incorporating uncertainty into backward erosion piping risk assessments

E3S Web of Conferences 7, 03007 (2016) FLOODrisk 2016 - 3rd European Conference on Flood Risk Management DOI: 10.1051/ e3sconf/2016 0703007 Incorporating Uncertainty into Backward Erosion Piping Risk Assessments 1,a Bryant A Robbins , Michael K Sharp 1 U.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 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 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 a shown in Figure 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 Illustration of BEP progressing beneath a levee Corresponding author: Bryant.A.Robbins@usace.army.mil © 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/) E3S Web of Conferences 7, 03007 (2016) FLOODrisk 2016 - 3rd European Conference on Flood Risk Management DOI: 10.1051/ e3sconf/2016 0703007 Literature Review 2.2 Critical Gradients and Uncertainty ,Q RUGHU WR GLVFXVV XQFHUWDLQW\ LQ %(3 ULVN DVVHVVPHQWV LW LV QHFHVVDU\ WR XQGHUVWDQG ERWK KRZ %(3 LV FXUUHQWO\ DVVHVVHG DQG KRZ XQFHUWDLQW\ LV FXUUHQWO\ FRQVLGHUHG 7KH IROORZLQJ VHFWLRQV SURYLGH D EULHI UHYLHZ RI FXUUHQW DVVHVVPHQW PHWKRGV DQG SUHYLRXV ZRUN WKDW KDV TXDQWLILHG XQFHUWDLQW\ UHJDUGLQJ FULWLFDO JUDGLHQWV 1XPHURXV DXWKRUV KDYH TXDQWLILHG WKH XQFHUWDLQW\ VXUURXQGLQJ WKH FULWLFDO FRQGLWLRQV IRU %(3 6FKPHUWPDQQ > @ PDGH DQ HVWLPDWH RI WKH YDULDQFH DVVRFLDWHG ZLWK WKH FRQGLWLRQDO GLVWULEXWLRQ RI FULWLFDO SRLQW JUDGLHQWV LSPW DV ZHOO DV DQ HVWLPDWH RI WKH 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WHVWV ZHUH FRUUHFWHG Figure Simplified event tree for BEP evaluation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eb of Conferences 7, 03007 (2016) FLOODrisk 2016 - 3rd European Conference on Flood Risk Management DOI: 10.1051/ e3sconf/2016 0703007 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, 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 The corrected, individual laboratory critical point gradients obtained from the 110 tests found in the literature are plotted in Figure 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 Suggested relationship for determining the critical point gradient as a function of Cu (from [6]) Points represent study averages of critical point gradient 9DULDEOH 5HIHUHQFH 9DOXH 6HHSDJH /HQJWK P /D\HU 'HSWK P /D\HU :LGWK P *UDLQ 6L]H G PP K Y 5HODWLYH 'HQVLW\ 6HHSDJH 3DWK ,QFOLQDWLRQ GHJUHHV 7DEOH 5HIHUHQFH YDOXHV XVHG IRU DGMXVWLQJ WKH ODERUDWRU\ IOXPH WHVW UHVXOWV WR FULWLFDO SRLQW JUDGLHQW YDOXHV VHH > @ IRU IXUWKHU GHWDLOV When using Figure (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 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] Figure Suggested relationship for determining the critical point gradient compared to the best fit, median line of all test results Points represent individual laboratory tests 3.2 Quantifying Uncertainty Visual observation of Figure also indicates that the individual laboratory test points exhibit increasing variance as the uniformity coefficient increases This While Figure is quite useful, it is difficult to estimate the uncertainty in the critical point gradient E3S Web of Conferences 7, 03007 (2016) FLOODrisk 2016 - 3rd European Conference on Flood Risk Management DOI: 10.1051/ e3sconf/2016 0703007 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 LSPW 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 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 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 The 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 Figure Critical point gradients from individual laboratory tests and best fit quantile regression lines 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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 > @ Acknowledgements 7KH DXWKRUV DUH YHU\ WKDQNIXO IRU WKH QXPHURXV UHYLHZV RI WKLV SDSHU ,Q SDUWLFXODU UHYLHZ FRPPHQWV DQG GLVFXVVLRQ SURYLGHG E\ -RKQ 6FKPHUWPDQQ ZHUH LQFUHGLEO\ KHOSIXO IRU XQGHUVWDQGLQJ SUHYLRXV ZRUN DQG GHFLGLQJ KRZ WR SRUWUD\ WKH LQIRUPDWLRQ LQ WKLV SDSHU 7KH DXWKRUV ZRXOG DOVR OLNH WR DFNQRZOHGJH WKH FRPPHQWV DQG VXJJHVWLRQV SURYLGHG E\ *UHJ DUORYLW] 0LQDO 3DUHNK 6FRWW 6KHZEULGJH 7LP 2¶/HDU\ DQG 7RP 7HUU\ 3HUPLVVLRQ WR SXEOLVK ZDV JUDQWHG E\ 'LUHFWRU *HRWHFKQLFDO DQG 6WUXFWXUHV /DERUDWRU\ > @ > @ > @ > @ > @ > 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