International Journal of Geosciences, 2011, 2, 204-213
doi:10.4236/ijg.2011.23022 Published Online August 2011 (http://www.SciRP.org/journal/ijg)
A Flexible Model for Moistu re-Suction Relationship for
Unsaturated Soils and Its Application
Nadarajah Ravichandran, Shada H. Krishnapillai
Civil Engineering Department, Clemson University, Clemson, USA
Received May 12, 2011; revised June 21, 2011; accepted July 25, 2011
Abstract
The mathematical equation for the moisture-suction relationship also known as soil water characteristic
curve (SWCC) is one of the constitutive relations necessary for the computational modeling of deformation
and flow problems of unsaturated soil using the finite element method. In this paper, a new empirical equa-
tion for the SWCC is developed that incorporates the actual air-entry suction and the maximum possible suc-
tion of the soil as input parameters. The capability of the new model is investigated by fitting the experimen-
tal data for twelve different soils that includes sands, silts, and clays. The model fits the experimental data
well including in high suction range which is one of the difficulties observed in other commonly used models
such as the Brooks and Corey, van Genuchten, and Fredlund and Xing models. The numerical stability and
the performance of the new model at low and high degrees of saturations in finite element simulation are in-
vestigated by simulating the dynamic response of a compacted embankment and the results are compared
with similar predictions made using widely used SWCC models.
Keywords: Soil-Water Characteristic Curve, Unsaturated Soils, SWCC for Low Degree of Saturation,
Moisture-Suction Relationship, Comparison of Soil-Water Characteristic Curves
1. Introduction
In recent years, the importance of unsaturated soil me-
chanics and its applications in the design and construc-
tions of safe and economical geotechnical engineering
structures is realized by not only the academic research-
ers but also by practicing engineers. The slope failure
after rainfall and the shrinking and swelling of clays are
two of the many examples that require a better under-
standing of unsaturated soil mechanics principles. In
contrast to saturated soil, variation in soil moisture con-
tent will have a great influence on the load bearing ca-
pacity, settlement and flow behavior of unsaturated soils.
For example, a drop in moisture content will result in an
increase in soil stiffness and strength, a decrease in soil
compressibility and a decrease in water permeability.
The increase/decrease in mechanical and flow behavior
depends upon the type of soil. For example, clayey soil
exhibits a greater change in stiffness and compressibility
compared to sandy soils for the same moisture content
change. From numerous observations, it is clear that the
amount of water present in the soil, measured in any
form such as degree of saturation or water content, has
great influence on unsaturated soil behavior.
Unsaturated soil is a three phase porous media con-
sisting of three bulk phases: solid skeleton, water, and
pore air. In addition to these three bulk phases, there ex-
ist three interfaces: solid-water interface, solid-air inter-
face, and water-air interface. Of the three, the water-air
interface (contractile skin) that does not exist in either
saturated or dry soil influences the unsaturated soil be-
havior (i.e., unsaturated soil behavior differs from satu-
rated soil not only because of the presence of air phase
but also by the presence of the water-air interface). The
contractile skin maintains the pressure balance between
water and air phases. The difference between the air
pressure and water pressure is known as matric suction,
which is a function of degree of saturation (amount of
water) and other properties such as void ratio, void dis-
tribution, particle size distribution and initial density.
The major difference between unsaturated and saturated
soil mechanics is the influence of matric suction on its
behavior, that is to say the mechanical and flow charac-
teristics of unsaturated soil are affected by matric suction
[1].
Numerical modeling of mechanical and flow problems

