Introduction

The basic Equation for the mechanicist study of water transfer processes in the unsaturated zone of the soil is established by combining the general principle of mass conservation and the Darcy-Buckingham law. Applying the transfer equation requires a representation of the hydraulic properties of the soil, with the volumetric water content (θ) expressed as a function of the water pressure head in the soil (ѱ; soil-water retention curve) and hydraulic conductivity (K), as a function of θ or ѱ (hydraulic conductivity curve). The bibliography contains analytical models to describe the hydraulic properties of the soil, which vary conceptually and in sophistication. Regarding this, ^{Assouline and Or (2013)} performed their revision. That set of models covers pedotransfer functions (^{Sobieraj et al. 2001}, ^{Wösten et al. 2001}, ^{Tóth et al., 2015}) and mechanicist models based on the Poiseuille and Laplace laws (^{Brooks and Corey 1964}; ^{van Genuchten 1980}; ^{Braddock et al., 2001}). The first group of models is established without explicitly considering the physical bases of water movement in the soil and the second group helps represent in detail the basic mechanisms of the process from laws of physics.

Mechanicist models for hydraulic properties are available in computer programs that simulate the water flow in the unsaturated zone (^{Diersch, 2014}). The descriptive ability of groups of models for hydraulic properties was also studied (^{Too et al., 2014}). However, the rationale behind the models in the context of the theory of infiltration was considered to a limited extent, since most analyses evaluate only their flexibility to describe experimental data in the field or the lab.

^{Fuentes et al. (1992)} analyzed the mechanicist models reported in the bibliography for some hydraulic properties of the soil. These authors showed that only the models by ^{Fujita (1952)} and ^{Parlange et al. (1982)} and van Genuchten’s soil-water retention curve (^{van Genuchten, 1980}), subjected to Burdine’s restriction (^{Burdine, 1953}), combined with Brooks and Corey’s conductivity curve (^{Brooks and Corey, 1964}), satisfy the integral properties of infiltration. Even if the hydraulic conductivity at a (K_s) saturation is known, the application of these models requires the estimation of two shape and one scale parameter, simultaneous with experimental series of soil-water retention and hydraulic conductivity to determine them.

^{Fuentes et al. (2001)} studied hydraulic conductivity based on the Poiseuille and Darcy laws in the context of fractal geometry, representing the porous space of the soil as a system of capillaries. They obtained a conceptual model that helped them establish, using the weight given to the radii of the capillaries in the resistance to the flow of water in the soil, the integral fractal relations named neutral pore, geometric pore, and large pore. The first model corresponds to Burdine’s integral relationship for hydraulic conductivity (^{Burdine, 1953}) with a correction factor; the second, to the Mualem’s relation (^{Mualem, 1976}), also with a correction factor; and the third was a new model, established by the authors of the study. Taking into account particular constraints between the parameters of the shape of van Genucthen’s soil-water retention curve (^{van Genucthen, 1980}), analytical models for hydraulic conductivity were derived. These relations between m and n, helped explain that the corrections of the analytical models for conductivity derived by ^{Van Genucthen (1980)} depend on the fractal dimension of the soil. The fractal models by ^{Fuentes et al. (2001)} for hydraulic conductivity satisfy the integral properties of infiltration and with known K_s only two parameters must be determined, one of which is form, and the other scale, and only the experimental soil-water retention curve is required to calculate them.

The aim of this study was to evaluate the descriptive ability of the fractal models by ^{Fuentes et al. (2001)} named neutral pore, geometric pore and large pore, with experimental information from the UNSODA database (^{Leij et al., 1996}; ^{Nemes et al., 2001}). The evaluation consisted in calibrating the parameters of van Genuchten’s retention model (^{van Genuchten, 1980}) subjected to its fractal restrictions, applying the analytical model for hydraulic conductivity and comparing its predictions with the experimental data. The hypothesis was that with the calibration of the two parameters that intervene in the soilwater retention fractal models, the relative hydraulic conductivity relations can be applied, accurately predicting the evolution of this variable.

