Published online 11 January 2008
Published in Soil Sci Soc Am J 72:25-32 (2008)
DOI: 10.2136/sssaj2006.0232
© 2008 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SOIL PHYSICS
Scaling Soil Water Characteristics of Golf Course and Athletic Field Sands from Particle-Size Distribution
Lalit M. Aryaa,*,
Daniel C. Bowmanb,
Bir B. Thapab and
D. Keith Casselc
a 770 El Caballo Dr., Oceanside, CA 92057
b Crop Science Dep., North Carolina State Univ., Raleigh, NC 27695
c Soil Science Dep., North Carolina State Univ., Raleigh, NC 27695
* Corresponding author (larya{at}sbcglobal.net).
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ABSTRACT
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The soil water characteristic (SWC) of sands is an important hydraulic parameter in designing golf courses and athletic fields. A modified version of the Arya–Paris model of the soil water characteristic was adapted to 14 golf course media that contained no to minor amounts of clay and silt. In this model, the particle-size distribution curve is divided into a number of fractions and the natural pore length, Li(n), is scaled using the diameter of spherical particles as the length unit. The scaled pore length is given by 2Rini
i, where ni is the number of spherical particles in the ith fraction, 2Ri is the particle diameter, and i is the scaling parameter, which is calculated using the relationship lognii= a + blogni. Although the model adapted well, there were concerns about the sensitivity of predicted SWCs to uncertainties in parameters a and b. Consequently, we developed and evaluated a procedure to predict Li(n) directly from straight pore lengths, Li(c) in counterpart cubic close-packed assemblages of spherical particles, using the relationship logLi(n) = c + dlogLi(c). Predicted pressure heads using both procedures were similar with best-fit parameters. When uncertainties were imposed on Parameters a, b and c, d, however, SWCs using the latter procedure showed far less sensitivity, as measured by the root mean square residuals (RMSRs). In addition, for sand materials grouped together on the basis of similarity in particle-size distribution and bulk density, replacing individual best-fit parameters by the group mean parameters did not have significant effects on predicted pressure heads.
Abbreviations: AP, Arya–Paris model • PSD, particle-size distribution • RMSR, root mean square residual • SWC, soil water characteristic
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INTRODUCTION
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The soil water characteristic (SWC), that is, the relationship between the soil water pressure head, h, and the volumetric water content, 
, is an important hydraulic property of sands and soil materials that is often required as input in soil water flow models. It is also a measure of soil quality that reflects the effects of textural composition, mineralogy, packing density, changes in organic matter content, and management practices. Direct measurement of the SWC, however, is difficult, time consuming, and subject to numerous errors.
As a result, indirect approaches to predict the SWC from routinely measured soil properties (e.g., texture, bulk density, and organic matter content) have become popular. Soil texture, alone or in combination with bulk density and organic matter content, has been used to predict selected points of the SWC using regression techniques (e.g., Ghosh, 1976; Gupta and Larson, 1979; Rajkai and Varallyay, 1992). A particle-size distribution (PSD) curve can also be translated into a corresponding SWC curve (e.g., Arya and Paris,1981; Haverkamp and Parlange, 1986). The basis for the translation lies in the shape similarity of the two curves, implying that pore-size distribution is closely related to particle-size distribution, modified by the packing density and particle aggregation. Because the entire SWC curve is more desirable than just a few select points, there has been a growing interest in the latter technique (e.g., Tyler and Wheatcraft, 1989; Shepard, 1993; Smettem and Gregory, 1996, Scheinost et al., 1997; Arya et al., 1999; Tomasella et al., 2000; Cornelis et al., 2001; Zhuang et al., 2001; Hunt and Gee, 2002; Wang, 2002; Hwang and Powers, 2003; Chan and Govindaraju, 2004; Vaz et al., 2005).
Although models to predict the SWC from the PSD have a physical basis, simplification is inherent when scaling pore attributes from simple formations to complex structures. Therefore, unknown parameters have to be introduced and evaluated empirically. Our research objective was to adapt the Arya–Paris (AP) model to golf course and athletic field sands. Sands are used on golf courses primarily to construct the root zone of putting greens and tees, and also to fill bunkers and topdress existing greens. Guidelines for putting green construction emphasize a narrow PSD with minimal or no amounts of silt and clay and relatively high hydraulic conductivity (Davis et al., 1988; U.S. Golf Association, Green Section Staff, 1993). To ensure proper performance, sands for root zone construction are routinely tested by commercial laboratories for PSD, hydraulic conductivity, and water-holding capacity. Because considerable variation in testing results can occur from laboratory to laboratory, a modeling approach may enhance analysis uniformity. Because sands used in golf courses and athletic fields are starkly different from field soils, we felt it necessary to develop model parameters unique to them. In addition, there also are possibilities that pore radii computed in the AP model in the present form may be relatively more sensitive to uncertainties in parameters (e.g., Arya and Dierolf, 1992). Thus, we felt the need to evaluate sensitivity issues and explore alternative formulations that might be less sensitive.
