Reliability assessment of offshore dynamic scour protections using copulas

Failure modes of scour protections

In offshore wind turbines, the most common type of protection consists of rock materials placed around the foundation (rip-rap). The design of a scour protection generally implies the definition of four key elements (Zaaijer and Tempel, 2004). First, the grading of the rock material; second, the definition of the thickness of the armour layer, the grading and thickness of the filter layer and finally, the protection’s extent.

Those elements can be associated with a different failure mode of the scour protection, as presented in De Vos (2008): (a) erosion of the top layer, (b) loss of subsoil through the scour protection, (c) edge scour and (d) flow slide (Figure 1).

OPEN IN VIEWER

Figure 1.

Failure modes of scour protections.

This research is focused on the first failure mode, that is, the erosion of the top layer. This failure mode depends on the dimensions of the rock material employed in the armour layer. Usually, the rock material is defined by the mean stone diameter, D₅₀, which is the side of the sieve with square openings through which 50% of stones pass (by weight). In fact, the stone size and the weight play a major role in preventing the displacement of the stones. The mean nominal diameter of the stones is applied as D_n50/D₅₀ = 0.84 (CIRIA et al., 2007).

Failure criterion of static and dynamic scour protections

When no movement is allowed for the top layer stones, the scour protection is considered as statically stable. The design of these protections is most commonly focused on empirical methodologies (De Vos et al., 2011; Whitehouse, 1998). However, one can opt to design a dynamically stable scour protection (Schoesitter et al., 2014; Whitehouse et al., 2014). In dynamic scour protections, the idea is to allow for some movement of the stones, without overcoming an acceptable damage number. This design enables one to use smaller stones, thus leading to the reduction of the material, the transportation and the construction costs, provided that the armour thickness remains the same.

There are several design methods that are suitable for steady current scour, often related to scour at bridge piers. For example, Melville et al. (2000), Agrawal et al. (2007) and Prendergast and Gavin (2014) provide an extensive review of such methods and the nature of scour protections under currents. However, for combined wave and current, the design of scour protections still presents a reduced number of methods fully developed (De Vos et al., 2011; den Boon et al., 2004). Usually, the design of a statically stable scour protection is based on the criteria of the threshold of motion, originally introduced by Shields (1936). Then, physical models are used to validate the proposed design. Shields (1936) defined the Shields’ critical parameter θ_cr for the threshold of motion as in equation (1)

θ c r = τ c r g ( ρ s − ρ w ) d s = u * σ 2 g Δ d s

(1)

This parameter depends on the critical shear stress (τ_cr), the sediments diameter (d_s), the gravitational acceleration (g), the density of the sediments (ρ_s) and the water (ρ_w). Alternatively, one can use ∆ = (ρ_s − ρ_w)/ρ_w and the critical shear velocity (u_*cr). The critical Shields’ parameter sets the threshold of motion at which non-cohesive particles are displaced by the flow. The same is to say that for a specific rock material, if the critical Shields’ parameter is achieved, then the protection’s units are displaced, leading to protection’s failure. This means that the maximum wave and current-induced shear stress (τ_wcmax) cannot exceed the critical shear stress of the armour material. The formulation of the previous criterion can be seen in equation (2) that leads to equation (3)

τ w c max < τ c r

(2)

τ w c max < θ c r g ( ρ s − ρ w ) d s

(3)

Shields’ formula has been adjusted by several authors for combined wave and current (Fredsøe and Deigaard, 1992; Soulsby, 1997, among others). The use of this criterion for static stability is not the aim of this article. A major difficulty related with this criterion is the assessment of the maximum bed shear stress under wave and current environment (τ_wcmax). This occurs because the resulting shear stress is not a linear sum of their separate influence. Further details on how to compute the combined wave- and current-induced shear stress are summarised in De Vos et al. (2012) and are fully presented in the works of Fredsøe and Deigaard (1992) and Soulsby (1997).

Looking for a cost-effective alternative to the statically stable scour protections, the OPTI-PILE project (E-Connection, 2002–2004) conducted a physical model study on a Froude scale 1/47.25 to analyse the feasibility of dynamic scour protections. In the OPTI-PILE project, the scour tests were classified into three damage categories (De Vos et al., 2012): no movement of the rocks (static scour protection), some movement but no failure (dynamic scour protections) and failure. The stability parameter (stab) was introduced in order to assess the damage level of protections (equation (4)). This stability parameter assumes the critical Shields’ parameter equal to 0.056, while the maximum Shields’ parameter is defined as in equation (5) (De Vos et al., 2011)

s t a b = τ w c max τ c r

(4)

θ max = τ max ρ w g ( s − 1 ) D 50

(5)

The transition between movement and no movement was obtained for stab = 0.415. The transition between movement without failure and with failure was obtained for stab equal to 0.460. Furthermore, De Vos et al. (2012) noticed that the stability parameter failed to accurately predict the damage levels of a new test series (Froude scale 1/50). The author pointed that this parameter tended to underestimate the damage in the protection, particularly for large current velocities (U_c > 0.2 m/s, scale 1/50), for large wave peak periods (T_p = 1.7 s, scale 1/50) combined with small stone sizes and for waves opposing currents. Underestimations were also found for high-density stones and small wave heights. The same study showed that overestimations occurred for the large stone sizes, depending on the wave peak period.

Although the stab parameter set the first steps to implement a dynamically stable design, the parameter was still very much based on the threshold of motion criterion. Therefore, this approach still presented some of the problems related with the approximations used to evaluate the shear stress under combined waves and currents or the limit considered for θ_cr which directly leads to differences in the assessment of the damage level.

More recently, De Vos et al. (2012) suggested a new predictive formula for dynamic scour protections based on the concept of the damage number (S_3D). Despite the small range of hydrodynamic conditions and the reduced number of armour stones tested (D_n50 = {3.5; 5; 7.2} mm, scale 1/50), a total of 85 scour tests showed promising results (see De Vos et al., 2012 and De Vos, 2008). Also, Schoesitter et al. (2014) and Whitehouse et al. (2014) conducted a physical model study with 23 new scour tests and obtained dynamically stable scour protections according to this parameter. The non-dimensional damage number, S_3D, is obtained as in equation (6)

S 3 D , p r e d i c t e d N b 0 = a 0 U m 3 T m − 1 , 0 2 g d ( s − 1 ) 3 2 D n 50 2 + a 1 ( a 2 + a 3 ( U c w s ) 2 ( U c + a 4 U m ) 2 d g D n 50 3 2 )

(6)

where N is the number of waves, w_s is the fall velocity, d is the water depth and U_m is the bottom orbital velocity calculated from the wave spectrum as in De Vos et al. (2012). The U_c is the current velocity. T_m-1,0 is the energy spectral wave period, calculated from the spectrum’s moments m₋₁ and m₀ or simplified by the relationship T_p = 1.107T_m−1,0 for a JONSWAP spectrum with a peak enhancement factor of γ = 3.3. The b₀, a₀, a₂ and a₃ parameters are determined through regression and are, respectively, equal to 0.243, 0.00076, −0.022 and 0.0079. The parameters a₁ and a₄ depend on the hydrodynamic conditions and are obtained by equations (7) and (8), respectively. U_r is the Ursell number

a 1 = { 0 f o r U c g D n 50 < 0 . 92 a n d c o l l i n e a r w a v e s a n d c u r r e n t s 1 U c g D n 50 ≥ 0 . 92 o r o p p o sin g w a v e s a n d c u r r e n t s

(7)

a 4 = { 1 f o r c o l l i n e a r w a v e s a n d c u r r e n t s U r 6 . 4 f o r o p p o sin g w a v e s c u r r e n t s

(8)

When developing the damage number formula, the protection was considered to have failed when the filter was exposed over a minimum area of four armour units 4 D 50 2 . This was the same criteria adopted for statically stable scour protections in den Boon et al. (2004). De Vos et al. (2012) found that for S_3D > 1, the scour protections had failed. Still, the original results showed that the acceptable damage level was not immediately clear. Thus, further research is needed to clarify the boundary between the acceptable S_3D and the failure of the protection. In the original study, the predictive formula is validated in comparison with the measured values of the damage number in each test.

The methodology used to measure the damage number in the physical model is fully detailed (De Vos, 2008). In summary, to obtain the measured damage number, the scour protection is divided into sub-areas, each one with an area equal to the cross section of the monopile, that is, A s u b = π D p 2 / 4 . The damage number in each sub-area is obtained as in equation (9), where V_e is the eroded volume, measured with a three-dimensional (3D) laser profiler, and Dp is the pile diameter. Then, the representative S_3D of the scour protection corresponds to the maximum measured damage number of all sub-areas (equation (10)). In Figure 2, the original comparison between the measured values after 3000 waves of S_3D (equation (10)) and the predicted ones (equation (6)) is provided. The damage number is non-dimensional

S 3 D , s u b = V e D n 50 π D P 2 4

(9)

S 3 D , m e a s = max i = 1 t o # s u b − a r e a s ( S 3 D , s u b , i )

(10)

OPEN IN VIEWER

Figure 2.

Measured and predicted damage numbers.

The damage number represents a recent optimisation of scour protections. Moreover, it has the advantage of avoiding the calculation of the combined wave- and current-induced bed shear stress. Also, note that the S_3D parameter is more ‘distant’ from the criterion of the threshold of criterion than the stab parameter. To have a meaningful estimate of the probabilities of failure, one would have to perform literally thousands of scour tests. This is unfeasible when dealing with physical models of marine scour protections, due to the cost and the number of hours needed per test, until the equilibrium profile is reached. Therefore, the failure criterion adopted in the present reliability study consisted of the comparison of the predictive formula (equation (6)) with the suggested limit S_3Daccept = 1, which leads to equation (11).

