Published in Vadose Zone Journal 4:127-133 (2005)
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Dynamic Water-Entry Pressure for Initially Dry Glass Beads and Sea Sand
Takeyuki Annaka* and
Susumu Hanayama
Faculty of Agriculture, Yamagata University, Tsuruoka, Yamagata 997-8555, Japan
* Corresponding author (annakt{at}tds1.tr.yamagata-u.ac.jp)
Received 6 April 2004.
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ABSTRACT
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Dynamic water-entry pressure has been considered an important factor in the mechanism of fingered flow formation. We measured and compared dynamic water-entry pressures of air-dried glass beads of 0.4-mm mean particle size and for 0.30- to 0.60-mm (30–50 mesh) sea sand. Two measurements were conducted. First, breakthrough pressures were measured by increasing the supply water pressure incrementally from a low water pressure at which water could not enter the porous medium. Second, the relationship between the supply water pressure and the initial infiltration rate was measured by keeping the supply water pressure constant at several values. Measured values for the 0.4-mm glass beads and the 30- to 50-mesh sea sand showed some differences. While the breakthrough pressure was similar to the advancing capillary pressure for both materials, only the 30- to 50-mesh sea sand showed almost the same value as the water pressure at the inflection point on the initial wetting curve. For the 0.4-mm glass beads, the breakthrough pressure was higher (less negative) than that of the inflection point. Furthermore, the 0.4-mm glass beads showed a sharp jump in the initial infiltration rate when the supply water pressure exceeded the breakthrough pressure, whereas the 30- to 50-mesh sea sand showed an almost linear increase in the initial infiltration rate. These differences are discussed in terms of oscillatory nature of the water-entry pressure. An expression for the dynamic water-entry pressure was fitted to the measured data. The expression was used for estimating the width of fingers formed in the materials during infiltration into layered conditions and was found to be useful for studying fingered flow formation, at least for the glass beads.
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INTRODUCTION
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WETTING FRONT INSTABILITY is one of the mechanisms that produce preferential flow, which causes nonuniform water distribution in a soil profile and potentially rapid transport of contaminants to the groundwater. According to the criteria of wetting front instability (Raats, 1973; Philip, 1975), wetting fronts formed during infiltration become unstable when the infiltration rate is less than the hydraulic conductivity of the soil, or when the gradient in the water pressure head behind the wetting front is opposed to the flow. When the wetting front becomes unstable, small perturbations on the front will grow into fingers, which then govern water and solute movement in the soil profile.
A large number of studies have been carried to decipher the important effects of finger size and water content within the finger on water and solute movement in soil profiles. Linear instability analyses (e.g., Philip, 1975; Parlange and Hill, 1976) and dimensional analyses in conjunction with systematic experiments (Glass et al., 1989a, 1989b) have been used to estimate finger size. Water content and water pressure distributions within the finger have also been analyzed (Selker et al., 1992; Nieber, 1996). However, more accurate prediction of such finger properties requires thorough understanding of the finger formation process, that is, the mechanism of fingered flow. Until now, this process has been considered mainly in terms of the Darcy–Buckingham flux equation, hysteretic moisture retention functions, and the water-entry pressure (Hillel and Baker, 1988; Glass et al., 1989c; Nieber, 1996; Geiger and Durnford, 2000).
Among those factors, the water-entry pressure may be the most important in the finger formation process because of its role in determining the boundary condition at the wetting front. Hillel and Baker (1988) treated the water-entry pressure as a single, characteristic value for air-dried porous media, and discussed the important role of the water-entry pressure in the formation of fingered flow. By means of infiltration experiments into fine-over-coarse layered sands, Baker and Hillel (1990) showed that the water-entry pressure has almost the same value as the inflection point on the wetting curve. Recently, Wang et al. (2000) measured the water-entry pressure with a water ponding method for water-repellent soil and with a tension-pressure infiltrometer method for both water-repellent and wettable soils.
