Published online 13 September 2005
Published in Vadose Zone J 4:908-914 (2005)
DOI: 10.2136/vzj2004.0158
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
The Pressure Head Regime in the Induction Zone During Unstable Nonponding Infiltration
Theory and Experiments
H. Choa,
G. H. de Rooijb,* and
M. Inouec
a Dep. of Agricultural Sciences, Saga Univ., Honjou 1, Saga-shi 840-8502, Japan
b Dep. Environmental Sciences, Soil Physics, Ecohydrology, and Groundwater Management Group, Wageningen Univ., Nieuwe Kanaal 11, 6709 PA Wageningen, The Netherlands
c Arid Land Research Center, Tottori Univ., 1390 Hamaska, Tottori 680, Japan
* Corresponding author (ger.derooij{at}wur.nl)
Received 28 October 2004.
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ABSTRACT
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Fingered flow rapidly moves water and pollutants from the root zone to the groundwater through a limited fraction of the unsaturated zone, limiting the possibilities for decay and adsorption. The onset of wetting front instability and the characteristics of the flow pattern under nonponding infiltration have received limited attention. We aim to theoretically and experimentally advance our understanding of pre-fingered flow, and contrast fingered flow under ponding and nonponding conditions. We developed a Green-Ampt based expression for the pressure head in a developing induction zone (from which fingers protrude) for the time before fingers developed. A uniform, nonponding water flux was applied to the surface of two-dimensional glass bead porous media with a dry region above a capillary fringe. Microtensiometers recorded pressure heads in the induction zone. The pressure head data confirmed both the theoretical early-time pre-finger model, and a model developed earlier for late-time lateral flow toward fully developed fingers. The physically more realistic constant flux boundary condition of our experiments gave larger finger spacings and travel times, compared to the frequently used set-up with ponding infiltration into a fine-over-coarse porous medium.
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INTRODUCTION
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UNSTABLE WETTING FRONTS produce fingers that rapidly remove water from the root zone and limit the soil's ability to neutralize or adsorb contaminants before they reach the groundwater. Therefore, wetting front instability has been extensively researched (see the reviews by Hillel, 1987; Glass and Nicholl, 1996; de Rooij, 2000; Hendrickx and Flury, 2001; Nieber, 2001). The physical principles governing fingered flow have been studied mainly by means of Hele-Shaw cells filled with idealized porous media (e.g., Hill and Parlange, 1972; Tamai et al., 1987; Diment and Watson, 1985; Glass et al., 1989, Baker and Hillel, 1990; Pendexter and Furbish, 1991; Wang et al., 1998, 2003).
At the onset of fingered flow, a few of usually many wetting front protrusions extending from the induction zone gain dominance and develop into full-grown fingers while the other protrusions come to a virtual stand-still. The interplay between this finger formation process and the pressure head dynamics in the induction zone has received relatively little experimental attention, as most studies focused on the properties of the fingers themselves. Baker and Hillel (1990), Selker et al. (1992a), and Cho and de Rooij (1999)( 2002) used rapid-response tensiometers in Hele-Shaw cells to observe pressure heads in the induction zone. Of those, only the latter had enough observation points in the induction zone to study lateral flow toward the fingers in an induction zone below a fine-over-coarse interface. Geiger and Durnford (2000) performed similar measurements in 12.7-mm diameter columns.
As indicated above, detailed lateral pressure head distributions in the induction zone have only been observed in ponded fine-over-coarse profiles. We therefore performed nonponding infiltration experiments with two-dimensional glass bead porous media. We studied all stages of finger development and compared finger behavior with earlier ponded infiltration experiments (Cho and de Rooij, 2002). Flow inside the fingers was theoretically treated by Selker et al. (1992b), while de Rooij (1995) and de Rooij et al. (1996) theoretically analyzed finger dissipation in the capillary fringe. Theories for the onset of instability (see de Rooij's [2000] review) do not target details of the flow. This paper introduces a simple Green-Ampt type model for flow in the induction zone under constant, nonponding infiltration before finger formation to further develop the finger formation theory by Hillel and Baker (1988).
The objectives of this paper are (i) to develop the theory for prefingered flow; (ii) to verify experimentally the model-predicted pressure head evolution shortly after infiltration; (iii) to highlight, theoretically and experimentally, some limitations of the conventional set-up in which fingers are generated by ponded infiltration through a poorly conductive layer into an initially dry, coarse-grained medium; (iv) to expand and adapt the findings of Cho and de Rooij (2002) for the early stages of finger development under ponded infiltration. In addition, we provide a physical explanation of the bottle-neck shape of fingers just above the capillary fringe, first reported by Liu et al. (1994).
