SPREADING
Oil is not only transported by winds and currents, but also spreads as a film of varying
thickness on the surface of the water. Current models estimate spread inaccurately. The
reasons for this are two-fold. One, the spreading mechanisms are extremely idealized,
and the actual forces, initial conditions and boundary constraints are usually imprecisely
determined.
WHY IT MATTERS
Given that oil spreading, and hence slick area, is uncertain, what is the affect on other
weathering processes?
Evaporation – Basic evaporation models usually are linear functions of slick area.
Doubling the slick area at a particular instant in time, doubles the mass loss to the
atmosphere. However, evaporative losses in one time step reduce the subsequent vapor
pressure estimate for the next time step so that excessive evaporation estimation early in
the spill will be partly compensated by underestimation later with the reverse case for
initial underestimation. Hence the cumulative evaporation error will be proportionately less
than the slick area error estimate. More advanced evaporation models that adjust for
vertical component stratification effects will give better answers if not only total slick area
but also location varying slick thickness is correctly predicted.
Dispersion – The standard Delvigne-Sweeney (Delvigne and Sweeney, 1988)
dispersion model is linear in surface ‘volume’ but only a weak function of surface area.
Hence, dispersion model predictions are less sensitive to slick area error than evaporation
models.
Cleanup – Generally, cleanup techniques (e.g. skimming, dispersants) are
concerned not with the slick area in general but with the area of thick oil (recoverable oil
for skimming operations). Fay spreading, discussed below, does not separate oil into
thick and thin portions but other modeled processes calculate a varying slick thickness.
However, the dominate process that produces recoverable oil for most spills is Langmuir
cell production and most spill models neglect it. Hence, present spreading models
provide little useful prediction for response.
Environmental impact – Bird oiling is the most significant environmental impact
from surface oil most spill incidents. Errors in surface area and patch location can result in
bird impact determination errors, but other uncertainties usually swamp this error term.
Hence, the main reasons to accurately model surface spreading are to provide input data
to evaporation and response calculations. Next generation spread models should consider
these requirements.
CURRENT KNOWLEDGE
Spreading of oil encompasses both the physical spreading of the oil on the top of the
water and the dynamics of the surface water that is contaminated by oil. The latter can
spread the surface film and also break it into pieces and allows for future coalescence.
This breakage a chaotic process and cannot be modeled deterministically. At best, future
models may be able to estimate the number and size of the patches.
Fay
Early oil spreading models used the three Fay spreading formulas. Fay divided the
spreading process into three separate phases, depending upon the major driving and
retarding force for spreading. When the slick is relatively thick, gravity causes the oil to
spread laterally; later, interfacial tension at the periphery will be the dominant spreading
force. The main retarding force is initially inertia followed later by the viscous drag of the
water. Fay therefore labeled the three phases: gravity-inertial, gravity-viscous, and surface
tension- viscous. The gravity-inertial phase happens rapidly, on the order of a few
minutes, except for the largest spills. The area grows as a linear function of time until the
relative thickness is small enough that the transition to gravity-viscous spreading occurs.
This time is calculated by the formula
 V 1/ 3
t  2.6
 .
g w w 
where  w is the viscosity of water(1 cSt),  w is the normalized density difference between
oil and water, and V= volume spilled. The area at this transition time is often taken as the
initial area 
for many spread models. The area then continues to grow according to the
gravity-viscous phase until it becomes so thin that surface tension becomes the driving
 force.

