Compressibility and Pore Pressure Parameters - Unsaturated Soil Mechanics in Engineering Practice - page 794

volume of the continuous air phase in zone 1 decreases due
to an increase in the pore-air pressure from u a 1 to u a 2 ,the
controlling radius may in fact increase from R s 1 to R s 2 .
The above postulate is possible for the nonspherical air
bubbles such as those in zone 1. The nonspherical air bub-
bles have nonuniform radii of curvature over their surface,
but the assumption is made that only the controlling radius
is found in the free pore-water. The radius of the air-water
interface in the free water should be in agreement with the
capillary equation. Nevertheless, the increase in total stress
will eventually cause the air bubbles to take on a spherical
form, as shown in zone 2 (Fig. 15.10). For spherical air bub-
bles, a decrease in volume must be followed by a decrease
in the radius. At this point, the above-mentioned anomaly
associated with the use of the capillary equation cannot be
resolved.
On the basis of the above conceptual difficulties, it would
appear that the capillary equation should not be incorpo-
rated into the equation for the compressibility of air-water
mixtures. The pore-air and pore-water pressures must be
measured in the continuous pore-air zone (i.e., zone 1 in
Fig. 15.10) or computed using B a and B w pore pressure
parameters that are derived later in this chapter. Air pressure
measurements are valid as long as the air phase is continu-
ous. In the occluded zone (i.e., zone 2 in Fig. 15.10), the air
bubbles do not interact with the soil structure. The presence
of air bubbles merely renders the pore fluid compressible.
The pore-air and pore-water pressures can be assumed to be
equal in the occluded zone (i.e., zone 2).

pore-air pressure in response to the increase in total confin-
ing stress. The pore pressure response to an increase in the
confining stress is referred to as the B pore pressure param-
eter. At a point during the loading of the soil (i.e., point 1),
the tangent and the secant pore pressure parameters can be
defined for both the air and water phases (Fig. 15.11). If
the increase in the isotropic pressure is referenced to the
initial condition, the B secant pore pressure parameter can
be expressed as follows for the air phase:

u a
σ 3

B a = (15.19)

where:

B a

= secant pore pressure parameter for the air phase

during isotropic, undrained compression,

u a = increase in pore-air pressure due to an increase in

isotropic pressure (i.e., u a − u a 0 ),

u a 0 = initial pore-air pressure, and

σ 3 = increase in isotropic pressure from the initial con-

dition.

Similarly, for the water phase, the B pore pressure param-
eter can be expressed as

u w
σ 3

B w = (15.20)

where:

B w =

secant pore pressure parameter for the water phase
during isotropic, undrained compression,

15.4 DERIVATION OF PORE PRESSURE
PARAMETERS

u w = increase in pore-water pressure due to an increase

in isotropic pressure,

The pore pressure response resulting from a change in
total stress during undrained compression can be expressed
in terms of pore pressure parameters. The pore pressure
parameters corresponding to various loading conditions
can be derived from the relative compressibility values for
the materials involved. One pore pressure equation can be
derived for the water phase and another equation can be
derived for the air phase. These equations can be used to
compute the pore pressure responses in an unsaturated soil
during undrained loading. More direct measurements of the
pore pressure parameters in the laboratory are required to
confirm the accuracy of the theoretically derived equations.

σ 3 = change in pore-water pressure measured from the

initial condition (i.e., u w − u a 0 ), and

u w 0 = initial pore-water pressure.

If an infinitesimal increase in the isotropic pressure is con-
sidered at point 1, the pore pressure response at point 1
can be expressed as the tangent pore pressure parameter B
(Fig. 15.11). The B tangent pore pressure parameters for the
air and water phases are written as follows:

du a
dσ 3

B a = (15.21)

15.4.1 Tangent and Secant Pore Pressure Parameters
The pore pressure parameters for an unsaturated soil can
be defined either as a tangent-type parameter at a point or a
secant-type parameter referenced back to an initial condition.
These definitions are similar in concept to the tangent and
secant modulus used in the theory of elasticity.
Consider the development of pore-air and pore-water pres-
sures during isotropic, undrained compression, as shown in
Fig. 15.11. The pore-water pressure increases faster than the

du w
dσ 3

B w = (15.22)

where:

B a = tangent pore pressure parameter for the air phase

during isotropic, undrained compression,

du a = increase in pore-air pressure due to an infinitesimal

increase in isotropic pressure, dσ 3