Calibration of Piping Assessment Models in the Netherlands

Transcription

1 ISGSR Vogt, Shuppener, Straub & Bräu (eds) Bundesanstalt für Wasserbau ISBN Calibration of Piping Assessment Models in the Netherlands J. Lopez de la Cruz & E.O.F. Calle Deltares, Unit Geo-engineering, Delft, Netherlands T. Shwekendiek Delft University of Tehnology, Department of Hydrauli Engineering, Delft, Netherlands Deltares, Unit Geo-engineering, Delft, Netherlands ABSTRACT: New insights into the failure mehanism piping (under-seepage) regarding the physial proess as well as reliability aspets have led to a revision of the Duth design and safety assessment rules for dikes. This paper desribes how the required fator of safety for piping is derived from a top level requirements formulated in terms of an aeptable probability of flooding. The main steps herein are (a) to aount for the length-effet to translate requirements on dike ring (system) level to admissible probabilities of failure on dike setion (element) level and (b) the alibration of safety fators as a funtion of the (element) target reliability. Keywords: ode alibration, piping, under-seepage, target reliability, length-effets Nomenlature D [m] : thikness of sand layer F F [-] : fore fator F G [-] : geometrial shape fator F R [-] : resistane fator F S [-] : sale fator H [m] : hydrauli head differene (aross struture) H [m] : ritial hydrauli head differene L [m] : seepage length RD [-] : relative density RD m [-] : mean relative density (small sale experiments ) [-] : erosion oeffiient d 70 [m] : 70-perentile value of grain size distribution of the piping-sensitive layer d 70m [m] : mean value of d 70 value in the experiments (small sale experiments -2.08e -4 ) h [m+ref]: waterside water level h b [m+ref]: landside water level p [N/m 3 ]: unit weight of partiles w [N/m 3 ]: unit weight of water [-] : Whites onstant [DEG]: Bedding angle of sand [m 2 ] : intrinsi permeability R [-] importane fator V R [-] oeffiient of variation p [-] safety fator h [m] the normative water level d [m] Blanket layer thikness h b [m] deimal height 587

2 1 INTRODUCTION Primary flood defenses in the Netherlands undergo a 5-yearly safety assessment. Based on eonomi as well as soietal risk aeptane riteria, the urrent safety standards are defined in terms of exeedane probabilities of hydrauli load onditions (see Figure 1). For failure mehanisms other than overtopping these are ommonly interpreted as admissible probabilities P f,adm,dr of (system) failure of a dike ring (polder). That implies that the riteria to be handled for individual dike setions and failure mehanisms need to be striter: P f,adm,ds,meh < P f,adm,dr. That is for two reasons: (a) eah mehanism an ause failure and (b) (b) failure of eah individual setion means system failure. We are dealing with a serial system. A third omponent is the length-effet. The probability of failure inreases with inreasing length of a dike setion with statistially homogeneous properties (Vrouwenvelder, 2006). Normative Exeedane Frequeny [1/yr] 1/10,000 1/4,000 1/2,000 1/1,250 high grounds North Sea Germany Belgium Figure 1. Normative exeedane probabilities of Hydrauli Load Conditions in the Netherlands All these aspets illustrate the need to establish higher target reliabilities loally and for eah failure mehanism in order to ahieve a suffiiently reliable dike ring (system). In order to translate these requirements into pratial terms, semi-probabilisti assessment and design rules need to be derived, using harateristi values and (partial) fators of safety instead of reliability analysis tehniques. This paper desribes how a loal assessment rule for piping (under-seepage) has been derived, the goal of whih is to be onsistent with the high level riteria in terms of probabilities of flooding. Before going into the details of the alibration ode, the revised piping model is desribed, inluding a onise disussion of the model unertainty. Subsequently, the format of the new safety assessment rules is presented and the derivation of the required fator of safety is disussed. The latter onsists of two main steps: (a) derivation of the aeptable probability of (piping) failure of a dike setion and (b) the alibration of the required fator of safety as funtion of the target reliability. 2 REVISED PIPING MODEL 2.1 The Equilibrium Model by Sellmeijer (1988) Sellmeijer (1988) proposed a omputational for piping with three main elements: groundwater flow, pipe flow through the erosion hannel and limit equilibrium of soil partiles in the hannel. For safety assessment purposes, the following equilibrium ondition was derived, whih desribes the ritial gradient over the struture. In other words, for lower gradients (H/L) the erosion pipe development stops aord- 588

