# Stochastic Modeling of Forces on Jacket-Type Offshore Structures Colonized by Marine Growth

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Requirements for a Meta-Model

#### 2.2. Description of Bio-Colonization Temporal Dynamic

#### 2.3. Description of Bio-Colonization Temporal Dynamic

#### 2.4. Initiation Phase and Propagation Phases

#### 2.5. Database Post-Treatment, Virtual Database, and Aggregation of Influencing Factors

#### 2.5.1. Environmental Data at the Case-Study Site

^{i}

_{t,τ}, is used for the initiation phase determination and C

^{i}

_{t,τ}for the modeling of the propagation phase. Four types of larval development combining the slow (S) and fast (F) growth possibilities are presented in the Table 1 for the three initiation times (corresponding to the three spawning periods) obtained from the database considering key factors and thresholds described in the previous section. The first larval development is always slow because the water temperature is below 14 °C during early spring, and the third one can be slow only if the second one is also slow (because the water temperature cannot fluctuate abruptly). These results come from the natural seasonal variations of temperature during one year. These frequencies will be considered as discrete probabilities for the modeling. At the end of this larval growth period, we considered that larvae settled on the structures, and that was the start of the propagation phase (macro-colonization) described in Table 1 for the 17 annual chronicles.

#### 2.5.2. Environmental Data at the Case-Study Site

^{−1}from growth data by Garen et al. (2004). Simulations started for 1 mm individuals (0.02 g of Dry Flesh Mass—DFM), a biometry corresponding to post-settled organisms. Results of the calibration are presented in Figure A2 in Appendix A. A good level of agreement between observations and simulations was obtained for shell length and dry flesh mass, a biological variable often used in bioenergetics models to assess the consistency of the simulations.

#### 2.6. The Relation between Environmental Factors, Growth, and the Start of Macro-Colonization

_{t,τ}). The parameterization of the function can integrate the environmental variables. Temperature is a variable of the DEB model, but Chl. a concentration is the main driver of growth. It was therefore decided to parameterize the Gamma process only with Chl. a. However, due to potential coupled effects between temperature and Chl. a, we analyzed the correlation between temperature and growth over the time-series. From ΔS obtained from DEB simulations, the scatter diagram of ΔS vs. temperature showed that there was no significant correlation between these two variables with a Pearson correlation coefficient ρ = 0.21 (Figure 4). That means that temperature is not a key driver of shell growth. On the contrary, there was a structured relationship between growth and Chl. a (Figure 5). It can be noted that uncertainty increases when Chl. a increases. Moreover, there is a ΔS plateau showing that the capability of an individual to grow is limited by the additional food supply: a concentration higher than 8 µg·L

^{−1}does not lead to a larger ΔS. This is due to a well-described physiological phenomenon of maximum somatic growth in bivalves [39,40].

#### 2.7. Chlorophyll Data Aggregation for Growth Computation

_{(.)}is the aggregated Chl. a; T

_{(i)}, is a 10 day period, and Chl(t), is the linear equation of Chl. a obtained from linear interpolation between adjacent measured values for a colonization period, and n is the number of 10 day time intervals after (i), in which the data aggregation is performed. The best correlation between ΔS and Chl. a has been obtained for a monthly aggregation (3 time intervals, n = 2 in (2)). This time-step preserved the spring bloom typical of the seasonal dynamic of phytoplankton at the study site latitude.

^{2}of 0.74 (Figure 6):

^{−1}was identified as the threshold beyond which ΔS remained constant at 0.235 cm/time interval. The Chl. a time-series was then truncated with the mentioned threshold to significantly improve the convergence of the Gamma process parameterization, without an important degradation of the database. Note that this threshold depends on the metabolism of the organism and is, therefore, species-specific.

#### 2.8. Non-Stationary Modeling of Shell Growth through Stochastic Gamma Process

#### 2.8.1. Growth Approximation through Gamma Processes Meta-Models

_{S}and a scale function β

_{S}(4). We discredited time horizon into equal intervals of length τ = 10 days. Then, the state-dependent non-stationary and bivariate Gamma process was represented as a series of state-stationary Gamma processes in each time interval. The rate of the deterioration process can thus be considered as the process resulting from the Gamma process variations from one time-interval to another. The deterioration increment in a given time interval ΔS

_{t,τ}has been considered to be a random variable with a shape function (α

_{S}) dependent of the present deterioration state S

_{t,τ}and a second variable, the state of chlorophyll-a concentration C

_{t,τ}. Thus, for each time step τ, we have:

_{t,τ}is the shell length for each time interval of τ and α

_{S}and β

_{S}are the shape and scale functions of the Gamma process, respectively. To simplify the modeling of this process, it has been assumed that the scale function β

_{S}was constant and Gamma process was only governed by the shape function [11].

