Published online 28 September 2007
Published in Soil Sci Soc Am J 71:1685-1693 (2007)
DOI: 10.2136/sssaj2006.0357
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SOIL PHYSICS
Explicit Formulae for Soil Water Diffusivity Using the One-Step Outflow Technique
J. D. Valiantzas,
P. Londra* and
A. Sassalou
Agricultural Univ. of Athens, Lab. of Agricultural Hydraulics, 75 Iera Odos Str., 11855 Athens, Greece
* Corresponding author (v.londra{at}aua.gr).
 |
ABSTRACT
|
|---|
One of the most widely used methods in determining soil water diffusivity, D, as a function of the volumetric water content, 
, is the one-step outflow method. Unfortunately, the calculation of D() according to direct methods usually requires the estimation of the first and second derivative of the outflow volume, V, dV/dt and d2V/dt2 of the original measured cumulative outflow data V(t). These derivatives are usually approximated by applying a finite difference technique to two or three consecutive outflow measurements. Therefore, the accuracy of the results depends essentially on the accuracy of local outflow measurements. Inaccuracies are more intensive in the last portion of the outflow curve, where small errors of measurement may yield large inaccuracies in the first and second derivatives of the outflow curve. Furthermore, the entire calculation procedure for estimating dV/dt and d2V/dt2 is rather laborious and complicated. We developed a direct method based on a simple curve-fitting procedure applied to the experimental outflow data for a simple power and an extended power form function, which leads to direct calculation of the soil water diffusivity function from explicit formulae. The only parameters involved in these formulae are the empirical parameters obtained by the curve-fitting procedure. The proposed method was verified from one-step outflow experiments performed in nine porous materials (one sand, two soils, and six substrate mixtures). Comparison with previous methods indicated a good performance of the proposed explicit algebraic formulae.
|
INTRODUCTION
|
|---|
Modeling water flow in unsaturated porous media for the management of irrigation water as well as for the movement of salts and contaminants in soil depends on the availability of representative values for soil hydraulic properties. Both the hydraulic conductivity (K), as a function of volumetric water content () or water pressure head (h), and the soil water retention function (h) influence the rate at which water and solute move in the vadose zone, thus affecting the travel time of contaminants to groundwater.
The soil water retention function can easily be determined by measuring water contents at different pressure heads. Unsaturated hydraulic conductivity measurement, on the other hand, is often time consuming and may require expensive equipment.
As reliable direct measurements of unsaturated hydraulic conductivity are often difficult to conduct, several researchers have proposed statistical pore-size distribution models that predict hydraulic conductivity from the more easily measured water retention data and the measured value of saturated hydraulic conductivity, Ks (Childs and Collis George, 1950; Burdine, 1953; Mualem, 1976). Furthermore, the introduction of continuous analytical functions for the soil water retention curve, combined with capillary theories, leads to different closed-form analytical models describing soil hydraulic properties (Brooks and Corey, 1964; van Genuchten, 1980). Unfortunately, in some cases, the calculated K() values using the soil water retention data and the saturated conductivity may deviate significantly from the actual K() (Talsma, 1985; Poulovassilis et al., 1988; Valiantzas and Sassalou, 1991).
The one-step outflow method is one of the most widely used laboratory techniques (being easily adopted for routine laboratory work) for determining the soil water diffusivity relationship D() or the hydraulic conductivity function K() (if water retention data are available). Moreover, the technique yields fast results and is relatively cheap.
The determination of soil hydraulic properties from outflow data is approached in two different ways. On the one hand, in the direct approach, Gardner (1962), Gupta et al. (1974), Passioura (1976), Valiantzas et al. (1988), and Valiantzas (1989, 1990) have proposed procedures for directly calculating D() from the outflow rate produced by a single large step in pressure. This experimental procedure for obtaining the diffusivity function has been termed the one-step outflow method (Doering, 1965). The direct-approach methods used by these researchers do not require assumptions of any mathematical form for the determination of hydraulic properties, i.e., (h) and K().
