Electronic Journal of Polish Agricultural Universities (EJPAU) founded by all Polish Agriculture Universities presents original papers and review articles relevant to all aspects of agricultural sciences. It is target for persons working both in science and industry,regulatory agencies or teaching in agricultural sector. Covered by IFIS Publishing (Food Science and Technology Abstracts), ELSEVIER Science - Food Science and Technology Program, CAS USA (Chemical Abstracts), CABI Publishing UK and ALPSP (Association of Learned and Professional Society Publisher - full membership). Presented in the Master List of Thomson ISI.
Volume 10
Issue 1
Environmental Development
Available Online: http://www.ejpau.media.pl/volume10/issue1/art-04.html


Marek Spychalski1, Cezary Kaźmierowski2, Zbigniew Kaczmarek3
1 Department of Soil Science and Land Reclamation, Agricultural University of Poznan, Poland
2 Department of Soil Science and Remote Sensing of Soils, UAM, Poznan, Poland
3 Department of Soil Science, Agricultural University of Poznan, Poland



The paper presents relations between drainage porosity (nd) and the saturated hydraulic conductivity (Ks). The undisturbed soil samples were characterised by the saturated hydraulic conductivity, water retention curve (WRC) and estimated sum of the soil macro and mesopores known as the drainage porosity. The latter sum was estimated from the pF curves as a difference between the total porosity (t) and the field capacity (FC). Two equations describing these relationships were proposed and the coefficients of the power function of Ahuja et al. [1, 2] were modified. The equations proposed were tested on the original data set and on the independent selected UNSODA data set [23] and compared with other previously published pedotransfer functions (PTFs). The results of the comparative analysis have shown that for 0.03 < d<0.30 m3 × m-3, the equation proposed in this paper provides a more accurate assessment of the saturated hydraulic conductivity. It has been also shown that the relation derived is universal and does not depend on the soil texture.

Key words: saturated hydraulic conductivity, drainage porosity, pedotransfer function.


The knowledge of saturated hydraulic conductivity of soil (Ks) is necessary for modelling the water flow in the soil, both in the saturated and unsaturated zone, and transportation of water-soluble pollutants in the soil. The knowledge of this conductivity is also needed in designing of the drainage of the area and in construction of earth dam and levee.

Direct measurements of the saturated hydraulic conductivity are time consuming [5], and a demand of their accuracy leads to a profound increase in their cost [28], therefore, much attention has been devoted to find out a reliable indirect method for estimation of this parameter. In early equations describing this parameter Ks was calculated on the basis of the soil texture – the equations proposed by Hazen, Selheim, USBR, after Wieczysty [41]. Sometimes along with the texture, other physical parameters describing the soil were used, such as the surface area or total porosity (the expression proposed by Krüger after Wieczysty [41]. Unfortunately, the expressions proposed by the above authors had limited applications and the final Ks value was charged with a significant error. In the last twenty years much progress has been made in indirect methods for estimation of different difficult to measure soil characteristics known as the pedotransfer function (PTF) [5]. In these solutions the hydraulic properties of the soil are determined on the basis of routinely, easily and cheaply measurable soil parameters. The pedotransfer function (PTF) has been most often given on the basis of the soil texture, total porosity and bulk density, sometimes along with other soil parameters using the method of multiple regression analysis [6, 9, 10, 14, 32] or the method of neural network analysis [28, 34].

The aim of the study reported was to test different relations between drainage porosity and the saturated hydraulic conductivity.


The soil is a porous medium with a developed network of pores. Its natural ability to form aggregates implies that the soil pore network forms structures of a few different pore sizes. The greatest pores are on the external margins of soil aggregates while the smallest form a network of micropores inside the aggregates [7, 8]. Water is much more mobile in the macro than in the micropore network, which is a consequence of the character and intensity of forces acting on the water molecule in these two types of network [13, 22].

The size and distribution of pores in the soil depend on the soil texture, structure and bulk density. Literature provides a number of relations between different texture characteristics and the saturated [4, 9, 10] or near-saturated hydraulic conductivity [16] and between the bulk density and the saturated hydraulic conductivity [42].

