24 oil & hydrocarbon spills, modelling, analysis & control · considered in those models...

Oil spill movement in coastal seas: modelling

and numerical simulations'

J. F. C. A. Meyer*, R. F. Cantao* & I. R. F. Poffo*

*Department of Applied Mathematics, UN 1C AMP - State

University at Campinas, Campinas, Brazil

Email: joni@ime. unicamp. br, cantao@ime. unicamp. br

^ State Agency for Environmental Technology, CETESB, Sao

Paulo, Brazil

Email: [email protected]

Abstract

In this paper we will locate our present research results for efforts in Brazil inthe realm of oil spill modelling and simulation. In the beginning of the nineties,a project was presented in our University for optimizing resources of clean-upactivities following an oil spill. In this global project the mathematical modelsfor numerical simulation should include several phenomena not previouslyconsidered in those models then in use by government agencies. TheBiomathematics Research Group, of the Applied Mathematics Department thenbegan to work in this direction, using successive variations of some classicalmodels. Besides the theoretical layout, this paper presents some case studies fora specific coastal region in southern Brazil.

1 Introduction

Several ports along the Brazilian coast have at several times been affectedmore or less seriously by oil spill accidents. A specially hazardous regionis that of the Island of Sao Sebastiao on the coast of Sao Paulo state,

' This work was sponsored by the Sao Paulo State Science Foundation:FAPESP.

Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

24 Oil & Hydrocarbon Spills, Modelling, Analysis & Control

known as the Beautiful Island ("Ilha Bela"), where well over 200 oil spill

accidents happened in the last twenty six years (Poffo et alii, 1996). On

the side of the continent, in the channel which separates the continent fromthis island, there is an oil terminal, the main Brazilian facility for oil

transfer, through which over fifty percent of all transported oil in Brazil is

processed. After several serious accidents, both state authorities

(CETESB) and the national petroleum company (PETROBRAS) have

joined efforts to constitute a legal set of guidelines in the case of suchaccidents, for clean-up activities and environmental protection andrestoration actions. Part of this effort has been the production ofMathematical Modelling and Computational Simulations, in charge of the

Biomathematics Group, in the Department of Applied Mathematics in the

State University at Campinas.hi previous work (Meyer

and Cantao, 1996) initial models

were presented, but the diffusive

aspects, due to numerical

limitations, surpassed those of the

advective behaviors. In thepresent work, these limitations

have been overcome by the use ofa relative sophistication in theapproximation methods, and acomparison with historical reports

is briefly commented. Theadopted strategy was to resort tofirst-order finite elements for the

Figure 1: San Sebastian Channel. spatial approximations, instead ofsecond-order, as in the above-

mentioned papers, with the introduction of upwinding terms (c/Brooks andHughes, 1982), while maintaining the second order Crank-Nicolson

method in time.

2 The Mathematical Model

The classical characterization of oil spills, due to Fay (1969), leads usquite naturally to the adoption of the so-called Diffiisive-Advective, orSecond phase, which describes the oil spill from a few hours after itsoccurrence to about two weeks later, a period in which most sightings


Oil & Hydrocarbon Spills, Modelling, Analysis & Control 25

happen, and during which clean-up activities are carried out. During this

phase, the oil spill can be of several square kilometers, while maintaining a

depth between 2 and 5 centimeters. These dimensions indicate the natural

adoption of only two space variables, a decision sustained by most of the

authors who work with the modelling during this period (Cuesta et alii,1990, Benque et alii, 1980, Psaraftis, Ziogas, 1985). Therefore, the

mathematical formulation of the described problem, for a concentration c

of oil at time f and at a point (xy) is given by c = c(x,y;t), for (x,y) eQ

and / e(0,7'] such that

dc— = div(aVc) - div(Vc) - cc + f(x,y\t) (1)

such that

&/<U = 0 and -a%L - f\c, for P,, wF, = m , Vf E(0,7] (2)

and

c(x̂ ;0) = c,(%,;;) for all (x,̂ ) eQ. (3)

Tlie involved parameters also describe the considered phenomena:

• The model considers effective diffusion and, for coastal situations, adiffusivity, a, which is constant in space and time (Stolzenbach et alii,

1977, Meyer, 1993).

• In the advective term, V describes the resultant velocity due to tides,

wind and currents, and accounts for transport of the oil, varying inspace and time, and div(V) is assumed to be null.

• The several decay processes are considered in the model as a linearfunction of the concentration of oil, oc - even though this can be

modified, in order to have a varying in space and time.

• Boundary conditions indicate that no oil crosses part 1^ of the

boundary, which describes the land boundaries, or the shores along the

channel margins, and that along r, the model accounts for oil leaving

the domain in proportion to the normal component of resulting oceancurrents.

• Finally, this formulation also includes possible oil sources (thissituation actually occurred in the studied region when one of the mainpipelines was accidentally perforated).

