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ORIGINAL RESEARCH

Modeling the Soil Water Retention Curve for Conditions of Variable Porosity

^a Institute of Plant Nutrition and Soil Science, University of Kiel, Olshausenstr. 40, 24118 Kiel, Germany
^b C.F Strange, Present address: Department of Soil Sciences, UFZ Centre for Environmental Research, Halle, Germany
^* Corresponding author (rhorn{at}soils.uni-kiel.de)

Received 7 October 2004.

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORETICAL ASPECTS MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Temporal changes in the pore space of unsaturated soils may result from such processes as mechanical compaction and tillage of agricultural soils. Moreover, soil pore space and soil water content are mutually interactive. For example, porosity and pore-space geometry control the unsaturated soil hydraulic properties such as the water retention curve, while conversely soil water and hydraulic stress often affects soil pore size. The experiments discussed in this paper focus on both of these two aspects and additionally consider how a previously applied mechanical stress can influence the effects of subsequent hydraulic stresses on the soil hydraulic properties. We show the effects of uniaxial volume changes during drying on the water retention curves of three sandy and silty soils. Results indicate that previously loaded soils differ in how drying affects volume changes. Uniaxial volume changes of all soils could be described well with a model similar to van Genuchten's soil water retention function. Four models were developed to describe soil water retention as a function of variable void ratio and then tested against measured data. The variably porosity models generally give much better descriptions of the observed retention data than the conventional van Genuchten function. Two models which express the van Genuchten shape parameters and n in terms of two adjustable coefficients were found to be particularly attractive. The models presented in this study permit one to consider changes in the void ratio induced both by externally applied stress and by internal (i.e., hydraulic) stress.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORETICAL ASPECTS MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
WATER FLOW AND SOLUTE transport processes in structured soils depend not only on soil porosity and soil texture, but also on pore shape and pore continuity or tortuosity (e.g., Horton et al., 1994). The influence of soil textural and structural properties on water flow can be described approximately using the pore-size distribution (PSD), which is often calculated from the corresponding water retention curve. However, soil pore systems (including soil structure) are generally not rigid; for example, the PSD often changes with time as a result of wetting and drying processes, chemical exchange, and human and biological activities (Leij et al., 2002; Horn, 2004). Consequently, associated water flow and solute transport conditions often also vary with time. Additionally, the effects of stress-induced soil structural changes on the unsaturated hydraulic properties must be considered. Unfortunately, such effects are seldom analyzed in detail (Horn et al., 1995), mostly because they are difficult to predict (Horton et al., 1994).

Describing and accurately quantifying dynamic changes in the PSD in response to internal and external stresses is challenging because of the complexity of the pore space and our incomplete understanding of the main processes involved. Development and testing models accounting for changes in the soil PSD is further hampered by a lack of comprehensive data about soil structural and hydraulic dynamics (Leij et al., 2002). The theory of one-dimensional water flow in saturated and unsaturated swelling soils, and the fundamental differences between flow in swelling and rigid matrix systems, is reviewed by Smiles (2000). Terzaghi (1923) already some 80 years ago described water flow in soils subject to volume changes. Most current approaches for calculating water flow are based on a mass balance equation and Darcy's law. The use of material coordinates, instead of fixed spatial coordinates such as normally used for nonswelling soils, leads to a flow equation analogous to the Richards equation (Smiles, 2000). Darcy's law relates the volume flux of water, q, to the gradient in the hydraulic potential and the soil hydraulic conductivity. Application of the Richards equation requires knowledge of both the water content, , and the hydraulic conductivity, K, as a function of the pressure head, h.

A popular equation for describing the soil water retention curve, (h), is the van Genuchten function (van Genuchten, 1980), which enables estimation of the unsaturated soil hydraulic conductivity if combined with the model of Mualem (1976). Empirical experiments to determine the hydraulic functions are generally time consuming and expensive. An alternative to direct measurements is the use of pedotransfer functions. To predict the water retention curve (such as the shape parameters and n in van Genuchten's function), as well as the hydraulic conductivity, a great number of pedotransfer functions have been developed which use soil texture, bulk density, or other soil variables as inputs (e.g., Tietje and Tapkenhinrichs, 1993). Wösten et al. (1999) developed the HYPRES database for this purpose, which provides van Genuchten parameters for a wide range of European soils.