Materials and methods

The Equation that describes the water transfer process in unsaturated soils is:

where H=ѱ-z is the hydraulic head [L]; ѱ is the water pressure head in the soil [L]; z is the positive vertical coordinate [L] in a descending direction; θ is the volumetric water content [L³L^-3] which is a function of the pressure head, a relation known as soil-water retention curve; K is the hydraulic conductivity curve of the soil [LT^-1], which is a function of the pressure or the volumetric water content; θ(ѱ) and K(θ) are also known as the hydraulic properties of the soil; S is the volume of water extracted by the plant roots per unit of volumetric soil in time unit [L3L^-3T^-1]; ∇=∂∂x,∂∂y,∂∂z is the gradient operator [L^-1]; and t is time [T].

In order to formally consider the existing relation between the geometry of the porous soil and the hydraulic properties of the soil, two fundamental laws must be taken into consideration:

1) Poiseuille’s law, which relates the mean water velocity (v) in a cylindrical capillary of a radius (r) with the energy gradient ∇H:

where ρ_w is water density [ML^-3]; µ is the dynamic viscosity [ML^-1T^-1]; and g is the acceleration of gravity [LT^-2].

2) Laplace’s law, which relates the pressure head with the radius of curvature r_c of the meniscus of water in a pore with a radius of r:

where σ is the surface tension in the water-air interface [MT^-2]. The radii of curvature and pore are related through the angle of contact a that exists between the meniscus of the water and the soil particles: r=r_c cos a.

The models by ^{Fuentes et al. (2001)} are established by representing the soil as a fractal object and considering Equations (2) and (3). The analytical representations obtained by introducing van Genuchten’s soil-water retention curve (^{van Genuchten, 1980}) in the general fractal models of hydraulic conductivity K(S_e) are:

Model of the geometric pore:

Model of the neutral pore:

Model of the large pore:

where m and n are parameters of shape; ѱ_d is the parameter of scale of the pressure [L]; S_e(ѱ)=[(θ(ѱ)- θ_r)/(θ_s- θ_r)] is the effective saturation; θ_s is the saturated water content [L³L^-3]; θ_r is the residual water content [L³L^-3]; Ks is the saturated hydraulic conductivity [LT^-1]; 0≤K_r≤1 is the relative hydraulic conductivity; s is the relative dimension 0≤s=D/E≤1, where D is the fractal dimension of the object (soil) and E is the dimension of the Euclidean space. The study established that the relative dimension of the soil is an implicit function within its volumetric porosity:

Equation (7) was solved with the Newton-Raphson numerical method (^{Burden et al., 2015}), and in order to accelerate the convergence of the calculation process for the relative dimension, the initial estimator of s=log12-1+5/log12≈0.6942 was used, which is the exact solution of the function for the particular value of the volumetric porosity ϕ =1/2.

The experimental evidence to calibrate and validate models (4)-(6) was taken from the UNSODA database version 2.0, presented by ^{Nemes et al. (2001)}, who improved the original version, developed by ^{Leij et al. (1996)}. Version 2.0 contains experimental information from 790 soils, covering the 12 textural classes defined by the United States Department of Agriculture (USDA), with 184 sands, 64 loamy sands, 133 sandy loams, 52 sandy-clay-loam, 3 silts, 142 silt loams, 36 clay loams, 70 loamy, 33 silty-clay-loam, 3 sandy clays, 21 silty clay and 39 clays.

From the experimental information in the UNSODA base, 208 soils were selected (Table 1). The soils selected presented experimental information on granulometry, soil-water retention θ(ѱ), water conductivity K(θ), volumetric porosity (ϕ), saturated water content (θ_s) and saturated hydraulic conductivity (K_s). Another criterion was to consider only soils with experimental data on soil-water retention and hydraulic conductivity estimated from the analysis of similar water flow processes in the soil (draying tests of unsaturated or saturated soils), so they were compatible between both experimental series and minimizing the effect of the capillary hysteresis that stands out in the soil-water retention curve.

