HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Wang, K. Articles by Wang, F. Search for Related Content PubMed Articles by Wang, K. Articles by Wang, F. Agricola Articles by Wang, K. Articles by Wang, F. Related Collections Structure and Properties Soil Physics Fractal Approaches

Soil Physics

Testing the Pore-Solid Fractal Model for the Soil Water Retention Function

^a F. Wang, State Key Lab. of Water Resources and Hydropower Engineering Science, Wuhan Univ., Wuhan 430072, China
^b Dep. of Renewable Resources, Univ. of Wyoming, Laramie, WY 82071-3354, USA

^* Corresponding author (renduo{at}uwyo.edu)

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The soil water retention curve is an important hydraulic function for the study of flow transport processes in unsaturated soils. To accurately describe and interpret the hydraulic property, a general soil water retention function was developed based on the pore-solid fractal (PSF) model. The objective of this study was to evaluate the general soil water retention function using data of 65 soils and to compare the PSF function with its special cases, that is, three other soil water retention functions. Defined from the parameters of the PSF, an index of ß/_s was used to quantify the relationship between the PSF and the other soil water retention functions. The PSF function fit all the data sets well, whereas the other retention functions only matched the retention data for some soils, ranging from 11 to 72% of the tested soils. Directly fitting these functions with the data sets showed that for 30 to 40% of the tested soils, these functions gave poorer results than the PSF.

Abbreviations: PSF, pore-solid fractal

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
THE SOIL WATER retention function defines the relationship between water content and soil matric potential, and is an important hydraulic property necessary to study water flow and chemical movement in unsaturated soils. Various models have been developed to characterize the soil water retention function (e.g., Brooks and Corey, 1964; Tyler and Wheatcraft, 1990; Rieu and Sposito, 1991). Applications of fractal geometry (Mandelbrot et al., 1984) in soil science have shown that the complicated porous medium can be characterized by the fractal representation. In recent years, fractal approaches have been developed to model soil aggregate, soil mass, soil pore space, and pore surface and particle-size distributions (e.g., Turcotte, 1986; Perfect and Kay, 1991; Bittelli et al., 1999; Perrier and Bird, 2002). Fractal models of soil hydraulic properties, such as the unsaturated soil hydraulic conductivity and water retention function, have also been proposed (Tyler and Wheatcraft, 1990; Rieu and Sposito, 1991; Gimenez et al., 1997). For example, Tyler and Wheatcraft (1990) derived an expression for soil water retention using the Sierpinski carpet model. Perfect et al. (1998) introduced a retention function related to the Euclidean dimension of space and a fractal dimension.

The PSF model developed by Perrier et al. (1999) and Perrier and Bird (2002) is an extension and generalization of the fractal approach to modeling soil structure. In its most general form, the model illustrates the symmetry between the solid and pore phases. In this respect, it is quite different from a conventional fractal model (Perrier and Bird, 2002). Based on the PSF theory, Bird et al. (2000) derived a soil water retention function. The PSF model yields a general expression for the water retention function. The retention functions of Brooks and Corey (1964), Tyler and Wheatcraft (1990), and Rieu and Sposito (1991) become special cases of the PSF water retention function. Bird et al. (2000) tested the PSF model with only two soils. The objective of this paper was to test the general PSF function with soil water retention data of 65 soils and to compare the PSF function with the models of Brooks and Corey (1964), Tyler and Wheatcraft (1990), and Rieu and Sposito (1991).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Based on the PSF theory, Bird et al. (2000) derived a soil water retention function. The PSF function is expressed as follows:

[1]

where is the volumetric water content (cm³ cm^–3), h is the suction head (cm), is the total soil porosity (cm³ cm^–3), p and s denote proportions of regions occupied by pores and solids (cm³ cm^–3), respectively, h_min and h_max (cm) are the suctions at which the largest and smallest pores explicitly defined within the PSF desaturate, d is the Euclidean dimension, and D is the fractal dimension of the PSF. Setting d = 3 and

[2]

we have

[3]

In the model of Bird et al. (2000), the fractal dimension is calculated using cumulative distributions of solid mass. Specifically the following equation is used for the calculation:

[4]

Here M_s(i) is the total mass of solid particle size r_s(i), which can be obtained directly by summing the solids volume of size r_s(i) multiplied by their density d_s, r_s(1) is the upper limit of the particle size, i is fractal iteration number, L is the linear size of the studying region, is a constant such that L^d equals the bulk volume of the PSF.

