Environmental Engineering Reference
In-Depth Information
slope of the piezometric surface can be estimated
from the piezometric heads measured at monitoring
wells. The effective hydraulic conductivity in isotropic
media can be expressed in terms of the statistics of
the hydraulic conductivity field by the relation (Dagan,
1989)
which can be written in vector notation as
∇ = −
h
0 0002337
.
i
+
0 0003426
.
j
where
i
and
j
are unit vectors in the coordinate
directions.
The dispersion coefficient in porous media can be
stated generally as a tensor quantity,
D
ij
(L
2
T
−1
), which
is typically expressed in terms of the magnitude of the
mean seepage velocity,
V
(LT
−1
), by the relation
2
σ
Y
K
1
−
1
-D flow
G
2
K
=
K
2
-D flow
(5.32)
eff
G
D
= α
V
(5.33)
2
σ
ij
ij
Y
K
1
+
3
-D flo
w
G
6
where
α
ij
is the
dispersivity
of the porous medium (L).
In general porous media,
α
ij
is a symmetric tensor with
six independent components and can be written in the
form
EXAMPLE 5.5
The piezometric heads are measured at three locations
in an aquifer. Point A is located at (0 km, 0 km), point
B is located at (1 km, −0.5 km), and point C is located
at (0.5 km, −1.2 km), and the piezometric heads at A,
B, and C are 2.157, 1.752, and 1.629 m, respectively.
Determine the hydraulic gradient in the aquifer.
α
α
α
11
12
13
α
ij
=
α
α
α
(5.34)
21
22
23
α
α
α
31
32
33
where
α
ij
=
α
ji
. In cases where the flow direction coin-
cides with one of the principal directions of the hydrau-
lic conductivity, the off-diagonal terms in the dispersivity
tensor are equal to zero, and
α
ij
can be written in the
form
Solution
The piezometric head,
h
, in the triangular region ABC
can be assumed to be planar and given by
α
0
0
h x y
( ,
) =
ax by c
+
+
11
α
ij
=
0
α
0
(5.35)
22
where
a
,
b
, and
c
are constants, and (
x
,
y
) are the coor-
dinate locations. Applying this equation to points A, B,
and C (with all linear dimensions in meters) yields
0
0
α
33
where
α
11
is generally taken as the dispersivity in the
flow direction, and
α
22
and
α
33
are the dispersivities in
the horizontal- and vertical-transverse principal direc-
tions of the hydraulic conductivity, respectively. The
component of the dispersivity in the direction of flow is
called the
longitudinal dispersivity
, and the other com-
ponents of the dispersivity are called the
transverse
dispersivities
.
The dispersivites used to describe the transport of
contaminants in porous media cannot be taken as con-
stant unless the contaminant cloud has traversed several
correlation length scales of the hydraulic conductivity,
or the contaminant cloud is sufficiently large to encom-
pass several correlation length scales of the hydraulic
conductivity. If none of these conditions exists, the dis-
persivity increases as the contaminant cloud moves
through the porous medium. As the contaminant cloud
moves and grows in size, it continually experiences a
wider range of hydraulic conductivities. Ultimately, as
the entire range of hydraulic conductivities is experi-
enced, the dispersivity approaches a constant value
2 157
.
=
a
( )
0
+
b
( )
0
+
c
1 752
.
=
a
(
1000
)
+
b
(
−
500
)
+
c
1 629
.
=
a
(
500
)
+
b
(
−
1200
)
+
c
The solution of these equations is
a
= −0.0002337,
b
= 0.0003426, and
c
= 2.157. From the planar head dis-
tribution, it is clear that the components of the head
gradient,
J
1
and
J
2
, are given by
=
∂
∂
h
x
=
∂
∂
h
y
J
=
a
, and
J
=
b
1
2
Therefore, in this case, the components of the head
gradient are
∂
∂
h
x
∂
∂
h
y
= −
0 0002337
.
and
=
0 0003426
.
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