Modelling the shrinkage curve of soil clay pastes

Geoderma 112 (2003) 71 – 95 www.elsevier.com/locate/geoderma

Modelling the shrinkage curve of soil clay pastes V.Y. Chertkov * Faculty of Agricultural Engineering Technion, Israel Institute of Technology, Haifa 32000, Israel Received 21 August 2001; received in revised form 27 August 2002; accepted 27 September 2002

Abstract The shrinkage properties of soil require an understanding of the volumetric changes to swelling clays, the presence of non-swelling minerals like silt and sand and the soil volume occupied by cracks and other voids. It is natural to consider first the matrix of clay particles only. The objective of this work is the further development of a basic model of the shrinkage curve of a soil clay based on its microstructure. The specific goals include developments of . .

a relation between the saturation degree and pore-size distribution; a physical condition determining the shrinkage limit of a soil clay paste.

The points of the basic model (the major concepts, definitions, assumptions, simple formulas and results) that are relevant to these goals are preliminarily briefly summarized. Application of the points supplemented by the above developments allows one: .

.

.

to reduce the number of independent macroparameters of clay paste that were introduced in the basic model; to express the residual water content, shrinkage limit and shrinkage curve of clay paste through the reduced number of its macroparameters; to show additional interconnection between micro- and macroparameters of the clay paste microstructure, besides those indicated in the basic model.

Comparison between model estimates and relevant available data on characteristic points of the shrinkage curve of soil clay pastes gives arguments in favour of the basic model supplemented here that enables one to predict the total shrinkage curve of a clay paste, including the residual water content, shrinkage and liquid limits. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Clay paste; Pore microstructure; Shrinkage curve; Residual water content; Shrinkage limit; Liquid limit

* Tel.: +972-4829-3331; fax: +972-4822-1529. E-mail address: [email protected] (V.Y. Chertkov). 0016-7061/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 6 - 7 0 6 1 ( 0 2 ) 0 0 2 9 7 - 5

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1. Introduction Most soils swell and shrink, and the shrinkage curve is one of their physical characteristics. It is necessary for the prediction of soil behavior when the soil interacts with water. A number of researchers have observed complex changes of pore space in shrink –swell soils with water content variation (e.g., Talsma, 1977; Stengel, 1981, 1984; Tessier, 1980, 1989; Tessier and Pe´dro, 1984; Fie`s, 1984, 1992; Bruand and Prost, 1987; Fie`s and Bruand, 1990, 1998; Cabidoche and Ruy, 2001). Silt, sand, cracks and voids of other origins disturb the clay continuity in soil, complicating its shrink– swell behaviour. It is natural to consider first a paste that consists only of clay particles. In an earlier paper (Chertkov, 2000), a model was proposed connecting the shrinkage curve of a disaggregated pure-clay paste with parameters of its probabilistic microstructure and pore-size distribution. In the following paper, that model is referred to as the basic one. The present work is a continuation and development of the basic model in an attempt to create some premises for the future quantitative understanding empirically observed swell –shrink phenomena (beyond the scope of this paper). The original paper on the basic model is very detailed. In order to make life easy on the reader and to prevent him/her from reverting to the original publication in the course of perusing this paper, I first briefly summarize the major concepts, definitions, assumptions, simple formulas and results of the basic model that are relevant to the objectives of this work. After summarizing I will formulate the objectives of the work more specifically. 1.1. General A clay paste consists of plate-like clay particles or quasicrystals constructed, in turn, of stacked clay layers (Wilding and Tessier, 1988). Interparticle porosity is understood as a pore space associated with a microstructural network formed by the clay particles. For a broad class of clays, ‘‘shrink –swell is minimally related to loss of interlayer water – interlayer porosity’’ and ‘‘mostly correlated with water loss and gain between clay particles – interparticle porosity’’ (Wilding and Tessier, 1988). It is the case that is considered in the basic model. Clay particles are assumed to be deformable but with no accumulation or loss of water (note that a modification of the basic model accounting for the interlayer porosity is possible, but is beyond the scope of this work). 1.2. The macroparameters characterizing the shrinkage – swelling of a clay We consider clay to be a deformable solid. From this viewpoint, the water content of a clay can vary from zero to the maximum value in the solid state of the clay, i.e., to its liquid limit. Indeed, by definition, ‘‘the liquid limit is empirically understood to be the water content at which the shear strength approaches that of a liquid’’ (Fam and Dusseault, 1999). Thus, in the wording of the basic model, ‘‘the maximum water content of the clay’’ means one ‘‘in the solid state’’ (the liquid limit). It is obvious that the maximum water content measured in any experiment (in a solid state) will be smaller than the corresponding liquid limit. A (dimensionless) relative water content f

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73

of the clay is defined as a ratio of gravimetric (w) water content to its maximum (wM) (i.e., the liquid limit) as fuw=wM ;

0VfV1:

ð1Þ

A more general definition of f is a water mass or volume associated with a unit mass of solid phase and divided by the corresponding maximum water mass or volume (in the solid state). A (dimensionless) relative volume of the clay (v) is defined as vuV =VM ;

vz VvV1

ð2Þ

where V is a clay volume (i.e., summary that of clay particles and pores filled with water and/or air) at a water content f V 1; VM is a volume magnitude of the same clay at the maximum water content in the solid state, f = 1 (the index ‘‘M’’ designates throughout the relation to the condition of the maximum water content in the solid state of the clay—the liquid limit), and vz defined as vz uVz =VM

ð3Þ

is the minimum relative volume of the clay paste (Fig. 1) to be determined by the clay volume V = Vz in the zero shrinkage area. The current volume V in Eq. (2) is the sum of a pore volume (Vp) depending on f and a solid phase volume (Vs). The vs value defined as vs uVs =VM

ð4Þ

is a volume fraction of the solid phase of the clay at the maximum water content (the liquid limit) (Fig. 1).

Fig. 1. The general form of the shrinkage curve of a clay matrix (a reproduction of Fig. 2 from Chertkov (2000)). ‘‘Clay’’ here means a system of clay particles and interparticle pores filled with water and/or air.

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The relative coordinates f and v can be transformed to customary coordinates: gravimetric water content w (g/g) and the specific volume of clay, V¯ (cm3/g) as w ¼ wM f;

v V¯ ¼ vs qs

ð5Þ

where wM ¼ ð1=ðvs qs Þ  1=qs Þqw

ð6Þ

is a mass of water at the maximum water content (the liquid limit) per unit mass of solid phase (or oven-dried clay); qw is the density of water; qs is the density of solid phase. 1.3. The microparameters of a clay paste The clay paste has a honeycomb-like structure (with possible gaps) (Wilding and Tessier, 1988). The volume of a single 3D pore is associated with outlining its clay particles. However, each of these clay particles also relates to an adjacent pore. For this reason, we associate with a given 3D pore only a half-thickness outlining its clay particles. A dimension of pore, together with this half-thickness, is referred to as an external pore dimension, unlike its (usual) internal dimension. There are pores of different dimension. Accordingly, a clay matrix microstructure is characterized by minimum (ro) and maximum (rm) external dimensions of pores, the average thickness of deformable plate-like clay particles (D) outlining the pores and by the pore dimension distribution (f(r/rm); here, r is an external pore dimension. I incorporate part of the solid phase (half-thickness of clay particles) in the definition of external dimensions of pores (r, ro, rm) because an approach to obtaining the f distribution that I use (see Section 1.4) is based on dividing the total volume into ‘‘elementary cells’’. In the case under consideration, the total volume means the total one of the clay pastes, including both clay particles and interparticle pores. Values of ro, rm and D are functions of a relative volume of clay (v) or relative water content (f). 1.4. Major assumptions of clay paste microstructure and their consequences The following assumptions set up connections between micro- and macroparameters of the clay paste in the frame of the basic model. The initial assumption of the basic model (essentially used in the previous section) is based on observations (e.g., Wilding and Tessier, 1988). It states that a clay paste is divided into pores by a network of deformable (but not accumulating or losing water) plate-like clay particles of a quasicrystal structure. An additional natural assumption—during shrinking – swelling, the volume of all pores changes proportionally to the volume of the clay paste—leads to expressions of ro and rm as ro ðvÞ ¼ roM v1=3 ;

