Environmental Engineering Reference
In-Depth Information
conductivity is in m/d. Estimate the effective hydraulic
conductivity and the macrodispersion coefficient in the
aquifer.
dominates, and for 0.1 ≤ Pe ≤ 10, both advection and
dispersion are important. In municipal well fields, values
of Pe within several meters of the wellheads tend to be
high, indicating that contaminant transport is advection-
dominated and dispersion effects are relatively small.
In contrast to defining the Péclet number in terms of
the longitudinal macrodispersion coefficient,
D
L
, as
given by Equation (5.39), a Péclet number, Pe
m
, can be
defined in terms of the molecular diffusion coefficient
as
Solution
From the given data, the hydraulic conductivity field is
described statistically by 〈
Y
〉 = 2.5,
σ
Y
= 0.5, and
λ
= 5 m.
The geometric mean hydraulic conductivity,
K
G
, is given
by Equation (5.30) as
K
=
e
Y
=
e
2 5
.
=
12
m/d
=
Vd
D
G
Pe
m
(5.40)
m
and the effective hydraulic conductivity, for three-
dimensional flow, is given by Equation (5.32) as
where
d
is the characteristic pore size and
D
m
is the
molecular diffusion coefficient. Previous investigations
have shown that the pore-scale longitudinal dispersion
coefficient is much greater than the molecular diffusion
coefficient when Pe
m
> 10, and the pore-scale transverse
dispersion coefficient is much greater than the molecu-
lar diffusion coefficient when Pe
m
> 100 (Bijeljic and
Blunt, 2007; Perkins and Johnson, 1963). Laboratory-
scale experiments and numerical simulations have indi-
cated that the ratio of longitudinal to transverse
dispersion,
D
L
/
D
T
, depends on the Péclet number, and
three
Pe
m
regimes can be identified. In the first regime,
Pe
m
< 1 and
D
L
/
D
T
= 1, since the only mechanism for
dispersion is by molecular diffusion. In the second
regime, 1 < Pe
m
< Pe
crit
, where Pe
crit
marks the start of
the advection dominated dispersion regime and is on
the order 100 (Bijeljic et al., 2004); both advection and
diffusion contribute to dispersion, and
D
L
/
D
T
increases
with
Pe
m
, with
D
L
/
D
T
increasing from about 1 to 7. In
the third regime, Pe > Pe
crit
, there is nearly linear depen-
dence on Pe
m
for both
D
L
and
D
T
with
D
L
/
D
T
≈ 7.
2
2
σ
=
0 5
6
.
=
Y
K
=
K
1
+
(
12 1
)
+
12 5
.
m/d
eff
G
6
The mean seepage velocity,
V
, in the aquifer is given
by Equation (5.31) as
K
n
eff
V
= −
J
e
where
J
= −0.001 and
n
e
= 0.2; hence,
12 5
0 2
.
.
V
= −
(
−
0 001
.
)
=
0 063
.
m/d
Since
σ
Y
= 0.5 and
λ
= 5 m, the longitudinal macro-
dispersivity,
α
11
, can be estimated by Equation (5.36) as
α
=
σ λ
2
=
( . ) ( )
0 5
2
5
=
1 25
.
m
11
Y
and, according to Equation (5.36), the theoretical trans-
verse macrodispersivities are both zero. The longitudi-
nal dispersion coefficient,
D
11
, is given by
EXAMPLE 5.7
The mean seepage velocity in an aquifer is 1 m/day, the
mean pore size is 1 mm, and the molecular diffusion
coefficient of a certain toxic contaminant in water
is 10
−9
m
2
is Determine whether molecular diffusion
should be considered in a pore-scale contaminant trans-
port model.
D
=
α
V
=
( .
1 25 0 063
)( .
)
=
0 079
.
m /d
2
11
11
The relative importance of advective transport to dis-
persive transport can be measured by the
Péclet number
,
Pe, defined as
Solution
=
VL
D
Pe
(5.39)
From the data given,
V
= 1 m/day,
d
= 1 mm = 0.001 m,
and
D
m
= 10
−9
m
2
/s = 8.64 × 10
−5
m
2
/d. The Péclet
number, Pe
m
, is given by
L
where
V
is the mean seepage velocity (LT
−1
),
L
is the
characteristic length scale (L), and
D
L
is the character-
istic longitudinal dispersion coefficient (L2T−1).
2
T
−1
). For
Pe > 10, advection dominates, for Pe < 0.1, dispersion
Vd
D
1 0 001
8 64 10
( )( .
)
Pe
m
=
=
=
12
−
5
.
×
m
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