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Soil Physics

The Shrinkage Geometry Factor of a Soil Layer

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

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

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 REFERENCES

 
The shrinkage geometry factor connects the changes of subsidence and the total crack volume at the shrinkage of a soil layer matrix. An available approach for estimating the factor does not account for the crack volume in samples, the stretching of a shrinking soil layer, and a possible change of the factor with water content. A recent new approach for estimation of the factor enables one to account for the factor's dependence on water content and the impact of cracks in a sample. The present work also accounts for the stretching of a shrinking soil layer. Corrected value of the factor is described in terms of the shrinkage curves of: a layer including unconnected solid cubes from the available approach, a stretched layer with cracks, a sample with cracks, and the soil matrix without cracks. This correction leads to changes in the soil subsidence and total crack volume compared with those obtained from the available approach. Two known physical conditions are used: the specific soil matrix volume without cracks is the same for the sample and the layer; and, the matrix deformations of an unlimited shrinking soil layer are longitudinal those of a thin quasi-elastic plate. Two available experimental examples are considered. The former relates to clay paste samples. The latter relates to an aggregated clay soil in situ and in samples. The geometry factor from the available approach, and that obtained after correction, differ significantly. The corrected factor also shows the appreciable change with water content.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 REFERENCES

 
THE GEOMETRICAL ASPECTS of hydraulic properties and water flow in swell-shrink soils are closely connected to crack network geometry and anisotropy of shrink-swell deformations in the soil matrix. An available approach to crack network geometry as crack width, depth, spacing, and volume distributions (Chertkov and Ravina, 1998, 1999a, 1999b, 1999c; Chertkov, 2000a) has already been applied to estimate the contribution of capillary interaggregate and interblock cracks into the hydraulic conductivity of swelling soils (Chertkov and Ravina, 2001, 2002). Recently, the approach was applied to explain the microrelief origin of a heavy clay soil surface (Chertkov, 2005). The effect of shrink-swell anisotropy was also considered to generalize flow equations in the case of the axially symmetric two-dimensional deformation of shrink-swell soil samples without cracks (Garnier et al., 1997a, 1997b).

A general feature of all the above works is the important role of the shrinkage geometry factor, r_s (Bronswijk, 1988, 1989, 1990, 1991a, 1991b) in considering the effects of both the crack network geometry and shrink-swell anisotropy. In addition, the r_s factor is used for experimental estimation of the total crack volume (Baer and Anderson, 1997).

Important applications of the r_s concept are based on Bronswijk's known presentation (Bronswijk, 1988, 1989, 1991a, 1991b) and measurement method (Bronswijk, 1990) of the r_s value. According to this presentation, the shrinkage geometry factor determines the relation between the variations of soil subsidence (z) and those of the matrix volume (V) of a soil layer in field conditions to be

[1]

where z and V are the thickness and matrix volume, respectively, of a soil layer at saturation (i.e., the maximum possible water content). According to the measurement method, the r_s value only relates to the r_s factor after oven drying. Following Bronswijk (1988)(1989, 1990, 1991a, 1991b), the value r_s = 3 was used by Baer and Anderson (1997) and Chertkov and Ravina (1998) as applied to layers of cracked clay soils. Garnier et al. (1997a)(1997b) used different r_s values, which were also constant with water content decrease in modeling water flow through swell-shrink cores that supposedly do not contain cracks.

Aside from the constant r_s value, the practical use of the exact Eq. [1] in the frame of Bronswijk's (1988)(1989, 1990, 1991a, 1991b) approach is based on a geometrical schematization of the crack system in a soil layer (Fig. 1) . Recently Chertkov et al. (2004) noted three implicit assumptions of Bronswijk's schematization and formulated them explicitly [below Assumption 1 is given in an equivalent but more visual form than in Chertkov et al. (2004)].

• – Assumption 1: Stretching a shrinking soil layer does not influence the soil subsidence and layer crack volume;
  
• – Assumption 2: Cracks do not appear and develop in drying soil samples;
  
• – Assumption 3: The r_s factor does not depend on soil moisture.
  

View larger version (15K): [in this window] [in a new window]   Fig. 1. Scheme of correspondence between a real layer and a model from Bronswijk (1988)(1989, 1990, 1991a, 1991b) (vertical cross-sections). (a) The real connected layer of thickness z with distributed cracks. Initial saturated state at the gravimetric water content w = wo. (b) Model layer of disconnected cubes of size z without cracks at w = wo. (c) Real subsidence z of the layer and the same cracks after drying to w < wo and opening (openings are symbolized by thicker lines; additional developing cracks are not shown). (d) Model layer after drying to w < wo. All cracks are concentrated as gaps (shaded strips) between shrinking parallelepipeds of horizontal size x' without cracks. Subsidence of the disconnected parallelepipeds, z' < z.

