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Published online 1 August 2007
Published in Vadose Zone J 6:436-445 (2007)
DOI: 10.2136/vzj2006.0060
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Modeling Water Movement in Heterogeneous Water-Repellent Soil: 1. Development of a Contact Angle–Dependent Water-Retention Model

J. Bachmanna,*, M. Deurerb and G. Aryec

a Institute of Soil Science, Univ. of Hannover, Herrenhaeuser Str. 2, 30419 Hannover, Germany
b Sustainable Land Use Group, HortResearch, Tennent Dr., P.O. Box 11030, Palmerston North, New Zealand
c Dep. of Environmental Sciences, Univ. of California, Riverside, CA 92521, USA
* Corresponding author (bachmann{at}ifbk.uni-hannover.de).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.


Received 19 April 2006.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Contact Angle on Soil...
 Wetting Behavior of Soil...
 A Contact Angle-Dependent Water...
 Conclusions
 REFERENCES
 
In the past, hydrophobic soils have been associated mostly with preferential flow phenomena. It has become increasingly evident that besides the phenomena of extremely (hydrophobic) water-repellent soils, soils with reduced wettability are more the rule than the exception. Despite the extensive literature on the hydraulic behavior of water-repellent soil, a conceptual model flexible enough to describe typical behavior of wettable and hydrophobic soils as well as for soils with intermediate wetting properties is still missing. We propose a water-content and time-dependent contact angle (CA) model that was used as an extension of the van Genuchten equation for the capillary pressure–saturation (CPS) relationship. This model is based on conventional retention parameterizations; that is, the hydrophilic soil is considered as a special case of the general wetting model. Hydrophobic soils as well as soils with subcritical (reduced wettability, not hydrophobic) water repellency are represented by this model. Conceptually, the proposed model links microscopic interfacial properties, indicated by the CA, with the macroscopic hydraulic model mainly by modifying the {alpha} of the van Genuchten equation. The modification basically accounts for hysteresis of the main drainage–main wetting branch of the CPS relationship. Compared with conventional hydraulic models, only a few more parameters are needed to describe the wettability extension of the model: mean maximum and minimum CAs and their autocorrelation functions. Additionally, characteristic rewetting time and a breakthrough pressure function are needed for a complete description of the hydraulic properties of the soil. This extended hydraulic model serves as a base for a simulation study in unsaturated soil with reduced wettability.

Abbreviations: CA, contact angle • CPS, capillary pressure–saturation • MED, molarity of an ethanol droplet test • REV, representative elementary volume • SOM, soil organic matter • WDPT, water drop penetration time • WDPTT, water drop penetration time test


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Contact Angle on Soil...
 Wetting Behavior of Soil...
 A Contact Angle-Dependent Water...
 Conclusions
 REFERENCES
 
Water movement in dry soil is the result of nonequilibrium conditions between the bulk of the liquid and the interfaces (Marmur, 1992a). Hence, capillary flow in the vadose zone is driven by gravity and by interfacial pressure differences. The magnitude of the capillary pressure between the liquid and the gaseous phase depends on the local curvature of the water menisci. This, in turn, is determined by the local wetting properties, the geometry of the pore walls and when considering flow processes (Hassanizadeh and Gray, 1993), and on the microscopic fluid velocity (Friedman, 1999). However, a hydraulic model for the larger scale usually used for soil-water simulations must consider soil properties more generally by averaging inherently microscale pressures of the numerous water menisci for a defined reference volume. A hydraulic model for an unsaturated hydrophobic soil should consider moisture-dependent wetting properties, a positive water-entry pressure for hydrophobic soil domains, and finally, a time-dependent CA, which is often labeled as persistence. Persistence should be closely related to the water content and the wetting history of the specific location itself and its neighborhood.

Most studies dealing with water movement in water-repellent soil have been made in extremely hydrophobic soils. During the past 15 yr it has become increasingly evident that besides the phenomenon of extremely water-repellent soils, reduced wettability is observed more often than commonly believed. Moreover, almost all soils, as well as ideal laboratory systems, like packing of glass beads (Lu et al., 1994) or glass slide surfaces (Hartge and Bachmann, 2000), are not completely wettable. This behavior is indicated by advancing CAs larger than 30°, measured during the transition from dry to wet at the three-phase boundary. In the past, hydrophobic soils have mostly been associated with preferential flow phenomena (Ritsema and Dekker, 1994; Wang et al., 2000a, 2000b). Preferential flow locally enhances the flux of water and therefore promotes the solute transport through fingers in the unsaturated zones of the soil (Clothier et al., 2000). This behavior is typically caused by the occurrence of a vertical series of three zones in the soil profile: the distribution, the fingering, and the redistribution zones (de Rooij, 2000). The hydraulic conductivity of actual severely water-repellent soils is extremely small, and their water-retention functions are often strongly hysteretic (Morrow, 1975, 1976; Bauters et al., 1998). The latter is the mechanism underlying the fingering of water during infiltration that is observed in water-repellent soils due to a small hydraulic gradient that is built up at the dry–wet soil interface (Ritsema et al., 1998). It was found that at some locations, more than 80% of the draining water is transported through preferential flowpaths, while bypassing large areas of the soil, which results in a reduced filter capacity of the soil to retain pollutants (Ritsema and Dekker, 1994). Generally, wettability of soil particles seems to be a time- and moisture-dependent phenomenon. However, the wetting behavior of these soils is not recognized as a continuous dynamic-state variable that appears under dry or moderate moist conditions at many sites (Doerr and Thomas, 2000). It changes primarily with the actual moisture content, {theta} (m3 m–3), and the matric head h (Pa), along with the wetting history and, with minor importance, soil temperature (Bachmann et al., 2002; Bachmann and van der Ploeg, 2002). Potential hydrophobicity has been defined as the maximum extent of water repellency, whereas hydrophobicity measured under varying field moisture contents has been defined as actual water repellency (Dekker and Ritsema, 1994). Water repellency has been found to be at a maximum near the permanent wilting point (Regalado and Ritter, 2005; Goebel et al., 2004), decreasing rapidly as moisture contents approach field capacity (King, 1981).

