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Using inverse analysis to estimate hydraulic properties of unsaturated sand from one-dimensional outflow experiments
Engineering Geology 164 (2013) 163–171
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Using inverse analysis to estimate hydraulic properties of unsaturated sand from one-dimensional outflow experiments Nam To-Viet, Tuk-Ki Min, Hosung Shin ⁎ Dept. of Civil Engineering, University of Ulsan, Republic of Korea
a r t i c l e
i n f o
Article history: Received 11 March 2013 Received in revised form 5 June 2013 Accepted 11 July 2013 Available online 20 July 2013 Keywords: Unsaturated hydraulic property Inverse analysis One-dimensional column test Finite element method
a b s t r a c t A one-dimensional (1-D) vertical draining test for Jumunjin sand was carried out in one-step and multi-step outflow conditions. An inverse analysis method was conducted to determine the hydraulic properties of unsaturated soil. A non-linear optimization method combining a finite element code and inversion analysis was used to minimize the objective function, defined by the difference between observed and predicted data. Unsaturated hydraulic parameters in van Genuchten model were estimated using soil suction measurements in 10 cm intervals and an outflow rate at the bottom of a sand column. The predicted hydraulic properties and the experimental results were in close agreement when the measurements were compared from the one-step and multi-step outflow experiments. The results also showed that the multi-step outflow experiment was more appropriate in determining the unsaturated hydraulic properties than the one-step outflow experiment. The comparison between predicted and measured results concluded that the inverse analysis based on the 1-D outflow experiment was reliable and useful to determine the hydraulic properties of unsaturated soils. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The hydraulic properties of saturated and unsaturated soils are very important parameters that determine movement of gas, water and solute in geological systems. These properties govern and affect many fundamental problems of soils such as seepage, slope stability, bearing capacity, consolidation and settlement and water contamination in various fields: geotechnical engineering, environmental engineering, soil science, agricultural engineering, groundwater hydrology, etc. However, obtaining reliable hydraulic properties is still difficult, especially unsaturated hydraulic conductivity and water contents at various suctions. Therefore, reasonable and simple methods to accurately determine the hydraulic properties of unsaturated soils are very necessary and essential. Up until now, many useful hydraulic measurements including direct, indirect, numerical and semi-empirical methods have been developed to assess the hydraulic properties of soils. Since direct measurements are usually time-consuming, expensive and especially difficult to obtain (Klute, 1972; Olson and Daniel, 1981; Hillel, 1982), indirect, semi-empirical measurements or numerical modeling were generally considered effective and widely used (Brooks and Corey, 1964; Green and Corey, 1971; Mualem, 1976; van Genuchten, 1980; Fredlund et al., 1994). The hydraulic properties of unsaturated soil are represented by two relationships: volumetric water content–soil suction (θ–ψ) and
⁎ Corresponding author. Tel.: +82 52 259 1723. 0013-7952/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.enggeo.2013.07.005
hydraulic conductivity–soil suction (k–ψ). The θ–ψ relationship is called the soil–water characteristic curve (SWCC) in geotechnical engineering or water retention curve (WRC) in agricultural engineering. The SWCC plays an important role in unsaturated soil mechanics because it is directly related to many mechanical properties such as volume change, diffusivity, shear strength functions, adsorption, etc. (Fredlund and Rahardjo, 1993). Numerous studies have described the effective use of SWCC to predict the hydraulic conductivity of unsaturated soils (Gardner, 1958; Brooks and Corey, 1964; Brutsaert, 1966; van Genuchten, 1980; Fredlund and Xing, 1994; Fredlund et al., 2011). The hydraulic conductivity in the k–ψ relationship of unsaturated soils can be displayed as unsaturated hydraulic conductivity (ku) or relative hydraulic conductivity (kr), which is expressed as the ratio between unsaturated hydraulic conductivity at a given soil suction and saturated hydraulic conductivity (van Genuchten, 1980; Fredlund et al., 1994; Ruana and Illangasekare, 1999). In recent decades, one of the most popular experimental methods to measure hydraulic properties of unsaturated soils in the laboratory is a one-dimensional (1-D) column test. Researchers typically use three types of methodologies: one-step outflow experiments (instantaneous application of one large pressure step), multi-step outflow experiments (application of several smaller pressure steps) and continuous flow experiments (application of the smooth continuous change in the pressure gradient) (Schultze and Durner, 1996). These experiments were performed under steady or transient conditions including wetting or drying processes. The hydraulic properties of unsaturated soils were determined from discharge water velocity, evolution of soil suction and water content during the test.
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Fig. 1. (a) The schematic diagram of column test, (b) tensiometer equipment, (c) electrical resistivity tester.
Nowadays, numerical methods combined with an optimization code have become promising tools to solve hydraulic problems easily and effectively. With these methods, the unknown parameters of hydraulic properties are estimated by minimizing the difference between the
Saturation (x 100%)
1.0 0.8 0.6 0.4 0.2 0.0 0
2000
4000
6000
8000
Electrical resistivity (Ohm.m) Fig. 2. Calibration relationship between electrical resistivity and saturation for Jumunjin sand.
predicted and observed measurements of flow rate, water content and soil suction. The application of the inverse solution technique to onestep outflow experiments was proposed early to estimate hydraulic properties using only cumulative outflow data (Kool et al., 1985a,b; Kool and Parker, 1988). Since soils are inherently heterogeneous and complex material, inversion analysis for hydraulic properties can lead to non-unique solution (Levasseur et al., 2009). Therefore, to overcome non-uniqueness in inversion analysis, it has been recommended to include additional θ(ψ) (Hudson et al., 1991; van Dam et al., 1992; Bohne et al., 1993; Simunek et al., 1998), or tensiometer measurements (Toorman et al., 1992; Eching and Hopmans, 1993) in the objective function (Crescimanno and Iovino, 1995). Thus, combination of outflow, volumetric water content and suction over time could be used for the inversion of parameters for SWCC (Durner et al., 1997). Table 1 Properties of Jumunjin sand. Sample
Gs
Cu
Cc
D50 (mm)
emax
emin
Jumunjin sand
2.66
1.65
0.94
0.55
0.992
0.596
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Fig. 3. a. Cumulative outflow over time in the one-step outflow experiment. b. Discharge rate over time in the one-step outflow experiment.
This study is to assess the applicability of an inverse parameter estimation method in a 1-D outflow experiment to determine the hydraulic properties of unsaturated soils. De-saturation experiment was carried out on a vertical sand column to measure cumulative outflow, water content and suction of soil over time. Simultaneously, a finite element code and inversion code were developed for hydraulic parameter estimations using measured data of soil suction and the outflow rate from outflow experiments as input data. In addition, a comparison between predicted hydraulic properties from the one-step and multi-step outflow methods was also performed to determine the optimal method for inverse parameter estimation of unsaturated hydraulic properties.