The predictive formula combines several variables related to the loads acting on the protection, for example, U_m, U_c or T_m−1,0, or to the structural resistance of the protection, for example, D_n50 or ρ_s. Therefore, and despite the fact that it is being compared with a somehow deterministic limit (S_3Daccept = 1), the formula does account for the variability in the loads and the resistance parameters of the protection, hence being a suitable starting candidate to assess the reliability of a dynamically designed scour protection

N b 0 ( a 0 U m 3 T m − 1 , 0 2 g d ( s − 1 ) 3 2 D n 50 2 + a 1 ( a 2 + a 3 ( U c w s ) 2 ( U c + a 4 U m ) 2 d g D n 50 3 2 ) ) < 1

(11)

The lack of data regarding the scour and other phenomena related to the design of offshore wind turbines, and their foundations always affect the choice, thus influencing the outcome of the probabilities. Likely due to confidential policies followed by the offshore wind industry, De Vos et al. (2012) point out that not many results are reported in the literature regarding the methods and the validation models, thus leading to a very restricted group of people who have the knowledge and experience to design these protections. In this sense, this article also contributes to the discussion of a common ground to build a framework for reliability assessment of rip-rap systems.

Modelling significant wave heights and peak spectral periods

A proper simulation of the limit state function implies that both the random wave heights and periods are generated according to their structure of dependence. These random variables must be modelled by a joint cumulative distribution function (CDF), thus leading to reasonable pairs of H_s and T_p. In order to build a multivariate (in this case, a bivariate) model, it is important to address the measures of dependence and the independence assumption.

Moreover, the joint model should be compared with other reference models, for example, the independent and the non-parametric models. Once the comparison is established, the designer should properly justify his choice, bearing in mind that the model suitable for a specific case might not be adequate in others.

Measures of dependence

In order to apply a copula-based model, it is common to use Kendall’s tau (τ_k) to represent the dependence between variables (Mahfoud, 2012). Details on this measure of dependence and alternative ones, as the Spearman’s rho, are given in Durante and Sempi (2015). Consider two independent and identically distributed random vectors (X1;Y2) and (X2;Y2), each with the same joint modelling CDF. Kendall’s tau (τ_k) can be defined as the difference of the probabilities of concordance and discordance (equation (16)) (Montes-Iturrizaga and Heredia-Zavoni, 2016)

τ K = P [ ( X 1 − X 2 ) ( Y 1 − Y 2 ) > 0 ] − P [ ( X 1 − X 2 ) ( Y 1 − Y 2 ) < 0 ]

(16)

Introduction to copula-based models

As in other cases of maritime engineering, the existent correlation between the spectral parameters (H_s; T_p) is one of the most pronounced. In this research, a straightforward application of the so-called Copulas-based models is presented to model the correlated sea-state parameters, in order to assess the probability of failure of dynamic scour protections. Full details on the existing numerous copula families existent can be found in Nelsen (2006).

Copulas are functions that couple multivariate distribution functions to their marginal distributions (Montes-Iturrizaga and Heredia-Zavoni, 2016). These functions have uniform one-dimensional (1D) margins on the interval [0;1] and are invariant under monotone increasing transformations of the marginals (Nelsen, 2006). The main advantage of copulas is that they enable one to separate the marginal behaviour and the dependence structure of the variables from their joint distribution function (Mahfoud, 2012).

Consider X as a vector of random variables with marginal distribution functions defined by F(X_i), with i = 1, …, d. The transformation U_i = F(X_i) is a dependent uniformly distributed vector of random variables, with u = (u₁, …, u_d) on the space [0,1]^d. If F(X_i) are continuous, the joint distribution function of X can be expressed as in equation (17) (Montes-Iturrizaga and Heredia-Zavoni, 2016)

F ( x ) = C ( F 1 ( x 1 ) , … , F d ( x d ) ) = C ( u 1 , … , u d )

(17)

where C(u) is the copula of the distribution, C:[0,1]^d → [0,1] and u = (u₁, …, u_d). Equation (30) was originally introduced as Sklar’s theorem (Nelsen, 2006). The copula C(u) and the correspondent copula density c(u) can also be defined as in equations (18) and (19), thus leading to the joint distribution of X given by equation (20)

C ( u ) = F ( F 1 − 1 ( u 1 ) , … , F d − 1 ( u d ) )

(18)

c ( u ) = ∂ d C ( u 1 , … , u d ) ∂ u 1 , … , ∂ u d

(19)

f ( x 1 , x 2 , … , x d ) = c { F 1 ( x ) , F 2 ( x 2 ) , … , F d ( x d ) } ∏ i = 1 d f i ( x i )

(20)

Note that f_i(x_i) is the marginal of x_i. Therefore, the joint distribution function of X corresponds to the combination of each marginal in X and the information on the dependence structure, which is retained by the copula function.

The presented study focused on two families of copulas that are commonly used for bivariate modelling, namely, the Archimedean and the Elliptical family of copulas. Moreover, an independent copula and the more flexible Tawn type 2 copula were included in the analysis of H_s and T_p. Copulas are built on parametric estimations, which in this study are based on τ_K. Equation (21) is used to estimate copula’s parameter (α) (Embrechts et al., 2001). C(u,v) depends on α and the copula families can be constructed as in Durante and Sempi (2015) based on φ(α), which is the so-called generator function

τ K = 4 ∫ 0 1 ∫ 0 1 C ( u , v ) d C ( u , v ) − 1

(21)

Independent copula

Considering two random independent variables X and Y with u = F(X) and v = G(Y), one can define the independent copula as in equation (22)

C ( u , v ) = ∏ ( u , v ) = u v

(22)

Elliptical copula

Although the class of elliptical distributions provides a wide range of multivariate distributions (Durante and Sempi, 2015), the present work used the Gaussian copula and the t-copula. If Φ⁻¹(u) and Φ⁻¹(v) define the quantile functions associated with the normal standard distribution function Φ, with the correlation matrix R, then the Gaussian copula is defined by equation (23)

C ( u , v ) = Φ R ( Φ − 1 ( u ) , Φ − 1 ( v ) )

(23)

The correlation matrix R is obtained with the Pearson coefficient between u and v. Similarly, the t-copula (equation (24)) can also be obtained from the multivariate version of the student t distribution

C t ( u ) = t R , v ( t v − 1 ( u ) , t v − 1 ( v ) )

(24)

Archimedean copulas

The Archimedean copulas are commonly used due to their straightforward application and many useful properties (Montes-Iturrizaga and Heredia-Zavoni, 2016). An Archimedean copula C(u,v) exists if there is a convex, decreasing generator function φ:[0,1]→[0, ∞[ such as φ(1) = 0. Moreover, this family presents the generic formulation (equation (25))

C ( u , v ) = φ − 1 { φ ( u ) + φ ( v ) }

(25)

In the Archimedean copulas, the relationship with Kendall’s tau is obtained as in equation (26)

τ K = 1 + 4 ∫ 0 1 φ ( t ) φ ( t ) d t

(26)

This work focuses on the application of Frank, Gumbel and Clayton copulas usually identified as the most common ones from the Archimedean family. Gumbel copula has upper tail dependence, and the Clayton one has lower tail dependence. Frank copula has neither lower nor upper tail dependence. The Gumbel copula (equation (27)) has a parameter α restricted on [1, ∞[, while Clayton copula’s parameter varies from]−1; ∞[\0 and is defined by equation (28). Finally, the Frank copula is obtained from equation (29) for any parameter α in ]−∞;∞[\0

C ( u , v ) = exp ( − [ ( − log ( u ) ) α + ( − log ( v ) ) α ] 1 α )

(27)

C ( u , v ) = [ max { u − α + v − α − 1 } ; 0 ] − 1 α

(28)

C ( u , v ) = − α − 1 log ( 1 + ( e − α ⋅ u − 1 ) ( e − α ⋅ v − 1 ) ( e − α − 1 ) )

(29)

Table 1 provides the association between the parameter and Kendall’s tau. Moreover, the upper and lower tail dependence coefficients (respectively, λ_U and λ_L) for each Archimedean copula are included.

OPEN IN VIEWER

Table 1.

Relation between Kendall’s tau and the tail dependence coefficients for Archimedean copulas.

Copula	τK	λU	λL
Gumbel	α/(α + 2)	0	2−1/α
Clayton	1 −α−1	2 − 21/α	0
Frank	1 + 4[D1(α) − 1]/α	0	0
D k ( α ) = k α k ∫ 0 α t k exp ( t ) − 1 d t f o r k = { 1 , 2 }

Source: Adapted from Mahfoud (2012).

Tawn type 2 copula

The Tawn type 2 copula can be considered as an asymmetric extension of the Gumbel copula with three parameters. Therefore, it is an extreme value copula. This copula becomes exchangeable when δ = ρ. When δ = ρ = 1, this copula matches the Gumbel one. The parameters are defined in the following domain α ∈ [1,+∞[, δ ∈ [0,1] and ρ ∈ [0,1]. Usually, Tawn type 2 copula presents two versions, which have one of the asymmetry parameters fixed to 1, so that the corresponding copula density is either left- or right-skewed (in relation to the main diagonal). In this study, the Tawn type 2 (right-skewed) copula was implemented (equation (30)) with R package ‘VineCopula’ (Schepsmeier et al., 2017)

C ( u , v ) = exp { ln ( u ) 1 − δ + ln ( v ) 1 − ρ − [ ( − δ ln ( u ) ) α + ( − ρ ln ( v ) ) α ] 1 α }

(30)

Significant wave heights and peak spectral periods

Figure 3 provides the scatter diagram of the 124-month dataset (Hs; T_p). A first look of the diagram shows that a log-normal could be expected, particularly regarding wave heights. When dealing with wave heights, as H_s increases, the number of occurrences tends to decrease, since the variable is moving towards an extreme value. However, in the present case, near H_s = 6 m, a concentration of occurrences seems to exist. The majority of the values are placed between 0.14 and 3 m for significant wave heights and between 2 and 15 s for the peak periods. Figure 3 indicates that smaller wave heights are correlated to smaller peak periods. The wave height increases with the period until a certain extent. The concentration of occurrences near the 6 m, with peak periods ranging from 10 to 15 s, leads to a small peak in the histogram’s upper tail of H_s. Such behaviour may contribute to a worse goodness-of-fit, when attempting to fit the theoretical distributions. There was no evidence that such concentration of occurrences corresponded to an outlier behaviour. Therefore, they were included in the analysis of the marginal distributions.

