WATER RESOURCES RESEARCH,
VOL. 34, NO. 8, PAGES 1935-1940, AUGUST
1998
Evaluation of hydrodynamic scaling in porous media using
finger dimensions
John S. Selker
Departmentof Bioresource
Engineering,OregonStateUniversity,Corvallis
Martin
H. Schroth
Departmentof Civil, Construction
andEnvironmental
Engineering,OregonStateUniversity,Corvallis
Abstract. The use of dimensionless
scalingis ubiquitousto hydrodynamicanalysis,
providinga powerfulmethodof extendinglimitedexperimetnalresultsand generalizing
theories.Miller and Miller [1956]contributeda scalingframeworkfor immisciblefluid flow
throughporousmedia that relied on consistency
of the contactanglebetweensystemsto
be compared.It is commonto assumethat the effectivecontactanglewill be zero in clean
sandmaterialwherewater is the wettingliquid. The well-documentedunstablewetting
processof fingeredflow is usedhere as a diagnostictool for the scalingrelationships
for
infiltrationinto sandymedia.Throughcomparisonof fingercrosssectionsproducedusing
three liquidsaswell asvariousconcentrations
of anionicsurfactant,it is shownthat the
zero contactangleassumption
is very poor evenfor laboratorycleanedsilicasand:
Experimentalresultsdemonstrate
effectivecontactanglesapproaching
60ø. Scalingwas
effectivefor a givenliquid betweensandsof differingparticlesize.Theseresultssuggest
that cautionshouldbe exercisedwhen applyingscalingtheory to initial wetting of porous
mediaby liquidsof differinggas-liquidinterfacialtensions.
for fingeredflow [Parlangeand Hill, 1976; Glasset al., 1989;
Selkeret al., 1992]. The equationspresentedby Parlangeand
The power and appeal of the scalinglaws introducedby Hill [1976] are readilyscaledusingthe methodsof Miller and
Miller and Miller [1956] is the ability to quantitativelyextend Miller [1956]to obtainthe scalingrelationshipfor fingerwidth
resultsof experimentsmade under one set of conditionsto given by
media of differingto grain size and to liquidsof varyingvispg,,t
cositiesand/or interfacial tensions.This systemof scalingis
d. =
d
(1)
applied most commonlyto computeunsaturatedhydraulic
O'g1
conductivityand liquid retentionbetweensetsof similarmewhere p is the liquid density,# is the constantof gravitational
dia. In this application,only the is characteristic
grain sizeof
X is the characteristic
medialengthscale,O-g•
is
the variousmedia is required.The samescalinganalysisis well acceleration,
the gas-liquidinterfacialtension,d is the fingerwidthin a given
suited to understandingthe time evolution of macroscopic
media, and d. is the scaledfinger width [Glassand Nicholl,
processes,
suchas infiltration,evaporation,unstablewetting.
1996].This expressionis very convenientin that it is explicitin
Althoughmany studieshavevalidatedscalingbasedon charmeasurableparameters.In thispaperwe testthisexpression
by
acteristiclength [e.g.,Hillel and Elrick, 1990], the predictive
varyingX throughthe useof four similarmedia, p throughthe
utility of scalingwith viscosity,density,and interfacialtension
byvaryingthe imbibing
has not been thoroughlyinvestigated.The objectiveof this useof threedifferentliquids,andcrg•
liquid and addingan anionicsurfactantto water.
paperis to explorethe quantitativepredictionsof Miller scalA noteworthyfeature of (1) is the lack of dependenceon
ing of macroscopic
dimensions
with respectto characteristic
liquid viscosity.This reflectsthe fact that the forcesaffecting
particlegrain size,liquid density,interfacialtension,and to a
the fingerdimensionsare thosegeneratedby gravity(tendsto
lesserdegreeviscosity.This was achievedusingthe Millermake fingerssmaller) and interfacial energies(as these inscalingresultsfor the dimensionsof unstablewetting in aircrease,fingersbecomewider). For (1) to hold,the flux through
dried sandymedia as a diagnosticmeasureof macroscopic
the fingermustbe muchlessthan the saturatedconductivityof
scalingpredictions.
