Determining the soil-water retention curve using mercury intrusion porosimetry test in consideration of soil volume change

Wen-Jing Sun, Yu-Jun Cui

a State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics,Chinese Academy of Sciences, Wuhan,430071,China

b Department of Civil Engineering, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200444, China

c Ecole des Ponts ParisTech, Laboratoire Navier/CERMES, 6-8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, Marne-la-Vallée cedex 2, 77455, France

d Institute for the Conservation of Cultural Heritage, Shanghai, 200444, China

Keywords:Soil-water retention curve (SWRC)Mercury intrusion porosimetry (MIP)Pore size distribution (PSD)Deformable soils

ABSTRACT It is well-known that a close link exists between soil-water retention curve (SWRC) and pore size distribution(PSD).Theoretically,mercury intrusion porosimetry (MIP)test simulates a soil drying path and the test results can be used to deduce the SWRC(termed SWRCMIP).However,SWRCMIP does not include the effect of volume change, compared with the conventional SWRC that is directly determined by suction measurement or suction control techniques.For deformable soils,there is a significant difference between conventional SWRC and SWRCMIP. In this study, drying test was carried out on a reconstituted silty soil, and the volume change, suction, and PSD were measured on samples with different water contents. The change in the deduced SWRCMIP and its relationship with the conventional SWRC were analyzed. The results showed that the volume change of soil is the main reason accounting for the difference between conventional SWRC and SWRCMIP. Based on the test results, a transformation model was then proposed for conventional SWRC and SWRCMIP,for which the soil state with no volume change is taken as a reference. Comparison between the experimental and predicted SWRCs showed that the proposed model can well consider the influence of soil volume change on its water retention property.

1. Introduction

A soil-water retention curve (SWRC) describes the soil-water amount (in terms of gravimetric water content w or volumetric water content θ or degree of saturation Sr)at a given suction s.This curve is essential in analyzing water transfer in unsaturated soils,and is also of paramount importance when modeling the coupled hydro-mechanical behaviors of unsaturated soils (Wheeler, 1996;Sun et al., 2007; Nuth and Laloui, 2008; Sun and Sun, 2012).

Conventional SWRCs are usually investigated using either suction measurement or suction control techniques. However, application of these techniques is usually time-consuming (Aubertin et al., 2003), especially for clayey soils. As volume change can occur when suction changes, the conventional SWRC includes the effect of volume change. There are numerous SWRC models available in the literature,such as Brooks and Corey model(Brooks and Corey,1964),van Genuchten model(van Genuchten,1980),and F-X model(Fredlund and Xing,1994),to name only a few.But these models generally do not account for the effect of volume change.Fredlund(2018)proposed mathematical algorithms combining the shrinkage curve and the SWRC,allowing for separating the effect of volume change from that of degree of saturation.

Based on the pore size distribution(PSD)obtained from mercury intrusion porosimetry(MIP)test, the SWRC in the drying path can be obtained by applying Laplace’s equation(Prapaharan et al.,1985;Delage et al.,1995; Romero et al.,1999; Aung et al., 2001; Simms and Yanful, 2002, 2005; Mu?oz-Castelblanco et al., 2012; Hu et al., 2013; Zhai et al., 2019). It is worth noting that the SWRC derived from MIP result represents the SWRC under a constant void ratio,which is termed SWRCMIP.Accordingly,the derived degree of saturation and suction relationship is termed SrMIP-s, the derived water content and suction relationship as wMIP-s, and the derived void ratio and suction relationship as eMIP-s.

Delage et al.(1995)analyzed the PSDs and the SWRCs of various geomaterials, i.e. siliceous sandstone, clayey sandstone, overconsolidated clay, and compacted silt. A good agreement was observed between SWRCMIPand conventional SWRC for sandstones, while this agreement was not observed for fine-grained soils. Mu?oz-Castelblanco et al. (2012) also reported a significant difference between SWRCMIPand conventional SWRC for loess.These differences were discussed in the literature, but no conclusive explanations were given. For example, Romero et al. (1999)suggested that the differences could arise from the different effects that water and dissolved salts produced on clay fabric compared to the process in mercury intrusion. Simms and Yanful(2002) mentioned the possible pore trapping effect, i.e. MIP only gives the radius of the entrance pore, thus it somewhat overestimates the porous volume associated with the estimated diameter.

