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Published online 11 April 2005
Published in Soil Sci Soc Am J 69:584-592 (2005)
DOI: 10.2136/sssaj2004.0146
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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Soil Physics

Modeling Soil Shrinkage Curve across a Wide Range of Soil Types

X. Penga,b,* and R. Horna

a Institute of Plant Nutrition and Soil Science, CAU Kiel, Olshausenstr. 40, D-24118 Kiel, Germany
b Institute of Soil Science, Chinese Academy of Sciences, P.O. Box 821, Nanjing 210008, People's Republic of China

* Corresponding author (xh.peng{at}soils.uni-kiel.de)


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
Knowledge about soil shrinkage improves the understanding and prediction of unsaturated hydraulic properties in nonrigid soils. Until now, there is no general model available, which is widely accepted and applied to quantify soil shrinkage curves. The objectives in this present paper are (i) to propose a new and simple model and to test it with a wide range of soil types; (ii) to mathematically distinguish the shrinkage zones; and (iii) to evaluate probable physical meaning of its parameters. The results show that the modified van Genuchten water retention curve model fits the data obtained from Reeve and Hall's six soils, Talsma's three soils and three Typic Chromexert' soils well and the correlation coefficients in the tested 12 soils are always higher than 0.995. The different shrinkage zones defined by the mathematical method agree well with actual soil shrinkage curve. The proportional shrinkage zone accounts for 30.9 to 79.9% of the total water loss and 63.5 to 93.9% of the total volume decrease, while the zero shrinkage zone only accounts for <2.7% of the total volume decrease. The {alpha}, n, and m parameters of the modified van Genuchten model control soil shrinkage curve.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
THE SOIL VOLUME for nonrigid soils generally decreases with water loss as a function of the rate and completeness of drying and the volume change causes rearrangements of soil particles and aggregates (Smiles, 2000). Swelling and shrinkage modifications of soil structure influence the movement of water and solutes in the soil matrix (Garnier et al., 1997). Consequently, the transport of water and solutes in nonrigid soils becomes more complex than in rigid soils, requiring additional simultaneous measurements of soil volume. For example, hydraulic conductivity measurements require knowledge of the relationships between water content and head pressure of the water retention curve as well as the soil volume change. Therefore, the best understanding and prediction of transport processes in nonrigid soils requires a full knowledge of the dynamic processes associated with soil shrinkage.

To determine how soil volume decreases during drying, soil shrinkage behavior can be characterized by its void ratio and moisture ratio (Bronswijk, 1991; Groenevelt and Grant, 2001; Kim et al., 1992; Tariq and Durnford, 1993), or by specific volume and water content (Braudeau et al., 1999; McGarry and Malafant, 1987). In this study, we consider the void ratio and the moisture ratio to best identify the shrinkage behavior. They are defined as:

[1]

[2]
where e and {vartheta} are the void and moisture ratios and Vf, Vw, and Vs are the volumes of pore, water and solid, respectively. Plotting the void ratio against the moisture ratio is often used to represent the soil shrinkage curve. This shrinkage curve portrays four distinct shrinkage zones (Braudeau et al., 1999; Tariq and Durnford, 1993). These zones from the wet side to the dry side are defined as: (i) structural shrinkage, (ii) proportional shrinkage, (iii) residual shrinkage, and (iv) zero shrinkage. In the structural and residual shrinkage ranges, soil volume decrease is smaller than the water loss. In the proportional shrinkage region of the graph, soil volume decreases are often assumed to equal water loss and the volume of air remains constant (McGarry and Malafant, 1987; Tariq and Durnford, 1993). In the zero shrinkage, the volume of soil does not change as the last of the residual soil water is removed. However, not all soil samples show the four shrinkage zones. In some cases, the shrinkage curve lacks a structural shrinkage segment (Kim et al., 1992, 1999). In other cases, the zero shrinkage zone is absent from the shrinkage curve (McGarry and Malafant, 1987). These different cases of shrinkage curves for different soils are difficult to model.

