Measurement and Estimation of State Variables - Unsaturated Soil Mechanics in Engineering Practice - page 174

accommodate hysteresis. The SWCC equations are written
in terms of gravimetric water content; however, each of the
rearranged equations would have the same form if written
in terms of volumetric water content.
The parameters for the SWCC must be known when using
a SWCC to compute soil suction. Sigmoidal shapes for the
SWCC can be defined using three fitting parameters, gener-
ally identified as a, n, and m .The a -type parameter in most
SWCCs is a soil suction value that is somewhat greater than
the actual air-entry value. The n -type parameter is related
to the rate of desaturation of the soil as suction exceeds the
air-entry value, and in some cases, a further m -type param-
eter is used to give greater flexibility in fitting SWCC data.
In addition, the saturated gravimetric water content w s must
be known.
Gardner (1958a). The Gardner (1958a) equation was
originally proposed to describe the coefficient of permeabil-
ity function for an unsaturated soil. However, the mathe-
matical form proposed for the permeability equation has
also been applied to the water content versus soil suction
relationship. The Gardner equation can be written as

and Corey equation can be rearranged to compute soil suc-
tion corresponding to the measured water content:
ψ = a w s
w

1 /n

(4.37)

Two fitting parameters, a and n , and the saturated water
content w s are required along with the measured water con-
tent for the calculation of soil suction.
Van Genuchten (1980). Van Genuchten (1980)
proposed a three-parameter SWCC equation which has the
following form:

w s

w (ψ) = 1

+ (aψ) n m

(4.38)

The van Genuchten (1980) equation can be rearranged to
solve for soil suction in terms of water content:

w s
w

1 1 /n

1 /m

1
a

ψ = − (4.39)

The soil suction of a soil can be calculated if the three fit-
ting parameters a , m , and n for the van Genuchten equation
are known along with the saturated water content w s .The
usage of this equation is again limited to the range between
the air-entry value and the residual suction due to the asymp-
totic nature of the equation.
Van Genuchten (1980)-Mualem (1976). In 1976,
Mualem had suggested that the n and m parameters
in the SWCC equation bear a fixed relationship with
m = (n −

w s

w ( ψ ) = (4.34)

+ aψ n 1

where:

ψ = any soil suction value, kPa,

w (ψ) = water content at any soil suction, %,

w s = saturated water content, %, and 1 )/n . This suggestion reduces the three-parameter

equation of van Genuchten (1980) a , n = fitting parameters. to a two-parameter

SWCC equation:
w (ψ) =

w s

Equation 4.34 can be rearranged such that soil suction ψ
is dependent upon the measured water content w :

(4.40)

1

+ (aψ) n 1 − 1 /n

1
a

1 1 /n

w s
w −

The van Genuchten (1980)-Mualem (1976) equation can
be rearranged to solve for soil suction in terms of water
content:

ψ = (4.35)

w s
w

1 1 /n

n/(n − 1 )

The soil suction can be calculated corresponding to any
other water content, w, if the fitting parameters for the Gard-
ner equation are known.
Brooks and Corey (1964). Brooks and Corey (1964)
divided the SWCC into two zones: one zone where the soil
suctions are less than the air-entry value and the other zone
where soil suctions are greater than the air-entry value. This
gives rise to equations of the following form:
⎧
⎨

1
a

ψ = − (4.41)

The vanGenuchten (1980)-Mualem (1976) equation can be
applied between the air-entry value and residual conditions.
Van Genuchten (1980)-Burdine (1953). In 1953, Bur-
dine suggested that the n and m parameters for a SWCC
equation could bear a fixed relationship with m = (n −

2 )/n .

This resulted in a two-parameter equation:

w (ψ) = w s ψ<ψ aev

w s ψ
a

− n

w s

(4.36)

w (ψ) = (4.42)

1

+ (aψ) n 1 − 2 /n

⎩ w (ψ) = ψ ≥ ψ aev

The van Genuchten (1980)-Burdine (1953) equation can
be rearranged to solve for soil suction in terms of water
content:

where:

ψ aev = air-entry suction, kPa.

w s
w

1 1 /n

n/(n − 2 )

1
a

It is not possible to use the proposed SWCC equation for
estimating suctions prior to the air-entry value. The Brooks

ψ = − (4.43)