Published in Vadose Zone Journal 3:592-601 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Small-Scale Features of Gravity-Driven Flow in Unsaturated Fractures
Grace W. Su*,a,
Jil T. Gellera,
James R. Huntb and
Karsten Pruessa
a Lawrence Berkeley National Laboratory, Earth Sciences Division, Berkeley, CA 94720
b Department of Civil and Environmental Engineering, University of California at Berkeley, Berkeley, CA 94720
* Corresponding author (gwsu{at}lbl.gov).
Received 24 April 2003.
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ABSTRACT
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Liquid flow through unsaturated fractures often proceeds as fingers or preferential flow paths. During the invasion of liquid fingers into an initially dry, nonhorizontal fracture, fingers may drain, forming a narrow thread of liquid called a rivulet that connects to a wider portion of liquid at the advancing front, defined as a blob. Experimental studies using idealized fractures were performed to investigate the effects of wettability, surface roughness, and aperture size on several important features of gravity-driven flow in fractures: liquid drainage, blob migration, and rivulet flow. The experiments demonstrate that the critical length of the blob before drainage occurred was significantly longer on surfaces with intermediate wettability and on surfaces with roughness on the order of 100 µm than on a smooth, flat water-wetting surface. However, drainage did not occur on surfaces with smaller-scale roughness on the order of 10 µm. Blob velocities were also measured and were always less than the saturated gravity-driven flow velocity, even when a liquid with a static contact angle of zero was used. This reduction in velocity was attributed to contact angle hysteresis. Rivulet widths measured as a function of flow rate between glass and acrylic parallel plates were generally larger on the acrylic plates than the glass plates for a particular flow rate, demonstrating the sensitivity of rivulet flow to wettability. In addition, the cubic law overpredicted the measured rivulet widths, except for the widths measured between the acrylic plates at 20°. The effect of aperture variability on rivulet flow was also examined. At a critical aperture ranging between 0.25 and 0.37 mm, the liquid in the rivulet did not completely span the aperture, forming two streamlets of liquid on either side of the fracture.
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INTRODUCTION
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LABORATORY AND FIELD experiments have provided considerable evidence that liquid flow through unsaturated fractures proceeds as fingers or along preferential flow paths (e.g., Nicholl et al., 1994; Su et al., 1999; Glass et al., 2002). Fingering is the focusing of flow into narrow channels (e.g., Chuoke et al., 1959) with seepage velocities dramatically increased compared with spatially uniform flow. Predictions of contaminant migration require an accurate representation of actual liquid velocities, particularly within the unsaturated zone.
Visualization experiments conducted on analog fractures have demonstrated the importance of gravity on the liquid distribution and flow dynamics during unsaturated fingered flow in analog fractures (Nicholl et al., 1994; Glass and Nicholl, 1996; Su et al., 1999). For instance, during the initial invasion of fingers into a dry, inclined analog fracture, the liquid finger may drain, forming a narrow thread of liquid called a rivulet that connects to a wider portion of liquid at the advancing front, defined as a blob in this study. This rivulet–blob structure has not been observed during fingered flow in horizontal fractures. Gravity also gives rise to liquid flow oscillations in unsaturated fractures, where rivulets along preferential flow channels undergo cycles of snapping and reforming (Su et al., 1999; Geller et al., 2000).
An improved understanding of small-scale processes affecting fingered flow in nonhorizontal unsaturated fractures is essential since they may significantly affect liquid velocities and contaminant transport predictions at larger scales. In this study, small-scale features of gravity-driven flow in fractures were examined in laboratory experiments conducted on idealized fractures with controlled apertures. The specific features investigated include drainage of a liquid finger into a blob–rivulet structure, the rate of blob advancement, and rivulet flow. Finger drainage and blob velocities are examined in analog fractures with different wettabilities, surface roughness, and aperture size. Contact angle hysteresis during blob advancement is calculated from the measured blob velocities in the different fractures. Rivulet flow is examined in parallel plates with different wettabilities and in an analog fracture with the apertures increasing linearly from the top to bottom.
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Wettability
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Wettability describes the tendency of a fluid to spread along a solid surface in the presence of another immiscible fluid (Dullien, 1992) and is often quantified through contact angle measurements. Contact angle is defined as the angle measured between the solid and the liquid–gas interface through the liquid. The contact angle of water on clean sand and glass is generally close to zero. However, relatively large contact angles have been measured even on natural mineral surfaces. Selker and Schroth (1998) measured contact angles as high as 58° for water on grains of sand, and Geller et al. (1996) measured contact angles of approximately 60° for water on a smoothed piece of granite.
