The role of contact angle on unstable flow formation during infiltration and drainage in wettable porous media


  • Rony Wallach,

    Corresponding author
    1. R.H. Smith Faculty of Agriculture Food and Environment Hebrew University of Jerusalem, Rehovot, Israel
    • Corresponding author: R. Wallach, R.H. Smith Faculty of Agriculture, Food and Environment, Hebrew University of Jerusalem, Hertzel St., Rehovot 76100, Israel. (

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  • Michal Margolis,

    1. R.H. Smith Faculty of Agriculture Food and Environment Hebrew University of Jerusalem, Rehovot, Israel
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  • Ellen R. Graber

    1. Institute of Soil, Water, and Environmental Sciences Volcani Center, Agricultural Research Organization, Bet Dagan, Israel
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[1] The impact of contact angle on 2-D spatial and temporal water-content distribution during infiltration and drainage was experimentally studied. The 0.3–0.5 mm fraction of a quartz dune sand was treated and turned subcritically repellent (contact angle of 33°, 48°, 56°, and 75° for S33, S48, S56, and S75, respectively). The media were packed uniformly in transparent flow chambers and water was supplied to the surface as a point source at different rates (1–20 ml/min). A sequence of gray-value images was taken by CCD camera during infiltration and subsequent drainage; gray values were converted to volumetric water content by water volume balance. Narrow and long plumes with water accumulation behind the downward moving wetting front (tip) and negative water gradient above it (tail) developed in the S56 and S75 media during infiltration at lower water application rates. The plumes became bulbous with spatially uniform water-content distribution as water application rates increased. All plumes in these media propagated downward at a constant rate during infiltration and did not change their shape during drainage. In contrast, regular plume shapes were observed in the S33 and S48 media at all flow rates, and drainage profiles were nonmonotonic with a transition plane at the depth that water reached during infiltration. Given that the studied media have similar pore-size distributions, the conclusion is that imbibition hindered by the nonzero contact angle induced pressure buildup at the wetting front (dynamic water-entry value) that controlled the plume shape and internal water-content distribution during infiltration and drainage.

1. Introduction

[2] Proper description of the movement of infiltrating fluids through the vadose zone is essential to understand the processes involved in root zone/ground water recharge, subsurface transport of solutes, remediation of contaminants, and efficient irrigation practices. Infiltration of water into wettable porous media can be either stable or unstable. During stable infiltration into a homogeneous porous medium at a steady surface flux lower than the saturated hydraulic conductivity of the medium, a planar wetting front propagates with a downward negative water-content gradient. In unstable infiltration, a nonplanar wetting front and preferential flow paths (fingers) are formed [Peck, 1965; Hill and Parlange, 1972; Glass et al., 1989a; Selker et al., 1992a; Kapoor, 1996; Yao and Hendrickx, 1996]. Once such fingers develop, the flow area is significantly reduced relative to the planar wetting front, and pore velocities increase. This increase in pore velocity can greatly decrease the residence time of solutes in the vadose zone, thus accelerating the rate at which they arrive at the water table. This type of preferential flow has been called gravity-driven fingering, as it is a result of an instability driven by the relatively greater density of water compared with that of the air it displaces. Despite many theoretical and experimental investigations, it remains difficult to predict when infiltration is likely to be unstable and produce preferential flow paths, or to be stable and laterally uniform.

[3] Numerous experimental studies examining the conditions in porous media under which unstable flow develops have been made over the last 50 years, particularly in soils where water and air have been used as displacing and displaced fluids, respectively. Reviews of experimental and theoretical studies concerned principally with unstable flow in unsaturated soils were given by Glass and Nicholl [1996] and de Rooij [2000]. Various parameters and processes at both micro and macroscopic scales are thought to influence the onset of unstable flow, as well as to dictate the development of flow fingers and to influence their dimensions. The ratio between the densities and viscosities of the displacing and displaced fluids, the pore-size distribution of the porous medium, entry values of the displacing and displaced fluids, and the flux of the displacing fluid are among the parameters most widely evaluated. However, a basic question regarding processes at the wetting front that lead to unstable flow with nonmonotonic water-content distribution in a uniform wettable porous media still remains open. Shiozawa and Fujimaki [2004] noted that “the fundamental reason why extremely nonuniform water flows can occur in uniform, hydrophilic media remains unclear, and the occurrence of gravity-induced finger flow is still a big mystery, at least from the point of view of the modern theory of unsaturated water flow in porous media.” Similarly, DiCarlo [2010] commented that “the imbibition step is the crucial step that determines whether or not the infiltration is monotonic. If one can understand the physics of the imbibition that causes (or does not cause) the overshoot, one will understand the whole infiltration pattern.”

[4] A parameter that has been frequently overlooked, particularly in studies on wetting and drainage of unsaturated soils, is the intrinsic contact angle (θ) between the water and soil particles. Generally, it is implicitly assumed that the contact angle of wettable soils is sufficiently close to zero that it can be neglected; this follows from the fact that contact angle is not routinely determined together with other soil hydraulic characteristics. Nevertheless, cases in which contact angle was measured revealed that soils frequently show varying degrees of wettability. Selker and Schroth [1998] noted that the assumption of zero water-solid contact angle is often violated, even in well-cleaned silica sands. Taylor et al. [2000] reported that water partially wets limestones and Culligan et al. [2005] found that cos θ < 1 for water and quartz sand. Aminzadeh et al. [2011] measured an apparent contact angle of 30° for the 20/30 sand that has been widely used for slab and column studies where unstable flow has been frequently observed. Marshall and Holmes [1979] speculated that “contact angles greater than zero probably affect the advance of water quite commonly during the wetting of dry soil, although the effects are easily observed only in severe cases of water repellence.” Soils for which a nonzero contact angle (0 < θ < 90°) was observed have been termed “subcritically repellent” [Tillman et al., 1989; Hallet et al., 2001]. Lamparter et al. [2006], who studied the effect of subcritical repellency in a sandy soil on water infiltration and preferential flow formation, found that water infiltration rates were reduced due to subcritical repellency by a factor of 3–170 compared with the rate of infiltration of ethanol (θ assumed to be 0°). Although preferential flow was not observed on their experimental scale (infiltrometer), such a flow regime was observed in a different tracer experiment at the same site, where a 10 m transect was examined under natural rainfall conditions [Deurer et al., 2001].

