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On-line version ISSN 2521-9766Print version ISSN 1405-3195

Agrociencia vol.52 n.8 México Nov./Dec. 2018



Fractal analytical models for the hydraulic properties of unsaturated soils

Manuel Zavala1  * 

Heber Saucedo2 

Carlos Fuentes2 

1Universidad Autónoma de Zacatecas. Avenida Ramón López Velarde 801, Centro, 98000, Zacatecas, México.

2Instituto Mexicano de Tecnología del Agua, Paseo Cuauhnáhuac 8532, Progreso, 62550. Morelos, México. (,


The predictive capability of the mechanicist models of the hydraulic properties of unsaturated soils must be known in detail before developing simulations of water transfer processes in the vadose zone of the soil. The aim of this study was to evaluate the flexibility of adjusting experimental data and the predictive capability of three fractal models for the soil-water retention curve and the hydraulic conductivity curve that satisfy the integral properties of infiltration (geometric pore, neutral pore and large pore). The models consider concepts of fractal geometry, the Laplace and Poiseuille equations and Darcy’s law; we established its level of description and performance in regard to two traditional mechanicist models. The evaluation included 208 soils selected from the Unsaturated Soil Hydraulic Database, which were used to cover nine texture types. The hydraulic conductivity experimental data were used to validate the conductivity fractal models. The linear correspondence analysis between theoretical and experimental data showed that the predictive capability of the fractal models for relative conductivity is good, since the coefficient of determination in the texture types was positive (>0.75 y <0.86). In this way, at least 75 % of the variability of the conductivity is explained by the adjusted regression model. The fractal models presented a higher prediction in regard to the combination of classic mechanicist models, used in soil physics.

Keywords: relative hydraulic conductivity; soil-water retention curve; fractal models; Laplace equation; Poiseuille equation and Darcy’s law


La capacidad predictiva de los modelos mecanicistas de las propiedades hidráulicas de los suelos no saturados debe conocerse a detalle antes de desarrollar simulaciones de procesos de transferencia de agua en la zona vadosa del suelo. El objetivo de este estudio fue evaluar la flexibilidad de ajustar datos experimentales y la capacidad de predicción de tres modelos fractales para la curva de retención de humedad y la curva de conductividad hidráulica que satisfacen las propiedades integrales de la infiltración (poro geométrico, poro neutro y poro grande). Los modelos consideran conceptos de geometría fractal, las leyes de Laplace, Poiseuille y Darcy, se estableció su nivel de descripción y su desempeño respecto de dos modelos mecanicistas tradicionales. La evaluación incluyó 208 suelos seleccionados de la base Unsaturated Soil Hydraulic Database, con los que se cubrieron nueve clases texturales. Series experimentales de retención de humedad se usaron para calibrar los parámetros de forma y escala de las funciones de retención. Los datos experimentales de conductividad hidráulica se usaron para validar los modelos fractales de conductividad. El análisis de correspondencia lineal, entre datos teóricos y experimentales, mostró que la capacidad predictiva de los modelos fractales para la conductividad relativa es buena, porque el coeficiente de determinación en las clases texturales fue positivo (>0.75 y <0.86). Así, al menos 75 % de la variabilidad de la conductividad lo explica el modelo de regresión ajustado. Los modelos fractales presentaron predicción mayor respecto a la de la combinación de modelos mecanicistas clásicos, usados en la física de suelos.

Palabras clave: conductividad hidráulica relativa; curva de retención de humedad; modelos fractales; ley de Laplace; ley de Poiseuille y ley de Darcy


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 (Ks) 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 Ks 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:

θt=·KH-dKdθθz-S (1)

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:

v=ρwg8μr2H (2)

where ρw 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 rc of the meniscus of water in a pore with a radius of r:

ψ=-2σρwgrc (3)

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=rc 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(Se) are:

Model of the geometric pore:

Seψ=1+ψ/ψdn-mKr=KSe/Ks=1-1-Se1/msm20<sm=1-2s/n<1 (4)

Model of the neutral pore:

Seψ=1+ψ/ψdn-mKr=kSe/Ks=Ses1-1-Se1/msm0<sm=1-4s/n<1 (5)

Model of the large pore:

Se(ψ)=1+ψ/ψdn-mKr=kSe/Ks=1-1-Se1/m2sm0<2sm=1-4s/n<1 (6)

where m and n are parameters of shape; ѱd is the parameter of scale of the pressure [L]; Se(ѱ)=[(θ(ѱ)- θr)/(θs- θr)] is the effective saturation; θs is the saturated water content [L3L-3]; θr is the residual water content [L3L-3]; Ks is the saturated hydraulic conductivity [LT-1]; 0≤Kr≤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:

1-ϕs+ϕ2s=1 (7)

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/log120.6942 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 (θs) and saturated hydraulic conductivity (Ks). 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.

Table 1 Soils selected from the UNSODA database. 

