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Keywords:

  • gravity-driven fingers;
  • infiltration;
  • nonmonoticity;
  • preferential flow;
  • saturation overshoot;
  • wetting front instability

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Materials and Methods
  5. 3. Results
  6. 4. Discussion
  7. Acknowledgments
  8. References

[1] Gravity-driven fingers in uniform porous medium are known to have a distinctive nonmonotonic saturation profile, with saturated (or nearly so) tips and much less saturation in the tails. In this work, constant-flux infiltrations into confined porous media (laterally smaller than the finger diameter, thus essentially one-dimensional) are found to produce saturation overshoot identical to that found in gravity-driven fingers. Light transmission is used to measure the saturation profiles as a function of infiltrating flux, porous media grain size, grain sphericity, and initial water saturation. Saturation overshoot is found to cease below a certain minimum infiltrating flux. This minimum flux depends greatly on the grain sphericity and initial water content of the media and slightly on the mean grain size. The observed saturation overshoot is inconsistent with a continuum description of porous media but qualitatively matches well observations and predictions from discrete pore-filling mechanisms. This suggests that pore-scale physics controls saturation overshoot and in turn gravity-driven fingering.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Materials and Methods
  5. 3. Results
  6. 4. Discussion
  7. Acknowledgments
  8. References

[2] Water infiltration into uniform water-wet porous media has been shown to produce preferential flow paths under certain boundary and initial conditions and media properties. This type of preferential flow has been called gravity-driven fingering, as it is a result of an instability driven by the density of the water being greater than that of the air it displaces. Despite many theoretical and experimental investigations, it remains unclear when infiltration is likely to be unstable and produce preferential flow, or stable and laterally uniform. Historically, almost all theoretical calculations are based on perturbing a flat wetting front and finding the most unstable wavelength of the perturbation [Chuoke et al., 1959; Parlange and Hill, 1976; Glass et al., 1991; Wang et al., 1998; Du et al., 2001], with key differences in the way capillary forces are handled at the interface.

[3] Experimental observations [Glass et al., 1989; Selker et al., 1992; Liu et al., 1994] have shown that gravity-driven fingers exhibit saturation overshoot, a characteristic water saturation pattern as depicted in Figure 1. This pattern consists of a region directly behind the wetting front with a high and uniform water saturation called the finger tip, followed by another region with a low and uniform water saturation called the finger tail. Selker et al. [1992] have shown that saturation overshoot is associated with pressure overshoot (decreasing water pressure from the tip to the tail of the finger), predicted to be a necessary condition for gravity-driven flow instability [Raats, 1973]. Additionally, it has been found that as gravity-driven fingering is eliminated with increasing initial water content [Diment and Watson, 1985], saturation and pressure overshoot are also eliminated [Bauters et al., 2000].

image

Figure 1. Cartoon of a preferential flow path and the associated saturation within the flow path. Saturation overshoot occurs when the tip saturation is greater than the tail saturation.

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[4] From these observations, it has been proposed that saturation overshoot is a necessary and sufficient prerequisite for gravity-driven fingering, i.e., gravity-driven fingering will occur if and only if saturation (or pressure) overshoot occurs [Geiger and Durnford, 2000; Eliassi and Glass, 2001]. Thus a correct understanding of the physics controlling saturation overshoot is necessary for understanding the larger question of gravity-driven fingers.

[5] Recently, Eliassi and Glass [2001] have argued that Richards' equation with standard nonmonotonic pressure saturation and relative permeability curves cannot produce solutions with saturation overshoot, although there is some discussion on defining monotonicity for flows in greater than one dimension [Braddock and Norbury, 2003; Eliassi and Glass, 2003b]. In subsequent papers, possible continuum additions to the Richards' equation that may lead to saturation overshoot are given [Eliassi and Glass, 2002, 2003a].

