Environmental Engineering Reference
In-Depth Information
Van Genuchten (1980) proposed the use of a continuous
function between soil suction and normalized water content:
where:
θ(R)
=
volumetric water content when all the pores with
radius less than or equal to
R
are filled with water,
1
n
=
1
(a
vg
ψ)
n
vg
m
vg
(5.21)
+
R
min
=
minimum pore radius in the soil, and
R
max
=
the maximum pore radius, and for the saturated
case,
θ(R
max
)
where:
=
θ
s
.
a
vg
,n
vg
,m
vg
=
three soil fitting parameters and
The capillary law indicates that there should be an inverse
relationship between matric suction and the radius of cur-
vature of the air-water interface (i.e., the meniscus). The
air-water interface should bear an inverse relationship to
the pore sizes that are desaturated at a particular suction as
indicated in the equation
n
=
normalized water content (i.e., normal-
ized between saturated water content,
θ
s
and residual water content,
θ
r
).
The van Genuchten (1980) equation provides additional
flexibility to the SWCC. In an attempt to obtain a closed-form
expression for hydraulic conductivity, van Genuchten (1980)
fixed the relationship between the
m
vg
and
n
vg
parameters
through use of the equations shown in Table 5.2. However,
the linkage between the
m
vg
and
n
vg
parameters reduced the
flexibility of Eq. 5.21. More accurate best fits to measured
laboratory data can be obtained by leaving
m
vg
and
n
vg
param-
eters without a fixed relationship.
C
ψ
r
=
(5.24)
where:
C
=
2
T
s
cos
α
1
=
constant,
T
s
=
surface tension of water, and
α
1
=
angle of contact between water and soil.
5.3.2 Theory of Pore-Size Distribution and the SWCC
Equations that have been proposed for the SWCC are empir-
ical in nature. Each equation was originally applied to a
particular group of soils. There are many different forms of
SWCC equations that could be tested to assess their fit of
experimental data. The pore-size distribution or slope of the
SWCC appears to have the form of a normal distribution
curve. Consequently, the following equation might be used
to approximate the SWCC:
Two particular suction conditions can be defined as
follows:
C
R
min
ψ
max
=
(5.25)
C
R
max
ψ
aev
=
(5.26)
where:
ψ
max
=
suction value corresponding to the minimum pore
radius and
a
5
e
−
(b
5
ψ)
m
n
=
(5.22)
ψ
aev
=
air-entry suction value of the soil.
where:
Using the capillary law, Eq. 5.23 can be expressed in
terms of soil suction:
a
5
,b
5
,m
=
curve-fitting parameters.
Equation 5.22 is not suitable as a general form for the
SWCC, but it might apply for some soils over a limited
range of suction values. The pore-size distribution curve for
soils provides a theoretical basis for the shape of the SWCC.
Soils can be regarded as a set of interconnected pores that
are randomly distributed. The pores can be characterized by
a pore radius
r
and described by a function
f(r)
, where
f(r)
dr
is the relative volume of pores between radius
r
to
r
f
C
h
d
C
h
f
C
h
C
h
2
dh
(5.27)
ψ
ψ
max
θ(ψ)
=
=
ψ
max
ψ
where:
h
=
a dummy variable of integration representing soil
suction.
dr
. In other words,
f(r)
is the density of pore volume
corresponding to radius
r
. Since
f(r)
dr
represents the pores
between radius
r
to
r
+
Equation 5.27 is a general equation form that can be used
to describe the relationship between volumetric water con-
tent and soil suction. The SWCC can be uniquely determined
using Eq. 5.27 if the pore-size distribution
f(r)
of a soil is
known. Several special cases can be defined as follows:
dr
that are filled with water, the
volumetric water content can be expressed as follows:
+
R
max
θ(R)
=
F(r)
dr
(5.23)
1. Case of a constant-pore-size function. The pore sizes
can be assumed to be uniformly distributed, that is,
R
min
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