WATERRESOURCES
RESEARCH,
VOL.28,NO.9, PAGES
2513-2521,
SEPTEMBER
1992
Fingered Flow in Two Dimensions
1. Measurement of Matric Potential
J. $ELKER
Department
ofBioresource
Engineering,
Oregon
StateUniversity,
Corvallis
P. LECLERQ,
J.-Y. PARLANGE,
ANDT. $TEENHUi$
Department
of Agricultural
andBiological
Engineering,
CornellUniversity,
Ithaca,New York
Precisemanagement
of the changing
matricpotentialduringinfiltration
into unsaturated
soil
requiredthe development
of miniature,high-speed,
planartensiometers.
A novel designwas
developed,
withresponse
timeof lessthan1 s. The applicability
of the devices
is shownthrough
measurements
of thematricpotential
ingrowing
instabilities,
bothin theinduction
zoneandalongthe
verticalfingerprofile.Tensiometry
is demonstrated
to be a practicalmethodof obtainingdatawith
hightemporaland spatialresolution
for the studyof dynamicflowfieldsandfacilitatestestingof
theoretical results for unstable flow fields.
INTRODUCTION
Unstablewetting fronts have been documentedwidely in
fieldand laboratorysettings[e.g., Saffmanand Taylor, 1958;
Hill and Parlange, 1972; Starr et al., 1978; Diment and
Watson, 1985;Glass et al., !989; Baker and Hillel, 1990;Van
Oremenet al., 1989].Wettingfront instabilitygenerallyleads
to the developmentof fingeredflow, which can significantly
increase the transport velocity of pollutants to aquifers
throughthe vadose zone [Glass et al., 1988; Kung, !988].
This has practical implicationsdue to the increaseduse of
agriculturalchemicalswhich rely on residencein the upper
soilprofile for degradation.In addition, fingeredflow may
give rise to increased recharge of groundwater in arid
environments.
of the matricpotentialin fine materialabovea coarsesoil
with an unstable wetting front were reported [Baker and
Hillel, 1990],althoughto datewe are not awareof measurementsof the matricpotentialtakenwithingrowinginstabilities.Thispaperinvestigates
the designof tensiometers
for
measurement
of the matricpotentialof growinginstabilities.
Twoof the primaryobjectivesof this studyare to facilitate
the observationof the horizontalgradientin the induction
zonebetweenfingersandto characterize
thepressure
profile
withingrowingfingers.
Copyright1992by the AmericanGeophysical
Union.
DESIGN AND TESTING OF PLANAR TENSIOMETERS
Design and Testing Considerations
In two-dimensionalexperiments,measurementsof unstable wetting fronts in porous media have been made by
making direct visual observation of the flow field through
transparentpanels [e.g., Hill and Parlange, 1972; Glass et
al., 1989]. Considerable insight has been gained through
theseinvestigationsregardingthe geometryof unstableflow
fields. Glass et al. [1990] showed that the diameter of
three-dimensionalinstabilities has the same dependenceon
soilpropertiesas in two dimensions,beinga factor of 4.8/•r
largerthan for two-dimensional instabilities.
The suggestionof using tensiometersto gain further understandingof unstable wetting fronts has been made by
otherauthors[e.g., Raats, 1973]. Recently, measurements
Papernumber92WR00963.
0043-1397/92/92WR-00963505.00
This papertreatstwo topics. The first topic is the fabrication and testing of the tensiometersrequired for measurement of matric potential in dynamic unsaturated flow fields.
The secondtopic is an assessmentof the devicesemployed
in the measurementsof mattic potential made in developing
unstable wetting fronts. A companion paper [Selker et al.,
this issue] addressesthe application of tensiometric measurementsin verifying an analytic expressionfor the unsaturated conductivity and vertical pressure profile of growing
fingers.
The smallest feature of the unstable flow field is often the
minimum depth of the induction zone (the wetted region
prior to fingered flow [see Hill and Parlange, 1972]). In
previous similar experiments the induction zone has been
from 1.0 to 2.5 cm in depth [Selker, 1991], indicating
tensiometers
should have a width of 1.0 cm or less to allow
representation of the variability in pressure within this
region. Required tensiometer response time is dictated by
the speed of growth in the unstable flow, which in similar
experiments
hasbeen0.4 cm s-• [Selker,1991].Takinga
characteristic dimension of 1.0 cm, dimensional analysis
indicates a required temporal resolution of approximately
2s.
