Oil spill contamination in the open sea has produced some of the worst environmental disasters in history [6, 22]. In the case of the Prestige accident (Galicia, Spain 2002), more than 10 million gallons of heavy/residual fuel oil were spilled [27] and thousands of kilometers of coastline in Spain, France and Portugal were polluted [5]. This spill is considered as the largest environmental disaster in the history of both Spain and Portugal, and the cost of the disaster was evaluated to be more than 770 million euros [20]. Because of its large geographical spread, the spill reached all types of marine habitat, from offshore depths to shallow creeks. The worst affected habitats were in coastal areas, and this includes land damage caused by clean-up operations. Furthermore, this had economic repercussions on the inshore fishing and shellfish sector (e.g., [31]). The Prestige disaster was similar to that of the Exxon Valdez spill in terms of bird mortality, and it has been considered as one of the non-natural events most deadly to wildlife ever to have occurred in Europe. It was estimated that more than 50% of the sea birds and otters of the area were killed.
Another case is the ship ‘Oleg Naydenov’ that sunk near the Canary Islands coast, Spain, on 14 April, 2015 [16]. The tanks of this ship were filled with around 1400 tons of oil. The oil spilled into the sea with a flow estimated between 5 to 10 litres per hour. During several weeks, this oil spill provoked the pollution of the ‘Gran Canaria’ Island with several oil spots reaching its coastline.
One of the cleaning-recovering techniques for oil spills with light- to medium-viscosity oils, is the use of skimmer ships [7, 10]. These ships might use various pumps distributed along their waterline to suck the oil remaining at the surface of the water directly into storage units. These vessels move inside the oil spots to clean them as quickly as possible.
In order to be able to improve the efficiency of this pumping process, we first simulate the movement of the oil spots in both space and time using our Eulerian mathematical model proposed in [2, 16]. We note that in the literature, the prevailing models to forecast the fate oil spills on the sea are based on Lagrangian approaches [29, 30]. However, in our case, we are considering in our model, additional effects related to the cleaning process, such as the transport and sink effects generated by the skimmer ships, which require to have a continuous (not particle wise) representation of the distribution of oil at each point of the considered area and at each time of the simulation. This explains why we have chosen an Eulerian approach. We then proceed to model the trajectory for a skimmer ship, and using a global optimisation algorithm, we optimize the trajectory to maximise the amount of oil pumped on a fixed period of time. Here, two scenarios are considered: cleaning on the whole area under study, or giving priority to prevent the oil from reaching the coast.
The technical ability of different pumps to suck several types of oil (from light to medium), have a large influence on the efficiency of the pumping process [15]. In this work, we present results comparing three types of pumps (considering only their pumping effective power), to show the advantage of creating optimal trajectories for the skimmers, independently of its pumping abilities. In particular, during the experiments proposed here, we consider the characteristics of the so called ‘Controlled Floating Skimmer’ pump, build by the Novetec company (see
We also highlight the fact that the proposed methodology may be applied to any other cleaning method requiring a trajectory planning.
The content of this article is as follows: In Section 2.1, we describe the main aspects of the Eulerian mathematical model we use to simulate the motion of the oil, taking into account the pumping process of a skimmer ship. In Section 2.2, we present the formulations of different objective functions to optimise the trajectory of the skimmer ship. In Section 3.1, we describe numerical experiments based on the 2016 Oleg Naydenov and the 2002 Prestige oil spills [16], citing the sources of data, to validate the forecast of the movement of the oil obtained with our model and to compare them with satellite images and discuss the results. Then, in Section 3.2, using the results obtained by the model, we solve several optimization problems to find optimal trajectories for the pumping process and analyse the behaviour of the solution regarding some key parameters (as the pumping power of the skimmer). Finally, some conclusions are made in Section 4
Mathematical modelling of the transport and diffusion of an oil spill in the sea is of high interest to remediate the environmental impact (e.g., [24, 25, 26]).
We have developed an Eulerian model for the case where the density of the pollutant is smaller than that of one of the sea water (so that it remains at the surface) and assuming that the layer-thickness of the pollutant
We denote by
As we are interested in this work in studying the effect of a skimmer ship capable of pumping the spilled oil, the evolution of
From a practical point of view, a skimmer ship can be composed of multiple pumps, cleaning the water along the vessel waterline. However, for simplicity, we assume that there is only one pump, neglecting the length of the ship (small compared to the size of), which is a circle of radius
In order to avoid the undesired modelling effect of diffusion propagating at infinite speed [11], we use a non-linear diffusion term. We have also included a boundary condition with suitable absorbing properties to simulate the behaviour of the solution near the boundary of the computational domain.
For a detailed description of this model, see Reference [16].
The mathematical model follows:
where:
The function
For the numerical solution of this equation, we use a finite volume discretization method coupled with an operator-splitting approach for the pumping term to reduce the computational time. Furthermore, to limit the artificial diffusion effect, typically produced by this kind of numerical model [11], we use second-order accurate time discretization schemes with nonlinear limiters to treat the transport. The full scheme of the considered numerical model can be found in Reference [16].
