Geoscience Reference
In-Depth Information
capillarity totally dominate those of gravity. Second, it has been useful as an essential
part, or building block, of solutions for the later stages, obtained by various techniques.
9.2.1
Diffusion formulation of this horizontal flow process
Consider a uniform soil profile, which has an initial water content
θ
i
; the flow is allowed
to start, when the water content at the soil surface is suddenly increased to
>θ
i
). The
problem may be formulated by the one-dimensional form of Richards's (1931) equation
without gravity term, that is Equation (9.3); to indicate that it describes horizontal flow,
in what follows it is expressed in terms of the
x
-coordinate. Upon substitution of (8.32)
this can be written in the form of a diffusion equation as
θ
0
(
D
w
∂θ
∂
∂θ
∂
∂
∂
t
=
(9.6)
x
x
The boundary conditions are still (9.2); these may be recast in terms of the horizontal
coordinate as
θ
=
θ
i
x
>
0
t
=
0
(9.7)
θ
=
θ
0
x
=
0
t
≥
0
in which
θ
i
is the initial water content and
θ
0
is the water content maintained at the
surface
x
0, where the water enters the soil. A simple experimental setup to study this
problem is illustrated in Figure 9.2, and the experimental data that can be obtained with
it are illustrated in Figure 9.3.
In the formulation of infiltration problems it is often convenient to normalize the water
content, as follows
=
θ
−
θ
i
θ
0
−
θ
i
S
n
=
(9.8)
With this normalized water content the governing equation and boundary conditions
become
D
w
∂
∂
S
n
∂
∂
∂
S
n
∂
t
=
(9.9)
x
x
and
S
n
=
0
x
>
0
t
=
0
(9.10)
S
n
=
1
x
=
0
t
≥
0
Similarity approach
By the application of Boltzmann's (1894) transformation, which combines the space and
time variables into one independent variable,
xt
−
1
/
2
φ
=
(9.11)
