Probability of dike failure due to uplifting and piping

Transcription

1 Probability of dike failure due to lifting and iing J.M. van Noortwijk KV Consultants, P.O. Bo 0, 803 AC Lelystad, The Netherlands A.C.W.M. Vrouwenvelder TNO Building and Construction Research, P.O. Bo 49, 600 AA Delft, The Netherlands E.O.. Calle Delft Geotechnics, P.O. Bo 69, 600 AB Delft, The Netherlands K.A.. Slijkhuis Ministry of Transort, Public Works, and Water Management, Civil Engineering Division, P.O. Bo 0000, 350 LA Utrecht, The Netherlands ABSTRACT: A methodology is resented for determining the robability of dikes failing due to lifting and iing using directional samling. The study concerns one dike section in the lower river area of The Netherlands. or this dike section, the three most imortant random quantities are: the North-Sea water level, the river Rhine discharge, and the critical head in the event of lifting and iing. Dike failure due to lifting and iing is defined as the event in which the resistance (the critical head) dros below the stress (the outer water level, a combination of both sea level and river discharge, minus the inner water level). Since the critical head is correlated over the length of a dike, the satial variation and correlation is modelled using a Markovian deendency structure. Three-dimensional directional samling on the basis of the olar coordinates of the sea water level, the river discharge, and the critical head is used to determine the failure robability. NTRODUCTON The aim of the aer is determining the robability of dikes failing due to lifting and iing. Ulifting occurs when the covering layer of a dike bursts due to the high water ressure, whereas iing under dikes occurs due to the entrainment of soil articles by the erosive action of seeage flow. This study concerns one dike section of the dike ring oeksche Waard (dike section 3.), which is situated in the lower river area of The Netherlands. The three most imortant random quantities reresenting inherent uncertainties are: the North-Sea water level, the river Rhine discharge, and the critical head in the event of lifting and iing. n this situation, dike failure due to lifting and iing is defined as the event in which the resistance (the critical head) dros below the stress (the outer water level, a combination of both sea level and river discharge, minus the inner water level). The statistical uncertainties reresent the uncertainties in the arameters of the robability distributions of the sea water level and the river discharge. Since the critical head is correlated over the length of a dike section, the satial variation and correlation of the critical head in the event of lifting and iing is modelled using a Markovian deendency structure. This means that the random quantities reresenting the inherent uncertainties in the critical head of one dike subsection only deend on the values of the corresonding random quantities in the two adjacent dike subsections. The robabilities of failure due to lifting and iing are calculated in three stes. irst, a dike section is subdivided into smaller subsections by assuming the limit-state function of lifting and iing to be a Gaussian stationary rocess and using the theory of the level-crossing roblem. Second, the failure robabilities of one dike subsection and two adjacent dike subsections are calculated using directional simulation. Third, the failure robability of one dike section is determined by aroimating the Gaussian stationary rocess by a Markov rocess with resect to failure of dike subsections. Three-dimensional directional samling is used to determine the robability of failure due to lifting and iing. An advantage of three-dimensional directional samling is that large samle sizes are not required. The results of directional samling are comared to the results of irst Order Reliability Method (ORM). The outline of the aer is as follows. The limitstate functions of lifting and iing are introduced in Section. The modelling of the satial correlation and variation of the critical head for lifting and iing, as well as determining the failure robability of a dike ring (series system of dike subsections) is studied in Section 3. The directional samling technique, which is used to obtain the robability of failure due to lifting and iing, is resented in Section 4. Results and conclusions can be found in the last section.