[6] Previously, pressure overshoot in one-dimensional (1-D) infiltrations has been observed by Stonestrom and Akstin [1994] and studied by Geiger and Durnford [2000] using tensiometry. Geiger and Durnford [2000] used soils that exhibit preferential flow but packed them into small columns such that the lateral dimension was smaller than the finger size and the displacements were one-dimensional. They found pressure overshoot for initially dry coarse sands at all flow rates tested, and for initially dry fine sands at moderate flow rates, but with no pressure overshoot for initially wet sands or for fine sands and low flow rates.

[7] In this study, 1-D saturation profiles are measured in sands susceptible to preferential flow by confining the sands in tubes smaller than the preferential flow size a la Geiger and Durnford [2000]. It is found that the saturation overshoot takes place and is comparable to that seen in slab studies [Selker et al., 1992; Bauters et al., 2000] for the same media. Saturation overshoot is measured in detail as a function of infiltrating flux, mean grain size, grain sphericity, and initial water content. It is discussed how the measured saturation overshoot is inconsistent with a continuum description of porous media but qualitatively matches well observations and predictions from discrete pore-filling mechanisms.

2. Materials and Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Materials and Methods
  5. 3. Results
  6. 4. Discussion
  7. Acknowledgments
  8. References

[8] Four different sands were used, and their characteristics are listed in Table 1. The three similarly shaped sands of varying grain sizes (called hereinafter by their trade name Accusand, mined in Ottawa, Minnesota; Unimin Corporation) all had smooth, almost spherical grains. The grey sand (called such because of its color, mined in Emmett, Idaho; Unimin Corporation) had a grain size between the 12/20 and 20/30 Accusands, but the grains were quite angular. Figure 2 shows the physical differences between the grains with the spherical 12/20 grains on the left and the angular grey grains on the right. The pressure-saturation drainage curves and hydraulic conductivity curves of the Accusands have previously been measured, and the sands were found to be Miller-similar [Miller and Miller, 1956; Schroth et al., 1996]. The conductivity of the grey sand was measured in a 30-cm-long, 2.54-cm-diameter column. Figure 3 shows the pressure-saturation drainage curves of the 12/20 and grey sands measured by draining the column until it came to capillary/gravity equilibrium [DiCarlo, 2003], followed by sectioning, weighing, and drying the sections. Figure 4 shows the primary imbibition curves of the 12/20 and grey sands measured by letting water imbibe vertically for 1 week in an initially dry column, followed by sectioning. Clearly the angular grey and spherical 12/20 sands have very similar macroscopic characteristic curves although the porosity (ϕ) was slightly greater in the grey sand (0.40 versus 0.35), due most likely to the angularity of the grains. The curves for the other two spherical sands are not shown for the sake of clarity, but the van Genuchten fitting parameters [van Genuchten, 1980] of all the sands are listed in Table 1.

image

Figure 2. Photograph of (left) the spherical grains of the 12/20 sand and (right) the angular grains of the grey sand. The length scale for both images is given below.

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image

Figure 3. Primary drainage pressure-saturation curves for the 12/20 and the grey sands.

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image

Figure 4. Primary imbibition pressure-saturation curves for the 12/20 and the grey sands.

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Table 1. Physical Parameters for the Four Sands Used in the Studya
Sandd50, mmShapeK, cm/minϕθrDrainageImbibition
α, cm−1nα, cm−1n
  • a

    The imbibition curves for all the sands, the drainage curves for the 12/20 sand, and all of the parameters for the grey sands are from measurements explained in the text. The rest of the data for the Accusand sands (12/20, 20/30, 30/40) are from Schroth et al. [1996].

