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This article was downloaded by: [Umeå University Library]On: 22 April 2014, At: 02:37Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

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A Lagrangian/Eulerian oil spill model for continental

watersCédric Goeury Engineer-Researcher

a, Jean-Michel Hervouet (IAHR Member) Senior

Researcherbc

, Isabelle Baudin-Bizien Engineerd & François Thouvenel Engineer

e

a Saint-Venant Laboratory for Hydraulics, Université Paris-Est (Joint Research Unit

Between EDF, CETMEF and Ecole des Ponts ParisTech), Chatou, Franceb EDF R&D, National Hydraulics and Environment Laboratory (LNHE), Chatou, France

c Saint-Venant Laboratory for Hydraulics, Université Paris-Est (Joint Research Unit

Between EDF, CETMEF and Ecole des Ponts ParisTech), Chatou, France

d VEOLIA Environnement Recherche et Innovation (VERI), Rueil-Malmaison, Francee VEOLIA Environnement Recherche et Innovation (VERI), Rueil-Malmaison, France

Published online: 03 Mar 2014.

To cite this article: Cédric Goeury Engineer-Researcher, Jean-Michel Hervouet (IAHR Member) Senior Researcher, IsabelleBaudin-Bizien Engineer & François Thouvenel Engineer (2014) A Lagrangian/Eulerian oil spill model for continental waters,Journal of Hydraulic Research, 52:1, 36-48, DOI: 10.1080/00221686.2013.841778

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Journal of Hydraulic Research Vol. 52, No. 1 (2014), pp. 36–48

http://dx.doi.org/10.1080/00221686.2013.841778

© 2014 International Association for Hydro-Environment Engineering and Research

Research paper

A Lagrangian/Eulerian oil spill model for continental waters

CÉDRIC GOEURY, Engineer-Researcher, Saint-Venant Laboratory for Hydraulics, Université Paris-Est (Joint Research Unit Between EDF, CETMEF and Ecole des Ponts ParisTech), Chatou, France

Email: [email protected]

JEAN-MICHEL HERVOUET (IAHR MEMBER), Senior Researcher, EDF R&D, National Hydraulics and Environment Laboratory(LNHE), Chatou, France; Saint-Venant Laboratory for Hydraulics, Université Paris-Est (Joint Research Unit Between EDF,

CETMEF and Ecole des Ponts ParisTech), Chatou, France

Email: [email protected] (author for correspondence)

ISABELLE BAUDIN-BIZIEN, Engineer, VEOLIA Environnement Recherche et Innovation (VERI), Rueil-Malmaison, France Email: [email protected]

FRANÇOIS THOUVENEL, Engineer, VEOLIA Environnement Recherche et Innovation (VERI), Rueil-Malmaison, France Email: [email protected]

ABSTRACTThe application of the European Water Framework Directive on water quality for human consumption and industrial activities creates a need for water quality assessment and monitoring systems. The MIGR’HYCAR research project was initiated to provide decisional tools for risks connectedto oil spills in continental waters. In this paper, the focus is set on a numerical model to simulate oil spill in continental waters and to estimate polycyclic aromatic hydrocarbons (PAHs) toxicity in the aquatic environment. A Lagrangian/Eulerian approach is proposed. The Lagrangian model

describes the transport of an oil spill near the free surface. The oil spill model enables to simulate the main processes driving oil plumes: advection,diffusion, evaporation, dissolution, spreading and volatilization. Though generally considered as a minor process, dissolution is important from the point of view of toxicity. To model PAHs dissolution in water, an Eulerian advection–diffusion model is used. Laboratory experiments were conductedto characterize the numerous kinetics of dissolution and volatilization processes. Model validation was carried out with the following test cases:transport processes based on the well-documented seawater spill (advection), weathering processes based on laboratory experiments cited in theliterature (spreading and evaporation processes) and artificial river test measurements (dissolution and volatilization).

Keywords: Continental waters; Lagrangian/Eulerian model; oil spill; PAHs dissolution; weathering processes

1 Introduction

Oil spills can be due to human error, accidental or voluntarydischarge of cargo residues, domestic or industrial tank over-

flows, leakage from fuel stations, traffic accidents, amongstother causes. When faced with hydrocarbon contamination of inland waterways, public authorities can seldom rely on ded-icated decision-making tools to intervene in an effective way.Whereas considerable management and monitoring resourcesare rapidly deployed for off- or inshore oil incidents, the morefrequent occurrence of continental water pollution is dealt withusing relatively modest means. A limited grasp of the natureand magnitude of such events often renders both industry and

government powerless in controlling their impacts (Goeury2012).

In the last three decades, many researchers have studied the behaviour and fate of oil spills. Amongst these oil spill mod-

els, two approaches are usually found: Lagrangian models andEulerian models. The Lagrangian models (Lonin 1999, Zhenget al. 2002) represent the oil slick by a large set of hydrocarbon packets. Each packet is advected by the action of currentand wind. However, the number of particles in these modelsmust be restricted in order to limit the computational time. InEulerian oil spill models (Tkalich et al. 2003, Papadimitrakiset al. 2006), the mass and momentum equations are solved for the oil slick. The main drawback of the Eulerian formulation

Received 15 May 2012; accepted 4 September 2013/Open for discussion until 31 August 2014.

