Environmental Engineering Reference
In-Depth Information
the longitudinal macrodispersivity,
α
11
, can be related to
the travel distance,
L
, by the relation
TABLE 5.5. Typical Longitudinal Dispersivities for Various
Length Scales
Longitudinal Dispersivity (m)
1 53
.
0 0169
.
L
,
L
<
100
m
α
11
=
m
(5.41)
Scale (m)
Average
Range
1 46
.
0 0175
.
L
,
100
m
<
L
<
3500
<1
0.001-0.01
0.0001-0.01
1-10
0.1-1.0
0.001-1.0
where
α
11
and
L
are both measured in meters.
Analyses by Al-Suwaiyan (1998) demonstrate that
the observed macrodispersivities reported by Gelhar
et al. (1992) are scattered about the mean (approxi-
mated by Eq. 5.41), with the upper limit of the scatter
at about five times the mean and the lower limit at about
one-fifth of the mean. These uncertainty limits should
be accounted for whenever Equation (5.41) is used in
contaminant transport predictions. For very small scales,
on the order of the pore size, dispersion is caused pri-
marily by pore-scale mechanical dispersion and molecu-
lar diffusion, where the longitudinal and transverse
dispersivities can be estimated by the relations
10-100
25
1-100
the longitudinal dispersivity for various ranges of length
scales are shown in Table 5.5.
EXAMPLE 5.8
A contaminant plume in an aquifer is approximately
50 m long, 10 m wide, and 3 m deep. The characteristic
pore size in the aquifer is 3 mm, the molecular diffusion
coefficient is 2 × 10
−9
m
2
is the tortuosity is 1.5, and the
mean seepage velocity is 0.5 m/day. Estimate the com-
ponents of the dispersion coefficient.
D
V
D
V
*
m
α
=
α
+
L
L
τ
(5.42)
m
α
=
α
*
+
Solution
T
T
τ
From the given data,
L
x
= 50 m,
L
y
= 10 m,
L
z
= 3 m,
D
m
= 2 × 10
−9
m
2
/s = 1.73 × 10
−4
m
2
/d,
τ
= 1.5, and
V
= 0.5 m/d. The length scale,
L
, of the contaminant
plume can be approximated by the relation
where
α
*
and
α
*
are pore-scale longitudinal and
transverse dispersivities (L), respectively,
D
m
is the
molecular diffusion coefficient in water (L2T−1),
2
T
−1
),
τ
is
the tortuosity (dimensionless) (which accounts for the
effect of the solid matrix on diffusion), and
V
is the
mean seepage velocity (LT
−1
). The molecular diffusion
coefficient divided by the tortuosity represents the
effective molecular diffusion coefficient in porous media
and is sometimes called the
bulk diffusion coefficient.
.
Values of
α
*
are typically on the order of the pore size
of the porous medium,
α
*
is typically on the order of
0.1-0.01
α
*
(Delleur, 1998),
τ
is typically in the range
2-100 (lower values are associated with coarse material,
such as sands; higher values are associated with finer
material, such as clays), and typical values of the molec-
ular diffusion coefficient are in the range of 10
−5
-
10
−3
m
2
/d at 25°C.
For travel distances longer than 3500 m (2 mi), the
longitudinal dispersivity tends to asymptote to an upper
limit that is consistent with a finite variability in the
hydraulic conductivity. In cases where the dispersivity
increases with travel distance, the Fickian assumption
of a constant dispersion coefficient is not supported
and the dispersion is termed
non-Fickian
. However, a
Fickian approximation to the mixing process is obtained
by adjusting the dispersion coefficient with length scale,
and the advection-diffusion equation can be used to
approximate the dispersion process. Typical values of
L
=
L L
x
=
(
50 10
)(
)
=
22 m
y
and since
L
< 100 m, the longitudinal macrodispersivity
can be estimated by Equation (5.41) as
1 53
.
1 53
.
α
11
=
0 0169
.
L
=
0 0169 22
.
(
)
=
1 9
.
m
The horizontal-transverse macrodispersivity,
α
22
, can
be estimated as 0.1
α
11
, which gives
α
=
0 1
.
α
=
0 1 1 9
. ( . )
=
0 19
. m
22
11
and the vertical-transverse macrodispersivity,
α
33
, can be
estimated as 0.01
α
11
, which gives
α
=
0 01
.
α
=
0 01 1 9
.
( . )
=
0 019
.
m
33
11
The local longitudinal dispersivity,
α
L
, is given by
Equation (5.42), where
α
*
is on the order of the pore
size (0.003 m), and hence
−
4
D
V
1 73 10
1 5 0 5
.
( . )( . )
×
m
α
=
α
*
+
=
0 003
.
+
=
0 0032
.
m
L
L
τ
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