On the other hand, in the indirect approach, the parameter estimation methods have been applied together with laboratory outflow experiments for determining hydraulic properties (Kool et al., 1985; Parker et al., 1985; Kool et al., 1987; Valiantzas and Kerkides, 1990; van Dam et al., 1992, 1994; Eching and Hopmans, 1993; Eching et al., 1994; and others). Parameter estimation techniques enable simultaneous determination of the soil water retention curve and hydraulic conductivity and diffusivity functions just from an outflow experiment. An assumption of particular mathematical forms for the hydraulic properties is required, however. The indirect method is certainly more flexible than the direct one, but nonetheless there are some disadvantages concerning the way the problem is posed, instability, and lack of uniqueness of the solution.
According to the one-step outflow method, a short disturbed or undisturbed soil sample of length L is placed on top of a saturated porous plate. The soil sample is allowed to wet gradually from the bottom until saturation is reached. The sudden application of a large increment of positive pressure at the top of the soil sample, or negative pressure from the bottom of the soil sample in a Haines-type assembly, marks the initiation of the outflow process by which the outflow volume V is recorded with time t until the soil water content reaches the final equilibrium value f.
The calculations of D() according to the direct methods, however, usually require the estimation of the first and second derivative of V, dV/dt and d2V/dt2, of the original measured cumulative outflow data V(t). These derivatives are usually approximated by applying a finite difference technique to two or three consecutive outflow measurements. Therefore, the accuracy of the results depends essentially on the accuracy of local outflow measurements. Inaccuracies are more intensive in the last portion of the outflow curve, where small errors of measurement may yield large inaccuracies in the first and second derivatives of the outflow curve. Furthermore, the entire calculation procedure for estimating dV/dt and d2V/dt2 is rather laborious and complicated.
In this study, a simplified procedure leading to the direct calculation of the D() [or K()] relationships by simple algebraic formulae was developed. The method requires a simple curve fitting of a power function or an extended power function to the original outflow V(t) data. Since, the fitted curve is used for the calculation of the first and second derivatives of the outflow volume, the accuracy of these derivatives is not affected by perturbations due to random experimental errors in the region of calculation. Using the equation proposed by Valiantzas (1989), simple algebraic formulae were derived for the direct calculation of D(). The only parameters required in these equations are the empirical curve-fitting parameters obtained by the curve-fitting procedure. This method, contrary to the method based on the parameter estimation technique, does not require any assumption concerning the mathematical forms of the hydraulic properties and is not affected by numerical instabilities (nonconvergence of the solution or nonuniqueness problem).
For the validation of the method, one-step outflow experiments were performed in the laboratory on sand, soils, and various substrate mixtures to determine the relationship between diffusivity or hydraulic conductivity and moisture content.
|
MATERIALS AND METHODS
|
|---|
Simplified Diffusivity Formulae
Valiantzas (1989) proposed a simple equation for determining D as a function of mean volumetric water content, 
, from the one-step outflow data. The proposed equation was derived from the systematic analysis of a weighting function that corrects Gardner's diffusivity function (Gardner, 1962). The proposed equation was validated for various types of soils (Valiantzas, 1989; Valiantzas and Sassalou, 1991).
The D() function of Valiantzas (1989) is calculated from the cumulative outflow volume V(t) as
 | [1] |
where is the mean volumetric water content at time t, = i – V(t)/Vc, i is the uniform initial volumetric water content, Vc is the core sample volume, and q is the rate of outflow, q = d /dt. From the plot of the measured V against the square root of time 
t, the nonlinear portion of the curve (Stage III), where the effect of impedence of the porous plate becomes negligible, is identified (Passioura, 1976; Valiantzas, 1990). From the data of this stage, using a finite difference scheme, the derivatives q = d /dt and dq/d are then calculated. This method (Valiantzas, 1989) was tested for accuracy by various simulated experiments representing typical situations as well as two laboratory experiments (Valiantzas and Sassalou, 1991).
The three stages of outflow, according to Passioura (1976), are clearly identifiable in all curves. In the first stage, corresponding to the initial part of the curve before the outflow becomes linear with t, it is the plate impedance only that determines the outflow rate. In the second stage, during which the outflow varies linearly with t, the effect of plate impedance has not yet become negligible. The third stage corresponds to the portion of the curve where cumulative outflow ceases to be linear with t and the effect of plate impedance is minimal.