It is assumed that soil pores can be full saturated. Then Eq. [3] is rewritten as

[5]

where _s is the saturated water content (_s = ). If ß = _s, Eq. [5] is equivalent to the equation of Tyler and Wheatcraft (1990):

[6]

The Brooks and Corey (1964) formulation is as follows:

[7]

in which _r is the residual water content (cm³ cm^–3), h_a is the air-entry suction (cm) and equivalent to h_min in Eq. [5], and is the pore-size distribution index. Comparing Eq. [5] with Eq. [7], we can see that they are essentially the same if ß = _s – _r and = 3 – D. In Eq. [1], when s = 0, ß = 1, the PSF reduces to a simple mass fractal structure and the water retention function becomes (Bird et al., 2000)

[8]

which is the model by Rieu and Sposito (1991).

To evaluate the applicability of the PSF function (Eq. [5]) and its special cases, the models of Brooks and Corey (1964), Tyler and Wheatcraft (1990), and Rieu and Sposito (1991), we utilized 65 data sets of soil water retention collected from the literature and our own measurements. Among the data sets, 41 were obtained from the Unsaturated Soil Hydraulic Database (UNSODA) (Leij et al., 1996). Ten data sets were collected from Huang and Zhan (2002), six were from Basile and D'Urso (1997), and eight were our measured data sets. Huang and Zhan (2002) measured the data of soil water retention using undisturbed soil samples with the pressure plate method (Hillel, 1998). Basile and D'Urso (1997) measured the soil water retention data using the classical wind method (Wendroth et al., 1993). We used undisturbed soil samples and the pressure plate method to measure the soil water retention curves. The data sets of soil water retention from the UNSODA (Leij et al., 1996) were measured using several methods, such as the pressure plate method, hanging water column and pressure plate method, pressure outflow method, and tensiometry and TDR method. For all the data sets, soils were classified to clay (<2 µm), silt (2–50 µm), and sand (>50 µm) based on the USDA classification scheme. Then soil texture was determined according to the USDA triangle. Clay contents of the soils varied between 0 and 72%. To warrant meaningful statistical analyses, each of the data sets included more than 10 data points. The soils were grouped based on texture in Table 1.

The detailed calculations and statistics were described as follows. First we fit the PSF (Eq. [5]) to the data sets of soil water retention to obtain values of the parameters _s, ß, h_min, and D. Then the values of ß/_s and ß were used to quantify the relationship between the PSF (Eq. [5]) and the other soil water retention functions (Eq. [6], [7], and [8]). With the confidence intervals of the fitted _s and ß, we calculated intervals of ß/_s of the soils. For a soil, if the interval of ß/_s included 1, we considered that ß/_s of the soil was not significantly different from 1 or ß _s. If all ß/_s values within the interval were greater or smaller than 1, we considered ß/_s of the soil was significantly different from 1, that is, ß > _s or ß < _s. It should be pointed out that the measured saturated water content is more accurately the "satiated water content" because of trapped air.

The fractal dimension D plays an important role in the soil water retention curve functions. Therefore, the models of Tyler and Wheatcraft (1990) and Rieu and Sposito (1991) were also used to fit the data of soil water retention so that we can compare the fractal dimensions by independently fitting with the models (Eq. [5], [6], and [8]). A t test was applied to assess the significant difference of the fractal dimensions (Karlen and Colvin, 1992). The differences between measured soil water retention data and fitted values with the models were evaluated using a statistical scheme (a 95% confidence interval band) similar to that described by Logsdon et al. (2002). For model comparison, we also utilized the root mean square errors between measured data and predicted retention functions with the different models.