2. Research methodology 2.1. Experimental study A 1-D column (H = 80 cm, D = 28 cm, wall thickness = 1 cm) made of Plexiglas was used to conduct the vertical drainage test. Eight tensiometers and eight electrical resistivity probes (ERP) were placed in the sample every 10 cm vertically (L1 ~ L8) as shown in Fig. 1a, and were connected to data acquisition systems. The tensiometers were used to measure the soil suction, which had a tiny porous ceramic cup 6.5 mm in diameter and 2.5 cm long, attached to the tensiometer body via long nylon tubing. To install tensiometer easily into the soil column, it was inserted into a plastic screw as shown in Fig. 1b. Soil suction at each height was measured using tensiometer, and it was displaced
2.2. Numerical study
0.16 0.14
Displacement (cm)
and automatically saved by data logger which is connected and controlled by a computer (Figure 1b). ERP was used to measure soil resistance to calculate water saturation (Kechavarzi and Soga, 2002). ERP was made of two rods (10 cm length, 2 mm diameter and 1.2 cm distance apart) inserted into the plastic screw and was connected to the switching box, which was used to eliminate the electrical interference caused by simultaneous measurement of ERPs (Figure 1c). Electrical resistance of unsaturated soil at different water saturation was measured to build calibration curve in the preliminary tests (Figure 2), and that was fitted with Archie's law (exponent m = 1.32 and Archie exponent na = 1.57). The plastic bag located on the top of the column minimized water evaporation from the sample surface in the column test. The cumulative outflow at the bottom of the column was measured by weighing and automatic recording. The properties of the soil used are shown in Table 1. The sand grains were mixed under water and scooped into the column to prepare a fully saturated sample. The temperature in the laboratory was maintained at 20 °C to prevent a change in water–air surface tension (Kechavarzi and Soga, 2002). Opening the valve located at the bottom of column drained the water out of the soil column via gravity. In the one-step outflow experiment, outflow was maintained at the bottom level of the column during the test. In the multi-step outflow experiment, the bottom valve opened stepwise to maintain an apparent water table in the column, with the following levels. The beginning water height was from 80.0 cm to 55.0 cm, at 24 h from 55.0 cm to 25.0 cm and at 48 h from 25.0 cm to 0.0 cm. Soil suction, soil electrical resistivity and total water outflow were measured every 10 s. Settlement in the soil column was measured using imageprocessing techniques (Shin and Santamarina, 2011).
2.2.1. Forward simulation In porous materials, water can exist in the form of liquid and vapor, and the governing equation of water is derived from a generalized law of mass conservation (Shin, 2011):
0.12 0.10 0.08
0.04
2 3 i ∂h w w w w w w ρl ϕ Sl þ ρg ϕ ð1−Sl Þ þ ∇ 4ρl ϕ Sl u˙ þρl q þ ρg ϕ ð1−Sl Þ u˙ þρg q 5 ¼ 0 ∂t e e e el eg
0.02
ð1Þ
0.06
0.00 0
1000
2000
3000
4000
Time (second) Fig. 4. Surface displacement over time from the one-step outflow experiment.
w where ϕ is porosity of the soil, Sl is saturation of liquid water, ρw l and ρg are mass of water per unit volume in the liquid and gas phases. u˙ is the velocity of soil particles and q and q are the liquid and gas floweflux in each phase, respectively. el eg
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80
1.0
70 60
Height (cm)
Degree of saturation
0.8
0.6
L2 (15 cm) L5 (45 cm) L8 (75 cm)
0.4
0 min 0.5 min 1 min 2 min 5 min 10 min 1 hr 24 hr 72 hr
50 40 30 20
0.2 10 0.0
0 0
100
200
300
400
500
600
700
0.0
0.5
1.0
Time (minute)
Degree of saturation
(a)
(b)
Fig. 5. a. Water saturation over time obtained from the one-step outflow experiment at different heights. b. Water saturation profile from the one-step outflow experiment.
For 1-D vertical transient water flow assuming no gas flow rate and no soil particle movement, Eq. (1) becomes the well-known Richards equation: " # ∂ w w ρl ϕ Sl þ ∇ ρl q ¼ 0: ∂t e el
2.2.2. Inverse simulation A non-linear optimization method was used to minimize the objective function using L2 error norm, which expresses the difference between observed and predicted data by using the least-square solution for the parameters in Eqs. (3) and (4) (Santamarina and Fratta, 2005).
ð2Þ
h iT h i T bm N bm N Γ ¼ e :e ¼ W :y −W :h:x 0 −W :h:Δx : W :y −W :h:x 0 −W :h:Δx
For the hydraulic constitutive model for unsaturated soils, this study uses van Genuchten model (1980) for the SWCC and unsaturated relative hydraulic conductivity.
Se ¼
krl ¼
S−Sr min ¼ 1−Sr min
1þ
1 !−λ Pg −Pl 1−λ P0
ð5Þ where x are unknown soil properties (x ¼ x 0 þ Δx), x 0 is initial guess of bm N unknown variables, y is the measurement data (saturation, pressure head, outflow, displacements, stresses, etc.), h is the transformation matrix which is hij = ∂yi/∂xj (y ¼ h:x), and W is the weight factor (where Wij = weight value, Wij = 0 for i ≠ j). The partial derivative of Eq. (5) with respect to Δx is zero to minimize the objective function:
ð3Þ
pffiffiffiffiffi 1=λ λ 2 Se 1− 1−Se
ð4Þ
where Se is effective saturation, Sr is the residual saturation, S is degree of saturation, Pg − Pl is soil suction, P0 and λ are the curve fitting parameter, krl(=kunsat/ksat) is relative hydraulic conductivity and ksat is saturated hydraulic conductivity. The Galerkin formulation of the governing equation was used to develop the finite element code. The initial condition for water pressure is static water gravity with zero water flux at the bottom boundary.
h i ∂Γ T T bm N T T −W :h:x 0 þ 2h :W :W :h:Δx ¼ 0 ¼ −2h :W W :y ∂Δx h i−1 h i T T T T bm N ⇒Δx ¼ h :W :W :h :h :W : W :y −W :h:x 0 : 80
8
Initial condition 0.5 min 1 min 2 min 5 min 10 min 1 hr 24 hr 72 hr Final condition
60
L2 (15 cm) L5 (45 cm) L8 (75 cm)
4 2
Height (cm)
Pore water pressure (kPa)
70 6
50 40 30
0 20 -2
10
-4 0
100
200
300
400
500
600
700
0 -7.5
-5.0
-2.5
0.0
2.5
5.0
Time (minute)
Pore water pressure (kPa)
(a)
(b)
7.5
Fig. 6. a. Pore-water pressure over time from the one-step outflow experiment at different heights. b. Pore-water pressure head profile from the one-step outflow experiment.
ð6Þ
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167
3. Experimental results Fig. 3 shows the measured cumulative outflow and discharge rate in the one-step outflow experiment. The cumulative outflow showed rapid changes during the first 10 min, rapidly reaching Q ≈ 24.21 cm after 2.5 h (Figure 3a). The maximum outflow rate just after starting the experiment was slightly greater than the saturated hydraulic conductivity of the soil column, but it was a very good initial pre-indicator, which was verified from the numerical simulations. Just after opening the bottom valve, the hydraulic gradient from outflow rate was very high so that the discharge rate at that time was very high. After 400 s, it converged to steady stage (Figure 3b). The sample surface began to settle immediately after starting the test, and nearly stopped after 10 min (Figure 4). Since the magnitude of the settlement was relatively small compared with the column height, we assumed constant void ratio when evaluating the unsaturated hydraulic properties. Figs. 5 and 6 show the variation in water saturation and suction head profiles over time during the drainage test. When opening the valve to allow free drainage at the bottom, the lower end of the soil column was exposed to atmospheric pressure at that moment. Both an instant pressure drop at the bottom of the column and water gravity were the main driving forces for this de-saturation process. Water saturation and suction at different locations changed dramatically in the beginning due to bottom drainage. But pore-water pressure took a long time to reach the final equilibrium state, so it did not reach hydrostatic pressure even after 700 min (Figure 6b; Shin, 2011). From the water saturation profiles, the minimum measured residual water saturation was close to 13%, and the water saturation at L2 was around 70% due to low capillary suction from water gravity (Figure 5). At final equilibrium, the distribution of saturation and pore pressure according to depth can be converted into the SWCC.
