Environmental Engineering Reference
In-Depth Information
2 d
−1
, calculate the concentration at the spill location
after 1 day and 1 week.
3
3
2
∂
∂
c
t
V
R
∂
∂
c
x
D
R
∂
∂
c
∑
∑
i
i
+
=
−
λ
c
(5.69)
x
2
d
i
d
i
i
=
1
i
=
1
Solution
where
R
d
is the retardation factor defined by Equation
(5.54). Equation (5.69) indicates that the fate and trans-
port of a tracer that is undergoing both sorption and
first-order decay is the same as if sorption is neglected,
but the mean fluid velocity and dispersion coefficients
are reduced by a factor 1/
R
d
. Equation (5.69) can be
further simplified by changing variables from
c
to
c
*
,
where
The concentration distribution (accounting for sorption
but prior to correction for decay) is given by
M
Hnt D D
(
x Vt
D t
−
)
2
y
D t
2
*
( ,
c x y t
, )
=
exp
−
−
4
4
4
π
L
T
L
T
where
M
= 3 kg,
H
= 1 m,
n
= 0.2,
D
L
= 0.05/
R
d
=
0.05/20 = 0.0025 m
2
/day,
D
T
= 0.005/
R
d
= 0.005/20 =
0.00025 m
2
/day,
x
= 0 m,
y
= 0 m, and
V
= 0.1/
R
d
=
0.1/20 = 0.005 m/day. Substituting these values into the
preceding equation yields
c
=
c e
*
−
λ
t
(5.70)
Substituting Equation (5.70) into Equation (5.69)
and simplifying yields
3
*
( ,
3
3
c
0 0
, )
t
=
*
*
2
*
∂
∂
c
t
V
R
∂
∂
c
x
D
R
∂
∂
c
x
∑
∑
i
i
+
=
(5.71)
4
π
( )( . )
1 0 2
t
( .
0 0025 0 00025
)( .
)
2
d
i
d
i
i
=
1
i
=
1
2
0 005
4 0 00
( .
t
)
exp
−
kg/m
3
( .
25
)
t
This is the advection-diffusion equation for a conser-
vative contaminant and demonstrates that the fate and
transport of a sorbing tracer undergoing first-order
decay can be modeled by (1) reducing the fluid velocity
and dispersion coefficients by the factor 1/
R
d
, (2)
neglecting both sorption and decay, and (3) reducing the
resulting concentration distribution by the factor
e
−λt
,
where
t
is the time since the release of the tracer mass.
Equation (5.71) assumes that the decay constant,
λ
, is
the same for both the aqueous phase contaminant and
the sorbed contaminant. In cases where these decay
coefficients differ, the “lumped” decay coefficient can be
taken as
1510
=
exp[
−
0 0025
.
t
]
kg/m
3
t
Correcting for decay requires multiplying by
e
−λt
,
where
λ
= 2 d
−1
, and therefore the actual concentration
as a function of time is given by
1510
c
( ,
0 0
, )
t
=
c
*
( ,
0 0
, )
t e
−
2
t
=
exp[
−
0 0025
.
t
−
2
t
]
kg/m
3
t
1510
=
exp
[
−2 0025
.
t
kg/m
]
3
t
Therefore, at
t
= 1 day,
c
(0, 0,
t
) = 205 kg/m
3
=
205,000 mg/L, and at
t
= 7 days
c
(0, 0,
t
) = 1.75 × 10
−4
kg/m
3
= 0.175 mg/L.
ρ
K
n
b
d
λ
=
λ
+
λ
(5.72)
aq
s
where
λ
aq
and
λ
s
are the decay coefficients in the aqueous
and sorbed phases (T
−1
), respectively. It is noteworthy
that some organic chemicals biodegrade rapidly in
water, but not when sorbed onto the solid matrix.
Although sorption and decay are used in most cases
to quantify the fate of contaminants in groundwater,
consideration of other fate processes, such as chemical
reactions that result in the complete consumption of a
pollutant, can lead to fundamentally different results.
For example, consideration of only sorption and decay
in a fate and transport model yields a plume that is
constantly increasing in size, while taking into account
complete consumption of a contaminant at the front of
a plume leads to a nongrowing plume, which has been
reported under some field conditions (e.g., Grathwohl
et al., 2001). Other commonly neglected processes, such
as the transfer of contaminants to immobile regions of
the aquifer and the spatial variability of mass transfer
EXAMPLE 5.12
Three kilograms of a contaminant is spilled over a
1-m depth of groundwater that is moving with an
average seepage velocity of 0.1 m/d. The longitudinal
and horizontal-transverse dispersion coefficients are
0.05 and 0.005 m
2
/d, respectively, vertical dispersion is
negligible, and the porosity is 0.2. If the retardation
factor is equal to 20 and the first-order decay factor is
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