There generally exists a difficulty regarding the scale of observation. According to Hassanizadeh and Gray (1993), it is problematic to define a macroscopic (i.e., "hydraulic") CA that is consistent with the microscopic CA observed along the contact lines of water menisci located around the contact points of the grains. Scheidegger (cited in Hassanizadeh and Gray, 1993) stated that a defined radius of curvature of a water meniscus is only defined when the porous medium is made of an regular assemblage of capillary tubes, which is not the case in real soils. Therefore, one should acknowledge that microscopic relations are useful aids in interpreting processes in porous media. One should also keep in mind that their validity in explaining macroscopic processes is limited. As pointed out in detail by Hassanizadeh and Gray (1993), the combination of variables derived from the micro- as well as the macroscale is conceptually problematic because (i) the macroscopic pressure at a given location in the soil matrix is not always in equilibrium with the values of the pressure applied outside (which in many studies is the basic assumption for the experimental set-up to measure soil CPS relationships), and (ii) the fluid pressure at the microscale cannot be extrapolated to an average macroscopic pressure due to deviations from the thermodynamic equilibrium state caused by local mechanical deformation of the interfaces. The latter effect is observed if one attempts to force one fluid to move against the other in a capillary while the meniscus and the contact line resist displacement. For a sufficiently small change in pressure, the curvature of the meniscus will adjust by changing the CA so that the change in pressure is accommodated without movement of the three-phase contact line. Hassanizadeh and Gray (1993) concluded that the existence of advancing and receding angles cannot be explained without taking fluid–solid interfacial stress into account. In a number of recent works, Hassanizadeh and Gray (1993) and later Or and Tuller (1999, 2000) and Dalla et al. (2002) developed a thermodynamic-based theory of two-phase flow. For the extended theory they defined Helmholtz free energy functions for the phases and interfaces in terms of state variables such as mass density, temperature, porosity, interfacial area density, and a solid phase strain tensor. This approach is helpful for obtaining a more appropriate representation of the functional dependence of capillary pressure. However, due to a lack of information on basic state variables like the interface area per unit volume as a function of saturation, this detailed theoretical model can be simplified for studies of practical interest. Considering less-detailed approaches, Arye et al. (2007) recently showed empirically that a linear scaling factor determined from independent data could successfully be used to explain the water-content distribution in capillaries filled with sand of various wettability. The scaling factor was interpreted as the cosine of the average CA. It was also shown that the general water-content distribution with height, which can be considered as an equilibrium CPS relationship, could be described well with a simple function known as the van Genuchten equation. It should be noted that for a time-dependent CA, as it was observed in numerous studies of surface chemistry (Engländer et al., 1996), the CA should have a considerable effect on the hysteresis of the CPS relationship. However, corresponding studies of wetting kinetics by using CA have rarely been made for natural soil material.

In this paper we propose a conceptual approach that combines microscopic interfacial properties as an additional parameter for the hydraulic model to link interfacial properties with a conventional hydraulic model. We used this semi-empirical approach to supply easy measurable quantities to a model that describes the typical behavior of water-repellent soil that has been observed in many studies. The conventional van Genuchten model, which was used by Nieber et al. (2003) for numerical simulations of water movement in hydrophobic soils, is in our proposed model extended with one additional parameter function, which is physically interpreted as the wetting coefficient cos(CA). This parameter theoretically allows the description of pressure-saturation relationships in unsaturated soil as well as the description of the positive breakthrough pressure needed for forced water penetration into dry hydrophobic soil. This extension of the van Genuchten model allows us to define wetting properties of the porous medium with independently measured CAs. Variability of the CAs also accounts for variability within a soil horizon, as well as for different average wettability attributed to different soil horizons. In a companion paper (Deurer and Bachmann, 2007) we present results from numerical simulation studies to demonstrate schematically the capability of the proposed hydraulic model extension to describe water transport in partly saturated soil under consideration of the wetting properties. The CA plays a major role for the parameterization of the hydraulic model described below. The following section presents some of the fundamental observations that have been made on the behavior of hydrophobic soil or with soil with reduced wettability and which were used to define the general behavior of the underlying CA model described below.


