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684
Estimation of hydraulic conductivity of
unsaturated soils using a geotechnical centrifuge
D.N. Singh and Sneha J. Kuriyan
Abstract: A saturated silty soil sample is centrifuged in a geotechnical centrifuge to create an unsaturated state. The
change in water content of the soil sample is recorded at different points along the length of the sample to obtain the
water-content profile, which is then used to obtain the unsaturated hydraulic conductivity of the soil sample. These hydraulic conductivity values are compared with those obtained and reported by previous researchers by conducting accelerated falling-head tests on this soil sample in a centrifuge. The study demonstrates the use of centrifugation
techniques to obtain hydraulic conductivities of unsaturated soils.
Key words: silty soil, saturated soil, unsaturated soil, hydraulic conductivity, centrifuge testing.
Résumé : Un échantillon de sol limoneux saturé a été centrifugé dans un centrifuge géotechnique pour produire un état
non saturé. Le changement en teneur en eau de l’échantillon de sol est enregistré à différents points le long de
l’échantillon pour obtenir un profil de teneur en eau qui a été ensuite utilisé pour obtenir la conductivité hydraulique
non saturée de l’échantillon de sol. Ces valeurs de conductivité hydraulique sont comparées avec les conductivités hydrauliques obtenues et rapportées antérieurement par des chercheurs qui ont conduit des essais accélérés à tête variable
sur ces échantillons de sol dans un centrifuge. Cette étude démontre l’utilisation des techniques de centrifugation pour
obtenir les conductivités hydrauliques des sols non saturés.
Mots clés : sol limoneux, sol saturé, sol non saturé, conductivité hydraulique, essai au centrifuge.
[Traduit par la Rédaction]
Singh and Kuriyan
Introduction
Unsaturated soils are encountered in the compacted clay
covers and linings of waste-management facilities (Fredlund
1995). The basic purpose of these covers and linings is to
minimize fluid flow and contain liquids that might contaminate the subsurface soil and the water table. Studies have revealed that one of the most important factors governing the
performance of these containment systems is the soil hydraulic conductivity (Fredlund 1995), which mainly depends
on the water content, dry density, and degree of saturation
(i.e., the compaction state) of the soil (Gourley and
Schreiner 1995). The movement of moisture, and hence the
spread of contaminant, takes place in the region surrounding
the waste-containment area, which is mostly unsaturated
(vadose zone). This necessitates estimation of unsaturated
soil hydraulic conductivity, which would help in designing
an efficient containment system (Rahardjo et al. 1995).
Various researchers have tried to evaluate hydraulic conductivity of soils by conducting either laboratory experiments (Bjerrum and Huder 1957; Daniel et al. 1985;
Received 11 June 2001. Accepted 6 November 2001.
Published on the NRC Research Press Web site at
http://cgj.nrc.ca on 22 May 2002.
D.N. Singh1 and S.J. Kuriyan. Department of Civil
Engineering, Geotechnical Engineering Division, Indian
Institute of Technology, Bombay, Powai, Mumbai 400 076,
India.
1
Corresponding author (e-mail: dns@civil.iitb.ac.in).
Can. Geotech. J. 39: 684–694 (2002)
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694
Chapuis 1990; Fleureau and Taibi 1995; Huang et al. 1995;
Uno et al. 1995) or in situ studies (Bouma et al. 1971; Daniel and Trautwein 1986; Little et al. 1995; Ankey et al.
1991). These methods have been classified as direct methods
and indirect methods. The direct methods are quite tedious
and time consuming and require expensive experimental setups (Stephens 1996). Indirect methods, which employ volumetric properties of the soil and the soil-water characteristic
curve (SWCC), are frequently adopted. Integration along the
SWCC provides a measure of the quantity of water in the
soil which can then be used to estimate the soil hydraulic
conductivity (Ray and Morris 1995; Takeshita and Kohno
1995).
Researchers have often conducted column studies (Richard and Weeks 1953; Bruce and Klute 1956) to study flow
through unsaturated soils. Both steady-state (Uno et al.
1995; Fleureau and Taibi 1995) and transient methods (Klute
1972) have been used for this purpose. The steady-state
methods are primarily laboratory methods and can be conducted on disturbed and undisturbed soil samples. These
methods are generally not adopted, however, as considerable
time (weeks or months) is required to attain steady-state
flow through a soil sample, particularly if it is unsaturated.
In such a situation, the transient (unsteady-state) methods is
adopted wherein the time dependence of some aspect of the
flow is used to obtain the soil hydraulic conductivity, or soilwater diffusivity. Although the transient or unsteady-state
methods do not require a complicated setup, they lead to
considerable difficulty in measuring soil parameters at different points inside the soil sample and interpreting the obtained data. For unsaturated soils, however, with low
DOI: 10.1139/T02-013
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Singh and Kuriyan
hydraulic conductivity, considerably more time is required
to attain a steady-state condition.
