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Published in Soil Sci. Soc. Am. J. 68:1807-1817 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

Division S-1—Soil Physics

An Approach for Estimating the Shrinkage Geometry Factor at a Moisture Content

V. Y. Chertkov*, I. Ravina and V. Zadoenko

Agricultural Engineering Division, Faculty of Civil and Environmental Engineering, Technion, Haifa 32000, Israel

* Corresponding author (agvictor{at}tx.technion.ac.il).


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSION
 APPENDIX
 REFERENCES
 
Soil shrinkage is characterized, along with the shrinkage curve, by a partition of the volume change of the soil matrix between contributions of cracks and soil subsidence. This partition is determined by the shrinkage geometry factor (rs). Knowledge of the value of rs is important for the consideration of water and solute transport in swelling and cracking soils. The rs concept was recently used for the generalization of flow equations in the case of the axially symmetric two-dimensional deformation of shrink-swell soils. Sufficient accuracy of the rs value is very essential for all these applications. However, the theoretical definition and available measurement method of the rs factor include some implicit assumptions that are disturbed in real conditions. These disturbances, which are not accounted for in rs measurements, can lead to distortion of the rs value. The objectives of the work are: to explicitly formulate the assumptions; to introduce a new presentation of the rs concept based on a comparison between different shrinkage curves of a soil; to suggest an approach for estimating the rs values corrected by taking into account the disturbance of one of the assumptions; and to experimentally illustrate the approach using the simplest case of pure-clay paste samples when they dry, shrink, and crack. The results show the necessity and practical possibility of considering the rs factor as a function of soil moisture and introducing to the factor the multiplicative correction that is connected with accounting for possible macrocracks in soil samples to be used for experimental estimation of the rs factor.

Abbreviations: CEC, cation exchange capacity


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSION
 APPENDIX
 REFERENCES
 
THE SHRINKING-SWELLING of a soil matrix, accompanied by vertical movements and cracking, essentially influences soil structure, hydraulic properties, and water flow, and their evolution with time. Two key dependencies determine the effects of soil shrinking-swelling. One of them, the so-called shrinkage curve, describes the soil volume change as a function of soil water content (e.g., Hillel, 1998). Still another key dependence determining effects of soil shrinking-swelling relates to the partition of the volume change of a soil matrix ({Delta}V) between cracks ({Delta}Vcr) and subsidence ({Delta}Vsub) contributions (Bronswijk, 1988). This partition is characterized by the so-called shrinkage geometry factor (rs) (Bronswijk, 1988). Rijniersce (1983) introduced the rs concept for cases of pure subsidence without cracking (rs = 1) and the isotropic shrinkage (rs = 3) of so-called unripened soils. Bronswijk (1988)(1989, 1990, 1991a, 1991b) generalized the rs concept to the case of an arbitrary combination of the possible {Delta}Vcr and {Delta}Vsub contributions at a given volume change of soil matrix, {Delta}V (i.e., for any combination of vertical and lateral soil deformations that is possible at a given {Delta}V). The total range of the generalized rs factor is 1 ≤ rs < {infty}. Bronswijk (1990) also suggested a measurement method for experimental estimation of the rs value.

The rs concept is not only used for experimental estimating of the crack volume (e.g., Baer and Anderson, 1997). Knowledge of the rs value is important for the consideration of water and solute transport in swelling and cracking soils. The rs concept was recently used to generalize flow equations in the case of axially symmetric two-dimensional deformation of shrink-swell soil samples without cracks (Garnier et al., 1997a, 1997b). These researchers remarked that sufficient accuracy of the rs value is very essential for all these applications. However, the theoretical definition and available measurement method of the rs factor include some implicit assumptions that are disturbed in real conditions. These disturbances, which are not accounted for in rs measurements, can lead to inaccuracy of the rs value.

The objectives of the work are (i) to explicitly formulate the assumptions of Bronswijk's approach; (ii) to introduce a new presentation and generalization of the rs concept based on a comparison between different shrinkage curves of a soil; (iii) to use the presentation and available data to illustrate the disturbances of the assumptions in real conditions; (iv) to suggest an approach (based on the new presentation and generalization of the rs concept) for estimating the corrected rs values by taking into account the disturbance of one of the assumptions; and (v) to experimentally illustrate the approach using the simplest case of pure-clay paste samples when they dry, shrink, and crack.

For the reader's convenience, we start with a brief summary of Bronswijk's (1988)(1989, 1990, 1991a, 1991b) approach and a remark of some different understanding of the rs factor from Garnier et al. (1997a)(1997b). In the experimental illustration of the approach to be suggested for correcting the rs value, we used Chertkov's (2000)(2003) model for prediction of the shrinkage curve of a clay matrix without cracks. For the reader's convenience, we give a brief summary of the model immediately before the description of the experimental part. Notation is summarized in Appendix.