The concept of dynamic water-entry pressure, which considers the water-entry pressure as an increasing function of flux, has often been used to better understand mechanism of the fingered flow generation (Geiger and Durnford, 2000; DiCarlo and Blunt, 2000). Geiger and Durnford (2000) measured the soil water pressure head near the wetting front for various flux rates for initially air-dried soil conditions and presented a mechanistic model that explained soil water pressure gradients of stable and unstable flow in homogeneous soils. Using the concept of dynamic water-entry pressure, DiCarlo and Blunt (2000) obtained a self-similar solution to a moving finger with a curved interface. From this solution they found analytical expressions for the shape and width of fully developed gravitationally driven fingers.
While the water-entry pressure can be estimated indirectly from the wetting curve (Wang et al., 2000), direct measurement of the water-entry pressure is still important to clarify the fingered flow processes because dynamic effects of the moving water–air interface in porous media are not very well understood at present. The objectives of this study were (i) to measure the dynamic water-entry pressure for initially air-dried glass beads and sea sand having almost the same mean particle size and (ii) to compare and discuss the measured values.
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MATERIALS AND METHODS
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Materials
Figure 1
shows the particle shapes of the glass beads and the sea sand (Wako Pure Chemical Industries, Ltd., Osaka, Japan) used in this study. Both materials were soaked in dilute hydrochloric acid, rinsed well with distilled water, and then air-dried. Physical properties of the materials are shown in Table 1. The particle size distributions, as well as the particle shapes as shown in Fig. 1, differ between the two materials. While the mean particle size of the 30- to 50-mesh sea sand is slightly larger than that of the 0.4-mm glass beads, the saturated hydraulic conductivity of the sea sand is lower than that of the glass beads.
Measurement of Dynamic Water-Entry Pressure
Water-entry pressures can be evaluated in two ways. One method is to consider it as a breakthrough pressure, that is, the critical water pressure at which water starts to enter the porous medium (Hillel and Baker, 1988; Baker and Hillel, 1990; Wang et al., 2000). Another method is to consider it as the relationship between water flux and water pressure at the wetting front, that is, as the dynamic water-entry pressure (Geiger and Durnford, 2000). In this study, we measured both the breakthrough pressure and the flux–pressure relationship for the materials to evaluate the dynamic water-entry pressure.
Figure 2
shows the apparatus used to measure dynamic water-entry pressure, as proposed by Idesawa and Annaka (1996). The apparatus consisted of an acrylic cell filled with air-dried particles, a Mariotte reservoir for water supply, and an electric balance to measure the cumulative amount of water supply. A filter was needed to establish contact between the porous medium in the cell and the supply water of negative pressure. A necessary property of the filter is that it have an air-entry pressure lower (more negative) than that of the porous medium and a permeability that does not substantially limit water-entry. For the 0.4-mm glass beads we used a 0.1-mm-thick cloth filter, which has good permeability and a higher (less negative) air-entry pressure. However, the filter could not be used for the 30- to 50-mesh sea sand because its air-entry pressure was higher (less negative) than that of the sand. Instead, a 0.02-mm-thick membrane filter, which has a lower air-entry pressure but is less permeable than the cloth filter, had to be used for the sea sand.
The acrylic cell was turned upside down and then immersed in a distilled water bath to fill the water pool with distilled water. After filling the water pool, the cell was connected to the Mariotte reservoir with a tube and removed from the water bath. Excess water on the filter was removed carefully with tissues, after which the cell was filled with air-dried material, either the 30- to 50-mesh sea sand or the 0.4-mm glass beads. A piece of gauze was placed on the surface of the sample, and the cell was capped. The cell was subsequently inverted, fixed with a stand, and placed on a laboratory jack, as shown in Fig. 2.
Using the apparatus, two types of measurement were conducted. With the supply water pressure increasing incrementally, we first measured the breakthrough pressure (i.e., the point at which water started to enter the dry sample). Our method for this measurement was similar to that of Wang et al. (2000), who first introduced tension-pressure infiltrometer method to measure the water-entry value. We began with a water pressure low enough for the supply water not to enter the sample: –9 cm for the 0.4-mm glass beads and –20 cm for the 30- to 50-mesh sea sand. If water did not appear to enter the sample for 1 min, the laboratory jack was operated to increase the supply water pressure by 1 cm. This process was repeated until the onset of water-entry was observed. When water-entry was observed, the vertical difference 
H (Fig. 2) was measured with a cathetometer, and the supply water pressure –H was considered to be the breakthrough pressure. This measurement was repeated six times for each material.