Chertkov et al. (2004) emphasized that the correcting M factor does not compensate for an inaccuracy of the r_s value in Bronswijk's approximation when using Assumption 1 and this inaccuracy should be addressed in a separate paper. In fact, Assumptions 1 and 2 are incorporated together into Bronswijk's approximation. One should therefore consider the corresponding correction that would compensate for the inaccuracy of the r_s value in Bronswijk's approximation because of the simultaneous use of Assumptions 1 and 2. The major objective of this work is namely to address this correction and thereby to find the totally corrected r_s value. Notation is summarized in Appendix 1.

Before proceeding to the main exposition, it is worth specifying the links between the concepts of cracking and of porosity in clay soils (clay content > 40%) and that we imply following Bronswijk (1988)(1989, 1990, 1991a, 1991b) and Hallaire (1984). The scale of clay soil aggregates or interaggregate (so-called structural) pores is simultaneously the scale of the smallest macrocracks (or simply cracks). We cannot distinguish between these smallest cracks and the interaggregate pores from the viewpoint of volume shrinkage. They coincide, and we say about the interaggregate cracks. Large subvertical cracks in a soil, small interaggregate cracks of different orientations, and all cracks of intermediate size, relate to one category of shrinkage cracks with varying volume. The r_s factor by definition (Eq. [1] and Bronswijk 1988, 1989, 1990, 1991a, 1991b) relates namely to the total volume of all these cracks in a clay soil layer or sample (r_s in these cases is different, see Chertkov et al. [2004] and below). In turn, the volume V of a clay soil matrix and its variation, V, in Eq. [1] are understood to be the summary volume of the intraaggregate matrix of all aggregates and the volume variation, respectively. The intraaggregate matrix includes solids (a network of clay particles embracing silt and sand grains), interparticle (or matric) pores, and possible microcracks. Scales of clay particles, matric pores, and the microcracks coincide by order of magnitude. The contribution of the microcracks to soil matrix volume is negligible for clay soils with clay content >40% by weight (Fiès and Bruand, 1990, 1998) and that we use in the following. Thus, the volume variation and shrinkage curve of the soil matrix are only determined by matric porosity.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 REFERENCES

 
Presentation of the Totally Corrected Shrinkage Geometry Factor
The purpose of this subsection is to express the r_s value for a connected soil layer with cracks in field conditions when Assumptions 1 and 2 are not fulfilled, through the r_s^' value in Bronswijk's approximation that accepts Assumptions 1 and 2. The derivation of the totally corrected r_s value is based on Chertkov et al.'s (2004) main results that are described by the titles of Appendices 2 and 3 and are briefly expressed by Eq. [A2.1], [A2.2], [A3.1], and [A3.2] from those appendices.

Figure 2 visually shows the difference among the shrinkage curve of the soil matrix without cracks [; _o is an initial value]; the shrinkage curve of the soil layer including the matrix and cracks in Bronswijk's approximation ; the true shrinkage curve of the soil layer including the matrix and cracks ; and the shrinkage curve of the soil sample with cracks . The simultaneous replacement, _s, _l^'_l, and r_sr^'_s in exact Eq. [A2.1] (that is equivalent to Eq. [1]) leads to the corresponding equation in Bronswijk's approximation,

[2]

Violation of Assumptions 1 and 2 in real conditions means that _s, ^'_l, and r^'_s in Eq. [2] differ from , _l, and r_s in Eq. [A2.1] at a given water content. Equations [A2.1], [A2.2], and [2] lead to the presentation of the totally corrected r_s value as

[3]

where

[4]

The correcting M factor (Chertkov et al., 2004) accounts for the violation of Assumption 2. The correcting L factor accounts for the violation of Assumption 1. Below the r_s values that only account for the violation of Assumption 2 (Eq. [A3.2]) will be designated as r_sM(w). Thus, all the corrected r_s values calculated by Chertkov et al. (2004) and designated as r_s values are in fact r_sM values. Note, however, that uncorrected r^'_s values from Chertkov et al. (2004) coincide with those from Eq. [2] and [3].

