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energies Article

A Simple Fractal-Based Model for Soil-Water Characteristic Curves Incorporating Effects of Initial Void Ratios Gaoliang Tao 1 Yuxuan Xia 3 1

2 3

*

ID

, Yin Chen 1

ID

, Lingwei Kong 2

ID

, Henglin Xiao 1 , Qingsheng Chen 1, * and

Hubei Provincial Ecological Road Engineering Technology Research Center, Hubei University of Technology, Wuhan 430068, China; [email protected] (G.T.); [email protected] (Y.C.); [email protected] (H.X.) State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China; [email protected] Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China; [email protected] Correspondence: [email protected]

Received: 25 April 2018; Accepted: 29 May 2018; Published: 1 June 2018

Abstract: In this paper, a simple and efficient fractal-based approach is presented for capturing the effects of initial void ratio on the soil-water characteristic curve (SWCC) in a deformable unsaturated soil. In terms of testing results, the SWCCs (expressed by gravimetric water content) of the unsaturated soils at different initial void ratios were found to be mainly controlled by the air-entry value (Ψ a ), while the fractal dimension (D) could be assumed to be constant. As a result, in contrast to the complexity of existing models, a simple and efficient model with only two parameters (i.e., D and Ψ a ) was established for predicting the SWCC considering the effects of initial void ratio. The procedure for determining the model parameters with clear physical meaning were then elaborated. The applicability and accuracy of the proposed model were well demonstrated by comparing its predictions with four sets of independent experimental data from the tests conducted in current work, as well as the literature on a wide range of soils, including Wuhan Clay, Hefei and Guangxi expansive soil, Saskatchewan silt, and loess. Good agreements were obtained between the experimental data and the model predictions in all of the cases considered. Keywords: soil-water characteristic curve; initial void ratio; air-entry value; fractal dimension; fractal model

1. Introduction The soil-water characteristic curve (SWCC) of a soil reflects the relationship between volumetric water content and matric suction. It is a useful tool for indicating the hydraulic properties of the soil and has various applications in the field of unsaturated soil mechanics, such as strength theory, percolation theory, consolidation theory, and constitutive theory [1–5]. The SWCC is also commonly employed in estimating the shear strength, stress–strain relationships, and permeability of unsaturated soils [6–8]. Over the past decades, a large number of studies have been carried out on the SWCCs of unsaturated soils [6–10]. It has been recognized that SWCCs are affected by various factors, such as pore-size distribution (PSD), particle-size distribution, and dry density [11,12]. One specific factor that affects the SWCC is the porosity of the soil, which can change considerably for the soils with variation of stress and suction states, as well as the stress and suction history of the soil [13]. Zhou et al. [14] Energies 2018, 11, 1419; doi:10.3390/en11061419

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reported that the samples of a given soil with different void ratios could be regarded as entirely different soils. As also stated by Assouline [15], a change of soil porosity can result in a significant change of SWCC, and such a change in soil porosity is a common feature of natural soils. In recent decades, the issue of the effect of soil porosity on the hydraulic properties of a soil has received great attentions of various researchers [16–24]. In the meanwhile, various models have been proposed to account for the effect of initial void ratio on SWCCs. For example, Van Genuchten [6] proposed an equation to empirically describe the SWCC data, in which there are three parameters and the parameter α is the main influence factor of air-entry value. Gallipoli et al. [17] modified the equation by assuming the parameter α to be related to the void ratio by a power law. Tarantino [25] presented a void ratio-dependent SWCC model. For high suction stage, it was assumed that the initial void ratio has insignificant influence on the SWCC data, with a result that the SWCC could be represented by a single equation. This model is very similar to the model proposed by Gallipoli et al. [17], but it incorporates one parameter less than Gallipoli’s model. Masin [23] proposed a hydraulic model that is capable of capturing the dependence of the degree of saturation on the void ratio and suction using the effective stress principle. In this model, three material parameters were required. Most recently, Zhou et al. [14] proposed a method to take account of the effects of initial void ratio on SWCC through introducing one more additional parameter in the existing empirical SWCC equations. Gallipoli et al. [26] formulated an SWCC model considering the effects of hysteresis and void ratio of the soils with various degrees of saturation. However, there are seven parameters in this model. Although substantial contributions have been made in the above models for predicting the water retention behavior of unsaturated soils, the application of these models are still largely restricted due to their complexity. As a result, it is of immense necessity to propose a simple and efficient approach with fewer parameters for estimating the SWCC of unsaturated soils, so that the engineers can readily apply it in practice. Recently, fractal theory has been increasingly recognized as a useful tool to analyze the physico-geometrical properties (e.g., particle-size distribution, pore volume, and pore surface) of porous media [27–31]. It was pointed out that the hydraulic characteristics of porous media are closely related to its pore characteristics [32], thus, fractal theory is also suitable to describe the hydraulic characteristics of porous media. Shi et al. [33] developed a fractal model for describing the spontaneous imbibition of wetting phase into the porous media. Most recently, some researchers have also devoted to fractal-based SWCC model [34–36]. It was reported that this approach is capable of combining the empirical models with the parameters with clear physical meaning, and it can also be readily applied in engineering practice [37]. Nevertheless, most of existing fractal-based SWCC models could not account for the effects of void ratios of soils. Most recently, Khalili et al. [38] proposed a fractal-based model considering volume change, and the fractal dimension of PSD in the model was taken to be void ratio dependent, however, this model has seven parameters, which are not easily determined. Meanwhile, it has been verified for clayey silty sand only. To address the above issues, this paper presents a simple and efficient SWCC model incorporating the effects of initial void ratios by employing the fractal theory. On the basis of experimental results, the SWCCs of the soils at different initial void ratios were found to be mainly controlled by air-entry value, while fractal dimension can be seen as a constant. Combined with fractal theory, a new model for predicting the air-entry value was proposed. Thereafter, a simple and efficient SWCC model capable of predicting the SWCCs at deformed state considering the effects of initial void ratio was established. In contrast to the complexity of existing models, only two parameters are needed in the proposed model. The applicability and accuracy of this model were demonstrated by comparing its predictions with four sets of independent experimental data from the tests conducted in current work, as well as the literatures. Good agreement was obtained between the experimental data and the model predictions in all of the cases considered.

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2. Proposed Model 2.1. Fractal Description of a Soil Assuming continuous pore-size distribution (PSD) over a range of pore sizes (r) between rmin and rmax , in which rmin and rmax are the smallest and largest pores, respectively, the probability density function of pore size r is written as [39] f (r ) = cr −1− D

(1)

where c is a constant, D is the fractal dimension, and r is the effective pore size of the connected pore channel. It is assumed that the minimum pore size rmin is near zero. Then, the total volume V (≤r) of pores having size less than or equal to r can be expressed as V (≤ r ) =

Z r 0

cr −1− D kV r3 dr =

ckV 3− D r 3−D

(2)

where kv is a pore volume shape-related constant corresponding to the volume of the pores. Assuming that pores having a size less than or equal to r are fully filled with water, then the gravimetric water content in the pores of soil particles weighing 1 g is w = ρw V (≤ r ) =

cρw kV 3− D r 3−D

(3)

where ρw is the mass density of water. The soil sample is regarded as fully saturated soil when the largest pores with rmax are filled with water. Then, substituting r with rmax in Equation (3) yields the following expression ws =

cρw kV 3− D r 3 − D max

(4)

where ws is gravimetric water content of the soil in saturated condition. 2.2. Fractal-Based Model for Variation of Soil-Water Characteristic Curve (SWCC) with Initial Void Ratio According to the Young-Laplace equation, the relationship between matrix suction Ψ and effective pore size r can be described as ψ = 2Ts cos α/r (5) where Ts is the surface tension and α is the contact angle. In the constant temperature condition, 2Ts cosα can be assumed as a constant. The matrix suction Ψ corresponding to maximum pore size can be approximately regarded as the air-entry value Ψ a , which can be captured by substituting r with rmax in Equation (5) ψa = 2Ts cos α/rmax (6) Then, substituting Equation (5) into Equation (3), and Equation (6) into Equation (4), respectively, yields the following expressions cρw kV 2Ts cos α 3− D w= ( ) (7) 3−D ψ ws =

cρw kV 2Ts cos α 3− D ( ) 3−D ψa

(8)

