Introduction

The design of coastal structures is a challenging task owing to the highly dynamic environmental conditions that control wave-structure interactions in coastal areas. Structural design often takes into account the significant wave height associated with a given return period. Nevertheless, structure failure also depends on the combination of wave conditions with other variables (e.g., wind and tides) that can increase or decrease coastal hazards. Therefore, to obtain reliable estimates of service life, probabilistic and multivariate analyses have been proposed (^{Cheung & Kyle, 1996}; ^{Guanche, Camus, Guanche, Mendez, & Medina, 2013}; ^{Camus, Mendez, Medina, & Cofiño, 2010}). More recently, methodologies have incorporated the role of uncertainty (e.g., ^{Palemón-Arcos, Torres-Freyermuth, Pedrozo-Acuña, & Salles, 2015}) and climate change in the life cycle of structures (^{Guanche et al., 2013}).

Hindcast information provides a means to obtain historical information at locations where data is scarce (^{Reguero, Menéndez, Méndez, Mínguez, & Losada, 2012}; ^{Appendini et al., 2013}). Meanwhile, great effort has also been devoted to developing methodologies to study the stability of coastal structures using numerical models (^{Losada, Lara, Guanche, & González-Ondina, 2008}; ^{Higuera, Lara, & Losada, 2013}; ^{Del Jesus, Lara, & Losada, 2012}). Furthermore, the increase in computational power enables performing detailed investigations of wave-structure interaction processes, including overtopping (^{Guanche, Losada, & Lara, 2009}) and sliding (^{Palemón-Arcos et al., 2015}) estimates. Nevertheless, it is still not practical to simulate a high number of cases to provide statistically robust information due to the computational effort required. Therefore, previous works has been devoted to investigating the application of algorithms that allow for selecting representative cases (^{Camus, Mendez, Medina, & Cofiño, 2011a}; ^{Snarey, Terrett, Willet, & Wilton, 1997}; ^{Willet, 1996}; ^{Polinsky, Feinstein, Shi, & Kuki, 1996}) to further reconstruct multi-year time series of nearshore wave conditions (^{Camus et al., 2010}), wave-structure interaction (^{Guanche et al., 2013}), and beach runup (^{Medellín et al., 2016}).

This study presents a downscaling approach to investigate detailed wave-structure interaction and to further conduct a probabilistic assessment of structural failure, considering the occurrence of extreme waves and water levels. The outline of this paper is the following. Section 2 describes the study area, including the environmental conditions and the structural characteristics. Section 3 provides an overview of the numerical models employed in this work. The methods are described in Section 4, and Section 5 presents the results. Finally, Section 6 presents the conclusions.

Study area

The study area is located in Ciudad del Carmen, Campeche, Mexico (Figure 1). The offshore bathymetry is characterized by a mild slope (1: 500). For the offshore bathymetry, we employed the ETOPO1 (^{Amante, 2009}), and a bathymetric survey conducted in 1999 was employed for the nearshore (Figure 2).

Figure 1 Study area location. 

Figure 2 Nearshore bathymetry in the study area. 

Wave and sea level conditions are fundamental to a structure failure analysis. The wave climate in this area shows a higher frequency of storms during winter months, associated with the passage of anticyclonic cold fronts through the Gulf of Mexico (^{Appendini et al., 2013}; ^{Appendini, Hernández-Lasheras, Meza-Padilla, & Kurczyn, 2018}). However, there are no available measured wave data for the study area. Therefore, we used 30-year wave hindcast data derived from a NARR reanalysis of the Gulf of Mexico and the Caribbean (^{Appendini, Torres-Freyermuth, Salles, Lopez-Gonzalez, & Mendoza, 2014}). The data used in the study corresponded to a node located at 18.8( N and -91.90( W (Figure 1), located at approximately 10 m deep and 15 km offshore of the study area. The hindcast data has a time step of 3 hours and covers the years 1979 through 2008, providing a total of 87 664 sea states. Each sea state consists of the significant wave height (H _s ), peak wave period (T _p ), and wave direction (θ). In addition, the astronomic tide (Z) was obtained using the harmonic constants for the global mesh from the numerical model MIKE 21 SW (^{Sørensen, Kofoed-Hansen, Rugbjerg, & Sørensen, 2004}), using the Global Tide Model by ^{Cheng and Andersen (2010)}.

