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SOIL PHYSICS

The Reference Shrinkage Curve at Higher than Critical Soil Clay Content

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

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

	ABSTRACT

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

 
The objective of this work was to construct and validate a model that shows how the clay shrinkage curve, under the influence of a silt–sand admixture as well as an inter- and intraaggregate structure, is transformed to the soil shrinkage curve. To meet this objective, we investigated (i) the reference shrinkage curve, that is, one without cracks, because cracks lead to a multivalued shrinkage curve; (ii) the rigid superficial (interface) layer of aggregates, with changed pore-size range and distribution, compared with the intraaggregate matrix; and (iii) soils with sufficiently high clay content when large pores inside the intraaggregate clay (so-called lacunar pores) are nonexistent. The methodology is based on detail accounting for contributions of the interface aggregate layer and intraaggregate matrix to the soil volume and water content during shrinkage. The reference shrinkage curve is determined by six physical soil parameters: oven-dried specific volume; maximum swelling water content; mean solid density; soil clay content; oven-dried structural porosity; and the ratio of aggregate solid mass to solid mass of the intraaggregate matrix. Only the last parameter was fitted for lack of data and compared with an unfitted estimate. The model was validated using data for eight soils. The important new conclusion is that the mere existence of the rigid superficial aggregate layer leads to a soil reference shrinkage curve that is convex upward in the structural shrinkage area, unlike the shrinkage curve of the clay contributing to the soil. The model can have numerous applications.

	INTRODUCTION

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

 
Shrink–swell is inherent to practically any soil (Hillel, 1998). A soil contains clay, silt, and sand fractions as well as the usual small admixture of organic matter (e.g., Bronswijk and Evers-Vermeer, 1990). If the mostly small contribution of the organic matter to soil shrink–swell is neglected, the only shrink–swell component is clay. At the same time, the experimental shrinkage curve of a pure clay (Fig. 1a ; e.g., Tessier and Pédro, 1984) and that of a soil containing the clay (Fig. 1b; e.g., Braudeau et al., 2005) differ essentially both qualitatively and quantitatively.

1. Unlike pure clays, in the overwhelming part of the structural shrinkage area (Fig. 1b), that is, between the maximum swelling point, W_h (i.e., the saturation point or initial point of shrinkage) and the transition point to basic (or normal or proportional) shrinkage, W_s, the soil shrinkage curve is convex upward.
   
2. Unlike pure clays, the slope of the soil shrinkage curve in the structural shrinkage area is essentially less than unity. Although in a water content range within the limits of that area the slope can look to be approximately close to a constant, in the general case that varies with water content, and the shrinkage curve can even have the inflection point in the structural shrinkage area (see Fig. 1b, Curves 1 and 2; e.g., Braudeau et al., 2005).
   
3. In the basic shrinkage area (Fig. 1b), that is, between W_s and the air-entry point, W_n, the shrinkage curve slope is constant for a given soil. Unlike the different clays (when the slope is always equal to unity), however, for different soils the slope varies between zero and unity.
   
4. Unlike pure clays, the maximum swelling point, W_h, of a soil can be situated at the pseudo-saturation line (Fig. 1b; e.g., Braudeau et al., 2005).
   

View larger version (20K): [in this window] [in a new window]   Fig. 1. The qualitative look of (a) the clay and (b) soil shrinkage curves. The points Wz, Wn, and Wh are the shrinkage limit, air-entry point, and maximum swelling point, respectively, which differ for (a) and (b) cases; Ws is the final point of the structural shrinkage area; Wm* is the soil water content not reached if the maximum volume, Yh, is reached at Wh on the pseudo-saturation (dash-dot) line. If Wh = Wm*, the volume Yh is reached at the true saturation (dashed) line. Curves 1 and 2 are the observed variants of the shrinkage curve. The curve slope in the basic shrinkage area for clay is always unity and for soil can be less than unity.

Soil shrinkage is accompanied by cracking (Bronswijk, 1989). Knowledge of crack-size distribution and crack volume is of crucial importance for the adequate physical modeling of soil hydraulic properties (Chertkov and Ravina, 2001, 2002; Chertkov, 2000b). Cracking on different scales and the total crack volume depend, however, on shrinkage conditions. The latter are essentially affected by sampling, sample preparation, sample size, and the drying regime (Yule and Ritchie, 1980a, 1980b; McGarry and Daniels, 1987; Crescimanno and Provenzano, 1999; Braudeau et al., 1999; Cabidoche and Ruy, 2001). As a result, in general, the measured shrinkage curve in itself is not a single-valued characteristic of a soil. To dependably estimate the crack volume contribution to a measured shrinkage curve, one should be able to predict the shrinkage curve that is stipulated only by soil shrinkage without cracking. We will refer to such a curve as the reference shrinkage curve because any measured shrinkage curve of the same soil should show equal or greater volume at a given water content (within the limits of experimental errors). The difference between the measured and reference shrinkage curve gives the crack volume contribution at a given water content. An attempt to estimate soil crack volume using such an approach was recently undertaken as applied to the unlimited cracked soil layer as a sample (field conditions) (Chertkov et al., 2004; Chertkov, 2005b).

