Published in Soil Sci. Soc. Am. J. 68:1154-1161 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-1—SOIL PHYSICS
A Conceptual Model to Predict the Deflation Threshold Shear Velocity as Affected by Near-Surface Soil Water
I. Theory
Wim M. Cornelis*,
Donald Gabriels and
Roger Hartmann
Ghent University, Dep. Soil Management and Soil Care, International Centre for Eremology, Coupure links 653, B-9000 Gent, Belgium
* Corresponding author (wim.cornelis{at}UGent.be).
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ABSTRACT
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A crucial parameter in predicting wind erosion is the deflation threshold shear velocity, which is highly dependent on near-surface soil water. The empirical and theoretical models to predict the deflation threshold as affected by near-surface water that have been reported in literature all suffer from a weak physical background and very large differences between the predicted results can be observed. The present study was conducted to develop a new conceptual model to predict the deflation threshold as affected by near-surface wetness and to contribute to a better understanding of the role of the latter on deflation of sediment. The model was developed by solving the moment balance equation for entrainment of soil particles by wind, including the moments associated with drag forces, lift forces, aerodynamic moment forces, gravitational forces, and interparticle forces due to dry and wet bonding. The wet bonding force, which represents the effect of near-surface soil water, is due to liquid bridge bonding or capillary forces, and adsorbed layer bonding or adhesive forces. It was related to the particle diameter squared, surface tension squared and the inverse of matric potential. The latter was then converted to water content by assuming a logarithmic relationship, which was shown to be valid between oven dryness and a matric potential of –1.5 MPa, a range that is of interest in the light of deflation of soil particles by wind. The conceptual model presented in this paper is relatively simple and predicts, for a given particle diameter and surface tension, the deflation threshold shear velocity as a function of the ratio between water content and the water content at a matric potential of –1.5 MPa.
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INTRODUCTION
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A CRUCIAL PARAMETER in predicting wind erosion is the threshold shear velocity. This is the minimal shear velocity, which has to be exceeded to initiate deflation or dislodgment of sediment by wind shear at the surface. Amongst the several factors that govern threshold conditions, near-surface water, which we define as the water in the few millimeter thick zone just below the soil surface, is one of the most significant. Through adhesion and capillary effects it strongly contributes to the binding forces keeping particles together (McKenna-Neuman and Nickling, 1989).
Several studies were conducted to determine the influence of water on entrainment of soil or sand particles by wind. These studies mainly resulted in empirical equations relating the near-surface water content to a deflation threshold, such as the threshold shear velocity (e.g., Chepil, 1956; Belly, 1964; Hotta et al., 1984; Saleh and Fryrear, 1995; Shao et al., 1996) or a threshold wind velocity measured at a certain height (Azizov, 1977; Chen et al., 1996). McKenna-Neuman and Nickling (1989) proposed a theoretical model, in which they solved the capillary force model of Fisher (1926) for cone-shaped sand particles, and simply related the threshold shear velocity of moist sand to the threshold value for dry sand multiplied with a term associated with the effect of near-surface water. The latter term was inversely proportional to the capillary potential. Fécan et al. (1999) extended this model to soils and incorporated the water retention model of Gardner (1970) into their parameterization. Gregory and Darwish (1990) considered in developing their theoretical model the particle-to-particle bonding force as the sum of the dry and the wet bonding force and incorporated it in a moment balance equation. They represented the capillary force by the tensile force resulting from diminished pressure as defined by Haines (1925) ignoring the correction introduced by Fisher (1926) for the tension exerted by the interface. As a consequence, their capillary force was proportional to the matric potential rather than to its inverse as in the models of McKenna-Neuman and Nickling (1989) and Fécan et al. (1999). This resulted in a negative contribution of the capillary potential to the wet bonding force, which is theoretically incorrect (Cornelis and Gabriels, 2003). Gregory and Darwish (1990) further related the matric potential exponentially to the inverse of the ratio between water content and water content at a matric potential of –1.5 MPa. Reviewing and evaluating the most widely used and the most recent of these models, Cornelis and Gabriels (2003), observed very large differences between the predicted results. At a water content of 0.5 times the value at –1.5 MPa matric potential, the increase in threshold shear velocity predicted with these models relative to the dry threshold shear velocity ranged from 117 to 171%. At –1.5 MPa matric potential, the increases ranged from 131 to 261%.
