gibson
etal. 2021). There are four selectable options of rheologi-
cal models for non-Newtonian flow simulations: Bingham,
Quadratic, Herschel-Bulkley, and Voellmy.
Back‑Analyzed Case Studies
The back-analyses of the 11 historical cases that we used
as baseline models in the FOSM analysis are detailed in
Adria (2022). The original database is provided in an
open-access data repository hosted at Borealis (Adria etal.
2023b). These cases are selected based on the availability
of information on the pre-and post-failure site and tailings
characteristics. Classifying the case studies based on the
type of breach process (CDA 2021), there are three ero-
sional breach and eight non-erosional breach case studies
(Table4). Overtopping and piping/seepage type of fail-
ure mechanisms commonly involve an erosional breach
process with a relatively large volume of supernatant
pond and a long breach duration, from several minutes to
hours. The non-erosional breach processes involve near-
instantaneous collapses and have characteristics other than
the erosional breaches mentioned above (CDA 2021). In
Table4, the Tapo Canyon Event 1 refers to the viscous
section of the Tapo Canyon tailings flow and the Cadia
Event 2 refers to the secondary liquefaction event, which
occurred on March 11, 2018; more details are provided in
Adria (2022).
The topographic data used in the models consisted of
a mix of publicly available and commercial sources, with
additional manual modifications as needed. The breach char-
acteristics and outflow volumes for each event were previ-
ously compiled in Adria (2022), Ghahramani etal. (2020),
and Rana etal. (2021a). The yield stress and viscosity in the
quadratic rheological model were calibrated in two steps.
First, the modelled inundation area was compared to the
observed inundation area as mapped by Ghahramani etal.
(2020) and Rana etal. (2021a) using a quantitative method
developed by Heiser etal. (2017). The modelled results
were then compared to available observations of arrival
time and runout depth within the inundation area to further
refine the calibrated yield stress and viscosity. The quadratic
rheological model as implemented in HEC-RAS also uses a
third term that relates shear stress to strain-rate squared, to
simulate dispersive effects. The coefficient for the dispersive
term is calculated with a combination of theoretical, empiri-
cal, and measurable sediment characteristics (e.g. particle
diameter). Only the particle diameter was varied in Adria
(2022) based on available data for each event. The calcu-
lated dispersive coefficients between all events ranged from
1.1 × 10–5 to 6.1 × 10–2, which aligns with the findings of
Julien and Lan (1991). The surface roughness was defined
with spatially varied Manning’s n values based on the land
cover observed on satellite/aerial imagery, as well as guid-
ance in Arcement and Schneider (1989) and Janssen (2016),
but it was not adjusted as part of the calibration process. The
best-fit model inputs and outputs are presented in Table5.
In the FOSM analysis, as described earlier, the number
of evaluation points is 2n + 1 (where n is the number of
model inputs) when the numerical approximation method
is used. Considering the five and six inputs for non-erosional
and erosional breach case studies, respectively, each non-
erosional breach case study was run 11 times, and each
erosional beach case study was run 13 times. Therefore, in
total, there were 127 evaluation points for all 11 case studies.
Refer to Supplementary Appendix A for all of the performed
runs.
To investigate the variation of sensitivity estimates along
the flow runout distance, three case studies were selected
(1985 Stava, 1998 Aznalcóllar, and 2019 Feijão). These
cases were selected because they have different released
volumes, breach processes, and topographic conditions, and
therefore the magnitude and runout path of these three cases
Table 4 Database of 11 back-analyzed tailings dam breaches (TDBs)
ID TDB case Country Year Breach process
1Stava Italy 1985 Non-erosional
2 Tapo Canyon Event 1 USA 1994 Non-erosional
3 Merriespruit (Harmony) South Africa 1994 Non-erosional
4 Aznalcóllar (Los Frailes) Spain 1998 Erosional
5 Ajka (Kolontar) Hungary 2010 Erosional
6 Kayakari Japan 2011 Non-erosional
7 Mount Polley Canada 2014 Erosional
8 Fundão Brazil 2015 Non-erosional
9 Tonglvshan China 2017 Non-erosional
10 Cadia Event 2 Australia 2018 Non-erosional
11 Feijão Brazil 2019 Non-erosional

Mine Water and the Environment
represent diverse morphological environments (see Tables4
and 5).
Probability Distribution Approximation
forPrediction
The FOSM method provides estimates of the mean and
standard deviation of the model outputs, which can be used
to make probabilistic forward predictions. However, since
the probability distributions of the inundation area, maxi-
mum flow velocity, maximum flow depth and frontal arrival
time are unknown and are not obtained directly through the
FOSM method, assumptions must be made. Normal and log-
normal approximations are typically used in geotechnical
problems (Kim etal. 2020; Nadim 2007). The log-normal
approximation is also considered reasonable in this applica-
tion, as a first approximation. It is used instead of the normal
distribution because the model input parameters cannot be
negative.
The Merriespruit case was used in this study to demon-
strate the application of the FOSM method for predicting the
probability of model outputs. Following Aaron etal. (2022),
we excluded any site-specific information about the breach
geometry, rheological parameters and observational data for
the numerical modelling. The Merriespruit case was also
excluded from the TDB calibration dataset for the purposes
of this demonstration. However, the total released volume
was not changed so that the observed and simulated results
could be roughly compared.
