A simple approach to predicting unsaturated hydraulic conductivity based on empirically scaled microscopic characteristic length

Abstract

A simple and empirical scaling approach is proposed for predicting the unsaturated hydraulic conductivity (K_us) based on the microscopic characteristic length (λ). The approach was tested on 19 different soils. The results showed that the proposed approach was useful for a wide range of soil water content, except at relatively high soil moisture. The predicted λ(θ) values were overestimated by 2%, which led to 5% overestimation in prediction of K_us(θ) at soil matric potentials of less than –0.1 MPa. However, the predictions of K_us(θ) were not satisfactory at soil matric potentials of greater than –0.1 MPa. The results also revealed that the proposed approach was accurate for predicting K_us(θ) in the dry range of soil water content. Future models should be developed based on the empirical and physical scaling approaches to improve prediction of soil hydraulic properties at high water contents. This means that new scaling factors need to incorporate more information about the pore geometry.

Editor D. Koutsoyiannis; Associate editor A. Carsteanu

Résumé

Nous proposons une approche de mise à l’échelle simple et empirique pour prévoir la conductivité hydraulique non saturée (K_us) à partir de la longueur caractéristique microscopique (λ). L’approche a été testée sur 19 sols différents. Les résultats ont montré que l’approche proposée était utile pour une large gamme de teneurs en eau du sol, sauf pour les humidité du sol relativement fortes. Les valeurs de λ(θ) prévues étaient surestimées de 2%, ce qui conduit à une surestimation de 5% pour la prévision de K_us(θ) pour des potentiels matriciels du sol de moins de ‒0,1 MPa. Toutefois, les prévisions de K_us(θ) n’ont pas été satisfaisantes pour des potentiels matriciels du sol supérieurs à ‒0,1 MPa. Les résultats ont également révélé que l’approche proposée est précise pour prévoir K_us(θ) dans la gamme des teneurs en eau des sols secs. Cependant, des futurs modèles devraient être élaborés en se basant sur des approches de mise à l’échelle empiriques et physiques afin d’améliorer la prévision des propriétés hydrauliques du sol pour de fortes teneurs en eau. Cela signifie que de nouveaux facteurs d’échelle devraient inclure plus d’informations sur la géométrie des pores.

Key words:

• microscopic characteristic length
• scaling
• soil water content
• unsaturated hydraulic conductivity

Mots clefs:

• longueur caractéristique microscopique
• mise à l’échelle
• teneur en eau du sol
• conductivité hydraulique non saturée

1 INTRODUCTION

In soil hydrology, the saturated hydraulic conductivity (K_s) and unsaturated hydraulic conductivity (K_us) are the essential and crucial parameters for practical and modelling purposes. Efficient soil and water management necessitates the knowledge of soil hydraulic behaviour at small and large scales, especially the soil water content–pressure head relationship, θ(h), and the soil hydraulic conductivity–soil water content relationship, K(θ) (Basile et al. 2006).

While θ(h) is simply measured in the laboratory, direct measurement of K(θ) is difficult, time-consuming and expensive. Many laboratory and field experiments have been done to directly and indirectly measure K_us, with laboratory measurements being more straightforward and accurate than those in the field (Eching et al. 1994, Zhuang et al. 2001). However, the major challenge faced in estimating K_us in recent decades has been how to deal with soil heterogeneity (van Genuchten and Sudicky 1999). Both field and laboratory studies have shown that K_us is spatially variable, such that it is difficult to use a single value to represent a specific site (Bertuzzi and Bruckler 1996, Zhuang et al. 2001, Basile et al. 2006).