In homogeneous materials, contact angle hysteresis, manifested by the difference in the receding and advancing contact angles, is affected by surface roughness and contamination (Dullien, 1992). The advancing contact angle is always larger than the receding one. Using measured wetting and drainage curves, Laroussi and DeBacker (1979) calculated an advancing angle of 66° and a receding angle of 46° for water on glass beads, and Bradford and Leij (1995) calculated an advancing angle of 32.7° and a receding angle of 0° for water on silica sand. For an intermediate wetting surface, the advancing contact angle may become >90° while the receding contact angle remains <90°. The advancing contact angle increases with increasing fluid flow velocity while the receding contact angle decreases with increasing velocity (Adler and Brenner, 1988). The radius of curvature at the advancing air–water interface also becomes larger as the gravity force increases (Iwata et al., 1995).
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Liquid Drainage in Porous Media and Fractures
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In homogeneous porous media, a reduction in the liquid saturation behind an advancing finger sometimes occurs, a phenomenon referred to as drainage. Measurements from laboratory experiments indicate that the drained region remains the same width as the advancing finger during invasion into an initially dry system (e.g., Glass et al., 1989). During fingered flow in unsaturated, nonhorizontal fractures, liquid drainage can also occur under continuous liquid application, but the subsequent distribution of the drained region is substantially different from the observations in porous media. When drainage occurs in fractures, a rivulet forms that connects an advancing blob to the source of liquid (Nicholl et al., 1993; Su et al., 1999).
Raats (1973) originally proposed a hypothesis to describe the physics of drainage in porous media. If the finger is viewed as a hanging column of liquid (Glass et al., 1989), the pressure head at the top of the finger decreases as the finger lengthens. When the pressure at the top of the finger falls below the air-entry pressure, the finger begins to drain. This hypothesis has been quantitatively verified in experiments conducted on homogeneous, coarse sand (Glass et al., 1989; Selker, 1992). If the rate at which liquid is supplied to the finger is slow, the critical length for drainage, Lcrit, is simply a balance of gravity and pressure forces:
 | [1] |
where ß is the angle of inclination relative to the horizontal direction, hb is the pressure head at the bottom of the finger, and ht is the pressure head at the top of the finger and is equal to the air-entry pressure head. Equation [1] can be used to determine the critical length for drainage in unsaturated fractures. In a homogeneous fracture modeled as parallel plates, the pressures at the bottom and top of the blob when drainage occurs can be obtained using the Young–Laplace equation (Adamson, 1990). Assuming the minor radius of curvature is negligible, the pressures are
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 | [3] |
where b is the aperture width between two parallel plates, 
is the surface tension, 
is the density of the liquid, g is the gravitational acceleration constant, 
a is the advancing contact angle, and r is the receding contact angle. Substituting Eq. [2] and [3] into Eq. [1] and solving for Lcrit, we obtain
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In unsaturated fractures, the blob that forms after drainage generally has a parallel-sided shape in the plane of the fracture (Su et al., 1999). Liquid drops on an inclined plane have also been observed to have similar shapes (Bikerman, 1950), and approaches to model the relation between the liquid drop shape, contact angles, and retentive force are similar to Eq. [4] (Bikerman, 1950; Furmidge, 1962; Extrand and Kumagai, 1995). The shape of the blobs in fractures may be affected by changes in wettability and heterogeneities.
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Blob Migration
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A cross section of a liquid blob advancing down an inclined parallel plate is shown in Fig. 1
. Assuming that the blob motion is controlled only by the gravitational force and the difference in capillary pressures at the bottom and top of the blob, the blob velocity can be described using Darcy's Law written in the following form:
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where ub is the blob velocity, Ks is the saturated hydraulic conductivity, and Lb is the length of the blob. The pressure heads at the bottom and top of the blob, hb and ht, are given by Eq. [2] and [3], respectively. Equation [5] assumes that the pressure gradient is attributed entirely to contact angle hysteresis and is similar to the expression Nicholl et al. (1994) used to analyze disconnected finger velocities in obscure glass plate fractures. Equations [1] and [5] are essentially a change of formula from Eq. [5] and [6] in Nicholl et al. (1994), which in turn follow Eq. [3] of Glass et al. (1989). The classic expression for the saturated hydraulic conductivity in parallel plates is given by (e.g., Bear, 1972):
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Equation [6] assumes low Reynolds number flow, which places an upper limit on b for a particular inclination angle. Substituting Eq. [2] and [3] into Eq. [5], the expression for the blob velocity becomes
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The difference in the cosines of the receding and advancing contact angles quantifies contact angle hysteresis, which is given by the following expression after rearranging Eq. [7]:
 | [8] |
The effect of the blob curvature in the plane of the fracture (Glass et al., 1998) is assumed to be negligible in Eq. [7] and [8]. Experimental data is used in this study to calculate contact angle hysteresis during blob advancement on different surfaces.
