Engineering
2013-2
ISSN 0169-6548
Effect of uncertainty in wave growth calculation on
overtopping over dikes
————————— a case study —————————
Henk Jan Verhagen
*
August 1, 2013
* Associate Professor, Section of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600 GA Delft, The Netherlands.
Tel. + 31 15 27 85067; Fax: +31 15 27 85124 e-mail: H.J.Verhagen@tudelft.nl
Communications on Hydraulic and Geotechnical Engineering
2013-2
ISSN 0169-6548
The communications on Hydraulic an Geotechnical Engineering have been published by the Department of Hydraulic Engineering at the Faculty of Civil Engineering of Delft University of Technology. In the first years mainly research reports were published, in the later years the main focus was republishing Ph.D.-theses from this Department. The function of the paper version of the Communications was to disseminate information mainly to other libraries and research institutes. (Note that not all Ph.D.-theses of the department were published in this series. For a full overview is referred to www.hydraulicengineering.tudelft.nl ==> research ==> dissertations).
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© 2013 TU Delft, Section Hydraulic Engineering, Henk Jan Verhagen
Introduction
After the storm surge disaster of 1953 in the Netherlands the political response was “this should never happen again”. Following the disaster, a scientific committee was formed to give advice on future design of sea defences. This commission concluded that dikes should be designed in such a way that during design conditions the probability of failure should be negligible small [DELTACOMMISSIE, 1961]. In subsequent years this has been elaborated [SCHIERECK, 1998]. For large lakes it means that under design conditions one has to determine the hydraulic boundary conditions and the prove that the dike can survive the given hydraulic load. A “negligible small” probability is later translated as less than 10%.
For each dike in the Netherlands the frequency of the design condition is given by law [Waterwet, 2013]. This frequency is between 1/250 per year to 1/10000 per year, depending on the area. In a book published by the national government the given hydraulic boundary conditions are given [RIJKWATERSTAAT, 2007].
At this moment for classical earth dikes failure is considered to take place when the wave overtopping is more than 1 l/s per running meter [VAN HIJUM, 1999]. Of course this includes some hidden safety because when a dike is overtopped by this amount of water damage of the inner slope is supposed to start, and this is not a full failure of the dike.
This report investigates the effect of uncertainty in wave prediction formulas for the design of dikes. In order to investigate the uncertainty in such wave growth formulae this has been worked out for a case. The selected location is along the Oostvaardersdijk in the Central part of the Netherlands. This dike is facing a NW fetch over a shallow lake (Markermeer), see Figure 1. According to the Waterwet, this dike has to be designed to withstand a storm with a probability of 1/4000 per year. According to RIJKWATERSTAAT [2007] the water level during design conditions is 1 m above NAP (NAP is National Datum, approx. Mean Sea Level. The wave height is not given in this report. Regulation require that for the large lakes the wave height/ water level combination is determined with the computer programme Hydra-M. This programme determines in a probabilistic way the relevant water-level/wave-height combination. However, the wave-height in this program is not determined in a probabilistic way, but using explicit growth function given a wind speed distribution.
In the example worked out in this report the Young and Verhagen formula (YOUNG & VERHAGEN, 1996] is used to determine the wave height from the wind speed.
Intermezzo:
Current research on grass inner slopes of dikes shows that: 1. The value of 1 l/s per running meter is too conservative
2. The effect of overtopping should not expressed in the “mean overtopping discharge” but as a cumulative overload of the critical velocity.