To calculate the in situ responsetime of a tensiometer, it
is necessary to determine whether the system is soil- or
tensiometer-limited [Townet, 1980]. Towner developed a
methodology for determining the rate-limiting factor for
sphericaltensiometers.To use Towner's model, four quantities must be known' C, the conductance of the cup (meters
squaredper second);S, the tensiometer's gauge sensitivity
(m-2);r, theradiusofthecup(meters);andK, thehydraulic
conductivity of the media (meters per second), which is a
function of moisturecontent and thus mattic potential. In the
present study we employ flat circular porous plates rather
than the sphericalgeometry considered by Townet. Towner's spherical result may be adjusted accordingly for the
2513
2514
SELKER ET AL.' FINGERED FLOW IN Two DIMENSIONS, 1
circular geometry, resulting in a factor of 4 decrease in
Column
Wall Silica
Sand
surface
areafora givenradius
device.
Theadjusted
formof
Towher's
[1981]relation
between
system
parameters
and
Approxin,ato
Scale
| 1 cm |
I
/
.......
.v.............
tensiometer
time
response
risgiven
by
r
CS •rSK
Porous
Stainless
Steel_
Boss
I
The
contribution
first
term
of
on
the
right-hand
tensiometer
side
to
of
the
{1)response
corresponds
time,
tothe
asAnalog Jam
Nut
• % 1
Output
De-aired
Water.•%%%•
••!
J•::•::•::•::•::
i.........................
..........
the flux limitation contribution of the soil. In five experiments, Townet found that (1) underestimated tensiometer
underprediction of 25% of response time IToh'her, 1981J.
Apractical
method
for
measuring
the
response
time
of ]
oped for a tensiometer submersed in a sinusoidally varying
pressure
field.Consider
anexternal
pressure
fieldXPex,
with
amplitude
Aand
frequency
w:
•e,r= 0
½,,.,=A
t< 0
sin wt
(2)
•
o
Transducer
arass
•
oustng
•½•;•:•½•
O-Ring
Seal
Fig.I. Cross
section
ofamounted
tensiometer.
t•0
For t >> r the pressurewithin a tensiometerxVt,.,,is given by
which yields the minimal hydraulic path connection. The
outer surl':accof the tensiometer body is threaded to fftcilitate
rigid placement of the device in the experimental chamberin
an acrylic boss bonded to the chamber with epoxy. An
= I + to2r 2 [sin tot- to, cos tot]
½3)
xlIten
O-ring groove is made in the exterior of the body of the
tensiometcr to provide a watertight seal to the chamber,
By considering the ratio of the peak amplitude of the precluding water loss in case of positive matric potential
[Khtte and Gardner, 1962' Watson and Jacl, son, 1967]
imposedpressurefield, max(q•,..,) = A, to the peak amplitude of the pressurewithin the tensiometer.max(q•t,,,), we
obtain a system attenuation 3' which may be related directly
conditions.
Two variants of this design were produced, sharing the
structural features noted above. Test results for both designs
to the tensiometer response time:
are presented to illustrate the importance of construction
technique in final device performance. In the first designthe
max (q• t,.,,) max (xP'ten)
porous stainless plate was cut using a circular die. The edges
3' =
=
(4)
max (•,.,)
A
of the disk were then beveled to 20ø, and the plate was
An expression for the peak pressure in the transducer is surface mounted in the brass body by crimping the brass
obtainedby settingthe first derivative of xVt,,,,with respect over the edge of the plate with a thin coating of 24-hour
epoxy at the brass-stainless interface. In the second design
to time equal to zero, showingthat •t,,,, is maximizedwhen
t = (-l/to) tan -! (l/tot) + nrr/to, for n = 0, I, 2,--' in the stainlessplate was cut with the 20ø bevel usinga vertical
milling machine, eliminating the compression of the porous
steady state conditions. The attenuation is then given by
plate required in the die cutting operation. In addition, the
plate was sealed to the bronze by a fine gauge O-ring
' 2 {sin[tan-• (l/tor)]
compressedbetween the back of the plate and the housing,
'Y= l + w-r
rather than by an adhesive (Figure 2). The O-ring seal was
adopted
to avoid loss of porosity of the stainless face plate
+ tot cos[tan-I (1/tot)]}
(5)
due to fluid migration of the epoxy resin. In addition to
By performingan experiment in which a sinusoidalpressure improvingthe average responsetime, it was anticipatedthat
is applied with frequency to and measuring3', (5) may be these changes would reduce variability among tensiometers.