View larger version (24K): [in this window] [in a new window]   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 a shrinking and cracking soil layer. 2—shrinkage curve 'l(w) of a shrinking soil layer with cracks in Bronswijk's approximation (see Fig. 1b and d, disconnected layer). 3—shrinkage curve l(w) of a shrinking soil layer with cracks in approximation of the present work (connected layer). 4—shrinkage curve s(w) of a shrinking soil sample with cracks. 5—shrinkage curve (w) of a soil matrix without cracks. 6—1:1 theoretical line. 'lz, lz, sz, and z designate values of corresponding specific volumes after oven drying. In general z < sz, that is, the oven-dried sample contains cracks. AB—the specific volume of the layer subsidence in Bronswijk's approximation at a given w; AC—the true specific volume of the layer subsidence at a given w; BE—the specific volume of cracks in the soil layer in Bronswijk's approximation at a given w; CE—the true specific volume of cracks in the soil layer at a given w; DE—the specific volume of cracks in the sample with free boundaries at a given w.

Since _l^'_lo (see Fig. 2, Curves 1 to 3) the additional logarithmic multiplier, L from Eq. [4] is in the range 0 < L 1. Because the correcting M factor in Eq. [3] has values M 1 (Chertkov et al., 2004), the L and M factors, in part, mutually compensate each other.

The first way to experimentally estimate an L value using Eq. [4] is to combine separate core sample measurements to find the specific volume of a soil layer in Bronswijk's approximation and separate measurements in situ to estimate the true specific volume of a soil layer . We could only find the available data on similar measurements (simultaneously on samples and in situ) permitting one to obtain independent estimates of ^'_l and _l, in the work of Hallaire (1984). A corresponding illustrative example will be given below.

The second way is to suggest some sound relation between ^'_l and _l and use the relation to exclude _l or ^'_l from Eq. [4]. The following theoretical subsection is devoted to the derivation of such a simple relation based on two assumptions that have repeatedly and successfully been applied in soil science literature. One can find ^'_l from core sample measurements and then _l using the relation (a corresponding illustrative example with data from Chertkov et al. [2004] will be given below), or one can find _l from in situ measurements and then ^'_l using the relation (this way is not illustrated because of the lack of data).

The Relation between the True Specific Volume of a Soil Layer and Specific Volume in Bronswijk's Approximation
In this subsection we graphically introduce the soil matrix displacements that are needed to derive the relation between _l and ^'_l. Then we accept two assumptions known in literature, which lead to the relation. After that we give the relation (Eq. [5]) and its connection to Poisson's ratio of the soil matrix. The interested reader can find all details of the derivation in Appendix 4.

Figures 1 and 3 schematically illustrate the vertical and horizontal displacements in the matrix of a soil layer and cube with a water content decrease. Because of the tensile stresses the layer subsidence, z is no less than the subsidence z' of an isolated soil cube (Chertkov et al., 2004), that is, z z' (Fig. 1). For the same reason, after shrinkage, the horizontal size x of a stretched layer matrix without cracks (per one initial imaginary cube of size z) is no less than the horizontal size x' of an isolated cube (Chertkov et al., 2004), that is, x x' (Fig. 3). Figure 4 visually shows these results.

View larger version (13K): [in this window] [in a new window]   Fig. 3. Scheme of horizontal shrinkage deformation (horizontal cross-sections) at the drying of an isolated cube (Bronswijk's approximation) and a cube that is part of a real connected layer and only mentally outlined in it. In Fig. 3b, c, and d the shrinkage of a soil matrix in the horizontal plane is considered to be isotropic. a. The horizontal basis at w = wo. b. The basis of the isolated cube without cracks and with free boundaries after shrinkage at w < wo. c. The basis of the mentally separated cube with fixed boundaries after shrinkage at w < wo. Internal tensile stresses developing at layer shrinkage lead to cracking (black strips). d. The area of the stretched soil matrix in Fig. 3c after the mental extraction of a crack area (area of black strips). The size of the stretched-matrix area, x > x'.

View larger version (16K): [in this window] [in a new window]   Fig. 4. Subsidence and horizontal deformation of the matrix of a shrinking soil layer with cracks compared with those of the matrix of an isolated shrinking soil cube without cracks (vertical cross-section).

The second assumption is the elastic model. The concepts from the theory of elasticity apply to soil only in an approximate way. However, many results of the consideration of shrinkage crack development in clay soils, especially in water saturated states, in a number of works (Haberfield and Johnston, 1990; Morris et al., 1992; Murdoch, 1993; Harison et al., 1994; Lima and Grismer, 1994; Konrad and Ayad, 1997a, 1997b; Ayad et al., 1997; Chertkov, 2002) in the frame of this simplest model are quite reasonable and give a sufficiently sound basis. Following these works we also assume the model in this work. That is, the deformations under the action of tensile stresses in the matrix of an unlimited shrinking soil layer (field conditions) are considered to be longitudinal deformations of a thin elastic plate.

