Geoscience Reference
In-Depth Information
presence of a number of terms which resemble the sorptivity (9.49) for the same power
diffusivity (8.40), provided the exponent
n
is not small. Therefore to a good approximation;
Equation (9.84) can be rewritten as
1
dS
n
−
b
−
1
+
1
n
A
0
2
P
P
k
0
S
n
t
p
=
(9.85)
−
0
Whenever
b
=
1 or both
n
and
b
are large, (9.85) can be integrated to yield the main
result
ln
P
p
P
p
−
k
0
A
0
2
P
k
0
t
p
=
(9.86)
where
P
p
is the precipitation rate at the time of ponding,
t
t
p
. Equation (9.86) was first
presented by Parlange and Smith (1976); they derived it from (9.83) in a somewhat different
way, namely without the specific expressions for
k
and
D
w
used here. Indeed, it can be seen
that this result is independent of the parameters in the two power expressions (8.36) for
k
and (8.40) for
D
w
.
=
Practical implementation
In many situations of practical interest, the rainfall intensity can be assumed to be
constant during the storm or at least prior to the onset of ponding; in this case one has
P
=
P
=
P
p
=
constant and Equation (9.86) can be written as
ln
P
P
A
0
2
Pk
0
t
p
=
(9.87)
−
k
0
As usual, Equation (9.87) can be made more general by expressing it in terms of
dimensionless variables. Its form suggests immediately the same scaling of the time
variable, already used in (9.21) and in (9.70); in addition, it suggests that the precipitation
rate be scaled with the hydraulic conductivity, so that
k
0
t
p
P
k
0
t
p
+
=
A
0
,
and
P
+
=
(9.88)
With these scaled variables (9.87) can be expressed as
ln
P
+
P
+
−
α
p
P
+
t
p
+
=
(9.89)
1
where
α
p
is a constant equal to 0.5. As expected, both (9.87) and (9.89) show how the
ponding is instantaneous at the beginning of the rainfall event, i.e.
t
p
=
k
0
,
that is when the rainfall intensity is much larger than the satiated hydraulic conductivity
of the soil. On the other hand, ponding never occurs, i.e.
t
p
→∞
0, when
P
, when the hydraulic
conductivity
k
0
is equal to or larger than the precipitation rate
P
. These features are
illustrated in Figures 9.15 and 9.16.
In the derivation of the results given in Equations (9.86), (9.87) and (9.89) it was
pointed out that the same would be obtained with any other functions for
k
and for
D
w
in (9.83), as long as they change rapidly in the vicinity of
S
n
=
1. This kind of
k
and