Geoscience Reference
In-Depth Information
small and for large
t
. This formulation can be obtained by making use of an approximate -
but quite accurate - method for solving the differential equations for
.By
using the power-form functions (8.36) and (8.41), for the asymptotic case of very large
b
(which in (8.15) represents soils with a narrow pore size distribution) one finds that
the infiltration rate (9.66) can be approximated quite closely by
φ,χ,ψ
and
ω
1
2
A
0
t
−
1
/
2
(1
3
y
2
4
y
3
f
c
=
k
0
+
−
2
y
+
−
···
)
(9.67)
k
0
t
1
/
2
where
y
β
0
is a constant that depends
on the pore size distribution of the soil, which for most soils is of the order of 2/3. The
main point of interest in Equation (9.67) is that for
y
2
=
β
0
/
A
0
and
A
0
is the sorptivity, as before;
1 it can be expressed in closed
form as a two-parameter algebraic infiltration equation, viz.
<
1
2
A
0
t
−
1
/
2
[1
+
β
0
(
k
0
t
1
/
2
A
0
)]
−
2
f
c
=
k
0
+
/
(9.68)
which does not diverge for large
t
but instead tends to the proper limit
k
0
,as
required by (9.4); also, for small
t
, Equation (9.68) approaches the proper limit, viz.
f
c
=
f
c
=
2)
A
0
t
−
1
/
2
, as required by (9.19). This correct behavior at low and high values
of
t
is also an indication that (9.68) is relatively insensitive to the exact value of
(1
/
β
0
. The
cumulative infiltration corresponding to (9.68) is
A
0
β
0
k
0
{
+
β
0
(
k
0
t
1
/
2
A
0
)]
−
1
F
c
=
k
0
t
+
1
−
[1
/
}
(9.69)
For a more general comparison, it is again convenient to express these results in terms
of dimensionless variables; Equation (9.68) confirms the scaling already formulated
in (9.21) for horizontal infiltration; accordingly for infiltration capacity one has the
following
k
0
t
f
c
k
0
k
0
F
c
A
0
t
+
=
A
0
,
f
c
+
=
and
F
c
+
=
(9.70)
Thus the scaled rate of infiltration can be written as
1
2
t
−
1
/
2
+
β
0
t
1
/
2
]
−
2
f
c
+
=
1
+
[1
(9.71)
+
+
and the corresponding cumulative infiltration as
t
+
+
β
−
1
0
+
β
0
t
1
/
2
)
−
1
]
F
c
+
=
[1
−
(1
(9.72)
+
Equation (9.71) is illustrated in Figure 9.14, where it can also be compared with the time
expansion expression (9.67), with the short time expression (9.19), i.e.
f
c
+
=
t
−
1
/
2
+
/
2,
and with the long time expression (9.4), i.e.
f
c
+
=
1.
1or
t
+
<β
−
0
, suggests that the
“small to intermediate values of time” as mentioned behind (9.66), for which (9.65) is
valid, should satisfy at least
The convergence criterion for (9.67), that is
y
<
k
0
)
2
t
<
(1
.
5
A
0
/
(9.73)
