1

A two-dimensional finite-volume model to simulate

laboratory swelling tests for unsaturated soils

Khaled Sebri*, Mehrez Jamei, and Houda Guiras Civil Engineering Laboratory

National Engineering Scholl of Tunis (ENIT)

E-mail: sebrikhaled@yahoo.fr

*Corresponding author

Abstract: Numerous laboratory swelling tests have been reported for the measurement of swelling pressure and the amount of

swell of an expansive soil. But these tests cost a lot of money, take a long period of time and need expensive equipments. Few

attempts, however, have been made to formulate a theoretical framework to simulate the testing procedures or to visualize the

different stress paths followed when using the various methods. A bi-dimensional theoretical model is proposed to descript the

matric suction pressures evolution with time and depth in a specimen as well as the volume changes (void ratio) during various

swell tests (œdometer and triaxial). The model is formulated based on equilibrium considerations, constitutive equations for an

unsaturated soil, and continuity requirement for the fluid phases. The transit water flow is coupled with the soil volume change

process. The model has been put into a finite volume formulation. All the parameters required to run the model can be obtained

by performing independent, common laboratory tests. The proposed numerical scheme was used to simulate the results from

free-swell œdometer tests and constant-volume œdometer tests. Computed values of volume change and matric suction are in

good agreement with measured values given from laboratory free swell and constant-volume tests maked by F. Shuai and D.G.

Fredlund (1998) and H. Guiras (1996). The difficulties related to the numerical tests were the give of a good permeability

function depending on suction and void ratio evolutions and convenient compressibility parameters. A lecture and

extrapolation technique based on the water retention curves was used.

After, some numerical tests with finite volume using an Euler implicit scheme to discrete time and space variables were

carried out to simulate different equilibrium suction conditions. All the tests lead to free-swell and constant-volume tests

prediction.

Keywords: swelling tests; œdometer laboratory tests; swelling path; finite volume method; prediction.

1 INTRODUCTION

The swelling compacted clays receive increasing attention as soils which lead to many stability structures problem (shallow

foundations, slope stability, etc.). In special applications, clays and bentonite are used as technical buffer materials in geotechnical barriers in order to isolate waste (Börgesson, L, (1985)). Therefore, the knowledge of the behaviour of the soil

material under a variety of hydro-thermo-mechanical conditions is of a special interest. The experimental evaluation of the clay

behaviour is necessary and requires a basic understanding of the occurring processes which include fluid flow, mechanical

reaction and sometimes heat transport. The complexity of the study results in thermo-hydro-mechanical coupling phenomena.

A basic experimental tests and a numerical modelling can be an effective ways to improve the understanding of those process

and to predict the long-term structures effects.

The swelling tests on compacted clay soils are essentially carried out in laboratory and rarely in site. Classically these tests

are used to determine two parameters which are: the swelling rate (ratio of the final and the initial volume variations) and the

swelling pressure (equal to the stress applied to the sample in order to obtain again the initial volume before swelling).

However, in practice in the swelling laboratory measurement of the kinetics of is also important and leads to quantify swelling

effects on structure. Unfortunately, the swelling phenomenon is often very slow and conducts to very long experiences. The

corresponded tests are often carried out during two to six months, which leads to some difficulties of management for the materials used. Other associated difficulties are related to the requirement of tests data. The swelling tests are carried out under

different possible paths. They are applied under controlled suction. They are also conducted under mechanical stress. The

difficulty leads in two points: first, it is not easy to assign a hydraulic path during loading (for example to control suction).

Second, it is difficult to reproduce the initial hydraulic (water content) and compacted conditions and to have exactly the same

path.

The aim of this paper is to show how the numerical approach should allow to reduce the number of experimental tests and

to predict different swelling paths. Basing on the finite volume technique, software was established using a convenient Euler

scheme. It can be possible with this numerical tool to predict some different free-swelling and constant-volume experiences

which were carried out on saturated and unsaturated bentonite. The originality of this study lies into the hydraulic coupling

with a nonlinear behavior of the soil which requires an integration of the behavior low in adaptive iterative computation.

In the literature, two fundamental models were proposed for the swelling prediction. The Barcelona model proposed by E. Alonso & al. (1992, 1993 and 1995) has the potentiality to predict the micro-structure and macro-structure swelling under

the suction and the total stress variations. The elasto-plastic model formulation can integrate separately the two components

of swelling: the micro and macro volumetric strains. It is formulated with the definition of a special potential function

which relates the irreversible swelling macro volumetric strain to the suction decreasing.