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Rivulet Flow
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Rivulets are another important feature of flow in unsaturated fractures. Assuming that flow though a rivulet obeys the cubic law for flow through smooth parallel plates, the flow rate can be calculated using
 | [9] |
where wr is the rivulet width.
For narrow rivulets, as has been observed in our experiments (Su et al., 1999, 2001), flow through the rivulet may not obey Eq. [9] since this is only valid when the rivulet width is much larger than the aperture width. In this study, rivulet widths as a function of flow rate will be measured and compared with predictions using Eq. [9] to examine whether this equation is valid for describing rivulet flow.
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Streamlet Flow
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During intermittent flow in unsaturated fractures, the rivulets are unstable and undergo cycles of snapping and reforming. The stability of rivulets may be affected by whether or not the liquid in a rivulet spans the aperture width. When liquid flows through sufficiently large apertures, streamlets may form, which are defined as rivulets that have one contact and one free surface. In the experiments that will be presented in this paper, the liquid will be supplied to a fracture with apertures increasing linearly from the top to bottom (wedge fracture) as a point source that forms only a single finger. The finger separates into two streamlets as the fracture aperture widens. A wedge fracture was used by White et al. (1977) to study fingering instability and measurements of the location where fingering began were predicted using the criterion derived by Philip (1975). The work presented here differs from White et al. (1977) since the transition of the rivulet into streamlets is examined rather than the onset of fingering.
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MATERIALS AND METHODS
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Laboratory experiments were conducted under idealized conditions to obtain qualitative and quantitative information on the factors affecting liquid drainage, blob migration, and rivulet flow in unsaturated fractures. Parallel plate fractures and a wedge fracture were used in this investigation to maintain control of the aperture widths. Four types of experiments were conducted, which were denoted as Exp. I, II, III, and IV. The experimental conditions are presented in Table 1, and a schematic of the experimental setup is shown in Fig. 2
. The fractures were assembled using 6.4-mm (1/4 inch)–thick flat plate glass, 4.8-mm (3/16 inch)–thick obscure plate glass (both from UC Glass Company Inc., Berkeley, CA) or 6.4-mm (1/4 inch)–thick acrylic plates (Polycast Technology Corp., Stamford, CT).
Characterization of Surfaces and Apertures
Acrylic was used to represent a fracture surface that had been altered by organic substances, while glass was used as an analog for a "clean" natural fracture surface. The aperture widths for the fractures in Table 1 indicate the size of the shims placed between two plates except for the obscure glass plates, which refers to a fracture that was assembled by mating an obscure glass plate to a flat glass plate. The sandblasted glass plates in Table 1 refer to an obscure glass plate sandblasted with 80-grit alumina silica that was mated to a flat glass plate. The small-scale surface roughness of the sandblasted obscure plate was on the order of 10 µm (Kneafsey and Pruess, 1998). Obscure glass is plate glass with a textured surface. A photograph of the aperture structure of the obscure glass plate mated to the flat glass plate is shown in Fig. 3
. The average aperture of the obscure-flat glass combination was estimated using the volume method by Hakami (1989). A drop of a known volume, 0.5 mL, was placed between the plates and the area covered by the drop was measured (28 cm2). The average aperture was 0.18 mm based on these values and was the same for the obscure-flat glass plate combination with and without sandblasting.
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Fig. 3. Photograph of the aperture pattern in the obscure–glass plate combination. Dyed water was injected between the plates for this photograph. The brighter regions indicate smaller apertures and the darker regions indicate larger apertures.
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A capillary rise experiment was also performed to estimate the aperture range in the obscure-flat glass plate combination. For vertical parallel plates, the aperture corresponding to the height of rise of the liquid is given by solving Eq. [2] for b. Water dyed with 0.2% Liquitint by volume was used for this measurement. The height of water rise in these plates varied between 5.1 and 7.0 cm, and the maximum and minimum static contact angles measured for the obscure plate (40 and 45°) were used to obtain the largest possible aperture range. Using these values, the apertures span between 0.13 to 0.20 mm. The obscure plate roughness is estimated from the difference between the range of apertures (70 µm) and is therefore on the order of 100 µm.
Side-view images of drops of dyed water on the glass, acrylic, and obscure glass were captured using the video camera and contact angles were measured from the images. The static contact angle of water dyed with 0.2% Liquitint was approximately 15° on the flat glass, 60 to 65° on acrylic, and 40 to 45° on the obscure plate. The contact angle measured on a rough surface is also referred to as the apparent contact angle (Dullien, 1992). Water with and without dye spread immediately upon contact with the sandblasted plates and therefore has a contact angle of zero.