Until now, numerous models have been proposed to describe the shrinkage curve; however, no general model has been proposed. Existing models describing the published shrinkage curves may be classified into three groups: (1) Models of the shrinkage curve devoid of a structural shrinkage zone (Chertkov, 2003; Giraldez et al., 1983; Kim et al., 1992; Olsen and Haugen, 1998). (2) Models of the shrinkage curve without a zero shrinkage zone (McGarry and Malafant, 1987). (3) A third group of models portray a four-zone shrinkage curve (Braudeau et al., 1999; Groenevelt and Grant, 2001; Tariq and Durnford, 1993). Obviously, the first two groups cannot cover all cases of the shrinkage curve. In Braudeau et al.'s (1999) model and Tariq and Durnford's (1993) models, the different zones of the shrinkage curve are modeled by separate equations requiring a lot of parameters. This mathematical approach not only complicates the modeling process but also generates additional problems during end-point estimation. Groenevelt and Bolt (1972) put forward one equation covering the entire shrinkage curve, which has recently been applied to a wide range of soil types and fits very well with measured data (Grant et al., 2002; Groenevelt and Grant, 2001, 2002; Groenevelt et al., 2001). However, this model does not take into account the ranges of the actual moisture ratios and void ratios, limiting the application of their model.

In this study our objectives are: (i) to propose a new model covering the entire shrinkage curve and to test it with a wide range of soil types with different shrinkage cases; (ii) to mathematically determine the shrinkage zones; and (iii) to evaluate probable physical meaning of its parameters.


   THEORY
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
Model Description
Soil shrinkage curves are generally sigmoidal in their shape (Fig. 1). Beginning at the dry side of the moisture ratio, small increases in the moisture ratio cause little changes in the shrinkage curve. As moisture ratios increase, void ratios increase rapidly until the moisture ratio approaches saturation. The shape of the shrinkage curve is opposite that of the water retention curve, where the volumetric water content decreases with increases in the matric potential (Fig. 2). The closed-form equation for predicting the water retention curve was first forwarded by van Genuchten (1980) as follows:

[3]
where {theta} is the volumetric water content, {psi} is the matric potential, and {theta}r and {theta}s are the residual and saturated volumetric water contents, respectively. The {alpha}, m, and n are fitting parameters, which control a portion of the S-shape soil shrinkage curve. The basic shape of the S-shape curve is controlled by the following equation, portrayed in Fig. 2.



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  		Fig. 1. Schematic diagram for the typical shrinkage curve with four shrinkage zones.

 


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  		Fig. 2. Schematic comparisons of the water retention curve modeled by van Genuchten model (Eq. [3]) and the shrinkage curve modeled by Eq. [10]. Equation [4] and Eq. [7] are the basic shapes of water retention curve and soil shrinkage curve, respectively.

 

[4]
where each parameter retains their previous definitions. Equation [4] fulfills the boundary conditions:

[5]

[6]

As mentioned above, the shrinkage curve is the inverse of the water retention curve. Therefore, we can replace the nth power in Eq. [4] with a negative sign and substitute {theta} and {psi} by the void ratio (e) and the moisture ratio ({vartheta}), respectively. Then Eq. [4] can be written as:

[7]
with the boundary conditions:

[8]

[9]

Thus the end points of the S-shaped sigmoid curve in Eq. [4] are replaced by Eq. [7] as portrayed in Fig. 2. However, the void ratio and the moisture ratio both depend on the soil porosity. The range of the {vartheta} value is only calculated from 0 to {vartheta}s, which is defined as oven-dry at 105°C and at saturation, respectively. Furthermore e is confined from the residual void ratio (er) to saturation void ratio (es). At saturation, {vartheta}s is equal to es. Taking the ranges of {vartheta} and e into account, the shrinkage curve may be expressed as follows:

[10]
where {alpha}, m, and n are fitting parameters. Equation [10] fulfills the boundary conditions and is portrayed in Fig. 2:

[11]

[12]

Determination for the End-Points of the Shrinkage Zones
The shrinkage curve normally is composed of four different zones (Fig. 1). Each zone has its own particular shape. According to the shrinkage curve shape, Groenevelt and Grant (2001) first reported that the end-points of the different shrinkage zones could be determined mathematically instead of visually.

The slope of the shrinkage curve, {sigma}({vartheta}), can be found by differentiating Eq. [10]:

[13]
The shrinkage curve has one inflection point ({vartheta}i, ei). The location, which can be defined by the extreme of Eq. [13], or differentiating Eq. [13] and finding its root(s) by using the "Root" function in MathCad (MathSoft, 2003):

[14]
The shrinkage curve usually has two maximum curvatures. One is located at its wet-side ({vartheta}w, ew); the other is at the dry-side ({vartheta}d, ed). Their locations also can be found by specific conditions at the extremes of Eq. [14].