ISSN 0022-1686 print/ISSN 1814-2079 onlinehttp://www.tandfonline.com

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38 C. Goeury et al. Journal of Hydraulic Research Vol. 52, No. 1 (2014)

Figure 1 Fate and transport of oil slick

∂hv

∂ t +

∂

∂ x(huv) +

∂

∂ y(hvv)

= − gh∂Z s

∂ y+ hF y + ∇ · (hνe ∇ (v)) (3)

where x and y are the horizontal Cartesian coordinates, t is time,u and v are the horizontal Cartesian components of the depth-averaged velocity, h is the water depth, νe is an effective diffusionrepresenting depth-averaged turbulent viscosity νt and disper-sion, Z s is the free surface elevation, F x and F y are the forcing

terms (e.g. friction).The depth-averaged turbulent kinetic energy (k ) and its

dissipation rate ( ) are given by the turbulent model k − :

∂k

∂ t + u

∂

∂ x(k ) + v

∂

∂ y(k )

= 1h

∇ ·

h

νt

σ k ∇ (k )

+ P − + P k v (4)

∂

∂ t + u

∂

∂ x() + v

∂

∂ y()

=

1h ∇ ·

h

νt

σ ∇ ()

+

k [C 1 P − C 2 ] + P v (5)

with P = νt

∂u

∂ y+

∂v

∂ x

∂ u

∂ y, P k v = C k

u3∗h

and

P v = C u4∗h2

(6)

where u∗ is the shear velocity on the bottom, C k = (1/

C f ) and

C = 3.6(C 2

C µ)/(C 3/4

f ) with C f the non-dimensionalfrictioncoefficient and νt = C µ(k 2/), C µ = 0.09, C 1 = 1.44, C 2 =1.92, σ k = 1.0 and σ = 1.3 (Hervouet 2007).

Telemac-2D solves the previous equation system using thefinite element method on a triangular element mesh. Telemac-2D can take into account the propagation of long waves, the bedfriction, the influence of the Coriolis force and meteorological

factors, the turbulence, sub- and supercritical flows, river andmarine flows, the influence of temperature or horizontal salinitygradients on density and dry areas in the computational domain,amongst other processes (Hervouet 2007).

2.2 Transport processes

Advection of particles is already included in the Telemac model.Therefore, here we discuss the advection field and how diffusionis uncorroborated.

Advection

On the free surface, the drifting of the oil slick is induced bythe flow velocity and by the action of wind. The oil slick driftvelocity is expressed as

uoil = uc + β uw (7)

where uoil is theoilslick velocity vector, uc is thecurrent velocityvector at the free surface, uw is the wind velocity vector above

the water surface and β represents the drift of the surface slick due to the wind.

In order to estimate the current velocity at the free surface,we assume a logarithmic profile for the vertical velocity. Con-sequently, the free surface velocity vector uc can be related tothe depth-averaged velocity vector u and the non-dimensionalfriction coefficient C f as

uc = u

1 +

1κ

C f

2

(8)

where κ is the von Karman constant (κ = 0.41).Inthisparagraph,thewindeffectontheoilspilldriftisstudied.

The force acting on a floating solid body with constant flow

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Journal of Hydraulic Research Vol. 52, No. 1 (2014) A Lagrangian/Eulerian oil spill model 39

Figure 2 Scheme of a solid body in water

velocity is

F = 12

ρSC d |U| U (9)

where U is the flow free surface velocity vector, S is the projected area of the solid body, ρ is the fluid density and C d is thedrag friction coefficient. If the solid moves with velocity v, it isnecessary to replace the fluid surface velocity vector U by therelative velocity vector ur = v − U in Eq. (9).

Newton’s second law at steady-state applied to the particleadvected by wind and current (Fig. 2) can be expressed as

ρwS cC d ,c|v − uc|(v − uc) + ρaS wC d ,w|v − uw|(v − uw) = 0(10)

where ρa is the air density, ρw is the water density, C d ,c is the

drag coefficient in thewater, C d ,w is the drag coefficient in the air,S w is the partial projected area of the solid body subjected to thewind, S c is the partial projected area of the solid body subjectedto the current, uc is the current water velocity vector and uw isthe wind velocity vector.

By decomposing the velocity vector of the solid body as v =αuc + γ uw, the following relationship is obtained:

v = uc + β uw

1 + βwith β =

ρaS wC d ,w

ρwS cC d ,c(11)

This approach can be applied to an oil slick by approximating thedrag forces with the friction forces exerted by the wind and thewater on the slick surface. Therefore, the partial projected areas,S w and S c, must be replaced by the complete slick surface. Thissurface is equal above and below the water surface. If the frictioncoefficients for water and air are also assumed to be equal, thewind drift factor can be simplified to

β =

ρa

ρw≈ 0.036 (12)

The oil spill drift velocity induced by wind action is 3.6% of the wind velocity vector. This theoretical result is close to theempirical drift factor usually used in oil spill models. Accordingto ASCE (1996), this drift velocity typically varies from 2.5 to

4.4% of the wind velocity vector, with a mean value comprised between 3 and 3.5%.