In this study, two simple empirical functions—a simple power form and an extended power form—are suggested to describe the measured cumulative outflow vs. t data. Combining these empirical equations with Eq. [1], simple algebraic formulae depending on the empirical curve-fitting parameters are derived for D().
Fitting the Power Form Equation
To identify the region of the measured outflow data V(t) that can be exploited to determine the D() function (Stage III), we represent the V(t) graph under its conventional form V(t). Instead of using the original variable V, a new dimensionless variable S, characterizing the fractional remaining outflow water volume, is introduced:
 | [2] |
where V
is the total volume of outflow. The measured V(t) data transformed to S(t) are used in the curve-fitting procedure. The following simple power form equation is proposed to fit the S(t) data (Fig. 1
):
 | [3] |
where A and B are the two empirical curve-fitting parameters of the power form Eq. [3]. The parameters to and b1 are related to A and B as b1 = –B and to = A–2/B. Equation [3] is valid for t 
to and –1 < B < 0. The physical interpretation of to is shown in Fig. 1. According to Eq. [3], S 
1 [V(t) 0] when t to and S 0 [V(t) V] when t . The portion of the S(t ) data used in the curve-fitting procedure corresponds to the third stage of the curve (Fig. 1), where t timp (timp is the time when the outflow curve becomes nonlinear and the effect of plate impedence becomes negligible) or S 
Simp or imp, where imp = Simp(i – f) + f. The values of timp and Simp can be identified from the S(t ) data curve (see Fig. 1).
Combining Eq. [2] and [3], the mean volumetric water content can be expressed as a function of time t:
 | [4] |
Then q = d/dt and the derivative dq/d are calculated as
 | [5] |
 | [6] |
Substituting Eq. [5] and [6] into Eq. [1] yields:
 | [7] |
To apply the proposed simplified algebraic formula Eq. [7], the power function Eq. [3] is fitted to the outflow experimental data S(t ), and the two fitting parameters, A and B, are obtained. Then D()is calculated directly from Eq. [7].
Extended Power Function
A similar procedure is applied using the following extended three-parameter power form equation to fit the S(t ) data:
 | [8] |
Since –1 < b < 0 when t , S(t ) c, and V(t) V(1 – c), Eq. [8] is valid for the values of t < (–c/a)2/b, since for the values of t > (–c/a)2/b the S values become negative. The extended power function does not comply with the physical phenomenon for t but, for the region of measurements, it offers a better fit (Fig. 2
).
The variable is expressed as a function of t by combining Eq. [2] and [8]:
 | [9] |
and the derivatives q and dq/d are calculated as
 | [10] |
and
 | [11] |
Substituting Eq. [10] and [11] into Eq. [1] yields:
 | [12] |
Calculation of Diffusivity as a Function of Mean Water Content
The procedure for determining D() is as follows:
1. Perform the one-step outflow experiment (the experimental procedure is developed below) and record V vs. t.
2. Plot the S = 1 – V(t )/V data values vs. t, and identify Stage III, i.e., the portion of the curve where S ceases to be linear with respect to t.
3. Fit the power functions S = A(t )B or S = a(t )b + c to Stage III of the S(t ) data. The A and B or a, b, and c fitting parameters are obtained. The least-square fitting procedure can be easily realized in a spread sheet (e.g., Excel) or another more sophisticated program.
4. Then the D() function is determined directly from Eq. [7] and [12].
The K() relationship is calculated according to the relationship K() = D()d/dh (Childs and Collis George, 1950). The slope d/dh is evaluated from the experimental retention curves.
Contrary to the above procedure, the inverse numerical analysis yields both K() and (h) from the same outflow data but this method has some disadvantages concerning instability and nonuniqueness of the solution.
Porous Materials
Experiments to determine the retention curves followed by the one-step outflow procedure were performed for nine samples of porous materials (one sand, two loam soils, and six substrate mixtures of peat, perlite, and coir), as they are presented in Table 1
. In addition, the results reported by Green et al. (1998) referring to the one-step outflow experiment of a red kandosol A horizon (RK) were used to test the method.
