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.)