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electromigration of the excess of charge contained in the
pore space of the material.
Another possibility is to write a generalized Richards
equation (Richards, 1931) that shows the influence of
the forcing term (we assume again that the air pressure
is constant). Starting with Equation (3.50) and replacing
the water pressure by the capillary head defined by
Ψ
α
b
Ψ
−
λ
,
Ψ≤
1
α
b
1,
s
e
=
3 55
Ψ
>1
α
b
∂
θ
∂
Ψ
α
b
Ψ
−
λ
+1
,
−
λα
b
ϕ
−
θ
r
Ψ≤
1
α
b
=
3 56
Ψ
>1
α
b
0,
2+3
λ
λ
e
=
−
2+3
λ
,
s
α
Ψ
Ψ≤
1
α
k
r
=
b
b
3 57
p
a
ρ
w
g (in m), the following generalized
Richards equation is obtained:
=
p
w
−
1,
Ψ
>1
α
b
θ
r
+
s
e
ϕ
−
θ
r
,
Ψ≤
1
α
b
θ
=
3 58
ρ
w
g
M
+
∂
θ
ϕ
,
Ψ
>1
α
b
Ψ
+
∇ −
K
h
∇ Ψ
+
z
∂
Ψ
3 51
Q
0
V
s
w
E
K
h
s
w
K
h
ρ
w
g
respectively, where
α
b
denotes the inverse of the capillary
entry pressure, which is related to the matric suction at
which pore fluid begins to leave a drying soil water sys-
tem. The pore size distribution index is represented by
=
∇
g
u
−
α
w
∇
u
−∇
C
e
∂
Ψ
∂
λ
t
+
∇ −
K
h
∇Ψ
+1
(a textural parameter), and
θ
r
represents the residual
3 52
Q
0
V
s
w
g
u
water content
. Sometimes, the residual water
saturation is not accounted for, and the capillary pressure
curve and the relative permeability are written as
θ
r
=
s
r
ϕ
=
−
α
∇
u
+
∇
K
h
−
ρ
w
g
E
w
s
w
K
h
=
k
r
k
0
ρ
w
g
η
w
=
k
r
K
s
3 53
−
λ
,
p
c
>
p
e
p
c
p
e
s
w
=
3 59
where
α
w
=
s
e
α
=
s
e
1
−
K
fr
K
S
and
K
=
K
fr
, denoting the
bulk modulus of the skeleton (drained bulk modulus).
Here,
K
h
denotes the hydraulic conductivity (in m s
−
1
),
K
s
denotes the hydraulic conductivity at saturation,
z
denotes the elevation above a datum, and
C
e
denotes the
specific storage term. This storage term is the sum of the
specific moisture capacity (in m
−
1
) (also called the water
capacity function) and the specific storage corresponding
to the poroelastic deformation of the material. This yields
1,
p
c
≤
p
e
2+3
λ
w
,
p
c
>
p
e
1,
s
k
r
=
3 60
p
c
≤
p
e
K
fr
K
S
, when the water satu-
ration reaches the irreducible water saturation, the two
source terms on the right-hand side of Equation (3.52)
are null. Therefore, there is no possible excitation below
the irreducible water saturation. In reality, this is not
necessarily true, and the model should be completed
by including film flow below the irreducible water
saturation.
The hydromechanical equations are defined in terms
of an effective stress tensor. As explained in details in Jar-
dani et al
.
(2010) and Revil and Jardani (2010), there is a
computational advantage in expressing the coupled
hydromechanical problem in terms of the fluid pressure
and displacement of the solid phase (4 unknowns in
total) rather than using the displacement of the solid
and filtration displacement (6 unknowns in total).
In saturated conditions, Newton
Because
α
w
=
s
e
α
=
s
e
1
−
C
e
=
∂
θ
∂
Ψ
+
ρ
w
g
M
3 54
Usually, in unsaturated conditions, the poroelastic
term is much smaller than the specific moisture capacity,
but the poroelastic term should not be neglected so that
we have a formulation that remains consistent with the
saturated state. The unsaturated hydraulic conductivity
is related to the relative permeability,
k
r
, and to
K
0
, the
hydraulic conductivity at saturation. With the Brooks
and Corey (1964) model, the porous material is saturated
when the fluid pressure reaches the atmospheric pres-
sure (
= 0 at the water table). The effective saturation,
the specific moisture capacity, the relative permeability,
and the water content are defined by
ψ
s law (which is a
momentum conservation equation for the skeleton,
partially filled with pore water) is written as
'