2

The second model was proposed by Fredlund and al. (1998). It is based on the constitutive state curves: soil structure and water content constitutive surfaces (figures 1 and 2).

Figure 1: characteristic curve for structure phase Figure 2: characteristic curve for water phase

1.1 Soil structure or void ratio constitutive surfaces

A schematic diagram of the soil structure for monotonic suction and stresses is shown in figure 1. The intersection curves

of the void ratio on the net total stress (σ-ua) (where σ=σv denotes the total vertical stress in oedometeric path) and suction

(s = ua-uw) planes (where ua and uw are respectively the air and water interstitial pressures) result in the definition of two

compressibility coefficients sm1 and

sm2 , where sm1 and

sm2 are respectively coefficient of compressibility of soil structure

respect to the net total stress and the suction on wetting or swelling according to the hydraulic path (wetting if suction

decreases). We note that the free-swelling path corresponds to the intersection curves for no loading path (just piston load

effect is taken into account). When metric suction is equal to zero, the curve on (e, (σ-ua)) plane is the same as the

conventional consolidation curve of saturated soil.

Clay can be over consolidated by desiccation and rebounded due to a suction decrease (HO D. Y. F,. Fredlund D.G, and Rahardjo H.(1992)).

1.2 Water phase constitutive surfaces

It’s not easy to have completely experimental information of the water constitutive surface. The water content versus matric

suction curves for the free-swelling path is essentially linear in a semi logarithmic scale (HO D. Y. F,. Fredlund D.G, and

Rahardjo H.(1992)).

In this paper, some available experimental information on the water content for the bentonite clay is given. The intersection

curves on the state surfaces of water content on the (σ-ua) and (s = ua-uw) planes results in two coefficients wm1 and

wm2 ,

where wm1 and

wm2 are coefficient of water content change respect to the net total stress and the suction (figure 2). It appears

that the two water content variation coefficients are approximately constant.

2 GOVERNING EQUATIONS

2.1 General theory

In this paper, we consider the free-swell path and volume-constant path. In general, there are two concepts to formulate the

balance equations: the transient water flow and the soil volume change. These two concepts are linked with each other and

governed by the following basic equations: (i) the force equilibrium equation for an element of soil, (ii) the constitutive

equations for an unsaturated soil, and (iii) the continuity equation for the pore fluids.

2.1.1 Equilibrium equation:

In the case of clay sample we have:

0)( div (1)

Whith σ σ σ : is the variation of the stress tensor components in Δt time.

2.1.2 Volume change constitutive relationships

Fredlund and Rahardjo (1993) proposed volume change constitutive relationships for an unsaturated soil as an extension of the

form of equation used for a saturated soil. For three-dimensional case the relationship for the soil structure can be written as:

smumV

V s

amean

svv

21

0

)( (2)

3

The relationship for the water phase can be written as:

smumV

V w

amean

ww

21

0

)( (3)

Where:

trmean3

1 ;

wa uus : the matric suction;

Δ : the increment of the stress states with time; V0 : the initial volume of the soil element;

ΔVv : the change in the volume of soil voids in the soil element;

ΔVw : the change in the volume of water in the soil element;

)1(

)]21)(1[(1

Ems

: The soil structure volume change modulus with respect to a change in stress ;

)1(

12

Hms

: The soil structure volume change modulus with respect to a change in matric suction;

E : the modulus of elasticity or Young’s modulus for the soil structure;

Μ : the Poisson’s ratio;

H : the elastic modulus with respect to a change in matric suction; wm1 : The water volume change modulus with respect to a change in stress; and

wm2 : The water volume change modulus with respect to a change in matric suction;

2.1.3 Continuity requirement for water phase:

The equation of continuity for the water phase in a soil takes the following form (Freeze and Cherry 1979):

)(0

ww qdiv

V

V

t

(4)

Where:

Vw :the water volume in the soil element ;

0V

Vw The net water volume change per unit volume f the soil element ;and

z

q

y

q

x

qqdiv wzwywx

w

)( ;

(qw)i (i=x,y,z) the water flow rate across a unit area of the soil element in the I direction.

