A Lagrangian Model for Shallow Water Bodies Contaminant Transport
INTRODUCTION
Mathematical models have been used to simulate circulation and dissolved substances transport in water bodies as rivers, coastal waters and estuaries.
Usually these models are composed by one or more partial differential equations, describing relevant physical processes evolution (water level, velocities, dissolved substances concentrations) that occurs in water bodies. The known mass conservation and momentum laws, state equation and state equation constituents transport equations are the equation system basis. In general is not possible to solve these equations analytically, so it's necessary to use numerical methods as finite difference and finite elements. An interesting alternative to the advective-diffusive transport equation numerical simulation is based on the particle tracking models. Despite of the other numerical methods that simulate the contaminant spreading by calculating concentration in the mesh nodes, the particle tracking model tracks several individual particles and then calculate the concentration.
As a Lagrangian model, it has the advantage of being independent of the mesh scale resolution, except in the velocity field details. Advantages of this kind of model are: (i) there is not mass loss, if it is not explicitly modeled by mass decay; (ii) it can deal with strong concentration gradients; (iii) the computation effort is restricted to areas where the particles slug is located; (iv) there is no numerical diffusion introduction; (v) neither numerical oscillations nor negative concentrations are produced.
These models have been used in the groundwater modeling
1988 1993) where the random
component represents the velocity spatial variation unsolvable due to the media heterogeneity properties. Particle models, with or without the random component, have also been used to simulate the residual circulation , 1980;
1982). But these are less usual then the models used to predict
pollutant concentration and 1964;
and 1992; 1993; 1995;
1997).
The model developed in this work simulate the tracking and the passive pollutant concentrations released in shallow water bodies, so that concentration estimative can be obtained for interest areas, for different forcing factors conditions. To validate the model, its results have been compared to the analytical solution results for a constant cross section area.
(KINZELBACH, DIMOU,
(AWAJI AWAJI,
(BUGLIARELLO JACKSON, AL-RABEH GUNAY, JIN, SANTOS, HORITA,
apud
et al.
MATHEMATICAL MODEL
NUMERICAL MODEL
The adopted mathematical model is formed only by the advection-diffusion equation, as the equations (Navier-Stokes
and continuity) that form the circulation model are solved apart.
Results from this model (velocity field and surface water level) are used by the transport model as input data. This uncoupled is possible only when the transported substance is passive, meaning that the substance does not interfere the local hydrodynamics.
The depth averaged transport equation (2DH) may be written as follows:
with and =1, 2, and where, for large scale flows, is the pollutant concentration, is velocity vector component in direction, is the turbulent diffusion coefficient and is the pollutant consumption or production rate tax.
In the particle-tracking model, several particles are released in the domain by a punctual source, having their path calculated by the sum of a deterministic component (advection) and an independent random component (turbulent diffusion). The two- dimensional (2DH) hydrodynamics model velocities are used to simulate the advective component, while the diffusive component is represented by a little random displacement in the particle mass center. The trajectories ensemble describes pollutant plume dispersion in a shallow water body.
The process begins with a finite number of particles being released per time step in a rectangular area (near field), and each particle is randomly placed in this area. The area dimensions must be big enough to enable a passive plume establishment in its interior. The described model is a far field model, meaning that simulates only processes that occur after the passive plume establishment.
The particles represent the released contaminant in the water body. The mass of each particle released is solved by the relationship between the number of particles released and the contaminant load per time unit (concentration source flow), as shown in the following equation.
where is the source flow, with a contaminant concentration , is the mass of each particle released, and / is the number of particles released per time unit.
i j C
u x
Eij R
Q
C M N t
i i
e
e pÄ
Journal of Coastal Research SI 39 1610 - 1613 ICS 2004 (Proceedings) Brazil ISSN 0749-0208
C. O. Horita and P. C. C. Rosman
† ‡†CTTMar UNIVALI, Itajaí 88302-202, Brazil cristina.horita@univali.br
HORITA, C. O. and ROSMAN, P. C. C., 2006. A lagrangian model for shallow water bodies contaminant transport.
Journal of Coastal Research, SI 39 (Proccendigs of the 8th International Coastal Symposium), 1610 - 1613. Itajaí, SC, Brazil, ISSN 0749-0208.
Particle tracking models have been developed and used to simulate tracking and concentration of the passive contaminants in water bodies. This paper presents the development of a depth average random tracking model for simulating dissolved contaminants transport. Comparing simulated concentrations with corresponding analytical solutions for a constant transversal area channel tests the model. Appropriate agreement is obtained.
