Published in Vadose Zone Journal 3:1439-1442 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Flow in Menisci Corners of Capillary Rivulets
Maria Inés Dragilaa,* and
Noam Weisbrodb
a Department of Crop and Soil Science, Oregon State University, 3017 Agriculture and Life Science Building, Corvallis, OR 97370, USA
b Department of Environmental Hydrology and Microbiology, Zuckerberg Inst. of Water Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Israel
* Corresponding author (maria.dragila{at}oregonstate.edu)
Received 25 December 2003.
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ABSTRACT
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Liquids that flow in unsaturated rock fractures exhibit a range of behavior distinct from flow within the porous matrix. Flow patterns observed in laboratory experiments include aperture-spanning droplets, snapping and continuous capillary threads or rivulets, islands of local saturation, and fingering. Knowledge of the contact area and flow rate for each mode is necessary to achieve successful modeling of the fracture–matrix solute transfer. A geometric model for a rivulet cross section that includes the shape of the curved meniscus between the two walls of a fracture is used in this paper to calculate (i) the flow rate within the menisci corners, (ii) a critical flow rate that determines whether a rivulet will snap or be continuous, and (iii) the width of the wetted contact against a fracture wall. For low flow rates that are near the critical value between snapping and continuous conditions, the contact area was found to be up to 18 times greater than that predicted using a simpler rectangular geometry for the rivulet cross section. A wider contact area also suggests a slower mean flow and larger time constant for solute transfer.
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INTRODUCTION
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FLUID MOTION in unsaturated fractures, which are of sufficiently small aperture to sustain fluid under capillary tension, is complex and still not well understood. The relative proportion of gravitational and capillary forces can cause various instabilities in fluid motion. When input flow rate to inclined fractures is lower than the fracture transport capacity, the resulting instability causes liquid to move as desaturating fingers (Nicholl et al., 1994), saturated droplets accompanied by snapping rivulets (Su et al., 1999; Wood et al., 2002; Glass et al., 2003; Dragila and Weisbrod, 2003; Or and Ghezzehei, personal communication, 2003), or continuous rivulets that do not snap (Su et al., 1999; Dragila and Weisbrod, 2004). Furthermore, fluid can also move as free-surface film flow along a single fracture wall sustaining a free-surface (Tokunaga and Wan, 1997). Regardless of orientation, and also occurring under pressure, two-phase flows form islands of local saturation that preferentially trap one phase as a function of aperture (Pruess and Tsang, 1990). If we are to successfully predict solute arrival times, fracture–matrix transfer, deposition of coatings, or fracture surface erosion, then it is necessary to understand the characteristic fluid velocity and matrix contact areas for each mode.
In inclined unsaturated fractures (<2 mm in aperture), rivulets are defined as continuous streams of liquid that are held under capillary tension and therefore sustain a pair of curved menisci between the two fracture walls (Fig. 1)
. The liquid moves in response to gravitational forces balanced by viscous resistive forces, and the cross-sectional shape of the rivulet depends on the flow rate and liquid–solid contact angle. The region between the fracture wall and curved meniscus as it nears the triple-phase contact line is defined herein as the meniscus corner (Fig. 2)
. Given that the shape of the meniscus is a function of only surface tension, contact angle, and aperture, the flow rate capacity of a meniscus corner does not change with input flow rate. However, increases in flow rate are accommodated by a thickening of the core of the rivulet (Fig. 2). As input flow rate is reduced, the relative proportion of flow in the corner increases. Therefore, the importance of corner flow in calculations of wetted contact area will be more significant at lower flow rates where the corner comprises a larger proportion of the wetted width. We developed and present an analytical expression for flow rate within a nonsnapping rivulet to quantitatively determine the flow rate within menisci corners and its fracture wall contact area. The solution permits a first-order predictive criterion to determine whether a rivulet will snap or be continuous and to determine the flow conditions in which the traditional and much simpler rectangular cross-sectional geometry for the rivulet is appropriate.
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Fig. 1. Schematic diagram of a capillary rivulet under capillary tension. Flow moves from top to bottom between two parallel plates (fracture walls).
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Fig. 2. Cross section of a rivulet and coordinate axes. Cross-hatching represents solid fracture walls, dark gray region is the rivulet core, light gray region comprises the menisci corners, R is the radius of curvature of the meniscus, b is the fracture half-aperture, and  is the contact angle.
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THEORY
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Meniscus Corner Flow Rate
To derive the flow equation, the rivulet is geometrically segmented into two flow regions, a core and four menisci corners (Fig. 2). Flow within the core is approximated by a Poiseuille parabolic velocity profile (Eq. [1]) (Bird et al., 1960). Flow in the corner is trapped between a wall and a free surface. Although no solution to the Navier–Stokes equations for this geometry has been suggested, an approximation can be made by subdividing the corner into strips and solving each as if it were a flat film of corresponding thickness and infinite extent in the plane of the wall (Fig. 3)
. Thus, each segment can be represented by a Stokes semiparabolic velocity profile (Eq. [2]) (Bird et al., 1960). The value of the two equations coincides at the location where the two regions (core and corner) merge (y = b).
 | [1] |
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where u(y)p is the Poiseuille steady-state velocity profile of a liquid flowing between parallel plates, uf[y(xi)] is the Stokes steady-state velocity profile of a flat film of thickness h(xi) sustaining a free surface at y = h, b is the half-fracture aperture, 
is the fluid density, g is the constant of gravitational acceleration, µ is the fluid dynamic viscosity, ß is the (dip) angle of the fracture plane to the horizontal, h(xi) is the height of the free surface from the wall at location xi, and y is the coordinate normal to the fracture wall. The mean velocity within the core region is found by integrating Eq. [1] from y = 0 to y = 2b to obtain the parallel plate law for saturated flow (Bird et al., 1960).
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The mean velocity within the ith rectangular element of the meniscus corner is found similarly, where Eq. [2] is integrated from y = 0 to y(xi) = h(xi).
 | [4] |
The flow rate (Qi) of the ith rectangular subsection is given by
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 | [6] |
where h(xi) is the film thickness of the ith element (centered at x = xi), and Ai is the area of the ith element (Fig. 3). Film thickness, h(xi), is given by the shape of the free surface (Eq. [7]) calculated by assuming the meniscus circumscribes an arc of radius R that intersects the plane of the wall (at y = 0, x = xo) where it subtends an angle equal to the contact angle ().
 | [7] |
where
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and the intersection of the meniscus with the fracture plane (xo) is given by
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The flow rate for the corner is derived by substituting Eq. [4], [6], [7], and [8] into Eq. [5] and summing over the range of elements.
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where x(i = 0) = 
x/2 and x(i = n) = xo – x/2. Using the flat film approximation limits the applicability of the results to contact angles that are well below 90°, thus forming a shallow meniscus corner. In the limit that x approaches zero, the summation can be approximated by an integral. It must be kept in mind that Eq. [11] is not a rigorous solution to the corner flow, but rather an approximation assuming a series of flat film solutions. The integral
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was solved using the tables of Gradshteyn and Ryzhik (1980)(p. 82, 2.262-1) to obtain the following expression for the flow rate in a single meniscus corner:
 | [12] |
where F() is defined as
 | [13] |
Critical Flow Rate
To sustain a continuous (nonsnapping) rivulet requires a flow rate greater than that accommodated by the corners. This first-order approach would suggest that the critical flow rate for snapping (Qcrit) is equal to 4Qcorner. Because of instabilities related to contact angle accommodation and inertial effects, rivulet snapping is likely to occur at slightly higher flow rates. A graph of the critical flow rate for a range of contact angles is shown in Fig. 4
. It is suggested that rivulets that have flow rates below these curves would snap, since at steady state the rivulet core would have zero thickness. Circles in Fig. 4 are published data of flow rates in fractured rock and parallel glass plates that resulted in snapping rivulets (Su et al., 1999; Dragila and Weisbrod, 2003; Or and Ghezzehei, personal communication, 2003). Squares represent flow rates in transition from snapping to continuous rivulets.
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Fig. 4. Flow rate through all four rivulet menisci corners as a function of aperture for two contact angles (0 and 80°). Curves are from Eq. [12]. Data is from Su et al. (1999), Dragila and Weisbrod (2003), and Or and Ghezzehei (personal communication, 2003). Circles represent flow rates that resulted in snapping rivulets; squares represent flow rates at the transition between snapping and continuous rivulets.
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Contact Area
The contact area between a rivulet and the fracture wall are important parameters in solute transfer models, but it is suggested here that a rectangular rivulet cross section may significantly underestimate the contact area. A more accurate calculation of the wetted contact width (i.e., the contact area per unit length of rivulet) can be made by noting that the wetted contact width for the rivulet (ww) is the sum of the width of the core (wb), plus the width of two menisci corners (xo).
 | [14] |
Assuming the central body sustains Poiseuille-type flow (Eq. [3]), the flow rate in that region (Qb) is given by
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and since
 | [16] |
Substituting from Eq. [3], [9], [15] and [16] into Eq. [14], the wetted width is
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By comparison, if menisci corners are not taken into account and the cross section of the rivulet is modeled by a rectangle, as is commonly done, the wetted width would be given by only the first term in the right-hand side of Eq. [17] (Su, 1999).
Comparing the ratio of the wetted width for the rectangular and curved-meniscus models reveals the magnitude of the potential underestimate in the wetted contact area when the meniscus shape is ignored (Fig. 5)
. This discrepancy increases as conditions for snapping are approached, with a maximum value of 18.4 at snapping. The two models converge as the flow rate is increased. For any aperture, the region of concern (corresponding to overestimates >2) occurs for flow rates <16Qcrit. Ironically, many unsaturated fractured rock studies focus on flow behavior near the snapping range (e.g., Su et al., 1999; Dragila and Weisbrod, 2003; and Or and Ghezzehei, personal communication), where there is the greatest disparity in calculation of wetted widths between the two models. Our results suggest that it may not be appropriate to assume a rectangular cross section if knowledge of matrix contact area is thought to be relevant to the investigation.
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Fig. 5. Ratio of the wetted contact width against fracture wall for two models: Model with rivulet corners (Eq. [17]) divided by rectangular model (first term on right-hand side of Eq. [17]). Four curves represent solution for four different apertures. "X" at the top of the curve represents the maximum value of 18.4 above which the rivulet should have snapped. As flow rate increases curves asymptotically approach a value of 1.
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DISCUSSION
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Equations [10] and [12] permitted first-order analysis of the flow rate that could be accommodated through a meniscus corner. This information can be especially useful when assessing under which conditions it is appropriate to assume the much simpler geometry of a rectangular rivulet cross section. When considering the amount of wetted wall contact area, the rectangular cross section is appropriate for flow rates that are >16Qcrit. Whereas, using a rectangular cross section for flow rates below 16Qcrit could underestimate the contact area by a factor of 2 to 18. The model is also useful to determine the approximate minimum flow rate that can sustain a continuous rivulet, and below which it will snap. In natural systems, steady-state values are achieved following a transient stage during which the rivulet core will progressively thin and rupture at some distance down the fracture. The distance required to establish steady state has not been calculated and is left for future study. Nevertheless, if the transient region were comparable to calculations made for the transient stage of a free-surface flat film, then the length before establishment of steady state would be expected to be on the order of centimeters (Dragila, 1999) and to increase with increasing flow rate. Although a quantitative assessment was not made, Su (1999) reported that as flow rate increased the distance before rivulet snapping also increased. Onset of snapping will also be sensitive to transient and inertial effects, surface tension variability, and contact angle changes.
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CONCLUSIONS
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Transport and matrix transfer characteristics of snapping and continuous rivulets are of considerable interest in studies of solute movement in fractured vadose zones. Snapping rivulets provide intermittent transport and formation of capillary droplets that make predictive modeling difficult, whereas continuous rivulets allow for very rapid transport while sustaining a relatively small contact area against the fracture wall. Theoretical analysis presented here proposes a critical flow rate, above which continuous rivulets will be sustained and below which snapping rivulets will snap. The critical flow rate was derived using a cross-sectional geometry for the rivulet shape that included the two concave menisci, and the model assumed that the transition between snapping and nonsnapping rivulets occurs when the width of the central portion of the rivulet is effectively reduced to zero.
Accurate determination of the contact area between the rivulet and fracture wall is important to determine solute transfer between the matrix and fracture. Flow within the menisci corners can be substantial and results in a widening of the contact area and reduction in the thickness of the rivulet core. For low flow rates that are near the critical value between snapping and continuous conditions, the contact area was found to be as much as 18 times greater than that predicted using a simpler rectangular geometry for the rivulet cross section. A wider contact area also suggests a slower mean flow and larger time constant for solute transfer. Consideration of the rivulet curvature is important for flow rates less than 16 times the critical value for snapping. The model indicates that assuming a rectangular cross section is sufficient for higher flow rates.
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ACKNOWLEDGMENTS
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The authors are grateful for critical and thorough reviews of this manuscript by the associate editor D. Or and by M. Nicholl and T. Ghezzehei.
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REFERENCES
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- Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 1960. Transport phenomena. John Wiley and Sons, New York.
- Dragila, M.I. 1999. A new theory for transport in unsaturated fractures: Free surface film flows. Ph.D. diss. University of Nevada, Reno.
- Dragila, M.I., and N. Weisbrod. 2004. Fluid motion through an unsaturated fracture junction. Water Resour. Res. 40:W02402. doi:10.1029/2003WR002356.
- Dragila, M.I., and N. Weisbrod. 2003. Parameters affecting fluid dynamics in large aperture fractures. Adv. Water Resour. 26:1219–1228.
- Glass, R.J., M.J. Nicholl, H. Rajaram, and T.R. Wood. 2003. Unsaturated flow through fracture networks: Evolution of liquid phase structure, dynamics, and the critical importance of fracture intersections. Water Resour. Res. 39(12):1352. doi:10.1029/2003WR002015
- Gradshteyn, I.S., and I.M. Ryzhik. 1980. Table of integrals, series, and products. Academic Press, San Diego, CA.
- 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.
- Pruess, K., and Y.W. Tsang. 1990. On two-phase relative permeability and capillary pressure of rough-walled rock fractures. Water Resour. Res. 26:1915–1926.
- Su, G.W. 1999. Flow dynamics and solute transport in unsaturated rock fractures. Ph.D. diss. Rep. LBNL-44440. Ernest Orlando Lawrence Berkeley Natl. Lab., Berkeley, CA.
- 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.
- Tokunaga, T., and J. Wan. 1997. Water film flow along fracture surfaces of porous rock. Water Resour. Res. 33:1287–1295.
- Wood, T.R., M.J. Nicholl, and R.J. Glass. 2002. Fracture intersections as integrators for unsaturated flow. Geophys. Res. Lett. 29(24):2191. doi:10.1029/2002GL015551
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ABSTRACT
INTRODUCTION