vz VvV1

ð7Þ

and rm ðvÞ ¼ rmM v1=3 ;

vz VvV1

ð8Þ

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75

where roM and rmM are values of the minimum and maximum external dimensions of the pores, respectively, at the liquid limit (v = 1). Still another natural assumption of the volume conservation of the solid phase of the clay in the shrinking – swelling process (together with the previous assumption) gives an expression of D as D ¼ DM v2=3 ;

vz VvV1

ð9Þ

where DM is a value of D at the liquid limit (v = 1). Based on Eqs. (7) –(9), one can define roz, rmz and Dz as the minimum and maximum (external) pore dimensions and thickness of clay particles, respectively, in the zero shrinkage states by roz uro ðvz Þ;

ð10Þ

rmz urm ðvz Þ;

ð11Þ

Dz uDðvz Þ:

ð12Þ

and

It is reasonable to assume that the pore number connected with a given mass of a solid phase is retained during shrinking – swelling. This results in an implicit dependence of the pore dimension distribution on the relative volume, v (or relative water content, f) only through rm(v). That is, a principally possible dependence f(r/ rm(v),v) is reduced to f(r/rm(v)). The pore dimension distribution, f(r/rm(v)) is determined as a particular case of distribution of volume-like objects. And that we are basing this on the following approach: a distribution of volume-like objects (e.g., soil blocks, sand – silt grains, clay matrix pores) originates by dividing a space (volume) by a large number of intersecting surface-like objects (shrinkage cracks, slits, clay particles, respectively) if the surface-like ones fulfill a number of simple conditions (Chertkov, 1986, 1995, 2000; Chertkov and Ravina, 1998, 1999). As a result, the f distribution is written as
 f ðr=rm Þ ¼ 1  exp  Iðr=rm Þ ; ð13Þ where Iðr=rm Þ ¼ lnðK* þ 1Þð4r=rm Þ4 expð4r=rm Þ;

ð14Þ

and K*i5 (Zhurkov et al., 1981) (note that the expression for I(r/rm) in Chertkov’s (2000) Eq. (4) contains two misprints corrected in Eq. (14)). Designating x u r/rm and taking K* = 5 in the following, we use Eqs. (13) and (14) as f ðxÞ ¼ 1  expðlnð6Þð4xÞ4 expð4xÞÞ:

ð15Þ

Two important relations connecting micro- and macroparameters flow out of definitions of vz and vs (Eqs. (3) and (4)) and roM, rmM, DM, roz, rmz and Dz (Eqs. (7) –(12)) as DM A; ð16Þ vs ¼ rmM

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vz ¼

roz 1 ; Dz c

ð17Þ

where A is a constant value to be determined by the f distribution from Eq. (15) and calculated numerically as A¼3

Z

1

0

1 df ðxÞ dx ¼ 13:57; x dx

ð18Þ

and the ratio c u roM/DM is practically constant for different clays as ð19Þ

curoM =DM i9: Finally, the ratio vz/vs will be needed below as vz rmz 1 : ¼ vs Dz A

ð20Þ

This follows from Eqs. (7), (8), (16), (17) and (19). 1.5. Shrinkage curve Fig. 1 schematically shows the shrinkage curve of a clay paste as a dependence v(f) or V¯(w). The main macroparameters determining the shrinkage curve are the volume fraction of the solid phase at the maximum water content equal to liquid limit (vs) (Fig. 1; Eq. (4)), the minimum relative volume of the clay paste (vz) (Fig. 1; Eq. (3)) and the maximum relative water content in the zero shrinkage area (fz), i.e., the shrinkage limit (Fig. 1) (note that actually, fz>0 always). The minimum relative water content in the normal shrinkage area (fn) (Fig. 1) is expressed through vs, vz and fz. The pore-volume fraction occupied by water at a given relative water content (degree of saturation), F(f) and a value of F(fz) u Fz plays a key part. Dependence F(f) has the following form FðfÞ ¼

1  vs f; vðfÞ  vs

0VfV1:

ð21Þ

Note that in the normal shrinkage area (1 z f z fn), v(f) = vs+(1  vs)f and from Eq. (21) F(f) u 1. In addition, according to Eq. (21) (at f = fz) fz is expressed through vs, vz and Fz as fz ¼

vz  vs Fz : 1  vs

ð22Þ

Thus, the shrinkage curve of a clay paste in the form v(f) is determined by parameters vs, vz and Fz. For prediction of the shrinkage curve in the form V¯(w) (except for liquid

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77

limit, w = wM, Eq. (6)), one should know vz/vs, qs and Fz. In the approximation of the basic model, V¯(w) is given by (Fig. 1)

V¯ ¼

8 > > > > > < > > > > > :

V¯ z ;

0VwVwz uwM fz

V¯ z þ a¯ ðw  wz Þ2 ; wz VwVwn uwM fn 1 w þ ; qs qw

ð23Þ

wn VwVwM

where V¯ z ¼ vz =ðvs qs Þ

ð23aÞ

wz ¼ ðvz =vs  1Þðqw =qs ÞFz ;

ð23bÞ

wn ¼ ðvz =vs  1Þðqw =qs Þð2  Fz Þ;

ð23cÞ

a¯ ¼

1 4ðvz =vs  1Þqw ðqw =qs Þð1  Fz Þ

ð23dÞ

and wM from Eq. (6). 1.6. Two remarks of the basic model to be developed in this work It was noted that the function F(f) could also be expressed through a pore dimension distribution. If pores are filled with water in order of increasing dimensions or lose water in order of decreasing dimensions, in the simplest approximation, we have FðfÞ ¼ f ðrw ðfÞ=rm ðfÞ  DðfÞÞÞ;

ð24Þ

where rw is the maximum internal dimension of water-filled pores, and f is a volume fraction of such pores at water content f. Thus, based on the simple model of clay paste microstructure (Sections 1.3 and 1.4), the first goal of this work is to consider in detail this connection between the pore-volume fraction occupied by water ( F(f)) and the pore-size distribution ( f(rw/(rm  D))). It was also noted that the following development of the basic model should express fz and Fz (see Eq. (22)) through vs and vz values and exclude Fz as an independent parameter determining the shrinkage curve in Eqs. (23b), (23c) and (23d). As a basis for this development, I noted some physical condition determining the point of the shrinkage limit (fz) of the clay matrix. Thus, the second goal of this work is to consider this physical condition in detail using what is stated above in the summary of the basic model.