As follows from the results of [11, 12, 15, 21, 22, 24, 25, 27, 32] the most important factor determining the value of Ks is the distribution of soil pores, and in particular the total content of macro and mesopores, known as the drainage porosity (d). This parameter is a difference between the total porosity (t) and field capacity (FC), assumed as water content at different soil water matric potentials (5 kPa, 10 kPa or 33 kPa). The drainage porosity depends mainly on soil structure and bulk density.

The idea of the saturated hydraulic conductivity calculation on the basis of non-capillary porosity was introduced by Baver [3]. On the basis of the equation proposed by Kozeny [20] Ahuja et al. [1, 2] derived the following relation between Ks and the drainage porosity (nd):

Ks = B(d)m         (1)

where (nd) is the difference between the total porosity (nt) and the field capacity (FC at 33 kPa), B is a constant depending on the type of soil and m is an empirical coefficient. The formula derived by Ahuja et al. [1, 2] was adapted by Suleiman and Ritchie [37], who determined Ks on the basis of the drainage porosity (nd) and the relative drainage porosity (dr), defined as d/FC , where FC is measured in laboratory at 33 kPa or in the field after 2 or 3 days of drainage. Unfortunately, the analysis of the accuracy of the proposed dependencies Ks(d) and Ks(dr) proved large errors in the estimation of Ks.

Minasny and McBratney [28] have adapted the drainage porosity determined for FC at 10 kPa to the equation proposed by Ahuja et al. [1, 2] obtaining the highest accuracy of estimation of Ks on inhomogeneous validation data sets. From among the earlier PTFs (either without nd or with d defined in a different way) a comparably high accuracy of estimation was provided only by the model of Rosetta (35). This result is an indirect confirmation of the correctness of determination of d for FC at 10 kPa.

Irrespective of the relationships proposed between the drainage porosity and saturated hydraulic conductivity, a number of models have been proposed in which Ks is estimated on the basis of the soil texture [9, 10, 33] or texture and bulk density [14, 34]. However, the hitherto proposed PTFs are not universal because, as shown by Minasny and McBratney [28] in particular textural groups (sandy, loamy and clayey) the highest accuracy of prediction was found for different earlier published multifactor dependencies in which drainage porosity was not taken into account: that by Dune and Puckett [10] for sandy soils, by Cosby et al. [9] for loamy soils and by Schaap et al. [34] for clayey soils.


The relationships between selected physical properties and the saturated hydraulic conductivity in soil were studied on 35 samples of which 28 represented the natural soil horizons and 7 came from model studies. The soil profiles studied belonged to Glossaquic Hapludalfs and Typic Endoaquolls developed from glacial till of Würm glaciation, located in Kleczew near Konin (4 profiles), and in Złotniki near Poznań (4 profiles) [17]. From these soil profiles undisturbed soil samples (100 cm3) and disturbed monoliths samples of soil with an interrupted structure were taken from selected genetic horizons and parent material up to the depth of 2,7 m. In the model study the field capacity and saturated hydraulic conductivity were determined for 7 bulk density levels of eolian sand deposits near Wielowies Klasztorna.

In the disturbed samples the particle size distribution was determined on a hydrometer [31], organic C content was determined by the Walkley-Black method [30] and the particle density by a picnometer [36]. The saturated hydraulic conductivity was determined in laboratory by the constant head method [18], as a mean value of four measurements. The water retention curves (WRC) were determined as mean values of four replicates using 1 bar ceramic plate [19]. The bulk density of the soil was measured in all samples used for Ks and WRC measurements. The sum of macro and mesopores (diameter > 30µm), called henceforth the drainage porosity (d) was found as the difference between the total porosity (t) and water content at 10 kPa (field capacity – FC):

d = t – FC         (2)

The regression equations describing the relations between nd and Ks were assessed by the statistical measures.