While the bulk density is known to have a strong effect on the van Genuchten parameters (Baumgartl and Horn, 1999), only a limited few studies have examined this effect in detail. Most of these investigated the influence of increasing bulk density by rainfall-induced soil surface sealing (e.g., Assouline, 2004). Thus far, different compression conditions of the same soil material have been assumed to lead to different soil materials, each with their own unique retention function; this even tough some functional relationship between the retention parameters and the compression condition is to be expected.

Interactions between hydraulic and mechanical stresses and stress-induced changes in the soil water retention and PSDs are still not well documented in the literature (Startsev and McNabb, 2001). Our main objectives in this paper are to address some of these interactions by quantifying (i) pore volume changes caused by hydraulic stress in sandy and silty soils, and (ii) the water retention curves as a function of previously applied mechanical stresses. Following (i) and (ii), we compare (iii) four new models for describing the hydraulic properties as a function of void ratio. These models should provide a better basis for predicting water, gas, and/or heat transport in soils.

	THEORETICAL ASPECTS

TOP ABSTRACT INTRODUCTION THEORETICAL ASPECTS MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Theory of Soil Shrinkage
Or and Ghezzehei (2002) distinguished between two causes for structural change in soils: capillary forces of water menisci between soil aggregates, and external static loads. They studied the combined effects of these two factors on samples that were subjected to repeated (transient) mechanical and hydraulic stress. Bruand and Prost (1987) investigated the shrinkage of aggregated clay samples due to soil water desiccation and identified four shrinkage phases in plots of the void ratio vs. moisture ratio: (i) structural shrinkage, (ii) normal shrinkage, (iii) residual shrinkage, and (iv) zero shrinkage. The void ratio, e, is defined as the ratio of pore volume, V_pore, to the volume of the soil matrix, V_matrix. The void ratio enables one to relate changes in the PSD to a constant soil volume, rather than to bulk density which is not constant in shrinking soils.

If the void ratio is plotted against the pressure head, h, the resulting plot resembles a soil water retention curve. Because of this similarity, the Genuchten function could be used also for mathematical description of soil shrinkage (Baumgartl and Koeck 2004):

[1]

where e is the void ratio, ê is the reference void ratio at saturation (i.e., at a pressure head of 0 kPa), e_r is the void ratio in shrinkage phase IV (no shrinkage), and n_s and _s are empirical shape parameters (with n_s > 1). Notice that Eq. [1] contains four independent parameters: ê, e_r, n_s, and _s.

Theory of Water Retention Changes with Varying Void Ratio
Soil hydraulic properties consist of the water retention curve, which describes the relationship between the soil pressure head, h, and the volumetric water content, , and the hydraulic conductivity function, which describes the influence of on the hydraulic conductivity, K(). According to van Genuchten (1980), the water retention curve is given by

[2]

where is the volumetric water content, _s and _r are the saturated and residual water contents, respectively, and n and are empirical shape parameters (with n > 1), and m = 1 – 1/n. The restriction that m = 1 – 1/n is necessary to permit direct integration if Eq. [2] is combined with Mulaem's hydraulic conductivity model (Mualem, 1976). If the parameters m and n are assumed to be mutually independent, much more complicated expressions for K(h) arise (van Genuchten et al., 1991). The use of van Genuchten's function is limited to a rigid soil matrix in that swelling and shrinking are assumed not to alter the water retention curve at various pore water pressures. Thus, in this paper we extend van Genuchten's approach by accounting for the dependence of the parameters _s, _r, n, and on the void ratio.

The saturated water content, _s, is generally assumed to be equal or close to total soil porosity. For our general model we assume that _s is linearly related to total soil porosity as follows

[3]

where P₁ and P₂ are empirical parameters, n_p is total soil porosity defined as V_pore/V_tot (with V_tot = V_pore + V_matrix), and e is the void ratio defined as V_pore/V_matrix.

The residual water content _r is considered to depend mainly on the surface area of soil particles, and thus is not affected by soil compaction if expressed on a mass basis (Assouline 2004). Following Assouline (2004)(Eq. [7]) the relationship between the volumetric water content and the void ratio can be derived:

[4]

in which P₃ is an empirical parameter.