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1.7. Subsequent objectives of this work The subsequent objectives of this work are based on the above developments of a basic model: 

to propose an approach for estimating the residual water content, shrinkage limit and corresponding F(f) values of a clay paste through vs and vz values;  to apply the approach for estimating the shrinkage curve of a clay paste in the frame of the basic model;  to show an additional interconnection between micro- and macroparameters of the clay paste microstructure, besides those indicated in the basic model;  to compare the model estimates of the characteristic water content values of the clay paste (the residual water content, shrinkage limit and liquid limit) with the available data. The brief summary of the basic model given above is sufficient for understanding the objectives, content and results of this work. Nevertheless, note that the basic model contains a number of supplementary details important for the most interested reader and not mentioned in the summary. It is also worth noting that the basic model predictions are in agreement with the available data, both on microparameters (including pore dimension distribution) and the shrinkage curve of a clay paste [the data on four clay pastes from Tessier and Pe´dro (1984) and a clayey paste from Bruand and Prost (1987)]. One final remark. The basic 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. It is likely to mean that in this particular case (of high clay content), microcracks that appear because of shrinkage-stress concentration on silt–sand grains (heterogeneities) give no essential contribution to volume change of the drying clayey paste. If so, the clayey paste should behave in drying similarly to a pure-clay paste. I intend to address the role of clay content and cracking based on the model in the following works.

2. The relation between the pore-volume fraction occupied by water and pore-size distribution of a soil clay paste Note that in the frame of the basic model, there is a mechanism allowing for description of hysteretical phenomena (beyond the scope of this work). However, because data on the shrinkage curve that I attract in the following for comparison with the model prediction relate to the drying (dehydration) process, I address this one here. I consider drying of a saturated clay paste from the liquid limit, f = 1 and v = 1 or w = wM (Eq. (6)) and V¯ = V¯M u 1/vsqs) (Fig. 1). Loss of water up to f = fn (Fig. 1) occurs due to decreasing pore dimensions without air penetration. Air-filled pores appear at f < fn. In the following, we assume that: 

the water-filled pores of a clay matrix lose water at f < fn in order of decreasing dimension (based on capillarity);

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79

the air-filled pores that appear keep on their surface an adsorbed water film of thickness that can decrease at sufficiently small water content close to zero.

Based on these assumptions in connection with water content variation, I introduce for description of the clay paste microstructure two additional characteristic functions of the relative clay volume, v (or relative water content, f) besides r0(f) (Eq. (7)), rm(f) (Eq. (8)) and D(f) (Eq. (9)). The first is the maximum of internal dimension, rw(f) of water-filled pores at water content f. The second is a thickness, l(f), of the adsorbed water film on the surface of air-filled pores at water content f. In drying a clay paste, the water content passes in the range 1 z f z 0 three characteristic points 1>fn>fz>fr z 0. Here, fn is the minimum relative water content in the normal shrinkage area (Fig. 1) (air-entry point), fz is the shrinkage limit (Fig. 1), and fr is the residual (relative) water content associated by definition with the air-dried state of totally air-filled pores and the maximum possible thickness of the adsorbed water film, lr. After further drying, when fr>f z 0, the thickness l(f) can decrease compared to lr. That is, I consider f = fr to be a point of transition from the hydration state when the smallest water-filled pores are still kept to adsorption state with the totally air-filled pores and the maximum possible thickness of the adsorbed water film. Accounting for the results of the basic model and physical meaning of the functions r0(f) (Eq. (7)), rm(f) (Eq. (8)), D(f) (Eq. (9)), rw(f) and l(f), those can be specified with the water content decrease in the four possible ranges of f: 1 z f z fn, fn z f z fz, fz z f z fr and fr z f z 0. For this specification, see Appendix A. Note that rw(f) in the first range decreases as water content decreases, coinciding with the maximum internal pore dimension at a given water content, rm (f)  D(f). In the second range, rw(f) decreases up to a characteristic value, rwz u rw(fz) (this value will be discussed in the next section). In the third range, rw(f) additionally decreases up to the minimum internal pore dimension in the zero shrinkage area, r0z  Dz. Finally, in the fourth range, the rw(f) value is kept. According to the above, the pore-volume fraction, F(f), occupied by water at a given relative water content, f, includes two contributions associated with the water of waterfilled pores ( Fp) and with the adsorbed water film of air-filled pores ( Fa), respectively FðfÞ ¼ Fp ðfÞ þ Fa ðfÞ:

ð25Þ

At a given relative water content, the contributions Fp and Fa can be written through a normalized pore dimension distribution, fn(r/(rm/(f)  D(f)),n(f)) connected with the f distribution from Eqs. (13) and (14). In this work, I use the f distribution with respect to the internal pore dimension, r, as in Eq. (24). By definition, the normalized distribution function, fn(x,n) is a volume fraction of pores with a relative internal dimension, r/(rm  D) < x (for details of the definition of fn and n functions, see Appendix B). Based on the definition of the distribution function, fn(x,n), one can write Fp(f) as  rw ðfÞ ; nðfÞ ; 1zfz0 ð26Þ Fp ðfÞ ¼ fn rm ðfÞ  DðfÞ

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V.Y. Chertkov / Geoderma 112 (2003) 71–95

with the specifications given in Appendices A and B. In addition, based on the definition of the distribution function, fn(x,n) and assuming the cubic pores, one can write Fa(f) as Z rm ðfÞDðfÞ 1 dfn ðr=ðrm ðfÞ  DðfÞÞ; nðfÞÞ dr; 1zfz0: ð27Þ Fa ðfÞ ¼ 6lðfÞ r dr rw ðfÞ The expression to be integrated represents the volume of the water film (with a thickness of l(f)) adsorbed at water content f by walls of cubic pores (having six sides, the factor 6 in Eq. (27) follows from here) with internal dimension between r and r + dr. Note that the relation between the pore-volume fraction occupied by water and poresize distribution of a soil clay matrix (Eq. (25)) degenerates to an identity in the normal shrinkage area (1 z f z fn). Indeed, in this area, rw = rm  D, l(f) = 0 (see Appendix A) and fn = 1 (see Appendix B). Therefore, from Eq. (26), Fp(f) u 1, and from Eq. (27) Fa(f) u 0. On the other hand, in the normal shrinkage area, F(f) u 1 (see the remark after Eq. (21)). Thus, Eq. (25) is reduced to the identity 1 u 1 at 1 z f z fn. Below, I am especially interested in the states of f = fr and f = fz.

3. A physical condition determining the shrinkage limit of a soil clay paste (~ z) I consider drying (dehydration) of a saturated clay paste starting from the liquid limit, f = 1 and v = 1 or w = wM (Eq. (6)) and V¯ = V¯M u 1/(vsqs) (Fig. 1). The maximum internal dimension of water-filled pores rw(f) also decreases (see Appendix A). At the stage of 1 z f z fn (normal shrinkage, Fig. 1) water loss occurs only due to deformation of the solid phase with decreasing pore volume and without penetration of air. At the air-entry point, f = fn (Fig. 1) the air phase appears. Here, it is worth reiterating the assumption accepted in the previous section that the waterfilled pores of a clay paste lose water at f < fn in the order of decreasing dimensions. It means that the air phase (appearing from the periphery of a sample) fills the pores in the same order beginning from the largest ones. Accounting for that, I take the following natural assumption: first the air phase appears as isolated (discontinued) bubbles. Then, with further water loss and air penetration at the fn z f z fz stage (Fig. 1) after accumulation of sufficiently high concentration of isolated air volumes (bubbles), part of those merges eventually forming a continuous air phase besides a discontinuous one keeping isolated volumes. Therefore, at this stage, water replacement occurs at the expense of both extension of continuous air phase (without clay paste shrinkage) and enlargement of isolated air volumes accompanied by air decompression inside them and some shrinkage of the clay paste. However, the process of merging and disappearing of the isolated air volumes continues. Then, one can state that for a given clay paste, eventually, at f < fn (Fig. 1), there is a certain water content, fz, at which the last isolated air-occupied pores of an internal dimension exceeding rwz u rw(fz) (see Appendix A), totally surrounded by water-filled pores of dimensions less than rwz, disappear. That is, at f < fz, the totally continuous air phase arises with the pressure of ambient air, and the clay

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81

paste macroshrinkage ceases. Note that the water-filled pores are still kept at fr < f < fz (cf. a definition of fr in the previous section). Note also that variations of the clay particle arrangement within the clay paste are possible at f < fz. I assume that it is this value of water content, f = fz, that determines the shrinkage limit location of the soil clay paste. Note that at f = fz, not all the air-occupied pores of the internal dimension more than rwz are totally surrounded by water-filled pores of internal dimensions less than rwz. However, it is critical that the last air-occupied pores isolated by the smaller water-occupied ones definitely disappear with water content decrease at f = fz, and the air phase becomes totally continuous. The maximum internal dimension rwz of water-filled pores at f = fz (or, which is the same, the minimum internal dimension of the last isolated air-filled pores surrounded by water-filled pores of smaller dimension) can be written as rwz ¼ aroz  Dz

ð28Þ

where ar0z is the corresponding external dimension presented in units of the minimum external dimension of the pores, roz. I estimate the a factor in Section 6.

