wetting phase permeability in a partially saturated...

Proc. Fifth Annual Int. High Level Radioactive Waste Management Conference, 2007-19, May 22-26, Las Vegas, NV, USA

WETTING PHASE PERMEABILITY IN A PARTIALLY SATURATEDHORIZONTAL FRACTURE

M.J. Nicholl and R.J. GlassGeohydrology Division 6115Sandia National Laboratories

Albuquerque, NM 87185

Abstract

Fractures within geologic media can dominate the hydraulicproperties of the system. Therefore, conceptual models used toassess the potential for radio-nuclide migration inunsaturated fractured rock such as that composing YuccaMountain, Nevada, must be consistent with flow processes inindividual fractures. A major obstacle to the understandingand simulation of unsaturated fracture flow is the paucity ofphysical data on both fracture aperture structure and relativepermeability. An experimental procedure is developed forcollecting detailed data on aperture and phase structure from atransparent analog fracture. To facilitate understanding ofbasic processes and provide a basis for development ofeffective property models, the simplest possible rough-walledfracture is used. Stable phase structures of varying complexityare created within the horizontal analog fracture. Wettingphase permeability is measured under steady-state conditions.A process based model for wetting phase relative permeabilityis then explored. Contributions of the following processes toreduced wetting phase permeability under unsaturatedconditions are considered: reduction in cross-sectional flowarea, increased path length, localized flow restriction, andpreferential occupation of large apertures by the non-wettingphase.

Introduction

Fractures within geologic media tend to dominate hydraulicbehavior of the system under both saturated and unsaturatedconditions, particularly when the matrix is of lowpermeability. When fully saturated with a flowing phase,individual fractures act as conduits through the media andallow free passage of that phase between the matrix blockswhich bound the fracture. As phase saturation decreases, flowbetween the bounding matrix blocks and in-plane flowthrough the fracture are both restricted. Permeability of thefracture to in-plane flow of a given phase is a discontinuousfunction of phase saturation. Under steady state conditions,flow in a single phase along the plane of a fracture will beconfined to the connected network of apertures filled by thatphase; if lowering phase saturation breaks the continuity ofthat network, then in-plane flow will cease. The effects ofphase saturation on fracture permeability are of significantinterest to a variety of disciplines within the geo-sciences,including petroleum extraction, geothermal energy, andground water contamination.

Exploration of two-phase fracture flow has been stimulated inrecent years by the proposed geologic isolation of high-levelradioactive waste in unsaturated fractured rock, such as thatlocated at Yucca Mountain, Nevada. Performance assessmentexercises to consider the possibility for off-site migration ofradio-nuclides are used to evaluate the suitability of apotential repository site1. Numerical models used inperformance assessment exercises are based on conceptualmodels of flow and transport through unsaturated fracturedrock. Exploration of flow in individual fractures is a necessarystep towards the construction of conceptual modelsconsistent with real world flow processes.

The vast majority of recent work on two-phase flow inindividual fractures has involved numerical simulation ofboth the aperture field and flow process; see Murphy andThompson2 for a concise review of previous work. Suchsimulation has been necessitated by a paucity of physical dataregarding both spatial structure of fracture apertures and flowprocesses within a variable aperture field. Two-dimensionalaperture fields measured at the sub-millimeter scale have onlyrecently been published3,4. In addition, to our knowledge,there are no published data of pressure/saturation or relativepermeability relations measured on single natural fractures.Relative permeability has been measured recently in atransparent natural fracture cast where phase saturationgeometry could be observed5.

We begin our study of fracture relative permeability with ananalog rough-walled fracture that is fabricated from two sheetsof textured plate glass. The resultant statisticallyhomogeneous, transparent analog fracture eliminates theeffects of matrix imbibition, aperture heterogeneity, anddifferential surface wettability while allowing data collectionthrough transmitted light visualization. Gravitational effectsare eliminated by orienting the analog fracture along thehorizontal plane. Through use of this simplest possible roughwalled fracture, we are able to develop techniques for spatialcharacterization of aperture and phase structure. Thosetechniques are then used to explore the effects of phasestructure on fracture relative permeability. Use of astatistically homogeneous analog fracture eliminatesmacroscopic control of the phase structure by the aperturefield. The techniques and understanding developed in thisinvestigation will be applied in future studies to statisticallyheterogeneous aperture fields as exhibited in cast replicas ofnatural fractures and designed into manufactured analogfractures. Such studies are necessary to build a properfoundation for valid simplifications of fracture behavior


required in effective property models for use in theperformance assessment exercises.