Procedure
Before each experiment, the plates were washed with distilled, deionized water. Excess water was wiped off, and the plates were allowed to air dry. The flow cells had dimensions of 20 by 33 cm and were loaded in a confining metal frame held by six bolts. The flow cell was then mounted over an inclined light table. The properties of the liquids used in the three series of experiments are summarized in Table 2. Observations were videotaped using a video camera (JVC KY-F55BU with motorized zoom lens JVC TY-10x6 MDPU) and recorded on a video recorder (Sony SVHS SVO-5800) with a time resolution of 1/30 s. Measurements from captured images were made using Adobe Photoshop (Adobe Systems Inc., San Jose, CA).
Experiment I was conducted to examine the effect of wettability and surface roughness on liquid drainage and blob advancement. Four analog fractures of varying wettabilities and roughnesses were used in this experiment (Table 1). Water was supplied to the initially dry plates at a constant flow rate by means of a syringe pump (Model 33, Harvard Apparatus, South Natick, MA). A hypodermic needle with a beveled end (26G1, Becton-Dickinson, Rutherford, NJ) was placed at the top of the plates to deliver water.
Experiment II was conducted to examine the significance of contact angle hysteresis on the blob velocity when a perfectly wetting liquid was used. In this experiment, blobs of n-dodecane were injected into acrylic plates with a constant aperture of 0.25 mm at two angles of inclination (20 and 85°). The n-dodecane was injected manually with a syringe since a constant flow rate did not have to be maintained for these measurements.
Changes in rivulet width as a function of input flow rate were examined in Exp. III using glass and acrylic parallel plates with a constant aperture of 0.25 mm. Two angles of inclination (20 and 80°) were used, and the flow rates applied using a syringe pump are summarized in Table 1.
In Exp. IV, the effect of aperture variability on rivulet flow was examined using a glass wedge fracture that had apertures increasing linearly from top to bottom. The wedge fracture was created by placing shims with thicknesses of 0.08 and 1.58 mm at the corners of the top and bottom of the plates, respectively. The plates were reassembled each time a different flow rate was used.
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RESULTS AND DISCUSSION
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This section presents experimental observations of liquid drainage, measurements of the blob velocities and the subsequent calculation of contact angle hysteresis, and the results of the rivulet width and flow dynamics experiments.
Visual Observations
Images of the liquid distribution in the four plates used in Exp. I are shown in Fig. 4
. The width of the initial advancing finger ranged between 0.5 to 1.0 cm in the glass, acrylic, and obscure plates and remained approximately constant while the length of the invading front increased, forming a column of liquid (Fig. 4a–4c). The top of this column of liquid began to drain after reaching a certain length in these three plates, resulting in the formation of a blob and rivulet. These observations are similar to the ones from the flow visualization experiments described in Su et al. (1999). The blob of water in the glass plates is considerably smaller than the blobs in the acrylic and obscure glass plates. In the obscure plates (Fig. 4c), water becomes trapped after drainage occurred across portions of the region wetted by the advancing front because of the roughness of the plates. The rivulet in the obscure plates is also more tortuous than the rivulet that forms in the glass and acrylic plates.
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Fig. 4. (a)–(c) Liquid distribution on three different surfaces, Q = 5 mL h–1 and ß = 85°. Before drainage (t1) and after drainage (t2). (d) Liquid distribution in sandblasted plates, Q = 2 mL h–1, ß = 15°. The saturated regions appear brighter than the films surrounding it since the water was not dyed.
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The liquid distribution in the sandblasted obscure plates (Fig. 4d) is considerably different than the distribution in the other three plates. In the sandblasted plates, the column of water did not drain regardless of the angle of inclination, resulting in the formation of a saturated finger. In addition, films of water spread adjacent to the saturated finger almost immediately after water was introduced in the plates because of the half-capillaries created by the small-scale surface roughness. The formation of films surrounding the finger is similar in appearance to the wetted regions that form adjacent to the nearly saturated fingers in porous media that do not fill the pores and move as a vapor or film (Glass and Nicholl, 1996; DiCarlo et al., 1999). The scale of surface roughness is an important factor determining whether or not a blob and rivulet will form. These features were observed in the obscure plates, which had roughness on the order of 100 µm, but were not observed in the sandblasted plates where smaller-scale roughness was present. In repeat experiments conducted on the same analog fractures, the blob–rivulet structure was always observed in the glass, acrylic, and obscure plates, but it was never observed in the sandblasted obscure plates. The blob lengths were also consistently smaller on the glass plates compared with the blob lengths in the acrylic and obscure plates.