These extremes can be used to distinguish the different zones of the shrinkage curve. In this study, we followed Groenevelt and Grant's method (2001) to determine the end-point between the structural shrinkage and the proportional shrinkage using the maximum of the curve at the wet-side. But the end-points between the proportional, residual and zero shrinkage are determined as follows.

The proportional shrinkage as suggested by Groenevelt and Grant (2001) has been referred to as "normal" shrinkage (McGarry and Malafant, 1987), because the volume decrease was assumed to be equal to the water loss. However, numerous experiments showed their relationships were not limited to a 1:1 line, but rather a linear correlation with divergent slope (Braudeau et al., 1999; Crescimanno and Provenzano, 1999; Kim et al., 1992). Braudeau et al. (1999) stressed the slope of them may be near 0.1 in some cases. Some authors (Groenevelt and Grant, 2001; Braudeau et al., 1999) reported that the proportional shrinkage should be defined as the slope of the volume decreased with respect to the water loss and this relationship was kept nearly constant but never zero. Therefore, the end-point from the proportional shrinkage to the residual shrinkage can be defined as the intersection ({vartheta}p, ep) between the tangent through the inflection point [x({vartheta})] and the tangent through the dry-side maximum of the curvature [y({vartheta})].


[15]

[16]
According to the definition of zero shrinkage, the end-point from the residual shrinkage to the zero shrinkage can be determined as the intersection ({vartheta}z, ez) between the tangent through the dry-side maximum of the curvature [y({vartheta})] and the horizontal line [z({vartheta})] through the residual point ({vartheta}0, er) when moisture ratio is equal to 0, which was defined as:

[17]
With the end-points of the differently characterized shrinkage zones, the extent of the different shrinkage zones can be expressed as a percentage of the total shrinkage according to the changes in the moisture ratio or the void ratio during their dehydration for each of their respective shrinkage zones. The structural, proportional, residual, and zero shrinkage zones, accounting for the total water loss or the total volume change can be written as follows:

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]
When a soil lacks structural shrinkage, the wet-side maximum of curvature is located away from the shrinkage curve. In this case, the proportional shrinkage starts from the saturated point ({vartheta}s, es) instead of the wet-side maximum of curvature ({vartheta}w, ew). On the other hand, when a soil lacks a dry-side maximum of shrinkage curvature, the tangent used to determine the end point between the residual shrinkage and the proportional shrinkage will go through the entire dry point ({vartheta}0, er) instead of the dry-side maximum of curvature ({vartheta}d, ed). The residual shrinkage subsequently finishes the residual point ({vartheta}0, er) instead of the intersection point ({vartheta}z, ez) as defined above.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
One data set consisting of nine different soil types was obtained from the different published literatures (Reeve and Hall, 1978; Talsam, 1977), which were reported again with Groenevelt and Bolt model (Grant et al., 2002; Groenevelt and Grant, 2001; Groenevelt et al., 2001). In Reeve and Hall's six soils (1978), the Wyre series and Fladbury are clayey alluvial soils and the Faulkbourne series and Ragdale are Chalky Boulder clay. The Wyre series and Faulkbourne series with angular and prismatic blocky structure was compared with the poorly structured Fladbury soil and Ragdale soil, respectively. Three Talsam's soils (1977) are exposed to three levels of applied stress (0, 6.3, and 11.2 kPa).

Furthermore, three new data sets were obtained from corresponding soil shrinkage tests for three Typic Chromexert soils (Soil Survey Staff, 1975) in Griffith, NSW, Australia. The three plots were subject to the FILTER project (Filtration and Irrigated Cropping for Land Treatment and Effluent Reuse). More detailed information is given by Jayawardane et al. (2001). Plot A, established for summer cropping in late 1994, was loosened to 1-m depth with a wide-blade deep ripper and subsequently irrigated with sewage effluent with an approximate salinity of 1.2 dS m–1. Plot B, established for winter cropping in 1998, was loosened to 0.65-m depth with a narrow-blade ripping implement and irrigated with sewage effluent with an approximate salinity of 1.2 dS m–1. In late 1999, before establishing the 1999–2000 summer crops, this plot was replowed to a depth of approximate 15 cm to improve crop establishment and to increase the infiltration rate. Plot C was established for summer cropping in 1998 and irrigated with sewage effluent with an approximate salinity of 10.8 dS m–1. For these undisturbed subsoils (10–25 cm) from the three plots the shrinkage was measured as a function of dehydration.