#### 2.8.2. Parameter Estimation of the Gamma Process (Learning Phase)

#### 2.8.3. Stochastic Simulation from Gamma Process (Propagation Phase)

_{S}, which controls the response dispersion of the Gamma process. Note that there is also a statistical bias when estimating standard deviation from the DEB time-series due to the limited amount of data (17 trajectories).

#### 2.9. Effect of Marine Growth and Hydrodynamic Forces on Jackets

- (i)
- (ii)
- evaluation of hydrodynamic forces by the physical modeling of marine growth characteristics obtained from in-situ measurements [6,48]. These studies were based on inspections carried out during survey campaigns. They advocate guidelines for the probabilistic modeling of hydrodynamic forces at a given time. The biofouling database has been analyzed to propose a model of marine growth evolution and to update the design criterion. A physical response surface matrix has been proposed in order to provide a probabilistic modeling of the environmental loading on jacket type offshore structures. The key parameter is the increase of the structural diameter due to the marine growth thickness.

#### 2.9.1. Effect of Marine Growth on Morison’s Equation

_{Morison}is the hydrodynamic force per unit length of the member (N/m), F

_{D}is the drag force per unit length of the member (N/m), F

_{I}is inertia force per unit length of the member (N/m), C

_{D}is the drag coefficient, C

_{M}is inertia coefficient, ρ is the density of water, D is member diameter (m), and u is velocity of wave’s water particles (m/s), $\dot{u}$ is the acceleration of wave’s water particles (m/s

^{2}). u and $\dot{u}$ are computed by Stoke’s model [50] from the knowledge of metocean data: wave height H and period T.

_{e}. The surface Roughness k is the average peak-to-valley height of hard growth organisms and the effective member diameter De can be obtained as:

_{c}is the outer diameter of the clean member and Th is the biocolonization thickness (i.e., the mean of distributed thickness around the diameter) obtained by circumferential measurements [3]. API [3] gives the relationship between De and the steady-flow drag coefficient (C

_{DS}) (9).

_{D}is then computed from the knowledge of C

_{DS}and the Keulegan-Carpenter number KC

_{mg}according to [3,6].

_{e}and Keulegan-Carpenter KC numbers are essential for characterizing the flow regime [6]. For most offshore jacket structures in extreme conditions, Reynolds numbers are put into the post-critical flow regime, where the steady-flow drag coefficient C

_{DS}for circular cylinders is independent of Reynolds number [3,53].

#### 2.9.2. Stochastic Modeling of Marine Growth and Hydrodynamic Parameters

_{t,τ}deduced from the simulated individual shell length time-series S

_{t,τ}for blue mussels in each time interval. For simplicity at this step of modeling, it has been assumed that the individual shell length time-series S(t) gives the average size time-series Th

_{t,τ}with a multiplying uncertain factor (10): it follows a uniform distribution with support [0.3; 0.6] at each of the i 10 day periods.

_{t,τ}based on individual shell length time-series S

_{t,τ}(11), with a random factor following a uniform distribution with support [0.2; 1]. The latter is a model error for modeling the uncertainty when quantifying the real effect of a randomly distributed roughness around the component. Note that intensive developments on underwater image processing are emerging [54,55,56], enabling one to envisage progress in on-site measurements. Recent works investigate the relationship between non-homogenous roughness and loading [57,58]. The wide range of uncertainty will, therefore, decrease in the next decade. Consequently, the error of computation of equivalent roughness is significant and the interval in (11) is large: it includes the stochastic distribution of shells around a tubular component and the error of model for computing the equivalent roughness.

_{DS}(t) time-series have been simulated according to (9).

_{t,τ}in each time interval τ from (4). Parameters of Gamma processes Th

_{t,τ}, and k

_{t,τ}are dependent of individual shell length St,τ and the hydraulic parameters (Re

_{mg}, KC

_{mg}), and therefore drag coefficients C

_{D}, depend on individual shell length S

_{t,τ}and the couple of wave height and period (H, T).

