Download Transport in streams and rivers

Гидрогеология Загрязнений
и их Транспорт в
Окружающей Среде
Yoram Eckstein, Ph.D.
Fulbright Professor 2013/2014
Tomsk Polytechnic University
Tomsk, Russian Federation
Fall Semester 2013
Transport and Fate
of Contaminants
in Surface
Waters
Nature of surface waters
Free surface in equilibrium with the
atmosphere
Open system; exchange with the
atmosphere, biosphere and the
lithosphere
Stratification
Flow velocity from 0 to x
Sources of
contaminants
Point sources
Non-point sources
Transport in streams and rivers
Manning’s velocity
The Manning formula is also known as the Gauckler–
Manning formula, or Gauckler–Manning–Strickler formula
in Europe. In the United States, in practice, it is very
frequently called simply Manning's Equation. The Manning
formula is an empirical formula estimating the average
velocity of a liquid flowing in a conduit that does not
completely enclose the liquid, i.e., open channel flow. All
flow in so-called open channels is driven by gravity. It was
first presented by the French engineer Philippe Gauckler in
1867, and later re-developed by the Irish engineer Robert
Manning in 1890.
Transport in streams and rivers
Manning’s velocity
𝑘
𝑣=
𝑛
2
3
𝑅ℎ
1
𝑆2
𝑣 is the cross-sectional average
velocity (L/T; ft/s, m/s);
k is a conversion factor of (L1/3/T), 1 m1/3/s for SI,
or 1.4859 ft1/3/s U.S. customary units, if required.
n is the Gauckler–Manning coefficient, it is unitless;
Rh is the hydraulic radius (L; ft, m);
dx
S =   the slope of the water surface or the linear hydraulic
 dl 
head loss (L/L) (S = hf/L).
Transport in streams and rivers
Manning’s velocity –
hydraulic radius
𝐴
𝑅ℎ =
𝑃
Rh is the hydraulic radius (L);
A is the cross sectional area of flow (L2);
P is the wetted perimeter (L).
The hydraulic radius is a measure of a channel flow
efficiency. Flow speed along the channel depends on
its cross-sectional shape (among other factors), and
the hydraulic radius is a characterization of the
channel that intends to capture such efficiency.
Transport in streams and rivers
Manning’s velocity
𝑘
𝑣=
𝑛
2
3
𝑅ℎ
1
𝑆2
n - the Gauckler–Manning coefficient
1
𝑑506
𝑛=
21.2
where d50 = median sediment diameter, m.
The Gauckler-Manning
roughness coefficient
0.012 < n < 0.050
smooth
rough
concrete
mountain
bed
streambed
Transport in streams
and rivers
Chezy’s velocity
v - velocity
dx 

v  C R 
 dl 
C – Chezy friction coefficient
 dx  - hydraulic gradient
 
 dl 
Transport in streams
and rivers
dx 

v  C R 
 dl 
Chezy friction
coefficient
1
1/6
𝐶 = (𝑅ℎ )
𝑛
Travel time
l - length
x2
l
1
t 
dx
v x1 v x 
v - velocity
Q = v∙A
Ji = Q∙Ci
Rate of flow (l3/t) flux of a chemical (M/t)
Miscible conservative tracer
D – dispersion coefficient
Miscible conservative
tracer
Gaussian (normal) curve – longitudinal D
1
C( x) 
e
 2
x2
 2
2
D

2
2t
where σ is the standard deviation and C(x) is
concentration of the transported chemical
Miscible conservative
tracer
Gaussian (normal) curve – longitudinal D
1
C ( x, t ) 
e
2 D t 2
x2

2 2 Dl t
l
C ( x, t ) 
1
4 D t
l
e
( x vt )2

4 Dl t
Miscible conservative
tracer
Gaussian normal curve – longitudinal D
M
C ( x, t ) 
e
4 D t
( x vt )2

4 Dl t
l
M – mass of the tracer
Miscible
non-conservative tracer
Gaussian normal curve – longitudinal D
M
C ( x, t ) 
e
4 D t
( x vt )2