the medium(infiltrationcannotbe conductivitylimited, under
Fingeredflow providesa very natural systemfor the inveswhichcircumstances
the wettingfront is stable).
tigationof macroscopic
scalingrelationshipsfor flow through
porousmedia. First, the processgivesrise to well defined
macroscopic
features(fingers)whichcan easilybe compared, 2. Materials
and Methods
with fingerwidth being the primary feature of the unstable
Two typesof experimentswereperformed.The primarydata
wettingprocess.Further, the linear stabilityanalysisof J.-Y.
used
to evaluatethe scalingrelationshipswere obtainedfrom
ParlangeandD. E. Hill hasprovideda robustpredictivemodel
thin slabinfiltrationexperiments.In theseexperimentsa series
Copyright1998by the AmericanGeophysicalUnion.
of fingersof flow were generatedin initially dry sandsusinga
variety
of fluid/sandcombinations.
A secondset of supporting
Paper number 98WR00625.
0043-1397/98/98 WR-00625 $09.00
experiments
wascarriedout investigating
the effectivecontact
1.
Introduction
1935
1936
SELKER
AND
SCHROTH:
EVALUATION
Table 1. SelectedPhysicalPropertiesof the PorousMedia
Organic
Texture
12/20
20/30
30/40
40/50
0.3
0.4
0.3
0.3
Porosity
(s.d.)
0.34 (0.02)
0.34 (0.02)
0.35 (0.01)
0.36 (0.03)
(s.d.),
SCALING
Table 2. SelectedPhysicalPropertiesof LiquidsEmployed
ds0
Carbon,
g/kg
OF HYDRODYNAMIC
Air/Liquid Interfacial
gsat,
mm
cm/min
Liquid
1.105(0.014)
0.713(0.023)
0.532(0.003)
0.359(0.010)
30.19
15.02
8.94
4.33
Pore water
Duoprime55
Soltrol220
Energy%],
Densityp,
71.3
30.0
25.9
1.00
0.86
0.81
dyn/cm
Viscosity,
g/cm3
cP
1.00
13.73
4.50
Propertiesare as measuredat 23øC.
All propertiesare as reportedby Schrothet al. [1996] exceptporosity, which was measured in each packing. The parameter ds0 is the
grain size of the particle for which half of the mass of media has
smallerparticles;Ksat is the saturatedhydraulicconductivity
to water.
angle of liquid-solid interfacesby applying surfactant/water
solutionsto singlegrainsof sand.
For the porous media, we selectedthe four Miller-similar
silica sandsdescribedby Schrothet al. [1996], with grades
identified as 12/20, 20/30, 30/40, and 40/50. These delineations
refer to the standard mesh sizeswhich bound the particle
dimensionsfrom above and below. For complete hydrodynamic characterizationand media preparation see work by
Schrothet al. [1996].The mediaconsistof commerciallysieved,
washed,nearly sphericalnatural silicasand.The organiccontent of the sandis -<0.04%for all media (Table 1).
To assess
the wettabilityof the sands,individualgrainswere
subjectedto wettingfrom aboveby dropletsgeneratedby a fine
gaugeneedle which was positionedusinga microscopestand
(Figure 1). The water dropletin thistesthadvolumeof 0.6 •L
as dispensedusing a micrometerplunger volumetricsyringe.
Individual grainsof the 12/20 sandwere attachedto micropipetteswith a solutionof 10% FD&C blue dye 2 that had been
spreadas a thin film on a mailing envelope'swater soluble
adhesivestrip. The pipettestips were touchedto the dye solution and then to the sand grains. The dried dye solution
provided a strong bond between the sand and the capillary
tube and allowedimmediatevisualdetectionof wettingof the
grain of sand through a 12x hand lens as evidencedby the
dissolutionof the blue dye from the bond. Contact anglewas
deemedto be nonzeroif there wasno point of wettingreaching
the blue dye bond within 30 s.