Fig. 7. Values of lambda (λ) and P0 vs. iterative steps.
Fig. 8. Comparison between predicted and measured outflow rate in the one-step outflow experiment.
4. Numerical results Numerical inversion analysis was conducted to evaluate unsaturated hydraulic properties using a 1-D de-saturation test. Hydraulic properties of unsaturated soils can be determined by two functions: the SWCC (relationship between pore-water pressure and water saturation) and unsaturated hydraulic conductivity (relationship between hydraulic conductivity of water and gas and degree of saturation). In this study, we chose the van Genuchten model for the SWCC and unsaturated hydraulic conductivity function (van Genuchten, 1980). Two unknown parameters in the hydraulic model, P0 and λ, were determined from inversion analysis using experimental measurements, outflow rate and soil suction over time. Numerical analysis is consisted of
Thus, updated soil properties x are given by h i h i−1 T T T T bm N x ¼ x 0 þ Δx ¼ x 0 þ h :W :W :h :h :W : W :y −W :h:x 0 :
ð7Þ
In the optimization process, the initial guess following van Genuchten parameters were: P0 = 0.8 and λ = 0.8. Simulations were repeated with updated parameter estimates until the objective function became smaller than the critical tolerance.
L2 (15 c m)
L5 ( 45 cm)
6 Measured data
4
Predicted data
2 0 -2 -4
8
Pore water pressure (kPa)
Pore water pressure (kPa)
Pore water pressure (kPa)
L8 (75 c m)
8
8
6 4 Measured data
2
Predicted data
0 -2
100
200
300
400
Time (minute)
500
600
700
4 Measured data
2
Predicted data
0 -2 -4
-4 0
6
0
100
200
300
400
Time (minute)
500
600
700
0
100
200
300
400
Time (minute)
Fig. 9. Comparison between predicted and measured pore water pressures in the one-step outflow experiment at locations L2, L5 and L8.
500
600
700
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Degree of saturation
Degree of saturation
0.8 0.6 Measured data
0.4
L8 (7 5 c m )
L5 ( 45 c m )
L 2 ( 1 5 cm )
1.0
Predicted data
0.2
1.0
1.0
0.8
0.8
0.6
Degree of saturation
(a)
Measured data Predicted data
0.4 0.2
0.6
0
100
200
300
400
500
600
700
0.2 0.0
0
100
Time (minute)
200
300
400
500
600
700
0
0.6 Measured data Predicted data
0.2
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.8
Measured data Predicted data
0.6 0.4 0.2 0.0 0.0
300
400
500
600
700
1.0
Degree of saturation
0.8
200
Time (minute)
1.0
Degree of saturation
Degree of saturation
1.0
0.0 0.0
100
Time (minute)
(b)
0.4
Predicted data
0.4
0.0
0.0
Measured data
0.5
1.0
Suction (-kPa)
1.5
2.0
2.5
3.0
3.5
0.8 Measured data Predicted data
0.6 0.4 0.2 0.0 0.0
0.5
Suction (-kPa)
1.0
1.5
2.0
2.5
3.0
3.5
Suction (-kPa)
Fig. 10. Comparison between predicted and measured hydraulic properties in the one-step outflow experiment at locations L2, L5 and L8: (a) Change in degree of saturation, (b) compiled SWCC.
(1) Discharge flow and suction measurement at 3 electrodes were used for inversion analysis to estimate unknown hydraulic parameters. (2) Measurement of degree of saturation at 3 electrodes is used to build compiled SWCC data which are compared with numerical results using estimated hydraulic parameters in the previous stage. Fig. 7 shows the iterative process used to determine the unknown parameters in the inversion analysis. After 10 iterations, initial guessed values λ = 0.8 and P0 = 0.8 converged to λ = 0.857 and P0 = 1.475. Numerical predictions using assessed hydraulic parameters produced good agreement with experimental observations (Figures 8 and 9). The evolution of water saturation at three locations in the soil column could be predicted through numerical analysis, and compared well with experimental observations using the electrical resistivity probe (Figure 10a). Soil suction and degree of water saturation were paired in the SWCC plot, and compiled SWCC data over time is plotted in Fig. 10b. Location L2 near the bottom of the column maintained a high degree of saturation during the tests, so all collected data were plotted near the saturation region. However, location L8, close to top of the column, experienced dramatic changes in the degree of saturation during the test, so most of the data were plotted in the residual region of
Table 2 Root mean square error from predicted and measured data in time domain (one-step outflow experiment).
Location RMSE Average
Pore-water pressure (kPa)
Saturation
L2 0.477 0.389
L2 0.035 0.041
L5 0.328
L8 0.363
the SWCC plot. For this reason, we had to install several tensiometers to measure soil suction, to cover the full range of the SWCC plot. Sources of the differences (errors) between measured and predicted data may come from the hydraulic model itself, and the measuring sensors (tensiometer, needle probe, electrical balance, etc.). Root mean square error (RMSE) was used to assess the differences:
RMSE ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX m p 2 u yi −yi u t i¼1
p where ym i represents experimental measurements, yi denotes predicted values using estimated soil properties such as pore-water pressure, degree of saturation and flow rate and N is total number of measurements. The comparison of the measured data (Figures 9 and 10) and the predicted data using assessed soil parameters from the one-step outflow experiment shows the relatively low errors for all three locations
8 6
L2 (15 cm) L5 (45 cm) L8 (75 cm)
4 2 0 -2 -4 0
L5 0.046
L8 0.042
ð8Þ
N
Pore water pressure (kPa)
two stages: inversion analysis and experimental verification using additional measurements.
1000
2000
3000
4000
Time (minute) Fig. 11. Pore-water pressure over time from the multi-step outflow experiment at three locations.
N. To-Viet et al. / Engineering Geology 164 (2013) 163–171
(b)
1.0
1.0
0.8
0.8
Degree of saturation
Degree of saturation
(a)
169
Measured data
0.6 0.4 0.2 0.0 0.0
0.6
Measured data
0.4 0.2 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0
0.5
Suction (-kPa)
1.0
1.5
2.0
2.5
3.0
3.5
Suction (-kPa)
Fig. 12. Compiled SWCC from locations L2, L5 and L8: a) one-step outflow experiment, b) multi-step outflow experiment.
Fig. 13. Values of lambda (λ) and P0 vs. iterative steps.