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MODEL DESCRIPTION
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The AP model translates the percentage of particles smaller than the diameter axis of the PSD curve to volumetric water content and the particle diameter axis to pressure head (Arya and Paris, 1981; Arya et al., 1999). A sample calculation is presented in Leij et al. (2002). First, the PSD is divided into n size fractions, and the solid mass in each fraction is assumed to be in the natural state of packing, with bulk and particle densities of the natural-structure sample uniformly applicable to each fraction. Pore volumes attributable to each size fraction are calculated using fraction solid mass and void ratio. Starting with the first (smallest particles) fraction, calculated pore volumes are progressively summed and considered filled with water. Each summation of filled pore volumes is divided by the bulk volume of the whole sample to obtain the potential volumetric water content at the upper bound of the successive particle-size fractions. These water contents are adjusted for effective saturation obtained from experimental SWC data. An equivalent pore radius is calculated for each particle-size fraction and converted to soil water pressure head using the capillary equation. Calculated pressure heads are sequentially paired with calculated water contents to obtain a predicted SWC.
Scaling of natural pore lengths using some attribute of the PSD is basic to estimating natural pore radii. The pore volume of each PSD fraction is considered to be represented by a single cylindrical capillary tube. Pressure heads for each calculated water content are inferred from the experimental SWC and converted to pore radii using the capillary equation. The pore radii are then paired with corresponding fraction pore volumes to obtain natural pore lengths. Subsequently, an empirical procedure is established to predict natural pore lengths using some characteristic of particles associated with each mass fraction on the PSD curve. Arya and Paris (1981) and Arya et al. (1999) used summation of diameters of spherical particles required to cover the natural pore length end to end.
The AP model may be expressed by developing several equations using the symbols summarized in the Appendix. After the PSD curve is divided into fractions, pore volumes are assigned to each fraction using the relationship
 | [1] |
where vi(p) is the pore volume associated with the ith fraction, Ri is the mean particle radius of the ith fraction, ni is the number of spherical particles of radius Ri that can be formed using the solid mass in the fraction, e is the void ratio, ri is the natural pore radius of the ith fraction, and Li(n) is the natural pore length. The number of spherical particles, ni, is calculated using
 | [2] |
where wi is solid mass of the ith fraction and 
s is particle density.
Pore volumes are progressively summed and converted to volumetric water contents of the ith fraction, i, using
 | [3] |
where 
is the total porosity of the sample and Sw is the ratio of measured saturated water content to theoretical porosity. In essence, the total of effective pore volume is distributed in the same proportion as the solid mass.
Rearranging terms in Eq. [1], ri can be expressed as
 | [4] |
Natural pore length, Li(n), is estimated by the number of spherical particles of radius Ri required to cover the natural pore length. Thus, if the scaled number of particles is nii, Li(n) in Eq. [4] can be replaced by 2Rinii, giving
 | [5] |
where i is the scaling parameter. Evaluation of i requires establishing a relationship between nii and ni. The experimental SWC and PSD are used. First, natural pore radius, ri, is found from the capillary equation
 | [6] |
where hi(m) is the measured pressure head that corresponds with the ith particle-size fraction, 
is the surface tension at the air–water interface, 
is the contact angle, w is the density of water, and g is the acceleration due to gravity. Subsequently, Eq. [6] and [1] give Li(n) as
 | [7] |
Since Li(n) = 2Rinii, dividing the right-hand side of Eq. [7] by 2Ri yields experimental values of nii. Replacing ni with the right-hand side of Eq. [2], substituting numerical values (using cm–g–s units) of constants, and simplifying transforms Eq. [7] to
 | [8] |
where 7.371 is a composite of constants and has the unit of cm–4. Once the nii and ni values are calculated for all fractions, an empirical relationship is established by fitting the data to
 | [9] |
where a and b are empirical parameters. Rearranging Eq. [9] gives
 | [10] |
Once the values of ni and i are compiled for each fraction, Eq. [5] is used to compute ri. The computed pore radii are converted to equivalent pressure head using Eq. [6], except hi(m) is now hi(p), the predicted pressure head. The predicted pressure heads are paired with the calculated water contents (Eq. [3]) to construct the AP model SWC.