N. RAVICHANDRAN S. H. KRISHNAPILLAI205
in unsaturated soil requires a set of governing differential
equations that represents the physics of the problem, a
stress-strain relationship that relates the deformation of
the soil body to the applied load and a moisture-suction
relationship. The moisture-suction relationship not only
affects the flow but also deformation characteristics of
the soil body because suction is one of the two stress
state variables widely used in the deformation analysis of
unsaturated soils. These mathematical relationships must
represent the true physics of the problem (material,
boundary condition and loading) more closely for accu-
rate predictions. It is obvious that mathematical models
will become complex when they represent the true be-
havior more closely. The complexity can arise in the
form of increased number of model parameters and/or
complex formulations. The complex mathematical equa-
tions that perform well in single element test or small
well defined problems may become numerically unstable
when real world problems are simulated with realistic
boundary and loading conditions. Also, when the number
of model parameters increase, some of them may not
have physical meaning and/or difficult to determine from
simple laboratory results. This paper focuses on devel-
oping a flexible and numerically stable moisture-suction
relationship for unsaturated soil.
The new empirical equation for the moisture-suction
relationship presented in this paper seems capable of
matching the measured data of various soils over a wide
range of degree of saturation (near dry to fully saturated
conditions). It can also be used either with residual water
content (the lower bound value for the water potential) or
with a maximum suction value (the upper bound value
for the suction) at near dry conditions. The capability of
the new model is verified by matching the experimental
data of twelve soils that includes sands, silts, and clays.
The performance and the stability of the proposed mois-
ture-suction model in the finite element modeling of un-
saturated soils is investigated by simulating the dynamic
behavior of a compacted embankment subjected to
earthquake shaking at a low degree of saturation.
2. Review of Widely Used Soil-Water
Characteristic Curves
The amount of water present in the soil can be expressed
in various forms such as degree of saturation (S), volu-
metric water content, or gravimetric water content. Also,
the SWCC for a given soil can differ depending upon the
wetting or drying process used to vary the moisture con-
tent in the soil sample. The portion of the SWCC ob-
tained by the wetting a dry sample is called the primary
wetting curve. Similarly, the curve obtained by drying a
wet sample is called the primary drying curve. The pri-
mary drying curve always exhibits higher suction com-
pared to the wetting curve (Figure 1) for a given degree
of saturation [2-9]. Although the wetting and drying
curves differ significantly and show hysteretic loops, most
of the mathematical models for the moisture-suction re-
lationship are represented by a single equation [10-12]
for easy use in finite element modeling.
2.1. Factors Influencing the SWCC and
Development Strategies
Various mathematical models have been developed to fit
the measured moisture-suction relationship of natural
soils using empirical, statistical and microscopic proce-
dures [10-16]. All of these models confirm an inverse
proportional relationship between the degree of satura-
tion and suction. This inverse relationship can be ex-
plained with the fundamental meniscus theory as follows.
When the degree of saturation increases, the radius (Rs)
of the meniscus will also increase. When Rs increases, the
pressure difference between the pore air pressure and the
pore water pressure (suction) will decrease as seen in
Equation (1).

gl 2
s
s
T
= ppR   (1)
where ψ is the suction, pg is the pore air pressure, pl is
the pore water pressure, and Ts is the surface tension.
The air-entry suction and the pore size distribution in-
dex are two of the basic parameters incorporated in most
of the widely used SWCC models and the effect of these
two basic properties are represented by two parameters a
and n, respectively [10-12]. In addition to these two pa-
rameters, Kawai et al. [17] showed that the initial void
ratio (e) affects the air-entry suction and proposed an
inverse relationship between a and e as shown in Equa-
tion (2).
2.51
160ea  (2)
Another study by Vanapalli et al. [18] showed that the
initial degree of saturation has significant influence on
the shape of SWCCs at lower suction range. For example,
a higher initial S makes the curves steeper, whereas the
effect of initial S is insignificant at higher suction values.
Through a series of experiments on undisturbed samples
of completely decomposed volcanic soil with net normal
stress levels of 0, 40 and 80 kPa, Ng and Pang [2]
showed the effect of net normal stress to be insignificant
for some soils. Fredlund and Rahardjo [1] and Vanapalli
et al. [18], studying the effect of total stress on the
SWCCs, found that the air-entry suction parameter a
increases with increasing equivalent pressure. The influ-
ence at high suction was investigated by Vanapalli et al.
[18], which showed that the SWCC exhibits similar be

