Theory of Water Flow through Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 334

a plot of effective degree of saturation S e versus matric
suction (Fig. 7.8b):

where

λ = pore-size distribution index defined as the negative

slope of the effective degree of saturation S e versus

S − S r

S e = (7.14)

matric suction u a − u w .

1 − S r

where:

Soils with a wide range of pore sizes have a small value
for the pore-size distribution index λ . Soils with a uniform
pore-size distribution have a larger value for λ . Figure 7.9
presents typical λ values for various soils which have been
obtained from matric suction versus degree of saturation
curves.
The coefficient of permeability with respect to the water
phase, k w , can be predicted from the matric suction versus
degree of saturation curves as follows (Brooks and Corey,
1964):

S e = effective degree of saturation,

S r = residual degree of saturation (decimal form), and

S = any degree of saturation (decimal form).

The residual degree of saturation S r was defined as the
degree of saturation at which an increase in matric suction
did not produce a significant change in the degree of satura-
tion. While the intent of the definition of residual degree of
saturation is clear, the numerical determination of this value
is left somewhat vague.
The effective degree of saturation S e can be computed by
first estimating a value for the residual degree of saturation
S r (Fig. 7.8a). The effective degree of saturation was then
plotted against the matric suction, as illustrated in Fig. 7.8b.
A horizontal and a sloping line can be drawn through the
data points. However, points at high-matric-suction values
may not lie on the straight line used for the first estimate of
the residual degree of saturation (Fig. 7.8b). It was reasoned
that the point corresponding to high matric suctions should
be forced to lie on the straight line by estimating a new value
for the residual degree of saturation S r . A second estimate of
the residual degree of saturation was then used to recalculate
values for effective degree of saturation S e . A new plot of
the matric suction versus effective degree of saturation curve
can then be obtained. The above procedure is repeated until
all of the points on the sloping line constitute a straight
line. This usually occurs after the second estimate of the
residual degree of saturation. The use of the above theory
is based on the assumption that coefficient of permeability
asymptotically approaches zero as the residual degree of
saturation of the soil is exceeded.

u w ≤ u a − u w b

k w = k s for u a − (7.16)

u w > u a − u w b k s S e

k w = for u a − (7.17)

where:

k s = coefficient of permeability with respect to the water

phase for the saturated soil (i.e., S = 100%) and

δ = an empirical constant for the permeability function.

The empirical constant δ can be related to the pore-size
distribution index λ :

2 + 3 λ

δ = (7.18)

λ

Table 7.1 presents several δ values and their corresponding
pore-size distribution indices λ measured for various soil
types.

u w b , is the matric
suction value that must be exceeded before air recedes into
the soil pores. The air-entry value is also referred to as
the “displacement pressure” in petroleum engineering or the
“bubbling pressure” in ceramics engineering (Corey, 1977).
The term “air-entry value” has been most commonly used
in the agricultural sciences and in geotechnical engineer-
ing. The air-entry value is a measure of the maximum pore
size in a soil. The intersection point between the straight
sloping line and the complete saturation ordinate (i.e., S e =
1.0) in Fig. 7.8b defines the air-entry value of the soil. The
sloping line passing through points where the matric suction
is greater than the air-entry value can be described by the
following equation:

The air-entry value of the soil, u a −

7.3.6 Relationship between Water Coefficient
of Permeability and Matric Suction
A closed-form equation for the permeability function can
be obtained by substituting Eq. 7.15 for the effective degree
of saturation into Eq. 7.17. The resulting equation defines
the coefficient of permeability of the soil at suction values

Table 7.1 Suggested Values of Constant δ and

Pore-Size Distribution Index λ for Various Soils

δ λ

Soils Source

Uniform sand 3.0 ∞ Irmay (1954)

u a − λ

u w b

Soil and porous rocks 4.0 2.0 Corey (1954)

for u a − u w > u a − u w b

S e = u a −

u w

(7.15)

Natural sand deposits 3.5 4.0 Averjanov (1950)