Returning to the abovementioned qualitative differences between the shrinkage curves of a soil and the clay contributing to it (Fig. 1), it should be emphasized that (i) the clay shrinkage curve in Fig. 1a does not contain the crack contribution (Tessier and Pédro, 1984; Chertkov, 2000a, 2003); and (ii) the reference shrinkage curve of a soil qualitatively has the same view as a measured shrinkage curve (Fig. 1b). The crack contribution can modify the reference shrinkage curve quantitatively, but not qualitatively. It follows that derivation of the reference shrinkage curve enables one not only to estimate the crack volume contribution to the measured shrinkage curve, but also to understand the origin of the qualitative differences between the shrinkage curves in Fig. 1a and 1b from the soil inter- and intraaggregate structures.

In general, in addition to the structural pores (associated with interaggregate space) and clay pores (associated with the space between clay particles), the soils contain the microcracks or lacunar pores inside intraaggregate clay (Fiès and Bruand, 1998). The lacunar pores inside the clay matrix can have sizes that essentially exceed those of usual clay pores and reach the sizes of intergrain spaces. Following Fiès and Bruand (1998), we refer to these relatively large pores (or microcracks) inside the intraaggregate clay matrix as lacunar pores (Fig. 2 ). These researchers observed lacunar pores in clay matrices entering the oven-dried artificial clay–silt–sand mixtures. At sufficiently small clay content (c), the lacunar pores are three dimensional, have an appreciable volume that changes with shrinkage, and form a network. As the clay content increases, the lacunar pore network gradually disappears, and only isolated (hidden) lacunar pores remain. At sufficiently high clay content (Fig. 2a), the lacunar pores have a crack-like shape and negligible volume, if any. It is obvious that, in this case of the practical lack of lacunar pores (Fig. 2a), the intraaggregate structure is relatively simple, and the derivation of the reference shrinkage curve should be simplified compared with the case in Fig. 2b.

View larger version (43K): [in this window] [in a new window]   Fig. 2. The illustrative scheme of the internal structure of aggregates at a clay content (a) c > c* and (b) c < c*, where c is the weight fraction of clay solids and c* is the critical soil clay content.

All the contemporary models of the soil shrinkage curve, despite all the differences between them, are based on the approximation of the experimental shrinkage curve data by some a priori taken mathematical expression (different for each different approach; Groenevelt and Bolt, 1972; Giraldez et al., 1983; Tariq and Durnford, 1993; Olsen and Haugen, 1998; Crescimanno and Provenzano, 1999; Braudeau et al., 1999, 2004; Groenevelt and Grant, 2001; Peng and Horn, 2005; Cornelis et al., 2006). Such expressions are not derived from considerations of the soil inter- and intraaggregate structures, but justified by fitting of their parameters (from three to 11 depending on the approach) to the experimental shrinkage curve data. The problem of these approaches, as applied to the derivation of the reference shrinkage curve, is not the fitting in itself, or the fitted parameters in themselves. The problem is that the obtained fitted approximations, by definition, contain the crack volume contribution that in an uncontrolled way enters the experimental data. Unlike approaches of this type, Braudeau et al. (1999, 2004, 2005) attempted to go from such the empirical approximation to the soil structure. That is, the approach of these researchers also included the crack volume effects (in an uncontrolled way) and is not suitable for the aims of the derivation of the reference shrinkage curve.

The objective of this work was to construct and validate a model that permits one to understand, on a fundamental level, how the shrinkage curve of a pure clay (Fig. 1a) under the influence of a silt–sand admixture as well as inter- and intraaggregate soil structures is transformed to the shrinkage curve of an aggregated soil (Fig. 1b), and what elementary physical parameters are necessary at the soil shrinkage curve prediction. As applied to this objective, one should stress the following reservations. We investigated (i) the development and validation of scientific understanding, but not immediate applications, for agriculture, environment, and structure engineering, although such the applications are inevitable and suggested (see below); (ii) the reference shrinkage curve, that is, without cracks, because cracks lead to a multivalued shrinkage curve; (iii) the new concepts of the intraaggregate soil structure (see below); and (iv) soils with sufficiently high clay content, c > c* (Fig. 2a) (the c* concept will be specified) that demonstrate a part of the abovementioned peculiarities of the reference shrinkage curve in Fig. 1b (the general case of soils with any clay content and all the abovementioned peculiarities of the reference shrinkage curve in Fig. 1b inherits a number of basic results of this work, but is beyond its scope and will be considered separately). Notation is summarized in the Appendix.

	THEORY

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

 
Conceptual Model
Critical Clay Content
Based on Fiès and Bruand's (1998) results, we make Assumption 1 about two soil groups with different intraaggregate structure depending on clay content. Group A is comprised of soils with such high clay content that (i) silt and sand grains inside aggregates do not come into contact at any possible water content (Fig. 2a); and (ii) clay fills in all the space between the grains at any water content (Fig. 2a). Group B is comprised of soils with such low clay content that (i) silt and sand grains inside aggregates come into contact at sufficiently low water content (at least at water content W = 0 kg kg^–1; Fig. 2b); and (ii) in the clay that fills in the intergrain space there are lacunar pores whose volume changes at shrinkage, at least at sufficiently low water content (Fig. 2b).

Assumption 1 implies the following quantitative aspect. For given characteristics of clay and silt–sand components, there is such a critical clay content value, c* that at a clay content c in the range c* < c 1, the corresponding soil relates to Group A (Fig. 2a), and at a clay content in the range 0 < c c* to Group B (Fig. 2b). In this work, we derived the expression for c* and compared its estimates with the available data. It is worth reiterating that Group A (Fig. 2a) is the only subject of this work.