N. RAVICHANDRAN S. H. KRISHNAPILLAI 206
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction
(
k
P
a
)
0
20
40
60
80
10
0
Deg
r
ee of satu
r
ation (%)
Wetting phase
Drying phase
Residual saturation
zone
Desaturation zone
Capillary
saturation
zone
ψ
air entry (aev)
Figure 1. Typical SWCC with different regions of saturation.
haviors at high suctions (20,000 - 300,000 kPa)
even if all other parameters differ, i.e . effect
of other factors being insignificant at high
suction. Recent studies sows that the initial density of
the soil also affects the SWCC [1]. Recent research pub-
lication shows that the uncertainty in the unsaturated soil
properties should also be taken into account for accurate
modeling of SWCC [19]. Although there are numerous
SWCC models available in the literature, we have used
the B-C [10], v-G [11] and F-X [12] models for further
investigation before developing our new model and for
comparing the applicability performance this model.
The B-C model shown in Equation (3) is one of the
earliest models widely used in many applications in-
cluding finite element codes. It is a non-smooth model
consisting of two parts. An inverse power law is used
beyond air-entry suction and 1.0 is used within the air-
entry suction for the effective degree of saturation. The
sudden change at the air-entry value may cause numeri-
cal instabilities especially when taking derivatives of the
curve as part of the development of general governing
equations for the dynamics of unsaturated soils at low S.

e1
n
if ψ a
Saif ψa








(3)
where is the effective degree of saturation given by
e
S

e1
rr
SSS S ,

is the suction and n is a fit-
ting parameter related to the pore size distribution index
of the soil, is the degree of saturation, r is the
residual saturation. Another widely used model is the
v-G model given in Equation (4) that provides a single
smooth equation for the entire range of S introducing
another fitting parameter m.
S S

e1
1 ()m
n
S
a

 (4)
where the fitting parameter m is related to the symmetry
of the model that increases the flexibility in fitting ex-
perimental data of many different soils. The other widely
used SWCC is the F-X model given in Equation (5) that
incorporates a correction factor,

C

. This correction
factor forces the model to pass through a prescribed suc-
tion value of 106 kPa at near dry condition while the B-C
and v-G models predict infinity that is considered unre-
alistic.


 
e*
ln e
m
n
C
S
a




(5)

ln 1
11000000
ln 1
ψ
Cr
Cψ
Cr




 



where is similar to but r is set to zero (used
with maximum suction concept). The variable e in Equa-
tion (5) is the natural logarithmic constant, and the pa-
rameter rin the correction factor is a parameter related
to the residual water content. Further investigation on the
F-X model by Leong and Rahardjo [20] revealed that the
correction factor
e*
S
C
e
S S


Cψ significantly affects the initial
portion of the curve (capillary saturation, desaturation
zones shown in Figu re 1 when the is relatively low.
r
C
2.2. Capabilities of the Widely Used Models and
Model Parameter Cali bration
Our evaluation of the existing SWCC models is based
upon 1) their capability to capture the moisture-suction
relation over the entire range of degrees of saturation,
especially in the low saturation ranges, and 2) their per-
formance in a fully coupled finite element simulation of
unsaturated soil. In addition to collecting information
from previously published papers [20,21], the authors
performed extensive studies on the influence of the mod-
el parameters in each of the widely used models. It
should be noted here that the authors’ intention is to use
the model in a fully coupled finite element simulation of
unsaturated soil. Therefore, the authors’ opinion about
the existing models may differ from others.
Although the elemental B-C model has two fitting pa-
rameters, it fails to fit the measured data in high suction
range. This failure is due to the fact that the SWCC
equation has no inflection point that is observed in many
of the measured SWCCs. The absence of this inflection
point limits its flexibility to match the experimental data.
In the case of three-parameter models such as v-G and
F-X models, the fitting parameters are not independent of
each other resulting in a random procedure to match the