Normally, soil microstructure is sensitive to the change of water content,especially for deformable soils.Delage et al.(1995)concluded that soil-water retention property was affected by the changes in microstructure. Mu?oz-Castelblanco et al. (2012) also showed the significant effects of changes in microstructure occurring at the level of clay aggregations and the growing importance of the water adsorption in the clay fraction at high suction. The hydro-mechanical responses of soil take place simultaneously when the soil is subjected to suction changes.That is to say, the total change in degree of saturation is induced by changes in both suction s and void ratio e (Simms and Yanful,2005; Maˇsín, 2010; Romero et al., 2011; Sun and Sun, 2012; Hu et al., 2013; Sun et al., 2014; Della Vecchia et al., 2015; Vaunat and Casini, 2017; Fredlund, 2018; Zhai et al., 2020). Therefore, it can be deduced that microstructural changes may be the reason for the difference between conventional SWRC and SWRCMIP,especially for deformable soils.

Recently,the coupled hydro-mechanical response due to suction changes was investigated by several authors (e.g. Gallipoli et al.,2003; Simms and Yanful, 2005; Sun et al., 2007; Nuth and Laloui,2008; Maˇsín, 2010; Hu et al., 2013; Tsiampousi et al., 2013;Fredlund,2018).Some of them proposed an approach based on the quantitative information derived from MIP data.Simms and Yanful(2005) developed a deformable pore-network model (DPNM) to predict the SWRC based on the evolution of measured PSDs for a compacted clayey soil under isotropic loading and/or desaturation.While in the DPNM model,pores were randomly mapped in space and idealized as a network. Hu et al. (2013) formulated a hysteretic SWRC model to account for the influence of deformation on the variation of saturation based on the changes in PSD function for deformable soils. In their model, the PSD at a deformed state can be obtained by horizontal shifting and vertical scaling of the PSD function from a referred initial state with void ratio e0.The premise of this model is that the shapes of the various PSDs can be considered to be insignificantly different from each other. This is obviously a too strong hypothesis for fine-grained soils, as illustrated by Sun and Cui (2018), testifying that the changes in the aggregate porosity were not negligible. Romero et al. (2011) and Della Vecchia et al. (2015) proposed a physically based conceptual framework for modeling the retention behavior of compacted clayey soils, which considers the PSD function evolution along the hydro-mechanical path. However,this framework contains a large number of calibrated parameters,which limits its application.

In this paper,drying tests were conducted on a reconstituted silty soil. The volume, suction and PSD were measured on samples with different target water contents. Based on the obtained results, the difference between conventional SWRC and SWRCMIPderived from PSD was analyzed.Particular attention was paid to the interrelationship between a series of SWRCMIPand conventional SWRC.Moreover,a transformation model between conventional SWRC and SWRCMIPwas established,allowing prediction of SWRC based on the SWRCMIPseries.In this study,the water retention mechanism associated with the volume change of soil was clearly identified.

2. Material preparation, testing program and calculating method

2.1. Material preparation and testing program

Aeolian Jossigny silt was used in this study.The liquid limit wLis 37%,the plastic limit wpis 19%,and the shrinkage limit wsis 12%.In the Casagrande diagram of plasticity,the soil is located close to the A-line, belonging to low-plasticity clay. The clay-size fraction of Jossigny silt is 34%.

Soil slurry,with a water content 1.5 times the liquid limit mixed with deionized water, was firstly poured into several small containers.Afterwards,the samples in the containers were air-dried to different target water contents,which were selected around wL,wpand ws. The air-drying intervals were short, i.e. 30 m, to avoid occurrence of macro-cracks in samples. After each drying operation, the container was covered for several hours for water homogenization. By repeating these steps, dried samples at different water contents were obtained.

After reaching the target water content, the sample was divided into 4 pieces:(1)one for water content measurement;(2)the second for the volume measurement based on the principle of buoyancy (Delage et al., 2007; Zeng et al., 2017); (3) the third(freeze-dried) for MIP investigation (Delage and Lefebvre, 1984;Delage et al.,1996) using an Autopore IV 9500 mercury intrusion porosimeter (micrometrics), which operates from 3.4 kPa(363.6 μm pore)to 227.5 MPa pressure(5.5 nm pore);and(4)the last for suction measurement using a chilled-mirror dew-point psychrometer (WP4C Dewpoint PotentiaMeter). To measure the low suction of soil,a test apparatus consisting of an odometer cell with 70 mm inner diameter, a porous ceramic disc with an airentry pressure of 50 kPa and a graduated tube with 6 mm inner diameter connected to a water tank was used.More details about this apparatus can be found in Feia et al. (2014) and Sun et al.(2020). Table 1 shows the indices of samples dried to different target states.