_{t,τ}affect the hydrodynamic coefficients through the relationships between the hydraulic parameters (R

_{e}, KC) and the diameter of the elements, which is dependent of the coefficient of Th

_{t,τ}itself [6]. The next section will explain how these cross-effects are accounted for.

#### 2.9.3. The Stochastic Modeling Wave Loading in the Presence of Marine Growth

- (#1) Statistical Identification: the employed parameters are the heights of extreme waves H and associated periods T. They are modeled with a random variable, whose probability is conditioned by the wave direction θ;
- (#2) A kinematic model for the fluid for computation of water particle velocity: the Stokes model [50] is used. It assumes that the fluid is Newtonian and irrotational and the trajectory of the fluid particles is elliptical. The kinematics field deduced from the velocity potential can be defined at any point M of coordinates x and z. The maximum velocity u
_{m}(#5) is deduced and is used in the computation of KC and Re (#6). - (#3) The fluid-structure Interaction model: this level is involved in the hydrodynamic coefficients determined by using the recommendation of [3].
- For the probabilistic modeling of CD, in order to avoid multiplying the case studies, only vertical elements under the wave crest are analyzed. This implies high horizontal speeds and accelerations that generate very small forces, which means that the inertia forces in (7) are very low and will be neglected in the following.
- (#4) The colonized diameter D
_{e}(t) is a stochastic process that results from the increase Th(t) of the initial radius of the clean component. Starting from (8), the diameter is computed by multiplying D_{c}by the factor θ_{mg}. The latter is computed from the thickness Th(t) (12):$${D}_{e}={D}_{c}+2\overline{th}={\theta}_{mg}{D}_{c}\mathrm{with}{\theta}_{mg}=\left(1+\frac{2T{h}_{t,\tau}}{{D}_{c}}\right).$$

_{e}> 5 × 105) by using 100 year-return wave characteristics, the drag coefficient does not depend on R

_{e}but rather on KC

_{mg}and C

_{DS}. Note that API ([3], section C2.3.1b7, p. 143 and p. 145) provides, in fact, a piecewise model on two intervals depending on KC or KC/C

_{DS}and the scales of these models are different. It results in two effects on the evolution of the drag force (C

_{D}): first for some values of C

_{DS}it is the cause of discontinuity of the model at KC = 12 and second, it is very difficult to analyze directly the effect of C

_{DS}. This is visible in Section 3.2.

_{mg}values. Moreover, it gathers wave and wind-sea values and the spectrum is very similar to the one in French Atlantic offshore sites. Using meteocean data from this region allowed us to cover a large range of KC

_{mg}to better illustrate the non-linear effects of marine growth on the drag coefficient evolution and hence on the load probabilistic distribution. This covers almost all configurations of Atlantic French offshore sites. Joint distribution of the extreme height and period for a return period of 100 years for the Gulf of Guinea are simulated based on [7]. It has been provided by recombination of sea states from the knowledge of the H-T scatter diagram. Representation of the joint distribution for wave height and the 100 year return period is presented in [59]. Note that breaking waves are not considered here.

_{DS}(C

_{D}in steady flow) time-series (C

_{DS}(t) = f(kt,τ/D

_{e}). In this paper, knowing C

_{DS}, a numerical fitting of the curve of C

_{D}= f(KC

_{mg}) given in [3] is used and is plotted in Section 3.2 (lower multi-linear curve for the smooth cylinder).

## 3. Results

#### 3.1. Simulation of the Drag Force Evolution from the Stochastic Time-Series of blue Mussels

_{t}is the numbers of time-series for the typical macro-colonization year of S(t), which should be selected randomly, N

_{s}is the sample size (here equal to 30,000), and P

_{t}is the occurrence probability of the typical macro-colonization year (Table 2). The simulation procedure is illustrated in Figure 13. The KC, C

_{D}, and drag forces are then computed according to the flowchart reported in Figure 10.

#### 3.2. Statistical Analysis of the Transfer of Distributions

_{D}along with its support due to the dependence of C

_{DS}to k/D

_{e}in (9). The mixing of sources of uncertainties due to independent macro-colonization inception time and independent growth builds finally a normal distribution as expected from the Central Limit Theorem.