View this table:
[in this window]
[in a new window]
|
Table 1. Conditional parameters of initial and final pressure head (hi and hf), initial and final volumetric water content (i and f), core sample length (L), and saturated hydraulic conductivity (Ks) of the experimental outflow procedure.
|
|
A sample of pure sand (SA) with grain sizes in the range between 210 and 500 µm was used. This sand material was cleaned with HCl to remove CaCO3.
Two undisturbed samples of a loam soil (22% clay, 34% silt, and 44% sand) were used, one sample taken before tillage (LOAMpt) and the other immediately after tillage (LEV) (Boubouka-Sassalou, 1996).
The six substrates used were selected to cover the range of porous materials used in container plant production. All substrates were based on peat, coir, and floriculture perlite. The peat was Lithuanian sphagnum peat moss. The coir was in compressed form with dimensions 20 by 10 by 5 cm and was a byproduct of coconut husk fiber treatment. The peat-based substrate mixtures were created with (on a volume basis): (i) 100% Lithuanian sphagnum peat moss (P100); (ii) 75% Lithuanian sphagnum peat moss and 25% perlite (P75); and (iii) 50% Lithuanian sphagnum peat moss and 50% perlite (P50). The coir-based substrate mixtures were created with (on a volume basis): (i) 100% coir (C100); and (ii) 50% coir and 50% perlite (C50) (Londra, 2001). The Levington F2 substrate was also used. This is a compost of fine texture with a medium nutrient level.
Experimental Procedure
Retention curve measurements and one-step outflow laboratory experiments were performed either (i) in pressure cell chambers, where a gas pressure step was applied to the top of the sample, or (ii) on a tension plate apparatus in a Haines-type assembly (Haines, 1930), where a negative (suction) pressure step was applied through the saturated tension plate. Pressure cell chambers were used for the undisturbed samples of the sand and for the two samples of the loam soil (pre- and post-tillage). The tension plate apparatus in a Haines (1930) type assembly was used for the disturbed samples of the substrate mixtures (P100, P75, P50, C100, C50, and the Levington compost). Since interest for the behavior of substrates is usually focused on the relatively low negative pressure region, the tension plate apparatus was used for the experimental procedure (plate air-entry value 190 cm H2O). Note that the plate impedance in the tension plate apparatus is significantly lower than that observed in classic pressure cell chambers. The pressure cell chambers that cover a larger region of pressures are used for soil samples (plate air-entry value 800 cm H2O).
Initially, the retention curves (h) were measured, followed by the one-step outflow experiment in the same apparatus. The sand sample was placed in a pressure cell chamber, after proper packing to assure homogeneity. The average length of the sample was 1.59 cm and the diameter was 7 cm. The sand sample was allowed to wet gradually until saturation. After that, the sample was subjected to a wetting–drying cycle and the primary drying curve data were obtained. To perform the one-step outflow experiment, the sample was initially under a small pressure head (hi = –4.2 cm H2O) corresponding to i = 0.295 m3 m–3, then a positive gas pressure step (hf = 279 cm H2O) was suddenly applied at the top of the sample and the cumulative outflow volume was recorded with time. The measured value of f was 0.021 m3 m–3 (Table 1). A similar experimental procedure in the pressure cell apparatus was also followed for the two soils, pretillage and post-tillage loam. For the substrate mixtures, samples were put on a tension plate apparatus in a Haines-type assembly (Haines, 1930). The air-entry value of the tension plate was –190 cm H2O. The samples were allowed to wet gradually from the bottom of the plate until saturation. After that, the samples were subjected to a wetting–drying cycle and the primary drying curve data were obtained. To perform the one-step outflow experiment, each sample was initially under a small pressure head hi corresponding to i. A negative pressure step hf was suddenly applied at the bottom of the sample and the cumulative outflow volume was recorded with time.
The saturated hydraulic conductivity, Ks, was determined independently by the constant-head method. Each sample was subjected to a wetting–drying cycle before the measurement.
The various conditional parameters involved in the experimental procedure are presented in Table 1.
|
RESULTS AND DISCUSSION
|
|---|
Figure 2 presents a comparison between the measured S(t) data and the predictions obtained using a curve-fitting procedure for the simple power form function, Eq. [3], and the extended power function, Eq. [8]. A detailed description of A and B or a, b, and c parameters and their statistical results for power functions are presented in Table 2
. The results indicated that there is a very good agreement between the experimental S(t) data and the prediction obtained using the extended fitted power form function (Eq. [8]).
