Environmental Engineering Reference
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of the relevant constitutive equation to the divergence of the
flow rate, as described by the flow law. The time derivative
of the constitutive equation controls the deformation that
occurs under various stress conditions while the divergence
of the flow rate controls the rate of flow of air and water.
The assumptions used in the derivation of equations for
consolidation and swelling are similar to those proposed by
Terzaghi (1943) for saturated soils, with some exceptions
and additions:
1. The air phase is assumed to be continuous. The coef-
ficient of permeability with respect to the air phase
approaches the diffusivity of air through water when
the air phase becomes occluded. At the other extreme,
the excess pore-air pressure is assumed to remain at
atmospheric pressure when the air voids are large. This
negates the need for solving the air phase PDE in
this case.
2. The coefficients of volume change for the soil (i.e.,
m
1
k
,m
2
,m
1
k
, and
m
2
) are assumed to remain constant
during the consolidation process. It is also possible to
make these coefficients a function of the stress state
during the solution of the PDEs.
3. The coefficients of permeability with respect to the air
and water phases are assumed to be a function of the
stress state or the volume-mass soil properties during
the consolidation process. It is also possible to consider
the case where the coefficients of permeability can be
assumed to remain constant during a process.
4. The effects of air diffusing through water, air dissolv-
ing in water, and the movement of water vapor are
ignored. The initial and final states of stress in the air
phase are the same (i.e., atmospheric) for most engi-
neering problems. Therefore, although air may go into
solution under increased pore-air pressure conditions,
it can also be assumed that it comes out of solution in
response to a return to the boundary conditions prior
to loading.
5. The soil particles and the pore-water are assumed to
be incompressible.
6. Strains occurring during consolidation are assumed to
be small.
The above assumptions are not completely accurate for all
cases; however, these assumptions are reasonable for the
derivation of a general theory of consolidation (or swelling)
for unsaturated soils.
Water flow
Air flow
∂
v
w
∂
y
J
a
+
∂
J
a
∂
y
v
w
+
dy
dy
Soil element
of thickness
dz
dy
dx
Y
v
w
J
a
X
Figure 16.2
Unsteady-state flow of air and water during one-
dimensional consolidation of unsaturated soil.
where:
dV
w
=
change in the volume of water in the soil
element over a specific time,
dt,
∂V
w
/∂t
=
net flux of water through the soil element,
v
w
=
water flow rate across a unit area of the soil
element in the
y
-direction, and
dx,dy,dz
=
infinitesimal dimensions in the
x
-,
y
-, and
z
-directions, respectively.
Considering the net flux of water per unit volume and
rearranging Eq. 16.7 yields
∂(V
w
/V
0
)
∂t
∂
v
w
∂y
=
(16.8)
where:
V
0
=
initial total volume of the soil element
(i.e.,
dx,dy,dz
) and
∂(V
w
/V
0
)/∂t
=
net flux of water per unit volume of the
soil.
Substituting Darcy's law for the flow rate of water,
v
w
,
into Eq. 16.8 gives
16.2.4 Water Phase Partial Differential Equation
Let us consider a referential element of unsaturated soil with
air and water flow during one-dimensional consolidation
(Fig. 16.2). The net flux of water through the element is
computed from the volume of water entering and leaving
the element within a period of time:
∂V
w
∂t
∂(V
w
/V
0
)
∂t
[
−
k
w
(∂h
w
/∂y)
]
∂y
=
(16.9)
where:
k
w
=
coefficient of permeability with respect to water
as a function of matric suction which varies
with location in the
y
-direction [i.e.,
k
w
(u
a
−
u
w
)
]
,
v
w
+
dy
dx dz
∂
v
w
∂y
=
−
v
w
dx dz
(16.7)
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