[5] Utilizing both 2-D flow chamber and capillary rise methods, we aimed at examining the effect of nonzero contact angle (0 < θ < 90°) on 2-D water-content distribution during point-source infiltration and subsequent drainage, and its effect on unstable flow formation. Moreover, we examined the effect of water flux (water application rate) on water-content distribution, plume shape, and development of unstable flow, with particular attention to the mutual interaction between static contact angle and flow rate in the subsequent development of the flow regime.

2. Materials and Methods

2.1. Preparation of Treated Silica Particles

[6] Quartz dune sand (0.3–0.5 mm fraction) was exhaustively cleaned of organic and inorganic contaminants as follows: (i) soaking 45 min in 30% H2O2; (ii) rinsing in deionized water (DW); (iii) soaking 30 min in 1:1 HCl:Methanol; (iv) rinsing in DW; (v) soaking 30 min in concentrated H2SO4; (vi) rinsing in DW; and (vii) drying at 105°C for 24 h. This sand (denoted S33) was stored in hermetically sealed jars until use, and also served as the basis for the four model substrates: (1) S33—According to Cras et al. [1999], concentrated H2SO4 soaking gives maximal hydrophilic sites on quartz, giving it the lowest possible contact angle. Since such clean quartz surfaces are readily contaminated by adsorption of organic molecules from the air during use, S33 underwent the full exhaustive cleaning procedure following each use. (2) S48—Following Lamb and Furlong [1982], fully hydroxylated, and therefore hydrophilic, quartz surfaces are rendered progressively more hydrophobic by condensation of surface hydroxyls to form siloxanes when the quartz is treated by heat. S48 was prepared by heating S33 to 450°C for 4 h. (3) S56—Fe-oxides on S48 were removed by a solution of tri-sodium citrate dihydrate (Na3C6H5O7−2H2O) and sodium hydrosulfite (Na2O4S2) in DW (following standard methods for removal of iron oxides from soil). The iron-oxide free sand was then heated at 1000°C for 4 h. (4) S75—Clean quartz sand (S33) surfaces were silanized following Hórvölgyi et al. [1996] using trimethylsilyl N,N-dimethylcarbamate (Me3SiC; Sigma-Aldrich, 98.0% purity). The sand was soaked in 1 M HCl for 80 min, and then rinsed well in DW, followed by a rinse in acetone and drying at 50°C for 24 h. Following this, the sand surfaces were silanized in a solution of Me3SiC (0.02% by volume) in n-hexane for 30 min. Silylation was arrested by adding an equal volume of ethanol, followed by three rinses in acetone, one rinse in n-hexane, and drying at room temperature. All substrates were stored in the laboratory under ambient conditions in sealed jars until use.

2.2. Capillary Rise Experiments and Contact Angle Measurements

[7] Packed glass columns (diameter 1.6 cm) were suspended from the bottom of an XT 6200C Precisa scale connected to a computer into an open Erlenmeyer flask containing the imbibing liquid. The weight of the columns during imbibition was recorded at set time intervals by the computer program Balint (V4.00). A cotton cloth fastened with twine served as a filter at the base of the column. A perforated piece of parafilm covered the top end of the column, reducing evaporation. A Mariotte bottle, connected through a rubber tube to the Erlenmeyer flask prevented the level of the imbibing liquid in the flask from decreasing as a result of infiltration into the column through capillary rise. The quartz grain packed glass column was lowered into Erlenmeyer flask containing the imbibing liquid until contact was made between the surface of the liquid and the base of the column, initiating the weight loss measurements. Two imbibing liquids were employed: double distilled water with a surface tension of 72.8 mN/m at 25°C (confirmed for each batch of water before each experiment), and n-dodecane (Sigma-Aldrich, 99%), with a surface tension of 25.4 mN/m at 25°C (also confirmed before use). Capillary rise experiments were repeated for each medium three to four times. Experiments were conducted in a temperature controlled room held at 25 ± 2°C.

[8] Both the Washburn equation

display math(1)

and Jurin law (the rise/fall of the liquid stops when the hydrostatic pressure balances that due to surface tension [Jurin, 1718]

display math(2)

were used to determine contact angle from the capillary rise experiments, where h(t) was the wetting front height, heq the maximum water front height (t → ∞), r the representative pore radius of the porous medium (usually defined by an auxiliary capillary rise experiment with a fluid of low surface tension and zero contact angle), σ the liquid surface tension, θ the contact angle between the liquid and the solid, and µ the liquid viscosity. The Washburn equation corresponds with the dynamic phase of capillary rise, where the wetting front height is a function of the square root of time. The Jurin equation refers to the equilibrium state, when wetting height does not change as a function of time.

[9] Due to the difficulty of directly measuring the wetting front height, weight measurements were used to indirectly derive this parameter, according to the equation:

display math(3)

where ε is the porosity of the glass beads, R the column radius, g the gravity coefficient, and h the wetting front height in the glass beads. Porosity was calculated according to the measured height of the sand in the glass columns and the known weight of sand added, and was determined to be 38.8%.

[10] Since θ and r are unknowns in the both the Washburn and Jurin equations, prior knowledge of r is necessary to calculate θ. Solving for the capillary radius, assumed to be equal for all four porous media, entailed performing capillary rise experiments with S33 using n-dodecane as the (assumed) totally wetting fluid (θ = 0), following Siebold et al. [1997]. By substituting the calculated radius, r, back into the Washburn and Jurin equations, contact angle was calculated from capillary rise experiments performed using water as the imbibing liquid. The Washburn approach (equation (1)) rendered an effective radius (r) of 14 µm, which gives a ratio between grain diameter to effective pore radius of 29 (grain diameter distribution was between 300 and 500 µm, such that average grain diameter was assumed to be 400 µm). The Jurin equation (equation (2)) rendered an r of 66 µm, which yields a grain diameter to effective pore radius ratio of 6.1. This value is very close to the theoretical value of 6.9 calculated by Glover and Walker [2009] for uncompacted spherical beads of uniform size, and as such, the Jurin approach was adopted for contact angle determinations.