12/201.105spherical300.350.0120.1299.700.3034.98
20/300.713spherical150.350.0160.099510.570.1776.23
30/400.532spherical9.00.350.0180.06713.100.17310.0
grey0.96angular320.400.010.128.870.3304.14

[9] For the infiltration experiments, the working sand was packed into slim cylindrical tubes (40 cm long) whose inner diameter was chosen such that the saturation within the column was uniform transverse to the flow direction at all times (i.e., there was no observable preferential flow in the columns, as the diameter was less than the finger diameter). A tube inner diameter of 1.27 cm was sufficient for the 20/30 and 30/40 Accusands, but an inner diameter of 0.95 cm was needed for the 12/20 Accusand and the grey sand. A one-holed rubber stopper and a screen were placed in the bottom of each tube to keep the sand in the tube while permitting airflow out the bottom. For initially dry media, an extension tube was attached to the top of the column, through which the sand was continuously poured from a funnel through two randomizer screens. The top 10 cm of the sand pack (which was within the extension tube) was then removed to ensure tight packing, with the porosities listed in Table 1. For initially wet media, the wet sand was dropped into the tube in 3-cm increments; after each increment the sand was tamped with a steel rod [Diment and Watson, 1985; Bauters et al., 2000]. This resulted in a uniform packing and a repeatable porosity of 0.37, slightly larger than the dry packs.

[10] Once packed, the tubes were placed in front of a light transmission chamber (explained below) for the infiltration experiment. Water was injected at the top of the tube through a 22-gauge needle. A syringe pump was used to provide water injection at uniform flow rates from 0.001 to 2 mL/min; at higher rates a peristaltic pump was used. Up to 8 mL of water was injected in each tube, and the progress of the wetting front and saturations behind the front were measured using light transmission.

[11] The light transmission chamber consisted of a light box (50 cm tall by 40 cm wide) and a black plywood mask of identical dimensions. The mask had three slits (42 cm tall by 0.6 cm wide) onto which the slim tubes were secured vertically. The center tube was a water-saturated tube of identical sand and provided an intensity reference for each image. On each side of the center tubes the infiltration tubes were run in duplicate. A digital camera (Canon S30) with time-lapse capability was used to record the light transmitted through each tube versus time.

[12] Saturations within each tube were calculated using two different methods. The first method involved measuring the front position (zf) versus time. If it is assumed that the saturation profile is self-similar and translates downward in time [Selker et al., 1992], the saturation equation image(z, t) can be written as

  • equation image

where z = 0 at the sand surface, and is positive downward, and equation image is a function of only distance (from the position of the front). The water saturation a distance z behind the front can be found using conservation of mass,

  • equation image

where q is the applied flux. This method works well for calculating the tip and tail saturations if the tip length is greater than roughly 5 cm (roughly the length needed to get a clear and measurable velocity of the front) but is unable to measure a snapshot of the whole profile.

[13] The second method involves using the intensity of the transmitted light versus z and converting it into θ versus z. From each image the intensity of the pixels along the tubes' vertical axis was obtained from the images and averaged over six pixels (0.37 cm) horizontally and 15 pixels (0.94 cm) vertically. From the measured intensity I(z, t), the normalized intensity In(z, t) was found using

  • equation image

as I(z, 0) is the intensity through the dry column (the first scan) and Is(z, t) is the intensity of the saturated center column. The normalized intensity was calibrated to θ by measuring θ in the flat tail region from the first method and an average In in the tail from the light transmission [Tidwell and Glass, 1994; Niemet and Selker, 2001]. This calibration was found to be linear for 20/30 and 30/40 sands in the large tubes but was noticeably nonlinear for the 12/20 and grey sand in the small tubes. Using these calibrations and an overall mass balance, θ(z, t) was obtained from In(z, t). Unsurprisingly, the θ(z, t) profiles had more noise in the larger, less uniform colored sands (12/20 and grey) than in the smaller, more uniform colored sands (20/30 and 30/40).

