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analytically or numerically. However, rainfall data are usually recorded as bar graphs, that
is as constant values over finite time intervals, hourly or daily. This simplifies the analysis
somewhat, as it allows the formulation of the input function in the same way as in the
previous example. In this example assume the following rainfall sequence
P = 0
for 0 t < t 1
P = 0 . 2 P c
for t 1 t < t 2
(10.127)
P = 0 . 9 P c
for t 2 t < t 3
in which P c is a reference intensity rainfall (e.g. 5 mm h 1 ).
Consider the case when the aquifer is dry at t = 0. Application of (10.118) with (10.127)
and (10.117) produces different expressions q ( t ) depending on the magnitude of t relative
to t 1 , t 2 and t 3 . For instance, when t 1 < t < t 2 the outflow is
exp (2 n 1) 2
d τ
t
2 k 0 η 0 ( t τ )
4 n e B 2
2 k 0 η 0 P c
n e B
π
q = q ( t ) =
0 . 2
(10.128)
n
=
1
,
2
,...
t 1
and after integration
2 1
exp (2 n 1) 2
8 BP c
(2 n 1) 2
2 k 0 η 0 ( t t 1 )
4 n e B 2
π
q =−
.
2 0
(10.129)
π
n = 1 , 2
Similarly, for the case when t > t 3 one has
exp (2 n 1) 2
d τ
t 2
2 k 0 η 0 ( t τ )
4 n e B 2
2 k 0 η 0 P c
n e B
π
q = q ( t ) =
0 . 2
n
=
1
,
2
, ...
t 1
exp (2 n 1) 2
d τ
t 3
2 k 0 η 0 ( t τ )
4 n e B 2
π
+
0 . 9
(10.130)
n
=
1
,
2
, ...
t 2
or, upon integration
0
2 exp (2 n 1) 2
8 BP c
(2 n 1) 2
2 k 0 η 0 ( t t 2 )
4 n e B 2
π
q =−
.
π
2
n = 1 , 2
exp (2 n 1) 2
9 exp (2 n 1) 2
2 k 0 η 0 ( t t 1 )
4 n e B 2
π
2 k 0 η 0 ( t t 3 )
4 n e B 2
π
+
.
0
exp (2 n 1) 2
2 k 0 η 0 ( t t 2 )
4 n e B 2
π
(10.131)
The outflow rate produced by (10.127) is illustrated in Figure 10.25, for time values t 1 =
2 days, t 2 t 1 = t 3 t 2 = 1 day. In Figure 10.25, for conciseness of notation, q is scaled
with BP c and it is expressed in terms of scaled time t + defined in (10.94). Note that t + = 0 . 1
corresponds to roughly 1 day here; this conversion is obtained for typical (Brutsaert and
Lopez, 1998; Eng and Brutsaert, 1999) field values for small catchments k 0 = 0 . 001ms 1 ,
n e = 0 . 02, η 0 = 2m,and B = 300 m.
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