ῐ
J. Jpn. Soc. Soil Phys.
῏῍ῌῐῑ῎
No. 3/, p. 1/ῌ3* ῍,**-῎Current and Prospective Applicationsof Zero Flux Plane (ZFP) Method
Magdi KHALIL*, Masaru SAKAI*, Masaru MIZOGUCHI* and Tsuyoshi MIYAZAKI** University of Tokyo, Graduate School of Agricultural and Life Sciences, Department of Biologicaland Environmental Engineering, Yayoi +ῌ+ῌ+, Bunkyo-ku, Tokyo ++-ῌ20/1, Japan
Abstract
Zero ﬂux plane (ZFP) method has been used to measure and estimate the evaporation,
evapotranspiration, some hydrological process etc. However, no systematic discussion has been
made concerning the ZFP method itself or its application to monitor and control the salt andcontamination movement. In this study, comprehensive analysis on existing data sets have been
performed to examine ZFP method applications and to give comments for further applications infuture. First, ZFP method deﬁnition and its mathematical derivation were stated. Second, theapplication of ZFP to evaporation, evapotranspiration and ground water recharge process werediscussed and analyzed. The di$culties that a#ect the application of ZFP method such as depthof soil, preciseness of measurements etc. have been listed and compared. Finally the potentialities
of ZFP method were discussed. The discussion showed that, the development of new techniques
related to measurement instruments, required experiments and simulations to implement ZFP ine#ective way is very important to monitor and control salt transportation and accumulation
control.
Key words : zero ﬂux plane, unsaturated soil, Hydrus-,D, salt accumulation, soil contamination+.

Introduction

The importance of the soil-water system innature and in the life of man has been realized
since the dawn of civilization and man’s awakening awareness of his relationship to his environment (Hillel, +31+). The liquid phase of a soil
is never just pure water. There are always
mineral salts and organic substances dissolvedin the water. Generally, in spite of the importance of its solutes, the liquid phase of soil iscalled soil water. (Kutilek and Nielsen, +33.).
Most of the process involving soil-water interaction in the ﬁeld occurs while the soil is in an
unsaturated condition (Hillel, +332), or in otherwords, in vadose zone, (Miyazaki, +33-). Therefore the movement of water in unsaturated soil
is of a great importance where there are many
applications especially related to the hydrolog-

ical cycle and plant growth and stress. Suchknowledge allows an estimate of the inﬂuenceof soil conditions on plant growth, determination of the schedule of irrigation and drainage.
Prior to the recent application of soil physics
techniques in water balance studies, the di#erence between the rainfall and the evapotranspiration was often used to determine the moisture balances and deﬁcit, e.g. Gardiner (+320).
Errors arise in both measurements and estimates, and give rise to considerable uncertainty (Cooper et al., +33*). Measurements or estimation of water ﬂux in unsaturated soil is
impractical because of the wide range of hydraulic conductivity values found in soils, their
spatial variation and hysteresis (Cooper et al.,+33*). One of the prospective and promisingtechniques used to determine the moisture balances in unsaturated soil is ZFP method.

῏ΐῐ῎῔῕ῑ ῒ 3/ ῏ ῌ,**-῍

76

The purposes of this review paper are to
make clear the ZFP concept physically andmathematically, to compare the published dataof the ZFP, and to discuss the accuracies,
di$culties and potentialities of this method.
,.

ZFP Method,. + Deﬁnition
ZFP is deﬁned as a plane, which separates
two zones of upward and downward movement of water in a thoroughly wetted soil with
evaporation and drainage occurring simultaneously. Water moves in an upward direction above this plane, and downward below it,i.e., there is no ﬂow across the boundary
separating the two zones. With negligible lateral soil variation, the “zero ﬂux” boundary
may be assumed to be planar. As the soil dries
the plane of “zero ﬂux” moves downward (Aryaet al., +31/). In other words, when evaporation
exceeds rainfall, it is expected that water in theupper part of the soil proﬁle moves upwards
towards crop roots or the soil surface as exempliﬁed in Fig. +. Water in the lower partcontinues to drain water table. The point
where water is neither ﬂowing up nor down isFig. +