A detailed description of the Merriespruit TDB event was
provided by Fourie and Papageorgiou (2001) and Wagener
(1997). The estimated total released volume was 0.615M
m3. For the purpose of demonstrating the probabilistic
method, we adopted a strategy of making reasonable model
and parameter selections that an experienced TDB practi-
tioner might make. A trapezoidal breach shape with side
slopes of 1V:1H was used. Assuming the breach height was
equal to the dam height (31m), and using an average breach
width to breach height ratio of 7 based on non-erosional
breach data in Adria (2022), the average breach width was
estimated to be 217m. The topographic data source is the
Airbus WorldDEM™ DTM with a 12m resolution. A con-
stant surface roughness (n) of 0.08 was used throughout the
runout path to account for both the suburban and wetland
areas that are observed in satellite imagery and aerial pho-
tographs. The Quadratic rheological model was selected for
this analysis. Since there is a lack of back-analyzed historical
case studies similar to Merriespruit, which could otherwise
inform rheological parameter selection, the yield stress and
viscosity in the present study were estimated by fitting expo-
nential curves to the yield stress and viscosity data provided
in Martin etal. (2022). Considering the volumetric solid
content of 50% for Merriespruit, the yield stress and viscos-
ity were estimated as 63Pa and 0.8Pas, respectively.
Results
Sensitivity andUncertainty Estimates
Figures2, 3, 4 and 5 illustrate the sensitivity of the mod-
elled inundation area, maximum flow velocity at 50%
runout, maximum flow depth at 50% runout, and flow
front arrival time at 50% runout, respectively, for each case
Table 5 List of the best-fit input values and the best-fit modelled outputs from the back-analyses of 11 TDBs
Numbers represent case IDs given in Table4
TRV reported total released volume, YS best-fit yield stress, V best-fit viscosity, R estimated surface roughness, TBW reported top breach width
a At the 50% observed runout distance
ID TRV (mm3) YS (Pa) V (Pas) R TBW (m) Inundation area (m2) Max
velocitya
(m/s)
Max deptha (m) Arrival timea (s)
1 0.1852 3.2 1.8 0.04 220 522,608 18.23 6.25 202
2 0.0275 4000 4 0.06 145 19,136 6.57 3.84 18.8
3 0.615 200 4 0.08 150 1,021,109 2.09 1.97 249
4 6.75 2.5 0.4 0.055 84 15,419,072 1.26 3.83 12,468
5 1.2 3.2 3.2 0.04 60 7,050,620 0.82 0.58 21,010
6 0.041 250 15 0.03 50 108,094 5.98 1.93 147
7 25 2 0.02 0.04 262 2,987,242 6.16 11.29 1980
8 32.2 3 0.05 0.07 650 10,216,937 2.49 14.16 6,460
9 0.5 250 2.5 0.04 260 278,761 6.27 3.18 34.6
10 0.16 1780 562 0.02 30 103,077 5.20 4.23 36
11 9.65 400 25 0.08 560 3,700,751 5.98 9.41 679

Mine Water and the Environment
study. Figure2 indicates that the inundation area was most
sensitive to total released volume in 9 out of 11 cases, with
the exceptions of Stava and Mt. Polley. Stava exhibited the
greatest sensitivity to surface roughness, while Mt. Polley
exhibited the greatest sensitivity to breach width. Yield
stress was one of the top two contributors to the sensitivity
of inundation area for more than half of the cases (6 out
of 11) (Fig.2).
Figures3 and 4 indicate that the maximum flow veloc-
ity and maximum flow depth at 50% runout were most
sensitive to surface roughness and total released volume,
respectively, in 8 out of 11 cases. For flow velocity, the
Fig. 2 Sensitivity of modelled
inundation area with respect to
the selected inputs
Stava
Ajka
Tonglvshan
Tapo Canyon Merriespruit
Kayakari Mt. Polley
Cadia II Feijão
Aznalcóllar
Fundão
Total Released Vo lume
Breach Width
Surface Roughness
Yield Stress
Viscosity
Breach Formation Ti
me
Fig. 3 Sensitivity of modelled
maximum flow velocity at
50% runout with respect to the
selected inputs
Stava
Ajka
Tonglvshan
Tapo Canyon Merriespruit
Kayakari Mt. Polley
Cadia II Feijão
Aznalcóllar
Fundão
Total Released Volume
BreachWidth
Surface Roughness
Yield Stress
Viscosity
Ti
me
Breach Formation

Mine Water and the Environment
exceptions are Cadia, Mt. Polley, and Ajka; Cadia and
Ajka exhibit the highest sensitivity to total released vol-
ume, while Mt. Polley exhibits the highest sensitivity
to breach width (Fig.3). For flow depth, the exceptions
were Stava, Mt. Polley, and Cadia; Mt. Polley and Cadia
exhibit the highest sensitivity to breach width, while
Stava exhibits the highest sensitivity to surface roughness
(Fig.4).
Figure5 indicates that the sensitivity results for flow
front arrival time display greater variability than the other
model outputs. Flow front arrival time is most sensitive to
surface roughness in 7 out of 11 cases, with the exceptions
Fig. 4 Sensitivity of modelled
maximum flow depth at 50%
runout with respect to the
selected inputs
Stava
Ajka
Tonglvshan
Tapo Canyon Merriespruit
Kayakari Mt. Polley
Cadia II Feijão
Aznalcóllar
Fundão
Total Released Volume
Breach Width
Surface Roughness
Yield Stress
Viscosity
Ti
me
Breach Formation
Fig. 5 Sensitivity of modelled
flow front arrival time at 50%
runout with respect to the
selected inputs
Total Released Vo lume
Breach Width
Surface Roughness
Yield Stress
Viscosity
Breach Formation Ti
me

Mine Water and the Environment
of Tapo Canyon, Ajka, Tonglvshan, and Cadia. Tapo Can-
yon exhibits the highest sensitivity to both total released
volume and breach width equally, while Ajka exhibits the
highest sensitivity to total released volume, and Tonglvs-
han and Cadia exhibits the highest sensitivity to breach
width.