To overcome these shortcomings, researchers have
adopted ultracentrifuges (Conca and Wright 1994) that can
measure hydraulic conductivities of the order of -10 –10 cm/s
at low water contents and have the added advantage of
achieving steady-state flow within a short span of time.
Also, good agreement has been observed between the results
obtained by this method and those estimated by the procedures of van Genuchten (1980) and Mualem (1976) using
the measured soil-water retention data of the soil (Nimmo et
al. 1987; Wright et al. 1994). During centrifugation, steadystate flow is achieved by open flow, wherein water is allowed to enter the sample during centrifugation. In closedflow centrifugation, however, where no fluid enters the sample during centrifugation, transient flow methods can be
used for estimation of hydraulic conductivity.
The potential of a geotechnical centrifuge as a modelling
tool has been tested in recent years to check clay liner hydraulic conductivity and compatibility and possible transport
mechanisms (Yanful et al. 1990; Airey 1993; Mitchell 1994;
Theriault and Mitchell 1997; Singh and Gupta 2000) and to
investigate different geoenvironmental engineering problems, particularly where transport mechanisms in soils are to
be studied (Zimmie et al. 1994). This motivated the authors
to demonstrate the utility of the centrifugation technique for
estimating the hydraulic conductivity of unsaturated soils.
As such, an effort is made in this study to use a geotechnical
centrifuge to create an unsaturated state of the soil, with
closed-flow centrifugation, for different periods of time and
acceleration levels. The results obtained and the principles
governing transient flow are used to determine unsaturated
soil hydraulic conductivity.
Basic principle
A geotechnical centrifuge can create an unsaturated soil
when a saturated soil is subjected to centrifugation.
Centrifugation offers an extraordinary saving of time, compared with other methods of measuring soil hydraulic conductivity, and has been extensively used to study flow
through saturated and unsaturated soils (Cargill and Ko
1983; Cooke and Mitchell 1991; Nimmo and Mello 1991;
Cooke 1994; Singh and Gupta 2000). Open-flow
centrifugation aims at establishing steady-state flow through
the sample while it is spinning in the centrifuge and computing the hydraulic conductivity using Darcy’s equation
(Stephens 1996).
Closed-flow centrifugation, where the water gradually
drains out the of the sample during spinning, is used to determine unsaturated soil hydraulic conductivity, as discussed
in the following. During centrifugation, since the centrifugal
force is in the outward direction on the water molecules, the
negative pressure gradient acts inward. Capillary pressure is
set up within the soil sample which varies along the length
of the sample, varying from zero at one end of the sample,
where it is open to the atmosphere, to a maximum value at
the inner end of the sample. At each rotational speed the
sample drains until the capillary force is equal and opposite
to the centrifugal force on the water molecules. After centrifuging the sample, the final water-content distribution along
685
Fig. 1. Schematic representation of a soil sample in a centrifuge
(after Corey 1977).
the sample length is determined by slicing it into six layers
and measuring the gravimetric water content of these layers.
Figure 1 gives a schematic representation of the test setup.
The suction pc, in centimetres of water column, created
within the sample when it is rotated in the centrifuge at an
angular velocity ω can be determined using (Corey 1977)
[1]
pc ≅
ρw ω 2 2
(R − r2 )
2g
where ρw is the density of water (in g/cm3), ω is the angular
velocity (in rad/s), R is the distance of the outer end of the
sample from the axis of rotation (= 29.5 cm), r is the distance of a point in the soil sample from the axis of rotation
(in cm), and g is Earth’s acceleration due to gravity
(= 981 cm/s2).
On the basis of literature reviewed (Stephens 1996), it is
understood that while analysing the flow through soils it is
also possible to use transient flow techniques, like the instantaneous profile method (Watson 1966) where disturbed
or undisturbed soil samples are subjected to known infiltration or drainage rates. By applying the law of mass conservation between two points a and b (a > b) along the soil
column, the Darcy velocity at a point a for time interval ∆t
(= t1 – t2), using the measured values of water content at
times t1 and t2, can be obtained using
[2]
V a , t = V b, t −
1
∆t
ra
∫ ∆θdr
rb
where Va,t and Vb,t are the Darcy velocities at r = ra and r =
rb (ra > rb), respectively, at time t; ∆t is the time interval between the measurements of the change in water content at
ra ; and ∆θ is the change in volumetric water content at r =
ra , for time interval ∆t.
Using this principle and extending it to the case of flow–
drainage of water from top to bottom of the centrifuged sample, applying eq. [2], we get
[3]
Va ,t = −
1
∆t
ra
∫ ∆θdr
rb
As the sample is draining from a state close to saturation,
and there is no inflow from the top of the sample, Vb,t = 0 in
eq. [2]. The change in volumetric water content, ∆θ, appearing in eq. [3] can be obtained with the help of following expression:
[4]
γ 
∆θ = ∆w  d 
 γw
