Hydro-Mechanical Properties of an Unsaturated Frictional Material

54 CHAPTER 2. STATE OF THE ART
1. The porous medium is assumed to be an assembly of randomly interconnected pores
with a certain statistical distribution.
2. The Hagen-Poiseuille equation is valid and used to determine the conductivity in a pore
channel.
3. The soil-water characteristic curve is a representation of the pore-size distribution func-
tion based on Kelvins capillary law.
Mualem (1986) provided an extensive study of statistical models for estimating unsaturated
hydraulic conductivity. The author reviewed the statistical models and found three general
pore-size functions, namely the Childs and Collis-George model (1950), the Burdine model
(1953) and the Mualem model (1976). Review of hydraulic conductivity functions utilizing
empirical expressions can be found in the geotechnical literature (Fredlund et al. 1994, Leong
& Rahardjo 1997a, Huang, Barbour & Fredlund 1998, Agus et al. 2003). Agus et al. (2003)
assessed statistical methods with different suction-water content equations for sands, silts and
clays. They found best agreement between experimental and measured results for sand when
using the modified Childs & Collis-George (1950) model together with the Fredlund & Xing
(1994) suction-water content model.
Numerous equations have been proposed by several researchers to predict the relative
hydraulic conductivity function from the soil-water characteristic curve. Among them are the
models by Marshall (1958), Millington & Quirk (1961), Kunze et al. (1968). An attractive
closed-form equation for the unsaturated hydraulic conductivity function was given by van
Genuchten (1980), who substituted the soil-water characteristic curve in the conductivity
model of Mualem (1976). Disadvantage of the closed form equation is, that the van Genuchten
soil-water characteristic curve parameter m is restricted to m = 1 − 2/n and thus reduces the
accuracy of the best fit. Therefore the closed form equation is not used in this thesis. Fredlund
et al. (1994) combined Fredlund and Xing’s model for soil-water characteristic curve with the
statistical pore-size distribution model of Childs & Collis-George (1950) and derived a flexible
relative hydraulic conductivity function. Three models, namely the Childs and Collis-George
model (1950), the Mualem model (1976) and the Fredlund et al. model (1994) are used in
this study. In the following the expressions are introduced.
- Childs & Collis-George (1950)
Based on randomly interconnected pores with a statistical distribution Childs & Collis-
George (1950) obtained the following expression:
k(θ) = M
� � ϱ=R(θ)
ϱ=Rmin
� r=R(θ)
r=ϱ
ϱ 2 f(ρ)f(r)drdρ +
� ρ=R(θ) � r=ϱ
ϱ=Rmin
r=Rmin
r 2 f(r)f(ϱ)dϱdr
�
(2.19)