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a Department of Environmental Sciences, University of California, Riverside, CA 92521
b Department of Environmental Sciences, Calif. State Univ. Fresno, CA 93740
* Corresponding author (wajury{at}mail.ucr.edu)
Received 7 August 2002.
ABSTRACT
A conceptual model is presented that represents the development of unstable flow in uniform soils during redistribution. The flow instability results in the propagation of fingers that drain water from the wetted soil matrix until equilibrium is reached. The model uses soil retention and hydraulic functions, plus relationships describing finger size and spatial frequency. The model assumes that all soils are unstable during redistribution, but shows that only coarse-textured soils will form fingers capable of moving appreciable distances. Once it forms, a finger moves downward at a rate governed by the rate of loss of water from the soil matrix, which can be predicted from the hydraulic conductivity function. Fingers are assumed to stay narrow as a result of hysteresis, which prevents lateral diffusion. The draining front in the soil matrix between the fingers is assumed to cease downward movement because water pressure drops below the threshold water-entry matric potential. The threshold potential is not present in the conventional Richards equation of soil water flow, which explains why unstable flow is not predicted by widely used simulation model codes.
UNSTABLE FLOW DURING INFILTRATION in unsaturated porous media has been studied for many years and is known to be associated with a number of existing conditions, including vertical flow from a fine-textured layer into a coarse one (Hill and Parlange, 1972; Baker and Hillel, 1990), vertical flow into a compressed air phase (Peck, 1965), infiltration into water-repellent soil (Hendrickx et al., 1993), and two-phase flow involving two fluids of contrasting density and viscosity (Chuoke et al., 1959). More recently, it has been demonstrated to occur during infiltration into homogeneous soil at flux rates substantially less than the saturated hydraulic conductivity (Selker et al., 1992b; Geiger and Durnford, 2000), and during redistribution following infiltration in homogeneous soil (Diment and Watson, 1985; Wang et al., 2003a). Unstable flow is distinct from other forms of so-called preferential flow in that it is a fluid phenomenon whose extreme flow location is not a consequence of permeability variations in the porous medium.
Unstable flow in porous media was first analyzed by Saffman and Taylor (1958), who showed that perturbations in a wetting front between two immiscible liquids will develop into instabilities whenever the velocity of the invading fluid exceeds a threshold value. Raats (1973) and Philip (1975) demonstrated that the condition for instability at a wetting front in a porous medium was that the total hydraulic gradient behind the front is less than unity (i.e., the water pressure decreases toward the surface). Any effort to describe the fingering process must begin by characterizing finger geometry, whose two most important attributes are finger diameter and the fraction of the total cross-sectional area occupied by fingers (hereafter called the finger flow fraction). Various attempts have been made to predict the finger diameter from soil properties and the water flow characteristics. They either involve an approximate equation that is a function of the sorptivity (Parlange and Hill, 1976) or include a macroscopic surface tension associated with the large-scale curvature induced by the radius of the finger (Chuoke et al., 1959; Glass et al., 1989a). Hydrostatic balance equations written at the wetting-front interface have periodic solutions that have a maximum unstable wavelength, and this is used to calculate the finger diameter. Both approaches involve associating one-half of the wavelength with the finger diameter. Parlange and Hill (1976) and Glass et al. (1989a) related the finger diameter to the soil's sorptivity S and the system flux ratio Rs by
 | [1] |
 | [2] |
* is the effective macroscopic surface tension at the interface between the finger and the soil ahead of the advancing front.
Wang et al. (1998) replaced the effective surface tension with an expression involving the water-entry pressure head hwe of the wetting curve by using the capillarity equation 
P = 2/R and defining an effective macroscopic curvature R* at the interface. This modification produces the expression
 | [3] |
The finger diameter is determined by two factors, the flow rate and the soil properties Ks and hwe. The fraction ß of the total area at the front that the fingers occupy is more difficult to estimate because it depends on the rate of supply of water from the matrix to the finger, as well as on the flux through the finger. Glass et al. (1989b) analyzed Hele–Shaw cell data and came up with the empirical relation
 | [4] |
In contrast, Hillel and Baker (1988) calculated ß at the interface between fine- and coarse-textured soil from a physical argument, assuming that all of the flux in the matrix is diverted to fingers after unstable flow begins, and that the flux through the finger is equal to K(hwe). With these assumptions, it is easy to show that
 | [5] |
Note that Eq. [4] and [5] predict very different behavior of ß as a function of Rs. In particular, the Hillel expression Eq. [5] is likely to underestimate the finger area fraction during redistribution when the matric potential opposes the flow.
