Geoscience Reference
In-Depth Information
defined in (10.66) and (10.74). Application of (10.158) and (10.159) yields the following
scaled time and the scaled outflow rate in terms of basin-scale parameters,
t
+
=
4
k
0
DL
2
t
/
(
n
e
A
2
)
(10.168)
Q
+
=
AQ
/
(4
k
0
D
2
L
2
)
Thus (10.155) assumes the form
dQ
+
dt
+
=
a
+
Q
b
+
(10.169)
where
a
+
is a (dimensionless) constant whose value depends only on
b
. As noted by Michel
(1999), the numerical value of
a
+
can be readily calculated for each of the theoretical values
of
b
=
1
,
3
/
2 and 3 from the respective expressions for
a
given in Equations (10.161),
(10.163) and (10.165). The following interpolation formula (Brutsaert and Lopez, 1999)
provides a close estimate of these theoretical values and may be useful for intermediate
values of
b
over that range,
a
+
=
10
.
513
−
15
.
030
b
1
/
2
+
3
.
662
b
(10.170)
Hydraulic aquifer characteristics at the basin scale
Equation (10.157) with (10.161)-(10.165) can be used to obtain an estimate of the
effective hydraulic parameters of the riparian aquifers in the basin (see also Brutsaert and
Nieber, 1977; Brutsaert and Lopez, 1998; Eng and Brutsaert, 1999). In the application
of this approach a decision must first be made whether (10.161) or (10.163) is the
more appropriate expression to describe the long-time outflow behavior of the basin.
In past applications, this was done by inspection of the slope of the lower envelope of
the low flows as they appear on a log-log plot of
vs
Q
. This is illustrated in
Figure 10.31, in which the slope of the envelope happens to be close to one, or
b
|
dQ
/
dt
|
1
in Equation (10.157). This has also been done by linear regression with all the data
points of log(
=
dt
) against log(
Q
). Neither procedure appears to be objective, and
at present, it is still not clear how an appropriate a priori value of
b
can be determined,
which describes the long-time behavior of a given basin. In the catchment studies by
Brutsaert and Nieber (1977) and Troch
et al
. (1993), it was concluded that
b
−
dQ
/
2,
whereas in Vogel and Kroll (1992), Brutsaert and Lopez (1998) and Eng and Brutsaert
(1999) it was decided to be
b
=
3
/
=
1. This will require further study.
Once the appropriate long-time outflow expression and its value of
b
have been
decided upon, the value of
a
1
and
a
3
(or
a
2
) can be determined from the lower envelopes
with slopes 3 and 1 (or 3
|
vs
Q
data. Examples of the procedure are shown in Figures 10.31 and 10.33. In what
follows, the determination of the basin-scale aquifer parameters is outlined for the linear
case with
b
/
2), respectively, on a log-log plot of the available
|
dQ
/
dt
=
1; however, the analogous analysis with
b
=
3
/
2 is straightforward, and
can be left as an exercise for the reader.
The value of
a
3
is related to the extinction coefficient of the exponential outflow
equation (10.153) by
K
−
1
a
3
=−
(10.171)