2 ALURE DUE TO UPLTNG AND PPNG Ulifting occurs when the covering layer of a dike bursts, due to the high water ressure, whereas iing under dikes occurs due to the entrainment of soil articles by the erosive action of seeage flow. The limit-state function of lifting and iing is given by (see also igure ) Z M ( + M ) b () and h z ma{ M, M } () with u u outer water level [m +NAP], b inner water level [m +NAP], critical head in the event of iing [m], u critical head in the event of lifting [m], critical head for lifting and iing [m], M h model factor water level [-], M model factor iing [-], M u model factor lifting [-], M z model factor water-flow model ZWENDL [m]. The two critical heads u and are functions of the following random quantities: the volume weights of sand and water; the angular rolling friction; the constant of White; the sizes of sand articles; the ermeability; the thickness of the covering layer and the sand layer; and the dike width. These random quantities are indeendent, lognormally distributed, and reresent inherent uncertainties in the critical head and are correlated over the length of the dike on the basis of the quadratic eonential correlation function in Eq. (3). The robability distribution of the outer water level is a miture of both the robability distribution of the North-Sea water level at oek of olland, denoted by S [m +NAP], and the robability distribution of the river Rhine discharge at Lobith, denoted by Q [m 3 /s]. The further down the river, the more the sea water level S affects the local water level, and the less the river discharge Q affects the local water level. Given a articular sea water level and a articular river discharge, the downstream water level can be obtained with the onedimensional water-flow model ZWENDL. On the basis of ZWENDL calculations, the local water level has been aroimated by a bilinear function of the sea water level S and the river discharge Q. The robability distribution of the annual maimum sea water level is a generalized Pareto distribution, whereas the robability distribution of the annual maimum river discharge is a iecewise eonential distribution. Besides the inherent uncertainties, the uncertainties in the statistical arameters of these two robability distributions are taken into acount. The subject of study is the robability of dike failure due to lifting and iing er two days. Therefore, the robability distributions of annual maimal sea level and discharge must be transformed to robability distributions of maima over two days. These transformed random quantities are denoted by S days and Q days, resectively, and they are indeendent. The other random quantities are distributed as follows. The inner water level b and the model factor of the water-flow model M z have a normal distribution; the model factor for lifting M u and the model factor for iing M have a lognormal distribution; and the model factor of the water level M h has a beta distribution. The two critical heads for lifting and iing, the outer and inner water level, and the four model factors are mutually indeendent. or further details about the robability distribution reresenting inherent and statistical uncertainties, we refer to Cooke & Van Noortwijk (998). 3 SPATAL CORRELATON AND VARATON n order to model the satial variation and correlation of the random quantities reresenting the inherent uncertainties in the critical head, Vrouwenvelder (993, Chater ) used the quadratic eonential correlation function ρ ( ) e, (3) d Clay Dike width Sand Clay igure. Cross-section of a dike. b Thickness covering layer Thickness sand layer where ρ() is the roduct moment correlation, is the distance between the two oints at which the stochastic rocess is studied [m], and d is the fluctuation scale [m]. The fluctuation scale reresents the satial variation: the larger the fluctuation scale, the less satial variation, and the larger the correlation coefficient. The arameters used to model the satial variation can be found in Cooke & Van Noortwijk (998). The failure robability of a dike section can be aroimated by regarding a dike section as a (se-

