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Effect of bimodal soil-water characteristic curve on the estimation of permeability function
Accepted Manuscript Effect of bimodal soil-water characteristic curve on the estimation of permeability function
Qian Zhai, Harianto Rahardjo, Alfrendo Satyanaga, Priono PII: DOI: Reference:
S0013-7952(17)30398-8 doi:10.1016/j.enggeo.2017.09.025 ENGEO 4662
To appear in:
Engineering Geology
Received date: Revised date: Accepted date:
12 March 2017 28 September 2017 28 September 2017
Please cite this article as: Qian Zhai, Harianto Rahardjo, Alfrendo Satyanaga, Priono , Effect of bimodal soil-water characteristic curve on the estimation of permeability function. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Engeo(2017), doi:10.1016/j.enggeo.2017.09.025
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ACCEPTED MANUSCRIPT Effect of bimodal soil-water characteristic curve on the estimation of permeability function
1,2,3
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Qian Zhai1 , Harianto Rahardjo2 , Alfrendo Satyanaga3 , and Priono4
School of Civil and Environmental Engineering, Nanyang Technological
Interdisciplinary Graduate School and Residues & Resource Reclamation Centre,
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4
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University, Nanyang Avenue, Singapore 639798
University, Singapore 639798 [email protected],
2
[email protected],
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[email protected],
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4
1
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E- mail:
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Nanyang Environment & Water Research Institute, Nanyang Technological
3
[email protected],
ACCEPTED MANUSCRIPT Abstract Soil- water characteristic curve (SWCC) defines the relationship between water content and suction in soil. Many unsaturated properties can be estimated from SWCC such as permeability function and unsaturated shear strength. Therefore, SWCC is considered as the key information of unsaturated soil properties.
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Permeability function, which defines the relationship between hydraulic conductivity
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and matric suction, is essential information for seepage analysis in understanding
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water flow in unsaturated soil. The permeability function is commonly determined indirectly from the SWCC rather than from direct measurement in the laboratory
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because the direct measurement is time consuming and costly. On the other hand, it has been reported that SWCC can have unimodal and bimodal characteristics. Many
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models have been proposed for the estimation of permeability function from a unimodal SWCC while limited numbers of model have been proposed for a bimodal
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SWCC. In this paper, experimental works in laboratory were carried out for measurements of the unsaturated permeability of soils with bimodal SWCC. Zhai and
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Rahardjo’s (2015) equation was used to estimate the permeability function of soils with bimodal SWCC. The experimental results show good agreement with the
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estimated permeability function. Therefore, Zhai and Rahardjo’s (2015) equation was recommended for the estimation of the permeability function from bimodal SWCC. This study also shows that the variability in bimodal SWCC has significant effect on the estimation of the permeability function.
Key words: unsaturated soil, bimodal soil-water characteristic curve, permeability function, variability
ACCEPTED MANUSCRIPT 1.
Introduction
Unsaturated soil mechanics has become a vibrant area of research in recent decades. Many research works focus on soil- water characteristic curve (SWCC) because the SWCC contains basic information of the unsaturated soil properties. SWCC is reported to be analogous to the pore-size distribution function which is an important
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property for understanding the water flow through a porous media (Marshall 1958,
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Kunze et al. 1968, Leong and Rahardjo 1997 and Zhai and Rahardjo 2015). SWCC
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has also been reported to have unimodal and bimodal shapes (Rahardjo et al. 2004, Zhang and Chen 2005, Satyanaga et al. 2013 and Li et al. 2014) and the different
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shapes represent different pore-size distribution functions. The sub-curves of a bimodal SWCC can be related to macro-pores and micro-pores of the soil (Othmer et
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al. 1991, Ross and Smettem 1993, Durner 1994 and Zhang and Chen 2005). It has been proven that permability function of soil can be estimated from SWCC by
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adopting the concept of pore-size distribution function. Based on Richards (1931)’ differential equation for seepage analysis, the solution to
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the differential equation is mainly governed by the hydraulic properties such as SWCC and permeability function. The variation in SWCC and permeability function
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(unimodal or bimodal) can significantly affect the seepage results. Most research works on the estimation of the permeability function were based on unimodal SWCC (Childs and Collis-George 1950, Marshall et al. 1958, Kunze et al. 1968, Mualem 1976 and Fredlund et al. 1994) and limited numbers of studies were reported for the bimodal SWCC. In this study, Zhai and Rahardjo’s (2015) equation was used to estimate the permeability function of soils with bimodal SWCC. The estimated bimodal permeability function was verified using the laboratory data obtained from the
ACCEPTED MANUSCRIPT experimental works carried out in this study. In addition, the uncertainty associated with the different equations used for best fitting bimodal SWCC on the estimation of the bimodal permeability function was investigated.
2. Literature review
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The study by Zapata (1999) indicated that mathematical equation for best fitting of
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SWCC data provides convenience to geotechnical engineers since the best fit equation
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is mathematically continuous and can be differentiated. Various best fit equations have been proposed by different researchers (Brooks and Corey 1964; Gardner 1958;
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Farrel and Larson 1972; van Genuchten 1980; Williams et al. 1983; Fredlund and Xing 1994; Kosugi 1994 and Satyanaga et al. 2013). As the fitting parameters in
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Satyanaga et al.’s (2013) equation can be related to soil properties, Satyanaga et a l. (2013) equation, as illustrated in Equation (1) was adopted as the best fit equation to
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represent the bimodal SWCC.
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a1 ln m1 a1 r s 1 s 2 1 erfc s1 ln 1 Cr w 1 10 6 a 2 ln 1 ln C r m 2 a 2 s 2 r 1 erfc s2 where: r= residual volumetric water content, a = air-entry value m = suction corresponding to the inflection point
(1)
ACCEPTED MANUSCRIPT s = geometric standard deviation of SWCC, as illustrated in Equation (2) Cr = input value same as the definition in Fredlund and Xing (1994) equation n
i
ln n
----- (2)
= the geometric mean of matric suction, 1 2 x
erfc = the complementary error function. erfc( x)
x2 1 exp dx 2 2
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n
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i 1
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s s exp
Subscripts 1 and 2 represent sub-curves 1 (associated with macro pores) and 2
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(associated with micro pores), respectively as illustrated in Figure 1.
Figure 1: Illustration of parameters in Satyanaga et al.’s (2013) equation (from Satyanaga et al. 2013)
ACCEPTED MANUSCRIPT Wijaya and Leong (2016) reviewed the equations for bimodal SWCC and divided these SWCC equations into three groups based on the approaches for determining the fitting parameters. The first approach is to arbitrarily determine the merging point and obtain the fitting parameters of two unimodal SWCCs separately, such as Smettem and Kirkby (1990), Wilson et al. (1992), Burger and Shackeford (2001). The seco nd
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approach is to separate the saturated water content into macro-pores and micro-pores
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and obtain the fitting parameters of two unimodal SWCCs simultaneously, such as
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Othmer et al. (1991), Ross and Smettem (1993), Durner (1994), Mallants et al. (1997) and Zhang and Chen (2005). The third approach is to incorporate some parameters
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determined graphically from the SWCC and obtain the fitting parameters by curve fitting technique, such as Gitirana and Fredlund (2004), Satyanaga et al. (2013) and Li
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et al. (2014).
Childs and Collis-George (1950) suggested that SWCC can be considered to be
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analogous to the pore-size distribution function. Adopting the concept of "cutting and rejoining" and the probability of random connection, Childs and Collis- George (1950)
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proposed the statistical method for the estimation of permeability function from SWCC. Subsequently, the statistical model was modified by Marshall (1958), Kunze
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et al. (1968), Mualem (1976), Fredlund et al. (1994) and Zhai and Rahardjo (2015). Zhai and Rahardjo (2015) pointed out that SWCC in the form of degree of saturation was most analogous to the pore-size distribution function. As the statistical model requires the SWCC to be divided into a finite number of segments, Kunze et al. (1968) proposed to evenly divide the SWCC along the volumetric water content and make each segment to have the same pore-size density. In order to have a good estimation of the permeability function using the statistical method, Zhai et al. (2017) recommended to evenly divide the SWCC along the matric suction rather than the
ACCEPTED MANUSCRIPT degree of saturation. Kunze et al.'s (1968) method was commonly adopted by researchers but inverse function is required to convert volumetric water content into matric suction as pointed out by Rahimi et al. (2015). To avoid using the inverse function, Zhai and Rahardjo (2015) suggested to divide the SWCC along the matric suction instead of the volumetric water content. The fitting parameters of the best fit
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Rahardjo's (2015) equation as illustrated in Equation (3).
