1 Introduction
Water movement in soils as described using hydrogeophysics (Cousin et al., 2009; Frangi et al., 2009) is related to their hydraulic properties which in turn are closely dependent on soil structure. Its high lateral and vertical variability in soils has led soil physicists to seek out physical indicators enabling the discussion of its characteristics, and more generally of the quality of physical properties (Dexter, 2004a, 2004b, 2004c; Dexter and Bird, 2001; Dexter and Czyz, 2007; Dexter et al., 2007; Reynolds et al., 2002). Among these indicators, the index proposed by Dexter and Bird (2001) and Dexter (2004a) enables the physical quality of soil (workability, permeability, structure stability, etc.) to be investigated and should be particularly effective for providing information on the soil hydric functioning. This index is the slope (S) of the soil-water retention curve (SWRC) at its inflection point. It is determined for the SWRC when the gravimetric water content (W), a function of soil-water suction (h) and expressed using the van Genuchten equation (Richard et al., 2001), is plotted with the natural logarithm of h. In this study we use W to denote the gravimetric water content, rather than θ, as in Dexter and Bird (2001), to be more consistent with the literature since θ usually represents the volumetric water content. As for the van Genuchten equation (van Genuchten, 1980) which was written for θ, it remains valid for W.
Dexter (2004a) derived the expression of the slope of the SWRC analytically to calculate the value of S, thus leading to the following expression:
$$S=-n\left({W}_{s}-{W}_{r}\right){\left[1+\frac{1}{m}\right]}^{-\left(1+m\right)}$$ | (1) |
Dexter and Bird (2001), however, noted that there were two possible inflection points depending on whether W is plotted against log(h) or against h. They reported that the two inflection points are in close proximity for soils with a narrow pore-size distribution. This explains why they used the inflection point of curves of W vs. log(h), believing this was an estimate of air entry into granular materials which were considered in their study. Another point not raised by Dexter and Bird (2001) concerned their choice for computing the slope in a graph W vs. ln(h) of the W curve as a function of h according to the van Genuchten equation (Dexter and Czyz, 2007; Dexter et al., 2007) instead of the slope of the W curve vs. ln(h) which would have been mathematically more consistent.
In this study we discuss the choice of Dexter and Bird (2001) and compare the S-index with the slope of the SWRC at its inflection point when it is expressed as a function of the independent variable h, ln(h) or log(h). The equations developed are applied to a non-compacted and compacted soil and the resulting values of the slope are compared to the S-index.
2 Theory
2.1 Expression of W according to h, ln(h) and log(h)
On the basis of the van Genuchten equation (van Genuchten, 1980), W can be expressed as:
$$W=\left({W}_{s-}{\stackrel{\u02c6}{W}}_{r}\right){\left[1+{\left(\stackrel{\u02c6}{\alpha}h\right)}^{\stackrel{\u02c6}{n}}\right]}^{-\stackrel{\u02c6}{m}}+{\stackrel{\u02c6}{W}}_{r}$$ | (2) |
$$W=g\left(\left.h\right|{W}_{s},{\stackrel{\u02c6}{W}}_{r},\stackrel{\u02c6}{n},\stackrel{\u02c6}{\alpha},\stackrel{\u02c6}{m}\right)$$ | (3) |
Similarly, W vs. ln(h), can be represented by f using Eq. (2) as:
$$W=f\left(\mathrm{ln}(h)\left|{W}_{s},{\stackrel{\u02c6}{W}}_{r1},{\stackrel{\u02c6}{n}}_{1}\right.,{\stackrel{\u02c6}{\alpha}}_{1},{\stackrel{\u02c6}{m}}_{1}\right)$$ | (4) |
