Estimation of unimodal water characteristic curve for gap-graded soil

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Estimation of unimodal water characteristic curve for gap-graded soil Alfrendo Satyanaga 1, Harianto Rahardjo ⇑, Qian Zhai 1 School of Civil and Environmental Engineering, Nanyang Technological University, Singapore Received 25 September 2016; received in revised form 1 June 2017; accepted 12 June 2017

Abstract Soils with a bimodal grain-size distribution (gap-graded soils) can be associated with unimodal or bimodal soil-water characteristic curves (SWCCs). Many equations have been developed to estimate SWCCs using grain-size distribution curves to overcome the high cost and long duration of SWCC experiments. Most of the equations are limited to the estimation of the SWCCs of soils with a unimodal grain-size distribution. Few studies have been conducted on the estimation of unimodal SWCCs for gap-graded soils. In this paper, procedures, equations and computer codes are proposed for estimating the unimodal SWCCs of gap-graded soils. The proposed equations are found to perform well in estimating the unimodal SWCCs of gap-graded soils. Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Grain-size distribution; Unimodal soil-water characteristic curve; Soil mixture; Gap-graded soil

1. Introduction Soil-water characteristic curves (SWCCs) describe the variations in water content with respect to soil suction (Fredlund and Rahardjo, 1993). Fredlund (2006) suggested plotting SWCCs in terms of the volumetric water content (hw) against the matric suction (ua-uw). The typical shape of SWCCs is sigmoidal; the curves are commonly plotted on a logarithmic scale over the entire range in suction. The main variables of SWCCs are saturated volumetric water content hs, residual volumetric water content hr, air-entry value wa and water-entry value ww. The air-

Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author at: School of Civil & Environmental Engineering, Nanyang Technological University, Blk N1, #1B-36, Nanyang Avenue, Singapore 639798, Singapore. E-mail addresses: [email protected] (A. Satyanaga), chrahardjo@ ntu.edu.sg (H. Rahardjo), [email protected] (Q. Zhai). URL: http://www.ntu.edu.sg/cee/ (H. Rahardjo). 1 Address: School of Civil & Environmental Engineering, Nanyang Technological University, Blk N1, N1-b1a-01a, Nanyang Avenue, Singapore 639798, Singapore.

entry value is defined as the matric suction at which air first enters the largest pore of the soil. The residual volumetric water content is defined as the volumetric water content at which further increases in matric suction do not result in any significant decreases in the volumetric water content. Matric suction at the inflection point is defined as the matric suction at which the volumetric water content of the soil specimen decreases rapidly. Soils with different textures and grain-size distributions have different SWCCs (Gallage and Uchimura, 2010; Rahardjo et al., 2012c). Sandy soils usually have a low air-entry value and an SWCC with a steep slope (Fig. 1). The air-entry value of silty soils is higher than that of sandy soils due to the presence of smaller pores. Grain-size distribution with bimodal characteristics is commonly observed in residual and colluvial soils (Rahardjo et al., 2012a; Zhang and Chen, 2005). These types of soils are known as gap-graded soils. Some gapgraded soils result in dual porosity soils where the soil-water characteristic curve (SWCC) shows bimodal behaviour (Rahardjo et al., 2014, 2012b; Stoicescu et al., 1998), while other gap-graded soils have an SWCC with

https://doi.org/10.1016/j.sandf.2017.08.009 0038-0806/Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Satyanaga, A. et al., Estimation of unimodal water characteristic curve for gap-graded soil, Soils Found. (2017), https:// doi.org/10.1016/j.sandf.2017.08.009

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A. Satyanaga et al. / Soils and Foundations xxx (2017) xxx–xxx

Gravimetric water content, w (%)

50

Indian Head Till

40

30

Processed Silt 20

Silty Sand 10

Beaver Creek Sand 0 10-2

10-1

100

101

102

103

104

105

106

Matric suction, ua-uw (kPa)

Fig. 1. SWCCs from different types of soil (modified from Fredlund et al., 2012).

unimodal characteristics (Priono et al., 2016). Continuous mathematical functions are required to represent the grain-size distribution of soils and SWCCs. A mathematical equation for a grain-size distribution curve can be used to classify the soil and to estimate the SWCC (Fredlund et al., 2012). On the other hand, a mathematical equation for SWCCs can be used to determine the SWCC variables, such as the air-entry value, the inflection point, the residual water content and the residual suction (Wijaya and Leong, 2016; Zhai and Rahardjo, 2012; Sheng and Zhou, 2011; Sheng, 2011; Fredlund and Xing, 1994; Kosugi, 1994). The mathematical equation for SWCCs can also be used to minimize the time required for the experimental works performed to determine SWCCs in the laboratory. Many equations have been proposed to estimate the SWCCs from the grain-size distribution since the measurement of SWCCs is relatively time-consuming and expensive. Huang et al. (2010) classified the equations used to estimate SWCCs into two groups: equations to estimate SWCCs based on the statistical relationship among the soil properties (Meskini-Wishkaee et al., 2014; Fredlund et al., 2012; Schaap and Leij, 1998; Vereecken et al., 1989) and equations to estimate SWCCs based on the physicoempirical approach (Chin et al., 2010; Hwang and Powers, 2003; Haverkamp et al., 1999; Arya and Paris, 1981). Arya and Paris (1981) proposed the idea of dividing the soil element into several fractions in a discrete domain with each fraction having the same porosity for estimating the SWCCs for soil as the unimodal grain-size distribution. There are two major assumptions made in this model: (i) that the solid volume in any given assemblage can be approximated as that of uniform-size spheres defined by the main particle radius for the fraction and (ii) that the volume of the resulting pores can be approximated as that of uniform-size cylindrical capillary tubes whose radii are related to the main particle radius for the fraction. Based on this model, a grain-size distribution will be translated into a pore-size distribution. Then, the pore radii can be converted into matric suction using the capillarity equa-