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Materials and Methods
  5. 3. Results
  6. 4. Discussion
  7. Acknowledgments
  8. References

[14] Figure 5 shows snapshots of the measured saturation profile using light transmission in dry 20/30 sand at six different fluxes. At the highest (11.8 cm/min) and lowest (7.9 × 10−4 cm/min) fluxes the profiles are monotonic with distance and no saturation overshoot is observed, while all of the intermediate fluxes exhibit saturation overshoot, with saturations at the wetting front (the tip) greatly exceeding those behind the tip (the tail). The slight oscillations in saturation at the tail of the higher fluxes are likely experimental artifacts related to light transmission variations near the end of the tubes. Going from high to low fluxes certain trends can be seen. Both the tip and the tail saturations decrease continuously with flux, but at different rates, as the tail saturation decreases quickly with flux, while the tip saturation only decreases at relatively low fluxes. The spatial structure of the overshoot also changes with decreasing flux. Defining the length of the tip to be from where the saturation is halfway between its initial value and the tip value, to where the saturation is halfway between the tip and the tail value, it is seen that at high fluxes the tips are longer and have more uniform saturation than at low fluxes.

image

Figure 5. Snapshots of the saturation profile versus depth for six different applied fluxes in initially dry 20/30 sand (Accusand) measured using light transmission. At the highest (11.8 cm/min) and lowest (7.9 × 10−4 cm/min) fluxes the profiles are monotonic with distance and no saturation overshoot is observed, while all of the intermediate fluxes exhibit saturation overshoot.

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[15] Figure 6 plots the measured tip and tail saturation as a function of flux for the same dry 20/30 sand. The tip and tail saturations are taken from the center 10 cm (vertical) of the column where the light transmission measurements are most accurate (unless, of course, the finger tip is longer than 20 cm, in which case the tail saturations are taken farther up as the tail never reaches the column center). Scatter in the data gives an estimate of the accuracy and repeatability of the saturations. The region where saturation overshoot is present is clearly demarcated where the tip saturation exceeds the tail saturation. The trends noticed in Figure 5 with the tip and tail saturations versus flux are evident. The continuous dependence of tail saturation on flux mirrors that of an unsaturated conductivity curve with a power-law-like dependence. By contrast, the tip saturation decreases only logarithmically with flux until roughly 0.05 cm/min, after which it drops off quickly, meeting the tail saturation at roughly 0.003 cm/min. This type of graph is the best way to observe the saturation overshoot dependence on flux for each medium used. In the following paragraphs we explore how the saturation overshoot changes as a function of grain size and sphericity.

image

Figure 6. The measured tip and tail saturation as a function of flux for the same dry 20/30 sand depicted in Figure 2. Saturation overshoot ceases at around a flux of 0.003 cm/min.

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[16] Figure 7 plots the tip and tail saturations versus flux for initially dry 12/20 sand (similar to Figure 6 except for the larger grained sand). In this case, the saturation overshoot region is larger (from roughly 0.001 to 15 cm/min) than seen for 20/30 sand. In addition, the tips are more unsaturated than in 20/30 sand.

image

Figure 7. The measured tip and tail saturation as a function of flux for the dry 12/20 sand. Saturation overshoot ceases at around a flux of 0.001 cm/min.

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[17] Figure 8 plots the tip and tail saturations versus flux for initially dry 30/40 sand. Here the saturation overshoot flux range (from 0.01 to 8 cm/min) is smaller than seen for 20/30 sand, and the tip saturation is higher than 20/30 sand.

image

Figure 8. The measured tip and tail saturation as a function of flux for the dry 30/40 sand. Saturation overshoot ceases at around a flux of 0.01 cm/min.

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[18] Figure 9 plots the tip and tail saturations versus flux for the angular grey sand. The saturation overshoot flux range (from 0.01 to 25 cm/min) is much smaller than the comparably sized 12/20 and 20/30 sands. Also, the difference between tip and tail saturation is much smaller than any of the round-grained Accusands, with the greatest saturation difference observed being a volumetric water saturation of 0.07. The tip saturation decreased quickly from fully saturated and seemed to follow the tail saturation much more closely.

image

Figure 9. The measured tip and tail saturation as a function of flux for the dry grey sand. Saturation overshoot ceases at around a flux of 0.01 cm/min, and the degree of saturation overshoot is much smaller than in comparably sized 20/30 and 12/20 Accusand.