Dividing Equation (8) by Equation (7) gives w ψa 3− D =( ) ws ψ

(9)

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Note that Equations (7) and (9) are only valid in the range of Ψ ≥ Ψ a . If matrix suction Ψ is less than Ψ a , the soil sample is assumed to be fully saturated. Then, the fractal model for soil-water characteristic curve is written as ( ψa 3− D w ψ ≥ ψa ws = ( ψ ) (10) w = ws ψ < ψa In the saturated condition, the water content of soil can be determined by ws =

e Gs

(11)

where Gs is the relative density and e refers to the initial void ratio. Then, the variation of SWCC with void ratio can be captured by substituting Equation (11) into Equation (10) ( ψ 3− D w = Ges ( ψa ) ψ ≥ ψa (12) e ψ < ψa w = Gs 2.3. Determination of Model Parameters The main objective of this study is to predict the SWCCs at the deformed state considering the effects of initial void ratio. It should be noted that the SWCC at the reference state is corresponding to the experimental soil sample with the maximum initial void ratio e0 , while the SWCCs at the deformed state refer to the experimental soil samples with the other arbitrary initial void ratio e1 (e1 < e0 ). As observed in Equation (12), the prediction of SWCCs at arbitrary initial void ratio is mainly governed by only two parameters: (i) fractal dimension D and (ii) air-entry value Ψ a at the deformed state. The procedure for determining the model parameters (i.e., D and Ψ a ) are now elaborated as follows. 2.3.1. Fractal Dimension at Reference State By taking the logarithm of both sides of Equation (7), it could be rewritten as ln w

w k v 2Ts cosα ) = ln cρ 3− D ( ψ

3− D

2Ts cosα w kv = ln cρ 3− D + (3 − D ) ln ψ w kv = (3 − D )[ln 2Ts cosα − ln ψ] + ln cρ 3− D 3− D w kv = (3 − D )(− ln ψ) + ln[ cρ ] 3− D (2Ts cosα )

Then, it is easy to obtain Equation (13) ln w ∝ (3 − D )(− ln ψ)

(13)

By plotting the lnw against (−lnΨ), (3 − D) can be evaluated from the gradient k of the graph, then the fractal dimension can be determined as D = 3 − k. 2.3.2. Fractal Dimension at Deformed State Based on the experimental data, it was found by Tao et al. [40] that SWCC expressed by gravimetric water content at different initial void ratios presents a type of “broom shape” distribution. More precisely, SWCCs at different initial void ratios are almost consistent when the high matrix suction is greater than air-entry value. As shown in Figure 1, SWCCs corresponding to initial void ratio e0 , e1 , and e2 (i.e., a-b-c-d, e-b-c-d, and f-c-d) almost overlap at the tail. It should be noted that the SWCC is expressed by gravimetric water content herein. As a result, the fractal dimension D1 for e1 at deformed state is approximatively equal to the fractal dimension D0 for e0 at reference state. That is, the fractal dimension D of saturated soil at deformed state can be assumed as a constant (i.e., D1 = D0 ).

gravimetric water content at different initial void ratios presents a type of “broom shape” distribution. More precisely, SWCCs at different initial void ratios are almost consistent when the high matrix suction is greater than air-entry value. As shown in Figure 1, SWCCs corresponding to initial void ratio e0, e1, and e2 (i.e., a-b-c-d, e-b-c-d, and f-c-d) almost overlap at the tail. It should be noted that the SWCC is expressed by gravimetric water content herein. As a result, the fractal Energies 2018, 11, 1419 5 of 20 dimension D1 for e1 at deformed state is approximatively equal to the fractal dimension D0 for e0 at reference state. That is, the fractal dimension D of saturated soil at deformed state can be assumed as a constant D1 = D0). It should be highlighted thatbeen similar ideas have beenetpresented by Bird et al. It should(i.e., be highlighted that similar ideas have presented by Bird al. [34] and Russell and [34] and [41]. Russell and Buzzi [41]. Buzzi initial void ratio e0 initial void ratio e1 initial void ratio e2 e0>e1>e2

a w=e1/GS

e b f c

w

d low matrix suction stageψ high matrix suction stage a1

ψ ：kPa Figure 1. Schematic sketch soil-water characteristic curves (SWCCs) terms gravimetric water Figure 1. Schematic sketch of of soil-water characteristic curves (SWCCs) in in terms of of gravimetric water content unsaturated soils with different initial void ratios. content of of unsaturated soils with different initial void ratios.

2.3.3. Air-Entry Value at at Reference State 2.3.3. Air-Entry Value Reference State If If thethe SWCC at at reference state is ismeasured, thethe corresponding fractal dimension D0Dcan bebe SWCC reference state measured, corresponding fractal dimension 0 can calculated by following the procedure described above, and the air-entry value Ψ a0 can be calculated by following the procedure described above, and the air-entry value Ψ a0 can be determined determined by best fitting (12) Equation to the experimental SWCCs. SWCC at state reference by best fitting Equation to the(12) experimental SWCCs. Then, the Then, SWCCthe at reference can be state can be expressed as expressed as ( e0 ψa0 3− D0 ( ψ )3− D ψ ≥ ψa0  w =e0Gs ψ a0 (14) 0 = ≥ ) w ψ ψ a0 ψ e0(  w =GGs ψ ψ< a0  s (14)  e 0  2.3.4. Air-Entry Value at Deformed State = w ψ < ψ a0



Gs

Drawing a horizontal line w = e1 /Gs , the horizontal line would have an intersection with the SWCC at reference state. The abscissa of this intersection can be approximately regarded as Ψ a1 2.3.4. Air-Entry Value State 1. By substituting w = e1 /Gs into the first formula in Equation corresponding to e1 , at asDeformed shown in Figure (14), the following expression is obtained Drawing a horizontal line w = e1/Gs, the horizontal line would have an intersection with the

SWCC at reference state. The abscissa of this intersection can be approximately regarded as Ψa1 e1 ψ 3− D0 = ( a0 ) (15) e0 ψ Equation (15) can be then simplified to ψ=

ψa0 e1 1/(3− D0 ) ( e0 )

(16)

The matrix suction expressed by Equation (16) is approximately considered as air-entry value Ψ a1 for e1 . 3. Model Validation To demonstrate the performance of the proposed fractal model (Equation (12)) for SWCCs of unsaturated soils considering effects of the initial void ratio, a series of simulations were carried out

3. Model Validation To demonstrate the performance of the proposed fractal model (Equation (12)) for SWCCs of unsaturated soils considering effects of the initial void ratio, a series of simulations were carried out using a set of independent laboratory tests conducted in the current work, as well as by 6others in the Energies 2018, 11, 1419 of 20 literatures [42–44] on a wide range of soils, including Wuhan Clay, Hefei and Guangxi expansive soil, Saskatchewan silt, and loess. The comparisons and discussions are presented as follows. using a set of independent laboratory tests conducted in the current work, as well as by others in the literatures [42–44] on a wide range of soils, including Wuhan Clay, Hefei and Guangxi expansive soil, 3.1. Wuhan Clay Saskatchewan silt, and loess. The comparisons and discussions are presented as follows.