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equation and permeability function could be directly correlated using Zhai and
2
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(3)
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where,
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nm
2
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k ( mi ) k ( m )
nm i
S ( mi ) S ( mi 1 ) 2 rm2i N 2 2 2 S ( mi ) S ( j ) S ( mi ) S ( j 1 ) r j j mi 1 2 2 S ( m ) S ( m1 ) rm N 2 2 2 S ( m ) S ( i ) S ( m ) S ( i 1 ) ri i m1
k( m+i)= calculated hydraulic conductivity,
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k( m ) = hydraulic conductivity at the reference point(i.e., = m), m = suction corresponding to the reference point,
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nm = porosity corresponding to the reference point, nm+i = porosity corresponding to the suction = m+i, S( m ) = degree of saturation corresponding to the reference point, S( m+i) = degree of saturation corresponding to the suction = m+i and rm, ri = equivalent pore radius corresponding to the suction of m and i, respectively. Zhai and Rahardjo’s (2015) equation can be used for any best fit equation and any shape of SWCC. Therefore, Zhai and Rahardjo’s (2015) equation was used to
ACCEPTED MANUSCRIPT establish new equation for the estimation of bimodal permeability function in this study.
3. Experimental works for direct measure ment of permeability function of soils with bimodal SWCC
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The unsaturated permeability tests were conducted using an unsaturated triaxial
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permeameter as shown in Figure 2. In this research, the unsaturated triaxial
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permeameters were assembled following the modifications recommended by Goh et al. (2015) and Rahimi and Rahardjo (2016). The unsaturated triaxial permeameter
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consists of a triaxial cell, modified top and bottom pedestals and a pair of high airentry ceramic discs. Two types of high air-entry ceramic disc namely 1-bar and 5-bar,
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with a thickness of 7.14 mm were used in this study. The 1-bar ceramic disc was used to measure the unsaturated permeability of the soil mixtures up to a suction of 100
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kPa, while the 5-bar ceramic disc was used to measure the unsaturated permeability of the soil mixtures up to a suction of 500 kPa. The top and bottom pedestals were
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composed of aluminum and stainless steel, respectively. Bo th the top and bottom pedestals were modified for the unsaturated permeability tests by creating a
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protruding air pressure outlet and spiral grooves in the water compartments, as indicated in Figure 3.
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Plexiglass Cover
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Top (Back) Pressure Tube
Specimen
O- rings
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Inlet/Transducers
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Valve/knob
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Inlet/Transducers
Figure 2. Unsaturated triaxial permeameter
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As presented by Goh (2012), the applied air pressure was less than the air-entry value of the ceramic disc and the original design of the ceramic disc only allowed water to
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pass through the disc (air could not pass through the disc). Therefore, it was impossible to apply and control both pore-air and pore-water pressures from one side
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of the soil specimen simultaneously. In this case, the ceramic discs have been modified by constructing an opening at the base of the grooves in the disc as shown in Figure 3. A protruding air pressure outlet was constructed in order to apply pore-air pressure to the specimen through the modified ceramic disc. Two water pressure outlets were placed at the spiral grooves to apply pore-water pressure into the water compartment and then into the specimen through the ceramic disc. The modified ceramic disc provided uniform distribution of water and air pressures to the specimen. Slow-setting epoxy was used to glue the sintered bronze to the modified ceramic disc
ACCEPTED MANUSCRIPT and also to glue the ceramic disc to the modified pedestal. Therefore, each of the top and bottom pedestals had three pressure outlets. The valves connecting the bottom pedestal and triaxial cell base were called the bottom pore-water pressure, bottom flushing line and bottom pore-air pressure valves. The top pedestal and triaxial cell
pressure, top flushing line and top pore-air pressure valves.
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base connection also had three corresponding valves, namely the top pore-water
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As pointed out by Ho and Fredlund (1982), air could diffuse through the ceramic
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discs and turned into air bubbles in the water compartments due to the high ratio between pore-air and pore-water pressures and long duration of the test. Accumulated
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air bubbles could affect the accuracy of the pore-water pressure and volume change measurements and impeded the flow of water into and from the specimen. In this case,
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the flushing system of the modified triaxial apparatus was used to remove the air bubbles which were trapped in the water compartments during the testing. The water
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compartments were flushed every 2 hours by applying 30 kPa of water pressure to valves A and F and the water with trapped air bubbles was flushed out from valves B
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and E and collected at DAVI. By adopting the flushing process, the accumulated air bubbles below the ceramic disc can be removed during the test without interrupting
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the test.
Another water outlet was located in the cell base to connect the cell pressure from the digital pressure volume control (DPVC). The unsaturated triaxial permeameter was connected to two DPVCs for pore-water pressure, a DPVC for cell pressure, a poreair pressure control system, four pressure transducers, a data acquisition system and a personal computer. The schematic diagram of the unsaturated triaxial permeameter used in this study is illustrated in Figure 4.
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Spiral Groove
Protruding Air Pressure Outlet
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Sintered Bronze
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Ceramic Disk
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Water Pressure Outlet
Figure 3. Modified pedestals and modified ceramic discs used in the unsaturated
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triaxial permeameter in the research
Figure 4. Schematic diagram of the unsaturated triaxial permeameter setup The unsaturated permeability tests involved saturation, consolidation, matric suction
ACCEPTED MANUSCRIPT equalization and unsaturated permeability measurement. Prior to the test, the ceramic discs on the top and bottom pedestals were saturated using de-aired distilled water with a pressure of 200 kPa for 24 hours. After the ceramic discs were saturated, the saturated coefficients of permeability of the top and bottom ceramic discs were then measured. A cell pressure of 500 kPa was applied to the ceramic disc and the water
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outflow through the disc was measured. The coefficient of permeability of the
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unsaturated soil was calculated from the quantity of outflow using Darcy’s Law,
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which is similar to that used for calculating the saturated permeability of the specimen. After equilibrium at a particular matric suction was achieved, the last stage
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of unsaturated permeability measurement was performed. Similar to the procedure used in the saturated permeability test, an upward flow of water through the specimen
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was created by applying a pressure head difference of 10 kPa between the top and bottom of the specimen via the base and back DPVCs, respectively. The unsaturated
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permeability test was stopped when a steady-state condition was achieved for a period of time. A steady-state condition, i.e. a condition where there is no change in water
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volume, is usually determined by graphical observation, similar to the determination of equilibrium time in the matric suction equalization stage.
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Laboratory tests were conducted on compacted soils which were produced by mixing L2-grade kaolin and 20-30 Ottawa sand. The compacted soils were used in the research to avoid the heterogeneity of the soils for better analyses. The saturated and unsaturated permeability tests of three soil samples named K50S50 (50% sand mixed with 50% kaolin), K50S50 with 9% mica (50% sand mixed with 50% kaolin and additional 9% mica) and sample K10S90 (90% sand mixed with 10% kaolin) were carried out in this study. Index properties of these three types of soil samples used in this study are summarized in Table 1.
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Table 1: Index properties of soils used in this study. K50S50 with 9% mica
K10S90
Dry Density, d (Mg/m )
1.75
1.77
1.84
Water Content, w (%)
12.1
16.0
8.3
Void Ratio, e
0.48
0.47
0.45
Liquid Limit, LL (%)
46.7
49.6
15.69
Plastic Limit, PL (%)
27.4
31.9
Plasticity Index, PI (%)
19.3
18.7
Specific Gravity, Gs
2.59
2.61
2.66
GSD – Sand (%)
50.0
59.8
90
GSD – Silt (%)
37.5
29.1
10
GSD – Clay (%)
12.5
11.1
0
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K50S50
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3
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Index properties
---
SM-ML
SM
SP
Saturated permeability ks (m/s)
1.12x10-8
1.12x10-7
2.65x10-5
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Unified Soil Classification System (USCS)
4. Comparison between the experimental data and the permeability function
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estimated using Zhai and Rahardjo’s (2015) method.
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Satyanaga et al.’s (2013) bimodal equation was adopted to best fit the SWCC data, as illustrated in Figure 5. An iterative nonlinear regression procedure that is provided in the Microsoft Excel software (Wraith and Or, 1998) was used to adjust all paramete rs
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to fit the equation for each soil. The complexity in best fitting the bimodal SWCC data was minimized by providing an appropriate initial value for each parameter. The initial value of m corresponds to matric suction at the midpoint of the SWCC slope obtained from the laboratory test. The initial value of r corresponds to the last matric suction applied in the SWCC test. The initial value for standard deviation () is assumed to be equal to 1. The initial value of r corresponds to the last volumetric water content obtained from the laboratory tests. The fitting parameters of the bimodal SWCC data in this study are summarized in Table 2. These fitting parameters
ACCEPTED MANUSCRIPT were subsequently used for the estimation of the permeability function using the modified Zhai and Rahardjo’s (2015) equation. The comparison between the experimental measured data and the estimation results are illustrated in Figure 6.