4. Relative residual water content of a soil clay paste (~ r) According to the physical definition of the (relative) residual water content (Section 2), there are no water-filled pores in a clay matrix at f = fr, i.e., Fp(fr) = 0 (it also follows from Eq. (27) at rw(fr) from Appendix A and definition of fn from Appendix B). Then, using Eqs. (17) – (20), one can obtain F(fr) from Eqs. (25) and (26) as Fðfr Þ ¼

lr Rðvz ; vs Þ: Dz

ð29Þ

The explicit view of the R(vz,vs) function entering Eq. (29) is considered in Appendix C. Equalizing F(fr) from Eq. (21) (note that v(fr) = vz) and from Eq. (29), we obtain an estimate of the (relative) residual water content fr by fr ¼

lr ðvz  vs Þ Rðvz ; vs Þ: Dz ð1  vs Þ

ð30Þ

This is yet another relation between micro- (lr/Dz) and macroparameters (fr, vz and vs), besides the relations of such kind reported in the basic model (Eqs. (16), (17) and (20)). The average thickness of plate-like clay particles (Dz) varies from 1 nm (10 9 m) to 40 nm (4  10 8 m) (Hillel, 1998, p.74), depending on the clay type. For the maximum thickness of a water film covering clay particles, we take that of a molecular layer, lri< 0.3 nm (3  10 10 m) (Hillel, 1998, p.35). Then, we have 0.0075i< lr/Dzi< 0.3. Thus, accounting for the ranges of variation of vz and vs (see Section 5), it usually follows from Eqs. (29) and (30) that: Fðfr Þb1 and fr b1:

ð31Þ

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5. The shrinkage limit of a soil clay paste (~ z) Taking f = fz, replacing ro(fz), rm(fz), D(fz), rw(fz) and l(fz) from Appendix A and using Eqs. (17) – (20), we obtain Fz u F(fz) from Eqs. (25) – (27) as a function of vz, vs, a factor and the lr/Dz ratio (see Appendix D) Fz uFz ðvz ; vs ; a; lr =Dz Þ:

ð32Þ

At a known Fz, the shrinkage limit fz is determined from Eq. (22). At 1Hfz>fr, when fr and fz are relatively close, the dependence of Fz on lr/Dz in Eq. (32) is essential (Appendix D). At 1>fzHfr, we can neglect the dependence of Fz on lr/Dz (Appendix D), i.e., Fz iFz ðvz ; vs ; aÞ:

ð33Þ

The vz parameter varies in the range 0.11i < vz < 1 (Appendix D). The lower boundary of the range corresponds with the case 1Hfz>fr and Eq. (32) (Appendix D). vz values essentially exceeding 0.11 correspond with the case 1>fzHfr and Eq. (33) (Appendix D). Data from Tessier and Pe´dro (1984) for four clay pastes and Bruand and Prost (1987) for clayey paste (containing 56% by weight of clay plus silt) illustrate the theoretically possible cases of 1Hfz>fr (see Chertkov’s (2000) Fig. 5d example) and 1>fzHfr (see Chertkov’s (2000) Figs. 5a –c and 6 examples). Finally, in connection with above range of vz values, it is worth mentioning that from estimates of the basic model, the vs parameter (which is always less than vz) varies in the range 0.03 –0.17. Some estimates of vs obtained below (Section 8 and Table 1) exceed this range and reach 0.23 (Table 1), keeping the same order of magnitude as 0.17.

Table 1 Shrinkage curve parameters of four clay pastes from Tessier and Pe´dro (1984) and the clayey paste from Bruand and Prost (1987) Clay paste

Experimental Best-fit estimate, values 1/qs vz/vs Fz (cm3/g)

(a) St. Austell kaolinitea (b) Illite from Le Puya (c) Kaolinite from Provinsa (d) Wyoming montmorillonitea variant 1 variant 2 Rhodic Luvisolb a b

Predicted values a

vz

vs

wz

V¯z wn (cm3/g)

V¯n wM V¯M wr (cm3/g) (cm3/g)

0.379

2.11 0.75 3.18 0.35 0.17 0.31 0.80

0.53 0.91

1.92 2.29

f 0.002

0.379

1.83 0.85 2.77 0.39 0.21 0.27 0.69

0.36 0.74

1.39 1.76

f 0.016

0.379

1.64 0.30 2.47 0.24 0.15 0.07 0.62

0.41 0.79

2.20 2.58

f 0.001

0.379 0.379 0.376

1.36 0 – 0.11 0.08 0 0.52 1.36 0.01 2.05 0.12 0.09 0.001 0.52 1.35 0.57 2.04 0.36 0.23 0.07 0.51

0.27 0.65 0.27 0.65 0.19 0.56

4.26 4.64 3.78 4.16 1.23 1.61

– < 0.001 f 0.01

Four clay pastes from Tessier and Pe´dro (1984). Clayey paste from Bruand and Prost (1987).

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6. Estimation of the a factor for a soil clay paste The a factor is a dimensionless value and for this reason, it can depend, according to basic model, on dimensionless vz, vs and Fz parameters (Section 1). However, Fz u F(fz) is a function of vz, vs, a and lr/Dz (Eq. (32)). Therefore, the a factor depends only on vz, vs and lr/Dz. In the case 1>fzHfr, according to Eq. (33), we can totally neglect the dependence of a on lr/Dz. In the case 1Hfz>fr, we can neglect the dependence of a on lr/Dz in the first approximation in force of the small value itself of the lr/Dz ratio (Eq. (31)). Thus, we can accept that the a factor always depends only on vz and vs. On the other hand, according to its definition (Eq. (28)), the a factor is only connected with the ‘‘rigid’’ state of the clay matrix in the area of zero (macro) shrinkage. At the same time, vz and vs are connected both with the ‘‘rigid’’ state and with the ‘‘swelling’’ state of the maximum water content (the liquid limit). It follows from their definitions (Eqs. (3) and (4)). Only the vz/vs ratio does not depend on the ‘‘swelling’’ state (see Eq. (20)). Thus, the a factor can only depend on vz/vs ratio. This critical point allows the derivation of an equation for the a factor as a function of the vz/vs ratio (Appendix E). This equation has the simple explicit solution (Appendix E) as a ¼ ðA=cÞðvz =vs Þ;

ð34Þ

where A and c are from Eqs. (18) and (19), respectively. Thus, Eq. (34) permits one to estimate the a factor for a clay paste with a known vz/vs ratio. According to this equation, the a factor is in the range 2i < ai < 3.2 (see Section 8 and Table 1) when the vz/vs ratio varies in the typical range, 1.3i < vz/vsi < 2.1, determined by the basic model.