Baumgartl and Horn (1999) found linear correlations between the van Genuchten parameters and the void ratio for a very expansive clay soil. To date, no good physical basis has been established for these correlations. Our approach to characterize changes in the water retention curve can be applied not only coarse-textured (sandy) and medium-textured (silty) soils, but also to fine-textured (clay) soils. We fitted water retention curves independently for each compression step and each soil using the RETC (version 6) nonlinear parameter optimization program (van Genuchten et al., 1991; www.ussl.ars.usda.gov/models/retc.htm [verified 11 Jan. 2005]). Based on results from Baumgartl and Horn (1999) and our present study, the parameter was related to e using

[5]

in which P₄ and P₅ are empirical parameters.

Conflicting evidence exists in the literature about the influence of soil compaction on the van Genuchten parameter n. Assouline et al. (1997) assumed a linear decrease in n with a change in bulk density. Horn et al. (1995) found that n can both increase or decrease with increased mechanical stress. Because n is limited to values greater than one (van Genuchten, 1980), we assumed heuristically an exponential function for the relationship between n and the void ratio:

[6]

where P₆ and P₇ are empirical parameters. We also tested a linear equation of the form n(e) = P₆ + P₇ e (results not presented here), but this approach was found inferior to the exponential equation in terms of fitting the data.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORETICAL ASPECTS MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Investigated Soils
Soil samples were collected at three sites in Germany: a Stagnic Chernozem site near Hildesheim (Lower Saxony), a Haplic Phaeozem site (a fallow site at the Julius-Kühn long-term experimental field in Halle, Saxony-Anhalt), and a Dystric Gleysol site close to Büsum (Schleswig Holstein). Table 1 provides some general physical and chemical properties of the three soils.

Haplic Phaeozem
This site was located near a loess–chernozem area typical for the eastern Harz mountain foreland at the research farm of the Martin Luther University of Halle-Wittenberg. The grain-size distribution of the soils inside this area varied considerably. Soil samples were taken from the A_p (0–25 cm), the A_xh (30–48 cm), and the B_vC_v (>70 cm soil depth) soil horizons of the grassland plot.

Dystric Gleysol
The Dystric Gleysol site is located in Speicherkoog along the North Sea coast south of Büsum. This site represents a typical alluvial marsh soil in its early stage of soil development. Dikes had been built in this area until 1978. The site is currently being used as sheep pasture. Due to the climatic conditions (annual average precipitation of 875 mm), both aerated and completely water-saturated conditions occur. Samples were taken from the A_h soil horizon (0–5 cm).

Preparation of Soil Samples
Great care was taken to prepare homogenized soil samples. Eight cores (each having a volume of 40 cm³) were placed on top of each other inside a cylinder (having an inside diameter identical to the outer diameter of the cores, and a height equal to the height of eight cores) as shown in Fig. 1 . The soils were sieved (<2 mm) and packed into the complete cylinder. The soils were subsequently compacted simultaneously from both sides to a predefined bulk density using a uniaxial compression device.

View larger version (37K): [in this window] [in a new window]   Fig. 1. Uniaxial compression device used to obtain evenly compacted homogenized soil samples. The equipment was installed within a frame, with the pressure of the compression pistons being transmitted via a hydraulic jack (from Gräsle, 1999).

The soil cylinders were saturated and desiccated to different pressure head values (for more than –3 kPa using a sand bed, from –6 to –65 kPa using ceramic plates, and for less than –100 kPa using air pressure). More detailed information is given by Hartge and Horn (1992). Gravimetric water contents and the volumes of the samples were determined after each equilibration to the defined pore water pressure, and then used to calculate volumetric water contents. The actual soil volume was determined using vernier caliper measurements at seven defined points at the surface. Detachment of the soil from the edge of the cores occurred in most samples only at pressure heads below –150 kPa. All experiments with the homogenized soils were performed using four replicates.

To test the applicability of the models to structured soils, we also used undisturbed soil cores (4 cm long, 100 mL volume) taken from six different depths of the Haplic Phaeozem soil profile in Halle. This profile had a very homogeneous soil texture (Table 1), while the bulk density varied considerably with depth. The soil samples from this site were hence ideal for testing the models and comparing results for homogenized and undisturbed structured soil cores.

Parameter Estimation
To simplify the analyses, parameter estimates were obtained for the soil shrinkage (Eq. [1]) and retention (Eq. [2]) curves using mean average values of four replicates at each compression step for each soil horizon. Comparative test, with and without averaging, showed only minor effects of averaging on the optimization parameters. Estimates at each compression step were obtained with RETC for the parameters ê, e_r, _s, and n_s in the shrinkage function (Eq. [1]) and for _s, _r, , and n in van Genuchten's function (Eq. [2]).

