Soil-Water Characteristic Curves for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 200

5.3 EQUATIONS FOR SWCC e

e 0

A large number of closed-form, empirical equations have
been proposed to best fit laboratory data for SWCCs. A list
of some of the equations appearing in the research literature
is shown in Table 5.2. The equations can be classified as
two-parameter SWCC equations and three-parameter SWCC
equations. Each of these equations can be best fit to labora-
tory data using a least-squares regression analysis (Fredlund
and Xing, 1994).
Each SWCC equation is written in terms of gravimetric
water content with the realization that the equations can also
be written in terms of volumetric water content or degree of
saturation. Some equations are written in terms of dimen-
sionless water content d , and others are written in terms
of normalized water content n . The saturated gravimet-
ric water content is designated as w s , and the gravimetric
residual water content is designated as w r .
Each of the SWCC equations shown in Table 5.2 has one
variable that bears a relationship to the air-entry value of
the soil and a second variable that is related to the rate
at which the soil desaturates. A third variable is used for
some equations and it allows the low-suction range near the
air-entry value to have a shape that is independent of the
shape of the SWCC in the high-suction range near resid-
ual conditions. The use of three parameters for the SWCC
provides greater flexibility for the best-fitting analysis. It
should be noted that the three-parameter equations may not
have parameters that are fully independent of one another.
Each of the SWCC equations can be best fit to either
the drying (desorption) branch or the wetting (adsorption)
branch. There are two difficulties common to the empirical
equations shown in Table 5.2. The first problem occurs
in the low-suction range where the SWCC equations
become asymptotic to a horizontal line. In other words, a
differentiation of the equation gives a water storage value,
written in terms of volumetric water content, m 2 that
approaches zero. This is not realistic and results in numeri-
cal instability when modeling unsaturated soil behavior in
the low-suction range. The last empirical equation shown in
Table 5.2 (i.e., Pham and Fredlund, 2006) does not provide
a continuous function for the SWCC, but it does provide
a more reasonable representation of the SWCC in the
low-suction range.
The second problem with the empirical SWCC equations
occurs at high-soil-suction values beyond residual condi-
tions where the results become asymptotic to a horizontal
line as soil suction goes to infinity. The Fredlund and Xing
(1994) equation overcomes this second problem by applying
a correction factor that directs the SWCC equation to a soil
suction of 10 6 kPa at zero water content. The Fredlund and
Xing (1994) equation is written as follows:

Virgin
compression
line

Air-entry value
Residual

C s

Yield

C c

log(soil suction)

wG s

Air-entry value

C s

Residual

C c

10 6

log(soil suction)

Figure 5.18 Limiting or bounding relationships for typical clayey
silt soil (from Fredlund, 2006).

equations that can represent the entire constitutive surfaces
for unsaturated soils. Once such equations are known, it
is possible to differentiate the equation to obtain the soil
properties at any stress state.
The volume-mass constitutive surfaces have distinctly dif-
ferent characteristics for clay soil and sand soil. Unsaturated
soils have distinct features such as the air-entry value and
residual conditions (i.e., residual suction and residual water
content), which are identifiable along the soil suction stress
state. Figures 5.4 and 5.5 previously showed typical volume-
mass constitutive surfaces (i.e., void ratio, gravimetric water
content, and degree of saturation) generated from measured
data on Beaver Creek sand and Regina clay (Pham, 2005).
Each of the constitutive surfaces are uniquely influenced by
the yield stress (or preconsolidation pressure), the AEV, and
the residual conditions of the soil. Each term is a function
of stress state.
The SWCC constitutes the water content-soil suction rela-
tionship portion of the overall volume-mass constitutive sur-
faces. The interpretation of the SWCC data on a clay soil
is quite different from the interpretation of SWCC data on a
sand soil. Either the water content or the degree-of-saturation
plot can be used to identify the air-entry value for sand; how-
ever, it is the degree-of-saturation plot that must be used
to identify the correct air-entry value for clay soil. Further
information on the volume-mass constitutive surfaces is later
provided during the discussion of volume change.

w s

w (ψ) = C(ψ) = ln e

(ψ/a f ) n f m f

(5.16)

+