called the ZFP.
The old term used for ZFP was static zone. It
was ﬁrst mentioned by Richards (+3/.), with
Fig. , where he used a new instrument, multiple unit tensiometer, to measure the hydraulichead and the hydraulic gradient of water inunsaturated ﬁeld. He used the term static zone
in connection with the soil-water system todesignate the locus of points, above whichwater movement is upward, and below whichwater movement is downward. Due to theinﬂuence of the evaporation at the surface ofthe soil, this static zone passes downward
through soil following the wetting. Each curvein Fig. , represents the total potential distribution with depth on the days in October indicated by the number on the curve. The curveof Oct. +3 shows the total potential distribution
while water was still ponded on the soil surface. Between Oct. ,+ and ,, there was areversal in the hydraulic gradient in the +*ῌ,*
cm soil interval. Between Oct. ,- and ,., theﬂow reversed in the ,*ῌ-* cm depth interval.
On Oct. ,-, there was +.*1 cm rain. This caused
a marked shift in the total potential values. By
Oct. ,1, the static zone has passed below the ,*Zero ﬂux plane concept.

ΐῒ : Current and Prospective Applications of Zero Flux Plane (ZFP) Method

Fig. ,First use of ZFP method. (from Richards, +3/.)

cm depth. By Oct. ,3, the static zone has passed
below the -* cm depth. Some time before Nov.
++, upward ﬂow was established from depths
greater than /* cm. Therefore, the static zone
corresponds to the depth at which the hydraulic gradient is zero. (Richards et al, +3/0). In
ﬁeld studies, the depth within the soil proﬁle atwhich the water ﬂux density is zero is of major
importance and relatively easy to measure(Kutilek and Nielsen, +33.).
,. , Mathematical derivation of ZFP
MethodStammers et al. (+31-) described the approximation of mathematical derivation of the correct expression of ZFP depth as shown in Fig.
-. Both evaporation and drainage could be
calculated at above and below ZFP.
z*ῌt+῍z*ῌt,῍

**EῐR῎ῑqῌt+῍dz῏ῑqῌt,῍dzz*ῌt,῍῎+ῌqῌt ῍῎qῌt ῍῎dz
,
῍ +
῏
,ῑ
z ῌt ῍
*

ῌ

+

Z

Z

Dῐῑqῌt+῍dz῏ῑqῌt,῍dz
z*ῌt+῍z*ῌt,῍z*ῌt,῍+῏ ῑῌqῌt+῍῎qῌt,῍῎dz῏, z ῌt ῍
῍
*

77

+

where :
E is evaporation over the period t+ to t,
R is rainfall over the same period
t is time῍

z is depth measured positively downwards
Z is the depth at which drainage is calculated
D is drainage over the period t+ to t, throughdepth Zq is volumetric water content
zo(t) is the ZFP depth at time t.
From Fig. -, it is clear that, by decreases the
moisture content in the soil, the ZFP is gettingmuch deeper as well. The procedures to obtainequations (+) and (,) based on Fig. - is given in
Appendix attached.
-.

Application of ZFP to Evaporationand Evapotranspiration

In dry periods, when evapotranspiration exceeds rainfall, the soil water in the upper partof the soil proﬁle moves upwards towards theroot zone and the soil surface. At lower depth
soil water moves downwards towards thewater table due to gravity. Above ZFP, any
reduction in the soil moisture content must be
due to the moisture loss caused by an excess ofevapotranspiration, below the ZFP, and assuming no uptake of moisture content by roots atthese depths, reduction in moisture content
must be due to drainage out of the soil, i.e.
recharge to the ground water table.
The ZFP moves downwards as loss toevapotranspiration increase. In using the ZFP,
it is assumed (Cooper et al., +33*) that :ΐῐ῎῔῕ῑ ῒ 3/ ῏ ῌ,**-῍

