Environmental Engineering Reference
In-Depth Information
Table 7.2 Relationships between Water Coefficient of Permeability and Matric Suction
Equations
Number
Source
Symbols
k
w
=
k
s
for
u
a
−
Brooks and Corey
(1964)
η
=
empirical constant
u
w
≤
u
a
−
u
w
b
=
2
+
3
λ
u
a
−
η
u
w
b
k
w
=
k
s
u
a
−
u
w
u
w
>
u
a
−
u
w
b
for
u
a
−
k
s
k
w
=
a
u
a
−
u
w
/
ρ
w
g
n
Gardner (1958a)
a
,
n
=
constant
1
+
k
s
n
=
k
w
=
Arbhabhirama and
Kridakorn (1968)
constant
(u
a
−
u
w
)
b
n
u
w
)/(u
a
−
+
1
In reality, the coefficient of permeability can change several
more orders of magnitude as the degree of saturation of the
soil is reduced.
An empirical mathematical form for a relationship between
coefficient of permeability and matric suction was proposed
by Gardner (1958a), and typical shapes for the function are
shown in Fig. 7.11. The equation provides a continuous per-
meability function extending from near-saturated conditions
under low matric suctions to high-suction conditions. The
permeability function is defined by two constants or fitting
parameters:
a
and
n
. The constant
n
defines the slope of the
permeability function while the
a
constant bears a relation-
ship to the breaking point on the permeability function (i.e.,
the air-entry value of the soil). Four typical functions with
differing values for
a
and
n
are illustrated in Fig. 7.11.
The Gardner permeability equation does not meet the
requirements for the estimation of a permeability function
since
a
and
n
are determined through use of a best-fit
regression analysis of permeability measurements. In
other words, coefficient-of-permeability data must first be
measured for various matric suction values. It is logical to
infer a relationship between Gardner's
a
parameter and the
air-entry value of the soil. However, it does not necessarily
follow that the
n
parameter bears a unique relationship to
the slope of the SWCC.
The term “permeability function” is used whenever the
coefficient of permeability is written as a function of any
volume-mass variable or stress state variable. In geotechnical
engineering, permeability function most commonly refers to
a mathematical relationship between coefficient of permeabil-
ity and soil suction. The permeability function term closely
fits the manner in which saturated-unsaturated seepage prob-
lems are solved using a numerical modeling procedure.
Numerous estimation procedures have been developed
for the calculation of a permeability function. Essentially
all estimation procedures for obtaining the permeability
function make use of the SWCC (i.e., the degree of
saturation versus soil suction relationship or the volumetric
water content versus soil suction relationship). A variety of
procedures are later discussed under the topic of “estimation
procedures” for obtaining the permeability function.
7.3.7 Hysteresis of Permeability Functions
The degree of saturation or volumetric water content shows
significant hysteresis when plotted versus soil suction
(Fig. 7.12a). Let us assume that there is no significant volume
change as soil suction is increased. The coefficient of perme-
ability is directly related to the volumetric water content (or
degree of saturation) and will also show significant hysteresis
when plotted versus soil suction. Figures 7.12a and 7.12b
demonstrate a similar hysteresis form for both the volumetric
water content
θ
and the coefficient of permeability
k
w
when
plotted versus soil suction. However, when the coefficient
of permeability is cross-plotted against volumetric water
content, there is essentially no hysteresis, as demonstrated in
Fig. 7.13.
The coefficient of permeability
k
w
is generally assumed
to be uniquely related to the degree of saturation
S
or the
volumetric water content
θ
. This assumption is reasonable
since the volume of water flow is a direct function of the
volume of water in the soil. The relationships between the
degree of saturation (or volumetric water content) and the
coefficient of permeability appear to exhibit little hysteresis
(Nielsen and Biggar, 1961; Topp and Miller, 1966; Corey,
1977; Hillel, 1982). Nielsen et al., (1972) stated: “The func-
tion
k
w
(θ)
is well-behaved, inasmuch as for coarse-textured
soils, it is approximately the same for both wetting and
drying.” However, this is not the case for the relationship
between the water coefficient of permeability
k
w
and matric
suction
u
a
−
u
w
. Since there is hysteresis in the relation-
ship between the volume of water in a soil and the stress
state (i.e., namely,
u
a
−
u
w
), there will also be hysteresis in
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