ADDITIONAL INDEX WORDS:Particle tracking, water quality model, numerical model . ABSTRACT
‡ AECo/PENO UFRJ, Rio de Janeiro 21945-970, Brazil pccr@peno.coppe.ufrj.br
x R hE C x h x u C t C
j ij i i
i
÷ ÷ +
ø ö ç ç
è æ
¶
¶
¶
= ¶
¶ + ¶
¶
¶ 1
(1)
p e e
N t C
M = Q D
(2)Journal of Coastal Research Special Issue 39, 2006,
The transport of launched particles in each instant is defined by the particle tracking, calculating its position to each time ( +1) , . For such a Taylor series expansion from the previous position is used, in instant, as indicated:
where H.O.T. are the despised higher order terms. The velocity field obtained from the hydrodynamic model as follows calculates the position time derivatives, :
The velocity for the particle advective transport, ( , ), follow the water body stream lines, that vary with time and space, according to local forcing factors. It is implicit in the method that the contaminant is passive, and that therefore its presence does not interfere with the environment hydrodynamics. In order to simulate the local turbulent diffusion in the particles scattering, a random velocity deviation is introduced, , which is added to the advection velocity determined by the hydrodynamic model.
The 2DH velocity field is suitable, when the substance transported is well mixed in the water column. When it is a floating substance or the contaminant plume is restricted to a water column level, free surface velocities or velocities in the appropriate level can be used, obtained from velocities profiles resulting from a 3D model.
To calculate the concentrations, the model defines a distribution grid inside of which the entire contaminant plume is contained with a break of at least 10% in each plume extremity.
Such grid can be independent of the hydrodynamic mesh.
Known the position of one given particle in one given time, the particle mass is distributed to each associated grid cell. For example, consider a particle of mass ( ) in one given instant located in ( , ), and grid cells, being the center of a generic cell defined by ( , ). The mass ( ) is distributed for the cells according to a specified distribution function. For example, in the simplest case, all the mass ( ) is placed in the cell where the particle is located. Such simplified procedure is suitable when the cell volume is big in relation to the plume associated to one given particle. When the plume volume is big in relation to the cells size, the model may use a Gaussian type function,
where the variances are related to the dispersion coefficients, or to the depth averaged turbulent diffusion, as indicated.
At one given instant after the particle of mass releasing, the mass portion in the generic point position ( ) would be defined by
The mass value of a given particle, at one given time depends on the defined kinetic reactions of production or consumption. For many substances a first order decay is verified, and it can be written as follows: ( ) = exp (- );
where is the mass particle in its launching instant, the decay constant e the particle life time. If the coliforms transport is simulated the decay constant can conveniently be written as = -1n(0.1)/ ;
where is the necessary time for 90% of decay.
In each time, adding all the portions of mass of contaminant allocated to the cell, and dividing the sum by the cell influence volume determine the effluent concentration in each grid cell. In the case of well-mixed substances in the water column, that volume can be time variable, in function of the water level variation.
With such procedure, different contamination scenarios can be obtained, due to different source flow conditions, source position, and environmental forcing factors. The probability of contamination over a determined control level would be function of the adverse forcing factors occurrence probability, which would transport the contaminant plume to undesirable places.
The model was applied to a simplified case with a known analytical solution in order to verify the model validity. The model was applied to a uniform rectangular channel with a continuous vertical line source.
It is assumed that the depth averaged longitudinal velocity ( ) is constant in direction; the velocity in direction is equal to zero; the depth is constant and the concentration variation in
n t P
P n t
P
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t
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MODEL VERIFICATION
A Lagrangian Model for Shallow Water Bodies Contaminant Transport
Figure 1. Channel discretization.
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2
22 2
1
H O T
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P
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(4)
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2
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) , (
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1
) , (
) , ( ) ) (
,
(
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1611
V
v
q q Esen E
E E
y x
=
= cos
(10)
) / tan(
) /
tan( v
*u
*a E
TE
La +
=
q
(11)direction is negligible. Considering these simplifications the analytical solution for this case is presented in the following
form 1986):
where is the source flow, is the channel width and the channel depth. The infinite series results from the necessary images to satisfy the zero mass flux at the channel sides.
The depth averaged diffusion coefficients and , are evaluated in flow function, as the equations that follow:
(HALERMAN,
q B h
Ex Ey
where and are calibration parameters, is the gravity acceleration, is the Chézy coefficient and is the bed roughness amplitude. and are respectively the horizontal dispersion coefficient, in the longitudinal and transversal directions to the flow streamlines.
The channel used in the numerical simulation has 100m x 10m, and has been discretized in elements of and equal to 5 and 1m, respectively, as show in Figure 1.
For the simulated case, it was used a 0.05m/s longitudinal velocity, channel depth of 5m, bed roughness amplitude of 0.03m/s and 0.02kg/s·m of flow per depth unit. In the numerical simulation 30 particles were launched in intervals of 25s, for 4100s. This simulation time was chosen in order to have a permanent flux, similar to the analytical solution conditions.