Figures6 and 7 display the uncertainty estimates, CoV
(coefficient of variation), values for four numerical outputs:
inundation area, maximum flow velocity, maximum flow
depth, and flow front arrival time at 50% of the observed
runout distance. These values are shown for both non-
erosional breach (Fig.6) and erosional breach (Fig.7) case
studies, considering the selected inputs individually as well
as all inputs together (all). The erosional breach case studies
include an additional input, breach formation time (BFT),
which is not applicable for the non-erosional breach case
studies (Fig.7). The findings suggest that for all case studies,
the uncertainties in the inundation area and maximum flow
depth with respect to the total released volume exceed 10%,
which is typically regarded as a high level of uncertainty in
practice. The uncertainties in the maximum flow velocity
and flow front arrival time with respect to surface rough-
ness exceed 10% for most of the case studies, specifically
10 out of 11 and 9 out of 11, respectively. Tables containing
sensitivity and uncertainty values are presented in Supple-
mentary Appendix B.
Sensitivity Variation Along Runout Path
Figure8 shows the sensitivity of the modelled maximum
flow velocity (a, b, c), maximum flow depth (d, e, f) and
flow front arrival time (g, h, i) at 10%, 25%, 50%, 75% and
90% of the Zone 1 runout distance for the Stava, Aznalcóllar
and Feijão cases. Sensitivity variation is observed along the
flow path. In most of the scenarios, the sensitivity to breach
width tends to decrease with distance from the breach. The
sensitivity to total released volume has an increasing trend
in most cases, which appears to plateau in some cases. The
sensitivity to surface roughness is largely case-dependent
without any discernible common trend. The sensitivity to
yield stress tends to increase with distance from the breach
in the case of Feijão. Sensitivity values along the runout path
are provided in Supplementary Appendix C.
Demonstration ofProbabilistic Prediction Approach
The Merriespruit demonstration case involves modelling the
probability distributions of two key parameters: the inundation
a)
c) d)
b)
Case Studies
l
l
A
l
l
A
l
l
A
l
l
Fig. 6 Non-erosional breach case study coefficient of variation (CoV)
values for four numerical outputs: a inundation area, b maximum
flow velocity, c maximum flow depth, and d flow front arrival time at
50% of the observed runout distance. These CoV values are presented
with respect to the five selected inputs, namely total released volume
(TRV), yield stress (YS), viscosity (V), surface roughness (R), and
breach width (BW), both individually and all inputs (All) together.
Note the different y-axis range for plot b)

Mine Water and the Environment
area and maximum flow velocity at 50% runout distance.
These parameters are modelled with respect to each input vari-
able individually as well as considering the total uncertainty
of all input variables together (Fig.9). The mean value of the
modelled inundation area is 1.54 km2, while the mean value of
the modelled maximum flow velocity at 50% runout distance
is 2.6m/s.
Since the model outputs cannot be negative, we used the
assumption of log-normal distribution. The observed Zone
1 inundation area for Merriespruit was estimated to be 0.89
km2 (Ghahramani etal. 2022), which is slightly outside of one
standard deviation from the mean of the probability density
curves for the inundation area, as shown in Fig.9b. Similarly,
the best-fit modelled maximum flow velocity at 50% of the
runout distance was ≈ 2.1m/s (Adria 2022), which is within
one standard deviation from the mean of the probability den-
sity curves for maximum flow velocity, as shown in Fig.9d.
Discussion
Sensitivity Analysis
The results of the sensitivity analysis suggest that modelled
inundation area and maximum flow depth are most sensitive
to total released volume, whereas modelled maximum flow
velocity and flow front arrival time are generally most sensi-
tive to surface roughness. These findings are conceptually
consistent with physical observations (Adria 2022; Ghah-
ramani etal. 2020) and agree with past findings that out-
flow volume is strongly correlated with inundation area and
runout distance (e.g. Concha-Larrauri and Lall 2018; Ghah-
ramani etal. 2020; Piciullo etal. 2022; Rico etal. 2008).
With regard to identifying primary contributors to the
sensitivity of model outputs, our results indicated similar
a)
c) d)
b)
Case Studies
A
l
l
A
l
l
A
l
l
A
l
l
Fig. 7 Erosional breach case study coefficient of variation (CoV) val-
ues for four numerical outputs: a inundation area, b maximum flow
velocity, c maximum flow depth, and d flow front arrival time at 50%
of the observed runout distance. These CoV values are presented
with respect to the six selected inputs, namely total released volume
(TRV), yield stress (YS), viscosity (V), surface roughness (R), breach
width (BW), and breach formation time (BFT), both individually and
all inputs (All) together. Note the different y-axis range for plot b)

Mine Water and the Environment
trends for most of the cases, with a few exceptions for each
model output. For Stava, the sensitivity of all four outputs
followed a similar pattern, with surface roughness as the
primary contributor and total released volume and breach
width among the top three contributors to the sensitivity
of those model outputs. This may be attributed to the steep
travel path at Stava, which is higher than all the other cases
in this study.