Diffusion

Eddies generated by turbulence affect the motion of petroleum particles in water and randomize their trajectory. Consequently, astochastic approach is adopted in order to take this phenomenoninto account. The non-conservative formulation of advection– diffusion equation (Eq. 13) is well adapted to modelling transportand dispersion of continuous contaminants, but, since the presentmodel uses a discrete particle description of contaminant transport and dispersion, a transformation must be applied to Eq. (13)to obtain a Lagrangian equation.

∂C

∂ t + u ∇ (C ) =

1h

∇ ·

hνt

σ c∇ C

(13)

where h isthewaterdepth, C is the depth-averagedpollutant con-centration, σ c represents the neutral turbulent Schmidt number and νt is the turbulent viscosity. The turbulent Schmidt number can be set to σ c = 0.72 (Violeau 2009).

The weathering process is dealt with in a further fractionalstep, and therefore not added to this equation which is used totreat only transport processes. A transformation must be appliedto Eq. (13) to obtain a Lagrangian equation. The first transfor-mation step consists of interpreting the concentration C (X, t ) asa probability P (X, t ) of finding a particle at a location X at a timet . Then, the development of the diffusion term in Eq. (13) leadsto the Fokker–Planck equation

∂ P

∂t = −

u −

1h

∇

hνt

σ c

∇ P + ∇ ·

νt

σ c∇ P

(14)

A stochastic solution to Eq. (14) is obtained by specifyingthe hydrocarbon particle position vector X(t ) according to thefollowing Langevin equation (Gardiner 2004):

X(t + δt ) = X(t ) +

u −

1h

∇

hνt

σ c

δt +

2νt σ c

δt ξ (t ) (15)

where δt is the time step, ξ (t ) is a vector with independent,standardized random components. In the above relationship,quantities h and νt are computed from Telemac-2D (Section 2.1)and the depth-averaged velocity vector u must be replaced bythe oil slick velocity vector uoil (Eq. 7).

2.3 Weathering processes

Spreading

Spreading is the most important weathering process. In fact, allmass transfer phenomena which occur during an oil spill areinfluenced by the area of the surface slick. Oil discharged into awater surface will immediately start to increase its surface area.This slick expansion is controlled by mechanical forces such

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as gravity, inertia, surface tension and viscosity (Fay 1971).However, the model proposed by Fay (1971) does not evenconsider the oil viscosity and would be more convenient for calm water. More recently, based on the layer averaged Navier– Stokes formulation proposed by Warluzel and Benque (1981)and Maroihi et al. (1992) have describedthespreading process asfollows:

S − arctan(aS ) = 4π µt (16)

with µ = σ wa − σ oa − σ ow

K and a =

23V o

2µ K ρo g

(17)

where σ wa is the water–air surface tension, σ oa is the oil–air sur-face tension, σ ow is theoil–water surface tension, K is thefrictioncoefficient at the oil water interface, V o is the volume of spilledoil, g is the gravity, is a parameter which relates the oil andwater densities (ρo, ρw, respectively): = (ρw − ρo)/ρw.

According to the same authors, experiments show that morethan 90% of the surface slick is controlled by gravity. This areais surrounded by a thinner oil slick controlled by surface tension.In this paper, the surface tension is neglected and the frictioncoefficient is K = (ρoνo)/e (where e is the slick thickness andνo is the oil kinematic viscosity). The slick surface formulation(Eq. 16) can be simplified to

S =27π

2V 3o g

νo t 1/4

(18)

The previous expression (Eq. 18) is used to determine each parti-clearea A p.Themainadvantageofthisequationisthatitcontainsfew parameters: the oil density ρo, the oil kinematic viscosity νoand the volume of spilled oil V o.

Mass transfer processes

The mass transfer between two phases is quantified theoreti-cally, based on the hypothesis that the mass transfer resistance

is located close to the interface between the two phases. In thenext sections, all processes are based on Whitman’s (1923) the-ory, which formulates the mass transfer flux for mass transfer phenomena.

Evaporation

Evaporation is the most important mass transfer process thatoil undergoes after a spill. In a few days, light crude or refined products can lose up to 75% of their volume (ASCE 1996). Anunderstanding of evaporation is important both from the practi-cal viewpoint of cleaning up spills and for developing predictivemodels. The evaporation model used is based on a pseudo-component approach. The change in mass of the petroleumcomponent i is characterized, using the molar flux expressionof Stiver and Mackay (1984) and the thermodynamic phase

equilibrium equation, by the following relationship:

dmidt

= − K evap A p P imi

j (m j / M wj ) RT

with P i = exp

H i RT Bi

1 − T BiT

(19)

where mi is the mass of the component i, K evap is the evaporationmass transfer coefficient, P i represents the vapour pressure of component i, M wj is the molar mass of component j , R is theuniversal gas constant, T is the ambient temperature, T Bi is the boiling point of component i and H i is the molar enthalpy of component i.

The Gray–Watson method (Boethling et al. 2000) is used todetermine the molar enthalpy H i in Eq. (19):

H i(T ) = T Bi R ln(82.06T Bi)

3 − 2 T T Bi

m

with m = 0.4133 − 0.2575 T

T Bi(20)

With the molar mass of component i calculated according toJones (1997) and the previous relationship (Eq. 20), all component parameters ( M wj , H i and T Bi) can be expressed as afunction of the component’s boiling point T Bi. Therefore, the parameters of the evaporation algorithm are the component’s boiling point T Bi and the initial petroleum composition, whichare characterized by the distillation curve, and the mass trans-fer coefficient K evap. In this model this coefficient is calculatedaccording to the theory of Mackay and Matsugu (1973).

