Geoscience Reference
In-Depth Information
extent. As shown, the soil water characteristic curve in Figure 10.12 describes drainage;
obviously, consideration of repeated draining and wetting cycles with hysteresis would
complicate the matter even more. This means that the drainable porosity
n
e
cannot be a
unique physical property of a given porous material, and that it must be considered as
a mere parameter that is to be adjusted and calibrated depending on the flow problem.
This is its main limitation, which should be kept in mind in practical applications of the
free surface approach in flow in porous media. But if this limitation is kept in mind, the
concept can yield useful results in the parameterization of groundwater flow processes
at the field and catchment scales.
Free surface condition
In principle, the condition at the free surface in a porous material is again Equation (5.1)
as presented in Chapter 5. If the function
F
=
0 describing the free surface is taken as
F
=
F
(
x
,
z
,
t
)
=
[
η
(
x
,
t
)-
z
]
=
0, in which
η
denotes the height of the free surface above
the reference level
z
=
0, this can be written as
u
∂η
∂
x
−
w
+
∂η
∂
t
=
0
at
z
=
η
(10.14)
in which, as before,
u
and
w
represent the
x
- and
z
-components of the true fluid velocity,
which is also that of the free surface.
The velocity of a fluid is its volumetric rate of flow per unit cross-sectional area occupied
by this fluid. Because the specific flux
q
in Darcy's equation is the volumetric flow rate
per unit cross-sectional area of total or bulk porous material, it does not represent the true
velocity of the fluid particles (cf. Section 8.3.1). Rather, with the assumption of a drainable
porosity, the actual velocity of the fluid particles must be taken as (
q
/
n
e
). Therefore, with
(
q
x
/
n
e
) and (
q
z
/
n
e
)asthe
x
- and
z
-components of the velocity of the fluid and also of the
free surface, Equation (10.14) becomes, in terms of the Darcy flux,
q
x
∂η
∂
x
−
q
z
+
n
e
∂η
∂
t
=
0at
z
=
η
(10.15)
With Darcy's law one obtains finally
n
e
k
0
∂η
∂
h
∂
x
∂η
∂
x
−
∂
h
∂
t
=
at
z
=
η
=
h
(10.16)
∂
z
There is also a second way of implementing Equation (5.1) in a porous material to
formulate the condition at the free surface. If the adopted free surface function is
F
=
F
(
x
,
z
,
t
)
=
[
h
(
x
,
z
,
t
)-
z
]
=
0, one has instead of Equation (10.14)
u
∂
h
∂
x
+
w
∂
h
∂
z
−
w
+
∂
h
∂
t
=
0
at
z
=
η
=
h
(10.17)
from which one obtains the free surface condition, as an alternative to Equation (10.16),
∂
h
∂
x
2
∂
h
∂
z
2
n
e
k
0
∂
h
−
∂
h
∂
z
∂
t
=
+
at
z
=
η
=
h
(10.18)
Either (10.16) or (10.18) can be used in the solution of the problem; the choice depends
usually on the specific mathematical aspects to be investigated.