Here we employ optical visualization techniques to exploresteady-state flow of the wetting phase through two-phasestructures of varying saturation and complexity. Water is usedas the wetting phase, and air as the non-wetting phase. Webegin by providing a short description of the experimentalapparatus, imaging system, and technique for measurement ofrelative permeability. A detailed description ofmethodologies for characterization of aperture and phasestructures through use of simple light absorption theory isthen presented. A course of experimentation in two-phasestructures of varying complexity and saturation is described.Measurements of apertures filled by the non-wetting phase areused to compare experimental phase structures to predictionsbased on current conceptual models of fracture-matrixequilibrium (e.g. Wang and Narasimhan6). Measurements offracture relative permeability are then presented and discussedrelative to a simple process oriented model. We conclude bysummarizing results and providing a short discussion offuture work directed towards the development of processoriented effective property models.

We note that strong analogies can be made from the two-phasesystem considered in these experiments consisting of a liquidwetting phase (water) and a gaseous non-wetting phase (air) toother systems of importance. Because experiments wereperformed under steady-state conditions, spatial structure ofthe entrapped phase is invariant, acting simply to preventwetting phase flow through the occupied regions. Thisscenario is directly analogous to single phase flow throughfractures exhibiting significant contact area. Experimentalresults are also applicable to steady state flow through astable two-phase structure of immiscible liquids, such asoil/water or more generally water/NAPL.

Experimental Approach

An analog rough-walled fracture is fabricated by holding twotextured glass plates (30.5 x 15.25 cm) in close contact.Application of a confining pressure (20 psi) assures that theresultant aperture field exhibits statistical homogeneity. Flowis directed along the long axis of the analog fracture throughimplementation of no-flow boundaries along the fracturesides. Fluid is uniformly supplied to one end of the fractureusing a pressure-stabilized peristaltic pump, creating aconstant flow boundary condition. At the downstream end ofthe fracture, a constant head boundary is implemented, withmass flow rate measured at the outflow. Manometers located atthe upstream and downstream boundaries are used to quantifydifferential pressure head; hydraulic conductivity (K) maythen be calculated through use of Darcy's law. As fluid densityand viscosity were held constant, relative permeability (kr)may be defined as the ratio of measured K to that obtainedunder saturated conditions (Ks).

kr = K

Ks(1)

The transparent nature of the analog fracture allows collectionof wetted structure data through the use of digital imagingtechniques. Hydraulic conductivity measurements wereconducted with the test fracture placed on a horizontallyoriented table that is back-lit with high-frequency fluorescentlamps. The imaging system focused on the fracture plane iscapable of obtaining data at 2048 x 2048 pixels of spatialresolution with a dynamic range of 4096 gray levels; for theanalog fracture considered in these experiments, pixel sizewas on the order of 2 x10-4 cm2. Optical contrast was enhancedby staining the experimental fluid; a mixture of 1g FD&Cblue #1 to 1 liter of de-ionized water was used. Wettingproperties of the experimental fluid were evaluated usingglass capillary tubes, and found to be consistent with de-ionized water.

aperture structure characterization

Mean aperture and saturated conductivity of the analogfracture were measured directly, while spatial variation in theaperture field was characterized using simple light absorptiontheory. The mean aperture (â = 0.0215 cm) was measured byvolumetric imbibition into the fracture under dry initialconditions. Measured saturated conductivity was 2.89 cm/s ata Reynolds Number (Re) of 3.1; in previous experiments7

using this analog fracture, Ks was found to remain constant forRe between 0.27 and 5.93.

Each pixel of the imaging array records the luminousintensity (I) of light transmitted through a small region of theanalog fracture. The absorption of light (initial intensity = Io)by a series of substances of thickness (di) with absorptioncoefficients ( α i ) is given by Lambert's Law (equation 2).

I = Io exp−(α 1d 1+α 2d 2+ ...) (2)

In order to measure the aperture, dap, it is necessary todecouple light absorption due to fluid within the aperturefrom that caused by the rest of the test cell and fracture plates.This is done by collecting an intensity image from the fracturesaturated with a clear fluid (Icl) and saturated with a dyed fluidof equivalent refractive index (Id). In order to minimizedispersion caused by the textured glass surface, a basesolution of 135% by weight sucrose was used to matchrefractive indices with the glass; the dyed solution alsocontained 0.25 gm per liter of FD&C blue #1 dye. Recordedlight intensity of the dye filled fracture is given by:

Id = Icl exp−α d dap (3)

Equation 3 is first solved for addap at each pixel locationwithin the aperture field. This result is then normalized usingthe measured mean of all addap values. Multiplication of thenormalized aperture field by the measured mean aperture (â) ofthe analog fracture yields the spatial structure of the aperturefield at the pixel scale; the resulting aperture distribution isshown as Figure 1. Subdividing the aperture domain intopixels implicitly averages the local data; therefore, the