3 ing to the model, whereas for higher gradients the erosion may reah the upstream side and thereby endanger the integrity of the struture by under-mining it. p FR tan H 1 F F F F 70 L R S G S 3 L F d w D D 1 G ln 0.25 FS L L The geometry and soil parameters are speified in the setion Nomenlature; oeffiient is omposed of three fators: - F R : resistane fator, being the strength of the sand - F S : sale fator, relating pore size and seepage size - F G : geometrial shape fator Notie that this riterion does not address the appearane of sand boils like exit gradient-based riteria do. Using this equilibrium riterion in safety assessment one impliitly allows sand boils to our, while pipe development until the upstream side is avoided. For more omplex geometries than a simple aquifer overed by a blanket layer, both with onstant thikness, the riterion (in a slightly simplified form) has been implemented in MSEEP, a numerial ode for groundwater flow omputations. (1) 2.2 Revision of the Sellmeijer Model Reently, a detailed experimental researh of the piping mehanism in The Netherlands (Lopez de la Cruz et al., 2010) has provided better insights into the underlying physial phenomenon and led to a revision of the Sellmeijer model. A multivariate regression analysis enabled re-alibrating the oeffiients in the model, by assessing the influene of eah measured variable on the ritial head simultaneously, resulting in H L F 1 F F F R S G 0.35 p RD R tan w RDm 0.6 d 70 d 70m S 3 L d70 F D 1 L (2) D FG 0.91 L Besides new oeffiients the revised model ontains a dependeny on the relative density (RD) of the piping sensitive sand layer (i.e. aquifer, usually the upper few deimeters). However, for safety assessment purposes, the influene of RD is not taken into aount. It is hard to determine in the field and of little influene despite the large unertainty. Therefore, it was preferred to inlude it in the model unertainty. Notie that for non-standard geometries the piping module in MSeep is also available for the revised model and reommended for determining F G. 2.3 Model Unertainty The model unertainty is aounted for by a multipliative model with fator m : H 1 m mfr FS FG (3) L 589

4 The experimental results from Beek et al. (2010) have been analyzed for determining the parameters of the model fator, whih is hosen to be modeled by a lognormal distribution. Its standard deviation is determined by a weighted variane-analysis, in whih more weight is given to the available data from prototype sale than to the small and medium sale laboratory experiments (Lopez de la Cruz et al., 2010). The resulting standard deviation of the model fator is m =0.12, the satter of the omparison of predited versus observed ritial head differene is illustrated in Figure H/L predited H/L observed IJdijk (prototype) Deltagoot (full sale) medium sale small sale observed failures Figure 2. Observed vs. predited (revised piping model) ritial piping gradients 3 SAFETY ASSESSMENT RULE The revised piping model an be used to assess the resistane against piping in terms of the ritial head differene: H m F F F L R S G (4) For safety assessments this value an be ompared to the head differene the struture experienes. For the Duth safety assessment rules it was deided to handle the same redution term as in the urrent guidelines: H hhb 0.3d (5) Consequently, the format of the new safety assessment rule is hosen suh that the ratio of the ritial head differene H (resistane) and the head differene inluding redution term (load) using harateristi values (5% respetively 95% quantiles) is required to be larger than the safety fator p : R HCk, mpk, FRk, F k Sk, FGk, Lk p S k H k hk hb, k 0.3d (6) k Notie that the harateristi fators F i,k are determined by using 5%/95%-quantiles for their input parameters. Furthermore, the harateristi (or, in fat, design value) for the water level h k is taken to be the normative water level (MHW) as defined in the Hydrauli Boundary Conditions (Rijkswaterstaat, 2007). The proedure to determine the harateristi values is beyond the sope of this study; referene is made to Euroode 0. 4 TARGET RELIABILITY AND FACTOR OF SAFETY The main goal of the alibration is to determine (partial) safety fators that, if onsequently used in design or safety assessment, lead to a struture that is at least as safe as the predetermined target reliability. 590