As mentioned above, the fractal dimension D was calculated by fitting Eq. [5] to the data of soil water retention. The fractal dimension D can also be evaluated by fitting Eq. [4] to particle distribution data. Taking log-transforms for both sides of Eq. [4] and fitting particle distribution data with the log-transformed equation, we can obtain the fractal dimension from the slope (3 – D) of the fitted line. Following the procedure by Logsdon and Karlen (2004), we conducted a paired comparison t test to check statistical significance for the fractal dimensions evaluated from Eq. [4] and [5].

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Table 2 summarizes the parameters (_s, ß, h_min, and D) from fitting Eq. [5] to the data sets of soil water retention for each group of the soils, including the mean, standard deviation, maximum, and minimum of the parameters. The PSF function fit the data well for all the soils. The values of goodness of fit (r²) between the measured and fitted soil water content values ranged from 0.928 to 0.999 (Table 2). Calculated values of ß/_s of the soils are also presented in Table 2, ranging from 0.67 to 1.64.

For soils with ß/_s > 1, using Eq. [6] (i.e., setting _s equal to ß in Eq. [5]) overestimated the soil water retention function as shown in Fig. 1a for a sand soil (ß = 0.35 and _s = 0.30). This category included 24 soils, accounting for 37% of the total tested soils. On the contrary, for the soils with ß/_s < 1, using Eq. [6] underestimated retention functions. As an example, Fig. 1b compares the measured soil water retention data with fitted curves of Eq. [5] and calculated curves of Eq. [6] for a clay soil with ß = 0.41 and _s = 0.56. There were 34 soils with ß/_s < 1, accounting for 52% of the total tested soils.

View larger version (16K): [in this window] [in a new window]   Fig. 1. Measured soil water retention data (circles), fitted curve of Eq. [5] (solid line), and calculated curve of Eq. [6] (dash line) for (a) a sand soil with ß/s > 1 and (b) a clay soil with ß/s < 1.

View larger version (20K): [in this window] [in a new window]   Fig. 2. Comparison of measured water retention data (circles) of a silty loam and fitted curves with Eq. [5] (solid line) and with Eq. [6] (dash line).

View larger version (20K): [in this window] [in a new window]   Fig. 3. Comparison of measured water retention data (circles) of a silty clay loam soil and fitted curves with Eq. [5] (solid line) and with Eq. [6] (dash line).

If (_s – _r)/ß = 1, the general PSF function reduces to Brooks and Corey (1964) model (Eq. [7]). Again we used ß/_s as an index to compare the applicability of these two models. For soils with ß/_s 1, the residual water content in Eq. [7] was calculated directly from _r = _s –ß and Eq. [7] works equally well as Eq. [5]. In Table 1, there were 47 soils satisfying the condition, accounting for 72% of the tested soils. These soils also included the soils with _r = 0, for which Tyler and Wheatcraft (1990) model also works well.

For soils with ß/_s > 1, using Eq. [7] should cause errors. When we used Eq. [7] to fit retention data of the soils with ß/_s > 1, we obtained negative vales of fitted residual water contents, which were physically meaningless. For instance, for a soil with _s = 0.44 and ß = 0.72, the fitted residual water content was equal to _s – ß = –0.28. If the residual water content was set to be zero, the fitter results were poor (Fig. 4). This category included 18 soils (two sand soils, two sandy loam soils, two silty clay loam soils, two silt soils, three clay soils, four clay loam soils, and six silty loam soils), accounting for 28% of the tested soils.

View larger version (18K): [in this window] [in a new window]   Fig. 4. Measured soil water retention data (circles) of a sand soil and fitted curves with Eq. [5] (s = 0.39, ß = 0.53) (solid line) and Eq. [7] (setting r = 0) (dash line).

We used Eq. [8] (i.e., fixing ß = 1 in Eq. [5]) to fit all the soil data sets. For 22 soils (one loamy sand soil, one loam soil, two silt loam soils, three sand soils, three silt soils, three sandy loam soils, four clay loam soils, and five clay soils), the fitted soil water retention curves were significantly different from the measured data. Figure 5 shows an example of the fitting curve with Eq. [8] (_s = 0.24, h_min = 8.50 cm, and D = 2.96) and compares with the fitting curve of Eq. [5] (_s = 0.24, ß = 0.21, h_min = 26.60 cm, and D = 2.49).