Scaling methods based on the similar media concept (SMC; Miller and Miller 1956) present a simple and powerful tool to approximately describe the field spatial variability of soil hydraulic properties (Ahuja et al. 1984). Examples of the application of scaling on some soil hydraulic properties have covered, but are not limited to K_us (Warrick et al. 1977, Simmons et al. 1979, Bertuzzi and Bruckler 1996, Poulsen et al. 1998, Zhuang et al. 2001, Basile et al. 2006), K_s (Ahuja and Williams 1991, Miyazaki 1996, Zhuang et al. 2000, Nakano and Miyazaki 2005), infiltration (Youngs and Price 1981, Ahuja et al. 1984, Rasoulzadeh and Sepaskhah 2003, Machiwal et al. 2006), and soil water retention curves (Kosugi and Hopmans 1998). In general, scaling is a method that relates the soil properties of different soil types or spatial locations by simple conversion factors called scaling factors (Ahuja and Williams 1991). Scaling factors can be derived by a dimensional analysis (physically-based) technique that is based on the existing physical similarity in the system, or by a functional normalization (empirical-based) method (Tillotson and Nielsen 1984). In this study, the empirical-based method is followed.

Generally, the literature that reports the application of the similar media concept is restricted to soils that have reasonably similar morphology, while studies in the last decade have mainly focused on approaches for non-similar media concepts (Comegna et al. 2000, Rahimi et al. 2011, Ahmadi and Sepaskhah 2012). This restriction is more pronounced in terms of hydraulic conductivity, especially K_us. Tyler et al. (1999) described the lack of accessible information on K_us and argued that the estimated values of K_us match fairly well with the observed values at high water contents; however, few data are available for comparison at very low water contents. Their argument clearly highlights the importance of development and validation of new theories that can be used to reliably estimate K_us over a wide range of soil moisture content.

So far, different scaled models have been presented to estimate K_us based on K_s. The Mualem (1976) and van Genuchten (1980) models are widely used because they are developed based on the soil water retention characteristics. However, several studies have recommended that soil hydraulic conductivity models should incorporate information about pore size distribution and geometry (Bird et al. 2000, Zhuang et al. 2000, Rahimi et al. 2011, Ahmadi and Sepaskhah 2012). Zhuang et al. (2001) stated that “new theories or concepts that mechanically include component of pore/particle arrangements are urgently needed in the simulation of soil flow”.

The estimation methods that are often used provide K_us data that match relatively well with the observations at high water contents; however, the datasets that include K_us measurements at very low water contents are limited (Tyler et al. 1999). Therefore, the objective of this study is to test a simple scaling approach for prediction of K_us based on the microscopic characteristic length, λ, that would be applicable for a wide range of soil moisture contents, and especially low soil moisture contents.

2 MATERIALS AND METHODS

2.1 Study area and soil characteristics

In this study 19 natural soils were collected from different locations in Fars province, southern Iran. Table 1 summarizes some of the physico-chemical characteristics of the soils. The soils were categorized into six soil textures: loam, loamy sand, sandy loam, silt loam, silty clay loam and silty clay; the sand, silt and clay contents were in the ranges 4–80%, 16–62% and 4–46%, respectively. The soil textures covered both coarse and fine textures, with relatively high amounts of silt particles in most soil samples. The soil texture classification was based on the particle size distribution of the United States Department of Agriculture (USDA), i.e. clay fraction: <2 μm, silt fraction: 2–50 μm and sand fraction: 50–2000 μm.

Table 1 Some physico-chemical and saturated soil characteristics of the soil texture/types used in the study.