We evaluated the proposed model with the published and/or new data from a wide range of soil types using a MathCad solve-block to determine the least sum of squared errors (MathSoft, 2003).


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
Modeling the Obtained Data from 12 Soils
The Reeve and Hall's, Talsma's, and Typic Chromexert' data modeled by Eq. [10] are shown in Fig. 3 through 5. The circle points are the measured data and the dotted line is the fitted data with the proposed model. All data seem to fit the model very well. The correlation coefficients between the measured and fitted data are always greater than 0.995 (Table 1). Table 1 lists the parameters of the model from all tested shrinkage curves. The er and es values are directly determined by the measured residual and saturated void ratio. The parameters {alpha} and m varied by a factor up to one thousand, while parameter n ranges from 0.811 to 4.285.



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  		Fig. 3. The shrinkage curves from Reeve and Hall's six soils modeled by Eq. [10]. (a) Faulkbourne Bw, Faulkbourne Btg and Ragdale, (b) Wyre Bw, Wyre Bg and Faldbury. The square point ({vartheta}z, ez) is the end-point from residual shrinkage to zero shrinkage; the cross point ({vartheta}d, ed) is the dry-side maximum of the curvature; the diamond point ({vartheta}p, ep) is the end-point from the proportional shrinkage to residual shrinkage; the plus point ({vartheta}w, ew) is the end-point from structural shrinkage to proportional shrinkage.

 


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  		Fig. 5. The shrinkage curves from three Typic Chromexert soils modeled by Eq. [10]: The square point ({vartheta}z, ez) is the end-point from residual shrinkage to zero shrinkage; the cross point ({vartheta}d, ed) is the dry-side maximum of the curvature; the diamond point ({vartheta}p, ep) is the end-point from the proportional shrinkage to residual shrinkage; the plus point ({vartheta}w, ew) is the end-point from structural shrinkage to proportional shrinkage.

 


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  		Fig. 4. The shrinkage curves from Talsma's soils with three levels of applied stress (P = 0; 6.3 and 11.2 kPa) modeled by Eq. [10]. The square point ({vartheta}z, ez) is the end-point from residual shrinkage to zero shrinkage; the cross point ({vartheta}d, ed) is the dry-side maximum of the curvature; the diamond point ({vartheta}p, ep) is the end-point from the proportional shrinkage to residual shrinkage.

 

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  		Table 1. The parameters of Eq. [10] and the correlation coefficient (r) for the measured and predicted data from all 12 tested soils (Reeve and Hall's six soils [1978], Talsma's three soils [1977], and three Typic Chromexert' soils).

 
In Fig. 3 through 5, the symbols like squares, crosses, diamonds, and pluses show the different end points of the shrinkage zone determined by mathematical method. The square points ({vartheta}z, ez) define the end point from residual shrinkage to zero shrinkage. The cross points ({vartheta}d, ed) show the maximum curvature of the dry-side part end. The diamond points ({vartheta}p, ep) are the end points from the proportional shrinkage to the residual shrinkage. The plus points ({vartheta}w, ew) represent the maximum curvature of the wet-side shrinkage curve, which determines the transition point from the structural shrinkage to the proportional shrinkage. Of all tested soils, only Faulkbourne Bw, Fladbury and three Typic Chromexert soils present the wet-side maximum of the curvature, indicating that they consist of the structural shrinkage, which does not occur for the other soils. Plot B of Typic Chromexert soils does not show the dry-side maximum curvature as well as the zero shrinkage.

Tables 2 and 3 sum up the water loss and the volumetric change for each shrinkage zone separated by end-points mentioned above. Of the four shrinkage zones, the proportional shrinkage zone accounts for 30.9 to 79.9% of the total water loss and 63.5 to 93.9% of the total volume change in all tested soils. The zero shrinkage zone accounts for <2.7% of the total volume change even when it accounts for as much as 34.7% of water loss. The slope of zero shrinkage is less than 0.05, which is in accordance with Reeve and Hall (1978). Volume decrease in residual shrinkage zone in all tested soils is much higher than zero shrinkage but water loss is smaller except for Wre Bw, Talsma's soil (P = 0 kPa) and three Typic Chromexert soils.