7. The shrinkage curve of a clay paste 7.1. The case of 1>fzHfr>0 In this case, the shrinkage curve is determined by qs, vs and vz according to Eqs. (23), (23a), (23b), (23c), (23d) and (6). Fz is estimated by the second term on the right side of Eq. (A4.1) (the first term is neglected) with the f(x) function from Eq. (15). 7.2. The case of 1Hfz >ifr >0 fr is always a small, but finite positive value (both in the case of Eq. (A4.2) and in the case of Eq. (A4.3)). Thus, in the case under consideration, fz and, according to Eq. (22), Fz are also small positive values. Therefore, in a good approximation, we can take fz = 0 and Fz = 0. In the case under study, we can take vzi0.11 (see Appendix D). That is, at small (but not zero) values of Fz and fz, we can use, for the description of the shrinkage curve of a clay paste, Eqs. (23), (23a), (23b), (23c), (23d) and (6) at fz = 0, Fz = 0 and vzi0.11. Thus, the shrinkage curve of a clay matrix with fzi0 is determined only by two parameters, 0.03i < vsi < 0.11 and qs.

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8. Model estimates The proposed approach, allowing for estimation of Fz by vs and vz parameters, was applied in the opposite direction to estimate vs and vz parameters for four clay pastes from Tessier and Pe´dro (1984) and clayey paste from Bruand and Prost (1987) (56% clay plus silt), using the least-square estimates of Fz and vz/vs ratio for these pastes obtained in validating the basic model. Initial data on 1/qs are from Tessier and Pe´dro (1984) and Bruand and Prost (1987) and these estimates are reproduced in Table 1. For the (d) clay (Table 1) I considered two variants. The first corresponds to approximations Fz = 0 and vzi0.11 (see Section 7.2). In the second one, I took the small, but finite value Fz = 0.01 obtained by the least-square procedure similar to that described in the basic model. In this variant, the consideration of the (d) paste was conducted similarly to the other four pastes. Preliminarily, I estimated the a factor (Table 1) for each of the five pastes using Eq. (34). The vz values (Table 1) for the five pastes with Fz p 0 (including the (d) clay at Fz = 0.01) were estimated using Eq. (A4.1) (neglecting the first term on its right side) at the known values of Fz, vz/vs and a (as well as c and A constants). Then, I estimated vs for these clays (Table 1) from the known value of the vz/vs ratio and the evaluated vz. In case (d) from Tessier and Pe´dro (1984) for the variant where Fz = 0, I used vzi0.11 (see Section 7.2) and also estimated vs (Table 1) from the known value of the vz/vs ratio. An additional estimation of the vs value, in this case by the least-square fitting Eqs. (23), (23a), (23b), (23c), (23d) and (6) to the twelve experimental points in Fig. 2 (not using the known vz/vs ratio), with variation of vs in the range 0.03i < vsi < 0.17, gave the same result. Then, using the

Fig. 2. Predicted shrinkage curve, data from Tessier and Pe´dro (1984) for Wyoming montmorillonite paste and 1:1 theoretical line.

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evaluated values of vs and vz as well as (Eqs. (5), (6), (23b) and (23c), I estimated wz, V¯z, wn, V¯n, wM (see Fig. 1 and Table 1) and V¯M (Table 1) for the five pastes, including two variants of the (d) clay. Finally, I estimated the gravimetric residual water content, wr (Table 1) for the five pastes. In all the cases, except for the (d) clay, I used Eqs. (30) and (A3.1) (accounting for wr = wMfr) and the estimates of lr and Dz cited after Eq. (30). For the (d) clay (the variant 2 when Fz p 0), I took advantage of the fact that with Fz being as small as Fz = 10 2, the corresponding estimate of wz (Table 1) by Eq. (22) (accounting for wz = wMfz) simultaneously gives an estimate of wri < wz. Because the estimates of lr and Dz in this case ((d) clay) were not used, one can also estimate the lr/Dz ratio using Eqs. (30) and (A3.1) and the estimated fr value as well as vs and vz. The estimation gives lr/ Dzi < 0.006.

9. Results and discussion Estimates of wz, V¯z and wn, V¯n (Table 1) found for the four pastes (except for the (d) clay) based on the vs and vz values coincide with those from Chertkov’s (2000) Table 3. It is natural because vs and vz were estimated using the basic model and the least-square Fz and vz/vs estimates from it. What is new here is that this coincidence allows us to consider these vs and vz values (Table 1) as best-fit ones found immediately by the least-square method for data from Tessier and Pe´dro (1984) and Bruand and Prost (1987). It implies that the vs and vz estimates are as well founded as the Fz and vz/vs estimates from the basic model. Comparison between estimates of wz, V¯z and wn, V¯n parameters (Table 1) for two variants of consideration of the (d) clay, shows that the small change of Fz (0 ! 0.01) and corresponding small changes of vz (0.111 ! 0.124) and vs (0.08 ! 0.09) do not practically influence these parameters. Thus, the results of the comparison validate the considerations in favor of the approximation of a shrinkage curve in Section 7.2 (at Fz = 0 and vzi0.11) for the case Fzb1, fzb1 (except for the liquid limit value, see below). Unlike the four other pastes in the case of the (d) clay, wn and V¯n given in Table 1 differ from corresponding estimates from Chertkov’s (2000) Table 3 (note that wz and V¯z estimates in Table 1 and Chertkov’s (2000) Table 3 coincide). This difference is connected with the use in this work of the description of the shrinkage curve at Fz>0 because Fz>Fr>0 always even though Fzb1. Comparison between data and prediction of the liquid limit of a soil clay paste, wM could serve as the best way to check the model. Unfortunately, I do not have data on wM immediately for clays presented in Table 1. I also could not find data for other clay pastes simultaneously on the shrinkage curve at f V fn and the liquid limit value. Note, however, that the predicted high values of wM of the pastes from Table 1 are similar to possible high values of water content of the clay pastes from Haines (1923) (1.02 g/g), Stengel (1981) (2 g/g), Tessier (1980, 1989) (2.4 and 1.4 g/g, respectively), Fam and Dusseault (1999) (up to f 7 g/g; these researchers cite liquid limit data collected from different authors) and Sivapullaiah et al. (2000) (3.44 g/g), according to variations in clay mineralogy. At the same time, the liquid limit of soils with low clay content and even clay soils (Warkentin, 1972; Fam and Dusseault, 1999; Sivapullaiah et al., 2000; Thomas et al., 2000) is, as a rule, appreciably lower than for clay pastes and usually varies between f 0.3 and f 0.9