OPEN IN VIEWER

Figure 3.

Scatter diagram and histograms of the 124-month dataset (H_s; T_p).

Several distributions were tested both for the significant wave height and the peak period. Also, the Kolmogorov–Smirnov and the Wasserstein distances (Villani, 2009) were used to assess the goodness-of-fit associated with each marginal. Table 6 provides the Kolmogorov–Smirnov (Basu et al., 2011) and the Wasserstein distances (Villani, 2009) referred to the theoretical distribution assumed and the empirical CDF.

OPEN IN VIEWER

Table 6.

List of parameters for the tested univariate distributions applied to the spectral parameters.

	Hs	Tp	Hs	Tp
	Kolmogorov–Smirnov		Wasserstein
Normal	0.1168	0.0948	0.2173	0.5983
Exponential	0.2133	0.3402	0.3464	2.5045
Rayleigh	0.1130	0.1385	0.1824	0.6344
GEV	0.2223	0.1711	0.3706	1.1154
GP	0.1716	0.2892	0.2285	1.6682
Log-normal	0.0071	0.0176	0.0162	0.1293
Weibull 2p	0.0594	0.0783	0.1149	0.4930
Weibull 3p	0.0594	0.0569	0.1249	0.4375

The log-normal distribution is the one providing the minimum distances for H_s and T_p, therefore, indicating this is the most reasonable distribution among the tested ones. Regarding the log-normal distribution, it was found that the best fit for the present dataset was obtained for the following parameters: log-mean (H_s) = 0.193; log-standard deviation (H_s) = 0.612; log-mean (T_p) = 1.901 and log-standard deviation (T_p) = 0.393.

A final note on this matter should be made concerning the limitations of the distributions’ ability to accurately describe the physics of waves at the location of the scour protection. When fitting a theoretical statistical model to describe the wave’s behaviour, it must be noted that the distribution chosen for the wave heights must be limited on the upper side. This is important because the random variables generated must still respect the physical constraints of the natural phenomena.

One of the most important aspects of ocean waves is the fact that their height might be limited by the water depth and bathymetry. In the present case, a simplified limit of H/d ⩽ 0.78 was adopted, in order to avoid unreasonable wave heights, randomly generated in the upper tail of the log-normal distribution. The factor of 0.78 is in fact considerably conservative, since Coastal Engineering Manual (CEM; see US Army Corps Engineers, 2002) recommends this value for regular waves. The same source applies a factor of 0.6 for irregular waves. Moreover, this limit is applicable to the maximum wave height at a certain depth d (m). Here, the analysis is focused on the significant wave height. Therefore, considering the 0.78 limit is expected to contribute to a conservative assessment of the probabilities of failure.

Current velocity

The area is dominated by the tidal currents, which present a bimodal north-south directional distribution and large coherence between surface and bottom currents with the lack of annual variations. In normal conditions, the surface currents range from 0.2 to 0.4 m/s (DMI, 2013). According to DMI (2013), the wind-induced currents in connection with surges are responsible for the extreme events. It was not possible to study the correlation or dependence measures between the waves and the current velocity due to the lack of data available. It was assumed that the currents were independent from waves, which is indeed a simplification of the probabilistic model.

Further studies should be performed to assess the influence of such correlation on the probabilities of failure. Moreover, the interaction between the wave direction and the currents or even their seasonality should be the aim of further research. Note that this could also contribute to a more accurate description of the spectral parameters. Since the interaction between strong currents and waves leads to changes in wave height and wave period (DNV GL, 2017a).

DMI (2013) reports that often the waves and current are orthogonal to each other. This is an important aspect, as the damage number proposed in DMI (2013), the waves and the currents are either collinear or opposing ones. Orthogonal directions are not considered in the present failure criteria. Nevertheless, research shows that scour and flow velocities might be enhanced in orthogonal cases (Miles et al., 2017). Here, a simplified assumption was made considering only a positive (collinear) or negative (opposing) U_c. The analysis could be further improved to consider the different directions associated with both variables.

A Weibull distribution was applied to model the variable U_c. In the existence of historical records, one should study the possible marginal distributions as previously done for the spectral parameters. Taking into consideration the values reported in DMI (2013), one could say that the annual mean would be between 0.2 and 0.6 m/s. The division between both classes was assumed as a suitable choice for the mean current velocity (µ_Uc = 0.4 m/s). Since some values occur above the 0.6 m/s threshold, the standard deviation was assumed as σ_Uc = 0.2 m/s. Note that more values occur between 0 and 0.2 m/s than above 0.6 m/s, and assuming these values, one can derive the correspondent parameters of the Weibull distribution. Here, the scale parameter is λ = 0.453 and the shape parameter is κ = 2.123. For dynamic scour protections, once the random current velocities are generated, the collinear or opposing situation is defined by attributing a random factor of 1 or −1, respectively.

Water depth

In this case, a reference level of 18 m was adopted based on Figure 4. In normal conditions, depending on the tide and the wind, the sea level may vary from −1 to 1 m. Nevertheless, DMI (2013) records showed that a maximum variation of 2.40 m has occurred during the 2003–2013 period, with several other occurrences exceeding 1.5 m. This includes the effects of the astronomical tides, which present a maximum range of 0.62 m, above and below the mean surface level relative to Geodetic Reference System 1980 (GRS80). The water depth influences not only the scour severity but also the hydrodynamic parameters used in the failure criterion. A detailed study on this variable should be performed in real design situations. However, in benefit of a more parsimonious model, a deterministic value of 18 m is used for the case study presented further.

OPEN IN VIEWER

Figure 4.

Water depth at several locations in Horns Rev 3 (DMI, 2013).

Rock size (D_n50)

Often, the rock size not only depends on the nearby availability or the transport costs, but it may also depend on technical aspects, such as the size of the available pipe vessels (Schoesitter et al., 2014). Therefore, the optimum solution is a result of multiple factors, which do not concern solely the size obtained according to a design criterion. Typically, the criterion establishes the minimum D_n50 that ensures the stability of the armour layer (static or dynamic), associated with a particular specific weight of the rubble mound material. In this case, this variable is modelled with a certain degree of variability since the rock material will not present the same size for all units. However, the rock material is ordered for a specified grading curve, which means that very large deviations are unlikely to occur.

This research assumes that the diameter of the rock material ranges from 0.3 to 0.5 m, as indicated in Energinet (2014). Another important aspect is the uniformity parameter of the material, here obtained as σ_U = D₈₅/D₁₅, which also influences the results of the design criterion (De Vos, 2008). The value of σ_U can be interpreted as a measure of the variability of the diameters used in the armour layer.

One can assume that the scour protection is designed for a target D₅₀ and that the rock material should be as close as possible to this target value. In this perspective, the natural choice relies on using a uniformity parameter equals to 1, implying a perfectly uniform material. However, note that if one assumes that the range [0.3; 0.5] m can correspond to the D₁₅ and D₈₅ limits, this would imply that σ_U = 1.67, which can still be considered close enough to assume that the material is uniform (Fazeres-Ferradosa, 2012). In the present case, σ_U is defined as 1.67. The D₅₀ is modelled for a µ_D50 = 0.4 m. In order to ensure that the uniformity parameter is respected, a triangular distribution was adopted so that [D₁₅; D₈₅] = [0.3;0.5] (m). This leads to a triangular distribution centred at 0.4, which has a lower limit of 0.179 m and an upper limit of 0.621 m.

Measures of dependence

Considering the 124-month dataset of H_s and T_p, the correlation between variables is given by τ_k = 0.3452, that is, there is a positive correlation, that is, large significant wave heights are generally associated with large peak periods. Note that small wave heights (say up to 2 m) present a larger range of peak periods when compared with the large wave heights (say above 4 m). As the significant wave increases, the variability of peak periods decreases. In terms of the scour protection, one expects that the small wave heights contribute less to failure situations, because they lead to low values of the orbital bottom velocity (U_m). However, the effect of peak periods might be harder to assess, because their influence is reflected in the numerator of the damage number and simultaneously in the denominator of U_m.

Despite the positive correlation showed by both dependence measures, the correlation at the upper tail (large wave heights and periods) may influence the probabilities of failure in a way which is not reflected by τ_k. Usually, when dealing with the tails’ behaviour, the tail dependence coefficients are a more reliable measure (Nelsen, 2006), as it will be discussed in further sections. The failure of the scour protection is usually associated with the occurrence of extreme wave heights. However, the occurrence of low wave heights with shorter or longer peak periods still influences the probability of failure. This occurs because the probability is dependent on the proportion of the extreme events and the so-called ‘normal sea-state conditions’. The first main idea to withdraw from the value of τ_k is that the magnitude of this correlation is being affected by the dispersion associated with T_p and H_s.