Other researchers have derived slightly different values for the empirical coefficients used
in early Fay, or have other small variations in the formulas for one of the phases. Mackay
suggested the concept of a thick-thin slick combination using the Fay formulas. This was
an attempt to correct the limitation of uniform thickness in the Fay calculations. This is a
gross oversimplification since real slicks can have thickness variations of several orders of
magnitude. Curiously, the Fay equations do not depend upon the viscosity of the spilled
oil, while common sense says that a heavy fuel oil would spread more slowly than a light
refined product like gasoline. Ross and Energetex Engineering (1985) modified the Fay
equation by inserting a correction factor that is equal to the oil-to-water viscosity ratio
raised to the 0.15 power. El-Tahan and Venkatesh (1994) introduced the concept of
‘velocity gradient’ that is inversely related to the oil viscosity and represents vertical shear
resistance in the oil slick. Chebbi (2000, 2001,2005) has modified the Fay formulas to
account for time dependent releases
Although the Fay formulas are theoretically sound for pure chemicals in laboratory tanks,
they have performed poorly in actual oil spills. In most reported cases, they have
underestimated spill area, whereas in at least one case, they significantly overestimated
the area. Also, Jeffery (1973) noticed no transition between the different spreading
phases. Lehr et al. (2002) and others have pointed out that only the gravity-viscous phase
happens in the time frame when spill response is typically occurring.
The Fay formulas theoretically allow a mechanism for spreading to stop. If the net radial
surface tension is negative, the oil should stop surface tension-viscosity spreading and,
instead, contract into a lens with its thickness controlled by gravity. In practice, alternative
stopping mechanisms, based on spill volume or thickness, are used instead by most
models. There is little accurate empirical information on final spill thickness from real spills
because techniques to perform the measurements are crude and unreliable, and are
seldom performed in any case.
Recommendation:
Research is needed to see what part of slick spreading is due to Fay as opposed to other
forces. Models using the Fay formulas for physical spreading need a term to account for
oil viscosity and some mechanism to account for variable slick thickness. For most spills,
only the gravity-viscous spreading is applicable. Fay spreading treats the oil as a
continuous fluid but most spill trajectory models represent the slick as a discrete
(Lagrangian) set of elements. Numerical studies should be done to ensure model
consistency.
Surface dynamics
The Fay formulas will not allow accurate prediction of slick area over any extended time
period because of the neglect of outside environmental factors and details of the initial
release. When a neutrally buoyant dye is dropped onto the ocean surface, the dye begins
to disperse due to water turbulence. The area of sea surface covered by the dye will grow
as a function of time and a suitably defined eddy diffusion coefficient, D(t) ,
A(t)  D(t)  t

Presuming that this same phenomena affects the dispersion
of positively buoyant
spilled oil, Ahlstrom has recommended simulating this process by dividing the oil into a
 of separate Lagrangian elements, and then randomly displacing these
suitable number
elements over time using a properly chosen eddy diffusion coefficient. Elliot concluded
that such a process is non-Fickian, i.e. a time-dependent diffusion parameter better
represents empirical results. Their diffusion parameter is (MKS units)
De dd y  0 . 03 3t 0 .1 6
.
It is possible to incorporate Fay spreading with diffusive spreading by treating the
former as a pseudo-diffusion process and matching twice the standard deviation of the
distribution of oil element locations with the radius of the oil slick as predicted by the
standard Fay formula. This yields a Fay pseudo-diffusion coefficient of
1/ 3
 g  V 2 
1/ 2
w
.
DFay  0.13
  
  t


w
Another approach is to allow the individual elements to spread according to Fay
spreading rules, treating these elements as spillets. These spillets are then dispersed and
movedby water turbulence and other forces. Care must be taken not to bias the results by
the number of elements selected and to handle interaction between the elements
properly. Otherwise, estimated area becomes a function of the arbitrary number of spillets.
This is complicated by the semi-empirical natures of both the Fay spreading coefficients
and the diffusion coefficient.
Recommendation
Since Fay spreading should be modified to include oil viscosity, the spillet approach
provides a cleaner method to incorporate the two distinct spreading phenomena.
However, as mentioned above, this approach has to be protected against spurious
numerics. A refinement for the next generation model would be to include a spatially
varying distribution of thickness values instead of a single number. The water surface area
of interest then can be assigned probabilities of oil volume contamination. This approach
works best if spreading is combined with transport, as it should be. It also requires
multiple simulations for a single modeled event.