Pressure plate tests were conducted to determine the SWCCs for samples of clay collected from 3.1. Wuhan Clay the bottom of a foundation ditch in Wuhan, China. The basic physical property index of soil is Pressure plate tests were conducted to determine the SWCCs for samples of clay collected from shown in Table experiment was completed at the Institute ofindex Rockof soil andis shown Soil Mechanics, the bottom 1. of aThe foundation ditch in Wuhan, China. The basic physical property in Wuhan Table Institute Chinese oftheSciences. plate instrument used 1. The of experiment wasAcademy completed at Institute ofThe Rock pressure and Soil Mechanics, Wuhan Institute of in the Chinese Academy of Sciences. The pressure instrument used invalue the experiment, which consists experiment, which consists of pressure cell,plate higher air-entry (HAE) ceramic disc,ofpressure pressure cell, higher air-entry value (HAE) ceramic disc, pressure gage, and nitrogen source, is shown gage, and nitrogen source, is shown in Figure 2. in Figure 2.

Table 1. Basic physical index Wuhan Table 1. Basic physicalproperty property index of of Wuhan clay. clay.

Natural Density Natural Water Natural Water Natural Density Relative Density Relative Density 3 Content(%) (%) (g/cm3) (g/cm ) Content 2.03 21.9 2.03 2.75 2.75 21.9

Liquid Limit Liquid Limit (%) (%) PlasticPlastic Limit (%)Limit (%) 38.9 38.9

20.4

20.4

Figure 2. Pressureplate plate apparatus instrument. Figure 2. Pressure apparatus instrument.

In the tests, seven samples were compacted using hydraulic jack to form different initial dry densities, ranging from 1.30 to 1.71 g/cm3 . The specific test procedure is as follows: (1) The sample with different dry densities, together with the HAE ceramic disc, was saturated. (2) The specimen, together with the stainless steel cutting rings, was placed on the HAE ceramic disc in the pressure cell. (3) The applied air pressure was imposed on the specimen when the pressure cell was sealed. (4) The water drained from the specimens was recorded during the whole process of the test. It was assumed to reach the equilibrium state at the current suction level when the water drainage of specimen was constant, then the next suction level would be imposed. (5) At the end of each suction level step, the drainage valve was closed and then the applied air pressure was released. Meanwhile, the weights of specimens needed to be measured. (6) The above-mentioned procedure was repeated until the whole test was completed. The initial void ratio of the specimen at the loosest state (e0 = 1.115) was regarded as the reference state. Following the procedure presented previously, the fractal dimension for the Wuhan Clay at

level step, the drainageof valve was needed closed to and the applied air pressure wasprocedure released. Meanwhile, the weights specimens be then measured. (6) The above-mentioned Meanwhile, weights of specimens needed to be measured. (6) The above-mentioned procedure was repeatedthe until the whole test was completed. was repeated until theratio whole completed. The initial void oftest thewas specimen at the loosest state (e0 = 1.115) was regarded as the The initial void ratiothe of procedure the specimen at the previously, loosest state 0 = 1.115) was regarded as the reference state. Following presented the(efractal dimension for the Wuhan Energies 2018, 11, 1419 7 of 20 reference state. Following procedure as presented previously, the fractal dimension for theof Wuhan Clay at reference state wasthe determined D0 = 2.869, with a fitting correlation coefficient up to Clay at reference state was determined as D 0 = 2.869, with a fitting correlation coefficient of up to 0.99, while the air-entry value at reference state was estimated to be Ψa0 = 1.66 kPa by fitting reference state was determined as D0 = 2.869, with a fitting correlation coefficient of up to 0.99, while the 0.99, while the value atSWCCs, reference state was estimated Ψa0 = 1.66 kPa by fitting Equation (12) to air-entry the experimental as shown in Figures 3 andto4,be respectively. air-entry value at reference state was estimated to be Ψ a0 = 1.66 kPa by fitting Equation (12) to the Equationexperimental (12) to the SWCCs, experimental SWCCs, as3shown in Figures 3 and 4, respectively. as shown in Figures and 4, respectively. -0.9 -0.9

ln wln w

-1.2 -1.2 -1.5 -1.5

experimental data linear fitting data experimental linear fitting y = 0.131x − 0.835 Ry 2==0.131x 0.99 − 0.835 D2==30.99 − 0.131 = 2.869 R D = 3 − 0.131 = 2.869

-1.8 -1.8 -2.1 -2.1-8 -8

-7 -7

-6 -6

-5 − ln-5ψ − ln ψ

-4 -4

-3 -3

-2 -2

Figure 3. Determination of the values of fractal dimension at reference state through plotting Figure 3. Determination of the values of fractal dimension at reference state through plotting Figure 3.experimental Determination of the (−lnΨ) valuesfor ofWuhan fractal Clay dimension reference state through plotting experimental data ofdata lnwofagainst with e0 =at1.115. lnw against (−lnΨ) for Wuhan Clay with e0 = 1.115. experimental data of lnw against (−lnΨ) for Wuhan Clay with e0 = 1.115.

0.4 0.4

experimental data fitting analysisdata experimental fitting analysis

w w

0.3 0.3 0.2 0.2 0.1 0.1 0.0 -1 0.010 10-1

100 100

101 101

102 103 2 3 10 ψ ：kPa10

ψ ：kPa

104 104

105 105

106 106

Figure 4. Determination of air-entry values at reference state through fitting Equation (12) to the

Figure 4. Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Wuhan Clay with e0 = 1.115. Figure 4. Determination of air-entry reference state through fitting Equation (12) to the = 1.115. experimental SWCCs for Wuhan Clayvalues with e0 at = 1.115.Clay at deformed state were determined using experimental forair-entry Wuhanvalues Clay with As a SWCCs result, the Ψ a ofe0Wuhan Equation (16), as shown in Table 2.

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As a result, the air-entry values Ψa of Wuhan Clay at deformed state were determined using Equation (16), as shown in Table 2. Energies 2018, 11, 1419

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Table 2. Predictions of fractal dimensions and air-entry values at deformed state.

Initial Fractal Air-Entry Initial Fractal Air-Entry Soil Type Table 2. Predictions of fractal dimensions and Soilair-entry Type values at deformed state. Void Ratio Dimension Value/kPa Void Ratio Dimension Value/kPa 1.037 2.89 Initial Void Fractal Air-Entry Initial Fractal Air-Entry 0.54 Void 15.50 Soil Type Soil Type 0.964 Ratio Dimension5.04 Value/kPa Ratio Dimension Value/kPa 0.528 16.50 0.897 1.037 8.74 2.89 Saskatchewan Wuhan clay 2.869 0.501 2.640 19.08 0.54 15.50 0.833 0.964 15.37 5.04 silt/T1 0.528 16.50 0.483 21.12 Saskatchewan 8.74 0.719 0.897 Wuhan clay 0.501 2.640 19.08 2.869 47.28 silt/T1 0.833 15.37 0.466 23.33 0.483 21.12 0.613 0.719 159.7447.28 0.466 23.33 0.613 159.74 0.513 18.77 0.513 18.77 0.490 21.08 Hefei 0.838 60.58 Saskatchewan 0.490 21.08 2.604 22.92 Saskatchewan 0.474 0.838 2.514 60.58 Hefei expansive 2.514 72.89 0.474 2.604 22.92 expansive soil soil 0.766 0.766 silt/T2 silt/T2 72.89 0.454 25.56 0.454 25.56 30.02 0.426 0.426 30.02 0.88 3.73 0.88 3.73 0.824 33.28 Guangxi XianXian ◦ Guangxi expansive soil0.824 2.589 33.28 2.825 9.29 0.750.75 2.825 9.29 0.753 2.589 41.44 Loess (5 C) 0.72 11.73 expansive soil 0.753 41.44 Loess (5 °C) 0.72 11.73

