© 3rd IAHR Europe Congress, Book of Proceedings, 2014, Porto - Portugal. ISBN 978-989-96479-2-3

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FLOOD DEFENCE DESIGN PARAMETERS CORRELATION INFLUENCE ON FAILURE PROBABILITY

– CASE STUDY OF BACKWARD EROSION PIPING

J.P. AGUILAR LOPEZ(1), J.J. WARMINK(1), R.M.J SCHIELEN(1,2) & S.J.M.H HULSCHER(1)

(1) Twente University, Enschede, The Netherlands

(2) Rijkswaterstaat, Arnhem, The Netherlands

j.p.aguilarlopez@utwente.nl, J.J.Warmink@utwente.nl,

R.M.J.Schielen@utwente.nl, S.J.M.H.Hulscher@utwente.nl

Abstract

Flood defences of particularly riverine deltaic areas are mainly composed of soil embankments artificially made. As any kind of structure, the materials, geometry and possible loads, might experience different deterioration processes that will compromise the stability of the structure. In order to estimate the probability of failure of such structures due to these processes, reliability theory is commonly used. The models in this kind of practice are equations that describe the state of failure of the structure. In this study, the possible correlation of parameters used in the limit state equations is going to be studied. Particularly for the failure process also known as backward erosion piping. To do that, bivariate correlation models known as “Copulas” where used to understand the effect of correlation in the failure probability estimation of a flood defence. The results show that the actual methodology where parameter correlation inside one single cross section is neglected might result in conservative estimation of the actual failure probabilities.

Keywords: Flood defence, Failure, Piping, Correlation, Copula.

1. Introduction

Flood defences are one of the most important parts of a flood risk management system, as they ensure the protection of human lives and valuable assets during an oncoming flood event. Normally they are located along rivers, lakes, coasts or any other particular scenario where fluctuation of water levels can be present. Consequently these structures are extensively long which in most of the cases obliges to build them with materials that are more cost effective compared to concrete. In principle Flood defences of riverine deltaic areas are constituted of soil embankments protected by different types of revetments. As any kind of structure, the materials, geometry and possible loads, might experience different deterioration processes that will compromise the stability of the structure during a flood event.

The occurrence of these processes are defined as “Failure mechanisms” or “Failure modes”. They describe in which way the structure might stop fulfilling intended purpose. Note that, the occurrence of a failure mechanism will not necessarily be traduced in the total collapse of the structure but more of the loss of stability itself. For design and safety assessment, the performance of the structure can be represented by limit state equations (Morris, Allsop, Buijs, Kortenhaus, Doorn and Lesniewska, 2009).

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They describe the residual strength of the structure for a particular event. It is calculated as the difference between the “resistance” of the structure under known design conditions and the strength demand also known as “load”.

For flood defences, the water load is calculated as a function of the wind speeds and/or water levels and possible discharges flowing over the structure depending on the failure mechanism to study. In contrast, the resistance depends on more factors as for each different failure mechanism their values will depend on the construction materials and the geometry of the flood defence. In the year 2007, the project FLOODSITE (Allsop, Kortenhaus and Morris, 2007) compiled a large inventory which contains limit state descriptions for more than 70 case specific possible failure mechanisms and their limit state equations with the aim to improve the available tools for flood safety assessment. For the design and reliability estimation of flood defences in the Netherlands, each of the main failure mechanisms has a predefined limit state function that is accepted by the Dutch national authorities. For them, the accurate estimation of the failure probability of the flood defence system is a major issue as almost 21% of the total extension of the country lies beneath sea level. Since the catastrophic flood of 1953, the government has major interest in the assessment and improvement of the flood defence system.

One of the latest Dutch studies of flood safety is the VNK2 (Rijkswaterstaat, 2012), which consists on a detailed probabilistic assessment study for the complete flood defence system that protects the country. One of the major conclusions of the project was that mainly 4 failure mechanisms (listed below) have the most impact in the total failure estimation for riverine flood defences.

• Overtopping • Backward erosion Piping (BEP) • Slope stability • Erosion

The omission of possible statistical dependence between different components is one major sources of error in the failure estimation of reliability of a system. Šimić , Vuković and Mikuličić (2003) showed that positive correlation of input variables have influence in the standard deviation of the failure probability density function. In their experiment, the mean limit state value remained almost constant while the increase in the standard deviation augmented proportionally to the spearman’s correlation coefficient. If so, the probability of occurrence of extreme values located in the tail might be higher than the ones estimated considering independence between parameters. The same effect but in the 95% quintiles of another limit state pdf for example can be observed in the results obtained for the reliability analysis of the power supply. The reliability study with correlated variables (Karanki, Jadhav, Chandrakar, Srividya and Verma, 2010), for the unavailability rates of the system might change in an order of magnitude of 10. In These kind of studies, the influence of correlation can be easily observed as they greatly affect events of low frequency.