tion. The cumulative pore volume with respect to the pore radii can be divided by the total volume to generate the volumetric water content with respect to the pore radii. The model developed by Arya and Paris (1981) performed well in estimating the SWCCs of soils with grain-size distributions that have unimodal characteristics. All other existing equations can only be used to estimate unimodal SWCCs for soils with a unimodal grain-size distribution (i.e., Perera et al., 2005; Fredlund et al., 2012; Zapata, 1999) or to estimate bimodal SWCCs for soils with a bimodal grain-size distribution (i.e., Alonso et al., 1987; Bagherieh et al., 2009). None of the existing equations can be used to estimate the unimodal SWCCs for soils with a bimodal grain-size distribution (gap-graded soils). Therefore, several equations are proposed in this study to describe the relationship between variables in a bimodal grain-size distribution of gap-graded soil and variables in a unimodal SWCC. In addition, the computer codes for the computation of unimodal SWCCs for gap-graded soils are developed. The scope of this work includes the measurement of SWCCs using a Tempe cell and a pressure plate in the laboratory for different compacted soils. The results of the SWCC tests in this study and the SWCC data from published literature are used to validate the proposed equations for the estimation of a unimodal SWCC for a gap-graded soil. 2. Mathematical equations for best fitting grain-size distribution and soil-water characteristic curve An appropriate mathematical equation with a clear physical definition is required to represent the grain-size distribution curve of soil and a soil-water characteristic curve (SWCC) since the equation will be used to relate the grain-size distribution curve and the SWCC. The term for the physical definition refers to the ability of the parameters in the equation to represent the variables of the grainsize distribution of the soil and the SWCC. In this study, Satyanaga et al.’s (2013) grain-size distribution equation was used to best fit the grain-size distribution of gapgraded soil since the parameters of the equation represent the variables of the grain-size distribution. The equation for the best fitting grain-size distribution with bimodal characteristics is as follows: 

 0:075 P ¼ 1  0:15 ln 1 þ d  1 2 0 d max 1 d m1 ln d max 1 d A  4W 1 @erfc sd1   13 0 2 d m2 ln ddmax d max 2 A5 þW 2 @ðb1 Þ þ ðb2 Þerfc ð1Þ sd2 where b1 = 1 when d  dmax2; b1 = 0 when d > dmax2 b2 = 0 when d  dmax2; b2 = 1 when d > dmax2

Please cite this article in press as: Satyanaga, A. et al., Estimation of unimodal water characteristic curve for gap-graded soil, Soils Found. (2017), https:// doi.org/10.1016/j.sandf.2017.08.009

Sand

Silt

Gravel

dmax2

Laboratory data Proposed equation

90

Subcurve 1

80 70

dm1

60

dmax1

50 dm2

40

sd1

30

Subcurve 2

20 10 sd2

ð2Þ

Subscripts 1 and 2 represent subcurves 1 and 2, respectively. All parameters in Eq. (1) are related to the variables of the grain-size distribution curve, as shown in Fig. 2. Eq. (1) consists of eight unknown parameters which require a numerical solution to obtain their values for best fitting the laboratory data of the grain-size distribution. Therefore, it is necessary to provide a suitable initial value for each parameter to minimize the complexity of Eq. (1). The initial value of each parameter can be obtained from the grain-size distribution curve in Fig. 2. Initial values W1 and W2 can be estimated from the mass of the soil particles in the first and second subcurves, respectively, as obtained from the laboratory data. An iterative nonlinear regression procedure that is provided in the Microsoft Excel software (Dodge and Stinson, 2007) can be used to adjust all the parameters to best fit the equation to the laboratory data of the grain-size distribution. The equation for best fitting the SWCC with the unimodal characteristics is as follows:  1 0 ln 1 þ ww r A hw ¼ @ 1   106 ln 1 þ w r 8 2 0 0  w w 1193 < = ln w aw a m AA 5  4hr þ ðhs  hr Þ@1  ðbÞerfc@ : ; s ð3Þ where b = 0 when w  wa; b = 1 when w > wa hw = calculated volumetric water content hs = saturated volumetric water content w = matric suction under consideration (kPa) wa = parameter representing the air-entry value of the soil (kPa) wm = parameter representing the matric suction at the inflection point of the SWCC (kPa) s = parameter representing the geometric standard deviation of the SWCC hr = parameter representing the residual volumetric water content

0 10-4

-3

10

-2

10 10-1 100 Particle diameter (mm)

101

102

Fig. 2. Grain-size distribution with bimodal characteristics (modified from Satyanaga et al., 2013).

0.40

Volumetric water content, w

erfc ¼ the complimentary error function
2 Z X 1 x pffiffiffiffiffiffi exp  ¼ dx 2 2p 1

Clay

100

Percent Passing (%)

P = cumulative grain-size distribution of soil d = soil particle diameter under consideration (mm) dmax = parameter representing the maximum diameter of the soil particle (mm) dm = parameter representing the geometric mean of the soil particle diameter (mm) W = parameter representing the percentage of the soil particle within coarse- or fine-grained particles sd = parameter representing the geometric standard deviation of the GSD curve

Lab data Proposed equation

0.35 0.30

s

0.25 0.20 0.15 0.10 0.05

a r

0.00 10-2

10-1

100 101 102 103 104 Matric suction, ua-uw (kPa)

105

106

wr = parameter representing the matric suction corresponding to the residual volumetric water content (kPa) All parameters in Eq. (3) are related to the SWCC variables shown in Fig. 3. Eq. (3) consists of six unknown parameters which require a numerical solution to obtain their values for best fitting the laboratory data of the SWCC. Therefore, it is necessary to provide a suitable initial value for each parameter to minimize the complexity of Eq. (3). The initial value for each parameter can be obtained from the SWCC in Fig. 3. An iterative non-linear regression procedure that is provided in the Microsoft Excel software (Dodge and Stinson, 2007) can be used to adjust all the parameters to best fit the equation to the laboratory data of the SWCC. 3. Soils used in this study A total of 16 data sets (Table 1) from the SoilVision database (SoilVision, 2002), Rahardjo et al. (2004),