As mentioned in Section 1, we address the problem of finding an optimal trajectory for the skimmer ship, for a particular oil spill scenario during a fixed time interval [0,
From a general point of view, we consider optimization problems of the form:
where
In this work, we have considered two particular optimisation problems associated with two different formulations of the objective function
For the given time where ( For the given time where coef(
These formulations take into consideration the evolution in time and space of the pollution concentration (obtained by solving the model), and specifically in the case of formulation (4), where the distances between the oil spots and the coast are variable.
In order to find numerically a smooth optimal pump trajectory (i.e., without sharp corners), we consider trajectories built by using cubic spline interpolation through
The set of interpolation points, denoted by
where
Given an interpolation point expressed in Cartesian coordinates
The resulting interpolated trajectory is denoted by
Furthermore, we need to avoid the ship leaving the domain of study Ω. To accomplish this, we project the trajectory
Thus, the numerical optimization problem that we solve, is of the form:
where
and if
A graphical representation of the function coef(.) is given in Figure 1.
Since Problem (5) exhibits many local minima[13], we need to use a global optimization method capable to find one global solution.
In order to solve optimisation Problem (2), we have developed an original hybrid global optimization method. This method is based on the combination of a particular Genetic Algorithm (GA) [12, 14, 28] and a Multi-layer line search algorithm [17, 18, 19] to improve the GA performances. In addition, to reduce the computational time required by the optimization process, we have designed a parallel version of this method [2]. A particular Matlab implementation of this optimisation method, called Global Optimisation Platform, has been used during this work to obtain the results presented in Section 3.2. It can be downloaded at:
In this section, we present the numerical experiments used to check the ability of the model to reproduce real observations. These experiments, presented in Sections 3.1.1 and 3.1.2, aim to validate the oil concentration evolution predicted by our model when no pumping process is considered.
To accomplish this, we compare the model results with real satellite images of the Prestige and the Oleg Naydenov hazards. Indeed, as done in other similar works (see, e.g., [4]), in case of Eulerian models, images give a reasonable representation of the continuous distribution of the oil spill at a given date and can be compared with the continuous distribution returned by the model. In the case of Lagrangian models, the use of buoys data for model validations seems to be also appropriate (see e.g., [1]).
On 13 November, 2002, the ‘Prestige’ ship started to spill oil in the open sea near the Galician coasts, Spain [27]. The authorities decided to send the ship far from the Spanish coasts. The ship sank in the Atlantic Ocean on 19 November 2002. Around 10 million gallons of crude oil were spilled, polluting thousands of kilometres of coastline in Spain, France and Portugal [5]. This spill is considered as the largest environmental disaster in the history of both Spain and Portugal, and the cost of this hazard was evaluated to be more than 770 million euros [20]. We use our mathematical model, without the pumping process, to simulate the oil concentration evolution from the beginning of the Prestige event on 13 November 2002 up to 17 November 2002 (the only available clear satellite image of the situation was taken this day, before the Prestige ship broke up). Considering this time interval, we use the following model parameters [9]:
The simulation area was taken as Ω ⊂ [−12.5, −7.5]×[42, 44.5] (in the longitude-latitude coordinate system) which is assumed to be large enough to avoid the oil concentration leaving this domain during the considered time interval, and the considered Spanish land is shown in Figure 2. The velocity fields of The trajectory followed by the Prestige ship was taken from the literature [8, 21]. To our knowledge, the exact amount of oil For the numerical finite volume scheme used to approximate the solution of model presented in Section 2.1, we consider a 100 × 100 spatial mesh and a time step of 1 hour. All other parameters are given in Reference [16].
Taking into consideration these values, we present in Figure 2 the solution given by our numerical model on 17 November 2002. In the same figure, we also show the satellite image taken by the Envisat ASAR satellite (property of the European Spatial Agency) at the same date (
We have also validated our method with the case of the Oleg Naydenov hazard in [16]. More precisely, the ship ‘Oleg Naydenov’ sank near the Canary Islands coasts, Spain, on 14 April 2015. The tanks of this ship were filled with around 1400 tons of oil. During several weeks, this oil spill provoked the pollution of the ‘Gran Canaria’ Island with several oil spots reaching its coastline. Taking into consideration this case, we present in Figure 3 the solution given by our numerical model on 12 April 2015 and the satellite image taken by a NASA satellite on the same date (
We are now interested in applying the optimisation method presented in this paper to solve several skimmer ship optimal trajectory problems by considering, as a study case, wind and sea currents data from the Prestige hazard. These problems are designed to study the efficiency of the optimal trajectories considering the objective function formulations (3) or (4) and several values for the power of the pump. Our main objective here is to show the advantage of using optimal trajectories for the cleaning ship. We note that this methodology can be applied for the design of any other cleaning method based on trajectory planning.
We study the Prestige hazard with the model parameters presented in Section 3.1.1. However, the model is now run from 13 November 2002 up to 19 November 2002 (i.e., the date when the Prestige ship broke up). The distribution of oil in the open sea on 19 November 2002 simulated by the model (without cleaning process) and the trajectory followed by the Prestige ship are presented in Figure 4. Furthermore, in the same figure, we also give a 3D representation of the oil concentration, in order to better visualise the distribution of oil contamination in Ω. For this reason, from now on, each time the oil concentration is shown, the 3D representation is also displayed.