78

Fig. -

Principle of the zero ﬂux plane method. (a) Hypothetical pressure potential proﬁles. (b) The
corresponding water content proﬁles.
The shading represents water that has evaporated or drained between t+ and t,. (fromCooper, +33*)+) Root extraction of soil moisture below
the ZFP is negligible, i.e. there is only drainage
below the ZFP,) Water inﬁltrating the soil surface moves
downward through the soil matrix ; if there ispercolation through macropores, such as
cracks or wormholes, evapotranspiration will
be overestimated and drainage will be underestimated-) There is no surface run-o#, if there is, it
should be taken into account.
Literature search revealed that most of authors have used ZFP method either to estimate
the evaporation, to calculate the drainage, or todi#erentiate between drainage and evaporation. (Van Bavel et al., +302 ; Stone et al., +31- ;Roman et al., +333). Therefore, we will discuss
two examples of estimating the short-term
evaporation and long-term evapotranspiration
by ZFP method in this section, followed byestimation of drainage in the next section.
Estimation of the evaporation in short termThony et al. (+313) estimated the evaporationfrom a +** m, plot of bare soil during a ﬁve-day
period of intense evaporation following heavy
rain, which had saturated the upper soil horizon. They evaluated the daily evaporation
rates for ﬁve days from ,/*2/11 to 1/*2/11, (a)by monitoring the moisture movement using
ZFP method and soil water balance methodand (b) by aerial measurements of the evaporative ﬂuxes into the atmosphere (energy balancemethod).
ZFP method has been used when ZFP existed. On the other hand, soil water balancemethod has been implemented during the absence of ZFP. The ﬁeld experiment is located in
Gernoble Mechanical Institute in France. The
soil proﬁle was as follows : from *.* to *.,/ mmuddy silt, from *.,/ to *./* m ﬁne sand andfrom *./* to +.* m clay soil. They used neutronprobe to measure water content and manometer tensiometers to measure the matric potentials. The total potential proﬁle and soil moisture content proﬁle during ﬁve days monitoring are shown in Fig. .. The total potential῍ῌ : Current and Prospective Applications of Zero Flux Plane (ZFP) Method

proﬁle of Aug. , shows that ZFP is blow ,* cm.
By Aug. 1, the ZFP has passed below the 0* cm
depth. The daily and cumulative evaporation
obtained by both ZFP and soil water balancemethods, and energy balance method werecompared in Table +.
They also studied the soil water dynamics inthe upper layer under natural cyclic evaporation conditions, it being found that the hydraulic head in this horizon is subject to periodic
variations. This phenomenon can be attributed principally to the cyclic nature of theevaporative process, which results in drying
up of the soil during the day followed by re-Fig. .

79

wetting at night as a result of internal redistribution. They concluded that the moisture
movement under these conditions is governed
principally by hydrodynamic e#ects, withthermal e#ects playing only a very minor role.
The results obtained by the two methods being
found in good agreement. Other literature concerning short-term evaporation were carriedout by Kalms et al. (+313) ; McGowan Williams(+32*) ; Inoue et al. (+32/) ; Yamamura et al.
(+322) ; Villegas and Morries (+33*) ; Payne etal. (+33*) ; Kanamori (+33/) ; Diez et al. (,***),
who all suggested the usefulness of ZFP
method to estimate short term evaporation.

The hydraulic potential proﬁle and soil moisture content proﬁle during ﬁve days monitoring. (from Thony, et al., +313)

Table +

Cumulative evaporation obtained by both ZFP and energybalance methods (from Thony et al., +313)

From

to

ZFP method andwater balance method

Energy balance method,ῌ*2ῌ11-ῌ*2ῌ11.ῌ*2ῌ11/ῌ*2ῌ110ῌ*2ῌ11-ῌ*2ῌ11.ῌ*2ῌ11/ῌ*2ῌ110ῌ*2ῌ111ῌ*2ῌ11-41/ mm-4,/ mm-4/* mm-4,/ mm-4,/ mm.4** mm.4,/ mm.4/* mm-4,/ mm-4/* mm