N. RAVICHANDRAN S. H. KRISHNAPILLAI207
experimental data by adjusting one or more of these pa-
rameters. For example, n can be increased by increasing
a and m simultaneously. When m increases, the slope of
the curve will also slightly increase, resulting in the
curve moving along the suction axis towards low suction
values. The other disadvantage of these three-parameter
models is the association of physical meaning to the pa-
rameter a. Although it is referred to as the air-entry suc-
tion related parameter, a in these models is generally
larger than the actual air-entry value of the soil. The val-
ue of a in these models falls within the desaturation zone,
although the initial point of the desaturation zone is de-
fined as the air-entry value. In the case of the F-X model,
a selection of relatively smaller r values also will af-
fect the initial portion of the curve. Such influence will
result in multiple combinations of model parameters that
are undesirable for finite element application in which
these parameters are coefficients of important terms
when derivatives are calculated.
C
The moisture-suction relationship of unsaturated soils
is observed by assuming either a reasonable lower bound
value for the water potential (residual water content) or
upper bound value for the suction (maximum suction).
Though the F-X model uses a maximum suction of 106
kPa, it has no theoretical basis and is expected to vary
from soil to soil. Although a maximum possible suction
of 106 kPa is shown as the theoretical maximum suction
[12], incorporating the maximum suction as part of the
model will increase its flexibility in fitting the measured
data well especially in low saturation ranges. In addition,
researchers may wish to use both lower bound and upper
bound concepts in a single numerical simulation with
multiple unsaturated soil layers (i.e. in real boring logs
for geotechnical engineering projects). For example,
there might be a need to specify residual water content
for clayey soils while preserving a maximum suction for
sandy soils in a single simulation. Therefore, a model
that can be used either with residual water content or
maximum suction is desirable. Also the fitting parame-
ters should be independent of each other so that a single
set of parameters can be obtained from calibration. A
realistic SWCC model must also include the actual air-
entry suction as a model parameter instead of a related
parameter. In this paper, a new flexible SWCC model is
presented that seems to have eliminated the shortcomings
found in the widely used models and includes desirable
features discussed above.
3. Proposed Equation for Moisture-Suction
Relationship (SWCC)
The new model is developed to solve the aforementioned
practical issues and to provide a SWCC model for use
with either residual water content concept or maximum
suction at near dry conditions. This new empirical equa-
tion is given in equation (6). The factors that influence a
SWCC and the modelling strategies described in Section
2 were closely followed to develop the new equation.
Also, one can understand by comparing the new and F-X
models that the proposed model is similar to the F-X
model in terms of the framework but differs significantly
with new parameters and correction function.


e*
0.5
1
1ln1 n
air entry
N
S
ma
m










(6)

0.5
1 1
r
rmax
N
NN






where a, n, m, and Nr are the fitting parameters; ψair-entry
is the actual air-entry suction, a is a non-dimensional
parameter that represents the ratio between the air-entry
suction and the suction at inflection point in the curve.
The parameter n is related to the pore-size distribution of
the soil, and m is related to asymmetry of the model in
moisture-suction plane similar to that of the v-G and F-X
models.

is the suction, ψmax is the maximum suction
or suction at near dry conditions, and Nr is a number re-
lated to residual water content. Though the function
()N

is similar to the function ()C

in the F-X model,
()N

does not affect the initial portion (portion in the
low suction range) of the curve. The effect of ()N

over
a range of Nr value (0 to 10) is discussed in the subse-
quent sections. The Equation (7) incorporating this re-
sidual water content (θr) is expressed below.

0.5
0.5
1 1
=
1
1 ln1
r
rrmax
sr n
air entry
N
N
m a
m


 












(7)
If the maximum suction concept is to be considered,
the Equation (7) can be simply reduced by setting θr to
zero (θr = 0) together with a maximum suction (ψmax) and
a calibrated Nr value. Conversely, if residual water con-
tent of the soil has to be considered, the factor Nr can be
set at zero (Nr = 0). When Nr is equal to zero, the func-
tion ()N

will be unity thusly reducing it to an Equa-
tion (8) that can handle the residual water content con-
cept.