   Contact Angle on Soil Particles: Theoretical Aspects
TOP
 ABSTRACT
 INTRODUCTION
 Contact Angle on Soil...
Wetting Behavior of Soil...
 A Contact Angle-Dependent Water...
 Conclusions
 REFERENCES
 
Due to their high surface energy, soil minerals are assumed to be completely wettable. The surface energy of hard solids, like metals and minerals, ranges from 5000 mJ m–2 (high-energy surfaces) to <15 mJ m–2 for surfaces composed of organic R–CF3 groups (Zisman, 1964). The hydrophilicity of minerals, their affinity for water, increases together with the density of their charges and polar groups, mainly OH–, on the surface (Ellerbrock et al., 2005. Therefore, it has also been argued that inorganic soil minerals are hydrophilic because their surfaces usually hold ions and polar groups (Tschapek, 1984). In contrast, measurements of interfacial tension in soil (Miyamoto and Letey, 1971) indicate surprisingly small values for mineral particles. Miyamoto and Letey (1971) reported interfacial tensions for quartz sand of about 43 mJ m–2, for water-repellent soil of 25 mJ m–2, and for water-repellent silane-treated soil about 10 mJ m–2. Later Bachmann et al. (2003) gave a range of 15 to 40 mJ m–2 for soils, varying from extreme water repellent to wettable.

The CA is a parameter that summarizes the relationships among the three interfaces (Eq. [1]) into one parameter. The intrinsic CA, {kappa} (°), is the microscopic angle, measured theoretically a short distance a few micrometers away from the solid surface of a few molecular layers (Marmur, 1992a, 1992b). Principally, this CA describes the basic relationship between the tangent to the liquid–vapor interface and the tangent to the solid interface. Introducing Young's equation yields

Formula 1[(1)]
where {sigma}sv, {sigma}sl, and {sigma}lv (mJ m–2) are the interfacial energies of the solid–vapor, solid–liquid, and liquid–vapor interfaces, respectively, and {pi} is the film pressure caused by adsorption of molecules originating from the liquid phase (Adamson, 1990), which for soil is water. It is notable that when {sigma}sv – {sigma}sl > {sigma}lv, cos {kappa} = 1; that is, {kappa} = 0 for any numerical value of this specific state indicating complete wetting (Padday, 1993). A CA >0° and <90° indicates reduced wettability, and a CA >90° indicates water repellency.

The effect of adsorbed water on the CA of soil particle surfaces is a combination of at least three different mechanisms: adsorption of water vapor (humidity <100%), where the film pressure concept may be valid (Good, 1993); filling of small pores through capillary condensation, which creates chemical composite surfaces with patches of solid and liquid interfaces (McHale et al., 2005); and conformational changes of the functional groups, which may account for time dependency of water repellency due to displacement of hydrophobic functional groups into pore spaces while avoiding contact with liquid water (Ma'shum and Farmer, 1985; Doerr and Thomas, 2003). The film pressure {pi} (mJ m–2) is the equilibrium film pressure of the adsorbate, which generally is at monolayer or lower coverage for a liquid with nonzero CA (Good, 1993). This reduces the surface tension {sigma}sv because the adsorption of liquid molecules is spontaneous. Busscher (1992) reported for chemically pure solid–liquid systems that the spreading pressure {pi} lowers the solid surface energy only for surfaces with {sigma}sv higher than 45 to 50 mJ m–2. For soils King (1981), de Jonge et al. (1999), Doerr et al. (2000), and Bachmann and van der Ploeg (2002) observed an increase of water repellency with increasing water vapor pressure until a critical value, after the wilting point. To ignore {pi} for in situ moisture contents lower than permanent wilting point seems for some soils appropriate (Miyamoto et al., 1972; Goebel, 2007), but this could also lead to incorrect results (Arye et al., 2006; Goebel et al., 2004; de Jonge et al., 1999).

Chemical heterogeneity of soil particle surfaces through coatings of soil organic matter (SOM) is considered one of the major sources for reduced wettability of high-energy mineral surfaces. Heterogeneity cannot be visualized as the effect of roughness. Considering the surfaces, CA is only a conceptual measure of wettability as the microscopic distribution across the surface cannot be quantified (Marmur, 1992a). Hysteresis of CA also seems to be strongly affected by the details of the arrangement of the patchy structure of the wettable parts of the surface. Using a conceptual and empirical approach, Cassie and Baxter (1944) proposed an equation where the surface contains two patches:

Formula 2[2]
where f1 and f2 are the area fractions (f1 + f2 = 1) with corresponding intrinsic angles {kappa}1 and {kappa}2, and {kappa} is the observed CA. Assuming heterogeneity at the microscopic scale, the CA will vary across such heterogeneous surfaces, especially when the distance between chemically heterogeneous patches is >1 µm. The minimum patch size that produces CA hysteresis is about 1 µm (McHale et al., 2005). In the case of dynamic CA at a moving three-phase boundary, the advancing CA is more affected by the hydrophobic fraction, whereas the receding CA is correlated with intrinsic CA of the wetting fraction (Good, 1993).