View larger version (21K): [in this window] [in a new window]   Fig. 5. Comparison of measured water retention data (circles) of a loam soil and fitted curves with Eq. [5] (solid line) and with Eq. [8] (dash line).

Fractal dimension values by fitting Eq. [4] to available particle distributions are listed in Table 4 for 13 soils. The largest mass fractal dimension was 2.93 for a clay soil, while the smallest value was 2.45 for a loamy sand soil. The fractal dimension was related to soil texture and showed an increasing tendency with the clay content. For these soils, the mean and variance of the fractal dimension values were 2.742 and 0.0147, respectively. For the same soils, the mean and variance of the fractal dimension values by fitting Eq. [5] to soil water retention data were 2.798 and 0.0226, respectively (Table 4). A paired comparison t test showed that the differences of the fractal dimension values fit with Eq. [4] and [5] were significant at a level of significance of 0.05. Since the fractal dimension is a sensitive parameter, a small difference of D can result in large deviations of estimated retention functions from measured data (Fig. 6).

View larger version (17K): [in this window] [in a new window]   Fig. 6. Measured water retention data (circles), fitted curve with Eq. [5] (solid line), and calculated curve with D from Eq. [4] (dash line) for (a) a sandy loam soil and (b) a sandy clay loam soil.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

The index of ß/_s, ranging from 0.67 to 1.64, was used to determine the applicability of Tyler and Wheatcraft (1990) model (Eq. [6]). Only for soils with _s/ß 1, does Eq. [6] describe the soil retention function well; otherwise, it either underestimates or overestimates retention functions. For the tested data sets, Eq. [6] described the soil retention curve well for 11% of the soils, whereas the equation underestimated the retention function for 52% of the soils and overestimated for 37% of the soils. Equation [6] was also used to fit the data sets independently. The results showed that the model fit the data reasonably well for 60% of the tested soils, including two sand soils with fitted fractal dimensions smaller than 2. For 40% of the soils, the equation provided less accurate results of fitted soil water retention curves than the PSF.

The index of ß/_s was also applied to compare the PSF retention function with the model of Brooks and Corey (1964) (Eq. [7]). As ß/_s 1, the general PSF function reduces to Eq. [7]. For about 72% of the tested soils, ß/_s 1, Eq. [7] matched the retention data of the soils as accurately as the PSF function. For the rest of the soils, ß/_s > 1, Eq. [7] resulted in negative vales of the residual water content, which was physically meaningless.

Theoretically, the PSF reduces to the model of Rieu and Sposito (1991) (Eq. [8]) when ß = 1. However, the fitted ß values from the PSF, ranging from 0.21 to 0.84, were not closed to 1 for all the soils. From this point of view, the model of Rieu and Sposito (1991) was not comparable with the PSF function for the tested soils. Directly fitting Eq. [8] with the data showed that for 37% of the soils, the equation gave poorer results than the PSF. For most of the soils, the fractal dimensions from the fitting process were great than 2.90.

The fractal dimension by fitting soil water retention data was compared with the fractal dimension evaluated using particle distribution data. For most of the soils, the fractal dimension values from particle distribution data were smaller than those from soil water retention data and the fractal dimensions from the two fitting processes were significantly different, which showed some inconsistencies in the PSF model.

	ACKNOWLEDGMENTS

 
This research was financially supported in part by grants of the National Science Foundation of China (No. 50279038 and 50239090) and BARD, the United States–Israel Binational Agricultural Research and Development Fund (No. US-3287-01R).

