Modelling hydraulic conductivity in a small centrifuge - NRC Research ...

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Modelling hydraulic conductivity in a small centrifuge Devendra N. Singh and Ashok K. Gupta

Abstract: Hydraulic conductivity modelling has been carried out using various conventional laboratory 1g tests and centrifuge tests for different compaction states of a silty soil. It has been shown that hydraulic conductivity is modelled adequately in the centrifuge. Key words: silty soil, hydraulic conductivity, centrifuge modelling, falling-head tests, consolidation tests, scale factor. Résumé : L’on a réalisé une modélisation de la conductivité hydraulique en utilisant divers essais conventionnels à 1-g en laboratoire et des essais au centrifuge pour différents états de compactage d’un sol limoneux. On a démontré que la conductivité hydraulique est modélisée adéquatement dans le centrifuge. Mots clés : sol limoneux, conductivité hydraulique, modélisation par centrifuge, essais à charge variable, essais de consolidation, facteur d’échelle. [Traduit par la Rédaction]

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Introduction Centrifuge modelling can provide valuable information regarding the movement of groundwater, the spread and growth of pollutant plumes, and the effectiveness of various strategies adopted for containment and remediation. In a centrifuge, a soil model is subjected to an inertial radial acceleration field which simulates gravitational acceleration that is greater than Earth’s gravity. As such, when model and prototype soils are the same and when the model with linear dimensions scaled down by a factor of N is subjected to a centrifugal acceleration N times greater than the Earth’s gravity, then the model experiences the same magnitude and distribution of self-weight stresses as those in its prototype. Geotechnical centrifuges have been used in recent years to understand and simulate transport mechanisms in soils in accelerated-gravity environments, validate existing mathematical models, and develop improved conceptual thinking for fundamental processes which are being simulated. It has been shown that the prototype stress regimes can be simulated in a centrifuge model with no loss of generality (Arulanandan et al. 1988). To assess the migration of contaminants and to design containment barriers, the hydraulic conductivity of geotechnical material is required. Centrifuge tests have been conducted to estimate the hydraulic conductivity of geotechnical materials in accelerated-gravity conditions, reproduce the prototype vertical effective stresses in

Received January 4, 2000. Accepted January 12, 2000. Published on the NRC Research Press website on October 25, 2000. D.N. Singh. Department of Civil Engineering, Indian Institute of Technology, Powai, Mumbai 400 076, India. A.K. Gupta. Department of Civil Engineering, North Eastern Regional Institute of Science and Technology, Nirjuli 791 109, Arunachal Pradesh, India. Can. Geotech. J. 37: 1150–1155 (2000)

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the soil samples, and reduce the testing time (Alemi et al. 1976; Nimmo et al. 1987; Mitchell 1994a, 1994b). The permeability behaviour of compacted clays (Daniel 1984; Harrop-Williams 1985), silty soils (Holtz 1985), mine tailings (Jessberger and Beine 1981), and soil–bentonite mixtures (Chapuis 1990) has been studied with respect to their use as prospective hydraulic barriers. Based on these studies, guidelines have been provided for selecting appropriate soil properties and compaction procedures that are likely to result in low hydraulic conductivity (Daniel 1990; Benson et al. 1994). Transient water flow in centrifuge models has been examined by Alemi et al. (1976) and Cargill and Ko (1983). Mitchell (1994a, 1994b) and Theriault and Mitchell (1997) have developed an apparatus to study clay liner – leachate compatibility under flexible, no lateral strain boundary conditions and have studied hydraulic conductivity behaviour of clays using a variety of permeants. An attempt has been made in this technical note to evaluate the hydraulic conductivity of a silty sand, at various void ratios, using a small geotechnical centrifuge. Various conventional laboratory 1g tests have also been conducted to obtain the scaling relationship for hydraulic conductivity between 1g and centrifuge tests.

Experimental investigations Physical properties of the soil and sample preparation A naturally available local soil was used in the present study and its physical properties are determined according to American Society for Testing and Materials (ASTM) test procedures. Specific gravity of the soil is 2.79. The gradation characteristics and Atterberg limits of the soil are shown in Fig. 1. The Standard Proctor compaction characteristics of the soil are presented in Fig. 2. To prepare soil samples, oven-dried soil was mixed with the required quantity of water and stored in polythene bags for at least 24 h to ensure proper mixing and maturing. Soil samples corresponding to © 2000 NRC Canada


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Fig. 1. Gradation characteristics of the soil.