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Can. Geotech. J. Vol. 39, 2002
Table 1. Properties of the silty soil used in the present study.
Specific gravity
Particle size characteristics (%)
Coarse sand (4.75–2.00 mm)
Medium sand (2.000–0.420 mm)
Fine sand (0.420–0.074 mm)
Silt (0.074–0.002 mm)
Clay (<0.002 mm)
Consistency limits (%)
Liquid limit
Plastic limit
Plasticity index
USCS soil classification
Standard Proctor compaction characteristics
Optimum moisture content (%)
Maximum dry unit weight (kN/m3)
2.79
Type
3.7
17.7
27.8
35.9
14.9
41
28
13
ML
20.8
17.0
where γ d is the dry unit weight of the soil, γ w is the unit
weight of water, and ∆w is the change in gravimetric water
content.
Further, the soil hydraulic conductivity K(ψ) under the effect of the hydraulic gradient i
[5]
i=
Table 2. Details of the geotechnical centrifuge used in the present study.
Arm radius
Max. outer radius
Centrifugation range
Max. acceleration
Payload capacity
Ramp-up time
Ramp-down time
Calibration of the centrifuge
418 rpm
512 rpm
592 rpm
662 rpm
Swinging buckets on
both sides of the arm
20 cm
31.5 cm
250–1000 rpm
300g
0.72 g-tons
20 s
80 s
50N
75N
100N
125N
Fig. 2. The centrifuge test setup.
∆ pc
lm
can be obtained by solving the following relationship:
[6]
K(ψ) = k u = −
V
i
where V is the Darcy velocity; ku is the unsaturated soil hydraulic conductivity; and ∆pc is the difference in suction
pressure at two points along the sample length, lm.
Experimental investigations
A silty soil was used in the present study and its physical,
gradational, and standard Proctor compaction characteristics
are presented in Table 1 (Singh and Gupta 2000).
Test methodology
An adequate amount of the oven-dried soil was mixed
with water to achieve a water content of 22.8%. The soil was
then stored for 24 h in airtight bags, for preconditioning and
maturing. The matured soil was compacted in a graduated
Perspex cylinder with an inner diameter of 35.0 mm and
length of 100 mm (as shown in Fig. 2). Graduations on the
cylinder helped in achieving (i) the desired unit weight, and
(ii) the volumetric deformation of the soil sample, if any,
during centrifugation. The compaction was completed in six
layers, giving 20 blows per layer, with the help of a flat bottom hand rammer to achieve a 60 mm high soil sample. The
sample (dry unit weight γ d = 16.5 kN/m3 and degree of saturation Sr = 92.2%) was then used for the centrifuge tests. To
ensure the homogeneity of the soil sample, in terms of its
water content, a sample was extruded onto a glass plate and
cut into six 10 mm thick slices. Gravimetric water content of
the slices, from top to bottom of the sample, was 21.8, 22.3,
22.4, 23.2, 23.5, and 22.9%. These values are very close to
the moulding water content (wo = 22.8%), and for all practi-
cal purposes it can be assumed that the water content in the
soil sample is homogeneous.
To create an unsaturated state of the soil sample, and
hence to estimate its unsaturated soil hydraulic conductivity,
identical soil samples were subjected to centrifugation at different acceleration levels, Ng (where N = 50, 75, 100, 125),
and different periods of time (15, 30, 60, 120, 240, 480, and
960 min). Details of the geotechnical centrifuge are presented in Table 2.
Through the observation window of the centrifuge, the
height of the soil sample during its flight, for various N values, was monitored continuously. The soil sample did not undergo any volumetric deformation (i.e., either swelling or
shrinkage). This observation is consistent with the fact that
for the soil used in the study the plasticity index is only 13%.
Results and discussions
After completion of the centrifugation, the soil sample
was sectioned in 10 mm thick slices and the water contents
of these slices were determined (Madhuri 1999). All water
contents have been normalized with respect to the initial
moulding water content of the sample (wo = 22.8%), as depicted in Table 3. In general, Table 3 shows that centrifugation
causes draining of the soil sample and hence the normalized
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687
Table 3. Normalized water contents (w/wo) for the soil sample due to centrifugation.
Centrifugation time (min)
N
50
75
100
125
r (cm)
a
24
25
26
27
28
29
24
25
26
27
28
29
24
25
26
27
28
29
24
25
26
27
28
29
15
30
60
120
240
480
960
0.9710
0.9671
0.9724
0.9858
0.9866
0.9876
0.9659
0.9771
0.9882
0.9977
0.9994
0.9933
0.9785
1.0093
0.9978
0.9737
0.9877
0.9814
0.9546
0.9749
0.9951
0.9808
0.9664
0.9650
0.9674
0.9819
0.9868
0.9849
0.9948
0.9790
0.9693
0.9738
0.9672
0.9894
0.9981
1.0361
0.9628
0.9718
0.9626
0.9665
0.9678
0.9479
0.9525
0.9671
0.9887
0.9876
0.9706
0.9698
0.9528
0.9495
0.9681
0.9643
0.9823
0.9808
0.9634
0.9695
0.9669
0.9853
0.9891
0.9844
0.9436
0.9567
0.9873
0.9914
0.9601
0.9432
0.9452
0.9470
0.9541
0.9611
0.9377
0.9632
0.9500
0.9515
0.9441
0.9701
0.9685
0.9681
0.9307
0.9386
0.9895
0.9863
0.9726
0.9766
0.9467
0.9708
0.9646
0.9701
0.9628
0.9591
0.9155
0.9426
0.9630
0.9697
0.9783
0.9640
0.9232
0.9280
0.9370
0.9176
0.9771
0.9351
0.9077
0.9175
0.9750
0.9839
0.9297
0.9320
0.8809
0.9060
0.9198
0.9358
0.9565
0.9505
0.6605
0.8617
0.8658
0.9096
0.9239
0.9250
0.8709
0.8830
0.8813
0.9138
0.9332
0.9257
0.8132
0.8915
0.9155
0.9187
0.9200
0.9312
0.6879
0.7912
0.8562
0.8551
0.8811
0.8900
0.6427
0.7514
0.7898
0.8095
0.8291
0.8324
0.7942
0.8019
0.8278
0.8250