The final relation between _l and ^'_l is as follows (see Appendix 4):

[5]

where

[6]

and

[7]

where is Poisson's ratio of the soil matrix.

Replacing _l(w) in Eq. [4] from Eq. [5] leads to the final expression for the value of the correcting L factor,

[8]

Thus, knowing the shrinkage curves (w), _s(w), and ^'_l(w) (Fig. 2) one can estimate r^'_s(w) in Bronswijk's approximation from Eq. [2], M(w) from Eq. [A2.2], L(w) from Eq. [8], and the corrected r_s(w) value from Eq. [3], if in Eq. [8] the evolution of the parameter with water content is also known for a given soil. As indicated above the parameter is a function of Poisson's ratio () of the soil matrix (see Eq. [6] and [7]).

The same two above assumptions eventually allow one to express the parameter through the sample measurement data. Detailed derivation of the expression can be found in Appendix 5. Here we only give the final result as

[9]

where

[10]

In Eq. [9] and [10] d_o is the initial cylindrical-sample diameter; d_m is an equivalent diameter of the soil matrix without cracks in a sample; h(w) is a current sample height; and (w) is the specific volume of the intraaggregate soil matrix that can be measured as the shrinkage curve of soil aggregates (e.g., Bronswijk and Evers-Vermeer, 1990). Possibility of the theoretical prediction of (w) for clay soils (clay content > 40% by weight) will be briefly considered below in illustrating the approach. Using values one can estimate the corresponding µ (Eq. [6]), (Eq. [6] and [7]), _l (Eq. [5]), L (Eq. [8]), and r_s (Eq. [3]) values.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 REFERENCES

 
To illustrate the approach for estimating the totally corrected r_s value we use the published data of a clay paste (Chertkov et al., 2004) and aggregated clay soil (Hallaire, 1984).

Additional Analysis of Data from Chertkov et al. (2004)
In this subsection we indicate parameters that were measured by Chertkov et al. (2004) and values that were estimated by these authors from the data. After that we consider an additional use of the parameter data and found values to estimate and _l values that were introduced above.

The data relate to clay (>95% of montmorillonite) extracted by standard methods from samples of the 0- to 30-cm layer of soil in Sarid, Israel. The clay paste water saturated nearly to the liquid limit was placed in small cylindrical containers of approximately 1.5-cm height and approximately 3.5-cm diam. Three parameters of the clay itself and three parameters of each clay sample were measured. The former included the clay particle density, _s, the specific volume of oven-dried clay matrix, _z, and the liquid limit, w_L (Table 2 from Chertkov et al., 2004). The latter include the diameter, d, height, h, and weight, m of a sample (Table 3 from Chertkov et al., 2004). The measurements were conducted once a day for 8 d with subsequent oven drying for four samples.

The clay properties, _s, _z, and w_L enabled the calculation of the specific volume of a clay matrix as a function of w from Chertkov (2000b)(2003). This dependence is reproduced in Fig. 5 as the solid line. It is worth noting that Chertkov's (2000b)(2003) model relating to pure-clay pastes fits pretty well with data from Bruand and Prost (1987) based on a clay-silt-sand mixture of high clay content, but not pure clay [see Chertkov (2003)(p.78)]. Except for that it can be shown that Chertkov's (2000b)(2003) model, after natural and simple generalization by introducing the content of non-clay solids as an additional parameter, describes (without fitting) the shrinkage curve data of aggregates relating to 21 clay soils (with clay content > 40% by weight) from Bronswijk and Evers-Vermeer (1990). In fact, the model can have rather broader applicability for (w) prediction than only for pure clay.

View larger version (26K): [in this window] [in a new window]   Fig. 5. The experimental specific volumes, 'l (asterisks) and l (squares), of an unlimited clay layer including cracks in Bronswijk's approximation and corrected accounting for violation of Assumption 1, respectively; the experimental specific volume, s (circles) of clay samples including cracks, and the specific volume (solid line) of a clay matrix without cracks, predicted from Chertkov (2000b)(2003) for the clay of the 0- to 30-cm depths of Sarid soil, Israel. The specific volumes 'l, s, and are from Chertkov et al. (2004). The maximum standard deviations of all the experimental points are less than 0.01 to 0.02 cm3 g–1. The inclined straight line is 1:1 theoretical one. Figures near the experimental points correspond to the measurement numbers in Table 1.