To determine the mean sea level (NM), it is necessary to include the storm surge (MT) contribution. Therefore, we obtained an estimation using (^{Dean & Dalrymple, 1991}):

Where:

where h ₀ is the water depth, in meters, n is the drag coefficient 1.3, A is a ratio of shear to hydrostatic forces, g is known as the acceleration of gravity, ρ _a is the air density (1.204 kg/m³), ( _w is the water density (1 028 kg/m³), u ₁₀ is the wind velocity from the NARR, and k is the friction factor equal to 0.0034.

Figure 3 shows the wave hindcast time series at the selected node, the astronomical tide (Z), and the storm surge (MT). The mean sea level, (NM) employed as an initial condition for each case, was obtained as the algebraic sum of the astronomical tide and the storm surge.

Figure 3 Time series of simulated data (1979-2008): Significant wave height, (b) peak period, (c) astronomical tide, (d) wave direction, and (e) storm surge. 

To illustrate the methodology, we chose a vertical structure located in Ciudad del Carmen (Campeche, Mexico). The structure was made of reinforced concrete with five steel bars per meter, and a total length of 13 m with 7.6 m buried in the seabed. The still water level in front of the structure was 3.4 m with a freeboard of 2 m with respect to the mean sea level (see Figure 4). The sheet piling system consisted of 25 cm thick and 60 cm wide elements that were joint together with a wale.

Figure 4 Concrete pile sheet wall, units in m. 

The sheet pilling was driven with a drop hammer to the depth specified in the design. Subsequently, a wale assembly was installed to brace the sheet members.

Numerical models

We coupled three numerical models to investigate the wave-structure interaction (Figure 5). The wave propagation model SWAN (^{Booij, Ris, & Holthuijsen, 1999}) was employed to propagate waves on the nearshore and for determining the boundary conditions of the hydrodynamic model SWASH (^{Zijlema, Stelling, & Smit, 2011}). The SWASH model estimates the wave overtopping and the pressures exerted on the vertical structure. The latter information was employed in a structural analysis model to determine if a structural failure would occur. The numerical models have been thoroughly described elsewhere, and hence only a brief description of each model is provided herein.

Figure 5 Conceptual diagram illustrating the coupling between the waves, hydrodynamics, and structural models. 

Methods

The framework employed in this work is illustrated in Figure 6. Firstly, we selected 600 representative cases to simulate wave propagation and wave-structure. With the hydrodynamic model, both the overtopping volume and forces for each case could be estimated. Furthermore, a structural analysis was carried out by coupling the hydrodynamic model with a structural dynamic model. Structural failure for the simulated cases could thereby be determined and subsequently employed to reconstruct the time series of wave overtopping and wave-induced structure displacement for every wave condition during the 1979 to 2008 period. In addition, a probabilistic assessment enabled us to determine the service life of the structure. Lastly, we used the numerical results as a tool to calibrate existing semi-analytical models in order to quickly assess structural failure.

Figure 6 Framework for the probabilistic assessment of structure failure. 

Downscaling

The wave hindcast data resulted in more than 80 000 combinations of wave parameters and mean sea levels (astronomical and storm surge) to employ as forcing conditions in the study area. Therefore, we used the Maximum Dissimilitude Algorithm according to previous work by ^{Camus Mendez, Medina, & Cofiño (2011b)} to select 600 modeling scenarios (blue circles in Figure 7).

Figure 7 Time series of 30 years of data:(a) significant wave height, (b) peak wave period, (c) tide, (d) mean wave direction, and (e) storm surge. The open blue circles correspond to the 600 selected cases with the Maximum Dissimilitude Algorithm (MDA). 

The selected wave conditions were propagated using the numerical model SWAN (^{Booij et al., 1999}) assuming a JONSWAP spectrum (γ=3.3) propagating across the nearshore bathymetry (Figure 2). The computational domain in SWAN is composed of 286 cells with a cell size of 55 m. The propagated spectra were employed for forcing the transient numerical model SWASH in the one-dimensional version. The computational domain was 2 000 cells with a uniform mesh of 0.25 m. Wave breaking was computed based on the wave steepness and hence no further calibration was required.

Functional and structural failure analysis

With the SWASH model results, overtopping volumes were then directly estimated for each simulated case. Overtopping calculations were given in m³/m/s. Furthermore, the numerical model enabled obtaining the dynamic pressure acting on the structure. Thus, by integrating the envelope of the maximum pressures, we obtained the acting forces over the structure. Figure 8a shows the overtopping time series for two simulated cases. The solid blue line corresponds to Hs = 3.3 m and Tp = 8.4 s, whereas the solid black line corresponds to Hs = 2.6 m and Tp = 14 s. Based on the modeling results, the overtopping rates can be estimated for each case. With regard to the forces obtained from the SWASH model, these were input into the structural model (StaadPro) for a detailed structural analysis. This numerical model enables investigating the detailed soil-structure interaction. Figure 8b shows the maximum pressures over the vertical wall for the two cases. The red line shows the diagram of the active pressures with its respective magnitudes. For these particular cases, the wave-induced forces were greater than the active pressure, and hence a structural failure would occur. This analysis was conducted for each of the 600 simulated cases.