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A. Satyanaga et al. / Soils and Foundations xxx (2017) xxx–xxx

Table 1 Soil data sets used for development of procedure to predict SWCC from bimodal GSD. Soil name

Saturated water content, wsat (%)

Dry density, qd (Mg/m3)

Gravel (%)

Sand (%)

Silt (%)

Clay (%)

Source

11372 11491 11492 11493 11494 11498 11520 11538 12462 Nakhonnayok Omkoi BTG-4 GM1a GM2a GM3a GRSa

18.86 32.02 39.6 25.52 37.4 44.54 30.57 45.54 25.07 33.00 60.25 35.7 18.33 22.26 26.48 32.08

1.49 1.43 1.29 1.31 1.19 1.17 1.46 1.80 1.61 1.41 1.80 1.14 1.74 1.68 1.62 1.44

0.81 0 1.1 1.6 0 0 0 0 2.6 6.20 3.00 0.00 50.00 50.00 50.00 5.00

0 22.2 54.9 56.0 71.4 29.5 53.0 28.9 66.3 21.40 30.50 33.8 0.00 0.00 0.00 36.00

17.5 61.8 34.4 31.8 13.2 45.2 37.3 48.5 27.6 42.40 43.89 39.4 36.70 36.90 31.10 39.90

56.9 16.0 9.6 10.6 15.4 25.3 9.7 22.6 3.5 36.20 25.61 26.9 13.30 13.10 18.90 19.10

SoilVision (2002) SoilVision (2002) SoilVision (2002) SoilVision (2002) SoilVision (2002) SoilVision (2002) SoilVision (2002) SoilVision (2002) SoilVision (2002) Jotisankasa et al. (2009) Jotisankasa et al. (2009) Rahardjo et al. (2004) Rahardjo et al. (2012c) Rahardjo et al. (2012c) Rahardjo et al. (2012c) Rahardjo et al. (2012c)

Jotisankasa et al. (2009) and Rahardjo et al. (2012c) was used to develop the relationship among the index properties, the grain-size distribution and the SWCC of the gapgraded soil. A total of 6 independent sets of soil data from published literature was used to evaluate the proposed equations for estimating the SWCC with unimodal characteristics for gap-graded soil (Table 2). The published data on the grain-size distribution and the SWCC were obtained either from the plots or from the tables of the original publications. The soils from the published data include undisturbed soil and residual soil with different percentages of sand, silt and clay (Table 2). A total of 6 independent sets of soil data from laboratory tests carried out in this study was also used to evaluate the proposed equations for estimating the SWCC with unimodal characteristics for gap-graded soil. Laboratory tests were carried out on 6 soil mixtures with different percentages of kaolin and sand and different initial conditions. Compacted soils were used in this study in order to ensure the homogeneity of the soil specimens for better analyses (Indrawan et al., 2006; Goh et al., 2010). The compacted soils were produced from mixtures with different percentages of coarse kaolin and Ottawa sand. These are required since the proposed equations for estimating the unimodal SWCC must be applicable to a wide range of gap-graded soils.

A characterization of the soil index properties was conducted in accordance with the ASTM standard, as presented in Table 3. The compaction curve was obtained from the Standard Proctor compaction test performed according to ASTM D0698-07E01 (2009). Rahardjo et al. (2012c) and Satyanaga et al. (2013) indicated that gapgraded soil compacted at dry density and a water content associated with the wet of optimum as well as that compacted at dry density and a water content associated with the maximum dry density and the optimum water content generated unimodal SWCCs. Therefore, the soil specimens in this study were also generated from sand-kaolin mixtures compacted at dry density and a water content associated with the wet of optimum. Identical specimens with the same initial conditions were prepared for different types of tests. Therefore, soil specimens were produced from static compaction in this study. The static compaction tests were carried out following the procedure adopted from Rahardjo et al. (2004). The static compaction was performed in three layers with a thickness of 10 mm for each layer. The measurement of the SWCC was performed using a Tempe cell and a pressure plate in accordance with ASTM D6838-02 (2008). The SWCC test using the Tempe cell was conducted on a soil specimen for matric suctions up to 100 kPa, whereas the SWCC test using the pressure plate was carried out on a soil specimen for matric suctions

Table 2 Properties of soil from published literature used to evaluate proposed equations for estimating SWCC of gap-graded soil. Soil name

Saturated water content, wsat (%)

Dry density, qd (Mg/m3)

Gravel (%)

Sand (%)

Silt (%)

Clay (%)

Source

SB9 SB11 BTG-8 BTG-14 GM1b GM2b

21.67 20.32 32.8 24.45 17.4 20.6

1.80 1.72 1.71 2.00 1.74 1.68

76.6 68.4 0 0.00 50 50

23.4 26.7 30 63.30 0 0

0.0 3.0 60.2 26.50 36.7 36.9

0.0 1.9 9.8 10.20 13.3 13.1

SoilVision (2002) SoilVision (2002) Rahardjo et al. (2004) Rahardjo et al. (2004) Rahardjo et al. (2012c) Rahardjo et al. (2012c)

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A. Satyanaga et al. / Soils and Foundations xxx (2017) xxx–xxx Table 3 Standards for index property tests. Index property tests

standards

Specific gravity Grain-size distribution Liquid limit Plastic limit Soil classification (Unified Soil Classification System)

ASTM ASTM ASTM ASTM ASTM

5

Table 4 Maximum dry density and optimum water content of sand-kaolin mixtures used in laboratory experiments conducted in this study.