, ]
solved numerically for the responsetime r.
Solid state pressuretransducershave been shownto give
fast and accurate measurements of tension in both field and
Tensiorneter
Constrltction
The tensiometersdesignedfor theseexperimentsemploya
porous stainless steel front plate, a bronze housing, and a
solid state pressuretransducer (Figure I). The front plate,
made from 20-/xm Mott Metallurgical Corporation flitted
stainless plate, is mechanically fastened to the tensiometer
body by rolling the lip of the bronze over the rim of the plate
in a lathe operation. The pressure transducer port is
threaded and fastened directly to the tensiometer body,
laboratory experiments [Klute and Peters, 1962; Watson,
1965;Long, 1984;Morrisonand Szecsody,1987;Dowdand
Williams, 1989]. In this work, pressureis measuredusing
Motorola MPX2010DP
temperature-compensated pressure
sensors,selectedon the basisof hysteresis(0.05% of full
scale), response time (1.0 ms), and range of pressure measurement (I 0 kPa).
The sinusoidal pressure field required for the measure-
mentof the tensiometerresponsetime was generatedusinga
modifiedperistalticpumpwhich was fed into a sealed10 x
SELKER
ETAL.' FINGERED
FLOWINTwoDIMENSIONS,
I
2515
0.1
Seal
% [-'-'-"•'"
""•-••;.i
":"
'' 1 Steel
0.01
]
Rolled
Lip 0.001
I
0.0001
Brass
0.001
I
0.01
I
0.1
K (cm/s)
Tensiometer
Housing
Fig. 4. Responsetime of the tensiometers as a function of soil
conductivity obtained using(I). The response time of the devices is
largelyindependentof conductivityto about 0.1 Ksat for 40-50 sand.
Scale
t
Meta!lurgical
to be 4.1 x 10-2 m s-t. The bubblingpres-
I mm
,
,,
Fig. 2. Cross section of prototype tensiometers employing an
O-ring seal in place of liquid adhesives.
30 cm cylindrical surge tank, which smoothed the alternating
pressureand suction generated by the pump. The outflow of
the surge tank was then attached to a 12-port testing reservoir (Figure 3). The pressure was monitored by a Motorola
MPX2010DP pressure sensor,following a 1.87-Hz sinusoidal
signal,far slower than the 1-msresponsetime of the sensor.
Signalsfrom the tensiometers and pressure monitor were
amplified,and peak amplitudes were measureddirectly from
an oscilloscope.
TensiometerTime Response Resttits
Time response results are first calculated using (1) and
then compared to the experimentally obtained values obtainedthroughapplicationof (5). The valuesof parametersin
(1) are computed from the geometry of the tensiometers,
withdata from componentmanufacturers.The conductivity
of the 20-g porous stainlesssteel is reported by Mott
Pressure
Monitor
_
Oscilloscope
Tensiometers
2A¾
I
II
_•
I Peristaltic
I se'•ø•ø•
....
I
sures of the tensiometers ranged from 70 to 180 cm, with a
mean of 111cm, indicating that the largest pore sizes ranged
from 8 to 20 /•m. The volumetric displacement of the
pressure
transducer
isrevorted
by Motorolato be 1.6x 10-8
m3, andareaof the porousplateis 3.8 x 10-5 m2. The
conductance C of the device is calculated as the product of
the porous plate conductivity and surface area divided by the
applied pressure head, here taken as 1 m of water, which
yieldsC = 1.6 x 10-6 m2 s-!. The gaugesensitivity
is the
TABLE
I.
