Environmental Engineering Reference
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and Equation (5.18) can be applied provided that
which at
t
= 7 days gives
2
4
π
t D D
T
A
(5.20)
10
( . )
( )( )
0 6 7
4 1 7
×
L
0
c
( ,
0 0 7
, )
=
exp
−
4
π
8
( )( )( . )
7 2 0 2
1 0 1
×
.
This relation is based on the requirement that the
size of the contaminated area is much greater than the
size of the initial spill area.
3
=
0 4
.
kg/m
=
480
mg/L
Hence, after 7 days, the concentration at the site of
the spill is approximately 53% of the maximum
concentration of 900 mg/L.
EXAMPLE 5.2
5.3.2 Continuous Point Source
Ten kilograms of a contaminant is spilled over the top
2 m of an aquifer. The longitudinal and horizontal-
transverse dispersion coefficients are 1 and 0.1 m
2
/d,
respectively; vertical mixing is negligible; the porosity
is 0.2; and the mean seepage velocity is 0.6 m/day.
(a) Estimate the maximum contaminant concentrations
in the groundwater 1 day, 1 week, 1 month, and 1 year
after the spill. (b) What is the contaminant concentra-
tion at the spill location after 1 week?
In the case where a conservative contaminant of initial
concentration
c
0
is injected continuously at a rate
Q
(L
3
T
−1
) into a uniform aquifer of depth
H
and mean
seepage velocity
V
, the concentration distribution
downstream of the source,
c
(
x
,
y
,
t
), is given by (Fried,
1975)
Qc
H D D
Vx
D
o
c x y t
( ,
, )
=
exp
[
W B W t B
( ,
0
)
−
( ,
)]
2
4
π
Solution
L
L
T
(a) From the data given,
M
= 10 kg,
H
= 2 m,
D
L
= 1 m
2
/d,
D
T
= 0.1 m
2
/d,
n
= 0.2, and
V
= 0.6 m/d.
According to Equation (5.18), the maximum con-
centration,
c
max
, occurs at
x
=
Vt
and
y
= 0 m; hence,
(5.21)
where the
x
coordinate is in the direction of the seepage
velocity;
y
is the transverse (horizontal) coordinate; the
source is located at the origin of the coordinate system;
D
L
and
D
T
are the longitudinal and transverse disper-
sion coefficients, respectively;
W
(
α
,
β
) is defined as
M
tHn D D
c
max
( ) =
t
4π
L
T
2
1
β
∞
∫
Substituting the given parameters gives
W
( ,
α β
)
=
exp
− −
y
dy
(5.22)
y
4
y
α
10
6 30
.
6300
3
c
max
( )
t
=
=
kg/m
=
mg/L
and
B
(dimensionless) is defined by
t
t
4
π
t
( )( . )
2 0 2
1 0 1
×
.
1
2
2
2
which yields the following results:
(
Vx
D
)
(
Vy
D D
)
(5.23)
B
=
+
4
2
4
L
L
T
t
c
max
(
t
)
(days)
(mg/L)
W
(
α
,
β
) is identical to the well function for a leaky
aquifer that is used in groundwater hydrology. To
facilitate the evaluation of Equation (5.21), values of
W
(
α
,
β
) are tabulated in Table 5.2. As
t
→ ∞, the
concentration distribution given by Equation (5.21)
approaches the steady-state solution (Bear, 1972)
1
6300
7
900
30
210
365
17.5
(b) The concentration at the spill location (
x
= 0 m,
y
= 0 m) as a function of time is given by Equation
(5.18) as
Qc
H D D
Vx
D
=
o
c x y
( ,
)
exp
1 2
/
2
π
(
)
2
L
T
L
(5.24)
1
2
2
V
D
2
x
D
2
y
D
2
M
tHn D D
(
Vt
D t
)
K
+
c
( ,
0 0
, )
t
=
exp
−
0
4
4
4
π
L
L
T
L
L
T
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