2.2. Calculating method

The mercury intrusion process is assimilated to a drying process,in which a non-wetting liquid is penetrating into a porous mediumfull of wetting fluid(Delage et al.,1996;Mu?oz-Castelblanco et al.,2012). The pore diameter can be deduced from the mercury pressure (Romero et al.,1999) as

Table 1Indices of soil samples dried to different target states.

where Tmis the surface tension of mercury (0.485 N/m), d is the pore entrance diameter(μm),θmis the mercury-soil contact angle(taken as 130°in this study), and p is the externally applied intrusion pressure (MN/m2).

The mercury intruded void ratio (eMIP) is calculated as

where Vsis the volume of soil, Vmis the volume of intruded mercury, msis the mass of soil, Gsis the specific gravity, and ρwis the water density.

From the derivative of the cumulative intrusion curve,the pore size density function is obtained:

Based on the PSD obtained from MIP test, the SWRCMIPcan be determined (Prapaharan et al., 1985; Romero et al., 1999; Aung et al., 2001; Simms and Yanful, 2002). The relationship between matric suction (ua-uw) and mercury intrusion pressure p can be deduced from following equation:

where Twis the surface tension of water(0.073 N/m),and θwis the water-soil contact angle (taken as 0°in this study).

Romero et al. (1999) suggested that the degree of saturation Srand water content w corresponding to the equivalent applied pressure should consider the hygroscopic water content related to the adsorbed water which was strongly attracted to the mineral surface,and the equivalent residual water content corresponding to the non-intruded porosity, as

where wsatstands for the saturated gravimetric water content,Srmstands for the non-wetting mercury degree of saturation, and wresis the equivalent residual water content corresponding to the maximum mercury intrusion pressure that the mercury porosimeter can reach.

Srmand wrescan be calculated as follows:

where eMIPmaxis the maximum mercury intruded void ratio,and e is the void ratio corresponding to different target drying states.

Finally, we have

where SrMIPis the degree of saturation obtained from MIP test,and wMIPis the water content derived from MIP test.

Based on the above measurements and calculations, the void ratio and the degree of saturation of soil samples at different target water contents were calculated, and the conventional SWRC and SWRCMIPwere also determined.

3. Experimental results

3.1. Shrinkage behavior and SWRC

Fig. 1 shows the results from the drying tests on the reconstituted Jossigny silt prepared at the initial water content of 1.5 wL.Fig.1a and c depicts the shrinkage behavior,i.e.the changes of void ratio with water content(e-w),and that of degree of saturation with water content (Sr-w), respectively. Fig. 1b depicts the volume change behavior under the effect of suction, i.e. void ratio with suction (e-s). Fig.1d shows Sr-s relationship of Jossigny silt.

The e-w relationship obeys a typical shrinkage characteristic curve of soil, as shown in Fig. 1a, which includes three stages of normal shrinkage, residual shrinkage and no shrinkage. The experimental results initially started from the stage of normal shrinkage, which coincided with the dashed full saturation line,and the samples kept fully saturated, as shown in Fig. 1c. Afterwards, when water content reached wae, the water content corresponding to the air entry value (AEV), the slope of the shrinkage curve decreased, and the stage of residual shrinkage began. At the air entry point,the degree of saturation began to decline,as can be seen from Fig.1c. From the Sr-s relationship in Fig.1d, the corresponding suction at wae, i.e. AEV could be determined at about 180 kPa. When the suction s exceeded the air entry value, Sr-s changed from the saturated to the unsaturated domain. After the water content reached the shrinkage limit ws, the void ratio remained unchanged with further drying,as shown in Fig.1a and b,and no shrinkage stage started.

