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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
Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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
Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
26 Oil & Hydrocarbon Spills, Modelling, Analysis & Control
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
Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
Oil & Hydrocarbon Spills, Modelling, Analysis & Control 27
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).
Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
28 Oil & Hydrocarbon Spills, Modelling, Analysis & Control
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
Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
Oil & Hydrocarbon Spills, Modelling, Analysis & Control 29
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
200 iterations 300 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.
Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
30 Oil & Hydrocarbon Spills, Modelling, Analysis & Control
Initial Condition 150 iterations
300 iterations 450 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.
Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
Oil & Hydrocarbon Spills, Modelling, Analysis & Control 31
References
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Streamline Upwind/Petrov-Galerkin Formulation Using Quadratic
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Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
32 Oil & Hydrocarbon Spills, Modelling, Analysis & Control
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Transactions on Ecology and the Environment vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541