Soil texture/type	Sand (%)	Clay (%)	Silt (%)	dg^(b) (mm)	σg ^(c)	ρ^(b) (g cm^-3)	OM (%)	pH	Ks (cm d^-1)	θs (cm^3 cm^-3)
Loam ^(a)	24	26	50	0.027	11.580	1.45	1.61	7.96	4.32	0.441
Loamy Sand1 ^(a)	80	4	16	0.431	6.208	1.55	0.34	7.79	35.57	0.431
Loamy Sand2 ^(a)	79	4	17	0.416	6.328	1.61	0.24	7.66	36.43	0.373
Sandy Loam1 ^(a)	63	7	30	0.210	9.136	1.60	0.37	7.73	11.23	0.466
Sandy Loam2 ^(a)	63	9	28	0.196	10.157	1.67	0.34	7.62	38.74	0.450
Silt Loam1	18	26	56	0.022	9.685	1.17	1.95	7.27	5.76	0.547
Silt Loam2	36	12	52	0.066	9.922	1.37	1.38	7.88	27.94	0.431
Silt Loam3	20	24	56	0.025	9.878	1.33	1.51	7.99	21.46	0.425
Silt Loam4^a	18	27	55	0.021	9.885	1.16	1.61	7.89	28.37	0.470
Silty Clay Loam1 ^(a)	8	30	62	0.013	7.020	1.39	1.91	8.01	6.48	0.458
Silty Clay Loam2 ^(a)	12	32	56	0.014	8.650	1.24	2.08	8.07	12.82	0.499
Silty Clay Loam3 ^(a)	10	28	62	0.015	7.436	1.30	2.18	7.95	10.94	0.523
Silty Clay Loam4	8	39	53	0.010	7.889	1.34	1.14	7.96	31.39	0.453
Silty Clay Loam5	6	34	60	0.011	6.725	1.35	1.31	7.82	7.20	0.446
Silty Clay Loam6	12	30	58	0.015	8.375	1.18	2.05	7.86	9.94	0.467
Silty Clay Loam7	18	28	54	0.020	10.082	1.18	1.91	7.82	31.82	0.457
Silty Clay Loam8	12	34	54	0.013	8.912	1.17	2.01	8.06	23.47	0.477
Silty Clay1	8	42	50	0.009	8.116	1.18	1.51	7.80	2.88	0.444
Silty Clay2	4	46	50	0.007	6.661	1.43	1.11	7.47	5.90	0.456

Notes

^(a) Used for validation,

^(b) Geometric mean particle diameter (Shirazi and Boersma 1984),

^(c) Geometric standard deviation (Shirazi and Boersma 1984).

Ten out of 19 soil types were selected for calibration and the remaining nine soil types were used for the validation (Table 1). The calibration and validation datasets were chosen such that the validation dataset included the soil textures/types that were not generally used in calibration. The calibration dataset was used to fit a scaled model according to the scaling factors, while the validation dataset was used to check the validity of the scaled model.

For each soil, K_s was measured in the laboratory on undisturbed soil samples using the falling head method (Klute and Dirksen 1986). The K_us of the undisturbed soil samples (height: 4 cm, diameter: 7 cm) was measured in the laboratory using the Gardner (1956) method. To measure K_us, the undisturbed saturated soil sample was placed in the pressure plate apparatus and then different pressure heads were applied on the sample and the volume of the outflow water was recorded at different time intervals. Each sample was exposed to five pressure heads: 0.03, 0.1, 0.5, 1 and 1.5 MPa. For each applied pressure, the experiment continued until equilibrium was reached. The Gardner method assumes that K_us and the hydraulic impedance are constant during the pressure increments (Rijtema 1959, Jackson et al. 1963).

The K_us associated with each pressure head is determined by using (Gardner 1956):

(1)

where K_us is the unsaturated soil hydraulic conductivity (cm d^-1); V is the volume of saturated soil (cm³); V₀ is the total outflow from the soil sample at each pressure head (cm³); h is the applied pressure head (cm); β is the slope between the logarithmic difference of the total outflow and outflow at any time, i.e. log(V₀ – V(t)) against time (d^-1); ρ is the density of water (g cm³); g is the acceleration due to gravity (cm s^-2), and L is the length of the soil sample (cm).

The volumetric soil moisture content at each pressure head was measured separately on the same soil sample. Dry bulk densities were determined by oven-drying the undisturbed soil samples at 105°C for 24 h. The particle size distribution (PSD) (not shown) of each soil was determined by hydrometer (particles <75 μm) and wet sieving (particles <75 μm) of the bulk soil sample, using the approaches described by Gee and Bauder (1986) and Gee and Or (2002).

2.2 Scaling theory and methodology

For scaling the K_s of a porous media, the microscopic characteristic length, λ_s, is defined as (Youngs 1990):

(2)

where λ_s and K_s are, respectively, the microscopic characteristic length [L] and saturated hydraulic characteristics [L T^-1] at saturated soil moisture content (θ_s); η is the viscosity of water [ML^-1 T^-1]; ρ is the density of water [ML^-3]; and g is the acceleration due to gravity [L T^-2].