Blob Velocities and Contact Angle Hysteresis
Plots of the distance to the tip of the advancing air–water meniscus as a function of time from Exp. I were made to obtain velocities of the advancing blob (ub) in the glass, acrylic, and obscure plates. These plots are shown in Fig. 5
for ß = 85°. The results from the sand-blasted plates were not evaluated since we were interested in comparing the rates of advancement where a blob and rivulet formed. Figure 5a contains plots of the blob advancement in the acrylic and glass plates with an aperture of 0.25 mm, and Fig. 5b presents the results from the obscure plates and the glass plates with an aperture of 0.15 mm. The break in the slope of the curve indicates the transition from the liquid accumulating into the blob to the migration of the blob–rivulet structure. All available measurements of the rate of blob advancement from Exp. I are summarized in Table 3. The blob velocity on the acrylic plates is 2.7 times slower than it is on the glass plates with the same aperture, demonstrating the effect of wettability on the blob velocity. The results from the parallel glass plates and obscure glass plates also demonstrate that the blob velocity is very sensitive to the aperture width. The velocity of the blob–rivulet structure in the glass plates inclined to 85° is nearly five times slower in the plates with an aperture of 0.15 mm compared with 0.25 mm (Table 3). The average blob velocity in the obscure glass plates at 85° is slightly less than the average blob velocity in the 0.15-mm parallel plates even though the average aperture of the obscure plates is 0.18 mm, indicating that the lower blob velocity is due to the aperture variability and roughness of the obscure plates.
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Fig. 5. Plots of the advancing front as a function of time on the (a) glass and acrylic plates and (b) glass and obscure plates. Q = 5 mL h–1 and ß = 85°.
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The critical length before a rivulet formed behind the blob on these different surfaces is also summarized in Table 3. The critical length is larger on the acrylic and the obscure plates compared with the glass plates because of the differences in wettability and surface roughness. Within each analog fracture, the critical blob length scales nearly with sinß. The average length of the blob as it advanced down the fracture was also measured and is summarized in Table 3. This length is longer than the critical length measured on the surfaces. If the magnitude of the contact angle hysteresis does not change substantially before and after the blob advances, Eq. [7] predicts that the length of the blob increases beyond the critical length once the blob begins to advance. In experiments repeated on the same set of plates, the critical blob length remained nearly the same, but the average length of the advancing blob varied in the different experiments. The difference in the advancing blob lengths is due to the sensitivity of flow to slight changes in the boundary conditions and possible contamination of the fracture surfaces between experiments. Even in a particular experiment, the advancing blob length varied, as indicated by the standard deviations of the advancing blob lengths in Table 3.
The magnitude of the contact angle hysteresis before and after the blob moves is calculated using Eq. [4] and [8], respectively (Table 3). The difference in the hysteresis before and after the blob advances is relatively small on all the surfaces. A slight increase in the hysteresis is usually calculated after the blob advances since the advancing contact angle increases with increasing velocity. For a particular surface, the increase in contact angle hysteresis from the low to high inclination angle was no larger than a factor of 2. Hysteresis was more sensitive to changes in the wettability and roughness of the fracture walls, increasing on the acrylic and obscure glass by a factor of 3 to 6 compared with the smooth glass. The hysteresis is always <1 on all the surfaces except for the acrylic plates, where a hysteresis of 1.25 was calculated for the advancing blob. A hysteresis value >1 indicates that the advancing meniscus had a contact angle >90°. Advancing contact angles of water drops on an inclined plane surface have been observed to be >90° on intermediate wetting surfaces (Extrand and Kumagai, 1997).
The contact angle hysteresis from the smooth and obscure glass plates and acrylic plates are summarized in Fig. 6
by plotting them as a function of the dimensionless velocity, defined as the ratio of the measured blob velocity to the saturated flow velocity, ub/(Kssinß). The dimensionless velocity decreases as the hysteresis increases, which is consistent with expectations since the measured blob velocity decreases as the hysteresis increases. If the data in Fig. 6 are consistent with Eq. [8], a linear relationship between the hysteresis and dimensionless velocity should exist that has an intercept and a negative slope equal to (Lbgbsinß)/(2). The results from the glass plates are qualitatively consistent with Eq. [8], where the data from the smooth and obscure glass plates have a linear relationship. The quantitative relationship between the hysteresis and dimensionless velocity must be obtained from the linear regression, however, since the slope and intercept of this line are not equal as predicted by Eq. [8]. The result from the acrylic plates deviates from the linear trend of the glass plates data, suggesting that hysteresis as a function of dimensionless velocity on an intermediate wetting surface may have a different relationship than it does for a strongly wetting surface. More data is needed, however, to determine this relationship for the acrylic plates.