Figure demonstrates that deformed state state using using the the proposed proposed model model Figure 55 demonstrates that the the calculated calculated SWCCs SWCCs at at deformed (Equation (12)) agree well with the experimental data of Wuhan Clay at various void ratios (i.e., e = 1.037, (Equation (12)) agree well with the experimental data of Wuhan Clay at various void ratios 0(i.e., e0 = 0.964, 0.897, 0.613, 1.037, 0.964, 0.833, 0.897,0.719, 0.833,and 0.719, andrespectively). 0.613, respectively). 0.4

0.4 experimental data predicted value

experimental data predicted value

0.3

0.2

w

w

0.3

0.1

0.2

0.1 e =1.037

0.0 10-2 10-1 100

e=0.964

101

102 103 ψ ：kPa

104

105

0.0 10-2 10-1 100

106

101

102 103

ψ ：kPa

(a)

105

106

(b) 0.4

0.4



0.3

w

0.3

w

104

0.2

0.2

0.1

0.1 e =0.897

0.0 10-2 10-1 100

e =0.833

101

102 103 ψ ：kPa

104

105

106

0.0 10-2 10-1 100

(c)

101

102 103

ψ ：kPa

(d) Figure 5. Cont.

104

105

106

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0.4

0.4


0.3

0.3

0.2

0.2

w

w


0.1

0.1 e = 0.719

0.0 10-2 10-1 100

e = 0.613

101

102 103

ψ ：kPa

104

105

106

0.0 10-2 10-1 100 101 102 103 104 105 106 ψ：kPa

(e)

(f)

Figure 5. 5. Measured Measured and Wuhan clay at deformed statestate at (a)ate = (b) e =(b) 0.964; Figure andpredicted predictedSWCCs SWCCsforfor Wuhan clay at deformed (a)1.037; e = 1.037; e= (c) e = 0.897; (d) e = 0.833; (e) e = 0.719; (f) e = 0.613. 0.964; (c) e = 0.897; (d) e = 0.833; (e) e = 0.719; (f) e = 0.613.

3.2. Hefei Hefei and and Guangxi Guangxi Expansive Expansive Soils Soils 3.2. Miao et et al. al. [42] [42] performed performed aa series series of of pressure pressure plate plate tests tests on on Hefei Hefei and and Guangxi Guangxiexpansive expansivesoil, soil, Miao where the specimens were compacted by the static compaction in steel rings at three dry densities, where the specimens were compacted by the static compaction in steel rings at three dry densities, 3, 3 1.54 3, 3respectively. 1.42 g/cm g/cm3,3 ,1.48 1.48 g/cm g/cm , 1.54g/cm g/cm , respectively.The Thephysical physicalproperties propertiesofofHefei Hefei and and Guangxi Guangxi 1.42 expansive soils are summarized in Tables 3 and 4, respectively. The void ratios of the specimens at the expansive are summarized in Tables 3 and 4, respectively. The void ratios of the specimens at loosest statestate (e0 =(e0.915 for Hefei expansive soil, e0soil, = 0.901 Guangxi expansive soil) weresoil) regarded the loosest 0 = 0.915 for Hefei expansive e0 =for 0.901 for Guangxi expansive were as the initial void ratios at reference state. regarded as the initial void ratios at reference state. Similarly, the parameters at reference state for 50.56kPa) kPa) Similarly, for Hefei Hefei expansive expansive soil soil (D (D00 == 2.514, Ψ Ψa0 a0 ==50.56 were obtained as shown air-entry values Ψ aΨata were shown in in Figures Figures66and and7,7,respectively, respectively,while whilethe thecorresponding corresponding air-entry values deformed state were determined using Equation (16), (16), as shown in Table As can be seen in Figure at deformed state were determined using Equation as shown in 2. Table 2. As can be seen in6, the fitting correlation coefficients for fractalfor dimension are up to 0.98, indicating that the SWCCs of Figure 6, the fitting correlation coefficients fractal dimension are up to 0.98, indicating that the Hefei specimens have obvious fractal features. SWCCs of Hefei specimens have obvious fractal features. Table 3. 3. Physical Physical properties properties of of Hefei Hefei expansive expansive soils soils (data (data after after Miao Miao et et al. al. [42]). [42]). Table

Relative Density Limit (%) Relative Density Liquid Liquid Limit (%) Plastic PlasticLimit Limit (%) (%) 2.72 58.6 26.4 2.72 58.6 26.4

Plasticity Index(%) (%) Plasticity Index 32.2 32.2

Table 4. Physical properties of Guangxi expansive soils (data after Miao et al. [42]). Table 4. Physical properties of Guangxi expansive soils (data after Miao et al. [42]).

Relative Density Liquid Limit (%) Plastic Limit (%) Relative Liquid Limit (%) 2.70 Density 61.4 Limit (%) Plastic 30.3 2.70

61.4

30.3

Plasticity Index (%) Plasticity31.1 Index (%) 31.1

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-0.5 -0.5

lnlnww

-1.0 -1.0 yy = = 0.486x 0.486x + + 0.811 0.811 22 R = 0.98 0.98 R2 = D D= = 33 − − 0.486 0.486 = = 2.514 2.514

-1.5 -1.5 -2.0 -2.0 -2.5 -2.5 -3.0 -3.0 -8 -8

-7 -7

-6 -6− ln ψ -5 -5 − ln ψ

-4 -4

-3 -3

Figure 6. 6. Determination Determinationof the values values of of fractal fractal dimension dimension at at reference reference state state through through plotting plotting Figure Figure 6. Determination ofofthe the values of fractal dimension at reference state through plotting experimental dataof of lnwagainst against (–lnΨ)for for Hefeiexpansive expansive soil with with e = 0.915. experimental experimental data data of lnw lnw against (–lnΨ) (–lnΨ) for Hefei Hefei expansive soil soil with ee0000== 0.915. 0.915. 0.4 0.4

experimental experimental data data fitting fitting analysis analysis

ww

0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 -1-1 10 10-1

00 10 100

11 10 101

22 33 10 102 10 103 ψ ψ ：kPa ：kPa

44 10 104

55 10 105

66 10 106

Figure 7. Determination of air-entry values at reference state through fitting Equation (12) to the Figure 7. of air-entry values at reference state Figure 7. Determination Determination air-entry values reference state through through fitting fitting Equation Equation (12) (12) to to the the experimental SWCCs for of Hefei expansive soilatwith e0 = 0.915. 00 = 0.915. experimental SWCCs for Hefei expansive soil with e experimental SWCCs for Hefei expansive soil with e0 = 0.915.