Parameter estimation for the four variable porosity models (Eq. [2] through [6]) were performed using the complete data set for each horizon. As shown in Table 1, specific models were extracted from the general models (Eq. [2] to [6]) by restricting the estimated parameters to 5 (Models 1 to 3) or 6 (Model 4). For example, if parameter P₇ in Eq. [6] is set to zero and only P₆ is estimated, then the equation simplifies to a constant value for n, as in the original van Genuchten function. Table 2 indicates which parameters were estimated and which were neglected in the four model approaches. Estimates of the parameters in Eq. [3] to [6] were obtained with the nonlinear parameter estimation procedure NLIN of the SAS software (SAS Institute Inc., Cary, NC). In this case, all compression steps for each soil horizon were used directly in one run. To compare the four models, we additionally calculated the adjusted r² (AR) according to (Hector et al., 2002)

[7]

where n is the sample size, and p is the number of explanatory variables. The adjusted r² assesses the efficiency of the different models by considering their complexity and cost in terms of the number of degrees of freedom.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORETICAL ASPECTS MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

View larger version (22K): [in this window] [in a new window]   Fig. 2. Shrinkage curves (void ratio vs. moisture ratio) for the homogenized soil samples of the Ap Stagnic Chernozem soil horizon. The different precompression steps are identified by reference void ratio between 1.38 and 1.69. Two shrinkage phases are shown: normal shrinkage for moisture ratios > 1.3 and residual shrinkage for moisture ratios < 1.3.

View larger version (24K): [in this window] [in a new window]   Fig. 3. Comparison of measured and fitted shrinkage curves for homogenized soil samples from the Stagnic Chernozem Ah horizon (Hildesheim site) involving five reference void ratios between 0.98 and 1.63 at saturation. Each point represents four replicates, with the vertical lines representing the standard deviation (SD) of these replicates.

View this table: [in this window] [in a new window]   Table 3. Optimized parameters in the shrinkage model (Eq. [1]) as fitted to hydraulic stress-induced soil volume changes for different precompressed soil samples from the different soil horizons at the four sites. The total shrinkage of the soil samples (e = ê – er) and coefficients of determination (r2) are given.

Total shrinkage of the soil samples (e = ê – e_r) increased with increasing reference void ratio (Table 3). An interesting feature of the Stagnic Chernozem A_h horizon is shown by the sample with the highest reference void ratio (ê = 1.63) in Fig. 3. This sample was compressed so strongly by hydraulic stress that it reached a smaller residual void ratio than the samples with reference void ratios ê of 1.53, 1.36, or 1.32. One possible explanation is stabilization of soil particles as a result of mechanical stress applied during sample preparation. It is possible that resistance against the hydraulic attractions (i.e., menisci forces) is more pronounced when the primary soil particles are blocked because of particle arrangement.

Water Retention Curves
The measured retention curves for the homogenized A_p horizon in Fig. 4 exhibit a pattern that is typical for soils having different void ratios. Lower reference void ratios lead to fewer macropores (more than –4 kPa; definition after SSSA, www.soils.org/sssagloss/pdf/table 2.pdf [verified 26 Jan. 2005]) and mesopores (–4 to –10 kPa), which can be calculated from reductions in the corresponding pores near saturation. On the other hand, ultramicropores (i.e., pores with diameters less than 5 µm, which drain at pressure heads below –60 kPa) appear to be unaffected by compression. The water contents at –150 kPa (pF 3.2) are nearly identical for the soil samples compacted to different void ratios. We hence conclude that the ultramicropores are primarily influenced by soil texture and only little by the applied compression stress. This is supported by the estimated residual water contents, _r, which varied only between 0.12 and 0.15 m³ m^–3 for the individual compression steps.

View larger version (20K): [in this window] [in a new window]   Fig. 4. Comparison of measured and fitted water retention curves for homogenized soil samples from the Stagnic Chernozem Ap horizon (Hildesheim site) involving five reference void ratios between 1.38 and 1.69 at saturation. Each point represents four replicates, with the vertical lines representing the standard deviation (SD) of these replicates.