The correlation degree can also be variable for different quintiles depending on the type of distribution and function present between the correlated variables. Correlation analysis is not only concerned about the degree of correlation but the time spatial distribution of the correlated random variables as well (Jongejan, Maaskant, Ter Horst, Havinga, Roode and Stefess, 2013). Extensive research has been done about the effect and assessment of spatial correlation of load and resistance estimation on flood defences in the Netherlands (Vrouwenvelder, 2006), (Kanning, 2012).

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However this correlations are analyzed considering how a variable depends on itself (Autocorrelation) along space and time. Flood defence systems are longitudinal structures that are analyzed as a system of parallel or series of elements from representative cross sections in the structure. The selection of the spacing of these elements highly depends on the spatial variance between parameters used in the calculations for the load and resistance estimation. Therefore the probability of failure of the whole system will increase proportionally to the separation between the representative cross sections. In reliability studies of flood defences is known as “The Length effect” (Vrijling, Schweckendiek and Kanning, 2011).

In this study it is intended to quantify the influence in failure probability by assuming correlated parameters when assessing of a “single” cross section in a riverine flood defence embankment. In particular for the BEP failure mechanism, as studies like VNK2 have concluded that this failure mode can be even more important than what it was expected by previous studies done in the Netherlands (Vrijling, 2010).

2. Backward erosion piping (BEP)

Structures such a flood defences are exposed to variable water loads that induce seepage flows through soil permeable layers. Levees which are built over sandy aquifers covered by semi-impervious clay layers are a good example. If sufficient pressure is built up inside the permeable layer, the structure will eventually be uplifted allowing water to move inside the aquifer. The occurrence of such process is considered as a failure mechanism itself. Afterwards the pressure is alleviated by a fracture in the semi-impervious clay layer which allows water to flow out to the hinter side of the structure. The water movement inside the porous media eventually will carry some of the finer particles creating voids in the granular soil structure. These particles will flow outside the aquifer creating sand boils around the exit point. As material is eroded, propagation of small channels towards the riverside of the structure will start occurring. If the water pressure does not come to an equilibrium state, the channels will be developed further in the direction of the point of entrance. If sufficient material is eroded from underneath of the aquifer, the flood defence structure will collapse originating a levee breach.

2.1 Sellmeijer limit state equation for BEP

Several empirical and numerical models for estimating the critical head of BEP have been developed since the early 19th century (Van Beek, Knoeff and Schweckendiek, 2011). The most common ones for flood defence assessment are Bligh and Lane. They are derived as empirical rules but the numerical model developed in the Netherlands by Sellmeijer (1991) is the most recent one. It combines theories of groundwater flow and water flow movement inside a pipe. Based on this model, a limit state equation was derived which includes the most important soil parameters correlated to the occurrence of the BEP.

The inclusion of local conditions inside this limit state equation such as particle representative diameter (d70) and hydraulic conductivity (K) makes it a more suitable approach for probabilistic flood risk assessment. Recently the model was re-calibrated with the obtained results of different experiments (Sellmeijer, de la Cruz, van Beek and Knoeff, 2011) in different scales.

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Zp � 56$��&$��&$�'&7 8 $h 8 :; 8 0.3d& [1]

Sellmeijer re-calibrated limit state equation (Zp) nomenclature:

η γ’sand γw θ d70 d70m ν K g D mp Hc FR FS FG L h hb d

[-] [N/m3] [N/m3] [deg.] [m.] [m.] [m2/s] [m/s] [m/s2] [m.] [-] [m.] [-] [-] [-] [m.] [m.] [m.] [m.]

: Sand drag force factor ( White’s coefficient) : Unitary weight of sand particles : Unitary weight of water : Bedding angle of sand grains : 70 percent quintile value grain size distribution of sand layer : Calibration reference value (2.08 x 10-4 m) : Kinematic viscosity of water at 20 °C : Hydraulic permeability of sand : Gravitational acceleration : Average thickness of sand layer : Modelling uncertainty factor : Critical hydraulic head difference : Resistance factor : Scale factor : Geometric factor : Seepage length from entrance point to sand boil water exit : Water level in the foreside of the flood defence : Water level at the hinter side of the flood defence sand boil location : Aquitard layer thickness at the sand boil exit point

Note: the product of the hydraulic permeability of soil and kinematic viscosity divided by the

gravitational acceleration is often also known as intrinsic permeability κ [m2].

