![Critical point gradients from individual laboratory tests and best fit quantile regression lines for the 10 th to 90 th quantiles. Open points are from a different testing configuration [24] than other points.](https://www.researchgate.net/profile/Bryant-Robbins/publication/309342658/figure/fig2/AS:419658298609667@1477065743949/Critical-point-gradients-from-individual-laboratory-tests-and-best-fit-quantile_Q320.jpg)
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a Corresponding author: Bryant.A.Robbins@usace.army.mil
Incorporating Uncertainty into Backward Erosion Piping Risk
Assessments
Bryant A. Robbins1,a, Michael K. Sharp1
1U.S. Army Engineer Research and Development Center, Vicksburg, MS, USA
Abstract. Backward erosion piping (BEP) is a type of internal erosion that typically involves the erosion of
foundation materials beneath an embankment. BEP has been shown, historically, to be the cause of approximately
one third of all internal erosion related failures. As such, the probability of BEP is commonly evaluated as part of
routine risk assessments for dams and levees in the United States. Currently, average gradient methods are
predominantly used to perform these assessments, supported by mean trends of critical gradient observed in
laboratory flume tests. Significant uncertainty exists surrounding the mean trends of critical gradient used in practice.
To quantify this uncertainty, over 100 laboratory-piping tests were compiled and analysed to assess the variability of
laboratory measurements of horizontal critical gradient. Results of these analyses indicate a large amount of
uncertainty surrounding critical gradient measurements for all soils, with increasing uncertainty as soils become less
uniform.
1 Introduction
Internal erosion refers to various processes that cause
erosion of soil material from within or beneath a water
retention structure such as a dam or levee. With regard to
flood risk, internal erosion is an issue of significant
concern. Approximately half of all historical dam failures
have been attributed to internal erosion [1]. While
internal erosion risk can be reduced through the use of
well-designed filters and drains, modifying the entirety of
existing infrastructure to meet modern filter standards is
not economically feasible. Therefore, it is of utmost
importance to be able to assess the likelihood of internal
erosion occurring on existing infrastructure such as dams,
levees, and canals.
Internal erosion processes can be subdivided into four
broad categories: concentrated leak erosion, backward
erosion piping (BEP), internal instability, and contact
erosion [2]. This paper will discuss solely BEP, which
accounts for approximately one third of all internal
erosion related dam failures [1], [3]. For discussion on
the other types of internal erosion, the authors suggest
reviewing Bonelli et al. [4].
The process of BEP is illustrated in Figure 1. For
BEP to occur, it is necessary to have an unfiltered
seepage exit through which soil can begin eroding. As
filters and drains are rare along levee systems in the
United States, the seepage exit condition is usually
unfiltered. This is evident by the numerous sand boils that
occur along the U.S. levee systems during each flood [5].
A sand boil is a small cone of deposited soil that occurs
concentrically around a concentrated seepage exit, as
shown in Figure 1. The presence of sand boils indicates
that the process of BEP has initiated at a particular site.
Whether the BEP process continues, ultimately leading to
structural failure, depends upon numerous conditions
being met (roof support, sufficient hydraulic gradients for
erosion propagation, and unsuccessful human
intervention). Estimation of the probability of failure due
to BEP should consider all of these factors, as well as the
uncertainty surrounding them. The focus of this paper is
on improving how uncertainty regarding critical gradients
for BEP is incorporated into risk assessments, with
particular emphasis on methods used in the U.S. While
the work reported is a simple extension of the
groundbreaking work of Schmertmann and Sellmeijer
([6] & [7]), the authors hope that the simple portrayal of
uncertainty presented leads to the objective quantification
of uncertainty in BEP risk assessments.
Figure 1. Illustration of BEP progressing beneath a levee.
DOI: 10.1051/
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© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative
Commons Attribution
License 4.0 (http://creativecommons.org/licenses/by/4.0/).
2 Literature Review
2.1 BEP Assessment Methods
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Figure 2. Simplified event tree for BEP evaluation.
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2.2 Critical Gradients and Uncertainty
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3 Estimating Critical Point Gradients
3.1 Current Practice
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Figure 3. Suggested relationship for determining the critical
point gradient as a function of Cu (from [6]). Points represent
study averages of critical point gradient.
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When using Figure 3 (or the equivalent figure from
[6]), to estimate critical point gradients, it is necessary to
correct for all factors described by Schmertmann using
the reference values in Table 1 to arrive at comparable
point gradient values. Neglecting to adjust the values to
equivalent point gradients will yield erroneous results;
the magnitude of these errors has been shown to be as
large as 100 percent for factors related to soil density and
problem geometry [15].
3.2 Quantifying Uncertainty
While Figure 3 is quite useful, it is difficult to
estimate the uncertainty in the critical point gradient
because study averages are presented. In order to
examine the uncertainty in greater detail, the individual
laboratory test results from each experimental series must
be examined. The authors have compiled all of the
laboratory test results from [19][25]. It was difficult to
establish the exact number of tests in each experimental
series from references [22] and [23]. However, [26] has
provided a very thorough overview of the experiments
conducted by de Wit and Silvis. This overview was used
to determine the individual tests conducted for each
experimental series conducted by de Wit and Silvis.