Mt. Polley is another exception in which the model is
highly sensitive to the breach width for the modelled inunda-
tion area, maximum flow velocity, and maximum flow depth
(Figs.2, 3, 4). This is likely due to the unique site conditions
related to the Mt. Polley failure. The Zone 1 extent of Mt.
Polley was truncated by Quesnel Lake 9km downstream of
the tailings facility. Without the presence of an intercepting
water body, a hypothetical failure of similar size and compo-
sition to Mt. Polley would be expected to travel farther than
9km. As a result, the 50% runout distance point considered
in this study for the Mt. Polley model may actually be more
representative of the 5–15% range if the event was not trun-
cated by the lake. From this perspective, the Mt. Polley sen-
sitivity results for all parameters are less exceptional to the
other events, as the Stava, Aznalcóllar, and Feijão results are
also consistently sensitive to breach width at about 5–15% of
their runout distances. Furthermore, for confined events like
Mt. Polley, the sensitivity of inundation area to outflow vol-
ume is primarily driven by the runout distance, with minor
changes in the flow width along the runout path. With the
truncated runout distance at Mt. Polley enforcing the same
runout distance for all sensitivity scenarios, along with a
predominantly channelized flow path, there was physically
little room for the inundation area to differ between input
variations.
Another consideration for Mount Polley is that the
released volume had a relatively low concentration of tail-
ings solids, and therefore could be reasonably approximated
a) b) c)
d) e) f)
h) i)
g)
Aznalcóllar, 1998
Stava, 1985 Feijão, 2019
Fig. 8 Variation of sensitivity with distance from the breach for modelled maximum flow velocity (a, b, c), maximum flow depth (d, e, f), and
frontal arrival time (g, h, i) in three selected case studies

Mine Water and the Environment
as Newtonian rather than non-Newtonian. The effect of low
solids concentrations is implicitly included in HEC-RAS
by using low values for the yield stress and viscosity. As
a result, one could expect the inundation area to have low
sensitivity to the low calibrated yield stress and viscosity
values for Mt. Polley, which in turn increases the relative
sensitivity of the other inputs. This rationale may apply to
the Aznalcóllar and Ajka cases as well, which also had rela-
tively low concentrations of tailings solids.
In the case of Cadia, the modelled maximum flow veloc-
ity was most sensitive to total released volume, while the
modelled maximum flow depth and flow front arrival time
were most sensitive to breach width. In general, surface
roughness acts as an external resisting force along the flow
runout path, and typically, changing the surface roughness
affects the modelled flow velocity and arrival time the most.
However, this was not the case for Cadia. However, the
breach width was one of the primary contributors to the sen-
sitivity of the modelled flow velocity, depth, and flow front
arrival time. One possible reason might be the proximity of
our measurement to the breach. The Cadia runout distance
was ≈ 480m, which is relatively short, and the sensitivity
analysis was done at 50% of the runout distance. Tonglvs-
han is the only other case that has a similar runout distance
to Cadia (≈ 500m), and breach width was also one of the
main contributors to the sensitivity of modelled maximum
flow velocity and flow front arrival time in that case. Also,
the Manning’s n value used for Cadia was relatively low,
as expected for barren land (Janssen 2016). The released
tailings had a solid concentration of ≈ 63% (Jefferies etal.
Inundation Area (m2)
Max. Velocity at 50% Observed Runout Distance (m/s)
Probability Density
Probability Density
0 2 4 6 8 10 12 0 2 4 6 8 10 12
0e+00 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 0e+00 2e+06 4e+06 6e+06 8e+06 1e+07
0e+00 1e-06 2e-06 3e-06 4e-06 5e-06
0e+00 2e-07 4e-07 6e-07
0.0 0.5 1.0 1.5
0.0 0.1 0.2 0.3
Probability Density
Probability Density
Breach Width
Surface Roughness
Total Released Volume
Yield Stress
Breach Width
Surface Roughness
Total Released Volume
Yield Stress
-1 SD
+1 SD
All Inputs Together
Mean
Besst-fit ModelledValue
-1 SD
+1 SD
All Inputs Together
Mean
Observed InundationArea
Fig. 9 The probability density of the modelled inundation area and maximum flow velocity at 50% of the Zone 1 runout distance are plotted
with respect to each input variable (a and c), as well as considering the total uncertainty of all input variables together (b and d)

Mine Water and the Environment
2019), and the calibrated yield stress and viscosity for the
tailings were among the highest used in Adria (2022), as
expected for a material that predominantly consisted of sol-
ids. The influence of surface roughness (external flow resist-
ance) should therefore be expected to be less consequential
than rheology (internal flow resistance), which is observed
for Cadia in Fig.5. Another possible reason might be related
to the selected rheological model. The Quadratic rheology
was used for the back-analysis of the Cadia case but con-
sidering the high solid concentration of the Cadia tailings,
rheological models developed for solid-dominated materials
(e.g. Voellmy rheology) might have been more appropriate.
The FOSM results presented in this study pertain spe-
cifically to the HEC-RAS numerical model. In Ghahramani
etal. (2022), the FOSM analysis revealed that each of the
four models (DAN3D, MADflow, FLO-2D, and FLOW-3D)
investigated in their study was sensitive to different input
parameters. However, the total released volume was identi-
fied as one of the top three contributors to the sensitivity of
modelled maximum flow velocity and depth at a specific
location for all four models. The results of the FOSM analy-
sis conducted in this study are consistent with those findings.