Now, we activate the pumping process by considering three levels of pumping power
This first value of The highest value of Although the global pumping system is one of the most powerful, due to technical restrictions this ship cannot perform pumping in movement. Furthermore, it has only been used once during the Gulf of Mexico Deepwater Horizon oil spill disaster in 2010 and was proven inefficient (see: Finally, the value of
In all these cases, we assume that the pumps are working in ideal conditions, as we omit the decrease in the pumping efficiency due to the type of oil and its physical transformation (e.g., emulsification).
For each value of the pumping power
The skimmer ship trajectory starts from the position (-9.4, 42.75) (longitude, latitude in degrees) and is parameterised by
In order to study the advantage of the optimised results obtained here, we compare the objective function values obtained with the computed optimal trajectories using the straightforward trajectory that simply follows the Prestige ship (this trajectory seems to be the first intuitive option taken by the authorities in the case of oil spill accidents). We make this comparison using different values of the pumping power
We first present and analyse the results obtained when considering the objective function formulation (3), which aims to reduce the amount of oil over the whole area. In this case, the considered experiments are
Objective function (3) value (Exp. OPT-0.3-(3) PT-4-(3) OPT-8-(3) 50,915 32,241 31,627 6 40 42 52,712 36,968 23,574 2 31 56
As we can see in Table 1, for the scenarios associated with pumping powers
We now focus on the results obtained with the formulation (4), that is cleaning the coast, and the experiments
Objective function (4) value (Exp. OPT-0.3-(4) OPT-4-(4) OPT-8-(4) 2.144e10 1.366e10 1.105e10 11 43 55 2.3632e10 2.2057e10 1.330e+10 2 9 45
Objective function (4) value (Exp. OPT-0.3-(4) OPT-4-(4) OPT-8-(4) 50,973 41,233 40,136 6 24 26 52,712 36,968 23,574 2 31 56
As we can observe in Figures 7 and 8, in all cases, the optimized trajectories remain near the coast, which was expected due to the definition of the cost function (4). Regarding the final values of the objective functions presented in Table 3, we see that all optimised trajectories give better results (i.e, lower objective function values) than the trajectory following the Prestige ship. Even in the scenario of using the most powerful pump
Focusing on the final oil concentration depicted in Figures 9 and 10, we observe that the contamination has been reduced near the coast but remains almost unchanged in the open sea. Thus, a better option should be to clean both coast and open sea. For instance, we can use a skimmer ship following the Prestige vessel and another ship following the optimal trajectory using objective function (4). In order to illustrate this idea, we consider two skimmer ships with
In this article, we have used an original Eulerian mathematical model, discussed in References [2, 13], to simulate the movement of oil spills in the open sea, taking into account the diffusion, the wind and sea currents, the motion of the contamination source and the effect of a skimmer ship to clean the oil. With this model, without considering the pumping process, we were able to approximate the movement of the spill produced by the Prestige vessel in 2002 and that of the Oleg Naydenov in 2016, both in Spain.
Then, we have introduced an optimisation procedure to design optimal trajectories of the considered skimmer ship, improving the amount of pumped oil. Furthermore, we have proposed two different objective functions: one for cleaning the whole domain, and a second formulation that prioritises the coast cleaning. We have used a suitable optimisation algorithm, to find the optimal trajectories for these two different formulations.
To illustrate the importance of our approach, we have solved numerical experiments on a study case created by using some data from the Prestige hazard information. We are aware that the type of oil spilled by the Prestige ship, was not suited for pumping (heavy/residual fuel oil), and, thus, we are not claiming that the results presented here are realistic for this specific scenario. Indeed, the skimmer may deal with light and medium viscosity oil. However, we have used the meteorological data and the trajectory of the ship to create test cases.
We have compared the amount of pumped oil when following the optimal trajectories or when following the Prestige vessel, using three different pumping power alternatives. The results show that when the general formulation of the objective function does not prioritise the coast: (i) for small or medium pumping power, the optimal trajectories remain near the coast anyway (as the highest concentration of oil has gone there because of the effect of the wind and sea currents) and (ii) for high pumping power, the optimal trajectory seems to follow closely the Prestige vessel. This can be explained by the fact that a high pumping power would clean the pollutant as it leaves the vessel.
For the case of the second formulation, independently of the pumping power, the optimal trajectories are much more efficient in cleaning the polluted coast areas than the trajectory that simply follows the polluting ship.
Furthermore, if the option of using two skimmer ships with a high pumping power is available, a good option is to use one skimmer ship near the coast following the optimal trajectory for the function defined in (4), and another one following the polluting source. In this case our results show that we can clean 88 percent of the pollutant oil, in the particular case studied here.
As a final consideration, we insist on the fact that the methodology proposed here is not only limited to skimmer ships, but can also be adapted to any other cleaning devices that require trajectory planning.
In future work, we will improve our model by including additional physical effects on the oil spill evolution, such as emulsification, evaporation or tide currents.