Maximum Soil Swelling Point
The sizes of structural (interaggregate) pores cannot change independently of the aggregate sizes. In turn, clay inside aggregates is the only shrink–swell agent (neglecting organic matter). For these reasons, we make Assumption 2: that the shrinkage of the intraaggregate clay, aggregates, and the soil as a whole starts simultaneously at total soil water content, W_h (Fig. 1b), corresponding to the initial point of soil shrinkage or the point of maximum soil swelling.

Interface Layer
Intraaggregate clay pores remain saturated at soil shrinkage up to soil water content W_n (Fig. 1b). With that, their dewatering at soil water content in the structural shrinkage area, W_s < W < W_h, is accompanied by a soil shrinkage curve slope that is <1 (Fig. 1b). This means that at W_s < W < W_h, soil shrinkage is connected with water loss not only from the saturated clay pores (in this case, the slope would be 1), but also from some additional water source that is subjected to rather weaker shrinkage. Except for the weaker than clay shrinkage, the additional water source is dewatered simultaneously with clay pores. For this reason (from capillarity), the water source should include the water-filled pores that are similar in size to the clay pores, i.e., less than approximately 10 to 15 m. Hence, the suitability of structural pores for the role of the additional water source is doubtful.

The superficial layer of any medium should have changed properties compared with the internal material (e.g., Landau and Lifshitz, 1980). Such surface effects can differ in nature. An example of similar effects, as applied to soils, is aggregate coating effects (Gerke and Köhne, 2002). Another example is a superficial aggregate layer of relatively small thickness compared with the intraaggregate matrix (Fig. 2a). Such a layer includes the same clay, silt, and sand fractions as in the intraaggregate matrix (Fig. 2a), but nevertheless has changed shrinkage properties and a pore structure. Already in the force of its superficial situation and small thickness, the matrix of the layer should be rather more rigid with a decrease in water content. In general, the pore structure is characterized by porosity, pore-size range, and pore-size distribution. The pore structure of the changed superficial aggregate layer should differ from that of the intraaggregate matrix (Fig. 2a), at least by one of the indicated three elements. Based on the above considerations of the additional water source in the soil and of the rigid superficial layer of aggregates with changed pore structure, we accept Assumption 3: the additional water source (i) has pores that are, at least in part, similar in size to the clay pores of the intraaggregate matrix, (ii) simultaneously empties with the clay pores, and (iii) contains a sufficient quantity of water to explain the soil shrinkage curve slope in the structural shrinkage area (W_s W W_h, Fig. 1b). This is water that is contained in the pores of a thin rigid superficial layer of aggregates (Fig. 2a) with changed pore structure. The pores in this thin layer are an interface between clay pores inside the aggregates (Fig. 2a) and structural pores between aggregates, and connected with both. For this reason we will term these pores interface pores. It is worth noting that the interface pores genetically originate from intraaggregate clay pores at aggregate fragmentation (Fig. 2a). The changed pore structure of the interface layer will be considered below. The interface and structural pores are different kinds of pores. The former are associated with the modification of the clay pore structure in a surface layer; the latter are associated with interaggregate space. These pore spaces are in immediate contact, and differ from the intraaggregate pore space.

General "Road Map" of This Study
The total (gravimetric) water content of a soil, W = w' + , includes the contributions of an intraaggregate clay (w'; Fig. 2a) and changed interface clay (; Fig. 2a). Note that the presentation W = w' + essentially relies on Assumptions 2 and 3 above. Structural pores are empty at shrinkage because of their size. Both and W = w' + vary as single-valued functions of w' because the interface clay pores giving and those of intraaggregate clay giving w' (Fig. 2a) are at the same varying suction. Thus, the w' and W axes in Fig. 3 are connected through W = w' + (w'). The values of w', W, and vary in the ranges 0 w' w_h', 0 W W_h (Fig. 3), and 0 _h, where w_h', W_h, and _h correspond to the state of the maximum aggregate and soil swelling (Assumption 2).

View larger version (40K): [in this window] [in a new window]   Fig. 3. Qualitative view of the reference shrinkage curve, Y(W) (specific volume of soil vs. total gravimetric water content) and that of aggregates, Ua(W) (for two options 1 and 2) where the clay content c > the critical value c* as well as a number of auxiliary curves: the specific volme of soil vs. intraaggregate contribution to total water content,Y(w'); specific volume of aggregates vs. intraaggregate contribution to total water content, Ua(w'); specific volume of intraaggregate matrix vs. its gravimetric water content, U(w); and intraaggregate contribution to the specific volume of aggregates vs. intraaggregate contribution to total water content, U'(w'). The water contents Wz, Wn, Ws, and Wh and water contents wz', wn', ws', and wh' are the values that correspond to the shrinkage limit, endpoint of basic shrinkage, endpoint of structural shrinkage, and maximum swelling of the soil, respectively; Wm is the maximum soil water content; water contents wz, wn, and wh are the values that correspond to the shrinkage limit, endpoint of basic shrinkage, and maximum swelling of the intraaggregate matrix, respectively; is the interface contribution to the total water content; K is the ratio of the aggregate solid mass to the solid mass of intraaggregate matrix; and Ui and Us are the specific volumes of an interface layer and structural pores, respectively. The 1:1 line denotes the saturation line for both the w and W axes.