Sensitivity Variation
The results in the “Sensitivity Variation Along Runout
Path" section indicate that the sensitivity of model outputs
to model inputs varies at different locations along the runout
path. This is consistent with a parallel complementary study
on the analogous problem of landslide runout that recog-
nized sensitivity variation over the extent of a landslide
runout model (Mitchell etal. 2022). Overall, the sensitivity
variations of the Stava, Aznalcóllar, and Feijão cases fol-
low a similar trend, except for the yield stress and surface
roughness curves, despite the different characteristics of
these three failures (Back-analyzed Case Studies section).
The sensitivity to breach width displays decreasing trends
in all of the plots. The breach width has a large influence on
the model outputs near the breach, but the influence gradu-
ally decreases with increasing runout distance (Fig.8).
The total released volume has a major influence on all
the model outputs at different locations along the runout
path (> 10% sensitivity values), with an increasing trend that
tends to plateau for some of the scenarios (e.g. Fig.8c, f,
g). In the case of Aznolcollar, there is a fluctuation in the
sensitivity of modelled maximum flow velocity to the total
released volume (Fig.8b). This may be attributed to the local
constriction of the runout path near a highway bridge that
crossed the inundation area near the 50% runout location,
where the physical constriction controls the velocity more
than any other model outputs.
For the Feijão case, the sensitivity to the yield stress dis-
plays an increasing trend, with the highest value at 90% of
the runout distance (the last measurement location). Model
outputs were not sensitive to yield stress for the Stava and
Aznalcóllar cases, which may be due to the steep travel path
slope along the Stava creek and the low solid concentration
of the Aznalcóllar tailings flow, respectively, nor to viscosity
for all three cases.
Comparing the three cases, the variation in sensitivity
to surface roughness for each model output have different
trends. For the Stava case, the sensitivity of the modelled
maximum flow depth to surface roughness displays an
increasing trend in the first 25% and a decreasing trend for
the rest. In contrast, there was a decreasing trend for the
Aznalcóllar case and an almost flat trend for the Feijão case
(Fig.8d–f). These differences might be due to distinct topo-
graphic conditions, such as steep terrain, sudden elevation
changes, or degree of confinement along the path.
In the case of Stava, the results indicate that the modelled
maximum flow depth and front flow arrival time were most
sensitive to breach width, while exhibiting very low to zero
sensitivity to other inputs at the 10% of runout distance.
This sensitivity pattern is similar to what was explained in
Sensitivity Analysis Section for the Cadia case, suggesting
that it may be due to the proximity of the measurement (10%
of runout) to the breach location where the dynamic effects
such as rapid changes in material behaviour can be signifi-
cant. When comparing the three cases, the first 10% of the
runout distance for the Stava is less than 500m from the
breach while this distance is about 3km for Aznalcóllar and
1km for Feijão. This sensitivity pattern changes at further
locations along the runout path (Fig.8d–g).
In this study, the 50% runout distance of Zone 1 was
selected as a reference point to compare general model sen-
sitivity estimates. However, analysis of sensitivity variation
along the path suggests that the 50% runout distance may
not necessarily be a key location of interest in every case.
Instead, the location for sensitivity analysis should be chosen
based on the specific purpose of the project, particularly
considering the locations of elements at risk.
Demonstration ofProbabilistic Prediction Approach
The Merriespruit demonstration case involved modelling
the probability distributions of two key parameters: the
inundation area and maximum flow velocity at 50% runout
distance. These parameters were modelled with respect to
each input variable individually and collectively (Fig.9).
In order to roughly compare the predicted results with the
observed ones, one of the main sources of uncertainty, the
total released volume, was kept as the reported value, as
mentioned in the Methodology section. Figure9b, d show
that the output results were over-predicted. The sensitivity
analysis results in Fig.2 showed that yield stress was the
top contributor to the sensitivity of inundation area for more

Mine Water and the Environment
than half of the cases. The over-prediction of the results
could be due to the lower yield stress selected (63Pa) com-
pared with the calibrated value (200Pa) provided in the
“Back-analyzed Case Studies” section. Another reason could
be the over-estimation of the average breach width value for
the simulation, compared to the reported value.
Selection of input parameters, such as total released vol-
ume, rheological parameters, and breach geometry, has been
a challenge for tailings dam breach-runout forward analysis.
Probability density curves can be used by practitioners and
modellers to constrain the ranges of estimated model out-
puts. For example, modellers may use the curves to identify
a range of values that are consistent with a certain level of
confidence, or to identify the most likely range of values for
the output. By doing so, the uncertainties associated with
each input variable can be accounted for, and more accurate
model predictions can be made.
However, the approximation methodology used to gener-
ate the probability density curves has some limitations and
assumptions that need to be considered when interpreting the
results. One limitation is the use of statistical distributions to
model the uncertainty of the input variables. While this can
be a useful approximation, it is important to recognize that
the choice of distribution may not always accurately reflect
the true uncertainty of the input variable. For instance, the
assumption of log-normal distribution may not always hold,
particularly for extreme events or rare occurrences, which
can lead to underestimation or overestimation of the prob-
ability of such events. Thus, the probability density curves
should be used with caution and in conjunction with other
information and expert judgement. Modellers should also be
aware of the limitations and assumptions of the approxima-
tion methodology, and carefully consider the potential effect
of correlated or extreme events that may not be accurately
captured by the probability density curves.