3. Correlation in flood risk

Evidence in data can show that events might have a higher degree of dependence for extreme values. Such effects can be included in statistical joint probability distributions if the physics of the process to study gives sufficient support to imply a certain and degree of correlation (Repko, Van Gelder, Voortman and Vrijling, 2004). Therefore bivariate models are a recommended tool for the consideration of correlated variables.

Figure 1. Flood defence embankment cross section.

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The degree of correlation in dike reliability studies considered for the calculation of the load term, might prevent the underestimation of failure probabilities (Diermanse and Geerse, 2012). Several studies that consider the possible correlation of parameters in the resistance estimation where found (Kanning, 2012), (Jongejan, Maaskant, Ter Horst, Havinga, Roode and Stefess, 2013) but to our knowledge none of them implement the Copula bivariate correlated methods for reliability of flood defences for a single cross section in the resistance term.

3.1 Copula bivariate correlation modelling

One of the different ways of implementing the correlation effect of two desired variables in a reliability study, is by using Copula methods during the random sampling process. Copulas are mainly mathematical functions that allow to build joint distributions two or more variables while maintaining the statistical properties of their marginal distributions (Biller and Gunes Corlu, 2012). All types of possible copulas are derived from the Sklar theorem which states that every probability function can be written as a Copula function of the uniformly transformed marginal values (Sklar, 1959). There are several different parametric copula family types that allow to model dependence between the univariate marginal distributions. The most common are the “Gaussian” which is built as a function of the normal distribution, the “Archimedean”(e.g Gumbel or Clayton) which allows to generate the samples correlated in different tails of the distribution (Figure 2) and the “Empirical” ones which allow to build the correlated models as a function of empirical univariate distributions. Copulas are spatially bounded in a [0,1] by n dimensions. Therefore, for stochastic modelling their implementation consists in generating a desired number of random samples for a given type of copula as a function of a desired degree of correlation and then calculating the parameter value with an inverse transformation. The inverse transformation can only be performed if distribution type and statistical parameters used of the studied variables are known beforehand.

Figure 2. Different types of Copulas with increasing degree of correlation.

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4. Cross correlation of K and d70

Several of the studies mentioned previously, showed that the estimation of the critical head by the use of the Sellmeijer model is highly sensitive to the sand characteristic parameters. For the latest re-calibration experiments, a multivariate analysis performed for small scale experiments showed that the hydraulic permeability K and grain size d70 parameters have the most strong influence in the critical head estimation of BEP (Sellmeijer, de la Cruz, van Beek and Knoeff, 2011). Therefore correlation between these two parameters will have an important effect in the output if modeled as bivariate jointly distributed.

A literature study was performed in order to find information that allowed us to find possible correlation functions between these two parameters. It was found that extensive research has been done concerning the estimation of hydraulic conductivity as a function of the grain size distribution (Chapuis, 2012). This allows us to ensure that the two parameters are indeed correlated. Furthermore, two major findings were extracted as the most important characteristics a K vs d70 bivariate model should have.

First, the model is positively correlated given the large evidence found in the literature while acknowledging that negative correlation is also possible as stated by other authors (Schweckendiek, Vrouwenvelder and Calle, 2014).

Second, the correlation should be stronger for the smaller values as most of the models describe the hydraulic conductivity based on the d10 quintile. However other authors have also found correlation between greater size quintiles and permeability (AlHomadhi, 2013) to a lower degree. While permeability and size grain show great dependence between each other, most of the models also include the porosity term as well. The inclusion of a third parameter in the model is out of the scope of this research and therefore the is not included in the present experiment.

5. Case study example

5.1 Input data

Data of a typical Dutch flood defence embankment presented for a benchmark workshop of reliability methods (Schweckendiek, 2010) by the geotechnical safety network task group 3 is was used as a case study.

Table 1 Input data for stochastic failure estimation of BEP with correlated K and d70

Variable Unit Distributed Mean Std.