According to Eq. [7], the blob velocity should be approximately equal to the gravity-driven flow velocity (Kssinß) when the contact angle hysteresis is minimal. To examine whether this occurs or not, Exp. II was conducted using n-dodecane, a perfectly wetting liquid under static conditions. The velocities of the blobs, which are summarized in Table 4, are about half the magnitude of the saturated gravity-driven flow velocity. These results indicate that the advancing contact angle becomes non-zero when the blob moves, which reduces the velocity from the saturated hydraulic conductivity corrected for gravity. Therefore, when flow proceeds as a series of blobs migrating down the fracture, contact angle hysteresis still constrains the velocity even if the liquid is perfectly wetting under static conditions.
Rivulet Widths
Experiment III was conducted to measure rivulet widths as a function of flow rate between parallel glass and acrylic plates inclined to 20 and 85°. The results are presented in Fig. 7 . At both angles of inclination, the rivulet width is generally larger on the acrylic plates than on the glass plates as the flow rate increases, demonstrating that wettability has a significant effect on rivulet flow. The differences in the widths on the acrylic and glass plates indicate that for a particular flow rate, the average velocity of a rivulet is smaller on an intermediate wetting surface compared with a strongly wetting surface. Comparison of the rivulet widths with Eq. [9] demonstrates that the widths are generally overpredicted using this equation, except for the widths measured at 20° on the acrylic plates. The rivulet was wide enough under those conditions such that Eq. [9] applies. A rivulet width smaller than predicted by Eq. [9] indicates that the average velocity through the rivulet is actually larger than the saturated gravity-driven flow velocity based on Darcy's Law. This is an important result since it demonstrates that rivulets can significantly enhance flow and transport.
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Fig. 7. Plots of measured and predicted rivulet widths as a function of flow rate on glass and acrylic plates (b = 0.25 mm) for inclination angles of (a) 20° and (b) 85°.
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Corner flow in the rivulet is not represented in Eq. [9] and may be important during rivulet flow. The curvature of the air–water interface is larger on the glass plates than on the acrylic plates, as shown in Fig. 8
. Corner flow plays a more important role as the curvature increases. The difference between the widths predicted by Eq. [9] and the measured rivulet widths is expected to be greater on the glass plates compared with the acrylic plates since more corner flow occurs.
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Fig. 8. Schematic of a rivulet cross section on a strongly wetting vs. an intermediate wetting surface. The corner flow regions are larger in the rivulet on a strongly wetting surface.
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Effect of Aperture Variability on Rivulet Flow
Experiment IV was conducted to better understand the conditions of rivulet and streamlet flows as a function of aperture. Figure 9 is the plan view of the liquid distribution in the wedge fracture. A wider region of water, or capillary pool, is present at the top of the fracture where the apertures are smaller; below this region, a rivulet forms. The liquid distribution is very similar to the distribution of water in the flow visualization experiments conducted on an epoxy fracture replica by Su et al. (1999). The rivulet underwent cycles of snapping and reforming at all the flow rates used in Exp. IV. A sequence of the liquid distributions in a wedge fracture is shown in Fig. 10
with a corresponding side view sketch of the liquid distribution in the rivulet. The plates were too narrow to obtain actual images of the liquid distribution from the side.
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Fig. 10. Rivulet dynamics in a wedge fracture and the corresponding liquid distribution from the side and across the aperture.
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An image of the rivulet and advancing blob is shown in Fig. 10a. Different numbered regions are shown in the figure to indicate changes in the liquid distribution across the rivulet as the apertures become progressively larger. In Region 1, the liquid at the top of the rivulet completely spans the aperture, as indicated in sketches of the cross section shown toward the bottom of Fig. 10. In Region 2, the liquid across the rivulet is barely connected by a thin liquid bridge. The rivulets do not span across the aperture in Region 3, and two streamlets exist. The advancing blob is located in Region 4, and the liquid spans the entire aperture in the blob. After the advancing blob migrates some distance, the rivulet detaches just above the blob when the downward velocity of the blob becomes larger than the rate at which the streamlets can lengthen (Fig. 10b). Once the connection with the blob is broken, the streamlets become unstable and the liquid spreads on the surface, as shown in Region 5 in Fig. 10c. The portion of the rivulet in Region 2 recoils into the smaller apertures due to surface tension (Fig. 10c–10d).
The distance traveled by the advancing blob before the rivulet detached from the advancing blob varied for a particular flow rate, but the distance where the streamlets began (ds) was reproducible. The distances measured from the top of the plates and the corresponding aperture widths are summarized in the Table 5 as a function of flow rate. For the range of flow rates used in this experiment (5–25 mL h–1), the corresponding distance where the streamlets began ranged from 3.7 to 6.1 cm, and the aperture widths ranged between 0.25 and 0.37 mm.