In this study, we propose and adopted the generalized form of equation (2) that implements K_us as:

(3)

where λ(θ) and K_us(θ) are the microscopic characteristic length and unsaturated hydraulic conductivity, respectively, at soil moisture content θ [L³ L^-3].

The sole objective of scaling is to coalesce a set of functional relationships into a single function using the scaling factors that describe the set as a whole (Machiwal et al. 2006). Using the functional normalization (empirical-based scaling) approach that aims to reduce the scatter of experimental data and concentrate them on a mean reference curve (Tillotson and Nielsen 1984, Comegna et al. 2000), the following scaling factors (θ^* and λ^*) are proposed to scale the measured data (θ, λ(θ)) into a reference curve (scaled model):

(4)

(5)

To test this scaling approach, the relationship (scaled model) between θ^* and λ^* is known (see Section 3.1), while the values of K_s and θ_s are measured a priori. At any known θ, equation (4) calculates the scaled θ^*, and λ^* is calculated based on the scaled model (θ^*,λ^*) (equation (10)). Knowing λ^*, equation (5) is used to give λ(θ). Therefore, K_us(θ) is predicted using equation (3).

In equations (2) and (3), η, ρ and g are constant. Therefore, K_s is scaled by the square of λ^*, i.e. λ^*2 = K_us(θ)/K_s, where λ reflects the pore geometry of the soil (Sposito and Jury 1990). This is indeed in agreement with the SMC of Miller and Miller (1956) and its physically-based scaling concept. Warrick (1990) and Sposito and Jury (1990) have also reported that the characteristic length is a scaling factor for unsaturated hydraulic conductivity. The λ^*2 relationship clearly shows that K_us is directly related to K_s by the square of the soil microscopic characteristic length, meaning that λ^* is the square root of relative hydraulic conductivity.

Corey (1977) introduced the following equation for the prediction of tortuosity at different values of θ:

(6)

where τ_s is the tortuosity at saturation; τ(θ) is the tortuosity of the unsaturated soil; Θ = (θ – θ_γ)/(θ_s – θ_γ) is the effective saturation, where θ_γ is the residual soil water content that is determined by fitting the van Genuchten model (van Genuchten 1980) to the soil water characteristic data.

Sposito and Jury (1990) pointed out that λ reflects the geometric arrangement of both the pore space and the solid particles. Since measurement of τ at different soil moisture contents is not a routine and simple task in soil hydrology, we investigated whether there would be significant mathematical relationships between τ and λ to describe the tortuosity at different values of θ as follows:

(7)

(8)

The accuracy of the scaled model is evaluated based on the root mean square error (RMSE) index as follows:

(9)

where K_us,measured and K_us,predicted are the measured and predicted K_us, respectively, and n is the number of observations.

3 RESULTS AND DISCUSSION

3.1 Calibration of the scaled model

The proposed scaling factors basically rely on the functional normalization techniques and empirical methods (Tillotson and Nielsen 1984) that aim to reduce the dispersion of experimental data to a reference scaled curve. Figure 1 shows the scatter plot of the calibration data, while Fig. 2 shows that the scaling factors (equations (4) and (5)) could reasonably coalesce the very scattered datasets into an exponentially scaled model as follows:

(10)

where R², SE, n and p stand for coefficient of determination, standard error of estimation, number of observations and probability level of significance of the scaled model, respectively.

Fig. 1 Scatter plot of the measured soil moisture content, θ, and the calculated microscopic characteristic length, λ, based on equation (2). Points represent the calibration data corresponding to θ_s, θ_0.03, θ_0.1, θ_0.5, θ₁ and θ_1.5.

Fig. 2 Diagram of the calibration data scaled according to the scaling factors of equations (4) and (5). The regression line shows the best fitted scaled curve for the whole scaled calibration dataset.