   THEORY
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSION
 APPENDIX
 REFERENCES
 
Two Viewpoints of the Shrinkage Geometry Factor
A Summary of Bronswijk's (1988)(1989, 1990, 1991a, 1991b) Approach
What the indicated works have in common is a relation between changes of the matrix volume and thickness of a soil layer when it dries, shrinks, and cracks. In real modeling, the soil matrix layer of thickness z in the water-saturated state is replaced by a water-saturated cube whose initial side length is z and volume V = z3 (Fig. 1a) . After a water loss, the cube volume and height decrease by {Delta}V and {Delta}z, respectively (Fig. 1a). In general, the decrease in lateral directions can be different, that is, x != y != z – {Delta}z (Fig. 1a). Then the volume decrease ({Delta}V) of the initial soil matrix layer related to one cube and the layer subsidence ({Delta}z) are connected as

[1]
where rs is by definition the dimensionless shrinkage geometry factor. Bronswijk borrowed the geometrical interpretation of the right part of Fig. 1a from Aitchison and Holmes (1953) and Fox (1964). The upper part of the initial dotted cube of thickness {Delta}z gives a contribution of subsidence ({Delta}Vsub) to the volume decrease of soil matrix {Delta}V. The lower part of the initial cube of thickness z – {Delta}z minus a current volume of the small parallelepiped, shown by solid lines, gives the contribution of the total crack volume ({Delta}Vcr) to {Delta}V. According to this interpretation, rs values in the three cases are obvious. In the case of subsidence without cracking when {Delta}V = z2{Delta}z, rs = 1. In the case of isotropic shrinkage when x = y = z – {Delta}z, rs = 3. In the case of cracking without subsidence when {Delta}z -> 0, rs -> {infty}. At 1 < rs < 3, the subsidence contribution to {Delta}V dominates the crack contribution. At 3 < rs < {infty}, the situation is opposite. Also, according to this interpretation one can write

[2]
and

[3]



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  		Fig. 1. Scheme explaining Bronswijk's (1988)(1989, 1990, 1991a, 1991b) model of soil layer shrinkage and cracking. (a) Shrinkage of elementary soil cube at initial layer thickness z and volume V = z3 as a result of subsidence {Delta}z and lateral reduction of initial cube sides to x and y; the volume supplementing the reduced cube [i.e., parallelepiped: x x y x (z – {Delta}z)] in lateral directions up to volume z x z x (z – {Delta}z) is interpreted as a crack volume per one initial cube. (b) The initial soil layer composed from unconnected elementary cubes and the layer after shrinkage composed from corresponding parallelepipeds and crack volumes; arrows symbolize the continuation of the three shown cubes to the unlimited layer.

 
Thus, if the rs factor is known, measurements of subsidence {Delta}z give {Delta}V, {Delta}Vsub, and {Delta}Vcr from Eq. [1] to [3]. These equations were used in modeling the role of continuously changing cracks in moisture transport in soil matrix and cracks (Bronswijk, 1988, 1989), estimating the changes of the soil matrix volume by measurements of soil subsidence (Bronswijk, 1991a), and estimating the changes of the total crack volume by soil subsidence measurements (Bronswijk, 1991b). All these applications relate either to field conditions (Bronswijk, 1988, 1989, 1991a) or to the so-called large core (lysimeter) (Bronswijk, 1991b). Hence, an unlimited soil layer with cracks is, in fact, meant as indicated in Bronswijk's works.

Bronswijk (1990) also suggested an approach for the experimental estimating of the rs factor by measurements of initial volume and volume as well as the subsidence of cylindrical soil samples after oven drying. Bronswijk (1988)(1989) noted that the rs factor should depend, in particular, on moisture content. However, the published measurements (Bronswijk, 1990) only relate to the rs factor after oven drying. The rs = 3 that was obtained for a clay soil of the central part of the Netherlands with clay content from 52 to 69% was used in other works of this researcher as well as in works of other authors (e.g., Baer and Anderson, 1997).

The viewpoint from Garnier et al. (1997a)(1997b)
These researchers formally regard the same model of the shrinking cube (Fig. 1a) and use the same definition of the rs factor (Eq. [1]) as applied to another situation, the limited soil sample that is considered as an anisotropically deformable solid without cracks, but not to an unlimited soil layer with developing cracks. The volume change of the limited sample of soil matrix ({Delta}V) also includes the subsidence or vertical deformation contribution ({Delta}Vsub) (Fig. 1a and Eq. [2]), but the volume {Delta}Vcr (Fig. 1a and Eq. [3]) that was associated with crack contribution in Bronswijk's approach is now interpreted as a contribution of lateral deformations ({Delta}Vlat). In the case of a limited deformable sample, the values of rs = 1, 3, and {infty} mean only vertical, isotropic, and lateral axially symmetric deformations, respectively. At 1 < rs < 3, the contribution of vertical deformations to {Delta}V dominates the contribution of lateral deformations. At 3 < rs < {infty}, the situation is the opposite. Garnier et al. (1997b) measured the evolution of the height and diameter of cylindrical samples with water content and noted a change in rs value. However, in modeling Garnier et al. (1997a)(1997b) also used rs values constant with drying. Specifications of the rs concept to be considered in this work are different for the two above viewpoints.