Next, we measured cumulative infiltration under constant supply water pressures. From the data involving temporal changes in the cumulative infiltration, we calculated a time-averaged flux for the first 1 min to obtain the dependence of the flux on the supply water pressure. For each measurement, we repacked the cell with the dry sample.
Characteristic curves for initial wetting from the air-dried condition were obtained using a soil column method (Nakano et al., 1995). The soil column for each material was allowed to wet by capillary rise until equilibrium was reached. The column was subsequently sampled at 1-cm intervals for the glass beads and at 2-cm intervals for the sea sand, and the water contents of the sections were determined by oven drying. In addition, we estimated the advancing capillary pressure head, that is, the water pressure head at the wetting front as it moves during infiltration (Green and Ampt, 1911), for the dry samples by fitting the Green–Ampt equation to measured data obtained from ponded infiltration experiments.
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RESULTS
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Breakthrough Pressure
Table 2 shows the ranges and means of the measured breakthrough pressure values. The advancing capillary pressure values are also shown. The breakthrough pressures of the 0.4-mm glass beads are clearly higher than those of the 30- to 50-mesh sea sand, even though the particle size of the glass beads is smaller than that of the sea sand. Furthermore, the value of the breakthrough pressure is almost the same as that of the advancing capillary pressure for both materials. This similarity between the breakthrough and the advancing capillary pressure values suggests that the filter effects were negligible for both materials.
Figure 3
shows initial wetting curves for both materials, obtained by wetting from air-dried condition. For the 30- to 50-mesh sea sand, the breakthrough pressure was about –10 cm, which indeed corresponded to the inflection point on the wetting curve, as reported by Baker and Hillel (1990). On the other hand, the –2.2-cm breakthrough pressure for the 0.4-mm glass beads substantially differed from the –7-cm water pressure at the inflection point.
Relationship between Supply Water Pressure and Initial Infiltration Rate
Figure 4
shows temporal changes in the cumulative infiltration for several supply water pressures. For the 0.4-mm glass beads at supply water pressures less than –2.3 cm, the cumulative infiltration increased very slowly at first, followed by a rapid increase. Such temporal change patterns for cumulative infiltration strongly depended on the supply water pressure. Elapsed time to the rapid increase in the cumulative infiltration was 8 to 9 min for the –2.6-cm supply water pressure, 3 to 4 min for –2.5 cm, and almost zero for –2.3 cm. This rapid increase in the cumulative infiltration was accompanied by the growth of a narrow finger in the medium. On the other hand, the cumulative infiltration for the 30- to 50-mesh sea sand, for all supply water pressures, showed an almost linear increase with time, and no temporal change as for the 0.4-mm glass beads. Finger growth was also observed for the sea sand, but the finger was wider than that for the glass beads.
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Fig. 4. Temporal changes in the cumulative infiltration for several supply water pressures: (a) for the 0.4-mm glass beads (a very slow increase is followed by a rapid increase at supply water pressures less than –2.3 cm) and (b) for the 0.30- to 0.60-mm (30–50 mesh) sea sand (a nearly linear increase in the cumulative infiltration with time).
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Figure 5
shows the relationships between the supply water pressure and a time-averaged infiltration rate during the first minute, which we refer to as the initial infiltration rate. This infiltration rate was calculated from the cumulative infiltration data at 1 min. For the 0.4-mm glass beads, the initial infiltration rate was almost zero when the supply water pressure was lower than –2.5 cm. However, the infiltration rate showed a rapid increase at a supply water pressure of –2.4 cm. By comparison, the initial infiltration rate for the 30- to 50-mesh sea sand increased almost linearly with the supply water pressure. The curves in Fig. 5 may be used to explain differences in the temporal change in cumulative infiltration shown in Fig. 4. When the supply water starts to enter the dry sample, a wetting front forms and begins to move downward. The water pressure at the wetting front then becomes higher due to the gravitational force. Thus, even when the supply water pressure is lower than the breakthrough pressure, the water pressure at the wetting front can exceed the breakthrough pressure. From this fact and the relationships shown in Fig. 5, we can expect a rapid increase in the infiltration rate, and hence cumulative infiltration, for the 0.4-mm glass beads. The cumulative infiltration for the 30- to 50-mesh sea sand, however, should not change similarly in time as for the glass beads since the relationship for the sea sand shows a linear increase in the infiltration rate with increasing supply water pressure.