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THEORY
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Raats (1973) and Hillel and Baker (1988) hypothesized that a wetting front penetrating a dry soil can only break up in distinct fingers if the downward water flux is insufficient to wet up the entire horizontal cross-sectional area of the soil. If the hydraulic conductivity is continuous for –
< h < 0 (where h is the pressure head, L), the soil can always adapt the pressure head to make the entire horizontal cross-section participate in the flow. Including a water-entry pressure head hwe (L) effectively makes flow below this threshold impossible. In that case, fingers can form if the hydraulic conductivity at hwe exceeds the infiltration flux density. From Darcy's Law it follows immediately that this corresponds to Philip's (1975) instability criterion, which stipulates that the pressure head gradient immediately behind the wetting front must oppose the flow. For coarse media, 
(hwe) is close to saturation, and a Green-Ampt type wetting front develops (see also Raats, 1973). Selker et al. (1992a)(1992b) and Cho and de Rooij (1999)(2002), among others, observed this type of wetting in the induction zone from which fingers emerged in the course of their experiments.
Jury et al. (2003) suggested that a -h relationship with a water-entry pressure head (below which the hydraulic conductivity during infiltration is zero) should allow Richards' equation to produce unstable wetting fronts. Egorov et al. (2003) conclude the same but warn that such a modification fundamentally has no fastest growing perturbation of the wetting front, while experiments typically show fairly uniform finger widths/radii. Based on Egorov et al.'s (2002) analysis they point to a nonequilibrium -h relationship as a mathematically more acceptable requirement for wetting front instability.
While the existence of a water-entry value during (near-) equilibrium conditions is physically realistic, hwe at equilibrium differs from that during infiltration (e.g., Geiger and Durnford, 2000). It is plausible that this dynamic nature of hwe is the way a nonequilibrium -h relationship manifests itself in tensiometer readings.
Both the dynamic hwe and the nonequilibrium -h relationship are macroscopic representations of the pore-scale infiltration process (see the observations of Lu et al., 1994a, 1994b). Both of them allow Richards' equation to generate unstable wetting fronts. According to Egorov et al. (2003), the latter has the advantage of being able to predict finger sizes.
In either case, flow in a drying soil should be stable. Film flow along pore walls occurs for h << hwe (see Lu et al., 1994a, 1994b), and the increased conductivity associated with uniformly wetted pore walls makes it likely that the memory effect in a nonequilibrium -h relationship is much smaller during drying than during wetting. Therefore, equilibrium -h relationships with a finite water-entry value as well as nonequilibrium -h relationships are likely to fulfill this requirement.
We analyze the flow in a developing induction zone by assuming Green-Ampt infiltration (sharp wetting front, uniform volumetric water content behind the front) until the induction zone reaches a thickness that allows fingers to form. The wetting front pressure head equals the dynamic water-entry value (dependent on the porous material and the flux density, see Geiger and Durnford, 2000). The nonponding, constant rainfall flux density is q0 (L T–1), and starts at time t = 0. In the early stages of infiltration (before finger formation), we have:
 | [1] |
where K (L T–1) is the unsaturated hydraulic conductivity, L (L) is the thickness of the induction zone (equal to the depth of the wetting front), and t (T) denotes time. The subscript 0 indicates the value is valid at the soil surface, the subscript "we" denotes a value corresponding to the water entry pressure head. Note that fingers can only form if q0 < Kwe. The gradient of the hydraulic head (the bracketed term in Eq. [1]) will then lie between 0 and –1, and consequently h0(t) can range from hwe to hwe – L(t).
Mass conservation dictates:
 | [2] |
where 0 is the initial volumetric water content, assumed uniform (which is reasonable in dry soils, even if h varies with depth).
With q0 constant, the downward velocity of the wetting front (Eq. [2]) as well the gradient of the hydraulic head behind the wetting front must necessarily be constant as well. Integrating Eq. [2] and combining the result with Eq. [1] leads to the following expression for the pressure head at the soil surface:
 | [3] |
The pressure head at the soil surface increases linearly with time for stable flows (q0 > Kwe), decreases linearly with time for unstable flows (q0 < Kwe), and remains constant when q0 = Kwe and unit gradient conditions prevail. It is important to note that this behavior of h0 under a nonponding constant flux upper boundary condition is fundamentally different from the poorly defined behavior of the pressure head at the textural interface of a fine-over-coarse profile that has often been used to produce fingered flow under a constant ponding depth (e.g., Hill and Parlange, 1972, and many subsequent experiments).