Explicit Formulation of Bronswijk's Model Assumptions
A real soil layer is always connected, even if the layer contains cracks. Because of this feature, lateral tensile stresses and cracks develop in the layer at shrinkage. Except for that, cracks in the layer are always distributed with an average spacing between them (e.g., Zein el Abedine and Robinson, 1971). Thus, Bronswijk's model in fact replaces the real connected layer with distributed cracks by a layer that is composed of contacting but unconnected water-saturated cubes (Fig. 1b, dotted cubes) which shrink as isolated deformable solids without cracks (Fig. 1b, parallelepipeds shown by solid lines). In other words, the model replaces the cracks distributed in a soil layer by boundaries of the cubes before drying and gaps between the parallelepipeds in the course of drying (Fig. 1b). That is, all actually distributed cracks are artificially concentrated as a crack volume {Delta}Vcr per one initial cube (Fig. 1). This replacement will be referred to below as Bronswijk's approximation for a shrinking and cracking soil layer. Such replacement is physically suitable for the shrinkage of a cylindrical sample without cracks and with free boundaries—the case from Garnier et al. (1997a)(1997b). Therefore, the {Delta}Vlat contribution of this case numerically coincides with the {Delta}Vcr contribution (Eq. [3]) for the layer shrinkage in Bronswijk's approximation, and values of {Delta}z, {Delta}Vsub, and rs are simply identical. However, for the shrinkage of a real unlimited cracked soil layer (field conditions) the replacement means an assumption. We formulate the assumption in two different, but equivalent forms.

Assumption 1.
(i) At a given {Delta}V, a link between {Delta}z (or {Delta}Vsub) and {Delta}Vcr through the rs factor (Eq. [1]–[3]) is the same in cases of a real connected layer with distributed cracks and a composed modeled layer (Fig. 1b); that is, the rs value is the same for these two cases; or (ii) in similar conditions the subsidence of a real connected soil layer with distributed cracks coincides with the subsidence of a modeled soil layer composed of unconnected water-saturated cubes that shrink with drying along the vertical and two lateral axes as isolated deformable solids without cracks.

Only in Assumption 1 in any form, that is, in Bronswijk's approximation, shrinkage and cracking of an unlimited layer (Bronswijk, 1988, 1989, 1990, 1991a, 1991b) and the shrinkage deformation of a limited sample (Garnier et al., 1997a, 1997b) are formally described by the same Eq. [1] through [3] and Fig. 1. Disturbance of this assumption qualitatively flows out of the simple physical considerations. The thickness of any connected layer decreases at tension because a lateral tensile deformation leads to the vertical compressive one. In the simplest case the ratio of the latter to the former is Poisson's ratio of a material (e.g., Landau and Lifshitz, 1986). Therefore, at a given {Delta}V, the thickness of a real stretched soil layer with distributed cracks will be smaller than the thickness of a layer that is composed of unconnected and hence unstretched parallelepipeds and gaps of crack volume between them (Fig. 1b). That is, the real subsidence of a cracked soil layer at a given {Delta}V will be larger than the subsidence in Bronswijk's approximation. If one knows the corrected subsidence {Delta}z at a given {Delta}V for a real connected layer with distributed cracks Eq. [1] to [3] give corrected values of rs, {Delta}Vsub, and {Delta}Vcr. Estimation of the corrected values accounting for the disturbance of Assumption 1 is beyond the scope of this work and will be addressed in the future.

Both the case of a composed layer (Bronswijk, 1988, 1989, 1990, 1991a, 1991b) and that of a limited sample (Garnier et al., 1997a, 1997b) consider an elementary cube or parallelepiped after deformation and the limited sample as a deformable solid without cracks. The method of the experimental estimation of rs values (Bronswijk, 1990) is based on measurements of only the external dimensions of soil samples (remolded or undisturbed) before and after oven drying. This available measurement method also implies that soil samples contain no cracks. Garnier et al. (1997a)( 1997b) also used a similar method of experimentally estimating the rs value. We formulate the corresponding implicit assumption of both Bronswijk's approximation and the viewpoint of Garnier et al. (1997a)(1997b) as Assumption 2.

Assumption 2.
Cracks do not appear and develop in drying soil samples.

Disturbance of this assumption in real conditions flows out of many images of shrinking soil samples (e.g., Hallaire, 1984, Fig. 4).

Finally, both Bronswijk (1988)(1989, 1990, 1991a, 1991b) and Garnier et al. (1997a)(1997b) noted a possible value variation of the rs factor with drying. However, in practical modeling, these researchers in fact used Assumption 3.

Assumption 3.
The rs factor does not depend on soil moisture.

New Presentation of the rs Concept
We account for V = z3 and replace the values that enter Eq. [1] for the drying layer with cracks by the corresponding specific volumes per unit weight of oven-dried soil. The initial volume of the soil matrix (V) is replaced by the initial value of the specific volume of the soil matrix . The current volume of drying soil matrix (V – {Delta}V) is replaced by the current value of the specific volume of the soil matrix . The summary volume of cracks and soil matrix (V – z2{Delta}z) is replaced by the specific volume of the layer , where cr.l is the specific volume of cracks in the layer. Then Eq. [1] can be rewritten as

[4]

Here, and l are the shrinkage curves of a soil matrix without cracks and soil layer with cracks, respectively. With that, rs = rs(w), where w is soil moisture.