3 ries) system of smaller dike subsections. Deendencies between failure events of the dike subsections can then be modelled on the basis of a Markovian deendency structure. or modelling the satial variation, this means that the random quantities in one dike subsection only deend on the values of the corresonding random quantities in the two adjacent dike subsections. The question arises how a dike section can best be subdivided into smaller subsections of length and which value of the fluctuation scale d should be chosen. n this subsection, we resent a methodology to obtain both the dike subsection length and the fluctuation scale. On the one hand, we assume the limit-state function of lifting and iing to be a Gaussian stationary rocess and we use the theory of the so-called level-crossing roblem [see, e.g., Vrouwenvelder (993), Paoulis (965, Chater 4), and Karlin & Taylor (975, Chater 9)]. On the other hand, we aroimate this Gaussian stationary rocess by a Markov rocess. The level-crossing roblem reads as follows. Let a Gaussian stationary rocess Z() be given with mean 0, standard deviation, and correlation function ρ(), where is the distance between the two oints at which the stochastic rocess is studied [m]. The correlation function must satisfy ρ (0) <. An er bound for the robability of eceeding the level β in a dike subsection of metres can be written as β ( ) e ρ (0). (4) π The smaller, the better this er bound can be used as an aroimation. or the quadratic eonential correlation function, Eq. (3) transforms into β ( ) e. (5) π d According to Vrouwenvelder (993), we can choose the length of the dike subsection, denoted by, as such that the corresonding robability of eceedence ( ) equals Φ( β): β ( ) Φ ( β ) e (6) β π or, similarly, d π. (7) β The aroimation for Φ( β) in Eq. (6) can only be alied when β >. or eamle, by using directional simulation, the β for one subsection of dike section 3. has the value (see Section 4) 6 β Φ (.650 ) (8) On the basis of the quadratic eonential correlation functions of the random quantities reresenting the inherent uncertainties in the critical head, the fluctuation scale d of the limit-state function or rocess Z() remains to be determined. A dike subsection is assumed to have a length of metres. Standard Monte Carlo simulation can then be used to calculate the correlation coefficient of Z( ) and Z( ), denoted by ρ( ). According to the quadratic eonential correlation function, the correlation coefficient ρ( ) equals ρ ( ) e (9) d or d ln( ρ(. (0) )) Using β 4.65, substitution of (7) into (9) results in π ρ( ) e () β The subsection length satisfying Eq. () can be determined with the aid of the following Picard iteration rocess: π n n+, n β ln( ρ( n )),,3,..., () where is the initial estimate. The subsection length can now be determined in the following seven stes: irst, we calculate the robability of failure due to lifting and iing of the first dike subsection using directional simulation (see Section 4) and substitute the resulting β into Eq. (). Second, we samle from the robability distributions of the outer water level (a bilinear function of the sea water level and the river discharge), the inner water level, and the four model factors using standard Monte Carlo samling. These random quantities do not deend on the dike subsection studied. Third, we make an initial estimate of the subsection length. ourth, we samle from the (lognormal) distributions reresenting the inherent uncertainties in the critical head of the first dike subsection and we calculate the corresonding samles of the critical head for lifting and iing denoted by. ifth, we samle from the conditional (lognormal) distributions reresenting the inherent uncertainties in the critical head of the second dike subsection given the corresonding samles of the first dike subsection and we calculate the corresonding values of the critical head for lifting and iing denoted by. Recall that for all inherent

4 uncertainties in the critical head, the correlation between the first and second dike subsection is defined by a quadratic eonential correlation function at the value. Sith, we estimate the samle correlation coefficient ρ( ) on the basis of the Monte Carlo samles of Z( ) and Z( ), and substitute it into Eq. (). Seventh, we determine a new estimate of the subsection length using Eq. (). As long as the (n+)-th estimate of differs from the n-th estimate, we reeat stes 5-7. or dike section 3., the Picard iteration rocess results in subsection length 50 metres. or convenience, the subsection length is rounded off to units of 5 metres. Probability lots of the Monte Carlo samles of and give evidence of lognormally distributed critical heads. Therefore, lognormal distributions can be fitted to the critical heads for the first and second dike subsection. Also the critical head for the combination of the first and second dike subsection, denoted by min{, }, (3) aears to be lognormally distributed. or further details about the arameters of these lognormal distributions, we refer to Cooke & Van Noortwijk (998). Given the subsection length, the robabilities of failure due to lifting and iing must be calculated both for