(Table 2). The hysteresis phenomenon, narrow range of saturation and lack of data near saturation of the SWCC contributed to the error in the evaluation of hydraulic properties. 5. Discussion In previous studies, multi-step outflow experiments have been known to reveal more accurate results than one-step outflow experiments (Crescimanno and Iovino, 1995; Schultze and Durner, 1996; Hwang and Powers, 2003). Although the one-step outflow experiment provided a good estimation of unsaturated hydraulic properties, a multi-step outflow experiment was also conducted to investigate the advantage of the experimental procedure. In the multi-step outflow experiment, pore-water pressure dropped three times, 8.0 to 5.5 kPa, 5.5 to 2.5 kPa and 2.5 to 0.0 kPa. The water pressures were maintained during 24 h for each pressure drop to
L2 (15 c m )
L5 (45 cm)
L 8 (7 5 c m )
Pore water pressure (kPa)
6
Measured data Predicted data
4 2 0 -2 -4
8
Pore water pressure (kPa)
8
8
Pore water pressure (kPa)
balance the water pressure inside the soil sample. Fig. 11 shows the three stages of pore-water pressure in the soil sample corresponding to each pressure drop. Compared to the one-step outflow experiment, the multi-step outflow experiment produced a larger number of experimental data within a narrow band in the compiled SWCC data from three locations, from the air entry to residual regions (Figure 12). This difference from onestep outflow experiment is due to smaller increments of hydraulic gradient at every pressure step. The same initial parameters were used to estimate the optimized soil parameters of unsaturated hydraulic properties (λ = 0.8 and P0 = 0.8). After 13 iterative steps, inversion results converged to λ = 0.883 and P0 = 1.508, which was slower than the one-step outflow experiment (Figure 13). Fig. 14 shows the evolution of pore-water pressure from the experimental observations and reproduced numerical results. Similar to the one-step outflow experiment, the degree of saturation and compiled SWCC at three locations in the soil column could be determined, and compared well with the measurements (Figure 15). SWCC data using the multi-step experiment agreed better with the model prediction than the one-step experiment (Figures 10b and 15b). In the multi-step outflow experiment, the change in pore-water pressure at each step was so slow that more hydraulic data at near saturation was collected and the hysteresis in the soil was smaller. The multi-step outflow experiment improved assessment of hydraulic properties (Figures 10 vs. 15, Tables 2 vs. 3). The two types of outflow experiments illustrated that the estimated soil parameters for unsaturated hydraulic properties were very close to actual measurements from both tests, as well as having good convergence to obtain optimized model parameters. The one-step outflow experiment required a shorter time to complete the experiment and smaller iterative steps in the inversion analysis. However, quick and fast changes in hydraulic pressure in the one-step outflow experiment can cause a loss in SWCC data between air-entry and residual regions
6 Measured data
4
Predicted data
2 0 -2 -4
0
1000
2000
Time (minute)
3000
4000
6 4
Measured data Predicted data
2 0 -2 -4
0
1000
2000
Time (minute)
3000
4000
0
1000
2000
Time (minute)
Fig. 14. Comparison between predicted and measured pore water pressure in the multi-step outflow experiment at locations L2, L5 and L8.
3000
4000
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N. To-Viet et al. / Engineering Geology 164 (2013) 163–171
L5 ( 45 c m)
L 2 ( 1 5 cm )
1.0
0.8
0.8
0.6 Measured data
0.4
Predicted data
0.2 0.0
L 8 ( 75 cm)
1.0
Degree of saturation
1.0
Degree of saturation
Degree of saturation
(a)
Measured data Predicted data
0.6 0.4 0.2 0.0
0
1000
2000
3000
4000
0.8 Measured data
0.6
Predicted data
0.4 0.2 0.0
0
1000
Time (minute)
2000
3000
4000
0
1000
Time (minute)
2000
3000
4000
Time (minute)
1.0
0.8
0.8
0.6 Measured data Predicted data
0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
1.0
Degree of saturation
1.0
Degree of saturation
Degree of saturation
(b) Measured data Predicted data
0.6 0.4 0.2 0.0 0.0
Suction (-kPa)
0.8
Measured data Predicted data
0.6 0.4 0.2 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Suction (-kPa)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Suction (-kPa)
Fig. 15. Comparison between predicted and measured hydraulic properties in the one-step outflow experiment at locations L2, L5 and L8: (a) change in degree of saturation, (b) compiled SWCC.
and high hysteresis in the soil column test, especially with coarsegrained soils. The multi-step outflow experiment required more time to conduct the experiment, along with the complex boundary condition in the numerical analysis. However, low changes in hydraulic pressure in the soil sample can help obtain more information about the hydraulic property of soil for the better prediction of hydraulic properties of unsaturated soils.
6. Conclusions Over the last several decades, the inverse analysis technique has been consistently used to determine unsaturated hydraulic properties. However, its reliability has yet to be sufficiently affirmed in comparisons with independently measured hydraulic properties of soils. A 1-D outflow experiment could be a good indirect experimental methodology to estimate the SWCC and unsaturated hydraulic conductivity function. In this study, one-step and multi-step outflow experiments were carried out to measure bottom discharge rates, soil suction and electrical resistances over time. Inverse analysis was performed to determine the unsaturated hydraulic function in the van Genuchten model using outflow rate and suction at specific depths over time. Compilations of soil suction and degree of saturation from electrical resistance provided
SWCC data was compared with model predictions from the inverse analysis. − The combined analysis of the finite element method and inversion technique was performed to evaluate hydraulic properties of unsaturated soils, which produced a good comparison of measurements from the one-step and multi-step outflow experiments. − The multi-step outflow experiment provided a wider suction range in experimental data than the one-step outflow experiment to fit measurements in the van Genuchten model. − However, the multi-step outflow experiment has more complex outflow control and needs more iteration for convergence in inversion analysis. − This study showed that inversion analysis of a 1-D de-saturation column test could be a good indirect method to evaluate hydraulic properties of unsaturated soils. Since we conducted laboratory experiments with uniform sand and with a simple initial boundary condition, the presented approach should be extended to further investigate various soils with different sedimentation. Acknowledgments This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (KRF-2011-0013210).
Table 3 Root mean square error from predicted and measured data in time domain (multi-step outflow experiment).
Location RMSE Average
Pore-water pressure (kPa)
Saturation
L2 0.282 0.313
L2 0.025 0.032
L5 0.3
L8 0.358
L5 0.040
L8 0.031
References Bohne, K., Roth, C., Leij, F.J., van Genuchten, M.Th, 1993. Rapid method for estimating the unsaturated hydraulic conductivity from infiltration measurements. Soil Science 155 (4), 237–244. Brooks, R.H., Corey, A.T., 1964. Hydraulic properties of porous media. Hydrology Paper No. 3. Colorado State University, Fort Collins, Colo (March). Brutsaert, W., 1966. Some methods of calculating unsaturated permeability. Transactions of the American Society of Agricultural Engineers 10, 400–404.
N. To-Viet et al. / Engineering Geology 164 (2013) 163–171 Crescimanno, G., Iovino, M., 1995. Parameter estimation by inverse method based on onestep and multi-step outflow experiments. Geoderma 68, 257–277. Durner, W., Schultze, E.B., Zurmuhl, T., 1997. State-of-the-art in inverse modeling of inflow/outflow experiments. Characterization and measurement of the hydraulic properties of unsaturated porous media. In: vanGenuchten, M.T., Leij, F.J., Wu, L. (Eds.), Proceedings of International Workshop, Riverside, CA. October 22–24, pp. 661–681. Eching, S.O., Hopmans, J.W., 1993. Optimization of hydraulic functions from transient outflow and soil water pressure data. Soil Science Society of America Journal 57 (5), 1167–1175. Fredlund, D.G., Rahardjo, H., 1993. Soil Mechanics for Unsaturated Soils. Wiley, New York. Fredlund, D.G., Xing, A., 1994. Equations for the soil–water characteristic curve. Canadian Geotechnical Journal 31 (4), 521–532. Fredlund, D.G., Xing, A., Huang, S., 1994. Predicting the permeability function for unsaturated soil using the soil–water characteristic curve. Canadian Geotechnical Journal 31 (3), 521–532. Fredlund, D.G., Sheng, D., Zhao, J., 2011. Estimation of soil suction from the soil–water characteristic curve. Canadian Geotechnical Journal 48, 186–198. Gardner, W.R., 1958. Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Science 85 (4), 228–232. Green, R.E., Corey, J.C., 1971. Calculation of hydraulic conductivity: a further evaluation of some predictive methods. Soil Science Society of America Proceedings 35, 3–8. Hillel, D., 1982. Introduction to Soil Physics. Academic, New York. Hudson, D.B., Young, M.H., Wierenga, P.J., Hills, R.H., 1991. Transient method for estimating soil water parameters of unsaturated soil cores. Agronomy Abstracts. ASA, Madison, WI, p. 221. Hwang, S.I., Powers, S.E., 2003. Estimating unique soil hydraulic parameters for sandy media from multi-step outflow experiments. Advances in Water Resources 26, 445–456. Kechavarzi, C., Soga, K., 2002. Determination of water saturation using miniature resistivity probes during intermediate scale and centrifuge multiphase flow laboratory experiments. Geotechnical Testing Journal, GTJODJ 25 (1), 95–103. Klute, A., 1972. The determination of the hydraulic conductivity and diffusivity of unsaturated soils. Soil Science 113 (4), 264–276. Kool, J.B., Parker, J.C., 1988. Analysis of the inverse problem for transient unsaturated flow. Water Resources Research 24 (6), 817–830.