Observations of the fingering process have repeatedly shown that fingers remain narrow as they propagate and persist for a long time after flow stops. Glass et al. (1989c) offered a physical explanation involving hysteresis in the matric-potential water content function (the water characteristic) for why the fingers don't disperse by lateral diffusion. The interior of the finger first wets to near saturation and then drains as the tip of the finger moves downward. At the same time, a narrow zone of wetting is created at the outside fringe of the finger adjacent to the inner core. The interior of the finger thus follows the drainage loop of the water characteristic curve, whereas the fringe is on the wetting curve. Since at the same water content the drying curve is at a lower matric potential than the wetting curve, equilibrium is reached at different water contents inside the core and at the fringe, and the fingers remain narrow.
Instabilities form at a wetting or draining front when a portion of the front moves ahead of the rest and is fed by lateral flow from the surrounding soil matrix. An important factor in maintaining fingered flow, for reasons that will be discussed below, is the water-entry pressure head hwe, which is defined as the minimum matric potential head at the wetting front interface required to allow fluid to enter the dry portion of the medium below the wetting front. It has been measured experimentally in layered media by Baker and Hillel (1990) and in homogeneous media by Geiger and Durnford (2000) and Wang et al. (2000). Baker and Hillel (1990) also showed that it corresponds approximately to the matric potential at the inflection point of the wetting loop of the moisture release curve.
Although most of the effort to characterize fingered flow has been experimental, recent attempts have been made to model the transport process during fingered flow instability. Nieber (1996)(Nieber et al., 2000) used a hysteretic form of the Richards equation together with a heavily biased numerical weighting scheme used to calculate hydraulic conductivity at the wetting front that essentially mimicked the water-entry matric potential by preventing water from moving into the dry region unless the wet zone was highly conducting. With this condition imposed, the Richards equation was able to predict the propagation of fingers. A different approach to modeling unstable flow was taken by Glass and Yarrington (1996), who used an invasion percolation algorithm that neglected viscous forces to create a complex pattern of gravity-induced fingers. The process is driven by pore filling in a medium composed of connected networks of pores.
The purpose of this paper is to use the known or postulated behavior of fingers during unstable flow to build a conceptual model of the fingering process during soil water redistribution. The emphasis of this approach is not simulation, but rather illustration of the main characteristics of fingering and its dependence on soil and management properties.
THEORY
Finger Shape and Propagation
Figure 1 shows a cross section of a portion of the draining front from a redistribution experiment conducted in a 1.0 m wide by 1.0 m deep by 0.1 m thick Hele–Shaw cell described in detail in Wang et al. (2003a). In this study, 5 cm of water was ponded on the surface of the cell containing a uniform dry coarse sand, and redistribution began at about t = 1 min. Several characteristics are evident in the figure. First, although perturbations in the shape of the front are present from the outset, instabilities do not form and cause fingers to grow until after redistribution begins. Second, once the fingering process starts, downward flow essentially stops in the matrix region between the fingers. Third, the initial finger area fraction ß is substantial (
50% in this experiment), but the fingers become thinner as the flow rate decreases.
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Figure 2 illustrates the formation and propagation of a finger during redistribution. During normal drainage, water pressure at the interface between the wet and dry zones is at the water-entry potential hwe, which allows water to enter the dry region below the entire draining front. As a perturbation forms, the depth of penetration becomes slightly greater at one point along the front, which shifts the water pressure distribution downward above that location. As a result, regions of the surrounding matrix begin to supply the zone above the finger, because the horizontal pressure distribution induces lateral flow. Subsequently, the water pressure in the surrounding matrix decreases, and the pressure at the wetting front drops below the water-entry pressure hwe, thereby stopping downward flow in the matrix region. The profile drainage then proceeds exclusively through propagating fingers and continues until flow stops. The finger may eventually dissipate, but this occurs on a time scale that is much longer than the propagation event (Glass et al., 1989c).