4 Dl t
e
 kt
l
M – mass of the tracer
M
C 
e
4 D t
max
l
 kt
Miscible
non-conservative tracer
Transversal dispersion
l
l - the length of the transverse mixing zone
Miscible
non-conservative tracer
Gaussian normal curve – transversal D
  2D t  w
t
t
w – width of the river
l - the length of the transverse
mixing zone
t = l/v
wv
l 
2D
2
t
Longitudinal and
transversal mixing
Longitudinal mixing is dominated by the
process of dispersion
Transversal mixing is caused only by flow
turbulence
Turbulence is predominantly by shear
velocity:
u* =
½
[gd(dx/dl)]
where d is the depth of the river
Longitudinal and
transversal mixing
for straight channels:
Dt = 0.15 d u*
for natural channels:
Dt = 0.6 d u*
Longitudinal and
transversal mixing
Dl = (0.011
2
v
w – width of the river
w )/(d u*)
Lakes & estuaries
Wind driven advection
2-d mixing
Stratification – thermocline
- halocline
Tidal effects in estuaries
Wind driven advection
The Lake Erie surface at the east end stands
ca. 1 m higher than at the west end
Wind driven advection
Summer lake stratification
Wind driven advection
Summer lake stratification
Wind driven advection
End of summer stratification
Wind driven advection
Fall mixing
Wind driven advection
Winter lake stratification
Thermocline and halocline
January 2010
temperature and
salinity profiles in
Arctic Ocean
Thermocline and halocline
Stream transport
Dissolved
load
Suspended load
Bed load
Solid particles in surface
waters
Suspended load
Bed load
Solid particles in surface
waters
Mineral – metal-hydroxides
- clay
ρ ≈ 2.6 g/cm3
Organic – bacteria & algae
ρ ≈ 1.3 g/cm3
Solid particles
in surface waters and air
Particle Settling
Stokes’ Law
 2  g    1 r
  

9  

 
s
f
f

f
Bottom sediments record
2
Particle Settling
Stokes’ Law
Bottom sediments record
Bottom sediments record
How do we establish the age of
layers in lake sediments record?
210Pb
dating is based on a relatively constant
atmospheric deposition of this radionuclide onto
surface waters and subsequent sorption of 210Pb on
particles in the water that eventually settle into a
chronostratigraphic deposit. The 210Pb concentration
(measured by its radioactivity) at any depth in the
sediment is equal to its concentration in freshly
deposited material (at the water/sediment interface) is
multiplied by exp(-λt), where λ is the 210Pb decay
constant = 0.03/year
How do we establish the age of
layers in lake sediments record?
Therefore:
−1
𝐴𝑑
𝑡=
× ln( )
𝜆
𝐴𝑜
Where:
Ad is 210Pb activity at a depth d
Ao is 210Pb activity at lake sediment surface
λ = 0.03/year
Concentration and Partial
Pressure of Gases in Air
Partial pressure = Percentage of
concentration of specific gas × Total
pressure of a gas
Dalton’s law
Total pressure = Sum of partial
pressure of all gases in a mixture
Concentration and Partial
Pressure of Gases in Air
Ambient Air
O2 = 20.93% = ~ 159 mm Hg PO2
CO2 = 0.03% = ~ 0.23 mm Hg PCO2
N2 = 79.04% = ~ 600 mm Hg PN2
Air-Water Exchange
Cequil = Ca/H
J = -kw(Cw – Ca/H)
kw is gas exchange coefficient
Cw is the gas concentration in water
Ca is the gas concentration in air
Air-Water Exchange
Solubility of CO2 and oxygen in
pure water
The tragedy at Lake
Nyos, Cameroon
Lake Nyos is a deep volcanic
crater lake, 5,900 feet (1,800
m) across and 682 feet (208
m) deep, that is thermally
stratified, with layers of warm,
less dense water near the surface floating on the
colder, denser water layers near the lake's bottom.
Over long periods, carbon dioxide gas seeping from
underground lava dissolve into the cold water at the
lake's bottom in great amounts. The amount of CO2
entering the lake is estimated to be about 90 million
kilogrammes annually.
The tragedy at Lake
Nyos, Cameroon
Lake Nyos fills a roughly
circular maar in the Oku
Volcanic Field, an explosion
crater caused when a lava
flow interacted violently
with groundwater. The
maar is believed to have
formed in an eruption about
400 years ago, and is 1,800
m (5,900 ft) across and 208
m (682 ft) deep
The tragedy at Nyos Lake,
Cameroon
Over time, the bottom layers of the lake become
supersaturated with CO2. When this occurs, the lake
becomes dangerously unstable, and an event such as an
earthquake or landslide can trigger a catastrophic
outgassing. This was the situation on August 21, 1986.
The tragedy at Nyos
Lake, Cameroon
On August 21, 1986 a small landslide in the lake
disturbed the lake stratification triggering
outgassing and up to a cubic kilometre of gas was
released. Because CO2 is denser than air, the gas
flowed down two valleys in a layer tens of metres
deep, displacing the air and suffocating all the
people and animals. The gas killed all living things
within a 15-mile (25km) radius of the lake,
suffocating 1,700 people and 3,500 livestock in
nearby towns and villages.
The tragedy at Nyos
Lake, Cameroon
The normally blue waters of
the lake turned a deep red
after the outgassing, due to
iron-rich water from the deep
rising to the surface and being
oxidised by the air. The level
of the lake dropped by about a
metre, representing the
volume of gas released.
The solution at Nyos
Lake, Cameroon
The method consists of a pipe
set up vertically between the
lake bottom and the surface. A
small pump raises the water in
the pipe up to a level where it
becomes saturated with gas, thus lightening the water
column; consequently, the diphasic fluid rises to the
surface. Therefore, once it has primed the gas lift, the
pump is not needed, since the process is self-powered:
above the saturation level, isothermal expansion of gas
bubbles drives the flow of the gas-liquid mixture as long
as dissolved gas is available for ex-solution and
expansion.
The solution at Nyos
Lake, Cameroon
Thin-Film Model
Ca
Molecular diffusion
Csa
Air film
Csw
Water film
Cw
Molecular diffusion
Thin-Film Model
Water-side control
J = -Dw(Cw – Ca/H)/δw
where δw is the thickness of the water film
if
Ca = 0
J = -DwCw/ δw = -kw Cw
Thin-Film Model
Air-side control
J = -(Da/δa) (CwH - Ca)
or
J = – (Da - H/δa ) (Cw - Ca /H)
where δa is the thickness of the air film
Thin-Film Model
for H ≈ 0.01




