Min Wng, G.N. Pnde, Stn Pietruszzk, Z.X. Zeng

a Zienkiewicz Centre for Computational Engineering, Swansea University, Swansea, SA1 8EN, UK

b Civil Engineering, McMaster University, Hamilton, Ontario, L8S 4L8, Canada

c école des Ponts ParisTech, Champs-sur-Marne, Marne-la-Vallée, 77455, France

Keywords:Soil water retention curve (SWRC)Gradation curve Pore size distribution (POSD)Unsaturated soil

A B S T R A C T The importance of soil water retention characteristics in modelling the hydro-mechanical response of unsaturated soils has been well recognised by many investigators in recent years. Determination of strain-dependent soil water retention curve(SWRC) is likely to be extraordinarily difficult. The first two authors have recently shown that SWRC can be computed from the gradation curve and the calculation result is consistent with the experimental results obtained from pressure plate tests.In this paper,based on a hypothesis related to change in the pore size distribution (POSD) due to volumetric strain of soil skeleton,a method to compute strain-dependent SWRC is presented.It is found that at initial degrees of saturation higher than 0.8,the influence of volumetric strain may be marginal whilst at initial degrees of saturation lower than 0.8, its influence is likely to be substantial. In all cases, the gradation curve of the soil affects the SWRC.

1. Introduction

The soil water retention curve(SWRC),also called the soil water characteristic curve(SWCC),is defined as the relation between the soil suction and the degree of saturation or volumetric water content. It has been used to model the hydro-mechanical behaviour and conductivity or permeability of unsaturated soils. On the experimental side of investigations, the most common technique for determination of SWRC is the pressure plate test(PPT).Finding SWRC for a particular soil experimentally is time-consuming and expensive.Moreover,most experiments for determining SWRC are conducted at zero strain whilst the results are likely to vary with imposed volumetric strain on the soil.

Over the past three decades, determination of SWRC of soils through analytical, semi-analytical and/or empirical procedures is extensively reported.Initial work was carried out by researchers in soil science, water management and agricultural engineering whose interest was in diverse topics such as plant growth, nutrition, flood control and river basin planning (Brooks and Corey,1964; van Genuchten, 1980; Arya and Paris, 1981; Huang et al.,1998; Gallipoli et al., 2003; Assouline, 2006). By assuming that the air-entry suction and the slope of the water retention curve are both dependent on void ratio, a four-parameter model was proposed by Huang et al. (1998) based on Brooks and Corey (1964)’s result. Gallipoli et al. (2003) developed a void ratio-dependent SWRC model by considering air-entry suction as a power function of void ratio based on the van Genuchten (VG) model (van Genuchten,1980). Moreover, Nuth and Laloui (2008) developed a model where the SWRC after deformation was obtained by shifting the intrinsic shape of SWRC along the logarithmic suction axis based on the strain.

It is obvious that the SWRC of a soil is dependent on its porosity,microstructure of pores as well as saturation (Romero and Simms,2008). The pore size distribution (POSD) measured from mercury intrusion porosimetry(MIP)was first used to predict the SWRC by Prapaharan et al. (1985). This method was then followed and improved by some researchers (e.g. Simms and Yanful, 2002;Beckett and Augarde, 2013). The correlation between SWRC and POSD was investigated and discussed by exploring the fundamentals of PPTs(Wang et al.,2015,2017).Compared to MIP,the PPT can be treated as a gas intrusion test, in which gas breaks through the pores and replaces water under certain pressure. An alternative way, which implicitly describes the relation between pores and SWRC, determines SWRC from the particle size distribution (PSD)(Gupta and Larso,1979;Haverkamp and Parlange,1986;Arya et al.,1999). It uses porosity and specific gravity of soil particles. In this method, the POSD is first computed from PSD using phase relationship and empirical equations for determining pore radii. The computed SWRC is then obtained from the approximate POSD.This method has been further validated through a series of experimental data(Chan and Govindaraju,2004)based on the assumptions that soil particles are spherical in shape.The above prediction methods based on POSD or PSD did not consider the changing POSD under deformation. However, in many cases, strain in unsaturated soils may be large(Ng and Pang,2000;Salager et al.,2013).To consider the effect of imposed strains on SWRC, some researchers tried to link SWRC to the average void ratio of soils after deformation based on the VG model(Tarantino,2009;Gallipoli et al.,2015;Pasha et al.,2016,2017).Some researchers accounted for the evolution of POSD during deformation by horizontal shifting and vertical scaling of the POSD function neglecting the change of intra-aggregate pores(Hu et al., 2013). Zhou and Ng (2014) proposed a simple stressdependent SWRC model where the pore structure was incorporated into a semi-empirical model (Gallipoli et al., 2003). Della Vecchia et al. (2015) proposed a methodology by linking the evolving pore size density function,which is a normalised form of POSD curve obtained from MIP and SWRC.Recently,another straindependent SWRC model was proposed based on shifting and scaling POSD. In addition, hysteresis between main drying and wetting paths was considered by pore non-uniformity(Cheng et al.,2019).