Samples of the A_h horizon (Fig. 5) show a redistribution of coarse to fine pores with increasing soil compaction. Soils with low compaction can store more water than heavily compacted soils near saturation, while at more negative pressure heads the water content in the more compacted soil is higher. Crossover pressure heads of the retention curves for the more or less compacted soils are between –6 and –65 kPa (Fig. 5). The measured water content of the most strongly compacted sample (ê = 0.98) at –1500 kPa differed significantly from the other samples (two-sided t test, level of significance 0.05%), whereas the estimated residual water contents (_r) increased for the A_h soil samples with increasing soil compaction from 0.17 to 0.23 (m³ m^–3).

View larger version (23K): [in this window] [in a new window]   Fig. 5. Comparison of measured and fitted water retention curves for homogenized soil samples from the Stagnic Chernozem Ah horizon (Hildesheim site) involving five different reference void ratios between 0.98 and 1.63 at saturation. Each point represents four replicates, with the vertical lines representing the standard deviation (SD).

View larger version (20K): [in this window] [in a new window]   Fig. 6. Linear relationship between the logarithm of the reciprocal value of [log(1/)] and the reference void ratio ê for all soils. The estimated van Genuchten parameter values are presented in Table 4.

The effect of water potential on changes in textural and structural porosity is described in detail by Bruand and Cousin (1995). They found that when soils are nearly water saturated (pressure head more than –1 kPa), even a small load (50 kPa) can cause a decrease in structural pores, leading to smaller values. A moderate load of 200 kPa resulted in a more pronounced decrease in structural porosity, and an increase in textural porosity. Bruand and Cousin (1995) explained the increase in smaller pores as being a relic of destroyed structural pores by compaction. This change in the PSD could also be obtained by increasing the parameter n. A high load (600 kPa) in a wet soil produced a much lower structural porosity but did not lead to changes in the textural porosity. In soils equilibrated at pressure heads between –63 and –1000 kPa, textural porosity was unaffected by compaction, while structural porosity decreased only at the higher loads (both at 200 and 600 kPa). In another study, Baumgartl and Horn (1999) observed a significant linear relationship between calculated van Genuchten parameters and the void ratio for a slightly expansive soil (kaolinitic clay). Both log (1/) and n decreased with increasing void ratio.

In contrast to , no uniform relationship between the van Genuchten parameter n and the reference void ratio ê has been identified in the literature. The behavior of n differs between soils (Table 4). While n of the Stagnic Chernozem A_p horizon, the Haplic Phaeozem B_vC_v horizon, and the Dystric Gleysol A_h horizon decreased with increasing reference void ratio ê, n of the Stagnic Chernozem A_h horizon and the Haplic Phaeozem A_xh horizon showed no significant correlation with ê, with n being relatively constant over the range of ê values. In contrast, n of the Haplic Phaeozem A_p horizon varied considerably without a clear trend. Our results are consistent with those of Horn et al. (1995) who found that n could increase or decrease with increased mechanical stress. Gräsle (1999) found n to decrease with increasing void ratio for three out of four soils, although the regression was significant only for pure sand. For the pure silt samples he found n to be independent of the void ratio. Our results, on the other hand, appear more consistent with those by Startsev and McNabb (2001) who found an increase in n from the third to the seventh skidding cycles, although this change was not significant.

Richard et al. (2001) published hydraulic properties (water retention curve and the unsaturated conductivity) for compacted and noncompacted soils. By using water retention data only to estimate the van Genuchten parameters with RETC, no significant differences were found between n of the compacted and the noncompacted soils. This was in contrast to when n was calculated from both the water retention data and unsaturated conductivity (n was 1.10 ± 0.08 SE for the noncompacted soils and 1.54 ± 0.08 SE for the compacted soils).

We also tested possible linear relationships between the soil shrinkage parameters _s and n_s and the van Genuchten parameters and n. No significant correlations were found for each of the four tests.

Water Retention Models at Variable Porosity
Stagnic Chernozem
Models 1 to 4 were first applied to the homogenized soil samples of the Stagnic Chernozem A_p horizon. Models 1 to 3 each contain five unknown parameters, while one of the unknown retention parameters _s, , or n is described interchangeably in terms of two coefficients. Model 4 is the most general in that six coefficients are fitted. This model also includes the approaches of Models 2 and 3. The general model with seven estimated coefficients, which includes Models 1 to 3, was not tested since results for Model 1 (linear function for _s) demonstrated only a very small effect of parameter P₂. For most soils P₂ in Model 1 was not significantly different from zero as indicated by the high estimated standard error. Good agreement was obtained between observed and simulated values for all models. Except for Model 1 for the A_h horizon, differences were negligible relative to the standard deviations of the measurements. Results of the model comparison for the Stagnic Chernozem are summarized in Table 5.

