Environmental Engineering Reference
In-Depth Information
k
w
=
unsaturated coefficient of permeability,
content axis and the permeability is expressed as a function
of volumetric water content. The van Genuchten (1980)
equation for the SWCC can be used in combination
with the Mualem (1976a) integration equation form for the
permeability function.
Van Genuchten suggested that the three fitting parameters
for the SWCC equation could be reduced to two fitting
parameters. The reduction to two fitting parameters was
accomplished by assuming a fixed relationship between
the
n
vm
k
s
=
saturated permeability coefficient,
ψ
=
soil suction, and
a
vb
=
fitting parameter related to the inverse of the air-
entry value.
Equation 8.12 is the result of substituting the van
Genuchten (1980) equation into the statistical integration
model proposed by Burdine (1953). The resulting closed-
form expression for the permeability function is commonly
referred to as the van Genuchten-Burdine permeability
equation. A closed-form equation for the coefficient of per-
meability has the advantage of simplicity over an integration
form for the permeability function.
The accuracy of the closed-form van Genuchten-Burdine
equation is in part dependent on the reasonableness of the
proposed relationship between the
m
vb
and
n
vb
variables.
It should be noted that Burdine (1953) did not write an
equation for the SWCC. Rather, van Genuchten (1980) made
an assumption related to the three parameters for the SWCC
equation, thereby reducing his equation to a two-parameter
equation that could yield a closed-form permeability func-
tion consistent with the Burdine (1953) integral form of the
equation.
1
/n
vm
with 0
<m
vm
<
1. It was now possible to produce a
closed-form permeability function for an unsaturated soil
based on soil parameters associated with the SWCC. By
substituting the van Genuchten (1980) equation into the
statistical integration model proposed by Mualem (1976a),
a closed-form expression was derived for the permeability
function:
and
m
vm
parameters; namely,
m
vm
=
−
1
1
+
a
vm
ψ
n
vm
−
m
vm
2
−
a
vm
ψ
n
vm
−
1
1
k
r
(ψ)
=
1
+
a
vm
ψ
n
vm
0
.
5
(8.14)
where:
8.2.4.3 Mualem (1976a) Model
Mualem (1976a) analyzed a conceptual model of a porous
medium similar to that of the Childs and Collis-George
(1950) model and derived the following equation for pre-
dicting the coefficient of permeability:
m
vm
=
fitting parameter set equal to 1
−
1
/n
vm
.
Equation 8.14 is commonly referred to as the van
Genuchten-Mualem permeability equation (van Genuchten,
1980). The closed-form equation for the coefficient-of-
permeability function has the advantage of simplicity over
an integration form for the permeability function when it
comes to solving saturated-unsaturated seepage problems.
The accuracy of the closed-form van Genuchten-Mualem
permeability equation is in part dependent upon the
reasonableness of the assumption relating the
m
vm
and
n
vm
variables.
There does not appear to have been an attempt by van
Genuchten (1980) to use laboratory test results to justify
the empirical relationship between the
n
vm
and
m
vm
soil
parameters. The relationship between
n
vm
and
m
vm
is a math-
ematical restriction and the
m
vm
parameter was limited to
varying between zero and 1. The
n
vm
variable also needed
to be a positive value. The mathematical restriction was nec-
essary in order to ensure that the “power” term was positive
and less than 1 (e.g., 0
<m
vm
<
1).
It is possible to test the reasonableness of the relationship
between
m
and
n
by performing best-fit regression analyses
on SWCC data on unsaturated soils. A large number of
test results were retrieved from the SoilVision database
and independent van Genuchten (1980)
m
vg
and
n
vg
parameters were back calculated using a best-fit regression
analysis. The results indicate that the empirical relationships
suggested by Burdine (1953) and Mualem (1976a) are not
representative of most experimental data. The Fredlund,
⎛
⎞
2
θ
r
θ
dθ/
[
ψ (θ)
]
⎝
⎠
n
k
r
(θ)
=
(8.13)
θ
r
θ
s
dθ/
[
ψ (θ)
]
where:
q
=
a constant set to 0.5,
n
=
normalized water content for the SWCC, and
k
r
(θ)
=
relative coefficient of permeability [i.e.,
k
r
(θ)
=
k(θ)/k
s
, where
k
(
θ
) is any coefficient of perme-
ability as a function of volumetric water content
and
k
s
is the saturated coefficient of permeability
of the soil].
The value of
q
depends on the soil-fluid properties and
may vary considerably for different soils. Based on the per-
meability data for 45 soils, Mualem (1976a) suggested an
optimal value of 0.5 for the
q
variable.
Soil suction must be expressed as a function of vol-
umetric water content when using a statistical model. The
integrations are performed along the volumetric water
Search WWH ::
Custom Search