By considering films on fracture surfaces as analogs to water in partially saturated porous ..... matric head at the inflow boundary was -6 mm for the rough- ened glass .... mm in each direction) with the motorized stage of the X26A end station.
WATER RESOURCES RESEARCH, VOL. 36, NO. 7, PAGES 1737-1746, JULY 2000
Transient film flow on rough fracture surfaces Tetsu K. Tokunaga and Jiamin Wan Earth SciencesDivision,LawrenceBerkeleyNational Laboratory,Berkeley,California
StephenR. Sutton Departmentof GeophysicalSciencesand Consortiumfor AdvancedRadiationSources,Universityof Chicago Chicago,Illinois
Abstract. Transientfilm flow in unsaturatedfractureswasinvestigatedconceptuallyand experimentally. By considering filmson fracturesurfacesas analogsto water in partially saturatedporousmedia,the film hydraulicdiffusivityand equationfor transientfilm flow are obtainedfrom their porousmediumcounterparts,the hydraulicdiffusivityand the Richardsequation.Experimentson roughenedglassshowthat the averagefilm thickness dependenceon matric potentialcan be approximatedas a power function.It is alsoshown that the film hydraulicdiffusivityincreaseswith increasedfilm thickness(and with
increased matricpotential). Fastfilmflow(average velocities greaterthan3 x 10-7 rns-• underunit gradientconditions)wasobservedfor averagefilm thicknesses greaterthan 2 /.tm and matric potentialsgreaterthan -1 kPa. 1.
Introduction The nature
of water
flow in unsaturated
fractured
rock is
complexandincompletely understood. In recentyearsthe phenomenonof fast,preferentialflow throughthe vadosezonehas beenidentifiedin a wide rangeof soiland rock types.Various mechanisms have been suggested, includingmacroporeflow [e.g.,Bevenand Germann, 1981; White, 1985], fracture flow [e.g.,Nitao and Buscheck,1991;Wanget al., 1993],flow fingering in soilsand alongfractures[Parlangeand Hill, 1976;Glass and Nicholl, 1996], and funneled flow [Kung, 1993]. Earlier conceptualmodelsof flow in partiallysaturatedfractureshave assumedthat local aperture segmentsmust be either fully water saturatedor fully desaturated[Wangand Narasimhan, 1985;Pruessand Tsang,1990].In suchmodels,local aperture saturationis assumedto be a strictfunctionof capillarypressure.In thesemodels,flow alongthe fractureplanecouldonly occurthrougha networkof locallysaturatedregions.Thus, in aperture-basedmodels,flow is essentiallyconceptualizedas saturatedflow within tortuouspathwaysin fractures. Recently,the conceptof film flow wasintroducedas a possibleprocessbywhichpreferentialflowcouldoccuralongtruly unsaturatedfractured rock. Kapoor [1994] investigatedthe case of free surfaceflow on a fracture surface.That study implicitlyconsiderswater filmsunder a small(positive)pressure potential and includesimbibitioninto the rock matrix. Our
work
concerns
water
films on fracture
surfaces under
near-zero(negative)matricpotentialsand examinesthe possibilityof fastunsaturatedflowunder"tension"[Tokunagaand Wan, 1997]. "Films" in this latter contextare a complexnetwork of thick pendularregionsthat form within topographic depressionsand thin films on topographicridges.Thus the thickness andconnectivity of pendularfilm regionsis expected to be importantin controllingfilm flow on individualfracture surfaces.We showedthat at matricpotentialsgreaterthan that needed to saturate the rock matrix, transmissivewater films Copyright2000 by the AmericanGeophysicalUnion. Paper number2000WR900079. 0043-1397/00/2000WR900079509.00
candevelopon fracturesurfaces. The matricpotentialdependenceof the averagefilm thicknessand film transmissivity of a BishopTuff fracturesurfacewere measuredusingequilibrium and steadystatemethods,respectively.The water films investigatedin the previousstudyaswell asthe presentonedevelop on roughsurfaces,rangein averagethicknessfrom about 1 to 50 •m, and flow in the laminarregime.In the presentpaper, additionalconceptsand experimentaltestsof film flow are presented.These new developmentsinclude introductionof the film hydraulicdiffusivityand transientfilm flow equation andnewmethodsfor measuringrelationsbetweenaveragefilm thickness, matricpotential,and film hydraulicdiffusivity. Sinceinterestin understanding film flow was largelymotivatedby the need to understandfast flow in unsaturatedfractured rock,it will be usefulto proposean approximatethreshold velocityabovewhichfilm flowmaybe considered"fast."As a referencepoint we will selectan averagefilm velocityof 3 x
10-? m s-1 (10 m yr-1) asthecriterionfor fastflow.Sucha velocitywould permit transportthrougha substantial100-mthickvadosezonewithin 1 decade,a relativelyshorttime from the point of view of seepagein arid and semiaridregions. Before describingexperimentalprocedures,aspectsof transientfilm flow will be considered.Having a contextfor characterizingnonsteadystate film flow is valuablenot only becausea wide range of vadosezone flow is transientin nature, but alsobecausea varietyof transientlaboratorymethodsfor quantifyingfilm flow canbe developed.
