Environmental Engineering Reference
In-Depth Information
called the
asymptotic macrodispersivity
or simply the
macrodispersivity
. In isotropic media, the correlation
length scale,
λ
, of the hydraulic conductivity is the same
in all directions, and the components of the macrodis-
persivity can be estimated using the approximate rela-
tions (Dagan, 1989; Chin and Wang, 1992)
TABLE 5.4. Variances and Correlation Length Scales of
Hydraulic Conductivity
σ
Y
λ
h
λ
v
Formation
(m)
(m)
Reference
Sandstone
1.5-2.2
-
0.3-1.0
Bakr (1976)
Sandstone
0.4
8
3
Goggin et al.
(1988)
α
=
σ λ
Y
,
2
α
=
α
=
0
(5.36)
Sand
0.9
>3
0.1
Byers and
Stephens (1983)
11
22
33
where it is interesting to note that the heterogeneous
structure of the porous medium does not create
transverse macrodispersion. Derivation of Equation
(5.36) assumes that the local mean seepage velocity is
statistically homogeneous, and the spatial correlation
of the hydraulic conductivity can be represented by
an exponential function of spatial separation. Assump-
tions in the theoretical approximations used in deriving
Equation (5.36) can be taken as valid up to
σ
Y
= 1.5
(Chin and Wang, 1992). In reality, transverse dispersivi-
ties are seldom equal to zero, and it has been shown that
such simple geologic structures as high hydraulic con-
ductivity lenses not aligned with the mean flow can
cause transverse dispersivities to be nonzero (Jankovi ´
et al., 2009).
In cases where the porous medium is stratified, iso-
tropic in the horizontal plane, and anisotropic in the
vertical plane, the correlation length scale of the hydrau-
lic conductivity in the horizontal plane can be denoted
by
λ
h
(L), and the correlation length scale in the vertical
direction denoted by
λ
v
(L). The
anisotropy ratio
,
e
(dimensionless), is then defined by
Sand
0.6
3
0.12
Sudicky (1986)
Sand
0.5
5
0.26
Hess (1989)
Sand
0.4
8
0.34
Woodbury and
Sudicky (1991)
Sand
0.4
4
0.2
Robin et al. (1991)
Sand
0.2
5
0.21
Woodbury and
Sudicky (1991)
Sand and
gravel
5
12
1.5
Boggs et al. (1990)
Sand and
gravel
2.1
13
1.5
Rehfeldt et al.
(1989)
Sand and
gravel
1.9
20
0.5
Hufschmied (1986)
Sand and
gravel
0.8
5
0.4
Smith (1978);
Smith (1981)
alternative to estimating spatial variations in hydraulic
conductivity with vertical spatial resolutions on the
order of 1.5 cm (0.6 in) (Bohling et al., 2012). neuman
et al. (2007) have proposed a methodology for estimat-
ing the geostatistics of the transmissivity from aquifer
tests in confined aquifer.
In estimating the (total) dispersivity in porous media,
the macrodispersivities calculated using either Equa-
tion (5.36) or (5.38) are added to the dispersivities asso-
ciated with hydrodynamic dispersion, which result from
pore-scale mixing and molecular diffusion.
e
=
λ
λ
v
(5.37)
h
and is typically on the order of 0.1 in most stratified
media. Gelhar and Axness (1983) have derived approxi-
mate relations to estimate the components of the mac-
rodispersivity in the case that the flow is in the plane of
isotropy. In this case, the longitudinal and transverse
components of the macrodispersivity tensor can be esti-
mated by
EXAMPLE 5.6
Several hydraulic conductivity measurements in an iso-
tropic aquifer indicate that the spatial covariance,
C
Y
, of
the log-hydraulic conductivity can be approximated by
the equation
2
α
=
σ λ
Y h
,
α
=
α
=
11
(5.38)
11
22
33
r
2
r
2
r
2
The relationships given in Equation (5.38) are
approximately valid for
σ
Y
< 1, but the exact range
of validity has not been established. Typical values
of
σ
Y
,
λ
h
, and
λ
v
in several formations are listed in
Table 5.4.
Estimating the geostatistics of the hydraulic con-
ductivity field from core samples is expensive and time
consuming. Direct push tools provide a less-expensive
1
2
3
C
=
σ
2
exp
−
−
−
Y
Y
λ
2
λ
2
λ
2
where
σ
Y
= 0.5,
λ
= 5 m, the spatial lags
r
1
and
r
2
are measured in the horizontal plane, and
r
3
is measured
in the vertical plane. The mean hydraulic gradient is
0.001, the effective porosity is 0.2, and the mean log-
hydraulic conductivity is 2.5 (where the hydraulic
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