Geoscience Reference
In-Depth Information
Fig. 9.13 Calculated
φ,χ
and
ψ
as
functions of water
content
θ
for Columbia
silt loam. The resulting
water content profiles
obtained with Equation
(9.65) are shown in
Figure 9.1. (After
Davidson
et al
., 1963.)
3
(cm min
−
1/2
)
φ
2
ψ
×
10
3
(cm min
−
3/2
)
2
1
(cm min
−
1
)
χ
×
10
0
0
θ
0.1
0.2
0.3
0.4
θ
0
0.5
i
Water content
problem; it has also stimulated subsequent advances in this field. This solution of (9.61),
which is equivalent to a perturbation series around the solution of (9.6) for a deep
homogeneous soil profile with the same boundary conditions, can be written as follows
z
t
1
/
2
t
3
/
2
t
2
=
φ
+
χ
+
ψ
+
ω
+···
t
(9.65)
where the functions
φ
=
φ
(
θ
) (see Equation (9.11)),
χ
=
χ
(
θ
),
ψ
=
ψ
(
θ
) and
ω
=
ω
), etc., are each governed by a separate ordinary differential equation, one of which
is (9.13) for
(
θ
φ
; for each of these equations Philip (1957b) presented a numerical method
of solution. The time expansion solution has been reported to produce good agreement
with experimental data in the laboratory by Davidson
et al.
(1963) (see Figure 9.1). The
functions
that were obtained in this calculation are shown in Figure 9.13.
Substitution of (9.65) into (9.63) produces the rate of infiltration
f
c
=
φ,χ
and
ψ
dF
c
/
dt
as follows
1
2
A
0
t
−
1
/
2
3
2
A
2
t
1
/
2
f
c
=
+
(
A
1
+
k
i
)
+
+
2
A
3
t
+···
(9.66)
θ
i
χ
θ
i
ψ
θ
0
θ
0
where
A
0
is
the
sorptivity
defined
in
(9.17),
and
A
1
=
d
θ,
A
2
=
d
θ
θ
i
ω
θ
0
and
A
3
=
d
θ.
The main shortcoming of any series solution like (9.65) is that eventually, i.e. for
large values of
t
, it fails to behave properly. This can be readily seen by comparing the
rate of infiltration obtained by using Equation (9.66) with the actual rate given by (9.4).
Although eventually the infiltration rate should approach the finite value
k
0
, the time
expansions (9.65) and (9.66) fail to converge for large values of
t
; therefore, they can
only be expected to be applicable for small and intermediate values of time.
9.3.3
A closed form of the series solution
Even though the series expansion (9.65) itself diverges, for some cases it can be shown
(Brutsaert, 1977) to lead to a formulation for infiltration which is well behaved both for