Sum +14** mm+34/* mmΐῐ῎῔῕ῑ ῒ 3/ ῏ ῌ,**-῍

80

Estimation of the evapotranspiration inlong term
Kirsch (+33-) used ZFP method for estimating the actual evapotranspiration (ET) in ﬁeld.
The experimental ﬁeld was located on the
grounds of the University of Illinois in U.S.A.
The soil at the site is silty clay loam. The
vegetation of the ﬁeld was fescue and red
clover. The rooting zone of the growing plants
extended to a depth of about .* cm. He used
about 0* tensiometers to monitor the total hydraulic potential, at depths of between 2 and+0/ cm. Neutron probe access tubes were used
to monitor the water moisture of the soil. He
carried out three ﬁeld experiments. First experiment was conducted for ,* days in Julyand August +321. The second experiment was
conducted for +, days in June and July. The
last one was conducted for +* days in October.
The movement of ZFP through the time during
the three experiments are shown in Fig. /. It isclear that the ZFP is getting deeper throughthe summer season and starts to be steady atthe beginning of winter. For the three ﬁeld
experiments, the errors ranged between +/-ῌ
for +-day ET estimates to .,ῌ for / days estimates. It is clear from the result that the
improper location of ZFP plays an importantrole to overestimation or underestimation ofwater content in the unsaturated area.
Other literature concerning long-term evapo
ration were done by Royer and Vachaud (+31.) ;Sharma (+32/) ; Dolman et al. (+322) ; Roman etal. (+333), who all obtained reasonable estimations of long-term evaporation by using ZFP

Fig. /

method.
..

Application of ZFP to Recharge Processes

For many dry climates, ground water is amajor source of water supply for irrigation,industrial, and domestic demands. Estimationof safe yield is an important basis for developing long-term management plans. This requires knowledge of recharge rates beneathvarious land use conditions, and their expectedvariability over a long periods.
Furthermore, the e$cient management ofground water resources requires knowledge ofthe quantity of water recharging the aquifer.
This allows safe and reliable pumping regimes
to be established without over-exploitation of
the aquifer or over-reduction of base ﬂows torivers fed by the aquifer. Ground water recharge can be easily estimated by ZFP method.
The reason for that is, below the maximum
depth reached by the ZFP, soil water ﬂow isalways downward ; hence, the water draining
through any depth below this will appear eventually as recharge at the water table. (e.g.
Cooper, +313 ; Sharma, +32/ ; Simmers, +321 ;Bouwer, +323).
The ZFP method is considered the mostdirect method for determining the recharge
rate of ground water. Therefore, several authors have used it to estimate the ground water
recharge (Royer and Vachaud, +31. ; Cooper,+32* ; Wellings, +32., Dreiss and Anderson, +32/ ;Healy, +323 ; Cooper et al., +33* ; Gardner et al.,+33* ; Sharma et al., +33+ ; Tang, +330 ; Ragab
et al., +331 ; Hosty and Mulqueen, +330 ; Ab-The movement of ZFP through time during the three experiments. (from Kirsch, +33-)῏῎ : Current and Prospective Applications of Zero Flux Plane (ZFP) Method

dallah et al., ,**+ ; Tsujimura et al., ,**+).
Table , gives the main features of groundwater recharge estimation done by some ofthese workers.
In the following section, a simple brief andanalysis will be mentioned about some of theseworks.
Sharma et al. (+33+) used the zero ﬂux plane
method and water balance method to estimate
the seasonal change in recharge to the under-

Year

81

laying sandy aquifer. They monitored down
the soil water dynamic of the unsaturated zone
to depth of ,* m of a period for three years.
They used neutron probes for moisture content
measurements. A simpliﬁed schematic of thehydrological cycle for the experiment is shownin Fig. 0. They deﬁned two references Z+ andZ,. The depth Z+ (῍+* m) is below the inﬂuenceof the root water uptake, and is the maximum
depth of the ZFP (Cooper, +313 ; Wellings andTable ,The main features of ground water recharge estimation done by some workers

Name &
location

Field &
soil surface

PurposeInstruments

Features
ZEP
Range

Exp.
TimingComments
Soil
proﬁle&Analysis+32*

CooperEnglandForested Evaluate ZFP N.P. & P.T.
and clearing as a tool tositesmeasure R
Sand lying
on Chalk

Max.
ZFP
-4, m

-.
months+32.

Wellings
EnglandGrass &
BarelySilty Clayloam lying
on Chalk

Estimate R
and E

N.P. & P.T.

-ῌ0 m

0 years.* m

ZFP is not valid
in winter time

+33* Cooper et al.
England

0 Grass
sites
Sand lying
on Chalk

Estimate R

N.P. & Ts.

Max.
ZFP
ῌ+40 m,ῌ/
years+*ῌ3* mE#ect of soil
properties on
E&R
is not clear+33+

SharmaAustralia

NativeWoodland&
Sandy soil

seasonalchange in R

N.P.

