Incorporating Uncertainty into Backward Erosion Piping Risk Assessments

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


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.

Quantifying Uncertainty
While Figure 3  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.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 n th 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 n th quantile approaches the n th 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 n th quantile, an n th 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 10 th and the 90 th 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 L SPW 90 th and 10 th 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 1 .
Figure 1.Illustration of BEP progressing beneath a levee.

Figure 3 .
Figure 3. Suggested relationship for determining the critical point gradient as a function of C u (from [6]).Points represent study averages of critical point gradient.
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].
The corrected, individual laboratory critical point gradients obtained from the 110 tests found in the literature are plotted in Figure4.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 .
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

Figure 5 .
Figure 5. 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.