Result of Sinusoidally Driven Tensiometer Response
Time Testing
Response
Tensiometer
1
2
3
4
6
7
8
9
I0
II
Pl
P2
P3
P4
P5
P6
P7
P8
P9
PlO
Pll
PI2
PI3
PI4
Attenuation
Time, s
0.32
0.0098
0.25
0.16
0.12
0.11
0.26
0.0019
0.28
0.29
0.48
0.51
0.63
0.42
0.63
0.40
0.40
0.37
0.83
0.50
0.63
0.53
0.50
0.18
0.26
9.1
0.33
0.54
0.68
0.79
0.31
46
0.61
0.28
0.16
o. 14
0.10
0.19
0.11
0.20
0.20
0.22
0.058
0.15
0.11
0.14
0.15
0.47
The attenuation between the surrounding pressure field and
pressuremeasuredwithin eachtensiometcr,with a 1.87Hz signal,is
listed along with the responsetime, as calculated from {5). Tensiometers with prefixes of '•P" are the prototypes which incorporate
Fig.3. Experimental
setupforbenchtestof tensiometer
response the improved porousplate cutting and mountingmethodsdiscussed
above, tested using a 1.82-Hz signal.
time usingsinusoidalpressurefield.
L•Pump
2516
SELKERET AL.' FINGEREDFLOWiN Two DIMENSIONS,1
responsetime illustratesthe need for as-built measurement
of tensiometer responsecharacteristics.
52.5 cm
1234
3.25 cm
Eliminating
the two slowdevices(tensiometers
2 and9),
567
5 crn
1.5 crn
8
the average epoxy-mounted tensiometer is 2.8 times slower
than the average O-ring designdevice of 0.17 s, which in turn
hasresponsetime 17 times the 0.0104 s predictedby (1). The
error of (1) for these devices is somewhat larger than that
foundby Towner[1981]but is consistent
in underpredicting
the responsetime. The standard deviation in responsetime
between the epoxy-mounted tensiometers is 37 times that of
the O-ring design for the full set, or 15 times if the two
35 cm
slowestepoxy-mounted
devicesarenotincluded.TheO-ting
designprovides a responsetime which is consistentlyless
than 0.5 s. Both designs satisfy the time requirementsof
measuringthe matric potential in growing instabilities,which
is demonstratedin the second section of this paper.
Air
MATRIC POTENTIAL IN GROWING INSTABILITIES
Materials
and Methods
Measurements of matric potential in growing instabilities
carried out in a two-dimensional
sand-filled
chamber are
presentedin this section. The chamber, holding a volumeof
sand 51.25 cm wide, 97 cm high, and 1 cm thick was built
from standard3/8 inch (0.953 cm) plate glass. The top of the
chamberwas open to allow packing and irrigation, with the
bottom of the chamber drained through 100 mesh stainless
steel screen.
Fig. 5.
Layout of two-dimensionalchamberwith tensiometerand
air pressure ports.
The front panel of the chamber was fitted to accommodate
tensiometers, as shown in Figure 5. Tensiometers were
mounted with O-rings and jam nuts to create watertight
measurementports, with the surface of the tensiometerflush
with the inside surface of the chamber (Figure 1). Air
pressuremonitorswere includedin each experiment.Tensiometer signalswere amplified using Analog Devices IncorporatedModel AD 524 precisioninstrumentationamplifiers
and read each 0.8 s using a Lab Master Incorporated DMA
analog-to-digitaldata acquisition board installed in an IBM
XT personal computer. A 0.50-s R-C low pass filter was
employedimmediatelyprior to the data acquisitionhardware
applied pressure head (1 m) divided by the volume of fluid
required to register this pressure changein the transducer,
45
whichyieldsS = 6.1 x 107 m-2.
o
It is straightforward to evaluate the effect of the experimental media on the response time. The following fingered
flow experimentswere carried out usinga white quartz sand,
referred to as 40-50 sand, which passedthrough a standard
U.S. number 40 sieve (opening of 0.425 ram) and which was
retained above a number 50 sieve (openings of 0.297 mm).
The saturated conductivity of 40-50 sand was measured in
ß•
30
E
25
•
2o
an upfiowcolumnexperimentto be 0.148 cm s-i. The
o.) 15
40
o
o
35
o
o
oo
o %o
8
predicted responsetime from (1) is approximately 0.01 s,
with the soil conductivity not affecting the predicted responsetime until conductivity of the mediais lessthan about
0.01cms-1 asshownin Figure4. Theseresultsagreewith
the conclusions obtained using Figure 1 of Townet [1980].