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CONCLUSIONS
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Observations of liquid advancement into inclined, unsaturated analog fractures indicated that drainage of a liquid finger into a blob–rivulet structure was an important feature of flow for a range of wettabilities and degrees of surface roughnesses. This feature was observed in glass and acrylic parallel plate fractures and in a fracture with roughness on the order of 100 µm. A continuous finger that did not drain was observed in an analog fracture with smaller scale roughness on the order of 10 µm. If the fracture roughness was partially 100 and 10 µm, the liquid distribution in the fracture would likely transition from rivulets in the regions with 100 µm roughness to wider fingers where the roughness is 10 µm.
On fractures with rough surfaces and intermediate wettability, the rate of blob advancement can be substantially less than the saturated gravity-driven flow velocity of the fracture. The reduction in the blob velocities from the saturated flow velocity is due to liquid–solid interactions during blob migration. The contact angle hysteresis of the blob decreased as the dimensionless velocity (the ratio between the measured blob velocity to the saturated flow velocity) increased on all the fracture surfaces. The contact angle hysteresis calculated for the blobs in the smooth and obscure glass plates had a nearly linear relationship as a function of dimensionless velocity.
The rivulet widths as a function of flow rate were measured between glass and acrylic parallel plates inclined to 20 and 85°. The rivulets were generally wider in the acrylic plates than in the glass plates for a particular flow rate, demonstrating the sensitivity of rivulet flow to the wettability. In addition, the cubic law consistently overpredicted the measured rivulet widths, except for the widths measured between the acrylic plates at 20°. Flow through the corner regions of the rivulet may play an important role during rivulet flow, but this is not accounted for using the cubic law. Rivulet flow was also examined in a variable aperture fracture with apertures increasing linearly from top to bottom. The rivulet underwent intermittent flow in this wedge fracture and the liquid in the rivulet did not completely span the aperture at a critical aperture ranging between 0.25 and 0.37 mm, forming two streamlets of liquid on either side of the fracture.
The results of this study demonstrate that rivulet and blob flows are complex and that flow velocities through fractures can either be enhanced or reduced depending on the type of flow occurring. When a rivulet is connected to an advancing blob, the velocity of the advancing finger is controlled by the rate at which the blob advances, which is less than the saturated gravity-driven flow velocity. When flow is entirely rivulet flow, the velocity through the rivulets can be larger than the saturated gravity-driven flow velocity calculated using Darcy's Law.
Developing conceptual models for unsaturated flow in fractures requires an understanding of how individual factors affect small-scale features flow since this information will help determine the relative contributions of these effects in more complicated systems. The experiments conducted in this study on analog fractures demonstrate that variations in the surface properties and fracture geometry have a significant impact on the types of flow processes that dominate during gravity-driven flow in unsaturated fractures. In natural fractured rock, other factors will also affect flow, such as fracture–matrix interactions, dissolution, precipitation, and heterogeneities in the physical and chemical parameters.
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ACKNOWLEDGMENTS
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This work was supported by the Director, Office of Energy Research, Office of Health and Environmental Sciences, Biological and Environmental Research Program, of the U.S. Department of Energy under Contract no. DE-AC03-76SF00098. Thanks are due to J.R. Nimmo, W. Herkelrath, and three anonymous reviewers for helpful comments on this paper. The work presented also benefited from discussions with M.I. Dragila.
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REFERENCES
|
|---|
- Adamson, A.W. 1990. Physical chemistry of surfaces. 4th ed. John Wiley, New York.
- Adler, P.M., and H. Brenner. 1988. Multiphase flow in porous media. Ann. Rev. Fluid Mech. 20:35–59.
- Bear, J. 1972. Dynamics of fluids in porous media. Elsevier, New York.
- Bikerman, J.J. 1950. Sliding of drops from surfaces of different roughnesses. J. Colloid Sci. 5:349–359.
- Bradford, S.A., and F.J. Leij. 1995. Wettability effects on scaling two- and three-fluid capillary pressure-saturation relations. Environ. Sci. Technol. 29:1446–1455.
- Chemical Rubber Company. 1994. CRC handbook of chemistry and physics. CRC Press, Boca Raton, FL.
- Chuoke, R.L., P. Van Muers, and C. Van der Poel. 1959. The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media. Pet. Trans. AIME 216:188–194.
- DiCarlo, D.A., T.W.J. Bauters, C.J.G. Darnault, T.S. Steenhuis, and J.-Y. Parlange. 1999. Lateral expansion of preferential flow paths in sands. Water Resour. Res. 35:427–434.