D0854-06E01 (2009) D0422-63R07 (2009) D4318-10 (2010) D4318-10 (2010) D2487-10 (2010)

between 100 kPa and 1500 kPa. The important part of the Tempe cell and the pressure plate is the porous ceramic disc that was saturated in a desiccator using de-aired distilled water. The measurement of the SWCC was carried out in small incremental steps to obtain accurate results for the SWCC. The soil specimen was weighed every day until the weight of the soil specimen reached equilibrium before the matric suction was increased to a higher value. The water compartment and the ceramic disc were flushed regularly to avoid the desaturation of the ceramic disc and the water compartment. The compaction curves for mixtures with different percentages of sand and kaolin are presented in Fig. 4. The maximum dry density and optimum water content for each soil mixture are presented in Table 4. Notations k and s in Table 4 stand for kaolin and sand, respectively. The soil mixture with 20% sand and 80% kaolin (M8) has the lowest dry density and the highest optimum water content. The maximum dry density increases and the optimum water content decreases with the increasing percentages of sand. This is due to the fact that clay particles absorb more water than sand particles. Soil mixtures at dry density and a water content of 95% of the maximum dry density on the wet of optimum were selected as the initial conditions for the SWCC tests on soil mixtures with different percentages of sand and kaolin. Table 5 shows the results of the index property tests on the selected soils. Soils M3, M4 and M5 were classified as

Soils

Maximum dry density, qd (Mg/m3)

Optimum water content, w (%)

M3 M4 M5 M6 M7 M8

1.95 1.94 1.84 1.67 1.61 1.53

9.40 10.00 14.00 16.50 19.50 21.00

(70S30K) (60S40K) (50S50K) (40S60K) (30S70K) (20S80K)

SM (silty sand), whereas soils M6, M7 and M8 were classified as MH (sandy silt with high plasticity). The amount of void is higher for the higher percentage of clay particles, resulting in the higher initial void ratio of the soil specimens. The grain-size distribution data of soil specimens M3 to M8 tested in this study were modelled with Eq. (1). This is because two distinct groups of soil particle size were observed in the grain-size distribution data for soil specimens M3 to M8 from laboratory tests carried out in this study (Fig. 5). Table 6 shows that the geometric means of soil particles dm1 and dm2 for specimens M3 to M8 are 0.706 mm and 0.007 mm for large and small particles, respectively. The maximum sizes of the large particles (dmax1) and the small particles (dmax2) are 2 mm and 0.015 mm, respectively. The geometric standard deviations of the large (sd1) and the small (sd2) particles are 0.007 and 0.86, respectively. Parameters dm1, dm2, dmax1 and dmax2 show that specimens M3 to M8 consist of medium sand and silt. The smaller value for parameter sd1, as compared to that for parameter sd2, shows that the medium sand is distributed more uniformly within soil specimens M3 to M8 than silt. In general, the parameters of Eq. (1) for best fitting the grain-size distribution curve are related to the physical properties of the soil as well as to the shape of the grain-size distribution for soils M3 to M8. 4. Development of model to estimate unimodal soil-water characteristic curve

2.0

M3 M4

Dry density,

d

(Mg/m3)

1.9

M5 M6

1.8

M7 M8

1.7

1.6

1.5

1.4

0

5

10

15

20

25

30

Water Content, w(%) Fig. 4. Compaction curves of kaolin-sand mixtures.

35

Measuring SWCCs in the laboratory and in the field is relatively time-consuming and expensive. Therefore, it is very useful to estimate the SWCCs, especially in the initial stages of a project, when the available information is limited. Previous studies showed that gap-graded soils may be associated with unimodal SWCCs. Rahardjo et al. (2012c) observed that gap-graded soils with no distinct differences between the large and the small particle sizes produced a uniform pore-size distribution and that their SWCCs have unimodal characteristics. Satyanaga (2015) indicated that gap-graded soils with a percentage of gravel higher than 5% generated unimodal characteristics of SWCCs. Satyanaga et al. (2013) concluded that a gapgraded soil compacted at dry density and a water content associated with the wet of optimum, as well as optimum

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Table 5 Summary of index properties of sand-kaolin mixtures. Index properties

M3

M4

M5

M6

M7

M8

Specific gravity, Gs Liquid limit, LL (%) Plastic limit, PL (%) Plasticity index, PI (%) Dry density, qd (Mg/m3) Water content, w (%) Saturated water content, wsat (%) Soil classification according to USCS

2.65 36.8 21.6 15.2 1.85 14.5 26.8 SM

2.62 41.2 24.8 16.4 1.84 15.0 27.6 SM

2.59 46.7 27.4 19.3 1.75 17.5 30.6 SM

2.60 48.9 28.2 20.7 1.59 22.8 36.3 MH

2.63 53.5 31.4 22.1 1.53 24.5 37.5 MH

2.62 57.8 33.6 24.2 1.45 27.0 39.2 MH

unimodal characteristics from the grain-size distribution with bimodal characteristics. Rouault and Assouline (1998) indicated that the relationship between the diameter of the soil particles (d) and the radius of the soil pores (r) is non-linear, as follows:

Clay

Silt

Sand

Gravel 100

Percent pasing (%)

90 80 70

50 40 30 20 10

ð4Þ

d ¼ urv

60

where u and v are parameters related to particle shape and soil packing, respectively. The soil pore radius in Eq. (4) can be related to the pressure head using the Young-Laplace equation, as shown in Eq. (5).

M3-Lab data M4-Lab data M5-Lab data M6-Lab data M7-Lab data M8-Lab data Best fitted using

Equation 4

0 102

101

100

10-1

10-2

10-3

10-4

Particle diameter (mm) Fig. 5. Grain-size distributions of the sand-kaolin mixtures.