Equations [1] and [4] as such are exact. However, in Bronswijk's approximation an exact value of cr.l entering l {equiv} + cr.l in the right part of Eq. [4] is replaced by gaps (Fig. 1b) based on (implicit) Assumption 1. Except for that, in the practical application an exact value of for soil matrix without cracks entering the left part of Eq. [4] is replaced by the specific volume of a cylindrical sample (Bronswijk, 1990), s {equiv} + cr.s, neglecting by the specific volume cr.s of possible (macro)cracks in the sample based on (implicit) Assumption 2. Here, s is the shrinkage curve of the sample with cracks. Note that, in general, cr.s depends on a sample size. Therefore, cr.s != cr.l. Approximations in the right and left parts of Eq. [4] lead to an inaccuracy in rs because they imply the replacement of exact Eq. [4] by

[5]
where l includes gaps (Fig. 1b) and r's is an uncorrected value of rs(w).

In this work we only address correction of rs connected with the disturbance of Assumption 2, that is, with the difference between and s entering the left parts of Eq. [4] and [5], respectively. In this work we consider that the difference between cr.l entering l in Eq. [4] and gaps (Fig. 1b) entering l in Eq. [5] is negligible according to Assumption 1. The correction connected with the disturbance of Assumption 1 will be addressed in the future.

We could only find a data combination of shrinkage curves l, s, and as a void ratio against the gravimetric water content in Hallaire's (1984) Fig. 2 . The data do not embrace a total range of water content. They relate to an aggregated clay soil with a clay fraction (52 to 56%) consisting mainly of montmorillonite and chlorite. Schematic curves in Fig. 2 qualitatively repeat these dependencies from Hallaire (1984) as the specific volume against the gravimetric water content. The difference between Curves 3 and 4 in Fig. 2 shows that, in general, rs and rs', as determined by Eq. [4] and [5], respectively, do not coincide.



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  		Fig. 2. Scheme of the different shrinkage curves of an aggregated clay soil based on Hallaire's (1984) Fig. 2. (1) Initial specific volume of shrinking and cracking soil layer. (2) Shrinkage curve l of shrinking soil layer with cracks; (3) shrinkage curve s of shrinking soil sample with cracks. (4) Shrinkage curve of soil matrix; in general, z < sz, that is, oven-dried sample contains cracks. (5) 1:1 theoretical line. AB, the specific volume of the layer subsidence at a given w; BD, the specific volume of cracks in the soil layer at a given w; CD, the specific volume of cracks in the soil sample with free boundaries at a given w.

 
Generalization of the rs Concept
By analogy to Eq. [4] that relates to an unlimited layer with developing cracks we can write the following relation for a shrinking sample of an arbitrary, in particular cylindrical, shape and with developing (macro)cracks as

[6]
where the M(w) factor determines partition of the matrix volume change, of the sample with possible (macro)cracks between the contribution of sample boundary displacement, s and the contribution of crack development, cr.s. In general, the introduced M factor can depend on the shape and size of the sample. We mean here the cylindrical samples of the usual laboratory heights and diameters of several centimeters. If cracks in the sample are absent, s = and M = 1. In the particular case of the sample with the shape of an unlimited layer with cracks, s coincides with l, the M factor coincides with the rs factor, and Eq. [6] coincides with Eq. [4]. Indeed, the contributions of subsidence and of the sample boundary displacement for an unlimited layer with cracks are identical.

It is worth emphasizing that unlike the partition of the matrix volume change of the sample with (macro)cracks, in particular, the cylindrical one to be described by Eq. [6], Bronswijk (1990) and Garnier et al. (1997a)(1997b) were interested in the partition of the volume change of a cylindrical sample without cracks between the contribution of height shortening and that of diameter shortening or lateral deformation.

Factorization of the rs Value for a Soil Layer
Replacing in Eq. [6] /o from Eq. [4] and s/o from Eq. [5] we come to the factorization of the rs value as

[7]

Thus, the factor M ≥ 1 of cylindrical samples with cracks plays the part of a multiplicative correction to the inexact estimate of rs' from Eq. [5]. If the (macro)cracks in the samples are absent, then M = 1.

An Approach for Estimating the Corrected rs Value
The advantages of presenting the rs concept in terms of soil subsidence ({Delta}z) and the volume change of a soil matrix ({Delta}V) (Bronswijk, 1988, 1989, 1990, 1991a, 1991b) are connected with the simplicity of measuring the external sample dimensions. We believe that presentation of the rs concept in terms of the different shrinkage curves of a soil using Eq. [4] through [6] permits one, possibly with some complication of measurements, to more exactly estimate the rs value as a function of soil moisture, accounting for the cracks in a sample. Equation [7] suggests an approach for estimating the corrected rs value, accounting for the disturbance of Assumption 2. Along with the experimental shrinkage curves l and s, the approach implies a knowledge of the shrinkage curve of the soil matrix []. The use of a physically modeled soil matrix shrinkage curve for estimating the multiplicative correction [M(w)] is the most convenient to experimentally illustrate the approach. For the present, the corresponding model is only for a pure-clay matrix (Chertkov, 2000, 2003). Therefore, we used the case of a pure-clay matrix to experimentally illustrate the estimation of the corrected rs value. It will be shown that the corresponding shrinkage curves are qualitatively similar to the curves in Fig. 2.