Fig. 2. Standard Proctor compaction characteristics of the soil. MDD, maximum dry density; OMC, optimum moisture content.

Table 1. Compaction state of soil samples. Sample No.

Moisture content (%)

Dry density (g/cm3)

Degree of saturation, Sr (%)

Void ratio, e

A B C D E F G H I J

10.6 12.8 14.2 16.0 18.3 20.1 21.8 23.4 24.8 27.6

1.55 1.59 1.61 1.64 1.68 1.70 1.69 1.66 1.64 1.54

37.2 47.1 54.0 64.2 77.0 87.2 94.0 95.2 98.3 94.6

0.80 0.76 0.73 0.70 0.66 0.64 0.65 0.68 0.70 0.81

moulding compaction states, as presented in Table 1, were tested for hydraulic conductivity using a geotechnical centrifuge. Conventional falling-head tests, oedometer falling-head tests, and consolidation tests were also conducted. Centrifuge tests Details of the geotechnical centrifuge used for this study are presented in Table 2. The setup for centrifuge tests consists of three detachable concentric perspex cylinders as shown in Fig. 3. The inner diameters of these cylinders are 66.1, 86.0, and 106.0 mm, respectively, and the height of the setup is 160 mm. The innermost cylinder is graduated (for recording the fall in water level) and holds the compacted soil sample. Soil samples 30 mm thick are prepared (in the innermost cylinder) corresponding to different moulding compaction states of the soil as shown in Table 1. These compaction states of the soil samples correspond to the Standard Proctor compaction states. Four small holes are provided close to the bottom of the inner and the middle cylinders to allow free drainage of water from one cylinder to the other. The annular space between the inner and middle cylinders can also be used to saturate the soil sample by closing the bottom holes and filling water up to the brim

Table 2. Details of the centrifuge. Type

Swinging buckets on both sides of the arm

Arm radius Max. outer radius Centrifugation range Max. acceleration

200 mm 315 mm 250–1000 rpm 300g

(with no water column on the top of the soil sample). Such an arrangement creates an upward, from bottom to top of the soil sample, hydraulic gradient of approximately five. A perforated brass plate is placed on top of the soil sample, if required, to check the swelling of the soil sample due to the application of this hydraulic gradient. This setup is left as is for a period of 24–36 h to ensure proper saturation of the soil sample. To remove any entrapped air in the soil sample the test setup is connected to a vacuum pump for about 2–3 h as required. This setup is later mounted in the centrifuge bucket for estimating hydraulic conductivity of soil samples. The centrifuge setup simulates a falling-head test in an accelerated environment, and hydraulic conductivity of the soil corresponding to the model, kcen, can be determined by eq. [1]: [1]

k cen =

L  h1  ln   t h2

where L is height of the soil sample; t is the time of centrifugation; and h1 and h2 are the initial and final heads, respectively, as measured during centrifugation through an observation window. The centrifuge test results corresponding to various levels of acceleration (N) are presented in Table 4. © 2000 NRC Canada



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Fig. 3. Centrifuge test setup. h1 and h2, initial and final water heads.

The pressure acting on a soil sample in an accelerated environment (Ng) is presented in Fig. 4. The pressure at the middle of the sample of height L caused by a water head, h, at Ng is equal to (hN γ w + γ satL /2N, where γ sat is the saturated unit weight of soil and γ w is the unit weight of water). Following this, the pressure acting on a sample at various g levels is obtained (see Table 3). The validity of Darcy’s law for fluid flow in the centrifuge model is checked by Reynolds number, Re, given by [2]

Re =

ρ vd µ

where ρ is density of the fluid, v is the specific discharge, d is the characteristic microscopic length of the soil (such as the effective particle size D10), and µ is the dynamic viscosity of the fluid. The term Re represents the dynamic similarity of fluid motion, which can be established by ensuring that the ratio of inertial forces to viscous forces in the fluid in the centrifuge model and prototype, respectively, remains invariant. But, since flow velocities are scaled in a centrifuge model, it is not possible to maintain Re constant if identical soils and fluid (water in the present study) are used in model and prototype. However, for fluid flow through soils Darcy’s law is valid as long as the Reynolds number does not exceed 1 (Bear 1972). For the present study, the Reynolds number was calculated on the basis of D10 (= 0.002 mm). The maximum value of Re, obtained for sample A at 200g, is 6.02 × 10–5 (i.e., Re < 1) which indicates the validity of Darcy’s law for fluid flow in centrifuge models. Conventional falling-head tests The tests were conducted according to Indian Standard Code IS 2720 (Part 17), 1986. Samples (A–J) for fallinghead permeability tests were prepared in a rigid mould 79.8 mm in diameter and 60 mm high for each of the compaction states, as indicated in Table 1. The soil samples were saturated in the mould by applying a positive hydraulic gradient (bottom to top) of four for 24 h and using a vacuum pump; the soil samples were observed to be 98–100% saturated. The corresponding hydraulic conductivity values (kf) are presented in Table 4.