The uniform volumetric water content ( = we) behind the wetting front leads to a pressure head profile of the form:
 | [4] |
where z (L) denotes depth below the soil surface and h(z,0) (L) is the initial, very low, pressure head. As indicated above, hwe – L(t) 
h0(t) hwe for unstable flows. Hence, in shallow induction zones, h will only vary a few centimeters. If hwe is located in the steep, wet-end range of the soil water characteristic, the water content will be nearly uniform over the entire depth of the induction zone, validating the Green-Ampt approximation. With L(t) derived by integrating Eq. [2] and h0(t) according to Eq. [3], Eq. [4] gives the pressure head at any depth within the induction zone [0 < z < L(t)]:
 | [5] |
Therefore, for unstable flows (q0 < Kwe), the pressure head anywhere in the induction zone decreases linearly with time as the induction zone thickens, and increases with depth. The latter is consistent with Philip's (1975) instability criterion mentioned above. The pressure head at the soil surface cannot be measured directly, but microtensiometers can be operated at very small depths, allowing experimental verification of Eq. [5].
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MATERIALS AND METHODS
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We performed nearly two-dimensional experiments in which a dye tracer visualized the flow pattern. The transparent, 1-cm-wide experimental chamber with drainage outlets in the bottom and microtensiometer/air outlet ports in one of its walls was described in detail by Cho and de Rooij (2002), who also described the rapid-response microtensiometers and their operating procedures. We experimented with glass beads of 0.120 to 150 mm (Run 1) and 0.220 to 250 mm (Run 2), homogeneously packed according to Cho and de Rooij (2002) to create a layer of 68-cm thickness. Table 1 gives relevant properties of the porous packs, and Fig. 1 gives the main drying and wetting curve of the fine material. The static water values for hwe were estimated from the height of the capillary fringe. The dynamic values were obtained from the highest (least negative) tensiometer readings. Saturated hydraulic conductivities were measured with a constant head permeameter (Klute and Dirksen, 1986, p. 694–700).
In both experiments, the chamber was placed in a water-filled reservoir with the water level at the same height as the chamber inner bottom. The water infiltrated through the drainage openings and created a capillary fringe. After 11 d the chamber was taken from the reservoir. A horizontal array of tensiometers (5-cm spacing) was installed 1 cm below the soil surface, with a few additional tensiometers at various depths (see Fig. 2 and 3)
, including the capillary fringe. Pressure heads were recorded every 10 s (Fig. 4)
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Fig. 2. Front view of the experimental chamber with the wetting fronts observed at the indicated times for the run with the 0.120- to 0.150-mm glass beads. Tensiometers (labeled with characters) are indicated by solid green circles.
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Fig. 3. Front view of the experimental chamber with the wetting fronts observed at the indicated times for the run with the 0.220- to 0.250-mm glass beads. Tensiometers (labeled with characters and numbers) are indicated by solid green circles.
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Fig. 4. Experimental set-up. Before the water application started, the chamber containing the porous material was taken out of the water reservoir that helped create the capillary fringe.
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A dye-solution (0.01 M KMnO4) in distilled water was applied to the porous medium surface by a low-flux rainfall simulator consisting of a nozzle oscillator and a micro-tubing pump (Fig. 4 and 5)
. Thirty-two nozzles (0.6-mm inner diam., 1.0-mm outer diam., 45-mm length) were mounted 25-mm apart on a 90-cm-long plate that oscillated horizontally with 10-mm amplitude at a frequency that could be varied between 1.7 and 17 cycles min–1. The micro-tubing pump (BVK MS/CA8-6, ISMATEC, Switzerland) had four blocks, with eight flow channels each. We fixed the application rate at 0.404 mm min–1 for the fine material and 0.420 mm min–1 for the coarse material, in both cases providing a constant-flux upper boundary condition without ponding. The wetting front advancement and finger development were videotaped and photographed (Fig. 4).
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RESULTS AND DISCUSSION
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Wetting Patterns
The run with the fine material produced a fairly flat wetting front after the initial perturbations caused by the individual nozzles damped out within the first minutes (Fig. 2). After about 75 min, three protrusions gained dominance over the other oscillations. At that time, the wetting front had already advanced to approximately 6 cm. Only the right-most oscillation developed into a finger. At the time the finger reached the capillary fringe (110 min after the start of infiltration), the wetting front halted at about 8-cm depth (excluding the protrusions).
Infiltration in the coarse material produced a more oscillating wetting front, which generated a single dominant feature after about 45 min. The finger that developed from this reached the capillary fringe after about 80 min. Again, the wetting front stopped advancing at that time, reaching a depth of about 5 cm.