0.5
1
=
1
1 ln1
r
sr n
air entry
m a
m

 









(8)

N. RAVICHANDRAN S. H. KRISHNAPILLAI 208
3.1. Derivatives of the New SWCC Model
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction (kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
a
= 1.0
a
= 2.5
a
= 7.5

aev
= 10 kPa
n = 2.5
m
= 2.5
Nr = 1

max
= 10
6
kPa
a
One of the instances in which the derivative of the mois-
ture-suction relation is required in the finite element
simulation of unsaturated soil is in the mass balance
equation for the water and air phases. The necessary de-
rivatives of the new model are given below in Equations
(9) and (10).

d
dd
sr
S  d


 (10)
3.2. Fitting Parameters in the New Model
Figure 2. Influence of the parameter a in the new SWCC
model.
Understanding the role of each parameter in the analyti-
cal model is important for adjusting these parameters to
fit the experimental data well to obtain the best set of
parameters. A detailed discussion based on the paramet-
ric studies performed upon the role of each parameter is
presented below.
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction (kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
n
= 1.0
n
= 2.5
n
= 5.0

aev
= 10 kPa
a = 2.5
m = 2.5
Nr = 1

max
= 10
6
kPa
aψ
aev
ψ
aev
n
n
3.2.1. Role of Parameter a
The parameter a is the ratio between the actual air-entry
suction of the soil and the suction at the inflection point of
the SWCC. The effect of parameter a on the shape of the
SWCC in the proposed model is shown in Figure 2. In the
case of B-C, v-G and F-X models, the curve can be
shifted along the suction axis by increasing a. However,
in the proposed model, a must be first adjusted until the
initiation point of the desaturation zone (Figure 1)
matches the air-entry suction (ψair-entry) of the soil. For
sandy soils, since the slope of the curve is steeper, the
value of a will be relatively small. From authors experi-
ence, a ranges between zero and two. For clayey soils the
slope is mild and the value of a is higher than five. The
value of a for silty soils, generally, falls between that of
clay and sand (approximately between 1 and 5).
Figure 3. Influence of the parameter n in the new SWCC
model.
increases the Sat a given suction will increase if the suc-
tion is greater than aψair-entry, and the Swill decrease if
the suction is less aψair-entry.
3.2.3. Role of Parameter m
Influence of m in the shape of the SWCC is shown in
Figure 4(a). As seen here, m in the proposed model does
not affect the curve when the suction is within 0 and
aψair-entry. This indicates that the parameter m does not
alter the shape of the curve that may require re-adjust-
ment of the parameters a and n. On the other hand, the
3.2.2. Role of Parameter n
The influence of the parameter n in the shape of the
SWCC is shown in Figure 3. As seen here, n changes the
slope of the curve about the inflection point. When n




 
d1
=
d1
1 ln1
11
21 1 ln1
m
n
air entry
n
air entry
nn
air entryair entry
S
ma
m
na N
dN
dψama
mm

 

 




















 



(9)

N. RAVICHANDRAN S. H. KRISHNAPILLAI 209
parameter m in the v-G and F-X models alters the slope
of the curve and the predicted air-entry suction as seen in
Figure 4(b). This requires readjustment of a and n to fit
the experimental curve. This not only requires a tedious
calibration procedure, but results in multiple possible
combinations of model parameters and difference in fi-
nite element simulation results for the same soil.
3.2.4. Role of Parameter Nr and ψmax
The influence of the parameter Nr, a parameter in the
correction factor in the new model, in reaching the speci-
fied maximum suction is shown in Figure 5. As men-
tioned previously, one of the advantages of the proposed
model is the use of maximum suction as a model pa-
rameter in which maximum suction must be obtained
from experimental results and used in the modeling. In
the example shown in Figure 5, the maximum suction of
106 kPa, a theoretical value obtained based on thermo-
dynamic principles for any soil [4], is used. As seen there,
because the parameter Nr does not affect the initial por-
tion (capillary saturation, desaturation zones) of the
curve, the effect of Nr on the other model parameters is
insignificant.
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction
(
k
Pa
)
0
20
40
60
80
10
0
Deg
r
ee of satu
r
ation (%)
m
= 1
m
= 2
m
= 8

