Geoscience Reference
In-Depth Information
an approximation; therefore, it is impossible that both (9.91) and (9.95) are valid. This
means that
t
p
, as obtained with Equation (9.96), is not valid either. The time to ponding
is such a crucial parameter in describing rainfall infiltration, that the first procedure
to determine the compression reference time should be used, if at all possible. Because
TCA is mainly concerned with mass conservation and cumulative infiltration, and less so
with infiltration rate, Equation (9.91) should have priority over (9.95). In other words, as
pointed out by J.-Y. Parlange (Liu
et al.
, 1998), the first procedure with
t
cr1
is preferable
over the second with
t
cr2
. The following example should give an idea of the practical
implementation of the TCA concept and of the error involved in the estimation of
t
p
by
means of the second procedure.
Example 9.3. Application with truncated time expansion
The above equations for rainfall infiltration can be applied with any of the available equa-
tions for infiltration capacity
f
c
or
F
c
discussed in Section 9.3. The calculations become
especially simple when time
t
can be expressed explicitly as a function of infiltration
capacity. Consider as an example the truncated time expansion (9.75) proposed by Philip
(1957a). By using the procedure, with the correct value of
t
p
determined independently,
one obtains from (9.94) for the compression reference time, implicitly
2
at
1
/
2
cr1
Pt
p
=
+
bt
cr1
and, after solution, explicitly
(
a
2
Pbt
p
)
1
/
2
]
2
b
2
t
cr1
=
[
−
a
+
+
/
(9.97)
in which
a
and
b
are the constants appearing in Equation (9.75).
With the second procedure, which is based on the infiltration rate constraint, by
combining (9.95) with (9.75) one obtains the following
b
)]
2
t
cr2
=
[
a
/
(
P
−
(9.98)
The time to incipient ponding can be calculated by substitution of (9.98) into (9.91),
which yields with the cumulative infiltration corresponding to (9.75),
a
2
(2
P
−
b
)
t
p
=
(9.99)
b
)
2
P
(
P
−
With the values of
t
p
and
t
cr
known, by either of the two procedures, the cumulative
infiltration (9.92) can now be written, for this example with (9.75), as
F
(
t
)
=
Pt
for
t
<
t
p
(9.100)
t
cr
)]
1
/
2
F
(
t
)
=
2
a
[
t
−
(
t
p
−
+
b
[
t
−
(
t
p
−
t
cr
)]
for
t
≥
t
p
and in a similar way, the rate of infiltration (9.93).
The error involved in the second TCA procedure can be determined by comparing the
resulting expression for
t
p
, i.e. Equation (9.99), with the more accurate (9.89) derived in
the previous section. As before, for the sake of generality, especially in this comparison
with another expression, it is useful to express (9.99) in dimensionless form. It should be
