International Journal of Geosciences, 2011, 2, 468-475
doi:10.4236/ijg.2011.24049 Published Online November 2011 (http://www.SciRP.org/journal/ijg)
Improving the Predictive Capability of Popular SWCCs by
Incorporating Maximum Possible Suction
Shada H. Krishnapillai, Nadarajah Ravichandran
Civil Engineering Department, Clemson University, Clemson, USA
Received July 12, 201 1; revised August 27, 2011; accepted October 6, 2011
Abstract
Soil-water characteristic curve (SWCC) that represents the relationship between the soil moisture and matric
suction is one of the important constitutive models required for numerical modeling of unsaturated soils. An
effective SWCC model should be capable of calculating the moisture-suction variation for the entire range of
degree of saturation. Applicability of popular SWCC models such as Brooks and Corey, van Genuchten, and
Fredlund and Xing is limited, especially in low (<20%) degree of saturation range. In this study, all these
models are modified by incorporating maximum suction as one of the model parameters, so that these mod-
els can be effectively used over the entire range of degree of saturation. The Fredlund et al. (1994) perme-
ability function is also modified based on the modification to the Fredlund and Xing SWCC model. The ap-
plicability of the improved models is investigated by calibrating the SWCC of various types of soil and pre-
sented in this paper. Based on this study it can be concluded that the modified models are flexible enough to
fit the experimental data for the entire range of degree of saturation.
Keywords: Unsaturated Soils, Soil Water Characteristic Curve, Permeability Function, Relative
Permeability of Unsaturated Soils, Relative Permeability Using Soil Water Characteristic Curve
1. Introduction
Unsaturated soil is a three phase porous media consisting
of three bulk phases: solid skeleton, water, and pore air.
In addition to these three bulk phases, there exist three
interfaces: solid-water interface, solid-air interface, and
water-air interface. Of the three interfaces, the water-air
interface also known as contractile skin that does not
exist in either saturated or dry soil influences the flow
and mechanical behavior of unsaturated soil. The con-
tractile skin maintains the pressure balance between wa-
ter and air phases. The difference between the air pres-
sure and water pressure is known as matric suction.
1.1. Soil Water Characteristic Curves
The Soil Water Characteristic Curves (SWCC) is a rela-
tionship between the amount of water present in the soil
(moisture) and the suction characteristics of the soil ma-
trix. The amount of water present in the soil can be ex-
pressed in terms of degree of saturation (S), volumetric
water content (θ), or gravimetric water content (w).
Many researchers have identified the factors which in-
fluence the shape of the SWCC and based on that, many
mathematical SWCC models were developed. These
basic soil properties include void ratio, void distribution,
particle size distribution and initial density. Gardner [1],
Brooks and Corey [2], van Genuchten [3], Kosugi [4],
and Fredlund and Xing [5] are some of the notable
mathematical models found in the literature. All these
models confirm an inverse proportional relationship be-
tween S and suction (ψ). This can be explained with the
fundamental meniscus theory shown in Equation (1).
When the S increases, the radius of the meniscus (Rs)
will increase. When Rs increases, the pressure difference
between the pore air pressure and the pore water pressure
(matric suction) will decrease.

2
gl
s
s
T
pp
R
  (1)
where m

is the suction, pg is pore gas pressure, pl is
pore liquid pressure, and Ts is surface tension.
The air-entry value and pore size distribution are two
basic parameters incorporated in widely used Brooks and
Corey (B-C), van Genuchten (v-G), and Fredlund and

469 S. H. KRISHNAPILLAI ET AL.
Xing (F-X) and are denoted by a an d n, respectively.
The B-C model shown in Equation (2) is one of the
basic SWCC models developed with two parameters.
Although this is widely us ed model, it doe s not provide a
continuous mathematical function for the entire range of
S.

 
1
=ln
r
m
n
sr eψa




(2)
where a and n are the fitting parameters. The parameter a
is related to the air-entry suction of the soil and the n is
related to the pore size distribution of the soil, θ is volu-
metric water content, θr is residual water content, and θs
is saturated water content.
The v-G model shown in Equation (3) provides a sin-
gle equation for the en tire range of S. This model has an
additional fitting parameter m, thereby making this mod-
el more flexible compared to the B-C model.