0.8378
0.8389
0.6950
0.7128
0.7256
0.7363
0.7411
0.7471
0.5953
0.6184
0.6508
0.6710
0.6824
0.6804
0.5140
0.5348
0.5725
0.5895
0.6002
0.6056
a
Distance of center of the slice from the axis of rotation.
water contents increase from top to bottom in the soil sample. However, the loss in water content is greater for higher
values of N and longer centrifugation times. A significant
loss in water content occurs only for centrifugation times
≥ 480 min. As such, the results obtained for 480 and 960 min
of centrifugation are analyzed (as presented in Table 4) further to obtain the normalized water content, the magnitude of
suction created, and the saturation of the soil sample.
The variation of normalized water content, w/wo, along
the length of the soil sample (prototype) for 480 and
960 min of centrifugation is shown in Figs. 3a and 3b, respectively. The normalized water content of the soil sample
decreases with an increase in centrifugation time and N value.
As shown in Fig. 3, the maximum effect of centrifugation is
at the top of the sample (bound A–B) as compared with that
at the bottom (bound A–C). These bounds can be used to estimate the normalized water content of the prototype soil
column of a certain thickness. A 10 mm thick soil slice represents an average water content over a prototype soil column of thickness 0.5, 0.75, 1.0, and 1.25 m corresponding to
N = 50, 75, 100, and 125, respectively.
Using eq. [1], the magnitude of suction (pc) created in the
soil sample is computed (as presented in Table 4). In general, the sample top and bottom exhibit maximum and minimum suction pressures, respectively. This may be attributed
to the capillary suction gradients imposed across the sample
length as it is rotated in the centrifuge. Incidentally, these
trends are similar to the findings of Corey (1977).
From the data presented in Table 4, a centrifugation time
of 480 min results in a state of the soil sample with satura-
tion varying from 59.3 to 86.0%. On the other hand,
960 min of centrifugation results in the state of the soil sample with saturation varying from 73 to 77%, 64 to 69%, 54
to 63%, and 47 to 56% for N values of 50, 75, 100, and 125,
respectively. The initial value of Sr for these soil samples
was 92.2%. As such, to obtain the unsaturated hydraulic
conductivity of the soil sample the normalized water contents corresponding to 960 min of centrifugation at N = 100
and 125 were used.
As presented in Table 5, the change in water content ∆w
corresponding to different values of r for different acceleration levels N and the average ∆w is computed for 960 min of
centrifugation of the soil sample.
Using eq. [4], for N = 50 and ∆w = 4.08%, the average
∆θ = 4.08 × 1.65 = 6.73.
Applying eq. [3], with ra = 29 cm and rb = 24 cm, V =
[1/(960 × 60)] × 6.73 × 5 = 5.84 × 10–6 m/s.
Using eq. [5] and the data presented in Table 4, the hydraulic gradient for the soil sample, centrifuged at N = 50,
can be obtained as (288.48 – 28.68)/5 = 51.96, and eq. [6]
gives an unsaturated soil hydraulic conductivity of ku =
(5.84 × 10–6)/51.96 = 1.12 × 10–7 m/s.
Similarly, the values of ku corresponding to N = 75, 100,
and 125 have been computed and are presented in Table 6,
which also gives the average values of Sr for different values
of N. Table 6 shows that except for N = 50, for which Sr =
75.68%, the obtained ku values are much lower (45–68%)
than the saturated soil hydraulic conductivity (ksat) values reported by Singh and Gupta (2000) for the same
centrifugation effort. This indicates that ku for the soil sam© 2002 NRC Canada
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Can. Geotech. J. Vol. 39, 2002
Table 4. Calculation of saturation and suction pressure in the soil sample due to centrifugation.
w/wo
N
50
75
100
125
Sr (%)
r (cm)
480 min
960 min
480 min
960 min
pc (g/(cm·s2))
pc (cm)a
pc (kPa)
24
25
26
27
28
29
24
25
26
27
28
29
24
25
26
27
28
29
24
25
26
27
28
29
0.8709
0.8830
0.8813
0.9138
0.9332
0.9257
0.8132
0.8915
0.9155
0.9187
0.9200
0.9312
0.6879
0.7912
0.8562
0.8551
0.8811
0.8900
0.6427
0.7514
0.7898
0.8095
0.8291
0.8324
0.7942
0.8019
0.8278
0.8250
0.8378
0.8389
0.6950
0.7128
0.7256
0.7363
0.7411
0.7471
0.5953
0.6184
0.6508
0.6710
0.6824
0.6804
0.5140
0.5348
0.5725
0.5895
0.6002
0.6056
80.3
81.4
81.2
84.2
86.0
85.3
75.0
82.2
84.4
84.7
84.8
85.8
63.4
72.9
78.9
78.8
81.2
82.1
59.3
69.3
72.8
74.6
76.4
76.7
73.2
73.9
76.3
76.1
77.2
77.3
64.1
65.7
66.9
67.9
68.3
68.9
54.9
57.0
60.0
61.9
62.9
62.7
47.4
49.3
52.8
54.3
55.3
55.8
282 999.3
235 872.8
186 822.8
135 849.3
82 952.2
28 131.6
424 498.9
353 809.2
280 234.2
203 773.9
124 428.3
42 197.43
565 998.5
471 745.6
373 645.6
271 698.5
165 904.4
56 263.24
707 498.2
589 682.0
467 057.0
339 623.2
207 380.5
70 329.04
288.48
240.44
190.44
138.48
84.56
28.68
432.72
360.66
285.66
207.72
126.84
43.01
576.96
480.88
380.88
276.96
169.12
57.35
721.20
601.10
476.10
346.20
211.40
71.69
28.28
23.57
18.67
13.58
8.29
2.81
42.42
35.36
28.01
20.36
12.44
4.22
56.56
47.15
37.34
27.15
16.58
5.62
70.71
58.93
46.68
33.94
20.73
7.03
a
Equivalent water column.
ple, for 50 < Sr < 60%, can be assumed to be equal to 1.10 ×
10–7 m/s.
have been used. For the sake of completeness, the fit equations are presented as follows and various parameters related
to these fits are presented in Table 8:
Validity of Darcy’s law in the centrifuge
[8]
w(ψ) =