The SMC of Miller and Miller (1956) was originally proposed for homogenous and similar geometry of porous media. Basically, the hydraulic conductivity and tensiometric data are scaled with degree of saturation rather than the soil water content in the SMC concept (Comegna et al. 2000), but equation (10) overcomes this rigorous concept by implementing the microscopic characteristic length of the porous medium as the pore geometry of the soil (Sposito and Jury 1990, Comegna et al. 2000). This is in agreement with Youngs (1990), who suggested that the SMC might be extended to non-similar media concepts (NSMC). Youngs and Price (1981) showed that it was possible to apply the SMC for one-dimensional infiltration on a range of soil materials composed of particles of different sizes and shapes. Nevertheless, Miyazaki (1996) extended the NSMC for scaling soil hydraulic conductivity based on the soil bulk density. Subsequently, the NSMC was applied successfully on K_us (Zhuang et al. 2001) and K_s (Zhuang et al. 2000, Nakano and Miyazaki 2005, Rahimi et al. 2011, Ahmadi and Sepaskhah 2012).

Zhuang et al. (2000) and Zhuang et al. (2001) suggested that the advantage of the NSMC to the SMC models in scaling the K_s and K_us is due to incorporation of bulk density and some parameters of soil texture. Zhuang et al. (2001) did not succeed in predicting K_us using the NSMC scaling approach for a wide range of soil textures in the UNSODA database (Nemes et al. 1999). Most likely the failure in prediction of K_us was due to the variety of K_us measuring methods reported in UNSODA. This highlights the limitation of applying publicly available datasets and databases (Ahmadi and Sepaskhah 2012). Zhuang et al. (2001) suggested that to improve the performance of the scaling models, it is necessary to develop theories or concepts that include soil structural parameters (pore size, shape and orientation of solids) that could results in pedotransfer functions for scaling purposes. However, an advantage of the simple scaling approach is that the K_us is explicitly dependent on θ and λ(θ), with the latter, in turn, dependent on the geometry of porous materials (Sposito and Jury 1990, Youngs 1990).

An advantage of the proposed scaling approach is its simplicity. In many cases K_s and θ_s are available or could be easily measured. Thus, with this basic information, and for practical aspects, it is possible to predict K_us for any assumed θ(h) within the same range of soil textures. Sadeghi et al. (2011) reported that the performance of scaling the soil hydraulic properties was sensitive to the soil textural range and initial soil water content conditions. They reported that, under such conditions, a classification of soil texture and initial conditions may alleviate the poor performance of scaling.

When this approach is coupled with the scaling approach for the water retention curve (Kosugi and Hopmans 1998, Sadeghi et al. 2011), it could make a solid base of scaling the soil hydraulic properties in different areas. However, in contrast to former studies indicating that hydraulic conductivity was related to the effective saturation, Θ (Mualem 1976, van Genuchten 1980, Comegna et al. 2000), we realized that the scaled hydraulic conductivities could be related to θ_s, as shown in equation (4), without any prior requirement for θ_γ as the residual soil moisture. In this regard, Mohammadi and Vanclooster (2011) and Comegna et al. (2000) proposed that, in scaling and modelling approaches, the microgeometry of the porous medium is a better soil physical representative than the effective saturation, since it includes the geometry of pore sizes and considers the presence of the water film in large and small pores. Lebeau and Konrad (2010) and Ippisch et al. (2006) have discussed the validity of the Mualem (1976) and van Genuchten (1980) models under various water flow patterns in the soil pores.

3.2 Validation of the scaled model

Figure 3 shows the results of validation of the scaled model (equation (10)) for predicting λ(θ). The scaled model predicted well the λ(θ) at soil moisture contents drier than field capacity (FC, or generally at soil matric potentials of less than –0.1 MPa), with overestimates of λ(θ) by approx. 2%. The RMSE of 7.93 × 10^-6 cm is very small and shows the accuracy of the model prediction for soil matric potentials of less than –0.1 MPa.

Fig. 3 Validation of the new scaling approach (scaled model) on λ(θ) using the validation data (Table 1).