2.3. Flow-Chamber Runs

[11] The experimental setup consisted of a transparent flow chamber, 25 cm high, 15 cm wide, and 0.5 cm thick (inner dimensions), packed uniformly prior to each run with the prepared substrates of different contact angles, and using a CCD camera to record the wetting and subsequent drainage processes. Two 500 W commercial halogen projectors were used to illuminate the flow chamber. A uniform highly diffuse light intensity on the chamber's frontal face ensured maximum distinction (in terms of gray values) between the dry and wetted soil.

[12] The flow cell was packed uniformly by pouring the media into the top of the cell and tapping it lightly 30 times. The chamber was then fixed to a wall-mounted clamp such that the cell could be adjusted to the same position for each experiment and leveled vertically and horizontally using a spirit level.

[13] A high-resolution, high-speed monochrome camera (PULNiX TM-2016-8, Sunnyvale, CA) was used to quantify light reflection from the flow chamber at a resolution of 1920 × 1080 active pixels with dimensions of 7.4 × 7.4 µm, whereby 2916 pixels represented 1 cm2 of the soil profile. The frames from the camera were loaded into a PC using a frame grabber (ALACRON® Fastmotion by Alacron Inc., NH, USA).

[14] Water was applied to the soil surface as a point source at six flow rates: 1.0, 3.0, 5.0, 10, 15, and 20 ml/min. The supply pipe was connected to a peristaltic pump (Masterflux L/S, Cole Parmer Instrument Company, Vernon Hills, IL). The pipe touched the soil surface to prevent the effect of drop kinetic energy on water penetration. The total water volume added to the soil surface for all runs was 5 ml (the first sets of runs were made with water volume of 3 ml, and then was changed into 5 ml to get a better view of the wetting and drainage process). Following the wetting stage (9 s to 5 min, depending on the water application rate), water was allowed to redistribute for an additional 30 min. Repeated experiments were performed for all runs (with the exception of 5 ml for S56, where only the rate of 1 ml/min was repeated). An image of the chamber filled with dry soil was taken prior to water application to be used as a reference in the image-analysis process.

[15] Digital analysis of the pictures was performed by ImageJ software (Wayne Rasband, National Institutes of Health, Bethesda, MD, Java 1.3.1_03). The input to the ImageJ was a matrix of 1920 × 1080 gray values for each image. To eliminate the effects of minor deviations from uniform illumination distribution and of spatial variations in the soil properties in the chamber, as well as of nonuniform packing, the gray-value matrix of the air-dried soil was subtracted from gray-value matrix of each image.

[16] Given that the relationship between the gray value and volumetric moisture content is linear [Wallach and Jortzick, 2008], the slope of the linear calibration curve was determined by the mass (water) balance. This slope was calculated as the ratio between the sum of change in water content at each pixel at a certain time (being the water volume added to the flow chamber up to this time) and the change in gray value at every pixel summed over the entire flow chamber. As the initial condition (initial dry image) was subtracted from subsequent images, it was assumed that the calibration line emanates from the axis origin. The slope for each run was determined as an average of the slopes for each image (taken at 1 min intervals for the low flow rate and every 30 s for the higher flow rates). Data from the flow-chamber experiment were analyzed by the “Image Processing Toolbox” of Matlab (MathWorks).

[17] The images of water-content distribution in the flow chamber were saved at different time intervals depending on the water application rate. For the lower water application rates, 1, 3, and 5 ml/min, images were saved every 4 s during the first 10 min, every 6 s during the next 15 min (already within the drainage stage), and every 10 s during the last 25 min. For the higher water application rates, 10, 15, and 20 ml/min, images were saved every 1 s during the first 2 min, every 4 s during the next 10 min (already in the drainage stage), 6 s during the next 5 min, and every 10 s during the last period. Representative results are given in the figures.

2.4. Contact Angle

[18] The Jurin analysis of the capillary rise experiments rendered contact angles for the four media as follows: S33 ∼33°, S48 ∼48°, S56 ∼56°, and S75 ∼75°. These results are consistent with the expected trend in contact angle for the four media based on their surface chemistry. The result for S75 was also consistent with preliminary tests of the silylation method on flat quartz surfaces. In those tests, contact angle of the silylated flat quartz surface as measured by sessile drop method was 70°, which is very similar to that obtained by the Jurin equation for the silylized quartz grains (S75, 75°).

3. Results

3.1. Flow Chamber

3.1.1. Infiltration Stage

[19] The water-content distribution in the flow chamber of S33, the medium having the lowest contact angle (∼33°) of the four studied media, is shown in Figure 1. The images in the first and second rows depict plumes containing 2 and 5 ml water, respectively, for the different water application rates (Q = 1, 3, 5, 10, 15, and 20 ml/min) during the infiltration stage. The images in third and fourth rows of Figures 1 and 2 depict the plumes containing 5 ml water after drainage (redistribution) of 5 and 30 min, respectively.

Figure 1.

Water-content distribution in the flow chamber for S33 medium at different water application rates. Rows 1 and 2 are for the wetting stage and rows 3 and 4 for the subsequent drainage stage. The first row is for 2 ml water in the chamber, the second row for 5 ml water in the chamber, the third row is for 5 min drainage and the fourth row for 30 min drainage.

Figure 2.

Water-content distribution in the flow chamber for S75 medium at different water application rates. Rows 1 and 2 are for the wetting stage and rows 3 and 4 for the subsequent drainage stage. The first row is for 2 ml water in the chamber, the second row for 5 ml water in the chamber, the third row is for 5 min drainage and the fourth row for 30 min drainage.

[20] Plumes in the S33 medium have uniform shapes (Figure 1) at all water application rates, as commonly obtained for a point water source in regular soils (θ → 0°). However, at low water application rates, the plumes are somewhat elongated in shape (prolate), while at higher rates, the shape becomes more bulbous (oblate). This can be seen very clearly at the 5 ml stage of wetting (Figure 1). Water application rate also affected plume size, with a smaller area and higher average moisture content being associated with higher water application rates (Figure 1). Moreover, at higher water application rates, the internal water-content distribution during the wetting stage (first and second rows in Figure 1), was relatively more uniform than at lower application rates.