$$W=k\left(\mathrm{log}(h){\left|{W}_{s},{\stackrel{\u02c6}{W}}_{r1},n{}_{1},\stackrel{\u02c6}{\alpha}\right.}_{2},{\stackrel{\u02c6}{m}}_{1}\right)$$ | (5) |
2.2 Derivation of the SWRC to obtain the inflection point
Taking Eqs. (3), (4), and (5) as general representations of the WRC and using Eq. (4), we can write the following derivatives:
$$\dot{g}=\frac{dW}{dh}=-\stackrel{\u02c6}{m}\stackrel{\u02c6}{n}\left({\stackrel{\u02c6}{\alpha}}^{\stackrel{\u02c6}{n}}\right)\text{\hspace{0.17em}}\left({W}_{s}-{\stackrel{\u02c6}{W}}_{r}\right)\left\{{\left[1+{\left(\stackrel{\u02c6}{\alpha}h\right)}^{\stackrel{\u02c6}{n}}\right]}^{-\stackrel{\u02c6}{m}-1}\right\}\mathrm{ln}\left({h}^{\stackrel{\u02c6}{n}-1}\right)$$ | (6) |
$$\dot{g}=\frac{dW}{d\left(\mathrm{ln}\left(h\right)\right)}={\stackrel{\u02c6}{m}}_{1}{\stackrel{\u02c6}{n}}_{1}\left({{\stackrel{\u02c6}{\alpha}}_{1}}^{{\stackrel{\u02c6}{n}}_{1}}\right)\left({W}_{s}-{\stackrel{\u02c6}{W}}_{r1}\right)\left\{\left[1+{\left({\stackrel{\u02c6}{\alpha}}_{1}\mathrm{ln}\left(h\right)\right)}^{-\stackrel{\u02c6}{m}-1}\right]\right\}\mathrm{log}{(h)}^{{\stackrel{\u02c6}{n}}_{1}-1}$$ | (7) |
$$\dot{k}=\frac{dW}{d\left(\mathrm{log}\left(h\right)\right)}=-{\stackrel{\u02c6}{m}}_{1}{\stackrel{\u02c6}{n}}_{1}\left({{\stackrel{\u02c6}{\alpha}}_{2}}^{{\stackrel{\u02c6}{n}}_{1}}\right)\left({W}_{s}-{\stackrel{\u02c6}{W}}_{r1}\right)\left\{{\left[1+\left({\stackrel{\u02c6}{\alpha}}_{2}\mathrm{log}{\left(h\right)}^{{\stackrel{\u02c6}{n}}_{1}}\right)\right]}^{-{\stackrel{\u02c6}{m}}_{1}-1}\right\}\mathrm{log}{\left(h\right)}^{{\stackrel{\u02c6}{n}}_{1}-1}$$ | (8) |
It is important to state that $\dot{f}=dW/d\left(\mathrm{ln}\left(h\right)\right)$ cannot be computed by simply applying the chain rule from Eq. (3) because the parameters determined by fitting either g (Eq. (3)) or f (Eq. (4)), subjected to Eq. (2), are not necessarily the same. This can be also said for functions g (Eq. (3)) and k (Eq. (5)), except that here the only difference between $\dot{f}$ and $\dot{k}$ is the magnitude of the scaling parameters ${\stackrel{\u02c6}{\alpha}}_{1}$ and ${\stackrel{\u02c6}{\alpha}}_{2}$.
It is known that any continuous and differentiable mathematical function has its inflection points located where the second derivative is null throughout its real domain. Thus, at the inflection points for function g, we can set:
$$\dot{g}=\frac{{d}^{2}W}{d{h}^{2}}=-\stackrel{\u02c6}{m}\stackrel{\u02c6}{n}\left({\stackrel{\u02c6}{\alpha}}^{\stackrel{\u02c6}{n}}\right)\left({W}_{s}-{\stackrel{\u02c6}{W}}_{r}\right)\left\{\left(-\stackrel{\u02c6}{m}\stackrel{\u02c6}{n}-\stackrel{\u02c6}{n}\right)\left({\stackrel{\u02c6}{\alpha}}^{\stackrel{\u02c6}{n}}\right){\left[1+{\left(\stackrel{\u02c6}{\alpha}h\right)}^{\stackrel{\u02c6}{n}}\right]}^{-\stackrel{\u02c6}{m}-2}{h}^{2\stackrel{\u02c6}{n}-2}+\left(\stackrel{\u02c6}{n}-1\right)\left({h}^{\stackrel{\u02c6}{n}-2}\right){\left[1+{\left(\stackrel{\u02c6}{\alpha}h\right)}^{\stackrel{\u02c6}{n}}\right]}^{-\stackrel{\u02c6}{m}-1}\right\}=0$$ | (9) |
$$\left(-\stackrel{\u02c6}{m}\stackrel{\u02c6}{n}-\stackrel{\u02c6}{n}\right)\left({\stackrel{\u02c6}{\alpha}}^{\stackrel{\u02c6}{n}}\right){\left[1+\left(\stackrel{\u02c6}{\alpha}h\right)\stackrel{\u02c6}{n}\right]}^{-\stackrel{\u02c6}{m}-2}\left({h}^{2\stackrel{\u02c6}{n}-2}\right)+\left(\stackrel{\u02c6}{n}-1\right)\left({h}^{\stackrel{\u02c6}{n}-2}\right){\left[1+{\left(\stackrel{\u02c6}{\alpha}h\right)}^{\stackrel{\u02c6}{n}}\right]}^{-\stackrel{\u02c6}{m}-1}=0$$ | (10) |