_{D}as a function of KC for the three above-mentioned 10 day periods. First, we plot the bounds of the relationship (CD)–(KC) with lower and upper lines that depict, respectively, the smooth and roughened cylinders’ drag coefficients. The discontinuity comes from the discontinuity of curves in the standards generated by the various scales (CD/CDS, CD, KC, KC/CDS) used around KC = 12. Note that this discontinuity for the smooth and roughed cylinders follows, respectively, a potential positive and a negative jump of the C

_{D}. Second, the scatter plots are reported in red, moving from the lower part to the upper part from 11th to 37th decade 10 day periods. Consequently, the distribution of C

_{D}is affected. An important point is that the distribution maintains two modes, the uppermost being around 1.2 and the lowermost following the shift of the non-linear transfer function, from 0.2 to 0.6 (see the 37th 10 day period). It demonstrates the evolution of the drag coefficients C

_{D}for the individual shell length from the non-linear transfer of the distribution of KC and during the probabilistic macro-colonization year (from the 11th to the 37th 10 day period). Finally, the probability of the highest values (typically 1.8) increases with time, which is a key result because it will potentially affect the distribution tail of the corresponding loading and decrease structural reliability. There is not a clear distinction between the macro-colonization inception times because of the mixing of all typical macro-colonization years. Indeed, the mixing of a large amount of potential macro-colonization inception times does not allow one to distinguish between the contribution of each year in terms of the mode in the distribution.

## 4. Discussion

_{e}and C

_{DS}(Figure 16). The distributions of D

_{e}are mono-modal because of the combination of all typical macro-colonization years. The distributions of C

_{DS}are bimodal and become mono-modal from the smooth to the ultra-roughened condition at the end of the macro-colonization period.

_{T}is plotted on the same Figure 16 to better illustrate differences in distribution (mode and tails) and the transfer of these distributions. The drag force is exponentially distributed. The right distribution tail moves to higher values according to time, thereby decreasing the reliability. We analyze this distribution tail after the computation of F

_{T_MAX}(note that distributions are bounded) and the fractiles F

_{T(90%)}, F

_{T(95%)}. Figure 17 shows the evolution of these statistics after each 10 day period. The latter increase smoothly with time except for the increase during one month and a half (from 10 day period 11 to 18). Finally, there is a great difference between the extreme values (F

_{T_MAX}) and the fractiles (F

_{T(90%)}and F

_{T(95%)}, confirming a long distribution tail that was observed already in Figure 16.

## 5. Conclusions

- -
- Environmental: due both to the physics of waves (height, period) and water parameters (temperature and chlorophyll-a).
- -
- Modeling: with an uncertainty of modeling from the shell size to the thickness and the roughness in the sense of API regulation.
- -
- Biological: accounting for the inter-individual variability.
- -
- Moreover, calculation of hydrodynamic forces due to the biocolonization using meteo-ocean data as well as biological data is a complex task and generates two types of difficulties.
- -
- First, the distribution of input variables that can be multi-modal (e.g., individual shell length) due to the various macro-colonization inception times.
- -
- Second, the nonlinear transfer from the Keulegan Carpenter number to drag coefficient generates bimodal distributions from mono-modal ones.

- -
- A single species was studied in a place where we can find barnacles and even algae. For the latter, relationships for the computation of drag coefficients are less developed and research is required.
- -
- There is uncertainty in the definition of roughness and its use by engineers, which is the reason why an uncertainty of modeling is added in this paper. Recent works [60] have proposed some improvements, but this is still an open area of study. Quantification from on site inspections is possible [54], thereby opening a new area for more representative tests in laboratories.
- -
- The probability of the occurrence of storms depends on seasons and could be introduced to reduce the conservatism.
- -
- Effects of the Cd variations on dynamics should be introduced to expand the method to fatigue assessment.
- -
- In the same manner, inertia forces and current could be added to get a more global influence of marine growth.

## 6. Patents

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Additional Information about the Growth of Blue Mussels

**Figure A1.**Schematic annual growth curve of individual blue mussels illustrating the acceleration and deceleration in the growth rate.

**Figure A2.**Calibration of the mussel DEB model used in this study to simulate shell length. (

**a**) Mussel growth in length (mm), (

**b**) Observed vs. simulated length; the dashed line corresponds to the 1:1 line, (

**c**) Mussel growth in dry flesh mass (DFM, g); Note the strong decrease of DFM corresponding to spawning in September, (

**d**) Observed vs. simulated DFM; the dashed line corresponds to the 1: 1 line.

**Figure A3.**The relationship between macro-colonization starting times (end of the initiation phase) expressed in 10 days; each color represents a year of the 1996–2012 time-series.