However, because this is not yet implemented in current design practice, this is not incorporated in this report. See also “Handreiking toetsen grasbekleidingen, 2010”
Figure 1: Location of the case study
Basic design of the dike height
The height of a dike is defined by the tolerable quantity of overtopping water. The overtopping discharge is computed with [EUROTOP 2007]:
1,0 1,0 0 0 3 0.067 exp tan c b m m m b z m R q A H gH γ ξ ξ γ γ α − − = − (1) in which:
q overtopping discharge (m3/s per meter)
Hm0 wave height calculated from the spectral energy slope of the dike
ξm-1,0 Iribarren number (=tanα/√H/L0) based on the first negative moment of the spectrum
Rc Crest height above water level
γb Correction factor for berm width
γz Correction factor for other geometrical effects (like approach angle)
A coefficient with a mean of 4.75 and a standard deviation of 0.5; for deterministic calculation a
value of 4.3 is recommended
The wave height is computed with the YOUNG AND VERHAGEN [1996] formula:
1 1 1 tanh tanh tanh n Y B A A A ε= (2)
in which:
ε non-dimensional total energy in the wave spectrum (=g2
E/u4) E total energy in the spectrum (Hm0=4√E)
u wind velocity (measured at 10 m height) A1 0.493 δ
0.75
B1 3.13∙10-3χ 0.57
δ dimensionless water depth = gd/u2 χ dimensionless fetch = gF/u2
n coefficient 1.75 AY coefficient 3.64∙10-3
The coefficients 1.75, 3.64∙10-3 , 0.493 and 3.13∙10-3 were fitted by Young and Verhagen during their
field experiments.
Both formulas are curve fitting formulas with experimentally determined coefficients. For formula (1) the developer of the formula (Van der Meer) has found that the inaccuracy in the formula can be expressed by a mean value for A of 4.75 and a standard deviation for A of 0.5. The value 0.067 can be considered exact. In fact also 0.067 is an approximation, but the uncertainty in this value has been incorporated in the standard deviation for A.
Normal procedure in the Netherlands is that the wind speed at design conditions is taken, the corresponding wave height is computed with formula (2) en subsequently the overtopping with formula (1), using a value of A = 4.3 , as recommended by Eurotop for deterministic calculations. In fact A = 4.3 means A = 4.75 – 0.45 = µ – 0.9 σ. In fact the multiplier 0.9 means that this value will be exceeded by 18% of the cases.
For formula (2) it has not yet been determined what is the standard deviations of the three coefficients. In the thesis work of BART [2013] this work has been done.
The conclusions of this work were:
• All uncertainties can be incorporated in the standard deviation of the coefficient AY; so the other coefficients should not be considered as stochasts in probabilistic calculations; • The variable AY has a normal distribution with mean 3.64⋅10-3 and a standard deviation of
1.08⋅10-3.
Boundary conditions
The 1/4000 wind condition can be determined from wind measurements. The data from Schiphol Airport (some 45 km SE of the site) can be used. A long term dataset is available. For extrapolation usually the Weibull-Rijkaart model is used. There are several variants of this model, as described by SMITS [2001] for Dutch wind data. Smits describes 9 variants. For Schiphol Airport the 1/4000 years wind speed is:
model A B C D E F G H I
speed 34.7 38.5 39.4 34.4 38.1 36.8 31.1 37.4 34.5 Table 1: 1/4000 wind speed in m/s at Schiphol airport.
Al these numbers are similarly good, but vary slightly in extrapolation method. So one may conclude that the expected 1/4000 wind speed has a mean value of 36.1 m with a standard deviation of 2.6 m/s.
The water level at design conditions is given as 1 m above NAP. For this example this value is considered an exact value. In fact this value is computed with a wind set-up formula in which also uncertainties are included. For clarity on the effect of wave growth on dikes, this variation is in first instance neglected.
The depth of the Markermeer is approximately 2.5 m below NAP. There is a small variation. Given the fact that during design conditions there is a wind set-up of 1 m, the average depth is 3.5 m. For both uncertainties in bed location as well as in wind set-up a standard deviation of 0.2 m is considered for the water depth.
The direct distance for the site to the other shore of the lake for waves from NW is 21.2 km. However, However, for NNW this is 22.7 km, and for WNW this is 23.8 km. For a short angle (around 122º) the fetch is 29 km. Although the fetch itself can be calculated very accurate, for the fetch length in the calculation a mean of 22 km and a standard deviation of 2 km is used. This standard deviation in fact includes the effect of variability in wind direction.
Deterministic calculation
In a deterministic calculation wind model I is used, because it is considered the best model. So the wind speed is 34.5 m/s. Using the Young and Verhagen formula this leads to a wave height of 1.49 m. Using a slope of 1:3, no reduction for roughness, berm, etc. and assuming a wave period of T = 4.7 seconds, this leads, using the deterministic overtopping formula to a crest height Rc of 3.85 m above design water level (so 4.85 m above NAP). (Note: in case the overtopping formula is used with a coefficient 4.75, this leads to a crest height of 3.49 m above design water level.)
