Air Flow through Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 462

coefficients D ay and k ay have similar functional relationships
with respect to soil suction.
The measurement of the air coefficient of permeability in a
triaxial cell is an application of one-dimensional steady-state
air flow. The air coefficient of permeability is assumed to be
constant throughout the soil specimen. The air flow equation
is reduced to a linear differential equation by neglecting
changes in the air coefficient of permeability with respect to
location:

dJ ay
dy

J ay

+ dy

dy

J ay

Soil
element of
thickness dz

∂ 2 u a
dy 2

y

= 0 (9.53)

dx

x

The pore-air pressure distribution in the y -direction is
obtained by integrating Eq. 9.53 two times:

z

Figure 9.8 One-dimensional steady-state air flow through unsat-
urated soil element.

u a = C 1 y + C 2 (9.54)

where:

The net mass rate of air flow can be written as
dJ ay
dy

dx dy dz

C 1 ,C 2 = constants of integration related to the boundary

conditions

= 0 (9.50)

y = distance in the y- direction.

Substituting Fick's law for the mass rate of flow into the
above equation yields a nonlinear differential equation:

Figure 9.34 illustrates the pore-air pressure distribution
within a soil specimen during an air permeability test. The
air pressures at both ends of the specimen (i.e., u a =

D ay (u a −

d {− u w ) du a / dy }

u ab at

= 0 (9.51)

y h s ) are the boundary con-

ditions. Substituting the boundary conditions into the flow
equation results in a linear equation for the pore-air pressure
along the soil specimen [i.e., u a =

= 0 . 0 and u a = u at = 0 . 0at y =

dy

where:

( 1 − y/h s )u ab ] .

D ay (u a − u w ) = air coefficient of transmission as a func-

tion of matric suction,

9.6.2 Two-Dimensional Steady-State Air Flow
Two-dimensional steady-state air flow is first formulated for
the case where the air coefficient of transmission is a func-
tion of matric suction. The air coefficients of transmission in
the x - and y -directions, D ax and D ay , are related to matric

du a /dy = pore-air pressure gradient in the y- direc-

tion,

u a = pore-air pressure, and

u w = pore-water pressure.

suction using the same transmission function, D a u a − u w .

The D ax /D ay ratio is assumed to be constant at any point
within the soil mass. An element of soil subjected to two-
dimensional air flow is shown in Fig. 9.9. Satisfying conti-
nuity for steady-state flow yields the following equation:
J ax +

The coefficient of transmission D ay (u a −

u w ) will be writ-
ten as D ay for simplicity, and it must be remembered that
the soil property is a function of soil suction. The spatial
variation of the variable, D ay , results in nonlinearity in the
mass flow equation:

J ax dy dz

∂J ax
∂x

dD ay
dy

∂ 2 u a
dy 2

du a
dy =

dx −

D ay

+ 0 (9.52)

J ay +

J ay dx dz

∂J ay
∂y

+ dy − = 0 (9.55)

where:

dD ay / dy

= change in the air coefficient of transmission

in the y- direction.

where:

J ax = mass rate of air flowing across a unit area of the

soil in the x -direction.

Equation 9.52 describes the pore-air pressure distribution
in the soil mass in the y- direction. The second term in the
equation accounts for the spatial variation in the coefficient
of transmission. The coefficient of transmission is obtained
by dividing the air coefficient of permeability k ay by gravi-
tational acceleration (i.e., D ay =

Rearranging Eq. 9.55 results in the following equation:
∂J ax
∂x

dx dy dz

∂J ay
∂y

+ = 0 (9.56)

k ay /g ). In other words, the