View this table: [in this window] [in a new window]   Table 1. Experimental estimates averaged by the four samples and standard deviations of the gravimetric water content (w), the rs factor value in Bronswijk's approximation , the correcting M factor, and the corrected in part rs factor value (rsM) (w, r's, M, rsM from Chertkov et al. (2004)) as well as the additional correcting factor (L), total correcting factor (ML), and the corrected rs factor value for the drying clay of the 0- to 30-cm layer of soil in Sarid, Israel.

The experimental data on m_d[ = m(0)], h(w), and d_o[= d(0)] as well as the values of (w) from Chertkov et al. (2004) permit us to additionally estimate the (w) function from Eq. [9] and [10]. Then using the values and the specific volume ^'_l we estimated the true specific volume of the soil layer, _l (Eq. [5]), the correcting L factor (Eq. [8]), and the totally corrected r_s value (Eq. [3]).

Data from Hallaire (1984)
In this subsection, we describe Hallaire's data as applied to the following analysis in the frame of the model under consideration.

The data relate to an aggregated clay soil. Clay content, mainly montmorillonite and chlorite, is 52 to 56% by weight. Saturated undisturbed core samples (15-cm diam., 7.2 cm height) were collected during the winter in the depth range from 20 to 120 cm. In the course of laboratory measurements, the water content of the samples decreased from w 0.3 g g^–1 (the field capacity). Figure 6 reproduces the data of three shrinkage curves from Hallaire's (1984) Fig. 2 in coordinates of void ratio versus gravimetric water content. The layer void ratio was measured by the vertical shrinkage of core samples, that is, in Bronswijk's approximation in terms of Chertkov et al. (2004). The sample void ratio (e_s corresponding to _s) was measured by the vertical and diameter shrinkage of core samples. The void ratio of aggregates (e corresponding to ) was also measured.

View larger version (26K): [in this window] [in a new window]   Fig. 6. Data on the shrinkage curves of an aggregated clay soil from Hallaire (1984). The el' values (asterisks) give the layer void ratio in Bronswijk's approximation. The es values (circles) give the sample void ratio. The e values (dots) give the aggregate void ratio. The e1 values (squares) give the true layer void ratio. The el values (solid line) give the least squares approximation (and extrapolation) of the square points that were used in the present work. Goodness of fit of the approximation is r2 = 0.975.

For numerical estimates we can directly use the experimental points e_l^' (asterisks), e_s (circles), and e (dots) in Fig. 6. The set of experimental e_l points (squares) only differ from the e_l^', e_s, and e point sets by the number of points and larger standard deviations. For this reason we need some fitted curve to approximate e_l data. Using the ten experimental e_l points at w < 0.26 g g^–1 (Fig. 6, squares) when e_l e^'_l e_s and the least-squares criterion, we found a squared e_l approximation and extrapolation to small water content values to be

[11]

Goodness of fit of the approximation is r² = 0.975. The e_l approximation meets two conditions: de_l/dw_|w=0 = 0 and at w = 0.262 g g^–1 e_l = 0.99 – the experimental value. The e_l approximation from Eq. [11] is shown in Fig. 6 by a solid line. This approximation describes real soil properties to the same extent, as do the experimental e_l points.

Hallaire (1984) noted that at maximum water content the interaggregate-crack void ratio, 0. We estimated the corresponding w and e values (Fig. 6) continuing the e^'_l and e trend lines up to their intersection at w_o 0.335 g g^–1 and e_o 1.07. A possible inaccuracy of this point that is connected to the _o value is very slightly reflected in the following estimates.

In general, aggregate water content and soil water content are different. However, as judged by the data presentation in Hallaire's (1984) Fig. 2 (sample measurements of e^'_l and e_s, aggregate measurements of e) and Hallaire's (1984) Fig. 5 (in situ measurements of e_l using reference marks in the ground), he implicitly accepted the approximate coincidence of the aggregate and field water contents. A possible reason for that are the sufficiently small soil, core, and aggregate water content that are below the field capacity. Thus, in the case of these particular data we, following Hallaire (1984), also accepted the equality of aggregate and soil water content. However, in general, the connection between field, sample, and aggregate water content is essential in estimating the r_s value in the frame of the approach that uses a number of different shrinkage curves (as in Fig. 2). This connection should be addressed in the future.

Analysis of Hallaire's (1984) Data
The specific volume and a void ratio e are connected by = /_s where _s is the density of solids. Therefore r^'_s, M, L, ML, and r_s factors are obtained from the data in Fig. 6 using the same Eq. [2], [A2.2], [4], and [A2.1] after replacements: _o (e_o + 1), (e + 1), _s (e_s + 1), _l (e_l + 1), and ^'_l . In addition, r_sM = Mr_s^'.