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Kool, J.B., Parker, J.C., vanGenuchten, M.Th, 1985a. Determining soil hydraulic properties from one-step outflow experiments by parameter estimation: I. Theory and numerical studies. Soil Science Society of America Journal 49, 1348–1354. Kool, J.B., Parker, J.C., vanGenuchten, M.Th, 1985b. Determining soil hydraulic properties from one-step outflow experiments by parameter estimation: II. Experimental studies. Soil Science Society of America Journal 49, 354–1359. Levasseur, S., Malecot, Y., Boulon, M., Flavigny, E., 2009. Statistical inverse analysis based on genetic algorithm and principal component analysis: method and developments using synthetic data. International Journal for Numerical and Analytical Methods in Geomechanics 33 (12), 1485–1511. Mualem, Y., 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. 12 (3), 513–522. Olson, R.E., Daniel, D.E., 1981. Measurement of the hydraulic conductivity of fine-grained soils. In: Zimmie, T.F., Riggs, C.O. (Eds.), Permeability and Groundwater Contaminant Transport. ASTMSTP746. ASTM, Philadelphia, pp. 18–64. Ruana, H., Illangasekare, T.H., 1999. Estimation of relative hydraulic conductivity of sandy soils based on a sheet flow model. Journal of Hydrology 219, 83–93. Santamarina, J.C., Fratta, D., 2005. Discrete Signals and Inverse Problems. Wiley, New York. Schultze, B., Durner, W., 1996. Onestep, multistep and continuous transient flow experiments for the determination of soil hydraulic properties. Bayreuther Forum Ökologie. , 28. Universität Bayreuth, Germany 194–197. Shin, H., 2011. Formulation of fully coupled THM behavior in unsaturated soil. Journal of the Korean Geotechnical Society 27 (3), 75–83. Shin, H., Santamarina, J.C., 2011. Desiccation cracks insaturated fine-grained soils: particlelevel phenomena and effective-stress analysis. Geotechnique 61 (11), 961–972. Simunek, J.R., Jaramillo, R.A., Schaap, M.G., Vandervaere, J.P., van Genuchten, M.T., 1998. Using an inverse method to estimate the hydraulic properties of crusted soils from tension disc infiltrometer data. Geoderma 86, 61–81. Toorman, A.F., Wierenga, P.J., Hills, R.G., 1992. Parameter estimation of hydraulic properties from one-step outflow data. Water Resources Research 28 (11), 3021–3028. Van Dam, J.C., Stricker, J.N.M., Verhoef, A., 1992. Inverse method for determining soil hydraulic functions from one-step outflow experiments. Soil Science Society of America Journal 56, 1042–1050. van Genuchten, M.T., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal 44, 892–898.
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Engineering Geology journal homepage: www.elsevier.com/locate/enggeo
Using inverse analysis to estimate hydraulic properties of unsaturated sand from one-dimensional outflow experiments Nam To-Viet, Tuk-Ki Min, Hosung Shin ⁎ Dept. of Civil Engineering, University of Ulsan, Republic of Korea
a r t i c l e
i n f o
Article history: Received 11 March 2013 Received in revised form 5 June 2013 Accepted 11 July 2013 Available online 20 July 2013 Keywords: Unsaturated hydraulic property Inverse analysis One-dimensional column test Finite element method
a b s t r a c t A one-dimensional (1-D) vertical draining test for Jumunjin sand was carried out in one-step and multi-step outflow conditions. An inverse analysis method was conducted to determine the hydraulic properties of unsaturated soil. A non-linear optimization method combining a finite element code and inversion analysis was used to minimize the objective function, defined by the difference between observed and predicted data. Unsaturated hydraulic parameters in van Genuchten model were estimated using soil suction measurements in 10 cm intervals and an outflow rate at the bottom of a sand column. The predicted hydraulic properties and the experimental results were in close agreement when the measurements were compared from the one-step and multi-step outflow experiments. The results also showed that the multi-step outflow experiment was more appropriate in determining the unsaturated hydraulic properties than the one-step outflow experiment. The comparison between predicted and measured results concluded that the inverse analysis based on the 1-D outflow experiment was reliable and useful to determine the hydraulic properties of unsaturated soils. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The hydraulic properties of saturated and unsaturated soils are very important parameters that determine movement of gas, water and solute in geological systems. These properties govern and affect many fundamental problems of soils such as seepage, slope stability, bearing capacity, consolidation and settlement and water contamination in various fields: geotechnical engineering, environmental engineering, soil science, agricultural engineering, groundwater hydrology, etc. However, obtaining reliable hydraulic properties is still difficult, especially unsaturated hydraulic conductivity and water contents at various suctions. Therefore, reasonable and simple methods to accurately determine the hydraulic properties of unsaturated soils are very necessary and essential. Up until now, many useful hydraulic measurements including direct, indirect, numerical and semi-empirical methods have been developed to assess the hydraulic properties of soils. Since direct measurements are usually time-consuming, expensive and especially difficult to obtain (Klute, 1972; Olson and Daniel, 1981; Hillel, 1982), indirect, semi-empirical measurements or numerical modeling were generally considered effective and widely used (Brooks and Corey, 1964; Green and Corey, 1971; Mualem, 1976; van Genuchten, 1980; Fredlund et al., 1994). The hydraulic properties of unsaturated soil are represented by two relationships: volumetric water content–soil suction (θ–ψ) and
⁎ Corresponding author. Tel.: +82 52 259 1723. 0013-7952/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.enggeo.2013.07.005
hydraulic conductivity–soil suction (k–ψ). The θ–ψ relationship is called the soil–water characteristic curve (SWCC) in geotechnical engineering or water retention curve (WRC) in agricultural engineering. The SWCC plays an important role in unsaturated soil mechanics because it is directly related to many mechanical properties such as volume change, diffusivity, shear strength functions, adsorption, etc. (Fredlund and Rahardjo, 1993). Numerous studies have described the effective use of SWCC to predict the hydraulic conductivity of unsaturated soils (Gardner, 1958; Brooks and Corey, 1964; Brutsaert, 1966; van Genuchten, 1980; Fredlund and Xing, 1994; Fredlund et al., 2011). The hydraulic conductivity in the k–ψ relationship of unsaturated soils can be displayed as unsaturated hydraulic conductivity (ku) or relative hydraulic conductivity (kr), which is expressed as the ratio between unsaturated hydraulic conductivity at a given soil suction and saturated hydraulic conductivity (van Genuchten, 1980; Fredlund et al., 1994; Ruana and Illangasekare, 1999). In recent decades, one of the most popular experimental methods to measure hydraulic properties of unsaturated soils in the laboratory is a one-dimensional (1-D) column test. Researchers typically use three types of methodologies: one-step outflow experiments (instantaneous application of one large pressure step), multi-step outflow experiments (application of several smaller pressure steps) and continuous flow experiments (application of the smooth continuous change in the pressure gradient) (Schultze and Durner, 1996). These experiments were performed under steady or transient conditions including wetting or drying processes. The hydraulic properties of unsaturated soils were determined from discharge water velocity, evolution of soil suction and water content during the test.
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Fig. 1. (a) The schematic diagram of column test, (b) tensiometer equipment, (c) electrical resistivity tester.
Nowadays, numerical methods combined with an optimization code have become promising tools to solve hydraulic problems easily and effectively. With these methods, the unknown parameters of hydraulic properties are estimated by minimizing the difference between the
Saturation (x 100%)
1.0 0.8 0.6 0.4 0.2 0.0 0
2000
4000
6000
8000
Electrical resistivity (Ohm.m) Fig. 2. Calibration relationship between electrical resistivity and saturation for Jumunjin sand.
predicted and observed measurements of flow rate, water content and soil suction. The application of the inverse solution technique to onestep outflow experiments was proposed early to estimate hydraulic properties using only cumulative outflow data (Kool et al., 1985a,b; Kool and Parker, 1988). Since soils are inherently heterogeneous and complex material, inversion analysis for hydraulic properties can lead to non-unique solution (Levasseur et al., 2009). Therefore, to overcome non-uniqueness in inversion analysis, it has been recommended to include additional θ(ψ) (Hudson et al., 1991; van Dam et al., 1992; Bohne et al., 1993; Simunek et al., 1998), or tensiometer measurements (Toorman et al., 1992; Eching and Hopmans, 1993) in the objective function (Crescimanno and Iovino, 1995). Thus, combination of outflow, volumetric water content and suction over time could be used for the inversion of parameters for SWCC (Durner et al., 1997). Table 1 Properties of Jumunjin sand. Sample
Gs
Cu
Cc
D50 (mm)
emax
emin
Jumunjin sand
2.66
1.65
0.94
0.55
0.992
0.596
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165
Fig. 3. a. Cumulative outflow over time in the one-step outflow experiment. b. Discharge rate over time in the one-step outflow experiment.
This study is to assess the applicability of an inverse parameter estimation method in a 1-D outflow experiment to determine the hydraulic properties of unsaturated soils. De-saturation experiment was carried out on a vertical sand column to measure cumulative outflow, water content and suction of soil over time. Simultaneously, a finite element code and inversion code were developed for hydraulic parameter estimations using measured data of soil suction and the outflow rate from outflow experiments as input data. In addition, a comparison between predicted hydraulic properties from the one-step and multi-step outflow methods was also performed to determine the optimal method for inverse parameter estimation of unsaturated hydraulic properties.
2. Research methodology 2.1. Experimental study A 1-D column (H = 80 cm, D = 28 cm, wall thickness = 1 cm) made of Plexiglas was used to conduct the vertical drainage test. Eight tensiometers and eight electrical resistivity probes (ERP) were placed in the sample every 10 cm vertically (L1 ~ L8) as shown in Fig. 1a, and were connected to data acquisition systems. The tensiometers were used to measure the soil suction, which had a tiny porous ceramic cup 6.5 mm in diameter and 2.5 cm long, attached to the tensiometer body via long nylon tubing. To install tensiometer easily into the soil column, it was inserted into a plastic screw as shown in Fig. 1b. Soil suction at each height was measured using tensiometer, and it was displaced
2.2. Numerical study
0.16 0.14
Displacement (cm)
and automatically saved by data logger which is connected and controlled by a computer (Figure 1b). ERP was used to measure soil resistance to calculate water saturation (Kechavarzi and Soga, 2002). ERP was made of two rods (10 cm length, 2 mm diameter and 1.2 cm distance apart) inserted into the plastic screw and was connected to the switching box, which was used to eliminate the electrical interference caused by simultaneous measurement of ERPs (Figure 1c). Electrical resistance of unsaturated soil at different water saturation was measured to build calibration curve in the preliminary tests (Figure 2), and that was fitted with Archie's law (exponent m = 1.32 and Archie exponent na = 1.57). The plastic bag located on the top of the column minimized water evaporation from the sample surface in the column test. The cumulative outflow at the bottom of the column was measured by weighing and automatic recording. The properties of the soil used are shown in Table 1. The sand grains were mixed under water and scooped into the column to prepare a fully saturated sample. The temperature in the laboratory was maintained at 20 °C to prevent a change in water–air surface tension (Kechavarzi and Soga, 2002). Opening the valve located at the bottom of column drained the water out of the soil column via gravity. In the one-step outflow experiment, outflow was maintained at the bottom level of the column during the test. In the multi-step outflow experiment, the bottom valve opened stepwise to maintain an apparent water table in the column, with the following levels. The beginning water height was from 80.0 cm to 55.0 cm, at 24 h from 55.0 cm to 25.0 cm and at 48 h from 25.0 cm to 0.0 cm. Soil suction, soil electrical resistivity and total water outflow were measured every 10 s. Settlement in the soil column was measured using imageprocessing techniques (Shin and Santamarina, 2011).
2.2.1. Forward simulation In porous materials, water can exist in the form of liquid and vapor, and the governing equation of water is derived from a generalized law of mass conservation (Shin, 2011):
0.12 0.10 0.08
0.04
2 3 i ∂h w w w w w w ρl ϕ Sl þ ρg ϕ ð1−Sl Þ þ ∇ 4ρl ϕ Sl u˙ þρl q þ ρg ϕ ð1−Sl Þ u˙ þρg q 5 ¼ 0 ∂t e e e el eg
0.02
ð1Þ
0.06
0.00 0
1000
2000
3000
4000
Time (second) Fig. 4. Surface displacement over time from the one-step outflow experiment.
w where ϕ is porosity of the soil, Sl is saturation of liquid water, ρw l and ρg are mass of water per unit volume in the liquid and gas phases. u˙ is the velocity of soil particles and q and q are the liquid and gas floweflux in each phase, respectively. el eg
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80
1.0
70 60
Height (cm)
Degree of saturation
0.8
0.6
L2 (15 cm) L5 (45 cm) L8 (75 cm)
0.4
0 min 0.5 min 1 min 2 min 5 min 10 min 1 hr 24 hr 72 hr
50 40 30 20
0.2 10 0.0
0 0
100
200
300
400
500
600
700
0.0
0.5
1.0
Time (minute)
Degree of saturation
(a)
(b)
Fig. 5. a. Water saturation over time obtained from the one-step outflow experiment at different heights. b. Water saturation profile from the one-step outflow experiment.
For 1-D vertical transient water flow assuming no gas flow rate and no soil particle movement, Eq. (1) becomes the well-known Richards equation: " # ∂ w w ρl ϕ Sl þ ∇ ρl q ¼ 0: ∂t e el
2.2.2. Inverse simulation A non-linear optimization method was used to minimize the objective function using L2 error norm, which expresses the difference between observed and predicted data by using the least-square solution for the parameters in Eqs. (3) and (4) (Santamarina and Fratta, 2005).
ð2Þ
h iT h i T bm N bm N Γ ¼ e :e ¼ W :y −W :h:x 0 −W :h:Δx : W :y −W :h:x 0 −W :h:Δx
For the hydraulic constitutive model for unsaturated soils, this study uses van Genuchten model (1980) for the SWCC and unsaturated relative hydraulic conductivity.