The objective of our study was to develop a new conceptual model to predict the deflation threshold shear velocity as affected by near-surface wetness and to contribute to a better understanding of the role of the latter on deflation of sediment. The main differences with the models of McKenna-Neuman and Nickling (1989) and Fécan et al. (1999) are that (i) we considered the interparticle force as the sum of the dry and wet bonding forces in solving the moment balance equation for entrainment of soil particles by wind, rather than multiplying the threshold shear velocity of dry sediment with a factor for near-surface wetness, (ii) we not only considered capillary forces between soil particles (using Fisher's model, 1926), but also took into account adhesive forces due to adsorbed water films (using Haines' model, 1925), (iii) we applied a water retention model that relates the matric potential exponentially to the inverse ratio between water content and the water content at a matric potential of –1.5 MPa, and (iv) we used a more general geometry for particle shapes rather than using cones. The main difference with the model of Gregory and Darwish (1990) is our approach of both the adhesive and the capillary force, using respectively the model of Haines (1925) and its correction by Fisher (1926).
The current paper is the first part of two papers on the new conceptual model. In a second part (Cornelis et al., 2004), the model is calibrated and verified using wind-tunnel data.
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The Interparticle Forces
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The most important forces that bind soil particles to one another are electrostatic (Coulomb) forces, van der Waals forces, and forces due to the presence of water, including liquid-bridge bonding (capillary forces) and adsorbed-layer bonding (adhesion forces) (Harnby, 1992).
Interparticle Forces between Dry Particles
For a detailed description of the electrostatic forces and the van der Waals forces for particles in the two-state phase solid-air, we refer here to Cornelis and Gabriels (2004). In expressing a conceptual model to predict the deflation threshold of dry particles, they considered the electrostatic force as negligible in comparison with the van der Waals force. The expression they presented for the interparticle force of dry platy to spherically shaped loose particles FvdW (N) was:
 | [1] |
where KvdW is a proportionality coefficient, and d is the particle diameter (m). Equation [1] can also be written as (Adamson and Gast, 1997):
 | [2] |
where AH is the Hamaker constant, and x is the distance between the particles (m). Tabor and Winterton (1968)( 1969) and Israelachvili and Tabor (1972) measured 10.7 x 10–20 and 13.5 x 10–20 J, respectively as the Hamaker constant for mica. For silica, Hough and White (1980) and Israelachvili (1992) give values of 6.5 x 10–20 and 6.55 x 10–20 J, respectively.
Interparticle Forces between Wetted Particles
Liquid-Bridge Bonding
The bonding forces between two particles in the three-phase state solid-air-liquid are, besides the electrostatic and the van der Waals forces, a result of liquid-bridge bonding and adsorbed-layer bonding. The liquid bridges exert an attractive force between particles once they are formed (Harnby, 1992). It is presumed that liquid surface tension causes particles to adhere to one another when water is present in the pore space between the particles (Kézdi, 1974). The surface of the water behaves as an elastic membrane as water molecules exert an attraction toward each other. According to Fisher (1926), the liquid bridge formed between two particles exerts a so-called capillary force that is the result of (i) the tension exerted by the air–water interface, and (ii) the resultant tensile force, owing to the hydrostatic pressure in the water being lower than that in the air, with which the remainder of the particle is in contact. The capillary force FC (N) that results in a pull between the particles can thus be written as:
 | [3] |
where R2 is the radius of the ‘waist’ of the water wedge (m), 
is the surface tension of the liquid (N m–1), and 
p is the pressure deficiency between the internal pressure of the water and the atmospheric pressure exerted on its free surface (Pa) (Fig. 1a). This pressure difference is given by the Young–Laplace equation:
 | [4] |
where R1 is the radius of the air–water interface curvature (m).