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Figure 3 shows the draining profile at the beginning and end of the finger propagation event. Initially, the wetted soil zone is filled with water from the infiltration process to a depth D, and has a water-content distribution 
(z) that depends on the soil and flux rate. A finger of diameter d forms and is supplied with water from the surrounding matrix, here represented as a zone of width L (or area L2 in three dimensions), which defines the spatial frequency of the finger. The finger moves downward, with its leading edge remaining at the water-entry pressure hwe, until flow ceases everywhere in the matrix and finger domain. At this time, the profile has reached hydrostatic equilibrium (dh/dz = -1).
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 | [6] |
mi is the initial water content in the matrix zone, 
(z) is the final water content in the finger and matrix zones, si is the soil water content below the wetting front, and Am and Af are the cross-sectional areas of the matrix and finger domains, respectively (Fig. 2).
The initial water content or pressure distribution between the surface and depth D can be determined by traditional modeling with the Richards equation. It may be estimated approximately for the case of constant infiltration at a rate i by solving the steady flow equation
 | [7] |
w(h) is the wetting branch of the water characteristic curve. The final water content distribution is the profile attained at hydrostatic equilibrium, and therefore is equal to
 | [8] |
d[h] is the drying curve of the water characteristic function.
The finger area fraction ß is equal to Af(D)/Am, the portion of the matrix cross section at the draining front that is occupied by fingers. We will use the representation in Eq. [4] from Glass et al. (1989b) in our model calculations. During redistribution, the finger diameter shrinks because the flow rate is decreasing, eventually reaching zero. According to the finger diameter representation in Eq. [3], this means that the ratio of the final finger area to the initial finger area is 1 - Rs. Assuming that the area declines linearly with depth (i.e., the finger is conical), we may model the area Af(z) in Eq. [6] by the expression
 | [9] |
With these assumptions, the final finger position may be calculated with Eq. [6] through [9] from the soil hydraulic properties, infiltration rate, and depth of wetting.
Dynamics of Finger Propagation
It is not clear a priori whether the rate of flow of water into the finger is limited by the rate of supply from the matrix or from the resistance within the finger domain. We tested the sensitivity of the rate of loss of water from the matrix to the finger geometry indirectly with the model HYDRUS-2D (Simunek et al., 1996), by using the profile from one-dimensional infiltration to depth L as the initial condition in a simulation where water was only allowed to drain through an aperture of width d. Simulations with different barrier widths produced nearly identical discharge rates from the aperture (Fig. 4), demonstrating that the rate of loss of water from the matrix governs the entry of water into the finger domain. Moreover, as shown by the line in the figure, this rate of loss is represented quite well by the gravity flow model
 | [10] |
a is the average water content in the profile. The volume flow rate entering the finger from the matrix is given by Amq(t), and this is equal to the rate of increase of water stored in the finger. Thus, at time t
 | [11] |
f(z,t) in the finger is determined by the supply rate q(t) and the constraint that f = we at the tip. As a rough approximation, we may assume that the water profile within the finger continuously adjusts to steady state, so that
 | [12] |
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d2 and using Eq. [3])
 | [13] |
Equations [11] through [13] are solved iteratively at each time t for x(t). The final position of x will correspond to the equilibrium position W calculated by Eq. [6] provided that drainage proceeds to equilibrium in a reasonable period of time.
