5 Physical sub-models
The purpose of the physical submodels is to estimate the process rates which are used for the prediction of the distribution of oil contaminants on the water surface, shoreline, river bed and concentration in the water column.
5.1 Surface advection
The advection or drifting oil on the water surface is supposed to take place in velocity field resulted from combined actions of wind, main water body currents and slowing down influence of ice cover, if any, as follows:
Drift Velocity Vector, DVV,
equals
vector sum of
Wind Drift Factor, WDF, times Wind Velocity Vector, WVV,
at 10 m above water surface and
Water Current Factor, WCF, times Water Velocity Vector, WtVV.
By default, the values of WDF and WCF used are being as 0.03 and 1.1 respectively ( Stolzenbach et al., 1977 ).
ROSA-EARTH uses experimentaly-based relationships between water current and ice covered slick-driving velocities:
1) Slick-driving Surface Water Velocity
is equal to
Ice Cover Slick Factor times Water Velocity.
2) Ice Cover Slick Factor
is equal to
Unity minus square root of Friction Amplification Factor
over Slick Velocity Factor.
3) Slick Velocity factor
is equal to
const1 times
the sum of
slick densimetric Froude number squarred and const2.
5.2 Horizontal diffusion
In rivers, the horizontal diffusion coefficient will be affected by the shear velocity, the depth of the flow and the wind condition as follows:
Horizontal diffusion coefficient
is equal to
an empirical coefficient times
depth of the flow times shear velocity.
The magnitude of the empirical coefficient in the former formula can be estimated as 0.15+-50% for the transverse mixing in straight rectangular channel and 0.6+-50% for irregular channels and natural streams ( Fisher et al., 1979 ).
5.3 Mechanical Spreading
The Fay's formulae for effective surface oil slick radius used in ROSA can be summarized as follows:
Spreading phase Effective oil slick radius, m
Gravity - inertia 1.14.(delta.g.V.t^2)^0.25
Gravity - viscous 0.98.(delta.g.V^2.t^1.5/enu^0.5)^0.167
Surface tension - 1.60.(sigma^2.t^3/(enu.row^2))^0.25
viscous
Final equilibrium (1.e05*V^0.75/3.14159)^0.5
Wherein, delta = relative density ratio;
V = volume of oil slick, m^3;
enu = kinematic viscosity of water, m^2/sec;
sigma = surface tension, N/m.
The following formula is used to predict the mechanical spreading of a continuous spill under ice ( Cox and Shultz, 1981 ):
The slick radius
is equal to
0.25 times 1/6 power of
the relative density ratio times the average volumetric oil
rate released squarred over half of the RMS roughness height
of ice cover times
2/3 power of the time from the beggining of the spill.
The above formulae are used to calculate surface oil slick areas to be used in mass balance of and to calculate average slick thickness in the whole range of time moments as explained in Section 6.2.
5.4 Shoreline hold-up
When oil reaches a shoreline, it will be deposited along it. However, the capability of a given segment of shoreline to hold oil is limited. Once this limit is reached, oil will remain offshore. The maximum amount of oil that can be deposited on various shoreline types depends upon surface oil thickness, shore slope etc. Based on observed and reported oil spills, it is accepted that the river shorelines are capable of absorbing oil at some predetermined volume per unit length.
By default, ROSA treats the shoreline oil-holding in terms of cubic meters of oil per linear meter of shorefront ( Gundlach, 1987 ). These values should be specified as a part of input data for shoreline types of river section in question. The amount of oil spill simulated is reduced by becoming trapped due to shoreline deposition.
5.5 Shoreline re-entrainment
The model, embodied in ROSA, simulates the net rate of oil removal from a shoreline as an exponential decay with a series of oil removal coefficients ( Gundlach, 1987 ). The latters describe the absorbency of the certain type of shoreline.
The first order decay function is as follows:
Mass of oil within each shoreline segment
is equal to
mass of oil originally deposited on the shore times
exponent of (minus the removal rate coefficient times
time in days since original deposition )
The range of removal coeficient magnitudes is from 0.001-0.01 for marshes up to 0.99 as for rocky shores under low wave conditions.
5.6 Evaporation
Evaporation accounts for the largest loss in oil volume. The volume fraction of evaporated oil is determined by Mackey et al, 1980, as
F = (1/C)*(lnPo + ln(C.Ke.t + 1/Po)),
where
ln Po = 10.6.(1 - To / Te),
where Te stands for ambient air temperature, K; and To is the initial boiling point, K.
For a crude oils:
C = 1158.9.API^(-1.1435)
To = 542.6 - 30.275 API + 1.565 API^2 - 0.03439 API^3 +
0.0002604 API^4
The API oil index and oil density are related, as follows
density = 141.5 / (API + 131.5)
The molar volume of oil is calculated from the oil molecular weight. Its value is varied between 150.0e-06 and 600.0e-06 m^3/mole, depending on the oil composition. For fuel oils the value is very close to 200.0e-06 m^3/mole.
5.7 Dissolution
The total dissolution rate is calculated as a dissolution mass transfer coefficient times slick area times the oil solubility in water ( Cohen et al, 1980 ).
The typical oil solubility is supposed to be equal to the solubility of fresh oil times the exponential function of minus decay constant times oil residential time.
5.8 Emulsification
The oil-in-water emulsification is treated as the entrainment of oil droplets into the water column due to mixing and dispersion. The simultaneous process of the latters is simulated by the exchange terms in conservation equation of surface oil based on rate coefficients ( Yapa et al, 1991 ) as follows:
Volumetric rate of oil-in-water emulsification
per unit water surface area
is equal to
empirical coefficient describing the rate
at which the surface oil is dispersed into
the water column times volume of surface oil per unit area.
5.9 Sedimentation
The oil sedimentation is treated as the oil deposition on the bed. It is is simulated by the exchange terms in conservation equation of surface oil based on rate coefficients ( Yapa et al, 1991 ) as follows:
Volumetric rate of oil deposition on the bed
per unit water surface area
is equal to
empirical coefficient describing the rate
of net oil deposition on the river bed per unit area times
depth averaged volumetric concentration of oil in the
suspended layer.
5.10 Vertical Mixing
In rivers, the mixing mechanism is dominated by turbulence intensity in the river stream. The useful formula which can be derived from the velocity profile is as follows:
Vertical diffusion coefficient is equal to an empirical coefficient times depth of the flow times shear velocity.
The value of an empirical coefficient has been found to be usually 0.067 ( Fisher, 1979 ).
In ROSA, the vertical mixing between surface and suspended layers is computed by the exchange source terms based on the rate coefficients.
5.11 Surface ice cover allocation
ROSA has its own built-in procedure to predict allocation of surface ice cover in a winter seasons.
To provide reasonable information for the distribution of ice cover on the water surface, the simple, but efficient, ice formation model is developed. The model nature is based upon the heat balance at water surface ( Michel, 1978 ) and returns the ice marker values either as unity for ice covered cells or as zero for open surface control volumes. The resulting marker distribution can be viewed by PHOTON as ice cover map.
It can be used if no information of real ice cover location exists. Alternatively, the model can be easily calibrated for specific meteorological conditions to fit the ice cover data available.