Fig. 4. Pressure acting on a soil sample in an accelerated environment.

Table 3. Pressure acting on the soil sample in the centrifuge. g level

Pressure (kN/m2)

50 100 150 200

50.5 101 151.5 202

Oedometer falling-head tests Soil samples (A–J) were prepared in an oedometer ring 75 mm in diameter and 25 mm thick for various compaction states of the soil as presented in Table 1. These samples were saturated by placing them in a fixed-ring consolidometer with a graduated and calibrated burette connected at the base, and a hydraulic gradient of about five was applied for about 24–36 h to ensure saturation of the samples. Falling-head permeability tests were conducted on these samples with the help of this burette. Porous stones were placed both on the top and bottom of the samples to provide uniform flow of water (Fig. 5). This setup behaves like a fixed-wall permeameter, since the soil sample is laterally constrained by the fixed wall of the oedometer ring, and has been termed the oedometer fallinghead test. Tests were conducted at 50.0, 100.0, 150.0, and 200.0 kPa normal pressures, in accordance with Table 3, and the obtained hydraulic conductivity values (ko) are presented in Table 4. Consolidation tests For consolidation tests, samples (A–J) were prepared in an oedometer ring 75 mm in diameter and 25 mm thick for various compaction states of the soil as presented in Table 1. These samples were placed in the consolidometer cell filled with water and allowed to saturate for almost 24–36 h under a suitable seating load. This setup simulates an indirect permeability test in which a sample of soil is compressed in the rigid oedometer ring at various vertical stress levels, with drainage allowed from both the top and bottom ends of the sample. The hydraulic conductivity is calculated using Terzaghi’s theory of one-dimensional consolidation: © 2000 NRC Canada



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Table 4. Hydraulic conductivity values (cm/s) and scale factors. Hydraulic conductivity

Scale factor

Sample No.

kf

ko

kc

kcen

N

xf

xo

xc

A

1.08×10–5

6.51×10–6 3.62×10–6 2.74×10–6 1.86×10–6

7.55×10–6 5.28×10–6 4.34×10–6 3.39×10–6

5.61×10–4 8.45×10–4 1.04×10–3 1.31×10–3

50 100 150 200

1.010 0.947 0.912 0.906

1.139 1.184 1.185 1.237

1.101 1.102 1.093 1.124

B

4.54×10–6

5.86×10–6 3.02×10–6 2.04×10–6 1.07×10–6

3.91×10–6 2.06×10–6 1.51×10–6 9.56×10–7

2.38×10–4 3.63×10–4 4.85×10–4 5.72×10–4

50 100 150 200

1.012 0.951 0.932 0.913

0.947 1.040 1.092 1.185

1.050 1.123 1.152 1.207

C

2.38×10–6

2.61×10–6 1.81×10–6 1.33×10–6 8.49×10–7

1.98×10–6 9.35×10–7 8.22×10–7 7.08×10–7

1.19×10–4 1.84×10–4 2.56×10–4 3.15×10–4

50 100 150 200

1.000 0.944 0.934 0.922

0.976 1.004 1.050 1.117

1.047 1.147 1.146 1.151

D

1.07×10–6

9.78×10–7 8.14×10–7 6.68×10–7 5.22×10–7

8.98×10–7 6.53×10–7 5.40×10–7 4.27×10–7

5.42×10–5 8.92×10–5 1.24×10–4 1.48×10–4

50 100 150 200

1.003 0.960 0.949 0.930

1.026 1.020 1.043 1.066

1.048 1.068 1.085 1.104

E

4.18×10–7

6.44×10–7 5.28×10–7 4.42×10–7 3.57×10–7

4.06×10–7 2.84×10–7 1.97×10–7 1.09×10–7

2.41×10–5 4.44×10–5 6.03×10–5 6.62×10–5

50 100 150 200

1.036 1.013 0.992 0.956

0.926 0.962 0.981 0.986

1.044 1.097 1.142 1.210

F

2.46×10–7

2.15×10–7 1.44×10–7 1.20×10–7 9.65×10–8

2.38×10–7 1.70×10–7 1.37×10–7 1.05×10–7

1.40×10–5 2.67×10–5 3.72×10–5 4.51×10–5

50 100 150 200

1.033 1.018 1.002 0.984

1.068 1.134 1.145 1.160

1.042 1.098 1.118 1.144

G

2.06×10–7

1.69×10–7 1.08×10–7 9.66×10–8 8.52×10–8

1.97×10–7 1.47×10–7 1.23×10–7 9.88×10–8

1.25×10–5 2.29×10–5 3.16×10–5 3.84×10–5

50 100 150 200

1.049 1.023 1.004 0.987

1.100 1.163 1.156 1.153

1.061 1.096 1.107 1.125

H

2.02×10–7

1.41×10–7 1.07×10–7 9.52×10–8 8.34×10–8

1.95×10–7 1.44×10–7 1.19×10–7 9.48×10–8

1.13×10–5 2.11×10–5 2.79×10–5 3.42×10–5

50 100 150 200

1.029 1.009 0.984 0.969

1.121 1.147 1.134 1.136

1.038 1.083 1.089 1.111

I

2.11×10–7

1.40×10–7 1.12×10–7 1.01×10–7 9.04×10–8

2.04×10–7 1.48×10–7 1.22×10–7 9.66×10–8

1.17×10–5 2.11×10–5 2.84×10–5 3.44×10–5

50 100 150 200

1.026 1.000 0.978 0.961

1.131 1.138 1.125 1.121

1.035 1.077 1.088 1.109

J

2.43×10–7

2.27×10–7 1.63×10–7 1.33×10–7 1.04×10–7

2.35×10–7 1.68×10–7 1.33×10–7 9.88×10–8

1.29×10–5 2.27×10–5 3.12×10–5 3.81×10–5

50 100 150 200

1.015 0.985 0.969 0.954

1.033 1.072 1.089 1.114

1.024 1.065 1.089 1.124

Note: kf, ko, kc, and kcen, hydraulic conductivity values corresponding to conventional falling-head, oedometer falling-head, consolidation, and centrifuge tests, respectively; xf, xo, and xc, scale factors of centrifuge tests with respect to conventional falling-head, oedometer falling-head, and consolidation tests, respectively.

[3]

k = cvmvγ w

where cv is the coefficient of consolidation, and mv is the volume compressibility of the soil. The coefficient of consolidation cv can be obtained by conventional t1/2 and log(t) relationships for the soil. Olson (1986) has demonstrated

that the calculated hydraulic conductivity values are less than the measured values and that the hydraulic conductivity values obtained from the t1/2 method are closer to the measured values than those obtained using the log(t) method. As such, the t1/2 method is used in this study to estimate cv and hence the hydraulic conductivity of the soil. The hydraulic © 2000 NRC Canada



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Fig. 5. Oedometer falling-head test setup.

conductivity values (kc) from consolidation responses for the normal pressures 50.0, 100.0, and 200.0 kPa, in accordance with Table 3, are presented in Table 4.

Scaling relationship for hydraulic conductivity The scaling relationship for hydraulic conductivity between centrifuge tests (kcen) and 1g tests (kp) can be written as [4]

k cen kp

= Nx

where kcen and kp are the hydraulic conductivity obtained from centrifuge tests and various 1g tests, respectively; N is the achieved acceleration level; and x is the scale factor for hydraulic conductivity whose value with respect to various 1g tests needs to be determined. To evaluate x, eq. [4] can be rewritten in the following form:

[5]

k  ln  cen   kp  x = ln(N )

Using eq. [5], the scale factors for hydraulic conductivity corresponding to different N values with respect to 1g laboratory tests have been calculated.

Discussion and conclusions The study demonstrates the usefulness of a geotechnical centrifuge for modelling hydraulic conductivity of compacted soils. Table 4 presents the results of various hydraulic conductivity tests and the obtained scaling factors xf, xo, and xc (corresponding to conventional falling-head tests, oedometer falling-head tests, and consolidation tests, respectively). Table 4 shows that the scaling factors for hydraulic conductivity in a geotechnical centrifuge are quite close to unity. This indicates that the hydraulic conductivity of soil samples in a geotechnical centrifuge is modelled N times that of the values obtained by other conventional (1g) tests.