The significance of the CA for indicating changes of the interfacial properties for higher water contents was demonstrated by Michel et al. (2001) and Chassin et al. (1986). Chassin et al. (1986) reported changes of surface free energy of Ca-montmorillonite as a function of water content. The main contribution of the surface energy came from siloxane groups originating from the surface oxygen groups of the silicon tetrahedra, whereas polar forces derived from the exchangeable cations. The adsorption of water molecules substantially modifies the surface free energy of the solid (Fig. 1 ). It is assumed that the changes in surface energy can be related to the hydration process of Ca-montmorillonite.


Figure 1
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  		FIG. 1. Dispersive {sigma}sD (circles) and nondispersive {sigma}sND (squares) surface free energy components as a function of the gravimetric water content (Fig. 1a) for a clay mineral (data from Chassin et al., 1986) and for a peat soil as a function of the water vapor humidity (Fig. 1b; data from Michel et al. 2001). Schematically shown is the equilibrium film pressure component {pi} as a function of moisture content.

 
By applying a modified sessile-drop method, Chassin et al. (1986) measured on pressed clay samples that the apparent surface free energy of the clay tends continuously from 180 mJ m–2, which is a zero CA wetting regime, to the value of water (72 mJ m–2) when the solid water content exceeds 50% (w/w). This result means, in turn, that the silicate surface itself no longer influences the surface properties with respect to wettability. Chassin et al. (1986) differentiated between three wetting regimes: a formation of a hydration shell of Ca2+ that masks the surface oxygen atoms for water content <15% w/w; a strong decrease of the dispersive component (15–50% w/w) due to the masking of the surface; and the structure of the adsorbed water is similar to bulk water.


   Wetting Behavior of Soil under Different Moisture Conditions
TOP
 ABSTRACT
 INTRODUCTION
 Contact Angle on Soil...
 Wetting Behavior of Soil...
A Contact Angle-Dependent Water...
 Conclusions
 REFERENCES
 
Generally, the CA–soil moisture relationship has not been well known up to now except in a few studies. Regalado and Ritter (2005) derived some statistical relations for the moisture-dependent wetting behavior of volcanic-ash soil where the soil-water repellency was quantified with the molarity of the ethanol droplet test (MED), as described by King (1981). Water repellency varied nonmonotonically between minimum and maximum CA within the water content at field capacity ({theta}FK) and the permanent wilting point ({theta}WP). The most significant parameter to differentiate between these soils, {Sigma}, was defined by integrating the relation of MED as a function of water content for the range between air dryness and field capacity. Useful correlations were found between the integrand and the soil-water content at minimum repellency. The location of the water content for minimum and maximum repellency was related through the water content at field capacity and at wilting point, respectively. The available data suggest further that the water content where the hydrophobic medium becomes wettable with increasing moisture content (CA <90°), increases with SOM content. Experimental results published by Dekker et al. (2003) indicate that the water content at which the soil becomes wettable varies for a dune sand from 25% v/v in the topsoil to <5% v/v in the subsoil. This is reasonably close to values that can be assumed for field capacity. Other authors (King, 1981; de Jonge et al., 1999; Goebel et al., 2004) found that maximum CAs are often found for matric potentials close to the permanent wilting point.

Most studies that investigated moisture-dependent wettability used the empirical water drop penetration time test (WDPTT). The common WDPTT consists simply of placing a water drop on the soil surface and recording the time required for the water to infiltrate. To relate the input parameter CA to the widely available WDPTT data, we plotted the corresponding data of many different soils taken from Bachmann et al. (2003) and Woche et al. (2005) and fitted a nonlinear empirical function to the data. The advancing CAs were measured with the Wilhelmy plate method.

Figure 2 shows a highly significant (r2 = 0.98) sigmoidal relation between the advancing CA and the logarithm of the WDPTT in the general form {kappa}(x) = c + xa(1 + me–x/t)/(1 + ne–x/tt), where x is the independent variable (CA) and c, a, m, e, t, n, and tt are empirical fitting parameters. Although the WDPTT is considered as a measure of the persistence of water repellency, and not for the intensity that occurs initially when a dry surface contacts water (which is attributed to the initial CA), we found a significant relationship for CA >90°. For smaller CAs, the WDPT is <5 s and must therefore be considered insensitive to detect differences in wettability for the whole range between hydrophobicity and complete wettability. This behavior may be supported by the positive hydraulic pressure at the solid–liquid interface due to the convex-shaped cross-section at the liquid–gas interface of the droplet. To convert water content–dependent WDPTT data to apparent CA, we applied the sigmoidal function to the WDPTT data of Täumer et al. (2005) for a humus sandy soil with SOM contents between <1 to 8% w/w (Fig. 3 ).