Summary of Chertkov's (2000)(2003) Model of a Clay Matrix Shrinkage Curve
In the form v({zeta}), where {zeta} is the relative water content of the clay, that is, the ratio of the current value of the gravimetric water content to the maximum possible value (the liquid limit), and v is the relative volume of drying clay, the shrinkage curve of a clay matrix is presented as (Fig. 3)

[8]
where vz and vs are indicated in Fig. 3,

[9]
is the shrinkage limit of the clay matrix, Fz is the pore volume fraction occupied by water at a water content corresponding to the shrinkage limit ({zeta}z), and

[10]
is the minimum water content in the normal shrinkage area (air-entry point). Using an approach from Chertkov (2003), one can estimate from vz and vs values the Fz and {zeta}z parameters of the pure clay, and thereby express the shrinkage curve of v({zeta}) (Eq. [8]–[10]) through parameters vs and vz only. Then, one can recalculate the dependence v({zeta}) as the shrinkage curve of the clay matrix in the form using the {leftrightarrow} v correspondency as

[11]
and the w {leftrightarrow} {zeta} correspondency as

[12]
where {rho}w is the density of water, and {rho}s is the density of the solid phase. The {rho}s, vs, and vz parameters can be found irrespective of the shrinkage curve. The density of clay particles, {rho}s, is measured by standard methods (Blake and Hartge, 1986). For a clay, the minimum relative volume vz and the relative volume vs of clay particles at the liquid limit, can be calculated from measured values of the liquid limit (wL) of the clay matrix and its specific volume in oven-dried state using Eq. [11] and [12] at v = vz and {zeta} = 1, respectively.



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  		Fig. 3. The general form of the shrinkage curve of a clay matrix [Chertkov's (2000) Fig. 2].

 

   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSION
 APPENDIX
 REFERENCES
 
Clays and Their Physicochemical Characteristics
All data relate to clay (>95% of montmorillonite) extracted by standard methods (Day, 1965) from samples of 0- to 30-cm and 30- to 60-cm layers of soil in Sarid, Israel. The samples were taken in the spring seasons of 2000–2003. The average clay content of the soil layers was 52%. The pH values, exchangeable cations, and cation exchange capacity (CEC) of the clay were measured by standard methods (Chapman, 1965; Peech, 1965). In the obtained values of pH, exchangeable cations, and CEC (Table 1), there was no significant difference between the two layers. The unit of a last decimal sign in Table 1 determines the accuracy of the averages.


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  		Table 1. Physicochemical characteristics of the clay of two layers of soil in Sarid, Israel.

 
Samples, Measured Parameters, Methods, and Data Presentation
The clay paste, water saturated nearly to the liquid limit, was placed in small metal containers of {approx}1.5-cm height, {approx}3.5-cm diam. The sample measurements for the clays of 0- to 30-cm and 30- to 60-cm soil layers were conducted using four and eight samples, respectively.

We measured three parameters of the clay itself and three parameters of each clay sample. The former include the clay particle density ({rho}s), the specific volume of oven-dried clay matrix, , and the liquid limit (wL). The latter include diameter (d), height (h), and weight (m) of a sample. The sample measurements for the clays of the 0- to 30-cm and 30- to 60-cm range were conducted for each sample once a day for 8 d with subsequent oven drying, and twice a day for 13 d plus once for the 14th day with subsequent oven drying, respectively.

The {rho}s density was measured by the standard pycnometer method (Blake and Hartge, 1986). The liquid limit wL was measured by the ASTM method (American Society for Testing and Materials, 1993). Observations showed that, after oven drying, the clay samples contained no cracks that could be seen by the naked eye. On the basis of these observations for the clay under consideration, we took that the specific volume of oven-dried clay matrix coincides with that of the oven-dried sample (see Fig. 2, case z = sz). Therefore, the specific volume z was determined as sz averaged by samples, that is, using the oven-dried volumes and weights of all clay samples (see below). Table 2 shows the experimental values of {rho}s, z, and wL, as well as the sample number (n1) and the number of experimental points (n2) for each clay sample of both soil depths. The standard deviation of the {rho}s value is 0.05 g cm–3. The unit of a last decimal sign determines the accuracy of other values.


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  		Table 2. Experimental values of {rho}s, z, and wL, the sample number (n1) and number of experimental points (n2) for each sample, and the vs, vz, Fz, and {zeta}z values calculated based on Chertkov's (2000)(2003) model for the clays of the 0- to 30-cm and 30- to 60-cm layers of soil in Sarid, Israel, using experimental values of {rho}s, z, and wL.