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[19] Now the dependence on initial water content of the sand is explored. Figure 10 plots the tip and tail saturations versus flux for 20/30 sand with an initial volumetric water saturation of 0.01. The saturation overshoot region (from 0.1 to 20 cm/min) is much smaller than observed for the dry 20/30 sand (Figure 6), and the saturation difference between tip and tail never exceeds 0.07. Figure 11 plots the tip and tail saturations versus flux for 20/30 sand with an initial volumetric water saturation of 0.02. Here we observe no saturation overshoot for any fluxes. Note that for both Figures 10 and 11 the initial water content lowered the tip saturation and did not affect the tail saturation, as the tail saturation is determined by the unsaturated conductivity. Table 2 shows that the saturation difference between tip and tail as a function of water content for this one-dimensional study and the two-dimensional slab study of Bauters et al. [2000] are similar.

image

Figure 10. The measured tip and tail saturation as a function of flux for the 20/30 sand with initial volumetric water saturation of 0.01. Saturation overshoot ceases at around a flux of 0.1 cm/min, much higher than in the dry sand, and the magnitude of the saturation overshoot is much less than in the dry sand. Note that the tail saturations are the same as a function of flux for the dry 20/30 sand, and the decrease in overshoot occurs because of the drop of tip saturations.

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image

Figure 11. The measured tip and tail saturation as a function of flux for the 20/30 sand with initial volumetric water saturation of 0.02. Saturation overshoot is not observed at any flux. Again, note that the tail saturations are the same as a function of flux for the dry 20/30 sand, and the lack of overshoot occurs because the tip saturations now equal the tail saturations.

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Table 2. Saturation Overshoot as a Function of the Initial Water Content in 20/30 Sand in the Two-Dimensional Slab Study of Bauters et al. [2000] and the Current One-Dimensional Column Studya
Initial Water ContentSaturation Overshoot (θtip − θtail)
Bauters et al. [2000]This Study
0.000.150.16
0.010.0650.06
0.020.0150.00

[20] In addition to determining the saturations of the tip and tail, the length of the tip can be calculated from data like that shown in Figure 5. Figure 12 plots the measured tip length versus flux for the initially dry 20/30 sand. If the difference between tail and tip saturation is below the noise level, no tip length is recorded, and it is assumed that the flow shows no overshoot. The tip length is longest at the highest fluxes that exhibit saturation overshoot, and the observed lengths decrease monotonically with flux. Similar dependence with flux is seen for the other dry sands, but with smaller tip lengths for the 12/20 and grey sand and longer lengths for the 30/40 sand, as shown in Table 3. For 20/30 sand the tip length shortened for sand with 0.01 initial water saturation and no tip was observed for sand with 0.02 initial water saturation (or, alternatively, the tip length became zero). Note that tip lengths of greater than 25 cm are unable to be observed in our columns which are 40 cm long.

image

Figure 12. The measured tip length as a function of flux for dry 20/30 sand. The tip lengths decrease monotonically with decreasing flux.

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Table 3. Tip Lengths at a Flux of 1 cm/min for All the Sands
SandTip Length, cm
12/206.5 ± 0.3
20/3011.2 ± 0.4
30/4016.3 ± 0.5
Grey5.0 ± 0.4
20/30 − 0.01 initial7.0 ± 1.0
20/30 − 0.02 initialno overshoot

[21] The results can be summarized as follows:

[22] 1. Saturation overshoot decreases quickly with increasing initial water content.

[23] 2. Saturation overshoot is much less for angular sand grains than for spherical sand grains.

[24] 3. Saturation overshoot varies for Miller-similar sands, with higher minimum fluxes and higher tip saturations for smaller-sized sands.