View this table:
[in this window]
[in a new window]
|
Table 2. Empirical parameters and their statistical results for simple power and extended power curve-fitting functions. The values in parentheses are the standard errors of fitted parameters and R2 is the coefficient of determination of the fitted curve.
|
|
On the other hand, there is a relatively good agreement between experimental S(t) values and the simple power form function predictions for the majority of the porous materials examined. For two of the materials (the loam soil and the Levington compost), however, the deviation of the power equation prediction from the experimental data became relatively large.
Figure 3
presents a comparison between the D() prediction obtained using the original (finite difference scheme) method of Valiantzas (1989) applied to the outflow data, the D() prediction obtained by algebraic Eq. [7] (based on a simple power function), and the prediction of Eq. [12] corresponding to the extended power function. For all the porous materials examined, there is a very good agreement between the original Valiantzas (1989) D() prediction and the D() values obtained by Eq. [12] using the extended power function. The agreement between the Valiantzas (1989) prediction and the simple power form prediction (Eq. [7]) is relatively good. Even in the worst-case scenarios of the loam soils and the Levington compost (Fig. 3b, 3c, and 3j), the maximum relative deviation between the two predictions of D() does not generally exceed 15% in the zone near the imp value.
View larger version (12K):
[in this window]
[in a new window]
|
Fig. 3. Soil water diffusivity as a function of volumetric water content, D(), using outflow data for various porous materials obtained by the Valiantzas equation (finite difference scheme), a simple power function, an extended power function, and Passioura's method. The short vertical line labeled "the effect of porous plate impedance" defines the region where the plate impedance is not negligible.
|
|
Passioura (1976), assuming the rate of change of water content, 
/t, to be effectively constant throughout the core at any given time, derived the following simple approximate equation:
 | [13] |
where L is the moisture content at the core top and q is the rate of outflow. To determine the unknown values of L that correspond to the estimated values of D, a graphical method was suggested by Passioura (1976).
For two porous materials (the sand and the red kandosol A horizon), D() was also calculated using Passioura's (1976) method (Fig. 3a and 3d). There is a good agreement between Passioura's D() predictions and the predictions of the methods used in this study.
Figure 4
presents a comparison between K() values obtained from the outflow data using (i) the method of Valiantzas (1989) using the finite difference scheme, (ii) the simple power function equation, (iii) the extended power function equation, and (iv) Passioura's (1976) method (for the sand and red kandosol samples). In all cases, K() values were obtained using measured V(t) and (h) curves as input. For all the porous materials examined, there is a good prediction of K() by the outflow methods. If the value of Ks is also measured, and combining the van Genuchten (1980) equation with a simplified form of the Mualem (1976) model, then K() can be predicted. The comparison between K() predictions according to the van Genuchten–Mualem model and the outflow prediction indicates that in some porous materials (the post-tillage loam, red kandosol, 100% peat moss, and 50/50 coir/perlite mixture), there is a very good agreement of the results (Fig. 4b, 4c, 4d, and 4h). In some other porous materials, there is a significant deviation between the van Genuchten K() prediction and the other methods.
View larger version (13K):
[in this window]
[in a new window]
|
Fig. 4. Soil hydraulic conductivity as a function of volumetric water content, K(), using outflow data for various porous materials obtained by the Valiantzas equation, a simple power function, an extended power function, Passioura's method, and the van Genuchten method.
|
|
This significant deviation of the two predictions may be due to: (i) the uncertainty of the experimentally obtained Ks values (van Dam et al., 1990; Tokunaga, 1988)—note that the value of Ks is crucial in the van Genuchten K() predictions—and (ii) the use of the simple form of the Mualem (1976) model where the tortuosity factor p is rather arbitrarily taken to be 0.5 (Valiantzas and Sassalou, 1991).
It is notable that the K() prediction according to the present method does not require the measurement of the Ks value.