η [-] Constant 0.25 -

γ’ sand [N/m3] Constant 17 - γw [N/m3] Constant 9.81 - θ [deg.] Constant 37 -

d70 [m.] Log-normal 0.0002 0.00003 d70m [m.] Constant 0.000208 - ν [m2/s] Constant K [m/s] Log-normal 0.00001 0.00001 g [m/s2] Constant 9.81 - D [m.] Normal 15 1.5 mp [-] Log-normal 1 0.12 L [m.] Normal 25 1.25 h [m.] Gumbel a=1.839 b=0.152 hb [m.] Normal -1 0.1 d [m.] Log-normal 3 0.9

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For the proper estimation of this kind of failure (BEP), it is also required to study the probability of heave and uplift. According to its conceptual model both need to occur in order to start the erosion process. For this particular study, we are only concern about the influence of correlation in the piping failure estimation. Therefore we are only going to study the effect in the probability of occurrence of piping limit state function which doesn’t necessary imply the eventual collapse of the defence.

5.2 Reliability method

The reliability study consisted in generating 2 million samples according to the stochastic input data presented in (Table 1). The values for mean and standard deviation where obtained from data reported in the VNK project (Rijkswaterstaat, 2012). For each generation, K and d70 where sampled with a certain copula model. Afterwards, the sampling process is repeated varying the rank correlation coefficient. Monte Carlo stochastic simulations where chosen as reliability estimation method despite the fact that common flood risk practice uses more efficient reliability methods (Steenberghen, Lassing, Vrouwenvelder and Waarts, 2004).

Still Monte Carlo also allows to have better control over the correlated variables while reproducing the rest of the parameters with the exact statistical parameters. Due to the high dimensionality of the problem it is not ensured that the Monte Carlo simulations will convey to a unique result with fix number of number of samples. Therefore a bootstrapping procedure was done in order to ensure a negligible error in the probability estimation.

For the estimation of the load random variable term, a Gumbel distribution is used with location and shape parameters a and b respectively. Note that the water levels in the back part of the flood defence (hb) are normally distributed as they depend on the artificial induced operation of the inner drainage system of the protected area.

5.3 Type of Copula for modelling K and d70 correlation

Correlation between K and d70 is modelled by the use of Copulas. Such methods have showed to be a powerful tool when field data is limited. For geotechnical reliability studies such methods have been studied in detail (Wu, 2013),(Tang, Li, Zhou, Phoon and Zhang, 2013), (Tang, Li, Rong, Phoon and Zhou, 2013). The application of such correlation models have the advantage of maintaining the statistical characteristics of the uni-variate random distributions while considering possible degrees of correlation between variables. It is also important to note that copulas are generated as a function of the Kendall (τ) rank correlation coefficient which can be calculated as a function of the often used Spearman’s correlation coefficient (rho). While Spearman’s estimate the correlation of values by assuming linearity, Kendall’s method estimates the correlation by the ranking of the different values. Consequently Spearman’s coefficient does not reflect the degree of nonlinearity.

One of the major concerns is the criteria for the selection of the type of Copula function. Given the two desired characteristics cited in section 4, three types of copulas Figure 3 where chosen as a way to estimate the influence of correlation in the failure estimation of BEP. One is the “Gumbel” copula from the Archimedean family which recreates a stronger dependence in the positive side of the tail of the random variable marginal distribution. The second one is the Gaussian copula which is capable of representing dependence in the two tails of random normally distributed variables. And a third one known as “Clayton” copula from the Archimedean family as well, which shows a strong dependence in the negative tail of the distribution.

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6. Results

The correlated variables K and d70 are log-normally distributed which implies that their variance can differ by several orders of magnitude with respect to their mean value. However when plotted as the logarithm of the values the Clayton copula reflects a better representation of the nature of permeability in sands (Figure 3Erro! A origem da referência não foi

encontrada.). As the d70 from the aquifer increases, the occurrence of voids between grains can be either filled by much more smaller grains, organic matter and/or air, while for an aquifer with smaller d70 particle diameter the voids are more easily filled with only mostly smaller particles. The reason for this, is because the smaller particles able to fill the smaller voids have a larger specific surface and therefore its permeability is very low like clay-sand materials for example. When modelling correlated random variables with copulas, an inverse transformation sampling method is required as the values generated represent the probability of occurrence of each random sample and not the value by itself.