[21] These same plume shape and spatial water-content distribution features as a function of water application rate can be seen in a particularly distinctive fashion in the plumes which developed in S75 (Figure 2), the medium with the highest contact angle (∼75°). Under low water application rates, the plumes in S75 are very thin and elongated; their shape becomes increasingly regular and bulbous as water application rates increases. While plumes in the S33 medium have regular shapes at every water application rate (Figure 1), irregular and tortuous shapes were obtained in the S75 medium, especially at the lowest rate of water application (Figure 2). Furthermore, in the S75 medium, the water-content distribution is distinctly nonuniform; water accumulates behind the wetting front and its gradient upward is negative. The negative water gradient is observed visually by the color gradients in the three leftmost images (Q of 1, 3, and 5 ml/min) in rows 1 and 2 (wetting stages) of Figure 2. The negative gradient is recognized as the trend in color from dark blue (higher water content) immediately behind the wetting front toward lighter blue and green (lower water content) in the direction of the upper surface of the chamber. Water accumulation behind the wetting front (tip) and a negative water gradient above it (tail) has been termed “saturation overshoot” in studies on unstable flow in soils typified by formation of flow fingers [Glass, 1989b; DiCarlo, 2004]. Overshoot is difficult to discern at high water application rates (Q of 10, 15, and 20 ml/min) in the S75 medium during the wetting stage (rows 1 and 2 of Figure 2), and at all water application rates, low and high, in the S33 medium (rows 1 and 2, Figure 1). The significant difference in plume shape developed at low water application rates in the two media, whose singular distinctive feature is their different contact angles (S33 ∼33° and S75 ∼75°), is attributed to the effect of contact angle on water imbibitions.

[22] To enable a more quantitative examination of water distribution in the plumes, water-content data at 5 min infiltration (solid lines, Figure 3) and 30 min redistribution (dotted lines, Figure 3) was plotted along a vertical cross section through the middle of the plumes (starting below the point water source) for the 1 and 10 ml/min water application rates. The inherently noisy data was slightly smoothed by exponential smoothing with damping factor of 0.7. The nonmonotonous water-content distribution with higher values at the bottom (termed saturation overshoot) during water infiltration (solid lines, Figure 3) could be recognized in both S33 (Figures 3a and 3b) and S75 media (Figures 3c and 3d) at both water application rates. The zigzag vertical water-content pattern at the upper part of the plume in Figure 3c is a result of the tortuous plume shape (note that the vertical cross section passes through dry and wet parts of the medium; Q = 1 ml/min, Figure 2).

Figure 3.

Water-content distribution along a vertical cross section underneath the point water source at the end of 5 min wetting (solid lines) and 30 min drainage (dotted lines) for (a) S100 1 ml/min, (b) S33 10 ml/min, (c) S75 1 ml/min, and (d) S75 10 ml/min.

[23] The average moisture content behind the wetting front (tip) was greater in the high contact angle medium (S75) than the low contact angle medium (S33; Figure 3). In S33, greater Q was also associated with higher water content at the tip. Like the overall length of the plumes (Figures 1 and 2), tip length was greater at the lower water application rate. The moisture content in the S33 medium is constant from the wetting front and up throughout most of the plume length (tip) and decreases only at the vicinity of the medium surface (tail) for the two water application rates (Figures 3a and 3b). A water-content decrease near the surface does not take place in the S75 medium (Figure 3d), indicating that the tip expands throughout the plume length in this medium. Note that the ratio between tip and tail lengths in Figure 3 is specific to the water volume within the plumes (5 ml in the current runs) and water application rate. Longer tails are apparently associated with longer water application periods that induce deeper wetting fronts (e.g., Figure 3c).

[24] Water-content data at 5 ml infiltration (solid line) and 30 min redistribution (dotted line) was similarly smoothed and plotted for horizontal cross sections at one-third the final plume depth for the infiltration stage for the 1 and 10 ml/min water application rates (Figure 4). The horizontal moisture content distribution during wetting (solid lines in Figure 4) was nearly invariant across the plumes, with plume edges being denoted by sharp drops in water content. The abrupt change between wet and dry porous media along the horizontal fronts is similar to that seen at the front of the downward moving plume in the vertical profiles (Figure 3).

Figure 4.

Water-content distribution along a horizontal cross section located about 1/3 of the final plume length from the surface at the end of the infiltration stage (solid lines) and 30 min drainage period (dotted lines) for (a) S33 1 ml/min, (b) S33 10 ml/min, (c) S75 1 ml/min, and (d) S75 10 ml/min.

[25] The plumes and internal moisture content distribution in S48 (CA ∼48o) and S56 (CA ∼56o) media at the end of the infiltration stage (5 ml) are depicted in Figure 5 (first and second rows, respectively), for the different water application rates. In general, the shapes of the plumes and internal water-content distribution in S48 medium are similar to those obtained in S33 (Figures 5 and 1, respectively), and in S56 to those in S75 (Figures 5 and 2, respectively) at the same respective water application rates.

Figure 5.

Water-content distribution in flow chamber for S48 (first and third rows) and S56 (second and fourth rows) media at different water application rates at the end of the infiltration stage and drainage stage, respectively. The water volume in the flow chamber for all cases is 5 ml.

3.1.2. Drainage Stage

[26] Following the active wetting stage, plume development and water redistribution was followed during free drainage for 30 min. The third and fourth rows in Figures 1 and 2 depict the shape and size of the plumes in the S33 and S75 media, respectively, as well as their internal water-content distribution throughout the drainage process. In S33, the plumes expanded laterally and vertically during drainage. The water-content profiles, which were comparatively uniform throughout the infiltration stage in S33 (solid lines in Figures 3a and 3b) along the vertical cross sections, became positive at their lower part and negative along their upper part after 30 min of drainage (dashed lines in Figures 3a and 3b). Note that the positive water-content gradient emanated from the depth that the wetting front had reached at the end of the wetting stage and continued downward from there (dashed lines in Figures 3a and 3b). Downward drainage was more pronounced in the plumes that developed under lower rates of water application (Figure 1 and dashed lines in Figures 3a and 3b). In contrast, the lateral expansion of the plumes through the course of drainage was greater in the plumes that had developed under higher water application rate regimes (Figure 1 and dashed lines in Figures 4a and 4b).