For our model liquidswe employeddistilleddeionizedwater, whichwe refer to as pore water (after prolongedcontact
with the solid matrix), and two non-aqueousphase liquids
(NAPLs), Soltrol© 220, which is a mixture of C•3 to C•7
branchedalkanes(PhillipsPetroleumCo., Bartlesville,Oklahoma),and Duoprime© 55 mineraloil (LyondellPetrochemical Co., Houston,Texas).Gas-liquidinterfacialtensionswere
measuredby the ring method [Du Noiiy, 1919],viscosities
were
measured using Ostwald viscometers[American Societyfor
Testingand Materials(ASTM), 1995], and densitieswere provided by the manufacturer.All measurementswere conducted
at 23øC_+0.5øC.The NAPLs provideinterfacialtensions
which
differed from that of water by almost a factor of 3, densities
which differedby about 20%, and viscosity's
differingby more
than a factor of 10 (Table 2).
The effectsof variation in the gas-liquidinterfacialtension
were exploredusinga rangeof sodiumdodecylsulfate(SDS)
surfactant
solutions.
width of the chamber.
0.46 mm OD needle
0.6I•1Droplet
All
dilutions
This funnel contained
3.
Measured
Surfactant
Solutions
Interfacial
Tensions
SingleSand Grain
,:•< Dye
Adhesive
I1•--•.•mm
Glass
Capillary
Figure 1. Experimental setupfor individualgrain measurement of wettability of sand.
maintained
a series of coarse
sieveswhich randomizedparticle motion as the sandfell into
the chamberto generatea homogeneouspacking.The chamber packingwas conductedin one continuouspouringto prevent local gradationsin particle size,which were observedto
form when particleflow was interruptedduringpacking.The
massof sand and the packingheight were recorded for each
experimentand usedto calculatethe averageporosity.
Fingerswere generatedby applyingthe liquidsat a point at
the sandsurfacewith constantflow providedusinga metering
Table
II
of SDS were
below the critical micelle concentration,beyondwhich little
changeis obtainedin the gas-liquidinterfacialtension[Downey
and Addison, 1937]. The range of concentrationsemployed
provided interfacial tensionsbetween 37.5 and 62.1 dyn/cm
(Table 3). Sincethe surfactantis anionic,the solid-liquidinterfacial interaction remained unaltered by the addition of
surfactantexceptfor the possibilityof removingsurfacecontamination, which would exposemore silica, potentially increasingthe solid-liquidcontactarea [Wahlgrenet al., 1993].
The experimentalchamberconsisted
of two glasspanels(50
cm wide x 60 cm high), separatedby a 0.95-cm-thickacrylic
spaceralongthe sidesand bottom.The chamberwas packed
with air-dried sand using a prismaticfunnel that fit the full
for Dilute
SDS
Molarity
SDS
Number of
Measurements
Air/Liquid Interfacial
Energyo-(s.d.),
dyn/cm
0.001
0.002
0.004
0.006
5
7
4
6
62.1 (1.4)
51.9 (1.7)
44.2 (0.5)
37.5 (0.3)
SELKER
Table 4.
AND
SCHROTH:
EVALUATION
OF HYDRODYNAMIC
SCALING
1937
Summaryof NAPL Experiments
Experiment/Run
NAPL
1
2/1
2/2
Sand
S
30/40
S
...
40/50
40/50
3
S
12/20
4
S
20/30
5/1
5/2
6/1
6/2
8/1
8/1
S
S
S
S
S
S
12/20
10
S
11
S
12
S
12/20
20/30
20/30
30/40
30/40
40/50
20/30
40/50
12/20
13/1
13/2
14/1
14/2
.-.
D
D
D
15
D
30/40
16
D
40/50
!7/1
17/2
18/1
18/2
19/1
19/2
20/1
20/1
D
D
D
D
D
D
D
D
12/20
12/20
20/30
20/30
12/20
20/30
20/30
30/40
30/40
40/50
40/50
Qw,
mL/min
QN,
mL/min
dw (s.d.),
cm
aN (s.d.),
cm
3.94
1.54
1.54
5.45
4.20
4.97
4.97
4.40
4.40
3.21
3.21
1.18
4.04
0.80
3.58
3.58
2.90
2.90
0.97
ß-.
