Environmental Engineering Reference
In-Depth Information
Comparing [9.21] and [9.22] shows that the relation between
K
l
and
k
is:
ρ
gk
l
−
8
k
K
=
−
≈
10
[mDarcy
]
[9.23]
l
η
If water content θ is preferred as the variable
10
governing liquid transfer,
because suction and water content gradients are related by:
∂
h
∇
h
=
∇
θ
[9.24]
∂
θ
putting [9.24] into [9.22] gives:
∂
h
J
=
−
ρ
K
∇
θ
=
−
ρ
D
θ
∇
θ
[9.25]
l
l
l
l
∂
θ
where
D
θ
[m
2
.s
-1
] represents the
diffusivity
of water under the effect of water content
gradients.
Equations [9.21], [9.22] and [9.25] are three different expressions of the same
liquid water flux, with three sets of variables and corresponding transfer coefficients
all related to capillary effects.
Gravity
effects should also be taken into account
when dealing with vertical infiltration (imbibition), for example. The corresponding
flux
J
lg
is given by:
J
=
±
ρ
K
u
[9.26]
lg
l
depending on the direction of the water transfer compared to vertical, indicated by
the
u
vector oriented upwards. Finally, the total water flux is the sum of
J
l
and
J
lg
.
9.3.3
. Water transfer in unsaturated porous materials
Darcy's law and other notions presented in the previous section were initially
developed considering
saturated
porous media. Of course, for a stone in a
monument this is not a normal situation. Fortunately, they have been generalized to
the unsaturated case [DAI 96] and [9.21], [9.22], [9.25] and [9.26] are still valid, but
10 This last variable rather than suction can be somewhat criticized because it does not have
the same thermodynamic value:
h
can actually be considered as a state variable but not
θ
which also depends on a material's properties as can be seen in equation [9.24]: the water-
content gradient is obtained by multiplying the suction gradient by
δθ/θh
, which is the
derivative of the retention curve.