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MODEL MODIFICATION
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Use of i as an exponent on ni is of concern because ri predicted by Eq. [5] is highly sensitive to Parameters a and b (e.g., Arya and Dierolf, 1992). Therefore, we explored an alternative for predicting Li(n). The alternative is
 | [11] |
where Li(c) is the total cylindrical pore length associated with the pore volume in a cubic close-packed assemblage of ni spherical particles (ith fraction). Parameters c and d are evaluated empirically by logLi(n) vs. logLi(c) regression. Predicted Li(n) from Eq. [11] may be used to find ri from Eq. [4], thus avoiding the use of i in Eq. [5]. Values of Li(n) to use in establishing the regression of Eq. [11] are calculated by Eq. [7] using experimental data. The procedure to calculate corresponding values of Li(c) is illustrated in Fig. 1
, which shows cross-sections of natural and cubic close-packed structures for a given particle-size fraction, assembled in the form of a cube. We can easily visualize and calculate the number of pores and pore radii associated with the cubic close-packed structure. The final result is
 | [12] |
as shown in Fig. 1.
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MATERIALS AND METHODS
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Sand Materials and their Physical Properties
Sand materials used in this study included seven commercially available sands that contained no clay or silt, Wagram loamy sand (an Arenic Hapludult), and six composites of select sands and Wagram loamy sand (Table 1
). The composites were intended to simulate a range of soil textures commonly found on golf courses and athletic fields. Particle-size distribution data were obtained on all materials, mainly by sieving, and where necessary by the hydrometer method (Gee and Or, 2002).
The SWC was determined for sands and composites at packing densities commonly observed in golf courses and athletic fields (Table 1). Polyvinyl chloride columns, 10-cm i.d. by 60 cm deep, were constructed by joining 2.5-cm-long sections with duct tape. The bottom end of each column was capped with a layer of porous fiberglass sheeting, held in place by a layer of metal screen and several layers of cheese cloth. Columns were filled with air-dry material by pouring preweighed amounts to fill each 2.5-cm section. While filling, columns were tapped from outside and stirred and tamped from inside with a suitably designed plunger to ensure uniform packing. Based on our previous experience with similar sands, we felt confident that this procedure did not result in significant variations in bulk density along the column. The standard deviation in bulk densities averaged about 0.057 g cm–3.
After filling, the bottom end of the column was connected to a water supply and each column was saturated slowly during a 1-h period by raising the water table in 5-cm increments. After the free water began to accumulate at the surface, the water table was held constant for 12 h. The columns were subsequently allowed to drain by gravity while covered to prevent evaporation. We allowed 24 h of drainage before sampling, although with these and similar media we have observed complete drainage in considerably less time. Certainly, absolute saturation will not be attained using this method, but we considered the method adequate since our ultimate goal was to develop a laboratory procedure that reasonably simulates actual wetting in place of the turf medium due to rainfall or irrigation.
After 24 h of drainage, columns were sectioned into 2.5-cm lengths and the water content of each section was determined by oven drying. Volumetric water content of the bottom section of the columns provided a measure of the near-saturation values for sands and composites. Ratios of the water content at the bottom of the column to porosity (computed from bulk and assumed particle density, s = 2.65 g cm–3) ranged from 0.926 to 1.051. These values are typical of those reported for a wide range of sands and soils (Leij et al., 1996). Ideally, the ratios of saturated water content to porosity should be 1.0, but variations occur due to sample preparation methods and handling, incomplete saturation due to air entrapment, use of one assumed value for particle density, and measurements of bulk density and water content on separate samples. The problem is unavoidable in soils that exhibit volume changes on wetting and drying.
Taking the bottom of the column as reference, the height to the middle of a section above the bottom of the drained column at equilibrium was considered equal to negative pressure head, expressed as negative centimeters. Thus, pairing the height above the bottom with its water content provided the experimental SWC data for the column.
