Environmental Engineering Reference
In-Depth Information
h
w
=
hydraulic head [i.e., gravitational plus pore-
water pressure head or
Y
The above equation is a general form PDE for the water
phase. The equation can be simplified for special soil con-
ditions, such as the fully saturated case, the completely dry
case, or special cases of unsaturated soil behavior. The water
phase PDE for the above-mentioned conditions is described
in the following sections.
+
u
w
/(ρ
w
g)
],
Y
=
elevation head,
u
w
=
pore-water pressures,
ρ
w
=
density of water,
g
=
gravitational acceleration, and
∂h
w
/∂y
=
hydraulic head gradient in the
y
-direction.
16.2.5 Transition to Saturated Condition
The coefficients of water volume change
m
1
k
and
m
2
become
equal to the coefficient of volume change
m
v
for a saturated
soil (i.e., degree of saturation
S
Rearranging the above equation gives
∂
2
h
w
∂y
2
∂(V
w
/V
0
)
∂t
∂k
w
∂y
∂h
w
∂y
=−
k
w
−
(16.10)
100%). The coefficient
of permeability
k
w
reverts to the coefficient of permeability
at saturation
k
s
. The saturated coefficient of permeability is
usually assumed to remain constant during the consolidation
process (i.e.,
∂k
s
/∂y
=
Let us substitute hydraulic head
h
w
in terms of the indi-
vidual components
(Y
+
u
w
/ρ
w
g),
Eq. 16.10:
=
0).
Substituting
m
v
and making
∂k
s
/∂y
equal to zero in Eq.
16.15 results in the Terzaghi formof the one-dimensional con-
solidation equation for saturated soils (Terzaghi, 1943). The
Terzaghi equation describes the pore-water pressure changes
during one-dimensional consolidation of a saturated soil. The
consolidation process of a saturated soil involves only the flow
of water, which in turn produces an equal volume change.
∂
2
[
Y
∂(V
w
/V
0
)
∂t
+
(u
w
/ρ
w
g)
]
∂y
2
=−
k
w
∂k
w
∂y
∂
[
Y
+
(u
w
/ρ
w
g)
]
∂y
−
(16.11)
∂
2
u
w
∂y
2
∂(V
w
/V
0
)
∂t
k
w
ρ
w
g
=−
1
ρ
w
g
∂k
w
∂y
∂u
w
∂y
∂k
w
∂y
−
−
(16.12)
16.2.6 Transition to Dry Soil Conditions
There is only a small amount of water present around the soil
particles when the soil approaches dry conditions (i.e., water
content less than residual conditions). Changes in matric
suction or net normal stress produce negligible change in
the volume of water when the soil is dry. The coefficients
of water volume change
m
1
k
and
m
2
and the coefficient of
permeability
k
w
go towards zero in dry soils. Any volume
change that occurs in a dry soil is the result of the soil
structure or air phase volume change. Setting the coefficients
of volume change and the coefficient of permeability to zero
(i.e.,
m
1
k
=
The water phase constitutive relation (Eq. 16.3) defines the
volume of water in an unsaturated soil element in terms of
net normal stress
d(σ
y
−
u
w
)
.
The flux of water per unit volume of the soil can be obtained
by differentiating the water phase constitutive relation with
respect to time:
u
a
)
and matric suction
d(u
a
−
∂(σ
y
−
u
a
)
∂(V
w
/V
0
)
∂t
∂(u
a
−
u
w
)
m
1
k
m
2
=
+
(16.13)
∂t
∂t
The coefficients of volume change are assumed to be con-
stant during the consolidation process. In other words, the
coefficients of volume change are only a function of the stress
state. The coefficients of volume change can be updated in
accordance with the stress state of the soil. The change in
total stress with respect to time is generally set to zero during
a consolidation process (i.e.,
∂σ
y
/∂t
m
2
0) in Eq. 16.15 causes the
PDE for the water phase to vanish. In other words, there is
no water flow during the consolidation of a dry soil.
=
0 and
k
w
=
16.2.7 Special Case of Unsaturated Soil
Air flow and water flow can take place simultaneously dur-
ing consolidation of an unsaturated soil (i.e., 0
<S<
100%).
The consolidation equation (i.e., Eq. 16.15) for the water
phase can be rearranged as follows:
0). Both equations for
water flux (i.e., Eqs 16.12 and 16.13) can be equated to yield
a PDE for the water phase:
=
∂u
a
∂t
∂u
a
∂t
∂u
w
∂t
m
1
k
m
2
m
2
−
+
−
∂
2
u
w
∂y
2
c
v
k
w
∂u
w
∂t
∂u
a
∂t
∂k
w
∂y
∂k
w
∂y
∂u
w
∂y
∂k
w
∂y
(16.16)
c
v
=−
C
w
+
+
+
c
g
∂
2
u
w
∂y
2
k
w
ρ
w
g
1
ρ
w
g
∂k
w
∂y
∂u
w
∂y
∂k
w
∂y
=−
−
−
(16.14)
where:
Rearranging the above equation gives
C
w
=
interactive constant associated with the water phase
PDE [i.e.,
(
1
∂u
w
∂t
m
2
)
∂u
a
∂t
m
2
(m
1
k
−
=−
m
2
/m
1
k
)/(m
2
/m
1
k
)
],
−
c
v
=
coefficient of consolidation with respect to the water
phase [i.e.,
k
w
/(ρ
w
gm
2
)
], and
∂
2
u
w
∂y
2
k
w
ρ
w
g
1
ρ
w
g
∂k
w
∂y
∂u
w
∂y
+
∂k
w
∂y
+
+
(16.15)
gravity term constant (i.e., 1
/m
2
).
c
g
=
Search WWH ::
Custom Search