The coefficient of determination, R2, was found from the classical equation (26):


Pi – predicted value
Oi – observed value
µm0, σ0 – mean value and standard deviation of observed values
µmp, σp – mean value and standard deviation of model predictions

The Akaike Information Criterion (AIC) in modification of Webster and McBratney [40] was used for intercomparison of equiparameter models and for selection of the best predictive soil hydraulic function:


N – number of data points, np – number of PTF parameters, Pi – predicted value, Oi – observed value

Taking into account the log-tailed distribution of Ks, lognormal transformation was used. To compare the absolute accuracy of the proposed and existing PTFs the root mean square residual (RMSR in ln(µm×s-1, eq. 5) was calculated as (28):


Following Titije and Hennings [38] the geometric mean of error ratio (GMER) and geometric standard deviation of the error ratio (GSDER) were calculated as:



εi = Pi / Oi

GMER indicates an average factor, the predicted values is overestimated if GMER >1 or underestimated if GMER < 1. GSDER indicates the deviation around the mean (GMER) and PTF is exact only if GSDER is 1 [28, 38].

Comparative analysis was performed on the independent set of data being a selected UNSODA set [23]. Analysis of the methods of determinations of Ks and the characteristics describing the physical properties of soils from this data set has shown that the set is genetically inhomogeneous and should not be used as a validation data set in the studies. Therefore, the analysis was performed on the selected UNSODA data set [23]. We took into account only the mineral soil horizons for which Ks was determined by the constant head, failing head and steady flux methods, and for which the water retention curves and particle size distribution was analysed; with bulk density higher then 1 Mg × m-3. We excluded also the soil horizons for which the volumetric water content was higher then total porosity. As a consequence we obtained a quasi-homogeneous validation data set consisting of 128 soil horizons, called henceforth the selected UNSODA data set [23].


The training data set (N=35) consists mainly of sandy loams and loamy sands with particle density in the range 2.64 ÷ 2.66 Mg/m3, bulk density in the range 1.49 ÷ 2.0 Mg/m3, and total porosity in the range 0.25 ÷ 0.43 m3/m3. A comparison of our data with the reduced set UNSODA [23] reveals considerable differences in the parameters of histograms of the corresponding physical properties of the soils to which they refer (boxplots in Fig.1). For example in 75% of our samples (soil horizons) the silt content (50-2 µm) does not exceed 18%, while in the selection of UNSODA data set this content of silt is exceeded in 75 % of soil samples. Still greater differences are noted between the distributions of the soil bulk density and total porosity. In the selection of UNSODA set over 75 % of samples have dry bulk density ρd < 1.57 Mg × m-3, while only 10% of our samples have bulk density in this range.

Fig. 1. Cumulative distribution of particle-size, bulk density, total porosity and saturated hydraulic conductivity for current and selection of UNSODA (Leij et al. 1996) data sets

As a result of the analyses we propose the use of the modified parameters of Ahuja et al., [2] (equation 10) or the two original equations for Ks(d) (eqn. 11 and 12; Table 1). Analysis of the accuracies of the Ks estimations obtained with the equations proposed (eqn. 10-12) and with the use of the hitherto published PTFs, on the current data set, has proved the advantage of the equations derived in this work (Table 2).

Fig. 2. Comparision of compatibility of Ks(fie) relationships described by power function of Ahuja et al. (1984) and proposed equations (egn. 20 and 21) on measured data from current and selection of UNSODA (Leij et al, 1996) data sets

It is worth emphasising that the plot of the power dependence of Ks(d) proposed by Ahuja et al. [2] is linear in the log-log coordinates and in the range of small drainage porosities it deviates significantly from the experimental data (Fig. 2). The results of measurements suggest a curvilinear character of the dependence Ks(d) also in the logarithmic system of coordinates. The character of this dependence is determined by the changes in the character of water flow at extremely low and extremely high values of drainage porosity. A significant reduction in the saturated hydraulic conductivity in the range of minimum drainage porosity (d< 0.03 m3 × m-3) is a consequence of a decreased rate of water flow through the soil following from diminished water flow in the network of macro- and mesopores and its increased flow through the network of micropores [ < 30 µm]. In the range of high drainage porosity (d > 0.2 m3 × m-3) a small increase in this quantity corresponds to a great increase in the saturated hydraulic conductivity, which follows from the effect of increased diameter of capillaries on the rate of the water flow and from the change in the character of this flow from the laminar, through intermediate to turbulent. In view of the above, the proposed curvilinear equations [11 and 12] seem more adequate for description of the dependence Ks(d). The use of the equations proposed is limited for (d > 0.033 m3 × m-3), but in the range of their applicability they provide estimation of Ks to a significantly better accuracy [RMSR = 0.73 ln(µm × s-1), table 2].