1
1
r
m
n
sr a

 


(3)
where the fitting parameter m is related to residual water
content. In some versions the parameter m is related to n
resulting in only two independent parameters a and n.
Other parameters are same as in the B-C model.
The F-X model is shown in Equation (4). The F-X
model assumes a maximum suction of 1,000,000 kPa at
dry condition, while the B-C and the v-G models assume
infinite value of maximum suction. This maximum pos-
sible suction of 1,000,000 kPa for any soil is based on
thermodynamic principles rather than actual measure-
ment. The F-X model is similar to the v-G model other
than the correction factor ()C

and the logarithmic
function in the equation. The logarithmic function is the
key in the F-X equation (Equation (4)). It keeps the de-
nominator a nonzero value for any suction value. The
corresponding correction function is shown in Equation
(5). This equation can be literally used for degree of
saturation that is below residual value. However, Fred-
lund and Xing [5] suggested another form of the same
model which is shown in Equation (5). This second form
can be used if a residual water content is known.



 
ln m
n
s
C
eψa





(4)
 

6
ln 1
1ln 110
r
r
ψψ
Cψ ψ

  (5)



1
ln
r
m
n
sr ea

 


(6)
where is the suction corresponding to the residu
r
ψ
water contental
r

. and other parameters ar the v-G odel
tion of the permeability is important for
s in
ty of unsatu-
e same as in
m
1.2. Relative Permeability Functions
Accurate evalua
accurate modeling of flow and deformation problem
unsaturated soils. Because the permeabili
rated soil is uniquely influenced by the degree of satura-
tion [6], the soil water characteristic curve (SWCC) of
the soil can be used to predict the permeability coeffi-
cient. The SWCC is a unique constitutive equation in
unsaturated soil that relates the degree of saturation to
the matric suction and it incorporates the basic soil prop-
erties associated with flow such as void ratio, pore size
distribution, void distribution, particle size distribution
and initial density. The major advantage of using the
SWCC is that the moisture-suction relationship can be
easily obtained experimentally than the moisture-per-
meability relationship.
Based on F-X SW CC model, the permeability functio n
shown in Equation (7) was proposed by Fredlund et al.
[6].







ln
d
y
b
y
y
eey
e







ln
d
aev
ry
bsy
y
Keey
e





(7)
where ψ is suction, Kr(ψ) is the relative perm
suction ψ, ψaev is the air-entry su ction, y is a dummy var-
iable of integration, b = ln(l,000,000), θ is volumetric
o
calcd
ils B-C and v-G models, a residual water con-
eability at
water content given in Equation (4) and θ′ is its deriva-
tive. a, n, m and Cr are fitting parameters of the F-X
model (Equation (4)).
1.3. Need for Modification to the Existing
Popular SWCC and Relative Permeability
Model
The B-C, v-G, and F-X models are being widely used t
ulate the moisture-suction relation of unsaturate
. For the so
tent value has to be specified. However these two models
calculate unrealistic suction when the normalized water
content is zero or less, i.e. water content of the soil is less
than or equal to the residual water content. In the F-X
model, the maximum suction is assumed to be 1,000,000
kPa. Although there are thermodynamic concepts to back
up this maximum suction, it is a concern to use a fixed
value for all types of soils. In addition, when the actual

S. H. KRISHNAPILLAI ET AL.
470
ts that there are three major
di
itional fitting parameter Cr. Incorporating the
m
lthough there are nu merous SWCC models available in
e the popu-
lar models. The B-C and v-G models
e modified primarily to make sure that these models no
he improved Brooks and Corey (I-B-C) model is given
C
moditional fitting parameter is introduced.
ven though the maximum suction ψmax is incorporated
maximum suction is low, usage of such larger maximum
suction value might over predict shear strength in nu-
merical simulations. Similar to the B-C and v-G models,
the second form of the F-X model (Equation (5)) also
calculates an unrealistic suction when the normalized
water content is zero or less. Therefore, to avoid an un-
realistic suction value at zero normalized water content,
the maximum suction value should be specified even
with a residual water content specified. In addition, the
fourth model parameter Cr in the F-X model is chosen
from a wide range (1 to 1,000,000 kPa) and it creates
difficulties in achieving a unique set of calibrated model
parameters. Also, the Cr affects the initial portion of the
curve when the value of Cr is relatively low and it is
considered as another disadvantage [6]. The primary
objective of this study is to increase the flexibility o f the
B-C and v-G models so that these models can predict
realistic high suctions in low degree of saturations with-
out causing numerical instabilities in finite element simu-
lations. That is, in the modeling the dynamic behavior of
unsaturated soils, there are numerous constitutive rela-
tions that can cause “numerical problems” during the
simulation. The best example is the nonlinear constitu-
tive model (elastoplastic stress-strain model). The other
one is the soil water characteristic cu rve which is used to
calculate the suction value at a given time increment for
the calculated degree of saturation or vise versa. If the
calculated suction is really large or really small, it can
introduce really large or really small number in one of
the element matrices (mass, damping or fluid stiffness).
This might result in crashing the program even with
really small time increment.
It is very challenging to model the soil behavior from
a fully dry condition to a fully saturated condition using
a single fully coupled finite element computer code. The
current state of the art sugges
fficulties in developing numerically stable simulation
capability. They are: difficulties in dealing with multiple
nodal/element variables in finite element formulation of
porous media at these extreme conditions, difficulties in
developing stress-strain behavior with appropriate stress
state variables at these extreme conditions, and difficul-
ties in accurately calculating the suction over the entire
range of degree of saturation. The modified models can
be incorporated in finite element simulation without in-
troducing numerical instabilities from SWCC. The idea
here is to modify the original curve in such a way it cal-
culates a finite number at very low degree of saturation.
In the original model, the curve is very steep at low de-
gree of saturation. A small change in degree of saturation
can result in unrealistic suction value at very low degree
of saturation. Since the modified curve calculates finite
values, numerical instability can be reduced or elimi-
nated.
In this study, the B-C and v-G models are modified by
incorporating correction factors. Also, the correction
factor in the F-X model is modified to avoid the effects
of add
aximum suction as part of the model increased its
flexibility in fitting measured data of various soils over
the full range of S. All three models are improved with
the feature to specify both residual water content and
maximum suction values. The capability of the improved
models is verified by matching with the experimental
data and prediction of original models. Based on the im-
proved F-X model, the permeability function proposed
by Fredlund et al. [6] is modified and presented.
2. Improved SWCC Models and
Comparisons
A
the literature, this study is intended to improv
B-C, v-G, and F-X
ar
longer calculate high suction when the normalized water
content is zero or less. And also the modified models
have the feature to specify both residual water content
an d maximum suction values.
2.1. The Improved Brooks and Corey (I-B-C)
Model
T
in Equation (8). To preserve the advantage of the B-
el, no add
E
in the equation, it cannot be considered to be a fitting
parameter, as the shape of the SWCC cannot be changed
by adjusting the ψmax. The I-B-C model does not provide
a continuous mathematical function for the entire range
of degree of saturation but gives a finite number of suc-
tion value at very low degree of saturation value.