1
C 
J  
C  



 
H

 

D DH
a
w
w
w
a
a
Estimating gas exchange
coefficient
MW
k D


k D
MW
A
A
B
B
B
A
Estimating gas exchange
coefficient
In absence of a tracer with
a known gas exchange
coefficient models are
constructed empirically
for each gas, e.g. the four
models for kO2 :
K O2


24.94  1  N  u *

d
Thackston & Krenkel, 1969
K O2
K O2
V 
 1.92   
d 
0.85
Neglescu & Rojanski, 1969
23.2  V 0.73
K O2 
d 1.75
Owens et al., 1964
103  V 0.413  w0.273

d 1.408
Bennet & Rathbun, 1972
Estimating gas exchange
coefficient
Similarly, gas exchange
coefficients for slowly flowing waters, lakes or
estuaries are approximated empirically:
for slowly flowing or stagnant waters:
kw[cm/sec] ≈ 4·10-4 + (4·10-5·u2w10)
or
ka[cm/sec] ≈ 0.3 + (0.002·uw10)
Using gas exchange
coefficient
The air-water flux
density is proportional to the difference
between a chemical concentration in water
[Cw] and the corresponding equilibrium
concentration [CwH].
Therefore:
C( x ,t )  Co  e
 k r t
Thin-Film of Air Model Above
a Slick of NAPL
P
C 
 MW 
RT
a
while δa is the thickness of
the stagnant air film above
the slick, the velocity of gas
transfer (vaporization) is
proportional to Da of that gas
J
 Da
a
Ca
Thin-Film of Air Model Above a
Slick of NAPL
the velocity of gas transfer (vaporization) is
also dependent on the size of the slick
v=
-0.11
-0.67
0.029·uw10·L ·Sc
where:
uw10 wind velocity 10m above the slick [m/hr]
L is the slick diameter [m]
Sc is the Schmidt number