View this table: [in this window] [in a new window]   Table 5. Parameter optimization results (estimated value ± standard error) for the four models given by Eq. [3] through [6] (see also Table 2) for the two Stagnic Chernozem soil horizons. Also indicated are parameters with the highest correlation, corresponding correlation coefficients, coefficients of determination (r2), and the adjusted r2 (AR). Conventional van Genuchten parameters are given in the row labeled VG (i.e., P1 = s, P3 = r, P4 = , and P6 = n).

The r² and adjusted r² (AR) values for Models 2 and 4 were essentially identical for the A_h horizon. As an example, Fig. 7 compares simulated and measured volumetric water contents as a function of pressure head and void ratio for Model 2. Clearly, a single conventional van Genuchten function is unable to accurately describe the water retention data of this horizon for all compression steps (r² was <0.9).

View larger version (77K): [in this window] [in a new window]   Fig. 7. Calculated water retention surface at variable porosity for homogenized soil samples from the Stagnic Chernozem Ah horizon using the estimated Model 2 parameters. Points represent the measured data. The estimated parameters are listed in Table 5.

As expected, the best fits for the Haplic Phaeozem soil samples were obtained with Model 4 (Table 6). Model 4, however, showed high correlation between the optimized parameters (Table 6). Because of this and the fact that differences in the goodness of fit of Models 2 and 4 were negligible, we prefer Model 2 for the Haplic Phaeozem.

View this table: [in this window] [in a new window]   Table 6. Parameter optimization results (estimated value ± standard error) for the three Haplic Phaeozem horizons. Also indicated are parameters with the highest correlation, corresponding correlation coefficients, coefficients of determination (r2), and the adjusted r2 (AR). Conventional van Genuchten parameters are given in the row labeled VG (i.e., P1 = s, P3 = r, P4 = , and P6 = n).

View this table: [in this window] [in a new window]   Table 8. Parameter optimization results (estimated value ± standard error) for the undisturbed soil samples from the five Haplic Phaeozem horizons having reference void ratio values between 0.50 and 0.84. Also indicated are parameters with the highest correlation, corresponding correlation coefficients, coefficients of determination (r2), and the adjusted r2 (AR). Conventional van Genuchten parameters are given in the row labeled VG (i.e., P1 = s, P3 = r, P4 = , and P6 = n).

View this table: [in this window] [in a new window]   Table 7. Parameter optimization results (estimated value ± standard error) for the Dystric Gleysol Ah horizon. Also indicated are parameters with the highest correlation, corresponding correlation coefficients, coefficients of determination (r2), and the adjusted r2 (AR). Conventional van Genuchten parameters are given in the row labeled VG (i.e., P1 = s, P3 = r, P4 = , and P6 = n).

View larger version (67K): [in this window] [in a new window]   Fig. 8. Calculated water retention surface at variable porosity for homogenized soil samples from the Haplic Phaeozem Ap horizon using the estimated Model 2 parameters. Points represent the measured data. The estimated parameters are listed in Table 6.

Dystric Gleysol
Because of the very uniform grain-size distribution of the Dystric Gleysol (31% coarse silt and 62% fine sand, while 93% of the soil consisted of particles between 20 and 200 µm), only very small differences in the reference void ratio could be achieved. The estimated model parameters (Table 7) were hence limited to only a small range of void ratios, while differences between the water retention model assuming variable porosity and the conventional van Genuchten function, which had one parameter less, were relatively small. Consequently, the question arises if the use of one additional parameter is justified, or if it is better to use the original van Genuchten function for this soil. Both modeling approaches describe the data for this soil very well, with differences in the coefficients of determination (r²) between the four variable porosity models being negligible small (Table 7).

Application to Structured Soil Samples
The above results show that the variable porosity models can be used to predict the influence of void ratio changes on the water retention curves of homogenized unstructured (repacked) soil samples. However, the soil hydraulic functions of field soils (the water retention and unsaturated hydraulic conductivity functions) are know to depend strongly on soil structure. The applicability of the models to structured soil samples was tested on undisturbed soil samples from the Haplic Phaeozem site (Halle, Germany) with different reference void ratios. We note here that not only the reference void ratio but also the C_org content varied with soil depth, while soil texture was very uniform throughout the entire soil profile (Table 1).