Water-Entry Matric Potential hwe
Following Baker and Hillel (1990), we may estimate the water-entry matric potential hwe by determining the inflection point of the moisture release curve. Using the van Genuchten (1980) parameterization
 | [14] |
 | [15] |
RESULTS
Figure 5 shows the predicted equilibrium depth as a function of Rs for fingers produced during redistribution following infiltration to a depth of 10 cm. Matric potential–water content curves taken from HYDRUS-2D (Simunek et al., 1996) for four soils representing characteristic textures are used in the simulations (Table 1). Several features are prominent. First, although by assumption all soils are unstable during redistribution, the finger depth is negligible in the finer-textured soils. Second, maximum finger depths are large both at low infiltration rates and high ones, reaching a minimum at intermediate values of Rs. The reason for this behavior is made clear in Fig. 6, which shows a cross section of the final finger position for the sand at various flow rates. At large Rs, most of the cross-sectional area is taken up by the finger (see Eq. [4]), but there is a large amount of water stored in the 10 cm matrix profile because the infiltration rate nearly saturated the soil. In contrast, there is much less water stored during infiltration at the lowest Rs, but the finger occupies only a small part of the cross section and therefore receives a lot of water from the surrounding matrix and penetrates to great depth.
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 | [16] |
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) function derived from the moisture release curve because the van Genuchten relation does not allow equilibrium to be reached in a physically reasonable time for coarse-textured soils, unless the medium is extremely wet. The values of ß and 
were adjusted so that K(hwe) = 0.125Ks and K(-100) = 0.1 mm d-1, which were deemed physically reasonable setpoints for purposes of illustration. With these criteria, the dynamic finger growth model approaches the equilibrium value calculated by Eq. [5] within a few days. DISCUSSION
Criteria for Instability and Finger Propagation
The notion that homogeneous soils may be unstable during redistribution is not widely accepted in mainstream soil physics, although it has been reported in the literature (Diment and Watson, 1985; Wang et al., 2003a). It has long been believed that capillary flow has a stabilizing influence on water movement, and that any perturbations in the advance of the wetting front that developed would be quickly damped out by lateral flow. In addition, prevailing theory would suggest that other parts of the draining front behind any finger in the process of formation would continue to advance, thereby depriving the finger of the water it would need to outpace the rest of the profile. Clearly that is not the case in Fig. 1, where the advancing fingers actually become narrower during their downward journey, and the matrix flow between the fingers virtually ceases. However, if we accept the hysteresis explanation for fingers remaining narrow (Glass et al., 1989c) and assume that the water pressure in the matrix drops below the water-entry pressure hwe, then fingers will form and propagate whenever the matric potential gradient opposes the flow. The water-entry pressure has been demonstrated to exist in soil in several experiments (Baker and Hillel, 1990; Geiger and Durnford, 2000; Wang et al., 2000) showing that water will not enter a dry medium until the liquid pressure at the point of entry is raised above a critical level, at which point infiltration suddenly begins.
However, the continuum description of water flow described by the Richards equation has no water-entry matric potential, because water will always flow from higher matric potential to lower, no matter what is the potential of the wetter region. To see that this is so, we need only examine Philip's solution to the equation for infiltration (Philip, 1969), where he shows that the sorptivity is a function of the difference in water content between the wet and dry regions, and that water will infiltrate into a completely dry medium at any wet region potential. Thus, in order for the Richards equation to model finger propagation correctly, it must include hysteresis to prevent finger widening and somehow insert a water-entry potential to prevent continued downward flow in the matrix. The only attempt to include the latter effect has been by Nieber (1996)(Nieber et al., 2000), who calculated the conductivity at the wetting front by a weighting procedure dominated by the low conductivity in the dry zone below the front. Whether this procedure provides an adequate substitution for using the threshold matric potential directly is a question that must await further study.
Soils and Instability
Our simulations are based on assumptions that predict that all soils will be unstable during infiltration. However, as indicated by Fig. 5, finger propagation is only important in coarser-textured soils, because fingers in finer media are wider, contain more water, and occupy more of the total cross section than in coarse-textured soil. As a result, the redistribution process will reach completion without much downward movement and the forward advance of fingers will be negligible. A key factor in a soil's susceptibility to deep penetration by fingers is the shape of the drying curve of the water content–matric potential function (Fig. 8). The equilibrium (h + z = const.) water content profile in the finger when movement stops is equal to the portion of the curve beginning at the tip of the finger where = we and extending upward. Since coarser-textured soils have little water remaining in them at relatively high matric potentials, most of the finger contains little water. The water content profiles for the finger shown in Fig. 5 have the same characteristic shapes measured in Selker et al. (1992a), with water content decreasing toward the surface.
