Environmental Engineering Reference
In-Depth Information
TABLE 5.9. Typical Values of Bulk Density and Porosity in
Porous Media
sion coefficients by a factor 1/
R
d
. In other words, the
fate and transport of a sorbing tracer can be modeled
by neglecting sorption but reducing the mean velocity
and dispersion coefficients by a factor of 1/
R
d
. Con-
taminants that have a higher partitioning coefficient
(
K
d
> 10
3
cm
3
/g) will move at a very slow rate, if at all.
On a cautionary note, the use of the (constant) retarda-
tion coefficient assumes that the sorption isotherm is
linear, partitioning reactions are very fast relative to
the rate of groundwater flow, and that equilibrium is
achieved between the aqueous and adsorbed phases of
the contaminant. In cases where these assumptions are
not valid, significant errors can occur in predictions of
contaminant fate and transport in groundwater.
Bulk Density
Porous media
(kg/m
3
)
Porosity
Limestone and shale
2780
0.01-0.20
Sandstone
2130
0.10-0.20
Gravel and sand
1920
0.30-0.35
Gravel
1870
0.30-0.40
Fine to medium mixed sand
1850
0.30-0.35
Uniform sand
1650
0.30-0.40
medium to coarse mixed sand
1530
0.35-0.40
Silt
1280
0.40-0.50
Clay
1220
0.45-0.55
Source of data
: Tindall and Kunkel (1999).
EXAMPLE 5.9
In applications to flow in porous media, the aqueous
contaminant concentration,
c
aq
, is commonly denoted by
c
, and hence Equation (5.51) can be written as
One kilogram of a contaminant is spilled over a 1-m
depth of groundwater and spreads laterally as the
groundwater moves with an average seepage velocity of
0.1 m/day. The longitudinal and transverse dispersion
coefficients are 0.03 and 0.003 m
2
/day, respectively; the
porosity is 0.2; the density of the aquifer material is
2.65 g/cm
3
; log
K
oc
is 1.72 (
K
oc
in cm
3
/g); and the organic
carbon fraction in the aquifer is 5%. (a) Calculate the
concentration at the spill location after 1 hour, 1 day,
and 1 week. (b) Compare these values with the concen-
tration obtained by neglecting sorption.
β
∂
∂
c
t
S
m
= −
(5.52)
n
Substituting this sorption model into the advection-
diffusion equation, Equation (5.17), yields
3
3
β
n
∂
c
t
∂
∂
c
x
∂
∂
2
c
∑
∑
1
+
+
V
=
D
(5.53)
i
i
∂
x
2
i
i
Solution
i
=
1
i
=
1
(a) The distribution coefficient,
K
d
, is given by
The term (1 +
β
/
n
) is commonly referred to as the
retardation factor
,
R
d
, where
K
=
f K
d
oc
oc
β
(5.54)
R
d
= +
1
where
f
oc
= 0.05 and
K
oc
= 10
1.72
= 52.5 cm
3
/g.
Therefore,
n
Dividing both sides of Equation (5.53) by
R
d
yields the following form of the advection-diffusion
equation:
K
d
=
( .
0 05 52 5
)(
. )
=
2 63
.
cm /g
3
The dimensionless constant
β
is given by
β ρ
=
K
=
(
1
−
n K
)
ρ
3
3
2
∂
∂
c
t
V
R
∂
∂
c
x
D
R
∂
∂
c
b
d
s
d
∑
∑
i
i
+
=
(5.55)
x
2
d
i
d
i
where
n
= 0.2,
ρ
s
= 2.65 g/cm
3
, and therefore
i
=
1
i
=
1
β=
(
1 0 2 2 65 2 63
−
. )( .
)( .
)
=
5 58
.
Comparing Equation (5.55), which accounts for the
sorbing of contaminants onto porous media, to the
advection-diffusion equation for conservative con-
taminants, Equation (5.17), it is clear that both equa-
tions have the same form, with sorption being accounted
for by reducing the mean seepage velocity and disper-
The retardation factor,
R
d
, is then given by
β
5 58
0 2
.
.
R
d
= +
1
= +
1
=
29
n
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