Limitations oftheFOSM Method
Although FOSM is a linearization technique, it can be
applied to models with non-linear output functions. The
FOSM method linearizes the non-linear output function by
approximating it as a Taylor series expansion around the
mean values of the input variables. Therefore, it assumes
that the output can be locally approximated as a linear func-
tion near the mean values of the input variables (Lee and
Mosalam 2005). However, the FOSM method comes with
limitations that should be considered when interpreting the
results. It is an approximate method that only considers the
first-order and second-moment (i.e. mean and variance),
rather than the distribution function, of the input variables.
Therefore, it may not work well for highly non-linear sys-
tems (Kunstmann etal. 2002). Better precision could be
achieved by using higher-order terms from the Taylor series
expansion. However, higher orders involve complex math-
ematics and require additional statistical information, such
as skewness and kurtosis, which are not easy to estimate due
to insufficient data. Another limitation is that the interac-
tion between input variables is not considered in the FOSM
method (Baecher and Christian 2005; Nadim 2007).
In this study, the FOSM approach was applied to three
erosional and eight non-erosional case studies. Although the
FOSM method is versatile and can be applied to other mod-
els, the FOSM results presented in this study are specific to
the HEC-RAS numerical model. Our interpretations provide
valuable information about HEC-RAS performance for each
case study. Although some similar trends were observed,
a larger sample size would be needed to draw broader and
more robust conclusions, particularly for the erosional
breach cases.
Conclusions
Our study highlights the importance of understanding the
uncertainty and sensitivity of model outputs to different
input variables for TDB runout modelling, which can help
improve the accuracy of risk assessments and mitigation
strategies in industry practice. In this study, the FOSM
methodology was applied to a database of 11 back-ana-
lyzed historical tailings flows to evaluate the uncertainties
in TDB runout modelling. Moreover, a sensitivity analysis
was conducted to determine the key factors contributing to
the sensitivity of the HEC-RAS model outputs, and sensitiv-
ity variations were analyzed at different locations along the
runout path. We also investigated the potential application
of the FOSM method to probabilistic runout modelling in
prediction scenarios.
Overall, the uncertainty results and sensitivity estimates
showed similar trends in most of the cases. To be able to
generate more reliable model results using HEC-RAS: (1)
researchers should develop better methods to predict poten-
tial release volumes; and (2) practitioners should use expert
judgment when estimating potential release volumes and
surface roughness values. However, there were some excep-
tions for each model output and the primary contributors to
the sensitivity of the model outputs varied depending on
the case study. The Mt. Polley case, for instance, was highly
sensitive to breach width for modelled inundation area,
maximum flow velocity, and maximum flow depth, poten-
tially due to the site conditions and the use of the Quadratic
rheology model, due to the relatively low solid concentration
of the Mt. Polley tailings flow. The Cadia Event 2 was also
sensitive to breach width for modelled flow velocity, depth,
and flow front arrival time. The influence of surface rough-
ness was observed to be less consequential than rheology,
potentially due to the high solid concentration of the Cadia

Mine Water and the Environment
tailings and the use of the Quadratic rheology model, instead
of other rheological models that were mainly developed for
solid materials. These results reinforce that considering site-
specific conditions and the selection of appropriate rheo-
logical models are crucial for accurate predictions in TDB
runout modelling.
We also found that the sensitivity variations along the
path for the Stava, Aznalcóllar, and Feijão cases followed
similar trends, with decreasing sensitivity to breach width
and increasing sensitivity to total released volume for all
three cases and increasing sensitivity to yield stress for Fei-
jão. The sensitivity of the model outputs to surface rough-
ness displayed a different trend for each case, which may
be due to different topographic conditions along the runout
path.
Lastly, the FOSM methodology was proposed as a proba-
bilistic approach to model-based tailings flow runout predic-
tion. A demonstration of the approach was presented to illus-
trate the potential usefulness of probability density curves in
constraining ranges of estimated model outputs in TDBAs.
Supplementary Information The online version contains supplemen-
tary material available at https:// doi. org/ 10. 1007/ s10230- 024- 00970-w .
Acknowledgements This work was funded by a fellowship (NG) from
the University of British Columbia Department of Earth, Ocean and
Atmospheric Sciences, as well as scholarships and grants from the Nat-
ural Sciences and Engineering Research Council of Canada (NSERC).
This work was part of the CanBreach Project, which is supported by
funding through an NSERC Collaborative Research Development
Grant and funding from the following industrial partners: Imperial Oil
Resources Inc., Suncor Energy Inc., BGC Engineering Inc., Golder
Associates Ltd., and Klohn Crippen Berger. The authors acknowledge
the constructive comments provided by Dr. Violeta Martin and Dr. Dirk
Van Zyl during the preparation of this study.
Data Availability All the model outputs, sensitivity analyses, and
uncertainty estimates have been included as supplementary material.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
References
Aaron J, McDougall S, Kowalski J, Mitchel A, Nolde N (2022) Prob-
abilistic prediction of rock avalanche runout using a numeri-
cal model. Landslides 19:2853–2869. https:// doi. org/ 10. 1007/
s10346- 022- 01939-y
Adria DAM (2022) Compilation and critical assessment of observa-
tions from a selection of historical tailings dam breach events for
numerical breach and runout modelling. MASc thesis, Univ Brit-
ish Columbia. https:// doi. org/ 10. 14288/1. 04217 82
Adria DAM, Ghahramani N, Rana NM, Martin V, McDougall S,
Evans SG, Take WA (2023a) Insights from the compilation and
critical assessment of breach and runout characteristics from
historical tailings dam failures: implications for numerical mod-
elling. Mine Water Environ 24:1–20. https:// doi. org/ 10. 1007/
s10230- 023- 00964-0
Adria DAM, Ghahramani N, Rana NM, Martin V, McDougall S, Evans
SG, Take WA (2023b) A database of tailings dam breach and
runout observations. Borealis Can Dataverse Reposit. https:// doi.
org/ 10. 5683/ SP2/ NXMXTI
Arcement GJ, Schneider VR (1989) Guide for selecting Manning’s
roughness coefficients for natural channels and flood plains.
