Environmental Engineering Reference
In-Depth Information
consolidation theory can be applied and relied upon in engi-
neering practice.
The theory of consolidation involving the application of
externally applied loads may not be as important in unsatu-
rated soil mechanics as the theory of consolidation has been
in saturated soil mechanics. However, transient processes
resulting from changing boundary conditions (i.e., pore fluid
pressures, hydraulic heads, or moisture fluxes) are of signifi-
cant importance to simulating unsteady-state unsaturated soil
processes. The theory of consolidation and unsteady-state
analyses are closely related in unsaturated soil mechanics.
The theory of consolidation for unsaturated soils can be
derived as an extension of the formulations used for satu-
rated soils. Continuity of a saturated soil element requires
that the volume change of a soil element be equal to the
change in the volume of water in the element. The the-
ory of consolidation equation derived by Terzaghi (1943)
for saturated soils satisfied the continuity requirements of a
two-phase system:
modified to take into account the compressibility of the pore
fluid.
Larmour (1966), Hill (1967), and Olson (1986) showed that
Terzaghi's equation with a modified coefficient of consoli-
dation c v can be used to describe the consolidation behavior
of unsaturated soils with occluded air bubbles. Scott (1963)
incorporated void ratio change and degree-of-saturation
change into the formulation of a consolidation equation for
unsaturated soils with occluded air bubbles. Blight (1961)
derived a consolidation equation for the air phase of a dry,
rigid, unsaturated soil. Fick's law of diffusion was used to
relate the mass transfer of air to an air pressure gradient.
Fredlund and Hasan (1979) presented two PDEs which
could be solved for the pore-air and pore-water pressure
changes during the consolidation process in an unsaturated
soil. The air phase was assumed to be continuous. Darcy's
law and Fick's law were applied to the flow of water and air,
respectively. The coefficients of permeability with respect to
both the water and air phases were considered to be func-
tions of matric suction or one of the volume-mass properties
of the soil. In other words, the two PDEs contained terms
which accounted for variations in the coefficients of per-
meability with respect to the air and water phases. Both
equations were solved simultaneously, and the method is
commonly referred to as a two-phase flow approach. The
formulation by Fredlund and Hasan (1979) was similar in
form to the conventional one-dimensional Terzaghi (1943)
derivation. The derivations also demonstrated a smooth tran-
sition between the unsaturated and saturated cases. Similar
consolidation equations have also been proposed by Lloret
and Alonso (1981).
The two partial differential, transient flow equations pro-
posed by Fredlund and Hasan (1979) have been used to
simulate total volume change and water volume change
behavior of compacted kaolin specimens when total stress
and matric suction were changed (Fredlund and Rahardjo,
1986). The pore pressure changes calculated using the two
differential equations of flow resulted in changes in the stress
state variables. The stress state variable changes were substi-
tuted into the soil structure and the water phase constitutive
equations to compute volume changes in the unsaturated
soil. Comparisons between the predicted volume changes
and experimental results showed reasonable agreement with
respect to time. However, pore pressure changes in the spec-
imens were not measured during these tests.
Rahardjo (1990) conducted one-dimensional consolida-
tion tests on an unsaturated silty sand using a specially
designed K 0 cylinder. The cylinder was developed to accom-
modate K 0 loading and allow for the simultaneous measure-
ment of pore-air and pore-water pressures throughout the
soil specimen. The total and water volume changes were
measured independently during the consolidation tests. The
results showed a rapid (i.e., essentially instantaneous) dissi-
pation of excess pore-air pressures for the unsaturated soil
used in the study. On the other hand, the excess pore-water
2 u w
∂y 2
∂u w
∂t
=
c v
(16.1)
where:
c v =
coefficient of consolidation [i.e., k s /(ρ w gm v ) ] ,
k s
=
coefficient of permeability with respect to water at
saturation (i.e., S
=
100 %),
ρ w =
density of water,
g
=
gravitational acceleration, and
m v =
coefficient of volume change for saturated soils.
The above equation describes changes in pore-water pres-
sures with respect to depth and time during the consolidation
process. Changes in pore-water pressure result in changes in
the effective stress
u w ) The effective stress changes can
be substituted into stress-deformation constitutive equations
to compute volume change. Volume changes are equal to
the volume of water flowing out of the soil as long as the
degree of saturation remains at 100%. The computed volume
changes can, in turn, be used to compute soil volume-mass
properties such as void ratio, water content, and density dur-
ing the consolidation process.
In 1941, Biot proposed a general theory of consolidation for
an unsaturated soil with occluded air bubbles. Two constitu-
tive equations relating stress state and strain were formulated
in terms of effective stress
u w ) and the pore-water pres-
sure u w . In other words, the need for separating the effects
of total stress and pore-water pressure was recognized by
Biot. One constitutive equation related the void ratio to the
stress state, and the other equation related the water con-
tent to the stress state of the soil. Other assumptions used in
Biot's theory were similar to those used in Terzaghi's the-
ory of consolidation for saturated soils. Biot's theory resulted
in an equation similar to Eq. 16.1 for one-dimensional con-
solidation; however, the coefficient of consolidation c v was
 
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