Geoscience Reference
In-Depth Information
9.3.1
Diffusion formulation of vertical infiltration of ponded water
The vertical downward movement of water into a dry soil, when the displaced air escapes
freely, is described by Equation (9.1), the one-dimensional version of Richards's (1931)
equation (8.55). Making use of Equation (8.32), one can express this as a diffusion
equation, or
D
w
∂θ
∂
∂θ
∂
∂
∂
−
∂
k
t
=
(9.61)
z
z
∂
z
To provide the maximal water supply rate at the point of entry
z
0, the soil surface is
considered to be covered by a thin layer of water, so that at the surface the soil water
pressure is atmospheric and the soil saturated. The initial water content is assumed to be
uniform throughout the profile. This situation is described by the boundary conditions
(9.2). Because the pressure
H
is eliminated in the diffusion formulation of the flow, these
are simply
=
θ
=
θ
i
z
>
0
t
=
0
(9.62)
θ
=
θ
0
z
=
0
t
≥
0
The solution of this problem is normally expressed as
θ
=
θ
(
z
,
t
). Once this solution is
known, it can be used in the form
z
t
) to obtain the cumulative infiltration volume.
As illustrated in Figure 9.4, this can be written as follows
=
z
(
θ,
θ
0
F
c
=
zd
θ
+
k
i
t
(9.63)
θ
i
where
k
i
is the capillary conductivity at
θ
=
θ
i
; the symbol
F
c
is given the subscript
c to indicate that it describes infiltration capacity. The second term on the right of
Equation (9.63) represents the downward motion of the water initially present in the
soil, under the influence of gravity; this is presumably negligibly small in most cases, if
the soil is initially dry enough. The infiltration rate can then be immediately calculated
as
f
c
=
dt
. Alternatively, as before, the infiltration rate can also be determined as
the Darcy flux at
z
dF
c
/
=
0, that is in its diffusion form,
k
D
w
∂θ
∂
f
c
=
−
z
+
(9.64)
z
=
0
For this problem, that is (9.61) subject to (9.62), numerous numerical methods of solution
have been presented in the literature. Again, however, in applications at the scale of a
catchment and of a region, it is often desirable to describe the phenomenon by a concise,
yet physically realistic, parameterization. Several such parameterizations are treated in
what follows.
9.3.2
Vertical infiltration as horizontal flow perturbed by gravity
The time expansion by Philip (1957b; 1969) is a method of solution that has received
much attention, as it was probably the first realistic attempt at solving the infiltration