+* m

- years,* m

No matric
potential
measuredRun o# isneglected+330

M. Hosty
IrelandDry
E & R tograssland groundwater
Sandy loam

N.P. & Ts.

*42//. m P.M.
ῌ+4/ m months
W.T. . m

Lowering watertable hasno e#ec on ZFP,**+

Abdallah
Grass land
ET & R to
M.A. et al.
Aquifergroundwater
Switzerland Sandy loam

N.P. & Ts.

*4//ῌ*41 + year Empincal
methodW.T. ,41
P.M.
ῌ-4, mSnow coverproblem &Particle size
distribution
e#ect* N.P. Neutron probe* Ts Tensiometers* E Evaporation* ET Evapotranspiration

No independent
estimate ofdrainage* P.T. Pressures Transducers* R Recharge* PM. Penman Model  3/

82

Fig. 0,**-A simpliﬁed schematic representation of the hydrological balance used in the analysis.
Here Z+ῌ+* m and Z,ῌ+2 m are arbitrary depths. Z+ is the maximum depth for the ZFP andrecharge below this depth is termed R+.
Z, is an arbitrary depth just above the water table and recharge below this depth is termedR,. (Sharma, et al., +33+)

Bell, +32*). The depth Z, (+2 m) is an arbitrary
depth just above the water table. Recharge
below Z+ and Z, are termed R+ and R, respectively. Four pore holes were drilled and soil
water content down the proﬁle was measured
for each.
It is interesting that the authors did not estimate evapotranspiration in computing the recharge and did not use the directly- measured
soil matric potentials in the proﬁle.
Foridentifying the location of ZFP in the proﬁle,matric potentials are obtained inversely frommeasured soil water content. However theirscheme could be valid for reasonably deep unsaturated zone exists between the maximum
rooting depth and the water table.
Thechanges in soil water storage for this depth
interval are used at appropriate times to compute water ﬂux.
There is some uncertainty in the exact loca-

tion of the ZFP particularly when the matricpotential is calculated not measured. However,with the start of the rainy seasons, ZFP develops at the surface and the wetting front moves
downward and ﬁnally cancels the original ZFP
below. The soil water proﬁle for the hole 0 at
various time during an annual cycle is shownin Fig. 1.Total water potential proﬁles for thesame hole are shown in Fig. 2. Approximate
location of the ZFP is shown for each proﬁle. It
is interesting that ZFP could be last to over +*meters and not for few centimeters below thesoil surface as known. Therefore, ZFP is dynamic not static.
Hosty and Mulqueen (+330) studied the soil
moisture for a free-draining podzolic soil witha water table at . m below ground level.
Evaporatranspiration and drainage to groundwater were computed using both zero ﬂux
plane and water balance methods. They used῎῍ : Current and Prospective Applications of Zero Flux Plane (ZFP) Method

Fig. 1

Total water potential for the same period shown in Figure. (Sharma et al.,+33+)

both neutron probes and tensiometers to measure the moisture content and hydraulic potential respectively. Total potential of pore waterpressure over the May to September period of+33, have been shown in Fig. 3, and Fig. +*.
The ﬁeld result showed that a ZFP developedin early May as shown in Fig. 3 and moved
intermittently downwards to a maximum of +./m as the summer progressed. On ,* August,the ZFP was +./ m below ground surface. Because of -0 mm rainfall between ,* and ,1August, hydraulic potentials increased. As
heavy rainfall continued (23.+ mm between ,1
August to +1 September), the hydraulic potential proﬁle moved toward the gravitational potential line and there was evidence of temporary saturation at the + m depth (Fig. +*).
Pumping of the aquifer took place from ,/August to +/ September at an average discharge of +// m- hῌ+. This had the e#ect oflowering the water table at the measurement

Fig. 2

83

Soil water proﬁle for hole 0 at various
time during an annual cycle. (Sharma etal., +33+)

site from . m to 1./ m below ground surface. It
is important to mention that, lowering thewater table had no e#ect on the soil moisturein this study, where it (the water table) was
initially deep and well below the root zone.
Although pumping of the aquifer was on going
during this period, it is signiﬁcant that the soil
moisture proﬁle above the water table was determined by the balance between rainfall andevaporation.
/.

