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1 . E + 0 2
S a n d
k ( c m d
−
1
)
1 . E + 0 0
L o a m
1 . E - 0 2
1 . E - 0 4
1 . E + 0 0
1 . E + 0 2 1 . E + 0 4
S u c t i o n , H ( c m w a t e r )
Fig. 8.29 Capillary conductivity
k
(cm d
−
1
) of Santa Ana river bottom sand and of Diablo loam
measured as functions of negative pressure
H
(cm water column) in the water during drying.
The curves represent Equation (8.37) of Gardner with
a
,
b
and
c
, respectively, equal to 1.7
×
10
8
, 2.5
10
6
, and 4 for the sand, and to 700, 1450, and 2 for the loam. (After Willis, 1960.)
×
where
β
is a constant,
D
wi
is the diffusivity at some initial or other reference moisture content
θ
i
, and
θ
0
is the moisture content at satiation. Gardner and Mayhugh (1958) used Equation
(8.39) in the numerical solution of the problem of sorption, or horizontal infiltration (see
Chapter 9). Reichardt
et al
. (1972) made use of Equation (8.39) to scale experimental data on
horizontal infiltration obtained from eight different air-dry soils, so that they could represent
the results by a single regression equation in terms of dimensionless variables (see Figure
8.30). Miller and Bresler (1977), who reconsidered the analysis of Reichardt
et al
. (1972),
showed that for many soil types
β
in Equation (8.39) may be fairly constant and not very
different from 8. They also found by linear regression that, if
θ
i
is taken as the air-dry water
content of the soil,
D
wi
is in fact related to the rate of advance of the wetting front during
horizontal infiltration or sorption. It was subsequently shown by Brutsaert (1979), how
theoretically this relationship is a direct consequence of the physical nature of sorption; in
addition, it was shown that
D
wi
must also be related to the infiltrated volume of water during
sorption and that the constants involved in these two relationships are unique functions of
β
.
It should be mentioned that the value of
β
=
8 was obtained with repacked laboratory soil
columns. Field measurements by Clothier and White (1981; 1982) yielded a much lower
value; actually, while on average the data could be represented by Equation (8.39) with
β
=
3, in the moisture range 0
.
20
≤
θ
≤
0
.
36,
D
w
was found to be nearly constant. In any
event, Equation (8.39) should be considered a two-parameter expression, as indicated. This
issue will be reexamined in Chapter 9.
A second diffusivity equation has a simple power form and follows directly from
Equation (8.32), implemented with (8.14) and (8.36). The result can be written as
D
w
=
k
0
α
S
e
(8.40)
where
α
=
H
b
[(
θ
0
−
θ
r
)
b
]
−
1
, and
β
=
(
n
−
b
−
1
−
1). A somewhat less simple form, which
has been used to parameterize soil properties for hydrologic purposes (Brutsaert, 1968b),
results from the similar combination of Equation (8.36) with (8.15) or (8.16), by means of