2. Transie_n_t_ Fi•,,, Flow and the Film Hydraulic Diffusivity Sincefilm flow in manywaysappearsto be a surfaceanalog to flow in bulk unsaturatedporousmedia, the macroscopic relationsuseful for describingfilm flow are expectedto be surfaceanalogsof the Darcy-Buckinghamlaw and the Richardsequation.Previously[Tokunagaand Wan, 1997],film flow wasconsideredfrom the perspectiveof steadystateflow,with the hydraulicpotential gradient as the drivingforce. In that contextthe appropriatesurfacetransportcoefficientlinking
1737
1738
TOKUNAGA
ET AL.: TRANSIENT
FILM FLOW ON ROUGH
volumetric flux per macroscopic fracturelength,J [L3 L-• T-•], to thehydraulic headgradient[L L-•] is thefilmtransmissivity T [L2 T-•]. In our previousstudyas well as the
FRACTURE
matrix
fracture surfaceQ(z +(Sz)
presentone,we considerthe film thicknessf[L] asa localareal thickness,averagedover thick capillary films within topographic depressionsand thin films on topographicridges. While film transportprocesses are expectedto be controlledby flowwithin interconnected channels, we attemptto maintaina macroscopic continuumpresentation.This macroscopicperspectiveis preferredhere sinceour currentunderstanding of microscalefilm flow over complexfracture surfacesremains qualitativeand sincewe lack experimentalmethodsfor quantitative
microscale
measurements.
A number
SURFACES
atxer filmj
of microscale
testsof flow on smoothsurfaces[e.g., Cazabatet al., 1990; Legerand Joanny,1992] and on surfaceswith simplechannel topographyhave been performed [e.g.,Pozrikidis,1988;Ransohoffand Radke, 1988; Zhao and Cerro, 1992; Romeroand Yost,1996;Ryeet al., 1996].In a macroscopic approachto film flow the appropriateaveragingdistancesfor continuumhydraulicpropertieswill generallydependon the topographyof the surfacein questionand can be complicatedby the scale dependenceof surfaceroughness,as noted in a later section. Figure 1. Reference volume for developingtransient film We assumehere that a localmeasurement encompasses a large flow relations. populationof hydraulicallysignificanttopographicoscillations. It may sometimesbe useful to considerfilm flow from the
-y
,--
x
by >
Q(z)
perspective of gradients in average film thickness [LL -•] rather than gradientsin hydraulichead as the drivingforce. in a mannerdirectlyanalogous to the RichardsequaThe appropriatetransportcoefficientis then the film hydraulic described diffusivity D(f) [L2 T-•], whichis relatedto the film trans- tion routinelyappliedto transientunsaturatedflow in porous missivityand the film thicknessdependence on matrichead, media. For simplicity,we will consideronly casesof onedimensionalflow alongthe z direction,without matrix imbibih,,, [L], through tion and evaporation-condensation. Volumetric fluxesrather T(f) than mass fluxes of water will be described since film flows over D(f) = (df/dhm) ' (1) largevariationsin absolutepressure,temperature,or solution chemistryare not consideredhere. The reference volume is As shownpreviously[Tokunagaand Wan, 1997],the relation depictedin Figure1. The instantaneous rate of changeof water betweenaveragefilm thicknessand mattic potentialfor frac- volume within the film is ture surfacesof wettableporousmedia is practicallynonhysOVw of teretic.Thus,for suchsurfaces, both T(f) andD(f) are also Ot = &y&z Ot' (3) nonhysteretic.The one-dimensionalfilm flux in the vertical direction
is
J = -D(f) and in the horizontal
direction
of
- T(f)
(2a)
oQ =&y T(f) oz &z
is
Of J =-D(f) 0•'
wheref is the averagefilm thicknessand Vw is the volumeof water. The changein water film volumetric flux along the macroscopic vertical flow path is
&z,
(4)
(2b)whereQ is the volumetricwater flow rate (positiveupward)
It shouldbe notedthat gradientsin averagefilm thicknesscan onlybe usefulfor predictingfilm flow on macroscopically homogeneoussurfaces,just asgradientsin volumetricwater content can only be applied to predict unsaturatedflow within macroscopically homogeneous regionsof porousmedia. Note from (2a) that the steady,unit gradient,gravity-driven film flow per transverselength of macroscopically homogeneousfracture surfaceis givenby - T(f), sincemacroscopic gradientsin film thicknessvanish, and that under such conditions the av-
eragefilm velocityis givenby -T/f. Sinceflow in natural systemsoften takesplace under transientconditions, it isusefulto haveconcepts aridanalyticaland experimentalmethodsfor characterizingtransient film flow. Under conditions of laminar flow, transient film flow can be
andH is the hydraulichead(H = hm + z). From continuity, (3) and (4) are equal,leadingto
Of 0 T(f)• Ot
Oz
Of 0 r(f)•-•-z +1 . ___[(Ohm)] ot
Oz
(5a)
(5b)
When gravity is negligible(horizontal film flow, and short distances,in general), transientfilm flow with variable T is describedby
Of_O[T(f)dhmOf] (6a) and with the chain rule gives
TOKUNAGA
•L(t)
ET AL.: TRANSIENT
FILM
FLOW
ON ROUGH
FRACTURE
SURFACES
hm•
-•[
Q = -WT -•--,
1739
(10)
where T is the wetted profile averagetransmissivity, L is the
wettedlengthalongthe directionof flow,and hm,f is the effectivewettingfront matrichead. In the aboveequationthe matric head at the inflow boundaryis not includedbecauseit was set to a nearly zero value in the experimentsdescribed later. Combining(9) and (10) gives
z
dL
Figure 2. Green-Ampt film model. Transient, one-dimensionalfilm flow is approximatedas an advancingstep-function
film, with instantaneous front positionat L(t), and with film uniform
Thm
d•-= faL'
(11)
whichupon integrationresultsin
thickness.