No peak over threshold was defined to limit the copulas’ application to the extreme waves’ occurrences, because the present research aims at quantifying the probabilities of failure of the scour protection for the 124-month scenario, without excluding any data. Note that the approximately 10 years of data available, in a typical approach as the annual block maxima, leads to a dataset of 10 pairs, which is an irrelevant quantity for reliability purposes. When looking for extreme wave heights associated with a specific return period, say H_100years, one could indeed filter the dataset to extreme data alone. This is not the case when looking for the probability of failure for a scour protection based on the 124 months.

This presents a τ_K, which is similar to the ones presented in other works; for example, Corbella and Stretch (2013), Dong et al. (2015), Vanem (2016), Montes-Iturrizaga and Heredia-Zavoni (2016) and Jane et al. (2016) reported values of τ_K for several locations, ranging from 0.21 to 0.8. Ultimately, the measures of dependence may vary depending not only on the site characteristics but also on the data available. For further analysis, the measures of dependence were chronologically calculated for blocks of 62 and 31 months (referred to as datasets), and the values are summarised in Table 7. One can see that the first chronological half (62a) of the sample presents higher dependence measures than the second one (62b). As expected, the overall sample (124) captures the influence of the data’s first and second half, meaning that the τ_K is placed between the ones presented by the several blocks.

OPEN IN VIEWER

Table 7.

Kendall’s τ per dataset.

No. of months	124	62a	62b	31a	31b	31c	31d
τK	0.3452	0.3642	0.3231	0.3625	0.3659	0.3174	0.3289

Since the measures of dependence are varying depending on the dataset, for the reliability assessment, this naturally means that the copula parameter will not be the same for each dataset. These differences may lead to changes in the assessment of the probability of failure. It is important to quantify such differences since for real design cases the data concerning the spectral parameters might not be abundant.

Reference models – independent copula and BKDE

As reference models, the independent copula and the BKDE method were applied. The independent copula assumes that H_s and T_p follow a log-normal distribution with the previously mentioned parameters, but without any correlation. This will be crucial to assess the probabilities of failure under the independence assumption. It is expected that this simplification provides different probabilities of failure, when compared to the other models that account for the correlation between the spectral parameters.

This research identified a literature gap in the reliability assessment and design of scour protections, namely, the ones designed for marine environment with a dynamic criterion. However, the independence of other environmental variables and parameters has been studied in several works related to current-induced scour. For instance, Chang and Tung (1994) and Muzzammil and Siddiqui (2009) performed a reliability assessment of scour phenomena at bridge piers. Both works concluded that considering the environmental variables as independent ones led to an overestimation of the probabilities of failure. Nevertheless, scour at bridge piers is current induced, while scour at marine structures can be either wave or current dominated. Therefore, it is important to assess the effects of correlation in the probabilities of failure to analyse whether the remarks made by Chang and Tung (1994) and Muzzammil and Siddiqui (2009) still hold for scour protections at offshore monopile foundations. Although in theory the independent copula is not a true model to represent the joint distribution T_p and H_s, it may lead to reasonable values of P_f, when compared to other copulas, hence providing a simpler and quicker evaluation of the protection’s reliability. The BKDE model is non-parametric and does not assume any theoretical distribution of H_s or T_p. However, BKDE provides a model as close as possible to the hindcast data sample.

Figure 5 provides a random sample of 10,000 pairs (H_s; T_p) for the independent copula generation, imposed over the 124-month dataset. The comparison does not correspond a perfect match, because the correlation between both spectral parameters is not being considered. Moreover, the copula generation is ‘blind’ in terms of the maximum significant wave height and the associated peak period. To account for the fact that the wave heights can be depth limited, the present generation algorithm accounts for a maximum wave height of 0.78 times the water depth (18 m). This is applied to all models for the sake of safety. Nevertheless, the main idea to retain is that the models fitted may provide wave heights that are overestimated, but from the practical point of view, this leads to a conservative assessment of the protection’s reliability. The independent copula presented a significantly different upper tail when compared to the 124-month dataset. Above H_s = 3 m, this copula provided larger values for the significant wave height and lower values for the peak period, when compared with the dataset. The direct effect of overestimated significant wave heights and underestimated peak periods on the damage number of the scour protection is not simple to analyse. Note that the damage number formula (equation (6)) is mainly related to the orbital bottom velocity, which indirectly depends on H_s and T_p. Intuitively, one could think that large wave heights with short periods tend to increase the stress on the armour layer, hence, producing a high damage rate and eventually failure.

OPEN IN VIEWER

Figure 5.

Sample of 10,000 pairs of H_s and T_p for the independent copula and the 124-month hindcast data.

The wave-induced shear stress depends on U m 2 . According to the linear wave theory, the orbital bottom velocity increases with H_s. However, the influence of the wave period on the wave friction factor (f_w) depends on a power law with respect to A = (U_m·T_p)/(2π). Several authors studied and proposed an equation of the type f_w = C(A/k_s)^B, where C is a constant and k_s is the bottom roughness as computed in De Vos et al. (2011). The B constant also assumes different values depending on the ratio A/k_s. For example, Fredsøe (1992) suggest −0.25 for A/k_s > 50, Soulsby (1997) uses B = 0.52, while Dixen et al. (2008) suggest −0.8 for 0.2 <A/k_s < 10. This leads to the conclusion that the influence of T_p in the wave-induced shear stress is very much dependent on the formulation applied to calculate the friction factor.

Due to this non-linearity in equation (6), it is harder to predict the influence of the wave parameters in the reliability of the scour protection. Moreover, several pairs appear outside the original data near the upper tail, where the failure domain is more likely to be. In Figure 6, a sample of 10,000 pairs (H_s; T_p) from the BKDE was plotted over the original hindcast data (124-month dataset). One can see that the kernel density estimation method provides close values to the 124-month dataset. Due to its empiric nature, the BKDE method is not able to provide values much higher than the 6 m for the significant wave height and the 22 s for the wave period. This method is important as it can be related to an empiric probability of failure.

OPEN IN VIEWER

Figure 6.

Sample of 10,000 pairs of H_s and T_p for the BKDE and the 124-month hindcast data.

Copula-based models

Measures of dependence

In addition to the reference models, two copulas from the elliptical family were applied: the Gaussian (Normal) and the t-copula. Three widely used Archimedean copulas were implemented, namely, the Frank, Gumbel and Clayton copulas. Due to their straightforward application, these copulas were chosen with the purpose of describing the joint probability of significant wave heights and peak periods. Moreover, the Tawn type 2 copula was applied due to its remarkable flexibility and non-symmetry when compared with the previously mentioned ones. This has proven to be an important factor to achieve a better goodness-of-fit with the datasets. It was concluded that the symmetric tested copulas did not capture the asymmetry present in the hindcast data.

One should note that more complex copulas could be used. However, the present research sets the first steps to perform a reliability assessment of scour protections, with a copula-based model applied to the sea-state parameters. Note that copula-based models have not been extended to the reliability analysis of scour protections. Therefore, it seems reasonable to start by using these examples, which rely on their simplicity and low number of parameters. The copulas were applied for the several datasets defined in Table 7, ultimately leading to different values of the probability of failure, mainly because the copula parameters are estimated with different quantities of data. The copula parameters were estimated by means of the maximum likelihood estimation (Genest and Favre, 2007), with the R software, using the ‘Icopula’ package. The parameters of each copula, by dataset, are presented in Table 8. Figures 7 and 8 provide a random sample of 10,000 pairs of significant wave heights and peak periods (H_s; T_p) from the Gaussian and t-copula compared with the original data (124 months).

OPEN IN VIEWER

Table 8.

Parameters, AIC and BIC, for each copula and dataset.

Copula		Dataset
		124 m	62m_a	62m_b	31m_a	31m_b	31m_c	31m_d
Gaussian copula	α	0.42	0.44	0.4	0.44	0.43	0.39	0.42
	AIC	−17,593	−10,190	−7439	−5315	−4873	−3593	−3858
	BIC	−17,584	−10,181	−7430	−5306	−4864	−3584	−3848
Student t-copula	α	0.46	0.47	0.44	0.47	0.47	0.45	0.44
	Df	5.75	6.05	5.45	6.02	6.07	4.84	6.42
	AIC	−21,933	−12,324	−9636	−6414	−5906	−5079	−4583
	BIC	−21,914	−12,305	−9617	−6395	−5888	−5060	−4564
Clayton copula	α	0.68	0.71	0.64	0.67	0.76	0.6	0.69
	AIC	−19,306	−10,839	−8501	−5438	−5433	−4299	−4235
	BIC	−19,296	−10,830	−8491	−5429	−5423	−4290	−4225
Frank copula	α	3.23	3.39	3.07	3.35	3.43	3.13	3.01
	AIC	−21,606	−12,279	−9375	−6233	−6046	−4808	−4569
	BIC	−21,597	−12,270	−9365	−6224	−6037	−4798	−4559
Gumbel copula	α	1.35	1.36	1.34	1.38	1.34	1.36	1.33
	AIC	−14,740	−8341	−6404	−4535	−3821	−3271	−3138
	BIC	−14,731	−8332	−6394	−4526	−3811	−3262	−3128
Tawn type 2	α	3.03	3.15	2.90	3.01	3.28	2.74	3.03
	δ	0.37	0.38	0.35	0.38	0.39	0.34	0.36
	ρ	1	1	1	1	1	1	1
	AIC	−40,312	−22,068	−17,959	−9989	−11,955	−8189	−9757
	BIC	−40,294	−22,051	−17,942	−9973	−11,939	−8173	−9741

AIC: Akaike information criterion; BIC: Bayesian information criterion.

OPEN IN VIEWER

Figure 7.

Sample of 10,000 pairs of H_s and T_p for the Gaussian copula and the 124-month hindcast data.