3.2. Microstructure

Fig. 2 presents the PSD of Jossigny silt during drying. Fig. 2a shows the cumulative intrusion curves. It can be seen that eMIPdecreased in the beginning and became almost unchanged after the water content reached the shrinkage limit. The PSD curves, as shown in Fig. 2b, are the derivative of the cumulative intrusion curves of Nos.(1)-(6)in Fig.2a,plotted in terms of δeMIP/δlog10d as a function of pore entrance diameter d. From Fig. 2b, all the PSD curves present a typical unimodal pattern(Fiès and Bruand,1998).When w ＞ws, significant pore refinement occurred after drying.However, with further drying, the curves began to shift to larger diameter. Sun and Cui (2018) explained this phenomenon as the development of possible micro-fissures of the clay part.Moreover,when w ＜ws,the shift trend of PSD curves ceased.In the meantime,the void ratio almost remained unchanged and reached the minimum value, emin.

3.3. SWRCMIP derived from MIP investigations

Fig.1. Results of the drying process of reconstituted Jossigny silt with initial water content of 1.5 wL. (a) e-w; (b) e-s; (c) Sr-w; (d) Sr-s.

Fig.3 presents the relationship between degree of saturation(Sr)and suction(s).The star marks show the SWRC results determined by suction measurement,and the other marks are corresponding to the samples with different target water contents.It can be noticed that the SWRCMIPsignificantly differs from the conventional SWRC.

SWRCMIPcan be divided into three segments on a semilogarithmic plot, i.e. boundary effect zone, transition zone and residual zone:

(1) In the boundary effect zone,SrMIPwas almost equal to 100%,where almost no mercury intrusion took place.

(2) In the transition zone, sudden drops occurred because of intrusion of the dominant pore diameters. It was also observed that the SrMIP-s curve shifted towards the Sr-s curve in the beginning; however, at the residual shrinkage stage,the SrMIP-s curve began to shift backwards due to the possible presence of drying-induced internal micro-fissures occurring in the clay fractions and in the interface between silt grain and clay particles. More details can be found in Sun and Cui(2018).

(3) In the residual zone,the SrMIP-s curves showed a shifting-up tendency with further drying, and were close to the Sr-s curve. The SrMIPrepresents the volume fraction of the nonintruded space and can be calculated by Eq. (10). The shifting-up of the SrMIP-s curve in the residual zone was the result of the changes of non-intruded void ratio (e - eMIP).The changes of SrMIP-s curve were also related to the microstructure change during drying. Moreover, according to the shifting-up trend, it could be deduced that the SrMIP-s curve of sample with the smallest void ratio (e = emin) almost reached the Sr-s curve. Meanwhile, the SWRCMIPfrom MIP test was the same as the conventional SWRC, which was in agreement with the observation of Delage et al. (1995).

3.4. Sr-s and SrMIP-s relationships

Fig.4 shows the sketch of Sr-s relationship(solid line from point A to B)and SrMIP-s relationship(dashed line from point A to C).Point A marked the coordinate(Sri,si)with void ratio eiand water content wi, and B (Sri+1, si+1) with void ratio ei+1and water content wi+1.From point A to B, when the suction increased from sito si+1, the degree of saturation decreased from Srito Sri+1.The absolute value of change in the degree of saturation when suction increased from

The change in degree of saturation at a constant void ratio eiwhen suction increased from sito si+1followed the SrMIP-s curve from point A to C, and could be described aswhich could be obtained by the SrMIP-s curve at a constant void ratio ei,i.e.dSr(s)e=ei= dSrMIP(s)e=ei.

Therefore, the change in degree of saturation caused by void ratio change under a constant suction(s=si)could bedeterminedwhich could be calculated asThrough drying tests, the relationship between degree of saturation and suction(Sr-s)and that between void ratio and suction(es) were obtained. Combining with the MIP results, the changes of|dSr(e)|／|dSr(s)|and|dSr|／|dSr(s)|with suction were determined,as shown in Fig.5.It can be seen from the changes of|dSr(e)|／|dSr(s)|(dashed line)that with increasing suction,its value decreased from 1 to 0 gradually,indicating that when suction is low,the reduction of degree of saturation is mainly caused by the changes of void ratio. By contrast, when the water content reached the shrinkage limit,the void ratio kept almost unchanged,and the contribution of void ratio to the change of degree of saturation |dSr| vanished.Conversely, with increasing suction, the value of |dSr|／|dSr(s)|increased from 0 to 1 gradually,indicating that when suction is low,the degree of saturation almost remains at 100%. At high suction,the void ratio tended to become unchanged and the reason for the change of degree of saturation is mainly induced by the change of suction, i.e. |dSr| = |dSr(s)|.

Fig.2. PSD of Jossigny silt during drying(data after Sun and Cui,2018):(a)Cumulative intrusion curves; and (b) PSD curves.