Probabilistic calculation
When distributions and standard deviations are known, one can also calculate the dike height in a probabilistic way. For this paper computations were carried out using a Matlab based program for probabilistic calculations [DEN HEIJER, 2012]. The source code of the relevant Matlab script is added as annex to this report.
Making a full probabilistic computation with a target of 50% failure leads to a crest height of 3.35 m above design water level. This value should be compared with the 3.49 m found in the deterministic computation. Setting the standard deviations of the coefficients in the overtopping and in the wave growth formula to zero, one finds with a crest height of 3.97 m a probability of failure of 10%. When including the standard deviation in the coefficient of the overtopping formula, this results in a crest height of 4,22 m. When including also the standard deviation of the wave growth formula, the required height becomes 4.37 m above design water level.
Figure 2: composition of the required dike height at this location A FORM calculation gives the following Alpha values for this calculation:
AY -60% u -26% d -11% F -4% A +69% T -17% tan α -22% Rc +8%
Large absolute values of the Alpha value indicate that this variable largely contributes to the uncer-tainty in the computation. The largest values in this table are AY and A, which are the “constants” in the formula. So it means that both “constants” are the weakest point in determining the dike height.
Some comments about the period
In the above calculation the period is considered a variable with a mean of 4.7 and a standard deviation of 0.1 seconds. Because the period and the wave height are partly correlated it is often considered not to use the wave period as a stochast, but the wave steepness. The wave steepness does not depend on the wave height, so one fulfils the requirement of independent variables in a probabilistic calculation. However, in this calculation neither the wave height nor the wave period should be input variables. The wave height has been calculated with formula 2. The wave period can also be calculated with a similar formula derived by YOUNG AND VERHAGEN [1996]:
2 2 2 tanh tanh tanh m Z B A A A ν = (3) in which: 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 deterministic probabilistic di ke h ei gh t a bo ve d es ig n w at er lev el (m )
incertainty wave growth formula
incertainty overtopping formula
incertain boundaries average value
ν non-dimensional peak frequency of the wave spectrum (=fpu/g)
fp peak frequency of the spectrum
u wind velocity (measured at 10 m height) A2 0.331 δ
-1.01
B2 5.215∙10-4χ 0.73
δ dimensionless water depth = gd/u2 χ dimensionless fetch = gF/u2
m coefficient -0.37 AZ coefficient 0.133
Like for the wave height, the coefficient Az can be considered as a stochast with a mean 0.133 and an (unknown) standard deviation. In the present research this standard deviation was not determined, therefore for this example T has been considered as a mean value 4.7 and a standard deviation 0.1 seconds.
Conclusion
Omission of the standard deviation of the constant in the wave growth formula for dike height calculations decreases the safety considerable. On the other hand, in dike height calculation, an overtopping of more than 1 l/s per meter is considered “failure of the dike”, which is also not the case. So, these effects partly compensate for each other to unknown extend. However, when in future failure of a dike is defined more precise, it is extremely relevant to include also the uncertainty in the wave growth formula.
References
BART, L. [2013] The accuracy of the Young and Verhagen formula for waves in water of finite depth. BSc thesis, Delft University of Technology
DELTACOMMISSIE [1961] Rapport Deltacommissie. Deel 1. Eindverslag en interimadviezen. Staatsuitgeverij, Den Haag.
DEN HEIJER, C. [2012] Reliability methods in OpenEarthTools. Comm. on Hydraulic and Geotechnical
Engineering, 2012-01, Delft University of Technology
EUROTOP [2007] Wave overtopping of sea defences and related structures, Environment Agency, Expertise Netwerk Waterkeren, Kuratorium für Forschung im Küsteningenieurswesen, Die Küste, RIJKSWATERSTAAT [2007] Hydraulische Randvoorwaarden primaire waterkeringen, RWS nota, Waterdienst, Lelystad
SCHIERECK, G.J. [1998] Grondslagen voor waterkeren, rapport TAW/ENW (English translation:
Fundamentals on Water Defences)
SMITS, A. [2001] Analysis of het Rijkaart-Weibull model. KNMI publication VAN HIJUM, E. [1999] Leidraad rivier- en meerdijken, rapport TAW/ENW WATERWET [2013] Waterwet
YOUNG, I.R., VERHAGEN, L.A. [1996] The growth of fetch limited waves in water of finite depth. Part I. Total energy and peak frequency, Coastal Engineering, 29:47-78.