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g/g, according to variations in clay content and clay mineralogy. The above data on the liquid limit of clay pastes provide no direct check of the model because they relate to clays not presented in Table 1. Nevertheless, the comparison between the predicted liquid limit of the clay pastes (Table 1) and the indicated data clearly speaks in favour of the feasibility of the model and stimulates its further direct experimental validation. Comparison between estimates of wM and V¯M parameters (Table 1) for two variants regarding the (d) clay shows that the small change of Fz (0 ! 0.01) and corresponding small changes of vz (0.111 ! 0.124) and vs (0.08 ! 0.09) appreciably influence these parameters of a shrinkage curve. That is, use of the accurate values of vz and vs is essential from the viewpoint of estimating the liquid limit of clay. The above estimates of wr are fairly small (see Table 1 and Eq. (31)). I have no immediate data on wr for clays presented in Table 1. Estimates obtained for different soils using van Genuchten’s (1980) approximation for the moisture retention curve (see, e.g., van Genuchten’s (1980) Table 1; Schaap et al.’s (1998) Table 3; Sˇimu˚nek et al.’s (1998) Table 1; Wildenshield et al.’s (2001) Tables 2 and 3) frequently contain large standard deviations. Nevertheless, as a whole, they show that the residual water contents for clays and clay soils are usually appreciably smaller than for soils with low clay content. Again, the comparison between the predicted residual water content for the clay pastes (Table 1) and the corresponding estimates available in the literature give no direct check of the model, but speaks in favour of its feasibility and gives impetus for a further direct experimental validation. It is also noteworthy that the estimate of the lr/Dz ratio found for the (d) clay (the end of Section 8) does not contradict the data indicated after Eq. (30). Finally, the feasibility of the model is reinforced by the abovementioned result: the values of vs (Table 1) are in a satisfactory agreement with the range estimated earlier (the end of Section 5). As a potential future application of the obtained results, I can propose three possible variants using the basic (microstructure and shrinkage curve) model of the clay paste in combination with these results for the prediction of its shrinkage curve. 9.1. A method of objective estimation of the liquid limit, wM and corresponding specific volume, V¯M of clay paste First, data on a part of the shrinkage curve in the area w1 V w V w2, where w1 < wz and w2>wn, are analyzed by an approach from the basic model, giving qs, vz/vs and Fz values. Then, one finds vs and vz as discussed in this work. After that, wM and V¯M can be found from Eqs. (6) and (5) (at v = 1) (Fig. 1). 9.2. A method of estimation of shrinkage limit, wz, water content, wn and the entire shrinkage curve of clay paste, using the specific volume in the zero shrinkage area, V¯z and the liquid limit, wM I assume that qs is known. After measuring the liquid limit, wM by the ASTM method D4318 (American Society for Testing and Materials, 1993) and V¯z as a ratio of oven-dried clay volume to its mass, one can find vs from Eq. (6) and vz/vs from Eq. (5) at v = vz. Then, we

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find wz as above. After that, wn and the entire shrinkage curve can be found according to the basic model. 9.3. A fundamental method of predicting the total shrinkage curve of soil clay paste Again, qs is assumed to be known. After measuring (or estimating) the microparameters of a clay paste, roz, Dz and rmz or roz, DM and rmM (methods of the measurements are beyond the scope of this work), one can find vs and vz values from Eqs. (16) and (17). wz is determined by qs, vs and vz as above. Then, wn and the total shrinkage curve can be found from the basic model.

10. Summary and conclusion A model of clay paste microstructure and shrinkage curve was suggested earlier. First, I briefly summarized a number of points of the basic model that are relevant to the objectives of this paper (Section 1). Then, as a development of the basic model, I considered: (a) a relation between the pore-volume fraction occupied by water and the pore-size distribution of a soil clay paste (Section 2); (b) a physical condition determining the point of the shrinkage limit of the paste (Section 3). Using the basic model and based on the relation and condition, the simple ways of estimation of the residual water content (Section 4), shrinkage (Sections 5 and 6) and liquid (Section 9) limits and shrinkage curve (Section 7) of the clay paste were suggested. At the same time, an additional relation between micro- and macroparameters, connecting residual water content, pore microstructure of the clay paste and thickness of adsorbed water film was obtained (Eqs. (30) and (A3.1)). Data for a clay simultaneously on the shrinkage curve (at water content lower than the liquid limit) and the liquid limit are not available. For this reason, I compared the model estimates of characteristic points of the shrinkage curve for a number of clay pastes (Section 8) with data on those for other clay pastes (Section 9). The obtained results give arguments that the basic model supplemented here allows one to predict the total shrinkage curve of a clay paste, including the residual water content, shrinkage and liquid limits based on their physical definitions and vs, vz and qs values. Nevertheless, the further validation of the model using data on the shrinkage curve, liquid limit, shrinkage limit and residual water content for the same clays is desirable. It should be emphasized that vs and vz can be expressed through the parameters of the clay matrix microstructure in the frame of the basic model. The basic model in combination with this work will be used as the basis of attempts to model the shrinkage curve of clay soils including silt, sand and cracks of different types.

Notation A = 13.57 constant value (Eq. (18)), dimensionless a factor entering a presentation of the minimum internal dimension, rwz, of the last isolated air-filled pores disappearing with water content decrease at f = fz (Eq. (28)), dimensionless

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coefficient of the square term in Eq. (23) (see Eq. (23d)), cm3/g pore-volume fraction occupied by water at a given relative water content, dimensionless Fa(f) contribution to pore-volume fraction occupied by water, F(f), associated with adsorbed water film of air-filled pores (Eq. (27)), dimensionless Fp(f) contribution to pore-volume fraction occupied by water, F(f), associated with water of water-filled pores (Eq. (26)), dimensionless Fz Fz u F(fz), dimensionless f(r/rm(v),v) general form of the cumulative pore dimension distribution of a clay –paste volume (or a volume fraction of pores of external dimensions < r) at a given relative volume v, dimensionless f(r/rm(v)) actual form of the cumulative pore dimension distribution of a clay –paste volume (or a volume fraction of pores of external dimensions < r) at a given relative volume v (at x u r/rm(v) f(x) is given by Eq. (15)), dimensionless f(x) function f(r/rm(v)) at x u r/rm(v) (Eq. (15)), dimensionless fn(x,n(f)) normalized distribution function (Eq. (A2.1)), dimensionless G(g) function to be determined by Eq. (A3.2), dimensionless I(r/rm) function entering an expression for f(r/rm) (Eq. (13)), dimensionless K* critical value of the ratio of the mean linear dimension of an area that one clay particle takes over to the mean dimension of the clay particle proper (Eq. (14)), dimensionless l(f) thickness of adsorbed water film remaining on walls of air-filled pores in drying at water content f, nm lr maximum possible thickness of adsorbed water film, nm R(vz,vs) function determining F(fr) (Eq. (29)) and to be determined by Eq. (A3.1), dimensionless r the largest external dimension of a pore, Am rm(v) the maximum external dimension of pores at a relative volume v of clay, Am rmM = rm(1) value of rm(v) at the liquid limit, Am rmz = rm(vz) value of rm(v) in the zero shrinkage area of clay paste, Am ro(v) the minimum external dimension of pores at a relative volume of clay v, Am roM = ro(1) value of ro(v) at the liquid limit, Am roz = ro(vz) value of ro(v) in the zero shrinkage area of the clay paste, Am rw the maximum internal dimension of water-filled pores, Am rwz maximum internal dimension of water-filled pores at f = fz (or, which is the same, the minimum internal dimension of the last isolated air-filled pores surrounded by water-filled pores of smaller dimension), Am V clay volume in the solid state at a water content, cm3 VM volume magnitude of the same clay at the maximum water content in the solid state (at the liquid limit), cm3 Vp pore volume of clay volume V, cm3 Vs solid phase volume of clay volume V, cm3 Vz clay volume in the zero shrinkage area, cm3 ¯ = v/vsqs specific volume of clay (per unit mass of solid phase), cm3/g V a¯ F(f)