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Figure 1: Aperture Distribution of Analog Fracture: Aperturedistribution of the analog aperture field is shown, along withdistributions of the aperture field as predicted by Lambert'sLaw and a numerical model based on profilometermeasurements as presented by Glass8.

distribution of dap given by equation 3 does not whollycapture the full range of apertures known to occur within theanalog fracture. The range should extend from 0 at contactpoints to approximately twice the mean aperture, rather thanthe range of 0.004 to 0.032 cm as obtained from equation 3. Inorder to correct for these discrepancies, the aperture field wasadjusted through multiplication by a stretch factor that variedlinearly with aperture between its lowest value (d = 0.004 cm)and its highest value (d = 0.032 cm), such that the expectedminimum and maximum values were preserved (Figure 1).Aperture distribution of a numerical model of this analogfracture based on profilometer measurements is shown forcomparison8. A gray-scale image, depicting spatial structureof the reconstructed aperture field is presented as Figure 2; forclarity, only a small section (~5 cm square) is shown.

phase structure delineation

In order to evaluate phase saturation and structure, it isnecessary to delineate the spatial distribution of each phase.Pixels were assumed to be occupied fully by a single phase; aspixel size is slightly smaller than the mean aperture, this is areasonable assumption. Differences in light attenuationbetween the phases were used to transform phase structureimages into binary arrays. In these arrays, pixels occupied bythe wetting and non-wetting phases are represented by 0's and1's, respectively. Light passing through a single phase ofvariable thickness produces a distribution of intensities.

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Figure 2: Spatial Structure of Model Aperture Field: A 5 cmsquare section of the model aperture field is shown in grayscale. Axes indicate size of the section in pixels, while thecolor bar indicates aperture size.

Assuming that light scattering at phase interfaces has aminimal effect, the distribution of intensities collected from atwo-phase structure will be the sum of the individualdistributions. Where the light absorbencies of the two phasesare significantly different, the composite distribution willshow two distinct local maxima, such as that seen in Figure 3;the dashed lines illustrate postulated distributions associatedwith individual phases. It is assumed that the centralminimum of the composite distribution (Ic in Figure 3)provides a minimum error estimate for delineating phaseoccupation on the basis of pixel intensity. For the two-phasesystem considered here (air and water), pixels with intensityvalues below Ic are assumed to be occupied by the wettingphase while those above Ic are assumed to be filled with thenon-wetting phase.

Preliminary investigations demonstrated that a single valueof Ic was not appropriate for the relatively large domainconsidered here (1.65 x 106 pixels). This problem wasresolved by operating on sub-regions of the image, 128 x 128pixels in size, such as that shown in Figure 4a. Whilecomprising less than 1% of the total image, each regioncontains sufficient information to form an appropriatehistogram (16384 pixels) as shown in Figure 4b. Prior todetermining the central minimum, an inverse distanceweighted moving average algorithm was used to smooth thehistogram, eliminating spurious local features. The smoothedhistogram was then searched to find the two most significant


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Figure 3: Distribution of Light Intensity in an Image of aTwo-Phase Structure: This histogram shows the distributionof intensities collected from experiment 3-3 (shown as Figure5c). The low intensity mode of this composite histogram isassociated with pixels filled by the wetting phase and thehigh intensity mode with the non-wetting phase. While theexact shape of the individual distributions is unknown, thedashed lines are used to illustrate likely forms for theindividual distributions.

local maxima and the associated central minimum, which wastaken to represent Ic for that sub-region. All pixels within thesub-region that are less than Ic are set to 0 and the remainderare set to 1 (Figure 4c). This procedure was then repeated forall such sub-regions within the image. Multiplication of thisbinary array by the reconstructed aperture field yields thethree-dimensional spatial structure of the non-wetting phase.By reversing 0's and 1's within the binary array, spatialstructure of the wetting phase is obtained.

Results and Discussion

Seven experiments were performed, each with a different phasestructure. As discussed below, three different methods wereused to form the phase structures. Steady-state flow of thenon-wetting phase was established prior to measurement of K.Images collected before and after permeability measurementswere used to verify stability of the wetted structure. Theseimages were also used to measure wetting phase saturationand spatial distribution of the phases; experimental data issummarized in Table 1. After each measurement, a series ofcomputer controlled solenoid valves were used to introduce asharp tracer front across the full width of the analog fracture.Subsequent images of tracer advancement serve to illustratechanneling of flow by the phase structure; thereby providingqualitative information regarding the effects of phasestructure on flow path length and spatial variation in thecross-sectional area available for flow.