[27] Unlike the plumes in S33, neither the shapes of the plumes nor the lateral or vertical water distribution inside the plumes changed significantly in the S75 medium throughout the drainage period stage (third and fourth rows in Figure 2 and dashed lines in Figures 3c, d and 4c, d).

[28] Plumes and internal moisture content distribution for S48 and S56 media after 30 min drainage are depicted for the different water application rates in Figure 5 (third and fourth rows, respectively). As in the wetting stage, the drainage plumes of S48 are similar to those of S33, and the drainage plumes of S56 are similar to those of S75. The similarity can be seen by comparing plume shapes and water-content distribution in the third row of Figure 5 with the fourth row of Figure 1, and the fourth row of Figure 5 with the fourth row in Figure 2 at the same water application rates.

[29] The data presented in Figures 1-5 are representative of the many replicate flow-chamber experiments performed.

4. Discussion

[30] The outcome from the flow-chamber experiments is that plume shape, size, and internal water-content distribution are significantly affected by both the static contact angle between the water and porous media and by the water application rate. We claim that this effect can be mainly related to the soil water sorptivity [Philip, 1955; Parlange, 1971] and its role on the spontaneous and forced water flow in the pores at different contact angles.

[31] Parlange and Hill [1976] related the finger's dimension to capillarity by including the sorptivity in the linear stability analysis used to predict formation of fingers by instability. Adjusting the parameters in the Parlange and Hill [1976] equation to air/water system yielded [Glass and Nicholl, 1996]:

display math(4)

where dc is the minimum finger size, S is the sorptivity evaluated between θs, the saturated moisture content, and θi is the initial moisture content, qs is the flux through the system, Ks is the saturated conductivity of the porous medium, and a is a constant equal to π for a 2-D system and 4.8 for a 3-D system [Glass et al., 1991].

[32] The inclusion of sorptivity in an expression to calculate the dimensions of fingers calls for an insight into the dependence of sorptivity on contact angle. Philip [1969] introduced the term intrinsic sorptivity, SI, which is the consequence of viscous flow induced by capillarity, to distinguish between the intrinsic properties of the porous media and those that depend on the imbibed liquid, so that

display math(5)

[33] Note that SI is independent of the properties of the liquid (including the contact angle between the liquid and the porous medium). A tacit introduction of the contact angle into equation (5) usually has been made (following equations (1) and (2)) by multiplying σ by cos θ, yielding that sorptivity may be scaled by (cos θ)½. Philip [1971] noted that this scaling is based on the over-simplified assumption that the media consist of bundles of long cylindrical capillaries, and suggested that S should depend on the contact angle in a more complicated way. Philip [1971] demonstrated his reservations on how the internal geometry affects the wetting properties by using conical pores with semi-apical angle α (0 < α < π/2) and showing the susceptibility of apparent contact angle to α [Philip, 1971, Figure 2]. A deviation from S = f [(σ cos θ)½] in equation (5) was obtained by Yang et al. [1988] and Selker and Schroth [1998]. Yang et al. (1988) found that the rates of horizontal 1-D flow of liquids exhibiting finite contact angles deviate from the values calculated for known contact angles using equation (1). Selker and Schroth [1998] noticed an inconsistency between the measured cross section of fingers formed in a sandy medium for three liquids and different concentrations of anionic surfactant, and the scaled fingers' width (cos θ)½. These result reveal that the synergistic effect between the contact angle and pore geometry induce a substantial decrease in sorptivity for a moderate increase in apparent contact angle.

[34] As soil water sorptivity and the lateral and vertical wetting front propagation are driven mainly by capillarity, they depend on the capillary pressure difference across the water-air interface, ΔPc = (2σ/r)·cos θ (equation (2)). A ΔPc at which the wetting front spontaneously propagates in a given porous medium is termed water-entry value. However, if an additional pressure is needed beyond a pinning pressure which must be overcome before the interface can move, or to accelerate the wetting front beyond the spontaneous movement velocity, the additional pressure is termed “dynamic soil water-entry value (pressure)” [Geiger and Durndorf, 2000; Annaka and Hanayama, 2005]. A static contact angle turns dynamic upon meniscus movement or when a pressure is applied to a pinned meniscus. The dynamic contact angle between moving water and solid particles, θ, was found to be in correlation with the capillary number (Ca = ηv/σ, the ratio of viscous forces that depends on flow velocity v, to interfacial forces at the water-air interface) for a uniform capillary [Hoffman, 1975]. A shift factor to that correlation is associated with the static contact angle [Hoffman, 1975]. Hilpert and co-authors [Hilpert, 2010; Hsu and Hilpert, 2011; Pellichero et al., 2012] found that the dynamic contact angle affects the flow in uniform capillary tubes and the analogous Green and Ampt model [Green and Ampt, 1911] used to simulate infiltration in soils. Weitz et al. [1987], on the other hand, found that pressure increase, ΔPc, that should be introduced at the wetting front of a wetting fluid that displaces a nonwetting fluid in porous media beyond the pinning pressure scales as v1/2. This scaling deviates from the linear dependence of ΔPc on v predicted by Washburn equation (equation (1)) for uniform capillary tubes. Postulating that modeling flow in porous media by uniform capillary tubes area is an oversimplification. Philip [1971] and Wang et al. [2013] developed a mathematical model to study the role of geometry, inertia, and dynamic contact angle on wetting and dewetting of uniform and nonuniform (with sinusoidal modulations of the radius) capillaries of different contact angles (0 < θ < 90°). Simulations using the dimensionless model showed that the inertial and dynamic contact angle terms can be neglected for narrow capillaries of uniform cross section but not for uniform, wide cross-section capillaries. Moreover, nonuniformity in cross-sectional area induced hysteresis, deceleration, blocking, and metastable equilibrium locations. An increase in contact angle further amplified the effect of geometry on wetting and dewetting processes. Wang et al. [2013, Figures 7 and 8] indicate that ΔPc should increase (forming a dynamic water-entry pressure) to maintain a certain meniscus propagation velocity as capillary nonuniformity increases and wettability decreases. This conclusion compares favorably with the conclusion made by Weitz et al. [1987] that a strong disorder in the porous medium and nonwetting fluids plays a major role in causing the velocity dependence of ΔPc. The flow-chamber results, starting with infiltration and following with drainage, are discussed in the following in light of this conception.