Cho and de Rooij (2002) used the same glass beads in the sublayer as were used in our Run 1, but generated fingers by ponding a fine-over coarse profile. Their Run 1 involved a nearly flat textural interface, compatible with our flat soil surface. Since their infiltration rate (0.373 mm min–1) is comparable to ours, we can compare their Run 1 with ours. In our constant flux experiments, finger spacing was three times larger. If larger flow chambers had been used, the difference might have been larger since the fingers in both of our experiments appeared near a chamber wall. The induction zone under prescribed infiltration was considerably thicker (8 cm compared to 3 cm). The cumulative infiltration required to reach the capillary fringe increased from 22.3 to 44.4 mm, even though the longer period of capillary rise (11 instead of 6 d) resulted in a thicker capillary fringe. Since the prescribed flux upper boundary condition reported here better represents field conditions than the constant head boundary condition of ponded infiltration experiments, the latter may underestimate solute residence times in the unsaturated zone and overestimate the fraction of the horizontal cross-section which delivers water and solutes to the groundwater.
The fingers dissipated in the capillary fringe, consistent with earlier observations (Cho and de Rooij, 1999, 2002; de Rooij and Cho, 1999; de Rooij et al., 2001). But just before diverging, the fingers strongly converged in the few centimeters above the capillary fringe. Liu et al. (1994) first reported this phenomenon in a prewetted layer, which was confirmed by the experiments involving a capillary fringe referenced above. It appears plausible that the smallest pores above the visible capillary fringe were wetted through capillary rise, thereby increasing the hydraulic conductivity immediately above the capillary fringe. If the wetting was such that the larger pores still required a water-entry pressure head to be exceeded, fingers could not dissipate in this region (the flow lines would not diverge). The increased conductivity would allow a smaller cross-section of the porous medium to conduct the water flux carried by the finger, thereby creating a region of convergent flow immediately above the nearly saturated region of diverging flow. This explanation is supported by the experiments by Bauters et al. (2000), who noted a reduced finger size when the porous medium was slightly wetted instead of air-dry.
Model Evaluation
The infiltration model represented by Eq. [5] predicts a linear decrease of the pressure head within the induction zone before finger formation, consistent with Selker et al.'s (1992a)(1992b) early time observations. At the wetting front, the pressure head equals the dynamic hwe. The rate of decrease is given by 
h/t = q0(we – 0)–1[(q0/Kwe) – 1]. Assuming saturation at the dynamic water entry value and 0 = 0, we calculated h/t from independently measured saturated water contents s and hydraulic conductivities Ks (L T–1) in Table 1. The time at which the wetting front reached the tensiometers was calculated by integrating Eq. [2]. Thus, the only parameter that was fitted to the observed h(t) curves was the dynamic value of hwe. The difference between the static and dynamic values of hwe (both determined from observations) in Table 1 clearly illustrates the unsuitability of the static value of hwe to predict pressure heads (and hence water contents and conductivities) prevailing behind unstable wetting fronts and in fingers. Figures 6 and 7
compare model results to observations for four representative tensiometers for both experiments. Even with the assumption of saturation the model performed remarkably well.
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Fig. 6. Observed and modeled pressure head trends in the induction zone for the 0.120- to 0.150-mm porous medium. The numbers indicate the distance between the tensiometers and the left chamber wall. All tensiometers were installed at 1 cm below the soil surface. Vertical black lines mark the times at which the dominant finger became visible and reached the capillary fringe.
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We compared our model to pressure head observations during forced one-dimensional infiltration in two silica sands and three sieve fractions of one of those (Geiger and Durnford, 2000). We estimated the observed h/t from the linear sections of the slopes of Geiger and Durnford's (2000) Fig. 2a and 2f, and calculated these slopes as outlined above using Geiger and Durnford's values of Ks and the porosity to approximate Kwe and we. The results (Table 2) show an excellent agreement between the observations and the model for relatively large infiltration rates (0.2 Ks), except for the fine sand (F-95) where the assumption of near-saturation at the dynamic hwe may have been invalid. For the coarse sand (F-14), the model performance for small flux rates was reasonable.
According to Eq. [5], the pressure head difference after passage of the wetting front between two vertically aligned points in the induction zone equals [1 – (q0/Kwe)]
z, where z (L) is the vertical distance between the points. We tested this relationship on tensiometer pair F and P (z = 10.0 cm) in Fig. 2. For the period between the moment the wetting front passed tensiometer P and the moment the finger reached the capillary fringe, the pressure head difference ranged without a clear trend from 9.6 to 9.9 cm, averaging 9.7 cm. The pressure head difference according to Eq. [5] should remain constant at 9.4 cm (97% of the observed value) based on the porous medium properties in Table 1 and the infiltration rate.