Dissolution

Dissolution is an important phenomenon from a toxicologicaland environmental point of view, although it only accounts for anegligible fraction of the oil mass. The oil quantity affected bythis process is about 1% of the initial mass. Due to their physico-chemical properties, only PAHs are assumed to be dissolved inthewater (Hibbs et al. 1999).Basedon Whitman’s(1923) theory,the evolution of the dissolved concentration is described by a

first-order differential equation

dC idt

= α (S i X i − C i) (21)

where S i is the solubility of the component i in water, X i is themolar fraction of component i and α = ( K diss A p)/V , where K dissis the mass transfer coefficient, A p is the particle area and V isthe node volume. Thereafter, δ t represents the time step.

By solving Eq. (21), the concentration of dissolved PAH iin the water column at time t (C n+1i ) as a function of the con-centration at the previous time step (t − 1) (C ni ) is given by the

following relation:

C n+1i = S i X n

i + [C ni − S i X

ni ] exp(−αδt ) (22)

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(a) (b)

Figure 3 Particle P inside an element (a) and node area (b)

The order of magnitude of the dissolved mass transfer coefficient K diss is of several cm h−1 (Yapa and Shen 1994, Hibbs et al.

1997). These values are given by these authors without justifica-tion, and therefore the dissolved mass transfer coefficient must be calibrated. Using the relationship linking mass and concen-tration, the mass loss at time t ≡ (n + 1)δt for each componenti can be deduced

massn+1i − massni = [1 − exp(−α j δt )](S i X

ni − C

ni )V j (23)

Volatilization

Dissolved oil components can be volatilized into the atmosphereonly in areas not covered by the surface slick. The volatilizationflux according to Whitman (1923) is expressed as follows:

F i = − K vol C i (24)

where F i is the mass flux of component i, C i is the concentrationof component i in the water and K vol is the overall volatilizationrate coefficient.

This flux expression contains only one parameter, thevolatilization rate coefficient K vol . As for the dissolution masstransfer coefficient, different values can be found in the literature(Hibbs et al. 1999), but they are not explained. This coefficientmust thus be calibrated.

2.4 Switching from Lagrangian to Eulerian formulation

As mentioned in Section 2.1, Telemac-2D is based on a vertexcentred finite element formulation, which means that variablesare defined on mesh nodes. If we consider a particle P inside anelement (Fig. 3a), its dissolvedmass must be distributed betweenthe element’s nodes.

Therefore, in order to compute the coefficient α at each node j , the area of each particle ( A p) is distributed between the nodes

of the local element using the following formula:

A pSHP ( j ) = Ar ( j ) (25)

where SHP ( j ) is the barycentric coordinate at node j and Ar ( j ) isthe partial particle area at node j . A node area is defined around

eachmeshnodebyaddingthequadrilateralsdefinedbythemedi-ans of each triangular element (grey area in Fig. 3 b). This isequivalent to the integral of test functions (Hervouet 2007). Thevolume V is obtained by multiplying the node area by the depthof the node.

The previous steps allow the coefficient α j to be calculated ateachnode j ofeachelementthatcontainsaparticle.Thedissolvedmass of PAH i in the water column is defined at each node j bythe following relation:

massn+1idiss = [1 − exp(α j δt )](S i X n

i − C ni )V j (26)

The total amount of dissolved mass for each particle componenti is

massn+1idisstot =

numnode j =0

massn+1idiss (27)

Thus, the quantity of tracer at the time step t , at node j , added bydissolution is defined by

C n+1 j = C n

j + massn+1idiss

V j (28)

The mass is conserved to machine accuracy in this process. Theadvection–diffusion equation (Eq. 13) can then be used to simu-late the transport and dispersion of dissolved PAHs in the water column. The parameters useful for the dissolution algorithm arethe solubility of the component i in water (S i), which is avail-able in the literature and the dissolution mass transfer coefficient( K diss), which must be calibrated.

3 Model calibration

As seen in the previous section, the dissolution and volatilizationmass transfer coefficients need to be calibrated. In this section,we propose a methodology to determine these coefficients using petroleum kinetics. Within the Migr’Hycar project, some tests

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Table 1 List of studied PAHs

Naphtalene N Chrysene CBenzo[b]thiophene BT Benzo[a]anthracene BAABiphenyl B Benzo[b]fluoranthene BBFAcenaphtylene ANY Benzo[k]fluoranthene BKFFluorene F Benzo[e]pyrene BEPPhenanthrene P Perylene PeAnthracene A Indeno(1,2,3,–cd)pyrene INDibenzothiophene D Dibenz[a,h]anthracene DBAFluoranthene FL Benzo(g,h,i)perylene BPEPyrene PY

were carried out by the Laboratoire de Chimie Agro-industriellelocated in Toulouse (France). The aim of these experiments wasto study the dissolution dynamics of the most significant PAHs(listed in Table 1) and their derived products.