OPEN IN VIEWER

Figure 8.

Sample of 10,000 pairs of H_s and T_p for the t-copula and the 124-months hindcast data.

Note that each random sample may provide slightly different values. This is also applicable to all copulas. Nevertheless, to assess the probabilities of failure, a stabilisation study of P_f regarding the generated sample size (n) is made in the ‘Reliability assessment’ section. In a similar way to the independent copula, the elliptical ones predict significant wave heights above the 6 m and peak periods that exceed the 25 s. It seems reasonable to assume that these cases may contribute for the failure of the scour protection.

However, it is possible to note that several waves above 3 m occur with lower peak periods than the ones presented by the original data. As seen in the previous section, it is hard to assess the combined effect of T_p and H_s in the damage number of the protection. Nevertheless, both copulas seem to provide several pairs that correspond to waves in the range of H_s = [0;4] (m) and T_p = [0;25] (s). These waves alone are not expected to contribute to the protection’s failure, although the cumulative effects of long sequences of waves are yet to be deeply understood. For example, a long-duration storm of waves with 4 m may lead to failure of the protection, even without the occurrence of a major wave, for example, 6 m or higher. Although this is not the focus of the present research, the authors note that the present concept of probability of failure concerns the general occurrence of extreme events within the overall population of waves. Still the probability of failure within the occurrence of a sequence of intermedium wave heights should be the aim of further research. The failure due to sequences of waves opposed to the one caused by extreme events is still closely related to the technical and scientific knowledge gaps regarding the long-term evolution of damage. Also note that the present research does not consider breaking waves, which may also lead to considerable damage levels at the protection.

De Vos (2008) and Whitehouse et al. (2014) performed experiments on dynamic scour protections, concluding that stabilisation of the damage number could be found after 5000 waves. However, in some tests, scour still appeared after 7000 waves and no tests were performed for even longer sequences of different intensity storms. Moreover, Schendel et al. (2014, 2016) performed scour tests with a wide-graded single layer until 9000 waves, but no scour stabilisation was found. The authors concluded that even longer tests were needed to understand the damage stabilisation.

The visual assessment of Figures 8 and 9 seems to indicate a poor fitting when the peak periods and the significant wave heights increase. The fit improves when the peak periods and the significant wave heights are small. In the right tail of the data (H_s > 4 m), the tendency to provide larger waves with shorter periods than the ones provided by the original data seems clear. This is evident for the Gaussian (Figure 7), the Clayton (Figure 9) and the Frank copula (Figure 10). This fact is not so evident in the t-copula (Figure 8) and the Gumbel copula (Figure 11), which present several points above and below the right tail of the original data. These models provided a considerable dispersion when compared with the original data (124-month dataset). The authors found that this dispersion could also be extended to the other datasets, that is, the datasets of 62 and 31 months.

OPEN IN VIEWER

Figure 9.

Sample of 10,000 pairs of Hs and Tp for the Clayton copula and the 124-month hindcast data.

OPEN IN VIEWER

Figure 10.

Sample of 10,000 pairs of H_s and T_p for the Frank copula and the 124-month hindcast data.

OPEN IN VIEWER

Figure 11.

Sample of 10,000 pairs of H_s and T_p for the Gumbel copula and the 124-month hindcast data.

The use of more complex models could lead to a better fit between the datasets and the simulated data (Vanem, 2016). Still, the copulas presented so far have been applied with a reasonable degree of success in other studies, namely, when modelling single storms with multivariate Archimedean copulas (Corbella and Stretch, 2013). Also, Jane et al. (2016) successfully used the Gaussian and t-copula to estimate the wave height records through spatial correlation at the south coast of England. Using the same Elliptical and Archimedean copulas (Clayton, Frank and Gumbel), Antão and Soares (2016) found that Gaussian and Gumbel copulas could accurately fit the empirical densities of the individual wave steepness and height.

However, as stated by Vanem (2016) and re-confirmed in this study, it is far from straightforward to find a good copula-based model for non-symmetric data, such as significant wave height and wave period. The graphical assessment of these two families of copulas indicates that they are closer to the independent copula than they are to the BKDE model. This was already expected because these copulas do not perform well under non-symmetric data.

Since the Archimedean and Elliptical copulas did not seem to fit the datasets in a satisfactory way, a final attempt was performed with the non-symmetric copula Tawn type 2. Due to its higher ‘flexibility’, when compared with the previous models, this copula seemed to be a reasonable candidate for fitting purposes. As it will be addressed in further sections, this model showed significant score improvements in the AIC and BIC. Nevertheless, as it is perceivable from Figure 12, that the fit between this copula and the 124-month dataset appears to be better than the previous ones.

OPEN IN VIEWER

Figure 12.

Sample of 10,000 pairs of H_s and Tp for the Tawn type 2 copula and the 124-month hindcast data.

The analysis performed with the Tawn type 2 copula showed that the asymmetry of the datasets with 31 and 62 months was also captured in a similar way to Figure 12. Unlike the previous copulas, the Tawn type 2 presented considerably low values of wave heights and peak periods. On the upper tail, the generated data showed a tendency to present higher significant wave heights and periods than the original data. This behaviour opposes the one showed by the Clayton, the Frank and the Gaussian copulas. In the lower tail, the generated values appear to be in a relatively good agreement with the original data, when compared with the remaining copulas.

The main idea to retain is that for each specific case, one should study several possible models and then make an informed decision. State-of-the-art indicates that modelling significant wave heights and peak spectral periods strongly depends on the copula and the dependence measures used. Therefore, a case-by-case approach should be adopted. Montes-Iturrizaga and Heredia-Zavoni (2015) pointed the relevance of selecting an appropriate model that best represents the dependence structure of the significant wave heights and peak periods. The visual analysis gave clues on the fact the Tawn type 2 copula seems to be a good candidate to model the spectral parameters for reliability analysis.

Models selection (AIC and BIC)

The models were compared by means of the AIC (Akaike, 1974) and the BIC (Schwarz, 1978). The best model is the one with the lowest AIC or BIC. Although there is no consensus about the criterion that should be followed, both AIC and BIC are widely used in the literature (Corbella and Stretch, 2013; Masina et al., 2015). The common formulation for the AIC and BIC is presented in equations (31) and (32), where k is the number of parameters and n is the sample’s size

A I C = − 2 l o g l i k e h o o d + 2 k

(31)

B I C = − 2 l o g l i k e h o o d + l n ( n ) × k

(32)

Table 8 provides the values obtained for the whole datasets by means of the maximum likelihood estimation method. The variable k is the number of parameters of the copula, while n is the number of observations in each dataset. The AIC and BIC results pointed the Tawn type 2 as the copula with the best goodness-of-fit for all the datasets. This fact confirms the visual analysis of the copulas plotted previously. The Frank copula provided the best scores for dataset 31b when compared with the other Archimedean and Elliptical copulas. However, the t-copula was the second best model for the datasets 62a, 62b, 31a, 31c and 31d, both with AIC and BIC.

In the present case, there is a consistency in the fact that the Tawn type 2 copula provided the best results. However, one should keep in mind that this may not hold for other samples from different locations than the one studied in this article. Regarding other applications of copula-based models and due to the lack of a ‘rule of thumb’ to choose the better one, a reasonable approach is to apply more than one criterion and to select the model that provides the overall best scores, as performed in this study.

Reliability assessment of dynamic scour protections

Validation and comparison of results

The values obtained from the hindcast data show that the maximum values for H_s and T_p were, respectively, 6.11 m and 21.61 s. However, the copula models may lead to considerably higher values than these. The following underlying questions seem to be reasonable: ‘Are the copula-based models wrong?’ or ‘Are the hindcast data unable to accurately represent the extreme values that may occur at the scour protection’s location?’ One cannot simply provide a straight answer to these questions. Therefore, hoping to understand the accuracy of the copula-based models, an application was performed by means of three theoretical examples. The first one considering a situation A where the probabilities of failure are expected to be ‘low’ (say in the order of 10⁻³) and the second one related to ‘high’ probabilities of failure (in the order of 10⁻¹), that is, situation B. Then, a third example was designed to obtain an intermediate P_f (in the order of 10⁻²), that is, situation C. The aim of these examples is to assess whether the copulas-based models are at least providing probabilities of failure which are in the same order of magnitude with the one presented by the non-parametric model (BKDE). Moreover, these examples enable one to analyse how the probability of failure varies depending on the model used. The situations considered in order to force the low, the high and the intermediate probabilities of failure are summarised in Table 9.

OPEN IN VIEWER

Table 9.

Simulation conditions for situations A, B and C.

Situation	A – low Pf	B – high Pf	C – intermediate
Simulation conditions
Hsmax (m)	4	6	6
Tpmax (s)	22	22	22
d (m)	10	10	10
Uc (m/s)	µ = 0.1; σUc = 0.05	µ = 2; σUc = 0.2	µ = 0.4; σUc = 0.2
D50 (m)	0.6	0.4	0.4
ps (kg/m3)	2650	2650	2650
pw (kg/m3)	1025	1025	1025
N (number of waves)	3000	3000	3000
g (m/s2)	9.81	9.81	9.81
S3Dacceptable (–)	1	0.25	1
g(x) (–)	1 − S3D	0.25 − S3D	1 − S3D

If the models are at least roughly accurate, the probabilities should have the same order of magnitude, after a considerable number of simulations of the function g(x), i.e. the limit state function. In situation A, the values of H_s and T_p are reduced and the acceptable limit for the damage number is 1. Also, the mean value of D₅₀ (0.6 m) is increased to ensure that harshest hydrodynamic conditions are needed in order to cause high values of the damage number. The current velocity is also dropped to 0.1 m/s with a standard deviation of 0.05 m/s. In situation B, the failure rate is increased by considering a high current velocity (2 m/s) with a standard deviation of 0.2 m/s. The adopted D₅₀ corresponds to 0.4 m. While the waves’ peak period is maintained as 22 s, the significant wave height is increased to a maximum of 6 m, which is roughly the maximum value expected from the hindcast data and the BKDE model. Figures 13–15 give the failure probabilities according to the number of Monte Carlo simulations (n) for the 124-month dataset.