Figure 2
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  		FIG. 2. Relation between the advancing contact angle {kappa}a and the common logarithm of the water drop penetration time, WDPT (a, m, n, t, tt, and c are empirical fitting parameters). Data from Bachmann et al. (2003) and Woche et al. (2005).

 

Figure 3
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  		FIG. 3. Water drop penetration time (WDPT) as a function of water content and organic matter content (SOM) in a sandy soil (top). Figures at bottom show the corresponding data after conversion of WDPT to contact angle with the function displayed in Fig. 4. Plot recalculated after data presented in Täumer et al. (2005).

 

Figure 4
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  		FIG. 4. Advancing and receding CA of air dry silt hydrophobized with dichlorodimethylsilane. Also shown are time-dependent contact angles (CAs) measured in a constant immersion depth. Soil was dried after wetting for 10 d in water, and CAs were measured with the Wilhelmy plate method (WPM).

 
Figure 3 shows the original WDPTT water-content data and a similar plot with the converted CA data. The general shape of the graphs, which follows a third-order polynom, is similar in shape for different SOM contents. It shows an increasing slope at the moist end and plateaulike behavior at the dry part of the curve for a higher carbon content. This behavior is similar to the proposed CA–soil moisture relationship described in the model below. Generally, this behavior is comparable to the water content–dependent MED relationships described by Regalado and Ritter (2005). The decrease of the CA for water contents below the permanent wilting point observed by some authors (King, 1981; de Jonge et al., 1999; Goebel et al., 2004; Regalado and Ritter, 2005) was not considered because the root water uptake was restricted to pF 4.2 in the simulation model (Deurer and Bachmann, 2007). It is also interesting to note that the measured CAs show a significant shift with time that is, in contrast to the WDPTT, also measurable for nonhydrophobic samples in the CA range <90° with the applied Wilhelmy plate method. Shown in Fig. 4 are the dynamic advancing and receding CAs, measured with the modified Wilhelmy plate method, which were attributed to samples moving into or out of the testing liquid (Bachmann et al., 2003). As expected, the advancing CA is larger than the receding angle with zero degree. Shown also are the CAs measured with a static Wilhelmy plate measured in a constant immersion depth. In this mode a linear decrease of the CA can be observed with the logarithm of time according to increasing wettability of the grains in contact with water. However, measurements about the wetting kinetics of soil particles are rare; hence, this behavior can only be rated as a preliminary result.

Hydrophobicity is believed to reappear when soil become dry again after wetting. Doerr and Thomas (2003) investigated the wettability of a sand and a sandy loam in central Portugal afforested with pine (Pinus) and Eucalyptus. Their study indicated that the soils remained hydrophilic as soil moisture content fell below the water content where water repellency was still observed during the wetting cycle. In some samples hydrophobicity did not recover after weeks. This behavior was explained by different effects, such as organic molecules that cause hydrophobicity to become completely reattached at the mineral surfaces after being dissolved in the soil solution, or the formation of new hydrophobic substances by microorganisms. Our studies performed with humus sandy soil from a forest site (beech [Fagus L.] and pine) show that the measured CA could be well reestablished after three consecutive wetting–drying cycles, which supports the model approach described below. Carbon content of the uppermost horizon (Fig. 5 , left) was 3.8 and 1.3% w/w in the second horizon (Fig. 5, right).


Figure 5
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  		FIG. 5. Cosine of the contact angle {kappa} of a humus sand from a topsoil of a forest stand after three repeated wetting–drying cycles. In each cycle samples were remoistured to approximately field capacity and dried either to oven-dryness, to air-dryness, to pF 4.2, or to freeze-dried. Contact angles (CAs) were measured with the capillary rise method. Average standard deviation of CA determinations approx. 4 to 5°.

 
Repeated wetting and drying cycles do not affect the cosine of the CA displayed in Fig. 5 considerably when the same drying state is established again in the next cycle. Different experiments with different drying intensities, soil dried at 105°C, air-dried, in equilibrium with a matric potential of 1500 kPa (field capacity, pF 4.2), or freeze-dried, gave characteristic CAs with the tendency to largest CAs (lowest cosine values) for the pF 4.2 variant.

All observations reported so far relate to already-wetted soil. For CAs >90°, infiltration into a pore or a soil domain cannot take place when the water pressure is lower than the water entry pressure, determined by the pore size, pore shape, and local CA. This pressure is referred to here as the breakthrough pressure head, {psi}p, which is related to the surface tnsion of the liquid {sigma}l and the 90° surface tension, {sigma}ND, that is, the liquid surface tension where the CA is 90°:

Formula 3[3]
where r is the equivalent pore radius, {rho} is the density of the liquid, and g is the acceleration due to gravity. Carillo et al. (1999) confirmed the validity of Eq. [3] for a sandy substrate hydrophobized with octadecylamine and with alcohol extracts from peat moss.