Table 2. Fitting parameters in Satyanaga et al.’s (2013) equation s1
a1
1
10
10.6
2.03
0.712
33
135
1 1
9.9 1.87
10.3 2.63
1.85 3.88
0.75 0.438
15 30
76 103.46
r
R2
1.06
500
0.025
99.37
0.8 1.5
600 750
0.124 0
99.0 99.02
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K50S50 with 9% mica K50S50 K10S90
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Fitting parameters in Satyanaga et al.'s (2013) equation m1 s1 s2 a2 m2 s2 r
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Soils
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0.8
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0.6
R2=99.4%
0.4
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Degree of saturation, S
1.0
0.2
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Satyanaga et al.'s (2013) equation Experimental data for K50S50 with 9% mica
0.0
10-2
10-1
100
101
102
103
104
105
106
Matric suction, (kPa)
(a) SWCC for the mixture of sand and kaolin (K50S50 with 9% mica)
1.0
Satyanaga et al.'s (2013) equation Experimental data for K50S50
0.8
R2=99.0%
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0.6
0.4
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Degree of saturation, S
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0.0 10-1
100
101
102
103
104
105
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10-2
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0.2
Matric suction, (kPa)
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(b) SWCC for the mixture of sand and kaolin (K50S50)
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R2=99.0%
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0.8
Satyanaga et al.'s (2013) equation Experimental data for K10S90
0.6
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Degree of saturation, S
1.0
0.4
0.2
0.0 10-2
10-1
100
101
102
103
104
105
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Matric suction, (kPa)
(c) SWCC for the mixture of sand and kaolin (K10S90) Figure 5. Best fit the experimental data with Satyanaga et al.’s (2013) equation
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R2=95.8%
10-7
10-8
10-9
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10-10
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10-11
10-12
Predicted using Zhai and Rahardjo's (2015) equation Experimental data for K50S50 with 9% mica
10-13 10-1
100
101
102
103
104
105
106
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10-2
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Coefficient of permeability, kw (m/s)
10-6
Matric suction, (kPa)
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(a) Permeability function of the mixture of sand and kaolin (K50S50 with 9% mica)
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Predicted using Zhai and Rahardjo's (2015) equation Experimental data for K50S50
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10-7
10-8
10-9
R2=89.1%
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Coefficient of permeability, kw (m/s)
10-6
10-10
10-11
10-12
10-13 10-2
10-1
100
101
102
103
104
105
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Matric suction, (kPa)
(b) Permeability function of the mixture of sand and kaolin (K50S50)
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10-1
R2=85.6%
10-3 10-4 10-5 10-6
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10-7 10-8 10-9
10-11 10-12
Predicted using Zhai and Rahardjo's (2015) equation Experimental data for K10S90
10-1
100
101
102
103
104
105
106
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10-13 10-2
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10-10
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Coefficient of permeability, kw (m/s)
10-2
Matric suction, (kPa)
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(c) Permeability function of the mixture of sand and kaolin (K10S90) Figure 6. Comparison between the experimental data and the predicted results using
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Zhai and Rahardjo’s (2015) equation on the permeability function. As illustrated in Figure 6, the coefficients of determination (R2 ) of the SWCC curve
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fitting for these three soils were computed as 99.4% for K50S50 with 9% mica, 99.0% for K50S50, and 99.0% for K10S90, respectively. High values of R2 indicate
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that Satyanaga et al.’s (2013) equation had good performance in best fitting the bimodal SWCC data . On the other hand, as illustrated in Figure 6, the coefficients of determination (R2 ) of predicted log (k w) for these three soils were computed as 95.8% for K50S50 with 9% mica, 89.1% for K50S50, and 85.6% for K10S90, respectively. The results indicated that the estimation on the permeability function using Zhai and Rahardjo’s (2015) equation agreed well with the experimental data. In other words, Satyanaga et al.’s (2013) equation and Zhai and Rahardjo’s (2015) equation can be used for the estimation of the permeability function from a bimodal SWCC.
ACCEPTED MANUSCRIPT 5. Effect of the variability in bimodal SWCC on the estimation of the permeability function If Fredlund and Xing’s (1994) and van Genunchten’s (1980) equations were used to best fit the experimental data following approach 1 as categorized by Wijaya and Leong (2016), then the fitting results can be obtained as illustrated in Figure 7. The
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fitting parameters for Fredlund and Xing’s (1994) and van Genunchten’s (1980)
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equations are summarized in Table 3. These fitting parameters were subsequently
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used for the estimation of the permeability function using Zhai and Rahardjo’s (2015) equation and the estimation results are illustrated in Figure 8. The estimation method
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on the permeability function using Zhai and Rahardjo’s (2015) method and Satyanaga et al.’s (2013) best fit equation is named as ZR-SA model. Similarly, ZR-FX which
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means using Zhai and Rahardjo’s (2015) method and Fredlund and Xing’s (1994) best fit equation and ZR-VG models which means Zhai and Rahardjo’s (2015) method and
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van Genunchten’s (1980) best fit equation can be named as illustrated in Figure 8. The coefficients of determination R2 for best fitting SWCC data and predicted log(k w)
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are illustrated in Table 4.
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Satyanaga et al.'s (2013) equation Fredlund and Xing's (1994) equation van Genuchten's (1980) equation K50S50 with 9% mica
0.8
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0.6
0.4
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Degree of saturation, S
1.0
0.0 10-2
10-1
100
101
102
SC
0.2
103
104
105
106
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Matric suction, (kPa)
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(a) Best fitted bimodal SWCCs for K50S50 with 9% mica
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0.8
Satyanaga et al.'s (2013) equation Fredlund and Xing's (1994) equation van Genuchten's (1980) equation K50S50 with 9% mica
0.6
0.4
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Degree of saturation, S
1.0
0.2
0.0
10-2
10-1
100
101
102
103
104
Matric suction, (kPa)
(b) Best fitted bimodal SWCCs for K50S50
105
106
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Satyanaga et al.'s (2013) equation Fredlund and Xing's (1994) equation van Genuchten's (1980) equation Experimental data for K10S90
0.8
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0.6
0.4
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Degree of saturation, S
1.0
0.0 10-1
100
101
102
103
104
105
106
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10-2
SC
0.2
Matric suction, (kPa)
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(c) Best fitted bimodal SWCCs for K10S90 Figure 7. Comparison of best fitted bimodal SWCC using three different best fit
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equations.
equations Soils
s1 1 1 1
a1
10.59 12.21 1.61
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K50S50 with 9% mica K50S50 K10S90
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Table 3. Fitting parameters in Fredlund and Xing’s (1994) and van Genuchten’s (1980)
K50S50 with 9% mica K50S50 K10S90
s1
a1
1 1 1
0.022 0.014 0.68
Fitting parameters in Fredlund and Xing’s (1994) equation n1 m1 s2 a2 n2 m2 Cr 14.96 0.14 0.71 71.4 5.45 0.59 1500 3.97 0.35 0.75 54.14 5.05 0.72 1500 17.14 0.15 0.57 43.71 4.11 0.59 1500 Fitting parameters in van Genuchten’s (1980) equation b1 c1 r1 s2 a2 b2 c2 3.85 2.55 50
7.67 7.68 0.037
0 0 0.586
0.715 0.75 0.59
0.016 0.023 0.005
5.98 4.39 1.76
0.151 0.2 3.66
r2 0 0 0
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Estimated using ZR-SA model Estimated using ZR-FX model Estimated using ZR-VG model K50S50 with 9% mica
10-7
10-8
10-9
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10-10
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10-11
10-12
10-13 10-1
100
101
102
103
104
105
106
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10-2
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Coefficient of permeability, kw (m/s)
10-6
Matric suction, (kPa)
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(a) Estimated permeability functions from bimodal SWCCs for K50S50 with 9% mica
ED EP T
10-7
Estimated using ZR-SA method Estimated using ZR-FX method Estimated using ZR-VG method Experimental data for K50S50
10-8
10-9
AC C
Coefficient of permeability, kw (m/s)
10-6
10-10
10-11
10-12
10-13 10-2
10-1
100
101
102
103
104
105
106
Matric suction, (kPa)
(b) Estimated permeability functions from bimodal SWCCs for K50S50
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10-1
10-3 10-4 10-5 10-6
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10-7 10-8
Estiamted using ZR-SA model Estiamted using ZR-FX model Estimated using ZR-VG model Experimental data for K10S90
10-10 10-11 10-12
10-1
100
101
102
103
104
105
106
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10-13 10-2
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10-9
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Coefficient of permeability, kw (m/s)
10-2
Matric suction, (kPa)
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(c) Estimated permeability functions from bimodal SWCCs for K10S90 Figure 8. Comparison of the estimation results on the permeability function from
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three different models
Table 4. R2 for best fitted SWCC and predicted hydraulic conductivity, log(k w)
K50S50
K10S90
Satyanaga et al. (2013) Fredlund and Xing (1994) van Genuchten (1980) Satyanaga et al. (2013) Fredlund and Xing (1994) van Genuchten (1980) Satyanaga et al. (2013) Fredlund and Xing (1994) van Genuchten (1980)
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K50S50 with 9% mica
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Best fit equations
Soils
R2 for best fitted SWCC 99.4% 99.7% 99.9% 99.0% 98.9% 99.9% 99.0% 99.6% 99.5%
Methods for prediction ZR-SA ZR-FX ZR-VG ZR-SA ZR-FX ZR-VG ZR-SA ZR-FX ZR-VG
R2 for log(kw ) 95.8% 95.3% 99.2% 89.1% 98.2% 98.3% 85.6% 85.2% 80.9%
As illustrated in Table 4, there was not much difference in R2 of best fitted SWCC obtained using three equations from Satyanaga et al. (2013), Fredlund and Xing (1994) and van Genuchten (1980). In other words, these three equations had similar performance in best fitting the bimodal SWCC data as illustrated in Figure 8. However, it appears that there was high variability in best fitted SWCCs from these
ACCEPTED MANUSCRIPT three best fit equations in the high suction range (i.e., greater than 200 kPa) due to the lack of the experimental data in this suction range. Therefore, the accuracy of the best fitted SWCC is much dependent on the suction range covered in the measurement. This observation agrees with the point presented by Zhai and Rahardjo (2013) and Rahimi and Rahardjo (2016). On the other hand, as illustrated in Table 4, the values
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of R2 of the predicted log (kw) were also high which indicated that all these three
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methods including ZR-SA, ZR-FX, and ZR-VG gave good prediction on the
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hydraulic conductivities. As illustrated in Figure 9, high variability in the predicted hydraulic conductivity was also observed in the high suction range (i.e., greater than
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200 kPa) due to the high variability in their best fitted SWCC in this suction range. Therefore, the SWCC measurement in the high suction range is recommended not
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only for a good best fitted SWCC, but also for a good estimation of the permeability
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function.