 
The clay samples in small metal containers slowly lost moisture at 23°C and at {approx}60% of the relative air humidity. The loss of water was measured by weighing the samples. In our illustrative experiments we used the simplest means to measure clay sample dimensions. Slide calipers measured the sample diameter and height during drying. Except for that, the loss of the sample volume after an initial shrinkage for 5 to 6 d was sometimes for control determined by the displacement of mercury as at shrinkage limit measurements (Das, 1979, p. 36–37). The data on diameter (d), height (h), and weight (m) of each sample at each measurement are presented in Table 3 for the clay of the 0- to 30-cm soil layer, and Tables 4 and 5 for the clay of the 30- to 60-cm soil layer. As in Tables 1 and 2, the unit of a last decimal sign determines the accuracy.


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  		Table 3. Diameter, height values (measured by slide calipers), and weight values of the four clay-paste samples for the clay of the 0- to 30-cm layer of Sarid soil.

 

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  		Table 4. Diameter and height values (measured by slide calipers) of the eight clay-paste samples for the clay of the 30- to 60-cm layer of Sarid soil.

 

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  		Table 5. Weight values of the eight clay-paste samples for the clay of the 30- to 60-cm layer of Sarid soil.

 
Data Analysis
Data from Tables 3, 4, and 5 on current diameter (d), height (h), and weight (m) of a sample were used for determining the gravimetric water content (w) and specific volumes l and s (corresponding to Curves 2 and 3 in Fig. 2) for each sample and each measurement. The gravimetric water content of each separate clay sample at measurements was determined by

[13]
where m is the current sample weight and md is the oven-dried weight of the sample. The specific volume of an unlimited clay layer with cracks (field conditions, Bronswijk's approximation) corresponding to a separate clay sample at a measurement was determined as

[14]
where Vso is an initial sample volume; do is an initial sample diameter; and {Delta}h is a sample height decrease at the measurement compared with initial height (ho). The specific volume of clay samples with internal cracks corresponding to a separate clay sample at a measurement was determined by

[15]
where Vs is a current sample volume for drying.

The data from Table 2 on {rho}s, z, and wL properties of the clays of the two layers were used to find vs and vz parameters of the clay matrix that determine its shrinkage curve from Chertkov's (2000)(2003) model. Then, using the model and vs and vz values, we found the Fz and {zeta}z values and eventually calculated from Eq. [8] through [12] the shrinkage curve of the clay matrix for two clays. In particular, we calculated the specific volume values of the clay matrix (), at averaged w values (by four or eight samples) corresponding to all nine or 28 consecutive measurements for clay from the upper and lower soil layers, respectively.

The (9 or 28) values found, together with corresponding l values from Eq. [14] gave the corrected value of the rs factor defined by Eq. [4] for a separate clay sample at a measurement as

[16]
where o = Vso/md. Similarly, the values found, together with corresponding s values from Eq. [15] gave the multiplicative correction M defined by Eq. [6] for a separate clay sample at a measurement as

[17]

Finally, the uncorrected r's factor defined by Eq. [5] and used in Bronswijk's approximation instead of the corrected rs factor, corresponding to a separate clay sample at a measurement, was determined as

[18]


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
SUMMARY AND CONCLUSION
 APPENDIX
 REFERENCES
 
Table 2 shows the calculated values of clay matrix parameters vs, vz, Fz, and {zeta}z using the above data on {rho}s, wL, and z. The accuracy of the estimates is determined by the unit of a last decimal sign (e.g., 0.01 for 0.83). Clay matrix shrinkage curves without cracks that were predicted from Chertkov's (2000)(2003) model ( and w values from Eq. [8]–[12]) using values vs, vz, {rho}s, and Fz from Table 2 are shown in Fig. 4 and 5 by solid lines.



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  		Fig. 4. The experimental specific volume of an unlimited clay layer (asterisks) including cracks, l (field conditions, Bronswijk's approximation when the layer before drying is composed of unconnected anisotropically shrinking cubes), and clay samples (circles) including cracks, s, and the specific volume of a clay matrix without cracks (solid line), predicted from Chertkov (2000)(2003) using data on {rho}s, z, and wL (Table 2) for the clay of 0- to 30-cm depths of soil in Sarid, Israel. The inclined straight line is a 1:1 theoretical. Figures near experimental points correspond to the measurement numbers in Tables 3 and 6.

 


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  		Fig. 5. As in Fig. 4 for the 30- to 60-cm depths, asterisks denote the experimental specific volume of an unlimited clay layer in Bronswijk's approximation, circles denote the experimental specific volume of clay samples with cracks, and the solid line represents the predicted specific volume of a clay matrix without cracks. Figures near experimental points correspond to the measurement numbers in Tables 4, 5, and 7.

 

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  		Table 6. Experimental estimates averaged by the four samples and standard deviations of the gravimetric water content (w), specific volumes of unlimited cracked clay-paste layer in Bronswijk's approximation (l) and cracked clay paste samples (s), and r's, M, and rs factors, as well as the model-predicted specific volume of the clay matrix () for the drying clay of the 0- to 30-cm layer of Sarid soil.