Equation [3] was solved by Fisher (1926) for an idealized soil consisting of spherical touching particles of uniform size. Assuming that the air–water interface is a portion of the surface of an anchor ring, the total tensile strength was expressed as a function of surface tension, particle diameter, and an angle between the normal and the axis of revolution of the water wedge at the point of contact. This was actually an expansion of the model of Haines (1925) who did not consider the tension due to the air–water interface. Fisher (1926) also made a correction to account for the noncircular profile of the air–water interface, expressing the so-called nodoid of Plateau (1873), that is, the characteristic profile of the liquid bridge between isodiametrical spheres (e.g., De Bisschop and Rigole, 1982, for a thermodynamic solution), as:
 | [5] |
where y is the distance at any point on the air–water interface curvature (m), and 
is the angle between the tangent to the curve and the axis of revolution (°). The model of Fisher (1926) was later verified through laboratory experiments by Alberry (1950), and solved for nontouching spheres.
Our purpose was not to express the capillary force as a function of the curvature of the water wedge, but rather as a function of the pressure deficiency p. Since the atmospheric pressure outside the water wedge is taken as a reference, p is equal to the water pressure in the water wedge pw. This pressure can be referred to as capillary potential 
c (Pa) and has hence a negative value. Note that the latter is not equal to matric potential m, which results from both capillary and adsorptive forces due to a soil matrix (Hillel, 1980). To write the capillary force as a function of c, y in Eq. [5] should be expressed as a function of c. In doing so, McKenna-Neuman and Nickling (1989) substituted cones, which they considered as simple, reasonable approximations of natural particle contacts, for spheres (Fig. 1b). However, we believe that the application of the cone model to predict FC is limited for several reasons. First, the model derivation is based on simplified geometrical assumptions, where the axes of revolution of two touching cones should be in line. In such a case
 | [6] |
and
 | [7] |
where y1 and y2 are the distance from the point of contact of the water–solid interface to the axis of revolution of the water waist for cone/particle 1 and 2, respectively, and 
1 and 2 are the cone/particle angles. If these axes are not in line, Eq. [6] and [7] do not hold. However, these equalities are the basis of the cone model for FC (McKenna-Neuman and Nickling, 1989). Second, when using Eq. [6] and [7] in combination with Eq. [5] to derive the geometrical factor G in the model of McKenna-Neuman and Nickling (1989), it appears that the latter is valid only if 1 
2 and if 1 + 2 > 
/2. In all other cases, G becomes zero and negative, respectively. This would imply that for those cases, no solution could be found for the capillary force. Third, when using cones, and when assuming that the contact angle between water and the surface of the particle is equal to zero, the angle 1 between the tangent to the water–solid interface at contact with cone/particle 1 and the axis of revolution is identical to the cone/particle angle 1. This is true for cone/particle 2 as well. For true shaped particles, the angle can consistently differ from , because (i) the contact angle at the water–solid interface is most likely different from zero for rough particle surfaces (Hillel, 1980), (ii) is affected by the particle geometry, and (iii) depends on the water content which is related to pw, the parameter that should be solved explicitly. For a given particle geometry, decreases as water content and hence R1 and R2 increase. Because of these limitations, we left the idea of representing soil particles as cones, and solved the Young–Laplace equation as such to R2.
The differential form of Eq. [4], developed first by Bashforth and Adams (1883) and valid for cylinders or spherical particles, implies that R1 is related to the third power of R2. In case of particles, the proportionality constant is a function of the particle diameter squared and hence:
 | [8] |
where a is a dimensionless proportionality coefficient depending on particle geometry. Substitution of Eq. [8] into Eq. [4] yields:
 | [9] |
The solution of Eq. [9] to R2 is a complex function, which can be simplified to read:
 | [10] |
where b is a factor depending on a, , and c (m–2). For c greater than –1 MPa, which is in the range of water conditions of interest for deflation of soil particles by wind, the second and third terms of Eq. [10] become negligibly small. Equation [10] can therefore be simplified to:
 | [11] |
where KC' is a function accounting for soil and fluid properties (m–2). Substituting Eq. [11] into Eq. [3] results in:
 | [12] |
where KC = 
is a coefficient accounting for soil and fluid properties (m–2). The final obtained Eq. [12] differs from the capillary-force model as deduced by McKenna-Neuman and Nickling model (1989) in that for a given surface tension and capillary potential the capillary force increases with the square of particle diameter. Furthermore, the geometric parameter KC is not merely related to the particle angle as in the case of the McKenna-Neuman and Nickling model (1989). It is on the contrary a complex function that can only be determined through curve fitting. The model is hence applicable for all kind of particle shapes (Fig. 1c) and does not pretend to be soluble if particle angle is known.