5 Commonly, in odes and standards (e.g., Euroode) the target reliability is hosen from safety lasses that reflet the severity of the onsequenes - the more severe the onsequenes the higher the target reliability. For the Duth flood defense system that basi onept is the same, exept that the target reliability is defined by the exeedane probabilities 1 (Figure 1) as probabilities of flooding. In other words, these are admissible probabilities of (system) failure. The probability that any of the elements of a dike ring (i.e., dike or other flood defenses) fails is defined as: P f,adm,dr [1/yr]. That means that this probability annot be used to define one target reliability for a partiular struture in the system diretly. The first step is a apragmati one: The probability of system failure is distributed over the failure mehanisms in the system that play a signifiant role. For piping the admissible probability of failure is 10% of the total: P f,adm,dr,p = 0.1 P f,adm,dr. This distribution over failure mehanisms an be treated as eonomi optimization problem, however, these onsiderations are beyond the sope of this paper. The seond step is to translate dike ring requirements into dike setion requirements. The latter beings the level at whih designs and safety assessments are arried out. The key element in this step is the length-effet (Vrouwenvelder, 2006). The probabilities of failure of the flood defenses that form a dike ring are partially orrelated. Usually, there is a large (spatial) orrelation between the loads on different setions, where also the resistane properties are highly independent. That implies that the probability of failure somewhere in the dike ring is larger than the probability of failure of one (or the weakest) element (i.e., dike setion): P f,adm,dr,p > P f,adm,ds,p. This is aounted for by inorporating the length effet. The details are further disussed in setion 5. TOP LEVEL REQUIREMENT Aeptable probability of failure of the dike ring: P f,adm,dr Aeptable probability of failure per mehanism (dike ring): P f,adm,dr,meh LOCAL TARGET RELIABILTY Aeptable probability of failure per mehanism (dike setion): P f,adm,dr,meh aount for ontributions of several mehanisms aount for length-effet Figure 3. Steps to determine Target Reliability Having determined the target reliability, the atual alibration ode is applied. A onvenient starting point is to pik standardized values for importane fators suh as given in the Euroode. For example, for a dominant load parameter: R = 0.8 with the standard formulae for partial resistane fators. However, from the FLORIS projet Rijkswaterstaat (2005) it is known that for piping the importane fators an vary signifiantly and even exeed R = 0.8. Therefore, an appropriate value for p for varying onditions is examined diretly by the analysis desribed in Figure 4. Further details are disussed in setion 6. Choose a value of p Find the seepage length L suh that the design rule (eq. 6) is fulfilled (H,k /H= p ) Determine the reliability index for this ondition (i.e., design): Figure 4. Steps to analyze the appropriateness of values for the required safety fator p 5 LENGTH EFFECTS Applying zero-level rossing theory for the input parameters to the piping model, a relationship between the admissible probability of failure for a dike setion and the admissible probability of failure on dike ring together is established. This inludes the influene of the dike setions lengths L dr,s that are sensitive to piping (i.e., potentially ontribute to the probability of failure). A detailed desription of this analysis is beyond the sope of this paper but more details an be found in Lopez de la Cruz (2010). The result is represented in Figure 5. 1 The normative exeedane probabilities are not exatly the admissible probabilities of flooding, but in the ontext of ode alibration for failure mehanisms other than overtopping are interpreted as suh. 591

6 6 5.5 req [1/yr] P f,adm,dr,p : 4 1/12,500 1/20, /40,000 1/100, L dr,s (km) Figure 5. Relation between target reliability, (sensitive) dike ring length and aeptable probability of piping for a dike ring The target reliability req =- -1 (P f,adm,ds,p ) inreases with the (piping-sensitive) umulative length of dike setions L dr,s forming the dike ring. For pratial purposes, the following formula is proposed whih fits the relations in Figure 5 very well: P 0.1P f, adm, dr f, adm, ds, p (8) 1 / leq Ldr, s where Calibration fator l Correlation length of the limit estate funtion for piping eq For harateristi Duth onditions a value of / l eq = is proposed. 6 UNCERTAINTIES AND CALIBRATION DATA SETS In order to hek the suitability of p as desribed in Figure 4, a set of onditions to analyze (parameter sets) are needed as well as their probability distributions per parameter. For the latter, it is reurred to the probabilisti modeling in the FLORIS projet. The distribution types and variation oeffiients (or standard deviations) per parameter are given in Table 1. Table 1. Probability distributions of piping load and resistane parameters Parameters Type Mean Spread (V=CoV, σ=std) D nominal V=5.0 k nominal V=1e-6 L nominal V=0.24 d70 nominal V=6e-5 eta Normal 0.25 V=0 theta Normal 37 σ=0 m_p 0.12 σ=0.12 h_b normal nominal σ=0.10 h Gumbel nominal σ =