Values w_z', w_n', and w_h' (Fig. 3) represent the shrinkage limit, air-entry point, and maximum swelling point, respectively, of the aggregated soil in terms of the intraaggregate contribution w' (Fig. 2a) to W per unit mass of oven-dried soil. By definition of the w_s' point (Fig. 3; the end of structural shrinkage), at w' < w_s' the contribution of the interface layer water content, (Fig. 2a) to W becomes negligible compared with w'. That is, in the range 0 w' w_s' the total water content W = w', and the presentation of the reference shrinkage curve as U_a(w') and Y(w') in Fig. 3 quantitatively coincides with U_a(W) and Y(W) at 0 W W_s. Only in the w_s' < w' w_h' and W_s < W W_h ranges of the structural shrinkage do the two presentations, U_a(w') and U_a (W) (Curves 1 or 2, Fig. 3) or Y(w') and Y(W) (Fig. 3), differ. The position of w_s' = W_s is discussed below.

The U_a(w') and Y(w') curves (Fig. 3) have an uncustomary view, namely because [unlike U'(w')] they were presented using an unusual (for U_a and Y volumes) w' coordinate. The presentation in usual coordinates, U_a(W) and Y(W) [see Fig. 3 at W > W_s, the Y(W), U_a(W) (Curve 1), and U_a(W) (Curve 2) curves) is determined by a W(w') dependence that changes the scale along the water content axis compared with w'. For the transition w' W, one needs to know the (w') dependence. Thus, to estimate the reference shrinkage curve, Y(W) [or U_a(W)], one should consequently find the dependences (w'), W(w') = w' + (w'), and U'(w') as well as constant additions U_i and U_s that give U_a(w') = U'(w') + U_i and Y(w') = U_a(w') + U_s. Then, parametric equations W(w'), U_a(w') or W(w'), and Y(w') give the reference shrinkage curve U_a(W) or Y(W) (Fig. 3). The total initial water content of the rigid interface layer, _h = W_h – w_h' (Fig. 3) is a reason for the deflection of U_a(W) and Y(W) in the structural shrinkage area (Fig. 3, W_s < W < W_h) of U_a(w') and Y(w') (Fig. 3, w_s' < w' w_h'), because a part of the water goes out of the rigid interface layer (Fig. 2a), and the slope of U_a(W) and Y(W) increases up to the total emptying of the rigid interface porosity at W = W_s (Fig. 3).

The interface contribution, (w'), to the total water content, W, flows out of the pore-size distribution in the interface layer (Fig. 2a) and is considered below, along with the constant additions U_i and U_s to U'(w').

The U'(w') dependence in itself (Fig. 3) is not a real shrinkage curve, namely because it gives only one contribution (of the intraaggregate matrix; Fig. 2a) to the specific soil volume, Y or U_a (two others are U_i and U_s). This is the reason why the U'(w') curve is situated below the 1:1 saturation line in Fig. 3; however, U'(w') is directly and simply connected with the shrinkage curve of the intraaggregate matrix in itself, U(w) (Fig. 3) and U'(w') and U(w) only differ by normalization: U' and w' are the specific volume and water content (contributions) of the intraaggregate matrix per unit mass of the oven-dried soil as a whole (including the solid mass of the interface layer; see Fig. 2a); U and w, however, are the specific volume and water content of the same intraaggregate matrix per unit mass of the oven-dried matrix itself (Fig. 2a). It follows that the water content values along w and w' axes in Fig. 3 are connected as w' = w/K and corresponding U and U' values as U' = U/K, where K is the ratio of the aggregate solid mass to the solid mass of the intraaggregate matrix (i.e., the solid mass of aggregates without the interface layer, see Fig. 2a). Hence, the relation between U'(w') and U(w) is

[1]

Thus, the auxiliary curve, U'(w') is expressed through the shrinkage curve U(w) and K ratio. Both are considered below. In connection with the situation of the U(w) curve in Fig. 3, it is important to emphasize that the w_h = Kw_h' and W_h values as well as U_h = U(w_h) and U_ah = U_a(W_h) values coincide (see Fig. 3) because the water contents and specific volumes of the two aggregate parts—the rigid interface layer and intraaggregate matrix (Fig. 2a)—coincide at the water saturation of the soil (W = W_h = w_h).

Shrinkage Curve of the Intraaggregate Matrix
Summary of a Clay Matrix Shrinkage Curve Model
In the form v(), where is the ratio of the gravimetric water content to its maximum—the liquid limit, and v is the relative volume of a drying clay, the shrinkage curve of a clay matrix is presented as (Fig. 4 ; Chertkov 2000a, 2003)

[2]

[3]

are the shrinkage limit and air-entry point, respectively, of the clay matrix, and F_z is the saturation degree at = _z. One can estimate F_z from the relative volume of a clay in the oven-dried state, v_z, and relative volume of clay solids, v_s, values, and then _z and _n (Eq. [3]) of the pure clay, and thereby express the shrinkage curve of v() (Eq. [2]) through parameters v_s and v_z only. In general, 0.11 < v_z < 1 and 0.03 < v_s < min(v_z and 0.2). If v_z 0.11 we have F_z 0 (more exactly F_z < 0.01 and _z, _n from Eq. [3]). If 0.11 < v_z < 1,

[4]

where f(x) = 1 – exp[–ln(6)(4x)⁴exp(–4x)], X v_z – 1, Y Av_z/v_s – 1, X (Av_z/v_s – )v_z, 9, and A 13.57 flow out of the clay microstructure consideration.

View larger version (26K): [in this window] [in a new window]   Fig. 4. The general form of the shrinkage curve of a clay matrix (modified from Chertkov, 2000a, Fig. 2). Subscripts z, n, and h indicate values at the shrinkage limit, endpoint of basic shrinkage, and maximum swelling of the clay, respectively; s and vs are the density and relative volume, respectively, of clay solids.