OPEN IN VIEWER

Figure 13.

Situation A for low probabilities of failure (H_smax = 4 m; T_pmax = 22 s).

OPEN IN VIEWER

Figure 14.

Situation B for high probabilities of failure (H_smax = 6 m; T_pmax = 22 s).

OPEN IN VIEWER

Figure 15.

Situation C for intermediate probabilities of failure (H_smax = 6 m; T_pmax = 22 s).

The three situations indicate that the obtained probabilities of failure seem to be fairly stable after n = 200,000. Although some variations may occur within each model, the P_f does not change in its order of magnitude. These variations seem to be more noticeable for situation B. However, one must note that this is not the usual domain of failure when dealing with engineering problems. Although situation B may serve as a theoretical case, having probabilities near 20% is an absurd value when dealing with reliability of offshore and marine structures. Still, it seems fair to admit that the probabilities of failure are indeed stabilised for situations A, B and C.

In these situations, the Tawn type 2 copula provides the highest values of P_f, somehow closer to the Gumbel copula. It makes sense that both copulas provide similar values, since the first is a non-symmetric version of the second. In the previous sections, it was noted that the Tawn type 2 copula presented a better fit to the original data. The fact that the Tawn type 2 copula is able to capture the upper tail behaviour of the hindcast data, better than the Gumbel copula, might be related to the higher values of P_f. Note that the Gumbel copula has upper tail dependence and zero lower tail dependence.

While the Tawn type 2 copula sets the higher limit of the obtained probabilities, the independent copula seems to set the lower one. Taking into consideration the discussion regarding the independence assumption of the spectral parameters, one can suspect that the independent copula might be providing unreasonable values of the probabilities of failure. However, it is interesting to note that this copula systematically provides the lowest values in situations A, B and C. The Clayton copula, which has lower tail dependence and zero upper tail dependence, is the closest one to the independent copula. It seems that the lack of ability to capture the upper tail behaviour of the hindcast data might be leading to lower values of the probability of failure.

Another interesting aspect is that the non-parametric BKDE and the Frank copula have a similar behaviour between situations. When the order of the probability of failure is 10⁻³ (situation A), they are closer to the independent copula, while for P_f in the order of 10⁻¹, these models gets close to the upper limit provided by the Tawn type 2 copula. For the intermediate case, the BKDE and the Frank copula get closer to the Gaussian copula, which is somehow in the middle of the limits set by the independent and the Tawn type 2 copulas.

Table 10 provides the probabilities of failure for the independent and the Tawn type 2 copulas after 1,000,000 simulations and the ratio between them.

OPEN IN VIEWER

Table 10.

Probabilities of failure for the Tawn type 2 and the independent copula for situations A, B and C.

Situation	Tawn type 2	Independent	Ratio Pftawn/Pfindependent
A	Pf = 0.006214	Pf = 0.000588	10.57
B	Pf = 0.22904	Pf = 0.20208	4.66
C	Pf = 0.016949	Pf = 0.003635	1.13

Since in typical engineering problems one is looking for very low probabilities of failure, say 10⁻⁴ or 10⁻⁵, the increasing gap between these models poses a source of uncertainty when dealing with the models’ choice. The present results are not conclusive regarding the best model to be adopted. There is no direct relation between the scores obtained in the AIC and BIC and the outcome of the probabilities between the models. The probabilities given by Tawn type 2 copula, the t-copula and the Frank copula vary from situation to situation. Although the Tawn type 2 copula presents the best AIC and BIC scores, this model does not provide similar probabilities of failure when compared with the BKDE. However, the independent copula gets closer to the BKDE for situation A (P_f ≈ 10⁻³). This is not so evident when the order of the probability of failure gradually increases, as in situations B and C.

However, due to the scores obtained in the BIC and AIC criteria and from a conservative perspective, it seems reasonable to use the model that provided the best fit to the hindcast data and that simultaneously gave the highest values of P_f, which in this case is the Tawn type 2 copula. Due to the inconclusive data, it is recommended that in practical situations, the designer uses several models before making a decision purely based on the information criteria. A conservative approach recommends that the model’s choice should rely on the one that provides the highest probabilities of failure. Moreover, it can be noted that considering the spectral parameters as independent may result in the underestimation of the probabilities, eventually leading to an unsafe scour protection design. This was better perceived in situations B and C.

Finally, the uncertainty of these predictions associated with the choice of the marginal distributions, that is, the log-normal for H_s and T_p, was not focused in this article and may also contribute to deviations from the BKDE result. Details on the marginal’s uncertainties are presented by Vanem (2015). The conservative limit chosen of H/d < 0.78 is also contributing for differences between the model’s probability of failure and the non-parametric estimation given by the BKDE.

Application to a case study

Consider now a somehow more real situation based on Horns Rev 3 offshore wind farm. Consider that no limits are imposed on the peak period and the significant wave height, with the exception of the depth limitation already mentioned. These reference values are provided in Table 11.

OPEN IN VIEWER

Table 11.

Reference values used to calculate the probabilities of failure for the case study.

Situation		Case study
Simulation conditions
Hsmax (m)	No limit imposed	pw (kg/m3)	1025
Tpmax (s)	No limit imposed	N (number of waves)	3000
d (m)	18	g (m/s2)	9.81
Uc (m/s)	µ = 0.4; σUc = 0.2	S3Dacceptable (–)	1
D50 (m)	0.4	g(x) (–)	1 − S3D
ps (kg/m3)	2650	–

This case study is similar to situations A and C previously defined. Nevertheless, the water depth is increased to 18 m and the values of the spectral parameters (H_s; T_p) are now able to achieve considerably higher values than the ones obtained from the hindcast data. In Table 12, the probabilities of failure based on the 124-month scenario are obtained for n = 300,000 simulations, which is above n = 200,000 simulations previously identified as stabilisation point for P_f. The correspondent annual probability of failure (P_f0) is also provided.

OPEN IN VIEWER

Table 12.

Probabilities of failure for the scour protection considered in the case study (D₅₀ = 0.4 m).

Model (124 months)	Pf (n = 300,000)	Pf0 (n = 300,000)
Independent	4.30E−04	4.16E−05
BKDE	4.97E−04	4.81E−05
Clayton	5.8E−04	5.62E−05
Frank	1.02E−03	9.88E−05
Gaussian	2.16E−03	2.10E−04
t-copula	3.16E−03	3.06E−04
Gumbel	44.5E−03	4.32E−04
Tawn type 2	4.69E−03	4.55E−04

BKDE: bivariate kernel density estimation.

The independent and the Tawn type 2 copula set the lower and upper limits for the probabilities of failure, respectively. In the present case, the increasing water depth contributed to reduce the values of the probability of failure, when compared with situation C. This is in agreement with the expected behaviour of scour phenomena, for which the severity tends to decrease with the increasing water depth (Whitehouse, 1998). From the independent copula to the Tawn type 2 copula, the magnitude of the probability of failure increases from 10⁻³ to 10⁻². Moreover, the BKDE is closer to the independent copula, as it occurred in situation A.

The case study and the previous situations indicate that the probability of failure is very much dependent on the copula proposed for the spectral parameters. This fact is similarly reported in other elements of offshore foundations. Montes-Iturrizaga and Heredia-Zavoni (2016) reported that for mooring lines, the probability of failure may vary one or two orders of magnitude, depending on the copula model.

To the authors’ knowledge, no studies are reported regarding the probabilities of failure in dynamic scour protections. This leads to an increased difficulty when it comes to assess whether the values obtained in Table 12 are acceptable or not. In fact these values are based on a 124-month dataset, which approximately corresponds to a scenario of 10.33 years. However, for a proper comparison with the common standards, one is more interested in the annual probability of failure (Pf₀). As a simplification, if one assumes that the failures of the scour protection are continuous-time stochastic process, with the failure events being independent from each other, that is, Poisson process, then the probability of failure can be converted to the annual values by means of the continuous exponential distribution F(t) = 1 − e^−vt, where t is the time associated with the probability desired, in this case 1 year, and v is the rate of the events occurring in the time interval associated with the reference scenario. Considering a constant rate over 10.33 years, one obtains the rate as v = −ln(1 − Pf_{10.33 years})/t, where t corresponds to 10.33 years. This conversion leads to the third column of Table 12, which provides the annual probability of failure associated with each model, based on the 124-month dataset as the reference scenario. Note, however, that the annual values in Table 12, consider that the failure rate is constant, which is already a questionable simplification, since it implies an independency between failure events.

In offshore wind structures, the IEC (2005, 2009) standards indicate a design lifetime of 20 years, for both the turbine generator and the foundation. DNV GL (2017b) recommends a 10⁻⁴ nominal annual probability of failure for the foundation design (in unmanned structures). In contrast, if these values are used in the approach proposed in ROM 0.0 (Puertos Del Estado, 2001), the correspondent return period exceeds by far the standard return period used in current practice (Negro et al., 2014).

If the failure probability considered is 10⁻⁵ (manned structures), the return period increases even more. Such values are extremely above the typical ones considered in common design standards. They are not in line with the statements from the offshore wind recommendations regarding the return periods. For example, for ultimate limit states (ULS), the characteristic loads required by DNV GL (2017b) correspond to the 98% percentile, and therefore, 50-year return period, which is completely different from the values mentioned before.

