Environmental Engineering Reference
In-Depth Information
Table 11.1 Classification and examples for hydraulic conductivities and permeabilities
K
f
[m/s]
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
10
12
k
[m
2
]
10
7
10
8
10
9
10
10
10
11
10
12
10
13
10
14
10
15
10
16
10
17
10
18
10
19
Pervious
Semi
”
Impervious
Gravel
Sand
Fine sand
Peat
Clay
9.81 m
2
/s is the acceleration due to gravity. Concerning the flow of water,
it has to be taken into account that the dynamic viscosity changes by a factor of
2 between 0
C and 25
C. The permeability varies across a relatively wide range,
which is shown in Table
11.1
.
Note that with the corresponding version of the
g ¼
-operator, the given formulation
of Darcy's Law (
11.12
) is valid in 1D, 2D and 3D situations. two- and three-
dimensional situations often make it necessary to distinguish between conductivities
in different directions. In the mathematical formulation this can be taken into account
by using a tensor (a matrix) K
f
instead of a scalar
K
f
value, or a permeability tensor k
instead of k. Some situations require an even more general formulation of Darcy's
Law. For example for variable density flow (Holzbecher
1998
) the head gradient on
the right side of the equation has to be replaced by the gradient of the dynamic
pressure:
∇
k
m
r p rgz
y
v
¼
ð
Þ
(11.14)
As additional variable besides velocities either hydraulic head
h
or pressure
p
appears. The concepts based on both variables are equivalent as long as there are
no density gradients, which is a general assumption here. The mathematical treat-
ment given here is similar to the derivations presented in textbooks on groundwater
flow (Bear
1972
; Bear and Verruijt
1987
). In both formulations of Darcy's Law,
(
11.12
) and (
11.14
), it is obvious that it is not the absolute value of pressure or head
that determines the velocity. Flow, in its size and direction, is induced by the
pressure or head gradient. For that reason it is irrelevant according to which refer-
ence value pressure and head are measured. From one application to the other the
reference frame often is chosen very differently taking the specific circumstances
in consideration.
In the mathematical formulation of porous media flow Darcy's Law replaces
the momentum conservation (
11.1
). The conservation of mass principle for porous
media can be formulated similarly to (
11.3
). The generalized formulation of the
steady state is
r
y
¼ Q
v
(11.15)