[5]

where _w and _s are the density of water and the clay particles, respectively. The _s, v_s, and v_z parameters can be found irrespective of the model (Eq. [2–4]) and experimental shrinkage curve. The solid density, _s, is measured by standard methods (Blake and Hartge, 1986). The minimum relative volume, v_z, and the relative volume of clay particles at the liquid limit, v_s, can be calculated from the measured specific volume in the oven-dried state, V_z, and the liquid limit, , of the clay matrix, using Eq. [5] at v = v_z and = 1, respectively. As applied to the shrinkage curve of a clay matrix, the _s, v_s, and v_z parameters reflect the specifics of different physical–chemical processes that are connected with clay mineralogy and concentration and type of cations in water.

The porosity of a clay matrix is given by

[6]

Connected or nearly connected clay particles outline clay micropores. The dimension of connected clay particles increases, and their thickness decreases, with water content. The model gives the following estimates:

[7]

where _z is the thickness of plate-like clay particles of a clay matrix in the zero shrinkage area, _M is the same at the liquid limit, r_mz is the maximum external size (including pore wall thickness) of pores (or the maximum size of clay particles) in the zero shrinkage area, and r_mM is the same at the liquid limit. The maximum internal size, r_m, of clay pores (excluding pore wall thickness) is determined to be

[8]

Shrinkage Curve of the Intraaggregate Matrix in Relative Coordinates
The parameter u is the ratio of the intraaggregate matrix volume (Fig. 2a) to the maximum volume in the solid state (at the liquid limit), while is the ratio of the gravimetric water content of the structure to its maximum value in the solid state (the liquid limit). The relative water content is similar for the pure clay matrix and the intraaggregate matrix in Fig. 2a because the pore space in these cases is the same. The transition from v() to u() for the simple intraaggregate matrix in Fig. 2a is determined to be

[9]

with u_z, u_s, and u_S being the relative volume of the intraaggregate matrix in Fig. 2a in the oven-dried state, the relative volume of the solid phase of the matrix (silt and sand grains and clay particles), and the similar relative volume of the nonclay solids, respectively. Replacing v, v_z, and v_s in Eq. [2] and [3] with Eq. [9], one can be convinced that the shrinkage curve u() for the intraaggregate matrix in Fig. 2a is again presented by Eq. [2] and [3], but after the replacement of all of the v symbols by u (Fig. 4). With that, , _z, _n, and F_z are the same, but u v, u_s v_s, u_z v_z, and u_n v_n. Note that the relative volume of the nonclay solid phase, u_S, falls out of the expressions for u() (Eq. [2] and [3] after replacement of v with u).

Shrinkage Curve of the Intraaggregate Matrix in Customary Coordinates and Its Slope in the Basic Shrinkage Area
One can transit from relative coordinates (, u) to customary ones (w, U) (specific volume vs. gravimetric water content) by the usual way as (cf. Eq. [5])

[10]

where w_M = w( = 1). According to Eq. [10], one can write the shrinkage curve slope as dU/dw = (du/d)/[_w(1 – u_s)]. Replacing du/d in the basic shrinkage area ( > _n) from Eq. [2] (after replacement of v with u) with du/d = (1 – u_s), we finally obtain for the intraaggregate matrix in Fig. 2a (see Fig. 3)

[11]

Maximum Swelling Point of a Clay Matrix and an Intraaggregate Matrix
The maximum swelling point of a clay matrix, _h, is between the air-entry point (Fig. 4, = _n) and the liquid limit ( = _M = 1). One can assume that at = _h the sizes of the majority of clay pores and the majority of clay particles outlining the pores are maximally possible for keeping the connections between the clay particles in their network, i.e., for keeping network integrity. At a water content in the range _h < < 1, at least a part of the connections is broken. We assume that at close to unity the separate disconnected clay particles make up the majority of them. Below, we estimate the _h value using Chertkov's (2000a) model concepts and the above summary, and assuming (i) the beginning of network destruction at > _h, and (ii) the majority of separate disconnected clay particles at close to unity. Figure 5a shows the two-dimensional illustrative scheme of a clay particle network before disconnections, i.e., at _h, where r is an external pore size (including pore wall thickness). At a given , r varies between the minimum and maximum values. The maximum external size, r_m, depends on . The maximum size of the separate clay particles after disconnections at > _h is r_mh = r_m( = _h). At a given after disconnections (Fig. 5b), the distance r between clay particle edges varies between the minimum and maximum values. The maximum at a given , r_m() as a function of reaches the maximum r_mM = r_m( = 1) at = _M = 1. Thus, the volume of the clay particle system increases between water contents = _h and = = 1 by [(r_mh + r_mM)/r_mh]³ times. That is (Fig. 4),

[12]

Let us estimate r_mM. One can represent the maximum distance between clay particle edges, r_mM, at = 1 as

[13]

where _h is the mean thickness of disconnected clay particles, (), at > _h. The clay particles are considered to be totally disconnected at the liquid limit, = _M = 1 if r_mM >> _h (Fig. 5b), i.e., k >> 1. A "much greater" sign means more than, at least by 10 times, i.e., k 10. Replacing r_mM in Eq. [12] with Eq. [13] at k 10, one obtains

[14]

View larger version (11K): [in this window] [in a new window]   Fig. 5. The two-dimensional scheme of a clay particle network (a) before disconnections at relative water content maximum swelling point h and (b) after disconnections at liquid limit M = 1 (r is a clay particle size; a clay particle surface is r2; r is the distance between the edges of the disconnected clay particles; is the mean thickness of clay particles).