During the kinetic experiments, representative hydrocarbonswere tested (heavy fuel, home heating oil and kerosene). In a beaker, the hydrocarbons are in contact with water during twodays. Water samples are taken at different times. An analysis of each sample allows the PAHs concentration to be defined andthe hydrocarbons kinetics are obtained.

These hydrocarbon kinetics in the aqueous phase are used todefine the volatilization mass transfer coefficient ( K vol ) and thedissolution mass transfer coefficient ( K diss). Based on the fluxexpressions of dissolution and volatilization, we assume an ana-lytical solution of the form a exp(bt ) + c exp(dt ) and find theconstants a, b, c and d that best fit the measurements in the senseof least squares. This procedure allows the mass transfer coef-ficient for each PAH to be calculated. For modelling purposes,the PAHs are classified according to their number of benzenerings and are thereforerepresented by four components. Thefour modelled pseudo-components are

• First class: two-ring PAHs (volatile and soluble aromatics);• Secondclass:three-ringPAHs(semi-volatileandsemi-soluble

aromatics);• Third class: four-ring PAHs (low volatility and solubility

aromatics);

• Fourth class: five and more ring PAHs (very low volatility andsolubility aromatics).

The mass transfer coefficients for each group of PAHs are cal-culated as a weighted average of the initial mass fraction groupcomponents. The results are summarized in Table 2. Values of the dissolution mass transfer coefficient found in the literaturerange from 6.5 × 10−7 to 1.15 × 10−5 m s−1 (Shen and Yapa1988). The dissolution mass transfer coefficients from heavyfuel and kerosene found previously are within this interval.However,thehomeheatingoilvalueismuchhigher:themaximalvalue (4.46 × 10−5 m s−1) is four times larger than the litera-ture value (1.15 × 10−5 m s−1). The volatilization mass transfer coefficient for the two benzene-ring PAHs found by the identi-fication parameter technique have the same order of magnitudethan the reference value of 1.2 × 10−5 m s−1 reported by Hibbset al. (1999). This coefficient decreases with the number of ben-zene rings. This can be explained by the low volatility of heavycomponents. It canbe concludedthat thecomplexity of hydrocar-

bons composition makes calibration essential before modellingthe hydrocarbons behaviour in the aquatic environment.

4 Numerical verification

Verification of the numerical model was carried out by simulat-ing benchmark test cases. First, the transport model is verifiedusing a well-documented real spill case and a theoretical casewith analytical solution. Second is affected to the weathering process verification (spreading and evaporation). The last part isdedicated to the verification of the dissolution algorithm using

an artificial river experiment.

4.1 Transport process verification

Real oil spill accident

Oil spills in continental waters have never been as well docu-mented as real seawater oil spills where aerial plans aregenerallydeployed to report the oil slick location. During the 1990s, atankerbrokein twosectionsand sank in seawater. Approximately15, 000 tons of oil were released into the marine environment.Aerial survey missions reported slick locations. The comparison

between the simulated trajectory and the observations is made inorder to validate the advection modelling.

Table 2 Mass transfer coefficient results

Kerosene First class Second class Third class Fourth class

Kerosene K diss (m s−1) 1.23 × 10−5 5.63 × 10−6 2.0 × 10−6 1.37 × 10−6

K vol (m s−1) 4 × 10−5 1.5 × 10−5 7.89 × 10−7 1.05 × 10−8

Home heating oil K diss (m s−1) 2.54 × 10−5 3.57 × 10−5 4.46 × 10−5 1.27 × 10−6

K vol (m s−1) 2.7 × 10−5 1.3 × 10−6 4.18 × 10−7 2 × 10−8

Heavy fuel K diss (m s−1) 5.54 × 10−6 6.52 × 10−6 3.47 × 10−6 1.33 × 10−6

K vol (m s−1) 1.08 × 10−5 3.66 × 10−7 5.18 × 10−8 1.2 × 10−8

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Figure 4 Trajectory of the oil slick

Thetransport model presented in Section 2.2 and the Telemachydrodynamic model (Section 2.1) are applied to simulate theoil spill. A mesh composed of approximately 90,000 nodes and150,000 triangles is used for this study. On offshore boundaries,the free surface elevation and/or velocity are imposed to repro-ducetidaleffects.Fortheshoreline,asolidwallconditionisused.Reanalysis of climate forecast system reanalysis surface winddata was carried out over the period of the oil spill to reproduce

wind conditions. The numerical prediction of the slick centroidis compared with the aerial observations, as shown in Fig. 4.

Simulation results agree with the observed trajectory exceptin the initial phase of the spill, where the spill is predicted moreto the east than the aerial observations with a maximal gap of 16 km. Moreover, there is a one day lag between the numerical prediction and the aerial observations. This lag is probably due touncertainty in the exact time of the oil release, and also to waveeffects, which are not considered in the model. The model is ingoodagreementwiththeobservationsandthistestcaseshowsthecapacity of the model to predict a real case accident. Moreover,

this test case allows the wind factor determined previously to bevalidated: since tide currents are relatively low in the offshorearea, wind action is the main driver of oil slick transport.