Measured outflow data of the 50/50 peat/perlite substrate mixture were analyzed using the HYDRUS-1D code (
im
nek et al., 1998). The inverse method analysis proposed by HYDRUS-1D leads to the estimation of the two curve-fitting parameters 
and n and the residual volumetric water content, r, of the van Genuchten equation for the prediction of the hydraulic properties (h) and K(). The HYDRUS-1D required input data are the outflow measurements, the hydraulic conductivity of the porous plate (Kp = 0.0093 cm min–1), and the Ks of the porous medium. Figure 5
shows the measured water retention curve of the 50/50 peat/perlite substrate along with the final water retention curve estimates obtained by the HYDRUS-1D computer program. As shown (see also Table 3
), the HYDRUS-1D applied with different initial values of , n, and r leads to different final estimates of these parameters. It is clearly shown that there is a nonuniqueness solution problem for the case of the 50/50 peat/perlite substrate. Furthermore, the two final solutions corresponding to two different local minima deviate significantly from the measured retention data. For values of > 0.09, the numerical solution is not convergent.
View this table:
[in this window]
[in a new window]
|
Table 3. Initial and final estimates of the curve-fitting parameters and n and the residual volumetric water content r obtained by the HYDRUS-1D code for the 50% peat moss, 50% perlite mixture and statistical results of the final parameter values. The values in parentheses are the standard errors of fitted parameters and R2 is the coefficient of determination of the outflow data fitted curve.
|
|
|
SUMMARY AND CONCLUSIONS
|
|---|
A simple power form function and an extended power function were fitted to the experimental outflow data. Simple algebraic equations were derived to calculate directly the D() function.
The comparison between D() predictions obtained by the extended power function corresponding to a D() algebraic equation and the original Valiantzas (1989) prediction as well as Passioura's (1976) results indicates a very good agreement among all the methods.
When the simple power function form was used, the obtained D() predictions were in relatively good agreement with the other two methods. Small deviations were observed near the region where the effect of plate impedance is significant. The results indicated that the method based on the extended power function can be used as a routine procedure to obtain D() predictions close to other methods such as Valiantzas (1989) or Passioura (1976). The simple power function method leads to less accurate predictions than the extended power function.
The comparison of the K() predictions based on the outflow data and the van Genuchten–Mualem model indicated that, for some porous materials, there is a significant deviation between the two methods. This deviation may be due to the uncertainty of the Ks measurement, which is a crucial value in the van Genuchten (1980) model, or to the use of a restricted Mualem (1976) model where the parameter p is a fixed value (0.5).
|
NOTES
|
|---|
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.
Received for publication October 17, 2006.
|
REFERENCES
|
|---|
- Boubouka-Sassalou, A. 1996. The effect of cultivation practices and climatological conditions on the soil hydraulic properties of the upper soil layer. Ph.D. diss. Dep. of Nat. Resour. Manage. and Agric. Eng., Agric. Univ. of Athens, Athens, Greece.
- Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrol. Pap. 3, Civ. Eng. Dep., Colorado State Univ., Fort Collins.
- Burdine, N.T. 1953. Relative permeability calculations from pore-size distribution data. Pet. Trans. Am. Inst. Min. Metall. Eng. 198:71–77.
- Childs, E.C., and N. Collis-George. 1950. The permeability of porous materials. Proc. R. Soc. London Ser. A 201:392–405.
- Doering, E.J. 1965. Soil water diffusivity by the one-step method. Soil Sci. 99:322–326.
- Eching, S.O., and J.W. Hopmans. 1993. Optimization of hydraulic functions from transient outflow and soil water pressure data. Soil Sci. Soc. Am. J. 57:1167–1175.[Web of Science]
- Eching, S.O., J.W. Hopmans, and O. Wendroth. 1994. Unsaturated hydraulic conductivity from transient multistep outflow and soil water pressure data. Soil Sci. Soc. Am. J. 58:687–695.[Web of Science]
- Gardner, W.R. 1962. Note on the separation and solution of diffusion type equations. Soil Sci. Soc. Am. Proc. 26:404.
- Green, T.W., Z. Paydar, H.P. Cresswell, and R.J. Drinkwater. 1998. Laboratory outflow technique for measurement of soil water diffusivity and hydraulic conductivity. Tech. Rep. 12/98. CSIRO, Australia.