[15]

To account for Eq. [7] and [15] one can rewrite Eq. [14] as 1.1³ < 1/v_h < 1.3³. As an estimate, we take 1/v_h = (1.1³ + 1.3³)/2 1.76 and v_h 0.57. Then, accounting for Eq. [2], one can write v_h = v_s + (1 – v_s)_h 0.57, i.e., _h depends on v_s as _h = (0.57 – v_s)/(1 – v_s). In the range 0.03 < v_s < 0.2 (see the above summary), this gives 0.46 < _h < 0.55. Accounting for the fact that k 10 is approximate, the variation of _h within the limits of this range is not significant, and in the following it is reasonable to take the estimate of (Fig. 4)

[16]

where _h=(_h) (Eq. [5]) is the gravimetric water content of the clay at maximum swelling. For the intraaggregate matrix (Fig. 2a), the ratio w_h/w_M of the maximum swelling point [w_h = w(_h)] to the liquid limit (w_M) is the same because pore spaces of the intraaggregate matrix (Fig. 2a) and the clay matrix coincide. That is, w_h = 0.5w_M.

Critical Clay Content
According to the definition of the critical clay content, c* (see above), it corresponds to the soil that has an oven-dried state with contacts between silt and sand grains inside aggregates (as in Fig. 2b), but with clay filling in the intergrain space (as in Fig. 2a; i.e., lacking lacunar pores). The system of grains that come into contact can be characterized by porosity, p. Then p is the summary volume of clay solids and clay pores (in the oven-dried state) per unit volume of oven-dried soil aggregates; v_s/v_z is the volume fraction of clay solids in the clay volume (in the oven-dried state); and pv_s/v_z is the volume of clay solids per unit volume of oven-dried soil aggregates. In addition, (1 – p) is the volume of silt and sand solids per unit volume of oven-dried soil aggregates. The clay content c* is determined to be the ratio of the volume of clay solids, pv_s/v_z, to the total volume of solids (pv_s/v_z + 1– p). Thus,

[17]

Hereafter, for simplicity, we do not take into account a possible small difference between the density of silt and sand grains and clay solids, and, correspondingly, the difference between the clay content by weight and the above volume fraction of clay solids. The corresponding modification is possible, but introduces a superfluous complication. (Note, that in the similar approximation u_S/u_s = 1 – c.)

Thus, the critical clay content, c*, of a soil is determined by the characteristics of the clay contributing to the soil (v_z/v_s) and the silt and sand component of the soil (p). If the clay content c > c* from Eq. [17], the grains inside aggregates do not come into contact in the oven-dried state (and, hence, at a nonzero water content), and the intergrain space is totally filled in with clay (Group A soils, which are the subject of this work; Fig. 2a). If the clay content c < c*, the lacunar pores occupy a part of the intergrain space in the oven-dried state (Group B soils; Fig. 2b). For the typical values p 0.5 and v_z/v_s 1.5, one obtains c* 0.4. This coincides with the result from Revil and Cathles (1999). For p, which can be between 0.3 and 0.8, and v_z/v_s, which can be approximately between 1.3 and 2 (Chertkov, 2003), however, c* can vary approximately between 0.3 (and even less) and 0.8. This range of c* is in the agreement with data from Fiès and Bruand (1998).

Reference Shrinkage Curve Slope in the Basic Shrinkage Area
According to Eq. [1], dU'/dw' = dU/dw (Fig. 3; 0 w' w_h'; 0 w w_h = Kw_h'). In turn, the reference shrinkage curve of a soil, Y(w') (Fig. 3) and aggregates, U_a(w') (Fig. 3) are similar in shape to the auxiliary U'(w') curve (Fig. 3), i.e., dU'/dw' = dY/dw' = dU_a/dw' at 0 w' w_h' because Y = U_a + U_s = U' + U_i + U_s. In particular, the basic shrinkage area, w_n w w_h, of U(w) (Fig. 3) where dU/dw = 1/_w (Eq. [11]) corresponds to the basic shrinkage area, w_n' = w_n/K w' w_h', of U'(w'), U_a(w'), and Y(w') (Fig. 3) where they have the similar slope, 1/_w. As noted above, at W W_s and w' w_s', W = w' (Fig. 3), U_a(W) = U_a(w'), and Y(W) = Y(w'). Hence, the slope of the observed reference shrinkage curve, dU_a/dW or dY/dW in the observed basic shrinkage area, W_n W W_s (w_n' w' w_s', see Fig. 3), is also given by 1/_w. Thus, for the structure in Fig. 2a, the model quantitatively predicts the reference shrinkage curve slope, 1/_w, in the basic shrinkage area (Fig. 3). Equation [1] leads to still another model prediction for the structure in Fig. 2a: the characteristic water contents, w_z' = W_z and w_n' = W_n., of an aggregated soil [Fig. 3, curves U_a(w') and Y(w')] are not equal to the characteristic water contents, w_z and w_n, of the corresponding intraaggregate matrix [Fig. 3, curve U(w)].