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Fig. 5. Relationships between supply water pressure and initial infiltration rate. The initial infiltration rate is the time averaged rate during the first minute after the sample had contact with the supply water.
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DISCUSSION
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Breakthrough Pressure
While the breakthrough pressure was nearly the same as the advancing capillary pressure for both materials (Table 2), the breakthrough pressure was essentially the same as the water pressure at the inflection point on the initial wetting curve only for the 30- to 50-mesh sea sand (Fig. 3). For the 0.4-mm glass beads, the breakthrough pressure was higher (less negative) than the corresponding pressure at the inflection point, which indicates that the glass beads have a relatively large water pressure range over which supply water cannot enter instantaneously. Figure 3 shows a sharp transition from completely dry to saturation over a fairly small change in the water pressure for the 0.4-mm glass beads, and a smooth continuous variation in saturation as the water pressure increases for the 30- to 50-mesh sea sand. This difference may be attributed to differences in particle shape and size distribution for the materials shown in Fig. 1, since these properties determine the behavior of the water–air interface and the breakthrough pressure. Lu et al. (1995) proposed a model for the capillary force acting on pore-scale water–air interfaces, where the capillary force oscillates from a minimum at the largest throat to a maximum at the neck as the water–air interface advances, to include entrapping the particle ahead of the interface and making the pore saturated. Provided that the interfaces at individual pores behave in such a manner, a continuous macroscopic water–air interface (i.e., a saturated wetting front) should be formed. This saturated wetting front may cause the creation of a breakthrough pressure, which is not identical to a temporal and spatial average of individual pore capillary pressures. On the other hand, the breakthrough pressure for the 30- to 50-mesh sea sand, which is identical to the water pressure at the inflection point, may be considered to be a temporal and spatial average of individual pore capillary pressures. However, the exact behavior of the water–air interfaces and the capillary pressures in these porous media, and what causes the two media to behave differently, needs more study.
Expression for the Dynamic Water-Entry Pressure and its Application to Finger Width Estimation
Weitz et al. (1987) measured the dynamic capillary pressure Pc vs. the interface velocity for a water–oil interface in 0.5-mm-diameter glass beads and found that an empirical expression fitted the data well. For the water–air interface, where Pc is considered to be the dynamic water-entry pressure, the expression can be written as
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where 
/rp is the breakthrough pressure for the medium, is the surface tension of water, rp is some characteristic pore radius, 
and ß are fitting constants, and NCa is the capillary number, which is defined as the ratio of the viscous force to the interfacial force (e.g., Hoffman, 1975). Following Friedman (1999), the capillary number for a saturated wetting front in a porous medium can be written as
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where µ is the viscosity of water, q is the water flux, and 
s is the saturated volumetric water content. Using the saturated hydraulic conductivity Ks and the expressions for the water-entry and breakthrough pressures, and modifying the constant term to include pore-entry pressure fluctuations (DiCarlo and Blunt, 2000), we obtain
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where hc is the dynamic water-entry pressure head, hbr is the breakthrough pressure head, 
gives the size of the entry pressure fluctuations, and ' = (µKs/s)ß. This equation shows the relationship between the dynamic water-entry pressure scaled by the absolute value of the breakthrough pressure and the infiltration rate normalized by the saturated conductivity of the porous medium. Values of the fitted constants as reported by Weitz et al. (1987) for the water–oil interface in sintered glass beads 0.5 mm in diameter were 
300 and ß 0.5. For the same experimental setup, except for the interface velocity, Stokes et al. (1988) obtained a similarly good relationship between Pc and the interface velocity at low velocities, for which fitted values were 1.0 x 104 and ß 1 (DiCarlo and Blunt, 2000). Figure 6
shows the normalized relationship between the infiltration rate and the supply water pressure for our experiment, originally shown in Fig. 5. By fitting Eq. [3] to the measured values, fitted constants were determined for both materials. For the 0.4-mm glass beads, we obtained = 0.18, ' = 0.53 ( = 11), and ß = 0.30. However, for the 30- to 50-mesh sea sand, we found = 0.10, ' = 5.5 ( = 2.4 x 104), and ß = 0.79. Differences in the values of fitted constants between the 0.4-mm glass beads and the 30- to 50-mesh sea sand are relatively large, especially for the value. At present, we are unable to give a good explanation for these differences.