Similarly, we have = ^1/2 from Eq. [5]. Note that, unlike the above case of a clay paste, for the aggregated clay soil is immediately obtained from e_l and e_l^' data and does not serve to find _l through ^'_l. Finally, is found from by Eq. [6] and [7].

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 REFERENCES

 
Results Flowing out of Chertkov et al.'s (2004) Data
In this subsection we formulate and discuss relevant results based on the experimental estimates and errors of (for a sufficiently exact and sound approach for calculation based on two known assumptions see Appendices 4 and 5).

Figure 7 shows the values (squares) corresponding to the measured gravimentic water contents of the clay and gives the quantitative illustration of the qualitative (w) dependence in Fig. A5.1.

View larger version (23K): [in this window] [in a new window]   Fig. 7. The experimental (squares) values in clay paste samples for the drying clay of a 0- to 30-cm layer of Sarid soil. The maximum standard deviations of all the experimental points are less than 0.01. Figures near experimental points correspond to the measurement numbers in Table 1.

1. Comparison in Fig. 5 between _l(w) (squares) and ^'_l(w) (asterisks) shows that the difference, (^'_l – _l) is beyond the limits of standard deviations , _l is appreciably less than ^'_l, and _l/^'_l 0.8. That is, Assumption 1 is violated.
   
2. Violation of Assumption 1 is also demonstrated by the difference between unity and the correcting L factor (Table 1). The (1 – L) difference is beyond the limits of standard deviations (L).
   
3. The difference between unity and the total correcting factor, ML (Table 1) demonstrates, along with the difference (1– M) (Chertkov et al., 2004), the violation of both Assumptions 1 and 2. The (1– ML) difference is also beyond the limits of standard deviations (ML).
   
4. Like r_s^' and r_sM factors (Chertkov et al., 2004), the totally corrected r_s factor (Table 1) changes with water content in the area 0.15–0.20 g g^–1 w w_L beyond the limits of standard deviations (r_s). That is, Assumption 3 is also violated.
   
5. The r_s^' and r_sM values (Chertkov et al., 2004) essentially exceed the final r_s values (Table 1). The latter are appreciably closer to unity. This is in agreement with the fact that we discuss the shrinkage of a pure clay paste that is in some measure close to so-called unripened soils (Rijniersce, 1983).
   

The corresponding experimental dependence of Poisson's ratio versus water content, (w) from Eq. [6] and [7] looks quite similar to that in Fig. 7 (squares), but with ordinate value range from 0.45 to 0.5. According to available data (Briones and Uehara, 1977; Haberfield and Johnston, 1990; Murdoch, 1993; Harison, 1994; Ayad et al., 1997; Lade, 2001) the Poisson's ratio of the clays changes with water content in saturated states only weakly, if at all, and decreases with water content decrease in the unsaturated states from 0.5 to 0.3. Thus, the available data in general correspond with the obtained data on (w).

Results Flowing out of Hallaire's (1984) Data
The aim of this subsection is to formulate and discuss relevant results based on immediate data on e_l^', e_l, e_s, and e from Fig. 6.

The r_s^', r_sM, and r_s values that were estimated using data from Fig. 6 are presented in Fig. 8 . The step between the values of the gravimetric water content was w = 0.01 g g^–1. The corresponding M, L, and ML estimates are shown in Fig. 9 . The obtained dependences are not quite smooth because numerical values e_l^', e_l, e_s, and e were directly taken from the trend lines in Fig. 6. However, this roughness does not influence the major results for the aggregated soil following Fig. 8 and 9.

1. At a given water content in the range 0.05 < w < 0.33 g g^–1 the r_s^', r_sM, and r_s values essentially differ confirming the violation of Assumptions 1 and 2.
   
2. The r_s^', r_sM, and r_s values essentially vary with water content decrease confirming the violation of Assumption 3.
   
3. In the overwhelming part of the water content range under consideration the r_s^', r_sM, and especially r_s values essentially differ from three. That is, in the broad area of water contents, the shrinkage of Hallaire's (1984) aggregated soil is anisotropic, even in Bronswijk's approximation as judged by r_s^'(w) behavior.
   
4. At w < 0.1 g g^–1 the r_s^' values, that is, r_s in Bronswijk's approximation, are equal to 2.84 and, hence, relatively close to Bronswijk's (1990) experimental estimate, r_s^' = 3, for his aggregated-clay-soil samples after oven drying.
   