Eq. (10) can be solved for h to obtain the precise location of its inflection point (h)_{i} as follows:
$${\left(h\right)}_{i}=\frac{1}{\stackrel{\u02c6}{\alpha}}{\left(\stackrel{\u02c6}{m}\right)}^{\frac{1}{{\stackrel{\u02c6}{n}}_{1}}}$$ | (11) |
Similarly, we can compute the second derivative of Eqs. (7) and (8) to obtain:
$${\left(\mathrm{ln}\left(h\right)\right)}_{i}=\frac{1}{{\stackrel{\u02c6}{\alpha}}_{1}}{\left({\stackrel{\u02c6}{m}}_{1}\right)}^{\frac{1}{{\stackrel{\u02c6}{n}}_{1}}}$$ | (12) |
$${\left(\mathrm{log}\left(h\right)\right)}_{i}=\frac{1}{{\stackrel{\u02c6}{\alpha}}_{2}}{\left({\stackrel{\u02c6}{m}}_{1}\right)}^{\frac{1}{{\stackrel{\u02c6}{n}}_{1}}}$$ | (13) |
2.3 Calculation of the slope at the inflection point of the SWRC
The slope, S_{h}, from function g (Eq. (3)) at its inflection point (Eq. (11)) is obtained by substituting Eq. (11) into Eq. (6), which yields:
$${S}_{h}=-\left(\stackrel{\u02c6}{\alpha}\right)\left(\stackrel{\u02c6}{n}-1\right)\left({W}_{s}-{\stackrel{\u02c6}{W}}_{r}\right){\left(\stackrel{\u02c6}{m}\right)}^{\stackrel{\u02c6}{m}}{\left(1+\stackrel{\u02c6}{m}\right)}^{-\stackrel{\u02c6}{m}-1}$$ | (14) |
Similarly, the slope, S_{ln(h)}, from function f (Eq. (4)) at its inflection point (Eq. (12)), is obtained by substituting Eq. (12) into Eq. (7):
$${S}_{\mathrm{ln}\left(h\right)}=-\left(\stackrel{\u02c6}{\alpha}\right)\left({\stackrel{\u02c6}{n}}_{1}-1\right)\left({W}_{s}-{\stackrel{\u02c6}{W}}_{r1}\right){\left({\stackrel{\u02c6}{m}}_{1}\right)}^{{\stackrel{\u02c6}{m}}_{1}}{\left(1+{\stackrel{\u02c6}{m}}_{1}\right)}^{-{\stackrel{\u02c6}{m}}_{1}-1}$$ | (15) |
The slope, S_{log(h)}, from function k (Eq. (5)) at its inflection point (Eq. (13)) is obtained by introducing Eq. (13) into Eq. (8):
$${S}_{\mathrm{log}\left(h\right)}=-\left({\stackrel{\u02c6}{\alpha}}_{2}\right)\left(\stackrel{\u02c6}{n}-1\right)\left({W}_{s}-{\stackrel{\u02c6}{W}}_{r1}\right){\left({\stackrel{\u02c6}{m}}_{1}\right)}^{{\stackrel{\u02c6}{m}}_{1}}{\left(1+{\stackrel{\u02c6}{m}}_{1}\right)}^{-{\stackrel{\u02c6}{m}}_{1}-1}$$ | (16) |
3 Application to a case study
3.1 The soil and methods used
The equations developed in this study were applied to samples from a cultivated soil where compacted layers were identified (Balbino et al., 2002). The soil studied was a clayey Oxisol (Typic Acrustox) (Soil Survey Staff, 1998), a Latossolo Vermelho according to the Brazilian Soil Classification (Santos et al., 2006) and a Ferralsol according to the IUSS-WRB (2006) soil classification. It was located on a private farm (latitude 16.493246 S, longitude 49.310337 W, and altitude 776 m), near the Embrapa Arroz e Feijão Agricultural Research Center, at Santo Antônio de Goiás, GO, Brazil. The native vegetation was a typical Cerrado until 1985. After clearing the land, the soil was occupied by annual crops with conventional tillage for two years and then by a pasture of Brachiaria decumbens cv. Basilisk stapf. The soil was managed according to intensive animal grazing without any addition of fertilizer. This management led to a compaction of the topsoil. In 2006, soil cores were collected with stainless steel 100 cm^{3} cylinders (diameter = 5.1 cm, height = 5.0 cm) in the compacted 0–5 cm and non-compacted 70–75 cm layers (Table 1). The higher bulk density found in the 0–5 cm layer is accounted for soil compaction since under native vegetation, this type of soil exhibits a uniform bulk density profile according to depth, with bulk density close to 1.0 g cm^{−3} (Balbino et al., 2002; Volland-Tuduri et al., 2005).