Received for publication July 18, 2004.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

• Basile, A., and G. D'Urso. 1997. Experimental corrections of simplified methods for predicting water retention curves in clay loamy soils from particle-size determination. Soil Technol. 10:261–272.[CrossRef]
• Bird, N., E. Perrier, and M. Rieu. 2000. The water retention curve for a model of soil structure with pore and solid fractal distributions. Eur. J. Soil Sci. 55:55–63.
• Bittelli, M., G.S. Campbell, and M. Flury. 1999. Characterization of particle-size distribution in soil with a fragmentation model. Soil Sci. Soc. Am. J. 63:782–788.[Abstract/Free Full Text]
• Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrology Pap. No. 3. Colorado State Univ., Fort Collins.
• Gimenez, D., E. Perfect, W.J. Rawls, and Y.A. Pachepsky. 1997. Fractal models for predicting soil hydraulic properties: A review. Eng. Geol. 48:161–183.
• Hillel, D. 1998. Environmental soil physics. Academic Press, San Diego, CA.
• Huang, G., and W. Zhan. 2002. Modeling soil water retention curve with a fractal method. (In Chinese.) Adv. Water Sci. 13:55–60.
• Karlen, D.L., and T.S. Colvin. 1992. Alternative farming system effects on profile nitrogen concentration on two Iowa farms. Soil Sci. Soc. Am. J. 56:1256–1259.
• Leij, F.J., W.J. Alves, M.Th. van Genuchten, and J.R. Williams. 1996. The UNSODA unsaturated soil hydraulic database. USEPA Rep. 600/R-96/095. USEPA, Cincinnati, OH.
• Logsdon, S.D., and D.L. Karlen. 2004. Bulk density as a soil quality indicator during conversion to no-tillage. Soil Tillage Res. 78:143–149.[CrossRef]
• Logsdon, S.D., T.C. Kaspar, D.W. Meek, and J.H. Prueger. 2002. Nitrate leaching as influenced by cover crops in large soil monoliths. Agron. J. 94:807–814.[Abstract/Free Full Text]
• Mandelbrot, B.B., D.E. Passaja, and A.J. Paulley. 1984. Fractal character of fracture surfaces of metals. Nature (London) 308:21–722.[CrossRef]
• Marquardt, D.W. 1963. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11:431–441.[CrossRef]
• Perfect, E., and B.D. Kay. 1991. Fractal theory applied to soil aggregation. Soil Sci. Soc. Am. J. 55:1552–1558.[Abstract/Free Full Text]
• Perfect, E., N.B. McLaughlin, B.D. Kay, and G.C. Topp. 1998. Reply to Comments on "An improved fractal equation for the soil water retention curve by E. Perfect et al." Water Resour. Res. 34:933–935.[CrossRef]
• Perrier, E., and N. Bird. 2002. Modeling soil fragmentation: The pore solid fractal approach. Soil Tillage Res. 64:91–99.[CrossRef]
• Perrier, E., N. Bird, and M. Rieu. 1999. Generalizing the fractal model of soil structure: The PSF approach. Geoderma 88:137–164.[CrossRef]
• Perrier, E., M. Rieu, G. Sposito, and G.D. Marsily. 1996. Models of the water retention curve for soils with a fractal pore size distribution. Water Resour. Res. 32:3025–3031.[CrossRef]
• Rieu, M., and G. Sposito. 1991. Fractal Fragmentation, soil porosity, and soil water properties. I. Theory. Soil Sci. Soc. Am. J. 55:1231–1238.[Abstract/Free Full Text]
• Turcotte, D.L. 1986. Fractals and fragmentation. J. Geophys. Res. 91:1921–1926.
• Tyler, S.W., and S.W. Wheatcraft. 1990. Fractal processes in soil water retention. Water Resour. Res. 26:1047–1054.[CrossRef]
• Wendroth, O., W. Ehlers, J.W. Hopmans, H. Kage, J. Halbertsma, and J.H.M. Wostem. 1993. Reevalution of the evaporation method for determining hydraulic functions in unsaturated soils. Soil Sci. Soc. Am. J. 57:1436–1443.[Abstract/Free Full Text]

This article has been cited by other articles:

				A. Cihan, E. Perfect, and J. S. Tyner Water Retention Models for Scale-Variant and Scale-Invariant Drainage of Mass Prefractal Porous Media Vadose Zone J., October 8, 2007; 6(4): 786 - 792. [Abstract] [Full Text] [PDF]

This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Wang, K. Articles by Wang, F. Search for Related Content PubMed Articles by Wang, K. Articles by Wang, F. Agricola Articles by Wang, K. Articles by Wang, F. Related Collections Structure and Properties Soil Physics Fractal Approaches