According to Fig. 6 , which shows data from Carillo et al. (1999) and Bauters et al. (1998), a linear relation between the cos(CA) and height of the positive water column, that is, the breakthrough pressure, was observed. A similar behavior was assumed for hydrophobic conditions in the hydraulic model, which is presented in the next section.


Figure 6
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  		FIG. 6. Water entry pressure as a function of the cosine of the contact angle for three different soils.

 

   A Contact Angle–Dependent Water-Retention Model for Soils
TOP
 ABSTRACT
 INTRODUCTION
 Contact Angle on Soil...
 Wetting Behavior of Soil...
 A Contact Angle-Dependent Water...
Conclusions
 REFERENCES
 
Concept of the Model
The quantities modeled are related to a representative elementary volume (REV) with a characteristic length. This model domain should consider inherently the typical properties of porous media varying from wettable to hydrophobic. Figure 7 shows the basic concept.


Figure 7
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  		FIG. 7. Schematic representation of the contact angle (CA) model. Location of node is indicated by i and j, time by t, contact angle by {kappa}; {alpha} is the water retention parameter of the van Genuchten equation, and {theta}WP indicates the water content at the permanent wilting point. REV = representative elementary volume.

 
Important properties should include the effect of the CA on the CPS relationship (Fig. 7a), a positive water-entry pressure for hydrophobic dry media (Fig. 7b), moisture-dependent wettability, and a time-dependent CA, which is often labeled as persistence (Fig. 7c). Persistence is closely related to the water content and the wetting history of the specific location itself and its neighborhood, which is schematically shown in Fig. 7d. In our companion paper (Deurer and Bachmann, 2007), we propose that the contact angle {kappa} for a location x within the soil matrix is not constant but is a continuous function that depends on small-scale interfacial properties and the local saturation of the neighboring REVs. It is assumed that {kappa}max(x) exhibits site-specific spatial variation with correlation lengths of about 20 to 50 cm (Dekker et al., 2003) down to undetectable repeated moisture pattern in the millimeter range (Hallett et al., 2004). Table 1 shows the input parameters of the model discussed in the following sections.


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  		TABLE 1. Variables that are needed for the parameterization of water repellency.

 
The characteristic features of the conceptual model shown in Fig. 7 are the water-content dependency of the CA and the hysteresis loop. The general shape of the {kappa}({theta}) relationship was derived from Fig. 3. The local CA {kappa}(S,x,t) is a function of water content ({theta}) and time (t) of the corresponding matrix; S is the normalized degree of sample saturation (Fig. 7d). At and above a specific water content {theta}FK(x) of a wet region, the CA decreases to the minimum CA, {kappa}min(x). The water content {theta}FK(x) denotes in the model the water content of the fully wettable region around field capacity, approximately at a matric head h of ~60 hPa in a sandy soil. At and below a critical water content, {theta}crit, the CA increases to the maximum CA, {kappa}max(x). The {kappa}max(x) denotes the potential severity of water repellency. For one continuous drying–rewetting cycle, the minimum and maximum CAs characterizing the boundary loop are assumed to be defined material functions for a given location, such as given by the advancing and receding CAs. The {theta}crit is assumed to be a site-specific value, which depends on the texture plus the quality and the quantity of the SOM (Ellerbrock et al., 2005; Doerr and Thomas, 2003; Täumer et al., 2005) and which is assumed to be located close to the permanent wilting point, {theta}WP. However, the specific values for {theta}crit and {theta}FK with respect to wetting properties are not a priori known. Examples can also be found of soils with an extreme degree of water repellency containing almost no organic matter (<1% w/w.; see Dekker et al., 2003).

The CAs at intermediate water contents between {theta}crit and {theta}FK are assumed to be hysteretic. Only a few observations can be used to support this assumption (Doerr and Thomas, 2003; Fig. 4). If a "wet" and wettable region at or above {theta}FK(x) dries out, the CA is not expected to increase for decreasing water contents above the critical water content. The CA is set to {kappa}min(x). In summary, we assume that the steep increase of the CA during drying is only a function of the water content and cannot be differentiated with respect to time. For very dry conditions, Engländer et al. (1996) reported a strong temporal effect of the dehydration of quartz surfaces, which corresponded to a strong increase in the CA. The time dependency was approximated with an exponential function cos{kappa} = a + be–t/to with a time constant to of about 8 d. In soil, the atmosphere for moisture content above {theta}WP is water vapor saturated so that such an effect cannot be expected. Further, the CA change during drying is accompanied by a receding CA, which also reduces the impact of the low receding CA on the CPS relationship. Doerr and Thomas (2003) recently presented data that emphasize a strong hysteretic effect of water repellency after wetting and a subsequent drying phase. They reported for the wetting process that a considerable increase in soil moisture does not necessarily lead to a significant reduction of water repellency. This effect may be caused by a mixture of wetted hydrophilic particles enclosed within a matrix of still hydrophobic surfaces. This would accordingly create chemically heterogeneous surfaces with zero degree domains for the wetted particles as indicated in Eq. [2]. Above a soil-specific water content, soil hydrophobicity indicated by WDPTT vanished, probably due to the breakdown of repellency as demonstrated by Fig. 4. During drying the soils remained hydrophilic even as the soil moisture content fell below the water content where water repellency was still observed during the wetting cycle. This specific "memory behavior" was also reported by Täumer et al. (2005).