Adsorbed-Layer Bonding
The adsorbed-layer bonding is caused by the overlapping of the adsorbed layers of neighboring particles. Its strength is proportional to the tensile strength of the adsorbed film. Packing density, particle shape, particle-size, and particle roughness will therefore play an important role (Harnby, 1992). This wet-layer bonding is an adhesion force that includes electrostatic forces, for example, between diffuse double layers, van der Waals and hydration forces, responsible for molecular interaction, and forces due to the overlapping of two interfacial regions, for example, mutual attraction between two clay plates across a slit-shaped pore space (Tuller et al., 1999). The result will be a mutual repulsion and attraction. These concepts were introduced independently by Derjaguin and Landau (1941) and by Verwey and Overbeek (1948), and are often referred to as the DLVO theory. Derjaguin (1957) has termed the repulsive force as the disjoining pressure. Because in many applications the van der Waals component dominates the other components, only van der Waals forces need to be considered. The above theory applies to smooth particles only. However, a soil consists mainly of rough particles, and the effect of surface roughness is to decrease the effective adsorbed layer thickness of the surface by an amount equal to half the average peak-to-trough height. This implies not only a decrease in the binding force but also a minimum adsorbed layer thickness below which interactions are limited to the peaks of roughness (Harnby, 1992). Therefore, the forces due to adsorbed-layer bonding are often considered as negligible for soil particles (Namikas and Sherman, 1995). It could hence be sufficient to consider only the van der Waals force between dry platy to spherically shaped loose particles (Eq. [1]). Notwithstanding this, and since the contribution of adsorbed-layer bonding to the attraction between two particles increases with water content due to an increase in contact area, we assumed here the force associated with adsorbed-layer bonding FA (N) to be equal to the tensile strength as defined by the first term in Fisher's equation (Eq. [3]):
 | [13] |
excluding the term for the pressure deficiency, which is associated with the effects of capillarity.
As water content increases, the width of the waist of the water wedge between the particles R2 increases. We can therefore assume FA to be inversely proportional to the suction in the adsorption films. Since FA further depends on the contact area between the water films, it is assumed to be proportional to the square of particle diameter and liquid surface tension as well. When using dimensional analysis, Eq. [13] becomes:
 | [14] |
where KA is another proportionality coefficient accounting for soil and fluid properties (m–2), and a is the potential, which is the negative of suction, in the adsorbed layers (Pa).
Total Wet Bonding Force
When combining Eq. [12] and [14], the interparticle force due to wet bonding FAC (N) can be written as:
 | [15] |
where KAC is a proportionality coefficient accounting for soil and fluid properties lumping both KC and KA together (m–2), and m is the matric potential (Pa). This is a simplified representation of the interparticle force due to wet bonding as it assumes the effect of the capillary force and adsorptive force to be equal. Of course, below a given matric potential, the adsorptive force dominates the capillary force, with the latter being negligible, and hence a 
m in that matric potential range. In the matric potential range where the capillary force is dominant, the adsorptive force is relatively negligible and there, c m. Using one single proportionality coefficient KAC over the whole matric potential range means that the effect of the adsorptive force is overestimated in our formulation in Eq. [15].
However, combining Eq. [12] and [14] into one formulation enabled us to disregard a minimum soil water as used by Fécan et al. (1999), below which the equilibrium between capillary and adsorbed water is shifted toward the latter. Further investigation is needed to model this minimum soil water. Note that the upper limit of FAC to play a role in keeping particles together is at the point where the individual water wedges at the contact points begin to coalescence as water content becomes sufficiently high. The lower limit is at the junction between adsorbed water and interstitial or structural water. According to Gardner (1986), it is difficult to distinguish between the two categories of water and the exact limit depends upon the soil mineralogy. For simplicity, we considered this limit as the water content at 105°C, that is, at oven dryness.