For non-deformable soils,the SrMIP-s curve is consistent with the Sr-s curve(Delage et al.,1995),and the value of|dSr|／|dSr(s)|can be approximately taken as 1.On the contrary,for deformable soils,the shapes of SrMIP-s curve and Sr-s curve differ significantly and the value of|dSr|／|dSr(s)|changes from 0 to 1 gradually with increasing suction.

3.5. Three-dimensional (3D) Sr-e-s and SrMIP-e-s surfaces

In order to better visualize the effect of void ratio on SWRC,two diagrams are proposed:one is the 3D Sr-e-s diagram and the other is the 3D SrMIP-e-s diagram,as shown in Fig.6.The SWRC is located on the Sr-e-s 3D surface with the change of void ratio, while the SWRCMIPwith a constant void ratio is located on the 3D SrMIP-e-s surface.

Fig. 3. SrMIP-s and Sr-s relationships.

The F-X equation(Fredlund and Xing,1994),with the correction factor for zero water content at 106kPa suction, was adopted in building the 3D surface for further investigation, as shown in Eq.(11).However,it is worth noting that other suitable models can also be used provided that they allow for the description of data over a full range of suction.

where sresis the suction corresponding to the equivalent residual water content; and a, n, and m are the parameters that affect the shape of the curve.

The 3D SrMIP-e-s surface can be obtained by the following steps:first, the SrMIP-s curve at a constant void ratio e = eiwas derived from the PSD curve obtained from MIP test; second, each SrMIP-s(e=ei)relationship was expressed by the F-X SWRC model,i.e.Eq.(11),and each curve had three corresponding parameters,i.e.a=ei,n = ei, and m = ei. Thus, the function of the parameter changing with the void ratio could be determined. Finally, the 3D SrMIP-e-s surface was built.

From the w-s relationship matched by the F-X SWRC model and the equation eSr= Gsw,the Sr-e-s surface was obtained. After that,several SWRCs at a constant void ratio were obtained using the F-X SWRC model.

Fig. 7 shows the projection of drying test results in Sr/SrMIP-e-s diagram. The thick solid curve is the SWRC in the drying path obtained in this study, and the thick dashed curve is the SWRC projection on the SrMIP-e-s surface. The projections of these two thick curves onto the Sr/SrMIP-o-s, Sr/SrMIP-o-e, and e-o-s surfaces were also shown.It is worth noting that,on Sr-o-s coordinate system,the projection of the thick solid curve is conventional Sr-s relationship in the drying path, with void ratio changing following the projection in the e-o-s coordinate system.

The projection of drying test results on the e-o-s coordinate system is shown in Fig.1b. It can be observed that the void ratio decreased with increasing suction.Fig.8 shows the sketch of the e-s relationship corresponding to the drying test results.It can be seen that each suction sihas a corresponding relationship with the void ratio ei. Combining the test results in Fig.1b and the sketch of e-s relationship in Fig. 8, it can be observed that the water content reached the shrinkage limit at ws= 12%, corresponding to suction ss= 1500 kPa and void ratio es= 0.52. Under further drying, the void ratio remained almost unchanged. When suction reached 106kPa, the void ratio reached the minimum value: e = emin(approximately 0.49).

Fig. 4. Sketch of Sr (or SrMIP)-s relationship.

Fig. 9 shows the projection of the test results on Sr/SrMIP-o-s coordinate system.The stars showed the SWRC results obtained in this study. Correspondingly, the thick solid curve represents the conventional Sr-s relationship.The dashed dot curve represents the SrMIP-s(e=ei)curve,which can be regarded as one of the MIP test results in the study,or as one of the curves selected from the SrMIPe-s surface at any void ratio ei. It is worth mentioning that the corresponding SrMIP-s (e = emin) curve, represented by a dashed curve, was obtained from the established SrMIP-e-s surface when e = emin.

As observed previously, the SrMIP-s curve moved rightwards with void ratio decreasing under the premise that there was no micro-fissure developed during drying. After the water content reached the shrinkage limit, the void ratio approached the minimum value emingradually.It can be reasonably assumed that the Sr-s (e = emin) curve on Sr-e-s surface, the SrMIP-s (e = emin)curve on SrMIP-e-s surface,and the projection of the conventional Sr-SWRC on Sr-o-s coordinate system coincided at high range of suction (see Fig.9). Therefore,the SrMIP-s(e=emin)curve can be taken as a reference, which connects the SrMIP-e-s and Sr-e-s surfaces.