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Finger Characteristics
The two most important features of fingering are the finger diameter d and the finger area fraction ß or spatial frequency. Finger diameter has been predicted reasonably well with the equations given here, and with similar formulations discussed in the review by de Rooij (2000). In particular, Eq. [3] requires only the water-entry matric potential and the infiltration rate, and has estimated finger diameter satisfactorily both in the laboratory (Wang et al., 1998, 2003a) and in the field (Wang et al., 2003b). In this paper, we have assumed that finger diameter shrinks according to Eq. [3] during redistribution, producing a conical shape that is roughly similar to what we have observed in our experiments. Thus, we feel that this characteristic is predictable.
We are less sanguine about the predictability of the finger area fraction ß. As we stated in the introduction, various authors have used quite different models for ß, and the form we used (Eq. [4]) for our calculations is an empirical relation derived from laboratory data (Glass et al., 1989b). As shown in Fig. 1, the area fraction declines with time as the fingers shrink during redistribution, and it is difficult to resolve the exact fraction occupied by fingers at the outset. Additional experimentation under different conditions will be required to produce more accurate models of this key parameter.
CONCLUSION
We have used a mixture of physical arguments and empirical relations to produce a conceptual model of the fingering process during redistribution. Although the model is simplistic, it was able to demonstrate some useful conceptual features of fingering, and to show which soils have a strong susceptibility to fingering. The most critical features of soil that create and maintain fingering are hysteresis, which allow fingers to stay narrow as they advance, and the water-entry pressure, which quickly blocks the water in the soil between the fingers from moving downward once fingering begins.
The finger apparently does not form the limiting resistance to water discharge from the wetted soil matrix, so that it is possible to develop a flux rate into the fingers that is a function only of the soil characteristics and initial wetting status at the end of infiltration. The finger will grow until hydrostatic equilibrium is reached in the soil profile, although it may be moving at a negligible rate before that time.
Capillary effects can overcome the barrier formed by the water-entry pressure, but apparently operate on a time scale longer than the finger propagation time. As a consequence, fingers are able to move substantial distances before dissipating into the surrounding matrix.
The conventional Richards flow equation cannot predict the occurrence, transport, or final extent of fingers unless both hysteresis and some representation of the water-entry pressure are included. Without these effects, capillary diffusion will smooth out fingers at the moment of formation and also allow the water in the matrix between fingers to move downward.
ACKNOWLEDGMENTS
The research was supported by Research Grant No. IS-2859-97 from BARD, the U.S.-Israel Binational Agricultural Research and Development Fund. Appreciation is expressed to Dr. Jirka Simunek for assisting us in adapting HYDRUS-2 to the simulation conditions we employed in this paper.
REFERENCES
This article has been cited by other articles:
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  T. Annaka and S. Hanayama Pressure Head Profile within Growing Fingers in Initially Dry Glass Beads Soil Sci. Soc. Am. J., May 16, 2007; 71(3): 901 - 908. [Abstract] [Full Text] [PDF]
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 M. J. Nicholl and R. J. Glass Infiltration into an Analog Fracture: Experimental Observations of Gravity-Driven Fingering Vadose Zone J., November 11, 2005; 4(4): 1123 - 1151. [Abstract] [Full Text] [PDF]
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H. Cho, G. H. de Rooij, and M. Inoue The Pressure Head Regime in the Induction Zone During Unstable Nonponding Infiltration: Theory and Experiments Vadose Zone J., September 12, 2005; 4(4): 908 - 914. [Abstract] [Full Text] [PDF]
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Z. Wang, W. A. Jury, A. Tuli, and D.-J. Kim Unstable Flow during Redistribution: Controlling Factors and Practical Implications Vadose Zone J., May 1, 2004; 3(2): 549 - 559. [Abstract] [Full Text] [PDF]
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Z. Wang, A. Tuli, and W. A. Jury Unstable Flow during Redistribution in Homogeneous Soil Vadose Zone J., February 1, 2003; 2(1): 52 - 60. [Abstract] [Full Text] [PDF]
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