Figure 8 (a) Wave overtopping time series and (b) dynamic forces for two simulated cases (solid black line: Hs = 2.6 and Tp = 14 s; solid blue line: Hs = 3.3 and Tp = 8 s). 

Results

This section presents the results from the structural and functional analyses carried out on the vertical structure considered in this work.

Wave overtopping

The wave overtopping results corresponding to the 600 simulated cases are presented in Figure 9a (black circles). This information was employed to reconstruct the 30-yr time series shown in Figure 9b (solid blue circles).

Figure 9 Reconstructed time series (blue dots) vs propagation results (black dots) corresponding to (a) overtopping volume and (b) resulting forces. 

Large overtopping events are strongly correlated with more energetic wave conditions. Based on this information, determining return periods and their relation to hazardous conditions is straightforward. For instance, the 1.2 m³/m value is reported as an unsafe value (^{USACE, 2002}). The current results show that for the structure considered in this work, such conditions are present with a return period of 17.5 years (Figure 10).

Figure 10 Overtopping return periods. 

Structural analysis

Knowledge of a structure’s capacity is required to determine structural failure. Therefore, structural characteristics need to be considered. More specifically, the structural resistance depends on the characteristics of the sheet piles. Furthermore, the specifications for the concrete and the steel employed are also necessary. Based on the existing norms for the design and construction of concrete structures, we obtained a resistance moment of 22 ton-m with a steel surface of 38.04 cm² corresponding to the vertical impermeable structure. Therefore, according to the structural review, the resulting force is 178.1 kN-m. The hydrodynamic model provided information on the forces acting on the structure. Hence, we were able to reconstruct the resulting forces from the 30-yr time series using the interpolation technique and the numerical results. Subsequently, the resulting forces were employed in the structural model to simulate the behavior of the sheet piles and determine whether structural failure occurs. The acting dynamic forces counteract the soil resistance. If the dynamic force is higher than the active pressure, a structural failure occurs. For this case, a return period of 3 years corresponded to the failure threshold (i.e., 178.1 kN/m) (see Figure 11).

Figure 11 Resulting forces as a function of the return period. 

Discussion: Parameterization

Wave-structure interaction results in wave overtopping and dynamic forces on the structure that are responsible for structure failure. In this work, we employed the coupling of numerical models to quantify different types of failures. However, this approach is not practical due to both the technical skills and computational time required. Therefore, we proposed the use of the present numerical results for the development of ad hoc parameterizations.

The overtopping parameterization is given as a function of the freeboard, f _b , peak period, and significant wave height. Thus, the formulation is given by:

The black dots in Figure 12 represent the numerical results, whereas the solid line represents the best fit to the model data described by Equation (9).

Figure 12 Overtopping based on 600 selected cases. 

Then, by analyzing the forces we can calibrate the semi-analytical model introduced by ^{Goda (2000)}. Thus, we used the numerical results to obtain the parameters that improve the fit in Goda’s model. The coefficient values for the vertical structure are the following:

In this particular case, the coastal structure does not present a rubble mound base and hence Equation (11) is zero. On the other hand, the values for α ₁ and α ₃ are dependent on the wave period. Figure 13a compares the values obtained from the best fit of the model data with the coefficients proposed by ^{Goda (2000)}. The main difference corresponds to α ₁, which improves the prediction for the current case (Figure 13b).

Figure 13. (a) Parameterization of coefficients α ₁ and α ₃ in ^{Goda (2000)} and (b) resultant force prediction using standard (solid blue line) and derived (solid black line) coefficients with respect to model results (black dots). 

Conclusions

This work presented a methodology to determine the service life of coastal structures using numerical and statistical tools. The current approach is particularly useful in coastal regions where data is scarce. To illustrate the methodology, we selected a vertical structure located in the Gulf of Mexico. The analysis of the results suggests that the structure presented overtopping failure with a return period of 17 years, and structural failure would occur with a return period of only 3 years. The numerical results were further employed to derive/revise parametric and semi-analytical models at this site. The derived formulation makes it possible to quickly estimate both forces and overtopping without the need for numerical models.