adopted.
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6. Estimation results on the permeability function if unimodal SWCCs are
In order to test the accuracy of the estimation results on the permeability function
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from unimodal SWCC, unimodal SWCC equations such as the original Fredlund and Xing (1994)’s and van Genuchten (1980)’s equations were used to best fit the experimental data of these three soils. The fitting parameters and best fitted SWCCs from these two equations are illustrated in Table 5 and Figure 9. The estimated permeability functions from the unimodal SWCCs are illustrated in Figure 10 as follows:
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Experimental data for K50S50 with 9% mica Fredlund and Xing (1994)'s equation van Genuchten (1980)'s equation
0.8
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0.6
0.4
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Degree of saturation, S
1.0
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0.2
0.0 10-1
100
101
102
103
104
105
106
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10-2
Matric suction, (kPa)
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(a) Best fitted unimodal SWCC for K50S50 with 9% mica
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0.8
0.6
0.4
0.2
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Degree of saturation, S
1.0
Experimental data for K50S50 Fredlund and Xing (1994)'s equation van Genuchten (1980)'s equation
0.0 10-2
10-1
100
101
102
103
Matric suction, (kPa) (b) Best fitted unimodal SWCC for K50S50
104
105
106
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Experimental data for K10S90 Fredlund and Xing (1994)'s equation van Genuchten (1980)'s equation
0.8
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0.6
0.4
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Degree of saturation, S
1.0
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0.2
0.0 10-1
100
101
102
103
104
105
106
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10-2
Matric suction, (kPa)
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(c) Best fitted unimodal SWCC for K10S90
Figure 9. Best fitted unimodal SWCCs for the soils
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Table 5. Fitting parameters of unimodal SWCC Fredlund and Xing (1994)'s equation a (kPa) n m Cr (kPa)
Soils
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K50S50 with 9% mica K50S50 K10S90 Soils
K50S50 with 9% mica K50S50 K10S90
836
130.81 1.48
0.88
13.5
R2 (%)
1500
99.17
1.19 4.02 1500 2.57 0.38 1500 van Genuchten (1980)'s equation
94.51 96.86
a
b
c
r
0.001
0.905
6.8
0.05
99.11
0.002 1.1
1.2 21.23
6.76 0.014
0.09 0.14
94.34 97.58
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(a) Estimated permeability function from unimodal SWCC for K50S50 with 9%
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mica
(b) Estimated permeability function from unimodal SWCC for K50S50
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(c) Estimated permeability function from unimodal SWCC for K10S90 Figure 10. Estimated permeability function from unimodal SWCC for the soils
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Figure 9 and Table 5 indicate that the unimodal SWCC has lower R2 as compared to the bimodal SWCC as shown in Figure 5 and Table 2. Lower R2 indicates that the
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unimodal SWCC has worse performance in best fitting the experimental data of these three samples than the bimodal SWCC. Consequently, the predicted permeability
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function from these unimodal SWCCs, as illustrated in Figure 10, are less accurate than those predicted from the bimodal SWCC, as illustrated in Figure 8. It can be concluded that the selection of unimodal or bimodal SWCC best fit equation has significant effect on the results of the estimated permeability function.
7. Conclusions. The modified Zhai and Rahardjo’s (2015) equation performed well in estimating the permeability function of soil with bimodal SWCC since the estimated bimodal permeability function had a good agreement with the laboratory data. Comparison
ACCEPTED MANUSCRIPT results of the estimation of the bimodal permeability function from the SWCCs that were best fitted using three different equations, i.e., Satyanaga et al.’s (2013) equation, Fredlund and Xing (1994)’s equation and van Genuchten’s (1980) equation showed a high variability in the high suction range due to the lack of SWCC experimental data in this suction range. Therefore, the SWCC measurements up to high suction values
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must be carried out in order to obtain an accurate best fitted SWCC for a good
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estimation of the permeability function.
Acknowledgement
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The authors gratefully acknowledge the assistance of Shi Xiang, Kevin Janiardy,
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Nickanor Tikno from School of Civil and Environmental Engineering, NTU,
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Singapore during the experiments and data collections.
ACCEPTED MANUSCRIPT References Brooks,
R.H., and Corey,
A.T. (1964) “Hydraulic properties of porous
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Gardner, W.R (1958) “Mathematics of isothermal water conduction in unsaturated soils.”Highway Research Board Special Rep. No. 40, International Symposium on physic-Chemical Phenomen in Soils, Washigton, D.C. 78-87. Gitirana Jr., G.F.N., and Fredlund, D.G. (2004) “Soil-water characteristic curve equation with independent properties” J. Geotech. Geoenviron. Eng. 130 (2), 209-212. Goh, S.G. (2012) “Hysteresis effects on mechanical behavior of unsaturated soil” Ph.D thesis. Nanyang Technological University. Goh, S.G., Rahardjo, H., Leong, E.C. (2015). “Modification of triaxial apparatus for
ACCEPTED MANUSCRIPT permeability measurement of unsaturated soils”. Soils and Foundations, 55(1):63-73. Ho, D. Y. F. and Fredlund, D. G. (1982) “Increase in Strength Due to Suction for Two Hong Kong Soils”, In Proceeding of ASCE Geotechnical Conference on Engineering and Construction in Tropical and Residuals Soils, Honolulu, pp. 263-295. Kunze, R.J., Uehara, G., and Graham, K (1968) “Factors important in the calculation
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Li, X., Li, J.H., and Zhang, L.M. (2014) “Predicting bimodal soil-water characteristic
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Kunze, R.J., Uehara, G., and Graham, K (1968) “Factors important in the calculation of hydraulic conductivity.” Proc., Soil Sci. Soc. of Am., 32, 760-765.
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ACCEPTED MANUSCRIPT unsaturated hydraulic conductivity.” Soil sci 152 (3), 139-149. Rahimi, A., H. Rahardjo and E.C. Leong (2015), “Effects of soil-water characteristic curve and relative permeability equations on estimation of unsaturated permeability function”, Soils and Foundations, Japanese Geotechnical Society, November, Vol. 55, No.6, pp. 1400-1411.
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Rahimi, A. and Rahardjo, H. (2016). “New approach to improve soil-water
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Richards L. A. (1931) “Capillary conduction of liquids through porous mediums.”
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Satyanaga, A. Rahardjo, H., Leong, E.C. and Wang J.Y. (2013) “Water characteristic curve of soil with bimodal grain-size distribution” Comput Geotech 2013;48:51–61.