Application di$culties of ZFP method/. + ZFP depth
It is well known that the depth of ZFP is notstationary or ﬁxed, it moves up and down
during the year and sometime, it does not exist
at all. ZFP depth ranges from few centimeters
to few meters below the soil surface. The
accurate determination of ZFP in the soil is noteasy and need special care and sensitive instruments. The misestimating of the location ofΐῐ῎῔῕ῑ ῒ 3/ ῏ ῌ,**-῍84

Fig. 3Fig. +*

Downward progression of the ZFP. (from Hosty et al., +330)

Hydraulic potential proﬁles before and during pumping tests. (from Hosty et al., +330)

ZFP leads to misestimating of soil water storage. This in turn, causes a considerable error in
calculating both recharge ground water andevaporation rate.
/. , Root zone
For its success, the ZFP method requires that
there be no extraction of water by roots beneath the ZFP since all losses of water in thisregion are assumed to be due to drainage andall losses of water above ZFP assumed to be
due to evaporation as well. Given that someroots are known to penetrate to at least 0 m,
this is unlikely to be strictly fulﬁlled.
However, and for drainage estimation, thefollowing points make it probable that extraction from below the ZFP is at most relatively῔ΐ : Current and Prospective Applications of Zero Flux Plane (ZFP) Method

small (Cooper, +313). :(a) Although no quantitative information isavailable on the distribution of roots, within
the proﬁle, the overwhelming majority has
been observed within the upper meter of soil.
Provided that water is fairly readily available,
it is expected that these roots will extract mostof the water with only a tiny proportion
coming from lower in the proﬁle.
(b) As the ZFP gets deeper, the fraction ofroots below it becomes steadily less.
(c) There is, at all times, at least one meterof soil above the ZFP at a tension of less than
one bar (+** kPa). Drying within the upper , m,where potentials reach several bars, proceeds
quite steadily to the end of the experiment, sothat water must be available even at thesetensions.
Moreover, and related to evaporation estimation, ZFP method gives a good agreement withtranspiration calculations based on meteorological data, when it was known that roots
penetrated below the ZFP, supporting the viewthat the root extraction from below the ZFP
does not introduce a major error in using themethod (Dolman et al. +322).
/. - Non-existence of ZFPZFP method is useful when accurate measurements at short time scales are required, andwhen frequent measurements can be made.
However, the method failed to estimate the soil
water during periods when rainfall exceeds
saturated hydraulic conductivity Ksat. In
other word, if ZFP is at land surface (no ZFP
existed), the method cannot be used, as only
downward movement throughout unsaturated
zone occurred.
The biggest limitation of the ZFP method isthat it can be only used under conditions
where a ZFP exists specially in dry season. In
wet areas, this may be infrequently. However,in many drier areas, ZFP may be observed forwell over half of the year. When ZFP cannot be
observed, the meteorological estimates of evaporation, although not totally satisfactory, isreasonable and drainage estimated during

85

these periods, for any depth below the groundsurface, may then be calculated from a simple
water balance of the soil proﬁle (Stammer etal., +31- ; Arya et al., +31/ ; Cooper, +313, +32* ;Cooper et al., +33* ; Kirsch, +33-) by using theequation.
z

DῑRῐEῐῒ῎qῌt+῍ῐqῌt,῍῏dz*/. . Preciseness of measurements
Although, ZFP method is based on measurements made directly in the controlling
medium, the soil, special care should be given
to the instruments in the ﬁeld. Therefore, highdensity of instruments and high frequency ofmeasurements needed to get relatively accurate results.
Sometimes errors are associated with moisture content measurements (Van Hylckama,+32*). This could be large relative to thechange of moisture content between successive
measurements ; therefore, this method is usually applied over a period of at least + week.
Using neutron probes near the soil surface
may lead to incorrect measurements, therefore,
accurate, sensitive tools should be used. As
shown in table ,, most of authors used neutronprobes to measure the moisture content. The
sources of error in soil water measurementswith a neutron probe can be summarized asfollow (McGowan et al., +32*) :a) Systematic errors that occurred as aresult of calibration, soil damage from accesstube installation and damage to surface soil
and vegetation.
b) Random errors which, happened becauseof random count error, relocation error andinherent soil variability errors as well.
Time Domain Reﬂectometry, TDR, may be
used more widely hitherto, because its preciseness and convenience to monitor both moisture content distribution and unsaturated soil
water movements.
Careful measurements of matric potential
and proper installation of tensiometers are crucial and important to detect ZFP in preciseway.