 

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  		Table 7. Experimental estimates averaged by the eight samples and standard deviations of the gravimetric water content (w), specific volumes of unlimited cracked clay-paste layer in Bronswijk's approximation (l) and cracked clay paste samples (s), and r's, M, and rs factors, as well as the model predicted specific volume of the clay matrix () for the drying clay of the 30- to 60-cm layer of Sarid soil.

 
Fig. 4 and 5 also present the experimental points of the shrinkage curves of the unlimited clay layer with cracks in Bronswijk's approximation (asterisks) and limited clay samples with cracks (circles). The points correspond to values of l from Eq. [14] (asterisks) and s from Eq. [15] (circles) averaged at a given measurement number, drying duration, and corresponding average clay moisture, w, by four or eight samples for the clay of 0- to 30-cm and 30- to 60-cm depths of the Sarid soil, respectively.

Tables 6 and 7 give the values and standard deviations for the averages by four or eight samples of w ± {delta}w (Eq. [13]), l ± {delta}l (Eq. [14]), s ± {delta}s (Eq. [15]), (Eq. [8]–[12]), r's ± {delta}r's (Eq. [18]), M ± {delta}M (Eq. [17]), and rs ± {delta}rs (Eq. [16]) at all measurements (9 or 28) for the clay of the two layers of Sarid soil. Here {delta}w, {delta}l, and so on mean corresponding standard deviations. Note that the number of significant figures of the predicted values in Tables 6 and 7 changes in connection with that of the experimental l and s values and their standard deviations. Note also that within the limits of the standard deviations the relation rs = M x rs' (Eq. [7]) is fulfilled for values of rs, M, and rs' averaged by four (Table 6) and eight (Table 7) samples. A number of points are worth noting in connection with the results presented in Fig. 4 and 5 and Tables 6 and 7.

  1. Experimentally obtained and predicted shrinkage curves (Fig. 4 and 5) for a clay layer with cracks (asterisks, l), clay samples with cracks (circles, s), and a clay matrix without cracks (solid curve, ), are qualitatively similar to corresponding experimental curves from Hallaire's (1984) Fig. 2 for an aggregated clay soil (clay fraction is 52 to 56%) and to the general schematic presentation of similar curves in Fig. 2 (in our experiments, cracks in clay paste samples closed at w = 0; that is, sz = z in Fig. 2).
  2. Results for the clay samples presented in Tables 6 and 7 show that the above-suggested approach allows one to experimentally estimate the shrinkage geometry factor before correction (r's) at a given soil moisture (w) as well as the corresponding multiplicative correction (M) of the clay layer with developing cracking. Qualitative similarity between experimental and predicted curves in Fig. 4 and 5, on the one hand, and experimental curves in Hallaire's (1984) Fig. 2 for an aggregated clay soil, on the other hand, permits one to assume a real possibility of using the approach for a general case of aggregated soil as well.
  3. Results for the clay samples presented in Tables 6 and 7 show that, unlike in Assumption 3, the variation of the shrinkage geometry factor before and after correction, r's and rs, respectively, with water content (w) is out of the limits of the standard deviations {delta}r's and {delta}rs, that is, statistically significant in the essential part of the water content range (0.15–0.2 g g–1 ≤ w ≤ wL {approx} 0.85–0.9 g g–1). Qualitative similarity between experimental and predicted curves in Fig. 4 and 5, on the one hand, and experimental curves in Hallaire's (1984) Fig. 2 for an aggregated clay soil, on the other hand, permit one to assume that the effect of r's and rs variation with water content will also take place for aggregated soils, although they already contain interaggregate cracks before drying.
  4. Results for the clay samples presented in Tables 6 and 7 show that, unlike in Assumption 2, a difference between the multiplicative correction (M) to the uncorrected shrinkage geometry factor (rs') and unity is out of the limits of the standard deviations ({delta}M), that is, statistically significant. Except for that, the difference varies in practically the total range of water content. The qualitative similarity that was used in the above points (ii) and (iii) also permit us to assume here that the effects of M > 1 and M = M(w) will also take place for aggregated soils. With that, interaggregate cracks and pores should increase these effects for aggregated soils.
  5. Thus, the results presented in Tables 6 and 7 and Fig. 4 and 5 show the necessity and practical possibility of considering the rs factor as a function of soil moisture and introducing to the factor the multiplicative correction M that is connected with accounting for possible (macro)cracks in soil samples to be used for experimental estimation of the rs factor.


   SUMMARY AND CONCLUSION
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSION
APPENDIX
 REFERENCES
 
The shrinkage geometry factor (rs) (Bronswijk, 1990) determines the relation between variations of soil subsidence and the total crack volume in field conditions with water content, or the relation between the vertical and lateral shrinkage of cylindrical soil samples without (macro)cracks. Both of these aspects of the rs factor have applications (Baer and Anderson, 1997; Garnier et al., 1997a, 1997b).

Bronswijk's (1988)(1989, 1990, 1991a, 1991b) model and measurement method are in fact based on three implicit assumptions. We formulate these assumptions explicitly. They are as follows: (i) In similar conditions, the subsidence of a real connected soil layer with distributed cracks coincides with the subsidence of a modeled soil layer composed of unconnected water-saturated cubes that shrink with drying along the vertical and two lateral axes as isolated deformable solids without cracks; (ii) cracks do not appear and develop in drying soil samples; and (iii) the rs factor does not depend on soil moisture.