1if
if
r
n sr
ψa
Cψa
a
 
 






(8)
The correction function for the I-B-C model is shown
below in Equation (9). A trial and error
followed to fit the data and there was no theoretical basis
for the equation for the correction function. An obvious
procedure was
condition needed to be satisfied was to obtain the correc-
tion function but to obtain

C

is equal to zero at the
residual water content.


1r C

  (9)

471 S. H. KRISHNAPILLAI ET AL.
where r

is suction at residual water content r

and
other parameters are same as i the B-C model. If the
residual water content in s set to zero, then r

becomes
max

.
2.2. Coparison of the B-C and the I-B-C
Models
ca
m
The pability of the Improved B-C (I-B-C) model in
redicting the moisture-suction relation is investigated
soils.
heof B-C and I-B-C Models for Columbia
ndy loam (data from [2]) is shown in Figure 1. The
ontent is assumed to be zero for all four so ils. As
sh
p
and compared with the B-C model for four different
comparison T
sa
Figures 2 and 3 show the comparison for Madrid clay
sand and Arlington soil, respectively. The Figures 4
shows the comparison for Indian head till (data from
[8]).
It should be noted that the experimental SWCC data
are not available for the full range of S (0% - 100%).
Based on the experimental data, the maximum suction of
1,000,000 kPa is chosen for all four soils. The residual
water c
own in these figures, the predictions by I-B-C model
shows slight improvement compared to the original B-C
model. The B-C, I-B-C models are not effective for
Figure 1. B-C and I-B-C SWCCs for Columbia sandy loam.
Figure 3. B-C and I-B-C SWCCs for Arlington soil.
Figure 4. B-C and I-B-C SWCCs for Indian head till.
sandy soils and it is eviden tly shown in Figure 1 as th e se
models failed to keep the shape of the SWCC without
reaching zero normalized water content in low suction
range.
2.3. The Improved van Genuchten (I-v-G) Model
The Improved van Genuchten (I-v-G) model is given in
Equation (10). Since the parameter a is related to the air-
entry suction, the model is revised so that the parameter
a has the unit of suction. The I-v-G model is developed
wit h the feature to specify both residual water content
and maximum suction value with no additional fitting
parameter.
Figure 2. B-C and I-B-C SWCCs for Madrid clay sand.



1n
sr a
  
quation (11) .
r
m
C 
  (10)
The modified correction function is shown below in
E