2.84
2.84
4.27
4.27
2.50
2.50
ß.ß.-
0.68
0.57
--.
1.39
1.19
1.34
1.34
1.09
1.09
0.65
0.65
0.25
0.97
0.20
...
0.57
0.25
.-.
0.20
0.20
0.45
0.45
0.25
0.25
0.30
.-.
0.22
0.22
1.69(0.13)
5.24 (0.50)
7.26 (0.75)
1.69(0.13)
2.88 (0.19)
0.84 (0.23)
1.12(0.23)
1.52(0.19)
1.42(0.24)
2.88(0.24)
2.80(0.17)
4.37 (0.83)
1.60(0.27)
5.81 (0.66)
0.89 (0.17)
0.81 (0.26)
2.57 (0.17)
2.40 (0.19)
2.90(0.39)
...
1.04(0.24)
0.92 (0.24)
1.85(0.17)
1.74(0.15)
3.22 (0.44)
3.19 (0.29)
.-.
..-
2.38(0.31)
4.70 (0.31)
..1.42(0.21)
2.12 (0.15)
1.44(0.36)
1.21(0.20)
1.59(0.21)
1.61(0.20)
2.36(0.28)
2.67(0.18)
4.01 (0.18)
1.61(0.16)
3.25 (0.28)
...
0.48 (0.11)
1.69(0.29)
...
2.43(0.44)
3.73 (0.84)
0.96 (0.24)
0.93 (0.14)
1.52(0.20)
1.71(0.22)
2.14 (0.29)
..2.82 (0.16)
3.55 (0.31)
D, Duoprime; S, Soltrol;Qw, flux of water employed;QN, flux of NAPL employed;dw and dN, water
and NAPL fingerwidths,respectively.
pistonpump. Flow rateswere establishedsuchthat the flux in
the fingerswould be approximatelyan order of magnitudeless
than the saturatedconductivityof the mediumfor that liquid in
a finger of the dimensionspredictedusing (1). Twenty-four
chamberpackingswere employed,with up to five fingersgenerated in each sandpack. In NAPL experimentsat least one
water fingerand one NAPL fingerwere formedin eachsand
pack. A total of 59 usablefingerswere generated(discarding
experiments
where adjacentfingersmerged)(Table 4). Finger
the unscaledmedia provideda rangeof finger diametersover
a factor of 5.45 betweensands,while media scalingreduced
this variabilityto a factor of 1.72 (Tables5 and 6). A simple
measureof the degree to which a given scalingmethod explainsobservedvariation is to considerthe value of the coefficientof variation(CV) betweenexperimentalfindings.In this
casewe note that the CV amongwater-generatedfingersin the
four media was 0.70 without media scaling,droppingto 0.24
when the effectsof characteristicgrainsizewere included.This
contours were traced onto acetate sheets attached to the front
reductionin variation betweenunscaledand scaledfinger diglasspanel of the chamberfor analysis.Fingerwidth wascharmensionswasequallydramaticfor the NAPLs, with CVs dropacterizedby measuringthe width at 10 equallyspacedintervals
ping by no lessthan a factor of 3.6 (Table 6), supportingthe
alongthe heightof eachfinger,from whichmean and standard
widelyusedscalingproceduresthat havebeenfoundto explain
deviation were calculated.
variabilityin hydrodynamicprocessesbetweensimilar media.
Scalingbasedon liquid density(whichdifferedonlyby 19%
3. Results and Discussion
betweenthe liquids),maintainedor improvedthe consistency
Water hasbeenthe liquid mostwidelyinvestigatedin Miller of the scaled dimensions in all but the 12/20 sand and thus
scalingto date. Consideringthe effect of media texture on the appearsto provide at least some of the expectedbenefitsof
dimensionsof fingersgeneratedwith pore water, we see that scaling(Table 7). There were no systematictrends in finger
Table 5. Summaryof WidthsObtainedin NAPL andWater FingeredFlow Experiments
12/20
20/30
30/40
40/50
Liquid
Width, cm
n
Width, cm
n
Width, cm
n
Width, cm
n
CV
Water (s.d.)
