Environmental Engineering Reference
In-Depth Information
consolidation process. This assumption is justifiable since
most unsaturated soils have a relatively rigid soil structure.
The current total volume of the soil, V, in Eq. 16.24 can be
assumed to be equal to the initial total volume of the soil,
V 0 . Substituting Eq. 16.24 into Eq. 16.23 yields
change are a function of the stress state variables and can
be updated accordingly, if needed. The total stress change
with respect to time is set to zero during the consolidation
process (i.e., ∂σ y /∂t
=
0). Equating both equations for the
air flux (i.e., Eqs. 16.28 and 16.29) gives rise to the PDE for
the air phase:
2 u a
dy 2
∂D a
∂y
∂u a
∂y
(16.25)
The density of air is a function of air pressure in accor-
dance with the ideal gas law:
∂(V a /V 0 )
∂t
S)n ∂ρ a
∂t
D a
ρ a
+
( 1
=−
D a
a /RT K )
2 u a
∂y 2
∂u a
∂t
∂u a
∂t
∂u w
∂t
m 1 k
m 2
m 2
+
=−
u a
¯
∂D a
∂y
( 1
S)n
∂u a
∂t
1
a /RT K )
∂u a
∂y
(16.30)
u a
¯
u a
¯
Rearranging the above equation gives
ω a
RT K ¯
ρ a =
u a
(16.26)
m 1 k
∂u a
∂t =
( 1
S)n
∂u w
∂t
m 2
m 2
u a
¯
where:
D a
a /RT K )
2 u a
∂y 2
∂D a
∂y
∂u a
∂y
(16.31)
Equation 16.31 is a general form of the air phase PDE.
The equation can be simplified for special cases such as the
fully saturated case, the dry soil case, or special cases of an
unsaturated soil. The air phase PDE for each of these cases
is outlined in the following sections.
1
a /RT K )
ω a
=
molecular mass of air, kg/kmol,
u a
¯
u a
¯
R
=
universal (molar) gas constant, [i.e., 8.31432 J/
(mol
·
K)],
T K
=
absolute temperature (i.e., T K =
T
+
273 . 16), K,
temperature, C,
T
=
u a
¯
=
absolute pore-air pressure (i.e.,
u a =
¯
u a
u atm ),
kPa,
u a
=
gauge pore-air pressure, kPa, and
u atm =
¯
atmospheric pressure (i.e., 101 kPa or 1 atm).
16.2.9 Transition toward Saturated Soil Conditions
The coefficients of air volume change m 1 k and m 2 become
equal to zero as the degree of saturation of a soil goes toward
100%. The coefficient of transmission D a approaches zero,
indicating the absence of air flow. Air may exist in the form
of occluded bubbles which are assumed to have a pressure
equal to that of the water phase, (i.e., u a
Replacing the air density ρ a in Eq. 16.25 with Eq. 16.26
gives
ω a
RT K
S)n ω a
RT K
∂u a
∂t
∂(V a /V 0 )
∂t
u a
¯
+
( 1
u w =
0). Air may
2 u a
∂y 2
∂D a
∂y
∂u a
∂y
move as air diffusion through the pore-water.
The water phase becomes more compressible than that
of pure water when occluded air bubbles are present in the
water phase. A rigorous solution of the case described by
Eq. 16.1 requires the use of appropriate soil properties (i.e.,
c v ,k s w , and m v ), which take compressible pore fluid
properties into consideration.
D a
=
(16.27)
Equation 16.27 can be further rearranged to give the air
flux per unit volume of the soil element:
D a
a /RT K )
2 u a
∂y 2
∂(V a /V 0 )
∂t
=−
u a
¯
∂D a
∂y
16.2.10 Transition toward Dry Soil Conditions
Achange inmatric suction produces negligible volume change
when the soil is in a dry condition (i.e., water content is less
than residual conditions). All coefficients of volume change
with respect to matric suction approach zero (i.e., m 2 =
( 1
S)n
∂u a
∂t
1
a /RT K )
∂u a
∂y
(16.28)
u a
¯
u a
¯
The air phase constitutive relation (Eq. 16.4) defines the
air volume change in the soil element due to changes in net
normal stress d(σ y
m 2
=
u w ) .
The derivative of the air phase constitutive relation with
respect to time is equal to the air flux per unit volume of
the soil element:
u a ) and matric suction d(u a
m 2 =
0). Volume change may still occur due to a change in
net normal stress if the soil structure is compressible. In this
case, the soil volume change is equal to the air phase volume
change (i.e., m 1 k =
m 1 k ) inFig. 16.1c. The coefficient of trans-
mission D a reverts to a constant coefficient that corresponds
to dry conditions. Therefore, Eq. 16.31 can be simplified to the
following form for a compressible soil under dry conditions:
m 1 k +
u a
u w
∂(σ y
u a )
∂(V a /V 0 )
∂t
m 1 k
m 2
=
+
(16.29)
∂t
∂t
The coefficients of volume change m 1 k and m 2 are assumed
to be constant during consolidation when differentiating
Eq. 16.4. The magnitudes of the coefficients of volume
∂u a
∂t =
D d
a /RT K )
2 u a
∂y 2
n
¯
(16.32)
u a
u a
¯
 
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