Principales caractéristiques physico-chimiques de l’horizon 0–5 cm compacté et de l’horizon 70–75 cm non compacté, sélectionnés dans le sol étudié.
Soil | Particle size distribution^{a} | Organic carbon^{a} | Bulk density^{b} | ||
Clay | Silt | Sand | |||
Compacted | 485 | 71 | 444 | 0.70 | 1.27 |
Non-compacted | 549 | 72 | 380 | 0.16 | 1.03 |
^{a } g kg^{−1}.
^{b } g cm^{−3}.
Gravimetric water contents (W in g g^{−1}) at −10, −30, −60, −100, −330, −800, −4000, −10 000, and −15 000 hPa were determined in triplicate for the two layers studied (Table 2) using the centrifuge method (Reatto et al., 2008; Silva et al., 2006). An SWRC was fitted using the van Genuchten equation (van Genuchten, 1980) (see Eq. (2)) to the different water contents measured for the compacted and non-compacted layers, using h, ln(h) or log(h) as independent variable. The Solver routine embedded in Microsoft Excel was used to obtain the fitting parameters ${\stackrel{\u02c6}{W}}_{r}$, $\stackrel{\u02c6}{\alpha}$, $\stackrel{\u02c6}{n}$, and $\stackrel{\u02c6}{m}$ (Table 3). During the fitting process, W_{s} was taken as the mean value of the three saturated water contents measured (Reatto et al., 2008): 0.367 g g^{−1} and 0.544 g g^{−1} for the compacted and non-compacted layer, respectively, and therefore was not adjusted.
Teneur en eau gravimétrique du sol (W g g^{−1}) des cylindres de sol provenant de l’horizon 0–5 cm compacté (C) et de l’horizon 70–75 cm non compacté (NC), en fonction de la succion (hPa).
Suction (hPa) | W – Compacted layer | W – Non-compacted layer | ||||
Replicate 1 | Replicate 2 | Replicate 3 | Replicate 1 | Replicate 2 | Replicate 3 | |
0 | 0.366 | 0.356 | 0.380 | 0.558 | 0.549 | 0.523 |
10 | 0.356 | 0.346 | 0.355 | 0.540 | 0.542 | 0.516 |
30 | 0.332 | 0.324 | 0.306 | 0.456 | 0.455 | 0.432 |
60 | 0.297 | 0.290 | 0.281 | 0.337 | 0.360 | 0.333 |
100 | 0.277 | 0.287 | 0.271 | 0.287 | 0.278 | 0.277 |
330 | 0.237 | 0.242 | 0.237 | 0.240 | 0.231 | 0.227 |
800 | 0.222 | 0.230 | 0.226 | 0.218 | 0.213 | 0.214 |
4000 | 0.199 | 0.206 | 0.204 | 0.201 | 0.195 | 0.195 |
10 000 | 0.185 | 0.190 | 0.190 | 0.193 | 0.184 | 0.187 |
15 000 | 0.178 | 0.180 | 0.181 | 0.182 | 0.176 | 0.175 |
Valeurs estimées des paramètres pour W en fonction de h, ln(h), ou log(h), et valeurs correspondantes des points d’inflexion et des valeurs S pour l’horizon 0–5 cm compacté (C) et de l’horizon 70–75 cm non compacté (NC).