aev
= 10 kPa
a = 2.5
n = 2.5
Nr = 1

max
= 10
6
kPa
m
(a)
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction
(
k
Pa
)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
In both v-G, F-X models
m
(b)
Figure 4. Influence of the m in the new and two other
widely used SWCC models.
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction
(
k
Pa
)
0
20
40
60
80
10
0
Deg
r
ee of satu
r
ation (%)
Nr = 0 ( N (  ) = 1)
Nr = 0.5
Nr = 2.5
Nr = 7.5

aev
= 10 kPa
a = 7.5
n = 1.5
m
= 2.5

max
= 10
6
kPa
Nr
Figure 5. Influence of the Nr in the new SWCC model.
As shown in Equation (8), the correction factor ()N

must be 1.0 to use the residual water content concept.
Setting Nr equals to zero (given that

is not equal to
max

) can yield this correction factor. Based upon our
experience, Nr varies within 1.0 and 5.0 for any soil.
Figure 6 shows the use of ψmax.
4. Predictive Capability of the New Model
The capability of the new model in predicting the mois-
ture-suction relation for twelve different soils that in-
clude sands, silt and clays is investigated. Due to limita-
tions in page number, comparisons for six soils are pre-
sented in this paper. The calibrated SWCCs together with
the model parameters are presented in Figure 7, in which
the ψair-entry is denoted as ψair. The predicted SWCC
without the correction factor

N

is also presented
for comparison purposes. Figures 7(a) and (b) show the
calibration of SWCC model for a Superstition sand (data
[22]) and Lakeland sand (data – [23]), respectively. Fig-
ures 7(c) and (d) show the calibration of SWCC model
parameters for Touchet silt loam (data – [10]) and Botkin
silt (data – [18]), respectively. Figures 7(e) and (f) show
the calibration of SWCC model parameters for
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
Suction (kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)

max
= 1 x10
5
kPa

max
= 1 x10
6
kPa

max
= 1 x10
7
kPa

aev
= 10 kPa
a = 7.5
n = 1.5
m
= 2.5
Nr = 2
Figure 6. Effectiveness of the ψmax in the new SWCC model.

N. RAVICHANDRAN S. H. KRISHNAPILLAI 210
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction (kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
Experimental
Nr = 1
Nr = 0

aev
= 2.25 kPa
a = 1.35, n = 7.25
m = 1

max
= 10
5
kPa
Superstition sand
(a)
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction(kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
Experi men tal
Nr = 1
Nr = 0

aev
= 2 kPa
a = 1.5, n = 7
m = 0.415

max
= 10
5
kPa
Lakeland sand
(b)
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction (kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
Experim ental
Nr = 1
Nr = 0

aev
= 7 kPa
a = 1.35
n = 7.5, m = 1.35

max
= 10
5
kPa
Touchet silt loam
(c)
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction(kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
Experimental
Nr = 3.5
Nr = 0

aev
= 15 kPa
a = 3.25
n = 2
m = 1.75

max
= 10
6
kPa
Botkin silt
(d)
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction (kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
Experim ental
Nr = 4.1
Nr = 0

aev
= 10 kPa
a = 5.7, n = 2
m = 0.375

max
= 10
6
kPa
Speswhite kaolin
(e)
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Suction (kPa)
0
20
40
60
80
100
Deg
r
ee of satu
r
ation (%)
Experimental
Nr = 1.25
Nr = 0

aev
= 1000 kPa
a = 18
n = 1, m = 1.1

max
= 10
6
kPa
Regina clay
(f)
Figure 7. Calibration of model parameters for various soils for both residual water content and maximum suction concepts.
Speswhite kaolin (data [24]) and Regina clay (data [18]),
respectively. It should be noted that the experimental
moisture-suction data for these soils are unavailable for
the full degree range of saturation (0 - 100%). The best
estimates for both the air-entry suction (ψair-entry) and
maximum suction (ψmax) for each soil are selected based
upon the variation of available experimental data.
These results show that the new model is effective and
flexible enough to fit the experimental data. The number
Nr in the new model can be chosen between 1 and 5 for
any soil and the correction factor

N

does not affect
the initial portion of the curve. The new model can be
effectively used with either the residual water content
concept or with a maximum suction value at near dry
conditions. In addition to eliminating this deficiency, the
new model includes all the advantages of the B-C, v-G
and F-X models. The performance of the new SWCC
model in finite element simulations is investigated by
simulating a compacted embankment at various initial
degrees of saturation subjected to earthquake shaking.

