Geoscience Reference
In-Depth Information
subscript 2. Using the Green-Ampt model, derive in terms of
θ
1
,
θ
2
and
θ
t
for both soil
columns the ratios of:
a) The distances of the wetting front at time
t
b) The cumulative iniltration at time
t
c) The times required for the wetting front to reach
s
d) The sorptivities
e) The iniltration rates at time
t
When gravity cannot be neglected, Eq. (
4.31
) can be written as:
k
s
sh
s
t
d
t
=
f
d
(4.37)
f
θθ
−
−
t
i
f
f
s
sh
h
sh
Integration, using
f
=+
−
1
f
, yields:
−
f
f
f
f
k
=+ −
( )
+
t
ts hs
ln
hC
(4.38)
f
f
f
f
θθ
−
t
i
The integration constant
C
can be found from the condition
s
f
= 0 for
t
= 0:
=− −
( )
Ch
ln
h
(4.39)
f
f
The inal result is (Koorevaar et al.,
1983
):
θθ
t
−
sh
h
−
−
t
=
i
sh
+
ln
f
f
(4.40)
f
f
k
t
f
The physical parameters
θ
t
,
θ
i
,
k
t
and
h
f
of a given soil must be found experimentally.
The values for
θ
t
and
k
t
are both near their values at saturation. The pressure head at
the wetting front
h
f
cannot be measured directly, but can be derived from Eq. (
4.36
)
by measuring
S, k
t
, and
θ
t
-
θ
i
. Values for
h
f
vary from about -0.05 to 0.8 m for differ-
ent soils. Once these parameters are known, the time needed for the wetting front to
reach a certain depth can be calculated directly with Eq. (
4.40
). To ind
s
f
for a certain
time
t
is more dificult, because
s
f
cannot be expressed explicitly as a function of
t
.
Two solution methods can be followed: (1) make a graph of
t
versus
s
f
and read for
various
t
from the graph; or (2) apply a numerical technique for root inding. Once
s
f
is known, the cumulative iniltration and the iniltration rate can be derived from Eqs.
(
4.28
) and (
4.29
).
Question 4.14:
For a ine sandy loam
k
t
= 1.38 cm d
-1
,
θ
i
= 0.1,
θ
t
= 0.5 and
h
f
= -40 cm.
a) Calculate
s
f
,
I
and
I
cum
for horizontal iniltration in this soil at
t
= 30 minutes.
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