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  		Table 2. Moisture ratio values of end-points defined mathematically and the percentages of their distinguished four shrinkage zones in total volumetric decrease.

 

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  		Table 3. Void ratio values of end-points defined mathematically and the percentages of their distinguished four shrinkage zones in total water loss.

 
Soil properties and management influence soil shrinkage behavior. In Reeve and Hall's six soils (1978), shrinkage capacities (= the total changes in void ratio) of angular and prismatic blocky structured Wyre series and Faulkbourne series are smaller than that of the poorly structured Fladbury and Ragdale, respectively, even if the saturated void ratio values are nearly identical. In Talsma's three soils (1977), higher mechanical stress produces higher soil bulk density and/or low void ratio. Soil shrinkage curves of three levels of applied stresses are different. The zero and residual shrinkage zones increase from 6.1 to 9.4% while the proportional shrinkage decreases from P = 0 kPa to P = 11.2 kPa. In three Typic Chromexert' soils, structural shrinkage accounts for 23.3% of total void ratio change and 49.8% of total moisture ratio change in Plot B after intermediate reploughing, which is much higher than that of the other two plots.

Parameter Estimation
The mathematical method allows a continuous fit of the void ratio versus moisture ratio relationship including the boundary conditions of parameters. The shape of soil shrinkage curve is based on the three parameters ({alpha}, n, and m) of Eq. [10]. Figures 6 through 8 show that the three fitting parameters control the different parts of the shrinkage curve. Increasing {alpha} values increase the range of structural shrinkage zone and decrease the ranges of zero and residual shrinkage zones of the shrinkage curve (Fig. 6). On the contrary, parameter m has an opposite function of parameter {alpha}. The larger the m values, the smaller the structural shrinkage and the greater are the zero and residual shrinkage ranges (Fig. 7). Figure 8 presents the effect of the n parameter on the slope of the proportional shrinkage zone. Increasing n values steepen the slope of the proportional shrinkage, indicating an increased volume change compared with the volumetric water loss. The boundaries of the three parameters are numerically limited from zero to infinite. When the parameters {alpha} and m are close to zero and/or infinite, there is a linear line through the saturation point and/or the residual point. Actually, soil shrinkage curve is always above the 1:1 linear line because the partial emptying water-filled pores are filled by air. Except the functions of three parameters mentioned above, parameters {alpha} and m also influence the slope of shrinkage curve and the parameter n affected the structural and/or zero shrinkages to some extent (Fig. 6–8). Table 4 shows a significant linear relationship between two parameters {alpha} and m. The effects of the three parameters on shrinkage curve are interactive and not completely independent, which complicates the analysis of the physical meaning of parameters. In this study, we do not find a good relationship between the shrinkage zones and the three parameters.



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  		Fig. 6. Effect of parameter {alpha} of the Eq. [10] on soil shrinkage curve at a given m = 1.0 and n = 2.0.

 


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  		Fig. 8. Effect of parameter n of the Eq. [10] on soil shrinkage curve at a given {alpha} = 2.0 and m = 1.0.

 


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  		Fig. 7. Effect of parameter m of the Eq. [10] on soil shrinkage curve at a given {alpha} = 1.0 and n = 2.0.

 

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  		Table 4. Linear relationship coefficients between the three parameters {alpha}, m, and n.

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
CONCLUSIONS
 APPENDIX
 REFERENCES
 