Scour protections are not typically designed by means of failure probabilities, as they would on a reliability-based methodology. Due to the lack of studies performed on the present subject, it is hard to evaluate what is an acceptable probability of failure for a specific scour protection. In Negro et al. (2014), the comparison between reliability design techniques and the return periods typically used in offshore wind industry is discussed with further detail. According to DNV GL (2017c), which for reliability purposes recommends the use of (DNV, 1992), one is able to gain a sense on the annual probabilities of failure acceptable for marine structures (Table 13).

OPEN IN VIEWER

Table 13.

Annual probability of failure (P_f0) for marine structures according to DNV GL (2017c) and DNV (1992).

Class of failure	Consequence of failure
	Less serious	Serious
I – Redundant structure	Pf0 = 10−3	Pf0 = 10−4
II – Significant warning before the occurrence of failure in a non-redundant structure	Pf0 = 10−4	Pf0 = 10−5
III – No warning before the occurrence of failure in a non-redundant structure	Pf0 = 10−5	Pf0 = 10−6

For a consequence of failure to be described as ‘less serious’, the risk of life upon failure must be considered as being relatively negligible. If one considers a scour protection at an offshore wind turbine, this is likely the case, since the structure is typically unmanned. Moreover, during extreme weather events, it is not expected that any maintenance or inspection operation is performed, which limits the consequence of failure in terms of life losses, thus being reasonable to admit that the consequence of failure can be considered less serious. Note, however, that this does not mean that the failure of the protection may not lead to considerable consequences, as the wind turbine can collapse due to scour occurrence (Prendergast et al., 2015). Moreover, the consequences scale up in terms of economic losses if the importance of the foundation in the capital expenditures (CAPEX) and the operating expenses (OPEX) parcels are considered (Gonzalez-Rodriguez, 2017).

The acceptable annual probability of failure is within the range of 10⁻³ and 10⁻⁵ (Table 13). The annual probabilities of failure obtained with the Tawn type 2 and the independent copulas range from 10⁻⁴ to 10⁻⁵. The failure of a scour protection is often noted long after the occurrence, due to the periodicity of maintenance and inspection operations. Since this is a submerged element of the offshore wind turbine, it is fairly reasonable to admit that there is no warning before failure occurrence. Thus, a probability of failure in agreement with the standards should be in the order of 10⁻⁵. A comparison between the standard values and the case study shows that depending on the model, the probability outcome may or may not be in agreement with the standards. This emphasises the importance of the model’s choice in the safety assessment.

Often refilling operations can be planned for scour protections after storm events, and the fact that the time scale of scour phenomena may not lead to the immediate collapse of the structure, as the backfilling process occurs, may somehow contribute to accept probabilities of failure which are in the order of 10⁻⁴.

Although not being the case, one must note that any model providing an order of 10⁻³ seems rather alarming due to the level of uncertainty encompassed in the systems’ response to scour phenomena. Although offshore wind farms frequently require maintenance operations (Chan and Mo, 2017), designing a scour protection for high values of P_f may undermine the cost-benefit of having a scour protection, as it will lead to very large OPEX costs for a lifetime of 20 years, which is considerably short in comparison with traditional structures. When dealing with other offshore structures and elements, for example, floating foundations (Zhang et al., 2016), mooring lines (Redón-Conde and Heredia-Zavoni, 2015) or structural elements of offshore wind turbines (Kim and Lee, 2015), higher orders can be found, often one deals with probabilities of failure close to 10⁻⁶ and sometimes higher. Thus, a balance between the acceptable consequences of failure and their influence on the cost of the scour protection must be achieved. Furthermore, the correlation between failure modes of the wind turbine can play a major role in the structural behaviour and must be analysed in the safety assessment. For instance, scour phenomena are often related to fatigue problems. The influence of the scour protection’s failure on the probability of failure related to the fatigue limit state is unknown for the present case. Facts like this may lead the designer to assume that the protection’s reliability is not compatible with probabilities of failure in the order of 10⁻⁴.

Tail dependence analysis

The model with best scores in the AIC and BIC (Tawn type 2 copula) did not present the closest probability of failure to the non-parametric model (BKDE). This is an interesting aspect that can be pointed, as one may tend to exclude certain models based on these criteria, even before performing the probability calculations.

The AIC and BIC provide an assessment of the overall fit of the model. However, when dealing with probabilities of failure, the domain of interest is typically related with the tails’ behaviour. It is the extremes of the joint distribution that may contribute to increase the failure rates. Therefore, the tail dependence is important to understand which model might be of better when estimating the probability of failure.

In situations A, B and C and the case study, as the probabilities become smaller and smaller, the Tawn type 2 copula deviated more and more from the BKDE. Table 14 provides the upper (λ_U) and lower (λ_L) tail dependence coefficients for each copula model, as calculated in Nelsen (2006). Furthermore, this non-parametric estimation of the tail dependence coefficients was performed according to Schmid and Schmidt (2007) considering the upper and lower quantiles of 10%, 5%, 2.5% and 1%.

OPEN IN VIEWER

Table 14.

Tail dependence coefficients for the 124-month dataset.

Tail dependence coefficients
124 months	λL		λU
Independent	0		0
Gaussian	0		0
t-copula	0.1591		0.1591
Clayton	0.3586		0
Gumbel	0		0.3316
Frank	0		0
Tawn type 2	0		0.3503
Non-parametric	10%	0.2887	10%	0.0256
	5%	0.2373	5%	0
	2.5%	0.1919	2.5%	0
	1%	0.1509	1%	0

H_s and T_p are considered to be upper (or lower) asymptotically dependent if λ_U (or λ_L) belongs to the interval]0;1]. If λ_U (or λ_L) = 0, the variables are considered to be asymptotically independent (Nelsen, 2006). Note that if the tails are asymptotically independent, that does not mean that the variables are actually independent. It solely means that as one moves to the upper or lower tail, the probability that T_p exceeds a certain quantile is independent from the probability that H_s exceeds the same quantile. In fact as shown in Table 14, the non-parametric estimation of the upper tail for the 10% quantile is already very low (0.0256). For even smaller quantiles, one is able to see that the spectral parameters show an asymptotically independent behaviour. Given this upper tail behaviour, it is understandable that the Tawn type 2 copula provides probabilities of failure which are not in the range of the ones provided by the BKDE. By definition, the Tawn type 2 copula imposes an upper tail dependence, which in this case is about λ_U = 0.3504. In order to obtain the same tail dependence with the non-parametric estimation, one would define the tail as starting in the 0.454 percentile, which is far from being the actual tail of the data.

One is able to conclude that the Tawn type 2 copula did not catch the asymptotical independence of the spectral parameters, as other models did, for example, Frank or Clayton copula. Note that the Gumbel copula, with λ_U close to 0.33, also provides probabilities near those of the Tawn type 2 copula.

The fact that the lower tail for the quantile of 1% has a positive dependence (in the non-parametric estimation), which in this case is closer to the t-copula model, also causes perturbations in models’ ability to provide closer values of the probability of failure to the BKDE. Note that the lower tail coefficient is not stabilised in Table 14. Further iterations should be made to access the value for which λ_L stabilises. This study emphasises the dependence of the probabilities on the tails’ modelling provided by each copula. An important conclusion arises from the tail dependence analysis: ‘when calculating a probability of failure, one should select a model that is not only based on the information criteria (AIC or BIC) but also based on the tail dependence behaviour’.

Regarding the dataset of 124 months, further research should be carried out to test more complex models, for example, mixed copulas, which are able to combine copulas with different tail dependences on the upper and lower tails (Zilko and Kurowicka, 2016).

Influence of Kendall’s τ

Another important aspect of the present research concerns the influence of the records’ duration, that is, the dataset, on the probabilities of failure. The same case of Table 11 was analysed for each dataset, and the probabilities were obtained for the Tawn type 2 copula. The results are provided in Figure 16. Although for a small range of τ_k, one is able to see that the probability of failure increases for an increasing dependence measure. Since the dependence measure is affected by the datasets’ properties, one can conclude that the dataset is of great importance when aiming to assess the safety of the scour protection. This points out an important limitation of the model, which relates to the quality of the hindcast or the observed data available. Nevertheless, this limitation is common to the majority of the statistical models available. Further investigation should be performed to analyse the influence of the dependence measure on final outcome of the probabilities. Despite the difference in the probabilities, the order of magnitude remained the same for the Tawn type 2 copula.

OPEN IN VIEWER

Figure 16.

Probability of failure versus Kendall’s tau per dataset for the Frank, the Tawn type 2, the t-copula and the BKDE (n = 300,000).

Despite the short range of tested datasets and respective values of τ_K, it is evident that both the copula and the measure of dependence influence the assessment of the probabilities of failure. The order of magnitude may vary between different models, which is in agreement with other works performed for other offshore foundations (Montes-Iturrizaga and Heredia-Zavoni, 2016). The present research emphasises that modelling the dependence structure of met-ocean data with copula-based models is very much dependent on the models’ selection. These results also agree with works performed for other locations and records (Corbella and Stretch, 2013; Jane et al., 2016; Vanem, 2017).

Although in the literature it is reported that depending on Kendall’s τ, the probability of failure may vary in its magnitude, this did not occur in the cases shown in Figure 16. The magnitude varies between models, but not within the same copula model. A reason for this is the fact that the present research only deals with a very short range of Kendall’s τ. Still the values of P_f are clearly influenced. One should also note that despite the large variation in duration of the datasets, for example, 50% from 124 to 62 months and 75% from 124 to 31 months, the maximum and minimum values of τ are only varying 12.3%.