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Fig. 6. Normalized relationship between the supply water pressure and the initial infiltration rate, with the infiltration rate being normalized by the saturated hydraulic conductivity of the medium, and the supply water pressure scaled by the absolute value of the breakthrough pressure as shown in Table 2. Fitted lines are also shown for both materials.
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The above difference in the dynamic water-entry pressure between the two materials can cause pronounced differences in the formation of fingers. Figure 7
shows two-dimensional fingered flow patterns formed in the 0.4-mm glass beads and the 30- to 50-mesh sea sand for similar fine-over-coarse layered conditions in which glass beads of 0.05-mm particle diameter were used for the top layer. From the figure we can estimate the width of fully developed fingers, which is about 1 cm for the glass beads and 5 to 7 cm for the sea sand. After DiCarlo and Blunt (2000), an expression for the finger width (d) can be written as
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Fig. 7. Fingered flow formed in the 0.4-mm glass beads and the 0.30- to 0.60-mm (30–50 mesh) sea sand during infiltration for fine-over-coarse layered conditions. Width and height of the layers were about 30 cm.
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Assuming q/Ks = 0.05, which is almost the same flux as for the experiments, Eq. [4] gives d = 1.1 cm for the glass beads and d = 11.3 cm for the sea sand using the fitted values of hbr, ', ß, and for each material. For the 0.4-mm glass beads, the estimated value agrees well with the measured one. However, the estimated value for the 30- to 50-mesh sea sand was about two times larger than the measured value. The first term of the right-hand side of Eq. [4] expresses the dynamic component of the finger width, while the second term expresses the static component (DiCarlo and Blunt, 2000). Since the dynamic and static components were estimated to be 9.2 and 2.1 cm, respectively, the dynamic component is considered to be overestimated. While it is not clear why the dynamic component was overestimated for the 30- to 50-mesh sea sand, the assumption of having a saturated wetting front may not have been satisfied for the sea sand. These results suggest that the expressions for the dynamic water-entry pressure, Eq. [3], and the finger width, Eq. [4], are useful for studying the mechanisms of fingered flow, at least for the air-dried glass beads in which saturated wetting front is formed during infiltration.
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CONCLUSIONS
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Dynamic water-entry pressures, based on both breakthrough pressures and relationships between the supply water pressure and the initial infiltration rate, were measured and compared for air-dried glass beads having 0.4-mm mean particle size and 0.30- to 0.60-mm (30–50 mesh) sea sand. The following differences were observed:- The breakthrough pressure, which was almost identical to the advancing capillary pressure for both materials, corresponded to the water pressure at the inflection point of the initial wetting curve for the 30- to 50-mesh sea sand, but not for the 0.4-mm glass beads.
- The 0.4-mm glass beads showed a sharp jump in the initial infiltration rate when the supply water pressure exceeded the breakthrough pressure, whereas the 30- to 50-mesh sea sand showed an almost linear increase.
The differences between the breakthrough pressures and the inflection points on the initial wetting curve were discussed in terms of the oscillatory nature of the water-entry pressure in air-dried glass beads. An expression for the dynamic water-entry pressure (after Weitz et al., 1987) was fitted to the experimental data. Following DiCarlo and Blunt (2000), fully developed finger widths were estimated for the infiltration experiments and compared with the experimental results; they showed good agreement for the 0.4-mm glass beads.