L(t) = x/2DGAt,
(12)
where the Green-Ampt(GA) film hydraulicdiffusivityis de-
Of_ 0 T(f ) -•-] Ot
Ox
'
fined as
(6b)
T
DGA = ( _fa/hf)'
(13)
By combiningthe definition of the film hydraulicdiffusivity (equation(1)) and (6b), we obtain the transientfilm flow The Green-Amptfilm hydraulicdiffusivitycan be determined diffusionequation: during a transientwetting processfrom the slopeof L versus the squareroot of time (equation(12)). Note thatthistransient O•=Ox D(f ) •xx' (7) film flow processis the surfaceanalogto one-dimensional infiltration or absorption,suchthat the film cumulativeinfilWhen D(f) is approximatelyconstant,(7) simplifiesto the trationI[L 3 L -•] is linearized
form
I = fax/2DGAt. Of
O2f
0--• = DaOX 2'
(8)
(14)
Sinceboth cumulativeabsorptionand cumulativeshort-term infiltrationin homogeneous porousmedia are givenby the
of thesorptivity S [L T-•/2] timesthesquare rootof where D a is the averagefilm hydraulicdiffusivity.While we product hasthe make use of transientfilm flow relationswhere gravityis neg- time [Philip,1969],the analogousfilm flow expression identical form ligible for purposesof experimentalcharacterizationof film hydraulicproperties,it is importantto recognizethe dominant influenceof gravityin larger-scaleflow.
butwithdifferent dimensions for thefilmsorptivity Sf [L2 3. Green-Ampt Analysis of Transient Film Spreading Onto Dry Surfaces Transientfilm flow can be studiedthroughmeasuringthe rate of film spreadingon initiallydry, hydrophilicsurfaces.We considersurfacesof impermeablesolidssuchthat matrix imbibition of water is unimportant,permittingisolationof film dynamics.Visual observationof wettingon suchsurfaceshas indicatedthat advancing wettingfrontsof filmsare often quite sharp,sothat theymaybe approximatedasstep-function wetting profiles.A simplemodelfor transientfilm flow under the step-functionapproximationcan be developedin a manner analogousto that of the early infiltrationmodel of Greenand Ampt [1911].The caseof transient,one-dimensional horizontal film flow will be consideredhere (Figure 2). The rate of film advance (dL/dt) in the step-function approximation is equatedwith the instantaneous flux at the inflow boundary (Q), suchthat
T-•/2]. Thisfilmsorptivity depends onsurface hydraulic properties(T andD), alongwith initial and boundaryfilm thicknesses(or matricpotentials),just as is the casefor transient wettingin porousmedia. Combining(14) and (15), the film
thickness, Green-Amptfilm hydraulicdiffusivity, andSf are related through
Do^ = 2fa' 4.
(16)
Film Hydraulic Diffusivities From Suction
Plate
Outflow
In additionto analysisbasedon the Green-Ampt approach,
we considered threemethodspreviouslydevelopedfor determininghydraulicdiffusivities of partiallysaturatedporousmedia that recognizeddependenceon volumetricwater content.
These methods alluseeithera pressure plateorsuctio n plate
deviceto imposestepchangesin potentialsalongone end of a samplewhich is in contactwith a porous,water-permeable plate [Klute, 1986]. The first method,proposedby Gardner Q- Wfadt ' (9) [1956],assumes that a constantD canbe assigned to relatively wheref, is the constantaveragefilm thickness, L is the wetted smallbut measurablechangesin water content(in our case, distancealongthe directionof flow, and W is the macroscopic film thickness).Sincethis assumption is somewhatrestrictive, wettedlengthtransverseto the flow direction.This flow rate is only some of its most basic aspectswill be examined and equatedwith product of W, T, and the magnitudeof the appliedto one set of data for the purposeof comparisonwith othermethods.This methodwasoriginallydevelopedusingthe effectivematric head gradient, dL
1740
TOKUNAGA
ET AL.: TRANSIENT
FILM
FLOW
matric potential as the time-dependentvariable,but as noted by Marshalland Holmes[1979],it canbe appliedto the water contentas well (saturationand potentialare assumedto be linearlyrelated over short incrementsmeasuredin individual tests). It appliesin the case of film flow to transient,onedimensionalflow on finite length(L) surfaces,initiatedby a stepchangein film thicknessat the boundaryin contactwith a pressureplate or suctionplate. In the caseof transientfilm flow in a suction plate device, the appropriate initial and boundaryconditionsare
5.