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van Genuchten, M.T. (1980) “A close form equation predicting the hydraulic conductivity of unsaturated soil.” Soil Sci. Soc Am. J 44, 892-898. Smettem, K.R.J., and Kirkby, C., (1990) “Measuring the hydraulic properties of a stable aggregated soil.” J. Hydrol. 117 (1-4), 1-13. Wijaya, M. and Leong E.C. (2016) “Equation for unimodal and bimodal soil-water characteristic curves” Soils and Foundations 56, 291-300.
ACCEPTED MANUSCRIPT William. J., Prebble.R.E., Williams W.T. and Hignett, C.T. (1983) “The influence of texture, structure and clay mineralogy on the soil moisture characteristics. ” Australian J. of Soil Res. 21, 15-32 Wilson, G.V., Jardine, P.M., and Gwo, J.P. (1992) “Modeling the hydraulic properties of a multi-region soil”Soil. Sci. Soc. Am. J. 56 (6), 1731-1737
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Zapata, C. E. (1999). "Uncertainty in Soil-Water Characteristic Curve and Impacts on
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Unsaturated Shear Strength Predictions", Ph.D. Dissertation, Arizona State University, Tempe, United States.
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Zhai Q, Rahardjo H. (2013) “Quantification of uncertainty in soil-water
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characteristic curve associated with fitting parameters” Engineering Geology 163: 144-152.
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Zhai Q, Rahardjo H. (2015) “Estimation of permeability function from the Soil-Water Characteristic Curve” Engineering Geology 199: 148-156.
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Zhai Q., Rahardjo H. and Satyanaga, A. (2017) “Effects of residual suction and residual water content on the estimation of permeability” Geoderma 303: 165-177
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Zhang L and Chen Q (2005). “Predicting bimodal soil–water characteristic curves.”
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ASCE J Geotech Geoenviron Eng;131(5):666–70.
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Highlight Equation for estimation of a bimodal permeability function. Verification of estimation results with experimental data. Comparison of estimation results using different best fit equations for a bimodal SWCC. Effect of variability in bimodal SWCC on the estimation results Measurement at high suction during SWCC test is recommended.
Qian Zhai, Harianto Rahardjo, Alfrendo Satyanaga, Priono PII: DOI: Reference:
S0013-7952(17)30398-8 doi:10.1016/j.enggeo.2017.09.025 ENGEO 4662
To appear in:
Engineering Geology
Received date: Revised date: Accepted date:
12 March 2017 28 September 2017 28 September 2017
Please cite this article as: Qian Zhai, Harianto Rahardjo, Alfrendo Satyanaga, Priono , Effect of bimodal soil-water characteristic curve on the estimation of permeability function. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Engeo(2017), doi:10.1016/j.enggeo.2017.09.025
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ACCEPTED MANUSCRIPT Effect of bimodal soil-water characteristic curve on the estimation of permeability function
1,2,3
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Qian Zhai1 , Harianto Rahardjo2 , Alfrendo Satyanaga3 , and Priono4
School of Civil and Environmental Engineering, Nanyang Technological
Interdisciplinary Graduate School and Residues & Resource Reclamation Centre,
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4
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University, Nanyang Avenue, Singapore 639798
University, Singapore 639798 [email protected],
2
[email protected],
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[email protected],
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4
1
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E- mail:
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Nanyang Environment & Water Research Institute, Nanyang Technological
3
[email protected],
ACCEPTED MANUSCRIPT Abstract Soil- water characteristic curve (SWCC) defines the relationship between water content and suction in soil. Many unsaturated properties can be estimated from SWCC such as permeability function and unsaturated shear strength. Therefore, SWCC is considered as the key information of unsaturated soil properties.
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Permeability function, which defines the relationship between hydraulic conductivity
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and matric suction, is essential information for seepage analysis in understanding
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water flow in unsaturated soil. The permeability function is commonly determined indirectly from the SWCC rather than from direct measurement in the laboratory
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because the direct measurement is time consuming and costly. On the other hand, it has been reported that SWCC can have unimodal and bimodal characteristics. Many
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models have been proposed for the estimation of permeability function from a unimodal SWCC while limited numbers of model have been proposed for a bimodal
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SWCC. In this paper, experimental works in laboratory were carried out for measurements of the unsaturated permeability of soils with bimodal SWCC. Zhai and
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Rahardjo’s (2015) equation was used to estimate the permeability function of soils with bimodal SWCC. The experimental results show good agreement with the
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estimated permeability function. Therefore, Zhai and Rahardjo’s (2015) equation was recommended for the estimation of the permeability function from bimodal SWCC. This study also shows that the variability in bimodal SWCC has significant effect on the estimation of the permeability function.
Key words: unsaturated soil, bimodal soil-water characteristic curve, permeability function, variability
ACCEPTED MANUSCRIPT 1.
Introduction
Unsaturated soil mechanics has become a vibrant area of research in recent decades. Many research works focus on soil- water characteristic curve (SWCC) because the SWCC contains basic information of the unsaturated soil properties. SWCC is reported to be analogous to the pore-size distribution function which is an important
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property for understanding the water flow through a porous media (Marshall 1958,
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Kunze et al. 1968, Leong and Rahardjo 1997 and Zhai and Rahardjo 2015). SWCC
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has also been reported to have unimodal and bimodal shapes (Rahardjo et al. 2004, Zhang and Chen 2005, Satyanaga et al. 2013 and Li et al. 2014) and the different
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shapes represent different pore-size distribution functions. The sub-curves of a bimodal SWCC can be related to macro-pores and micro-pores of the soil (Othmer et
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al. 1991, Ross and Smettem 1993, Durner 1994 and Zhang and Chen 2005). It has been proven that permability function of soil can be estimated from SWCC by
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adopting the concept of pore-size distribution function. Based on Richards (1931)’ differential equation for seepage analysis, the solution to
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the differential equation is mainly governed by the hydraulic properties such as SWCC and permeability function. The variation in SWCC and permeability function
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(unimodal or bimodal) can significantly affect the seepage results. Most research works on the estimation of the permeability function were based on unimodal SWCC (Childs and Collis-George 1950, Marshall et al. 1958, Kunze et al. 1968, Mualem 1976 and Fredlund et al. 1994) and limited numbers of studies were reported for the bimodal SWCC. In this study, Zhai and Rahardjo’s (2015) equation was used to estimate the permeability function of soils with bimodal SWCC. The estimated bimodal permeability function was verified using the laboratory data obtained from the
ACCEPTED MANUSCRIPT experimental works carried out in this study. In addition, the uncertainty associated with the different equations used for best fitting bimodal SWCC on the estimation of the bimodal permeability function was investigated.
2. Literature review
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The study by Zapata (1999) indicated that mathematical equation for best fitting of
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SWCC data provides convenience to geotechnical engineers since the best fit equation
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is mathematically continuous and can be differentiated. Various best fit equations have been proposed by different researchers (Brooks and Corey 1964; Gardner 1958;
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Farrel and Larson 1972; van Genuchten 1980; Williams et al. 1983; Fredlund and Xing 1994; Kosugi 1994 and Satyanaga et al. 2013). As the fitting parameters in
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Satyanaga et al.’s (2013) equation can be related to soil properties, Satyanaga et a l. (2013) equation, as illustrated in Equation (1) was adopted as the best fit equation to
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represent the bimodal SWCC.
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a1 ln m1 a1 r s 1 s 2 1 erfc s1 ln 1 Cr w 1 10 6 a 2 ln 1 ln C r m 2 a 2 s 2 r 1 erfc s2 where: r= residual volumetric water content, a = air-entry value m = suction corresponding to the inflection point
(1)
ACCEPTED MANUSCRIPT s = geometric standard deviation of SWCC, as illustrated in Equation (2) Cr = input value same as the definition in Fredlund and Xing (1994) equation n
i
ln n
----- (2)
= the geometric mean of matric suction, 1 2 x
erfc = the complementary error function. erfc( x)
x2 1 exp dx 2 2
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n
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i 1
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s s exp
Subscripts 1 and 2 represent sub-curves 1 (associated with macro pores) and 2
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(associated with micro pores), respectively as illustrated in Figure 1.
Figure 1: Illustration of parameters in Satyanaga et al.’s (2013) equation (from Satyanaga et al. 2013)
ACCEPTED MANUSCRIPT Wijaya and Leong (2016) reviewed the equations for bimodal SWCC and divided these SWCC equations into three groups based on the approaches for determining the fitting parameters. The first approach is to arbitrarily determine the merging point and obtain the fitting parameters of two unimodal SWCCs separately, such as Smettem and Kirkby (1990), Wilson et al. (1992), Burger and Shackeford (2001). The seco nd
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approach is to separate the saturated water content into macro-pores and micro-pores
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and obtain the fitting parameters of two unimodal SWCCs simultaneously, such as
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Othmer et al. (1991), Ross and Smettem (1993), Durner (1994), Mallants et al. (1997) and Zhang and Chen (2005). The third approach is to incorporate some parameters
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determined graphically from the SWCC and obtain the fitting parameters by curve fitting technique, such as Gitirana and Fredlund (2004), Satyanaga et al. (2013) and Li
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et al. (2014).