Formal Description of the Model
For each REV the following rules were formulated. In cases in which a "dry" region with {kappa}max(x) is rewetted, the CA will be a function of the local system state, f(S,x). The corresponding time of rewetting is the local rewetting time trw(x) (h). The rewetting process can be initialized under two conditions that (re)start trw(x) at the initialization time for rewetting, tirw(x) (h):

  1. The local water content increases from a water content below or equal to the {theta}crit to a value greater than {theta}crit but below {theta}FK.
  2. The water content at x is equal to or below {theta}crit, but the water content in the immediate vicinity of x is higher than the {theta}crit. In this case it is assumed to be driven by a very slow diffusion process, comparable in magnitude to the isothermal water vapor diffusion coefficient D{psi}v of about 10–10 m s–1 (Bachmann et al., 2003).

If one of the conditions apply, then tirw(x) > 0. Further, the local rewetting time trw(x) is defined by

Formula 4[4]
where t (h) is the time since the start (t = 0) and {theta}(x, t) is the water content at the coordinate x at time t. The water content in the immediate vicinity of x, denoted as {Phi}(x), at time t is written as {theta}[{Phi}(x),t]. In the two-dimensional simulation model, we defined, for example, that the vicinity of a grid node is represented by the eight neighboring grid nodes. Therefore, the size of the vicinity is defined by the grid resolution that we choose, for example, 2.5 cm. When the local water content fulfills {theta}(x,t) ≤ {theta}crit È {theta}[{Phi}(x),t] < {theta}crit or {theta}(x,t) ≥ {theta}FK, then tirw(x) and the total rewetting time trw(x) are (re)set to zero. The contact angle {kappa}(S,x) can now be calculated according to

Formula 5[5]
During rewetting [trw(x) > 0], the contact angle {kappa}(S,x) is a function of the system state, f(S,x), which is defined as

Formula 6[6]
with

Formula 7[7]
where the factors ß{theta}(x) and ßh(x) (dimensionless factors) describe the dependency of {kappa}(S,x) on the local water content and the local matric head, respectively. The time t0 (h) is the characteristic rewetting time that is assumed to be site-specific. This parameter describes the persistence of water repellency and is frequently expressed as the WDPTT. A clear distinction in the initial CA and its time dependency is still missing. According to WDPTT data, it may reach infinite time. Consequently, this relationship is conceptual and must be experimentally evaluated in further studies. In general, the CA and the time constant are the equivalent parameters of the MED and the WDPTT, respectively.

The normalized factor ß{theta}(x) describing the water content dependency of the CA ranges from close to zero to one. It is defined as follows:

Formula 8[8]
where {theta}r is the residual water content. The higher the water content, the smaller is ß{theta}(x) and the more rapid is the CA decrease.

Figure 8 shows the reduction of the CA in the vicinity of a wetting front according to an exponential decrease with time (Eq. [6]). Wetting starts when a neighboring point in the matrix is wetter than {theta}crit but still under negative capillary pressure. The exponential decrease is accelerated to a large extent when the hydraulic pressure returns to positive values. When the pressure exceeds the breakthrough pressure, the CA is immediately set to zero.


Figure 8
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  		FIG. 8. Schematic representation of the dynamic decrease of the water content-dependent contact angle with time t (Fig. 7a) and positive pressure h (Fig. 7b) in the vicinity of a wetting front. Parameters used were {theta}crit = 2 Vol.-%, {theta}FK = 18 Vol.-%, {kappa}max = 110°, {kappa}min = 0, characteristic rewetting time to = 30 d, breakthrough pressure ho = 30 cm.

 
For CAs greater than 90°, local ponding or, inside the soil matrix, saturation occurs if the water pressure of the surrounding REVs is lower than the water entry pressure of the hydrophobic REV.

An additional energy source enhances the decline of the CA and is formulated by the factor ßh(x):

Formula 9[9]
where hcrit({kappa},x) is the air entry value at x with the contact angle {kappa}(S,x). This is calculated according to

Formula 10[10]
where {alpha}vG' is a pore structure–related parameter according to the Miller and Miller similarity theory (Miller and Miller, 1956) and {alpha}h(x) is the location-specific scaling factor for the water-retention function. This is outlined in our companion paper (Deurer and Bachmann, 2007).