ON ROUGH
FRACTURE
SURFACES
Experimental Samples
To isolatefilm flow effectsfrom flow within the underlying matrix, it is desirableto conductexperimentson surfacesof impermeablesolids.Although it is possibleto conductfilm flow experimentson permeablerock, additionaltestsare required to characterizethe matrix contributionto flow [Tokunagaand Wan, 1997]. Subtractionof the matrix flow component for purposesof isolatingfilm flow on permeablesurfaces limitsthe accuracyof this approach,becauseof uncertaintyin the matrix flux. Thus, for purposesof testingconcepts,two types of impermeableglasseswere used: a glasscast of a f = fi, O --0. granite fracture. The castingmethod, photographsof glass For constantD, the solutionto (8) and (17) is givenby a casts,and resultsof varioushydraulicexperiments on castsare Fourier seriesin which only the first term is significantafter givenby Wan et al. [2000].The toughenedglasssurfacecont > O.1L2/D.After suchtime,the average film thickness is sistedof a 3.2-mm-thickglassplate, abradedon one sidewith givenby 80-gritsiliconcarbide.Both the glasscastand toughenedglass samplesused in the Green-Ampt wetting experimentswere 120 x 160 mm in bulk surfacearea. Suctionplate experiments were only done on a roughenedglasssamplesincethe analythomogeneoussurThe D obtained by fitting the above equationwill later be ical procedureassumesa macroscopically referred to as a result of Gardner's method 1. face. For these latter experiments,a much smaller, 6-mmA second,more generalmethodfor determininghydraulic tall x 15-mm-widex 3-mm-thick,samplewaspreparedin the diffusivitiesin soilswasalsoproposedby Gardner[1962],again sameway as the larger, 80-grit abradedglassplate. Roughnessmeasurementswere obtained with an atomic applicableto outflow from a pressureplate, pressuremembrane, or suctionplate device.A varyinghydraulicdiffusivity, forcemicroscope (AutoprobeM5, Park ScientificInstruments, dependenton water content(film thickness), is obtainedwith Sunnyvale, California)and a laserprofilometer(UBM, Sunnythis method and will be referred to as Gardner's method 2. It vale, California)for fine-scalefeaturesand with a laser proappliesin the caseof film flow to transient,one-dimensional filometry(LK-081 CCD laser displacement sensor,Keyence flowon finitelength(L) surfaces, initiatedby a stepchangein Corp., Woodcliff Lake, New Jersey)on coarser,long-range film thicknessat one boundaryin contactwith a pressureplate features.The AutoprobeM5 has a resolutionof 0.025nm and or suctionplate.The solutionto (7), subjectto the sameinitial a lateral rangeof 100 x 100/am. However,sincethe maximum measurablez variationof this atomicforce microscope is only andboundaryconditions(equation(17)), is 8 /am, scanson rough surfacesmust be confined to even 4L 2 df smallerareasthan the instrument's100-/amx 100-/amrange. The UBM laserhasa spotsizeof 1/am, a z measurement range The final method appliedin this studyis that of Passioura of 100/am, and a z resolutionof 0.06/am. The LK-081 laser has rangeof 30 mm,a spotdiameterof [1976],whichevaluatesthe hydraulicdiffusivityat the end of a vertical(z) measurement about 70/am, and an uncertainty of _+35/amin z. The View the soil column(x = L), basedon the changein the rate of water loss with change in averagewater content. Passioura Precis3000coordinatemeasuringmachineusedon both of the showedthat his analysisagreedwell with numerical simula- lasersystemshas resolutionand repeatabilityin the x-y plane tionsof outflowexperiments (typicallycalculating D to within of _+3 /am. Like natural fractures [Brownand Scholz,1985; rough5% of true valuesbut overestimating by about 25% as equi- Powerand Tullis,1991],the castexhibitsscale-dependent roughness librium is approached).In termsof a transient,suctionplate ness,with measuredvaluesof the root-mean-square from 600 to experiment,the Passioura[1976] methoddeterminesthe film (rmsr)rangingfrom 4 to 7/am (over20-/amscans), 800/am (over 10-mm scans), and from 2 to 3 mm (over 100-mm hydraulicdiffusivitythrough scans).The siliconcarbide-roughened glasssurfacehad a staL2dO tionary (scale-independent) rmsr of 9 /am for scanlengths greaterthan 500/am. Examplesof roughness profilesof glass castsand roughenedglassare givenby Wan et al. [2000]. where
( -4•-• ,r2Dt• f(t)=fs+ •8(fi- f) exp •. (18)
D(f) = (f _f•),rr 2dt'
(19)
D(fL)= 2 dfa'
(20)
of
Q= -
oL
ot
(21) 6. Measurement of the Green-Ampt Film Hydraulic Diffusivity
fL
- 0.61dInD(fD'
of horizontalfilm advanceover two initially (22) dryMeasurements surfaces,one of a glasscastof a granitefracturesurface
where fa is the averagefilm thickness,measuredwithin the and the other of a toughenedglassplate, were performed central region of a test surface.In the experimentalsection, within a sealableacrylicplasticchamber(Figure3a). The infilm hydraulicdiffusivitieswill be determinedusing each of flow water sourceconsistedof a saturated,0.5-bar, high-flow thesemethods,i.e.,by applying(16), (18), (19), and(20)-(22). ceramicblock(Soilmoisture EquipmentCorp.,SantaBarbara,
TOKUNAGA
ET AL.: TRANSIENT
FILM
FLOW
ON ROUGH
FRACTURE
plasticenclosure
SURFACES
1741
(a)
0.5 bar ceramic
b.l Iøckfilter •
free water
/paperglasssample L(t)
surface
•
supportblocks
.,:....,... ,:.:
•-••..; '•;• ......... •-•:::' ..... ......................... -,Figure 3. (a) Systemused'to measureGreen-•pt film parameters.At time-zero the inflow edge of the sampleis contactedto the water rese•oir, andL(t) is visuallyrecorded.(b) Photographof a wettingfront :• ...................
