Environmental Engineering Reference
In-Depth Information
skeleton, and permeabilities to water and air as a function of the degree of
saturation. Some results are presented below.
3.4.3.
Three-phase modeling
The handling of three-phase mixtures has been proposed by several authors
[HAS 79, HUT 99, KLU 99, LEW 98] and only the broad outlines are covered here.
The conservation of the mass of the three phases is written:
(
)
∂−
1
n
ρ
s
⎡
(
)
s
si
⎤
+∇⋅ −
1
nv
ρ
=
0
⎣
⎦
∂
t
∂
nS
ρ
w
rw
⎡
⎤
+∇⋅
nS
ρ
v
=
0
rwi
⎣
⎦
[3.14]
∂
∂−
t
nS
(
)
1
ρ
r
a
a
⎡
(
)
⎤
+∇⋅
nS
1
−
ρ
v
=
0
r
ai
⎣
⎦
∂
t
with
n
the porosity and
ρ ,ρ ,ρ
the density of grains, water and air, reciprocally.
s
wa
The equilibrium equation (conservation of momentum) is written simply:
∇+ =
σρ
g
0
[3.15]
ij
i
with
σ
ij
the total stress given by equation [3.10]. The constitutive relations include
the compressibility of air and water, the constitutive law for the deformation of the
solid skeleton, the suction-saturation relation (equation [3.11]) and the law of
permeability (equation [3.12]). Using the example of the elastoplastic law developed
by Geiser [GEI 99], yield surfaces with multi-mechanisms are defined in the stress
space
p-q-s
.
To complete the equation system, it is accepted that the flow of water
and air is governed by a Darcy-type law. The resolution of the equation system calls
upon the finite elements [LEW 98, KLU 99] and on Galerkin-type discretizations in
space and time by the Θ-method.
3.4.4.
Applications
Small-scale tests were carried out by Klubertanz and Gachet [KLU 99] that made
it possible to bring a slope of non-saturated soil to failure while precisely measuring
the pressure and displacement fields. An example of a displacement field is given in
Figure 3.12 (using image-processing techniques). In this case, an initial flow is
imposed parallel to the slope and instability is caused by the introduction of
increasing pore-water pressure with increasing depth.