Macroscopic hydraulic properties combine pore surface and pore geometry properties. To combine the surface properties with the CPS relationship, we used the van Genuchten (1980) retention model, which is extended to account for the CA. The parameters {alpha}vG and nvG are assumed to depend explicitly on the CA {kappa}(S,x). The CPS characteristic function of each location x is now represented by a set of CA-dependent van Genuchten parameters, {alpha}'vG({kappa},x) and n'vG({kappa},x):

Formula 11[11]

Formula 12[12]
where p is a site-specific factor >1. For example, the value of p = 1.6 was evaluated by the analysis of the rewetting CPS curves displayed by Bauters et al. (1998). The special case p = 1 indicates a linear scaling of the CPS relationship as indicated by Morrow (1976) and Ustohal et al. (1998). The factor p in Eq. [11] is probably soil dependent and needs further attention. In summary, the factor p signifies that nvG is less sensitive to a change in the contact angle {kappa}(S,x) than {alpha}vG because it is not for all soils different from one. Therefore in our simulation model the possible range of the factor n'vG({kappa},x) is discretized into a specific number of values (e.g., 10), where each value represents a CA range (e.g., 10–20°). Now, the CA dependency of the factor {alpha}'vG({kappa},x) can be conveniently included into the location-specific scaling factor {alpha}h(x). In summary, the complete CA-dependent CPS function h({Theta},{kappa},x) may be expressed as

Formula 13[13]
with

Formula 14[14]
where nvG' is a short-form notation for n'vG({kappa},x) and {alpha}'vG({kappa},x) is now an effective scaling factor. Using data from Bauters et al. (1998) and Arye et al. (2007) shows that describing imbibition characteristics for a sandy substrate with varying CA can only be made when two water-retention parameters (n'vG({kappa},x) and {alpha}'vG({kappa},x)) are scaled (Fig. 9 , figures at bottom).


Figure 9
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  		FIG. 9. Water tension–saturation relations (imbibitions) for a wettable soil ({kappa} = 0) used as reference soil (Ref. soil) to calculate the contact angle (CA) for hydrophobic soils with various contact angles. The 3.1% and 5% are the ratio of mixing of a bulk artificially hydrophobized quartz sand (100% hydrophobic) with a pure wettable quartz sand ({kappa} = 0) (Recalculated after data from Bauters et al., 1998). The ORC and EQL soils are naturally occurring hydrophobic sandy soils located under orange orchard and eucalyptus cover, respectively. Ignition of these soils (400 °C for 8 h) was used to establish a completely wettable reference soil (data taken from Arye et al., 2007). (VG function = van Genuchten function.)

 
In Fig. 9, the scaling was done with the fitted van Genuchten function of the zero CA soil as reference soil by multiplying the van Genuchten parameter {alpha} with the cosine of the independently measured CA. Alternatively, capillary pressure of the measured data points of the reference soil were multiplied directly with cos(CA). Since the van Genuchten estimation for the reference soil is to some extent restricted in its quality to fit the data properly, scaling of data pairs seemed to be better for single data rather than for the whole function. However, in general, the estimation of the CPS with a independent contact angle was sufficient to describe the CPS relation of soil with a nonzero CA in the wetting mode.


   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Contact Angle on Soil...
 Wetting Behavior of Soil...
 A Contact Angle-Dependent Water...
 Conclusions
REFERENCES
 
To describe the hydraulic behavior in soil water in general, wettability effects must be taken into account. Relevant processes that must be included in such a model include a water content–dependent CA, a positive water entry pressure for hydrophobic dry media, and time-dependent CAs interacting with the actual moisture content and the wetting history of the soil. The special situation in soils emphasizes that observations are, at best, apparent CAs that cannot be related directly to equilibrium CAs at interfaces within the medium (Philip, 1971). Our model concept is therefore based on apparent or conceptual (nonequilibrium) CAs, representing an advancing or a receding wetting front according to the wetting or dewetting process of the porous medium. It must be concluded that only relative values, rather than precise and thermodynamically sound CAs, are appropriate for soils. Hydrophobic soils as well as soils with only subcritical water repellency can be represented by this model, when advancing and receding CAs and a parameter characterizing the wetting kinetics of the soil are known. The literature shows that the concept of a critical soil water content is too rigid and may not be correct for most field sites. Often a transition range in soil water contents can be found where the soil changes from wettable to water repellent, depending for example, on wetting history and particle-size distribution of the soil. This is considered in our model for the drainage branch of the model. For rewetting, a shift of the critical water content may also be taken into account, probably as a function of duration of the wetted state. However, extending the CA model will require much more field observation focused particular on the hysteretic behavior of the wetting properties of soil with adequate experimental techniques.

The reestablishment of soil water repellency will be highly dependent on the rate and extent of the root water uptake and the evaporation rates. In particular, the root–soil interaction is drawing attention again to the plant–soil interactions with regards to the impact on the wettability of the soil. Systematic investigations that combine pore structural and pore-surface properties are needed to generalize our water-retention model. In a second paper we apply this model to demonstrate explicitly the impact of a dynamic contact angle on two-dimensional water movement in soils without limitation of the CA range (Deurer and Bachmann, 2007).


   ACKNOWLEDGMENTS
 
We thank Drs. B. Clothier and I. Vogeler for their review of an earlier draft of this paper.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Contact Angle on Soil...
 Wetting Behavior of Soil...
 A Contact Angle-Dependent Water...
 Conclusions
 REFERENCES
 




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