Solving Saturated/Unsaturated Water Flow Problems - Unsaturated Soil Mechanics in Engineering Practice - page 387

et al. (1994b) permeability function retains independence
among the a f ,n f , and m f fitting parameters when using the
SWCC in the calculation of the permeability function. It is
suggested that retaining independence of the SWCC fitting
variables leads to superior estimations of the permeability
functions.

concluded that the accuracy of the prediction is significantly
improved when the SWCC extended at least to residual
conditions. Therefore, the x integration should be carried
out at least over the interval from θ s to θ r :

ψ 2 (x) dx

θ θ s

θ − x θ s − x

ψ 2 (x) dx

k r (θ) = (8.18)

8.2.4.4 Fredlund et al. (1994b) Model
Fredlund et al. (1994b) used the Fredlund and Xing (1994)
SWCC equation along with the Childs and Collis-George
(1950) physical model to compute a water permeability
function. The procedure involved numerical integration
along the SWCC. Independent permeability functions can
be computed for the drying and wetting SWCC. It is
assumed that the volume change of the soil structure is
negligible when soil suction is changed. Equation 8.8 can
be expressed in the following integration form:

θ r θ r

Most proposed models for the relative coefficient of per-
meability k r (e.g., Eqs. 8.11, 8.13, and 8.18) require the
designation of residual water content and the SWCCs are
rarely used over the entire range of water contents. Brooks
and Corey (1964), Mualem (1976a), and van Genuchten
(1980) suggest different methods for the extrapolation of
the measured soil-water characteristic data and the deter-
mination of the residual water content. The conventional
procedure for the prediction of the coefficient of permeabil-
ity using Eq. 8.18 consists of two steps. First, the residual
water content of the soil under consideration was estimated
from experimental data. Then the measured soil-water char-
acteristic data are fitted using a mathematical equation for
the interval between saturated water content θ s and residual
water content θ r . Subsequent integrations were performed
using the best-fit curve between these two water content
conditions. It is not clearly known under what water content
condition the coefficient of permeability goes to zero. There-
fore, it is better to estimate the coefficient-of-permeability
function without being restricted to an empirical estimation
of zero water flow.
The total suction corresponding to zero water content (i.e.,
oven-dried water content) is essentially the same for all
types of soils. A value approaching 10 6 kPa has been sup-
ported by thermodynamic considerations (Richards, 1965)
and experimental results (Croney and Coleman, 1961). A
general equation describing the SWCC over the entire suc-
tion range (i.e., 0

ψ 2 (x) dx

θ θ s

θ − x x

ψ 2 (x) dx

θ s −

k (θ) = k s (8.15)

θ L θ L

where:

ψ = soil suction as a function of volumetric water con-

tent θ ,

x = dummy variable of integration representing water

content,

θ = volumetric water content,

θ s = volumetric water content at saturation, and

θ L = lower limit for the volumetric water content.

Equation 8.15 can also be expressed as a relative coeffi-

cient of permeability, k r θ i , in summation form:

m

m

k r θ i =

2 (j − i) + 1 2 j − 1

10 6 kPa) was proposed by Fredlund and −

(8.16)

ψ j ψ j Xing (1994):

j = i j = i

θ s

The integration form of Eq. 8.16 can be written as fol-
lows:

θ = C (ψ) (8.19)

ln e

+ ψ/a f n f

m f

ψ 2 (x) dx

where: θ θ s

θ − x θ s − x

ψ 2 (x) dx

k r (θ) = (8.17)

e = the natural number 2.71828,

θ L θ L

a f = inflection point, related to the air-entry value of

the soil,

The residual volumetric water content θ r is the water
content below which large increases in suction are required
to remove additional water. The residual water content can
generally be observed as a break in the SWCC; however, the
break in curvature is not always visible for high-plasticity
clay soils. It is commonly assumed that the coefficient of
permeability of a soil is essentially zero when its water con-
tent is below the residual water content. Kunze et al. (1968)
investigated the effect of using only a portion of the SWCC
for the prediction of coefficient of permeability. It was

n f = parameter that controls the slope at the inflection

point in the SWCC,

m f = parameter that is related to the residual water

content,

C (ψ) = correction function defined as

ln 1 ψ ψ r

+

C (ψ) = 1 −

ln 1 1 , 000 , 000 /ψ r

+

ψ r = constant related to the suction at residual water

content, θ r .