The measuredtensiometerresponsetimes are reportedin
Table 1. Two of the deviceshad a responsetime of over 1 s,
and the remainder of the tensiometer responsetimes are at
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
VolumetricMoistureContent(cm^3/cm"3)
Fig. 6. Resultsof three repetitionsof drainageexperiments
least an order of magnitudegreater than the times predicted using40-50 sand.The datapointsshownare averages
of thethree
using (1). This discrepancy between predicted and actual values obtained for each pressure.
SELKER
ETAL.'FINGERED
FLOW
INTwoDIMENSIONS,
1
TABLE 2. Parametric
ValuesObtained
in FourFingered
Flow Experiments
Experiment
1
2
3
4
Velocity,
Flux,
cm s--
hwe,cm
10-2 cm s-•
0.165
0.157
0.151
0.149
-5.2
-5.0
-4,0
-5.7
2.93
2.9*
3.26
0.624
2517
an applicationrate of 0.32 mL s-]. The results of these
experiments
are discussed
in this paperandin someof the
theoretical
implications
in thecompanion
paper.
In Figures7-10, Figures7 and9 showthreedistinctstages
of matricpotential
measurements
in experiments
1 and3 (the
positions
of thewettingfrontsare givenin Figures8 and 10,
respectively).Prior to infiltration, all of the tensiometers
showveryhighsuction,beingexposedto air-drysand.Upon
Fingergrowth velocity was calculatedfrom video recordsof the water contact the tensiometersmomentarily come to a
experiments.
Water enti'ypressures
(hwe)weretakenas the maxi- commonpotentialof approximately-4 cm (Table 2), the
so-called"water entry" pressureof the sand,as discussed
mum30-saveagepressureattainedduring infiltration.
*Not measured;
equipment
at settingsusedin experiment1.
by Hillel andBaker[1988],similarto the resultsfoundby
Baker and Hillel [1990].
The pressurerecordedfor the subsequent
400 s showsa
toeliminate
high-frequency
noisegenerated
by lightingand monotonicdecline in matric potential, to a minimum of
-21.0 cm at the uppermostpoint on the finger.From Figure
6 it isapparent
thatthispressure
corresponds
to theregionof
rapiddeclinein moisturecontent.The declinein pressure
in
experiment
1 slowsthroughthe400-550-sregion,priorto the
fingerreaching
the bottomof the chamberat 640s. Experiment3 maintaineda highertip velocity,reachingthe bottom
of saturated
conductivity
(varyingbetween
experimental
runs,asdiscussed
later)usinganoscillating
shuttledrivenby of the chamber after 587 s. Infiltration was halted after 1300
a feedback-controlled
dc motor. A singleinstabilitywas s in experiment1 and 1400s in experiment3, at which time
pointsdrops.
triggeredto develop directly over the central tensiometers the matricpoter•tialat all measuring
pumps.
The chamberwas filled with 40-50 sand,followingthe
methoddescribedby Glasset al. [1989],usinga randomized
gravitypackingtechnique.
Waterwasappliedto theupper
surface
of thesandslabat a constantrateapproximately
10%
The pressuremeasurements
exhibit an oscillationof about
1 cm pressureuntil the drainagephaseof the experiment,as
Theglasschamberwasilluminatedfrom behindby a bank shownin Figure 11. This cyclic variation is the result of the
system,whichhasa cycletime of approxof 25-kHz fluorescentlights.When sandbecomeswet, the waterapplication
transmission
of light is enhanced,which may be used to imately2.8 s; the oscillationin pressurestopswhen the
calculate
the moisturecontentthroughouta two-dimensional irrigationis halted, as expected. The oscillationdoes not
with slow responsetime, as shown
experiment[Hoa, 1981;Glasset al., 1989;Bell et al., 1990]. showup in tensiometers
by
the
data
obtained
from
port 2 which held tensiometer2 in
Therelationship
betweenmoisturecontentandpressure
was
obtained
for the media(Figure6) froma segmented
draining experiment1 (seeFigure7 andTable 1, wherethe response
time of tensiometer2 is measuredto be 9.1 s), verifyingthat
columnexperiment,as describedby Bell et al. [1990].
the signalreflectsvariationin matric potential rather than
experimental
noise.The 1-cmvariationsin matricpotential
Results and Discussion
whicharisein theseexperimentsare smallcomparedto the
The applicationrates in experiments1, 2, and 3 were 7-cm variationsin pressureseen in experiment4, which
between
1.50and1.77mL s-•, whilein experiment
4 the resultedfrom a 10-sdelay betweenpassesof the irrigation
applicator
shuttlewas run.at a 20% duty cycle, resultingin shuttle (Figures 12 and 13).