Childs and Collis-George (1950) suggested that SWCC can be considered to be
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analogous to the pore-size distribution function. Adopting the concept of "cutting and rejoining" and the probability of random connection, Childs and Collis- George (1950)
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proposed the statistical method for the estimation of permeability function from SWCC. Subsequently, the statistical model was modified by Marshall (1958), Kunze
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et al. (1968), Mualem (1976), Fredlund et al. (1994) and Zhai and Rahardjo (2015). Zhai and Rahardjo (2015) pointed out that SWCC in the form of degree of saturation was most analogous to the pore-size distribution function. As the statistical model requires the SWCC to be divided into a finite number of segments, Kunze et al. (1968) proposed to evenly divide the SWCC along the volumetric water content and make each segment to have the same pore-size density. In order to have a good estimation of the permeability function using the statistical method, Zhai et al. (2017) recommended to evenly divide the SWCC along the matric suction rather than the
ACCEPTED MANUSCRIPT degree of saturation. Kunze et al.'s (1968) method was commonly adopted by researchers but inverse function is required to convert volumetric water content into matric suction as pointed out by Rahimi et al. (2015). To avoid using the inverse function, Zhai and Rahardjo (2015) suggested to divide the SWCC along the matric suction instead of the volumetric water content. The fitting parameters of the best fit
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Rahardjo's (2015) equation as illustrated in Equation (3).
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equation and permeability function could be directly correlated using Zhai and
2
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(3)
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where,
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nm
2
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k ( mi ) k ( m )
nm i
S ( mi ) S ( mi 1 ) 2 rm2i N 2 2 2 S ( mi ) S ( j ) S ( mi ) S ( j 1 ) r j j mi 1 2 2 S ( m ) S ( m1 ) rm N 2 2 2 S ( m ) S ( i ) S ( m ) S ( i 1 ) ri i m1
k( m+i)= calculated hydraulic conductivity,
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k( m ) = hydraulic conductivity at the reference point(i.e., = m), m = suction corresponding to the reference point,
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nm = porosity corresponding to the reference point, nm+i = porosity corresponding to the suction = m+i, S( m ) = degree of saturation corresponding to the reference point, S( m+i) = degree of saturation corresponding to the suction = m+i and rm, ri = equivalent pore radius corresponding to the suction of m and i, respectively. Zhai and Rahardjo’s (2015) equation can be used for any best fit equation and any shape of SWCC. Therefore, Zhai and Rahardjo’s (2015) equation was used to
ACCEPTED MANUSCRIPT establish new equation for the estimation of bimodal permeability function in this study.
3. Experimental works for direct measure ment of permeability function of soils with bimodal SWCC
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The unsaturated permeability tests were conducted using an unsaturated triaxial
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permeameter as shown in Figure 2. In this research, the unsaturated triaxial
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permeameters were assembled following the modifications recommended by Goh et al. (2015) and Rahimi and Rahardjo (2016). The unsaturated triaxial permeameter
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consists of a triaxial cell, modified top and bottom pedestals and a pair of high airentry ceramic discs. Two types of high air-entry ceramic disc namely 1-bar and 5-bar,
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with a thickness of 7.14 mm were used in this study. The 1-bar ceramic disc was used to measure the unsaturated permeability of the soil mixtures up to a suction of 100
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kPa, while the 5-bar ceramic disc was used to measure the unsaturated permeability of the soil mixtures up to a suction of 500 kPa. The top and bottom pedestals were
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composed of aluminum and stainless steel, respectively. Bo th the top and bottom pedestals were modified for the unsaturated permeability tests by creating a
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protruding air pressure outlet and spiral grooves in the water compartments, as indicated in Figure 3.
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Plexiglass Cover
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Top (Back) Pressure Tube
Specimen
O- rings
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Inlet/Transducers
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Valve/knob
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Inlet/Transducers
Figure 2. Unsaturated triaxial permeameter
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As presented by Goh (2012), the applied air pressure was less than the air-entry value of the ceramic disc and the original design of the ceramic disc only allowed water to
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pass through the disc (air could not pass through the disc). Therefore, it was impossible to apply and control both pore-air and pore-water pressures from one side
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of the soil specimen simultaneously. In this case, the ceramic discs have been modified by constructing an opening at the base of the grooves in the disc as shown in Figure 3. A protruding air pressure outlet was constructed in order to apply pore-air pressure to the specimen through the modified ceramic disc. Two water pressure outlets were placed at the spiral grooves to apply pore-water pressure into the water compartment and then into the specimen through the ceramic disc. The modified ceramic disc provided uniform distribution of water and air pressures to the specimen. Slow-setting epoxy was used to glue the sintered bronze to the modified ceramic disc
ACCEPTED MANUSCRIPT and also to glue the ceramic disc to the modified pedestal. Therefore, each of the top and bottom pedestals had three pressure outlets. The valves connecting the bottom pedestal and triaxial cell base were called the bottom pore-water pressure, bottom flushing line and bottom pore-air pressure valves. The top pedestal and triaxial cell
pressure, top flushing line and top pore-air pressure valves.
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base connection also had three corresponding valves, namely the top pore-water
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As pointed out by Ho and Fredlund (1982), air could diffuse through the ceramic
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discs and turned into air bubbles in the water compartments due to the high ratio between pore-air and pore-water pressures and long duration of the test. Accumulated
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air bubbles could affect the accuracy of the pore-water pressure and volume change measurements and impeded the flow of water into and from the specimen. In this case,
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the flushing system of the modified triaxial apparatus was used to remove the air bubbles which were trapped in the water compartments during the testing. The water
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compartments were flushed every 2 hours by applying 30 kPa of water pressure to valves A and F and the water with trapped air bubbles was flushed out from valves B
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and E and collected at DAVI. By adopting the flushing process, the accumulated air bubbles below the ceramic disc can be removed during the test without interrupting
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the test.
Another water outlet was located in the cell base to connect the cell pressure from the digital pressure volume control (DPVC). The unsaturated triaxial permeameter was connected to two DPVCs for pore-water pressure, a DPVC for cell pressure, a poreair pressure control system, four pressure transducers, a data acquisition system and a personal computer. The schematic diagram of the unsaturated triaxial permeameter used in this study is illustrated in Figure 4.
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Spiral Groove
Protruding Air Pressure Outlet
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Sintered Bronze
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Ceramic Disk
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Water Pressure Outlet
Figure 3. Modified pedestals and modified ceramic discs used in the unsaturated
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triaxial permeameter in the research
Figure 4. Schematic diagram of the unsaturated triaxial permeameter setup The unsaturated permeability tests involved saturation, consolidation, matric suction
ACCEPTED MANUSCRIPT equalization and unsaturated permeability measurement. Prior to the test, the ceramic discs on the top and bottom pedestals were saturated using de-aired distilled water with a pressure of 200 kPa for 24 hours. After the ceramic discs were saturated, the saturated coefficients of permeability of the top and bottom ceramic discs were then measured. A cell pressure of 500 kPa was applied to the ceramic disc and the water
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outflow through the disc was measured. The coefficient of permeability of the
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unsaturated soil was calculated from the quantity of outflow using Darcy’s Law,
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which is similar to that used for calculating the saturated permeability of the specimen. After equilibrium at a particular matric suction was achieved, the last stage
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of unsaturated permeability measurement was performed. Similar to the procedure used in the saturated permeability test, an upward flow of water through the specimen
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was created by applying a pressure head difference of 10 kPa between the top and bottom of the specimen via the base and back DPVCs, respectively. The unsaturated
ED
permeability test was stopped when a steady-state condition was achieved for a period of time. A steady-state condition, i.e. a condition where there is no change in water
EP T
volume, is usually determined by graphical observation, similar to the determination of equilibrium time in the matric suction equalization stage.
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Laboratory tests were conducted on compacted soils which were produced by mixing L2-grade kaolin and 20-30 Ottawa sand. The compacted soils were used in the research to avoid the heterogeneity of the soils for better analyses. The saturated and unsaturated permeability tests of three soil samples named K50S50 (50% sand mixed with 50% kaolin), K50S50 with 9% mica (50% sand mixed with 50% kaolin and additional 9% mica) and sample K10S90 (90% sand mixed with 10% kaolin) were carried out in this study. Index properties of these three types of soil samples used in this study are summarized in Table 1.