Table 1. The equations proposed for estimation of Ks from drainage porosity and the limitations of their use


size *



Boundary conditions



Ahuja et al. (1984)

Ks = Bem
B= 4031.57; m = 3.295





Ks = a + b e1.5 + c e2.5 + d e3
a = -2.52; b = 581.598;
c = -6966.14; d = 11693.78

e > 0.033




Ks = a + b e1,5 + c e2ln e + d e/ln e
a = -3.51; b = -18154.6;
c = -12213.8; d = -6925.7

e > 0.034

* the number data of meeting the boundary conditions taken into account in determination of goodness-of-fit criteria

Table 2. Goodness-of-fit criteria for equations 10 -12 and published PTF in estimation Ks on current data set (n = 35)


Original training
data set






Proposed equations

equation 8 modification of Ahuja et al. 1984)




± 2.823



equation 9(a




± 2.381



equation 10(a




± 2.044



Previously published PTFs

Cosby et al. (1984)




± 11.888



Saxton et al. (1986)




± 9.206



Jabro (1992)




± 40.578



Dune and Pucket (1994)




± 8.180



Schaap et al. (1998) (b




± 4.654



Schaap et al. (1998) (c




± 5.130



Minasny and McBratney (2000), modification of Ahuja et al. (1984)




± 2.944



(a goodness-of-fit criteria determined for 31 data meeting the boundary conditions
(b Rosetta 1.2/2 – input data: sand, silt and clay contents and bulk density
(c Rosetta 1.2/3 – input data: sand, silt and clay contents, bulk density and vol. water content at 33 kPa

A recent analysis of the accuracy of Ks estimations according to different PTFs by Minasny and McBratney (28) has shown that on an independent data set the highest accuracy of Ks estimation (RMSR = 2.09 ln(µm × s-1) provided the model of Ahuja et al. (1984) with the parameters proposed by Minasny and McBratney [28], and for the model Rosetta 1.2/2 [35].

When applied to the current data set, the solution proposed by Minasny and McBratney [28] provides the best accuracy of Ks estimation [(RMSR = 1.07 ln(µm × s-1), Table 2] and along with the model of Rosetta 1.2/3 (35) gives the best value of the geometric mean of error ratio (GMER = 0.73 and 1.3; respectively). However, still these accuracies are worse than those achieved with the equations proposed in this work (eqn. 11 and 12, Table 3).

Table 3. Goodness-of-fit criteria for equations 10-12 and published PTFs in estimation Ks on selection of UNSODA data set (n = 128) (Leij et al. 1996)







increase of RMSRd

Proposed equations

equation 8 (modification
of Ahuja et al. 1984)



± 4.629




equation 9



± 3.503




equation 10



± 3.731




Previously published PTFs

Cosby et al. (1984)



± 6.186




Saxton et al. (1986)



± 11.361




Jabro (1992)



± 7.989




Dune and Pucket (1994)



± 8.106




Schaap et al. (1998) (b



± 5.044




Schaap et al. (1998) (c



± 4.285




Minasny and McBratney
(2000), modification of
Ahuja et al. (1984)



± 5.414




(b, (c – as in table 3
(d – relative increase of RMSR (%) = 100% × (RMSRi – RMSR min)/ GRMSR min

It should be noted that a comparative study has been performed only for those PTFs, whose quality has been established in earlier analyses [28, 34, 38]. The exception is the model proposed by Jabro [14], hitherto not verified, but used in the model study Vieira et al. [39].

The hitherto proposed PTFs have been also tested on the independent selection of UNSODA data set [23], and the corresponding measures of the goodness-of-fit are given in Table 3.

The results of analogous analyses performed on the current data set (Table 2) proved a higher accuracy of Ks estimation with equation 12, while the accuracy of predictions with equation 12 was somewhat lower. The accuracy of Ks estimation by both these equations on an independent data set clearly points to a higher goodness-of-fit of equation 12 (Table 3). The fact that the best fits were obtained with the use of equation 11 suggests its universal character, however, under the above-mentioned limitations of its use.