advancingon a glasscastfracture surface.
California) in contactwith a free water reservoir.A layer of coarsefilter paper is placedover the ceramicto providegood hydrauliccontactbetweenthe ceramicand the test sample. The filter paper interfaceprobablyalso significantlylowered the boundarysourcehydraulicresistancesince it also wicks water up directlyfrom the free water reservoir.The sampleis supportedhorizontally,abovethe free water surface,with plastic blocks.At time zero,thex = 0 edgeof the sampleis placed in contactwith the filter-coveredceramicplate, andthe visually observedwettingfront position(minimum,average,and maximum) is recordedas a functionof time (Figure 3b). The matric head at the inflowboundarywas -6 mm for the roughened glassplate and -12 _+ 5 mm for the glasscast. The variabilityin the latterwasdue to topographicvariationsalong
the x = 0 edge of the cast.At the end of the experimentthe samplewas quicklyplacedon an electronicbalancewithin an acrylicplastic,humidifiedbox, to determine the cumulative mass of water
7.
imbibed
Measurement
onto the test surface.
of Film
Thickness
and Its
Relation to Matric Potential and Hydraulic Diffusivities
With
a Suction
Plate
Device
Since a wide variety of suctionplate and pressureplate methodshave been successfully usedfor determinationof hydraulicpropertiesof partiallysaturatedporousmedia, a modifiedversionof the suctionplatewasdevelopedfor purposesof measuringfilm hydraulicpropertieson rough surfaces.This
1742
TOKUNAGA
airatlocal
ET AL.:
TRANSIENT
FILM
FLOW
toregulated vacuum
atmospheric Pß t air atlower pressure
Mylar
window
x-rays
i
i
water
10 mm
front view
reservoir
side view
approx. scale
ON
ROUGH
FRACTURE
SURFACES
of the air headspacein the solution reservoir.This is done through removal or injection of air via a syringewhile its hypodermicneedleis insertedinto a septumstopperandwhile monitoringthe pressuretransducerdisplay. Experimentswere conductedat beamlineX26A of the National SynchrotronLight Source (NSLS), BrookhavenNational Laboratory[Suttonet al., 1994;SuttonandRivers,1999]. Overviewsof synchrotronX-ray applicationsin Earth sciences are given by Brown and Parks [1989], Schulzeand Bertsch [1995], and Smith and Rivers[1995]. A brief descriptionof pertinentoperatingconditionsfor this experimentis provided here. The monochromator was set to 12.70 keV, about 34 eV
fluorescence / monochromatic x-ray x-rays/ beam from synchrotron x-ray .:•'"' •':":-
detector
abovethe Se(VI) K absorptionedge.Althoughthisbeamlineis routinelyusedas an X-ray microprobeand for microspectroscopywith spot sizesdown to 5 /am, the focusingopticswere removedduringtheseexperimentsin order to utilize a much broaderincidentX-ray beamof about200/am. The largerspot sizewaspreferredin the presentapplicationsinceit permitted averagingover greater areasof the rough surfaceand maximized the incident X-ray flux. X-ray fluorescencefrom the selenatein the water film was measuredwith an energydispersivedetector(lithium-driftedsilicon).Blankmeasurements on a quartz glassblock prior to exposureto seleniumcon-
'"'•'• ................... •' •water fracture surface I • /•/•• film
fluorescence
x-raysfrom
•
•'-••./•••_• •"•'•
withtracer
•
I
I
•
solute tracer
incident x-rays
close-up of x-ray interaction
with film
(plan view)
firmed
that matrix
Se concentrations
were below
detection.