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V¯M

specific volume of clay (per unit mass of solid phase) at f = 1 (at the liquid limit), cm3/g ¯n V specific volume of clay (per unit mass of solid phase) at f = fn, cm3/g ¯Vz specific volume of clay (per unit mass of solid phase) at f = fz, cm3/g v u V/VM relative volume of a clay, vz V v V 1, dimensionless vn minimum relative volume of the clay matrix in the normal shrinkage area, dimensionless vs volume fraction of the solid phase at the maximum water content (at the liquid limit), dimensionless vz the minimum relative volume of the clay matrix (realizing in the zero shrinkage area) dimensionless w = wMf gravimetric water content (per unit mass of solid phase), g/g wM maximum gravimetric water content or mass of water at the maximum water content (at the liquid limit) per unit mass of solid phase (or oven-dried clay), g/g wn gravimetric water content (per unit mass of solid phase) at f = fn, g/g wz gravimetric water content (per unit mass of solid phase) at f = fz, g/g w1 gravimetric water content < wz, g/g w2 gravimetric water content >wn, g/g X value to be determined by Eq. (A3.2), dimensionless x variable of Eq. (15) and integration variable of Eqs. (18) and (A3.2), dimensionless Y value to be determined by Eq. (A3.2), dimensionless m, n, o, s, w, z, M subscripts c ratio roM/DM (see Eq. (19)), dimensionless D(v) thickness of plate-like clay particles at a relative volume of clay v, Am DM = D(1) value of D(v) at the liquid limit, v = 1, Am Dz = D(vz) value of D(v) in the zero shrinkage area of clay paste, Am DX value to be determined after Eq. (A4.1), dimensionless f relative water content—a water mass or volume associated with a unit mass of solid phase and divided by the corresponding maximum water mass or volume in the solid state of clay (i.e., at the liquid limit), dimensionless fn the minimum relative water content in the normal shrinkage area, dimensionless fr relative residual water content associated by definition with the air-dried state of totally air-filled pores and the maximum possible thickness of the adsorbed water film, lr, dimensionless fz the maximum relative water content in the zero shrinkage area, dimensionless g u X/Y variable in Eqs. (A3.1) and (A3.2), dimensionless qs density of solid phase, g/cm3 qw density of water, g/cm3

Acknowledgements The research was supported in part by the German – Israeli Foundation for Scientific Research and Development and the Technion Water Research Institute. The author

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expresses his gratitude to Ary Bruand (Institut des Sciences de la Terre d’Orle´ans) and Paul Hallet (Scottish Crop Research Institute, Invergovrie, Dundee) for constructive review comments.

Appendix A . Characteristic functions of clay paste microstructure as functions of the relative water content The characteristic functions ro(f), rm(f), D(f), rw(f) and l (f) entering the right parts of Eqs. (26) and (27) can be specified in the four possible ranges of f as follows: at 1 z f z fn: ro ðfÞ ¼ roM vðfÞ1=3 ; rm ðfÞ ¼ rmM vðfÞ1=3 ; DðfÞ ¼ DM vðfÞ2=3 ; rw ðfÞ ¼ rm ðfÞ  DðfÞ; lðfÞ ¼ 0;

ðA1:1Þ

at ~ n z f z fz: ro ðfÞ ¼ roM vðfÞ1=3 ; rm ðfÞ ¼ rmM vðfÞ1=3 ; DðfÞ ¼ DM vðfÞ2=3 ; rm ðfn Þ  Dðfn Þzrw ðfÞzrwz ; lðfÞ ¼ lr ;

ðA1:2Þ

at ~ z z f z fr: ro ðfÞ ¼ roz ; rm ðfÞ ¼ rmz ; DðfÞ ¼ Dz ; rwz zrw ðfÞzroz  Dz ; lðfÞ ¼ lr ;

ðA1:3Þ

at fr z f z 0: ro ðfÞ ¼ roz ; rm ðfÞ ¼ rmz ; DðfÞ ¼ Dz ; rw ðfÞ ¼ roz  Dz ; lr zlðfÞz0:

ðA1:4Þ

Appendix B . The normalized pore dimension distribution The internal pore dimension r is in the range ro(f)  D(f) V r V rm(f)  D(f). Although in the basic model, the inequalities robrm and f((ro  D)/(rm  D))b1 take place, both ro and f((ro  D)/(rm  D)) are not equal to zero. Therefore, for greater accuracy, we replace the f function by a normalized distribution function, fn as  fn

  r ro ðfÞ  DðfÞ f f r rm ðfÞ  DðfÞ rm ðfÞ  DðfÞ  ; nðfÞ u ; 0VfV1 ro ðfÞ  DðfÞ rm ðfÞ  DðfÞ 1f rm ðfÞ  DðfÞ ðA2:1Þ

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where n(f) u (ro(f)  D(f))/(rm(f)  D(f)). At r = ro  D and r = rm  D, we have fn = 0 and fn = 1, respectively.

Appendix C . The R function in Eq. (29) At Fp(fr) = 0, according to Eqs. (25) and (26), the function R(vz,vs) in Eq. (29) can be written as Rðvz ; vs Þ ¼

2 GðX =Y Þ Y ð1  f ðX =Y ÞÞ

where f(x) is from Eq. (15), and accounting for Eqs. (17) – (20), we have: Z g vz 1 df ðxÞ X ucvz  1; Y u A  1; and GðgÞuA  3 dx: dx vs 0 x

ðA3:1Þ

ðA3:2Þ

Usually g u X/Y has values gi < 0.1, and in the simplest approximation, we obtain GðgÞiA;

f ðgÞi0:

ðA3:3Þ

Appendix D . Fz as a function of vz, vs, a factor and the lr/Dz ratio Fz can be presented as Fz ¼

2lr =Dz GððX þ DX Þ=Y Þ f ððX þ DX Þ=Y Þ  f ðX =Y Þ þ ð1  f ðX =Y ÞÞ 1  f ðX =Y Þ Y

ðA4:1Þ

where DX u (a  1)cvz (for X, Y and G(g) see Eqs. (A3.2) and (A3.3), f(x) from Eq. (15)). Two simple considerations permit us to simplify the estimation of Fz from Eq. (A4.1) and to determine the range of possible vz values of soil clay matrices. The first is as follows: the shrinkage limit always exceeds the residual water content, fz>fr. Because usually frb1 (Eq. (31)), the inequality, fz>fr can be rewritten as 1Hfz > fr

ðA4:2Þ

1 > fz Hfr :

ðA4:3Þ

or

It is clear from physical considerations that in the case of Eq. (A4.2), when fr and fz are relatively close, the first term on the right side of Eq. (A4.1) is essential (cf. Eq. (A3.1)). On the contrary, in the case of Eq. (A4.3) we can neglect the first term on the right side of Eq. (A4.1). The second consideration is connected with the following obvious physical and geometrical limitation: the internal dimension of the smallest pores at f = fr (when the

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adsorbed water film has yet been kept) coincides with or exceeds the doubled thickness of the adsorbed water film (Fig. 3), i.e., roz  Dz z2lr :

ðA4:4Þ

Using Eqs. (17) –(20) and accounting for Eq. (A3.2) for X, it follows that X ucvz  1z2lr =Dz :

ðA4:5Þ

Accounting for Eq. (19) and lr/Dzb1, Eq. (A4.5) together with inequality, vz < 1 (see Fig. 1) determine the range of possible vz values of soil clay pastes as 0:11i < vz < 1:

ðA4:6Þ

The lower boundary of the range corresponds with the condition roz  Dzi2lr (see Eq. (A4.4)) when the smallest pores are water filled at f = fr. It is clear that in this case, the separate air-filled pores, isolated by smaller water-filled pores, must disappear at the minimal possible water content, i.e., fr and fz should be as close as possible. Therefore, the lowest vz values (vzi0.11) can be related to the condition to be described by Eq. (A4.2). On the other hand, according to Eq. (A4.6), possible vz values can essentially exceed 0.11. Such values correspond with the condition to be described by Eq. (A4.3). In this case (at relatively large vz), Fz (and fz) is expressed through vz, vs and a only because, as was noted above, the first term on the right side of Eq. (A4.1) is negligible.

Fig. 3. Sketch of two opposite sections of the boundary of a minimal pore: ro(f) is the external dimension of the pore; D(f) is the thickness of plate-like clay particles; l(f) V lr is the thickness of an adsorbed water film.

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Appendix E . An equation for estimating the a factor Let us consider the transition from the shrinkage curve of a clay paste v(f) (Fig. 1) to be characterized by vz and vs to the limit of vz = 1 at vz/vs = constant, i.e., to a rigid matrix vðfÞ ! vz ¼ 1:

ðA5:1Þ

Note that because the a factor depends only on the vz/vs ratio (Section 6) and vz/vs = constant, the a value does not change at this transition. That is, a coincides with an initial value for the clay paste under study. In the limit of Eq. (A5.1), we have (Fig. 1) fz ! 1:

ðA5:2Þ

That is, in this limit, fz corresponds to the water-saturated state. At vz ! 1 and fz ! 1, Eq. (22) gives Fz ! 1:

ðA5:3Þ

This is natural because in the water-saturated paste the pore-volume fraction occupied by water ( Fz) should tend to unity. On the other hand, in the limit under consideration, a contribution of adsorbed water film can be neglected. Thus, from Eq. (A4.1), we have in this limit Fz !

f ððX þ DX Þ=Y Þ  f ðX =Y Þ 1  f ðX =Y Þ

ðA5:4Þ

(for X and Y see Eq. (A3.2); for DX see line after Eq. (A4.1)). Comparing Eqs. (A5.3) and (A5.4), we come to the following condition for finding the a factor of the clay paste f ððX þ DX Þ=Y Þ ¼ 1;

ðA5:5Þ

where vz/vs is a given constant and vz = 1. For the f(x) function, we use Eq. (15). This function reaches the maximum, f(x) = 1 at x = 1. Thus, replacing X, Y and DX by their expressions (indicated above) and taking vz = 1 we have from Eq. (A5.5) the relation: ðX þ DX Þ=Y uðac  1Þ=ððvz =vs ÞA  1Þ ¼ 1

ðA5:6Þ

that leads to a simple expression for the a factor (Eq. (34)).

References American Society for Testing and Materials, 1993. Standard test method for liquid limit, plastic limit, and plasticity index for soils (D4318). 1993 Annual Book of ASTM Standards, vol. 4. ASTM, Philadelphia, PA, pp. 554 – 563. Sect. 08. Bruand, A., Prost, R., 1987. Effect of water content on the fabric of a soil material: an experimental approach. J. Soil Sci. 38, 461 – 472.

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Cabidoche, Y.M., Ruy, S., 2001. Field shrinkage curves of a swelling clay soil: analysis of multiple structural swelling and shrinkage phases in the prisms of a Vertisol. Aust. J. Soil Res. 39, 143 – 160. Chertkov, V.Y., 1986. Chip development during multiple crack formation in a brittle rock. Sov. Min. Sci. 21, 489 – 495. Chertkov, V.Y., 1995. Mathematical simulation of soil cloddiness. Int. Agrophys. 9, 197 – 200. Chertkov, V.Y., 2000. Modeling the pore structure and shrinkage curve of soil clay matrix. Geoderma 95, 215 – 246. Chertkov, V.Y., Ravina, I., 1998. Modeling the crack network of swelling clay soils. Soil Sci. Soc. Am. J. 62, 1162 – 1171. Chertkov, V.Y., Ravina, I., 1999. Tortuosity of crack networks in swelling clay soils. Soil Sci. Soc. Am. J. 63, 1523 – 1530. Fam, M.A., Dusseault, M.B., 1999. Determination of the reactivity of clay – fluid systems using liquid limit data. Can. Geotech. J. 36, 161 – 165. Fie`s, J.C., 1984. Analyse de la re´partition du volume des pores dans les assamblages argile-squelettes: comparaison entre un mode`le d’espace poral textural et les donne´es fournies par la porosime´trie au mercure. Agronomie 4 (9), 891 – 899. Fie`s, J.C., 1992. Analysis of soil textural porosity relative to skeleton particle size, using mercury porosimetry. Soil Sci. Soc. Am. J. 56, 1062 – 1067. Fie`s, J.C., Bruand, A., 1990. Textural porosity of a silty clay soil using pore volume balance estimation, mercury porosimetry and quantified backscattered electron image (BESI). Geoderma 47, 209 – 219. Fie`s, J.C., Bruand, A., 1998. Particle packing and organization of the textural porosity in clay – silt – sand mixtures. Eur. J. Soil Sci. 49, 557 – 567. Haines, W.B., 1923. The volume-changes associated with variations of water content in soil. J. Agric. Sci. 13, 296 – 310. Hillel, D., 1998. Environmental Soil Physics. Academic Press, New York. 771 pp. Schaap, M.G., Leij, F.J., van Genuchten, M.Th., 1998. Neural network analysis for hierarchical prediction of soil hydraulic properties. Soil Sci. Soc. Am. J. 62, 847 – 855. Sˇimu˚nek, J., Wendroth, O., van Genuchten, M.Th., 1998. Parameter estimation analysis of the evaporation method for determining soil hydraulic properties. Soil Sci. Soc. Am. J. 62, 894 – 905. Sivapullaiah, P.V., Sridharan, A., Stalin, V.K., 2000. Hydraulic conductivity of bentonite – sand mixtures. Can. Geotech. J. 37, 406 – 413. Stengel, P., 1981. Relation entre le retrait et le potentiel de l’eau dans les me´langes smectites-limon. C.R. Acad. Sci. Paris, Se´rie D 293, 465 – 468. Stengel, P., 1984. Influence des contraintes hydriques et me´caniques sur l’espace poral. Se´minaire du De´partment Science du Sol: Comportements physique et me´caniques des sols, Pech-Rouge, Oct. 1982. INRA, Versailles, pp. 25 – 42. 107 pp. Talsma, T., 1977. A note on the shrinkage behaviour of a clay paste under various loads. Aust. J. Soil Res. 15, 275 – 277. Tessier, D., 1980. Sur la signification de la limite de retrait dans les argiles. C.R. Acad. Sci. Paris, Se´rie D 291, 377 – 380. Tessier, D., 1989. Behavior and microstructure of clay water systems. Management of Vertisols for Improved Agricultural Production: Proceedings of an IBSRAM Inaugural Workshop, 18 – 22 February 1985. ICRISAT Center, Patancheru, India, pp. 73 – 81. Tessier, D., Pe´dro, G., 1984. Recherches sur le roˆle des mine´raux argileux dans l’organisation et le comportement des sols. Livre Jubilaire du Cinquantenaire. AFES, pp. 223 – 234. Thomas, P.J., Baker, J.C., Zelazny, L.W., 2000. An expansive soil index for predicting shrink – swell potential. Soil Sci. Soc. Am. J. 64, 268 – 274. van Genuchten, M.Th., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892 – 898. Warkentin, B.P., 1972. Use of the liquid limit in characterizing clay soils. Can. J. Soil Sci. 52, 457 – 464. Wildenshield, D., Hopmans, J.W., Sˇimu˚nek, J., 2001. Flow rate dependence of soil hydraulic characteristics. Soil Sci. Soc. Am. J. 65, 35 – 48. Wilding, L.P., Tessier, D., 1988. Genesis of vertisols: shrink – swell phenomena. In: Wilding, L.P., Puentes, R.

V.Y. Chertkov / Geoderma 112 (2003) 71–95

95

(Eds.), Vertisols: Their Distribution, Properties, Classification and Management. Technical Monograph no. 18. Soil Management Support Services, College Station, TX 77843. Texas A&M University Printing Center. pp. 55 – 81. Zhurkov, S.N., Kuksenko, V.S., Petrov, V.A., 1981. Physical principles of prediction of mechanical disintegration. Sov. Phys. Dokl. 26, 755 – 757.