5. With water content decrease the r_s factor increases to approximately 2.9 and then smoothly decreases to approximately 2. Such behavior of the r_s(w) dependence qualitatively corresponds to two stages in the cracking process near the shrinking surface that were observed by Hallaire (1984) (see his Fig. 1): "At first, thin cracks (less than 5 mm wide) appeared, with about 3 cm spacing. Then some of these cracks opened wider (to more than 1 cm), with about 20 cm spacing while the remaining cracks were partially or even totally closed." One can assume that the rapid growth of r_s to 2.9 at drying (Fig. 8) corresponds to the rapid increase of the total crack volume at the first stage with the appearance of a dense network of small and thin cracks. Further lowering of r_s at drying to approximately 2 (Fig. 8) corresponds to a gradual decrease of the total crack volume at the second stage to an approximately stable value, in spite of the appearance of large and wide, but more seldom cracks, because of the closing of many small cracks.
   
6. The r_sM = Mr_s^' factor relates to samples with possible cracks. Therefore, the r_sM maximum, approximately 2.9, shows that many small cracks with the appreciable total volume appear immediately after the start of drying, not only in the soil layer, but even in the undisturbed aggregated samples, in spite of their free boundaries, relatively small sizes, and the shrinking in their height and diameter. Some recession of r_sM with drying is connected with the partial closing of the cracks in the samples. The subsequent r_sM growth with drying is connected with increasing the role of lateral compared with vertical shrinking of the samples.
   

View larger version (24K): [in this window] [in a new window]   Fig. 8. Three different approximations of the rs factor for Hallaire's (1984) aggregated clay soil as functions of water content. The rs' values (diamonds) of the rs factor for the soil sample or layer in Bronswijk's approximation; the rsM values (squares) of the rs factor corrected in part for the soil layer; and the totally corrected rs factor values (circles) for the soil layer.

View larger version (24K): [in this window] [in a new window]   Fig. 9. Three correcting factors, M, L, and ML for Hallaire's (1984) aggregated clay soil as functions of water content.

Based on Fig. 9, one can note that the essential variation of M, L, and ML factors with water content and their differences of unity in the overwhelming part of the w range also confirm violation of Assumptions 1 to 3 for the aggregated clay soil under consideration. In physical terms the difference between M and unity means the essential impact of cracks in the samples on the estimate of the r_sM factor relating to the samples and determining the total crack volume inside them. The difference between L and ML and unity means the essential impact of soil stretching in a layer, under the action of shrinkage tensile stresses, on the estimate of the r_s factor relating to the layer and determining the total crack volume in it.

Finally, Fig. 10 shows a slight change of Poisson's ratio for Hallaire's (1984) aggregated clay soil with water content decrease as in the above case of the clay paste where 0.45 < 0.5. The small variation of Poisson's ratio is in agreement with available data from the abovementioned works (Briones and Uehara, 1977; Haberfield and Johnston, 1990; Murdoch, 1993; Harison et al., 1994; Ayad et al., 1997; Lade, 2001). Dependence (w) (immediately found from experimental e_l^' and e_l) for the clay soil looks quite similar to that in Fig. 10 with the ordinate range being from 0.96 to 1.

View larger version (22K): [in this window] [in a new window]   Fig. 10. Poisson's ratio of Hallaire's (1984) aggregated clay soil as a function of water content.

1. For core samples the dependencies ^'_l(w), _s(w), and (w) (Fig. 2) are sufficient to estimate r_s^', M, and r_sM. Then, using the M, and r_sM values we estimate three separate contributions to the matrix volume change of the sample—the contribution of cracks developing in the cores, that of core height decrease, and that of core diameter decrease. For a cracked soil layer, to estimate the exact r_s value and exact contributions of cracks and subsidence into the matrix volume change, one should know either the (w) dependence and in situ _l(w) (Fig. 2) or the ^'_l(w) and _s(w) dependencies for samples (along with (w)) (Fig. 2) as well as the parameter as a function of water content. The parameter is estimated as suggested above (for details see Appendices 4 and 5).
   
2. In situ measurements can be time-consuming and may not have a high degree of accuracy. In such cases for estimating the true r_s value of a soil layer, the above approach permits one to use the measurements on core samples and then to find correcting L and M factors as described above.
   
3. The difference between r_s values relating to a soil layer, r_sM values relating to a soil sample, and r_s^' values in Bronswijk's approximation (Table 1 and Fig. 8) is connected with the restrained conditions of the shrinking layer unlike the soil sample, but not with scale effects. The difference leads to essentially different total crack volumes in the layer and undisturbed samples of the same soil (per unit of oven dried mass). For this reason, the results of hydraulic properties and flow measurements for swell-shrink soils on samples, being applied to soil layers (field conditions), can lead to inaccurate implications. However, as noted above, the correct r_s factor for a soil layer can be estimated from sample measurements using the ML multiplicative correction.
   