U.S. Geological Survey Water Supply Paper 2339. https:// doi.
org/ 10. 3133/ wsp23 39
Baecher G, Christian J (2005) Reliability and statistics in geotechni-
cal engineering. Wiley, West Sussex
Blight GE (2009) Geotechnical engineering for mine waste storage
facilities. CRC Press/Balkema, Leiden
Borgonovo E, Plischke E (2016) Sensitivity analysis: a review of
recent advances. Eur J Oper Res 248:869–887. https:// doi. org/
10. 1016/j. ejor. 2015. 06. 032
Brunner G (2020) HEC-RAS User’s Manual, version 6.3. Hydrologic
Engineering Center, Davis
Burges SJ, Lettenmaier DP (1975) Probabilistic methods in stream
quality management. J Am Water Resour Assoc 11:115–130.
https:// doi. org/ 10. 1111/j. 1752- 1688. 1975. tb006 64.x
Canadian Dam Association (CDA) (2021) Technical bulletin: tailings
dam breach analysis
Chow VT (1959) Open channel hydraulics. McGraw-Hill Book Co.,
New York City
Concha Larrauri P, Lall U (2018) Tailings dams failures: updated sta-
tistical model for discharge volume and runout. Environments
5:1–10. https:// doi. org/ 10. 3390/ envir onmen ts502 0028
Dettinger MD, Wilson JL (1981) First order analysis of uncertainty
in numerical models of groundwater flow part: 1. Mathematical
development. Water Resour Res 17:149–161. https:// doi. org/ 10.
1029/ WR017 i001p 00149
Fourie AB, Papageorgiou G (2001) Defining an appropriate steady
state line for Merriespruit gold tailings. Can Geotech J 38:695–
706. https:// doi. org/ 10. 1139/ T00- 111
Froehlich DC (2008) Embankment dam breach parameters and their
uncertainties. J Hydraul Eng 134:1708–1721. https:// doi. org/
10. 1061/ (ASCE) 0733- 9429(2008) 134: 12(1708)
Ghahramani N, Mitchell A, Rana NM, McDougall S, Evans SG,
Take WA (2020) Tailings-flow runout analysis: examining the
applicability of a semi-physical area–volume relationship using
a novel database. Nat Hazards Earth Syst Sci 20:3425–3438.
https:// doi. org/ 10. 5194/ nhess- 20- 3425- 2020
Ghahramani N, Chen HJ, Clohan D, Liu S, Llano-Serna M, Rana
NM, McDougall S, Evans SG, Take WA (2022) A benchmark-
ing study of four numerical runout models for the simulation of
tailings flows. Sci Total Environ 827:154245. https:// doi. org/ 10.
1016/j. scito tenv. 2022. 154245
Gibson S, Floyd I, Sánchez A, Heath R (2021) Comparing single-
phase, non-Newtonian approaches with experimental results:
validating flume-scale mud and debris flow in HEC-RAS. Earth
Surf Process Landf 46:540–553. https:// doi. org/ 10. 1002/ esp.
5044
Gibson S, Moura LZ, Ackerman C, Ortman N, Amorim R, Floyd I,
Eom M, Creech C, Sánchez A (2022) Prototype scale evalua-
tion of non-newtonian algorithms in HEC-RAS: mud and debris

Mine Water and the Environment
flow case studies of Santa Barbara and Brumadinho. Geosciences
12:134. https:// doi. org/ 10. 3390/ geosc ience s1203 0134
Heiser M, Scheidl C, Kaitna R (2017) Evaluation concepts to com-
pare observed and simulated deposition areas of mass move-
ments. Comput Geosci 21:335–343. https:// doi. org/ 10. 1007/
s10596- 016- 9609-9
Janssen C (2016) Manning’s n values for various land covers to use
for dam breach analyses by NRCS in Kansas. https:// rashms. com/
wp- conte nt/ uploa ds/ 2021/ 01/ Manni ngs-n- values- NLCD- NRCS.
pdf. Accessed 26 June 2023
Jefferies M, Morgenstern NR, Van Zyl DV, Wates J (2019) Report on
NTSF embankment failure, Cadia Valley operations, for Ashurst
Australia
Julien PY, Lan Y (1991) Rheology of hyperconcentrations. J Hydraul
Eng 117:346–353
Kim JS, Kim SY, Han TS (2020) Sensitivity and uncertainty estimation
of cement paste properties to microstructural characteristics using
FOSM method. Constr Build Mater 242:118159. https:// doi. org/
10. 1016/J. CONBU ILDMAT. 2020. 118159
Kleijnen JPC (1995) Sensitivity analysis and optimization of system
dynamics models: regression analysis and statistical design of
experiments. Syst Dyn Rev 11:275–288. https:// doi. org/ 10. 1002/
SDR. 42601 10403
Kunstmann H, Kinzelbach W (2000) Computation of stochastic
wellhead protection zones by combining the first-order second-
moment method and Kolmogorov backward equation analysis.