Darcy’s law can be applied to the flow of water through an unsaturated soil (Buckingham E.(1907) ; Richards L.A.(1931) ;

Childs E.C., and Collis-George N. (1950)):

)()( hgradskq ww (5)

Où

)(skw : the tensor of permeability with respect to the water phase (which is a function of matric suction).

hw : the hydraulic head )/()2/( 2 gsgvz ww ;

z : the elevation ;

vw : the velocity ;

uw : the pore water pressure ;

ρw : density of water ;

g : the gravitational acceleration ; and

)(hgrad : the hydraulic head gradient.

(5) in (4) gives :

)(0

hgradkdivV

V

tw

w

(6)

Assuming the thickness of the sample is small and the water velocity head in the sample is negligible, the continuity

equation for the water phase can be written as:

)(1

0

sgradkdivgV

V

tw

w

w

(7)

4

2.2 Derivation of the governing equations for swelling tests:

The following assumptions are made to simplify the derivation of the governing equation for swell tests:

i) Isotropic soil

ii) Infinitesimal strain

iii) Linear constitutive relations for a small charge in stress or matric suction

iv) The coefficient of permeability of water is constant for a small charge in matric suction

v) The pore air pressure is always equal to the surrounding air pressure (i.e.,ua=0).

The constitutive equations for an unsaturated soil become:

smmV

V s

mean

svv

21

0

(8)

smmV

V w

mean

ww

21

0

(9)

2.2.1 Governing differential equation for constant-load oedometer test:

During constant-load oedometer test is maintained constant (i.e., 0 mean ) . As result, the constitutive equations for an

unsaturated soil (8) and (9) can be simplified as:

smV

V svv

2

0

(10)

smV

V ww

2

0

(11)

Noting that the incremental charge in pore-water pressure with time is equal to the actual pore-water pressure charge (i.e.,

tutu ww //)( ) (7) in (11) gives :

)(1

2

sgradkdivgmt

sww

w

(12)

Equation (12) is a transient water flow equation for the constant-load test. It can be used to compute the negative pore-water

pressure at different depths and times during the constant-load swelling process.

2.2.2 Governing differential equation for constant-volume oedometer test: (3) in (7) gives :

)(1

21 sgradkdivgt

sm

tm w

w

wmeanw

(13)

During constant-volume test 00

V

V

t

v

; (2) gives:

t

s

m

m

t s

s

mean

1

2 (14)

(14) in (13) :

)(1

1

212 sgradkdiv

gt

s

m

mmm w

w

s

sww

(15)

Equation (15) is a transient water flow equation for the constant-volume test. It can be used to compute the negative pore-water

pressure at different depths and times during the constant-volume swelling process.

2.3 Discrete equations:

We use the finite volume method as conservative numerical scheme of non-linear diffusion equation (Masters I.,and al.,

(2004)) which is the result of some balance equations in the multiphase flow and for swilling tests. The deformation of the

porous media is described via the permeability which depends of the void ratio. The numerical scheme requires the iterative

integration algorithm. Indeed, an implicit Euler iterative scheme is used for the time and for the space coordinate variables.

Different numerical tests allow to conclude that the semi-implicit Euler scheme (with α=0.5) is better convenient. This scheme has been proved to be practical for the non-linearity form.

The equation to solve is:

),())()(( tsfssDs

(16)

5

yi

xi xi+1 xi+1/2 xi-1/2

Δxi+1/2

Δxi

Δyi

Ki,j Ki+1,j

Where :

)(1

)(2

sKm

sD ww

w : for constant-load swelling test; and

)(1

)(

1

212

sK

m

mmm

sD w

s

sww

w

: for constant-volume swelling test.

The implicit Euler scheme for the time variable and in a basic volume element (Ki,j) is written as in equations (16a);(16b) and

(16c):

t

sss

nntt

1

(16a)

1)1( nntt sss

(16b)

dxdyttsfdxdyssDdxdys ttttttt

),())()((

(16c)

Where f(s,t) is a function of numerical testing (it was chosen with different analytical non-linearity forms).

The implicit Euler scheme for the space dimensions (for example for the variable x) (figure 3) leads to:

2/1

1

1

11 ),(),(),(

i

i

n

i

nn

x

yxsyxsyx

x

s (16d)

Figure 3: basic volume element (Ki,j) and (Ki+1,j)

3 PREDICTIONS OF THE EXPERIMENTAL RESULTS

3.1 Validation of theoretical and experimental results of tests proposed by Fredlund D.G., and al. (1998)

Fredlund D.G., and al. (1998) proposed one-dimensional theoretical model (finite element model) to simulate the results from

several oedometer swilling tests (i.e., free-swell and constant-volume tests) on the compacted Regina clay.