Predictions of Soil Water Characteristics
Both the PSD and experimental SWC data were plotted and smooth curves were hand drawn for further use in the model. The PSD curve was divided into about 30 fractions to ensure that shapes of the PSD curves were adequately represented. Water content distributions were computed using Eq. [3], but pressure heads were predicted using two different scaling procedures: (i) the method of Arya et al. (1999), in which the scaling Parameter i (Eq. [10]) was applied to Eq. [5] to compute pore radii, and (ii) the modified method described in this study, in which the natural pore length, Li(n), predicted from cubic close-packed pore length, Li(c) (Eq. [11]), was applied to Eq. [4] to compute pore radii. Four versions of each method were assessed. First, i and Li(n), computed using the best-fit Parameters a and b (Eq. [9]) and c and d (Eq. [11]), were applied to each sand material. Second, the sand materials were pooled in two groups: (i) those that contained no clay or silt and (ii) those that contained some clay and silt, and the best-fit mean parameters were computed for each group. The group mean parameters were then applied to each medium to assess how the individual predictions of SWC would vary from predictions based on their own best-fit parameters. For the third and fourth versions, Parameters a, b and c, d were altered by ±10% of their best-fit value to assess the effect that errors of this magnitude might have on SWC predictions.
For each set of predictions, goodness of overall agreement between the predicted and experimental pressure heads was expressed in terms of the root mean square residuals (RMSRs), using the relationship
 | [13] |
Values of RMSRs are influenced by the range of pressure heads across which the comparisons are made. Comparisons that include large values of pressure heads generally produce larger RMSRs. Because the range of pressure heads may vary from one sand medium to another (or one data set to another for the same medium), one needs to be cautious in using RMSRs to compare the two scaling procedures. Comparison should be made only for the same medium across the same range of pressure heads. In some cases, poor agreement between one data pair in the dry range may offset good agreement between data pairs in the wet range of the moisture scale.
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RESULTS AND DISCUSSION
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Equations to predict natural pore lengths from PSD and packing density information are the key elements in the AP model. Therefore, primary attention needs to be directed to relationships and associated parameters in Eq. [9] and [11]. An example of the fit of Eq. [9] (r2 = 0.994) and [11] (r2 = 0.990) is presented in Fig. 2
for Commercial Sand 2. Similar linear relationships were found for the remaining 12 sand materials. Equation parameters (a, b and c, d) and r2 values of the fit are presented in Table 2
for all sand materials. Close examination of graphic results for each of the 14 sands indicates that, where deviations occur from linearity, they generally occur at very low logni and logLi(c) values. This region corresponds with near saturation on the SWC curve and large particle sizes on the PSD curve, both of which are difficult to define accurately. In addition, hand-drawn SWC and PSD curves and manual interpolation of values have their own errors.
It is expected that Parameters a, b and c, d developed for these sand materials might be used for similar sand materials to produce SWC data, but similarity often cannot be established with sufficient certainty. Therefore, we decided to estimate mean parameters and evaluate their effect on predictions of SWCs. Sand materials were divided in two groups: (i) those with no clay or silt and (ii) those with minor amounts of clay and silt. For each group, log nii vs. logni and logLi(n) vs. logLi(c) data were pooled to evaluate group mean parameters (a, b and c, d) and r2 values (Table 2). Although there is scatter in the data, a strong linear relationship is indicated for both Eq. [9] and [11] (Fig. 3a–3d
). Filter sand appears different from the others (Fig. 3a and 3c). Excluding this material from the group should further improve the goodness of fit. While the slopes (b and d values) are similar for both the groups, the intercepts (a and c) differ appreciably. We do not understand this behavior completely, but note that it implies that adding finer materials to golf course sands will alter the SWC of the turf media, and possibly the flow and water depletion patterns. This result may have ramifications in the selection of sands for golf course construction and the subsequent management of golf course turf.
The sensitivity of predicted SWCs to uncertainties in Parameters a, b or c, d is a concern. In practice, errors and uncertainties are unavoidable when choosing parameters for unknown media. We therefore ran a scenario of uncertainties in Parameters a, b (Eq. [9]) and c, d (Eq. [11]). Both pairs were altered by ±10% of their best-fit values. Also, we assessed the effect of using just the group mean values of these parameters. Predicted and experimental SWCs for each scenario for Bunker sand are presented in Fig. 4a
through 4d. Results show that with best-fit parameters both Eq. [9] and [11] produce SWC curves in excellent agreement with experimental SWC (Fig. 4a). When a, b and c, d are altered by 10% of their best-fit values, however, the agreement deteriorates, with the effect of error in a, b being greater than the error in c, d (Fig. 4b). Altering the best-fit values of a, b and c, d by –10% appears to have relatively small effect on agreement between the predicted and experimental SWCs (Fig. 4c). It is concluded that overestimating parameters will have greater negative effect on predictions than underestimating them. When group mean parameters were applied to an individual medium, the agreement between predicted and experimental SWCs remained excellent, and the difference in predicted curves using the two equations virtually disappeared (Fig. 3d). Data suggest that group mean parameters will do as well as the best-fit values when applied to similar media.
