Proceedings on the Second International Conference on the Application of Physical Modelling to Port and Coastal Protection
PROBABILISTIC DISTRIBUTIONS OF SEA WAVE PARAMETERS IN THE DIAGNOSIS OF RUBBLE-MOUND BREAKWATERS JOÃO ALFREDO SANTOS (1), RUI CAPITÃO (2) and ISAAC SOUSA (3) (1) Research Officer, LNEC, Av. do Brasil, 101, 1700-066 Lisboa, Portugal.
[email protected] (2) Research Officer, LNEC, Av. do Brasil, 101, 1700-066 Lisboa, Portugal.
[email protected] (3) Research Trainee, LNEC, Av. do Brasil, 101, 1700-066 Lisboa, Portugal.
[email protected]
The probabilistic procedure to simulate armour layer evolution under wave attack is presented. This procedure can be used to design the armour layer of rubble-mound breakwaters and to help in scheduling repair and / or maintenance works in these structures. This paper focuses on one of the key issues in this simulation: the selection of the most adequate distributions for the wave heights and periods to be used in the armour damage evolution formulations. An example of the application of such procedures to the breakwater that protects the fishing harbour of Sines (Portugal) is presented. Keywords: damage evolution; sea state distributions; breakwaters.
1. Introduction The design of the armour layer of rubble-mound breakwaters, as well as the decision-making process on the schedule of the repair and / or maintenance works in the same armour layer, can now be based upon a probabilistic approach. In fact, using the formula proposed by Melby and Kobayashi (1999) for the armour layer damage evolution,
[S (t )] = [S (t )] + (0,011 N ) 4
5 4 s
4
n
t − tn Tm
[1]
where, S (t ) is the damage at time t contained in the interval [tn , t n+1 ] , during which it can be assumed that wave conditions are constant (i.e. significant wave height, H s , and average zero up-crossing period, Tm , do not change) and N s = H s / (∆ Dn 50 ) is the stability number, ∆ being the submerged density of the armour layer material and Dn 50 the median nominal diameter of the same material, it is possible to simulate the armour layer response to any sequence of sea-waves incident on it. If one uses a large number of samples made of sea states 1
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with a given duration, one may estimate the probability of the damage at the end of that period reaching a pre-set level. For the design of the armour layer, the simulation period should be the desired structure life-span and the structure initial status should be the one expected after its construction, whereas for scheduling of repair and / or maintenance works on the structure, the structure's initial status should be the one measured in the last survey of the armour layer envelope and the simulation period should go from that instant up to the next expected survey.
Figure 1. The breakwater of Sines fishing harbour
From the above, it is clear that one of the key points in the use of Melby and Kobayashi formula is the correct simulation of the sea-state parameters' sequences. This paper illustrates the use of that formula in the simulation of the damage evolution of the armour layer of Sines fishing harbour, Figure 1. 2. Damage evolution simulations The first problem to be addressed in the damage evolution simulations is the unrealistic continuous damage increase predicted by equation [1] for very low energy sea-states that may last for long periods. Sousa (2007) solved that problem by using the critical stability number concept presented in Smith et al. (1999). Then equation [1] becomes 4 t − tn 4 S (t ) = [S (tn )] + 0,011 N s5 Tm
(
)
S (t ) = S (tn ) if N s ≤ N c
0.25
if N s > N c
[2]
So, when the stability number, N s , falls below the critical value, N c , there is no damage increase. This critical stability number depends on the armour layer porosity and slope as well as on the wave slope at the structure location. It is not difficult to infer from the above equation that the damage increase at the end of 2
Proceedings on the Second International Conference on the Application of Physical Modelling to Port and Coastal Protection
any period is independent from the order by which the sea-states acted on the structure during that period. This somewhat simplifies the simulation procedure. By transferring the wave characteristics measured offshore with a directional waverider buoy deployed at the SINES 1D location (Figure 1) up to the breakwater location, Sousa (2007) determined the three-hourly time series of significant wave heights, H s , and of average zero up-crossing periods, Tm . Probability distributions were fitted to these variables and so using those distributions and a sampling procedure, several sequences of significant wave heights and of average zero up-crossing periods were produced and the damage evolution curves for five of those sequences as well as the average of the results for the one hundred sequences generated are presented in Figure 2. This figure shows that the damage increment occurs at discrete events. It is particularly remarkable that most of the damage is the result of outstanding events. It can be also seen in that figure, specially in the average curve, that there is a trend for the damage to slow down as the armour layer becomes damaged. This can be interpreted as approaching an equilibrium state that, however, is never attained.
Figure 2. Damage evolution in five 100-year simulations and average of the 100 simulations performed with an armour layer made of 45 kN rock elements
3.
Distribution fitting
The definition of multivariate probability distribution functions for a better representation of the sea states necessary to correctly simulate the damage evolution of the rubble-mound structures is a problem still to be solved. Usually, the probability distributions that are fitted to the significant wave height and the average zero upcrossing period assume absence of any correlation between wave periods and the wave heights. This is not always true since there usually exists some degree of dependency between wave heights and wave periods - the greater the wave height the greater the wave period (positive correlation). Therefore, the use of multivariate probability distributions based on the adopted univariate 3
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(marginal) distributions of both variables is necessary to enable a better description of the correlation that usually exists between those variables. Guedes Soares & Cunha (2000) have studied, and generalized, the application of univariate models of the long-term time series of significant wave height to the case of the bivariate series of wave height and mean period. This was done considering measured data from Figueira da Foz buoy, to which bivariate auto-regressive models were successfully applied enabling them to describe the time evolution of the occurrence of pairs of values of wave heights and mean periods, taking into account memory effects they identified in the process. The concept of copula is another possible answer to this problem since it may be defined as a function to describe dependencies among variables and to provide a way to create distributions that model correlated multivariate data will be considered. De Michele et al. (2007) have defined a multivariate model of sea storms, characterized in terms of wave height, peak period, storm direction and storm duration, using copulas. In order to produce a good model of the multivariate probability distribution of the variables wave height and wave period, a good choice of the distribution functions of each variable has to be achieved. Therefore, univariate probability distributions are firstly fitted to the wave variables of interest. Then, a sampling procedure is defined and univariate distributions are found by fitting the available distributions to the several sequences of the significant wave heights and of the average zero up-crossing periods. To do this, use of wave data of Sines offshore buoy “SINES_1D” is considered. References Guedes Soares, C., Cunha, C. 2000. ‘Bivariate autoregressive models for the time series of significant wave height and mean period’. Coastal Engineering 40, pp 297-311. De Michele, C., Salvatori, G., Passoni, G., Vezzoli, R. 2007. ‘A multivariate model of sea storms using copulas’. Coastal Engineering 54, pp 734-751. Melby, J. A.; Kobayashi, N. 1999. ‘Damage Progression and variability on´breakwaters trunks’, Proc. Coastal Structures ‘99, pp. 309-315. Smith, W. G.; Kobayashi, N.; Kaku, S. 1992. ‘Profile Changes of Rock Slopes by Irregular Waves’, Proc. 23rd Coastal Engineering Conference, Volume 2, pp. 1559-1572. Sousa, I.A. 2007. ‘Damage progression of rubble-mound breakwaters armour layer. Evaluation using level III probabilistic methods’ (in Portuguese), Instituto Superior Técnico, Lisboa.
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