Exp #

Sw kr aw

aw

a ττττ Type

1-1 .906 0.715 .0209 .975 .831 11-2 .955 0.887 .0212 .989 .949 12-1 .554 0.109 .0183 .852 .272 22-2 .617 0.148 .0189 .881 .310 23-1 .676 0.197 .0194 .907 .355 33-2 .640 0.145 .0191 .891 .285 33-3 .663 0.160 .0194 .902 .297 3

Table 1

formation of phase saturations and structures

Three general classes of structure were considered; the first(Type 1) was intended to be consistent with assumptions ofthermodynamic equilibrium between phases, while the others(Type 2 & 3) reflect non-equilibrium phase invasionprocesses. Current conceptual models of fracture/matrixinteraction assume that at equilibrium, matric pressureintroduces an aperture cutoff defined by capillary bundletheory (e.g. Wang and Narasimhan6); all apertures smaller thanthis cutoff will be filled with the wetting phase, while the non-wetting phase will occupy all apertures larger than the cutoff.Since phase structures reach equilibrium through matrix flow,film flow, and evaporative redistribution, the physical andtemporal constraints imposed by our experimental systemprohibited creation of true equilibrium phase structures.However, it was hypothesized that alternative means could beused to distribute the non-wetting phase such that it occupiesonly the largest apertures, thereby capturing the salientfeature of equilibrium theory. Type 1 phase structures werecreated starting with a two-phase structure containing asignificant air fraction. A transient wetting phase flow wasthen induced thorough intermittent application of fluidpressure. This procedure acted to drive the non-wetting phaseout of the small apertures and allowed it to lodge in the largeapertures. The resultant wetted structure consisted of small,compact inclusions of the non-wetting phase which, like thelarge apertures, were distributed throughout the fracture (seeFigure 5a). Physical constraints on redistributing the airfraction limited exploration of such structures to largewetting phase saturation (Sw = 0.9-0.95).

Non-equilibrium processes are expected to affect wettedstructure genesis, fluid entrapment, and thus kr. Glass andNorton9 have shown that in-plane trapping mechanisms act toprevent aperture filling from the matrix according to capillarybundle theory. In order to explore the effects of phasestructure established by non-equilibrium processes on kr,more complicated Types 2 and 3 phase saturation structureswere created. Type 2 structures were formed by first draining asaturated fracture and then establishing liquid flow. Drainageof a saturated fracture through application of suction leavesthe residual wetting phase distributed about the fracture asdisconnected blobs. Subsequent application of steady wettingphase flow across the full fracture width creates a


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Figure 4: Delineation of Phase Structure: a) Sub-region of experiment 3-3 measuring 128 x 128 pixels, dark regions are filled withthe wetting phase and light regions with the non-wetting phase; b) Histogram of Figure 4a, solid line shows results of smoothingalgorithm. Note that the minimum of this sub-region is better defined than that for the entire image (Figure 3); c) Binarytransformation of Figure 4a showing the wetting phase as black and the non-wetting phase as white.

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Figure 5: Types of Wetted Structure Considered: a) Type 1 (exp. 1-1); b) Type 2 (exp. 2-1); c) Type 3 (exp. 3-3).


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Figure 6: Distribution of Air in Entrapped Regions:Normalized distribution of air filled apertures in each type ofstructure is shown. Distribution of the model aperture field isshown for comparison. a) Type 1; b) Type 2; c) Type 3.

structure containing pervasive entrapped non-wetting phaseregions (Figure 5b). Type 3 structures were formed by slowgas injection from one end of a saturated fracture followed byestablishing liquid flow. Slow gas injection along the planeof a saturated fracture under quasi-static conditions creates avery complex, fully connected non-wetting phase. Subsequentapplication of steady wetting phase flow across the fullfracture yields increased connectivity within the non-wettingphase as compared to the Type 2 structures (Figure 5c).Saturations for both Type 2 and 3 structures weresignificantly lower than for the Type 1 structures. No attemptswere made to systematically vary saturation; phase structureswere formed and then saturation measured.

analysis of phase aperture residence and saturationstructure

Normalized distributions of apertures filled with the wettingand non-wetting phases for Type 1, 2, and 3 structures areshown in Figures 6a, 6b, and 6c respectively; the aperturefield distribution is shown for comparison. As expected, forall structural types, the non-wetting phase preferentiallyoccupies the largest apertures. However, in contrast toequilibrium theory, in all phase structures considered, there issignificant overlap between the phase distributions. In theType 1 structures, which were intended to mimic equilibriumtheory, the non-wetting phase did not occupy all of the largestapertures and filled a relatively broad range of aperture sizes.Thus, our procedure for creating Type 1 phase structures