Then we introduce a new presentation and generalization of the rs factor, using the different shrinkage curves of a soil for unlimited cracked layer (field conditions) and for the limited sample with cracks. The use of the presentation, together with available data, shows that the above assumptions implicitly introduced in the theoretical and experimental determination of the rs factor cannot be exactly fulfilled in real conditions. Using the new presentation and generalization of the rs factor, we suggest an approach for correcting the approximated r's values in connection with the disturbance of the second assumption. Addressing a correction of the approximated r's values in connection with the disturbance of the first assumption is beyond the scope of this work. Finally, we consider an experimental example relating to clay samples to illustrate the approach for finding the corrected rs values and variation of the latter with water content. This example is the simplest and most convenient because of the absence of interaggregate pores in the pure-clay matrix and the possibility to use an available model for the shrinkage curve of a clay matrix without cracks. At the same time, the qualitative similarity between the different shrinkage curves that we obtained for clay samples and the shrinkage curves that are available for an aggregated clay soil enable one to assume that effects of the rs factor observed on clay samples in connection with disturbing the second and third assumptions should also be statistically significant for the general case of an aggregated soil. That is, for an aggregated soil the rs value should, even if weakly but statistically significantly, depend on the soil moisture, and the suggested approach for finding the corrected rs values should be applicable.

The more precise definition and determination of the shrinkage geometry factor are important for the modeling hydraulic properties of swelling soils in the light of results from Garnier et al. (1997a)( 1997b). We continue this research to address the finding of corrected rs values in connection with the disturbance of the first assumption.


   APPENDIX
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 SUMMARY AND CONCLUSION
 APPENDIX
REFERENCES
 
Notation
d, clay sample diameter, cm

do, initial diameter of the clay paste sample, cm

Fz, saturation degree of a clay matrix at {zeta} = {zeta}z, dimensionless

h, clay sample height, cm

ho, initial height of the clay paste sample, cm

M, multiplicative correction to the r's value, dimensionless

m, weight of the clay paste sample, g

md, oven-dried weight of the clay paste sample, g

rs, shrinkage geometry factor, dimensionless

rs', approximated or uncorrected value of rs, dimensionless

V, initial layer volume in Bronswijk's model, cm3

Vs, current sample volume for drying, cm3

Vso, initial volume of the clay paste sample, cm3

, specific volume of a soil matrix without cracks, cm3 g–1

o, initial specific volume of a soil matrix, cm3 g–1

cr.l, specific crack volume in a soil layer, cm3 g–1

cr.s, specific volume of (macro)cracks in a soil sample, cm3 g–1

l, specific volume of the soil layer with cracks, cm3 g–1

lz, specific volume of the soil layer with cracks at w = 0, cm3 g–1

s, specific volume of soil sample with cracks, cm3 g–1

sz, specific volume of the oven-dried soil sample with cracks, cm3 g–1

z, specific volume of the oven-dried soil matrix without cracks, cm3 g–1

v, relative volume of drying clay without cracks, dimensionless

vs, relative volume of clay particles at the liquid limit, dimensionless

vz, relative clay volume in the oven-dried state, dimensionless

w, soil water content, g g–1

wo, initial water content of a clay sample close to the liquid limit, g g–1

wL, liquid limit of the clay paste, g g–1

x, lateral size of a cube of initial z3 volume in Bronswijk's model, cm

y, lateral size of a cube of initial z3 volume in Bronswijk's model, cm

z, initial soil layer thickness in Bronswijk's model, cm

{Delta}h, height decrease of the clay sample at drying, cm

{Delta}V, layer volume decrease for drying in Bronswijk's model, cm3

{Delta}Vcr, contribution of the total crack volume to {Delta}V, cm3

{Delta}Vlat, contribution of the lateral deformation of the solid sample to {Delta}V, cm3

{Delta}Vsub, contribution of the layer subsidence to {Delta}V, cm3

{Delta}z, decrease in z for drying in Bronswijk's model, cm–3

{delta}M, standard deviation of the averaged M value, dimensionless

{delta}rs, standard deviation of the averaged rs value, dimensionless

{delta}r's, standard deviation of the averaged r's value, dimensionless

{delta}l, standard deviation of the averaged l value, cm3 g–1

{delta}s, standard deviation of the averaged s value, cm3 g–1

{delta}w, standard deviation of the averaged w value, g g–1

{zeta}, relative water content of the clay (w/wL), dimensionless

{zeta}n, minimum clay water content in the normal shrinkage area, dimensionless

{zeta}z, shrinkage limit of the clay in terms of {zeta}, dimensionless

{rho}s, clay particle density, g cm–3

{rho}w, density of water, g cm–3


   ACKNOWLEDGMENTS
 
The research was supported in part by the Technion Water Research Institute. The constructive criticism of the reviewers is gratefully acknowledged.

Received for publication June 10, 2003.


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