Taking into account the discontinuity of initial conditions, as well asthat of many cases of sources, the above formulation in terms of a



classical Partial Differential Equation is transformed into a variational, or"weak" formulation, which continues to hold in a generalized sense. It isalso very appropriate for the use of Galerkin's Method for Finite Elements.

We then have, with convenient use of Green's Theorem as well as of theboundary conditions:

L + (ac|v)« + < /?V c|v >,- = (/|v)e (4)

for c,v e5 with 5 = (v e I' ((0, T],H* (Q)):%̂ = 0}, and sufficiently

regular parameters.

3 The Discretized Model

In previous work, the authors had chosen approximating functions for

space variables that were second-order finite elements, but the presence of

the advective term forced the discretization to respect certain thresholds in

order to avoid numerical oscillations. These limitations led to models in

which the diffusion often surpassed advection in numerical importance. Inthis present work, this is avoided by the use of an upwinding term, which,if doesn't completely eliminate the undesired oscillations, greatlydiminishes their effect. In fact, the remaining oscilations are due to thetransient or acomodation time of the method near / = 0.

Galerkin's Method consists of redefining problem (4) in a finite-

dimensional subspace S^ d S, generated by a set of N^ functions

{(jK,7' = 1,...,7V,,}. Instead of c'(jt,j>;/), then, the discretized problem

seeks

Cfr =*̂ .Cj(t)<pj(x,y) (5)

in which, besides separating space and time variables, functions qr

include the linear part as well as the upwinding terms; coupled withCrank-Nicolson for the time variable, this leads us to an approximation of

(4) of order o(Ax̂ ) , o(Ay*) and o(Af *) given by a linear system:

A<T' - Be + d (6)

where



Af

Tand

Some special aspects of the above formulation should be mentioned.

1. Trial functions, namely \j/., are defined by \\r. = (p. + iVVcp., where i

stands for the upwinding factor. Its calculation is stressed in Codina

(1992). If T = 0 we turn back to the traditional Galerkin method.

2. When this kind of upwinding is used, a divergence free condition mustbe observed (Brooks and Hughes, 1982).

3. Source terms depending on their magnitude may induce oscillation, aswell.

4. The available velocity field is defined pointwise, not a very convenientsituation for evaluating the inner products in (6). So, an interpolation of

V using the same base functions of S is performed.

4 Numerical Simulations

For numerical simulations, the chosen region was that of the Sao Sebastiaochannel mentioned above. The domain was discretized using severalsoftware resources available in the Applied Mathematics Department atUNICAMP: Borland C++, PDEase2, Mathematica, besides a specialprogramme developed locally (cf D'Afonseca, 1996 - colaborator) forvisualizing numerical output.

The vector field, descriptive of the circulatory dynamics in thechannel is either in accordance to Furtado (1978), or approximatelydescribe a historical register of local information (PoflFo et alii, 1996).



Three main situations are modelled: the most "common" scenario for

the region, with a current which moves from southwest to northeast, a

completely inverted situation, in which currents flow towards southwest,

and a "combination" of both, resulting in an almost circulatory pattern.

4.1 Scenario 1

The first scenario consists of a source and an initially visible spot of oil,

both located near the oil terminal. The current moves from southwest tonortheast (the most common situation). The obtained results can be

appreciated in the following figures.

Initial Condition 150 iterations

300 iterations 450 iterations



4.2 Scenario 2

Here we have no pollutant source, but only an initial condition that enters

the channel by the north opening. The current is inverted relatively to theprevious scenario: it moves from northeast to southwest.

Initial condition 100 iterations


4.3 Scenario 3

Finally, we have a situation very similar to the World Galla ocurrence (cfPoffo et alii, 1996). A big spillage in the oil terminal, a small artificialpollutant source, and a velocity profile based on Furtado (1978), composethis scenario.



Initial Condition 150 iterations


5 Conclusion

These results greatly surpass those of previous publications by theauthors, in the sense that there is a great coincidence between reported oil

spill accidents and numerical simulation, as well as very well-behavedoscillations. Future work points toward the challenge os eliminating these

undesired instabilities.



References

1. Benque, J P., Hauguel, A. & Viollet, P. L. in Eng. Applications of

Comp. Hydraulics //, Pitman Advanced Publ. Programme, 1980.

2. Brooks A. & Hughes, T. J R Streamline-Upwind/Petrov-GalerkinSimulations for Convection Dominated Flows with particular

emphasis on the Incompressible Navier-Stokes Equations, Comp.Meth. In App. Math, and Eng., 32, pp. 199-259, 1982.

3. Codina, R, Onate, E & Cervera, M. The Intrinsic Time for the

Streamline Upwind/Petrov-Galerkin Formulation Using Quadratic

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9. Meyer, J. F. C A. & Cantao, R. F An Introductory MathematicalModel and Numerical Analysis of the Evolutionary Movement of OilSlicks in Coastal Seas: a case study, Development and Application ofComputer Techniques to Environmental Studies, eds. Zanetti, P. and

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Science, 1985, 31, pp. 1475-1491.

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