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did not fully simulate aperture filling according to capillarybundle theory. Since transient pressures were applied to breakup and mobilize entrapped non-wetting phase regions,pressures may not have been of sufficient magnitude todislodge the non-wetting phase from moderately largeapertures or fully break-up the non-wetting phase. In the Type2 and 3 phase structures, which were formed under non-equilibrium conditions, the non-wetting phase occupies a


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1-1 (S = 0.91) 1-2 (S = 0.96) 2-1 (S = 0.55) 2-2 (S = 0.62) 3-1 (S = 0.68) 3-2 (S = 0.64) 3-3 (S = 0.66)equil. (S = .57)equil (S = .87)

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Figure 7: Distribution of Entrapped Phase: Connected regions occupied by the non-wetting phase are identified as individualclusters. Normalized cumulative distributions of cluster sizes are shown for each experimental phase structure. For comparison,cluster size distributions for phase structures predicted by capillary bundle theory (Sw = 0.65, 0.89) are also shown.

broader range of aperture scales than observed for the Type 1structures.

Connected apertures filled with the non-wetting phase areidentified as individual clusters, which then can becharacterized with respect to size, shape, and spatial location.The non-wetting phase cluster size distribution for each of theexperiments is shown in Figure 7. Type 1 structures show aspan of nearly 2 orders of magnitude while Types 2 & 3 spannearly 3 orders. In addition, size of the largest clusterincreases as we pass from type 1 to 2, and on to 3. It isinstructive to compare the cluster size distributions formed inour experiment to those that would be formed on our aperturenetwork assuming equilibrium theory. A series of equilibriumphase structures were modeled by defining all apertures on thenetwork smaller than an arbitrarily selected cutoff as filledwith the wetting fluid. Non-wetting phase cluster sizedistributions for two different equilibrium structures (S =0.57, S = 0.87) are also shown on Figure 7. When compared toequilibrium structures at similar saturations, our experimentstend to larger clusters, and a larger range of cluster sizes.Structures formed under non-equilibrium processes (Types 2& 3) show order of magnitude differences in the cluster sizedistribution when compared to an equilibrium structureformed at a similar saturation. Cluster size distributions forType 1 structures are similar in shape to the equilibriummodel, however both examples plot to the right of anequilibrium structure at lower saturation, indicating thatentrapped clusters are generally larger in the Type 1 structuresthan predicted by capillary bundle theory. Even thoughaperture filling in Type 1 structures was not strictly inaccordance with equilibrium theory and non-wetting phaseclusters were larger than expected, both spatial distribution ofthe clusters and phase connectivity appeared to be very

similar to that predicted by the equilibrium model. Therefore,we assume that for the purposes of this study, Type 1structures provided a reasonable approximation ofequilibrium conditions at high liquid saturation.

relative permeability

Before considering relative permeability, we first discusssaturated permeability of the analog fracture. The simplestmodel for single phase flow through a fracture of aperture aand cross-sectional area A (A = a • fracture width) is to useDarcy's Law, with hydraulic conductivity approximated by theparallel plate law (e.g., Bird et al.10):

Q = −AK∇ φ (4a)

K = ρga2

12µ(4b)

where Q, ρ,µ ,g, and φ represent volumetric flow, fluiddensity, fluid viscosity, gravitational constant, and pressure-head, respectively. Hydraulic conductivity of the analogfracture used in these experiments was measured at 2.89 cm/s;equation 4b gives the equivalent hydraulic aperture as 0.0178cm, which is approximately 17% smaller than the measuredmean aperture (0.0215 cm). The observed discrepancy occursbecause the parallel plate model does not fully describe thephysics of flow through a variable aperture field11. Primarymechanisms unaccounted for in the parallel plate model aredeviation from a linear path and converge/divergence of flownormal to the fracture plane. Flow through a rough walledfracture does not follow a perfectly linear trace along thefracture plane as assumed in the parallel plate model. If allother hydraulic parameters are held constant, resistance toflow will increase linearly with length of the flow path. In the


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Figure 8: Relative Permeability as a Function of WettingPhase Saturation

analog fracture considered here, tracers were used to visualizein-plane variation in the flow path. Under saturatedconditions, macroscopic deviation from a linear path wasobserved to be relatively small. However, small wavelengthvariations may add significantly to the path length; therebyaccounting for a large portion of the observed deviation fromequation 4b. It is expected that flow resistance resulting fromthe convergence and divergence of flow paths normal to thefracture plane as flow passes through the variable aperturefield is responsible for most of the remaining discrepancy.Mean aperture does not contain sufficient information to fullycharacterize flow within a variable aperture field, but can beregarded as a reasonable first order approximation.