Tensiometer Data
Figures 6 and 7 clearly show three phases in the pressure head behavior related to the development of the wetting pattern. The first phase (development of the induction zone before finger formation; fine medium: 0 to 75 min, coarse medium: 0–45 min) confirms the linear pressure head decrease with time according to Eq. [5]. During the second phase (finger growth until the finger tip reached the capillary fringe; fine medium: 75 to 110 min, coarse medium: 45–80 min), relatively small pressure head differences emerged, creating the lateral gradients needed to supply the growing finger with water. The differences increased more rapidly in the coarse medium, possibly because the decreasing pressure head resulted in air intrusion, which reduced the hydraulic conductivity. The third phase started when the fingers reached the capillary fringe. In the fine medium, the pressure head dropped within a few minutes, the differences increased (indicating air intrusion), and the overall decreasing trend continued. In the coarse medium, the decreasing trend slowed down and reversed after about 90 min, when the desaturation of the induction zone and even the top of the finger could be visually confirmed (only in the coarse medium).
Figures 8 and 9
give the pressure head patterns in the induction zone at the times for which the wetting fronts are given in Fig. 2 and 3. Clearly, significant lateral pressure head gradients only developed after the fingers had firmly established themselves. The wetting pattern drove the pressure head pattern. This is completely opposite to the findings of Cho and de Rooij (2002) for ponded infiltration experiments, in which a sinusoidal pressure head pattern developed within minutes, and fingers later formed at locations of pressure head minima.
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Fig. 8. Lateral pressure head distribution in the induction zone of the 0.120- to 0.150-mm porous medium at 1 cm below the soil surface. Numbers indicate the time in minutes since the start of infiltration. The widths of the main protrusions are indicated by the black lines. The right-hand-side protrusion developed into a finger. Solid lines indicate theoretical fits.
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Fig. 9. As Fig. 8, for the 0.220- to 0.250-mm porous medium. Dominant protrusions were absent, and only the finger is indicated by black lines.
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The thick, nearly saturated induction zone is highly conductive. Consequently, the lateral flow toward the fingers required only small lateral pressure head gradients, which allowed the large finger spacing compared to the observations of Cho and de Rooij (2002) discussed above.
Modeling Lateral Flow
Cho and de Rooij (2002) developed a simple model for the pressure head pattern in the induction zone after finger formation:
 | [6] |
where x* (L) denotes the horizontal distance to the nearest finger center, W (L) is the distance between the finger center and its catchment's boundary, and C (L) is a constant.
The model assumes a uniform infiltration flux density and uniform and constant hydraulic conductivity and water content in the induction zone and should therefore be applicable to the experiments reported here. We identified the center of the fingers in both experiments to determine W for the left-hand side of the catchment. For each curve in Fig. 8 with t 
75 min, and for all curves with t 55 min in Fig. 9, we set C equal to the lowest readings of one of the tensiometers adjacent to the finger center. For Kwe, L, and q0 we used the values reported above. The calculated lateral pressure head gradients were generally larger than those observed for the fine porous medium. The mean absolute error (Nielsen and Wendroth, 2003, p. 152) between observed and calculated pressure heads in the fine medium ranged from 14 to 18 cm for the different times considered. The model performed much better for the coarse medium (mean absolute error between 0.78 and 3.6 cm). We also determined LKwe and C by least-square fitting Eq. [6] to the observed curves in Fig. 8 and 9. Since the curvature clearly changed after 105 min (fine medium) and 70 min (coarse medium), we allowed C to vary between curves (separate fits) to fit the height, but kept LKwe constant before and after these times of curvature change to obtain two different shapes for each experiment. The fits were excellent after the fingers had reached the capillary fringe.
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CONCLUSIONS
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In summary, we completed the work by Cho and de Rooij (2002) on flow and pressure head patterns in the induction zone by adding an analysis of the early-time behavior, before finger formation. We also quantified the differences in the flow characteristics emerging from a constant flux vs. a constant head upper boundary condition (the latter for a fine-over-coarse porous medium): the former leads to larger finger spacings and travel times to the capillary fringe than the latter. The model for lateral flow in the induction zone that Cho and de Rooij (2002) found to be of limited use for constant head experiments performed very well for a constant flux boundary condition after the fingers reached the capillary fringe.
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ACKNOWLEDGMENTS
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The first author thanks Dr. P. R. Berliner (Ben-Gurion Univ., Israel) for his valuable suggestions.
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ABSTRACT
INTRODUCTION
