Geoscience Reference
In-Depth Information
by integration of (
zd
θ
); as illustrated in Figure 9.4, this can also be accomplished by
integration of (
θ
dz
), so that one can write instead
z
f
F
=
(
θ
−
θ
i
)
dz
+
k
i
t
(9.79)
0
where
z
f
is the position of the wetting front. In the present situation
F
is also the time
integral of the precipitation rate, or
t
F
=
Pdt
(9.80)
0
These two expressions for the cumulative infiltration, namely (9.79) and (9.80) can be
combined to yield
t
θ
s
Pdt
=−
(
θ
−
θ
i
)(
∂
z
/∂θ
)
d
θ
(9.81)
0
θ
i
in which it is assumed that
k
i
, the conductivity at the initial water content, is negligibly
small, and the limits are
θ
s
at
z
=
0 and
θ
i
at
z
=
z
f
. Substitution of (9.77) (or (9.78)) into
(9.81) produces
t
θ
s
(
θ
−
θ
i
)
D
w
(
P
Pdt
=
d
θ
(9.82)
−
k
)
0
θ
i
This is an important result in that it represents a relationship between the water content at
the soil surface and time,
θ
s
=
θ
s
(
t
) prior to ponding.
The time to ponding
t
=
t
p
can be derived from Equation (9.82), by considering it as
the time necessary for the surface soil moisture content to reach satiation, or
θ
s
=
θ
0
. Thus
one has
t
p
θ
0
(
θ
−
θ
i
)
D
w
(
P
−
k
)
Pdt
=
d
θ
(9.83)
0
θ
i
Both
D
w
and
k
are functions of the water content
θ
, so that, when they are known, it should
in principle be possible to carry out the integration in Equation (9.83) for any arbitrary time
distribution of the precipitation
P
=
P
(
t
). The integration becomes especially simple if it
can be assumed that
D
w
and
k
change rapidly in the vicinity of
θ
=
θ
0
. Take for example
(8.36) for
k
and (8.40) for
D
w
; these are simple power functions, but the result would be the
same with any other functions, such as for example exponential functions, which exhibit a
similar behavior near
θ
=
θ
0
. After normalization of the water content with Equation (9.8)
and substitution of these two functions, (9.83) becomes
1
θ
0
−
θ
i
)
k
0
P
b
S
n
−
1
/
n
dS
n
H
b
(
P
−
k
0
S
n
t
p
=
(9.84)
0
in which
P
is the average precipitation rate during the event until the onset of pond-
ing. After bringing the term
S
n
−
1
/
b
n
inside the differential, one recognizes immediately the