Figure 3. Probability density functions of limit state equation for different copulas with different correlation

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For instance, a stronger dependence near to the value 0 (Clayton case) means stronger dependence in the values located in the left side of the marginal distribution. In the case of permeability for example they will correspond to the low permeable values which are more characteristic of clay sands. For the case of BEP, failure probability increases proportionally to the conductivity as the erosion resistance of the grains is over topped. Each copula type has a different degree of local linear correlation. Therefore, the rate of change of probability of failure is different. For example , the Gumbel copula gives a higher probability of failure for a low rank correlation value the Clayton copula will only achieve at a high correlation value (Figure 4). Copulas are built by functions known as generators. Generators are functions at the same time that can be expressed as a functions of correlation coefficients. The three copulas used are generated as a function of Kendall’s rank coefficient (τ). The correlation degree in conceived as the degree concordance in the ranking of one value with its correlated value of the second variable. For this reason is more suitable for copulas as it can estimate the degree of correlation of nonlinear functions in a more accurate way than the more commonly used ‘Spearman’s’ correlation coiffing (ρ). However as high degree τ coefficient doesn’t imply high linear correlation. For the case of Clayton copula for example, a τ = 0.999 still shows some scattering (Figure 3) in the positive side of the tail and therefore the probability of failure is not convergent to the same point as for the other 2 types of copulas (Figure 4).

Figure 4. Failure probability for BEP as function of rank correlation coefficient (τ).

For the case when estimating the probability of BEP failure with d70 and K completely uncorrelated (Figure 4), the probability of failure is 7.5e-5. For the Dutch probabilistic approach, the regulation states that the minimum total allowable failure return period is 1250 years (Pf = 8e-4) for Riverine flood defences. For BEP is estimated that around 1% from the total failure probability is attributed to this failure mechanism. Consequently the calculated failure probability will be 10 times higher than the expected. However, it was already stated that the present study considers only the probability of occurrence of piping on its own and therefore the real failure probability must be calculated as a conditional of the occurrence of uplift/heave failure mode. Based on the results of this study, when the correlation increases between k and d70, the probability density function changes as the area in the tails is more populated. This means that the frequency for values located along the pdf will change. Still validation of the correlation models is required for better assessments of tail frequency estimation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

x 10-5

Kendal coefficient (Tau)

Pro

bab

ilit

y o

f fa

ilure

Failure vs Rank correlation coefficient

GumbelGaussClayton

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7. Discussion

For the case of BEP, a higher correlation degree between the two studied variables will always imply a lower probability of failure of the flood defence. As the degree of correlation increases, the three types of copulas tend to unite. More kinds of Copulas are available but for the present study, only the ones that showed higher dependency degrees in the tails where selected. Although large amounts of samples are required in order to have a reliable estimate of the degree of correlation in the aquifer. For the flood defence reliability practice, correlation between variables is more important in the spatial and time scales as they are conceived as systems in parallel of representative cross sections. The type of soil that compose the Dutch aquifers is highly anisotropic and non-homogenous. Hence the method is also sensitive to the predefined statistical parameters and distribution shape used for the inverse transformation from the copula values. In addition most of the actual literature demonstrates that the inclusion of porosity is needed to have an accurate empirical estimation of permeability. Also big discontinuity’s can be found which are not possible to include in a simple limit state function.

On the other hand flood defences can include local transitions or structures that due to their small size can be assessed by a single representative cross section. Different forensic flood risk studies have shown that transitioning points in longitudinal flood defences are the most probable to fail during a flood event (Kanning, Van Baars, Van Gelder and Vrijling, 2007). A flood defence is only a safe as its most vulnerable component. In such case, special attention should be taken when estimating the probability of failure of these kind of point located structures. In the case of BEP the probability of occurrence might increase as impermeable structures such as roads, pipes or even houses embedded in the flood defences might increase the pore pressure accumulation and the possible presence of large voids in the soil. For these cases, the method presented in this study can help to bound the probabilities of failure to have a more precise estimation for the reliability of systems at point locations.

8. Conclusions

Different copula correlation models have been applied in the probabilistic assessment of Backward erosion piping for a single flood defence cross section.

As a general main conclusion, the present study showed that the assumption of uncorrelated variables of the actual safety assessment performed in the Netherlands might is conservative and safe. The assumption of uncorrelated values will be traduced in higher probabilities of failure for the BEP Sellmeijer Revised model.

No matter which copula type is chosen, the Sellmeijer limit state equation shows a reduction in the failure probability due to BEP for a higher degree of correlation.

Correlation assessment is recommended as the influence of a low degree of correlation can change the failure probability by even a factor of 5 deepening on the Copula model chosen.

Acknowledgements

This work is part of the research program of the Technology Foundation STW, financially supported by the Netherlands Organization for Scientific Research (NWO). The authors would like to thank STW foundation, The flood risk department of Deltares and the Multi-functional flood defences program for the funding and support.

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