The parameter estimation results indicate that differences in water retention between the Haplic Phaeozem horizons can be explained very well by changes in the void ratio, and hence that the new models do allow predictions of the water retention curves of the 6 horizons using only five or six parameters. Differences between the different models were negligibly small, with all models showing accurate descriptions of the measured water retention curves (r² > 0.964). The accuracy of the models was comparable to the accuracy obtained when the data were described with horizon-specific van Genuchten functions (requiring a total of 24 parameters for the six horizons). The coefficients of determination for the conventional van Genuchten functions ranged from 0.909 to 0.982 for the various horizons.

Figure 9 compares measured and simulated volumetric water contents as a function of both pressure head and void ratio for the structured Haplic Phaeozem soil samples for both Model 2 (Fig. 9a) and Model 3 (Fig. 9b). This example can be applied to all investigated soils since the differences between Models 2 and 3 are negligible small. The best r² values were found for Model 4, and the best adjusted r² for Model 2. The insignificance of the differences is also reflected by the fact that the estimation error of parameter P₅ was higher than its value (Table 8), which indicates that P₅ is not significantly different from zero. This shows that this parameter is not really needed and that Model 4 can be simplified to Model 3. Additionally the correlation between P₄ and P₅ was very high (Table 8), which means that one parameter can be explained by the other.

View larger version (34K): [in this window] [in a new window]   Fig. 9. Comparison of the calculated water retention surfaces with the observed retention data for soil samples from the different Haplic Phaeozem horizons. The calculated retention surfaces were obtained with estimated (A) Model 2 and (B) Model 3 parameters. The estimated parameters are listed in Table 8.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORETICAL ASPECTS MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

Leij et al. (2002) illustrated changes in the water retention curve of a soil after a sequence of wetting and drying cycles. They observed a decrease in the saturated water content from 0.57 to 0.51 after the first drying and rewetting cycle, and to 0.50 after five cycles, which confirms our results. This shows that changes in the hydraulic functions cannot be neglected. Our proposed soil water retention model for variable porosity provides a possible basis for considering such changes in the hydraulic functions. Alternatively, the Fokker–Planck equation could also be used to predict the changes in the PSD (Leij et al., 2002) and to deduce the soil water retention curve from the PSD. The data and the results in this paper should improve our understanding of the systematic influence of the void ratio on hydraulic parameters, which in turn could be useful for extending pedotransfer functions to conditions involving variable porosity.

What model is best? Our results suggest that no unique answer exists. With the exception of the Haplic Phaeozem A_xh horizon, the coefficients of determination for Models 2 and 3 were essentially identical. Both models accurately described the water retention data using variable void ratios, and with less correlation between fitted parameters as compared with Model 4. Additionally, the relationship between the void ratio and the parameter was clearer than the relationship between the van Genuchten parameter n and the void ratio (see also Startsev and McNabb 2001, Gräsle 1999, Baumgartl and Horn 1999). This corresponds with the view that for PSDs derived from the water retention curve, n reflects the slope and the location of the maximum of the distribution. On the other hand, if the maximum is not in near the macropores, and the macropores are mostly affected by compaction (Bruand and Cousin, 1995), the parameter n should be expected to change also. Additional measurements of the influence of pore-size changes on the unsaturated hydraulic conductivity may be needed to further improve and validate the modeling approaches outlined in this paper. Maybe such data would allow one to distinguish better between Models 2 and 3.

In contrast, Model 1 produced lower r² and adjusted r² values for all experiments. The residuals (observed vs. predicted values) for this model were generally less randomly distributed that those of Models 2, 3, and 4. Also, while four of seven optimizations with Model 4 showed the highest adjusted r², this model is not favored because of an additional parameter. This additional parameter is not necessary to obtain accurate results, as was demonstrated with Models 2 and 4.

	ACKNOWLEDGMENTS

 
The authors thank Mrs. D. Rexilius for her technical assistance in the laboratory and the reviewers not only for their valuable comments but also for the time-consuming corrections of the English text. We also thank Mrs. P. Clowes for improving the English text and style. We thank the German Research Foundation (DFG) for the financial support of this research by the grant HO 911/24.

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