Soltrol(s.d.)
Duoprime(s.d.)
1.04(0.31)
1.36(0.13)
0.79(0.27)
7
3
3
1.99(0.55)
1.73(0.26)
1.64(0.1)
8
4
3
2.78(0.56)
2.47(0.17)
2.29(0.02)
6
3
2
5.67(1.21)
3.99(0.73)
3.37(0.48)
4
3
1
0.70
0.49
0.54
CV
0.27
0.10
0.10
The number of repetitionsof each measurementis listed as n.
0.27
0.54
1938
SELKER
AND
SCHROTH:
EVALUATION
Table 6. NAPL and Water Finger Width ResultsScaled
With Media CharacteristicLength, Relative to 12/20
Liquid
Water
Soltrol
12/20
20/30
30/40
40/50
1.04
1.36
1.29
1.12
1.33
1.19
1.81
1.28
Duoprime
0.79
1.07
1.10
1.08
CV
0.27
0.10
0.10
0.27
CV
Table 8. NAPL and Water Finger Width ResultsScaledWith
Gas-LiquidInterfacial Tension,Relative to Pore Water
Liquid
Pc =
stated as
cos =
Pc =
(3)
O'gI
2(sg-
(4)
t'
which dependsonly upon the interfacial tensionbetweenthe
gas-solidand solid-liquidpairs.SDS, being an anionicsurfactant, doesnot affectthe gas-solidinterfacialtension[Wahlgren
et al., 1993]. Thus from (4) the capillarypressureacrossthe
gas-liquidinterface is expectedto be unaffected.Since it is
preciselythis capillary pressurewhich influencesthe finger
width, as long as the contactangleis nonzero,the gas-liquid
interfacial tensionis unrelated to finger width. Remarkably,
this fact is overlookedin mosttreatmentsof (2) in standard
texts,where o-and 3,are consideredto be essentiallyindependent parameters[e.g.,Juryet al., 1991].
Table 9. NAPL and Water Finger Width ResultsScaledWith
Media CharacteristicLength, Liquid Density,and GasLiquid Interfacial Tension,Relative to Pore Water
0.70
0.49
Water
Soltrol
0.54
-.-
O-sg- O-sl
where the subscriptsg indicatesthe solid-gasinterfacialtension,and sl indicatesthe solid-liquidinterfacialtension.Substitutingthe expression
for cos3,from (3) into (2), we seethat
the pressuredrop acrossthe capillaryinterfaceis givenby
5.67
4.93
0.18
(2)
t'
the gas-liquidinterfacialtension,and 3' is the liquid-solid-gas
contact angle. Young's equation for contact angle may be
2.78
3.05
3.92
20-g•cos3,
wherer is theeffective
radiusof thecapillary
aperture,
o-g•
is
1.99
2.14
0.07
0.70
0.49
0.54
...
densitydecreasesthe fingerwidth. Conversely,increasingthe
energydissipatedthroughwettingby increasinggas-liquidinterfacialtensionincreasesfingerwidth.