Variables | Independent variable^{h} | Dexter (2004a) | |||||||
h | ln(h) | log(h) | |||||||
C | NC | C | NC | C | NC | C | NC | ||
W_{s}(g g^{−1}) | 0.367 ± 0.012 | 0.544 ± 0.018 | 0.367 ± 0.012 | 0.544 ± 0.018 | 0.367 ± 0.012 | 0.544 ± 0.018 | 0.367 ± 0.012 | 0.544 ± 0.018 | |
W_{r} (g g^{−1}) | 0.160 ± 0.010 | 0.192 ± 0.004 | 0.147 ± 0.012 | 0.188 ± 0.004 | 0.147 ± 0.012 | 0.188 ± 0.006 | 0.147 ± 0.010 | 0.188 ± 0.004 | |
n ^{a} | 1.314 ± 0.045 | 2.057 ± 0.088 | 3.182 ± 0.254 | 6.396 ± 0.364 | 3.182 ± 0.254 | 6.396 ± 0.364 | 3.182 ± 0.045 | 6.396 ± 0.088 | |
α ^{b} | 0.057 ± 0.009 | 0.032 ± 0.002 | 0.227 ± 0.006 | 0.263 ± 0.003 | 0.524 ± 0.013 | 0.606 ± 0.008 | |||
m ^{a} | 0.239 ± 0.025 | 0.514 ± 0.020 | 0.686 ± 0.023 | 0.844 ± 0.008 | 0.686 ± 0.023 | 0.844 ± 0.008 | 0.686± 0.025 | 0.844 ± 0.020 | |
Suction at the inflection point^{c} | 5.876 | 22.421 | 3.948 | 3.699 | 1.696 | 1.606 | 1.715 | 1.632 | |
Slope at the inflection point^{d} | 0.0020 | 0.0046 | 0.035 | 0.142 | 0.0805 | 0.3261 | 0.0816 | 0.329 | |
Equivalent pore diameter at the inflection point^{e} | 510 | 134 | 60 | 74 | 60 | 74 | 58 | 70 | |
Water content at the inflection point^{f} | 0.300 | 0.394 | 0.266 | 0.373 | 0.266 | 0.365 | 0.266 | 0.365 | |
RMSE^{g} | 0.0065 | 0.0114 | 0.0065 | 0.0106 | 0.0065 | 0.0106 | 0.0065 | 0.0114 | |
R^{2} | 0.987 | 0.991 | 0.988 | 0.992 | 0.988 | 0.992 | 0.987 | 0.988 |
^{a } Dimensionless.
^{b } Units in hPa^{−1} for h; ln hPa^{−1}for ln h; and log hPa^{−1} for log h.
^{c } Units in hPa for h; ln hPa for ln h; and log hPa for log h.
^{d } Units for S_{h} (g g^{−1} hPa^{−1}); S_{ln h} (g g^{−1} ln hPa^{−1}); or S_{log h} (g g^{−1} log hPa^{−1}).
^{e } Unit in μm.
^{f } ^{g} $RMSE=\sqrt{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$N$}\right.\sum _{i=1}^{N}{\left({\stackrel{\u02c6}{W}}_{i}-{W}_{i}\right)}^{2}\text{\hspace{0.28em}}}\text{in}\text{\hspace{0.28em}}\text{g}{\text{\hspace{0.28em}g}}^{-1}\text{.}$
^{h } The standard errors for W_{s} were calculated directly from the measured values. Those for W_{r}, n, α, and m originated from the analysis of variance of errors due to regression when fitting these parameters.
3.2 Comparison of the different S-index values obtained
At this point, it should be remembered that Dexter and Bird (2001) and Dexter (2004a) derived the S-index formulation from the slope of SWRC plotted in an ln scale, and the result was transformed to a log scale by multiplying it by ln 10; this log scale was then used afterwards. In order to compare and discuss the location of the inflection point according to the independent variable used, we applied the equations developed here and those of Dexter and Bird (2001) and Dexter (2004a) to water retention properties found for compacted and non-compacted soils (Table 2).
The S-index computed using Eq. (1) and multiplied by ln 10 according to Dexter (2004a) was 0.082 and 0.329 for the compacted and non-compacted soils. Using Eq. (16), the slope at the inflection point of the SWRC expressed according to log(h) as independent variable was 0.081 and 0.326 for the compacted and non-compacted soils. These values are very close to the S-index computed as described by Dexter (2004a). Thus, using an equation of W fitted with h as independent variable and plotted with log(h) as abscissa, or an equation of W fitted with log(h) as independent variable and plotted according to log(h), the slopes of the two curves at the inflection point are very similar. This could be expected since the experimental points remain at the same place in the W – log(h) graph regardless of the independent variable used for the equation to describe the SWRC. Consequently, the slope at the inflection point of the SWRC computed according to Dexter (2004a) to lead to the S-index and used by many authors would have been similar using Eq. (16) instead of Eq. (1).