Contribution of the Interface Layer to Total Water Content
For the structure in Fig. 2a, we can write the interface-pore contribution (w') as

[18]

where _w is the water density; U_i is the interface layer specific volume; F_i(w') is the volume fraction of water-filled interface pores at a given water content contribution, w', of the intraaggregate matrix to W; is the porosity of the rigid interface layer, which coincides with the porosity of the intraaggregate matrix (Fig. 2a) at w' = w_h' as

[19]

where u_h = u(w_h). The 1 – u_s/u(w) expression for the intraaggregate matrix porosity is obtained from Eq. [6] at v and v_s from Eq. [9]. Denoting by R, R_m, and R_min the current, maximum, and minimum sizes, respectively, of interface pores in such a rigid matrix and using the intersecting-surfaces approach to soil structure (Chertkov, 2005a), we can write the simplest cumulative size distribution of the interface pores F_i(R, ) as

[20]

where (R – R_min)/(R_m – R_min) (R_min R R_m). To estimate R_m and R_min as well as a possible boundary R value of the water-filled interface pores at a given w', we first turn to some characteristics of intraaggregate clay pores (Fig. 2a). The interface and intraaggregate pore structures (Fig. 2a) are different, but genetically connected. For instance, the interface porosity (Eq. [19]) coincides with intraaggregate porosity at w' = w_h'. Ranges of pore sizes differ, but there is some correspondence between them. Among intraaggregate pores at a given w' there are pores of the maximum internal size r_m. Figure 6 shows the qualitative view of r_m(w') dependence. This dependence flows out of Eq. [8, 2, 3, 5] (Chertkov [2004] used the similar quantitative dependence for other aims) as

[21]

where r_mM is the maximum external size of intraaggregate pores at the liquid limit.

View larger version (13K): [in this window] [in a new window]   Fig. 6. A qualitative view of the maximum internal size of intraaggregate pores, rm, vs. water content, w'. Subscripts n, s, and h indicate the w' and rm values at the endpoint of the basic shrinkage, endpoint of the structural shrinkage, and maximum swelling of the soil, respectively; Rmin is the minimum size of interface pores; and Rm2 is the larger value of two possible maximum sizes of interface pores.

[22]

The smallest interface pores (R = R_min) should be emptied at w' = w_s'. Hence the R_min interface pore size corresponds to w' = w_s' (see Fig. 3 and 6). Since w_n' < w_s' < w_h', according to Fig. 6 and Eq. [21] we have

[23]

In the following section, the R_min value is specified. For a known R_min, Eq. [23] determines the w_s' point. Finally, for the maximum R_m size of interface pores, one can write the following obvious inequality (Fig. 6) as

[24]

Indeed, the R_m pore size in an interface layer cannot be less than the maximum intraaggregate pore size at the maximum aggregate swelling (r_mh). On the other hand, R_m cannot exceed r_mM. According to Eq. [24], two cases are possible:

[25]

and

[26]

For the latter case, the R_m value is specified below.

Specifications of the Interface Pore Sizes and Some Predictions

1. The maximum size of the interface pores, R_m = R_m1 (Eq. [25]), coincides with the maximum size of intraaggregate pores, r_mh, at aggregate saturation (Fig. 6). Unlike that, in the case of Eq. [26], the pores exceeding r_mh (between r_mh and R_m) can be in the interface layer. Such pores (>r_mh) can be considered to be microcracks. In this study, however, we consider soils of clay content c > c* with negligible volume of intraaggregate microcracks (lacunar pores). Therefore, in the case of Eq. [26], the maximum interface pore size R_m > r_mh is a formal one. The contribution of pores with sizes >r_mh to interface porosity is negligible, and the actual maximum interface pore size is r_mh. The R_m value from Eq. [26] (R_m > r_mh) influences the actual size distribution of the interface pores in this case, however, redistributing the same pore volume (U_i) between the pores of the same range (R_min < R < r_mh) in favor of larger pores close to r_mh compared with the distribution from Eq. [20]. The modified distribution F_i*(R,) corresponding to the case from Eq. [26] is

[27]

with F_i(r_mh,) = F_i(R = r_mh,). Thus, the model predicts two qualitatively different types of reference shrinkage curve in the structural shrinkage area (Fig. 3, U_a(W) Curves 1 and 2) that correspond to distributions from Eq. [20] and [27], respectively.

2. The r_mM value entered in Eq. [21] and [26] is usually between 6 and 20 µm. Estimates of w_n and w_h (from Eq. [2, 3, 5, 16] and then r_mn and r_mh (from the same equations and Eq. [21]), using these r_mM values, show that differences between r_mn and r_mh in Eq. [23] and between r_mh and r_mM in Eq. [26] (see Fig. 6) are in a small range, from <1 to <4 µm. For this reason, one can estimate R_min from Eq. [23] as (see Fig. 6)

[28]

and R_m for the case of Eq. [26] as (see Fig. 6)

[29]

These estimates of R_min and R_m (together with the estimate from Eq. [25]) lead to an important model result. According to Eq. [21], r_m(w) r_mM, in particular r_mn = r_m(w_n) r_mM and r_mh = r_m(w_h) r_mM. Then, according to Eq. [28, 29, 25], R_min r_mM and R_m r_mM. Therefore, replacing in (Eq. [20]) the R value from Eq. [22], R_min from Eq. [28], and R_m from Eq. [25] or [29], one can be convinced that the parameter together with the volume fraction of water-filled interface pores, F_i(w') (from Eq. [20] or [27]) do not depend on the maximum external dimension of the intraaggregate clay pores, r_mM. One can use for r_m, r_mn, r_mh, R_min, and R_m the corresponding ratios, r_m/r_mM, r_mn/r_mM, r_mh/r_mM, R_min/r_mM, and R_m/r_mM. Thus, unlike the case of a water retention curve (Chertkov, 2004), the r_mM value is not one of the physical parameters that determine the reference shrinkage curve.

