Environmental Engineering Reference
In-Depth Information
λ
i
, in the
x
i
coordinate directions. The geometric mean
hydraulic conductivity,
K
G
, is related to 〈
Y
〉 by
x
c
(
x
, 0, 0, 3650)
c
(
x
,0, 0, 3650)
(m)
(kg/m
3
)
(mg/L)
10
0.036
36
e
Y
K
G
=
(5.30)
100
8.66 × 10
−4
0.87
Data from a variety of geologic cores and the
statistics of the measured hydraulic conductivities are
tabulated in Table 5.3. These data indicate relatively
high values of
σ
Y
, which reflect a significant degree
of variability about the mean hydraulic conductivity.
The variance of the hydraulic conductivity is inversely
proportional to the magnitude of the support scale,
with larger support scales resulting in smaller variances
in the hydraulic conductivity. Consequently, when-
ever values of
σ
Y
are cited, it is sound practice also
to state the corresponding support scale. The support
scale of the data shown in Table 5.3 is on the order
of 10 cm (4 in.). The spatial covariance of
Y
must
also be associated with a stated support scale, since
both
σ
Y
and the correlation length scale,
λ
i
, depend
on the support scale. Larger support scales generally
yield larger correlation length scales. Porous media in
which the correlation length scales of the hydraulic con-
ductivity in the principal directions differ from each
other are called
anisotropic
media, and porous media
where the correlation length scales of the hydraulic con-
ductivity in the principal directions are all equal are
called
isotropic
media. The mean seepage velocity,
V
i
(LT
−1
), in isotropic porous media is given the the Darcy
equation,
Replacing
x
by
x
0
(= 833 m) in the
z
term yields
c
(1000, 0, 0, 3650) = 7.48 × 10
−14
kg/m
3
≈ 0 mg/L.
The results of this example show that biode-
gradation will have a significant effect on the contami-
nant concentrations downstream of the source. At
x
= 1000 m, the biodegraded contaminant concentra-
tion is negligible.
In practical applications of both analytic solutions
and numerical models of contaminant transport in
groundwater, it is important to keep in mind that the
model parameters, and sometimes the input data, are
seldom known with certainty. This uncertainty in model
parameters and input data generally translates into
uncertainty in model predictions, and so any predicted
concentrations are best presented as a probability dis-
tribution, or at least as a most likely value bounded by
upper and lower confidence limits.
5.4 TRANSPORT PROCESSES
Dispersion of contaminants in groundwater is caused by
spatial variations in the hydraulic conductivity of the
porous medium and, to a much smaller extent, by pore-
scale mixing and molecular diffusion. Pore-scale mixing
results from the differential movement of groundwater
through pores of various sizes and shapes, a process
called
mechanical dispersion
, and the combination of
mechanical dispersion and molecular diffusion is called
hydrodynamic dispersion
. Dispersion caused by large-
scale variations in hydraulic conductivity is called
macrodispersion
. Consider a porous medium in which
several samples of characteristic size
L
are tested for
their hydraulic conductivity,
K
. The hydraulic conductiv-
ity (
K
) is then a random space function (RSF) with
support scale
L
. Assuming that
K
is log normally dis-
tributed, it is convenient to work with the variable
Y
defined as
K
n
eff
V
= −
J
(5.31)
i
i
e
where
K
eff
is the effective hydraulic conductivity
(LT
−1
),
n
e
is the effective porosity (dimensionless), and
J
i
is the slope of the piezometric surface in the
i
direction (dimensionless). In practical applications, the
TABLE 5.3. Hydraulic Conductivity Statistics
〈
Y
〉 = 〈ln
K
〉
K
G
Formation
(
K
in m/d)
(m/d)
σ
Y
Sandstone
−2.0
0.13
0.92
Sandstone
−0.98
0.38
0.46
Sand and gravel
-
-
1.01
Sand and gravel
-
-
1.24
Sand and gravel
-
-
1.66
(5.29)
Y
= ln
K
Silty clay
−0.15
0.86
2.14
Loamy sand
0.59
1.81
1.98
where
Y
is a normally distributed RSF, characterized by
a mean, 〈
Y
〉; variance,
σ
2
; and correlation length scales,
Source of data
: Freeze (1975).
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