Table 6 Summary of grain-size distribution parameters of sand-kaolin mixtures based on Eq. (1). GSD parameters

M3

M4

M5

M6

M7

M8

Cu %sand (%) %silt (%) %clay (%) dmax1 (mm) dmax2 (mm) dm1 (mm) dm2 (mm) sd1 sd2 W1 W2

100 70 25.3 4.7 2 0.015 0.706 0.007 0.007 0.86 0.70 0.30

175 60 32.8 7.2 2 0.015 0.706 0.007 0.007 0.86 0.58 0.42

233 50 43.7 6.3 2 0.015 0.706 0.007 0.007 0.86 0.49 0.51

170 40 52.4 7.6 2 0.015 0.706 0.007 0.007 0.86 0.45 0.65

25 30 63.4 6.6 2 0.015 0.706 0.007 0.007 0.86 0.35 0.75

12.5 20 72.8 7.2 2 0.015 0.706 0.007 0.007 0.86 0.15 0.85

conditions, generated a unimodal SWCC. In addition, Satyanaga et al. (2013) carried out parametric studies on gap-graded soils. They observed that a gap-graded soil with a dry density higher than 1.44 Mg/m3 and a saturated water content between 18% and 33% is associated with the unimodal characteristics of SWCCs (Satyanaga, 2015). In this section, procedures for the estimation of unimodal SWCCs for gap-graded soil are presented. Eqs. (1) and (3) are used to establish the relationship between the grain-size distribution of soil and the SWCC. In addition, statistical analyses were carried out to understand the important factors affecting the estimation of SWCCs with

h¼

2/ cos h A ¼ qw gr r

ð5Þ

where h = pressure head r = pore radius / = surface tension between water and air h = contact angle qw = density of water g = acceleration of gravity A = (2/ cos h)/(qwgr) Brutsaert (1966) proposed using the value of A = 0.149 cm2 for the air-water-soil system. Therefore, Eq. (5) can be rewritten as follows: ln h ¼ ln

0:149 r

Substituting Eq. (4) into Eq. (6) yields
1=v d ln h ¼ ln 0:149  ln u Eq. (7) can also be written as follows:
 ln d  ln u ln h ¼ ln 0:149  or v 
 ln d  ln u h ¼ exp ln 0:149  v

ð6Þ

ð7Þ

ð8Þ ð9Þ

In order to obtain the corresponding matric suction, Eq. (9) can be rearranged as follows: 

u exp ln 0:149  ln dln v ð10Þ w¼ 10

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7

where w is in kPa and d is in cm Since the soil particle diameter is commonly measured in terms of the mm unit, Eq. (10) can be written as follows: h  i u exp ln 0:149  lnðd=10Þln v ð11Þ w¼ 10 Rearranging Eq. (11) yields Eq. (12) that can be used to estimate parameter wm in Eq. (3). " m !# ln 0:1d u wm ¼ 0:1 exp lnð0:149Þ  ð12Þ v where dm is the parameter representing the geometric mean of the soil particle diameter (mm) In this study, parameters u and v are proposed to be functions of the soil properties. Eq. (1) is used to best fit each piece of grain-size distribution data in Table 1, whereas Eq. (3) was used to best fit each SWCC of the soil data in Table 1. The regression analyses were carried out to find the suitable relationships among the parameters in Eq. (1), the parameters in Eq. (3), parameters u and v and the soil properties. The goodness of the fit of the proposed equation, for describing the relationship between the parameters in Eqs. (1) and (3) with the experimental data, was quantified using R-squared (R2). A value of R2 close to 1 indicates a good fit between the data from the experiments and the proposed equation. The regression analysis results indicated that parameter v is always equal to 0.5. Parameter u for coarse-grained soils is a function of the coefficient of uniformity (Cu) of the soil, as shown in Eq. (13) (Fig. 6). On the other hand, parameter u for fine-grained soils is a function of dry density, as shown in Eq. (14) (Fig. 6). uðcoarseÞ ¼ 0:028C u þ 9:21

ð13Þ

uðfinesÞ ¼ 1:031qd þ 19:53

ð14Þ

where u(coarse) = parameter related to the shape of the soil particles for soil with a percentage of passing less than or equal to 50% (P200  50%) u(fines) = parameter related to the shape of the soil particles for soil with a percentage of passing larger than 50% (P200 > 50%) Cu = d60/d10 = coefficient uniformity of the soil d60 = diameter of the soil particles corresponding to 60% finer in the grain-size distribution curve of the soil (mm) d10 = diameter of the soil particles corresponding to 10% finer in the grain-size distribution curve of the soil (mm) qd = dry density of the soil (Mg/m3) The results of the regression analyses also indicated that parameter wa of the coarse-grained soil is a function of wm and Cu (Fig. 7 and Eq. (15)), parameter wa of the finegrained soil is a function of wm, dm1 and dm2 (Fig. 7 and

Fig. 6. Relationships between parameter u and coefficient of uniformity for coarse-grained soil and relationship between parameter u and dry density for fine-grained soil.

Fig. 7. Relationship between wm/wa and Cu for coarse-grained soil and relationship between wm/wa and dm2/dm1 for fine-grained soil.

Eq. (16)), parameter wr is a function of the diameter of the soil particles corresponding to 10% finer in the grainsize distribution curve of the soil (Fig. 8 and Eqs. (17) and (18)), parameter s for coarse-grained soil is a function of the dry density of the soil (Fig. 9 and Eq. (19)) and parameter s for fine-grained soil is a function of the coefficient uniformity of the soil (Fig. 9 and Eq. (20)). wa ðcoarseÞ ¼

wm 0:429ðC 0:613 Þ u

wa ðfinesÞ ¼ 3:167

w
 m   0:5

ð15Þ ð16Þ

d m2 d m1

1:42 Þ wr ðcoarseÞ ¼ 0:512ðd 10

ð17Þ

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8

A. Satyanaga et al. / Soils and Foundations xxx (2017) xxx–xxx 106 105

-1.42 rcoarse = 0.512 d10 2 R = 0.839

104 r kPa

• Best fit grain-size distribution using Equation 1 (determine dmax1, dmax2, dm1, dm2, W1, W2, sd1 and sd1) • P200 50 % (coarse-grained soil) or P200 > 50 % (fine-grained soil) • Determine D10, D30, D60, Cu, d, wsat