3. The value of R_min (Eq. [28]) is in the middle between r_m = r_mn and r_m = r_mh (Fig. 6). Deflection of the r_m(w'K) dependence (Fig. 6) of linearity at w_n' w' w_h' exists, but is negligible [quantitative dependence r_m(w) see Chertkov (2004,Fig. 5)]. That is, the w' = w_s' point that corresponds to r_m = R_min (Fig. 6) is also practically in the middle between w' = w_n' and w' = w_h' (Fig. 6). Then, in the force of the linearity of the Y(w') [and U_a(w')] dependence (Fig. 3) at w_n' w' w_h', the Y(w_s') [and U_a(w_s')] value is also in the middle between the Y(w_n') [U_a(w_n')] and Y(w_h') [U_a(w_h')] values.

Relations between the Aggregate/Intraaggregate Mass Ratio, Oven-Dried Structural Porosity, and Specific Volumes of the Interface Layer and Structural Pores
By definition, oven-dried structural porosity P_z = U_s/Y_z = U_s/(U_s + U_az) (see Fig. 3). It follows that U_s = U_azP_z/(1 – P_z) (U_s = constant). After the successive replacements: U_az = U_z' + U_i (Fig. 3), U_z' = U_z/K (Eq. [1]), and U_z = u_z/(u_ss) (Eq. [10]), one obtains

[30]

One can write w_h – w_h' = w_h – w_h/K. In addition, accounting for w_h = W_h (Fig. 3; see above), one can write w_h – w_h' = W_h – w_h' = _h = U_iw (Eq. [18] at w' = w_h' and F_i(w_h') = 1). Equalizing the two expressions for w_h – w_h', one obtains

[31]

Another useful presentation of K is K = U_h/(U_h – U_i). This expression is identical to the right part of Eq. [31]. One can be convinced of that accounting for the expressions for w_h = w(_h) and U_h = U(_h) from Eq. [10], u_h = u(_h) from Eq. [2] (after replacing v with u), and from Eq. [19]. The identity reflects the position of the U_h = U(w_h) point in Fig. 3.

	MATERIALS AND METHODS

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

 
Input Parameters and Reference Shrinkage Curve Prediction
To predict the reference shrinkage curve, one needs six soil parameters: the oven-dried specific volume, Y_z (Fig. 3); maximum swelling water content, W_h (Fig. 3); mean solid density, _s; soil clay content, c; oven-dried structural porosity, P_z; and the ratio of aggregate solid mass to solid mass of intraaggregate matrix (Fig. 2a), K (or the ratio of the aggregate volume in the saturated state to the volume of the intraaggregate matrix in the same state). Let us assume that we know all six parameters. Then, the relation U_z' = U_z/K (Eq. [1]) after replacements: U_z' = Y_z – U_i – U_s (Fig. 3) and U_z = u_z/(u_ss) (Eq. [10]) gives

[32]

The relation W_h = w_h (Fig. 3) after replacement of w_h from Eq .[10] at _h = 0.5 gives

[33]

The presentation of K indicated after Eq. [31] gives still another relation as

[34]

at U_h = u_h/(u_ss) = 0.5(1 + u_s)/(u_ss) (Eq. [10] and [2] after replacing v with u). Solving Eq. [30] and [32–34] (given Y_z, W_h, _s, P_z, and K), one can find four values: u_s, u_z, U_i, and U_s. Then, given c, _s, u_s, and u_z, one can successively find v_s, v_z (using u_S/u_s = 1 – c and Eq. [9]), F_z (Eq. [4]), v() (Eq. [2] and [3]), r_m[v(w)]/r_mM (Eq. [8] or [21]), the interface layer contribution, (w') into W (Eq. [18–20, 22, 23, 25, 27–29]), and W(w') = w' + (w') (Fig. 3). Then, given u_s, u_z, U_i, U_s, K, and F_z, one can successively find u() (Eq. [2] after replacing v with u), U(w) (Eq. [10]), U'(w') (Eq. [1]), and Y(w') = U'(w') + U_i + U_s (Fig. 3). Finally, given Y(w') and W(w'), one can plot Y(W) (Fig. 3).

For data in the form of e() (void ratio vs. moisture ratio) before shrinkage curve prediction, the moisture ratio _h corresponding to W_h and void ratio (at the shrinkage limit ) e_z corresponding to Y_z are replaced by W_h and Y_z values as W_h = (_w/_s)_h, Y_z = (e_z + 1)/_s, and after the prediction the W and Y values are again replaced by the and e values.

The Y_z, W_h, _s, c, and P_z input parameters can be found or measured independently of the model and an observed shrinkage curve. Let us consider the possible ways of estimating the K ratio.

First, K can be connected with an intraaggregate structure (Fig. 2a). For this reason, K is quite a fundamental parameter, i.e., in general, it can be measured independently of the model and a soil shrinkage curve; however, the corresponding measurement methods and K data are currently unavailable.

Second, the general approach is as follows: K is calculated through U_i (Eq. [31]); in turn, the specific volume, U_i, of the interface layer (Fig. 2a and 3) can be expressed through its small thickness and surface per unit aggregate volume, and eventually through aggregate-size distribution and soil texture. In this case the number of elementary input parameters will be more than six; however, all those will be directly measured. In this study, such data were unavailable, but, in general, they can be obtained.

Third, the K ratio can be directly found using the measured shrinkage curve as K = W_h/w_h' (