In the preceding paragraph, factors contributing to thesaturated hydraulic conductivity of a variable aperturefracture were considered. We now assemble a model of wettingphase relative permeability in a horizontal fracture understeady state conditions. In the experiments presented here, thenon-wetting phase was immobile, and except at the phaseinterface, the non-wetting phase fully spans the aperture gap.As a result, apertures occupied by the non-wetting phase act asimpermeable barriers within the aperture field, analogous toareas within the fracture that are in direct contact. In order toexplain the observed rapid decrease in kr with Sw (Figure 8,Table 1), we first consider processes acting on the distributedor average properties of the two-phase structure.

Lowering wetting phase saturation reduces the cross-sectionalarea available for wetting phase flow. Darcy's Law definespermeability as a proportionality constant relating volumetric

flow to pressure gradient and cross-sectional flow area.Changing phase saturation does not alter the cross-sectionalarea of the fracture, but actually controls the portion of thatarea that is available for flow. As this modifier acts on theflowing phase, not the fracture, it is typically included in kr.From the geometric properties of the system, it is logical toexpect that a first order approximation for kr would take thefollowing form:

kr = Sw (5)

Fractured-reservoir simulators used by the petroleum industrytypically make this assumption12 which is shown as the solidline in Figure 8. While this simple model does not fit the datawell, it does account for a significant portion of the observeddecrease in kr with Sw. For the homogenous, isotropic analogfracture considered here, continuity implies that Sw willprovide an upper bound on kr. As it is possible toconceptualize aperture structures where Sw will not provide anupper bound on kr, this observation should be generalizedwith caution.

A second process acting on kr is the preferential occupation ofsmaller apertures by the wetting phase. Under two phaseconditions, the mean aperture occupied by the wetting phase(âw) is expected to be smaller than the mean aperture of thefracture (â). As hydraulic conductivity is assumed to beproportional to the aperture squared (equation 4b), thisobserved redistribution will have a different effect on kr thanthe simple reduction in cross-sectional area discussed above.Substitution of mean aperture into equation 4b provides afirst-order estimate of Ks for a rough-walled fracture; therefore,mean aperture occupied by the wetting phase is expected todisplay a similar relationship for permeability under two-�phase conditions. Assuming that all quantities in equation 4bexcept mean aperture are constant, solution of equation 1 forkr yields:

kr ∝aw

2

a2 (6)

As discussed above, convergence and divergence of theaperture field contributes to a discrepancy between mean andhydraulic apertures of the analog fracture. If the distributionsof a and aw are similar, taking the ratio of the means, as donein equation 6 is expected to cancel out much of thisdiscrepancy. Equations 5 and 6 may then be combined toproduce an improved model for relative permeability basedsolely on average properties of the phase structure (equation7).

kr ∝aw

2

a2 Sw (7)

If apertures within the fracture are filled in accordance withcapillary bundle theory, the ratio of âw to â will equal Sw,collapsing equation 7 to:

kr ∝ Sw3 =

aw3

a3 (8)


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Figure 9: Flow Path Visualization: Movement of a tracer pulse released from the left side of the fracture into the wetted structureshown as Figure 5a (experiment 1-1).

We note that equation 8 is identical to one presented by Wangand Narasimhan6 except for their inclusion of a tortuosityfactor (τ).

The model given by equation 8 is observed to fit the datareasonably well (Figure 8), particularly at large saturations.However, in the experiments presented here, aperture fillingwas not strictly in accordance with capillary bundle theory;hence Sw is not a good approximation for the ratio of âw to â,as assumed in equation 8. Predictions based on equation 7 donot fit the data nearly as well as equation 8 (Figure 8). Thisdoes not imply that equation 8 is a more appropriate model,merely that factors unaccounted for in either equation 7 or 8are significant and must be included.

As models based on distributed properties of the phasestructure (equation 7) are unable to fully account for observedbehavior, we must consider the effects of phase structure.To explore channelization of flow induced by phase structure,at the conclusion of each experiment a tracer consisting ofpure de-ionized water was released as a sharp pulse across thefull width of the analog fracture. Figures 9, 10, and 11illustrate tracer advancement through the phase structuresseen in Figure 5a, b, and c, respectively. Even the relativelysimple Type 1 phase structures act to channelize flow (Figure

9), breaking up the front into individual channels andretarding advancement in areas of relatively large non-wettingphase saturation (along the bottom edge). In the morecomplex Type 2 structure (Figure 10), channelization becomesmore distinct but the front still advances more or lessuniformly across the full fracture width. In Type 3 structures(Figure 11), flow appears constrained to two or three primarychannels that follow tortuous paths through this complicatedphase structure.