Regina clay is highly swelling. Regina clay was statically compacted to produce a specimen with à initial void ratio of 0.96 and

a molding water content of 26%. The initial matric suction at a water content of 26% was 575KPa.

We propose in the following to simulate and compare results given by the finite volume model and results given by Fredlund

and al. (1998).

The solution of governing equation requires the following soil properties: coefficient of permeability function kw; coefficients

of soil structure volume changesm1 and

sm2 ; and coefficients of water volume change wm1 and

wm2 .

Several types of laboratory tests were performed to independently measure the above soil properties. These tests are the

falling-head permeability test, the free-swell oedometer test, the pressure-plate test, the shrinkage test, and the constant-suction

consolidation test.

The values of the coefficient of permeability listed in table 1 were used in the computer program to simulate the various

oedometer swilling tests (i.e free-swell and constant-volume swell tests).

Coefficient Functions Values of parameters

Kw

n

w

wa

b

w

g

uua

ekK

1

0

K0=0.4x10-10 m/s

b=18.5 a=0.01

n=1.1

Table1 : The values of the coefficient of permeability

6

The values of soil structure volume changesm1 and

sm2 ; and coefficients of water volume change wm1 and

wm2 are given

from figure 4 and figure 5

Figure 4: Measured soil structure constitutive surface for

compacted Regina Clay.

Figure 5: Measured water phase constitutive surface for

compacted Regina Clay

3.1.1 Free-swell oedometer test

Computer simulation of the free-swell oedometer testing process were carried out by specifying a zoro flux boundary for the

top, right and left of the specimen and a zero pore-water pressure and zero displacement for the bottom. A uniform initial

matric suction of 575KPa was specified for all basics volume elements.

Numeric results:

Figure 6 shows the special profile (2D) of matric suction in the specimen.

Figure 6:The special profile (2D) of matric suction in the specimen at 3300000s

Figure 7 shows the measured and computed (Fredlund D.G.,and al.(1998) and finite volume model) deflection versus time

curves for 100 mm high specimen. A comparison between finite volume computed and measured curves shows good

agreement for the full length of the tesr for the specimen. The predicted and measured total heaves are almost the same.

A comparison between the computed and measured matric suction profiles for the 100 mm high specimen is presented in

figure 8. Good agreement was found between the measured and computed finite volume simulation matic suction for all stages

of the test (i.e., t=1500min; t=5300 min and t=55000min).

3.1.2 Constant-volume oedometer test:

To simulate the constant-volume oedometer test, a zero flux and zero displacement boundary were specified for the top, right

and left of the specimen and a zero pore-water pressure and zero displacement for the bottom. A uniform initial matric suction

of 575KPa was specified for all basics volume elements. The initial stress for all elements was equal to a token load 1KPa.

The measured and computed (i.e., finite volume model and Fredlund D.G.,and al.(1998) model) vertical total stress versus time

for the 100mm high specimen are shown in figure 9. A comparison of finite volume computed and measured curves show a

good agreement for the full duration of the test for the specimen.

7

Figure 7: Computed and measured deflection versus time curves for free-swell oedometer test

Figure 8: Computed and measured matric suction profiles for free-swell oedometer test.

-12

-10

-8

-6

-4

-2

0

0 10000 20000 30000 40000 50000 60000 70000

def

lect

ion

(m

m)

Time (min)

Simulation Fredlund(1998) Experimental points Simulation finite volume

0

20

40

60

80

100

0 100 200 300 400 500 600

elev

atio

n y

(m

m)

matric suction (KPa)

Experimental points 1500 min Simulation 1500 min (Fredlund 1998) Simulation 1500 min (finite volume)

Exprimental points 5300 min Simulation 5300 min (Fredlund 1998) Simulation 5300 min (finite volume)

Experimental points 55000 min Simulation 55000 min (Fredlund 1998) Simulation 55000 min (finite volume)

8

Figure 9: Computed and measured vertical net normal stress versus time curves for constant-volume oedometer test

The measured and computed profiles of matric suction are presented in figure 10. The correlation between measured and finite

volume model computed matric suction are reasonably good.

Figure 10: Computed and measured matric suction profiles for constant-volume oedometer test.