Wind Stress, Langmuir Cells and Sub-surface Oil Effects
Even including water turbulence, Fay spreading is still incomplete. Probably the most
important cause of long term oil spreading is wind stress on the slick and surface water.
Unfortunately, this is a complicated phenomenon that is only partly understood. Oil is
driven into the water column by breaking waves and broken into droplets of different size.
The larger droplets quickly resurface while the smaller droplets remain subsurface for
longer time periods and trail the moving main slick. Elliot (Elliot,1986,; Elliiot and Hurford,
1989; Elliot, Hurford, and Penn, 1986) hypothesized that this is the cause for the ‘comet’
shape of many slicks where a thicker part of the oil at the upwind part is encompassed in
a larger sheen which trails out behind the thick part. The smallest droplets never resurface
and are thus permanently removed from the surface slick. Proper modeling of such
phenomena requires a numerical simulation of the near surface water flow. However, an
often adequate alternative is to use a boundary layer approach where the horizontal
current velocity follows a logarithmic profile and the vertical velocity is calculated by using
a suitably modified form of Stokes Law. One recent application of this approach, using a
different velocity profile, is found in the model by Wang et al. (2007). Similarly, Galt and
Overstreet (2009) have modified the Delvigne-Sweeney (1988) dispersion formulas,
combined with Ding and Farmer’s (1994) breaking wave formulation to produce an Elliotstyle algorithm. Both of these recent models used dichotomic formulations of Stokes law
for vertical buoyancy velocity, uvb , based upon droplet size:

2
gddrop
w  o 
uvb 
18 w
w
8
  o 
gddrop w
3
w
uvb 
ddrop  dc
ddrop  dc
9.52 w2 / 3
dc 
3
g
w  o 
w
As mentioned earlier, oil slicks often break into windrows due to the effects of Langmuir
circulation. It is generally believed that Langmuir circulation is produced by the interaction
 and Stokes drift due to waves. Thorpe (2000) and Farmer and Li
of surface currents
(2000) have sketched an outline of how to include Langmuir effects in weathering models.
Galt and Overstreet (2009) have modified and implemented formulas suggested by
Leibovich (1997). Langmuir cell separation is set to three times the mixed layer depth
(approximated from the Ekman depth). The oil flows into the windrows with a
characteristic time inversely proportional to the wind speed at 10 meters. It is unclear from
the documentation why all the oil does not become totally linearly aligned in windrows of
minimum width, since, unlike real Langmuir cells, these cells are not transitory.
Recommendation
Elliot –style wind spreading is part of the standard ADIOS2 spill model and the Shell ROC
spill calculator but has never been actually checked in laboratory experiments or field
trials. While theoretically sound, empirical validation would be comforting. Certainly some
non-symmetric spreading factor should be included to account for wind-induced
spreading.
Langmuir effects are important in determining cleanup methods. It remains to be seen if
the prototypes suggested above are adequate. Further study is recommended in this area
before incorporating any of these prototypes into fate and behavior models.
Current affects and shorelines
Oil spilled from a fixed position will be affected by surface currents, particularly in strong
tidal current regimes. Oil spilled nearshore may have its spreading interrupted by shore
boundaries. These can most easily be included in the fate and behavior model by
connection to an oil transport module.
MODELLING CONSIDERATIONS
Spread rate is non-linear in time. Most fate and behavior models use a fixed time step for
their numerical calculations. This can result in either in poor representation of area during
the early phases of the spill because of too long a time step or an unnecessary number of
small steps later on in the event, increasing computational time. While more complicated
to program, there is no theoretical barrier to variable steps; small during the initial release
and large when the spill approaches its spreading limit.
Many modified Fay spreading models provide minimum thickness formulas when
spreading stops but formulas for the other processes typically do not have such limits.