Theoretical case with analytical solution

This simple theoretical case verifies the stochastic diffusionmodelling. In this case, a square basin with a 20m side ( L) isconsidered. A uniform triangular mesh composed of 9000 nodeswith a grid size of 0.3 m in both x and y directions is used. Thereis no flow velocity in the channel and the water level is constant.The dispersion coefficient ( D) is 0.05m2 s−1 along the longitu-dinal axis x. A number ( N p) of oil particles are instantaneouslydischarged at point x = 10.0 m, y = 10.0 m. The particle evolu-tion is only due to the stochastic term of Eq. (15) and particleconcentration evolves according to Fick’s law.

0 200 400 600 800 1 000 1 200 1 400 1600 1 8 00 2 000 2 2000

0.01

0.01

0.02

0.02

0.03

time (s)

a r e a ( m 2 )

measurementssimulated results

Figure 5 Spreading evolution of heavy fuel (“fuel #6”)

The numerical model results are in good agreement with theanalytical results obtained by solving Fick’s lawand thus validatethe stochastic diffusion algorithm.

4.2 Weathering process verification

Whereas all three major weathering processes (spreading, evap-oration and dissolution) occur simultaneously, evaporation hasa significant effect on dissolution, and the magnitude of bothevaporation and dissolution is linked to the slick area. More-over, dissolution has a small effect on the total oil mass balance.For this reason, spreading and evaporation modelling will beeach verified separately before the complete weathering processmodelling is verified using the artificial river tests.

Spreading

A spreading experiment has been presented by Osamor and

Ahlert (Goeury 2012). During these experiments, a volume of hydrocarbons was spilled in a rectangular Plexiglass tank (1.5 mlong × 1 m wide). The evolution of the slick surface is fol-lowed with a camera which allows to quantify the increasingarea of the surface slick. In this paper, the result for the heavyfuel (called “fuel #6”) is used. During this experiment, 25 ml of heavy fuel was spilled on still water in the tank. The oil param-eters used are the density ρo = 951 kgm−3 and the kinematicviscosity νo = 0.0028 m2 s−1. The simulation time and spatialsteps are t = 0.5 s and x = 0.015 m, respectively. A comparison between numerical results and these experimental data

is presented in Fig. 5.The spreading model results are within the values obtainedfor the experiment, although the model over-estimates the ini-tial spreading phase: 2 min after the release, the percentage of relative error between the predicted and measurement resultsis 65%. After 4min, it becomes less than 10%. This can beexplained by the fact that during the experiments the heavy fuelwas released gradually during 90 s and therefore inertial forcesseemtopredominatewhereasinthesimulationsthefuelreleaseisinstantaneous, andthe inertial forcesareneglected in the adoptedspreading equation (18) (Maroihi et al. 1992).

Evaporation

In order to evaluate the accuracy of Eq. (19), a laboratoryexperiment led by Mackay and Matsugu (1973) is simulated.

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0 10 20 30 40 50 60 7 0 80 90 100 110 1200

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%

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Figure 6 Gasoline evaporation. Reproduced with permission of D. Mackay and R.S. Matsugu (1973), Evaporation rates of liquid hydrocarbon spills on land and water , Canadian Journal of ChemicalEngineering, © 1973 Canadian Society for Chemical Engineering.

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Figure 7 Heavy fuel (Intermediate Fuel Oil (IFO) 300) evaporation

A gasoline volume of 15l was spilled in a wooden pan(1.22 m long × 1.22 m wide). Then, the oil volume was mea-sured periodically using the depth measurement. The outdoor conditions are known (wind velocity 6.7m s−1, temperature288K). In this numerical study, the ADIOS data base (http://response.restoration.noaa.gov/ADIOS) supplies the experimen-tal data for the gasoline composition. The simulation time andspatial steps are t = 1 s and x = 0.015 m, respectively.

Theevolution of themodelevaporationreproduces theexperi-mental results satisfactorily (Fig. 6). Subsequently, a comparison between the evaporation model and the ADIOS prediction for the heavy fuel (IFO 300) is carried out. For this test, an instanta-neous oil release in a tank is considered. A tank surface area of

500m2 has been chosen. An uniform south wind of 1 m s−1 anda water temperature of 288 K are assumed. The simulation time

and spatial steps are t = 0.5 s and x = 1.5 m, respectively.The results are shown in Fig. 7.

The results provided by ADIOS and by the present model arefairly close. After 120 min, an evaporated fraction difference of 0.5% can be observed between the models. This difference can be explained by the vapour pressure determination. In this paper,the Gray Watson (Boethling et al. 2000) method is used to deter-mine the vapour pressure (Section 2.3), whereas the Antoine’sequation (Jones 1997) is used in the ADIOS model.

4.3 The artificial river case

Experiments

An artificial river test campaign was conducted by Veolia Envi-ronnement Recherche et Innovation in order to observe thecapacity of the pollutant to dissolve PAHs.

The Umweltbundesamt German Federal Agency for Environ-ment has on its site 16 identical systems of artificial rivers. Thechannel is about 100 m long. Amongst these rivers called FSA(acronym for Fliess und StillgewassersimulationsAnlage: simu-lator rivers and lakes), eight are located outdoors. A water flow isgenerated in these flumes with a screw pump. A system for con-tinuous measurement of physical parameters is installed for eachriver, and there is one weather station. Two flumes were linkedtogether to increase the installation length and sinuosity (Fig. 8).