Although not often appreciated,for casesof nonzerocontact angle the energy of wetting is only a function of the
solid-liquidand solid-gasinterfacialenergies,while the gasliquid interfacialtensionis irrelevant.This can be seenimmediatelyby combiningtwo of the mostcommonresultsof capillary theory:the Laplaceequationfor pressureacrosscurved
interfaces,andYoung'sequationfor equilibriumcontactangle
[e.g.,Bear, 1972,pp. 442-445]. The Laplaceequationfor pressure acrossa curvedcapillaryinterface,Pc, is givenby
1.04
1.68
2.66
5.67
1.45
1.42
0.86
0.69
CV
1.91
2.78
0.90
0.96
0.69
0.70
40/50
0.06
1.99
0.63
0.33
30/40
0.92
CV
0.60
20/30
0.34
40/50
CV
12/20
CV
30/40
Duoprime
Table 7. NAPL and Water Finger Width ResultsScaled
With Liquid Density, Relative to Pore Water
1.04
0.49
20/30
0.15
---
alteringthe gas-liquidinterfacialtensionof the water through
addition of the anionic surfactant SDS. Experimentally,
changesin gas-liquidinterfacialtensionhad little effect on the
fingerdimension(Table 10). For example,in 20/30sandfinger
dimensionsvaried by only 15% while the interfacial tension
varied by 66%, with the largest finger being found in the
experimentwith the lowestinterfacialtension,the oppositeof
that predictedby (1). If we considerthe observedvariabilityin
finger width throughthe eight combinationsof conditions,the
CV for the raw fingerwidth data is 0.25. By scalingaccording
to (1) for media characteristic
lengthonly,the CV reducesto
0.10, indicatingthat media scalingaccountsfor about 60% of
the variability observed.On the other hand, scalingfor interfacial tensionactuallyincreasesthe CV for the scaledfinger
dimensions,as observedpreviouslyfor the NAPL fingers,indicatingthat the relationshipbetweeninterfacial tensionand
fingerdimensionprovidedin (1) is not observedin thesedata.
The sourceof this discrepancy
between(1) and the experimental data is deceptivelysimple: the gas-liquidinterfacial
tensionscalingrelationshipprovidedin (1) dependsupon the
contactanglebeingconstantbetweenexperiments[Millerand
Miller, 1956], an assumptionwhich appearsto be violated in
our experiments.The scalingrelationshipprovided by (1)
arisesbecauseof the competingforcesof capillarypressure
acrossthe wettingfront and gravitationalpotential.Increasing
the energydissipationdue to gravityby increasingthe fluid
Duoprime
12/20
Water
Soltrol
interfacial tension, we now consider the results obtained from
Water
Soltrol
SCALING
0.24
0.08
dimensionwith liquid viscosity,which varied by more than a
factor of 10 betweenwater and Duoprime, aspredictedby (1)
(Table 6).
Quite different resultswere obtainedwhen scalingwas carried out basedon the variationin gas-liquidinterfacialtension.
Consideringfirst the comparisonof "pure" liquids,Table 8
showsthat inclusionof the gas-liquidinterfacialtensionexaggerated the discrepanciesbetween experimentsrather than
explaining the variation in finger dimension.The CVs increasedby a minimum of a factor of 2 comparedto the unscaleddata. Inclusionof both liquid propertiesfound in (1)
densityand interfacial tension) improved upon the scaling
basedsolelyon interfacialtension,supportingthe aboveobservationthat densityeffectsappearto be correctlyidentified
in (1) (Table 9).
To shedlight on the shortcomingof scalingwith respectto
Liquid
OF HYDRODYNAMIC
Liquid
12/20
20/30
30/40
40/50
CV
1.04
0.61
1.29
0.50
1.33
0.53
1.81
0.57
0.24
0.08
Duoprime
0.39
0.52
0.54
0.53
0.15
CV
0.49
0.58
0.58
0.75
...
SELKER
AND
SCHROTH:
EVALUATION
In a heterogeneousporousmedium there is no singlecontact angle but rather a statisticaldistribution associatedwith
the varied surfaces found in the medium.