4.1. Infiltration

[35] The sharp change in water content along the perimeter of the plumes (high value behind the wetting front and dry soil ahead of the wetting front (Figures 1-4) can be rationalized by a substantial decrease in media sorptivity by virtue of an apparent contact angle that is higher than the equilibrium contact angle [Philip, 1971]. Note that the sharp change in water content along the wetting front perimeter persisted during the drainage process in the S75 plumes (Figures 2 and 4) while it disappeared along the perimeter of the S33 plumes (Figures 1 and 4).

[36] Fürst et al. [2009] argued whether the equation that arises by inserting the Darcy Law into the continuity equation (Richards equation) can deal with capillarity effects that take place at the common boundary of all three phases, liquid (water), solid (the porous matrix), and gas (air), and disappear once any of the phases is removed. They note that averaging of the discontinuous nature of capillary effects may lead to a meaningless description. Being a parabolic partial differential equation, the diffusion-like behavior of the wetting profile predicted by the Richards equation contradicts the nonmonotonic moisture content profiles in fingered flow [Eliassi and Glass, 2001; Fürst et al., 2009; DiCarlo, 2010] and observed in the current study as well (Figures 2-5). The interdependence between the resistance to imbibition induced by the contact angle at the wetting front and both saturation overshoot and wetting front propagation (Figures 1 and 2), is analyzed by a virtual mass balance for a horizontal layer located behind the downward moving wetting front. The lower side of the layer coincides with the moving wetting front and its upper side is defined at the location where the water content steeply decreases. This layer is the finger tip. The thickness of the layer and the average water content are controlled by the difference between inward and outward fluxes to the layer. The inward flux to the layer is the vertical water flow in the plume (along the downward directed z axis) above it and is assumed to be driven mainly by gravity. The outflow flux is Δθ · Δzt, where Δz is the distance that the wetting front proceeded during Δt, and Δθ is the difference between the average water content in Δz after and prior (initial water content) to the wetting front propagation.

[37] At the early stage of infiltration into a dry medium whose contact angle approaches zero, the higher capillary driving force compared to the gravity tends to smear the wetting front. Note that the capillary pressure at which spontaneous wetting of this dry medium takes place is termed static water-entry value. The extent of wetting front smearing with depth depends on the structure of the porous medium; more extensive smearing occurs in fine media having lower water-entry values, whereas sharper wetting fronts are associated with coarser media having a relatively uniform particle size distribution and higher water-entry values. In later stages of infiltration, when capillary and gravity driving forces at the wetting front vicinity become similar to each other, the shape of the wetting front turns constant and it moves downward as a traveling-wave type flow. If the capillary driving force is reduced by a contact angle significantly different from zero (as in the current study), the gravity-driven inward flux to the layer will prevail over the capillary force-driven outward flux, the dynamic contact angle will decrease and the wetting front will be sharper with accumulation of water behind it. Given that the contact angle at the wetting front is an advancing one, while for the already wetted soil it is the receding one, and that the advancing contact angle is inherently higher than the receding one particularly on rough surfaces [Morrow, 1976], the imbalance between the outward and inward fluxes through the layer, and hence the water accumulation (saturation overshoot), increases as the equilibrium and further on the dynamic contact angles increase. Water accumulation behind the wetting front is inherently accompanied by a pressure buildup that forms a positive matric potential gradient above it, ∂h/∂z [DiCarlo et al., 1999; Geiger and Durnford, 2000]. The additional pressure exerts additional force on the sharp wetting front and enhances its downward and lateral movement beyond the movement that would be induced by the spontaneous water-entry value. The bulbous plume obtained for the higher application rate and higher contact angles indicates that the additional force on the wetting front is larger than the gravity force, at least during short infiltration periods (as in this study), otherwise elongated plumes should develop. Since the pressure that builds up behind the wetting front increases with water application rates, wider plumes are expected for higher water application rates in media of a given contact angle. By the same token, contact angle considerations mean that at a given water application rate in media of different contact angles, plumes in lower contact angle media will be wider than in higher contact angle media. Seeing that, at a given volume of water in our 2D experimental chamber setup, plume width and length are interrelated, shorter plumes are associated with higher water application rates and lower contact angles.

[38] The effect of flow rate on plume expansion vertically and laterally is depicted in Figures 6 and 7, respectively. The plumes length and width were evaluated along the cross sections used for Figures 3 and 4, respectively. From Figure 6, it can be seen that the rate of plume propagation downward in the S33 medium was initially high and then became constant, while the rate of plume propagation downward in the S75 medium was apparently constant throughout the infiltration stage. The saturation profiles in Figure 3 and propagation patterns in Figure 6 may signify that the constant downward propagation rate was driven by the hanging water column formed by the saturation overshoot. For a given water application rate, higher downward propagation rates are associated with higher contact angles, and for a given contact angle, with higher water application rates (Figure 6). In contrast, in the lateral direction, the rate of plume expansion in the low contact angle medium is greater than in the high contact angle medium at all flow rates (with the exception of the highest flow rate, where the differences in lateral plume expansion rates between the two media are not significant) (Figure 7). The data thus correspond to the above analysis regarding the effect of contact angle on the resistance to imbibition and flow rate on saturation and pressure buildup behind the wetting front. Support for this conceptualization of the simultaneous effect of contact angle and water application rate on saturation overshoot is provided by DiCarlo [2010, Figures 2 and 3], where the effect of water application rate on both water content at the flow finger tip and the length of the tip in 1D columns were presented. Note that the measured contact angle of the 20/30 sand used in DiCarlo [2010] was 30° [Aminzadeh et al., 2011].