Matric Potential-Water Content Relationship
Equation [15] shows that it is the matric potential m that will determine the magnitude of the interparticle force FAC between particles of moist sediment. However, requirement of matric potential data forms a key limitation from a practical perspective (Namikas and Sherman, 1995). Standard equipment for measuring matric potential such as tensiometers are not able to record this potential in the uppermost millimeters of the soil that are exposed to the wind and where the matric potential can be much beyond the tensiometer's measuring range. Yet it is the water condition at low potentials in the uppermost layer of the soil that is of primarily concern here. Therefore, matric potential should be related to water content, a parameter that can be determined with much higher precision at the soil surface and at low water contents, with such techniques as the gravimetric method (dry and weigh method), radio spectrometry, infrared photography, microwave techniques, or through modeling of soil water transport as a function of meteorological parameters and soil type.
In literature, many functions are given to relate water content to matric potential m, such as the well-recognized models of Brooks and Corey (1964), Campbell (1974), van Genuchten (1980), and Kosugi (1994)(1997). These models are intended to describe the water retention curve over the whole range of water contents, but often give poor results at low water contents. In his recently published book on the physics and modeling of wind erosion, Shao (2000)(p. 297) suggests to use the Brooks and Corey (1964) equation to convert the matric potential in Eq. [15] to water content:
 | [16] |
where w is the gravimetric water content (kg kg–1), ws is the water content at saturation (kg kg–1), wr is the residual water content (kg kg–1), mb is the bubbling pressure or air-entry value (Pa), and 
is a dimensionless pore-size distribution index. As an example, the Brooks and Corey model is fitted in Fig. 2 to water retention data for the 300 to 500-µm sized sandy loam soil aggregates that were used by Cornelis et al. (2004). Although the model fits the data very well (R2 = 0.994), the matric potential is not defined if water content becomes smaller than 0.053 kg kg–1, which is very close to the water content at a matric potential of –1.5 MPa. Chepil (1956), Gregory and Darwish (1990), and Saleh and Fryrear (1995) reported deflation to cease once the water content exceeds the water content at –1.5 MPa. The problem with equations such as those of Brooks and Corey (1964) is that the residual water content is defined as the water content where dw/dm becomes zero. When determining wr by curve fitting, this results in m = –
MPa at w = wr (where wr > 0) in the case of coarse to medium fine soils or well-sorted soils, which is not realistic. For fine or not-well sorted soils, wr often becomes negative in the curve-fitting procedure. As a negative water content is undefined, wr is then forced to converge to zero, and this results as well in an unrealistic path of the retention curve at low water contents. The Shao's (2000) idea that "It is possible advantageous for wind-erosion modeling to use the well-established data sets for the soil hydraulic parameters ws, wr, mb, and ..." in the context of predicting the effect of soil water on deflation of sediment is therefore premature. Shao (2000) further assumed wr to be equal to the air-dry water content, which is, from a physical point of view acceptable. Indeed, at water contents below the air-dry value, deflation will probably be strongly reduced depending on the ambient relative humidity. However, air-dry water content is in most cases different from the wr value when the latter is simply determined by curve fitting. The air-dry water content of the 300- to 500-µm sized sandy loam soil aggregates was 0.014 kg kg–1 (Cornelis et al., 2004), which is about a factor of four lower than the residual water content as obtained from curve fitting the Brooks and Corey (1964) equation to the data given in Fig. 2. It should be mentioned that the curve-fitted value for wr strongly depends on the data range used. In most studies, the matric potential ranges between 0 and –1.5 MPa, which is the range on which wr values found in databases are generally based.
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Fig. 2. Observed water retention data of the 300- to 500-µm sized sandy loam aggregates, and the fitted Brooks and Corey model (1964; Eq. [16]), and the Rossi and Nimmo model (1994; Eq. [17]).