4. Transformation from SrMIP-s curves to Sr-s relationship

From above analyses, it can be noticed that a SWRCMIPcorresponds to a fixed pore structure; however, a real SWRC is affected by the changes of soil volume. It can also be deduced that SrMIP-s curves would move continuously towards the Sr-s curve under the condition of no micro-fissures, i.e. theoretically,a SWRC is a combination of a series of SWRCMIPat different suctions. Based on the finding that the SrMIP-s (e = emin) curve can be taken as a reference, connecting the SrMIP-e-s and the Sre-s surfaces, a transformation model could be established to predict the SWRC by a series of SWRCMIP, accounting for the effect of soil volume change on the soil-water retention property.

4.1. Transformation model

As the Sr-SWRC curve coincides with the SrMIP-s curve when e = emin, as shown in Fig. 9, i.e. Sr(s = si, e = ei) = SrMIP(s = si,e = emin), the difference between Sron the Sr-e-s surface and SrMIPon the SrMIP-e-s surface when s=si,combining with Eq.(10),can be expressed as

Fig. 5. Changes of |dSr(e)|／|dSr(s)| (or |dSr|／|dSr(s)|) with suction.

where eMIP(s=si, e=ei) represents the amount of mercury intrusion for soil sample with void ratio e=ei,s ≤si,and d ≥di. Fig.10 shows the eMIP-s relationship.The solid curve represents that under the condition of e=ei, eMIP(s = si, e = ei) is the mercury intrusion porosity ratio when e=ei,s ≤si,and d ≥di;while the dashed curve represents the case that under the condition of e=emin,eMIP(s=si,e = emin) is the mercury intrusion porosity ratio when e = emin,s ≤si, and d ≥di.

Fig.11 shows the change of PSD function when e decreases from eito emin. eMIP(si, ei) and eMIP(si, emin) can be expressed as

where Ad≥di(e = ei) represents the porosity proportion when d ≥di, which is the proportion of the shaded area with cross grain on the PSD curve when e = ei; and Ad≥di(e = emin) represents the porosity proportion when d ≥di, which is the proportion of the shaded area with vertical stripe on the PSD curve when e = emin. When the void ratio decreased to ei+n, the pore entrance diameter decreased to di+n, and the corresponding suction increased to si+n. Thus, eMIP(s = si+n, e = ei+n) and eMIP(s = si+n, e = emin) can be obtained using the above method,combining with Fig. 11.

Fig. 6. Three-dimensional (3D) Sr (or SrMIP)-e-s diagram.

Fig. 7. Projection of drying test results in Sr (or SrMIP)-e-s 3D diagram.

Fig. 8. e-s relationship.

Fig. 9. Comparison and connection between SrMIP-s and Sr-s relationships.

Therefore, using Eq. (12), the difference between Srand SrMIPwhen s = sican be further expressed as

It can be seen from Fig. 6 that the difference between Srand SrMIP, i.e. Sr-SrMIP(s=si), simplified as Yi, represents the distance between point A on Sr-e-s surface and point AMon SrMIP-e-s surface.When s=si+1, the variation between Srand SrMIPat s = si+1,simplified as Yi+1,represents the distance between point B on Sr-e-s surface and point BMon SrMIP-e-s surface.

Fig.10. eMIP-s relationship.

Fig.11. Change of PSD function with the void ratio decreasing from ei to emin.

The variation between Yi+1and Yi,i.e.ΔY represents the change of(Sr-SrMIP)from point s=sito point s=si+1.It can be expressed as

To summarize,according to Eq.(15),ΔY can be obtained by the following steps. First, the SrMIP-e-s surface can be obtained from at least three MIP experiment results of samples with different void ratios. Second, the SrMIP-s relationship for any void ratio eiand the minimum void ratio emincan be obtained from the deduced SrMIP-e-s surface. Then, the relationships of eMIP-s at e = eiand e = emincan be deduced by the obtained SrMIP-s relationship. The proportional “A” value in Eq. (15) can be obtained from the eMIP-s curves at e =eiand e=emin, or their PSD curves.Finally, the variation Yibetween Srand SrMIPat s = siin Eq. (14)can be obtained. Given the suction increasing step, Yi+1between Srand SrMIPat s=si+1can also be obtained by repeating the above procedures.