−1

ψ
mf

ln 1 +     
nf   

h r    
ψ  


 ln  exp(1) +     
w s 1 −

a  

6
 f   
 ln 1 + 10     
 






hr 



[9]
w (ψ) = w r + (w s − w r ){[1 + (a vgψ) nvg ]mvg }−1
[10]
a 
w (ψ) = w r + (w s − w r )  c 
ψ
The validity of Darcy’s law in the centrifuge experiments
can be checked with the help of the Reynolds number (Re):
[7]
Re =
Vd
v
where d is the characteristic microscopic length (= 3 µm),
and v is the kinematic viscosity of water (= 1.01 × 10–6 m2/s).
Using eqs. [3] and [7], Re values for the soil samples centrifuged at different values of N are obtained and are presented
in Table 7. For all the samples, Re is less than unity, which
indicates the validity of Darcy’s law in the centrifuge experiments.
Development of the soil-water
characteristic curves (SWCCs)
As depicted in Fig. 4, the soil-water characteristic curves
(SWCCs) have been developed, using the SoilVision 2.04
database (SoilVision Systems Ltd. 2000), for the results obtained for the soil sample centrifuged for 960 min corresponding to N = 125. For the sake of brevity, however, the
SWCCs for soil samples centrifuged for 960 min at other N
values are not being reported here. For developing the
SWCCs, eq. [8] (Fredlund and Xing 1994), eq. [9] (van
Genuchten 1980), and eq. [10] (Brooks and Corey 1964)
nc
where w(ψ) is the gravimetric water content at any suction
ψ, wr is the residual water content (RWC), ws is the
gravimetric water content at saturation, af and avg are soil
parameters primarily dependent on the air entry value
(AEV), nf and nvg are soil parameters dependent on the rate
of extraction of water from the soil beyond the AEV, mf is
the soil parameter which is a function of the RWC, hr is the
suction (in kPa) corresponding to the RWC, mvg is a fitting
parameter, ac is the bubbling pressure (in kPa), and nc is the
pore-size index.
Figure 4 that shows that, in general, the water content versus suction data yielded by the centrifugation technique match
very well with the theoretical predictions. These SWCCs also
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689
Fig. 3. Variation of normalized water content (w/wo) along the
length of the soil column due to its centrifugation.
Table 5. Changes in the water content of the soil sample after
960 min of centrifugation.
N
50
Average
75
Average
100
Average
125
r (cm)
pc (kPa)
w/wo
1 – w/wo
24
25
26
27
28
29
28.28
23.57
18.67
13.58
8.29
2.81
0.7942
0.8019
0.8278
0.8250
0.8378
0.8389
0.2057
0.1980
0.1722
0.1749
0.1622
0.1611
24
25
26
27
28
29
42.42
35.36
28.01
20.36
12.44
4.22
0.6950
0.7128
0.7256
0.7363
0.7411
0.7471
0.305
0.2872
0.2744
0.2637
0.2589
0.2529
24
25
26
27
28
29
56.56
47.15
37.34
27.15
16.58
5.62
0.5953
0.6184
0.6508
0.6710
0.6824
0.6804
0.4047
0.3816
0.3492
0.3290
0.3176
0.3196
24
25
26
27
28
29
70.71
58.93
46.68
33.94
20.73
7.03
0.5140
0.5348
0.5725
0.5895
0.6002
0.6056
0.4860
0.4652
0.4275
0.4105
0.3998
0.3944
Average
∆w (%)
4.69
4.52
3.93
3.99
3.70
3.67
4.08
6.95
6.55
6.26
6.01
5.90
5.77
6.24
9.23
8.70
7.96
7.50
7.24
7.29
7.99
11.08
10.61
9.75
9.36
9.12
8.99
9.82