Se ¼
krl ¼
S−Sr min ¼ 1−Sr min
1þ
1 !−λ Pg −Pl 1−λ P0
ð5Þ where x are unknown soil properties (x ¼ x 0 þ Δx), x 0 is initial guess of bm N unknown variables, y is the measurement data (saturation, pressure head, outflow, displacements, stresses, etc.), h is the transformation matrix which is hij = ∂yi/∂xj (y ¼ h:x), and W is the weight factor (where Wij = weight value, Wij = 0 for i ≠ j). The partial derivative of Eq. (5) with respect to Δx is zero to minimize the objective function:
ð3Þ
pffiffiffiffiffi 1=λ λ 2 Se 1− 1−Se
ð4Þ
where Se is effective saturation, Sr is the residual saturation, S is degree of saturation, Pg − Pl is soil suction, P0 and λ are the curve fitting parameter, krl(=kunsat/ksat) is relative hydraulic conductivity and ksat is saturated hydraulic conductivity. The Galerkin formulation of the governing equation was used to develop the finite element code. The initial condition for water pressure is static water gravity with zero water flux at the bottom boundary.
h i ∂Γ T T bm N T T −W :h:x 0 þ 2h :W :W :h:Δx ¼ 0 ¼ −2h :W W :y ∂Δx h i−1 h i T T T T bm N ⇒Δx ¼ h :W :W :h :h :W : W :y −W :h:x 0 : 80
8
Initial condition 0.5 min 1 min 2 min 5 min 10 min 1 hr 24 hr 72 hr Final condition
60
L2 (15 cm) L5 (45 cm) L8 (75 cm)
4 2
Height (cm)
Pore water pressure (kPa)
70 6
50 40 30
0 20 -2
10
-4 0
100
200
300
400
500
600
700
0 -7.5
-5.0
-2.5
0.0
2.5
5.0
Time (minute)
Pore water pressure (kPa)
(a)
(b)
7.5
Fig. 6. a. Pore-water pressure over time from the one-step outflow experiment at different heights. b. Pore-water pressure head profile from the one-step outflow experiment.
ð6Þ
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167
3. Experimental results Fig. 3 shows the measured cumulative outflow and discharge rate in the one-step outflow experiment. The cumulative outflow showed rapid changes during the first 10 min, rapidly reaching Q ≈ 24.21 cm after 2.5 h (Figure 3a). The maximum outflow rate just after starting the experiment was slightly greater than the saturated hydraulic conductivity of the soil column, but it was a very good initial pre-indicator, which was verified from the numerical simulations. Just after opening the bottom valve, the hydraulic gradient from outflow rate was very high so that the discharge rate at that time was very high. After 400 s, it converged to steady stage (Figure 3b). The sample surface began to settle immediately after starting the test, and nearly stopped after 10 min (Figure 4). Since the magnitude of the settlement was relatively small compared with the column height, we assumed constant void ratio when evaluating the unsaturated hydraulic properties. Figs. 5 and 6 show the variation in water saturation and suction head profiles over time during the drainage test. When opening the valve to allow free drainage at the bottom, the lower end of the soil column was exposed to atmospheric pressure at that moment. Both an instant pressure drop at the bottom of the column and water gravity were the main driving forces for this de-saturation process. Water saturation and suction at different locations changed dramatically in the beginning due to bottom drainage. But pore-water pressure took a long time to reach the final equilibrium state, so it did not reach hydrostatic pressure even after 700 min (Figure 6b; Shin, 2011). From the water saturation profiles, the minimum measured residual water saturation was close to 13%, and the water saturation at L2 was around 70% due to low capillary suction from water gravity (Figure 5). At final equilibrium, the distribution of saturation and pore pressure according to depth can be converted into the SWCC.
Fig. 7. Values of lambda (λ) and P0 vs. iterative steps.
Fig. 8. Comparison between predicted and measured outflow rate in the one-step outflow experiment.
4. Numerical results Numerical inversion analysis was conducted to evaluate unsaturated hydraulic properties using a 1-D de-saturation test. Hydraulic properties of unsaturated soils can be determined by two functions: the SWCC (relationship between pore-water pressure and water saturation) and unsaturated hydraulic conductivity (relationship between hydraulic conductivity of water and gas and degree of saturation). In this study, we chose the van Genuchten model for the SWCC and unsaturated hydraulic conductivity function (van Genuchten, 1980). Two unknown parameters in the hydraulic model, P0 and λ, were determined from inversion analysis using experimental measurements, outflow rate and soil suction over time. Numerical analysis is consisted of
Thus, updated soil properties x are given by h i h i−1 T T T T bm N x ¼ x 0 þ Δx ¼ x 0 þ h :W :W :h :h :W : W :y −W :h:x 0 :
ð7Þ
In the optimization process, the initial guess following van Genuchten parameters were: P0 = 0.8 and λ = 0.8. Simulations were repeated with updated parameter estimates until the objective function became smaller than the critical tolerance.
L2 (15 c m)
L5 ( 45 cm)
6 Measured data
4
Predicted data
2 0 -2 -4
8
Pore water pressure (kPa)
Pore water pressure (kPa)
Pore water pressure (kPa)
L8 (75 c m)
8
8
6 4 Measured data
2
Predicted data
0 -2
100
200
300
400
Time (minute)
500
600
700
4 Measured data
2
Predicted data
0 -2 -4
-4 0
6
0
100
200
300
400
Time (minute)
500
600
700
0
100
200
300
400
Time (minute)
Fig. 9. Comparison between predicted and measured pore water pressures in the one-step outflow experiment at locations L2, L5 and L8.
500
600
700
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Degree of saturation
Degree of saturation
0.8 0.6 Measured data
0.4
L8 (7 5 c m )
L5 ( 45 c m )
L 2 ( 1 5 cm )
1.0
Predicted data
0.2
1.0
1.0
0.8
0.8
0.6
Degree of saturation
(a)
Measured data Predicted data
0.4 0.2
0.6
0
100
200
300
400
500
600
700
0.2 0.0
0
100
Time (minute)
200
300
400
500
600
700
0
0.6 Measured data Predicted data
0.2
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.8
Measured data Predicted data
0.6 0.4 0.2 0.0 0.0
300
400
500
600
700
1.0
Degree of saturation
0.8
200
Time (minute)
1.0
Degree of saturation
Degree of saturation
1.0
0.0 0.0
100
Time (minute)
(b)
0.4
Predicted data
0.4
0.0
0.0
Measured data
0.5
1.0
Suction (-kPa)
1.5
2.0
2.5
3.0
3.5
0.8 Measured data Predicted data
0.6 0.4 0.2 0.0 0.0
0.5
Suction (-kPa)
1.0
1.5
2.0
2.5
3.0
3.5
Suction (-kPa)
Fig. 10. Comparison between predicted and measured hydraulic properties in the one-step outflow experiment at locations L2, L5 and L8: (a) Change in degree of saturation, (b) compiled SWCC.
(1) Discharge flow and suction measurement at 3 electrodes were used for inversion analysis to estimate unknown hydraulic parameters. (2) Measurement of degree of saturation at 3 electrodes is used to build compiled SWCC data which are compared with numerical results using estimated hydraulic parameters in the previous stage. Fig. 7 shows the iterative process used to determine the unknown parameters in the inversion analysis. After 10 iterations, initial guessed values λ = 0.8 and P0 = 0.8 converged to λ = 0.857 and P0 = 1.475. Numerical predictions using assessed hydraulic parameters produced good agreement with experimental observations (Figures 8 and 9). The evolution of water saturation at three locations in the soil column could be predicted through numerical analysis, and compared well with experimental observations using the electrical resistivity probe (Figure 10a). Soil suction and degree of water saturation were paired in the SWCC plot, and compiled SWCC data over time is plotted in Fig. 10b. Location L2 near the bottom of the column maintained a high degree of saturation during the tests, so all collected data were plotted near the saturation region. However, location L8, close to top of the column, experienced dramatic changes in the degree of saturation during the test, so most of the data were plotted in the residual region of
Table 2 Root mean square error from predicted and measured data in time domain (one-step outflow experiment).