ΐῐ῎῔῕ῑ ῒ 3/ ῏ ῌ,**-῍86/. / Hysteresis
Almost all the soil moisture characteristic
curves show hysteresis due to the ink bottle
e#ect. Since the sizes of soil pores change
successively, the existing of the ink bottle
e#ect is generally recognized (Miyazaki, +33-).
Care must be exercised in incorporating thee#ect of hysteresis, by using the appropriate
part of sorption or desorption relationship. In
most cases where the e#ective rainy seasons isdistinct (Wellings and Bell +32*) the desorption
curve could be used without introducing mucherror (Sharma et al., +33+).
0.

Potentialities of ZFP method

Although many researchers used ZFP to estimate the water ﬂux in soil to measure recharge
to ground water, to compute drainage orevapotranspiration, and to calculate unsaturated hydraulic properties, but there was no study
that focused on the ZFP phenomena itself.
Since ZFP is governed with many parameters,
the studying and analyzing these factors could
contribute much to implement several new
aspects of applications. The developing of ZFP
could be achieved by improving measurements
tools and technique, implementing new experiments in laboratory to monitor and study ZFP
and using numerical model and simulation foranalysis as well.
Measurement
Although it could be concluded that ZFP is
uncertain for some ﬁeld uses, the errors
associated with estimation of soil moisture
content may be minimized through use of moreaccurate methods for measuring water content
or increasing the number of neutron probe
access tubes sampled at the ﬁeld both action
would likely make estimation of ET more reliable.
More accurate methods of quantifying soil
water content may include, for example, timedomain reﬂectometry. Errors associated improperly location of ZFP may be minimized byusing time-domain reﬂectometry method.
There is no doubt the fact that accurate deﬁni-tion of the total hydraulic head distribution
with the depth, making identiﬁcation of the
ZFP more precise.
Laboratory Experiments
Careful experiments should be carried out inthe laboratory to monitor and observe the ZFP
in soil column. The factors a#ecting the ZFPmovement will be tested and analyzed. Salt
movement and accumulation in soil should be
deeply investigated using di#erent types ofsoils.
Numerical model
One of the powerful tools, which may help
much in dealing with ZFP phenomenon, is numerical model. HYDRUS-,D model is used for
simulation of water ﬂow and solute transport
in two-dimensional variably saturated media.
The program numerically solves the Richards’
equation for saturated-unsaturated water ﬂow
and the Fickian-based advection-dispersion
equation for solute transport. Using HYDRUS,D model could give good assistance to analyze
the experimental results and to simulate thecomplex problems. For example, Hydrus-,D
could simulate di#erent topographical problems, such as sloped soil, which are di$cult tobe simulated at the laboratory.
Salt transportation and accumulation
In fact, the maintaining of ZFP higher than
the salt accumulated subsoil, nuclear disposals,
contaminated layers.... etc., is practically very
important. Therefore the control of ZFP in
unsaturated soil is of a great interest where
there are many applications especially related
to the environmental, agricultural and civil engineering. In addition to hydrological ﬁelds.
Such knowledge allows us to select the suitable
depths in the soil to bury the contaminateddisposals in safe way. This knowledge is also
important to determine when to irrigate, tocalculate the amount of water to apply, and to
estimate drainage below the root zone. Furthermore, preserving ZFP above the salt accumulated subsoil is very important to agricultural ﬁeld. Selecting the convenient depths
into the soil to establish an oil pipes network is῍ῌ : Current and Prospective Applications of Zero Flux Plane (ZFP) Method

a#ected as well by the ZFP. Protecting the
ground water from contamination will be achieved partially by ZFP control.
References
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῕῔ : Current and Prospective Applications of Zero Flux Plane (ZFP) Methodor ;Appendix
Water balance equation above ZFP can be
written as :RῑE῏ΐqῌ
where :
R is the rainfall over the period t+ to t,,
E is the evaporation over the period t+ to t,,
ΐq is the volumetric water content change
above ZFP.
Since the decrease of water content (q) between z* (t+) and z* (t,) due to evaporation isz*῍t,῎approximated by+ῐq῍t ῎ῐq῍t ῎ῒdz
+
ῑ ,
ΐ
,ῒ
z ῍t ῎*+Volumetric water content change ΐq above
ZFP is given byz*῍t+῎z*῍t+῎**ΐqῑῒq῍t,῎dzῐῒq῍t+῎dz+
ῐq῍t ῎ῐq῍t ῎ῒdz
+
ῑ ,
ΐ
,ῒ
z ῍t ῎*z*῍t,῎z*῍t+῎ῐῒq῍t+῎dz῏
*