Mueller-Kirchenbauer conducted more tests than
documented in [25]. However, for the sake of
consistency with [6], only the single test reported in [25]
was considered. In total, 110 laboratory piping tests were
found in the references previously mentioned. Of these,
9 of the piping tests were right censored (did not fail) [19,
21]. For all of the tests, the test results were corrected
using the correction factors provided in [6] to correct the
individual test results to the common reference values in
Table 1.
The corrected, individual laboratory critical point
gradients obtained from the 110 tests found in the
literature are plotted in Figure 4. For comparison
purposes, the no-test default line proposed by
Schmertmann and the best-fit median line are plotted as
well. From visual observation, it is seen that the no-test
default line (dashed) proposed by Schmertmann [6] more
closely approximates a lower bound than an average
trend for low values of uniformity coefficient.
Figure 4. Suggested relationship for determining the critical
point gradient compared to the best fit, median line of all test
results. Points represent individual laboratory tests.
Visual observation of Figure 4 also indicates that the
individual laboratory test points exhibit increasing
variance as the uniformity coefficient increases. This
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non-constant variance, called heteroscedasticity, indicates
that ordinary least squares (OLS) linear regression is not
a suitable means of estimating the uncertainty
surrounding the expected value (mean trend). OLS will
not capture the heteroscedasticity due to the constant
variance, Gaussian residual models typically associated
with linear regression. Transformations (e.g. log
transforms when dealing with exponential data) are
commonly used to transform the data to a distribution of
a particular form such that OLS regression techniques can
be used to estimate the conditional probability
distribution of the dependent variable. Another approach
to capturing the heteroscedasticity is the use of
generalized linear regression models, which attempt to
model the changing variance across the range of
covariates. In order to keep the results as simple and as
visual as possible, the data quantiles were used as
estimates of the conditional probability distribution.
The nth quantile of a sample of data is the value in the
data set for which the proportion n of the sample is lower,
and the proportion (1-n) is higher. As the size of the
sample increases, such that the empirical probability
density function (PDF) of the data more closely
approximates the underlying probability distribution, the
nth quantile approaches the nth percentile of the underlying
probability distribution. For small samples, the empirical
quantiles will exhibit less variance than the equivalent
percentiles due to the influence of the sample size. For
this particular application, the consequences of this are
minor compared to the influence of the many correction
factors used in arriving at the estimates of critical point
gradients for each test. For this reason, the authors
consider the quantiles to be an adequate estimate of the
conditional probability distribution of critical point
gradients.
To estimate the quantiles of the individual laboratory
test data, first order quantile regression was performed. In
other words, a linear equation was fit through the data
such that, for each nth quantile, an nth fraction of the data
lies below the line and a (1-n) fraction of the data lies
above the line. For an excellent (and quite humorous)
introduction to quantiles and quantile regression, the
authors recommend reviewing [27]. Quantile regression
was used to estimate each quantile between the 10th and
the 90th quantiles in increments of 10 percent. While
statistically incorrect to include the censored observations
in the regression, inclusion results in a conservative bias
and still provides useful information. For this reason, and
given the sparse data at high values of uniformity
coefficient, the censored data was included in the
regression analyses. The resulting quantiles are plotted in
comparison to the data in Figure 5. The 6 observations
obtained from [24] are distinguished from the rest of the
data as these observations were obtained from a different
testing configuration and exhibit more variability than the
other test series. While these observations were included
to provide a direct comparison to [6], they should be
carefully evaluated when using the results of this study.
From the linear quantiles plotted in Figure 5, it is
readily seen that the variance in the conditional
distribution of the critical point gradient increases with
increasing uniformity coefficient. It is also observed that
the spread in the data is quite large. At a uniformity
coefficient of 2, the difference between the 90th and
10th percentiles is 0.26. At a uniformity coefficient of 6,
the difference is 0.82. In both cases, the spread in the
distribution is large and should be considered in risk
assessments. Figure 5 can readily be used to inform
estimates of the conditional distribution of critical point
gradients for estimating the probability of BEP
progression in the appropriate node of an event tree
analysis.
Figure 5. Critical point gradients from individual laboratory
tests and best fit quantile regression lines for the 10th to 90th
quantiles. Open points are from a different testing configuration
[24] than other points.
4 Discussion
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5 Conclusions
A compilation of laboratory measurements of
critical point gradients for backward erosion piping is
presented. The probability distribution of critical point
gradients is characterized as a function of uniformity
coefficient through first order quantile regression on the
sample of 110 laboratory test results. Results of these
analyses indicate a large amount of uncertainty
surrounding critical gradient measurements for all soils,
with increasing uncertainty as soils become less uniform.
The results of this study can be used in risk assessments
to estimate the probability of progression for backward
erosion piping.
6 Acknowledgements
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