Simultaneously, from Fig. 6, ΔY can also be expressed geometrically as

where SrMIPi+1-SrMIPi= ΔSrMIP(i+1)-(i)corresponds to the variation of degree of saturation on the SrMIP-e-s surface, and includes two parts: one caused by the change of suction and the other caused by the change of void ratio,which can be expressed by the following integral:

Finally, using the values of the degree of saturation and water content at suction siand combining Eqs. (15)-(17), the Sri+1and wi+1at si+1can be deduced as follows:

Subsequently, the SWRC can be predicted from a series of SWRCMIPby using the transformation model.

In summary, in the transformation model, the Sr-e-s surface where SWRC is located, and SrMIP-e-s surface where the SWRCMIPwith unchanged void ratio is located, are defined. Based on the finding that the Sr-s(e=emin)curve,the SrMIP-s(e=emin)curve,and the conventional Sr-SWRC coincided at high suction, the soil state with no volume change was taken as a reference, i.e. SrMIP-s(e=emin)curve,which connected the SrMIP-e-s and Sr-e-s surfaces.After that,based on the evolution of PSD curve due to the change of porosity and the variation of SrMIPdeduced from the SrMIP-e-s 3D surface, the degree of saturation can be determined. Finally, the conventional Sr-SWRC was obtained. It is worth noting that the transformation model introduces no more parameters than those in the F-X model.

The transformation model is suitable for saturated samples undergoing drying test, no matter what kind of stress has been imposed on them before saturation.Upon wetting,the SrMIP-s curve would shift leftwards due to soil swelling (increase of porosity).Theoretically,the same mechanism of analysis can be applied.This will be verified in future studies when experimental data are available.

Fig.12. Transformation of SWRCMIP to SWRC in drying of reconstituted Jossigny silt: (a) Sr-s; and (b) w-s.

4.2. Application of the transformation model

By applying the proposed approach, the transformation from a series of SWRCMIPto the SWRC was completed. Fig.12 shows the comparison between the experimental and predicted SWRCs from a series of SWRCMIPof reconstituted Jossigny silt in drying,including Sr-s relationship in Fig. 12a and w-s relationship in Fig.12b.

According to the shrinkage curve of reconstituted Jossigny silt,the minimum void ratio eminis 0.49.In Fig.12a,the star marks show the SWRC test results;the solid curve represents the predicted Sr-s curve by the transformation model;the square marks represent the SrMIP-s relationship, which was obtained as follows: given the suction increasing step,the corresponding void ratio was obtained by the e-s curve,and each point on the e-s curve corresponded to a point on the obtained SrMIP-e-s surface. Then, these points were projected on the Sr/SrMIP-o-s coordinate system, and the SrMIP-s relationship can be obtained.

It can be seen that the Sr-s and the w-s curves predicted by the transformation model were in good agreement with the measured SWRC results, testifying the validity of the proposed model and indicating that the proposed model can satisfactorily account for the influence of soil volume change on its water retention property.

5. Conclusions

In order to analyze the difference between the conventional SWRC and SWRCMIPderived from PSD due to volume change,drying test was conducted on a reconstituted silty soil, together with the volume,suction,and PSD measurements.The changes of a series of SWRCMIPand their relation with conventional SWRC were analyzed.It can be concluded that deformation of thesoil is themain reason for the difference between the conventional SWRC and SWRCMIP.

A transformation model was proposed.The model was based on the finding that the Sr-s(e=emin)curve on Sr-e-s surface,the SrMIP-s(e = emin) curve on the SrMIP-e-s surface, and the projection of the SWRC on Sr/SrMIP-o-s coordinate system coincided at high suction.This model took the soil state with no volume change as a reference,and took the SrMIP-s(e=emin)curveas a reference in theprediction of Sr-SWRC,which connected the SrMIP-e-s and Sr-e-s surfaces.It suggests that the proposed model is suitable for undisturbed and compacted saturated samples undergoing drying path, no matter what kind of stress has been imposed on them before saturation.

Declaration of competing interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

The authors are grateful for Shanghai Key Innovative Team of Cultural Heritage Conservation and the financial support from the National Sciences Foundation of China (Grant Nos. 41977214 and 41572284) and the Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences(Grant No.Z013008).All the tests were conducted at UR Navier-CERMES,Ecole des Ponts-ParisTech.