[25] 4. The tip is only fully saturated at the highest fluxes, with tip saturation decreasing continuously with decreasing flux.

4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Materials and Methods
  5. 3. Results
  6. 4. Discussion
  7. Acknowledgments
  8. References

[26] Clearly, saturation overshoot is present in a 1-D system and is not just a function of preferential flow in 2-D or 3-D systems. In addition to the overshoot saturations matching up well for initially saturated 20/30 sand in 2-D experiments as shown in Table 2, Hendrickx and Yao [1996] found that gravity-driven fingers in 3-D show a dramatic increase in diameter as the flux is decreased from 0.01 to 0.002 cm/min, and fingering is completely suppressed below fluxes of 0.001 cm/min for their 14/20 sand. This is precisely the flux range where the saturation overshoot ceases in 12/20 sand in the 1-D system (Figure 7). Thus it can be argued that these 1-D results are directly applicable to the dynamics of gravity-driven preferential flow in 3-D systems.

[27] In addition, that saturation overshoot occurs in a 1-D system is clear evidence that the 1-D Richards' equation cannot describe the behavior at the wetting front [Eliassi and Glass, 2001] and consequently cannot describe the behavior that causes gravity-driven preferential flow [Jury et al., 2003; Wang et al., 2003]. Other continuum descriptions appear to be inadequate from comparing the grey sand results to the 12/20 sand results. The sands have very similar macroscopic continuum properties (drainage and imbibition characteristic curves) but very different overshoot properties. Continuum descriptions of unsaturated flow are based on the concept of a representative elementary volume (REV) over which properties are slowly varying and averaging over many pores can take place [Bear, 1972]. For saturation overshoot, the fronts are abrupt at the pore scale [Lu et al., 1994], so it is not surprising that the Richards' equation, or any other continuum description, would likely be physically inappropriate.

[28] On a physical level, why does saturation overshoot occur (i.e., why is the tip saturation greater than the tail)? The tail saturation is controlled by the applied flux. as asymptotically the hydraulic conductivity must equal the applied flux (the only gradient left is the gravitational gradient). In contrast, the tip saturation is basically the minimum saturation available for a certain wetting front velocity and media. If this saturation is less than the tail saturation, the saturation increase will be continuous, but if it is greater than tail saturation, saturation overshoot will take place.

[29] It is suggested that the minimum saturation behind a sharp wetting front is controlled by the pore-scale filling processes. Using micromodels of porous media, Lenormand and Zarcone [1984] observed that at high frontal velocities the wetting front is very sharp, with almost all pores filled directly behind the front. At very low frontal velocities the front is diffuse, with many pores and throats remaining unfilled behind the front. Lenormand and Zarcone [1984] explained this phenomenon in terms of a competition between piston-like collective filling at the main front and filling ahead of the front due to conduction through water layers in the corners of the pore space [see also Blunt and Scher, 1995]. At high frontal velocities, piston-like collective filling preferentially occurs, as conduction through the water layers cannot match the speed of the front, leading to a sharp jump in saturation at the front. At low frontal velocities, conduction through the layers can match the speed of the front, allowing pores to fill ahead of the main front, and on a macroscale, a relatively smooth increase of the saturation at the front. This picture has been observed in pore-scale observations on infiltration into glass bead porous media [Lu et al., 1994]. In dry media the front was abrupt on the pore scale, while if the media had an initial saturation, the front was diffuse as water moved through wetting layers. Finally, pore-scale network models with layers have been able to reproduce these observations [Hughes and Blunt, 2000, 2001].