Figure 6.

Plume propagation downward in the S33 and S75 media at different water application rates.

Figure 7.

Plume expansion along a horizontal cross section located at about one-third of the final plume length from the surface in the S33 and S75 media at different water application rates.

[39] Glass et al. [1989b] and Nicholl et al. [1994] postulated that the length of the saturation overshoot layer is the difference between the water and air-entry values determined from hysteretic water retention curves. The head loss by viscous dissipation as finger propagates was not included in this postulation. While the wetting front moves downward, the matric potential at a given point within the plume tip continuously decreases until it reaches the air-entry value, at which time major pores begin to empty according to the drainage scanning pressure saturation curve [Selker et al., 1992b]. Since the water-entry value at the wetting front, ΔPc, is dynamic rather than static (singled valued) and depends on the imbibition velocity [Weitz et al., 1987; Geiger and Durndorf, 2000, Annaka and Hanayama, 2005], lower capillary pressure/higher water-entry values are needed to wet a given porous media at higher imbibition velocity. Considering both the dynamic water-entry value at the wetting front and the above concept for the dependence of the length of saturation overshoot length on the water-entry value, it can be understood that longer tips will be associated with higher wetting front velocities. Indeed, DiCarlo [2004] measured longer tip lengths and higher moisture content behind the wetting front at higher infiltration rates in 1-D experiments. The tip length therefore cannot be a priori determined by measured characteristic curves (wetting and drainage scans) for subcritical repellent media as it depends on external factors as well (e.g., water application rate that affect the imbibition rate). Note that the uniform moisture content distribution along a vertical cross section for the plumes obtained for the high water application rates are probably too short to comprise an entire saturation overshoot profile (tip and tail) as in for the higher application rates in first and second rows in Figures 1 and 2. This implies that the entire plume is a tip of a yet undeveloped saturation overshoot profile (Figures 3b and 3d). It is presumed that a tail would eventually develop were these plumes to elongate further, as seen in the narrow plumes associated with the low water application rates (Figures 3a and 3c) and lower propagation rates (Figure 6a), that induce higher dynamic water-entry (capillary pressure) values at the wetting front and an a correction for the viscous dissipation of head due to permeability at the flow rate of the finger.

[40] As discussed above for the downward moving wetting front, a sharp decrease in water content from saturation behind the wetting front to initially dry medium beyond it in the horizontal direction (Figures 4b–4d) indicates that the pressure behind the wetting front is likely higher than the spontaneous (static) water-entry value (dynamic water-entry value). For a given contact angle, the high pressure behind the side wetting front forms a dynamic water-entry value that increases the rate at which the plume expands compared to the rate that would be obtained by the spontaneous water-entry value. Indeed, the plume's lateral expansion versus time at 3 cm below the surface (the depth where the water-content distribution in Figure 4 were quantified) depicted in Figure 7 indicates that the plume expansion rate increases as water application rate increases and contact angle decreases. The bulbous-shaped plumes obtained for all four media at high water application rates (15 and 20 ml/min) (Figures 1, 2, and 5) can be explained by the supposition above. Moreover, the second row of Figures 1 and 2 and the first and second rows of Figure 5 illustrate nicely the role of water application rate on the water content behind the side wetting front and the plume width for each studied medium (contact angle), namely, the plume width increases with water application rate owing to the associated pressure increase. It is apparent from these figures that for the low water application rate in S33 and S48, the water content at the plume sides is lower than at its core leading to a less bulbous-like shape. Note that at low water application rates, the pressure behind the downward moving wetting front that is induced by the tip length is higher than the pressure at the side wetting front, which brings about longer and narrower plumes.

4.1.1. Drainage Stage

[41] Youngs [1958a, 1958b] showed that the shape of the soil moisture profile during drainage in one-dimensional flow is not necessarily the same as that of infiltration. Subsequently, Youngs and Poulovassilis [1976] identified two forms of drainage profile. In the first, the moisture profile shape remains similar to that of infiltration, maintaining the highest water content at the soil surface and the lowest at the wetting front. None of the experiments performed here exhibited this form of drainage profile. However, water-content variation during drainage in S33 and S48 media, where a positive water-content gradient was developed at the upper part of the plume and a negative gradient at the lower part of the plume, does recall the second profile form discussed in Youngs and Poulovassilis [1976]. Note that positive z points downward. These drainage profiles follow Peck [1971] who conceptualized that at each depth during drainage, moisture content increases to a maximum and then decreases. Thus, when the maximum water-content value is at z = z* (the transition plan), the soil in the upper zone 0 ≤ z ≤ z* is drying, and in the lower region z > z*, wetting, forming a positive water-content gradient above z = z* and a negative water-content gradient below it. The z = z* in the S33 and S48 media is the location of the wetting front at the end of infiltration.

[42] Water-content distributions in the S56 and S75 media, having the highest value of moisture content at the moving wetting front and lower values above it, correspond to yet a third drainage form which is similar to that measured by Youngs [1958b] in glass beads. For air to enter at the surface, the length of the wetted zone must be larger than the difference between the water and air-entry values of the media. According to Youngs [1958b], if the length of the wetted zone is sufficiently small, water will not move (remain “frozen”) near the soil surface. Seeing that the only difference between the four media used herein was indeed their contact angle, it can be concluded that the differences in drainage form are due to the effect of contact angle on water- and air-entry values.