J Hydrol 237:127–146. https:// doi. org/ 10. 1016/ S0022- 1694(00)
00281-X
Kunstmann H, Kinzelbach W, Siegfried T (2002) Conditional first-
order second-moment method and its application to the quantifica-
tion of uncertainty in groundwater modeling. Water Resour Res
38:6-1–6-14. https:// doi. org/ 10. 1029/ 2000W R0000 22
Lee T-H, Mosalam KM (2005) Seismic demand sensitivity of rein-
forced concrete shear-wall building using FOSM method. Earthq
Eng Struct Dyn 34:1719–1736. https:// doi. org/ 10. 1002/ EQE. 506
Llano-Serna MA, Farias MM, Pedroso DM, Williams DJ, Sheng D
(2018) An assessment of statistically based relationships between
critical state parameters. Géotechnique 68:556–560. https:// doi.
org/ 10. 1680/ jgeot. 16.T. 012
Martin V, Al-Mamun M, Small A (2019) CDA technical bulletin on
tailings dam breach analyses. Sustainable and safe dams around
the world. CRC Press, Ottawa, pp 3484–3498
Martin V, Adria D, Wong H (2022) Inundation modelling of non-new-
tonian tailings dam breach outflows. In: Proc. 27th world con-
gress of the international commission on large dams, question
105, response 28
McDougall S (2017) 2014 Canadian geotechnical colloquium: land-
slide runout analysis—current practice and challenges. Can Geo-
tech J 54:605–620. https:// doi. org/ 10. 1139/ cgj- 2016- 0104
Melo M, Eleutério J (2023) Probabilistic analysis of floods from tail-
ings dam failures: a method to analyze the impact of rheological
parameters on the HEC-RAS Bingham and Herschel–Bulkley
models. Water 15:2866. https:// doi. org/ 10. 3390/ w1516 2866
Mitchell A, Zubrycky S, McDougall S, Aaron J, Jacquemart M, Hübl
J, Kaitna R, Graf C (2022) Variable hydrograph inputs for a
numerical debris-flow runout model. Nat Hazards Earth Syst Sci
22:1627–1654. https:// doi. org/ 10. 5194/ nhess- 22- 1627- 2022
Nadim F (2007) Tools and strategies for dealing with uncertainty in
geotechnics. In: Griffiths DV, Fenton GA (eds) Probabilistic meth-
ods in geotechnical engineering, CISM Courses and Lectures.
Springer, Vienna, pp 71–95
Piciullo L, Storrøsten EB, Liu Z, Nadim F, Lacasse S (2022) A
new look at the statistics of tailings dam failures. J Eng Geol
303:106657. https:// doi. org/ 10. 1016/j. enggeo. 2022. 106657
Pirulli M, Barbero M, Marchelli M, Scavia C (2017) The failure of the
Stava Valley tailings dams (northern Italy): numerical analysis of
the flow dynamics and rheological properties. Geoenviron Disas-
ters 4:3. https:// doi. org/ 10. 1186/ s40677- 016- 0066-5
Rana NM, Ghahramani N, Evans SG, McDougall S, Small A, Take WA
(2021a) Catastrophic mass flows resulting from tailings impound-
mentfailures. J Eng Geol 292:106262. https:// doi. org/ 10. 1016/j.
enggeo. 2021. 106262
Rana NM, Ghahramani N, Evans SG, McDougall S, Small A, Take WA
(2021b) A comprehensive global database of tailings flows. Bore-
alis Can Dataverse Reposit. https:// doi. org/ 10. 5683/ SP2/ NXMXTI
Razavi S, Jakeman A, Saltelli A, Prier C, Iooss B, Borgonovo E, Plis-
chke E, Piano SL etal (2021) The future of sensitivity analysis:
an essential discipline for systems modeling and policy support.
Environ Model Softw 137:104954. https:// doi. org/ 10. 1016/J.
ENVSO FT. 2020. 104954
Rico M, Benito G, Díez-Herrero A (2008) Floods from tailings dam
failures. J Hazard Mater 154:79–87. https:// doi. org/ 10. 1016/j.
jhazm at. 2007. 09. 110
Santamarina JC, Torres-Cruz LA, Bachus RC (2019) Why coal ash and
tailings dam disasters occur. Science 364:526–528
Tonkin M, Doherty J (2009) Calibration-constrained Monte Carlo anal-
ysis of highly parameterized models using subspace techniques.
Water Resour Res. https:// doi. org/ 10. 1029/ 2007W R0066 78
Wagener F (1997) The Merriespruit slimes dam failure: overview and
lessons learnt. J S Afr Inst Civ Eng 39:11–15
Wahl TL (1998) Prediction of embankment dam breach parameters:
literature review and needs assessment. U.S. Bureau of Recla-
mation. https:// www. usbr. gov/ ssle/ damsa fety/ TechD ev/ DSOTe
chDev/ DSO- 98- 04. pdf. Accessed 14 June 2023
Wahl TL (2014) Evaluation of erodibility-based embankment dam
breach equations. Hydraulic Laboratory Report. https:// www. usbr.
gov/ tsc/ techr efere nces/ hydra ulics_ lab/ pubs/ HL/ HL- 2014- 02. pdf.
Accessed 14 June 2023
Wang S-J, Hsu K-C (2009) The application of the first-order second-
moment method to analyze poroelastic problems in heterogeneous
porous media. J Hydrol 369:209–221. https:// doi. org/ 10. 1016/j.
jhydr ol. 2009. 02. 049