Air Flow through Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 461

h = Henry's volumetric coefficient of solubility, V ad /

V w ,

9.6 PARTIAL DIFFERENTIAL EQUATIONS FOR
AIR FLOW THROUGH UNSATURATED SOILS

V ad = volume of air dissolved in pore-water, and

Air is compressible and, as such, changes in density must
be taken into consideration when describing the flow of air
through a referential element. The mass flow of air through
a referential element takes the form of a partial differential
equation for compressible fluid flow.
A series of partial differential equations are presented
that describe the components of air flow under various
conditions. Steady-state air flow is first presented followed
by transient or unsteady-state flow. The partial differential
equations associated with one-, two-, and three-dimensional
situations are presented.

density of air, kg/m 3 . ρ a =

Equation 9.45 can be rewritten as follows:

D ad
ρ a

∂C ad
∂

D ad ∗
ρ a

∂ u a

∂y

¯ ∂ u a

∂y

¯

v a y

=− =− (9.46)

u a ¯

where:

D ad ∗ =

coefficient of transmission, (kg m)/(kN s), and

u a ¯ = absolute air pressure, kPa.

9.6.1 One-Dimensional Steady-State Air Flow
The bulk flow of air (i.e., free air) can occur through an
unsaturated soil when the air phase is continuous. Air is
compressible and the derivation of the partial differential
equations for air flow must take into consideration changes
in density as a result of pressure changes.
The air coefficient of transmission D a or the air coefficient
of permeability k a (i.e., D a g ) is a function of the volume-
mass properties (or stress state) of the soil. The relationship
between the air coefficient of permeability k a and matric suc-
tion [i.e., k a (u a −

The soil properties D ad and D ad ∗ have been defined and
can be measured or estimated. The diffusion of dissolved
air through liquid pore-water decreases as the degree of
saturation of a soil decreases. Dissolved air movement
becomes insignificant when compared with the flow of free
air through the soil. The dissolved air component υ ad can
be incorporated into the prediction of the coefficient of
transmission D ad ∗ through use of a tortuosity coefficient.
The flow of dissolved pore-air carried by water flow (i.e.,
advection) can be described using Darcy's law for water
flow and taking the amount of dissolved air into account:

u w ) ] or degree of saturation [i.e., k a (S e ) ]
was previously described. The value of k a or D a may vary
with location, depending upon the distribution of the pore-
air volume in the soil. The air coefficient of permeability in
an unsaturated soil can vary with direction and location. The
air coefficient of permeability at a point can be assumed to
be constant with respect to time during steady-state air flow
provided soil suction remains constant.
Steady-state formulations for one-dimensional air flow are
first presented using Fick's law. Heterogeneous, isotropic,
and anisotropic conditions are taken into consideration in
the derivation of the flow equations. Steady-state air flow
is analyzed assuming that the pore-water pressures have
reached equilibrium and are not influenced by the pore-
water pressures. One-dimensional air flow equations can be
solved using numerical methods such as the finite difference
method or the finite element method.
Consider an unsaturated soil ( i.e., heterogeneous), ele-
ment with air flow in the y -direction (Fig. 9.8). The air flow
has a mass rate J ay under steady-state conditions. The mass
rate is assumed to be positive for upward air flow. The prin-
ciple of continuity requires that the mass of air flowing into
the soil element must be equal to the mass of air flowing
out of the element:
J ay +

∂h
∂y

v a y

=−

hk w (9.47)

where:

v a y

= dissolved pore-air flow rate in the y- direction across

a unit area of the soil due to bulk pore-liquid water
flow, m/s.

The total flow of pore-air, v y , is the summation of the
three flow mechanisms described by Eqs. 9.42, 9.45, 9.46,
and 9.47:

k ad
γ a

k a
γ a

∂u a
∂y

∂u a
∂y

∂h
∂y
(9.48)

v a y

v y = v a y v a y + + =− − − hk w

where:

air coefficient of permeability, γ a D a ∗ /ρ a ,m/s,

k a =

k ad diffusion coefficient of air through water, γ a D a ∗ /

¯

=

u a , m/s, and

unit weight of air, kN/m 3 .

γ a =

Equation 9.48 provides a smooth transition between unsat-
urated and saturated conditions. The soil becomes saturated
as soil suction decreases and k a decreases toward zero. How-
ever, the flow of air does not completely cease for satu-
rated conditions. Pore-air conductivity by diffusion within
the pore-water and the flow of dissolved air carried by bulk
water flow increase as the pores become filled with water.

dy dx dz

dJ ay
dy

− J ay dx dz = 0 (9.49)

where:

J ay = mass rate of air flow across a unit area of the soil

in the y- direction.