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Fig. 4. Experimental and predicted soil water characteristic for Bunker sand: (a) best-fit parameters, (b) Parameters a, b (Eq. [9]) and c, d (Eq. [11]) altered by +10% of their best-fit values, (c) Parameters a, b and c, d altered by –10% of their best-fit values, and (d) best-fit mean parameters for the sand materials with no clay or silt.
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Sample results provided above using data for Bunker sand indicate that both Eq. [9] and [11] are likely to produce similar results when best-fit parameters for both are applied. This general conclusion is further examined in Fig. 5
, where predicted SWCs for all 14 media using one method are compared with those using the other. Data show excellent agreement between the two methods, but the relationship is not exactly 1:1. Both methods produce nearly identical results in the wet range of the SWC, but the differences progressively widen in the dry range. Absolute values of predicted pressure heads using Eq. [9] are generally smaller than those using Eq. [11].
A more quantitative estimate of the deviations between experimental and predicted pressure heads was obtained using RMSRs, which were calculated using Eq. [13]. Results for the 14 sand materials are presented in Table 3
, where the columns are numbered from 1 to 15 for easy reference. As expected, RMSRs show variation from one sand material to another. When best-fit parameters were applied, RMSRs for both Eq. [9] and [11] were generally small (see Columns 2 and 9), but those for Eq. [9] tended to be slightly smaller, indicating overall superior agreement with experimental data. When predictions of SWC were made using the group mean parameters, RMSRs tended to increase (see Columns 3 and 10), but not in a dramatic way. A dramatic increase in RMSRs occurred for both Eq. [9] and [11] when Parameters a, b and c, d were altered by 10% of their best-fit values (see Columns 4 and 11), with increases being far more dramatic with Eq. [9] than with Eq. [11]. These effects are illustrated in Fig. 4b for Bunker sand. The increases in RMSRs, when expressed as a percentage of best-fit values, indicate that Eq. [9] is likely to perform far more poorly when parameters are overestimated (see Columns 7 and 14). Use of group mean parameters and parameters altered by –10% of their best-fit values exhibited smaller changes in RMSRs (see Columns 6 and 13, and 8 and 15), but those for Eq. [11] were far gentler than those for Eq. [9].
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Table 3. Overall deviation between predicted and experimental SWCs, measured by the root mean square residuals (RMSRs).
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Based on the above results, we conclude that both Eq. [9] and [11] are effective in estimating natural pore lengths from properties of counterpart cubic close-packed structures. Predictions of SWCs using Eq. [11], however, are likely to be less sensitive to uncertainties in equation parameters. In this study, group mean parameters appear to have worked well (Fig. 4d, Table 3), but close similarity between individual sands in the group should be noted (Table 1). In the traditional textural classification of the USDA (e.g., loam, sandy loam, silt loam, etc.), use of the textural triangle permits inclusion of soils with widely differing PSDs, bulk density, and organic matter content, and the idea of using textural class mean parameters may not work so well.
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CONCLUSIONS
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This study was conducted to adapt the Arya–Paris model (Arya et al., 1999) to golf course and athletic field sands. The model has adapted well, and the required parameters were developed for 14 sand materials. Because of the nature of scaling of natural pore lengths in the original AP model, however, there were concerns about the sensitivity of the predicted SWCs to uncertainties in model parameters. Consequently, we developed and evaluated an alternative procedure to scale natural pore lengths from straight pore lengths in counterpart cubic close-packed structures. Both of the procedures produce similar results when best-fit parameters are applied. The modified procedure is simpler to apply, however, and predicted SWCs are far less sensitive to uncertainties in model parameters. This modified Arya–Paris model shows promise for use in laboratories that test sands for athletic turf and golf course construction.
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APPENDIX
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ACKNOWLEDGMENTS
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This research was supported by funds provided by the Turfgrass Council of North Carolina.
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NOTES
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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 June 16, 2006.
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TOP
ABSTRACT
INTRODUCTION
Ri3ni)/3; pore volume in assemblage = 8Ri3ni – (4
