Environmental Engineering Reference
In-Depth Information
S gw between successive measurements of water-
content profile, Δ S gw , is equivalent to drainage,
D , for that period:
0
Day 0
1
Day 20
D
=−
S
=− −
(
S
S
) /(
tt
)
gw
gw
gw
(5.8)
2
il
il
il
i1
il
i1
ZFP
where il is an index of time, t ; ( t il - t i-1 ) is the
time interval between the two profile measure-
ments; and D and Δ S gw have dimensions of L/T.
In actuality, measurements of water-content
profile seldom extend all the way to the base
of the underlying aquifer; few studies even
attempt measurements down to the water
table. Most applications of the ZFP method rely
on water-content measurements to some fixed
depth within the unsaturated zone; the only
requirement regarding that depth, as discussed
in the following paragraphs, is that the inter-
val between the ZFP and that depth be of suf-
ficient thickness to fully capture both leading
and trailing edges of individual pulses of down-
ward-moving water.
For estimating evapotranspiration rates with
the ZFP method, the storage term of interest is
S uz , the amount of water stored between land
surface and the zero-flux plane:
3
4
5
6
0
0.1
0.2
0.3
0.4
Water content (m 3 /m 3 )
Figure 5.3 Example of water-content profiles obtained
over an interval of 20 days. The amount of water stored
between the zero-flux plane, ZFP (3 m depth), and the 6 m
depth at time t il is calculated from the formula:
5
S
= ×× +
0.5
z
{
θ
(
t
)
θ
(
t
)}
1
il
il
1
i+1
1
i = 0
where il is the depth index ( il = 0 at 3 m and il = 5 at 6 m),
Δ z il is the distance between measurement points (0.5 m),
and θ il is measured water content at depth il . At day 0,
950 mm of water was stored in this interval, and at day 20,
894 mm was stored. So over the 20 day period, 56 mm of
drainage is calculated. Similar calculations between land
surface and the ZFP indicate that 69 mm of water was lost
to evapotranspiration during the same period, assuming no
precipitation fell.
0
= θ
ZFP
(5.9)
S
dz
uz
z
The change in water storage in that depth inter-
val, Δ S uz , is determined as:
profile for measurements made 20 days apart.
By assuming that water content varies linearly
with depth between sampling points, the equa-
tion given in Figure 5.3 can be used to calculate
the amount of water stored between the ZFP
and the 6 m depth for each measurement date.
The drainage rate for the time period between
the two measurement dates, 2.8 mm/d, was
determined with Equation ( 5.8 ). The equation
given in Figure 5.3 is derived by applying the
trapezoid rule (Anton, 1984 ) for evaluation of
the integral in Equation ( 5.7 ). Analogous dis-
crete formulas can be derived for evaluation of
Equations ( 5.6 ) and ( 5.9 ). In the event of a shift
in depth of the ZFP between two profile-meas-
urement dates, the water storage change within
the interval between the two ZFP depths must
somehow be apportioned between drainage and
S
uz
= −
(
S
uz
S
uz
) /(
tt
)
(5.10)
il
il
i1
il
i1
A simple water-budget equation can then be
derived for estimating evapotranspiration, ET ,
over the measurement time interval:
=− −∆ uz
off
ET
P
R
S
(5.11)
where P is precipitation and irrigation, R off is
runoff, and it is assumed that all applied water
infiltrates, runs off, or is evaporated.
Calculating the amount of water stored
in a vertical interval of the unsaturated zone
(i.e. evaluating the integrals in Equations ( 5.6 ),
( 5.7 ), and ( 5.9 )) is straightforward, regardless of
the method used to measure water content, if
water-content measurements are available at
discrete depths. Fig u re 5.3 shows water contents
in a hypothetical 6 m thick unsaturated zone
 
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