On the other hand, the location of the inflection point of the curve of W vs. h, and the slope of the curve at this point, have more physical meaning than the corresponding values computed by Dexter (2004a). The value of h at the inflection point can be considered as the “breakthrough” matrix potential at which air penetrates throughout the soil as discussed by White et al. (1972) and Dullien (1992). The slopes at the inflection point of the SWRC using Eq. (14) (using h as independent variable) were 0.0020 and 0.0046 for the compacted and non-compacted soil. These values are 41 and 72 times smaller than the corresponding S-index values (Table 3). Suction at the corresponding inflection point using Eq. (11) was 6 and 22 hPa for the compacted and non-compacted soil, while according to Dexter (2004a) it was 52 and 43 hPa (Table 3).
Using Jurin's law (Bruand and Prost, 1987), we computed the equivalent pore diameter corresponding to the suction at the inflection point of the SWRC (Table 3). The results showed a close equivalent pore diameter for compacted and non-compacted soil at the inflection point when the SWRC was plotted with ln(h) or log(h) as independent variable (60 and 74 μm) and according to Dexter (2004a) (58 and 70 μm). On the other hand, the equivalent pore diameter at the inflection point of the SWRC was about four times higher for compacted soil (510 μm) than for the non-compacted soil (134 μm) when the SWRC was plotted with h as independent variable (Table 3).
In contrast to what is indicated by the S-index, however, air would penetrate throughout the soil at a smaller suction, and consequently for a larger equivalent pore diameter for compacted than for non-compacted soil. This result may appear surprising since compaction leads to smaller porosity with a shift of the inflection point on the SWRC to larger suction (Bruand and Cousin, 1995; Cui et al., 2010; Richard et al., 2001). The effects of compaction on pore geometry are difficult to understand since they depend on the structure and related pore types prior to compaction, on soil composition and water content, and on the intensity of compaction.
Beneath native vegetation, the soil studied had a weak macrostructure and a pronounced granular structure at the micrometer scale (Balbino et al., 2002; Volland-Tuduri et al., 2005). Since the structure of non-compacted soil is considered as similar to the structure under native vegetation, its theoretical SWRC would be a bimodal curve with two inflection points:
- • the first inflection point would correspond to pore draining resulting from the assemblage of the micro-aggregates and occurring for a very low suction of several hPa such as for coarse sandy soils;
- • the second corresponding to pore draining resulting from the assemblage of elementary particles in micro-aggregates and occurring for values of several hundred hPa. Because of the difficulty to correctly measure water retention of the soils studied at several hPa, the second inflection point is the only one that is usually measured (Balbino et al., 2002).
When soil is compacted, the pores resulting from the assemblage of micro-aggregates are transformed into smaller pores (Bruand and Cousin, 1995; Richard et al., 2001). The resulting SWRC contains one inflection point, which is related to a continuous distribution of equivalent pore diameters from the smaller pores, which were distorted by compaction into those resulting from the assemblage of the elementary particles in micro-aggregates. Fig. 1, based on the results of several studies on similar soils (Balbino et al., 2001; Volland-Tuduri et al., 2004, 2005), illustrates how using such a transformation of porosity makes it possible to pass from a SWRC with a given inflection point and its related equivalent pore diameter for a non-compacted soil, to another SWRC with an inflection corresponding to a larger equivalent pore diameter for compacted soil.
Finally, our results question the value of S as a possible index to determine the physical quality of soil. The values of h at the inflection point determined for compacted and non-compacted soil are low, thus corresponding to a water content close to saturation which should not be optimal for soil tillage.
4 Conclusion
Our results show that the expression of the SWRC according to ln(h) or log(h) instead of h as independent variable leads to different values of the S-index. Computing the S-index when the SWRC is expressed with h as independent variable is both mathematically and physically consistent. We also show that independently of the consistency of the approach, the discussion of the physical properties of the soil can thus be limited according to the independent variable used. For the soil selected, our results in fact show that calculation of the S-index when it is expressed with h as independent variable significantly increases the relevance of the analysis compared to the range of the S-indices when it is expressed as proposed by Dexter (2004a). Further work will aim at determining in which proportion the S-index is affected for a large range of soils and verifying if the use of h as independent variable effectively increases sensitivity of the analysis.
Acknowledgments
This research is part of the project Embrapa Cerrados-IRD, No. 0203205 (Mapping of the Biome Cerrado Landscape and Functioning of Representative Soils). This work was supported by a grant awarded to G.G. Santos by the CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil)