Shrinkage anisotropy characteristics from soil structure and initial sample/layer size. V.Y. Chertkov*

Transcription

1 Srinkage anisotropy caracteristics from soil structure and initial sample/layer size V.Y. Certkov* Division of Environmental, Water, and Agricultural Engineering, Faculty of Civil and Environmental Engineering, Tecnion, Haifa 32000, Israel Abstract. Te objective of tis work is a pysical prediction of suc soil srinkage anisotropy caracteristics as variation wit drying of (i) different sample/layer sizes and (ii) te srinkage geometry factor. Wit tat, a new presentation of te srinkage anisotropy concept is suggested troug te sample/layer size ratios. Te work objective is reaced in two steps. First, te relations are derived between te indicated soil srinkage anisotropy caracteristics and tree different srinkage curves of a soil relating to: small samples (witout cracking at srinkage), sufficiently large samples (wit internal cracking), and layers of similar tickness. Ten, te results of a recent work wit respect to te pysical prediction of te tree srinkage curves are used. Tese results connect te srinkage curves wit te initial sample size/layer tickness as well as caracteristics of soil texture and structure (bot inter- and intra-aggregate) as pysical parameters. Te parameters determining te reference srinkage curve (relating to te small samples) and initial sample size/layer tickness are needed for te prediction of te above soil srinkage anisotropy caracteristics. Using te available data on two soils and samples of two essentially different sizes, illustrative estimates of te relative sample sizes, new caracteristic of te srinkage anisotropy, and srinkage geometry factor are given as te pysically predicted values. Keywords: soil structure, srinkage anizotropy, cracking, critical sample size, lacunar factor, crack factor. *Corresponding autor. Tel.: address: agvictor@tx.tecnion.ac.il; vycert@ymail.com (V.Y. Certkov).. Introduction Te anisotropy of soil srinkage essentially influences soil ydraulic properties, water flow, and transport penomena, even toug tere are no cracks (e.g., Garnier et al., 997a, 997b). It is obvious tat cracking additionally complicates tis influence (e.g., Coppola et al., 202). For tis reason te prediction of te soil srinkage anisotropy caracteristics based on te soil texture and structure (bot interand intra-aggregate) as well as te initial sample/layer sizes as pysical parameters, is important for te pysical understanding ydrological processes in srink-swell soils. Srinkage anisotropy reflects te possible difference between te vertical and orizontal srinkage of a sample. Starting from Bronswijk (988, 989, 990, 99a, 99b), accounting for srinkage anisotropy is usually conducted in terms of te srinkage geometry factor, r s. Tere are a number of works on te srinkage geometry factor, its measurement, and use for different aims (Bronswijk, 988, 989, 990, 99a, 99b; Garnier et al., 997a, 997b; Baer and Anderson, 997; Crescimanno and Provenzano, 999; Cornelis et al., 2006; Peng and Horn, 2007; Boivin, 2007; Coppola et al., 202, among oters). Also available are works on te necessary and essential corrections of te srinkage geometry factor estimated by Bronswijk's approac (Certkov, 2005, 2008a). However, r s remains te empirical engineering parameter tat is a complex function of soil water content, W (Certkov, 2005, 2008a)

2 2 and determined (at a given W) from measurements of soil subsidence and soil matrix srinkage (or soil crack volume). Tus, it is not currently possible to pysically predict tis engineering parameter from soil structure, pysical properties, and sample size or layer tickness. Te objective of tis work is to suggest suc prediction of r s for soils wit relatively small organic matter content ( %) and, in addition, to introduce a more immediate presentation of soil srinkage anisotropy troug te sample/layer size ratios, also as a pysically predicted function of W. First, we briefly review te results of a recent work (Certkov, 202a) wit respect to te pysical prediction of tree different srinkage curves of a soil, relating to small samples (witout cracking at srinkage), sufficiently large samples (wit internal cracking), and layers of similar tickness (Section 2). Tese results are used in te following. We, ten, consider te connection between te variation of sample/layer sizes wit water content and tree srinkage curves (Section 3), new presentation of te soil srinkage anisotropy concept (Section 4), pysical prediction of r s and some of its applications (Section 5), some specifications connected wit water content profile and orizontal cracks (Section 6). Finally, we give and discuss illustrative estimates for two soils and two initial sample sizes from Crescimanno and Provenzano (999), wit respect to te current relative sample/layer sizes, new caracteristic of soil srinkage anisotropy, and srinkage geometry factor as pysically predicted functions of water content (Section 7). Notation of te values tat repeat is summarized at te end of te paper. 2. Srinkage curves as functions of soil structure and initial sample/layer size In a recent work Certkov (202a) considered an approac to te pysical prediction of a soil srinkage curve, including te crack volume contribution, depending on sample size or layer tickness as well as soil texture and structure for aggregated soils wit negligible organic matter content. Tree key points of te approac are: (i) accounting for te recently suggested intra-aggregate structure (Fig.) including lacunar pores and an aggregate surface layer wit specific properties (Certkov, 2007a, 2007b, 2008b); (ii) te consideration of all te contributions to te soil volume and water content based on te inter- and intra-aggregate soil structure; and (iii) te use of new concepts of te lacunar factor (k), te crack factor (q), and critical sample size (*). Te calculation of k as a function of te ratio, c/c * (c being te soil clay content; c * being te critical clay content (Certkov, 2007a); in particular, c * depends on clay type) in te case were te cracks do not exist in small samples (Certkov, 200), was generalized to te case of large samples wit internal cracking (Certkov, 202a). Te crack factors, q s for samples and q l for layers as functions of te ratio, /* ( being te initial sample size or layer tickness at maximum swelling) are calculated as (Certkov, 202a) q s (/*)=0, 0</* (a) q s (/*)=b (/*-) 2, /* +δ (b) q s (/*)=-b 2 /(/*-), /* +δ (c) and q l (/*)=b (/*) 2, 0 /* δ (2a) q l (/*)=-b 2 /(/*), /* δ (2b)

3 3 were δ=(/(3b )) /2, b 2 =2δ/3=(2/3)(/(3b )) /2. (3) wit teoretical values of universal constants, b 0.5, b 2, and δ.5. Te critical sample size, * is calculated as a function of minimum (X min ) and maximum (X m ) aggregate sizes, and structural porosity (P ) at maximum swelling (Certkov, 202a). Te srinkage curves (specific volume vs. gravimetric water content) of te sample of a given size, Y s (W, /*) and of te layer of a given tickness, Y l (W, /*), including crack volume contribution, are expressed troug te reference srinkage curve, Y r (W) (Certkov, 2007a, 2007b), and te q factor as (Certkov, 202a) Y s (W, /*)=(-q s (/*))Y r (W)+q s (/*)Y r, 0 W W (4) Y l (W, /*)=(-q l (/*))Y r (W)+q l (/*)Y r, 0 W W (5) were Y r Y r (W ). Te pysical parameters determining te reference srinkage curve, Y r (W) (including te k factor) ave been discussed in detail (Certkov, 2007a, 2007b, 200, 202a). 3. Sample and layer size evolution at srinkage Te found specific soil volumes, Y s and Y l (section 2) allow one to estimate te variation of te different sample and layer sizes at srinkage. Te layer sizes can be used to predict te subsidence of layer surfaces and orizontal deformation of te soil matrix inside te cracked layer. Te sample sizes can also be used to consider srinkage anisotropy at cracking (section 4). First, we define and regard te current sizes of a sample. In te case of an initially cubic sample tere are tree sizes of interest (Fig.2): te current orizontal sample size, x'(w) (wit crack contribution at >* or witout it at <*); te current sample eigt, z'(w); and te current orizontal size, x''(w) of te matrix inside te sample if te summary crack volume is mentally excluded (at <* x'=x''). In te initial state z'(w )=x'(w )=. Below we also use te layer volume in Bronswijk's approximation (Certkov, 2005) wen te layer at maximum swelling is constructed of a set of contacting, but disconnected cube samples. In suc an approximation gaps appear between te initial cubes at te layer srinkage process. Hence, te latter can be illustrated by te same Fig.2 for te separate cube sample. Te specific soil volume of te layer in tis approximation is designated as Y l ' (unlike Y l ). We take advantage, below, of te link between Y l ' and te specific soil volume, Y l of a real connected and cracked layer (Certkov, 2005) as Y l =Y l '(+x'/) 2 /4. (6) In tis link (Eq.(6)) x' is te current orizontal sample size (Fig.2a). In addition, te specific volume of te layer in Bronswijk's approximation, Y l ' is simply connected to te current sample eigt, z' (Fig.2c). Indeed, in suc an approximation te current layer volume of te tickness z' (per surface area of te layer) can be written as z' 2 (see Fig.2c) and by definition of Y l ' as Y l ' 3 /Y ( 3 /Y m is te oven-dried layer mass per surface area of te layer). From te equality of tese values one obtains Y l '=(z'/)y. (7)

4 4 Still anoter necessary relation between sample eigt, z' and orizontal size, x' (Fig.2), (x'/) 2 (z'/)=y s /Y, (8) follows from te equality x' 2 z'=y s 3 /Y between te two different expressions of te current sample volume. Finally, we use te relation between te sample eigt, z' and orizontal size x'' of te soil matrix inside te sample (after te mental exclusion of possible cracks) (Fig.2), (x''/) 2 (z'/)=y r /Y, (9) tat follows from te equality x'' 2 z'=y r 3 /Y between te two different presentations of te current volume of te soil matrix inside te sample. Te last four relations (Eqs.(6)-(9)) between te values tat we seek: z', x', x'', and Y l ', enable one to find tose troug Y s (W, /*), Y l (W, /*), Y r (W), Y, and as z'/=(y s /Y )[2(Y l /Y s ) /2 -] 2, (0) x'/=[2(y l /Y s ) /2 -] -, () x''/=(y r /Y s ) /2 [2(Y l /Y s ) /2 -] -, (2) Y l '/Y =(Y s /Y )[2(Y l /Y s ) /2 -] 2. (3) Note tat at Y s =Y r (wen tere are no cracks in te sample at < * ), x' (Eq.()) coincides wit x'' (Eq.(2)) as it sould be. In addition, z'/=y l '/Y (Eqs.(0) and (3)), also as it sould be. Unlike te sizes of te sample (z', x', x''; Fig.2), two sizes caracterizing layer srinkage (Fig.3) are only expressed troug Y l, Y r, and Y. Tese two sizes are (Fig.3) te current tickness of te real srinking and cracking layer, z and te current orizontal size x of te soil matrix volume (per surface area of te layer) after te mental exclusion of te crack volume. Te expressions z/=y l /Y (4) and x/=(y r /Y l ) /2 (5) follow, respectively, from te equality of te two different presentations of te current layer volume (per surface area of te layer), 2 z=y l 3 /Y, and te equality of te two different presentations of te current volume of te soil matrix inside te layer (per surface area of te layer) after te mental exclusion of crack volume, x 2 z=y r 3 /Y. Since te srinkage curves, Y s (W,/*) and Y l (W,/*), depend on te initial sample/layer size, (see Eqs.()-(5)), te variation of Y l '/Y and te relative sample/layer sizes, z'/, x'/, x''/, z/, and x/ wit te water content also depend on (see Eqs.(0)-(5)). Note tat te above sample/layer sizes are pysically predicted

5 5 troug te specific soil volumes (Eqs.()-(5)) witout using te concept of te srinkage geometry factor of te sample or layer (see section 5). 4. Te anisotropy of soil srinkage Te srinkage geometry factor, r s is an indirect caracteristic of soil srinkage anisotropy. Te rater more immediate and clear presentation and prediction of soil srinkage anisotropy, a(w, /*) can be given by te tree possible ratios of te vertical and orizontal sample and layer sizes as a(w, /*): z'/x', z'/x'', z/x. (6) Te sizes are known from Eqs.(0) - (2), (4) and (5) as pysically predicted functions of te water content, W. Figure 4 sows te possible qualitative view of te above ratios for a soil sample or layer. Te view also depends on te /* ratio. In addition to te "external" srinkage anisotropy, z'/x', connected wit te external sample sizes (Fig.2), te ratios z'/x'' and z/x of sample (z') and layer (z) vertical sizes to "internal" orizontal sizes of te soil matrix in sample (x'') and layer (x) (Figs.2 and 3), can be useful for te understanding and prediction of te anisotropy of cracking and te corresponding anisotropy of ydraulic conductivity in real soils. Examples of te a(w) function (i.e., te z'/x', z'/x'', z/x ratios) prediction, at te "small" (/*<) and "large" (/*>) values for real soils in sample and layer geometry, are considered in Section Te srinkage geometry factor and some of its applications Te corrected srinkage geometry factor, r s (compared to Bronswijk's (990) approximation) for te sample is written as (Certkov, 2005, 2008a) r s (W, /*)=log(y r (W)/Y )/log(z'(w, /*)/)=log(y r (W)/Y )/log(y l '(W, /*)/Y ). (7) Te pysically predicted Y r (W), Y l '(W, /*), and z'(w, /*) (sections 2 and 3) mean te similar prediction of r s. Suc a pysically predicted r s value, for te sample case wit possible cracks, can be used, for instance, in te problem of water flow in swelling soils in framework of Garnier et al.'s (997a, 997b) approac. One sould note te difference between te srinkage of a small sample (/*<) and a large one wit cracks (/*>) as applied to te r s concept. In te former case te decrease of te matrix volume of te soil sample consists of two contributions, sample volume decrease at te expense of vertical size decrease (subsidence) and at te expense of orizontal size decrease (lateral deformation). For tis reason, knowing r s and, for instance, te vertical size, one can estimate te current orizontal size using Bronswijk's (990) known formula. In te latter case te crack volume, occurring inside te sample, gives te additional tird contribution (tat is negative since crack volume grows at srinkage) to te decrease of te matrix volume of te soil sample. Terefore, in te case of large samples (/*>) te knowledge of r s and te current vertical sample size are not sufficient for te separation between te contribution of te internal cracks and tat of te lateral deformation. Note tat te pysical prediction of te sample z', x', and x'' sizes (Eqs.(0)-(2)) decides tis issue. Te corrected srinkage geometry factor, r s (compared to Bronswijk's (990) approximation) relating to te layer case is expressed troug te different specific volumes of te soil as (Certkov, 2005, 2008a) r s (W, /*)=log(y r (W)/Y )/log(y l (W, /*)/Y ) (8)

6 6 (Y is te specific soil volume at maximum swelling). All values entering tis expression are already known (see above). Since te values entering Eq.(8) depend on a number of pysical soil parameters (Certkov, 202a), tis equation pysically predicts te r s (W) dependence for a cracked soil layer (wit water content being omogeneous witin te layer). Note tat unlike te sample case (see above), in a layer of any tickness tere are only two contributions to te decrease in te matrix volume of te soil layer at srinkage, layer tickness decrease (a positive contribution) leading to soil subsidence and crack volume increase in te layer (a negative contribution). Terefore, in te layer case (field conditions), knowing r s (e.g., from Eq.(8)) and te soil subsidence along te vertical profile, one can predict te crack volume in te layers using Bronswijk's (990) formula. However, te crack volume can be pysically predicted witout te use of te r s concept (Certkov, 202a). In ligt of tat, it is wort noting anoter possible and important application of te pysical prediction of te r s value (Eq.(8)) to estimating some aspects of crack network geometry in a layer. Indeed, te pysical prediction of te srinkage geometry factor as a function of W in te layer case enables a similar prediction of te crack-widt distribution in te layer (Certkov, 2008a) Te crack-widt distribution is important for pysical estimating te soil ydraulic properties. Examples of te r s (W) dependence prediction at te small and large /* values for real soils in sample and layer geometry are considered in Section Te effects of water content profile variation and orizontal cracks We implied above tat water content is omogeneous witin te limits of a layer and sample. For tis reason, wen estimating te crack network caracteristics in te vertical cross-section of a soil, one sould divide te cross-section into orizontal layers wit water content being omogeneous witin te limits of eac layer. Suc a division is carried out using te observed (or predicted) water content profile. However, tis profile can vary wit time. As a result, a layer wit initially omogeneous water content sould be divided after some time into two tinner layers or, on te contrary, tis layer can enter a ticker layer. Terefore, variations in water content profile can lead to suc observable effects as, e.g., te increase of te total crack volume in a layer at drying wit te occurrence (or opening) of larger cracks and te closing of many small cracks tat appeared earlier (Hallaire, 984). In any case for te dependable prediction of te evolution in tickness of soil layers, and crack volume or geometry (i.e., different crack distributions) inside tem, one needs to use te sufficiently accurate data on te vertical water content profile (or prediction of te profile) and its variation wit time. Vertical cracks were considered above. Horizontal cracks are secondary ones since tey start from te walls of te vertical cracks as a result of additional drying and srinkage of te soil matrix along te walls (Certkov and Ravina, 999). Te orizontal cracks contribute little to te total crack volume (Certkov, 2008a). By definition of te reference srinkage curve, Y r (W) te orizontal cracks do not influence te latter. In addition, te development of orizontal cracks neiter canges te soil (Y l ) and vertical crack (U cr l ) volumes, nor te soil surface subsidence (Certkov and Ravina, 999; Certkov, 2008a). For tis reason te orizontal cracks ave little influence on te srinkage geometry factor, r s (Eq.(8)) and distributions of te vertical crack caracteristics. Note, owever, tat contribution of orizontal cracks to te soil ydraulic conductivity (tat is beyond te scope of tis work) can be essential.

7 7 Finally, it sould be empasized tat te above noted vertical and orizontal cracks are macroscopic tose (up to tens of centimeters in size) and form te srinkage crack network (for te 2D illustrative example see Guidi et al., 978) unlike te microcracks tat are observed on te images of te tin soil sections and can ave any orientation (e.g., Bui and Mermut, 989; Velde et al., 996). Merging of te microcracks under srinkage stresses leads to te network formation of te quasi vertical and quasi orizontal macrocracks troug te mecanism of multiple cracking (Certkov and Ravina, 998; Certkov, 2008a). 7. Illustrative estimates of te relative sizes, srinkage anisotropy, and srinkage geometry factor of samples and layers Te aim of tis section is to use two soils from Crescimanno and Provenzano (999) tat were considered in Certkov (202a) as examples, in order to illustrate te possible beavior of te soil srinkage anisotropy caracteristics, regarded above for sample and layer geometry, as functions of water content and initial sample/layer size. Te illustrative dependences in Figs.5-7 rely on te srinkage curves, Y r (W) (Certkov, 2007a, 2007b), Y s (W, /*) and Y l (W, /*) (Eqs.()-(5)) (Certkov, 202a) tat were found for Delia a and Delia 6 soils. Delia 2a and Delia 4 soils from Crescimanno and Provenzano (999) tat were also considered in Certkov (202a), lead to similar dependencies. Te soil pysical caracteristics tat are necessary to predict Y r (W), Y s (W, /*), and Y l (W, /*) for te two soils are indicated in Table. Te curves in Figs.5-7 relating to samples and layers are marked by solid and dased lines, respectively. For eac srinkage caracteristic (except for x''/ and z'/x''; see below) two curves are sown tat correspond to small (=3. cm <*) and large (=.5 cm >*) sample or layer size (for * see Table ). Tese sizes coincide wit tose from Crescimanno and Provenzano (999). Te relative sample sizes from Eqs.(0)-(2) (see also Fig.2) and relative layer sizes from Eqs.(4) and (5) (see also Fig.3) predicted for Delia a and Delia 6 soils are sown in Fig.5. At =3. cm <* x''=x'. For tis reason x''/ is only sown at =.5 cm >*. Scrutinizing Fig.5 one notes te following points.. Te different caracteristic sizes of te sample or layer can coincide wit eac oter at some water content values (except for te point W=W ). 2. Te relative caracteristic sizes of small (<*) samples (curves 4 and 5) and tin (<*) layers (curves 8 and 9) meet te inequality z/<z'/<x'/<x/. (9) Tat is, in te case of small samples and layers: (i) te vertical srinkage (relative subsidence -z'/ or -z/) dominates te lateral (-x'/ or -x/) one (wic corresponds to cracking in te layer case); and (ii) tis domination is stronger for layers tan for samples: x/-z/>x'/-z'/. (20) Tis is natural since te internal tension or stretcing occurs in a srinking layer (Certkov, 2005). Te values of x'/, z'/, x/, and z/ (or positions of curves 4, 5, 8, and 9 in Fig.5) temselves depend on te specific features of te particular soil (see Table ). 3. In te case of large samples and layers (>*) tere are two general laws: (i) curve is iger tan curve 2, i.e., x'/>x''/, tis is just te consequence of te x' and x'' definitions in section 3 (Fig.2); and (ii) curves 3 and 7 lie iger tan curves 5 and 9,

8 8 i.e., te relative subsidence, -z'/ or -z/ of small samples (curve 5) or tin layers (curve 9) exceeds tat of large samples (curve 3) or tick layers (curve 7). In all oter relations te mutual arrangement of curves -3, 6, and 7 of large samples and layers (>*) and values of teir relative caracteristic sizes are determined by te specific features of te particular soil (see Table ) and te relative initial size (/*). Te srinkage anisotropy caracteristics, z'/x', z'/x'', and z/x, predicted for Delia a and Delia 6 soils are sown in Fig.6. z'/x'' is only sown at =.5cm >* as z'/x''=z'/x' at =3.cm < *. Te following points about te srinkage anisotropy curves in Fig.6 are wort noting.. Different srinkage anisotropy curves can intersect (tis is te consequence of te similar property of te curves in Fig.5). 2. Deflection of curves in Fig.6 wit respect to te isotropy case (z'/x'=z'/x''=z/x=) is maximum for tin layers (curve 5) and ten, for small samples (curve 3). In addition, tis deflection for small samples and tin layers increases wit drying. At qualitative similarity between te anisotropy curves of small samples and tin layers (curves 3 and 5, respectively) for te different soils, te curves quantitatively and appreciably differ from eac oter depending on te particular soil features (see Table ). 3. Unlike in te case of small samples and tin layers, te beavior of srinkage anisotropy curves for large samples (curves and 2) and tick layers (curve 4) qualitatively varies from soil to soil. Indeed, bot z'(w)<x'(w) and z'(w)>x'(w) are possible (cf. curves for Delia a and Delia 6 soils) and bot z(w)<x(w) and z(w)>x(w) are possible (cf. curves 4 for Delia a and Delia 6 soils). At te same time, for bot soils z'(w)>x''(w) (see curves 2 for Delia a and Delia 6 soils). In addition, for bot soils curve 2 of a large sample is iger tan anoter curve of a large sample (curve ) since, by definition, x'>x''. Te similar mutual arrangement of curves 2 (for a large sample) and 4 (for a tick layer), i.e., te inequality, z'(w)/x''(w)>z(w)/x(w) is also observed for te two soils, but tis result is not some trivial consequence of a definition. Te srinkage geometry factors for samples (Eq.(7)) and for layers (Eq.(8)) predicted for Delia a and Delia 6 soils, are sown in Fig.7. A number of points sould be noted in connection wit Fig.7.. Te mutual arrangement of te srinkage-geometry-factor curves for small (curve ) and large (curve 2) samples as well as tin (curve 3) and tick (curve 4) layers, is similar for te two soils. 2. Judging by curves and 3 te volume srinkage of small samples (curve ) and tin layers (curve 3) principally occurs at te expense of te soil subsidence, since all curves and 3 in Fig.7 correspond to r s values close to unity. Note tat suc predicted beavior is natural and expectable. Wit tat te relative subsidence of a tin layer is larger tan tat of a small sample (for bot soils) because curves lie iger tan curves 3 in Fig.7. Tis prediction is also reasonable. 3. For eac of te two soils te appreciable fraction ( 50%) of volume srinkage of a tick-layer matrix or large-sample matrix turns into te crack volume since curves 2 and 4 in Fig.7 correspond to r s Te specific course of eac r s (W) dependence in Fig.7 is determined by te particular pysical features of a corresponding soil (Table ), sample/layer geometry, and relative size, /*. In general, one can state tat te above considered soil srinkage anisotropy caracteristics, as functions of water content, can and sould be predicted from te particular pysical soil features of (i) te texture and intra-aggregate structure, (ii)

9 9 inter-aggregate structure, and (iii) initial sample size or layer tickness (see Table and Certkov (202a)). 8. Results and conclusion In tis work we considered te pysical prediction of soil srinkage anisotropy caracteristics based on soil texture and structure (bot inter- and intra-aggregate) as well as te initial sample/layer size. Te results are as follows. (i). Sample and layer size evolution at srinkage. Te srinkage curves of sample, Y s (W, /*) and layer, Y l (W, /*) (Certkov, 202a) togeter wit te reference srinkage curve Y r (W) (Certkov, 2007a, 2007b) allowed us to find te evolution of a number of caracteristic sizes of soil sample and layer at srinkage (vertical and orizontal size, and size of soil matrix witout cracks). Figure 5 and te points noted in Section 7 illustrate te size dependencies on water content for two particular soils. (ii). Direct presentation of te srinkage anisotropy troug te sample size ratios. Te predicted evolution of vertical and orizontal sizes permitted us to suggest a new, more direct and visual presentation of te srinkage anisotropy concept as te ratios of te sizes. Te illustrative numerical estimates of srinkage anisotropy (Fig.6) and te points noted in Section 7 sow te potential usefulness and informativeness of te presentation. (iii). Presentation of te srinkage geometry factor of samples and layers troug soil structure and initial sample/layer size. Knowing te Y s (W, /*) and Y l (W, /*) curves, one can predict te srinkage geometry factor r s to be a function of pysical soil caracteristics. Te illustrative numerical estimates of te srinkage geometry factor for two particular soils (Fig.7) and te points noted in Section 7, sow te evidence in favor of tat. In turn, te predicted r s value determines te pysically realizing combination of te crack volume and subsidence (as functions of water content) for a given soil layer and local ydrological conditions. According to Certkov (2008a) te srinkage geometry factor and crack network geometry (i.e., different distributions) in a layer are closely connected at srinkage. Tis means tat te crack network geometry can also be totally pysically predicted using te soil structure and initial layer tickness. Te obtained results enable te pysical prediction of all of te srinkage anisotropy caracteristics based on indicated soil and sample features (see Table ). A recent pysical prediction of te soil swelling curve (Certkov, 202b) suggests future extension of te above results to te cases of swelling and srink-swell cycling. Notation a srinkage anisotropy (Eq.(6)) (dimensionless) b, b 2 universal constants in Eq.(3) (dimensionless) c clay content (dimensionless) c * critical clay content (dimensionless) initial sample size of approximately cubic sape at maximum swelling (cm) * critical sample size at maximum swelling (cm) k lacunar factor (dimensionless) m oven-dried layer mass per surface area of te layer (kg) q crack factor (dimensionless) q l crack factor of te layer of initial tickness (dimensionless) q s crack factor of te approximately cubic- or cylindrically-saped sample (wit close eigt and diameter) (dimensionless) r s srinkage geometry factor (dimensionless) W gravimetric water content of soil (kg kg - )

10 0 W W value at maximum swelling (kg kg - ) x current orizontal size of te soil matrix at layer srinkage inside te basis (Fig.3) (cm) x'(w) current orizontal size of te initial cubic sample (cm) x''(w) size of soil matrix inside te sample (see Figs.2b and 2c) (cm) Y specific volume of te soil wit cracks (dm 3 kg - ) Y specific soil volume at maximum swelling (dm 3 kg - ) Y l specific soil volume in te case of cracked, but connected layer (dm 3 kg - ) Y r (W) reference srinkage curve (dm 3 kg - ) Y rz minimum specific volume of te reference srinkage curve (dm 3 kg - ) Y s specific soil volume in te case of a sample wit or witout cracks (dm 3 kg - ) Y l ' specific soil volume of te layer in Bronswijk's approximation (dm 3 kg - ) z(w) current tickness of te real srinking and cracking layer (Fig.3) (cm) z'(w) current eigt of te initially (approximately) cubic sample (Fig.2) (cm) δ universal constant in Eq.(3) (dimensionless) References Baer, J.U., Anderson, S.N., 997. Landscape effects on desiccation cracking in an Aqualf. Soil Science Society of America Journal 6, Boivin, P., Anizotropy, cracking, and srinking of vertisol samples. Experimental study and srinkage modeling. Geoderma 38, Bronswijk, J.J.B., 988. Modeling of water balance, cracking and subsidence of clay soils. Journal of Hydrology 97, Bronswijk, J.J.B., 989. Prediction of actual cracking and subsidence in clay soils. Soil Science 48, Bronswijk, J.J.B., 990. Srinkage geometry of a eavy clay soil at various stresses. Soil Science Society of America Journal 54, Bronswijk, J.J.B., 99a. Relation between vertical soil movements and water-content canges in cracking clays. Soil Science Society of America Journal 55, Bronswijk, J.J.B., 99b. Drying, cracking, and subsidence of a clay soil in a lysimeter. Soil Science 52, Bui, E. N., Mermut, A. R., 989. Orientation of planar voids in vertisols and soils wit vertic properties. Soil Science Society of America Journal 53, Certkov, V.Y., Te srinkage geometry factor of a soil layer. Soil Science Society of America Journal 69, Certkov, V.Y., 2007a. Te reference srinkage curve at iger tan critical soil clay content. Soil Science Society of America Journal 7, Certkov, V.Y., 2007b. Te soil reference srinkage curve. Open Hydrology Journal, -8. Certkov, V.Y., 2008a. Te geometry of soil crack networks. Open Hydrology Journal 2, Certkov, V.Y., 2008b. Te pysical effects of an intra-aggregate structure on soil srinkage. Geoderma 46, Certkov, V.Y., 200. Te soil lacunar factor and reference srinkage. International Agropysics 24, Certkov, V.Y., 202a. An integrated approac to soil structure, srinkage, and cracking in samples and layers. Geoderma 73-74, Certkov, V.Y., 202b. Pysical modeling of te soil swelling curve vs. te srinkage curve. Advances in Water Resources 44,

11 Certkov, V.Y., Ravina, I., 998. Modeling te crack network of swelling clay soils. Soil Science Society of America Journal 62, Certkov, V.Y., Ravina, I., 999. Morpology of orizontal cracks in swelling soils. Teoretical and Applied Fracture Mecanics 3, Coppola, A., Gerke, H. H., Comegna, A., Basile, A., Comegna, V., 202. Dualpermeability model for flow in srinking soil wit dominant orizontal deformation. Water Resources Researc, Vol. 48, W08527, doi:0.029/20wr0376. Cornelis, W.M., Corluy, J., Medina, H., Diaz, J., Hartmann, R., Van Meirvenne, M., Ruiz, M.E., Measuring and modeling te soil srinkage caracteristic curve. Geoderma 37, Crescimanno, G., Provenzano, G., 999. Soil srinkage caracteristic curve in clay soils: Measurement and prediction. Soil Science Society of America Journal 63, Garnier, P., Perrier, E., Angulo-Jaramillo, R., Baveye, P., 997a. Numerical model of 3-dimensional anisotropic deformation and -dimensional water flow in swelling soils. Soil Science 62, Garnier, P., Rieu, M., Boivin, P., Vauclin, M., Baveye, P., 997b. Determining te ydraulic properties of a swelling soil from a transient evaporation experiment. Soil Science Society of America Journal 6, Guidi, G., Pagliai, M., Petruzzelli, G., 978. Quantitative size evaluation of cracks and clods in artificially dried soil samples. Geoderma 9, Hallaire, V., 984. Evolution of crack networks during srinkage of a clay soil under grass and winter weat crop. In J. Bouma and P.A.C. Raats (Eds.) Proc. ISSS Symp. Water and solute movement in eavy clay soils, Wageningen, te Neterlands Aug ILRI, Wageningen, te Neterlands, pp Peng, X, Horn, R., Anisotropic srinkage and swelling of some organic and inorganic soils. European Journal of Soil Science 58, Velde, B., Moreau, E., Terribile, F., 996. Pore networks in an Italian Vertisol: quantitative caracterization by two dimensional image analysis. Geoderma 72,

12 2 Figure captions Fig.. Te scematic illustration of te accepted soil structure (Certkov, 2007a, 2007b). Sown are () an assembly of many soil aggregates and inter-aggregate pores contributing to te specific soil volume, Y; (2) an aggregate, as a wole, contributing to te specific volume U a =U i +U'; (3) an aggregate indicated wit two parts: (3a) interface layer contributing to te specific volume U i and (3b) intra-aggregate matrix contributing to te specific volumes U and U'=U/K; (4) an aggregate indicated wit te intra-aggregate structure: (4a) clay, (4b) silt and sand grains, and (4c) lacunar pores; and (5) an inter-aggregate pore leading, at srinkage, to inter-aggregate crack contributing to te specific volume U cr. U is te specific volume of te intra-aggregate matrix (per unit mass of te oven-dried matrix itself). U' is te specific volume of te intra-aggregate matrix (per unit mass of te oven-dried soil). U i is te specific volume of te interface layer (per unit mass of te oven-dried soil). U cr is te specific volume of cracks (per unit mass of te oven-dried soil). U a is te specific volume of aggregates (per unit mass of te oven-dried soil). K is te aggregate/inter-aggregate mass ratio. Fig.2. Te cubic sample srinkage of initial size at maximum swelling. (a) Horizontal cross-section. x' is te current orizontal sample size at W<W. Possible cracks (black strips) are distributed in te srinking sample. (b) Te same orizontal cross-section. x'' is te current orizontal size of te matrix inside te sample at W<W. Saded bands correspond to te summary crack volume (cracks are mentally collected togeter compared to Fig.2a). (c) Te vertical cross-section of te srinking sample. z' is te current vertical sample size at W<W. Te vertical saded band corresponds to te summary crack volume (cracks are mentally collected togeter). Fig.3. Te layer srinkage of initial tickness at maximum swelling. (a) Te orizontal cross-section of te basis at W<W. Black strips symbolize cracks. (b) Te same orizontal cross-section of te basis at W<W. x is te current orizontal size of te soil matrix inside te basis at W<W. Saded bands correspond to te summary crack volume (cracks are mentally collected togeter compared to Fig.3a). x>x'' in Fig.2. (c) Te vertical cross-section of te srinking layer witin te limits of te basis (overlapping Fig.2c for comparison; x', z' and x'' are as in Fig.2c). x is as in Fig.3b. z is te current tickness of te real srinking and cracking layer at W<W. z<z' in Fig.2. Fig.4. Possible qualitative view of te a(w) dependence (z'(w)/x'(w) or z'(w)/x''(w) or z(w)/x(w)) of a soil sample or layer. Solid line: a>. Dased line: a. Fig.5. Te relative sample/layer sizes of te two soils. Te solid lines correspond to sample case: - x'(w)/ dependence, =.5 cm>*; 2 - x''(w)/, =.5 cm; 3 - z'(w)/, =.5 cm; 4 - x'(w)/, =3. cm<*; 5 - z'(w)/, =3. cm. Te dased lines correspond to layer case: 6 - x(w)/ dependence, =.5 cm; 7 - z(w)/, =.5 cm; 8 - x(w)/, =3. cm; 9 - z(w)/, =3. cm. Fig.6. Te srinkage anisotropy of te two soils. Te solid lines correspond to sample case: - z'(w)/x'(w) dependence, =.5 cm>*; 2 - z'(w)/x''(w), =.5 cm; 3 - z'(w)/x'(w), =3. cm<*. Te dased lines correspond to layer case: 4 - z(w)/x(w) dependence, =.5 cm; 5 - z(w)/x(w), =3. cm. Fig.7. Te srinkage geometry factor of te two soils. Te solid lines correspond to sample case: - =3. cm<*; 2 - =.5 cm>*. Te dased lines correspond to layer case: 3 - =3. cm; 4 - =.5 cm.

13 3a 2 3b 4a 4b 5 4c Fig.

14 a b c x' x' x'' x' x' x' z' x'' x'' Fig.2

15 a b c x z' x' z x x'' x Fig.3

16 Srinkage anisotropy, a 0 0 W z W Gravimetric water content, W (kg kg - ) Fig.4

17 Relative caracteristic sample/layer size Delia a W Total water content, W (kg kg - ) Relative caracteristic sample/layer size Delia Total water content, W (kg kg - ) W Fig.5

18 . 2 Delia a Srinkage anisotropy Total water content, W (kg kg - ) W Srinkage anisotropy Delia 6 W Total water content, W (kg kg - ) Fig.6

19 6 2 Delia a Srinkage geometry factor W Total water content, W (kg kg - ) Srinkage geometry factor Delia W Total water content, W (kg kg - ) Fig.7

20 Table. Input parameters* for pysical prediction of te reference srinkage curve, Yr(W), sample srinkage curve, Ys(W, /*), and layer srinkage curve, Yl(W, /*) for two soils. Te clay content values are from Crescimanno and Provenzano (999); te oter parameters were estimated in Certkov (202a) based on different data from Crescimanno and Provenzano (999) Data source c ρs (kg dm -3 ) Yrz (dm 3 kg - ) W (kg kg - ) k Pz Ulp (dm 3 kg - ) Xmin (mm) Xm (mm) * (cm) Crescimanno and Provenzano (999, Table, Fig.4), soil Delia a As above, soil Delia * Clay content (c), mean solid density (ρs), minimum specific volume of te reference srinkage curve (Yrz), water content at maximum swelling (W), lacunar factor (k), structural porosity in oven-dried state (Pz), specific lacunar pore volume at maximum swelling (Ulp), minimum aggregate size (Xmin), maximum aggregate size at maximum swelling (Xm), critical sample size (*).