Geoscience Reference
In-Depth Information
p
w
=
0
WT
p
w
>
0
D
z
D
c
x=B
x
Fig. 10.3 Initial state for the unconfined riparian aquifer, shown in Figure 10.2. The water table (WT) is
assumed to be at the soil surface and the entire aquifer is saturated.
formulated as
∇
2
h
=
0
S
e
=
1
.
00
≤
x
≤
B
0
≤
z
≤
D
(10.3)
h
=
D
0
≤
x
≤
Bz
=
D
Since
k
0, which is assumed to be an effective hydraulic conductivity that
is constant over the whole flow domain, Equation (10.1) reduces to the Laplace equation,
as indicated in Equation (10.3). This initial state of the aquifer is shown in Figure 10.3.
=
k
0
at
S
e
=
1
.
Similarity criteria
By scaling the variables, it can readily be seen, that the only relevant (dimensionless)
parameters in this problem are: (i) those related to the soil, viz.
n
and
b
; (ii) those related to
the geometry of the flow, viz.
B
+
=
B
/
D
and
D
c
+
=
D
c
/
D
; and (iii) one related to both soil
and geometry, viz. (
aD
). Because
a
−
1
in Equation (8.15) can be considered a measure of
the thickness of the capillary fringe, (
aD
)
−
1
expresses the relative importance of capillarity,
and thus of the partly saturated flow zone, with respect to the vertical dimensions of the
aquifer.
Results of some example calculations for
D
c
+
=
0
,
B
+
=
1
.
0
,
with
n
=
3, (
aD
)
−
1
=
and
b
=
1
.
5 are shown in Figure 10.4, with
n
=
3, (
aD
)
−
1
0
.
36
,
=
0
.
1
,
and
b
=
3in
Figure 10.5, and with
n
=
8
,
(
aD
)
−
1
=
0
.
36
,
and
b
=
1
.
5 in Figure 10.6. The values of
the parameters (mainly (
aD
)
−
1
and
b
) for Figures 10.4 and 10.6 could represent, for exam-
ple, a loam soil in an aquifer of approximately 3 m depth; those of Figure 10.5 a somewhat
coarser material in an aquifer of roughly the same depth. These calculations show that the
water table tends to fall faster for higher values of (
aD
)
−
1
, and for smaller values of
b
;
but the value of
n
does not appear to affect this very much. This illustrates that in real
situations where capillary effects are important, it may be deceiving to use the water table
as an upper boundary of the flow domain, because a large amount of water may be left in
the unsaturated zone above the water table. It can also be observed that for large values
of
n
and after large
t
the lines of equal hydraulic head are close to horizontal outside the
zone of saturation, whereas they are more nearly vertical within this zone. This refraction
phenomenon is mainly the result of the fact that the upper boundary of the saturated zone
constitutes a boundary between a zone of high and a zone of low conductivity.
The results of these numerical experiments can be used to derive similarity criteria to
determine different regimes of flow in unconfined aquifers by comparing them with the
results of special solutions for each of these regimes (Verma and Brutsaert, 1971a, b). In
principle, all three parameters
n
,
b
and (
aD
)
−
1
, which involve soil characteristics, should
