Compressibility and Pore Pressure Parameters - Unsaturated Soil Mechanics in Engineering Practice - page 800

These pore pressure parameters are defined as tangent-type
parameters referenced to a particular stress point:

Load, d s y

O-ring
seals du a

dσ y

Rigid cap

B ak = (15.48)

Rigid
metal
ring

Compacted
soil

du w
dσ y

B wk = (15.49)

Rigid base

Changes in pore-air and pore-water pressures can be
expressed in terms of pore pressure parameters and written
as follows:

Figure 15.14 Conditions simulated in Hilf (1948) derivation of
pore pressure parameters for K 0 -undrained loading.

B wk = R 1 k B ak + R 2 k (15.50)

(See Fig. 15.14.) The reduction in volume can be assumed to
be the result of the compression of free air and air dissolving
into water. The soil solids and the water are assumed to be
incompressible. Vapor pressure and temperature effects are
also assumed to be negligible.
The amount of dissolved air in water is computed in accor-
dance with Henry's law. Free and dissolved air is regarded
as a single total volume of air at a particular pressure. The
change in pore-air pressure between initial and final loading
conditions is computed using Boyle's law.
The initial and final conditions considered in Hilf's anal-
ysis are shown in Fig. 15.15. The total volume of air associ-
ated with the initial condition, V a 0 , can be written as follows:

B ak = R 3 k B wk − R 4 k (15.51)

Substituting Eq. 15.41 into Eq. 15.51 results in an equation
for the B ak pore-air pressure parameter:

R 2 k R 3 k − R 4 k
1

B ak = (15.52)

− R 1 k R 3 k

Similarly, Eq. 15.51 can be substituted into Eq. 15.50 to
give an equation for the B wk pore-water pressure parameter:

R 2 k − R 1 k R 4 k
1

B wk = (15.53)

− R 1 k R 3 k

V a 0 = [ ( 1 − S 0 )n 0 + hS 0 n 0 ] V 0 (15.54)

The pore-air and pore-water pressure responses at any
point during K 0 -undrained loading can be computed using
the B ak and B wk pore pressure parameters.

where:

V a 0 =

initial volume of free and dissolved air,

S 0 = initial degree of saturation,

15.5.3 Hilf's Analysis
Hilf (1948) outlined a procedure to compute the change
in pore pressure in compacted earth fills as a result of an
applied total stress. The pore pressure analysis can be rear-
ranged to take on the form of a pore pressure parameter
equation. The derivation is based on the results of a one-
dimensional oedometer test on a compacted soil, Boyle's
law, and Henry's law.
The Hilf (1948) analysis gives rise to a nonlinear relation-
ship between total stress and pore pressure. A secant-type
pore pressure parameter can be derived from this relation-
ship. This method has been extensively used by the U. S.
Bureau of Reclamation (i.e., USBR method) and has proven
to be quite satisfactory for estimating pore pressures in com-
pacted fills (Gibbs et al., 1960). The same formulation for
the estimation of pore pressures in compacted soils has also
been advanced by Bishop (1957).
Hilf (1948) stated: “To illustrate the role of air in the
relation between consolidation and pore pressure, consider
a sample of moist earth compacted in a laboratory cylinder.
If a static load is applied by means of a tight-fitting piston,
permitting neither air nor water to escape, it is found that
there is a measurable reduction in volume of the soil mass.”

n 0 = initial porosity, and

V 0 = initial volume of the soil.

The first and second terms in the above equation represent
the free and dissolved air volumes, respectively. The ini-
tial absolute pore-air pressure is denoted as

u a 0 and can be
assumed to be at atmospheric conditions (i.e., 101.3 kPa). An
increment in the major principal stress, σ y , is then applied
to the soil specimen. The total volume of air decreases and
the air pressure increases in accordance with Boyle's law.
The air volume change is equal to the void volume change
V v since the soil solids and water are assumed to be incom-
pressible. Therefore, the air volume change can be written
as a change in porosity (i.e., n = V v /V 0 ) times the ini-
tial volume of soil, V 0 , as illustrated in Fig. 15.15. The total
volume of air under final conditions, V af , can be expressed
as

V af = [ ( 1 − S 0 )n 0 + hS 0 n 0 − n ] V 0 (15.55)

where:

V af = final volume of free and dissolved air and

n = change in porosity (i.e., decrease).