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According to Rossi and Nimmo (1994), the water content becomes approximately proportional to the logarithm of matric potential in the dry range of the water retention curve, where the above-mentioned water retention models seem to be inappropriate. When including the Ross et al. (1991) correction that forces the model to converge to zero water content at a matric potential of about –103 MPa at oven dryness, their "three parameter sum model" for matric potential as a function of water content was written as (Rossi and Nimmo, 1994):
 | [17] |
where mi is the matric potential at the junction point where the two curves join (Pa), md is the matric potential at oven dryness (–103 MPa), and and ß are shape parameters that are determined by the conditions that ensure the continuity of both Eq. [17] and its first derivative to mi. The reader is referred here to Rossi and Nimmo (1994) for more details. The main advantage of Eq. [17] is that it gives a realistic path of the water retention curve at matric potentials lower than those at –1.5 MPa. In Fig. 2, the "three parameter sum model" as proposed by Rossi and Nimmo (1994) is fitted to the data and R2 was 0.994 as well. Such a curve is, however, not widely used because of its rather complex shape.
Since particle entrainment by wind already stops at rather low water contents (Chepil, 1956; Gregory and Darwish, 1990), we propose only to use the logarithmic part of the Rossi and Nimmo model (1994) that is applicable in the dry range of the curve. Campbell and Shiozawa (1992) and Schofield (1935) measured water contents of soils ranging from sand to silty clay at matric potentials far beyond m = –1.5 MPa. Inspection of their data indicate a loglinear relationship between matric potential and water content for matric potentials smaller than approximately –0.03 and –1 MPa for sand and silt loam, respectively, which were the extreme values in their data set. Their silty clay soil showed an intermediate value. Since, for the given soils, these extreme matric potential values correspond to volumetric water contents ranging from, respectively, approximately 0.05 m3 m–3 to approximately 0.15 m3 m–3, it is plausible to represent that matric potential that determines the effect of water on the threshold shear velocity, by a single logarithmic equation.
The expression we propose for matric potential as a function of water content, and which should be valid within the above-mentioned water content range that is of importance to predict the deflation threshold shear velocity, is:
 | [18] |
where c is a dimensionless regression coefficient accounting for soil properties. Since Eq. [18] is valid for matric potentials smaller than at least –1 MPa, the coefficient c can be related to the water content at –1.5 MPa w1.5, a reference water content that is, though rather arbitrary when considering deflation of soil particles, often available in soil data bases such as those mentioned by Shao (2000), or that else can be predicted from pedotransfer functions. Cornelis et al. (2001) demonstrated that the functions of Vereecken et al. (1989) are relatively accurate to estimate water content at a matric potential of –1.5 MPa. With
 | [19] |
Eq. [18] becomes:
 | [20] |
Use of Eq. [15], rather than using Eq. [12] and [14] separately, has the advantage that a single relationship between matric potential and water content such as Eq. [20] can be used. However, further investigation is needed to find simple equations to relate capillary potential and adsorptive potential to water content separately. Promising work on this has been undertaken by Or and Tuller (1999). Their theories were not considered here to keep the threshold shear velocity model as simple as possible.
A General Model for the Deflation Threshold Shear Velocity of Dry and Wetted Particles
A particle resting on a surface bed and exposed to a fluid stream experiences several forces: a horizontal-drag force FD, a lift force FL, an aerodynamic-moment force FM, the gravitational force FG and an interparticle force FIp (Iversen et al., 1976). The interparticle force FIp can be written as:
 | [21] |
which is the sum of Eq. [1] and [15].
The gravitational force FG, or the particle weight, should be augmented with the weight of the water film that is formed around the particles. The weight of the particle with its water film will increase with water content:
 | [22] |
where KG is a dimensionless proportionality coefficient (= /6 for ideally spherical, smooth particles), 
s is the particle density (Mg m–3), f is the fluid density (Mg m–3), and g is the gravitational acceleration (m s–2).
The aerodynamic forces can be defined as (Cornelis and Gabriels, 2004):
 | [23] |
 | [24] |
 | [25] |
where KD, KM, and KL are dimensionless proportionality coefficients, and u*t is the threshold shear velocity (m s–1).