Textura Código del suelo
Arena (75) 1014 1041 1042 1043 1050 1052 1053 1054 1060 1061 1063 1070 1071 1072 1073 1074 1075 1140 1141 1142 1240 1241 1460 1461 1462 1463 1464 1465 1466 1467 2100 2120 2121 2122 2123 2124 2125 2126 2210 2211 2212 2213 2217 2220 2221 2310 2540 2550 3070 3080 4000 4001 4021 4051 4052 4060 4061 4132 4140 4141 4142 4150 4151 4152 4440 4441 4443 4444 4445 4480 4490 4650 4651 4660 4661
Franco limoso (40) 1280 1281 1282 1490 2491 2492 2493 3240 3242 3250 3252 3253 3260 3261 3262 3263 3264 4030 4031 4032 4033 4040 4041 4042 4043 4070 4071 4080 4082 4090 4091 4092 4180 4181 4182 4183 4570 4671 4672 4673
Franco arenoso (31) 1091 1120 1121 1130 1131 1161 1380 1381 1390 1391 1392 2111 2130 2131 2132 2532 2541 2551 3050 4100 4110 4111 4112 4160 4161 4162 4170 4171 4172 4470 4500
Arena franca (26) 1010 1011 1012 1013 1015 1051 1062 1090 1143 1160 2101 2102 2103 2104 2105 2110 4010 4011 4020 4050 4062 4130 4580 4581 4582 4583
Arcilla (10) 1162 1163 1181 1182 3281 3282 4120 4121 4680 4681
Franco arcilloso arenoso (10) 1092 1132 1164 1165 1166 1184 1382 2542 2552 4460
Franco (8) 1260 1261 2530 2531 4101 4102 4780 4790
Franco arcilloso limoso (7) 1362 1371 2021 2022 3241 3251 4450
ranco arcilloso (1) 1180

Results and discussion

Parameters m and ѱd were determined for each soil-water retention model θ(ѱ)=(θs- θr) Se+θr, 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 θr=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 σxy is the covariance of xy (experimental-adjusted), σx is the standard deviation of experimental data, and σy 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, θs=0.312 cm3/cm3 and θr =0 cm3/cm3

Table 2 Results of the non-linear regression analyses on the experimental data of soil-water retention of the first soils analyzed. 

ID ϕ θs s Poro geométrico Ecuación (4) Poro neutro Ecuación (5) Poro grande Ecuación (6)
cm3/cm3 cm3/cm3 ѱd
1010 0.390 0.384 0.670 -23.10 0.325 -17.56 0.167 -17.18 0.148
1011 0.433 0.411 0.680 -25.38 0.406 -19.38 0.212 -18.91 0.184
1012 0.474 0.381 0.688 -22.03 0.369 -17.37 0.196 -17.04 0.171
1013 0.442 0.376 0.682 -24.19 0.463 -18.56 0.243 -18.04 0.206
1014 0.446 0.361 0.682 -29.16 0.521 -21.99 0.273 -21.09 0.226
1015 0.346 0.262 0.662 -14.15 0.134 -11.71 0.067 -11.66 0.064
1041 0.429 0.330 0.679 -39.01 0.564 -28.40 0.290 -26.95 0.236
1042 0.395 0.340 0.672 -41.50 0.663 -30.45 0.346 -28.44 0.271
1043 0.390 0.320 0.671 -48.36 0.783 -36.74 0.424 -33.70 0.314
1050 0.396 0.384 0.672 -13.49 0.383 -10.98 0.204 -10.77 0.178
1051 0.400 0.389 0.673 -8.80 0.353 -7.25 0.188 -7.19 0.166
1052 0.415 0.365 0.676 -8.08 0.379 -6.88 0.207 -6.87 0.182
1053 0.419 0.366 0.677 -9.67 0.512 -8.13 0.289 -8.02 0.242
1054 0.419 0.367 0.677 -12.19 0.682 -9.79 0.390 -9.30 0.303
1060 0.415 0.361 0.676 -16.62 0.410 -13.31 0.219 -13.07 0.189

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 (Kr) 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 (θs) (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).

Table 3 Coefficient of determination r2 by type of texture (Kr_experimental vs Kr_modelo). 

Textura Modelos de conductividad hidráulica relativa
Poro geométrico (4) Poro neutro (5) Poro grande (6)
Arena (75) 0.8641 0.8629 0.8626
Franco limoso (40) 0.7946 0.7893 0.8008
Franco arenoso (31) 0.7482 0.7503 0.7529
Arena franca (26) 0.8697 0.8708 0.8710
Arcilla (10) 0.8850 0.8863 0.8858
Franco arcilloso arenoso (10) 0.7470 0.7621 0.7629
Franco (8) 0.8604 0.8495 0.8613
Franco arcilloso limoso (7) 0.7998 0.7829 0.8010
Franco arcilloso (1) 0.9953 0.9952 0.9953

Figure 2 Validation of the fractal conductivity models (4)(6) considering the experimental data for relative hydraulic conductivity (Kr=K(θ)/Ks) of the Grenoble soil 4444: Ks=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:

Seψ=1+ψ/ψdn-m0m=1-2/n1Kr=KSeKs=Sen (8)

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:

η=2s2mn+1 (9)

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 ѱd; 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).

Table 4 Coefficient of determination r2 by type of texture (Kr_experimental vs Kr_modelo). 

Textura Ecuaciones (8) y (9)
Arena (75) 0.8425
Franco limoso (40) 0.8021
Franco arenoso (31) 0.7498
Arena franca (26) 0.8697
Arcilla (10) 0.8538
Franco arcilloso arenoso (10) 0.7760
Franco (8) 0.8381
Franco arcilloso limoso (7) 0.7592
Franco arcilloso (1) 0.9953


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.

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Received: January 2018; Accepted: September 2018

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