It is apparent from Figures 9-11 that phase structure acts tochannelize flow, and that the degree of flow modification isrelated to complexity and connectivity of the non-wettingphase structure. Two processes that act to decreasepermeability are path length and localized variation inchannel width. At the low saturations exhibited by Type 2 and3 structures, path length will be significantly longer thanexpected under fully saturated conditions. As permeability isexpected to demonstrate a linear relationship with pathlength, this process is believed to account for much of thediscrepancy between equation 7 and measured data. Theincreased complexity of the Type 3 structures creates longerwavelengths in the flow path than in the Type 2 structures.However, average path length is not necessarily any longer inthe Type 3 structures.


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Figure 10: Flow Path Visualization: Movement of a tracer pulse released from the left side of the fracture into the wetted structureshown as Figure 5b (experiment 2-1).

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Figure 11: Flow Path Visualization: Movement of a tracer pulse released from the left side of the fracture into the wetted structureshown as Figure 5c (experiment 3-3).


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nal

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trib

utio

n

1-1 1-2 2-1 2-2 3-1 3-2 3-3

Figure 12: Saturation Normal to Imposed Gradient: Saturationnormal to the imposed gradient provides a first orderapproximation of local cross-sectional area open to flow.Fractional distributions show that Type 3 structures exhibitthe most variation and are hence most likely to contain chokepoints.

The channels created by the two-phase structure are not ofuniform width. Therefore, relatively large amounts of fluid areforced to pass through narrow "choke points" that act torestrict flow, causing large localized pressure drops. Pyrak-Nolte et. al.13 concluded that fracture permeability iscontrolled by the largest connected channel within the systemand that flow through that channel is controlled by its'narrowest aperture, or choke point. Measurement of saturationalong a linear trace perpendicular to the imposed gradientprovides a first order approximation of cross-sectional flowarea; distributions of all such traces are presented as Figure12. It is apparent that variation in cross-sectional flow areaincreases with structural complexity and connection of theentrapped phase. Therefore, Type 3 structures will be the mostlikely to experience flow constriction at choke points. Thisobservation is affirmed by qualitative evaluation of tracerpulses (Figures 9-11) and illustrated by displaying variationin cross-sectional area along the Type 3 structure ofexperiment 3-3 (Figure 13, see also Figure 5c). This analysisdoes not imply that Type 3 structures have the narrowestchannels, simply that flow along those channels are likely toexperience the most variation in area and hence local pressuredrop. The significance of choke points will decrease withincreasing number of flow channels and interconnectionbetween the channels.

Further analysis of phase structure must be accomplished incombination with numerical simulation before the relativeimportance of choke points and channelization can be

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Downgradient Distance (cm)

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urat

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Figure 13: Variation in Cross-Sectional Flow Area: Saturationintegrated along a cross-section normal to the imposedgradient is shown for experiment 3-3.

determined and quantified; for now we take the conventionalapproach and bundle the effects of phase structure into asingle parameter, tortuosity (τ).

kr = Swτ aw2

a2 (9)

This relationship is a modification of one presented by Wangand Narasimhan6 with differences in the assumptionsregarding aperture filling, as stated previously. Equation 9was used to calculate τ for the experiments presented here;values are given in Table 1. As expected, phase structure has amuch smaller effect on kr in Type 1 structures than in eitherType 2 or Type 3 structures. This is most likely due to thelarge difference in Sw between the structural types, rather thanany effects induced by the specific structural forms. Eventhough Type 2 and 3 structures appear to be very different,there was no distinguishable difference in measured τ betweenthe two. Therefore, it may be possible to write τ as a functionof Sw for some broader class of structures that wouldencompass Types 1, 2 and 3. Development of such arelationship will require an in depth exploration intostructural control of path length and choke points, as well asthe effects of these processes on kr.

While the experiments presented here imply that τ is a strongfunction of saturation and a weak function of phase structure,that conclusion may not be universally true. Non-equilibriumprocesses, such as the gravity-driven wetting front instabilitydemonstrated by Nicholl et al.7, may act to reduce τ bycreating oriented wetted structures. In inclined fractures, theeffects of gravity can destabilize a wetting front, leading to


the development of fingers oriented in the direction ofgravity. At times short with respect to matrix imbibition andevaporative redistribution, moisture left by the passage ofindividual fingers is expected to create preferential pathwaysfor subsequent flow14. These pathways may then lead todevelopment of a wetted structure oriented in the direction ofgravity, analogous to a series of parallel channels. In such ascenario, spatial structure of the wetted phase is expected toexhibit significant, and perhaps dominant control over τ.Wetted structures formed by other non-equilibrium processesmay also significantly effect τ, and hence kr.

