Solving Saturated/Unsaturated Water Flow Problems - Unsaturated Soil Mechanics in Engineering Practice - page 378

The normalized water content n and the effective degree
of saturation S e are the same variable when volume changes
are negligible as soil suction is increased. The above-
mentioned water content variables (i.e., volumetric water
content, gravimetric water content, and degree of saturation)
are used somewhat interchangeably along with the dimen-
sionless and normalized forms, and it is important to note
the conditions under which errors can be introduced when
using the improper designation of the amount of water in
a soil for the estimation of the unsaturated permeability
function. On the other hand, it should be noted that volu-
metric water content θ must be used when differentiating
the SWCC (i.e., expressed as volumetric water content
versus soil suction) to obtain the water storage modulus.

SWCC through use of a curve-fitting procedure. The Brooks
and Corey (1964) equation is one example of an empirical
model. The equation is sometimes also referred to as a
macroscopic model because of consideration given to the
pore-size distribution within the soil. The Brooks and Corey
(1964) permeability function consists of two parts: the per-
meability below the air-entry value of the soil and the per-
meability above the air-entry value. The permeability below
the air-entry value is equal to the saturated coefficient of
permeability:

u w ≤ u a − u w b

k w = k s for u a − (8.6)

Above the air-entry value of the soil, the Brooks and
Corey (1964) function can be written as

8.2.2 Definitions Related to Designation of Water
Coefficient of Permeability
The coefficient of permeability depends on the amount of
water in the soil, which in turn depends on soil suction
ψ . Soil suction may be in terms of either matric suction
(i.e., u a −

u a − 2 + 3 λ

u w b

u w > u a − u w b

u a −

u w

k w = k s for u a −

(8.7)

u w , where u a is pore-air pressure and u w is pore-
water pressure) or total suction (i.e., matric plus osmotic
suctions). The term “permeability function” for unsaturated
soils is generally used to represent the relationship between
the coefficient of permeability and soil suction. The relative
coefficient of permeability, k r (ψ) , is equal to the coefficient
of permeability at a selected soil suction, k w (ψ) , referenced
to the saturated coefficient of permeability, k s , as follows:

where:

λ = pore-size distribution index defined as the negative

slope of the effective degree of saturation.

The pore-size distribution index is obtained from the
SWCC. Figure 8.3a shows a typical SWCC while Fig. 8.3b
shows the same plot as degree of saturation versus soil
suction. Figure 8.3c shows the same information plotted as
the effective degree of saturation (logarithm scale) versus
soil suction (logarithm scale). The degree of saturation
has been normalized between full-saturation conditions
and a point representing residual saturation. The slope of
the normalized log-log plot using the effective degree of
saturation yields the pore-size distribution index λ .The
pore-size distribution index along with an understanding of
the air-entry value and the saturated coefficient of perme-
ability of the soil provides the necessary information for the
estimation of the water permeability function.
The Brooks and Corey (1964) empirical permeability
functions allow the unsaturated coefficient of permeability
of a soil to be computed without measuring the unsaturated
coefficient of permeability. All estimation procedures for
the permeability function have some similar features. First,
all estimated permeability functions start at fully saturated
soil conditions. Second, all estimated permeability functions
are also intimately related to the air-entry value of the
soil. As a result, it is to be anticipated that all estimation
procedures provide similar estimations of the unsaturated
coefficient of permeability in the low soil suction range.
It is to be expected that different permeability estimation
procedures will begin to show divergence in the estimated
permeability functions as soil suctions increase well beyond
the air-entry value.

k w (ψ)
k s

k r (ψ) = (8.5)

The relative coefficient of permeability can also be written
as a function of volumetric water content, k r ( θ ) . However,
for geotechnical engineering applications the permeability
function is generally used as a scalar quantity which is a
function of soil suction, k r (ψ) .

8.2.3 Empirical Equations for Water Permeability
Function
All permeability functions appear to provide essentially the
same approximations of the coefficient of permeability from
saturated conditions through the air-entry value of the soil
and into the start of the transition zone. All equations pro-
duce a similar overall functional form that responds to the
air-entry value and the rate of desaturation of the soil. All
empirical and statistical estimation procedures for the pre-
diction of the water permeability function involve the use
of the SWCC.
Empirical permeability functions describe the variation
in the water coefficient of permeability with soil suction,
k w (ψ) [or volumetric water content, k w (θ) ]. The parame-
ters for the equations are generally determined from the