- Gupta, S.C., D.A. Farrel, and W.E. Larson. 1974. Determining effective soil water diffusivities from one-step outflow experiments. Soil Sci. Soc. Am. J. 38:710–716.[Abstract/Free Full Text]
- Haines, W.B. 1930. Studies in the physical properties of soils. V. The hysteresis effect in capillary properties and the modes of moisture distribution associated therewith. J. Agric. Sci. 20:97–116.
- Kool, J.B., J.C. Parker, and M.Th. van Genuchten. 1985. Determining soil hydraulic properties from one-step outflow experiments by parameter estimation: I. Theory and numerical studies. Soil Sci. Soc. Am. J. 49:1348–1354.[Abstract/Free Full Text]
- Kool, J.B., J.C. Parker, and M.Th. van Genuchten. 1987. Parameter estimation for unsaturated flow and transport models—A review. J. Hydrol. 91:255–293.[CrossRef]
- Londra, P. 2001. Moisture–aeration balance of substrates as developed under different irrigation treatments. Ph.D. diss. Dep. of Nat. Resour. Manage. and Agric. Eng., Agric. Univ. of Athens, Athens, Greece.
- Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513–522.[CrossRef]
- Parker, J.C., J.B. Kool, and M.Th. van Genuchten. 1985. Determining soil hydraulic properties from one-step outflow experiments by parameter estimation: II. Experimental studies. Soil Sci. Soc. Am. J. 49:1354–1359.[Abstract/Free Full Text]
- Passioura, J.B. 1976. Determining soil water diffusivities from one step outflow experiments. Aust. J. Soil Res. 15:1–8.
- Poulovassilis, A., M. Polychronides, and P. Kerkides. 1988. Evaluation of various computational schemes in calculating unsaturated hydraulic conductivity. Agric. Water Manage. 13:317–327.[CrossRef]
- imnek, J., M. Sejna, and M.Th. van Genuchten. 1998. The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media. U.S. Salinity Lab., Riverside, CA.
- Talsma, T. 1985. Prediction of hydraulic conductivity from soil water retention data. Soil Sci. 140:184–188.
- Tokunaga, T.K. 1988. Laboratory permeability errors from annular wall flow. Soil Sci. Soc. Am. J. 52:24–27.[Abstract/Free Full Text]
- Valiantzas, J.D. 1989. A simple approximate equation to calculate diffusivities from one-step outflow experiments. Soil Sci. Soc. Am. J. 53:342–349.[Abstract/Free Full Text]
- Valiantzas, J.D. 1990. Analysis of outflow experiments subject to significant plate impedance. Water Resour. Res. 26:2921–2929.[CrossRef]
- Valiantzas, J.D., and P.G. Kerkides. 1990. A simple iterative method for simultaneous determination of soil hydraulic properties from one-step outflow data. Water Resour. Res. 26:143–152.[CrossRef]
- Valiantzas, J.D., P.G. Kerkides, and A. Poulovassilis. 1988. An improvement to the one-step outflow method for the determination of soil water diffusivities. Water Resour. Res. 24:1911–1920.[CrossRef]
- Valiantzas, J.D., and A. Sassalou. 1991. Laboratory determination of unsaturated hydraulic conductivity using a generalized-form hydraulic model. J. Hydrol. 128:293–304.[CrossRef]
- van Dam, J.C., J.N.M. Stricker, and P. Droogers. 1990. From one-step to multi-step: Determination of soil hydraulic functions by outflow experiments. Rep. 7. Dep. of Hydrol., Soil Phys. and Hydraul., Wageningen Agric. Univ., Wageningen, the Netherlands.
- van Dam, J.C., J.N.M. Stricker, and P. Droogers. 1992. Inverse method for determining soil hydraulic functions from one-step outflow experiments. Soil Sci. Soc. Am. J. 56:1042–1050.[Abstract/Free Full Text]
- van Dam, J.C., J.N.M. Stricker, and P. Droogers. 1994. Inverse method to determine soil hydraulic functions from multistep outflow experiments. Soil Sci. Soc. Am. J. 58:647–652.[Web of Science]
- van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898.[Web of Science]
TOP
ABSTRACT
INTRODUCTION
b1] and extended power [S = a(