Thehydrocarbonisspilledthrougharingonwatersurface,the pollutant is injected inside (Fig. 9a). Then, the ring is removedto allow the pollutant transport.

To observe the evolution of the concentration of dissolvedPAHs, a fluorescence probe is used. Every morning a blank sample is made to know the initial concentration of PAHs already present in the channel. When the signal (%) is approximately onthe peak, a water sample is taken during 30 s using an automaticdevice located near the probe (Fig. 10a). The samples are thensent to the Center of Documentation, Research and Experimen-tation (CEDRE on accidental water pollution) for the analysis of dissolved concentrations of PAHs in samples. For each sample,there is therefore a concentration of total PAHs (ng l−1) and a probe signal (%).

With the various tests carried out the same day and with thesame hydrocarbon, it is possible to draw a calibration curve

Figure 8 Artificial river sketch (source: Umweltbundesamt)

http://response.restoration.noaa.gov/ADIOShttp://response.restoration.noaa.gov/ADIOShttp://response.restoration.noaa.gov/ADIOShttp://response.restoration.noaa.gov/ADIOS

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(a) (b)

Figure 9 Release device (a) and heavy fuel around obstacle (b)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 950

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water sample

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R2 = 0.6462

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r a w s i g n a l f l u o r e s c e n c

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(c)

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concentrationsmoothing

Figure 10 Experimental results of dissolved PAHs in the artificial river: (a) fluorescence measurement of dissolved petroleum, (b) PAH calibrationcurve and (c) concentration of dissolved petroleum after the home heating oil spill

(Fig. 10 b). This curve will allow to convert the signal probe in% into a total PAH concentration in ng l−1 (Fig. 10c). Then, thereal concentration as function of time is obtained by subtractingthe value of the corresponding blank.

Theprofilesobtainedare used to validate thenumerical model.

Numerical verification

Themeso-scaleartificialriverexperiments allowto test thePAHsdissolution model prediction capacity. The numerical model pre-sented in Section 2 is used to simulate oil slick transport andweathering processes in the artificial river presented in Fig. 8.The finite element mesh consists of 23,234 nodes and 43,000 tri-angles of average size 0.08 m (Fig. 11c). The flow velocity andsurface elevationare imposed respectivelyon inflow andoutflow

boundary conditions. For shoreline nodes, solid wall conditionsare considered.

In this work, the artificial river test compaign is reproducednumerically (heavy fuel, home heating oil and kerosene spills).The product characteristics such as the petroleum composition,the distillation curve and physical properties (heavy fuel (ρo =950, νo = 4465), kerosene (ρo = 795, νo = 2) and home heatingoil (ρo = 840, νo = 7) with ρo (at 20◦C) the oil density (kgm−3)and νo (at 20◦C) the kinetic viscosity (mm2 s−1)) amongst other properties are supplied by the French oil company TOTAL andthe CEDRE.

Inthefirstsimulation,akerosenespillwhichoccursinthefirstartificial river curve is considered. A volume of 2 × 10−5 m3 has been spilled into the channel. The flow velocity is imposed to0.1ms−1 on inflow boundary. A velocity of 0.2 m s−1 was also

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(a)

(b) (c)

Figure 11 Artificial river model: (a) global view of the model, (b) oil spill simulation and (c) unstructured mesh

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Figure 12 Kerosene concentration evolution in the water column

tested, without notable impact on the results. There are obstaclesin the channel (Fig. 9 b).

A simulation result is shown in Fig. 11( b). The particles rep-resent the oil surface slick whereas the Eulerian tracer representsthe dissolved petroleum in the water column. The numerical andexperimental concentrations of the petroleum dissolved in thewater column are shown in Fig. 12.

The numerical dissolved and dispersed hydrocarbon concen-trations (32 ng l−1) in the water column has the same order of magnitude and compare well with experiments (26 ng l−1).However,thereisadelay(about160s)betweenthemodelresults

and the experimental expected values. This lag can be explained by the outdoor conditions which cannot be modelled, such asgusts of wind.

Then, heavy fuel and home heating fuel spills have been stud-ied. These two cases concern an oil spill which occurs in the firstartificialrivercurve, with a riverflowimposedto0.1m s−1.Thereis no obstacle in the first case (Fig. 13a) in contrary of the secondcase (Fig. 13 b).

The algorithm reproduces qualitatively the dissolved hydro-carbons in the water column, though the dissolved petroleumconcentration in the water column can be twice smaller or

twice higher than measurements. The model over-estimatesthe dissolved home heating oil concentration whereas itunder-estimates the dissolved heavy fuel concentration. An openquestion is whether the kinetic data obtained in beaker withsmall spills (1.5 × 10−4 kg) (Section 3) are still valid at larger scales and different concentrations. There is also a time shiftthat could be explained by the gusts of wind that occurredduring the experiments. Given the already established valida-tion of hydrodynamics and in view of the state-of-the-art of numerical analysis of advection and diffusion, the uncertain-ties here are in the estimated kinetic data of dissolution andvolatilization.