Table 6. Comparison of the unsaturated and saturated soil hydraulic conductivities.
N
Average
∆w (%)
50
75
100
125
4.08
6.24
7.99
9.82
Average
∆θ (%)
6.73
10.30
13.18
16.20
Average
Sr (%)
i
ku
(m/s, ×10–7)a
ksat
(m/s, ×10–7)b
75.68
66.96
59.90
52.50
51.96
77.94
103.92
129.90
1.12
1.15
1.10
1.08
1.13
2.11
2.79
3.42
a
Present study.
Singh and Gupta (2000).
b
show that the water content decreases as the suction increases. These trends are in accordance with the results presented by Leong and Rahardjo (1997) and demonstrate the
usefulness of a geotechnical centrifuge in creating an unsaturated state of the soil and developing SWCCs for the soil.
Estimation of unsaturated soil hydraulic
conductivity
To estimate the unsaturated hydraulic conductivity ku of
the soil from the SWCCs, the SoilVision 2.04 (SoilVision
Systems Ltd. 2000) database was used to estimate the saturated soil hydraulic conductivity ksat. For this purpose,
eqs. [11]–[16] proposed by Kozeny, Terzaghi, Kruger,
Zamarin (Vukovic and Soro 1992), Rawls and Brakensiek
(1983), and Rawls et al. (1993), respectively, have been
used:
n3
(d10) 2
(1 − n) 2
[11]
k sat = 5400
[12]
ksat = 200e2(d10)2
[13]
k sat = 0.278
[14]
k sat = 8.07
n
(d10) 2
(1 − n) 2
n3
C nτ (d10) 2
(1 − n) 2
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690
Can. Geotech. J. Vol. 39, 2002
Fig. 4. Soil-water characteristic curves (SWCCs) for the soil sample centrifuged for 960 min at N = 125.
ksat = 100 [ exp (19.52348n – 8.96847
Table 7. Values of the Reynolds number Re
for soil samples.
– 0.028212C + 0.00018107 S 2 – 0.0094125C 2
[15]
− 8.395215n2 + 0.077718 Sn − 0.00298 S 2 n2
2
2
+ 0.001434 S 2 n − 0.000035C 2 S ) 2.77 × 10 − 6 ]
[16]
 nx 
k sat = 4.41 × 109  2  R12
N 
where ksat is in centimetres per second; d10 is the effective
grain diameter (in mm); n is the porosity; e is the voids ratio; τ is the temperature correction factor; Cn is an empirical
coefficient, which depends on the porosity; S and C are the
sand and clay fractions, respectively, as per the U.S. Department of Agriculture classification; x is a soil-dependent constant; N is the total pore size classes; and R1 is the average
pore radius (in cm).
The ksat values estimated from eqs. [11]–[16] are presented in Table 9, which shows that they vary from 1.38 ×
10–7 to 26.60 × 10–7 m/s. As expected, the unsaturated soil
hydraulic conductivity (= 1.10 × 10–7 m/s) is much lower
than the saturated soil hydraulic conductivity. Further, the
saturated soil hydraulic conductivity, estimated with the help
of an equation suggested by Rawls et al. (1993) and equal to
2.06 × 10–7 m/s, has been linked to different “pedo” transfer
functions (represented by eqs. [17]–[22] and termed the
Kunze (KCAL), modified Campbell, Brooks and Corey, van
Genuchten, Fredlund and Xing, and Campbell PTFs, respectively), available in SoilVision 2.04 (SoilVision Systems Ltd.
2000), to establish the variation of relative permeability (kr)
with changes in soil suction.
V (m/s, ×10–6)
5.84
8.94
11.44
14.07
N
50
75
100
125
− 0.019492C n + 0.0000173 S C + 0.02733C n
2 2
[17]
kr =
T 2s ρwg θ qs
k sc N 2 2 µ w
M
j =i