From the point of view of geotechnical engineers, the PSD or gradation curve is a fundamental characteristic of soil and is routinely obtained from standard soil tests. This paper aims at developing a methodology to derive strain-dependent SWRC from PSD based on certain assumptions related to change in pore sizes as a result of imposed volumetric strain(Shin et al.,2015).It should be mentioned that this work only considers the drying phase; to consider the wetting phase, the effect of pore non-uniformity(Cheng et al., 2019) and/or contact angle should be accounted for.

Table 1 Void ratios and specific gravities of studied soils.

2. Determination of POSD and SWRC from PSD

In this section,a method of converting PSD to SWRC as proposed by Arya et al. (1999) is outlined for continuity and clarity. Little known to geotechnical engineers, this method has been used by soil scientists for a considerable time.

(1) Pore volume, degree of saturation and water content

To obtain POSD from the cumulative PSD curve, the latter is firstly divided into many (say N) fractions. In each size fraction,particles are assumed to have the same shape(sphere)and size i.e.diameter. It is further assumed that the specific gravity of solid particles (Gs) for all N fractions is the same, and then the pore volume associated with each size fraction can be obtained from the phase relation (Arya and Paris,1981):

where Viis the pore volume per unit mass in the ith particle size range,Wiis the solid mass per unit mass in the same range,and e is the void ratio.

Next,the cumulative degree of saturation Srand water content w of the sample can be respectively determined by

(2) Pore size and suction

In order to determine the pore size, the volume of resulting pores corresponding to each size fraction in the particle assembly is further approximated as cylindrical capillary tubes whose radii are closely related to the particle size in the fraction. In each size fraction,the particle and void volume can be computed from Vpi=andwhere Riand riare the particle and pore radii in the ith size fraction, respectively; niis the number of spherical particles;and hiis the total pore length.In a cubic packing of uniform-size spherical particles, the total pore length is 2niRi.Given that the real particles are non-spherical and have orientations for each soil,the actual number of particles should exceed ni.Then,the total pore length is assumed in the form ofwhere α is a model parameter related to particle shape and orientation.

Fig.1. Grain size distribution curves of three soils.

Fig. 2. Computed POSD curves of three soils.

Fig. 3. Comparison of computed SWRCs and PPT results.

Fig. 4. Cumulative POSD curves under different strains.

Dividing Vviby Vpi, the equation characterising the pore diameter of each range can be obtained as follows:

where diand Diare the pore size and mean particle diameter in the size fraction i, respectively. nican be determined from the definitions of particle volume (Vpi== Wi/ ρp) and particle density ρpas

By analysing a series of experimental data for many various soils, it was found that α ranged from 1.3 to 1.5 for most of soils(Arya and Paris,1981).As mentioned before,α is a model parameter reflecting the actual particle shape and orientation.The suggested α value ranges from 1.35 to 1.39.

It is well known that the suction Picorresponding to each pore size can be obtained from the Washburn equation as

where Tsis the surface tension;and θ is the contact angle between pore fluid and solid particles,and θ=0°is selected for water during the drying phase.

Based on the aforementioned formulations, a programme is written for converting PSD to SWRC from the gradation curve.Besides the PSD,specific gravity and void ratio for a specific soil are primary input data.

Fig. 5. (a) PSD curve and (b) strain-dependent SWRCs of Tsukuba sand.

3. Comparison of SWRCs computed from gradation curve with physical experiments

In this section,experimental data of different soils will be used to validate the aforementioned method of converting PSD to SWRC.In this study,only experiments based on PPT using‘a(chǎn)xis translation’concept(Vanapalli et al.,2008)are adopted and it is considered to be reliable for matric suctions up to a maximum value of 1500 kPa in PPT.For comparison, three soils are selected.Two low plasticity clay (CL) and silty sand (SM)) samples are from Nam et al. (2010)where PSD curves and SWRCs of a number of river bank sands were obtained from PPT. The third soil (ES) chosen is the one for which PSD, POSD and SWRC results are presented in Wang et al.(2015, 2017). The void ratios and specific gravities of these soils are summarised in Table 1.

Figs.1 and 2 show the PSD and computed POSD curves of these soils. Comparison of computed and experimentally observed SWRCs is shown in Fig. 3, where the surface tension Ts= 72 × 10-6kN/m is assumed, and a close correlation can be observed.For soils CL and ES,the computed SWRCs match the PPT results very well;while,for soil SM,the overall shape of computed SWRC matches that obtained from PPT except for a couple of points.

It is noticed that the proposed model is only applicable to in situ samples. We tried to predict SWRC for reconstituted samples, but the results were not good. This is one of the limitations of this method.

Fig. 6. Strain-dependent SWRCs of low-plasticity clay CL.