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Table 1: Index properties of soils used in this study. K50S50 with 9% mica
K10S90
Dry Density, d (Mg/m )
1.75
1.77
1.84
Water Content, w (%)
12.1
16.0
8.3
Void Ratio, e
0.48
0.47
0.45
Liquid Limit, LL (%)
46.7
49.6
15.69
Plastic Limit, PL (%)
27.4
31.9
Plasticity Index, PI (%)
19.3
18.7
Specific Gravity, Gs
2.59
2.61
2.66
GSD – Sand (%)
50.0
59.8
90
GSD – Silt (%)
37.5
29.1
10
GSD – Clay (%)
12.5
11.1
0
RI
PT
K50S50
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3
SC
Index properties
---
SM-ML
SM
SP
Saturated permeability ks (m/s)
1.12x10-8
1.12x10-7
2.65x10-5
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Unified Soil Classification System (USCS)
4. Comparison between the experimental data and the permeability function
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estimated using Zhai and Rahardjo’s (2015) method.
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Satyanaga et al.’s (2013) bimodal equation was adopted to best fit the SWCC data, as illustrated in Figure 5. An iterative nonlinear regression procedure that is provided in the Microsoft Excel software (Wraith and Or, 1998) was used to adjust all paramete rs
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to fit the equation for each soil. The complexity in best fitting the bimodal SWCC data was minimized by providing an appropriate initial value for each parameter. The initial value of m corresponds to matric suction at the midpoint of the SWCC slope obtained from the laboratory test. The initial value of r corresponds to the last matric suction applied in the SWCC test. The initial value for standard deviation () is assumed to be equal to 1. The initial value of r corresponds to the last volumetric water content obtained from the laboratory tests. The fitting parameters of the bimodal SWCC data in this study are summarized in Table 2. These fitting parameters
ACCEPTED MANUSCRIPT were subsequently used for the estimation of the permeability function using the modified Zhai and Rahardjo’s (2015) equation. The comparison between the experimental measured data and the estimation results are illustrated in Figure 6.
Table 2. Fitting parameters in Satyanaga et al.’s (2013) equation s1
a1
1
10
10.6
2.03
0.712
33
135
1 1
9.9 1.87
10.3 2.63
1.85 3.88
0.75 0.438
15 30
76 103.46
r
R2
1.06
500
0.025
99.37
0.8 1.5
600 750
0.124 0
99.0 99.02
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SC
K50S50 with 9% mica K50S50 K10S90
PT
Fitting parameters in Satyanaga et al.'s (2013) equation m1 s1 s2 a2 m2 s2 r
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Soils
MA
0.8
ED
0.6
R2=99.4%
0.4
EP T
Degree of saturation, S
1.0
0.2
AC C
Satyanaga et al.'s (2013) equation Experimental data for K50S50 with 9% mica
0.0
10-2
10-1
100
101
102
103
104
105
106
Matric suction, (kPa)
(a) SWCC for the mixture of sand and kaolin (K50S50 with 9% mica)
1.0
Satyanaga et al.'s (2013) equation Experimental data for K50S50
0.8
R2=99.0%
PT
0.6
0.4
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Degree of saturation, S
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0.0 10-1
100
101
102
103
104
105
106
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10-2
SC
0.2
Matric suction, (kPa)
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(b) SWCC for the mixture of sand and kaolin (K50S50)
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R2=99.0%
EP T
0.8
Satyanaga et al.'s (2013) equation Experimental data for K10S90
0.6
AC C
Degree of saturation, S
1.0
0.4
0.2
0.0 10-2
10-1
100
101
102
103
104
105
106
Matric suction, (kPa)
(c) SWCC for the mixture of sand and kaolin (K10S90) Figure 5. Best fit the experimental data with Satyanaga et al.’s (2013) equation
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R2=95.8%
10-7
10-8
10-9
PT
10-10
RI
10-11
10-12
Predicted using Zhai and Rahardjo's (2015) equation Experimental data for K50S50 with 9% mica
10-13 10-1
100
101
102
103
104
105
106
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10-2
SC
Coefficient of permeability, kw (m/s)
10-6
Matric suction, (kPa)
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(a) Permeability function of the mixture of sand and kaolin (K50S50 with 9% mica)
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Predicted using Zhai and Rahardjo's (2015) equation Experimental data for K50S50
EP T
10-7
10-8
10-9
R2=89.1%
AC C
Coefficient of permeability, kw (m/s)
10-6
10-10
10-11
10-12
10-13 10-2
10-1
100
101
102
103
104
105
106
Matric suction, (kPa)
(b) Permeability function of the mixture of sand and kaolin (K50S50)
ACCEPTED MANUSCRIPT
10-1
R2=85.6%
10-3 10-4 10-5 10-6
PT
10-7 10-8 10-9
10-11 10-12
Predicted using Zhai and Rahardjo's (2015) equation Experimental data for K10S90
10-1
100
101
102
103
104
105
106
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10-13 10-2
RI
10-10
SC
Coefficient of permeability, kw (m/s)
10-2
Matric suction, (kPa)
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(c) Permeability function of the mixture of sand and kaolin (K10S90) Figure 6. Comparison between the experimental data and the predicted results using
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Zhai and Rahardjo’s (2015) equation on the permeability function. As illustrated in Figure 6, the coefficients of determination (R2 ) of the SWCC curve
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fitting for these three soils were computed as 99.4% for K50S50 with 9% mica, 99.0% for K50S50, and 99.0% for K10S90, respectively. High values of R2 indicate
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that Satyanaga et al.’s (2013) equation had good performance in best fitting the bimodal SWCC data . On the other hand, as illustrated in Figure 6, the coefficients of determination (R2 ) of predicted log (k w) for these three soils were computed as 95.8% for K50S50 with 9% mica, 89.1% for K50S50, and 85.6% for K10S90, respectively. The results indicated that the estimation on the permeability function using Zhai and Rahardjo’s (2015) equation agreed well with the experimental data. In other words, Satyanaga et al.’s (2013) equation and Zhai and Rahardjo’s (2015) equation can be used for the estimation of the permeability function from a bimodal SWCC.
ACCEPTED MANUSCRIPT 5. Effect of the variability in bimodal SWCC on the estimation of the permeability function If Fredlund and Xing’s (1994) and van Genunchten’s (1980) equations were used to best fit the experimental data following approach 1 as categorized by Wijaya and Leong (2016), then the fitting results can be obtained as illustrated in Figure 7. The
PT
fitting parameters for Fredlund and Xing’s (1994) and van Genunchten’s (1980)
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equations are summarized in Table 3. These fitting parameters were subsequently
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used for the estimation of the permeability function using Zhai and Rahardjo’s (2015) equation and the estimation results are illustrated in Figure 8. The estimation method
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on the permeability function using Zhai and Rahardjo’s (2015) method and Satyanaga et al.’s (2013) best fit equation is named as ZR-SA model. Similarly, ZR-FX which
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means using Zhai and Rahardjo’s (2015) method and Fredlund and Xing’s (1994) best fit equation and ZR-VG models which means Zhai and Rahardjo’s (2015) method and
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van Genunchten’s (1980) best fit equation can be named as illustrated in Figure 8. The coefficients of determination R2 for best fitting SWCC data and predicted log(k w)
AC C
EP T
are illustrated in Table 4.
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Satyanaga et al.'s (2013) equation Fredlund and Xing's (1994) equation van Genuchten's (1980) equation K50S50 with 9% mica
0.8
PT
0.6
0.4
RI
Degree of saturation, S
1.0
0.0 10-2
10-1
100
101
102
SC
0.2
103
104
105
106
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Matric suction, (kPa)
MA
(a) Best fitted bimodal SWCCs for K50S50 with 9% mica
ED EP T
0.8
Satyanaga et al.'s (2013) equation Fredlund and Xing's (1994) equation van Genuchten's (1980) equation K50S50 with 9% mica
0.6
0.4
AC C
Degree of saturation, S
1.0
0.2
0.0
10-2
10-1
100
101
102
103
104
Matric suction, (kPa)
(b) Best fitted bimodal SWCCs for K50S50
105
106
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Satyanaga et al.'s (2013) equation Fredlund and Xing's (1994) equation van Genuchten's (1980) equation Experimental data for K10S90
0.8
PT
0.6
0.4
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Degree of saturation, S
1.0
0.0 10-1
100
101
102
103
104
105
106
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10-2
SC
0.2
Matric suction, (kPa)
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(c) Best fitted bimodal SWCCs for K10S90 Figure 7. Comparison of best fitted bimodal SWCC using three different best fit
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equations.