of this behavior
here
and
therefore
must
solution
(i.e.,thatO'sg
-- O'sl
• O'gl)
, thecontactanglefor pore
water may be computedusing(3) to be greater than or equal
to 58ø, significantlydepartingfrom the typicallyassumedvalue
1939
Sand and Pore Water
d,b
dX,c
dirt,d
•d/o-,e
SDS
Sand
mL/min
Q,
t/a
cm
cm
cm
cm
0.001
0.002
0.004
0.006
0.001
0.002
0.004
20/30
20/30
20/30
20/30
30/40
30/40
30/40
4.1
4.1
4.5
4.5
3.6
3.6
3.6
2
2
2
2
1
1
1
1.09
1.08
1.15
1.26
1.92
1.81
1.72
0.71
0.71
0.75
0.82
0.92
0.89
0.83
1.25
1.48
1.85
2.40
2.20
2.49
2.77
0.81
0.96
1.21
1.56
1.06
1.19
1.33
30/40
3.2/3.6f
2
conclude that the effective contact angle for the media was
0.006
nonzero at least to the lowest gas-liquidinterfacial tension CV
tested.If we were to supposethat the contact angle was zero
for solutions with surface tension lower than the 0.006 M SDS
SCALING
Table 10. SDS-Fingered Width Results, Scaled to 12/20
An ascribed contact
anglefor a medium mustbe recognizedas an effectiveparameter. We know that as the gas-liquidinterfacial tension decreases,the contact angle also decreases.Had the effective
contact angle become zero beyond some critical interfacial
tension value, it would have been expected that the Millerscalingresultwould hold for all lower interfacial tensions.We
see no evidence
OF HYDRODYNAMIC
......
1.82
0.87
3.46
1.66
0.25
0.10
0.32
0.23
aNumber of replicates.
hAverage
measured
fingerdiameter.
CAveragemeasuredfinger diameter scaledto the 12/20 characteristics length.
dAverage
measured
fingerdiameterscaled
to thesurface
tensionof
of zero.
pore water.
Evaluation of the effectivecontactangle of the fluids on the
sandswas undertaken using the set-up shown in Figure 1.
Twelve of the 26 grainstestedhad nonzero contact anglewith
pure water. It was also observedthat the time required to
achievefull wetting of the 14 grainswhich did have zero contact anglewas typicallybetween5 and 15 s. This suggeststhat
the effective contact angle is a function of the rate of infiltration, which could explain someof the enlargingfinger dimensionsunder low infiltration behavior found by Hendrickxand
Yao [1996]. Of the 12 grains that had nonzero contact angle
with pure water, six continuedto have nonzero contactangle
with 0.001 M SDS, and five had non zero contact angle with
0.006 M SDS. This was determined by repeating the experiment with the same sand grains using 0.001 M and 0.006 M
solutionssuccessively,
allowingthe grain to dry betweenexper-
eAveragemeasuredfinger diameterscaledto the surfacetensionof
pore water and the 12/20 characteristiclength.
iments.
Since the surfactant
in the 0.001 M solution
remained
on the sand grains, the effective solution strengthduring the
0.006 M trials could have been closer to 0.007 M, which does
fThetwofingershaddifferingfluxes,asindicated.
angle is typically far from zero and related to the rate of
advanceof the wetting process.Previousresearchby Glasset
al. [1989] found that finger dimensionsstabilizedat low infiltration rates, which is at odds with this conclusion. The recent
result of Hendrickxand Yao [1996], however, suggeststhat
finger dimensionsincreasewhen infiltrationis extremelyslow,
which may be explainedby the expectedreduction in contact
angle as dynamiceffectsare minimized.
Acknowledgments. We thank Arron Burkhart and Darren Lerner,
who provided invaluable assistancein completingthe experiments
described;the U.S. Department of Energy SubsurfaceScienceProgram for their support of this work under contract DE-FG06-92ER61523;andthe constructive
reviewsprovidedby J. M. H. Hendrickx
not affect the qualitative conclusionsof these tests. For the and R. A. Birchwood.
grainsthat did not wet, the advancementof the liquid surface
appeared to be obstructedby physicaldefects on the sand
References
surface. These results are consistentwith the fingered flow
experiments,which find that contactangleis nonzerofor pure American Societyfor Testingand Materials, Standardtest method for
kinematicviscosityof transparentand opaqueliquids (the calculawater and that a decreasinggas-liquidinterfacial tensiondoes
tion of dynamicviscosity),DesignationD 445-94, Am. Natl. Stand.,
reduce the effective contactangle.
Annu. Bk. ASTM Stand., 05.01, pp. 160-167, Washington,D.C.,
1995.
4.