[43] Nicholl et al. [1994] demonstrated how the interplay between water- and air-entry values and flux affect the stability of the movement of slugs in a natural fracture. Their study supports the role of the difference between water- and air-entry values in the formation of profiles with unchanged geometry obtained for S56 and S75 media. The fate of the profile of unchanged geometry and internal water distribution during drainage is similar to that of a falling slug in a vertical capillary tube. The difference between water and air-entry values is analogous to the case where the front meniscus of the slug is flatter than the rear one. Two effects contribute to this effect (i) an advancing contact angle (θa) sets up at the front, which flattens the meniscus and (ii) the film left behind reduces the radius of the rear meniscus forming a receding contact angle (θr). The difference in Laplace pressures between the top and rear menisci do not compensate and a force opposing the motion is generated. The balance between capillary, gravitational, and viscous forces along the slug is [Bico and Quéré, 2001]

display math(6)

where ρ and η are the liquid density and viscosity, respectively, R is the tube radius and L is the slug length. Equation (6) indicates a slug/drainage profile will move downward if the driving force (the forward capillary force + gravity) will be higher than the holding force (capillary force at the tail + viscous friction). A slug length L is therefore needed to initiate flow at a constant velocity v, where both L and v depend on the difference between the advancing and receding contact angle, more practically on the advancing contact angle. The “freezing” of the plumes in S75 and S56 media indicates that a critical plume length L has not been reached by the amount of 5 ml that was added during the infiltration stage. On the contrary, the 5 ml water volume was sufficient to induce a draining plume movement in the S33 and S45 media where shorter lengths were needed due to their lower water-entry values. Or [2008] generalized the slug flow equation for unsaturated porous media using scaling relationships between the primary forces influencing flow in porous media (capillary, viscous, and gravitational) toward development of criteria for onset of unstable flows.

[44] While the shape of the plumes and internal water-content distribution were similar for all four media during the course of infiltration at higher water application rates independent of contact angle, differences between the media were evident upon drainage. Plumes spread laterally and vertically during drainage in the lower contact angle media while in the higher contact angle media, spreading in any direction was negligible. According to Weitz et al. [1987] and Geiger and Durnford [2000], spreading takes place as long as the dynamic water-entry value at the wetting front is higher than the static one, and wetting front propagation ceases when the two become equal. Thus, it can be concluded that the dynamic water-entry value at the wetting front in the higher contact angle media was probably close to the static value, and that once water application ceased, the pressure behind the wetting front decreased and the dynamic water-entry value approached the static one, freezing the plume shape. In contrast, the lateral and vertical spreading of the wetting front in the lower contact angle media following the cessation of water application indicates that the pressure at the wetting front was higher than the static water-entry value. With continued drainage, the dynamic water-entry value at the wetting front approached the static value, thus resulting in diminishing elongation and lateral expansion velocities with time. These effects disclose the effect of the contact angle on the water-entry value.

[45] The similarity between the plumes in S33 and S45 during the infiltration and drainage stages, on one hand, and the substantial difference between them and the plumes which developed in S56 and S75, on the other hand, may signify that the effect of the contact angle on the plume shape, size, and internal water-content distribution is not linear with θ or cos θ. This conclusion coincides with the analysis made by Philip [1971] and Ransohoff and Radke [1988] regarding the effect of the contact angle on the sorptivity and imbibition, respectively.

5. Conclusions

[46] The current experimental setup that compels a 2-D flow and a point water source of different application rates provides a unique tool to study the synergistic effect of the contact angle and water application rate on stable and unstable flow and finger formation during the infiltration and subsequent drainage stages. Given that the studied media have similar pore-size distributions, an evident conclusion is that the differences in the wetting and redistribution patterns are owing to the nonzero contact angles. At low water application rates, the water-solid phase contact angle has a significant impact on the shape of water plumes, their size, and their internal water-content distribution during infiltration and drainage stages. At high water application rates, the influence of the static and dynamic contact angles on the shape of the plumes and the internal water-content distribution is strongly diminished during the active infiltration stage, but in the drainage phase, contact angle again becomes a major determining factor. These effects are due to the balance between the pressure buildup behind the wetting front along the plume perimeter and the resistance to imbibition induced by the local dynamic contact angle. The enhanced pressure, (dynamic water-entry value) controls the propagation of the wetting front in general, and the downward moving wetting front induced by saturation overshoot in particular. The results of this study indicate that the dependence of the sorptivity on contact angle is not linear with (cos θ)½. The effect of the contact angle on the sorptivity and imbibition cannot be decoupled from the pressure buildup behind the wetting front that forms a dynamic water-entry value that is larger than the one determined by the wetting retention curve. This pressure build up sharpens the transition from wet to dry along the wetting front and affects its propagation rate. Note that the “classical” shape of the plumes and the internal water-content distribution in the S33 and S48 media for the infiltration stage could be interpreted as if changes in water flow are associated only with higher contact angles (56° and 75°). However, the nonclassical shapes and internal water-content distribution obtained for the drainage stage in S33 and S48 indicate that lower contact angles affect the flow pattern as well, although to a lesser extent than higher contact angles.

[47] The results of this study have practical as well as theoretical significance. For example, the dependence of the plume shape in general, and the ratios of the vertical to horizontal dimensions of the wetted soil in particular, on the contact angle and water application rate should be considered when a drip irrigation system is designed and managed. An imbedded assumption in the equations and models used for drip irrigation design is that the soil has a zero contact angle. These models are used inter alia to obtain a wetting strip along the laterals to prevent poor seed germination and increase irrigation uniformity. However, considering the increasing evidence that soils have contact angles higher than zero (but still wettable—subcritically repellent), the models' predictions may overestimate the radial wetting pattern at the soil surface and underestimate the wetting front depth. The result will be dry spots between adjacent drippers and extended deep percolation beyond the active root zone depth. This phenomenon has already been observed in a number of instances (personal experience). The distance between the drippers along the laterals and between laterals would therefore need to be adjusted to take into account the effect of the contact angle on water flow in the soil profile. This issue is even more critical in water repellent soils, such as those rendered water repellent by prolonged irrigation with effluents [Wallach et al., 2005]. Although the contact angle of these soils is initially larger than 90°, the soils become subcritically repellent as the contact angle decreases due to contact of the soil particles with water. The dynamics of water penetration into soils of differing water repellency degree was studied by Wallach and Jortzick [2008], Wallach [2010], and Xiong et al. [2011], and the results show plumes that are unstable and similar in a general manner to those obtained in the current study.


[48] This study was financed by the Binational Agricultural Research and Development Fund (BARD), project IS-3962-07. The authors thank the AE and anonymous reviewers for their useful comments.