Coarse soil Fine soil

-1.53 rfine = 0.771 d10 2 R = 0.954

103

Check Cu,

and wsat

Unimodal SWCC

102 101 100 10-4

d

10-3

10-2 d10 (mm)

10-1

Coarse Soil (P200 ≤ 50 %)

Fine Soil (P200 > 50 %)

Measure wsat calculate

Measure wsat calculate

100

m

funcon of dm, Cu

a

funcon of

m,

Cu

s

m

a

funcon of dm,

funcon of

m,

s

d

dm1, dm2

Fig. 8. Relationship between wr and d10 for coarse- and fine-grained soils. r

Cu 7

101

102

funcon of D10

s funcon of

103

104 s = -0.771

funcon of D10

s funcon of Cu

105 r

6

d

r

d

+ 9.716

=0

r

=0

Fig. 10. Procedures to estimate SWCC with unimodal characteristics for gap-graded soils.

R2 = 0.967

5 4

s

5. Evaluation of proposed model using soil data from published literature

3 2 s = 9.89 c -0.17 u

1

R2 = 0.913

0 8

12

Dry density,

16 d

20

(kN/m3)

Fig. 9. Relationship between s and dry density of coarse-grained soil and relationship between s and coefficient uniformity of fine-grained.

The performance of the proposed equations in estimating SWCCs with unimodal characteristics for gap-graded soils is evaluated using two criteria. The first criterion is the root mean squared error (RMSE). Smaller values for the RMSE indicate a better predictive capability of the proposed equations. RMSE is defined as follows: rffiffiffiffiffiffiffiffi SSE RMSE ¼ n

ð21Þ

where wr ðfinesÞ ¼

1:53 0:771ðd 10 Þ

ð18Þ

sðcoarseÞ ¼ 0:471qd þ 9:716

ð19Þ

sðfinesÞ ¼ 9:89ðC 0:17 Þ u

ð20Þ

Based on the relationships among the soil properties, the parameters in the grain-size distribution equation and the parameters in the soil-water characteristic equation, procedures for estimating the SWCCs were established. The procedures for estimating the SWCCs with unimodal characteristics for gap-graded soil are summarized in Fig. 10. The computer codes for the computation of the estimated unimodal SWCCs are presented in Appendix A. The computer codes were written using the Visual Basic platform. After estimating the parameters in the SWCC equations, the SWCC for gap-graded soils can then be established.

SSE ¼

n X ðy li  y mi Þ2

ð22Þ

i¼1

SSE = Sum of Square Error Yli = SWCC data from laboratory tests Ymi = estimated SWCC n = number of investigated data The second criterion uses the average relative error (ARE). In this study, the performance of the proposed equations in estimating the SWCC successfully is considered good if the ARE is less than or equal to 10%. The ARE is defined as follows: Pn

y li y mi

i¼1 y li ARE ¼  100 ð23Þ n

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A. Satyanaga et al. / Soils and Foundations xxx (2017) xxx–xxx

Gravel

100

Sand Coarse Medium

Fine

Clay

SB9-published data SB11-published data BTG14-published data Best fitted using Equation 1

90 80

Percent pasing (%)

Silt

70 60 50 40 30 20 10 0 102

101

100

10-1

10-2

10-3

10-4

Particle diameter (mm) Fig. 11. Grain-size distributions of soils SB9, SB11 and BTG14 from published literature with bimodal characteristics.

Gravel

100

Sand Coarse Medium

Fine

Silt

Clay

BTG8-published data GM1b-published data GM2b-published data Best Fitted using Equation 1

90

Percent pasing (%)

The estimated SWCCs using the proposed model in this study were also compared with the estimated SWCCs using Arya and Paris’s (1981) model to evaluate the performance of the proposed model. Prior to the estimation of the SWCC, the grain-size distribution of the selected soil was best fitted using Eq. (1) (Figs. 11 and 12). The parameters of Eq. (1) for best fitting the gap-graded soils are presented in Table 7. Then, the parameters of Eq. (1) in Table 7 were incorporated into the proposed equations in this study (Eqs. (16)–(22)) to estimate the SWCCs of the corresponding soils. The estimation of SWCCs with unimodal characteristics was carried out following the procedures presented in Fig. 10. Comparisons of the estimated SWCCs and the SWCC data from published literature are presented in Figs. 13– 15. The parameters of the estimated SWCCs are shown in Table 8. The values of parameter wa for soils SB9, SB11, BTG14, BTG8, GM1b and GM2b are 800, 1, 150, 100, 70 and 60 kPa (Table 8), respectively, which correspond to the air-entry values of those soils (Figs. 13–15). The values of parameter wm for soils SB9, SB11, BTG14, BTG8, GM1b and GM2b are 11,316, 0.57, 1851, 1098, 1107 and 1069 kPa (Table 8), respectively, which correspond to the suction at the inflection point (diameter of the pore with the highest frequency among the other pores), as shown in Figs. 13–15. The values of parameter wr for soils SB9, SB11, BTG14, BTG8, GM1b and GM2b are 10,000, 300, 20,000, 100,000, 20,000 and 20,000 kPa (Table 8), respectively, which correspond to the matric suction for the residual volumetric water content if it is plotted manually from the SWCCs in Figs. 13–15. Other parameters in Table 8 are also compared with the properties of the SWCCs, as shown in Figs. 13–15. It is concluded that the parameters in Table 8 are related to the mechanical properties of soil (air-entry value, suction at the inflection point and residual suction). In other

9

80 70 60 50 40 30 20 10 0 102

101

100

10-1

10-2

10-3

10-4

Particle diameter (mm) Fig. 12. Grain-size distributions of soils BTG8, GM1b and GM2b from published literature with bimodal characteristics.