k 
k
k r = 1 − min  Θ P ψ + min


k sat 
k sat

[19]
5λ

2+
 ψb 2
kr =   
ψ
 
 1
kr =
(ψ > ψ b)
(ψ ≤ ψ b)
{1 − (αψ) n−1[ 1 + (αψ) n ] − m}2
m
[ 1 + (αψ) n ] 2
ln (1 000 000)
[21a] k r =
1.74
2.65
3.39
4.17
∑ [( 2 j + 1 − 2 i) ψ −j2 ]
[18]
[20]
Re (×10–5)
∫
θ (e y ) − θ (ψ)
θ ′ (e y ) dy
ey
∫
θ (e y ) − w θ
θ ′ (e y ) dy
ey
ln ( ψ )
ln (1 000 000)
ln ( ψ AEV )
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691
Table 8. Details of various parameters used in the
fitting functions Brooks and Corey (1964), van
Genuchten (1980), and Fredlund and Xing (1994).
Brooks and Corey
ac (kPa)
nc
Error
RWC (%)
AEV (kPa)
Maximum slope
van Genuchten
avg (kPa–1)
nvg
mvg
Error
RWC (%)
AEV (kPa)
Maximum slope
Fredlund and Xing
af (kPa)
nf
mf
hr (kPa)
Error
RWC (%)
AEV (kPa)
Maximum slope
30.99
0.19
0.9672
0.6
30.90
0.42
0.015
4.68
0.23
0.9809
0.1
52.13
1.30
76.23
1.29
0.67
699 005.8
0.9941
29.5
31.35
0.41
Table 9. Estimated saturated soil hydraulic conductivities using
SoilVision 2.04 (SoilVision Systems Ltd. 2000).
Proposed equation
ksat (m/s, ×10–7)
Kozeny (Vukovic and Soro 1992)
Terzaghi (Vukovic and Soro 1992)
Rawls and Brakensiek 1983
Kruger (Vukovic and Soro 1992)
Zamarin (Vukovic and Soro 1992)
Rawls et al. 1993
26.60
1.38
2.55
11.50
13.20
2.06


ψ 
ln 1 +
 

θs
 Cr  
[21b] θ = 1 −
m


n
 1000 000 
  ln  e +  ψ  
 ln 1 +
  
Cr

  