Differences Between the Two above Experimental Examples
Besides using an aggregated clay soil unlike a pure-clay paste, Hallaire's (1984) data differ from Chertkov et al.'s (2004) data in two relations. First, Hallaire's (1984) data on the true void ratio of a soil layer, e_l (connected to _l) were obtained based on in situ measurements. Unlike that, Chertkov et al.'s (2004) data do not immediately contain the values of the true specific volume, _l of a layer. We estimated the _l values for a clay paste layer using the parameter that was in turn estimated based on Chertkov et al.'s (2004) data of sample height and diameter evolution with drying. Second, the data on an aggregate void ratio, e (connected to ) from Hallaire (1984) were also immediately measured using the aggregates. Unlike Hallaire (1984), Chertkov et al. (2004) estimated using the data of clay paste properties (_s, _z, m_L) and a model from Chertkov (2000b)(2003).

Estimating Poisson's Ratio
Comparison between the estimates (Fig. 7, square points) that were obtained for the clay paste using the quasi-elastic model and those (0.96 < 1) obtained for the aggregated clay soil directly from the e_l and e_l^' data (from Eq. [5] ² = /), speaks in favor of the feasibility of the quasi-elastic model of a soil layer as applied to the connection between its true specific volume (_l) and the specific volume in Bronswijk's approximation (^'_l).

Poisson's ratio is directly connected to the parameter (Eq. [6] and [7]). For this reason the feasibility of the quasi-elastic model as well as similarity between estimates of Poisson's ratio for the clay paste (0.45 < 0.5) and in Fig. 10 for the aggregated clay soil, speaks in favor of the possibility of estimating the Poisson's ratio of a swell-shrink soil either from the _l/^'_l ratio as it was made for the aggregated clay soil or from the sample diameter and height measurements as was made for the clay paste. Thus, the approach for r_s estimation simultaneously gives a method for experimentally estimating Poisson's ratio for swelling soils at different moisture contents.

	SUMMARY AND CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION SUMMARY AND CONCLUSIONS APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 REFERENCES

 
The shrinkage of clay soil samples is accompanied by the appearance and development of cracks. Many images show evidence of that (see e.g., Hallaire's [1984] Fig. 4). The shrinkage of a clay soil layer in the field is accompanied by the appearance and development of tensile stresses and deformations because the layer is in restrained conditions. Both the crack volume in samples and stretching in a shrinking soil layer are not accounted for in the known approach (Bronswijk, 1988, 1989, 1990, 1991a, 1991b) for estimating the shrinkage geometry factor r_s that connects changes in subsidence and the total crack volume at the shrinkage of a soil layer matrix. The possible change of r_s with water content is also not accounted for. At the same time the r_s concept is used to describe the crack network geometry (Chertkov and Ravina, 1998, 1999a, 1999b, 1999c; Chertkov, 2000a, 2005) and contribution of capillary cracks into the hydraulic conductivity of swelling soils (Chertkov and Ravina, 2001, 2002), to estimate the soil crack volume (Baer and Anderson, 1997), and to model water flow in swell-shrink soils without cracks (Garnier et al., 1997a, 1997b). For these applications the accuracy of the r_s value is very important. Chertkov et al. (2004) recently suggested an approach to account for the impact of the water content on the r_s factor and crack presence in a sample on r_s estimation. In the frame of the approach the r_s factor as a function of water content is described by the different shrinkage curves of a soil: the shrinkage curve of a layer with cracks in Bronswijk's approximation, ^'_l(w), the shrinkage curve of a sample with cracks, _s(w), and the shrinkage curve of a soil matrix without cracks, (w).

In the present work, we consider correction of the r_s factor value accounting for the stretching of a shrinking soil layer. This correction leads to changes in the soil subsidence and crack volume compared with Bronswijk's approximation. To estimate the corrected r_s value, starting from the initial estimate, r_s^' (Bronswijk, 1990) we introduce, along with (w), _s(w), and ^'_l(w), the shrinkage curve of a stretched layer with cracks, _l(w), based on two known physical conditions:

1. following Bronswijk (1988)(1989, 1990, 1991a, 1991b) we also consider that the soil matrix volume without cracks (per unit mass of oven-dried soil) only depends on water content and not on stresses in the soil; that is, at a given water content the volume of a soil matrix without cracks is the same for a sample and a layer; and
   
2. following a number of authors (Haberfield and Johnston, 1990; Morris et al., 1992; Murdoch, 1993; Harison et al., 1994