Figure 8 indicates that that the SWCCs SWCCs computed by by the current current model compared compared well with with the Figure Figure 88 indicates indicates that the the SWCCs computed computed by the the current model model compared well well with the the laboratory data data of of the the unsaturated unsaturated Hefei Hefei expansive expansive soils soils at at all all initial initial void void ratios. ratios. laboratory laboratory data of the unsaturated Hefei expansive soils at all initial void ratios. 0.4 0.4

0.4 0.4

0.2 0.2

0.3 0.3

ww

ww

0.3 0.3

0.1 0.1

0.2 0.2 0.1 0.1

e=0.838 e=0.838

0.0 0.0 -1-1 10 10-1

experimental experimental data data predicted predicted value value

experimental experimental data data predicted predicted value value

00 10 100

e=0.766 e=0.766 11 10 101

22 10 102 ψ ψ ：kPa ：kPa

(a) (a)

33 10 103

44 10 104

55 10 105

0.0 0.0 -1-1 10 10-1

00 10 100

11 22 10 10 101 ψ ：kPa 102 ψ ：kPa

33 10 103

44 10 104

55 10 105

(b) (b)

Figure 8. Measured and predicted SWCCs for Hefei expansive soils at deformed state at (a) e = 0.838; Figure 8. Measured and predicted SWCCs for Hefei expansive soils at deformed state at (a) e = 0.838; Figure Measured andMiao predicted (b) e = 8. 0.766 (data after et al. SWCCs [42]). for Hefei expansive soils at deformed state at (a) e = 0.838; (b) e = 0.766 (data after Miao et al. [42]). (b) e = 0.766 (data after Miao et al. [42]).

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11 of 20 11of of20 20 11

The parameters at reference state for Guangxi expansive soils soils (D (D00 == = 2.589, 2.589, Ψ 26.78 kPa) kPa) were were The == 26.78 Theparameters parametersat atreference referencestate statefor for Guangxi Guangxi expansive expansive soils (D 2.589, Ψ Ψa0a0 a0 = 26.78 kPa) were 0 obtained as shown in Figures 9 and 10, respectively, while the corresponding air-entry values Ψ at obtained obtainedasasshown shownininFigures Figures99and and10, 10,respectively, respectively,while whilethe thecorresponding correspondingair-entry air-entryvalues valuesΨΨaaaat at deformed state were determined using Equation (16), as shown in Table 2. As can be seen in Figure deformed state were determined using Equation (16), as shown in Table 2. As can be seen in Figure deformed state were determined using Equation (16), as shown in Table 2. As can be seen in Figure 9, 9,the thefitting fittingcorrelation correlationcoefficients coefficientsfor forfractal fractaldimension dimensionare areup upto to0.97, 0.97,indicating indicatingthat that the SWCCs of 9, the fitting correlation coefficients for fractal dimension are up to 0.97, indicating thatthe theSWCCs SWCCsof of Guangxi specimens have obvious fractal features. Guangxi Guangxispecimens specimenshave haveobvious obviousfractal fractalfeatures. features. -0.5 -0.5 -1.0 -1.0

ln lnw w

-1.5 -1.5

0.411x++0.294 0.294 yy==0.411x 2 0.97 RR2 ==0.97 0.411==2.589 2.589 DD==33−−0.411

-2.0 -2.0 -2.5 -2.5 -3.0 -3.0 -8 -8

-7 -7

-6 -6

-5 -5 − ln − ln ψψ

-4 -4

-3 -3

-2 -2

Figure9. 9. Determination Determinationof thevalues valuesof fractaldimension dimensionat referencestate statethrough throughplotting plotting Figure 9. Determination ofofthe the values ofoffractal fractal dimension atatreference reference state through plotting Figure experimental data of lnw against ( − lnΨ) for Guangxi expansive soil with e = 0.901. experimental data of lnw against (−lnΨ) for Guangxi expansive soil with e 0 = 0.901. 0 experimental data of lnw against (−lnΨ) for Guangxi expansive soil with e0 = 0.901.

0.4 0.4 experimentaldata data experimental fittinganalysis analysis fitting

w w

0.3 0.3

0.2 0.2

0.1 0.1

0.0 0.0 10-1-1 10

1000 10

1011 10

1022 10 1033 10 ：kPa ψψ：kPa

1044 10

1055 10

1066 10

Figure 10. 10. Determination Determinationofofair-entry air-entryvalues valuesatatreference referencestate statethrough throughfitting fittingEquation Equation(12) (12)to tothe the Figure Figure 10. Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Guangxi expansive soil with e = 0.901. 0.901. experimental SWCCs SWCCs for for Guangxi Guangxi expansive expansive soil soil with with ee000==0.901. experimental

Figure11 11indicates indicatesthat thatthe theSWCCs SWCCscomputed computedby bythe thecurrent currentmodel modelcompared comparedwell wellwith withthe the Figure 11 indicates that the SWCCs computed by the current model compared well with the Figure laboratory data of the unsaturated Guangxi expansive soils at all initial void ratios. laboratory data of the unsaturated Guangxi expansive soils at all initial void ratios. laboratory data of the unsaturated Guangxi expansive soils at all initial void ratios.


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experimental data predicted value experimental data predicted value

0.4

0.3

0.3 0.3 w w

w

experimental data predicted value experimental data predicted value

0.4

0.3

0.2 w

12 of 20

0.4

0.4

0.2

0.2 0.2

0.1

0.1

0.1

0.1 e=0.824 e=0.824

(b) e=0.753

0.0 -1 10 100 102 103 101 0.0 -1 0 1 2 10 10 10 103 10 ψ ：kPa ψ ：kPa

104 104

0.0 (b) e=0.753 -1 100 0.010

105 105

10-1

100

101

101

102

2 ψ10 ：kPa ψ ：kPa

103

103

104

104

105

105

(a) (b) (a) (b) Figure 11. Measured and predicted SWCCs for Guangxi expansive soils at deformed state at Figure 11. Measured and predicted SWCCs for Guangxi expansive soils at deformed state at (a) e = 11. Measured and predicted SWCCs for[42]). Guangxi expansive soils at deformed state at (a) e = (a)Figure e = 0.824; (b) e = 0.753 (data after Miao et al. 0.824; (b) e = 0.753 (data after Miao et al. [42]). 0.824; (b) e = 0.753 (data after Miao et al. [42]).

3.3.Saskatchewan SaskatchewanSilt Silt 3.3. 3.3. Saskatchewan Silt The change of theSWCCs SWCCs withthe the variationofofinitial initial voidratios ratiosfor forpressure pressureplate platecells cellstests tests TheThe change of of the change the SWCCswith with thevariation variation of initial void void ratios for pressure plate cells tests performed by Huang [43] on the Saskatchewan silt was simulated in this section for further model performed byby Huang [43] ononthe in this thissection sectionfor forfurther furthermodel model performed Huang [43] theSaskatchewan Saskatchewansilt silt was was simulated simulated in validation.The The physical properties of Saskatchewan silt shown are shown in Table 5. specimens The specimens validation. physical properties of Saskatchewan silt are in Table 5. The were validation. The physical properties of Saskatchewan silt are shown in Table 5. The specimens were were compressed in stainless sample rings by conventional odometer under the same consolidation compressed in in stainless under the the same sameconsolidation consolidation compressed stainlesssample samplerings ringsby by conventional conventional odometer odometer under pressures. Two groups of testswere were conducted. In first the group first group of(T1), teststhe (T1), thevoid initial void pressures. Two groups ofof tests conducted. of tests tests (T1), the initial void ratios pressures. Two groups testswere conducted.In Inthe the first of initial ratios ofspecimens the specimens were 0.692, 0.540, 0.528, 0.501, 0.483, andrespectively, 0.466, respectively, while thevoid initial ofratios the specimens were 0.692, 0.466, respectively, whilethe the initial void of the were 0.692,0.540, 0.540,0.528, 0.528,0.501, 0.501,0.483, 0.483, and and while initial void ratios of specimens in the second group of tests (T2) were 0.525, 0.513, 0.490, 0.474, 0.454, ratios of specimens the secondgroup groupofoftests tests(T2) (T2) were were 0.525, 0.513, ratios of specimens in in the second 0.513, 0.490, 0.490,0.474, 0.474,0.454, 0.454,and and0.426, 0.426, and 0.426, respectively. respectively. respectively. The void ratio the specimen the looseststate stateeee000===0.692 0.692for forT1 T1was was regarded the initial The void ratio the specimenatat atthe theloosest state 0.692 regarded asasas the initial The void ratio ofofof the specimen for T1 was regarded the initial void ratio at reference state. The calibrated SWCC parameters of T1 at reference state for the first tests void ratio at reference state. The calibrated SWCC parameters of T1 at reference state for the first void ratio at reference state. The calibrated SWCC parameters of T1 at reference state for the first (D = 2.640, Ψ = 7.78 kPa) were obtained as shown in Figures 12 and 13, respectively. The corresponding tests = 2.640, = 7.78kPa) kPa)were wereobtained obtained as as shown shown in in Figures 0 (D0(D a0 Ψa0Ψa0 tests =0 2.640, = 7.78 Figures 12 12 and and13, 13,respectively. respectively.The The corresponding air-entry values at deformed state are presented in Table 2. air-entry values at deformed state are presented in Table 2. corresponding air-entry values at deformed state are presented in Table 2. Table5.5.Physical Physical properties properties of of Saskatchewan [43]). Table Saskatchewansilt silt(data (dataafter afterHuang Huang [43]). Table 5. Physical properties of Saskatchewan silt (data after Huang [43]).