Location RMSE Average
Pore-water pressure (kPa)
Saturation
L2 0.477 0.389
L2 0.035 0.041
L5 0.328
L8 0.363
the SWCC plot. For this reason, we had to install several tensiometers to measure soil suction, to cover the full range of the SWCC plot. Sources of the differences (errors) between measured and predicted data may come from the hydraulic model itself, and the measuring sensors (tensiometer, needle probe, electrical balance, etc.). Root mean square error (RMSE) was used to assess the differences:
RMSE ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX m p 2 u yi −yi u t i¼1
p where ym i represents experimental measurements, yi denotes predicted values using estimated soil properties such as pore-water pressure, degree of saturation and flow rate and N is total number of measurements. The comparison of the measured data (Figures 9 and 10) and the predicted data using assessed soil parameters from the one-step outflow experiment shows the relatively low errors for all three locations
8 6
L2 (15 cm) L5 (45 cm) L8 (75 cm)
4 2 0 -2 -4 0
L5 0.046
L8 0.042
ð8Þ
N
Pore water pressure (kPa)
two stages: inversion analysis and experimental verification using additional measurements.
1000
2000
3000
4000
Time (minute) Fig. 11. Pore-water pressure over time from the multi-step outflow experiment at three locations.
N. To-Viet et al. / Engineering Geology 164 (2013) 163–171
(b)
1.0
1.0
0.8
0.8
Degree of saturation
Degree of saturation
(a)
169
Measured data
0.6 0.4 0.2 0.0 0.0
0.6
Measured data
0.4 0.2 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0
0.5
Suction (-kPa)
1.0
1.5
2.0
2.5
3.0
3.5
Suction (-kPa)
Fig. 12. Compiled SWCC from locations L2, L5 and L8: a) one-step outflow experiment, b) multi-step outflow experiment.
Fig. 13. Values of lambda (λ) and P0 vs. iterative steps.
(Table 2). The hysteresis phenomenon, narrow range of saturation and lack of data near saturation of the SWCC contributed to the error in the evaluation of hydraulic properties. 5. Discussion In previous studies, multi-step outflow experiments have been known to reveal more accurate results than one-step outflow experiments (Crescimanno and Iovino, 1995; Schultze and Durner, 1996; Hwang and Powers, 2003). Although the one-step outflow experiment provided a good estimation of unsaturated hydraulic properties, a multi-step outflow experiment was also conducted to investigate the advantage of the experimental procedure. In the multi-step outflow experiment, pore-water pressure dropped three times, 8.0 to 5.5 kPa, 5.5 to 2.5 kPa and 2.5 to 0.0 kPa. The water pressures were maintained during 24 h for each pressure drop to
L2 (15 c m )
L5 (45 cm)
L 8 (7 5 c m )
Pore water pressure (kPa)
6
Measured data Predicted data
4 2 0 -2 -4
8
Pore water pressure (kPa)
8
8
Pore water pressure (kPa)
balance the water pressure inside the soil sample. Fig. 11 shows the three stages of pore-water pressure in the soil sample corresponding to each pressure drop. Compared to the one-step outflow experiment, the multi-step outflow experiment produced a larger number of experimental data within a narrow band in the compiled SWCC data from three locations, from the air entry to residual regions (Figure 12). This difference from onestep outflow experiment is due to smaller increments of hydraulic gradient at every pressure step. The same initial parameters were used to estimate the optimized soil parameters of unsaturated hydraulic properties (λ = 0.8 and P0 = 0.8). After 13 iterative steps, inversion results converged to λ = 0.883 and P0 = 1.508, which was slower than the one-step outflow experiment (Figure 13). Fig. 14 shows the evolution of pore-water pressure from the experimental observations and reproduced numerical results. Similar to the one-step outflow experiment, the degree of saturation and compiled SWCC at three locations in the soil column could be determined, and compared well with the measurements (Figure 15). SWCC data using the multi-step experiment agreed better with the model prediction than the one-step experiment (Figures 10b and 15b). In the multi-step outflow experiment, the change in pore-water pressure at each step was so slow that more hydraulic data at near saturation was collected and the hysteresis in the soil was smaller. The multi-step outflow experiment improved assessment of hydraulic properties (Figures 10 vs. 15, Tables 2 vs. 3). The two types of outflow experiments illustrated that the estimated soil parameters for unsaturated hydraulic properties were very close to actual measurements from both tests, as well as having good convergence to obtain optimized model parameters. The one-step outflow experiment required a shorter time to complete the experiment and smaller iterative steps in the inversion analysis. However, quick and fast changes in hydraulic pressure in the one-step outflow experiment can cause a loss in SWCC data between air-entry and residual regions
6 Measured data
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Fig. 14. Comparison between predicted and measured pore water pressure in the multi-step outflow experiment at locations L2, L5 and L8.
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L5 ( 45 c m)
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Fig. 15. Comparison between predicted and measured hydraulic properties in the one-step outflow experiment at locations L2, L5 and L8: (a) change in degree of saturation, (b) compiled SWCC.
and high hysteresis in the soil column test, especially with coarsegrained soils. The multi-step outflow experiment required more time to conduct the experiment, along with the complex boundary condition in the numerical analysis. However, low changes in hydraulic pressure in the soil sample can help obtain more information about the hydraulic property of soil for the better prediction of hydraulic properties of unsaturated soils.
6. Conclusions Over the last several decades, the inverse analysis technique has been consistently used to determine unsaturated hydraulic properties. However, its reliability has yet to be sufficiently affirmed in comparisons with independently measured hydraulic properties of soils. A 1-D outflow experiment could be a good indirect experimental methodology to estimate the SWCC and unsaturated hydraulic conductivity function. In this study, one-step and multi-step outflow experiments were carried out to measure bottom discharge rates, soil suction and electrical resistances over time. Inverse analysis was performed to determine the unsaturated hydraulic function in the van Genuchten model using outflow rate and suction at specific depths over time. Compilations of soil suction and degree of saturation from electrical resistance provided
SWCC data was compared with model predictions from the inverse analysis. − The combined analysis of the finite element method and inversion technique was performed to evaluate hydraulic properties of unsaturated soils, which produced a good comparison of measurements from the one-step and multi-step outflow experiments. − The multi-step outflow experiment provided a wider suction range in experimental data than the one-step outflow experiment to fit measurements in the van Genuchten model. − However, the multi-step outflow experiment has more complex outflow control and needs more iteration for convergence in inversion analysis. − This study showed that inversion analysis of a 1-D de-saturation column test could be a good indirect method to evaluate hydraulic properties of unsaturated soils. Since we conducted laboratory experiments with uniform sand and with a simple initial boundary condition, the presented approach should be extended to further investigate various soils with different sedimentation. Acknowledgments This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (KRF-2011-0013210).
Table 3 Root mean square error from predicted and measured data in time domain (multi-step outflow experiment).
Location RMSE Average
Pore-water pressure (kPa)
Saturation
L2 0.282 0.313
L2 0.025 0.032
L5 0.3
L8 0.358
L5 0.040
L8 0.031
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