This conclusion is of great importance in evaluating the hydraulic conductivity of compacted, fine-grained soils which exhibit low hydraulic conductivity. This also highlights the significance of centrifuge modelling in estimating hydraulic conductivity of compacted liner materials in a short period of time. A comparison of scale factors presented in Table 4 shows that the xf values are closer to unity than the values of xo and xc. This indicates that the agreement between the conventional falling-head test results and the centrifuge test results is very good and is better than that of the other test methods. The study shows that centrifuge testing can be developed as a valid means of determining soil hydraulic conductivity.

References Alemi, M.H., Neilsen, D.H., and Biggar, J.W. 1976. Determining the hydraulic conductivity of soil cores by centrifugation. Soil Science Society of America Journal, 40: 212–218. Arulanandan, K., Thompson, P.Y., Kutter, B.L., Meegoda, N.J., Muraleetharan, K.K., and Yogachandran, C. 1988. Centrifuge modelling of transport processes for pollutants in soils. Journal of Geotechnical Engineering, ASCE, 114(2): 185–205. Bear, J. 1972. Dynamics of fluids in porous media. American Elsevier Inc., New York. Benson, C.H., Zhai, H., and Wang, X. 1994. Estimating hydraulic conductivity of compacted clay liners. Journal of Geotechnical Engineering, ASCE, 120(2): 366–387. Cargill, K.W., and Ko, H.Y. 1983. Centrifuge modelling of transient water flow. Journal of Geotechnical Engineering, ASCE, 109(4): 536–555. Chapuis, R.P. 1990. Sand–bentonite liners: predicting permeability from laboratory tests. Canadian Geotechnical Journal, 27: 47–57. Daniel, D.E. 1984. Predicting hydraulic conductivity of clay liners. Journal of Geotechnical Engineering, ASCE, 110: 285–300. Daniel, D.E. 1990. Summary review of construction quality control for earthen liners. In Waste containment systems: construction, regulation, and performance. Edited by R. Bonaparte. Geotechnical Special Publication 26, American Society of Civil Engineers, New York, pp. 175–189. Harrop-Williams, K. 1985. Clay liner permeability: evaluation and variation. Journal of Geotechnical Engineering, ASCE, 111: 1211–1225. © 2000 NRC Canada



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Holtz, W.G. 1985. Predicting hydraulic conductivity of clay liners: Discussion. Journal of Geotechnical Engineering, ASCE, 111: 1457–1459. Jessberger, H.L., and Beine, R.A. 1981. Impermeabilisation of disposal sites by impervious blankets consisting of mine refuge. In Proceedings of the 10th International Conference of Soil Mechanics and Foundation Engineering, Stockholm, Vol. 4, pp. 745–746. Mitchell, R.J. 1994a. A flexible, no lateral strain apparatus for clay liner – leachate testing. In Centrifuge 94. Edited by C.F. Leung, F.H. Lee, and T.S. Tan. A.A. Balkema, Rotterdam, The Netherlands, pp. 351–355. Mitchell, R.J. 1994b. Centrifuge techniques for testing clay liner samples. Canadian Geotechnical Journal, 31: 577–583. Nimmo, J.R., Rubin, J., and Hammermeister, D.P. 1987. Unsaturated flow in a centrifugal field. Measurement of hydraulic conductivity and testing of Darcy’s law. Water Resources Research, 23(1): 124–134. Olson, R.E. 1986. State of the art: consolidation testing. In Consolidation of soils: testing and evaluation. Edited by R.N. Yong and F.C. Townsend. American Society for Testing and Materials, Special Technical Publication 892, pp. 7–70. Theriault, J.A., and Mitchell, R.J. 1997. Use of a modelling centrifuge for testing clay liner compatibility with permeants. Canadian Geotechnical Journal, 34: 71–77.

List of symbols cv coefficient of consolidation

d D10 e g h h1, h2 k kcen kp kf, ko, kc

L mv N Re Sr t v x xf, xo, xc γw γsat µ ρ

characteristic microscopic length of the soil effective particle size void ratio acceleration due to gravity water head initial and final water heads hydraulic conductivity hydraulic conductivity in the centrifuge hydraulic conductivity at 1g level hydraulic conductivity values with respect to conventional falling-head tests, oedometer falling-head tests, and consolidation tests sample height volume compressibility of the soil centrifuge acceleration level Reynolds number degree of saturation (%) of the soil sample time specific discharge scale factor scale factors with respect to conventional falling-head tests, oedometer falling-head tests, and consolidation tests unit weight of water saturated unit weight of soil dynamic viscosity of the fluid (water, in the present study) density of the fluid

© 2000 NRC Canada