- Dullien, F.A.L. 1992. Porous media: Fluid transport and pore structure. Academic Press, New York.
- Extrand, C.W., and Y. Kumagai. 1995. Liquid drops on an inclined plane: The relation between contact angles, drop shape, and retentive force. J. Colloid Interface Sci. 170:515–521.
- Extrand, C.W., and Y. Kumagai. 1997. An experimental study of contact angle hysteresis. J. Colloid Interface Sci. 191:378–383.[Medline]
- Furmidge, C.G.L. 1962. Studies at phase interfaces. 1. The sliding of liquid drops on solid surfaces and a theory for spray retention. J. Colloid Sci. 17:309–324.
- Geller, J.T., H.Y. Holman, G. Su, K. Pruess, and J.C. Hunter-Cevera. 2000. Flow dynamics and potential for biodegradation of organic contaminants in fractured rock vadose zone. J. Contam. Hydrol. 43:63–90.
- Geller, J.T., G. Su, and K. Pruess. 1996. Preliminary studies of water seepage through rough-walled fractures. LBNL Report 38810. LBNL, Berkeley, CA.
- Glass, R.J., and M.J. Nicholl. 1996. Physics of gravity fingering of immiscible fluids within porous media: An overview of current understanding and selected complicating factors. Geoderma 70:133–163.[ISI]
- Glass, R.J., M.J. Nicholl, A.L. Ramirez, and W.D. Daily. 2002. Liquid phase structure within an unsaturated fracture network beneath a surface infiltration event: Field experiment. Water Resour. Res. 38(10). 1199. doi:10.1029/2001WR000167.
- Glass, R.J., M.J. Nicholl, and L. Yarrington. 1998. A modified invasion percolation model for low-capillary number immiscible displacements in horizontal rough-walled fractures: Influence of local in-plane curvature. Water Resour. Res. 34:3215–3234.
- Glass, R.J., J.Y. Parlange, and T.S. Steenhuis. 1989. Mechanism for finger persistence in homogeneous, unsaturated, porous media: Theory and verification. Soil Sci. 148:60–70.
- Hakami, E. 1989. Water flow in single rock joints. SKB, Stripa Project, Technical Report 89–08.
- Iwata, S., T. Tabuchi, and B.P. Warkentin. 1995. Soil-water interactions: Mechanisms and applications. Marcel Dekker, New York.
- Kneafsey, T., and K. Pruess. 1998. Laboratory experiments on heat-driven two-phase flows in natural and artificial rock fractures. Water Resour. Res. 34:3349–3367.
- Laroussi, C., and L.W. DeBacker. 1979. Relations between geometrical properties of glass beads media and their main
(
) hysteresis loops. Soil Sci. Soc. Am. J. 43:646–650.[Abstract/Free Full Text]
- Nicholl, M.J., R.J. Glass, and H.A. Hguyem. 1993. Small-scale behavior of single gravity-driven fingers in an initially dry fracture. p. 2061–2070. In Proceedings of the Fourth International High-Level Radioactive Waste Management Conference, Las Vegas, NV. 26–30 Apr. 1993. Am. Soc. Civ. Eng., New York.
- Nicholl, M.J., R.J. Glass, and S.W. Wheatcraft. 1994. Gravity-driven infiltration instability in initially dry nonhorizontal fractures. Water Resour. Res. 30:2533–2546.
- Philip, J.R. 1975. Stability analysis of infiltration. Soil Sci. Soc. Am. Proc. 39:1042–1049.
- Raats, P.A.C. 1973. Unstable wetting fronts in uniform and nonuniform soils. Soil Sci. Soc. Am. Proc. 37:681–685.
- Selker, J. 1992. Fingered flow in two dimensions. 2. Predicting finger moisture profile. Water Resour. Res. 28:2523–2528.
- Selker, J.S., and M.H. Schroth. 1998. Evaluation of hydrodynamic scaling in porous media using finger dimensions. Water Resour. Res. 34:1935–1940.
- Su, G.W., J.T. Geller, K. Preuss, and J.R. Hunt. 2001. Solute transport along preferential flow paths in unsaturated fractures. Water Resour. Res. 37:2481–2491.
- Su, G.W., J.T. Geller, K. Pruess, and F. Wen. 1999. Experimental studies of water seepage and intermittent flow in unsaturated, rough-walled fractures. Water Resour. Res. 35:1019–1037.
- White, I., P.M. Colombera, and J.R. Philip. 1977. Experimental studies of wetting front instability induced by gradual change of pressure gradient and by heterogeneous porous media. Soil Sci. Soc. Am. J. 41:483–489.[Abstract/Free Full Text]
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