῏

Z

ῐ

z*῍t,῎z*῍t,῎

*

*

*

*

+

Z

z*῍t+῎z*῍t,῎

**

DῑRῐRῐῒq῍t+῎dz῏ῒq῍t,῎dzz*῍t,῎ῐ++ῐq῍t ῎῏q῍t ῎ῒdz
,
ῑ +
ΐ
,ῒ
z ῍t ῎*ῑῐῒq῍t+῎dz῏ῒq῍t,῎dz+z*῍t,῎Zῐῌq῍t,῎dz῏ῒ῍q῍t,῎῎dz῎῍ῒ῏*z ῍t ῎z*῍t,῎*+ῐ ῒῐq῍t+῎῏q῍t,῎ῒdzῌΐ, z ῍t ῑ῎*

z*῍t,῎ῐῒῌq῍T,῎῎dz῏ῒ῍q῍t+῎῎dz
῍
῏
*
*

+

z*῍t,῎z*῍t+῎z*῍t+῎Z+ῐῐῒq῍t,῎dz ῐq῍t,῎ῐq῍t+῎ῒ
ῑ ῒ
ΐdzΐ
,ῑῒ
z ῍t ῎z ῍t ῎*Z+ῐq῍t ῎῏q῍t ῎ῒdz
,
ῑ +
ΐ
,ῒ
z ῍t ῎**+Zz*῍t,῎ῑῐῒq῍t+῎dz῏ῒq῍t,῎dz*+

RῑE῏D῏ῒῌq῍t,῎ῐq῍t+῎῎dz῍῏*+ῐq῍t ῎ῐq῍t ῎ῒdz
+
ῑ ,
ΐ
,ῒ
z ῍t ῎z*῍t,῎+ῐq῍t ῎῏q῍t ῎ῒdz
,
ῑ +
ΐ
,ῒ
z ῍t ῎*z*῍t,῎**

Since water balance equation for whole soil
proﬁle isRῑE῏D῏ΐq῎where :
R is the rainfall over the period t+ to t,,
E is the evaporation over the period t+ to t,,
D is the drainage over the period t+ to t,,
ΐq is the volumetric water content change
from zero to arbitral depth Z.

+*z*῍t+῎*z*῍t,῎ῑῐq῍t,῎dzῐῒq῍t,῎dzῒῑῒΐ*z ῍t ῎*z*῍t,῎

DῑRῐRῐῒq῍t+῎dz῏ῒq῍t,῎dz+z*῍t,῎z*῍t+῎

EῑR῏ῒq῍t+῎dzῐῒq῍t,῎dz

DῑRῐEῐῒῌq῍t,῎dz῏ῒ῍q῍t+῎῎dz ῏῍῏**z*῍t,῎῏

z*῍t+῎῍*

z*῍t+῎z*῍t,῎

**RῑEῐῒq῍t+῎dz῏ῒq῍t,῎dzz*῍t,῎ῐ+ῐq῍t ῎῏q῍t ῎ῒdz
,
ῑ +
ΐ
,ῒ
z ῍t ῎
*

+

,

Z

῏ῌ
q῍t+῎dz῏ῒ῍q῍t+῎῎dz῎῍ῒ῏*z ῍t ῎+Substitution of (,) into (+) gives

89

Z

+

Z

Dῑῒῌq῍t+῎῎dz῎ῐῒ῍q῍t,῎῎dz῎῍῏z ῍t ῎z ῍t ῎*+*z*῍t,῎,+ῐ ῒῐq῍t+῎῏q῍t,῎ῒdzῌΐ, z ῍t ῑ῎*+124c~ ® 3/ ¯ ,,**--

90 ZFPῌ*ῌ
*ῌ῍

*ῌ ῌ

*

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