Conclusions
We have shown that Miller scalingfor media texture is an
effective means of explainingthe hydrodynamicbehavior of
liquidsin porousmedia, and the effectsof liquid densityand
viscosityare well predicted. In accountingfor the effects of
interfacial tension,on the other hand, Miller scalingbasedon
the typical assumptionof zero contact angle did not provide
accuratepredictionof observedbehavior.It appearsthat the
effectivecontactangle of the liquid-solidpair was variable in
the infiltration process,violating a key assumptionof Millerscaling.This was unexpectedfor a very clean, low-organicmatter silica sandbut was confirmedin wetting testson single
sandgrains.Further experimentswill be required to fully understandthese observations,but these resultswould suggest
that the use of interfacial tension scalingis not advisablein
many casesand that the widely employedassumptionof zero
contactangle is not justified. The observationof slowwetting
of individual sand grains suggeststhat the effective contact
Bear, J., Dynamicsof Fluids in PorousMedia, Dover, Mineola, N.Y.,
1972.
Downey,J., and C. C. Addison,The propertiesof detergentsolutions,
II, The surfaceand interfacialtensionsof aqueoussolutionsof alkyl
sodiumsulphates,Trans.FaradaySoc.,33, 1243-1260, 1937.
Du Nofiy, P. L., A new apparatusfor measuringsurface tension,
J. Gen. Physiol.,1,521-524, 1919.
Glass,R. J., and M. J. Nicholl, Physicsof gravityfingeringof immiscible fluids within porous media: An overviewof current understandingand selectedcomplicatingfactors,Geoderma,70, 133-163,
1996.
Glass,R. J., T. S. Steenhuis,and J.-Y. Parlange,Wetting front instability, 2, Experimentaldeterminationof relationshipsbetween system parametersand two-dimensional
unstableflow field behaviorin
initially dry porousmedia, WaterResour.Res.,25, 1195-1207,1989.
Hendrickx,J. M. H., and T.-M. Yao, Predictionof wetting front stability in dry field soilsusingsoil and precipitationdata, Geoderma,
70, 265-280, 1996.
Hillel, D., andD. E. Elrick (Eds.),Scalingin SoilPhysics:
Principles
and
Applications,Spec.Publ. 25, Soil Sci. Soc. of Am., Madison, Wis.,
1990.
Jury,W. A., W. R. Gardner, and W. H. Gardner, SoilPhysics,5th ed.,
John Wiley, New York, 1991.
1940
SELKER
AND
SCHROTH:
EVALUATION
Miller, E. E., and R. D. Miller, Physicaltheory for capillaryflow
phenomena,J. Aœœl.Phys.,27, 324-332, 1956.
Parlange,J.-Y., and D. E. Hill, Theoreticalanalysisof wettingfront
instabilityin soils,Soil Sci., 122, 236-239, 1976.
Schroth,M. H., S. J. Ahearn, J. S. Selker, and J. D. Istok, Character-
izationof Miller-similarsandsfor laboratoryhydrologicstudies,Soil
Sci. Sac. Am. J., 60, 1331-1339, 1996.
OF HYDRODYNAMIC
SCALING
M. H. Schroth,Departmentof Civil, ConstructionandEnvironmental Engineering,Oregon State University,AppersonHall, Corvallis,
OR 97331.
J. S. Selker,Departmentof Bioresource
Engineering,OregonState
University,GilmoreHall, Rm. 240,Corvallis,OR 97331-3906.(e-mail:
selkerj@engr.orst.edu)
Selker,J. S., T. S. Steenhuis,and J.-Y. Parlange,Wetting front instability in homogeneous
sandysoilsundercontinuousinfiltration,Soil
Sci. Sac. Am. J., 56, 1346-1350, 1992.
Wahlgren,M. C., T. Arnebrant, A. Askendal,and S. Welin-Klinst6m,
The elutabilityof fibrinogenby sodiumdodecylsulphateand alkyltrimethylammoniumbromides,ColloidsSurf.A, 70, 151-158, 1993.
(ReceivedFebruary25, 1997;revisedFebruary10, 1998;
acceptedFebruary18, 1998.)