Environmental Engineering Reference
In-Depth Information
Figure 3.11. Suction curve
- second, the water permeability is modified to take into account the effect of
non-saturation, for example by a relationship such as:
k
K
r
k
=
ρ
g
[3.12]
w
µ
w
with K the geometrical permeability, µ the dynamic viscosity of the water and k r the
relative permeability given by [SEK 83]:
b
Sa
r
k
=
[3.13]
r
1
a
where the material constants are
a=S
the minimum degree of saturation
r,min
(discontinuous water phase) and
b=2log(ψ + . Figure 3.10 shows the
development of the relative permeabilities to water and air depending on the degree
of saturation;
)
3
0
− finally, the plasticity criterion and possibly the elastic parameters are expressed
as a function of the suction (for example equation [3.9]).
A still more complete formulation consists of adding a state variable in the
analysis, e.g. the air pressure, and thus considering a three-phase mixture (solid,
water and air). The advantage, aside from greater rigor, is that we are freed from the
hypothesis of a static air phase ( u a = 0) and it allows the full potential of advanced
elastoplastic laws for non-saturated soils to be used.
Klubertanz [KLU 99] proposed a modeling of non-saturated slopes by
incorporating the three-phase nature of the soil, the coupling between the three
phases, a three-dimensional formulation, non-linear material laws for the solid
 
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