equations Soils
s1 1 1 1
a1
10.59 12.21 1.61
AC C
K50S50 with 9% mica K50S50 K10S90
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Table 3. Fitting parameters in Fredlund and Xing’s (1994) and van Genuchten’s (1980)
K50S50 with 9% mica K50S50 K10S90
s1
a1
1 1 1
0.022 0.014 0.68
Fitting parameters in Fredlund and Xing’s (1994) equation n1 m1 s2 a2 n2 m2 Cr 14.96 0.14 0.71 71.4 5.45 0.59 1500 3.97 0.35 0.75 54.14 5.05 0.72 1500 17.14 0.15 0.57 43.71 4.11 0.59 1500 Fitting parameters in van Genuchten’s (1980) equation b1 c1 r1 s2 a2 b2 c2 3.85 2.55 50
7.67 7.68 0.037
0 0 0.586
0.715 0.75 0.59
0.016 0.023 0.005
5.98 4.39 1.76
0.151 0.2 3.66
r2 0 0 0
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Estimated using ZR-SA model Estimated using ZR-FX model Estimated using ZR-VG model K50S50 with 9% mica
10-7
10-8
10-9
PT
10-10
RI
10-11
10-12
10-13 10-1
100
101
102
103
104
105
106
NU
10-2
SC
Coefficient of permeability, kw (m/s)
10-6
Matric suction, (kPa)
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(a) Estimated permeability functions from bimodal SWCCs for K50S50 with 9% mica
ED EP T
10-7
Estimated using ZR-SA method Estimated using ZR-FX method Estimated using ZR-VG method Experimental data for K50S50
10-8
10-9
AC C
Coefficient of permeability, kw (m/s)
10-6
10-10
10-11
10-12
10-13 10-2
10-1
100
101
102
103
104
105
106
Matric suction, (kPa)
(b) Estimated permeability functions from bimodal SWCCs for K50S50
ACCEPTED MANUSCRIPT
10-1
10-3 10-4 10-5 10-6
PT
10-7 10-8
Estiamted using ZR-SA model Estiamted using ZR-FX model Estimated using ZR-VG model Experimental data for K10S90
10-10 10-11 10-12
10-1
100
101
102
103
104
105
106
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10-13 10-2
RI
10-9
SC
Coefficient of permeability, kw (m/s)
10-2
Matric suction, (kPa)
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(c) Estimated permeability functions from bimodal SWCCs for K10S90 Figure 8. Comparison of the estimation results on the permeability function from
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three different models
Table 4. R2 for best fitted SWCC and predicted hydraulic conductivity, log(k w)
K50S50
K10S90
Satyanaga et al. (2013) Fredlund and Xing (1994) van Genuchten (1980) Satyanaga et al. (2013) Fredlund and Xing (1994) van Genuchten (1980) Satyanaga et al. (2013) Fredlund and Xing (1994) van Genuchten (1980)
AC C
K50S50 with 9% mica
EP T
Best fit equations
Soils
R2 for best fitted SWCC 99.4% 99.7% 99.9% 99.0% 98.9% 99.9% 99.0% 99.6% 99.5%
Methods for prediction ZR-SA ZR-FX ZR-VG ZR-SA ZR-FX ZR-VG ZR-SA ZR-FX ZR-VG
R2 for log(kw ) 95.8% 95.3% 99.2% 89.1% 98.2% 98.3% 85.6% 85.2% 80.9%
As illustrated in Table 4, there was not much difference in R2 of best fitted SWCC obtained using three equations from Satyanaga et al. (2013), Fredlund and Xing (1994) and van Genuchten (1980). In other words, these three equations had similar performance in best fitting the bimodal SWCC data as illustrated in Figure 8. However, it appears that there was high variability in best fitted SWCCs from these
ACCEPTED MANUSCRIPT three best fit equations in the high suction range (i.e., greater than 200 kPa) due to the lack of the experimental data in this suction range. Therefore, the accuracy of the best fitted SWCC is much dependent on the suction range covered in the measurement. This observation agrees with the point presented by Zhai and Rahardjo (2013) and Rahimi and Rahardjo (2016). On the other hand, as illustrated in Table 4, the values
PT
of R2 of the predicted log (kw) were also high which indicated that all these three
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methods including ZR-SA, ZR-FX, and ZR-VG gave good prediction on the
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hydraulic conductivities. As illustrated in Figure 9, high variability in the predicted hydraulic conductivity was also observed in the high suction range (i.e., greater than
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200 kPa) due to the high variability in their best fitted SWCC in this suction range. Therefore, the SWCC measurement in the high suction range is recommended not
MA
only for a good best fitted SWCC, but also for a good estimation of the permeability
ED
function.
adopted.
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6. Estimation results on the permeability function if unimodal SWCCs are
In order to test the accuracy of the estimation results on the permeability function
AC C
from unimodal SWCC, unimodal SWCC equations such as the original Fredlund and Xing (1994)’s and van Genuchten (1980)’s equations were used to best fit the experimental data of these three soils. The fitting parameters and best fitted SWCCs from these two equations are illustrated in Table 5 and Figure 9. The estimated permeability functions from the unimodal SWCCs are illustrated in Figure 10 as follows:
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Experimental data for K50S50 with 9% mica Fredlund and Xing (1994)'s equation van Genuchten (1980)'s equation
0.8
PT
0.6
0.4
RI
Degree of saturation, S
1.0
SC
0.2
0.0 10-1
100
101
102
103
104
105
106
NU
10-2
Matric suction, (kPa)
ED
MA
(a) Best fitted unimodal SWCC for K50S50 with 9% mica
EP T
0.8
0.6
0.4
0.2
AC C
Degree of saturation, S
1.0
Experimental data for K50S50 Fredlund and Xing (1994)'s equation van Genuchten (1980)'s equation
0.0 10-2
10-1
100
101
102
103
Matric suction, (kPa) (b) Best fitted unimodal SWCC for K50S50
104
105
106
ACCEPTED MANUSCRIPT
Experimental data for K10S90 Fredlund and Xing (1994)'s equation van Genuchten (1980)'s equation
0.8
PT
0.6
0.4
RI
Degree of saturation, S
1.0
SC
0.2
0.0 10-1
100
101
102
103
104
105
106
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10-2
Matric suction, (kPa)
MA
(c) Best fitted unimodal SWCC for K10S90
Figure 9. Best fitted unimodal SWCCs for the soils
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Table 5. Fitting parameters of unimodal SWCC Fredlund and Xing (1994)'s equation a (kPa) n m Cr (kPa)
Soils
AC C
EP T
K50S50 with 9% mica K50S50 K10S90 Soils
K50S50 with 9% mica K50S50 K10S90
836
130.81 1.48
0.88
13.5
R2 (%)
1500
99.17
1.19 4.02 1500 2.57 0.38 1500 van Genuchten (1980)'s equation
94.51 96.86
a
b
c
r
0.001
0.905
6.8
0.05
99.11
0.002 1.1
1.2 21.23
6.76 0.014
0.09 0.14
94.34 97.58
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
MA
(a) Estimated permeability function from unimodal SWCC for K50S50 with 9%
AC C
EP T
ED
mica
(b) Estimated permeability function from unimodal SWCC for K50S50
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SC
RI
PT
ACCEPTED MANUSCRIPT
MA
(c) Estimated permeability function from unimodal SWCC for K10S90 Figure 10. Estimated permeability function from unimodal SWCC for the soils
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Figure 9 and Table 5 indicate that the unimodal SWCC has lower R2 as compared to the bimodal SWCC as shown in Figure 5 and Table 2. Lower R2 indicates that the
EP T
unimodal SWCC has worse performance in best fitting the experimental data of these three samples than the bimodal SWCC. Consequently, the predicted permeability
AC C
function from these unimodal SWCCs, as illustrated in Figure 10, are less accurate than those predicted from the bimodal SWCC, as illustrated in Figure 8. It can be concluded that the selection of unimodal or bimodal SWCC best fit equation has significant effect on the results of the estimated permeability function.
7. Conclusions. The modified Zhai and Rahardjo’s (2015) equation performed well in estimating the permeability function of soil with bimodal SWCC since the estimated bimodal permeability function had a good agreement with the laboratory data. Comparison
ACCEPTED MANUSCRIPT results of the estimation of the bimodal permeability function from the SWCCs that were best fitted using three different equations, i.e., Satyanaga et al.’s (2013) equation, Fredlund and Xing (1994)’s equation and van Genuchten’s (1980) equation showed a high variability in the high suction range due to the lack of SWCC experimental data in this suction range. Therefore, the SWCC measurements up to high suction values
PT
must be carried out in order to obtain an accurate best fitted SWCC for a good
SC
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estimation of the permeability function.
Acknowledgement
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The authors gratefully acknowledge the assistance of Shi Xiang, Kevin Janiardy,
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Nickanor Tikno from School of Civil and Environmental Engineering, NTU,
AC C
EP T
ED
Singapore during the experiments and data collections.
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Highlight Equation for estimation of a bimodal permeability function. Verification of estimation results with experimental data. Comparison of estimation results using different best fit equations for a bimodal SWCC. Effect of variability in bimodal SWCC on the estimation results Measurement at high suction during SWCC test is recommended.