Geoscience Reference
In-Depth Information
In light of (9.90), the rate of infiltration is
f
(
t
)
=
P
for
t
<
t
p
(9.93)
=
−
(
t
p
−
≥
f
(
t
)
f
c
(
t
t
cr
))
for
t
t
p
In Equations (9.92) and (9.93) the time to ponding and the compression reference time
remain to be determined. These two variables are related by (9.91); as this is one equa-
tion with two unknowns, additional information is needed to solve for
t
p
and
t
cr
. The
time to ponding is a real physical quantity, whereas the compression reference time
t
cr
is essentially a parameter, arising in the TCA approximation. There are two possible
procedures of estimating
t
cr
. In the first, it is obtained from the time to ponding; the
latter is estimated independently, from measurements or with expressions like Equations
(9.86) or (9.89). In the second procedure, use is made of the precipitation intensity to
solve for both
t
p
and
t
cr
by means of the TCA approximation.
Estimation with the correct time to ponding
In the first procedure the value of
t
p
is determined independently and with a known
or tolerable accuracy. The time to ponding can be measured directly, as in controlled
situations during irrigation, or by appropriate observations; also, as reviewed in the
previous section, there are reliable expressions available for this purpose, which are
based on the solution of Richards's equation (see Parlange and Smith, 1976; Broadbridge
and White, 1987). With
t
p
known, one obtains then as the inverse of Equation (9.91)
t
cr1
=
t
(
F
c
=
Pt
p
)
(9.94)
in which the subscript 1 indicates that
t
cr
is obtained by the first alternative procedure.
Note that with this procedure the infiltration rate has a discontinuity at
t
t
p
(see Figure
9.17). This is unavoidable, and is a result of the approximate nature of TCA. However,
the basic assumption of TCA, expressed in Equation (9.90), is satisfied.
=
Estimation from the precipitation intensity
In past applications of TCA, it has usually not been assumed that the time to ponding
t
p
can be determined independently. Rather,
t
p
has usually been estimated by assuming, that
under potential conditions
t
cr
is the time after the start of rainfall that would be required
to produce not only the same infiltrated volume, i.e. (
Pt
p
) as at the time of ponding, but
also the same infiltration rate, i.e.
P
. Thus one has the additional equation,
P
=
f
c
(
t
cr2
)
(9.95)
from which the compression reference time is obtained as the simple inverse function
t
cr2
=
P
); the subscript 2 indicates that it is estimated by this second alternative
procedure. The time to ponding can then be calculated by means of Equation (9.91) as
t
(
f
c
=
t
p
=
F
c
(
t
cr2
)
/
P
(9.96)
This procedure is illustrated in Figure 9.18.
In the real world, it is certainly the case that at the time of ponding both
F
=
Pt
p
and
f
=
P
must be satisfied. But it should be kept in mind that the TCA concept is only
