Geoscience Reference
In-Depth Information
additional constant parameter
c
as follows
S
e
=
(1
+
(
aH
)
b
)
−
c
(8.16)
The choice of the particular function will mostly have to depend on the desired balance
between parsimony and flexibility of the parameterization. The price of greater flexibility is
usually a larger number of parameters: Equations (8.14) and (8.15) require three parameters
whereas Equation (8.16) needs the estimation of four parameters for its application.
8.3
WATER TRANSPORT IN A POROUS MATERIAL
8.3.1
Dynamics of pore-filling fluids: the law of Darcy
The original experiments
In a report on the public fountains and water supply for the City of Dijon in Burgundy,
Darcy (1856) presented the results of his experiments on the seepage of water through
a pipe filled with sand, with a 0.35 m inside diameter and a 3.00 m effective length (see
Figure 8.21). In brief, he found that the rate of flow
Q
through the sand layer was directly
proportional to the cross-sectional area
A
of the sand column and to the difference of
hydraulic head
h
across the layer, and inversely proportional to the length
L
of the
sand column. In this notation his result can be formulated as
Q
=
kA
(
h
1
−
h
2
)
/
L
(8.17)
in which the subscripts 1 and 2 refer to the entrance and the exit section of the column,
respectively. The symbol
k
represents a constant of proportionality, which is now com-
monly referred to as the
hydraulic conductivity
, and which has the dimensions [L T
−
1
].
In the experiments of Darcy the water had essentially a constant specific weight and the
hydraulic head can be defined as usual, namely
p
w
γ
w
h
=
z
+
(8.18)
where
z
is the vertical coordinate. When the negative pressure is expressed as equivalent
water column, Equation (8.18) can also be written concisely as
h
=
−
H
.
Any instrument used to measure the hydraulic conductivity
k
, that is similar to the set-
up originally used by Darcy, is often referred to as a
permeameter
. Over the years many
different designs have been developed, but they are all nearly the same in principle, in
that they provide the measurements of
Q
and (
h
1
−
z
h
2
) needed to invert Equation (8.17)
in order to estimate
k
. Some types of permeameters are also available commercially.
Formulation at a point
Under the assumption that the porous material can be treated as a continuum, both
A
and
L
can be allowed to become infinitesimally small, so that Equation (8.17) describes
the flow at a point and can be written concisely in common vector notation as
q
=−
k
∇
h
(8.19)
where
q
q
z
k
is the specific volumetric flux, that is the volumetric rate of
flow per unit area of porous material,
=
q
x
i
+
q
y
j
+
∇=
(
∂/∂
x
)
i
+
(
∂/∂
y
)
j
+
(
∂/∂
z
)
k
the gradient