Sensitivity to mean diameter (D₅₀)

The literature has demonstrated by physical model studies that dynamic scour protections tend to be more stable for large armour units (Schoesitter et al., 2014; Whitehouse et al., 2014). For non-cohesive sediments and combined wave and currents, scour phenomena increases for finer sediments. In dynamic scour protections, one is aiming at a mean stone diameter that simultaneously gives the required stability to the armour layer, but does not exceed the equivalent value of a statically stable design. Several known offshore wind farms use values of D₅₀, which may vary from D₅₀ = 0.3 m to D₅₀ = 0.5 m, for example, North Hoyle, Egmond aan zee and Arklow Bank (Matutano et al., 2013) are in this range. Other offshore wind farms may present slightly lower values, for example, Scroby Sands has a D₅₀ = 0.15 m. However, in offshore structure literature, the reports on such technical details of scour protections are often limited. The present sensitivity analysis was performed for an interval between 0.2 m, which is likely to be a very low size for the armour units in common cases, and 1 m. Note that 1 m is indeed a theoretical value for the sake of analysis, as the pipe vessel installation of the protection rarely deals with values above 0.5 m.

Figure 17 shows the variation of the probabilities of failure with the mean stone diameter of the armour material. After D₅₀ = 0.5 m, the changes are largely reduced. One is able to state that all models are sensitive to changes in this variable, which was already expected, as the diameter is an important resistance variable of the protection (De Vos et al., 2012). The major effects on the probability of failure are registered between 0.2 and 0.5 m, which are the common sizes found in the industry. This emphasises that this is the region where the analysis of P_f is most important for practical issues.

OPEN IN VIEWER

Figure 17.

Probabilities’ sensitivity to the mean stone diameter of the armour layer stones modelled with triangular distribution with lower and upper limits of µ_D50 = ±0.1 m, for example, for µ_D50 = 0.4 m, D₅₀ = [0.3;0.5] m.

Although the order of P_f may change with the D₅₀ value, it should be noted that the reduction of P_f for D₅₀ above 0.5 m may not compensate from the economical point of view. Moreover, using such large values for the armour units does not comply with the dynamic stability of the protection, as the static stability is achieved with large diameters and weights of the rock material. For the simulated conditions, with constant water depth, provided that the distribution of U_c and the copula model for H_s and T_p remain the same, it was noted that P_f follows an hyperbolic function of D₅₀ of the type P f = A × D 50 λ , where A and λ are constants that depend on the copula model. Considering the Tawn type 2 and independent copulas, which set the highest and lowest values of P_f, the constants obtained were, respectively, A = 0.0006, λ = −2.083 and a squared correlation coefficient of R² = 0.9976 and A = 8 × 10⁻⁶, λ = −3.285 and R² = 0.9807. One can also note that if the mean diameter increases, the differences between the models get smaller and smaller.

An important aspect, which is not the focus of the present research, is the influence of the armour layer thickness on the probability of failure. Further research should be performed to adapt the failure criterion to include this variable and its effects in the probability of failure. Nevertheless, the main idea that seems useful to retain is the fact that the probability of failure tends to decrease with increasing D₅₀ and that the D₅₀ values of interest are lying between 0.2 and 0.5 m. This idea is in agreement with other works performed on the stability of dynamic scour protections (Fazeres-Ferradosa and Taveira-Pinto, 2015).

Sensitivity to current velocity (U_c)

To analyse the effect of the current velocity, the simulations were performed for U_c ranging from 0.1 and 1 m/s. The standard deviation used was 0.2 m/s. The range of variation is based on the minimum bottom velocity reported in DMI (2013). The value associated with the bottom velocity at Horns Rev 3 location is 0.9 m/s for a return period of 50 years. However, surface values were reported to be above 1 m/s (DMI, 2013). Although one is only interested in the bottom velocity, the upper limit of 1 m/s was defined in order to be slightly above the reported bottom value for T_r = 50 years. Often the design values for current velocity deal with a return period of 10 years (DNV GL, 2017a).

Figure 18 shows that for the tested models, the probability of failure increases with increasing current velocity. The results are in agreement with expected behaviour of the scour protection. Increasing current velocity tends to increase the bed shear stress, thus increasing the instability on the armour layer. For scour in marine environments, the presence of currents may induce scour occurrence for Keulegan–Carpenter (KC) numbers below 6, which is the minimum value for scour occurrence with waves alone (Sumer and Fredsøe, 2002). Moreover, the increase in the probability of failure approximately follows a linear function, for which a vertical translation is obtained depending on the model used for the spectral parameters. As in the previous case, the independent and the Tawn type 2 copulas tend to set the lowest and highest limits for the probabilities domain. When the current velocity increases, the differences in the model’s probabilities also increase. For a linear approximation, one obtains for the Tawn type 2 copula P_f = 0.0017U_c + 0.004 and R² = 0.9928, while in the independent copula the approximation is given by P_f = 0.0002U_c + 0.0003 and R² = 0.946.

OPEN IN VIEWER

Figure 18.

Probabilities’ sensitivity to the currents mean velocity (µ_Uc = 0.4 m/s; σ_Uc = 0.2 m/s).

The present failure criterion only considers unidirectional and opposing wave-current environment. Further research is needed to improve the failure criterion for multi-directions between waves and currents. For example, in De Vos et al. (2012), it is pointed out that the damage number increases under waves opposing currents, although no other information is found for other angles between flow components. However, it is expected that the probability of failure depends not only on the magnitude of the current velocity but also on the relative direction between current and waves.

Sensitivity to the acceptable damage number (S_3Daccept)

The acceptable damage number can be interpreted as a resistance variable of the scour protection. If one considers that the acceptable limit can be increased, then the probability of failure should be reduced, given the fact that the other variables remain the same. Note that increasing the acceptable damage number translates to a less restrictive failure criterion. As discussed in De Vos et al. (2011, 2012), the statically stable scour protections were obtained for S_3Daccept = 0.25, while dynamic ones were obtained for a limit of S_3Daccept = 1. Although some cases were reported to be stable at S_3Daccept = 1.25. The criterion based on equation (6) is developed for dynamic scour protections. Therefore, the probabilities of failure obtained for 0.25 (statically stable protections) may not be in agreement with the ones evaluated, for example, using equation (3). Nevertheless, studying the probabilities’ sensitivity in the interval [0.25; 1.25] is important to understand how a conservative criterion (S_3Daccept = 0.25) or a less conservative one (S_3Daccept = 1.25) can influence the probabilities of failure.

Figure 19 provides the probability of failure as a function of the acceptable damage number. When the acceptable damage number increases, the probability of failure decreases, because increasing the acceptable damage number means that the designer assumes that the protection is able to endure large damage quantities, thus meaning that failure is only considered to occur for larger displacements at the armour layer.

OPEN IN VIEWER

Figure 19.

Probabilities’ sensitivity to different limits of acceptable damage number.

However, assuming that the acceptable limit is 0.25 means that the scour protection must present an equivalent static stability. This leads to larger probabilities of failure, since the protection is designed to be dynamically stable, which implies that the actual S_3D is likely to be higher than 0.25. If the acceptable limit decreases, the differences between the models increase (Figure 19). This behaviour is in agreement with the situations previously analysed.

The analysis showed that the sensitivity to the acceptable damage number is similar to the one showed for the mean diameter of the armour stones. An approximation to the hyperbolic function of the type P f = A × S 3 D a c c e p t λ can also be adopted here. For the Tawn and the independent copula, one obtains A = 0.004, λ = −0.744 and R²=0.9996; A = 0.0004; λ = −1.206 and R² = 0.9975, respectively. The probabilities of failure obtained for S_3Daccept = 0.25 should be further compared with a statically stable criterion, in order to clarify whether the consistency of the values obtained. The authors recognise that this should be the aim of further research. If Figure 3 is recalled, it seems fairly safe to adopt a limit of S_3Daccept = 1. From Figure 19, it seems that the potential gains obtained from adopting a value of 1.25 might be somehow risky and does not lead to a considerable reduction in the probabilities. Conversely, values near to 0.25 seem to express a very conservative approach that seems more suitable for static stability than for dynamic one. Considering an S_3Daccept = 0.25 may require a reformulation of the failure criterion inspired in the threshold of motion, that is, equation (3) instead of equation (6).

Sensitivity to the water depth (d)

Taking into consideration the depth-limited wave heights and the bathymetry at Horns Rev 3, a new series of H_s and T_p were generated. Then, the probability of failure based on a 124-month scenario was calculated for a water depth ranging from 10 to 22 m. This is roughly the minimum and maximum values expected at the location. For large water depth to pile diameter ratio, the local scour effect tends to decrease (Fazeres-Ferradosa, 2012). Here, the pile diameter is not considered in the damage number calculation. Still one can fairly assume that if the other variables are kept constant, then increasing the water depth is expected to generate smaller damage numbers, thus leading to smaller probabilities of failure.

Figure 20 shows that the models were able to capture the expected relation. As the water depth increases, the influence on the probability of failure decreases. In the tested range of water depths, the order of magnitude of P_f can change. This emphasises the influence of the water depth for locations where the bathymetry is shallower and shallower. A decrease in the water depth limits the possible wave heights at the scour protection. However, this also affects the orbital bottom velocity and increases the wave’s related term in equation (6). Therefore, the damage number increases for the remaining conditions. Similar to D₅₀ and S_3Daccept, an approximation between the water depth and the probability of failure can be obtained by means of a hyperbolic function of the type P_f = A × d^λ. For the Tawn and the independent copula, one, respectively, obtains A = 3.5526, λ = −2.288 and R² = 0.9979; A = 24.481; λ = −3.842 and R² = 0.9946.

OPEN IN VIEWER

Figure 20.

Probabilities’ sensitivity to the water depth (m).