Apparently, the scaled model cannot predict λ(θ) at relatively high soil moisture contents; here, the predicted λ(θ) values are seriously underestimated. The reason for this discrepancy is not clear but we speculate that, at high soil water contents, the scaled model cannot effectively take into consideration the relatively free movement of water in the soil pores and probably other soil structure parameters dominate this phenomenon. It is noteworthy that at high soil water content there exists a two-fluid phase system such that the water-filled pores are bounded by water–solid and water–air interfaces, which have different surface tension and thus may lead to different geometries of the water-filled pore spaces that influence the performance of models (Tuller and Or 2001, Lebeau and Konrad 2010, Mohammadi and Vanclooster 2011, Tokunaga 2011). However, this may reveal the mismatch between the calibration and validation data for high soil moisture contents.

To check if the scaled model is applicable for predicting λ(θ) at high water content on other datasets, the data reported by Basile et al. (2006) for two sandy loam soils were used. The data of two sites (3 and 15 that basically held the primary specifications for this verification purpose) were chosen because the soil matric potential varied between –0.001 and –0.03 MPa, implying high moisture content limited to FC. The soil texture at both sites is sandy loam. Figure 4 shows the results of this analysis and it is clear that the data are scattered markedly below the 1:1 line and yet the scaled model underestimates λ(θ) for high soil moisture contents. Therefore, we suggest that the scaled model might be used for predicting λ(θ) at soil water contents lower than FC, preferably at soil matric potentials of less than –0.1 MPa.These results confirm the argument of Sadeghi et al. (2011) and Lebeau and Konrad (2010), who mentioned that the performance of scaling models strongly depends on the initial conditions, such as soil water content, pore size distribution and geometry, and the domain of soil textures. In such conditions, classification of the soil traits and then applying the scaling approach to each class may overcome the poor performance of the models (Sadeghi et al. 2011).

Fig. 4 Comparison of the measured and predicted λ(θ) with the 1:1 line at high soil moisture contents in our study (FC) and the literature (Basile et al. 2006).

The results of validation of the scaled model on predicting K_us are shown in Fig. 5. The K_us is overestimated by around 5%. Similar to the results for λ(θ), the predicted K_us at high soil water contents (soil matric potential > –0.1 MPa) are considerably underestimated (open circles in Fig. 5), but the predicted K_us at low soil water contents (soil matric potentials < –0.1 MPa) are more satisfactory, with an overestimation of about 5% (solid circles in Fig. 5). The RMSE is 9.46 × 10^-4 cm d^-1, which is small and demonstrates the accuracy of the model prediction for soil matric potentials of less than –0.1 MPa. This reveals that the scaled model is more capable of predicting K_us in the dry range of soil water content. Therefore, this simple scaling approach is successful in predicting the K_us at water contents lower than FC. However, Zurmühl and Durner (1998) showed the effect of bimodal pore size distributions on the behaviour of K_us at high soil water contents, which is different from the unimodal pore size distribution hypothesis of this study. So it seems that the disagreements between the measured and simulated K_us are partly affected by pore size distribution. It should be mentioned that bimodal pore size distribution affects the λ(θ) through the geometries of the water-filled pore spaces (Tuller and Or 2001), which lead to estimation of K_us.

Fig. 5 Validation of the new scaling approach (scaled model) on K_us(θ) using the validation data (Table 1).

More studies are needed to improve the accuracy of the scaling factors by developing pedotransfer functions and incorporating other soil physical parameters that govern soil water movement at high water potentials in sandy to clayey soils (Sadeghi et al. 2011). Important factors, such as soil water potential and soil water content, could potentially serve as useful parameters for scaling soil hydraulic conductivity (Sadeghi and Ghahraman 2010, Sadeghi et al. 2011). They reported that for an improved scaling process for K_s, not only similarity in soil structure is necessary, but also the soil water retention characteristics should be similar. Accordingly, Zhuang et al. (2001) reported that it is not possible to find a good match between observed and predicted K_us when applying different scaled hydraulic conductivity models on a wide range of soil type data in the UNSODA database (Nemes et al. 1999). Thus it remains a challenging issue in the prediction of K_us which should be addressed in future research to improve the accuracy of K_us models by incorporating more determining soil factors that affect K_us (Zhuang et al. 2000, 2001).