0

50

100

150

200

250

300

0 10000 20000 30000 40000 50000 60000

vert

ical

str

ain

(M

Pa)

Time (min)

Finite volume simulation Simulation (Fredlund1998) Experimental points

0

20

40

60

80

100

120

0 100 200 300 400 500 600

Ele

vati

on

y(m

m)

Matric suction (KPa)

Finite volume simulation 1500min Experimental points 1500min Simulation 1500min (Fredlund1998)

Experimental points 54000min Simulation 5400min(Fredlund 1998) Finite volume simulation 54000min

9

3.2 Validation of the experimental results of constant-load swilling test proposed by Guiras, H. 1996

The soil used in this studu is a Ca2+ bentonite reconstituted in laboratory. Guiras, H.(1996), presents the experimental results of

the constant load swelling oedometer test. The parameters introduced in this model are presented in table 2 and 3:

Composition of Ca2+

bentonite : Values

Specific gravity (kN/m3) 27,4

Liquid limit Wl (%)

Plasticity index Ip (%) 100

Shrinking limit Wr (%) 8 - 13

Clay <2mm (%) 89

Silt (%) 3

Sand (%) 8

Monmorillonite (%) 80

Unified soil classification CL

Table 2 : Composition of Ca2+ bentonite

Sample Designation: Path A Path B

Height (cm) 1 2,39

Diameter (cm) 7 7

Water content (%) 22,5 23,5

Dry density (kN/m3) 14,5 14,5

Suction (*) (kPa) 1300 3900

Table 3: Parameters of samples

3.2.1 Constant load swelling oedometer test

In order to determine the coefficients sm2 and

wm2 , wetting path on one dimensional free swell test is carried out (Guiras H.,

(1996), Skandaji H. & al.(2000)). To compare the kinetic of swelling rate, applying incremental suction and wetting path to saturated state, two paths are carried out. Figure 11 shows the stress path in (σ-ua) and (ua-uw) plane. All samples started

initially from the same initial suction (near s=3900 kPa). For the first path (A), suction were changed stepwise in order 1,2,3, 4

and 5 from initial suction of si= 3900 kPa to full saturation (s=0 kPa). The increments of suction are: 3900(1)-1300(2)-1000(3)-

400(4) –200(5) and 0(6)kPa. A vertical stress is equal to 11 kPa corresponding to the self weight of loading cap. At each

suction step, stresses is kept constant, however, when all measuring components (soil deformation and water volume change)

had reached equilibrium (near 15 days), it proceeded to next step.

Contrarily, for the other path, sample is soaked on the two faces with distilled water using casagrande oedometer cell. The

wetting path is directly carried from initial suction 3900 kPa to zero (path B). In this case, the humidification is not controlled

and the exchange of water is made freely until vertical deformation is stabilized. We suppose that the sample is fully saturated

when we reach stabilities (near one month). Note that for the second path, the self weight of loading cap is 5 kPa. We suppose

that there is no difference to these two stresses, if compared to swelling pressure or insitu loading.

Figure 11: Different wetting paths under not and controlled suction

0

1000

2000

3000

4000

5000

0 5 10 15 20

Vertical net stress v-ua (kPa)

Su

ctio

n

s =

ua-

u w (

kP

a)

Path A

Path B

(1)

(2)

(3)

(4)(5)

(6)

10

The permeability function used here is given by Jamei M. et al (2005).

Coefficient Functions Values of parameters

Kw o

w n

w

w

kk

u

g

b

a

e

u1+a

ko = 0,45 10-12(m/s)

b = 8 10-4

a = 10-4

n = 2,15

Table4 : The values of the coefficient of permeability

In the following we proposed to simulate the second level of path A. In the second level matric suction decreases from

1300KPa to 1000KPa.

Figure 12 shows measured and computed volume change versus time curves. A comparison of finite-volume computed and

measured curves show a good agreement for the full duration of the test for the specimen.

Figure 12: computed and measured volume change for constant-load oedemeter test

Computed finite volume simulation of the path B was proposed in the following. Figure 13 shows computed volume change

versus time; figure 14 the computed permeability coefficient versus matric suction and figure 15 the computed matric suction

profiles for constant-load oedometer test.