(Figure5) by applying 0.5 mL of water to this area immediately prior to initiation of infiltration.
-2
I
-4
I
-6
-8
Port 6
Port 7
Port 5
-10
Port2
-12
Port 4
•
-16
•
-18
a. -20
-22
-24
-28
-30
0
I
I
i
'I
200
400
600
800
'
i
1000
-
I
1200
'"t
1400
1600
Time(seconds)
Fig. 7. Plot of matric potentialversustime measuredin ports2, 4, 5, 6, and7 in experiment1.
2518
SELKER
ETAL.:FINGERED
FLOW
INTwoDIMENSIONS,
I
52.5 cm
o
52.5
o
o
.7rl
o
o
T2T3
o
cm
T4T5
o
o
o
550
550
i
Fig. 8. Development
of the unstablewettingfrontfor experiment !. Tracingsof the wettingfr0nt's positionafter 50, 150,250,
350, 450, and 550 s are shown.
Fig. 10. Development
of the unstable
wettingfrontin experiment3. Tracingsof the wettingfront'spositionafter 50, 150,250,
350, 450, and 550 s are shown.
-2
-4
-6
-8
Port 1
-10
Port 2
.--
-12
Port 3
•
-14
Pdrt 4
Port 5
E -16
•
-18
m -20
-22
-24
-28
-30
0
200
400
600
800
1000
1200
1400
1600
Time(seconds)
Fig.9. Plotof'matticpotential
versus
timemeasured
inportsI, 2, 3, 4, and5 in experiment
3.
SELKER
ETAL.'FINGERED
FLOW
INTwoDIMENSIONS,
I
-4
52.5
2519
cm
-5
;,, /'
-5.5
-6
.........
I I I¾,I,'!!;
I,
Port 3
Port 4
-6.5
Port
50
55
,
60
t55
70
75
Time (seconds)
Fig. 11. Pressure measured in experiment 3 in the tensiometers
mountedin ports 3, 4, and 5.
Determinationof the horizontalgradientof matricpotential in the induction zone between fingersand characteriza-
tionof the pressureprofilein .theupperportionof a growing
I
instabilitymay be obtainedfrom the data shownin Figures
7
and9 for experimentsI and 3. During the period of finger
growththepressure
horizontally
throughtheinduction
zone
appeareduniform.After the geome.tryof the flow fieldwas
established
(Figures8 and 10), however, a significantgradient toward the finger developed in the inductionzone, with
tensiometersshowing a monotonic increase in tension towardthe central finger: in experiments I and 3, the horizon-
Fig. 13. Developmentof unstablewetting front in experiment4.
talpotential
gradient.
in theinduc.tjon
zonetowardthecentral approximately-21 cm at port 4 and -17 cm at port 1. Thus
finger was about half of the magnitudeof the vertical thereis a 4-cmdifferencein matricpotentialin the 9.75 cm
potentialgradient due to gravity. For example, in experi- separatingthese points or a pressure gradient of about 0.4
ment 3 (Figure 9), after 1000 s the matric potential is cmcm-• . Thisgradient
required
several
hundred
seconds
to
,
-6-
-8-10
-
-12 -
-14 -16 -!8
-
-22
-
-24
0
I
t
200
400
,
'
t..........
6,00
I.
800
'......
t
! 000
".
I
1200
.....
I
1400
1600
Time (.seconds)
Fig. 12. Measurements
of matricpotentialin port4 fromexperiment
4, wherepasses
of the irrigationshuttlewere
separated
by 10-sdelays,givingriseto significant
variation
in matricpotential
in thefinger.
