Environmental Engineering Reference
In-Depth Information
r
σ
2
2
2
∆
m
(
t
)
=
ρ
r
z
(
t
)
=
ρ
r
t
=
π
r
A
t
[9.28]
l
l
2
η
which shows a √t dependency. Nevertheless, this highly simplified mechanical
model is clearly far from translating the complexity of infiltration. To improve this
approach, let us first formulate the differential equation for water transfer in porous
media. First, a conservation equation for water fluxes can be written:
∂
θ
1
∂
J
l
=
−
[9.29]
∂
t
l
∂
z
ρ
[9.29] simply translates the fact that the difference between the fluxes entering -
or leaving - a volume element of width ∂z is what is stored in - or released from -
it. Then, combining [9.29] and the liquid flux expressions [9.22] and [9.26], the
well-known Richards equation (1931) is obtained:
⎡
⎤
∂
θ
∂
θ
∂
h
∂
h
∂
⎛
∂
h
⎞
=
=
C
=
K
+
1
[9.30]
⎜
⎝
⎟
⎠
⎢
⎣
⎥
⎦
l
∂
t
∂
h
∂
t
∂
t
∂
z
∂
z
where
C
[m
-1
] is the
hydraulic capacity
(inverse of the derivative of the retention
curve). If [9.25] instead of [9.22] is combined with [9.29] and [9.26], the following
“Fokker-Planck”-type equation is obtained:
∂
θ
∂
⎡
∂
θ
⎤
=
D
+
K
[9.31]
⎢
⎣
⎥
⎦
θ
l
∂
t
∂
z
∂
z
[9.30] and [9.31] are two basic formulations of water transfer under the
combined effects of capillarity and gravity. For the case of the capillary absorption
test considered, the effect of gravity is negligible compared to capillarity and [9.31]
simplifies to:
∂
θ
∂
⎡
∂
θ
⎤
=
D
[9.32]
⎢
⎣
⎥
⎦
θ
∂
t
∂
z
∂
z
which is a “classical” partial differential equation (diffusion parabolic-type
equation). This is strongly non-linear because of the strong dependency of
D
θ
on θ.
Its resolution requires adapted numerical techniques [LAU 96b]. In most cases,
[9.22] does not have any analytical solution but can be simplified using
Boltzmann's variable:
