Published online 16 November 2005
Published in Vadose Zone J 4:1123-1151 (2005)
DOI: 10.2136/vzj2004.0110
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Infiltration into an Analog Fracture
Experimental Observations of Gravity-Driven Fingering
M. J. Nicholla,* and
R. J. Glassb
a Geoscience Dep., Univ. of Nevada, Las Vegas, NV 89122-4010
b Flow Visualization and Processes Lab., Sandia National Laboratories, Albuquerque, NM
* Corresponding author (michael.nicholl{at}ccmail.nevada.edu)
Received 27 August 2004.
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ABSTRACT
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The infiltration of water into unsaturated geologic media is an immiscible displacement process that is unstable with respect to gravity and can thus lead to the formation of gravity-driven fingers. Where the geologic media (e.g., rock, soil) is fractured, gravity-driven fingers within the fractures may lead to extremely rapid vertical migration of waterborne contaminants. We designed analog fractures to facilitate the competition between viscous, gravity, and capillary forces that is expected to control finger behavior, then conducted an extended experimental investigation to observe and measure finger behavior. Results show that the spatially variant two-dimensional nature of fracture geometry leads to different behavior than is reported for the related problem of gravity-driven fingers in porous media. Observations of finger behavior are presented, along with a simple scale analysis used to relate the key measures of finger velocity, finger width, and fingertip length. We also present a series of illustrative experiments designed to guide future research.
Abbreviations: DNAPLS, dense nonaqueous phase liquids
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INTRODUCTION
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INFILTRATION is an immiscible displacement process in which water seeps downward into unsaturated soil or rock, displacing the resident air phase from pores and fractures. Simple linear stability theory suggests that gravity will act to destabilize infiltration, while viscous and capillary forces will provide a stabilizing influence (e.g., Saffman and Taylor, 1958; Chouke et al., 1959). Gravity-driven fingers will form in situations where infiltration is unstable. The occurrence and subsequent behavior of gravity-driven fingers in granular porous media (i.e., sands of various textures) has received considerable attention (see reviews in Chen et al., 1995; Chen and Neuman, 1996; Glass and Nicholl, 1996; Scanlon et al., 1997; de Rooij, 2000; Eliassi and Glass, 2002). Conversely, few investigations have considered gravity-driven fingers within individual fractures (Nicholl et al., 1992, 1993a, 1993b, 1994; Glass and Nicholl, 1996; Su et al., 1999, 2001, 2004). Gravity-driven fingers can also occur in free-surface flows on large aperture fractures (e.g., Benson, 2001); however, we restrict our discussion to those that locally saturate the fracture aperture.
The need to consider gravity-driven fingers in fractures as separate from those formed in granular porous media rises from the basic differences in topology of the void spaces between the two. Fracture void space differs from that found in granular porous media in terms of dimensionality (two vs. three), connection, size, isotropy, and homogeneity. In turn, those differences fundamentally alter the balance between capillary, gravity, and viscous forces that controls the occurrence and behavior of gravity-driven fingers. The two-dimensional nature of fractures forces infiltration to occur at an arbitrary angle with respect to gravity and places constraints on accessibility that lead to greatly enhanced phase entrapment over porous media. Finally, granular porous media commonly exhibits micro-rough grain surfaces and closely spaced intergranular contacts. In fractures, contact points are likely to be widely spaced with respect to the fracture aperture, and in many instances fracture surfaces will be locally smooth. These differences will affect both capillary properties along the air–water interface and the nature of residual moisture content following gravity drainage.
The occurrence of gravity-driven fingers will have a substantial influence on infiltration within an individual fracture. For a given infiltration event, fingers will move much faster and further than would be predicted for a flat (stable) displacement front. In addition, fingers will occupy a much smaller cross-sectional area than a flat front, making them difficult to detect and greatly restricting contact between infiltrating water and the fracture walls. This latter characteristic is critical, as imbibition of water into the adjacent rock matrix and film flow along the fracture walls would slow or perhaps halt advancement within the fracture. Contact with the fracture walls also facilitates processes that inhibit the migration of waterborne contaminants, such as adsorption and chemical or biological degradation. Thus, for infiltration into otherwise low permeability units such as that shown in Fig. 1
, the formation of gravity-driven fingers in fractures may lead to transport velocities that are orders of magnitude more rapid than would be expected for capillary (i.e., matrix) dominated flow. Gravity-driven fingers may also be a more ubiquitous occurrence in fractures than in natural porous media, where the presence of fine materials and initial moisture content can act to suppress fingering.
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Fig. 1. Cliff face showing an approximately 10-m high exposure of fractured basalt located in southeastern Idaho (Schaefer, 2002). Extensive vertical fractures formed during cooling of the molten lava dominate the fracture network, which also includes a smaller number of less extensive subhorizontal fractures. The horizontal recess is a sedimentary interbed.
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Here, we present laboratory experiments designed to explore gravity-driven fingers formed during infiltration into single fractures. Insight from related areas (Background and Theoretical Network) was used to develop a systematic investigative approach. The experimental design and analog fracture used to control the balance between viscous, capillary, and gravitational forces are presented in Experimental Design. Our results begin with experimental observations of fingers formed during the redistribution of flow that occurs after ponded infiltration (Experimental Observations: Multiple Fingers as Generated via Redistribution Following Ponding Events). Next (Experimental Observations: Single Fingers from Point Sources), we consider individual fingers initiated from steady flow to a point source. Measured data are presented in Quantitative Evaluation of Finger Behavior, followed by the development of simple predictive relations for finger width and fingertip length as functions of finger velocity, with comparison with measured data. In Experimental Observations: Illustrative Extensions to our Principal Results, we present a series of illustrative experiments intended to guide future work. We consider (i) a modification to our principal analog fracture that alters the balance between forces, (ii) micro-roughness on the fracture surfaces, (iii) natural fracture aperture fields, (iv) imbibition by the adjacent matrix, and (v) buoyant nonwetting fingers. We then conclude with a summary of our key observations, followed by an introduction to the related areas of network-scale flows and non-wetting displacement by dense nonaqueous phase liquids (DNAPLs).
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BACKGROUND AND THEORETICAL FRAMEWORK
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This section outlines results from related areas that guided our investigation. We begin with the general topic of immiscible displacement in fractures. Then we review linear stability analysis of rectilinear displacement in the Hele-Shaw geometry and customize relations developed in that context to the infiltration of water into an air-filled system.
Fractures and Immiscible Displacement
Fracture flow occurs within the void space, or aperture between two uneven surfaces that result from brittle failure. This variable-aperture field determines the permeability of the fracture. Aperture variability also places important controls on the immiscible displacement of one fluid by another, and progress has been made with respect to capillary dominated (i.e., slow and horizontal) displacements.
Fracture Aperture Fields
Fracture surfaces can span from very rough (such as tectonic fractures in plutonic granites) to relatively smooth (such as columnar cooling fractures in volcanic rocks as seen in Fig. 1). A number of measurements on rough fracture surfaces have suggested a self-affine fractal topography at length scales from the submillimeter upward (e.g., Brown and Scholz, 1985; Poon et al., 1992; Schmittbuhl et al., 1995). However, the smoother surfaces of cooling fractures in volcanic rock or fractures in glasses tend to lack a fractal quality at the smaller scales (e.g., Throckmorton and Verbeek, 1995). Plastic deformation and disaggregation following brittle failure will usually prevent the surfaces from mating perfectly, as will small displacements between the two surfaces (e.g., Plouraboue et al., 1995). A gap or aperture field is thus created that varies from point to point and may include contact areas where the aperture, a (L), goes to zero. Initial fracture topology may be subsequently modified by additional movement, the dissolution and precipitation of minerals, and/or mechanical erosion of the fracture surfaces (e.g., National Research Council, 1996; Hanna and Rajaram, 1998; Weisbrod et al., 1998, 2000; Streit and Cox, 2000; Durham et al., 2001).
Although the topology of natural fractures exhibits extreme variability, there are expected commonalities that can be used to constrain fracture aperture fields for the study of gravity-driven fingering in the laboratory. Displacement of one fracture surface with respect to the other will disrupt long-range correlations between the surfaces and impose a maximum length scale on spatial correlation in the aperture field (e.g., Brown, 1995; Plouraboue et al., 1995). Detailed aperture characterizations are scarce; however, the aperture spatial correlation length, 
, has been estimated in the laboratory through X-ray tomography (Keller, 1998) of naturally fractured granite cores ( 
0.08–1.2 cm), destructive sectioning (Hakami and Larsson, 1996) of an epoxy filled natural granite fracture ( 1 cm), and light absorption within cast replicas (Lee et al., 2003) of artificially fractured sandstone ( 0.6–3.8 cm). Additionally, below a certain length scale, fracture surfaces, and thus aperture, will become smooth. This lower limit will not only vary with the rock material, but also be greatly influenced by precipitation and dissolution processes. Finally, because most fractures require some propping to remain open (such as occurs in translation along the rough surface), many fractures have mean apertures in the range where capillary forces are important (i.e., below 0.1 cm). The investigations referenced above estimated mean aperture, <a>, at 0.0639 to 0.0825 cm (Keller, 1998), 0.36 cm (Hakami and Larsson, 1996), and 0.0259 to 0.0384 cm (Lee et al., 2003).
Fracture Permeability
Steady flow through a fracture at low Reynolds number (Re <1–10) is commonly assumed to follow Darcy's Law. For a fracture filled with a single fluid phase, flux, q (L T–1), will be linearly dependent on the hydraulic gradient, with the constant of proportionality given by the saturated hydraulic conductivity, Ks (L T–1):
 | [1] |
where k represents intrinsic permeability of the fracture (L2), 
is fluid density (M L–3), g is the gravitational constant (L T–2), and µ is the fluid kinematic viscosity (M/LT). Attempts to parameterize k for rough-walled fractures have confirmed that aperture variability leads to a deviation from the Hele-Shaw result where k = a2/12 (e.g., Brown, 1989; Zimmerman and Bodvarsson, 1996; Nicholl et al., 1999).
The presence of two or more fluids within a fracture leads to phase interference, and permeability to each will be less than k (e.g., Fourar et al., 1993; Murphy and Thomson, 1993; Persoff and Pruess, 1995; Nicholl et al., 2000). The reduction in permeability is often represented by the relative permeability, kr, a dimensionless fraction that varies between 0 and 1. Under most natural hydrologic conditions, water either displaces the air phase as it invades a fracture, or air reinvades as water drains. In both cases, incomplete displacement and an absence of film flow may lead to a phase structure in which only one phase is free to flow, while the other occupies entrapped zones that obstruct flow. Because the connectivity of fracture void space is essentially two-dimensional, entrapped zones have a much larger influence on permeability than in three-dimensional porous media.
Capillary Displacement in Fractures
Capillary forces impose a pressure jump, Pc (F L–2), across the interface between two immiscible fluids (e.g., air and water). In the absence of viscous and gravity forces (e.g., slow horizontal flow), Pc along the interface provides the sole control on displacement, and thus fluid phase structure. Under these quasistatic conditions, Pc is dependent on the local interfacial curvature as given by the Laplace–Young equation:
 | [2] |
where 
is the interfacial tension (M T–2), r1 is the first principal radius of interfacial curvature (L), and r2 is the second principal radius of curvature (L). For a variable-aperture fracture, r1 spans the two walls of the fracture, while r2 lies in the plane of the fracture (Fig. 2)
. Thus, r1 is constrained by the fluid–fluid–solid contact angle, 
, while r2 is not (National Research Council, 1996). Where fracture aperture varies symmetrically about a mean plane (Fig. 2b), r1 can be written as
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where ß describes the local change in aperture. The second principal radius of curvature evolves as the interface grows (Fig. 2a). Within a spatially correlated random field, r2 can be approximated as (Glass et al., 1998)
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where 
is the angle between two vectors that approximate the local interface, as measured from the displacing fluid side (Fig. 2b).
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Fig. 2. (a) Sketch illustrating the fluid–fluid interface in plan view (not to scale). In-plane curvature, r2, is shown at two different locations, while the inset depicts the approximation of r2 in Eq. [4]. (b) Cross-sectional view along the line A–A' (Fig. 2a) depicts the definition of r1. Inset shows and ß from Eq. [3].
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Modified Invasion Percolation (MIP) simulations that embody Eq. [2] to [4] have shown that phase structure during capillary displacement in a variable-aperture field is controlled by the competition between interfacial roughening due to random aperture (r1) variations and interfacial smoothing due to in-plane (r2) curvature (Glass et al., 1998). To first order, phase structure is determined by the Curvature number, C, a dimensionless parameter formed as a ratio of representative values for each influence:
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Later, Glass et al. (2003) found that the ratio C/
more comprehensively controls phase structure, where represents the coefficient of variation (L L–1) for the aperture field. For C/ well below 1, aperture-induced curvature dominates, and the phase structure is controlled by capillary fingering within the spatially correlated field. As C/ increases, capillary fingers widen to above the spatial correlation length. For C/ well above 1, the interface is dominated by r2 curvature, and thus behaves as though in a Hele-Shaw cell, with little or no entrapment of the defending phase.
Finally, we note that, just as in porous media, capillary invasion pressures for fractures are different for wetting and nonwetting fluids. A number of processes lead to hysteresis. The most important is that nonwetting invasion is controlled by the large apertures within the field, while wetting invasion is controlled by the smaller ones. Second-order influences come from contact angle hysteresis and dynamics, where the wetting angle is often larger than the drainage angle (e.g., Dussan, 1979; de Gennes, 1985). The combination of these influences changes the distribution of apertures that are sampled between nonwetting and wetting invasion and, through interaction with r2, yields differences in the displacement phase structures. In essence, wetting phase invasion has a higher C/ and a correspondingly more macroscopic phase structure than nonwetting invasion (Glass et al., 2003).
Results from Linear Stability Theory in a Hele-Shaw Cell
Linear theory has been used to consider the stability of rectilinear displacement in a Hele-Shaw cell (e.g., Saffman and Taylor, 1958; Chouke et al., 1959; Saffman, 1986; Homsy, 1987), which is essentially a constant aperture fracture. In the absence of capillarity, rectilinear displacement of one incompressible viscous fluid (subscript 1) by another (subscript 2) along the plane of a Hele-Shaw cell is expected to be unstable to perturbations of all wavelengths when the following inequality is satisfied (after Chouke et al., 1959):
 | [6] |
where U is interfacial velocity (L T–1) in the direction of displacement and 
is the included angle between the direction of displacement and the vertical downward direction (i.e., cos is positive for downward displacement). This simple linear analysis suggests that gravity-driven fingers will form where a destabilizing gravitational force (i.e., right-hand side of Eq. [6] is positive) is not offset by a stabilizing viscous force of equal or greater magnitude.
Assuming displacement in a Hele-Shaw cell where the fluid–fluid interface is perturbed at all wavelengths, the linear analysis used to develop Eq. [6] predicts that the shortest wavelength perturbations will experience the most rapid growth (e.g., Homsy, 1987). However, for immiscible fluids, surface tension will act to preferentially damp short wavelengths. Competition between these two wavelength-dependent influences (growth rate and dampening) will act to select a single wavelength for maximum growth rate, m (Chouke et al., 1959):
 | [7] |
Noting that fingers are expected to form at widths on the order of m/2, Eq. [7] suggests that finger width will be determined by both interfacial tension and the degree of instability as expressed by the bracketed term.
Unstable Infiltration
Specialization of Eq. [6] to infiltration is straightforward. Water closely approximates an incompressible viscous fluid, while air may be treated as an in viscid fluid of negligible density that escapes freely during displacement. Under these conditions, Eq. [6] may be restated as
 | [8] |
which suggests that gravity-driven fingers are likely to form at infiltration velocities <Kscos. Because Kscos gives the flux for saturated flow influenced by gravity alone, we see that local capillary forces, heterogeneity within the field, or external pressure can act to stabilize or destabilize infiltration. Simplification of Eq. [7] to displacement of air by water leads to
 | [9] |
We note, however, that the simple action of capillarity put forth in Eq. [7] is only strictly valid within a Hele-Shaw cell. There, the aperture induced component of the capillary force, r1 in Eq. [2], is of equal value everywhere along the interface, thus leaving the in-plane component, r2 in Eq. [2], to smooth or stabilize the front. In a variable-aperture field, r1 related capillary forces will vary along the fluid–fluid interface, yielding some deviation from relations such as Eq. [7] and [9] that assume constant r1. Analogous discrepancies are also relevant for porous media (e.g., Chouke et al., 1959; Parlange and Hill, 1976).
Experiments in granular porous media provide additional information that is germane to our study of gravity-driven fingering in fractures. Infiltration into narrowly distributed dry sands has been found to be unstable for (i) steady supply (e.g., rainfall) at q < Kscos (e.g., Selker et al., 1992a), (ii) steady supply where Ks increases with depth (e.g., Hill and Parlange, 1972; Diment and Watson, 1985; Glass et al., 1989b), and (iii) redistribution following viscous controlled infiltration (e.g., Raats, 1973; Philip, 1975; Jury et al., 2003; Wang et al., 2003a, 2003b). It has also been found that individual fingers in initially dry sands consist of a saturated tip, followed by a partially desaturated zone (Glass et al., 1989c; Selker et al., 1992b). Glass et al. (1989c) related length of the saturated fingertip, Ltip, to fingertip velocity, v (L T–1), through Darcy's Law:
 | [10] |
where 
w is the wetting pressure head (L) for the media, and d is the drainage pressure head (L). Due to capillary hysteresis, d will be more negative than w in water-wettable porous media; thus, the bracketed tem in Eq. [10] will not exceed one, and v 
0 when Ltip = w – d (see also Wang et al., 2004).
Hysteresis also causes finger locations to persist from one infiltration cycle to the next (e.g., Glass et al., 1988, 1989a; Liu et al., 1994). Conversely, uniform moisture fields have been found to widen, or even suppress the formation of gravity-driven fingers (Diment and Watson, 1985; Glass and Nicholl, 1996; Wang et al., 2003a, 2003b). While the exact value of moisture content required to suppress instability remains open, it is likely to be defined by the point at which liquid films can be maintained on the surface of grains (Glass and Nicholl, 1996).
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EXPERIMENTAL DESIGN
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Exploration of gravity-driven fingering in fractures requires that we be able to control and/or vary the influence of capillary, gravity, and viscous forces in a systematic fashion. It must also be possible to measure and observe behavior without perturbing the flow field. To meet these objectives, we used textured glass plates to fabricate analog fractures that were transparent, water wettable, and reproducible. Between experiments we held the fracture geometry constant, and altered the balance between capillary and gravity forces by varying the angle of the fracture within the gravitational field. We considered two different boundary conditions at the top of the fracture: (i) ponding followed by redistribution and (ii) constant supply to point sources where single fingers initiate from the controlled perturbation. In both cases, we varied viscous forces between experiments by altering the fluid application (pond volume for redistribution, supply rate for single fingers). Given the strong influence of antecedent moisture observed in porous media, we also varied initial conditions within the analog fracture from dry to partially saturated, with particular emphasis on structure of the initial moisture field. In this section, we present details of the experimental system, the measured properties of the analog fracture, and outline the experiments performed.
Experimental System
Aperture fields were fabricated in two sizes, 30 by 60 cm and 15 by 30 cm; the larger size was used to consider systems characterized by multiple fingers, while the smaller one was used to focus in on the behavior of individual fingers. For both sizes, displacement was directed in the long axis. Confinement cells constructed to hold the experiment (Fig. 3a)
were pressurized with gas to place the analog fractures under a 0.138 MPa (20 psi) normal load. The normal pressure held the textured glass plates in close contact, eliminating long wavelength disturbances in the aperture fields. Windows in each cell allowed us to view almost the entire aperture field (except for a border of 1 cm along the edges). The cell design also allowed us to implement various boundary conditions and to reassemble the aperture fields in a repeatable alignment. In preliminary trials to test reproducibility, we found it necessary to carefully clean and dry the glass surfaces between experiments. It was also important to follow a strict protocol while first assembling the cell, and then torquing the bolts that hold it together (Fig. 3a). In the course of our experiments the textured plates broke several times and were replaced.
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Fig. 3. a) Sketches of our 15- by 30- and 30- by 60-cm test cells in plan and cross-sectional views (not to scale). Each of the textured glass plates is separated from a 19-mm (3/4") glass plate window by a thin rectangular gasket. A needle inserted through the gasket allows pressurization of the intervening space, pushing the two textured plates into close contact. Note that a larger number of bolts than shown were used to assemble the 30- by 60-cm cell. (b) Sketch of our test stand (not to scale). Light intensity is controlled through a feedback circuit. The light box also contains a cooling system to prevent warming the experiment.
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Data were obtained from sequential images collected during each experiment. The confinement cells were clamped onto a light table that includes a steel superstructure designed to hold a CCD camera (512 x 512 pixel, 8-bit resolution) at a fixed location above the experiment (Fig. 3b). The whole apparatus (light table, confinement cell, and CCD camera) could be rotated as a unit to allow variation of the gravity force through fracture inclination (i.e., cos). The amount of light transmitted through the analog fracture was modified locally by the presence of air, deionized water, or dyed water. Dyes (FD&C Blue #1 and Red #3) were added to deionized water in concentrations of 1 g L–1 or less; simple tests assured that addition of the dye imposed a negligible influence on fluid properties.
Analog Fracture Properties
The textured glass plates used in our experiments were selected from a category of decorative materials known as obscure glass. We examined various textures and selected one that would produce homogenous, isotropic aperture fields within the range of natural fractures, and exhibit C/ between the extremes of r1 or r2 domination. The analog fractures were water-wettable, with a static contact angle of 35 to 55°. The pattern of the textured glass plates used in our experiments can be clearly seen in the aperture field produced when one textured plate is pressed up against a smooth glass plate (Fig. 4a)
as measured by transmitted light imaging (e.g., Detwiler et al., 1999). For our experiments, two textured plates were assembled in face-to-face contact to produce an aperture field as seen in Fig. 4b. In both instances, the 3- by 3-cm segments shown in the figure are representative of the entire aperture field. Aperture distributions for both fields are shown in Fig. 4c. Aperture statistics are provided in Table 1, along with hydraulic and capillary measurements. Vertical capillary rise was measured as an estimate of the wetting pressure head (w), while the drainage pressure head (d) was taken as the fluid height after free drainage. By design, the characteristic length scale of the aperture variability (Fig. 4d) is much smaller ( 0.08 cm) than the experiments (15 by 30 and 30 by 60 cm). The resulting analog fractures are homogenous and isotropic at the macroscopic scale, thus assuring that observed behavior is controlled by the processes under study rather than heterogeneity-driven channeling. Finally, we note that the statistical measures of our analog fractures (<a>, , ) fall within the range of those reported for natural fractures (e.g., Johns et al., 1993; Hakami and Larsson, 1996; Keller, 1998; Su et al., 1999).
Experiments Conducted
Our experimental investigation of gravity-driven fingering in fractures included a large number of experiments conducted during approximately 10 yr. A small portion of these experiments were published in Nicholl et al. (1994); additional preliminary results were presented at conferences (Nicholl et al., 1992, 1993a, 1993b), and mentioned in Glass and Nicholl (1996). Given the extent of our experimental investigations, we have chosen to present a synthesis of results that highlights the major observations across experiments. Experimental Observations: Multiple Fingers as Generated via Redistribution Following Ponding Events considers gravity-driven fingers formed following ponded infiltration of water into fractures both with and without antecedent moisture content. Experimental Observations: Single Fingers from Point Sources focuses on gravity-driven fingers generated from single point sources, again with and without antecedent moisture content. In Quantitative Evaluation of Finger Behavior, measured data are evaluated, and a simple local force balance is employed to develop relationships for fingertip length and finger width as functions of finger velocity. Finally in Experimental Observations: Illustrative Extensions to our Principal Results, we extend our principal results with a set of illustrative experiments designed to guide future work.
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EXPERIMENTAL OBSERVATIONS: MULTIPLE FINGERS AS GENERATED VIA REDISTRIBUTION FOLLOWING PONDING EVENTS
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One of the most ubiquitous situations that can lead to gravity-driven fingers is the redistribution of flow that follows a rainfall or irrigation event (e.g., Raats, 1973; Philip, 1975; Jury et al., 2003; Wang et al., 2003a, 2003b). We simulated such a situation by rapidly ponding a finite volume of water in a shallow reservoir that covered the top edge of our 30- by 60-cm analog fracture. All other edges of the fracture were left open to atmospheric pressure so air could freely escape. Over a series of experiments, the volume of water added to the pond was varied to explore the influence of viscous forces. Fracture inclination (i.e., cos) was varied to consider gravity forces. First, we consider infiltration into a dry fracture, then look at the influence of initial moisture.
Initially Dry Fractures
We conducted 24 experiments to consider ponded infiltration into an initially dry fracture. Four values of cos were considered (1.0, 0.75, 0.50, and 0.25). Input volume to the pond, Vp (L3), was varied over a range from 3.4 to 28.4 cm3; note that total aperture volume of the analog fracture was about 41 cm3. The analog fracture was disassembled, cleaned, and dried between trials.
Digital images of fluid advancement (Fig. 5)
show the transition from initially stable to unstable infiltration where gravity-driven fingers form. Water is added to the pond from 16 evenly spaced point sources. This slight nonuniformity introduces finite-amplitude perturbations to the fluid–fluid interface. With water in the pond (Fig. 5a) capillary forces act only along the leading edge of the fluid slug, where they reinforce gravity. As a result, interfacial velocity is high (U > Kscos) and viscous forces preferentially focus flow into the least advanced portions of the leading edge, damping initial perturbations (i.e., flow is stable). When fluid in the pond is exhausted, gravity acting on the fluid slug exerts tension along the trailing edge of the slug and initiates drainage (Fig. 5b). Drainage pressures along the trailing edge of the fluid slug act to oppose gravity and are of greater magnitude than the wetting pressures that reinforce gravity along the leading edge. Thus, capillary forces within the fluid slug reverse direction and oppose flow. Interfacial velocity decreases (U < Kscos), and flow is preferentially redistributed into the furthest advanced portions of the front (perturbations grow) (i.e., flow is unstable).
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Fig. 5. The formation of gravity-driven fingers following ponded infiltration (Vp = 4.29 cm3) into the initially dry 30- by 60-cm cell at cos = 1.0.
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At the onset of instability, gravity-driven fingers begin to develop from the most dominant perturbations to the front and take on the characteristic width of those perturbations (Fig. 5c). While the fluid slug is intact (i.e., laterally connected) individual fingers compete for fluid, such that larger and more advanced fingers grow at the expense of their smaller neighbors. Fingers eventually separate from one another to fragment the slug and become distinct entities (Fig. 5d) that move as compact bodies, leaving behind a trail of small fluid blobs pinned within the aperture field. Fingertip length decreases in response to fluid loss during advancement. Because the capillary force that resists movement is inversely proportional to fingertip length, fingers slow as they lose fluid, and eventually stop when gravitational forces are balanced by capillary forces. The smallest two fingers seen in Fig. 5d stalled within the fracture. If the fracture had been longer, the other three fingers would eventually have stopped.
Although much smaller than their parent fingers, the fluid blobs left behind following passage of the fingertip (Fig. 5b–5d) are often considerably larger than the spatial correlation length of the aperture field, . This observation suggests that water is retained in apertures of all sizes. The drainage process involves a reinvasion along the trailing edge of the saturated fingertip by the nonwetting air phase. The reinvasion process entraps water and pins it within the aperture field in accordance with the local balance between capillary, viscous, and gravitational forces. When compared with the smooth fingertip, the differences between wetting and nonwetting invasion at similar C/ (Glass et al., 1998, 2003) produces a much more complicated interface along the trailing edge.
Across all experiments, the finite amplitude perturbations present at the time of redistribution control the width of subsequent fingers. In our experiments, viscous damping of the initial perturbations increased with distance traveled before redistribution. In Fig. 5b, the 16 perturbations induced by fluid application damped into five in the short distance between the top edge of the fracture and the onset of redistribution. As illustrated in Fig. 6
, larger pond volumes increase the viscous damping before redistribution, which acts to increase finger width and reduce the number of fingers. Variation of fracture inclination showed little if any effect on finger width. However, increasing cos did lead to faster and somewhat smoother fingers that more closely follow the gravitational vector (Fig. 7)
.
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Fig. 6. The effects of pond volume, Vp, on fingers formed in the initially dry 30- by 60-cm cell at cos = 1.0. Image time in seconds after starting the experiment (t) gives an indication of relative velocity. (a) Vp = 4.29 cm3, t = 23 s; (b) Vp = 6.17 cm3, t = 23 s; (c) Vp = 10.23 cm3, t = 23 s; and (d) Vp = 14.06 cm3, t = 15 s.
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In order for redistribution to proceed after the pond is depleted the fluid slug must exert sufficient tension to initiate drainage. Otherwise it will remain pinned within the fracture. Note that tension applied by the slug is given by the vertical distance between the leading and trailing edges of the slug. The fluid slug shown in Fig. 8
appears to be pinned in place after the pond is depleted (Fig. 8a). However, for the next 10 min drainage along the top of the fracture proceeds on a very slow ("pore-by-pore") basis. Tension applied by the fluid slug is about 1.4 cm, much less than the average drainage pressure of –5.1 cm (Table 1). As a result, drainage is restricted to the largest apertures (see top edge of Fig. 8b), with the drained fluid being redistributed to the tip of the most advanced perturbation (see circled region in Fig. 8b). The perturbation grows slowly for about the next 10 min (Fig. 8c), then accelerates and widens substantially during the next 5 min (Fig. 8d). At this point, tension exerted by the finger (4.9 cm) is of similar order to the drainage pressure, and thus drainage involves a wider range of apertures. Locally, tension generated within the growing finger is sufficient to initiate drainage within the narrow neck connecting it to the original fluid body (see circled region in Fig. 8d).
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Fig. 8. Slow growth of a single finger following ponded infiltration (Vp = 6.23 cm3) into the initially dry 30- by 60-cm cell at cos = 0.25. Image times give an indication of relative velocity. (a) t = 39 s; (b) t = 619 s; (c) t = 1211 s; and (d) t = 1518 s.
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Influence of Initial Moisture
As discussed above, experimental studies in hydrophilic sands suggest that gravity-driven fingers formed under initially dry conditions will persist to create preferential pathways for subsequent infiltration events. To explore this possibility, we conducted 31 experiments where the fracture contained a structured initial moisture field created by a previous unstable infiltration event. Experiments were conducted at cos = 1.0, 0.75, and 0.50; Vp was varied over a range from 3.75 to 26 cm3. We also conducted five experiments in which the fracture contained a uniformly distributed initial moisture field.
The structured initial moisture field created by unstable infiltration was found to guide subsequent events. Figure 9a
shows the moisture structure present at the conclusion of the experiment shown in Fig. 5; three fingers have passed through to the bottom of the cell, while two have stalled and are pinned within. A second ponding event of similar magnitude (Vp = 4.2 cm3) to the first (4.3 cm3) follows the initial moisture structure (Fig. 9b and 9c). The second event entraps significant amounts of air to form fingers that are not only much longer than were observed under dry initial conditions, but also exhibit a complex internal structure that was not present in the dry fracture (Fig. 5d). The second event loses little, if any, fluid as it passes through the initial moisture structure to restart the stalled fingertips (Fig. 9d). Fluid loss begins when the second event exits the initial moisture structure and continues to advance, as was observed under dry initial conditions.
In experiments where the second ponding event was of substantially smaller magnitude than the initial event, we observed fingers that were much narrower than those that created the initial moisture field. This observation conflicts with experimental results for granular porous media, where it has been found that a uniform initial moisture field provides a stabilizing influence that widens or suppresses gravity-driven fingers (Diment and Watson, 1985; Glass and Nicholl, 1996; Wang et al., 2003a, 2003b). To explore this apparent conflict, we conducted five additional redistribution experiments for "uniform" initial moisture fields. We saturated the 30- by 60-cm cell, then rotated it to the desired inclination and allowed gravity drainage to a residual moisture content. To drain the capillary fringe, we used absorbent towels to apply a gentle suction along the bottom boundary. At steep inclinations, this procedure produced a residual moisture structure that consisted of small disconnected fluid blobs distributed pretty much uniformly about the cell (Fig. 10)
. At lower inclinations, fluid blobs left behind after gravity drainage were generally larger and less uniform (shape, size, spatial distribution), leading to higher average residual moisture content.
Contrary to results in porous media, initial moisture appeared to enhance fingering relative to dry initial conditions. Figure 11 shows a typical experiment in which dyed water was ponded above a uniformly distributed initial moisture field of undyed water. Infiltration (Fig. 11a) entraps significant amounts of air to form a complicated front characterized by sharp, short wavelength perturbations. This contrasts significantly with the smoother, longer wavelength perturbations observed during infiltration into the same cell under dry initial conditions (Fig. 5a). Gravity-driven fingers form when fluid in the pond is exhausted and further advancement requires drainage along the trailing edge of the fluid slug (Fig. 11b). Fingers (Fig. 11c and 11d) form long, complicated structures that entrap significant amounts of air. With respect to dry initial conditions in the same cell, fingers appeared to be faster, longer, narrower, more complex, and more numerous. We also observed that the fingers tended to meander more on their way downward, merging with adjacent fingers and splitting into smaller fingers. In doing so, the advancing fingers leave little fluid behind. However, they do rearrange and interact with the initial moisture field. In Fig. 11, the dyed fluid in the fingers transitions from dark to shades of gray as it mixes with the undyed water used to create the initial moisture field.
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Fig. 11. Formation of gravity-driven fingers following ponded infiltration (Vp = 6.98 cm3) into a uniform initial moisture field at cos = 0.25. Water in the initial moisture field was not dyed, and is nearly transparent.
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EXPERIMENTAL OBSERVATIONS: SINGLE FINGERS FROM POINT SOURCES
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The number, size, and behavior of individual fingers initiated under redistribution were tied closely to the size of perturbations present at the onset of instability. To more clearly understand finger behavior, we generated individual fingers from the controlled perturbation produced by steady fluid supply to a point source at the top of the fracture. The supply rate was varied between experiments to alter viscous forces, and fracture inclination was changed to control gravity forces. To begin this section, we consider infiltration into a dry fracture. We then look at the influence of initial moisture.
Initially Dry Fractures
We conducted 76 experiments in which individual fingers were generated from steady supply to a point source at the top of a dry fracture. To improve image resolution, experiments were conducted in the 15- by 30-cm cell (10 cm3 aperture volume). Four values of cos were considered (1.0, 0. 5, 0.25, and 0.125). Supply rate, Q (L3 T–1), was varied over a range from 0.022 to 13.3 cm3 min–1. The cell was disassembled, cleaned, dried, and reassembled between trials.
At low Q, infiltration from a point source at the top of a vertical fracture is initially pinned to the upper boundary by capillary forces (Fig. 12a)
. The finger grows preferentially in the direction of gravity and eventually reaches sufficient length to break free of the top boundary and move on its own. The size of this initial fingertip is controlled by the balance between capillary, gravitational, and viscous forces rather than an initial perturbation; otherwise it is directly analogous to those seen in the redistribution experiments. The saturated fingertip is separated from the fluid source by a partially desaturated zone (Fig. 12b). Fluid structure within the desaturated zone results from a combination of drainage behind the saturated fingertip and resupply from the fluid source. Fluid from the source continues to move through the desaturated zone to resupply the fingertip, which loses fluid as it wets the fracture behind it (Fig. 12c and 12d). At low Q and large cos, connection between the fingertip and the fluid source was intermittent. Flow from the source would follow behind the tip, reconnect when the tip slowed due to fluid loss, and disconnect again when the fingertip was replenished.
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Fig. 12. Formation of a gravity-driven finger from steady supply (Q = 1.36 cm3 min–1) to a point source at cos = 1.0.
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The influence of cos on finger behavior at low Q is shown as Fig. 13
. At cos = 1.0 the finger is narrow, closely aligned with the gravitational vector, and exhibits substantial desaturation behind a saturated tip that is fully disconnected from the fluid source (Fig. 13a). Desaturation behind the fingertip decreases and connection to the fluid source increases as cos is reduced (Fig. 13b–13d). Fingers also become wider and slower, and the saturated tip is longer. Fingers meandered more at lower cos, where the increased importance of capillary fingering leads to a rougher and more complex structure that entraps air within the finger (Fig. 13c and 13d). It is important to note that gravity-driven fingering is an important process even at small fracture inclinations. Comparison of Fig. 13d and Fig. 14
clearly shows that infiltration at cos = 0.125 (82.9° from vertical) is very different than horizontal imbibition (cos = 0).
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Fig. 14. (a, b) Slow horizontal (cos = 0.0) invasion of the air-filled 15- by 30-cm cell. Invasion is completely controlled by capillary forces (quasistatic displacement), (c) eventually filling the cell to a satiated condition where air is fully entrapped at a variety of length scales.
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The effects of Q on finger behavior at cos = 1.0 are illustrated in Fig. 15
. Desaturation behind the fingertip was small at large Q (Fig. 15a) and increased dramatically as Q was decreased (Fig. 15b–15d). Fingertips became narrower, shorter, and slower with decreasing Q. They also tended to meander more in response to local heterogeneity, and exhibit a more complicated outline than at large Q. Flow within the desaturated zone becomes more complex at lower values of Q, with continuous flow along narrow channels (Fig. 15c) replaced by a dripping mechanism that intermittently connects multiple pools (Fig. 15d). The increased dynamics associated with intermittent flow structures can lead to switching between pathways and the formation of dendritic secondary fingers (see Fig. 15d).
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Fig. 15. Effect of Q on finger behavior under dry initial conditions at cos = 1.0. Image times give an indication of relative velocity. (a) Q = 13.3 cm3 min–1, t = 32 s; (b) Q = 1.25 cm3 min–1, t = 36 s; (c) Q = 0.254 cm3 min–1, t = 108 s; and (d) Q = 0.0243 cm3 min–1, t = 690 s.
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In our system, local aperture variability (i.e., r1 curvature) subjects fingertips to finite amplitude perturbations at wavelengths on the order of the aperture spatial correlation length, . Figure 13d shows the effects of such a perturbation on a fingertip at small cos. After bifurcating, both tips advanced in parallel, then rejoined to entrap a blob of air. At steeper inclinations, we observed bifurcations in which one finger clearly dominated (Fig. 16)
. A finite amplitude perturbation splits the fingertip (Fig. 16a) to form two distinct tips that continue downward and compete for the available fluid (Fig. 16b). Capillary and gravity forces favor growth of the most advanced fingertip (Fig. 16c). Eventually, the dominant finger starves the other and continues to advance. The failed bifurcation is left behind as a saturated lobe on the side of the moisture structure (Fig. 16d). Dendritic secondary fingers commonly developed from such appendages to the moisture structure (Fig. 17)
. Shortly after the initial bifurcation (Fig. 17a), the more advanced fingertip dominates, while the smaller fingertip is left behind as an appendage to the moisture structure (Fig. 17b). Later, flow through the desaturated zone joins with this appendage (Fig. 17c) to form a second, dendritic finger (Fig. 17d).
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Fig. 16. Macroscopic instability of a gravity-driven finger initiated by steady supply (Q = 2.51 cm3 min–1) to point source at the top of the 15- by 30-cm air-filled cell at cos = 1.0.
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Fig. 17. Dendritic secondary finger formed in the 15- by 30-cm cell (cos = 0.50) from steady supply (Q = 0.0254 cm3 min–1) under dry initial conditions.
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Where flow within the desaturated zone became disconnected, small fluid blobs were pinned within the aperture field by capillary forces. Individual blobs would grow in size as fluid was added. The stored fluid is eventually released as a discrete pulse. Figure 18a
shows a poorly connected desaturated zone that turned steady supply from above into intermittent drips. To examine temporal behavior at a single fluid blob (Fig. 18b), we measured changes between rapidly collected sequential images. In Fig. 18c, white zones repeatedly drained and filled, while the rest (black) remained unchanged. Beyond simple pulsation, this system exhibits a potential for highly complicated behavior, particularly if one considers hysteresis, viscous energy loss during discharge, and dynamic contact angle. As such, it bears a strong resemblance to the "dripping faucet" problem considered by Shaw (1984), where nonlinear coupling between state variables leads to a rich variety of behavior from steady to chaotic. Beyond the scale of a single fluid blob, it may be possible for fluid cascades to rapidly span the system. Finally, the example shown in Fig. 18 exhibited pulsation along a single pathway; however, we also noted pulsed flow to switch pathways within the desaturated zone as connections snapped and reformed (Fig. 19)
. Such small-scale reconfigurations can lead to the formation of macroscopic dendrites as seen in Fig. 15d.
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Fig. 18. Unsteady flow resulting from steady supply to an inclined fracture. (a) Desaturated zone behind a fingertip in the 15- by 30-cm cell at steady supply (Q = 0.025 cm3 min–1) under dry initial conditions (cos = 0.50). (b) Enlargement of the boxed area focuses on a single fluid blob. (c) Dynamic behavior within the boxed area is explored by recording changes in saturation with time. White represents repeated drain and fill cycles (e.g., drip points), while black indicates no change.
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Fig. 19. Four different fluid configurations observed within the desaturated zone behind a finger formed from steady supply (Q = 0.0198 cm3 min–1) to a point source. As illustrated in these enlargements, pulsed flow imposed by the desaturated zone led to changes in connection.
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Influence of Initial Moisture
To explore the influence of initial moisture on fingers formed by steady supply to a point source, we conducted one experiment in a structured moisture field, then moved on to systematic experimentation in uniform initial moisture fields (27 experiments). Experiments were conducted at cos = 1.0, 0.75, 0. 5, and 0.25; Q was varied over a range from 0.35 to 2.5 cm3 min–1. Experiments were performed in the 30 x 60 cm cell (41-cm3 aperture volume) to allow sufficient room for fingers to interact with the in situ moisture field.
We began with an experiment designed to determine if steady supply from a point source would follow an existing moisture structure. To create the structured moisture field outlined in Fig. 20a
, we inverted the cell slightly and imbibed undyed water. The apparatus was then quickly rotated to cos = 0.25, placing the imbibed water at the top of an otherwise dry fracture. Immediate redistribution of the imbibed water led to the formation of one large finger, which wetted the outlined zone as it passed through. Subsequent steady flow from a point source formed a near vertical finger in the uniformly distributed portion of the initial moisture field, then diverted laterally to stay within that field (Fig. 20b and 20c). The finger follows a narrow and complicated pathway that abruptly widens near the bottom as it enters the saturated capillary fringe left behind by the previous event (Fig. 20c). Accumulation of fluid above a lobe in the initial moisture field (Fig. 20b and 20c) led to a new dendritic finger that invaded the dry region beneath (Fig. 20d). In time, all flow switched into the new finger (Fig. 21a)
, providing an excellent opportunity to observe the difference between dry and prewetted initial conditions. As seen in Fig. 21b, the area swept by the finger on the left (prewetted zone) exhibits a structure very similar to what we see in the desaturated zone behind a fingertip in an initially dry fracture. It is much narrower and more complicated than the relatively wide and smooth region swept by the finger to the right (initially dry zone).
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ABSTRACT
INTRODUCTION