one dike subsection searately and for two adjacent dike subsections combined. On the basis of these two robabilities, the robability of failure for a (series) system of dike subsections can be aroimated by regarding the failure of dike subsections as a Markov rocess. n mathematical terms, the robability of failure of a dike section of length l, denoted by (l), can now be written as a function of the failure robability of one subsection ( ) and the failure robability of two adjacent subsections ( ) in the following manner: ( l) Pr{no failure #} [Pr{no failure # no failure #}] ( l [ ( )] ( l / [ ( )] / ) ), ( l / ) (4) where Pr{failure #i} denotes the robability of failure in the i-th dike subsection. By alying Eq. (4), the robability of failure of a dike section can be easily comuted. 4 DRECTONAL SMULATON The aim of this section is to calculate the robabilities of dike failure due to lifting and iing both for one subsection searately and for two adjacent subsections combined. n calculating these failure robabilities using standard Monte Carlo simulation, the roblem arises that there are not enough samles in the failure region to obtain reliable results. To seed Monte Carlo simulation, we use directional samling. Roughly seaking, directional simulation means the following. Rather than samling straight from the robability distributions of the sea water level S, the river discharge Q, and the critical head, we samle the directional angle and the directional radius in the (s, q, h )-lain. This ays off when we have the fortune of being able to calculate the conditional robability that the length of the radius belongs to the failure region in elicit form. or eamle, when n random quantities have a multivariate normal distribution it is well-known that the conditional distribution of the squared radius, when the value of the directional vector is given, is a chisquare distribution with n degrees of freedom (see Ditlevsen & Madsen, 996, Chater 9). The number of random quantities emloyed in the directional samling rogram have been reduced to three on the basis of grahical steering. Cooke & Van Noortwijk (999) visualised the effects of the residual random quantities using scatter lots. Using these lots they argued that the most imortant random quantities are the river discharge, the sea water level, and the critical head. n order to calculate the robability of failure due to lifting and iing, we now aly directional simulation to three standard eonentially distributed random quantities that are statistically indeendent. The reason for this is that we can easily transform the river discharge Q days, the sea water level S days, and the critical head in the event of lifting and iing to three standard eonentially distributed random quantities. This can be achieved by alying the transformation Q S days days ( q) ( s) ( h) e{ }, or, equivalently, ln( y ln( w ln( e{ y}, e{ w}, Q S days days ( q)) r sin( θ )sin( ψ ), ( s)) r cos( θ )sin( ψ ), ( h)) r cos( ψ ), (5) (6) where r is the directional radius, and θ and ψ are the directional angles. Since the random quantities Q days, S days, and are indeendent, the standard eonentially distributed random quantities X, Y, and W are also indeendent. ence, since the Jacobian of transformation (6) equals r sin(ψ), the joint robability density function of the directional coordinates (R,Θ,) can be written as

5 f R, Θ, with ( r, θ, ψ ) r sin( ψ ) e{ g( θ, ψ ) r} [0, ) ( r) [0, π / ] ( θ ) [0, π / ] ( ψ ), (7) g ( θ, ψ ) sin( θ )sin( ψ ) + cos( θ )sin( ψ ) + cos( ψ ), (8) where A () if A and A () 0 if A for every set A. Accordingly, the joint robability density function of Θ and becomes sin( ψ ) f Θ, ( θ, ψ ) [0, / ] ( θ ) [0, / ] ( ψ ) 3 π π. (9) rom the robability density function of the random vector (R,Θ,), and the robability density function of the random vector (Θ,), the conditional robability density function of R for fied values of Θ and writes as f R Θ, ( r θ, ψ ) 3 r e{ g( θ, ψ ) r} [0, ) with cumulative distribution function R Θ, ( r θ, ψ ) θ, ψ ) r] + g( θ, ψ ) r + ( r) (0) () e{ g( θ, ψ ) r}. The conditional distribution of R when (Θ,) (θ,ψ) is given, can be recognised as a gamma distribution with shae arameter 3 and scale arameter g(θ,ψ). Given the value of (Θ,), the robability of failure can now be written as Pr{ R > r ( θ, ψ ) Θ θ, ψ} R Θ, ( r ( θ, ψ ) θ, ψ ), () where r (θ,ψ) is the zero of the limit-state function z(q,s,h) described in Section : z( Qdays Sdays (e{ r (e{ r (e{ r ( θ, ψ )sin( θ )sin( ψ )}), ( θ, ψ ) cos( θ )sin( ψ )}), ( θ, ψ )cos( ψ )}) ) 0. (3) Unfortunately, the zero r (θ,ψ) cannot be obtained in elicit form and has to be