Geoscience Reference
In-Depth Information
Because the series (9.66) diverges for large values of time, Philip (1957a) proposed
at
−
1
/
2
f
c
=
+
b
(9.75)
where,
a
and
b
are constants. On the basis of the analysis leading to Equation (9.66),
these constants can be estimated as
a
=
A
0
/
=
(
A
1
+
k
i
) at least for short to
intermediate values of time. However, these values of
a
and
b
cannot serve to describe
the phenomenon for large values of time; indeed, calculations of
A
1
for different soils
(Brutsaert, 1977; see also Equation (9.67)) show that it is usually of the order of
k
0
/
2 and
b
3;
thus with
b
k
i
) in (9.75),
f
c
will also approach this value, rather than
k
0
,as
required by Equation (9.4). This means that, strictly speaking, (9.75) can be applicable
only over a limited time range, and that the values of
a
and
b
depend on that range. But
in many situations of practical interest this should not be a serious obstacle, provided
the constants
a
and
b
are considered curve-fitting parameters to suit the time range of
interest.
=
(
A
1
+
Exponential decay equation
Horton (1939; 1940) proposed an empirical equation that has received wide attention in
hydrology, in the form of an exponential decay function,
a
)
e
−
ct
f
c
=
+
−
a
(
b
(9.76)
where
a
,
b
and
c
are constants, which have to be estimated (Horton, 1942). Clearly,
b
is
the initial infiltration rate and
a
should be equal to
k
0
. Although the exponential function
is mathematically convenient in practical applications, this time dependency is hard to
reconcile with the results of the theoretical analyses based on Richards's equation.
9.4
RAIN INFILTRATION
Observed rainfall rates in nature only rarely exceed the initial infiltration capacity of
the soil. Therefore, in most situations, for a certain initial period at least, all the rainfall
that reaches the ground surface without being intercepted infiltrates into the soil profile.
During this initial phase, the surface water content gradually increases and the absorptive
capacity of the soil decreases. There are two possible scenarios for what happens next,
depending on the intensity of the precipitation (see Figure 9.15). Consider the simple case
of a constant rate of precipitation on the surface of a deep homogeneous soil profile. If the
rainfall intensity is smaller than the satiated hydraulic conductivity, i.e.
P
k
0
, it will
never exceed the ability of the soil to absorb the rainwater. Eventually, the surface water
content will tend to reach a value
≤
θ
s
, such that the hydraulic conductivity at that water
content is equal to the precipitation rate, or
k
(
θ
s
)
=
P
. In the limiting case, if
P
=
k
0
,
the soil surface will eventually, as
t
→∞
, reach full satiation, or
θ
s
→
θ
0
. The second
scenario occurs when
P
k
0
. Although initially following the onset of precipitation all
the rainwater infiltrates, after a finite period of time
t
>
=
t
p
the soil surface becomes fully
satiated, i.e.
θ
s
=
θ
0
. From that moment onward, conditions change markedly: as the
surface soil is satiated and the rainfall intensity exceeds the infiltration capacity, ponding
takes place and the excess precipitation may be evacuated by overland flow.
