Environmental Engineering Reference
In-Depth Information
Equation 15.8 is a slight modification of the compress-
ibility equation for air-water mixtures proposed by Fredlund
(1976). The total stress change
dσ
is used as the reference
pressure in Eq. 15.8, while the pore-water pressure change
du
w
was used as the reference pressure in the compressibil-
ity equation by Fredlund (1976). The term
d(V
w
−
V
d
)/dσ
in Eq. 15.8 is considered to be equal to
dV
w
/dσ
since the
dissolved air is a fixed volume internal to the water. As such,
its volume does not change. The total volume of water,
V
w
,
is therefore used in computing the compressibility of water
The isothermal compressibility of air,
C
a
,
is equal to the
inverse of the absolute air pressure:
SC
w
du
w
dσ
(
1
−
S
+
hS
)(
du
a
/dσ )
u
a
C
aw
=
+
(15.12)
15.3.2.1 Use of Pore Pressure Parameters
in Compressibility Equation
The ratio between pore pressure change and total stress
change
(
du
/dσ )
is referred to as a pore pressure parameter.
This parameter indicates the magnitude of the pore pressure
change in response to a total stress change. The pore pres-
sure parameter concept was introduced by Skempton (1954)
and Bishop (1954). The pore pressure parameters for the air
and water phases are different (Bishop, 1961b; Bishop and
Henkel, 1962) and depend primarily upon the degree of sat-
uration of the soil. The parameters also vary depending on
the loading conditions.
The pore pressure parameters can also be directly mea-
sured in the laboratory. For isotropic loading conditions, the
parameter is commonly called the
B
pore pressure parame-
ter, and it can be substituted into Eq. 15.12 as follows:
[i.e.,
C
w
=−
1
/V
w
(
dV
w
/
du
w
)
].
The change in air volume occurs as a result of the com-
pression of the free air in accordance with Boyle's law and a
further dissolution of free air into water in accordance with
Henry's law. The total air volume change can be obtained
directly using Boyle's law by considering the initial and
final pressures and the volume conditions in the air phase.
The free-air compression and the air dissolving in water are
assumed to be complete under final conditions. The free and
dissolved air can be considered as one volume with a uni-
form pressure. Although the volume of dissolved air,
V
d
,
is a fixed quantity, it is maintained in the formulation for
clarity. Therefore, the dissolved air volume
V
d
also appears
in the equation.
Applying the chain rule of differentiation to the compress-
ibility equation gives
(
1
−
S
+
hS
)B
a
C
aw
=
SC
w
B
w
+
(15.13)
u
a
where:
dV
w
du
w
1
V
w
+
V
a
−
du
w
dσ
+
d(V
a
+
V
d
)
du
a
du
a
dσ
B
w
=
pore-water pressure parameter for isotropic loading
(i.e.,
du
w
/dσ
3
),
C
aw
=
(15.9)
σ
3
=
isotropic (confining) total stress, and
where:
B
a
=
pore-air pressure parameter for isotropic loading
(i.e.,
du
a
/dσ
3
).
dV
w
/
du
w
=
water volume change with respect to
a pore-water pressure change,
The compressibility of the pore fluid in an unsaturated
soil takes into account the matric suction of the soil through
the
B
w
and
B
a
parameters. In the absence of soil solids,
B
a
,
B
w
=
du
w
/dσ
=
water pressure change with respect to
a total stress change,
d(V
a
+
V
d
)/
du
a
=
air volume change with respect to a
pore-air pressure change, and
1. In the presence of soil particles, however, the sur-
face tension effects will result in
B
a
,B
w
<
1
.
0
,
depending
upon the matric suction of the soil. The pore-air and the
pore-water pressures change at differing rates in response to
the applied total stress. The
B
w
value is greater than the
B
a
value. The
B
a
and
B
w
parameters are low at low degrees of
saturation, and both parameters approach a value of 1.0 at
saturation. At this point, the matric suction of the soil goes
to zero.
du
a
/dσ
=
air pressure change with respect to a
total stress change.
Rearranging the above equation gives
V
w
V
w
+
V
a
du
w
dσ
1
V
w
dV
w
du
w
C
aw
=−
V
a
+
V
d
V
w
+
V
a
du
a
dσ
1
V
a
+
V
d
d(V
a
+
V
d
)
du
a
−
(15.10)
15.3.2.2 Components of Compressibility of Air-Water
Mixture
The first term in the compressibility equation (i.e.,
Eq. 15.13) accounts for the compressibility of the water
portion of the mixture, while the second term accounts for
the compressibility of the air portion. The compressibility
of the air portion is due to the compression of free air [i.e.,
(
1
Substituting the volume relations in Fig. 15.7 and
Eqs. 15.2 and 15.7 into Eq. 15.10 yields the compressibility
of an air-water mixture:
SC
w
du
w
dσ
hS
)C
a
du
a
dσ
C
aw
=
+
(
1
−
S
+
(15.11)
−
S)B
a
/ u
a
] and air dissolving into water (i.e.,
hSB
a
/ u
a
).
Search WWH ::
Custom Search