The proposed single mathematical equation model provides an excellent description of the shrinkage curves for all soils reported. The equation fits all cases, even if the structural and/or zero shrinkage zones are missing or if the four shrinkage zones can be distinguished. The correlation coefficients (r) between the measured and the predicted data are always higher than 0.995. Although some existing models can also fit the measured data, for example, Groenevelt and Bolt's model (Groenevelt and Bolt, 1972) and Braudeau's model (Braudeau et al., 1999), our model has several advantages as compared with others. (1) The model clearly expresses the shrinkage curve, using one equation, which not only simplifies the model, but also offers the possibility of exactly distinguishing the different characteristic shrinkage zones mathematically instead of manually. (2) It takes into account the ranges of the void ratio and the moisture ratio constrained by soil porosity. Most of the existing models do not define the physical boundary conditions through the parameters. Consequently, in some cases, these models unrealistically predicted void ratio and moisture ratio values even for values exceeding the saturated condition (Groenevelt and Grant, 2002). (3) This new single equation model is also more flexible as it can model shrinkage curves for soils exhibiting four shrinkage zones as well as soils lacking structural shrinkage and missing zero shrinkage zones. The models of Chertkov (2003), Giraldez et al. (1983), Kim et al. (1992) and Olsen and Haugen (1998) describe the shrinkage curves of soils missing structural shrinkage zones. McGarry and Malafant's model best fits the shrinkage curve of soils missing the zero shrinkage zone. The single equation model generalizes all zones of the shrinkage curve. (4) Only three parameters are needed to describe the shrinkage curve. Those models of Braudeau et al. (1999), Chertkov (2003), McGarry and Malafant (1987), and Tariq and Durnford (1993) require more input data or parameters, greatly reducing their applicability.

Parameters in our model present an interactive effect on shrinkage behaviors. Parameters {alpha} and m have an analogous but opposite function on soil shrinkage curve. Increasing {alpha} values increased the structural shrinkage while decreasing if zero and residual shrinkages exist. The opposite is true for the parameter m. Parameter n controls the slope of the soil shrinkage curve. Higher values of the parameter n steepen the slope of the shrinkage curve, indicating a more pronounced volume loss than the volumetric water loss. Peng et al. (unpublished data, 2004) found a significant relationship between parameter n and the saturated hydraulic conductivity in a saline-sodic soil. In this study, however, we do not find a good relationship between the parameters and the shrinkage zones, which may be ascribed to the wide range of the tested soils and data limitation. The Talsma's soils were exposed to loaded shrinkage while Reeve and Hall's soils and Typic Chromexert' soils were not compacted. Meanwhile, the parameters' values are determined not only by the shape of the shrinkage curve, but also by the ranges of the void ratio and the moisture ratio. These factors modify the values of the parameter simultaneously. A significant linear relationship between parameters {alpha} with m suggests an interactive effect on soil shrinkage curve patterns, which confirms they are not totally independent each other.

Groenevelt and Grant (2001) determined the transition point from the structural shrinkage to the proportional shrinkage using the wet-side maximum of the curvature. The point was corresponded well with the water contents at the Atterberg lower plastic limits of the soils they tested (Groenevelt and Grant, 2002), and was defined as swelling limit (McGarry and Malafant, 1987) or macropore shrinkage limit (Braudeau et al., 2004). They also used the dry-side maximum of the curvature as a shrinkage limit to separate the residual shrinkage from the zero shrinkage. Actually, the dry-side maximum curvature still shows the volumetric shrinkage to some extent, which was confirmed by this study. Figures 3 through 5 present the intersection point (square point) between the tangent through the dry-side maximum of the curvature and the horizontal line through the point ({vartheta}0, er), which is more consistent with the definition of the zero shrinkage than the dry-side maximum of the curvature. The volumetric change in the zero shrinkage separated from the residual shrinkage by the intersection only accounted for <2.7% of the total volume decrease. The volume decrease in the proportional shrinkage is often defined as being equal to the water loss (Bronswijk, 1991; McGarry and Malafant, 1987). However, the slope does not always correspond with parallel shrinkages (Groenevelt and Grant, 2001; Mitchell, 1992; Tariq, 1993), as their range of ratios can be even lower than 0.1 (Braudeau et al., 1999). Also the Talsma's soils exhibited a slope value much lower than 1.0. Therefore, we use the delta moisture ratio and void ratio ranges to define the proportional shrinkage as long as the decrease in water volume is kept constant but does not approach zero. Additionally, the proportional shrinkage can be separated from the residual shrinkage by considering the intersection between the tangent through the inflection points of the shrinkage curve and the tangent through the dry-side maximum of the curvature. This transition point from proportional shrinkage to residual shrinkage is similar to the air entry point proposed by Sposito and Giráldez (1976). The proportional shrinkage accounts for 30.9 to 79.9% of the total water loss and for 63.5 to 93.9% of the total volume decrease. Thus, the proportional shrinkage is the main contributor of the four zones during the entire shrinkage process. This agrees fully with the reports by Braudeau et al. (1999) and Kim et al. (1999). Braudeau et al. (2004) developed a conceptual model of pore structure-water medium corresponding to the four characterized shrinkage zones. Water pools in structure and residual shrinkage zones present the part of no and/or little swelling macropores and micropores, respectively. A large volume decrease in proportional residual shrinkage range indicates the function of swelling micropores while water loss in zero shrinkage represents the segment of non-swelling micropores. The hierarchical pores and their function based on water loss and volume change improve to understand the soil shrinkage behavior.