4. Strain-dependent POSD and SWRC

Currently, most of the SWRC tests are conducted on soil samples without initial strain or overburden pressure. Here, it is emphasised that it is only the volumetric component of the total imposed strains on which the SWRC depends. These strains may be imposed on a soil sample or are obtained at a point from the analysis of a boundary value problem. The strain-dependent SWRCs are required for computational models in investigation of mechanical behaviour of unsaturated soils. Recently, a hypothesis related to change in POSD as a result of volumetric strains was proposed for computing permeability of rocks and rock-like materials using a pipe network model(Shin et al.,2015).The same hypothesis for the change of pores diameters is used here.

Hypothesis.A volumetric strain (compaction/dilation) causes a decrease or increase in the diameters of the pores, which is assumed to be proportional to their original diameter of the pores prior to imposition of strain.The overall change in porosity(or void ratio) is such that it corresponds to the magnitude of imposed strain.

The justification of above hypothesis can be supported by the following work. Based on the theory of elasticity, an analytical solution, indicating the change in pore radius under hydrostatic stress/strain is linearly proportional to pore radius,was derived (Chau, 2012). Gor et al. (2015) found a similar scenario that a linear relation exists between the pore size and isothermal compressibility of fluid filled pores from experimental tests.

Fig. 7. Strain-dependent SWRCs of low-plasticity clay ES.

Fig. 8. Strain-dependent SWRCs of silty sand SM.

It is noted that the variation of void ratio e at a certain level of volumetric strain εvcan be obtained by

The above hypothesis can be directly employed to compute the evolution of POSD from PSD of soils under external loading. Recognising that the SWRC is closely related to the POSD, the straindependent SWRC can be determined from the evolving POSD curves based on Eqs.(2)and(6).Fig.4 shows the POSD of the soil CL chosen for study at εv= 3%, 6% and 10%.

To validate the proposed method,the Tsukuba sand(Gallage and Uchimura, 2010) with initial void ratio e = 1.25 and specific gravity (Gs= 2.75) was first selected to perform the numerical simulation of strain-dependent SWRC.Fig.5a shows the gradation curve of Tsukuba sand.The SWRCs of initial Tsukuba sand and the one under strain of ε = 0.187 are shown in Fig.5b.As few papers reported the PSD and SWRC under various strains at the same time,only the Tsukuba sand under strain magnitude of ε = 0.187 was used to compare the simulated one. It can be found that both the SWRCs of initial sample with void ratio (e = 1.25) and the one under strain of ε = 0.187 agree well with experimental data except for a few points.

To investigate the strain effect on the water retention characteristic of different soils,the SWRCs of low-plasticity soils CL and ES and silty sand SM are given in Figs.6-8,respectively.It is observed that the whole shape of SWRCs is shifted to the right when soils undergo deformation.It means that the breakthrough of air into the pores of the soil is more arduous in denser states, which was recently reported by Salager et al. (2013). The proposed method is capable of capturing the strain-dependency of SWRC. In addition,by comparing the SWRCs of different soils under volumetric strain loading, it is found that the effect of strain on the water retention shape of coarse-grained soils (SM) is more significant than that of fine-grained soils (CL and ES). When the degree of saturation exceeds 0.8,the gas is likely to be presented as discrete bubbles.It is first experimentally discovered and mathematically formulated by Hong et al.(2019,2020) that the gas bubble can either enhance or suppress the volumetric strain behaviour of the soil,depending on the back pressure. These additional effects may further affect the water retention behaviour of the soil.

5. Conclusions

This paper presents a method for computing volumetric strain-dependent SWRCs of unsaturated soils from the gradation curve.The POSD obtained from gradation curve,initial void ratio or porosity and specific gravity of soil grains is modified to account for the imposed volumetric strain assuming that the change in pore diameters is proportional to current diameter.The strain-dependent SWRC is then computed in the usual way from the relation between suction and corresponding pore sizes.It is seen that at initial degrees of saturation higher than 0.8 in the Sr-suction relation, the influence of volumetric strains may be marginal; whilst at initial degrees of saturation lower than 0.8, its influence is likely to be substantial. It is noticed that the effect of strain on the SWRC of coarse-grained soils is more pronounced than that of fine-grained soils. Moreover, results of this study indicate that the influence of strains on SWRC is likely to be of little significance if imposed volumetric strain is less than 5%.

This work only accounts for the effect of volumetric strain on the drying phase of SWRC.How to consider the shear strain effect and the hysteresis will be our next step.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

List of notation

N Number of fractions

GsSpecific gravity of solid particles

ViPore volume per unit mass in the ith particle size range

WiSolid mass per unit mass in the ith particle size range

e Void ratio

SrDegree of saturation

w Water content

DiMean particle diameter corresponding to each size range

diPore size corresponding to each size range

α Model parameter

niNumber of spherical particles in the ith particle size range

PiSuction corresponding to each pore size

TsSurface tension

θ Contact angle between soil particles and the fluid

εvVolumetric strain