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Figure 13 Oil concentration evolution in the water column

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5 Discussion

Since the model developed in this paper will be deployed in awarning system for emergency situations, its main applicationsare:

• to simulate oil spills in continental waters (lakes, rivers andestuaries),

• to estimate PAHs toxicity in the aquatic environment and• to become a decision support tool for water intake operators

and public authorities.

The Lagrangian oil surface slick model presents a reasonable performance on a real tanker accident. This benchmark showsthe model’s capability to simulate oil spills whose predominantdriving force is the wind (in offshore area, tidal currents areweak, and wind becomes the main driver of oil slick transport).Validationofthemodellingofweatheringprocesseshasbeencar-ried out, first by separately verifying spreading and evaporation,and then by testing the complete weathering process. Spreadingand evaporation modelling are in satisfactory agreement withobservations. Nevertheless, the model over-estimates the initialspreading phase. This can be explained by the fact that during theexperiments the heavy fuel was released gradually, thus inertialforcespredominate,whereasinthespreadingequationdevelopedin this work these forces are neglected. The oil fate model is usedto predict the general outcome and concentrations of the variousPAHs in the water column resulting from oil spills in the arti-

ficial river. In general, the predicted PAH concentrations are of the same order of magnitude as the expected results. Differencesobserved can be explained by the following.

First, the absence of hydrocarbon component interactions inthe slick, for reason of the very complex petroleum mixtureeffects on oil weathering predictions, constitutes a difference between model and experiment results.

Next, uncertainties in the experiment measurements werecaused by the linear hypothesis of PAH probe response in resulttreatment (Section 4.3).Acalibrationcurveisdrawneverydayinorder to take into account air and water temperatures, solar radi-

ation, amongst other properties and minimize the measurementuncertainties.Furthermore, another reason of this model failure may be a

scaling factor because the kinetic data used for the model calibration (Section 3) have been obtained for relatively small spill(1.5 × 10−4 kg).

Finally, in the river experiment, the temperature ranges from18◦Cto23◦Cwhereasthecalibratedmasstransfercoefficientsaredetermined at the temperature of 20◦C. This reason constitutes adifference between model and experiment results.

6 Conclusion

A Lagrangian/Eulerian model has been developed to simulateoil spills in continental waters. The Lagrangian model describes

the transport and fate processes of an oil surface slick. EachLagrangian hydrocarbon particle is considered as a mixtureof discrete non-interacting hydrocarbon components. In thismodel,particlesarerepresentedbycomponentcategories(PAHs, pseudo-componentscharacterized by distillation curves), and thefate of each component is tracked separately. To model PAHsdissolution in water, a Eulerian advection–diffusion model isused. This approach is useful when an environmental studyrequires the coupling of an Eulerian hydrodynamic model and awater quality simulation. This oil slick model has been coupledwith the Telemac hydrodynamic model for flow propagation.The dissolution and volatilization mass transfer coefficients have been calibrated using fuel kinetics test results. Model validationwas carried out with the following test cases: transport processeswere validated using the well-documented seawater spill (advec-tion)andtheanalyticalFick’slawequation(stochasticdiffusion),

weathering processes were validated using laboratory experi-ments (spreading and evaporation processes) and artificial river testmeasurements(dissolutionandvolatilization).Agoodagree-ment between these cases and the numerical prediction has beenobtained. It can be concluded that the algorithms reproduce sat-isfactorily the values expected, although more experimental dataarestill neededfor a better validation, particularly for dissolution processes.

Acknowledgements

The authors would like to thank all Migr’Hycar partners whohave contributed significantly to this paper (EDF, Saint-VenantLaboratory for Hydraulics (Michel Benoit), VERI (EmmanuelSoyeux), Laboratoire de Chimie Agro-industrielle (MireilleVignoles, Caroline Sablayrolles and Pascale de Caro), CEDRE(Vincent Gouriou and Julien Guyomarch), Artelia (Olivier Bertrand) and TOTAL (Nicolas Lesage and Yann Fortin)).

Funding

The Migr’Hycar research project is supported by the French

Research Agency ANR as a result of the PRECODD 2008 callfor proposals. The PRECODD Eco-technology and SustainableDevelopment Programme is a research scheme aimed at support-ing emerging techniques, procedures and concepts that can helpcontrol theenvironmental impactof industrial andurban activity.

Notation

A p = particle areaC = depth-averaged pollutant concentrationC d = drag coefficient

C f = non-dimensional friction coefficienth = water depth H i = molar enthalpy of component ik = turbulent kinetic energy

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K diss = dissolution mass transfer coefficient K evap = evaporation mass transfer coefficient K vol = volatilization mass transfer coefficientmi = mass of the petroleum component i

M wi = molar mass of component i

P i = vapour pressure of component iS = slick surfaceS i = solubility of component iT Bi = boiling point of component iu = depth-averaged velocity vector uc = current velocity vector at the free surfaceuoil = oil slick velocity vector uw = wind velocity vector above the water surfaceU = flow free surface velocity vector V o = volume of spilled oilX = vector of particle location

X i = molar fraction of component iβ = drift coefficient of the surface slick due to the windε = turbulent energy dissipation rateνo = oil kinematic viscosityνt = depth-averaged turbulent viscosityρ, ρw = density of water ρo = density of oil

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