Annex
Matlab script for calculation of dike height
function [resultFORM resultMC] = Young_Overtop(varargin)
%YOUNG_OVERTOP probabilistic calculation of overtopping, including %wave height generation using Young&Verhagen formula
%
% Example of probabilistic Monte Carlo and FORM routines applied to the % van der Meer formula.
%
% Syntax:
% [resultMC resultFORM] = Young_Overtop %
% Output:
% resultMC = structure with Monte Carlo results % resultFORM = structure with FORM results
%
% Example
% [resultMC resultFORM] = prob_Young_Overtop %
% See also %% Copyright notice
% ---% Copyright (C) 2009 Delft University of Technology
% Kees den Heijer Henk Jan Verhagen %
% H.J.Verhagen@TUDelft.nl %
% Faculty of Civil Engineering and Geosciences % P.O. Box 5048
% 2600 GA Delft % The Netherlands %
% This library is free software: you can redistribute it and/or % modify it under the terms of the GNU Lesser General Public % License as published by the Free Software Foundation, either % version 2.1 of the License, or (at your option) any later version. %
% This library is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU % Lesser General Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public
% License along with this library. If not, see <http://www.gnu.org/licenses/>. %
---stochast = struct(...
'Name', {
'A'... % [-] Coefficient of Young formula
'u10'... % [m/s] wind speed at 10 m
'd'... % [m] Depth of lake
'F'... % [m] Fetch length
'b'... % [-] coefficient in overtopping formula
'T'... % [s] wave period
'tanalfa'... % [-] slope of structure
'hk'... % [m] crest height of dike
},...
'Distr', {
@norm_inv... % Young coeff
@norm_inv... % wind speed
@norm_inv... % depth of lake
@norm_inv... % fetch lenght
@norm_inv... % coeff in overtop formula
@norm_inv... % slope of structure
@norm_inv... % crest height of dike
},...
'Params', {
{3.64E-3 1.08E-3}... % Young coeff
{36 2.6}... % wind speed
{3.5 0.20}... % depth of lake
{22000 2000}... % fetch length
{4.75 0.5}... % coeff in overtop formula
{4.7 0.1}... % wave period
{0.3333 0.01}... % slope of structure
{4.30 .05}... % crest height of dike
} ...
);
%% main matter: running the calculation % run the calculation using Monte Carlo
resultMC = MC(stochast,...
'NrSamples', 3e4,...
'x2zFunction', @Young_Overtop_x2z);
% run the calculation using FORM
resultFORM = FORM(stochast,...
'x2zFunction', @Young_Overtop_x2z);
%% Z-function
function z = Young_Overtop_x2z(x, varnames, Resistance, varargin)
%% retrieve calculation values
for i = 1:size(x,2) samples.(varnames{i}) = x(:,i); end %% g = 9.81; %[m/s2] n=1.74;
% loop through all samples and derive z-values
for i = 1:size(x,1) AA=samples.A(i); % AA=3.64E-3; depth=samples.d(i); % depth=3.5; fetch=samples.F(i); % fetch=21200; TT=samples.T(i); % TT=4.7; u=samples.u10(i); % u=35.5; delta=(g*depth)/u^2; chi=(g*fetch)/u^2; A1=0.493*delta^0.75; B1=3.13E-3*chi^0.57; epsilon=AA*(tanh(A1)*tanh(B1/tanh(A1)))^n; E=epsilon*u^4/g^2; H=sqrt(16*E); L=1.565*TT*TT; slope=samples.tanalfa(i); % slope=0.3333; freeboard=samples.hk(i); % freeboard=3.85; coeff=samples.b(i); % coeff=4.75; ksi=slope/sqrt(H/L); QQ=0.067/sqrt(slope)*ksi*exp(-coeff*freeboard/(ksi*H*1.0)); q=QQ*sqrt(g*H^3);
z(i,:) = (0.001-q); %failure function for 1 l/s overtopping
% z(i,:) = real(H); % this statement gives distribution of wave height