However, actual spills often do reach a fixed area with separate surface pieces breaking
and separating but not individually increasing in area. Viscosity plays a major role in this
process. Diesel and other low viscosity oils can spread to extremely thin sheens while
heavy fuel oils do not.
Recommendation
Numerical schemes should use variable time steps. Field trial should be done to develop
and verify spread cessation for slicks based upon viscosity and/or other factors.
RECOMMENDED APPROACH
To summarize recommended approach, the next generation of spreading module should;
1. Represent spill as a list of moveable and area varying spillets
2, Use as driving factors physical (Fay gravity-viscous) spreading, turbulent diffusion,
wind–induced spreading, and Langmuir affects (if possible). For fixed spill locations,
surface currents need to be included
3.Utilize variable time steps and spreading termination algorithms.
DATA CONSIDERATIONS
Input data
Spill release rate
Oil density
Oil viscosity
Water density
Wind speed
Surface current
Sub-surface current profile
Horizontal and vertical dispersion coefficients
Mixed layer depth
Droplet size distribution
Wave regime
Output data
Spatially varying thickness
Oil surface area.
REFERENCES
Chebbi, R. (2000) Inertia-gravity spreading of oil on water. Chemical Eng. Sci., 55, 49534960.
Chebbi, R. (2001) Viscous-gravity spreading of oil on water, IACHE Journal, 47, 288-294.
Chebbi, R. (2005) Dynamics of unidirectional gravity-inertial spreading of oil on water,
Chem. Enegineering Sci., 60, 6806-6813.
Delvigne, G., and C. Sweeny (1988) Natural dispersion of oil, Oil and Chemical Poll., 4,
281-310.
Ding L., and D. Farmer (1994) Observations of breaking wave statistics, J. of Phys.
Ocean., 24, 1368-1387.
Elliot, A. J. (1989) The Influence of wind and wave shear on the spreading of a plume at
sea, Oil and Chemical Poll, 5, 347-363.
Elliot, A.J., {1986).Shear Diffusion of the Spread of Oil in the Surface Layers of the North
Sea, Dt. Hydrog. Z., 39: 113-137
Elliot,A.J., and N.Hurford (1989) The Influence of Wind and Wave Shear on the Spreading
of a Plume at Sea”, Oil & Chemical Pollution, 5: 347-363
Elliot, A.J. , N. Hurford, and C. J. Penn(1986). Shear Diffusion and the Spreading of Oil
Slicks, Marine Pollution Bull., 17: 308-313
El-Tahan, H. and S. Venkatesh (1994). Behavior of Oil Spills in Cold and Ice-infested
Waters- Analysis of Experimental Data on Oil Spreading, Proc. of the Seventh Arctic and
Marine Oil Spill Program Technical Seminar, 337-354
Farmer, D. Li, M. (1994.)Oil dispersion by turbulence and coherent circulations. Ocean
Engineering, 21, 575-586.
Fay, J.A., (1969).The Spread of Oil Slicks on a Calm Sea”, Fluid Mechanics Laboratory,
Dept. of Mech. Eng. , MIT, Cambridge, Mass.
Galt, J. and R. Overstreet (2009) Development of spreading algorithms for MARO,
unpublished.
Jeffery, P.G., (1973).Large Scale Experiments on the Spreading of Oil at Sea and
Disappearance by Natural Factors”, Proc. of the Joint Conf. on Prevention and Control of
Oil Spills, 469-474
Lehr, W., R. Jones, M. Evans, D. Simecek-Beatty, R. Overstreet (2002) Revisions of the
ADIOS oil spill model, Env. Modelling and Software, 17, 191-199.
Ross, S. L. and Energetic Engineering, “In Site Burning of Uncontained Oil Slicks”, Env.
Canada Report EE 60 (1985).
Thorpe, S.A., (2000) Langmuir circulation and the dispersion of oil spills in shallow seas.
Spill Science and Technology, this volume.
Wang, S., Y. Shen, Y. Guo, J. Tang (2007) Ocean Engineering, 35, 503-510.