determined numerically. n order to samle values of (Θ,), both the marginal robability density function of Θ and the conditional robability density function of, when the value Θ θ is given, remain to be determined. After some algebra, the marginal robability density function of Θ can be written as f Θ ( θ ) tan( ψ ) 0 [0, π / ] ( θ ) v( θ ) tan( ψ ) + θ )] θ ) tan( ψ ) + ] θ )] where π ψ 0 sin( ψ ) tan( ψ ) θ ) tan( ψ ) + ] [0, π / ] ( θ ), 3 [0, π / ] 3 ( θ ) dψ [0, π / ] ( θ ) d tan( ψ ) tan( ψ ) 0 (4) v ( θ ) sin( θ ) + cos( θ ) (5) with cumulative distribution function Θ( θ ). (6) + tan( θ ) Accordingy, the conditional robability density function of when the value Θ θ is given follows from Eqs. (9) and (4): sin( ψ ) θ )] f Θ( ψ θ ) [0, / ] ( ψ ) 3 π (7) with cumulative distribution function Θ ψ ( ψ θ ) f f ( θ ) ~ Θ Θ, ψ 0 v( θ ) tan( ψ ) +. θ ) tan( ψ ) + ] ( θ, ~ ψ ) d ~ ψ (8) Let P and P be indeendent and standard uniformly distributed random quantities, then values of the random vector (Θ,) can be generated as follows: θ arctan, ψ arctan [sin( θ ) + cos( θ )]. (9) Note that for two-dimensional directional samling, the robability density function of the directional angle Θ is also Eq. (4). The conditional distribution of the directional radius R, when the directional angle θ is given, is a gamma distribution with shae arameter and scale arameter v(θ). 5 RESULTS AND CONCLUSONS n this section, the results are resented for dike section 3. of the Dutch dike ring the oeksche Waard. The robability of dike failure due to lifting and iing er two days can be calculated using

6 0 5 Directional samling for dike section 3. According to Vrouwenvelder (998), ORM results in a robability of failure due to lifting and iing of ence, both results are quite close to each other. Probability P0 P Number of samles 0 4 igure. The robability of failure of one subsection ( P0 ) and two subsections ( P ) of dike section 3. determined on the basis of 0,000 samles using three-dimensional directional samling. directional samling on the basis of the sea water level, the river discharge, and the critical head in the event of lifting and iing. urthermore, the satial variation and correlation of the critical head in the event of lifting and iing can be modelled using a Markovian deendency structure. This means that the random quantities, reresenting the inherent uncertainties in the critical head, of one dike subsection only deend on the values of the corresonding random quantities in the two adjacent dike subsections. The results of the three-dimensional directional simulation (on the basis of 0,000 samles) and the subsequent dike ring reliability calculations are as follows: REERENCES Cooke, R.M. & van Noortwijk, J.M Uncertainty analysis of inundation robabilities (Phase ). Delft: Delft University of Technology & KV Consultants. Cooke, R.M. & van Noortwijk, J.M Grahical methods for uncertainty and sensitivity analysis. To aear in A. Saltelli, editor, Mathematical and statistical methods for sensitivity analysis of model outut. New York: John Wiley & Sons. Ditlevsen, O. & Madsen,.O Structural reliability methods. Chichester: John Wiley & Sons. Karlin, S. & Taylor,.M A irst Course in Stochastic Processes; Second Edition. New York: Academic Press, nc. Paoulis, A Probability, Random Variables, and Stochastic Processes. Tokyo: McGraw-ill. Vrouwenvelder, A.C.W.M Rekenmethoden voor een robabilistische dijkringanalyse [Calculation methods for a robabilistic dike-ring analysis], TNO-Reort 93-CON- R33. Rijswijk: TNO Building and Construction Research. Vrouwenvelder, A.C.W.M Project onzekerheidsanalyse inundatiekans [Project uncertainty analysis inundation robability], TNO-Reort 98-CON-R0646. Rijswijk: TNO Building and Construction Research. the eected robability of failure due to lifting and iing of one subsection of length 50 metres is ( ) ; the eected robability of failure due to lifting and iing of two adjacent subsections is ( ) ; since the length of dike section 3. is l 00 metres, the eected robability of failure due to lifting and iing of the dike section 3. (l) (note that if dike section 3. would be,000 metres of length, Eq. (4) would lead to a failure robability of ). The iteration rocess which resulted in this failure robability is shown in igure. t clearly illustrates how fast the three-dimensional directional simulation converges! The advantage of three-dimensional directional samling is that large samle sizes are not required (samle sizes of about 0,000 samles already sly satisfactory results). The directional simulation results above have been comared to results obtained using irst Order Reliability Method.