Soil shrinkage is a result of rearrangement of soil particles induced by internal stress (e.g., hydraulic; tensile; capillary). Therefore, soil physical and chemical properties and managements play an important role in soil shrinkage behavior. In this study, angular and prismatic blocky structured soils present smaller changes in void ratio as compared with poorly structured soils in Reeve and Hall's soils. The angular and prismatic blocky structured soils were stronger than poorly structured soils. At a given dehydration stress, poorly structured soils was easily to deform (Horn and Baumgartl, 1999). A mechanically stressed soil (Talsma's soils) showed an increased bulk density and/or a decreased void ratio, which coincides with more contact points between soil particles and higher soil strength but also smaller spaces to rearrange soil particles (Hartge, 2000). Soil homogenization due to reploughing always resulted in an altered shrinkage pattern when compared with the identical soil without any intermediate homogenization. In the three Typic Chromexert soils, Plot B showed a very intense void ratio or moisture ratio decline in the structural and proportional shrinkage ranges while the declines in the residual and the zero shrinkage zones were much smaller than in the corresponding horizons of the other plots. This could be explained by the reploughing, which was originally performed to increase the hydraulic conductivity. However, it remains unsolved so far, it is possible to correlate physical soil properties like aggregate strength, predrying stress and actual pore water pressure values with the parameters of the van Genuchten model to derive a coupled (hydraulic and mechanical) model about the soil shrinkage process. Furthermore the effect of the complete collapse of the pore system on the shrinkage curve patterns requires more investigations.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
APPENDIX
 REFERENCES
 
The modified van Genuchten model demonstrates an excellent fit with obtained data for several soil types subjected to field soil conditions. The model has several advantages as compared with others: (i) the entire shrinkage curve can be expressed by one equation with three parameters; (ii) it takes into account the actual magnitude of the soil void ratio and moisture ratio; (iii) it can model different cases of soil shrinkage curve. Therefore, different shrinkage zones can be distinguished and quantified by a single equation, which will more accurately predict unsaturated hydraulic conductivity in nonrigid soils. Although this new mathematical approach provides new and more quantifiable predictions of soil shrinkage changes across the full range of soil water content, the universal application of this model to predict shrinkage-induced changes in the hydraulic and mechanical properties of nonrigid soils requires additional research. Furthermore, the physical meaning of these parameters in relation to soil physics need additional analysis.


   APPENDIX
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 APPENDIX
REFERENCES
 
List of Parameters
e, void ratio, volume of soil solid per volume of soil solid (cm3 cm–3)

er, residual void ratio (cm3 cm–3)

es, void ratio at saturation (cm3 cm–3)

{vartheta}, moisture ratio, volume of soil water per volume of soil solid (cm3 cm–3)

{vartheta}0, moisture ratio at oven-dry at 105°C ({vartheta}0 = 0)

{vartheta}s, moisture ratio at saturation, (cm3 cm–3)

Vf, volume of soil pore (cm3)

Vw, volume of soil water (cm3)

Vs, volume of soil solid (cm3)

{theta}, volume of soil water per total volume of soil (cm3 cm–3)

{theta}r, residual volumetric water content (cm3 cm–3)

{theta}s, saturated volumetric water content (cm3 cm–3)

{psi}, matric potential (kPa)

{alpha}, m, and n, parameters of modified van Genuchten model.


   ACKNOWLEDGMENTS
 
This research was supported by the German Research Foundation (DFG) through the grant Ho911/2433. Dr. X. Peng gratefully thanks to the Max-Planck Foundation providing the fellowship for his postdoctor research at Christian-Albrechts University, Kiel, Germany. We also thank Profs. A. Smucker and C.D. Grant for their kind comments, which are helpful to improve the paper.

Received for publication April 24, 2004.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 




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