N. RAVICHANDRAN S. H. KRISHNAPILLAI211
5. Performance of the New SWCC in Finite
Element Simulations
One of the primary applications of mathematical repre-
sentation of soil-moisture relation in geotechnical engi-
neering is in the finite element modelling of unsaturated
soils subjected to various loadings and boundary condi-
tions. In general, mathematical equations become com-
plex when they represent the true behaviour while the
complex equations simultaneously limit the application
of such equations in boundary value problems via nu-
merical instabilities. This statement is true for governing
differential equations that are spatially discretized to
form finite element equations, stress-strain relationship
and moisture-suction relationship. Because of the fact
that different elements in a problem domain (finite ele-
ment mesh) can be at different levels of degrees of satu-
ration when subjected to flux and traction boundary con-
ditions, a single moisture-suction model should have the
flexibility and stability to be used in finite element simu-
lations.
5.1. Finite Element Simulation Tool for
Unsaturated Soils
Although complete and partially reduced finite element
formulations are available [25], a simplified finite ele-
ment formulation for unsaturated soil [24] is used in this
study. The simplified formulation is free of numerical
instabilities arise from other sources for the problems
analyzed in this paper [25] and can easily capture if there
is any numerical instabilities occur due to proposed
SWCC. The complete governing differential equations
representing the physics of unsaturated soil derived using
mass balance, momentum balance and laws of thermo-
dynamics was simplified by neglecting the relative ac-
celerations and velocities of the pore air and water
phases. The corresponding finite element equations are
solved considering the solid displacement as the primary
nodal unknowns, and the element fields (e.g. water pres-
sure and air pressure) are calculated using the mass bal-
ance equations. This simplified formulation represents
the undrained soil condition since the relative accelera-
tions and velocities are neglected. Therefore, the change
in degree of saturation in a finite element occurs due to
the deformation (volumetric deformation) of the solid
skeleton (a finite element). When the degree of saturation
changes due to deformation, corresponding matric suc-
tion is calculated using the SWCC. The reason for se-
lecting the reduced formulation is to isolate the numeri-
cal instabilities arising from other sources such as gov-
erning equations (as in the case of complete formulation).
Also, because the permeability coefficient of water in
unsaturated state is much smaller than in saturated soil
states, we may assume that unsaturated soils behave in
an undrained condition especially under earthquake
shaking.
5.2. Simulation Results and Discussion
To show the influence of various SWCC models, the
dynamic behavior of a compacted earthen embankment
made of Speswhite Kaolin subjected to earthquake shak-
ing at the base was simulated using the finite element
code described previously. The finite element mesh
shown in Figure 8 consists of 292 quadrilateral elements.
The base of the embankment was assumed to be imper-
meable and fixed in all directions throughout the analysis,
whereas all other sides of the embankment were assumed
to be traction free. The embankment was numerically
shaken with the acceleration time history shown in Fig-
ure 9, and the stress-strain relationship of the soil was
assumed to be linear elastic. Again, the reason for se-
lecting the simplest stress-strain relationship is to isolate
the numerical instabilities arising from the complex
elasto-plastic constitutive model. The linear elastic
model parameters and other soil properties are shown in
Table 1.
Simulations were performed using all four models,
B-C, v-G, F-X and our proposed model (S-R). The
SWCC model parameters were calibrated against the
experimental data published by Sivakumar [23]. The
40 m
E1
185 m
96 m8m
N1
Figure 8. Finite element mesh of the compacted embankment.
051015 20 25 30
Ti
m
e (s)
-0.4
-0.2
0
0.2
0.4
Base accele
r
ation (g)
Figure 9. Base acceleration time history.