Figure 13: Computed volume change for constant-load oedemeter test (path B)

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16

vo

lum

e c

han

ge %

Time (day)

dVw/V0:Experimental points dVw/V0 :Finite volume simulation

dVv/V0 Exprimental points dVv/V0:Finite volume simulation

11

Figure 14: Computed permeability for constant-load oedemeter test (path B)

Figure 15: Computed matric suction profiles for constant-load oedometer test (path B)

Summary and conclusions

A three dimensional theoretical model has been formulated to describe the matric suction and volume-change behavior during

various swelling oedometer tests. The model is based on the equilibrium equation, the constitutive equations for unsaturated

soils, and the continuity equation for the pore fluids. The transit water flow process is coupled with the soil volume change process in this model. The model can be used to describe the volume-change behavior, both swelling and collapse, and total

stress and matric suction development of an unsaturated soil during an oedometer test.

A tow dimensional finite volume formulation has been proposed as a numerical solution for the theoretical model. A computer

program was developed based on the proposed formulation.

The proposed theoretical model was used to simulate the results from free-swell, constant-volume, and constant-load

oedometer tests. In general good agreement was found between the computed and measured values of volume change, total

stress, and matric suction.

The finite volume method is used for the conservative equations to save time in the simulation (i.e., 30 second max per

simulation of oedometer test).

The work presented in this paper concentrated on a theoretical simulation of the swelling-pressure measurements commonly

performed in a laboratory. However, a three-dimensional model can be developed on the basis of this model. This model would greatly assist in the prediction of in situ total heave or collapse, the in situ swelling pressure, and the rate of

swell or collapse.

References

Alonso E.E., Gens A. and JOSA A. (1992) ‘A unified model for expansive soil behaviour’. 7th int. conf. on expensive soils,

Dallas U.S.A., vol.1, p.24-29.

12

Alonso E.E. and Gens A. (1993) ‘On the mechanical behavior of arid soils’. Proc. Of the 1st int. symp. on eng.

Characteristics of arid soils, London, Fookers & Parry eds, p.173-2.5.

Alonso E.E, LORET A., Gens A. and YANG D.Q. (1995) ‘Experimental behaviour of expansive double-structure clay’ 1st

int. conf. of unsaturated soils, 1995, Paris, vol.1, p.11-16.

Börgesson L. (1985) ‘Water flow and swelling pressure in non-saturated bentonite-based clay barriers’. Engineering

geology (314),p.229-237.

Buckingham E. 1907 ‘Studies of the movement of soil moisture’. U.S. Department of Agriculture, Bureau of Soils, Bulletin

No.38.

Childs E.C., and Collis-George N.,(1950) ‘The permeability of porous material’. Proceeding of the Royal Society of

London, A,201: 392-405. Fredlund D.G., and Rahardjo H. (1993) ‘Soil mechanics for unsaturated soils’ Johon Wiley & Sons, New York.

Freeze R.A., and Cherry J.A. (1979) ‘Groundwater’. Prentice Hall inc., Englewood Cliffs, N.J.

Gens A. and Alonso E.E. (1992) ‘A framework for the behaviour of unsaturated expansive clays’. Can. Goetch. J.

29.p1013-1032.

Guiras H. (1996) ‘Déformabilité des sols argileux non saturés : Etude expérimentale et application à la modilisation’. Thése

de doctorat INPL France.

Ho D.Y.F., Fredlund D.G., and Rahardjo H. (1992) ‘Volume change indices during loading and unloading of unsaturated

soil’. Can. Goetch. J. 29.p.195-207.

Jamei M., Sebri K., Saada A., and Guiras H. (2005) ‘Appoche par volumes finis pour la résolution d’une equation de

diffusion dans une argile gonflante non saturée’. 30ième journées du GFHN, Tunis 21-22 novembre.

Masters I., Rees I., and Lewis R.W. (2004) ‘Advanced programming methods and data structures for computational modeling using edge-based finite volume methods’. Int. J. for Numer Anal Meth. Geomech. 28.p.1521-1532.

Richards L.A. (1931) ‘Capillary conduction of liquids through porous medium’. Journal of Physics,1.p.318-333.

Shuai F., and Fredlund D.G. (1998) ‘Model for simulation of swelling-pressure measurement on expansive soils’. Can.

Geotch. J. 35,p.96-114.

Skandaji H., Mansouri F., Jamei M., and Touil O. (2000) ‘A new model to predict the swelling parameters on deformable

unsaturated soil under controlled suction’. Geneng 2000.