The averagewater film thicknessfrom a givenmeasurement was calculated
from a linear calibration
curve constructed
from
standardswith knownSe surfaceconcentrations (Se massper unit area). Thesestandardsampleswere preparedby micropipettingselectedamountsof selenatesolutionsonto smallfilter
Figure 4. Sample cell for synchrotronX-ray fluorescence paperpads(ranging in areafrom4 to 25mm2).Theresulting measurementsof film thicknessversusmatric potential.
arealSeconcentrations rangedfrom7.8to 94 ngmm-2, and
equivalentfilm thicknesses rangedfrom 0.82 to 9.8/am for the 242-mM solution. Standard sampleswere scannedwith the device(Figure4) is designedto measureequilibriumrelations X-ray microprobeto obtain the areal distributionof Se within betweenmatric potential and averagefilm thicknessand to eachpad. Edgesof filter padsshowed20-30% enrichmentof measurefilm hydraulicdiffusivitieson surfacesof imperme- the Se tracer due to enhancedevaporationand about 10% able solidsthroughX-ray fluorescenceof a solutetracer. A relative depletionfrom centralregions.The spatialmap of Se smallblockof the roughenedglassis placedon top of a high- concentrations on eachpad was averagedfor use in later film conductanceceramicplate (0.5 bar, SoilmoistureEquipment thickness calculations. A range of matric potentialsfrom -0.3 to -50 kPa was Corp., Santa Barbara, California), with the roughenedglass surfaceorientedvertically.Filter paper is placedbetweenthe examined.For an equilibrationat a givenmatricpotential,the sampleandceramicto improvehydrauliccontact.The aqueous samplechamberwasrasteredin front of the stationaryincident solutionusedin this deviceis spikedwith a tracer excitedby X-ray beam in a seriesof horizontaland/orverticalsteps(0.5 X-ray absorptionand detectableby X-ray fluorescencespec- mm in eachdirection)with the motorizedstageof the X26A trometry. The tracer used in this experimentwas selenate end station.This procedurepermittedareal averagingof X-ray (SEO42-), addedasa sodium saltat a concentration of 242mM. fluorescencewithin the central 3-mm x 3-mm region of the The selectionof Se as a tracer wasbasedon its large K-edge roughenedsurface.Measurements at a givenspoton the rough X-ray fluorescence yield of 60% [Kostrounet al., 1971]and its glasssurfacewere obtained within 5-15 s. Energy balance very low fluorescentX-ray absorbance by the Mylar film used calculationsindicatethat the local temperatureincreaseunder for thesamplechamber window.The choiceof SeO42in par- theseoperatingconditionswaslessthan IøC. Rasteringof the ticularwas basedon its very low affinityto most mineral sur- samplerelativeto the X-ray beampermittedareal averagingof faces[Nealand Sposito,1989].The sample,ceramicplate, and film thickness measurements as well as minimization of local solutionreservoirare containedwithin an acrylicplasticblock, thermal loading from the incident beam. For a given step sealedto minimizeevaporation,but with the samplehousing changein boundarymatricpotential,the samplewasperiodipinhole-ventedto maintain the air in equilibriumwith local cally scanneduntil the averageSe X-ray fluorescencesignal atmospheric pressure.The Mylar film windowsealingthe front becameconstant,indicatingequilibrium.The observedstable face of the samplehousingpermits incident and fluorescent equilibriumfluorescence indicatedthat thermal loadingfrom X-ray fluxeswith insignificantattenuationat the photonener- the X-ray beam had only a minor effect on thesemeasuregiesused.The regulatedpressureof the air headspaceoverly- ments. The equilibrium averagefilm thicknessat the preing the tracerreservoirsolutionis monitoredcontinuously with scribedmatric potential was calculatedfrom the equilibrium a pressuretransducer(Tensimeter, Soil MeasurementSys- Se X-ray fluorescenceintensity(normalized to the incident tems, Tucson,Arizona). The matric potential at the ceramic photon flux and count time) by referencingthe calibration surfaceis controlledby regulatingthe subatmospheric pressure samples.Transient film thinningresultswere used to deter-
TOKUNAGA
lOO
ß
i
ß
i
ß
i
ß
i
'
ET AL.: TRANSIENT
i
ß
i
'
i
'
i
ß
i
'
FILM
i
ß
I
FLOW
ON ROUGH
E E
SURFACES
1743
'
90
f= 1.13(-hm)-0.370 , r2=0.955
glass cast ofgranite fracture T.-'' •
80
FRACTURE
10
D(Green-Ampt) =3.2x10' 7m2/s ([.-•
70 60 50
ß
40 30 20
lO o
••
•
i
10
.
i
20
.
i
30
.
- D(Green-Ampt) =1.4x10"m'/s I
40
•
I
50
,
I
60
,
I
70
,
I
80
,
I
90
.
i
.
i
100 110 120
10
0.1
0.01
0.001
-matrichead, -h m (m)
square-root seconds
Figure 5. Wetting front distanceversussquareroot of time for film flowover a toughenedglasssurface,and a glasscastof a granite fracture surface.Data points indicateaveragefront positions.Range bars indicateapproximaterangesin wetting front positions.
1
Figure 6. Film thicknessversusmattic head, for the roughened glasssurface,measuredwith synchrotronX-ray fluorescence of a selenate
tracer.