N. RAVICHANDRAN S. H. KRISHNAPILLAI 212
Table 1. Linear elastic model parameters for Speswhite
kaolin.
Properties Value
Solid grain density Mg/m3 2.62
Liquid density Mg/m3 1.0
Gas density (×10–3) Mg/m3 2.1
Bulk modulus of liquid (×106) kPa 2.2
Bulk modulus of gas kPa 101.3
Viscosity of liquid (×10–6) kPa·s 1.0
Viscosity of gas (×10–8) kPa·s 1.0
Young’s Modulus (×105) kPa 0.3
Poisson’s ratio 0.2
calibrated model parameters for the B-C model are: a =
17 kPa, n = 0.18, irreducible S = 0; for the v-G model: a
= 0.058 kPa-1, n = 2.85, m = 0.063, irreducible S = 0; for
the F-X model: a = 28 kPa, n = 1.65, m = 0.365, Cr =
5,000 kPa ; for the new model: ψair-entry = 10 kPa, a = 5.7,
n = 2, m = 0.375, Nr = 4.1, and ψmax = 106 kPa. In the
new model, the air-entry suction (ψair-entry) and the
maximum suction (ψmax) values were selected by looking
at the trend of the experimental curve.
As mentioned previously, all the models predict iden-
tical responses when the degree of saturation falls in the
mid-range and show significant differences or become
inapplicable in low and/or high degree of saturation
range. Therefore, in this paper, the initial degree of satu-
rations of 10%, 25% and 90% were selected for the finite
element simulations. The S corresponding to the residual
water content (irreducible S) is set to be zero in the B-C
and v-G models to simulate identical soil condition in all
four models. The predicted incremental matric suction
time histories in element E1 (Figure 8) are presented in
Figure 10. As shown in the figures, while the initial S
increases the initial suction and the suction variation due
to external loading decrease. The comparison study
shows that the results predicted using the v-G, F-X mod-
els are close for an initial S of 90%, the B-C model pre-
dicts a slightly lower suction variation as shown in Fig-
ure 10(b). As seen in Figure 10(a), significant differ-
ences are observed when the initial degree of saturation
is 10 %. At 10% initial S, the new and the F-X models
also predict close responses, but the other two models
could not be used in this analysis. The accuracy of the
predicted response could not be verified due to the lack
of experimental results for the finite element problem
shown. However, this finite element simulation study
shows that the new model is numerically stable and can
be effectively used to capture the moisture-suction varia-
tion with wide range of initial S, especially at low de-
grees of saturation.
01234567891
Ti
0
m
e (s)
-10
-5
0
5
10
Inc
r
emen
t
al suc
t
ion (MPa)
B-C
v-G
F-X
S-R
(a)
dos = 10%
01234567891
Ti
0
m
e (s)
-3.0
-1.5
0.0
1.5
3.0
Inc
r
emen
t
al suc
t
ion (kPa)
B-C
v-G
F-X
S-R
(b)
dos = 90%
Figure 10. Predicted suction variation with different initial dos
for the dynamic analysis of the embankment (dos-degree of
saturation).
6. Conclusions
The proposed empirical equation for moisture-suction
relationship in unsaturated soil seems flexible enough to
match the experimental data at low degrees of saturation
and numerically stable in finite element simulations of
dynamic problems. The new model can be used either
with a residual water content (lower bound value for the
water potential) or with a maximum suction value (upper
bound value) for dry case. If the maximum suction and
air entry suctions are available for a soil, this data can be
directly used in the proposed model. The performance of
the new model is verified by fitting the experimental data
of twelve different soils. The calibration results show
that the new model can be successfully used to model
various types of soils over a wide range of degree of
saturation without any numerical difficulties. The factor


N

that is introduced to ensure the above capability
is effective and numerically stable. Unlike the range of
correction factor, Cr, in the F-X, the correction factor,
(Nr), in the new model has a small range (it is recom-
mended to use 1-10). The initial portion of the SWCC is
not affected by the correction factor

N

as opposed
to F-X model and the


N

enables the control of high

N. RAVICHANDRAN S. H. KRISHNAPILLAI 213
suction portion independently.
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