Natural Water Natural Water Relative Density Water Content (%) Limit Liquid(%) LimitPlastic (%) Plastic Limit (%) Plastic Relative DensityNatural Liquid Limit (%) Plastic IndexIndex (%) (%) Content (%) Liquid Limit (%) Plastic Limit (%) Relative Density Plastic Index (%) Content0.86 (%) 2.682.68 22.2 0.86 22.2 16.6 16.6 5.6 5.6 2.68 0.86 22.2 16.6 5.6 -1.0 -1.0 -1.5 -1.5

y = 0.360x − 0.614

ln w ln w

2

= 0.99 − 0.614 y R= 0.360x D = 3 − 0.360 = 2.640 R2 = 0.99 D = 3 − 0.360 = 2.640

-2.0

-2.0 -2.5

-2.5 -3.0

-3.0

-6

-5

-4 − ln ψ

-3

-2

-6 -5 -4 -3 -2 − ln ψ Figure 12. Determination of the values of fractal dimension at reference state through plotting Figure 12. Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (−lnΨ) for Saskatchewan silt (T1) with e0 = 0.692. experimental data of lnw against (−lnΨ) for Saskatchewan silt (T1) with e0 = 0.692. Figure 12. Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (−lnΨ) for Saskatchewan silt (T1) with e0 = 0.692.


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0.4 experimental data fitting analysis experimental data fitting analysis

0.4

0.3

0.2

w

w

0.3

0.2

0.1 0.1 0.0 10-1 100 0.0 10-1 100

101 101

102 103 10 ψ 2：kPa103 ψ ：kPa

104 105 106 104 105 106

Figure Figure 13. 13. Determination Determination of of air-entry air-entry values values at at reference reference state state through through fitting fitting Equation Equation (12) (12) to to the the Figure 13. Determination of air-entry values at with reference state through fitting Equation (12) to the experimental SWCCs silt (T1) with experimental SWCCs for for Saskatchewan Saskatchewan silt (T1) ee00 == 0.692. 0.692. experimental SWCCs for Saskatchewan silt (T1) with e0 = 0.692.

Figure presents the comparison of the experimental SWCCs and their predictions, Figure 1414presents the comparison of the experimental SWCCs and their predictions, Figure 14 presents the comparison of the experimental SWCCs and their demonstrating predictions, demonstrating that the SWCCs Saskatchewan of unsaturatedsilt Saskatchewan silt (T1) with the variation of initial that the SWCCs of unsaturated (T1) with the variation of initial void were demonstrating that the SWCCs of unsaturated Saskatchewan silt (T1) with the variation ratios of initial void ratios were captured well. captured well. void ratios were captured well. 0.4

0.4 0.4

0.4

0.3 0.3

w

ww

0.30.3

w

experimental data

experimental data predicted value predicted value

experimental data experimental data predicted value predicted value

0.2 0.2

0.10.1

0.1 0.1

0.20.2

e=0.528 e=0.528

e=0.540 e=0.540

0.00.0 -1 -1 0 0 1 1 22 103 3 10 1044 1010 1010 1010 1010 ：kPa10 ψψ：kPa

1055 10

0.0 0.0 -1 0 1 2 3 4 5 6 10 10-1 10 100 10 101 10102 10103 10104 10105 10 106 ：kPa ψψ ：kPa

1066 10

(a) (a)

(b) (b)

0.4 0.4

0.4

0.4

experimental data experimental data predicted value predicted value

0.3 0.3

experimental data experimental predicted value data

0.3

0.2 0.2

0.2

ww

w

w

predicted value

0.3

0.2

0.1 0.1

0.1

0.1 e=0.501

0.0 e=0.501 -1 100 0.0 10 -1

10

0

10

e=0.483

101 1

10

102 103 ψ2 ：kPa 3

10 10 ψ ：kPa (c) (c)

104 4

10

105

106

5

6

10

10

0.0 e=0.483 -1 100 0.010 -1

10

Figure 14. Cont.

0

10

101 1

10

102 103 104 105 106 ψ ：kPa 102 103 104 105 106 (d) ψ ：kPa

(d)

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0.4 0.4 0.4

experimentaldata data experimental predictedvalue value predicted experimental data predicted value

0.2 0.2 0.2

w

ww

0.3 0.3 0.3

0.1 0.1 0.1 e=0.466 e=0.466

0.0 e=0.466 0.0 0.0 10-1-1 10 1000 10 1011 10 1022 10 1033 10 -1 0 1 2 10 10 10 10 ψ：kPa ：kPa103 ψ ψ ：kPa

1044 10 1055 10 1066 10 4 5 10 10 106

(e) (e) (e) Figure 14. Measuredand and predicted SWCCs Saskatchewan silt specimens (T1) at deformed state Figure14. 14. Measured predicted SWCCs forfor Saskatchewan silt specimens (T1) atat deformed state at Figure and predicted silt specimens (T1) deformed state Figure 14.Measured Measured and predictedSWCCs SWCCsfor forSaskatchewan Saskatchewan silt specimens (T1) at deformed state at (a) e = 0.540; (b) e = 0.528; (c) e = 0.501; (d) e = 0.483; (e) e = 0.466 (data after Huang [43]). (a) e = 0.540; (b) e = 0.528; (c) e = 0.501; (d) e = 0.483; (e) e = 0.466 (data after Huang [43]). at (a) e = e0.540; (b)(b) e =e 0.528; (dataafter afterHuang Huang[43]). [43]). at (a) = 0.540; = 0.528;(c)(c)e = e =0.501; 0.501;(d) (d)ee==0.483; 0.483; (e) (e) ee == 0.466 0.466 (data

The void ratio of the specimen at the loosest state ee000 == 0.525 0.525for forT2 T2was wasregarded regarded as the initial The void ratio of the initial The void ratio thespecimen specimenat atthe theloosest loosest state state asas the initial The void ratio ofofthe the specimen at the loosest state e0 =0.525 0.525for forT2 T2was wasregarded regarded as the initial void ratio atreference reference state. The calibrated SWCC parametersof ofSaskatchewan Saskatchewansilt silt of T2 at reference void ratio at state. reference void ratio reference state.The Thecalibrated calibratedSWCC SWCC parameters parameters ofof T2T2 atat reference void ratio atat reference state. The calibrated SWCC parametersof ofSaskatchewan Saskatchewansilt silt of T2 at reference state for the second tests (D 0 = 2.604, Ψa0 = 17.71 kPa) were obtained as shown in Figures 15 and 16, state for the second tests 0 = 2.604, Ψ a0a0== 17.71 kPa) were obtained as shown in Figures 15 and state the second tests(D (D 0== 2.604, Ψ 17.71 kPa) were obtained as shown in Figures 15 and 16,16, state forfor the second tests (D 2.604, Ψ = 17.71 kPa) were obtained as shown in Figures 15 and 16, 0 a0 respectively. The corresponding air-entry values atdeformed deformedstate stateare arepresented presentedinin in Table 2. respectively. The corresponding air-entry values at deformed Table 2. respectively. The corresponding air-entry values at state are presented Table 2. respectively. The corresponding air-entry values at deformed state are presented in Table 2. -1.0 -1.0 -1.0

lnlnww ln w

-1.5 -1.5 -1.5 -2.0 -2.0 -2.0

y =0.396x 0.396x −− 0.493 yyR===0.396x 0.99 − 0.493 0.99 RRD ==0.99 = 3 −0.396 0.396 = = 2.604 2.604 DD==33−−0.396 = 2.604

-2.5 -2.5 -2.5 -3.0 -3.0 -6 -3.0 -6 -6

-5 -5 -5

-4 -4ψ − ln -4 −−lnlnψψ

-3

-2

-3 -3

-2 -2

Figure 15. Determination of the values of fractal dimension at reference state through plotting Figure 15. Determination of of the the values values of of fractal fractal dimension dimension at at reference reference state state through through plotting Figure 15. Determination Determination plotting Figure 15. of the valuesfor ofSaskatchewan fractal dimension atwith reference state through plotting experimental data of lnw against (−lnΨ) silt (T2) e0 = 0.525. experimental data of lnw against ( − lnΨ) for Saskatchewan silt (T2) with e = 0.525. 0.525. experimentaldata dataof oflnw lnwagainst against(−lnΨ) (−lnΨ)for forSaskatchewan Saskatchewansilt silt(T2) (T2)with withee000==0.525. experimental 0.4 0.4 0.4

ww w

0.3 0.3 0.3

experimental data fitting analysis data experimental experimental data fittinganalysis analysis fitting

0.2 0.2 0.2 0.1

0.1 0.1 0.0 10-1

100 101 102 103 0.0 ψ ：kPa 3 0.0 10-1-1 10 1000 10 1011 10 1022 10 103 10 ：kPa ψψ：kPa

104

105

1044 10 1055 10

106

1066 10

Figure 16. Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Saskatchewan silt (T2) with e0 = 0.525. Figure 16. 16. Determination ofair-entry air-entryvalues values atreference reference statethrough throughfitting fittingEquation Equation(12) (12)to to the the Figure 16. Determination Determination of air-entry values at reference state through fitting Equation (12) to the Figure of at state experimental SWCCs for Saskatchewan silt (T2) with e = 0.525. 0 = 0.525. experimental SWCCs for Saskatchewan silt (T2) with e 0 experimental SWCCs for Saskatchewan silt (T2) with e0 = 0.525.

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Figure presents the comparison of the experimental SWCCs and their predictions, Figure1717 presents the comparison of the experimental SWCCs and their demonstrating predictions, that the SWCCs of unsaturated Saskatchewan silt (T2) with the variation of initial void were demonstrating that the SWCCs of unsaturated Saskatchewan silt (T2) with the variationratios of initial captured well. void ratios were captured well. 0.4


experimental data predicted data

0.3

w

w

0.3

0.2

0.1

0.2

0.1 e=0.490

e=0.513

0.0 10-1

100

101

102 103 ψ ：kPa

104

105

106

0.0 10-1

100

101

102 103 ψ :kPa

(a)

106


experimental data predicted data

0.3

0.3

0.2

0.2

w

w

105

(b)

0.4

0.1

0.1 e=0.474

0.0 10-1

104

100

e=0.454

101

102 103 ψ ：kPa

104

105

0.0 10-1

106

100

101

102 103 ψ ：kPa

(c)

104

105

106

(d) 0.4 experimental data predicted value

0.3

w

0.2

0.1 e=0.426

0.0 10-1

100

101

102 103 ψ：kPa

104

105

106

(e)

Figure Figure 17. 17. Measured Measured and andpredicted predictedSWCCs SWCCsfor forSaskatchewan Saskatchewansilt siltspecimens specimens(T2) (T2)atatdeformed deformedstate stateat (a) e = 0.513; (b) e = 0.490; (c) e = 0.474; (d) e = 0.454; (e) e = 0.426 (data after Huang [43]). at (a) e = 0.513; (b) e = 0.490; (c) e = 0.474; (d) e = 0.454; (e) e = 0.426 (data after Huang [43]).


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3.4. Loess

Wang et al. [44] conducted a series of laboratory tests to investigate the effect of dry density and 3.4. Loess temperature 3.4. Loess on the SWCC of Xi’an loess. The physical properties of Saskatchewan silt are shown in Wang et al. [44] conducted a series of laboratory tests to Table 6. The compacted samples with different densities ofinvestigate 1.2 g/cm3,the 1.4effect g/cmof3, dry 1.5 density g/cm3, and and 1.6 Wang et on al. [44] conducted a series of laboratory testsproperties to investigate the effect of dry density and temperature the SWCC of Xi’an loess. The physical of Saskatchewan silt are shown 3, respectively, were prepared and tested by the high-speed centrifuge method. The g/cmtemperature void on compacted the SWCC of Xi’an loess. The physical properties Saskatchewan shown inratio 3 , 1.4 g/cm3silt 3 , and in Table 6. The samples with different densities of 1.2 of g/cm , 1.5are g/cm of a specimen at compacted the loosestsamples state (ewith 0 = 1.23) was regarded as the initial ratio reference state. 3, 1.4void 3, 1.5 at 3, and 1.6 Table 6. The different of 1.2 g/cm g/cm g/cm 3 , respectively, 1.6 g/cm were prepared and testeddensities by the high-speed centrifuge method. The void ratio 3, respectively, The experimental temperatures wereand controlled atthe 5 high-speed °C, 15 °C, centrifuge 25 °C, andmethod. 35 °C, The respectively. For g/cm tested of a specimen at the were loosestprepared state (e0 = 1.23) was by regarded as the initial void ratio at referencevoid state.ratio The model validation, experimental data at 5 °C was employed. The parameters of the loess at reference of a specimen temperatures at the loosest were state controlled (e0 = 1.23) was ratio at reference state. ◦ C, initial ◦ C, respectively. experimental at 5 ◦regarded C, 15 ◦ C, as 25the and 35void For model experimental were controlled at 5 °C, 15 °C, 25 °C, 35 °C, respectively. For the state The (D 0 = 2.825, Ψ a0 = temperatures 0.55 kPa) were obtained as shown in Figures 18 and 19, respectively, while ◦ validation, experimental data at 5 C was employed. The parameters of the loess at reference state model validation, experimental data at 5 °C was employed. The parameters of the loess at reference corresponding values at deformed with in various void (i.e., e0while = 0.88, (D0 = 2.825,air-entry Ψ a0 = 0.55 kPa) were obtainedstate as shown Figuresinitial 18 and 19, ratios respectively, the0.75, state (D 0 = 2.825, Ψa0 = 0.55 kPa) were obtained as shown in Figures 18 and 19, respectively, while the and 0.72, respectively) arevalues shown in Table state 2. Itwith is shown Figure that the fitting correlation corresponding air-entry at deformed various in initial void 18 ratios (i.e., e0 = 0.88, 0.75, and corresponding values deformed state with various initial void ratios (i.e.,significant e0 = coefficient 0.88, 0.75, 0.72, respectively) arewhich shown in at Table 2. It is shown in Figure thatspecimens the fitting correlation coefficient is up to air-entry 1.00, highlights that the SWCCs of 18 loess have fractal and 0.72, respectively) are shown in Table 2. It is shown in Figure 18 that the fitting correlation is up to 1.00, which highlights that the SWCCs of loess specimens have significant fractal features. features. coefficient is up to 1.00, which highlights that the SWCCs of loess specimens have significant fractal features. Table 6. Physical properties of Xi’an loess (data after Wang et al. [44]). Table 6. Physical properties of Xi’an loess (data after Wang et al. [44]). 6. Physical properties loess (data after Wang etIndex al. [44]). Depth Liquid Limit Plastic Limit (%) Plastic (%) (%) DepthTable Liquid Limit (%)(%)of Xi’an Plastic Limit (%) Plastic Index 2.68 Liquid 30.7 (%) 18.4 12.3 2.68 30.7 12.3 (%) Depth Limit Plastic18.4 Limit (%) Plastic Index

