Hydro-Mechanical Properties of an Unsaturated Frictional Material

2.5. CONSTITUTIVE MODELS FOR HYDRAULIC FUNCTIONS 53
Empirical Models
Empirical models (Richards 1931, Gardner 1958, Brooks & Corey 1964) incorporate the sat-
urated hydraulic conductivity and a sufficient set of experimental data of unsaturated hy-
draulic conductivity measurements, that are best fitted. One of the earliest equation, a linear
function, was proposed by Richards (1931). The equations are either related to the suction
(Richards 1931, Wind 1955, Gardner 1958) or to the volumetric water content (Gardner 1958,
Campbell 1973, Dane & Klute 1977). Because water is only flowing through the water phase
in soils unsaturated hydraulic conductivity and soil-water characteristic curve are similar in
shape. Thus soil-water characteristic curve parameters of well established soils can also be
used to determine unsaturated hydraulic conductivity function. Advantages and disadvan-
tages of direct and indirect methods for obtaining the unsaturated hydraulic conductivity
have been given by Leong & Rahardjo (1997a).
Macroscopic Models
Macroscopic models were developed on the assumption that laminar flow occurs on a micro-
scopic scale that obeys Darcy´s law at the macroscopic level. Generally macroscopic models
are based on the following general form:
kr(ψ) = Se(ψ) δ
(2.17)
where Se is the effective degree of saturation, ψ is the suction and the exponent δ is a
fitting parameter. Depending on the type of soil the parameter δ is varying between 2 and
4. Averjanov (1950) suggested δ = 3.5, Irmay (1954) suggested δ = 3.0 and Corey (1954)
suggested δ = 4.0. Brooks & Corey (1964) showed that the exponent δ = 3 is valid for
uniform soils only. To account for the effect of pore-size distribution Brooks & Corey (1964)
expanded the exponent to δ = (2 + 3λ)/λ, where λ is a pore-size distribution index. A
knowledge of the soil-water characteristic curve and the saturated hydraulic conductivity is
required when using macroscopic models. The main disadvantage of macroscopic models is,
that the effect of pore-size distribution is neglected.
Statistical Models
Statistical models use the best-fitted suction-water content relation in connection with the
saturated hydraulic conductivity to derive the relative hydraulic conductivity function. The
unsaturated hydraulic conductivity function k(ψ) can be calculated from the following rela-
tionship:
kr(ψ) = k(ψ)/ks; with kr(ψ) = 1 for ψ ≤ ψaev
(2.18)
where: kr is the relative hydraulic conductivity and ks is the saturated hydraulic conductivity.
Statistical models are based on the following assumptions (Mualem 1986):