a 

[22]
θ
k r =  s 
θ
2+
2
b
where ksc is the calculated saturated permeability; Ts is the
surface tension of water; ρw is the density of water; µw is the
viscosity of water; M is the total number of intervals between the saturated volumetric content on the SWCC; N is
the total number of intervals computed between the saturated
volumetric water content and zero water content; q is a constant that accounts for the interaction of pores of various
sizes; kmin is the minimum permeability; w is the water content; Θ is the normalized water content; P is the fitting pa-
Table 10. Unsaturated soil hydraulic conductivity for the soil
sample estimated using the Kunze (KCAL), modified Campbell,
Brooks and Corey, van Genuchten, Fredlund and Xing, and Campbell “pedo” transfer functions (PTFs).
pc (kPa)
Kunze (KCAL)
40.44
47.45
52.39
56.42
59.95
63.19
66.23
69.15
Modified Campbell
7.59
15.85
22.91
33.11
47.86
69.18
Brooks and Corey
7.59
15.85
22.91
33.11
47.86
69.18
van Genuchten
7.59
10.96
22.91
33.11
47.86
69.18
Fredlund and Xing
7.59
10.96
15.85
22.91
33.11
47.86
69.18
Campbell
7.59
10.96
15.85
22.91
33.11
47.86
69.18
kr
ku (m/s, ×10–7)
0.990
0.835
0.724
0.635
0.560
0.496
0.440
0.390
2.04
1.72
1.49
1.31
1.15
1.02
0.91
0.80
0.980
0.951
0.925
0.886
0.834
0.767
2.02
1.96
1.90
1.83
1.72
1.58
1.000
1.000
1.000
0.849
0.341
0.137
2.06
2.06
2.06
1.75
0.70
0.28
0.999
0.997
0.961
0.851
0.529
0.114
2.06
2.05
1.98
1.75
1.09
0.23
0.534
0.450
0.362
0.273
0.190
0.121
0.068
1.10
0.93
0.75
0.56
0.39
0.25
0.14
0.923
0.881
0.819
0.731
0.618
0.483
0.344
1.90
1.82
1.69
1.51
1.27
1.00
0.71
rameter; ψb and λ are Brooks and Corey SWCC fitting parameters, respectively; α, n, and m are the van Genuchten
SWCC fitting parameters; ψ is the soil suction; ψAEV is the
soil suction corresponding to the AEV; y is a variable of integration representing the logarithm of suction; θ′ is the derivative of eq. [21b]; Cr is a constant related to the suction
corresponding to the RWC; m is a parameter related to the
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692
Can. Geotech. J. Vol. 39, 2002
Fig. 5. Variation of relative permeability (kr) with suction for the soil sample centrifuged for 960 min at N = 125.
RWC; n is a parameter that controls the slope at the inflection point in the SWCC; a is the air-entry value of the soil; e
is the natural number (=2.71828); θs is the saturated volumetric water content and θ is the volumetric water content at
any particular suction, obtained from the SWCC curve; and
b is the fitting parameter.
Variation of kr with changes in soil suction for the soil
sample centrifuged for 960 min at N = 125 is shown in
Fig. 5. For the sake of brevity, however, the trends for the
soil samples centrifuged for 960 min corresponding to other
N values are not reported here. Figure 5 shows that, in general, kr decreases as the suction increases. For N = 125, however, various PTFs yield curves that deviate slightly from
one another, with the modified Campbell and Fredlund and
Xing PTFs yielding the two extremes. These curves can be
used to predict the unsaturated soil hydraulic conductivity,
ku, for a known suction value by multiplying kr with the saturated hydraulic conductivity, ksat (2.06 × 10–7 m/s), obtained using the equation proposed by Rawls et al. (1993).
As centrifugation of the soil sample, for 960 min at N =
100 and N = 125, yields 5.62 ≤ pc ≤ 70.71 kPa (Table 5),
when ku for the soil is estimated in this range, with the help
of different PTFs as shown in Table 10, ku varies from
0.14 × 10–7 to 2.06 × 10–7 m/s. It is interesting to note that
ku for the soil, obtained by the centrifugation method, as depicted in Table 6 (= 1.10 × 10–7 m/s) falls in this range. As
such, the study highlights the usefulness of centrifugation, in
conjunction with the transient flow technique, for estimation
of unsaturated soil hydraulic conductivity.
Conclusions
The following conclusions are drawn from the present
study:
(1) The unsaturated state of the soil depends on the N
value and on the time of centrifugation. A uniformly unsaturated soil state could only be obtained for N ≥ 100 after
960 min of centrifugation.
(2) Centrifugation results in the creation of suction
(≤ 71 kPa) in the soil sample. The suction is highest at the
top of the sample and lowest at the bottom of the sample.
(3) The unsaturated soil hydraulic conductivity, k u, for
5 ≤ p c ≤ 71 kPa can be assumed to be equal to 1.10 ×
10–7 m/s.
(4) The centrifugation method yields a ku value, for Sr <
60%, that is 45–68% less than the ksat value reported by
Singh and Gupta (2000) for the same centrifugation effort.
(5) Theoretical (best) fits to the data obtained from
centrifugation yield soil-water characteristic curves (SWCCs)
for which the coefficient of regression is close to unity.
(6) The obtained SWCCs can be used to estimate the unsaturated hydraulic conductivity of a soil sample with the
help of relative permeability versus suction curves.
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List of symbols
ac
af, avg
Cn
d
d10
e
g
hr
i
kmin
kr
ksc
ku, ksat
bubbling pressure
soil parameters
empirical coefficient that depends on porosity n
characteristic microscopic length
effective grain diameter
voids ratio
acceleration due to gravity (= 981 cm/s2)
suction corresponding to the residual water content
hydraulic gradient
minimum permeability
relative permeability
calculated saturated permeability
unsaturated and saturated soil hydraulic conductivity,
respectively
K(ψ) hydraulic conductivity at suction ψ
lm length of the soil sample (model)
mf soil parameter
mvg fitting parameter
n
nc
nf, nvg
N
N
pc
∆pc
P
r
ra, rb
R
R1
Re
Sr
∆t
Ts
V
Va,t, Vb,t
∆w
w, wo
w/wo
wr
ws
w( ψ )
x
ψ
γd
γw
µw
ν
ρw
τ
∆θ
Θ
ω
porosity
pore size index
soil parameters
total number of pore size classes
acceleration level
capillary suction
difference in suction pressure at two points along the
sample length
fitting parameter
distance of a point in the soil sample from the axis of
rotation
radial distance of two points within the sample (ra > rb)
distance of the outer end of the sample from the axis of
rotation (=29.5 cm)
average pore radius
Reynolds number
degree of saturation (%)
time interval between the measurements of the change
in water content at ra
surface tension of water
Darcy velocity
Darcy velocities at r = ra and r = rb, respectively, at
time t
change in gravimetric water content (%)
final and moulding water contents (%)
normalized water content
residual water content
gravimetric water content at saturation
gravimetric water content at suction ψ
soil-dependent constant
suction
dry unit weight of the soil sample
unit weight of water
viscosity of water
kinematic viscosity of water
density of water
temperature correction factor
change in volumetric water content (%) at r = ra for
time ∆t (= t1 – t2)
normalized water content
angular velocity
© 2002 NRC Canada
I:\cgj\Cgj39\Cgj-03\T02-013.vp
Thursday, May 16, 2002 9:35:29 AM