Results and discussion

Parameters m and ѱ_d were determined for each soil-water retention model θ(ѱ)=(θ_s- θ_r) S_e+θ_r, using a non-linear regression analysis with retention functions (4)-(6) and the corresponding experimental information. These analyses considered the restrictions between specified parameters m and n and we assumed that θ_r=0 (^{Haverkamp et al., 2002}).Figures 1a-1c show the results of the calibration performed for sandy soil named Grenoble (code 4444), which is a soil with experimental data obtained under controlled conditions in the laboratory, and it is widely used in the bibliography (^{Fuentes et al., 1992}; ^{Berlotti and Mayergoys, 2006}). The greatest data adjustment is obtained using the large pore model (6), which is confirmed with the coefficient of determination (r²), calculated with r2=σxy2/σx2σy2, where σ_xy is the covariance of xy (experimental-adjusted), σ_x is the standard deviation of experimental data, and σ_y is the standard deviation of calibrated data. We can also notice that the three models have a high ability to adjust experimental data, since they present values for r² higher than 0.97. This ability to adjust to the fractal models of soil-water retention is similar to that displayed by the classic model by ^{van Genuchten (1980)} subjected to ^{Mualem’s restriction (1976)}, 0<m=1-1/n<1. ^{Yang and You (2013)} used this model and obtained, for four soil classes, types of intelligent optimization algorithms, adjustments that varied between 0.960≤r²≤0.997. The parameters of non-linear regression models obtained with the data of the first soils analyzed are shown in Table 2.

Figure 1 Adjustment of the experimental data of water retention of the soil 4444 with φ=0.370 cm³/cm³, s=0.667, θ_s=0.312 cm³/cm³ and θ_r =0 cm³/cm³. 

The fractal models of hydraulic conductivity (4)(6) were applied with the calibration parameters, their prediction was compared to experimental data, and r² was determined by textural class. Relative conductivity (K_r) was used instead of hydraulic conductivity (K) since the former varies between 0 and 1 in any type of soil, which simplifies analysis. In the case of the validation of soil 4444, the best prediction was obtained with the geometric pore model, although it came close to the result for the large pore model (Figures 2a-2c). The validation by textural type of the 208 soils of Table 1 showed that the predictions of the models of relative conductivity of the geometric, neutral and large pores were stable and the differences between them were minimum (Table 3). The three fractal models displayed an adequate predictive ability in most of the textural classes, although the values for volumetric porosity (ϕ) and saturated water content in UNSODA (θ_s) (Table 2) are different. This affects the prediction of the models evaluated because its parameters m and n depend on parameter s, which is a function of ϕ (Equation 7).

Figure 2 Validation of the fractal conductivity models (4)(6) considering the experimental data for relative hydraulic conductivity (K_r=K(θ)/K_s) of the Grenoble soil 4444: K_s=368.9 cm/d and s=0.667. 

The calibration and validation procedure described in this study was applied to contrast the prediction levels of the fractal models, considering van Genuchten’s classical water retention model (^{van Genuchten, 1980}), subjected to Burdine’s restriction (^{Burdine, 1953}) combined with Brooks and Corey’s hydraulic conductivity model (^{Brooks and Corey, 1964}). This set of functions satisfies the integral properties of the infiltration (^{Fuentes et al., 1992}) and it is used by analysts and modelers of water flows in the vadose zone of the soil:

where η is a parameter of shape that must be calibrated considering the experimental data for hydraulic conductivity. In this way, this exponent constituted an additional parameter to be determined in regard with the fractal models. To eliminate η from the calibration process and to make the comparison of results compatible, we pick up the analysis developed by ^{Fuentes et al. (2001)}, who established that the exponent is related to van Genuchten’s soil-water retention curve shape parameters as follows:

The water retention Equation (8) was applied in order to adjust soil-water retention experimental data (Table 1) and to calculate the values of m and ѱ_d; Equation (9) was used to estimate η, and using the relative hydraulic conductivity equation by Brooks and Corey (8), we determined, by texture class, the linear correspondence between modeled and experimental data (Table 4). The fractal models (Table 3) improved their description in eight of the nine textural classes. Only in the silt loam texture did the classic models (8) present a greater r², but with a small difference in comparison with large pore model (0.8021 vs 0.8008).

Conclusions

The analytical functions of the soil-water retention curves of the geometric, neutral, and large pore fractal models, with two parameters to calibrate, adjust the experimental data of the soils and similarly adjust to van Genuchten’s classic retention model (^{van Genuchten, 1980}), subject to Mualem’s restriction (^{Mualem, 1976}). The fractal models for relative hydraulic conductivity satisfactorily describe the behavior of this hydraulic variable in a range of soil textures, and even present a better level of prediction than Brooks and Corey’s classic model (^{Brooks and Corey, 1964}). All three fractal models are reliable and can be taken into account in water flow simulations of the unsaturated area of the soil. The large pore model provides the greatest predictions of relative hydraulic conductivity.