Geoscience Reference
In-Depth Information
0.7
0.6
0.5
q
+
3
0.4
0.3
0.2
1
0.1
1,2
0
048 2 6 0
t
+
Fig. 10.14 Scaled outflow hydrographs
q
+
=
q
+
(
t
+
) from an aquifer with rectanglar cross section on a
horizontal impermeable bed, calculated with the free surface water table assumption for
D
c
+
=
0. The scaled rate of flow is defined as
q
+
=
q
/
(
Dk
0
) and the scaled time variable as
t
+
=
(
n
e
D
). Curve 3 describes the result of the two-dimensional analysis (see Section 10.2)
with Laplace's equation and the free surface condition (10.16) or (10.22) for
B
+
=
6. Curves 1
and 2 are the results obtained with the one-dimensional hydraulic approach (see Section 10.3)
with Boussinesq's equation (10.30) for
B
+
=
6 and 8, respectively. The circles and the triangles
are experimental results for
B
+
=
6 and 8, respectively. (After Verma and Brutsaert, 1971b.)
k
0
t
/
drainage. They derived the positions and shapes of a falling water table as a succession of
steady flow conditions. Thus the distribution of the hydraulic head
h
for an initially known
water table position was found by a numerical solution of Laplace's equation by means of a
relaxation procedure. The next water table position, for a time
δ
t
later, was then determined
by means of Equation (10.22), and so on. The method was extended in Brutsaert
et al
. (1961)
by the inclusion of a partly saturated zone above the water table, and Laplace's equation
was solved by an electrical network analog. Subsequently, other methods have been used to
solve this and similar free surface flow problems in porous materials, namely perturbation
techniques allowing linearization of Equation (10.16) (see Dagan, 1966; VandeGiesen
et al
.,
1994), finite difference methods (see Verma and Brutsaert, 1971a) and boundary integral
methods (Liggett and Liu, 1983).
The dimensionless number Ca can be used as the decisive criterion for the applicability of
the free surface approach, as compared to the complete description of saturated-unsaturated
flow. For example, Figures 10.7-10.10 illustrate how, unless Ca
=
(
aD
)
−
1
is small, the
neglect of the capillary zone results in an overestimate of the outflow rate for small times,
but in an underestimate for long times. As mentioned earlier, the other two parameters,
B
+
=
B
/
D
and
D
c
+
=
D
c
/
D
, represent the criteria for geometric similarity. If the breadth of
an unconfined aquifer
B
, that is the distance from stream to divide, is at least 10 times larger
than the depth
D
, the outflow from a saturated aquifer can be satisfactorily reproduced by
the application of hydraulic groundwater theory (see Section 10.3). Figure 10.14 illustrates
this for an aquifer with
B
+
=
(
B
/
D
)
=
6 and 8, in which the partly saturated zone above the
water table is neglected; the experimental points were obtained with a Hele-Shaw viscous