The particles take approximately 1900s to leave the channel, so with a simulation time a little bigger than the double time necessary to the particles leave the channel; a permanent flux is guaranteed, where equal number of particles enters and leaves the channel.
According to Figure 2 to Figure 5, the model is more sensitive to variations in parameter than in parameter. This can be explained by the presence of the channel sides, which due to the zero mass flux condition at closed contours, make that the mass that would be distributed to points out of the domain to be reflected to the channel interior, making the distribution graph more smooth. This effect does not occur to parameter because the channel is long enough and the contours in this direction are open.
As seen in the presented Figures, values of 1.5 to and gave more accurate results. Taking these values as a start, a
“gentle adjustment” was made, in order to try to improve the calibration of these parameters, as shown in Figure 6 to values of 1.7 and 1.6 to and , respectively.
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á
á á
á á
L T
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L T
T L
L
T L
T L
g C
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x y
CONCLUSIONS
The particle-tracking model, when compared to an analytical solution has presented consistent results. Showing that, despite the fact that it does not solve the advective diffusive transport equation directly, it gives equivalent results. The little differences observed can be eliminated by the calibration parameters adjustment.
Horita and Rosman
Figure 3. Comparison between results from analytical and numerical solution. (a) Cross section; (b) Longitudinal profile, for L equal to 1.5 and T equal to 1.0.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00
Channel length (m) Concentration(kg/m3)
Numerical Analytical
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
Channel width (m) Concentration(kg/m3)
Numerical - 25m Numerical - 55m Numerical - 85m Analytical - 25m Analytical - 55m Analytical - 85m
Journal of Coastal Research Special Issue 39, 2006, 1612
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+¥-¥
=
ú ú
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Figure 2. Comparison between results from analytical and numerical solution. (a) Cross section; (b) Longitudinal profile, for L and T equal to 1.0.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00
Channel length (m) Concentration(kg/m3)
Numerical Analytical
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
Channnel width (m) Concentration(kg/m3)
Numerical - 25m Numerical - 55m Numerical - 85m Analytical - 25m Analytical - 55m Analytical - 85m
LITERATURE CITED
AL-RABEH, GUNAY,
AWAJI,
AWAJI, IMASATO, KUNISHI,
BUGLIARELLO, JACKSON,
DIMOU, ADAMS,
HARLEMAN,
HORITA,
JIN,
SANTOS, A. H. and N., 1992. On the application of a
particle dispersion model. , vol. 17,
195-210.
T., 1982. Water mixing in a tidal current and the effect of turbulence on tidal exchange through a strait.
, vol. 12, 510-514.
T., N. and H., 1980. Tidal exchange through a strait a numerical experiment using a simple
model basin. , vol.10,
1499-1508.
G. and E. D., 1964. Random walk study of convective diffusion.
, ASCE 90 (EM4), 49-77.
K. N. and E. E., 1993. A random-walk, particle
tracking model for well mixed estuaries and coastal waters.
, vol. 37, 99-110.
D., 1986. Transport Processes in Environmental Engineering. Notes for lectures in 1977 Water Control, Massachusetts Institute Technology.
C. O., 1997. Estudos de Validação e Aplicação de um Modelo Lagrangeano para Transporte de Contaminantes em Corpos D'água Rasos. Rio de Janeiro, Rio de Janeiro:
Federal University of Rio de Janeiro, Master's thesis, 106p.
X. Y., 1993. Quasi-three-dimensional numerical modeling of flow and dispersion in shallow water. Amsterdam, The Netherlands: Delft University of Technology, Ph. D. Thesis, 174p.
L. H., 1995. Um modelo para a trajetória de partículas em corpos d'água rasos. Rio de Janeiro, Rio de Janeiro:
Federal University of Rio de Janeiro, Master's thesis, 97p.
Coastal Engineering
Journal of Physical Oceanography
Journal of Physical Ocanography
Journal of Engineering Mechanics Division
Estuarine, Coastal and Shelf Science
Figure 5. Comparison between results from analytical and numerical solution. (a) Cross section; (b) Longitudinal profile, for L and T equal to 1.5.
A Lagrangian Model for Shallow Water Bodies Contaminant Transport
Journal of Coastal Research Special Issue 39, 2006,
1613
Figure 6. Comparison between results from analytical and numerical solution after the parameters calibration. (a) Cross section; (b) Longitudinal profile, for L equal to 1.7 and T equal to 1.6.
Figure 4. Comparison between results from analytical and numerical solution. (a) Cross section; (b) Longitudinal profile, for L equal to 1.0 and T equal to 1.5.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00
Channel length (m) Concentration(kg/m3)
Numerical Analytical
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
Channel width (m) Concentration(kg/m3)
Numerical - 25m Numerical - 55m Numerical - 85 m Analytical - 25m Analytical - 55m Analytical - 85m