1990;Rieu and Sposito,1991a,b; Bird et al., 1996;Perrieret al., 1996].The mannerin whichthe exponentof a powerlaw fit to film thickness-potential relationsmay be related to fracture surfacestructureand to thesepreviousstudiesof unsaturated porousmedia is not yet known. 8. Results and Discussion An important differencebetween the singlefracture surWetting of the horizontallyorientedglassfracturecastand faces investigatedhere, and porous media, is that the latter toughenedglassplate proceededwithoutocclusionof residual have well-definedupper limits for volumetricwater content dry regions.Cumulativewater film uptake on the two glass (i.e., the porosity),whereassinglefracturesurfacesgenerally surfaces proceededin accordance with the expected(equation do not.The lackof a well-definedupperlimit on film thickness, on (12)) squareroot of timeproportionality(Figure5). The cause hencealsoa similarlackof a "saturated"film transmissivity of the initiallysteeperslopefor wettingof the toughenedglass unbounded(effectivelyinfinite aperture) fracture surfaces, (fortimes-
1E-13
._
1.0
ß
X-ray fluorescencedata
0.5
Gardner
Method
E •
1E-14
E
1E-15
._
1
,
1E-16 0.0
,
0
I 1000
,
I 2000
,
I 3000
•
I 4000
1
,
5000
time sincestep potentialchange, s
Figure 7. An exampleof transientwater film thinningon roughenedglass,in responseto a stepchangein the matric potentialat the boundaryfrom -1.2 to -20 kPa.
I
I
I
2
3
4
,
5
averagefilmthickness, f (pm)
Figure 9. Film transmissivity functionT(f) for the roughenedglasssurface,obtainedby multiplying D(f) by df/dhm. ent approaches.The effectiveD for the 1.0- to 1.2-/•m film
thicknessrange,calculatedfrom Gardner'smethod1 (dea constantdiffusivity,whereasD decreases with decreasing scribedpreviously),is alsoincludedin Figure8. Note that it film thickness.
alsoagreeswell with resultsobtainedfor variableD.
In order to investigatea wider rangeof film thicknessdependentdiffusivities, segments of the continuous D(f) rela-
monotonic decreases with decreased film thickness. In the limit
The measuredD(f)
shownin Figure8 collectively exhibit
tion were obtained with Gardner's method 2 and with Passioura's method. Results of such calculations are shown in
of thinnerfilmsit mightbe possiblethat increased valuesof D develop, aswasfirstproposed for unsaturated porousmediaby Figure8, wheretime-dependent film thinningdata are com- deGennes[1985],whodescribed sucha trendas"hyperdisperbined with (19) and (20)-(22) to obtain the film hydraulic sion."Hyperdispersive D waspredictedby de Gennesfor pardiffusivityas a functionof averagefilm thickness. Both Gard- tially saturatedporous media in which water was retained ner'ssecondmethodandthe Passioura analyses areplottedfor primarilyin isolatedpendularringsand pits and interconone of the outflowstepsin orderto permitcomparisons. The nectedthroughthin films.Hyperdispersion hasbeen observed two methodsyieldedsimilarD(f) relations.The Passioura- for someporousmediabyBacriet al. [1985,1990]and Toledo basedanalyses wereappliedto thetwootheroutflowstepsalso et al. [1993]. In the presentcasethe lack of observationof shownin Figure 8, indicatingfair reproducibilityas well as hyperdispersive D(f ) suggests that film flowis controlledby continuityof resultsobtainedfrom adjacentsegmentsof flowalonginterconnected surfacechannels ratherthanbyflow D(f). The Green-Amptintegraldiffusivityfilm from the ex- in truethinfilms.Microscopic observations andprofilometry of perimenton initiallydryroughglassis alsoplottedin Figure8, the test surfaceslend supportto the expectationthat flow for its associated gravimetrically measuredaveragefilm thick- occursprimarily through such interconnectedsurfacechanness.Good agreementis obtainedfrom thesetwo very differ- nels,but future experiments with higherspatialresolutionare neededto gainan improvedunderstanding of thisaspect. In order to obtainthe film transmissivity andvelocityrela-
tionsfor the roughglasssurface,the composite D(f) curve (Figure8) was first multipliedby the first derivativeof the powerlaw fit to thef(hm) relation(Figure6). Thisresultsin thefilmtransmissivity relationshownin Figure9, alongwithits estimated rangeof uncertainty of 50%, obtainedbyestimating therelativeuncertainties in D anddf/dhmto be 30% and40%, respectively. Dividingthis film transmissivity functionby its corresponding averagefilm thicknessgivesthe averagefilm velocity,shownas a functionof f andhm in Figures10a and 10b,respectively. The relativeuncertaintyin v(f) was estimatedto be 54%,basedon relativeuncertainties for T andf of 50% and 20%, respectively. Theseresultsshowthat average unit gradientfilm velocitieson thisimpermeable glasssurface
• 167 .>
/
Passiouramethod i
._._o •
Gardner
method 1
8
;
;
Gardner method 2
ß E
Green-Amptmethod
......
compositeD(f)
+30% limits(composite)
169 0
....
i.•/ ....
1
i
2
....
i
....
3
i
4
....
