Geoscience Reference
In-Depth Information
1
0.8
0.6
1
0.4
0.2
2
0
1
1.5
2
2.5
3
P
+
Fig. 9.19 Curve 1 shows the scaled time to ponding
t
p
+
=
(
k
0
/
A
0
)
2
t
p
against scaled rainfall rate
P
+
=
(
P
/
k
0
), as calculated with the second procedure in the TCA approximation, i.e.
Equation (9.101); the rainfall rate is assumed to be constant during the rainfall event. Curve 2
represents the negative error of this result relative to Equation (9.89) with
α
p
=
0
.
55 (which is
shown in Figure 9.16), and thus indicates that
t
p
+
is underestimated with this procedure.
recalled that the constants in the infiltration equation (9.75), used in this example, can be
taken as
a
3; with these values of the constants and in terms of the
scaled variables shown in (9.88), Equation (9.99) can be written as
=
A
0
/
2 and
b
=
k
0
/
0
.
5 (
P
+
−
1
/
6)
P
+
(
P
+
−
1
/
3)
2
t
p
+
=
(9.101)
This result is illustrated in Figure 9.19, and can be compared with the more accurate result
shown in Figure 9.16. Also shown is the error inherent in Equation (9.101) relative to (9.89);
it can be seen that the error in this second procedure can be considerable, and that it results in
underestimates of the time to ponding, ranging between roughly 10% and 70%, depending
on the rainfall intensity. As seen in Equations (9.92) and (9.100), an underestimated time
to ponding
t
p
produces an underestimate of the infiltration
F
.
Accuracy of the TCA approximation
Several studies allow an assessment of the time compression approximation. One is the
analysis of intermittent infiltration by Ibrahim and Brutsaert (1968), on the basis of the
numerical solution of Equation (9.1) for conditions representing alternating potential infil-
tration and drainage (or redistribution) cycles. The hysteresis in the soil water characteristic
was taken into account by means of the concept of independent domain. Inspection of
the results shows that the cumulative infiltration, after restarting it following a drainage
period of a given duration, can be obtained by merely time-shifting the initial (i.e. prior to
the drainage) cumulative potential infiltration curve over a certain time period; however,
the required time-shift period tends to be shorter than the drainage period, which is the
time-shift assumed under TCA. This means that TCA usually underestimates infiltration.
Similar results were obtained by Reeves and Miller (1975); although in some extreme cases
the reported error was as large as 15% to 20%, in most cases it was considerably smaller.