2520
SELKERET AL.' FINGEREDFLOW IN TWO DIMENSIONS, I
0.5
0.4
0.3-
0.2
0.1
0
-0.1.
t
,I
I
t
I
I
200
400
600
800
1000
1200
-0,2
'
I
1400
16o0
Time (seconds)
Fig. 14. Air pressuremeasuredin experiment3.
and fabrication and that published tensiometer designequations can lead to unrealisticallylow predictionsfor response
wetted area of the flow field was established, No lateral
time. A simple response time measurement method was
spreadingof fingers was observed in the period of these developed,the resultsof which were consistentwith experexperiments.This corresponds
well with the observations
of imental observations.The application of these deviceswas
Glass et al. [1989], where very little spreadingwas observed demonstrated in the context of unstable wetting in two
dimensions with illustrative results.
up to 4 hours after infiltration under much higher flux.
The air pressurein the chamber duringinfiltrationfluctu-
become established, indicating that the matric pressurein
the induction continued to change after the geometry of the
ated between -0.2
and 0.6 cm of water head over the 1600-s
experiments,about 2% of the variation observedin the
matric potential (Figure !4). Consideringthe first 500 s of
this experiment,the matric potential abovethe fingerexperiencesa rapid transitionfrom -4 cm to -21 cm, a changeof
17 cm, while air pressure changesonly 0.35 cm. It appears
that entrappedair is not required to create instabilityunder
uniform nonpondinginfiltration [Philip, 1975].
The measurementsof matric potential allow the testingof
criteria which have been proposedfor wetting front instability. Raats [1973] and Philip [1975] statethat a flow field will
be unstableif the gradientin the matric potential opposesthe
flow. In these experiments,instability is observed,and the
gradientin matric potential appearsto opposethe direction
of flow, in agreement with this criterion. Here we infer the
gradient in matric potential by noting that the tip of the
Acknowledgments. Researchsupportedby the U.S. Geological
Survey, Department of the Interior, under USGS award 14-PO0001-G1749. The views and conclusions contained in this document
are thoseof the authorsand shouldnot be interpreted as necessarily
representing
the officialpolicies,either expressedor implied,of the
U.S. government.
REFERENCES
Baker, R. S., and D. Hillel, Laboratory testsof a theory of fingering
duringinfiltrationinto layered soils, Soil Sci. $oc. Am. J., 54,
20-30, 1990.
Bell, J., J. S. Selker, T. S. Steenhuis,and R. J. Glass, Rapid
measurement of moisture in two-dimensional sand slabs, ASAE
Pap. 90-263,30 pp., Am. Soc. of Agric. Eng., St. Joseph,Mich.,
1990.
Diment, G. A., and K. K. Watson, Stability analysis of water
fingersare at XI•we(•-4
cm) while the pressuredropsto
movement
in unsaturated
porousmaterials,3, Experimental
studies, Water Resour. Res., 21(7), 979-984, 1985.
•-20 cm above the finger after •500 s (e.g., Figure 7). It is
interestingto note that while Raats predictsthat flow gener- Dowd, J. F., and A. G. Williams, Calibration and use of pressure
transducers
in soilhydrology,Hydrol.Processes,
3, 43--49,1989.
ated by continuousnonpondingrainfall (as is generatedin Glass,R. J., T. S. Steenhuis,and J.-Y. Parlange,Wettingfront
this experiment)shouldbe unstable,Philip [!975] concludes
instability
as a rapidandfar-reaching
hydrologicprocess
in the
vadosezone, J. Contarn. Hydrol., 3, 207-226, 1988.
that this configurationshouldbe stable. From our experience
for
the criterion for stability as presented by Raats [1973] Glass,R. J., T. S. Steenhuis,and J.-Y. Parlange,Mechanism
finger
persistence
in
homogeneous,
unsaturated,
porous
media:
provides an accurate predictor of flow field stability.
Theory and verification, Soil Sci., 148(1), 60-70, 1989.
CONCLUSIONS
The tensiometer design and measurement of tensiometer
Glass,R. J., S. Cann,J. King,N. Bailey,J.-Y. Parlange,
andT. S.
Steenhuis,
Wettingfrontinstability
in unsaturated
porous
media:
A three-dimensional
study,Transp.PorousMedia, 5, 247-268,
1990.