Finally, we note that while results presented here imply that kris a weak function of phase structure, spatial structure of thephases will almost certainly have a significant effect ontransport properties. Type 3 structures such as that shown inFigure 11, act to channelize flow into discrete channels, whichin turn are expected to act as rapid pathways. Furthermore,complexity of the structure creates large "dead" zonesanalogous to dead-end pores in porous media. As a result, theresidence time distribution for tracer movement through aType 3 structure is expected to differ significantly from thatassociated with a Type 2 structure of similar permeability.Similar situations are seen to occur when consideringsaturated flow through heterogeneous porous media; structureis observed to exhibit minimal control on effectivepermeability while exhibiting significant control on transportproperties (e.g. Smith and Schwartz15; Yeh16).

Conclusions

We have used simple light absorption theory to develop amethodology for characterization of aperture and phasestructure within a transparent analog fracture. In addition toproviding detailed spatial characterization of the aperturefield, this technique provides a means of measuring phasecomplexity in the fracture plane and exploring thedistribution of apertures occupied by both wetting and non-wetting phases. The techniques developed were used toexplore wetting phase relative permeability under steady-stateflow conditions in the horizontally oriented analog fracture.

As expected, in all phase structures considered, large apertureswere preferentially occupied by the non-wetting phase. Inphase structures formed by mimicking potential non-equilibrium field processes, the non-wetting phase occupied asignificantly wider distribution of apertures than predicted byequilibrium theory. Detailed measurements were used toexplore a simple process oriented model for relativepermeability. Parameters incorporated in the model includewetting phase saturation (Sw), mean aperture occupied by thewetting phase (âw), and a structural complexity term. Phasesaturation was shown to be an inadequate estimator of âw,except for large wetting phase saturation (Sw > 0.9) structuressimilar to those expected to form under the equilibriumassumptions of capillary bundle theory. Distributedparameters (Sw and âw) were shown to be insufficient to modelrelative permeability in the low wetting phase saturation (Sw =

0.55-0.66) structures considered. Tortuosity and localizedflow restriction were observed to reduce relative permeabilityby as much as 70% in such phase structures. Effects of spatialstructure on kr were not observed to vary greatly betweenphase structures of similar saturation but differingcomplexity; however phase structure is expected to have amarked effect on transport properties of the system.

Field scale numerical models for use in performanceassessment in unsaturated fractured rock at Yucca Mountaincannot explicitly model individual fractures in detail;therefore, effective property models which properly includetwo-phase flow in fractures must be developed. Theinvestigation presented here describes the first steps towardsdevelopment of process oriented models of fracture hydraulicproperties under two-phase flow conditions. Subsequentdevelopment and testing involves both physical andnumerical experimentation. Cast replicas of natural tufffractures from Yucca Mountain will be used inexperimentation similar to that accomplished here in a simpleanalog rough-walled fracture. Since we will never be able tounderstand the behavior of all fractures at Yucca Mountainfrom a small set of cast natural replicas, we are alsodeveloping the technology to directly manufacture analogfractures through numerically controlled milling techniques.This will allow systematic exploration into the hydrauliceffects of various aperture properties, including meanaperture, correlation length, roughness length scales, apertureheterogeneity, and anisotropy. Milled fractures will bereplicated using transparent casting compounds to allow flowvisualization; by varying the casting technique, surfacewetting properties can also be varied. To limit the number ofmanufactured fractures required, aperture scale numericalmodels, once properly tested, will be used to obtainadditional data. This experimental and numericalinvestigation will identify the essential fracturecharacteristics required for modeling hydraulic behaviorunder two-phase flow conditions. Such characteristics andvalid simplifications of models for two-phase fracture flowwill then be incorporated into a properly formulated effectiveproperty model for inclusion in the performance assessmentexercises at Yucca Mountain.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the assistance andsupport provided by the following: W.C. Ginn in design andfabrication of the experimental apparatus; L. Orear indevelopment of image acquisition software; P.B. Davies, C.H.Ho, and S.W. Webb for their careful and constructive reviews.This work was supported by the U.S. Department of Energy,Office of Civilian Radioactive Waste Management, YuccaMountain Site Characterization Project Office, under contractDE-AC04-94AL85000.


Nomenclature

a - fracture apertureâ - mean apertureaw - aperture occupied by wetting fluidâw - mean aperture occupied by wetting fluidA - cross-sectional flow aread - thickness of absorptive materialdap - thickness of apertureg - gravitational constantI - luminous intensityIc - intensity used to delineate between phasesIcl - intensity associated with clear fluidId - intensity associated with dyed fluidIo - initial intensitykr - relative permeabilityK - hydraulic conductivityKs - saturated hydraulic conductivityQ - volumetric flow rateRe - Reynolds numberSw - wetting fluid saturationα - light absorption coefficientαd - absorption coefficient for dyeµ - fluid viscosityρ - fluid densityφ - pressure headτ - tortuosity

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