[30] This simple conceptual picture qualitatively fits all of the results. The greater the initial water content, the greater the water layer conductivity, which in turn allows pores ahead of the main front to fill [Lu et al., 1994]. This allows the saturation increase at the front to be more gradual (even at high frontal velocities) and saturation overshoot to lessen or disappear (result 1). In initially dry media, there are no initial micron-sized layers, but the layers will form slowly ahead of the front as long as the medium is water-wetting. (If the media were water-repellent, it seems likely that saturation overshoot would continue to arbitrarily low fluxes. This has not been tested as of yet.) That saturation overshoot is much less for angular grains (result 2) follows because it is easier to form water layers and the layers are more conductive in an angular rather than a smooth pore space [Lenormand and Zarcone, 1984]. The higher minimum fluxes for saturation overshoot with smaller grain sizes (result 3) follows because the distance between pores over which the layers must transport water scales with grain size, while the amount of water needed to fill each pore (or throat) scales as the grain sized cubed. Thus, for smaller grain sizes, less water needs to be moved shorter distances ahead of the front to maintain a smooth front, and saturation overshoot is suppressed at higher fluxes. The smooth decrease in tip saturation with decreasing flux (result 4) follows because the transition between the wetting front being dominated by collective pore-filling to snap-off and associated pore-filling ahead of the front will likely be smooth due to the variations in pore and throat size.

[31] Other approaches have been undertaken to explain saturation overshoot. Eliassi and Glass [2002, 2003a] propose alternate continuum terms to be added to the Richards' equation that would produce saturation overshoot. Conceivably, the coefficients of these terms can be adjusted to match some (or maybe all) of the experimental results presented here. From the arguments above, it is suggested that adding another continuum term to deal with a noncontinuum phenomenon (sharp fronts at the pore scale [Lu et al., 1994]) will misrepresent the actual physics. Since all large-scale models are continuum models, though, the suggested additional terms may be a useful numerical kluge for large-scale models.

[32] Along these lines, it is noted that in discrete network models of unsaturated flow where cells are either filled or unfilled with water, saturation overshoot occurs naturally [Glass and Yarrington, 1996; Hughes and Blunt, 2000, 2001; Glass and Yarrington, 2003]. Again, this is due to the collective nature of the filling (and sharp fronts) for these models. Any correct discrete model must also have a physical mechanism which smoothes the front and eliminates overshoot at low fluxes, as seen in the experiments. Studies are currently under way using pore-scale models with both corner and piston-like flow to see if the match between the measured and modeled can be made quantitative.

[33] Finally, note that the behavior behind the sharp wetting front is very well described by the Richards' equation [Selker et al., 1992]. Thus it appears, at least for the measurements presented here, that the traditional unsaturated flow equation only fails at the sharp wetting front, a regime where it was never meant to work in the first place [Bear, 1972].

[34] In summary, saturation overshoot can occur in a one-dimensional system. This result highlights the inadequacies of the Richards' equation for describing overshoot phenomena. In contrast, saturation overshoot can be qualitatively understood in terms of the pore-filling processes. In addition to the pore-scale modeling studies that are mentioned above, studies are currently under way to correlate the saturation overshoot results to specific observations of gravity-driven fingering (e.g., finger spacing, finger width, whether fingering occurs at all). Along with the results of this study, these future studies aim to determine whether gravity-driven fingering can be understood in terms of a continuum model or understood only as a result of the discrete nature of the porous media.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Materials and Methods
  5. 3. Results
  6. 4. Discussion
  7. Acknowledgments
  8. References

[35] The author thanks Robert Smith for experimental assistance, Sean Bennett, Martin Blunt, Richard Hughes, Tammo Steenhuis, and Yves Parlange for helpful discussions, and the three anonymous reviewers for their insightful comments.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Materials and Methods
  5. 3. Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
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  • Eliassi, M., and R. J. Glass (2001), On the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state, Water Resour. Res., 37, 20192035.
  • Eliassi, M., and R. J. Glass (2002), On the porous-continuum modeling of gravity-driven fingers in unsaturated materials: Extension of standard theory with a hold-back-pile-up effect, Water Resour. Res., 38(11), 1234, doi:10.1029/2001WR001131.
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