Environmental Engineering Reference
In-Depth Information
S
a
=
saturation associated with the adhesive component,
and
0
.
86
h
1
.
2
c
0
ψ
r
(cm)
=
(5.89)
m
=
0
.
00003
(5.90)
· =
Macaulay brackets defined as
x
=
0
.
5
(x
+|
x
|
)
,
used to describe a ramp function.
a
c
=
0
.
0007
(5.91)
where:
ρ
s
The capillary component of saturation is empirically
related to soil suction using the following equation:
density of the soil solids, kg/m
3
,
=
h
c
0
ψ
1
m
w
L
=
liquid limit, %.
2
exp
−
m(h
co
ψ)
2
S
c
=
1
−
+
(5.82)
5.12.8 Description of Ayra and Paris (1981) Method
Arya and Paris (1981) presented the first physio-empirical
method to estimate the SWCC. The model made use of basic
information from the grain-size distribution curve. Volumet-
ric water contents were calculated based on an estimation
of the pore sizes in the soil. The pore radii were converted
to equivalent soil suctions through use of the capillary the-
ory. The estimation method used empirical factors to account
for uncertainties in the procedure. The pore radius estima-
tion was based on the assumption of spherical particles and
cylindrical pores. Arya and Paris (1981) assumed the pore-
size distribution and the grain-size distribution of soils to
be approximately congruent. In other words, larger particles
produce larger interparticle voids than smaller particles and
vice versa.
Various models have been proposed to estimate the ran-
dom packing nature of spherical particles in an attempt to
improve on the estimation of the pore-size distribution in
a heterogeneous system (Iwata et al., 1998). The Arya and
Paris (1981) model was later modified by Haverkamp and
Parlange (1986), who applied the concept of shape sim-
ilarity between the SWCC and the cumulative grain-size
distribution for sand soils. Gupta and Ewing (1992) applied
the Arya-Paris model to (i) the grain-size distributions in
order to model intra-aggregate pores and (ii) the aggregate-
size distributions to model inter-aggregate pores. Nimmo
(1997) presented a method of accounting for the influence
of fabric and soil structure through the use of aggregrate-size
distribution.
where:
h
c
0
=
equivalent capillary height which is related to an
equivalent pore diameter and the solid surface area,
ψ
=
soil suction represented as a head or length, and
m
=
pore-size coefficient, unitless.
The adhesive component of saturation is empirically
related to soil suction through the following equation:
a
c
1
(h
c
0
/ψ
n
)
2
/
3
e
1
/
3
(ψ/ψ
n
)
1
/
6
ln
(
1
+
ψ/ψ
r
)
S
a
=
−
(5.83)
ln
(
1
+
ψ
0
/ψ
r
)
where:
a
c
=
adhesion coefficient, unitless,
=
e
void ratio,
ψ
n
=
normalization parameter introduced to maintain
consistency in the units,
ψ
n
=
1 cm when
ψ
is in
cm, and
suction head equal to 10
7
cm of water correspond-
ing to a dry soil condition.
ψ
0
=
Four parameters,
h
co
,ψ
r
,
m
, and
a
c
, are required when
solving the MK model. The parameters are defined as fol-
lows when the soil is granular in nature:
0
.
75
[1
.
17 log
(C
u
)
h
c
0
(cm)
=
(5.84)
+
1]
eD
10
0
.
86
h
1
.
2
c
0
ψ
r
(cm)
=
(5.85)
5.12.9 Description of M.D. Fredlund (2000) Model
The physio-empirical method forms the basis for the develop-
ment of the M.D. Fredlund (2000) model. It was hypothesized
that the grain-size distribution provides a physical descrip-
tion that could be used as the basis for the SWCC estimation
technique. The grain-size distribution is limited, however, in
that it does not provide an indication of the in situ density
(or porosity) of a soil or the fabric of the soil. In addition,
the packing arrangement of various grain sizes constitutes
another important factor. No attempt was made to represent
complex soil fabrics in the estimation of the SWCC.
The M.D. Fredlund (2000) method first divides the grain-
size distribution into small particle groupings of relatively
uniform particle sizes. It is hypothesized that for each uni-
form group of particles there exists a somewhat unique
1
C
u
m
=
(5.86)
a
c
=
0
.
01
(5.87)
where:
D
10
=
diameter corresponding to 10% passing on the
grain-size curve and
C
u
=
uniformity coefficient equal to
D
60
/D
10
.
The four parameters are defined as follows when the soil
is plastic and essentially incompressible:
0
.
15
ρ
s
e
w
1
.
45
L
h
c
0
(cm)
=
(5.88)
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