The model Rosetta 1.2/3 [35], developed on UNSODA data set, gives in fact the highest coefficient of determinations (R2 = 0.499, Table 3), however, the absolute error of this model is 36 % higher than that of equation 12 (RMSR = 1.55 and 1.25 ln(µm × s-), respectively; Table 3).

In general, the models taking into account only the soil texture, bulk density and total porosity [9, 35, 14, 10] provide predictions with poor accuracy (Tables 2 and 3). These models have been developed on large data sets (Table 2) and their low accuracy is related to the fact that they use only parameters indirectly related to Ks. According to the data shown in Fig. 1 the distributions of Ks in both data sets analysed are similar in spite of significantly different distributions of the soil texture, bulk density and total porosity. The absolute error of the above-given models on both data sets (RMSR in Tables 2 and 3) clearly shows that their accuracy of Ks estimation is lower than that of PTFs using the drainage porosity concept Ahuja et al. [2] and proposed eqn. 11 and 12. The only exception is the model Rosetta, developed on the basis of physical parameters of the soil by the method of neural network analysis. This model is recommended when the drainage porosity data are not available or when the values of this parameter are beyond the range of the equation applicability.

Analysis of the accuracies of predictions by different PTF, on the current and the validation data sets (Tables 2 and 3) indicates that for the soils of high density (the current data set) the majority of PTF overestimate Ks (GMER>1, Table 3), except that proposed by Minasny and McBratney [28]. For the soils of smaller densities (selection of UNSODA data set) [23] only the models proposed by Saxton et al. [33] and Jabro [14] overestimate the value of Ks. This comparative analysis also shows (Table 3) that for the soils of intermediate bulk densities eqn. 11 proposed in this work provides the best accuracy of predictions [RMSR = 1.25 ln(µm × s-1) table 3], negligible error ratio (GMER = 1.03) and the lowest geometric deviation of error ratio (GSDER = 3,50), whereas for the soils of high densities more accurate estimation of Ks is provided by eqn. 12 (RMSR = 1.33 ln(µm  ×  s-1), GMER = 0.808 and GSDER = 3.71; Table 3).

The practical application of the equations presented is only determined by the knowledge of FC at 10 kPa, the parameter necessary for estimation of the drainage porosity. The availability of these data is often limited. The procedure of determination of FC is time consuming and needs special equipment and undisturbed soil samples, which is a significant drawback in the use of the relations proposed. Hence, a natural continuation of the study will be the search for a possibility of indirect assessment of Ks and FC on the basis of the standard analytical data. It seems that an increase in the accuracy of Ks estimation by PTFs can be achieved by analysis of Ks(d) relationships on large and genetically homogeneous data sets or by applying the ANN method for this data sets.


  1. The results presented have fully confirmed the thesis that the drainage porosity has determining effect on the saturated hydraulic conductivity of water in soil. Therefore, the drainage porosity can be treated as a universal parameter describing the physical soil condition, its relative with soil compaction and soil structure.

  2. The relation Ks= f(d) has been described by the equation proposed by Ahuja et al. (2), which, in the light of the results reported in this work, is insufficient and not fully correct as it assumes a linear character of this relation in the log-log system of coordinates. The relation proposed in this paper (Ks = a + b d1.5 + c d2.5 + d d3 where: a = -2.52; b = 581.598; c = 6966.14; d = 11693.78) curvilinear in the log-log system of coordinates, has enhanced the accuracy of estimation of Ks. The proposed Ks(d) relations has been derived on the basis of a relatively small data set (N = 35), however, their verification on the independent validation data set indicates accurate estimation of Ks(d).


Financial support from the state budget under project No 2 PO4G 094 28 is gratefully acknowledged.


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Accepted for print: 2.01.2007

Marek Spychalski
Department of Soil Science and Land Reclamation,
Agricultural University of Poznan, Poland
Piatkowska 94, 61-691 Poznan, Poland
Phone: 61-846-6440
email: marspych@ au. poznan. pl

Cezary Kaźmierowski
Department of Soil Science and Remote Sensing of Soils,
UAM, Poznan, Poland
Dziegielowa 27, 61-680 Poznan, Poland

Zbigniew Kaczmarek
Department of Soil Science,
Agricultural University of Poznan, Poland
Mazowiecka 42, 60-623 Poznan, Poland

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.