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**Figure 2.**Inter-annual variations (1996–2012) of water temperature (°C) (

**a**), and chlorophyll-a (Chl. a, µg·L

^{−1}) (

**b**), at Le Croisic sampling station (Loire-Atlantique, France). Data from Ifremer/Quadrige/Rephy

^{©}.

**Figure 3.**(

**a**) Individual annual shell length trajectories simulated by a mussel Dynamic Energy Budget (DEB) model and (

**b**) corresponding final length.

**Figure 4.**Scatter diagram of the variations of mussel shell length (ΔS) vs. temperature for 10 days periods during the 17 year time series (each color represents a year of the 1996–2012 time-series).

**Figure 5.**Scatter diagram of the variations of mussel shell length (ΔS) vs. Chlorophyll a for 10 days periods during the 17 year time series (each color represents a year of the 1996–2012 time-series).

**Figure 7.**Individual growth trajectories obtained with the DEB model for each year of the 17 year time series (solid red lines) compared with simulated individual growth obtained with the Gamma process approach (dotted blue lines). The time unit represents 10 day periods.

**Figure 8.**Comparison of average (solid lines) and standard deviation (dashed lines) of shell length from Gamma process (blue lines) and DEB model (red lines). Time unit represents 10 day periods.

**Figure 11.**Simulated individual shell length of blue mussels for the 2nd typical macro-colonization (11-12-17). 200 simulations are presented.

**Figure 13.**Schematic procedure of probabilistic individual shell length time-series from the typical macro-colonization year.

**Figure 14.**Evolution of shell length distribution as a function of time for three selected 10 day periods: (

**a**) 15th, (

**b**) 18th, (

**c**) 37th.

**Figure 15.**Distributions of the drag coefficients (C

_{D}) as a function of (KC) values for three selected 10 day periods and Monte-Carlo simulations (cloud of red points): (

**a**) 11th, (

**b**) 18th, (

**c**) 37th.

**Figure 16.**Comparison of the distribution of hydrodynamic parameters for the selected 10 day periods: (

**a**) 11th, (

**b**) 18th, (

**c**) 37th.

**Table 1.**Inter-annual development types for three main spawning events (S: slow initiation phase, F: fast initiation phase).

Development Type | Occurrence | Probability |
---|---|---|

SSS | 2 | 0.12 |

SSF | 10 | 0.59 |

SFS | 0 | 0.00 |

SFF | 5 | 0.29 |

**Table 2.**Date of start of macro-colonization, expressed in 10 day periods, for three main spawning events of blue mussel.

Start of Macro-Colonization | Occurrence | Probability | ||
---|---|---|---|---|

11 | 14 | 15 | 2 | 0.12 |

11 | 14 | 17 | 2 | 0.12 |

11 | 15 | 16 | 1 | 0.06 |

12 | 15 | 16 | 3 | 0.18 |

13 | 14 | 17 | 1 | 0.06 |

13 | 16 | 17 | 3 | 0.18 |

14 | 15 | 18 | 1 | 0.06 |

14 | 17 | 18 | 1 | 0.06 |

15 | 16 | 19 | 2 | 0.12 |

16 | 17 | 20 | 1 | 0.06 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ameryoun, H.; Schoefs, F.; Barillé, L.; Thomas, Y.
Stochastic Modeling of Forces on Jacket-Type Offshore Structures Colonized by Marine Growth. *J. Mar. Sci. Eng.* **2019**, *7*, 158.
https://doi.org/10.3390/jmse7050158

**AMA Style**

Ameryoun H, Schoefs F, Barillé L, Thomas Y.
Stochastic Modeling of Forces on Jacket-Type Offshore Structures Colonized by Marine Growth. *Journal of Marine Science and Engineering*. 2019; 7(5):158.
https://doi.org/10.3390/jmse7050158

**Chicago/Turabian Style**

Ameryoun, Hamed, Franck Schoefs, Laurent Barillé, and Yoann Thomas.
2019. "Stochastic Modeling of Forces on Jacket-Type Offshore Structures Colonized by Marine Growth" *Journal of Marine Science and Engineering* 7, no. 5: 158.
https://doi.org/10.3390/jmse7050158