Hill, D. E., andJ.-Y.Parlange,
Wetting
frontinstability
in layered
responsetime conductedin this study has yielded a reprosoils, Soil Sci. Soc. Am. Proc., 36, 697-702, 1972.
ducible design for compact high-speedplanar tensiometers Hillel, D., and A. Baker,Descriptivetheoryof fingering
during
which may be appliedin a wide rangeof studies.We have
infiltrationinto layeredsoils,Soil Sci., 146(1),51-56, 1988.
allowing
the measurement
of rapid
shown that tensiometer performance is sensitive to design Hoa, N. T., A newmethod
SELKER
ETAL.:FINGERED
FLOW
INTwoDIMENSIONS,
1
variationson the water contentin sandyporousmedia, Water
Resour. Res., 17(1), 41-48, 1981.
Klute,A., and W. R. Gardner,Tensiometerresponsetime, Soil
Sci., 93, 204-207, 1962.
Klute, A., and D. B. Peters, A recordingtensiometerwith a short
response
time, Soil Sci. $oc. Am. Proc., 26, 87-88, 1962.
2521
dimensions,2, Predictingfingermoistureprofile, Water Resour.
Res., this issue.
Starr, J. L., H. C. DeRoo, C. R. Frink, and J.-Y. Parlange,
Leachingcharacteristics
of a layeredfield soil, Soil Sci. Soc.Am.
J., 42, 386-391, 1978.
Towner, G. D., Theory of time responseof tensiometers,
J. Soil
Kung,K-J. S., Groundtruthaboutwaterflowpatternin a sandysoil
Sci., 31,607-621, 1980.
andits influenceon solutesamplingand transportmodeling,in Towner, G. D., The responsetime of tensiometersembeddedin
Validationof Flow and TransportModelsfor the Unsaturated
saturatedsoil pedsof low hydraulic conductivity,J. Agric. Eng.
Zone,InternationalConference
and Workshop
Proceedings,
RuRes., 26, 541-549, 1981.
idoso,New Mexico,May 23-26, 1988,editedby P. J. Wierenga Van Ommen, H. S., R. Dijksma, J. M. H. Hendrickx, L. W.
andD. Bachelet,pp. 224-230,New MexicoStateUniversity,Las
Dekker, J. Hulshof, and M. Van den Heuvel, Experimental
Cruces, 1988.
Long,F. L., A field systemfor automatically
measuring
soilwater
potential,Soil Sci., !37, 227-230, 1984.
Morrison,R. D., andJ. E. Szecsody,A tensiometer
andporewater
samplerfor vadosezone monitoring,Soil Sci., 144(5), 367-372,
1987.
Philip,J. R., Stability analysisof infiltration,Soil Sci. Soc. Am.
Proc., 39, 1042-1049, 1975.
Raats,P. A. C., Unstable wetting fronts in uniform and nonuniform
soils, Soil Sci. Soc. Am. Proc., 37, 681-685, 1973.
Richards,L. A., Methodsof measuringmoisturetension,Soil Sci.,
assessmentof preferential flow paths in a field soil, J. Hydrol.,
105,253-262,
1989.
Watson,K. K., Someoperatingcharacteristicsof a rapid response
tensiometer system, Water Resour. Res., I(4), 577-586, 1965.
Watson, K. K., and R. D. Jackson, Temperature effects in a
tensiometer-pressuretransducer system, Soil Sci. Soc. Am.
Proc., 31, 156-160, 1967.
P. Leclerq, J.-Y. Parlange, and T. Steenhuis, Department of
Agriculturaland BiologicalEngineering,Cornell University, Ithaca,
NY
14853.
68, 95-112, 1949.
J. Selker,Departmentof BioresourceEngineering,Gilmore Hall,
Saffman,P. G., and G. Taylor, The penetration of a fluid into a Oregon State University, Corvallis, OR 97331.
porousmedium of Hele-Shaw cell containinga more viscous
liquid, Proc. R. Soc. London, Set. A, 312-329, 1958.
Selker,J. S., Unstable wetting in homogeneoussoilsunder continuous infiltration, Ph.D. dissertation, Cornell Univ., Ithaca, New
(Received April 10, 1991;
York, 1991.
revised April 1, 1992;
Selker,J. S., J.-Y. Parlange,and T. Steenhuis,Fingeredflow in two
acceptedApril 21, 1992.)