At the instant of particle motion, the aerodynamic forces FD, FL, and FM will exceed the retarding forces FG and FIp, and the particle will pivot about a downstream point of contact. At the instant of deflation, the moments are balanced:
 | [26] |
where a', b', and c' are moment arm lengths (m). When substituting Eq. [1], [15], and [22] to [25] into Eq. [26] and with a' = a1d, b' = b1d, and c' = c1d, the threshold shear velocity u*t or u*tw in the case of moist sediment, becomes:
 | [27] |
which can be simplified to
 | [28] |
where
 | [29] |
When grouping the different coefficients in Eq. [29] that are constant for a given soil as:
 | [30] |
 | [31] |
 | [32] |
and applying Eq. [20], the coefficient A can be written as:
 | [33] |
The values for the parameters A1 and A2 were determined by Cornelis and Gabriels (2004) using dry-sediment data and were 0.013 and 1.7 x 10–4 N m–1, respectively. Both the coefficients KAC and KvdW depend on particle shape, particle diameter, particle roughness, and interparticle distance, and A3 is therefore a geometry factor. Unless it is possible to quantify particle shape, particle diameter, particle roughness, and interparticle distance of a given bulk soil, A3 should be determined by curve fitting. Cornelis et al. (2004) found A3 to be equal to 3 x 1014 N–1 m–1.
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CONCLUSIONS
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Near-surface soil wetness is generally considered as an important factor in determining the onset of wind erosion. However, until now its effect was not well understood and when consulting international literature, very large differences can be observed between experimental results. In this study, existing theories on liquid bridge bonding and adsorbed layer bonding were reanalyzed. The model of Fisher (1926) for capillary forces FC, which are responsible for liquid-bridge bonding, was applied to express these forces as a function of capillary potential c. The Young–Laplace equation was solved to relate the radius of the waist of the water wedge between two particles to capillary potential. By imposing some assumptions, the capillary force was expressed as FC = KC 2 d2/|c|, where KC is a proportionality coefficient accounting for soil and fluid properties, is the liquid surface tension of the liquid and d is particle diameter. Although adsorbed-layer bonding is often considered as negligible for soil particles, a function similar to the capillary-force model was proposed for the adsorption force FA by considering tensile strength only: FA = KA 2 d2/|a|, where KA is another proportionality coefficient accounting for soil and fluid properties, and a is the potential due to adsorptive forces. The interparticle force due to wet bonding was then written as FAC = KAC 2 d2/|m|, where KAC is a proportionality coefficient accounting for soil and fluid properties lumping both KA and KC together, and m is the matric potential. Although this expression is a simplified representation of the interparticle force between particles of wetted sediment, we believe that this hypothesis is sufficiently appropriate for wind-erosion prediction.
Since determination of matric potential is a key limitation from a practical perspective, it was related to gravimetric water content w. A logarithmic relation was assumed including a correction to force the model to converge to zero water content at oven dryness (m –103 MPa). It was shown by considering data from literature that such relation described the water retention curve very well at matric potentials that are of interest in the light of deflation of soil particles. Matric potential was then expressed as a function of water content including only one unknown parameter, viz. water content at –1.5 MPa w1.5.
The wet-bonding force was incorporated into a moment balance equation for entrainment of soil particles by wind, including the moments associated with a drag force, a lift force, an aerodynamic moment force, the gravitational force, and an interparticle force due to dry and wet bonding. A conceptual model for threshold shear velocity, which included only one unknown proportionality coefficient A3 for the effect of wetness, was finally established. A primary advantage of our approach was that water content was indexed to matric potential by using w1.5 only, a parameter that can be rather easily determined. It was also pointed out that widely accepted equations for the water retention curve such as the models of Brooks and Corey (1964) are not applicable to predict deflation of soil particles as they fail to describe the water retention curve at water conditions below –1.5 MPa.
The advantage of relating the interparticle force to matric potential rather than to the capillary force and the adsorptive force separately, was that a minimum water content below which this force is dominant, could be disregarded. Furthermore, it allowed considering a single equation to relate potential to water content, which resulted in a relatively simple threshold shear velocity model.
Received for publication April 4, 2003.
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REFERENCES
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ABSTRACT
INTRODUCTION