5
averagefilmthickness, f (pm)
Figure 8. Film hydraulicdiffusivityfunctionsfor the roughenedglasssurface,calculatedfrom suctionplateoutflowmeasurements. Uncertaintiesin the Passioura[1972]and Gardner [1956] method2 are shownas the boundingdashedlines. Uncertaintiesin the Gardnermethod1 are indicatedby the symbolsize.Also shownis the film hydraulicdiffusivityobtainedwith the Green-Amptapproach.
begin to exceedour selectedthresholdfor fast flow of 10 m
yr-• whenaverage filmthicknesses aregreater thanabout2.5 /•m andwhenmatricpotentialsare greaterthan about- 1 kPa. Thisresult,alongwith our previouswork [Tokunaga and Wan, 1997],indicatesthat near-zeromatricpotentialsare required for fastfilm flow.It shouldbe notedthat our currentlyavailable databaseis still very limited,so that comprehensive delineationof conditions neededfor fastflowis stillpremature. In consideringthe presentresults,it shouldbe remembered
TOKUNAGA
ET AL.: TRANSIENT
FILM
FLOW
ON ROUGH
FRACTURE
SURFACES
1745
selectedvaluesof mean aperture) will also be conductedin order to begin integratingsurfaceand aperture contributions
1 E-05
in unsaturated
fracture
flow.
1 E-06
E
1E-07
ß --
1E-08
9.
1 E-09
1E-10
I
I
I
2
3
4
filmthickness, pm
(a) 1E-05
1 E-06 E
1E-07
Summary
Aspectsof film flow in unsaturatedfracturesexaminedin thisstudyincludedconstitutiverelationsfor transientfilm flow and measurements of film hydraulicparametersunder transient conditions.By consideringfilms on fracturesurfacesas analogsto water in partially saturatedporousmedia, the film hydraulicdiffusivityand equationfor transientfilm flow were obtainedfrom their porousmediumcounterparts,the hydraulic diffusivityand the Richards equation. Several different measurementtechniques,based upon previouslydeveloped soilphysicsmethods,were developedfor applicationto studies of film flow.The experiments describedin the presentstudyon a toughenedglasssurface(9-•m root-mean-square roughness) showedthe averagefilm thicknessdependenceon mattic potential is approximatedas a powerfunction.It was alsoshown that the film hydraulicdiffusivityincreaseswith increasedfilm thickness (andwith increasedmatticpotential).Fastfilm flow, definedhere asflow with averagevelocitiesgreaterthan 10 m
yr- • underunitgradient conditions, wasobserved for average 1 E-08
film thicknessesgreater than 2.5 •m and matric potentials greater than -1 kPa on the toughenedglasssurface.
1 E-09
Acknowledgments. We thank Tom Orr, Andrew Mei, Jim O'Neill, Robert Conners,and Keith Olson,of LBNL, for assistance in partsof this work; Grace Shea-McCarthy,of the University of Chicago, for 1E-10 • , • , • , • , • , • , assistance at NSLS X26A; and Vivek Kapoor for providinga copyof -12000 -10000 -8000 -6000 -4000 -2000 0 hisreport.The helpfulinternalreviewcommentsby GarrisonSposito (b) matricpotential, Pa (Universityof California,Berkeley,andLBNL), SrinivasVeerapaneni Figure 10. Unit gradientaveragefilm velocityfor the rough- (LBNL), and RobertW. Zimmerman(Imperial College,London,and We alsothank Mike Nicholl and enedglasssurface,obtainedby dividingT(f) byf. Average LBNL) are gratefullyacknowledged. This work film velocitiesare relatedto (a) averagefilm thickness and (b) two anonymousreviewersfor their very helpful suggestions. was carried out under U.S. Departmentof Energy (DOE) contract film matric head. DE-AC03-76SF00098,with fundingprovidedby the DOE, BasicEnergy Sciences,GeosciencesResearchProgram. This work was also supportedin part by DOE-Geosciencesgrant DE-FG02-92ER14244 that mostof the measurements were obtainedfor a roughened (S.R.S.).Researchwascarriedout (in part) at the NationalSynchroglasssurfacethat is much smoother(9-tamroot-mean-square tron Light Source,BrookhavenNational Laboratory,which is supported by the U.S. Departmentof Energy,Divisionof Materials Sci-
roughness) thanmanynaturalfracturedrocksurfaces (at com- ences and Division of Chemical Sciences under contract DE-AC02parablemeasurement lengthsin the millimeterto centimeter 98CH10886.We thankthe staffof NSLS for providingthe synchrotron range). Furthermore, since our test surfacewas artificially radiation. roughenedby abrasion,its topographyis not stronglyposition dependentand is bestcharacterizedasstationaryor Euclidian. Naturalrockfracturesurfaces,on the otherhand,haveprofiles that are typicallystrongly position dependent and are best describedas nonstationaryor fractal [e.g.,Powerand Tullis, 1991].Surfaceswith higherroughnessare expectedto permit thicker films only at very near zero matric potentials,since filling of larger-amplitudefracturesurfacedepressions will be associatedwith larger radii of air-water interfacialcurvature. Note that the inverserelation betweenmatric potential and interfacialcurvature,and the scale-dependent fracturesurface roughness imply that the area overwhichcontinuumfilm hydraulicpropertiesneedsto be averagedincreasesas zero matric potentialis approached. Further experimentsare beingdesignedto improveour understandingof surfaceroughnessinfluenceson film flow, applyingthe methodspresentedhere to glasses and othermineral surfaces with varyingsurfaceroughness. Experimentson single glassfracturecast surfacesand mated glasscastsurfaces(at
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(ReceivedAugust3, 1999;revisedMarch 21, 2000; acceptedMarch 22, 2000.)