7 The assessment rule should hold for the range of onditions that is expeted to be enountered in pratie. To establish these onditions, for eah parameter with nominal mean value in the table a low, medium and high value (for typial Duth onditions) have been hosen. The resistane parameters are summarized in Table 2. Table 2. Nominal Values of Piping Resistane Parameters for Calibration Sets Variable Unfavourable Average Favourable D [m] k [m/s] 1E-04 1E-05 1E-06 d70 [m] 1,2E-04 2,0E-04 4,0E-04 d [m] 0,1 2,5 6,0 For the load parameters (influening the head differene H) three parameter sets are hosen to represent different hydrauli load regimes together with the aording aeptable probability of flooding (see Table 3). Table 3. Parameter Sets of Hydrauli Load Variables Coast River Estuary P(h>MHW) h k = MHW [m+nap] 3,3 6,3 3,1 h 10 (deimate height) [m] 0,75 0,7 0,35 hb [m+nap] -2,5 4,2-1,0 The first three parameters in the table above an be used to define a the Gumble distribution for the water level. The deimate height is the water level differene that inreases the exeedane probability of the water level by a fator 10 with respet to the normative water level with known exeedane probability: Ph ( MHW) 10 Ph ( h ) (9) 10 Combining eah load parameter set with eah of the resistane variables, we obtain 243 (=3^5) alibration parameter sets. 7 CALIBRATION RESULTS The alibration analysis has been arried out for eah of the 243 parameter sets (previous setion) and for six different values of the partial resistane fator γ p : 1.0, 1.2, 1.4, 1.6, 1.8. This setion presents a summary of the alibration results. Eah of the points in the figure 6 shows whih value of the reliability index (horizontal axis) is found for the underlying parameter set and the p (vertial axis). The results present some lusters due to the hydrauli load regimes and the seleted blanket layer thikness, whih ats in this ase as a load redution term. In order to illustrate this, the data points are plotted with different olors and shapes (see legend). From Figure 6, it an be seen that the load onditions together with the blanket layer thikness determine the performane of the safety fator. 593

8 Hydrauli loads O oast (1/10000yr)? upper river (1/1250)? lower river (1/4000) Blanket layer thikness no blanket layer (red) thin blanket layer (blue) thik blanket layer (green) relevant range (inluding length-effet) 4.2 < req,lo < 5.2 p 1.60 level I: = 0.8, V=0.3 level I: = 0.7, V=0.25 level I: = 0.6, V= () = << Figure 6: Overview Calibration Results: required safety fator p vs. realized reliability index The blak line presents the proposed linear relationship between the required reliability and the partial resistane fator: ( ) (10) p req p The line is hosen on the left of the satter points, approximately through the 95%-quantiles per safety fator level. That means that for 95% of the (alibration) ases the reliability ahieved by applying the safety fator is higher than the target reliability, while 5% of the (rather extreme) ases would result in a slightly unonservative design, whih was supposed to be aeptable. The lower limit of p was a politial deision maintaining safety fator of 1.2 that is urrently in use as a lower limit. A point in the upper left region of the satter implies a region with under-performane. However, all of these points are triangles, meaning that they represent river load onditions. In the Netherlands, those areas have safety requirements that do not exeed req =4.7 even for the longest dike rings. Therefore, these points are irrelevant. The green in the figure 6, indiates the range of γ p -values expeted to be applied using the proposed assessment rule in the Netherlands, based on the relevant range of required reliability index inluding the length-effet. The resulting range is 1.2 < γ p < 1.6 with the remark that values of 1.4 are expeted to be exeeded only in rare ases suh as long dike rings with low aeptable probability of flooding. The red dashed and dotted lines indiate the resulting safety fator values after the standard level-i equations for lognormal distributed resistane is applied for different ombinations of importane fator ( R ) and oeffiient of variation (V R ). 8 DISCUSSION A target reliability-dependent safety fator for piping is derived for design and safety assessment in the Netherlands. The inorporation of the presented results in the design guidelines is still pending. The target reliability is derived from speifi flood protetion norms in the Netherlands instead of onsequene lasses suh as used in the Euroode. In the derivation, it is aounted for system reliability aspets suh as aounting for several mehanisms as well as for the length-effet. Taking into aount the satter of the alibration points around the proposed relation between the safety fator and the target reliability shows a typial aspet of semi-probabilisti design and safety assessments. For a signifiant range of the hosen onditions, the approah leads to over-design. This an 594

9 be avoided by more differentiation. For example, by establishing different safety fators for different load regimes or by reliability-based design ( reliability analysis in safety assessment). REFERENCES Beek, V. M. v., J. G. Knoeff, J. Rietdijk, J. B. Sellmeijer, and J. Lopez De La Cruz, Influene of sand and sale on the piping proess - experiments and multivariate analysis, Physial Modelling in Geotehnis, Zürih, Taylor and Franis Group. Euroode 0. EN 1990 'Euroode: Basis of strutural design. Lopez de la Cruz, J., Shwekendiek, T., Mai Van, C. & Kanning, W Calibration of Partial Resistane Fators for Piping and Uplift. Deltares report no GEO Rijkswaterstaat (2005). Main report of Flood Risks and Safety in the Netherlands (Floris). Rijkswaterstaat (2007). Hydraulishe Randvoorwaarden primaire waterkeringen (in Duth). August Vrouwenvelder, Ton (2006). Spatial effets in reliability analysis of flood protetion systems. Pro. International Forum on Engineering Deision Making, Seond IFED Forum, April 26-29, 2006, Lake Louise. 595