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REFERENCES
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- Baker, R.S., and D. Hillel. 1990. Laboratory tests of a theory of fingering during infiltration into layered soils. Soil Sci. Soc. Am. J. 54:20–30.[Abstract/Free Full Text]
- DiCarlo, D.A., and M.J. Blunt. 2000. Determination of finger shape using the dynamic capillary pressure. Water Resour. Res. 36:2781–2785.
- Friedman, S.P. 1999. Dynamic contact angle explanation of flow rate-dependent saturation-pressure relationships during transient liquid flow in unsaturated porous media. J. Adhes. Sci. Technol. 13:1495–1518.
- Geiger, S.L., and D.S. Durnford. 2000. Infiltration in homogeneous sands and a mechanistic model of unstable flow. Soil Sci. Soc. Am. J. 64:460–469.[Abstract/Free Full Text]
- Glass, R.J., J.-Y. Parlange, and T.S. Steenhuis. 1989a. Wetting front instability: 1. Theoretical discussion and dimensional analysis. Water Resour. Res. 25:1187–1194.
- Glass, R.J., T.S. Steenhuis, and J.-Y. Parlange. 1989b. Wetting front instability: 2. Experimental determination of relationships between system parameters and two-dimensional unstable flow field behavior in initially dry porous media. Water Resour. Res. 25:1195–1207.
- Glass, R.J., T.S. Steenhuis, and J.-Y. Parlange. 1989c. Mechanism for finger persistence in homogeneous, unsaturated porous media: Theory and verification. Soil Sci. 148:60–70.
- Green, W.H., and G.A. Ampt. 1911. Studies on soil physics. Part 1. The flow of air and water through soils. J. Agric. Sci. 4:1–24.
- Hillel, D., and R.S. Baker. 1988. A descriptive theory of fingering during infiltration into layered soils. Soil Sci. 146:51–56.
- Hoffman, R.L. 1975. A study of the advancing interface. 1. Interface shape in liquid-gas system. J. Colloid Interface Sci. 50:228–241.
- Idesawa, S., and T. Annaka. 1996. Growth of fingers due to wetting front instability and the mechanism of finger persistence. 2. Physical features of water penetration into packed glass beads and water entry threshold. Trans. JSIDRE 186:1–6.
- Lu, T.X., D.R. Nielsen, and J.W. Biggar. 1995. Water movement in glass bead porous media. 3. Theoretical analysis of capillary rise into initially dry media. Water Resour. Res. 31:11–18.
- Nakano, M., T. Miyazaki, S. Shiozawa, and T. Nishimura. 1995. Physical and environmental analysis of soils. University of Tokyo Press, Tokyo.
- Nieber, J.L. 1996. Modeling finger development and persistence in initially dry porous media. Geoderma 70:207–229.[ISI]
- Parlange, J.-Y., and D.E. Hill. 1976. Theoretical analysis of wetting front instability in soils. Soil Sci. 122:236–239.
- Philip, J.R. 1975. Stability analysis of infiltration. Soil Sci. Soc. Am. Proc. 39:1042–1049.
- Raats, P.A.C. 1973. Unstable wetting fronts in uniform and nonuniform soils. Soil Sci. Soc. Am. Proc. 37:681–685.
- Selker, J., J.-Y. Parlange, and T.S. Steenhuis. 1992. Fingered flow in two dimensions: 2. Predicting finger moisture profile. Water Resour. Res. 28:2523–2528.
- Stokes, J.P., A.P. Kushnick, and M.O. Robbins. 1988. Interface dynamics in porous media: A random-field description. Phys. Rev. Lett. 60:1386–1389.[Medline]
- Wang, Z., L. Wu, and Q.J. Wu. 2000. Water-entry value as an alternative indicator of soil water-repellency and wettability. J. Hydrol. (Amsterdam) 231–232:76–83.
- Weitz, D.A., J.P. Stokes, R.C. Ball, and A.P. Kushnick. 1987. Dynamic capillary pressure in porous media: Origin of the viscous-fingering length scale. Phys. Rev. Lett. 59:2967–2970.[Medline]
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ABSTRACT
INTRODUCTION