2.68

30.7

18.4

12.3

0.0 0.0

-0.5 -0.5

-1.0

0.175x − 0.892 R2 =y2=1.00 R 3=−1.00 D = = 2.825 -1.5 D = 3 0.175 − 0.175 = 2.825

-1.5 ln w

ln w

-1.0 y = 0.175x − 0.892

-2.0 -2.0 -2.5 -2.5 -3.0 -3.0 -7 -7

-6-6

-5 -4 -5 -4 ln ψ ψ −−ln

-3-3

-2 -2

Figure 18. 18. Determination Determinationofof the the values values of of fractal fractal dimension dimension at at reference reference state state through through plotting plotting Figure

Figure 18. Determination of the values of fractal dimension at reference state through plotting experimental data data of of lnw lnw against against (−lnΨ) (−lnΨ)for forloess loesswith withe0e0= =1.230. 1.230. experimental experimental data of lnw against (−lnΨ) for loess with e0 = 1.230. 0.5

0.5

0.4

experimental data fitting analysis data experimental

fitting analysis

0.4 0.3

w

w

0.3 0.2

0.2 0.1

0.1

0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 ψ ：kPa 0.0

10-3 10-2 10-1 100 101 102 103 104 105 106

Figure 19. Determination of air-entry values at reference state through fitting Equation (12) to the Figure 19. Determination of air-entry values at ψ reference state through fitting Equation (12) to the ：kPa experimental SWCCs for loess with e0 = 1.230. experimental SWCCs for loess with e0 = 1.230.

Figure 19. Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for loess with e0 = 1.230.


17 of 20 17 of 20

The prediction in interms termsofofvariation variation initial void ratio is shown in Figure A very The model model prediction ofof initial void ratio is shown in Figure 20. A20. very good good agreement is obtained with the experimental data,demonstrates which demonstrates the capability of the agreement is obtained with the experimental data, which the capability of the model in model in capturing theofchange SWCCs ratio for unsaturated soils. capturing the change SWCCsofwith voidwith ratiovoid for unsaturated soils. 0.4



0.3

0.2

w

w

0.3

0.1

0.2

0.1 e=0.75

e=0.88

0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 ψ ：kPa

0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 ψ ：kPa

(a)

(b) 0.4 experimental data predicted value

w

0.3

0.2

0.1 e=0.72

0.0 10-3 10-2 10-1 100 101 102 103 104 105 106 ψ ：kPa

(c) Figure 20. 20. Measured SWCCs forfor Xi’an loess specimens at deformed statestate at (a) = 0.88; Figure Measuredand andpredicted predicted SWCCs Xi’an loess specimens at deformed ate(a) e= (b) e = 0.75; (c) e = 0.72 (data after Wang et al. [44]). 0.88; (b) e = 0.75; (c) e = 0.72 (data after Wang et al. [44]).

4. Discussion Discussion 4. There are of SWCC expression, including gravimetric water water content–suction There are generally generallythree threeforms forms of SWCC expression, including gravimetric content– curve (w–Ψ curve), volumetric water content–suction curve (θ–Ψ curve), and degree of saturation–suction suction curve (w–Ψ curve), volumetric water content–suction curve (θ–Ψ curve), and degree of curve (Sr –Ψ curve).curve In the(Scurrent study, the theoretical principle of the proposed model is illustrated saturation–suction r–Ψ curve). In the current study, the theoretical principle of the proposed in Figure 1, where the SWCC presents typeSWCC of “broom shape” distribution in terms of distribution the gravimetric model is illustrated in Figure 1, wherea the presents a type of “broom shape” in water content. If it is necessary to obtain the SWCCs expressed by volumetric water content or degreeby of terms of the gravimetric water content. If it is necessary to obtain the SWCCs expressed saturation, the gravimetric water content can be converted to the volumetric watercan content or degreeto of volumetric water content or degree of saturation, the gravimetric water content be converted saturation. Respectively, the transformation formulas are expressed as the volumetric water content or degree of saturation. Respectively, the transformation formulas are

expressed as

θ = wGS /(1 + e)

= SwG = θ SrwG / (1S /e + e)

(17) (18) (17)

In the drying process, water drains through large pores when the suction is low, which means the Sr = wGto e S /capillary particles around large pores are gradually subjected pressure. As a result, the size of (18) large

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pores decreases gradually. As the suction increases, small pores begin to drain water and the pore size is reduced correspondingly, leading to the shrinkage of the macroscopic volume of soils. The larger the initial void ratio, the more obvious this phenomenon. In the above process, it is only when the water is drained that the pore size can be reduced, and the drainage of water occurs firstly at large pores, and then at small pores. Hence, the radius of pores starting to drain water under a certain suction can be approximatively seen as its initial size. In contrast, in the wetting process, the drainage of water has been completed, and the volume shrinkage of large and small pores has also finished. So, the pore-size distribution characteristics at this stage have great difference from that in saturated state, which leads to the phenomenon of SWCC hysteresis. It should be stated that only the influence of initial void ratio on SWCC in drying path was investigated in this work, while the scenario for SWCC in wetting path will be studied in the future study. 5. Conclusions A simple and efficient fractal-based approach capable of capturing effects of initial void ratio was presented for the SWCC of unsaturated soils. In terms of the experimental data, the SWCCs of the unsaturated soils at different initial void ratios were found to be mainly controlled by the air-entry value, while the fractal dimension could be assumed to be constant. In contrast to the complexity of existing models, only two parameters (i.e., fractal dimension D and air-entry value Ψ a ) were employed in the current model. Determination of the model parameters with clear physical meaning were elaborated. The application of the model to a wide range of experimental data from the tests conducted in the current work, as well as the literatures, was examined. Good agreement was obtained between the experimental data and the model predictions in all of the cases considered. Author Contributions: The research study was carried out successfully with contribution from all authors. The main research idea and manuscript preparation were contributed by G.T. and Q.C.; Y.C. contributed on the manuscript preparation and performed the correlative experiment. L.K. and H.X. gave several suggestions from the industrial perspectives. Y.X. assisted on finalizing research work and manuscript. All authors revised and approved the publication of the paper. Acknowledgments: The financial support from the National Key R&D Program of China (No. 2016YFC0502208), National Natural Science Foundation of China (No. 51708189 and 51409097), and Research project of Hubei Provincial Education Department (No. D20161405) are gratefully acknowledged. Conflicts of Interest: The authors declare no conflict of interest.

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