3.3 Relationship between tortuosity and microscopic characteristic length

Soil hydraulic conductivity depends on the soil pore geometry and tortuosity is an identification of this geometry, revealing the path of water movement in the soil. However, measurement of tortuosity is not easy and any indirect calculation of tortuosity in terms of the normalized τ given by equations (11) and (12) is useful in modelling soil hydraulic properties. Our analysis revealed that there is a significant relationship (Fig. 6) between τ (calculated by equation (6)) and λ (calculated by equations (2) and (3)), as follows:

(11)

The logarithmic model fitted the data well which implies that most of the variation in tortuosity happens at relatively low water content, i.e. where the slope of the curve is steep, which is in agreement with Moldrup et al. (2001). However, as the soil moisture content increases, the curve approaches a plateau, meaning that τ does not change considerably at high water content (Moldrup et al. 2001). The pooled coefficients of variation of all τ_s/τ(θ) and λ(θ)/λ_s values are 0.824 and 1.477, respectively, showing that the variance of the normalized λ is greater than of the normalized τ. This may imply that incorporating λ in the hydraulic conductivity models could be a better choice than τ, since it only holds the information of soil pore characteristics (Moldrup et al. 2001). This result is also confirmed by previous studies in which λ was a better representative of soil similarities in terms of pore size distribution (Tuli et al. 2001, Zhu and Mohanty 2006).

Fig. 6 Logarithmic relationship between λ(θ)/λ_s and τ_s/τ(θ) and based on the all data (Table 1).

There are various laboratory and field methods for measuring K_us (a brief review of methods may be found in Eching et al. 1994, Hillel 1998, Tyler et al. 1999, Basile et al. 2006). Each of the available methods gives different K_us values for a specific soil sample, depending on the measurements set-up. Therefore, the reported K_us values in databases such as UNSODA and other publicly available datasets are subject to some degree of systematic variation, and hence no predictions can be unbiased. So, for modelling purposes the data should preferably originate from a specific method of measurement to minimize the influence of systematic variations in the data during the scaling process as much as possible. Mohammadi and Vanclooster (2011) and Ahmadi and Sepaskhah (2012) have extensively discussed the data concerns, their quality and their consistency in scaling methods. However, such scaled models might be validated on additional datasets to extend their applicability. As already reported by Tyler et al. (1999), datasets similar to that used in this study are scarce and other popular international databases, such as UNSODA (Nemes et al. 1999), are not applicable and could lead to inconsistent results (Zhuang et al. 2001).

Measurement of K_us is costly and time-consuming, especially when it is required to know its values over a large area that has high spatial variation. The proposed scaling approach is very helpful and applicable in such areas where a primary and accurate knowledge of K_us is required for planning soil and water management scenarios (e.g. irrigation and drainage, groundwater recharge, soil water redistribution and infiltration).

4 CONCLUSIONS

This study introduced a simple scaling approach based on the scaled microscopic characteristic length (λ) to predict K_us in the dry range of soil water contents. The calibration results (10 soil types) showed that the introduced scaling factors were successful in coalescing the scattered observations of λ(θ) and θ into a reference equation for the scaled model. The validation results on the second dataset (nine soil types) showed that the scaled model could be successfully used to predict λ with 2% overestimation for different soil moistures, which resulted in prediction of K_us at the same soil moisture contents with around 5% overestimation. However, the scaled model was not accurate for predicting K_us at soil moisture contents around field capacity and slightly drier (soil matric potential > –0.1 MPa), representing the wet range of soil moisture. Further analysis revealed that including λ as an indication of soil structure within the formulation of soil hydraulic conductivity could be more informative than use of tortuosity, τ, because λ was more sensitive to soil moisture variations than τ.

Overall, this study has suggested a simple method that could be effectively used in predicting K_us in the dry range of soil water contents. Future research on developing soil hydraulic conductivity models should concentrate in detail on the specific soil physical parameters that affect K_us at relatively high soil moisture contents. Developing pedotransfer functions which include the fundamental soil structure and architecture parameters, such as pore geometry, would help in deriving more accurate scaling factors.