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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