Table 7 Summary of parameters of Eq. (1) for best-fitting gap-graded soil from published literature. Variables

SB9

SB11

BTG8

BTG14

GM-1b

GM-2b

dmax1 (mm) dmax2 (mm) dm1 (mm) dm2 (mm) sd1 sd2 W1 W2 dmin (mm) dr (mm)

2 0.1 0.121 0.009 0.095 0.031 0.8 0.2 0.0001 0.075

30 1 16 0.6 0.35 0.25 0.79 0.21 0.0001 0.075

2 0.1 0.07 0.008 0.1 0.05 0.42 0.58 0.0001 0.075

25 0.1 0.442 0.010 0.02 0.11 0.79 0.21 0.0001 0.075

30 0.24 16 0.001 0.05 0.023 0.48 0.52 0.0001 0.075

30 0.02 10 0.001 0.034 0.30 0.47 0.53 0.0001 0.075

words, the parameters of the proposed SWCC equation represent the variables of the SWCC. Figs. 13–15 show that the estimated SWCC agreed closely with the measured SWCC data from the published literature. The value of the AREs of the proposed equation for estimating the SWCCs of soils SB9, SB11, BTG14, BTG8, GM1b and GM2b are 1.8%, 9.8%, 9.5%, 4.5%, 1.1% and 1.3%, respectively. The proposed equations perform well in estimating the SWCCs for gap-graded soils since the AREs of the estimated SWCCs are less than 10%. In addition, the RMSEs of the proposed equation for the selected soil data are very small and close to zero. This indicates a good performance of the proposed equations in estimating the SWCCs of gap-graded soils. The estimated SWCCs using the proposed equations in this study were also compared with the estimated SWCCs using Arya and Paris’s (1981) equation. Figs. 13–15 show that the estimated SWCCs using Arya and Paris’s (1981) equation had higher RMSEs than the estimated SWCCs using the proposed model in this study. In addition, the estimated SWCCs using Arya and Paris’s (1981) equation had AREs higher than 10%, which indicated that the estimated SWCCs using Arya and Paris’s (1981) equation did

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A. Satyanaga et al. / Soils and Foundations xxx (2017) xxx–xxx Table 8 Parameters of proposed equation for estimating SWCC of soils SB9, SB11, GM1b, GM2b, BTG8 and BTG14. Parameters

SB9

SB11

GM1b

GM2b

BTG8

BTG14

hs wa (kPa) wm (kPa) S wr (kPa) hr RMSE ARE (%)

0.448 800 11,316 5.1 10,000 0.1 0.008 1.8

0.204 1 0.57 5.7 300 0.0 0.017 9.8

0.303 70 1107 1.9 20,000 0.0 0.004 1.1

0.346 60 1069 2.1 20,000 0.0 0.006 1.3

0.560 100 1098 1.6 100,000 0.0 0.021 4.5

0.49 150 1851 3.9 20,000 0.08 0.042 9.5

6. Evaluation of proposed model using soil data from laboratory tests carried out in this study Fig. 13. Comparison of the experimental data of soils SB9 and SB11 with the estimated SWCC using the equation proposed in this study.

Fig. 14. Comparison of the experimental data of soils BTG8 and BTG14 with the estimated SWCC using the equation proposed in this study.

The SWCC data from the experimental works in this study indicated the unimodal characteristics of specimens M3 to M8 with one air-entry value, one inflection point and one residual matric suction as well as one residual suction (Figs. 16–18). The unimodal SWCCs of specimens M3 to M8 were in agreement with the criteria for the SWCC characteristics for gap-graded soils, as studied by Satyanaga et al. (2013), where qd should be higher than 1.44 Mg/m3 and wsat should be between 18% and 33%. Figs. 16–18 also show the estimated data for the SWCCs which were estimated using the procedure in Fig. 10. The parameters of the SWCC equation (Eq. (3)) were estimated based on their relationship with the soil properties and the parameters (Eqs. (15)–(20)) in the grain-size distribution equation (Eq. (1)). The performance of the proposed equations in estimating the SWCCs is evaluated using the ARE and RMSE criteria. The AREs of the proposed equations in estimating the SWCCs of gap-graded soils M3 to M8 are 6.7%, 9.6%, 8.5%, 8.0%, 7.7% and 5.5%. Since the values of the AREs for specimens M3 to M8 are less than 20%, the proposed equation is considered successful for predicting SWCCs from bimodal GSD. In addition, the RMSEs of the proposed equation for estimating the

Fig. 15. Comparison of the experimental data of soils GM1b and GM2b with the estimated SWCC using the equation proposed in this study.

not agree with the laboratory data on the SWCCs. This could happen since Arya and Paris’s (1981) equation was developed to estimate the SWCCs of soils with a unimodal grain-size distribution.

Fig. 16. Comparison of the experimental data of soils M3 and M4 with the estimated SWCC using the equation proposed in this study.

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Table 9 Parameters of proposed equation for estimating SWCC of soils M3 to M8.

Fig. 17. Comparison of the experimental data of soils M5 and M6 with the estimated SWCC using the equation proposed in this study.

Fig. 18. Comparison of the experimental data of soils M7 and M8 with the estimated SWCC using the equation proposed in this study.

SWCCs of soils M3 to M8 are small. This indicates a good performance of the proposed equation in estimating the SWCCs of gap-graded soils. The estimated SWCCs using the proposed equations in this study were also compared with the estimated SWCCs using Arya and Paris’s (1981) equation. Figs. 16–18 show that the estimated SWCCs using Arya and Paris’s (1981) equation did not agree with the laboratory data on SWCCs since Arya and Paris’s (1981) equation was developed to estimate the SWCCs of soils with a unimodal grain-size distribution. Based on the estimated SWCCs, the air-entry value of soil specimens M3 to M8, as obtained from parameter wa, are 20, 20, 2, 2, 3 and 15 kPa, respectively (Table 9). Parameter wm shows that the matric suction at the inflection point of soil specimens M3 to M8 are 160, 245, 2.9, 40, 131 and 347 kPa, respectively (Table 9). The air-entry values of soil specimens M3 to M8 from the parameters of the estimated SWCC and those manually plotted in the SWCC graph are similar. Therefore, parameters wa and wm, from the proposed equation, are able to represent the air-entry value and the inflection point of the soil, respectively.

