Fourth Study Conference on BALTEX Scala Cinema Gudhjem

3. Hydrodynamic model of nutrient loads in
the Western Dvina/ Daugava River
The one-dimension hydrodynamic model of pollution
load in rivers used in this study is based on the common
set of equations for the movement of water and for the
movement of a pollutant.
Governing equations for the calculation of hydraulic
parameters (water levels, discharges, average velocities
and velocity distribution in each cross section of the
river) are based on the Saint-Venant generalized equations.
The unknown values η(x,t) (water level) and Q(x,t) (water
discharge) are found by solving the following hyperbolic
system of equations
∂η ∂Q
B + = q
(1)
0
∂t
∂x1
∂Q
∂Q
2 2 ∂η
+ 2V
+ (CV
−V
)B1
= Ф (2)
t ∂x
∂x
∂ 1
1
which satisfies the following conditions :
- initial Q(x1,t)=Q0(x1); η(x1,t)= η0(x1) (3)
- boundary Q(0,t)= ϕ(t);Q(L,η)=ψ(z) (4)
⎡ ⎛ ∂ω
⎞ ⎤ 2 Q Q
Ф = ⎢B1I
0 + H const ⎥V
− gω
;
where :
⎜
⎣ x ⎟ =
2
⎝ ∂ 1 ⎠ ⎦ K
Q
V = ; C
ω
V
=
ω
g ; K = ωC
B
For the calculation of the velocity distribution in each
cross-section the real flow is assumed to be replaced by
a superposition of two hypothetical flat rectangular
flows (vertical with depth H and horizontal with width
B). The velocity distribution results from solving of the
nonlinear system of equations taking into account the
mentioned hypothesis and a logarithmic law for the distribution
of a velocity in the vertical.
The one-dimensional hydrodynamic model of the nutrient
transport are based on the equations of turbulent
diffusion.
∂ ( ωC)
∂ ( ωVC
) ∂ ⎛ ∂C
⎞
+ = ωD
+ ωf
∂t
∂x
∂x
⎜
∂x
⎟ (5)
1
1 ⎝ 1 ⎠
which satisfies the following conditions :
- initial C( x1,
t0
) = ϕ(
x1)
(6)
- boundary C( 0,
t)
= ψ(
t)
(7)
where :
Q -water discharge in cross section, m 3 /s;
η - water level, m;
V -average velocity in cross section, m/s;
B -width of transitional part of cross section, m;
B0 -width of common of cross section, m;
ω - area of cross section, m 3 ;
Q -inflow discharge, m 3 /s;
I0 - bottom gradient;
C -average concentration of pollutant in cross
section, mg/l:
D -coefficient of longitudinal dispersion, m 2 /s;
R -hydraulic radius, m;
CR -Chezi coefficient;
L -length of river site, m;
q -water influx (per unit length), m 3 /s;
H –depth, m.
1
R
R
- 189 -
Modeling Criteria
The one-dimensional hydrodynamic model can be
used for the calculation of long-distance pollution transport
in rivers only in cases where the pollution gradients
across the river are negligibly small compared to those
along the river. This is in particular important for mean
calculations of cross-boundary transport.
Model description
The method of finite differences is used to solve the onedimensional
hydrodynamic model based on equations
(1), (2) and (5). The one-dimensional hydrodynamic
model for the Western Dvina/Daugava River was developed.
The one-dimensional hydrodynamic model allows
to calculate the distribution of average concentration of
pollutants along the river and to forecast the latter for a
time period of 30 days in case of accidental loads from a
point source. Also, an average pollution discharge is
calculated for locations where the river crosses state
boundaries.
The procedure of modeling the nutrient components
transport in Western Dvina/Daugava has been designed
as a convenient “Menu-driven system”. Users may apply
the model by providing required model input information
with a convenient easy-to-use dialog system (see
fig.3). A relevant User Manual was developed for this
purpose.
Figure 3. Example of the hydrodynamic prognostic
model user menu output for the Western Dvina/Daugava
river showing results for 2 hours (left) and 18 hours
(right) after injection of loads from a point source.
Model Calibration.
The calibration of the model’s hydraulic variables was
made based on observed hydraulic data. The calibration
parameters in this case are the local stress coefficients
along the perimeter of the cross sections. The calibration
parameters for the chemical parameters are coefficients
characterizing the distribution of “self cleaning” of the
river, included in the right-hand side of equation (6).
Experimental data of nitrogen ammonia distribution in
the Western Dvina/Daugava were used.
References
Korneev, V., Rogunovich V., Stanckevich, A., Water
regime of the river Pripiat and its runs at incomplete
realization of the design of guard of basin from deluging.
International <strong>Conference</strong> “Modern problems
of study, usage and protection of natural recourses of
Polesje-region”, Minsk (Russian). p.120.
Rogunovich, V.P. Automatisation of a mathematical
modeling of flow and substance transport in the
water-stream systems, St.-Petersburg,1989 (Russian),
p 263.