INTRODUCTION

Several equations are available to describe the relation between matric potential and soil water content, the soil water retention function. The most frequently applied ones are the Brooks and Corey (1964) equation (BC) and the ^{van Genuchten (1980)} equation (VG). When expressing effective saturation Θ, a quantity scaling water content from 0 to 1 between residual and saturated values, the BC equation contains two parameters, one of which (*h*_{b}) explicitly represents the matric potential corresponding to air-entry. On the other hand, the VG equation, to be applied using the ^{Mualem (1976)} or ^{Burdine (1953)} parametric restriction, holds also two parameters (α and *n*), but none of them has a clear physical meaning.

In Brazilian soil physics literature, several authors relate the inverse of parameter α to the air-entry pressure, in analogy to the BC parameter *h*_{b}. Some use this kind of identification when describing VG parameters (Souza et al., 2008a; ^{Lima et al., 2014}; Oliveira Júnior et al., 2014), others when explicitly interpreting results of α values to the air-entry pressure (Souza et al., 2008b; ^{Silva et al., 2009}; ^{Mota et al., 2017}). In international literature, a similar description can sometimes be found (^{Pollaco and Mohanty, 2012}; ^{Aschonitis and Antonopoulos, 2013}; ^{Aschonitis et al., 2015}; ^{Dokoohaki et al., 2017}).

Here, we demonstrate that this interpretation of the VG α parameter is incorrect and should therefore be avoided.

DEVELOPMENT

The air-entry pressure, or “bubbling pressure” *h*_{b} (m), of a soil or porous material is defined as the matric potential at which the first (largest) pore starts draining its water (Brooks and Corey, 1964). Considering the Young-Laplace capillary equation (Equation 1), it is determined by the radius of the largest pore *r*_{m} (m) as:

in which σ (J m^{-2}) is the surface tension of water, φ the contact angle between the water surface, the surrounding air, and the pore walls, r (kg m^{-3}) the density of water, and *g* (m s^{-2}) the gravity. In some water retention models, *h*_{b} (m) is an explicit fitting parameter, notably in the Brooks and Corey (1964) model (Equation 2):

in which *h* is the matric potential, Θ = (θ – θ_{r})/(θ_{s} – θ_{r}) is the effective saturation, θ, θ_{r}, and θ_{s} are water content, residual water content, and saturated water content, respectively, all on a volume base (m^{3} m^{-3}). The air-entry pressure corresponds to the onset of water content reduction with further decreasing matric potentials. As such, the water content at the air-entry pressure θ_{b} (m^{3} m^{-3}) equals the saturated water content θ_{s} and Θ = 1 at *h* = *h*_{b} (Equation 2).

The air-entry pressure is not explicitly present in the frequently used ^{van Genuchten (1980)} water retention equation (VG, Equation 3):

in which a, *n*, and *m* (function of *n*) are fitting parameters, a having the inverse dimension of *h* (e.g. m^{-1}). The VG equation is defined together with the theory presented by ^{Mualem (1976)} or ^{Burdine (1953)}, and when applying the respective parametric restrictions (defining *m* as a function of *n*), it can be used to estimate the hydraulic conductivity function from retention parameters. The Mualem restriction is as follows (Equation 4):

while the Burdine restriction is (Equation 5):

Consequently, 0< *m* <1. As mentioned, many authors assume that α is the inverse of the absolute value of the air-entry pressure *h*_{b}, i.e. (Equation 6):

This assumption has its origin in a comparison between equation 2 and 3. If |*h*| becomes very large, equation 3 reduces to equation 7:

Equation 7 is equal to equation 2, with λ = *mn* (or, *l* = *n* - 1) with the Mualem restriction, equation 4, and λ = *n* - 2 with the Burdine restriction, equation 5 and a given by equation 6. However, this does not justify the interpretation of a as the inverse of the bubbling pressure, as equation 7 is only valid for very large values of |*h*|, whereas |*h*_{b}| is, in fact, a relatively small value. Fitting soils with several textures from the Hydrus package (^{Šimůnek et al., 2016}), values of |*h*_{b}| range between 0.05 and 0.4 m (Figure 1), corresponding to pore diameters of 5.87 ∙ 10^{-4} and 7.35 ∙ 10^{-5} m, respectively.

Moreover, if the interpretation of a as the inverse of the bubbling pressure were true, then combining equation 6 to equation 3 would result in the following expression (Equation 8) for the effective saturation, corresponding to the bubbling pressure Θ_{b}:

Equation 8 yields values for Θ_{b} between 1 (at *m* = 0) and 0.5 (at *m* = 1), as shown in figure 2, and in obvious disagreement with the notion that θ_{b} = θ_{s}; consequently, Θ_{b} = 1. The value of *m* = 0 implies in *n* = 1 (Mualem restriction, equation 4) or *n* = 2 (Burdine restriction, equation 5). Such values are physically unrealistic, as *m* = 0 results in Θ = 1 for any value of *h* (Equation 7). Equation 8 implies in the fact that the greater the value of *m* (and, from equations 4 and 5, the greater the value of *n*), the larger the deviation between the inverse of a and the air-entry pressure. This can also be seen in figure 3, showing the retention curves (Θ as a function of the matric potential) for two values of parameter a and *n* = 2 (top) and *n* = 5 (bottom). In this figure, 1/a indicates the supposed values of the air-entry pressure according to equation 6, with corresponding Θ_{b} given by equation 8. Figure 2 also clearly demonstrates the effect of a on the shape of the retention curve, with a being a mere scaling parameter relative to the matric potential axis.

CONCLUSION

We showed, mathematically and graphically, that the Van Genuchten retention equation parameter a is not equal to, nor simply correlated to the (inverse of) air-entry matric potential, as frequently alleged. Instead, a is a scaling parameter relative to the matric potential axis. Recognizing this mathematical fact may improve the interpretation of soil hydraulic properties based on water retention parameters and prevent the error of using the relationship shown in equation 6 to correlate parameters from equations 2 and 3.