Parameters

M3

M4

M5

M6

M7

M8

hs wa (kPa) wm (kPa) S wr (kPa) hr RMSE ARE (%)

0.296 20 160 4.7 4000 0.02 0.019 6.7

0.386 20 245 4.1 1000 0.10 0.023 9.6

0.455 2 29 1.1 100 0.02 0.015 8.5

0.636 2 40 2.2 1000 0.00 0.025 8.0

0.635 3 131 2.5 3000 0.00 0.026 7.7

0.653 15 347 2.9 20,000 0.10 0.033 5.5

In general, the proposed procedure and equations are able to estimate the SWCCs of soils M3 to M8 accurately because the proposed procedure and equations for estimating SWCCs consider the effect of the soil properties on the SWCCs. In addition, the parameters of the proposed equations are able to define the variables of the SWCCs, such as the air-entry value of the soil, the inflection point of the slope in the SWCCs, the residual suction, and the saturated and the residual water content of the soil. Figs. 16–18 indicate that the estimated SWCCs using Arya and Paris’s (1981) equation did not agree with the laboratory data on the SWCCs since the estimated SWCCs using Arya and Paris’s (1981) equation had AREs higher than 10%. In addition, the estimated SWCCs using Arya and Paris’s (1981) equation had higher RMSEs than the estimated SWCCs using the proposed model in this study. This might occur since Arya and Paris’s (1981) equation was developed to estimate the SWCCs of soils with a unimodal grain-size distribution. 7. Conclusions The conclusions from the research in this study are as follows: 1. The characteristics of SWCCs for gap-graded soils are unimodal if the soil has a percentage of gravel higher than 5%, a dry density higher than 1.44 Mg/m3 and a saturated water content between 18% and 33%. In addition, the characteristics of SWCCs for gap-graded soils are also unimodal if the soil is compacted at dry density and a water content associated with the wet of optimum as well as at the optimum conditions. These criteria were established based on the index properties of gap-graded soils. Further studies must be carried out to investigate the effect of the soil structure on the estimation of SWCCs for gap-graded soils. 2. Procedures, computer codes and equations for estimating unimodal SWCCs for gap-graded soils have been proposed and evaluated using soil data from published literature and laboratory tests carried out in this study. The results of the evaluation show that the proposed equations could be used to estimate the unimodal SWCCs successfully with AREs less than 10% and RMSEs close to zero.

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A. Satyanaga et al. / Soils and Foundations xxx (2017) xxx–xxx

3. The variables of unimodal SWCCs for gap-graded soils can be estimated from the variables and best-fitting parameters of the grain-size distribution curves. The air-entry value is a function of the inflection point of the grain-size distribution curve. The inflection point of SWCCs for coarse-grained soils is a function of the coefficient of uniformity of the soil. On the other hand, the inflection point of SWCCs for fine-grained soils is a function of the dry density of the soil. Residual suction is a function of the diameter of the soil particles corresponding to 10% finer (D10) of the soil. The standard deviation of unimodal SWCCs for a coarse-grained soil is a function of the dry density of the soil, while the standard deviation of unimodal SWCCs for a fine-grained soil is a function of the coefficient of uniformity of the soil.

Acknowledgements This work was supported by a research grant from a collaboration project between the Housing and Development Board and Nanyang Technological University (NTU), Singapore. The authors gratefully acknowledge the assistance of the Geotechnical Laboratory staff, School of Civil and Environmental Engineering, NTU, Singapore during the experiments and data collection. Appendix A. Computer codes for computation of parameters in estimated unimodal SWCC for gap-graded soils

Public Function sat_vol(satw, rd) ‘to calculate saturated volumetric water content sat_vol = satw * rd / 1000 End Function Public Function inflec(D10, D60, P200, rd, dm1) ‘calculate suction at inflection point cu = D60 / D10 If P200 > 50 Then u = (-1.031 * rd / 10) + 19.53 Else u = (0.028 * cu) + 9.21 End If v = 0.5 a = -1.9038 – ((Application.WorksheetFunction.Ln(0. 1 * dm1 / u)) / v) inflec = 0.1 * Exp(a) End Function Public Function AEV(D10, D60, P200, ym, dm1, dm2) ‘calculate the air-entry value cu = D60 / D10 If P200 > 50 Then AEV = ym / (3.167 * ((dm2/dm1)^2)) Else

AEV = ym / (0.429 * (cu^0.613)) End If End Function Public Function stdev(rd, D10, D60, P200) ‘calculate standard deviation cu = D60 / D10 If P200 > 50 Then stdev = (-0.471 * rd / 10) + 9.716 Else stdev = 9.89 * (cu ^ -0.17) End If End Function Public Function res_vol(P200, D10) ‘calculate residual volumetric water content If P200 > 50 Then res_vol = 0.771 * (D10 ^ -1.53) Else res_vol = 0.512 * (D10 ^ -1.42) End If End Function Public Function volwat(x, yr, qr, qs, ya, ym, s) ‘to calculate volumetric water content for every suction c1 = Application.WorksheetFunction.Ln(1 + (x / yr)) c2 = Application.WorksheetFunction.Ln(1 + (1000000 / yr)) cr = 1 - (c1 / c2) slopea = Application.WorksheetFunction.Ln((ya - x) / (ya - ym)) / s If x > ya Then slopeb = 1 - Application.WorksheetFunction.NormS Dist(slopea) Else slopeb = 1 End If volwat = cr * (qr + ((qs - qr) * slopeb)) End Function

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