Theory of Water Flow through Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 341

The diffusion coefficient D v ∗ is equal to zero when the soil is
saturated and increases as air starts to occupy part of the soil
pore space. On the other hand, the hydraulic conductivity k w
is at its highest value when the soil is saturated and starts
declining as air begins entering the soil pore spaces. As the
soil dries, the water coefficient of permeability k w eventually
becomes lower than the vapor diffusion coefficient k v .At
this point, vapor flow will become equal to liquid water flow.

Phase transfers are usually assumed to be instantaneous in
soil systems. Assuming local thermodynamic equilibrium,
neglecting the effects of the osmotic suction, and assuming
that the air pressure is equal to the atmospheric pressure, the
following relationship can be written (i.e., Lord Kelvin's
equation):

u w gW v
γ w RT K

u air
v

u air =

v 0 e

(7.24)

where:

7.3.9 Lower Limit for Water Coefficient
of Permeability
The liquid water coefficient of permeability k w decreases
with increasing soil suction. There is, however, a “shutoff”
of liquid water flow at some value of suction. The chal-
lenge is to determine the smallest possible water coefficient
of permeability. Most of the proposed procedures for the
calculation of the water permeability function result in an
indefinite decrease in the water coefficient of permeability
on a logarithmic scale. Liquid water flow may occur at rel-
atively high soil suctions, but there should be some point
where there is a transfer from liquid water flow to predom-
inantly vapor water flow.
Extremely small values of the liquid water coefficient
of permeability k w can create serious difficulties when
undertaking numerical modeling of water seepage. Ebrahimi-
Birang et al.(2004) presented two approaches for the deter-
mination of a minimum value for the water coefficient of
permeability k w . The first approach used residual conditions
to calculate minimum values for the liquid coefficient of
permeability k w . The second approach was based on water
vapor flow theory.

u air
v 0

= saturation vapor pressure, kPa, of the soil-water at

temperature T K and

acceleration of gravity, 9.81 m/s 2 . g =

Values of saturation soil vapor pressure have been exper-
imentally obtained by Kaye and Laby (1973) for various
temperatures. Other parameters in Eq. 7.24 were previously
defined. A relationship between the vapor pressure gradients
u ai v and the gradients of the other two variables, u w and T k ,
are determined by differentiating Eq. 7.24 with respect to
space, y :
∂u air
v
∂y

∂u w
∂y −

gW v u air u w

T K

∂T K
∂y

v
γ w RT K

= (7.25)

The total moisture flow (liquid and vapor), q y [kg/(m 2 s)],
is obtained by summing Eq. 7.25 and Darcy's liquid flow
equation and the water vapor equation 7.23. Equation 7.25
can then be used to express vapor flow in terms of pore-
water pressure and temperature gradients:

g k w +

γ w D m

1 ∂u w

∂y

∂Y
∂y

q y =− − ρ w k w

7.3.9.1 Liquid Water Permeability at Residual
Conditions
Ebrahimi-Birang et al.(2004) selected 45 soils from the lit-
erature and compiled the measured SWCCs, saturated coef-
ficients of permeability, and volume-mass properties. The
soils were classified as sand, silty sand, sandy silt, and
silt. Several predictive permeability models were used to
calculate the water permeability coefficient at residual con-
ditions for each soil. The unsaturated coefficient of perme-
ability corresponding to residual water content conditions
was determined. The determination of the water coefficient
of permeability corresponding to residual conditions is illus-
trated in Fig. 7.15.
Water coefficients of permeability corresponding to resid-
ual conditions were determined using the procedure pro-
posed by Fredlund et al. (1994b), as shown in Fig. 7.16.
Similar distributions of the water coefficient of permeability
were also found for other proposed procedures for predict-
ing the water permeability functions. The results show a
considerable variation in the water permeability coefficient
near residual conditions. In general, the water coefficients
of permeability corresponding to residual suction appeared
to be too large.

u air
v

D v ∗ ∂u air

ρ w D m u w
T K

∂T K
∂y

u a + ¯

v
∂y

+ − (7.26)

u a ¯

where

D v ∗
ρ w

u air
v

W v u air u a + ¯ g

γ w

v
RT K

D m (m/s)/(kN/m 3 )

=

u a ¯

Equation 7.26 governs one-dimensional flow of liquid
water and water vapor flow based on a single gradient.
The comparison of the terms k w and k v (where k v =

γ w D m )
gives a comparison between the amount of liquid and vapor
water flow. The terms k w and k v can be compared if isother-
mal conditions exist, and the gravitational component is
neglected since both terms are then based on the gradient of
pore-water pressure.
Temperature gradients are required for the solution of
Eq. 7.26 for nonisothermal conditions. The nonisothermal
solution would require that the heat flow partial differential
equation also be solved (Gitirana and Fredlund, 2003b).
Equation 7.22 shows that the diffusion coefficient of water
vapor through soil, D v ∗ , is a function of degree of saturation
S and porosity n , which in turn are functions of soil suction.