0.5
max
1
1m Cm 







(11)
where ψmax is maxi mum suction and other par ameters are

S. H. KRISHNAPILLAI ET AL.
472
2.4. Predictive Capability of the I-v-G Model
Capability of the Improved v-G (I-v-G) model in pre-
dicting the moisture-suction relation is presen ted for Co-
lu Figures 5-8, respectively. Similar
to the I-B-C model, maximum suction of 1,000,000 kPa
and residual water content of zero are us
soils. As shown in Figures 5-8, the I-v-G model is capa-
same as in the v-G model.
mbia sandy loam, Madrid clay sand, Arlington soil,
and Indian head till in
ed for all four
Figure 5. v-G and I-v-G SWCCs for Columbia sandy loam.
Figure 6. v-G and I-v-G SWCCs for Madrid clay sand.
Figure 8. v-G and I-v-G SWCCs for Indian head till.
ble of calculating the moisture-suction relation for full
range of S, whereas the v-G model is not effective. As
shown in Figure 5, the v-G, I-v-G models are also not
suitable for sandy soils as these models also failed to
keep the SWCC without reaching zero normalized water
content in low suction range.
2.5. The Improved Fredlund and Xing (I-F-X)
Model
The Improved Fredlund and Xing (I-F-X) model is given
in Ehe quation (12). The I-F-X model is developed with t
feature to specify both residual water content and maxi-
mum suction value without the parameter Cr, i.e. with
only three fitting parameters. Therefore, the effect of Cr
in the initial portion of the F-X model [7] is avoided in
the I-F-X model.
Figure 7. v-G and I-v-G SWCCs for Arlington soil.




=
rC

 
ln
sr ea
 
he modified
m
n   (12)
T correction function is shown below in
quation (13) . E


0.5
max
1
1m Cm 






(13)
where all the parameters are same as in the I-v-G model.
2.6. Predictive Capability of the I-F-X Model
The predictive capability of the I-F-X mo
ing the moisture-suction relation is presented in Figures
9-12. Similar to the I-B-C, I-v-G models, 1,000,000 kPa
m
e of S. However the I-F-X model can be
considered better as it has only three fittin
whereas the F-X model has four.
del in predict-
aximum suction and zero residual water content are
used. It can be noted that the I-F-X model is also effec-
tive in full rangg parameters,

473 S. H. KRISHNAPILLAI ET AL.
Figure 9. F-X and I-F-X SWCCs for Columbia sandy loam.
Figure 10. F-X and I-F-X SWCCs for Madrid clay sand.
Figure 11. F-X and I-F-X SWCCs for Arlington soil.
3. Modified Permeability Function and
Comparisons
Based on F-X SWCC model, a permeability function is
proposed by Fredlund et al. [7] and has been widely used.
Therefore, it is important to modify the Fredlund et al
permeability function (F-All model) based on the I-F-X
SWCC model. Th e F-All model is modified based on the
I-F-X SWCC model, and presented as I-F-All model in
Equation (14). One can immediately see that the im
prore
-
ved and original relative permeability equations a
Figure 12. F-X and I-F-X SWCCs for Indian head till.
basically the same but the correction factors . The
correction factor for the improved model in in
Equation (16) .
()Cψ
s show






 

ln
ln
d
d
aev
y
b
y
y
ry
bsy
y
eey
e
Keey
e










(14)
The function

is given by the following equation
(E quation (15) ).
 


ln
s
m
n
Cψ
eψa

 

(15)
The modified correction factor

C

is given by
Equation (16) .


0.5
max
1
1m Cm









where
(16)

max
e same as in the-All model.
is maximum suction and other parameters
ar F
3.1. Predictive Capability of the Improved
Fredlund et al. Model (I-F-All Mo
The permeability coefficients of water in four different
so I-F-A
res 13-16. The Figure 13 illustrates the
predictions for Superstition sand and the comparison
with experimental data (data from [9]). As
ure 13, the F-All and I-F-All models show better match
ecy of these two models
the higher suction range could not be verified. ted re-
sultsloam
xperimental data from [2]). Similar to the Superstition
del)
ils are predicted with F-All and ll models and
presented in Figu
shown in Fig-
with thexperimental data. However, because of the lack
of experimental data, the accura
in The Figure 14 shows the comparison of predic
and experimental data for Columbia sandy
(e

S. H. KRISHNAPILLAI ET AL.
474
Figure 13. F-All and I-F-All models for Superstition sand.
Figure 14. F-All and I-F-All models for Columbia sandy
loam.
Figure 15. F-All and I-F-All models for Touchet silt loam.
sand, the predictions of F-All and I-F-All models match
well with the experimental data in the lower suction
range. As shown in Figure 15, similar predictions are
obtained for the Touchet silt loam (experimental data
from [2]). The Figure 16 shows the prediction and com-
parison for Yolo light clay (data from [10]). As shown
there, the difference between the experimental data and
the predictions of F-All and I-F-All models increases
the suction increases. In addition, the prediction of F-All
model slightly deviates from the prediction of I-F-Al
as
l
Figure 16. F-All and I-F-All models for Yolo light clay.
model at higher suction range.
4. Conclusions
The widely used and most popular soil water characteris-
tic curves (Brooks and Corey, van Genuchten, and Fred-
lund & Xing) are modified to capture the high suctions at
low degree of saturation. New correction functions are
introduced in Brooks-Corey and van Genuchten models
and the correction function is the original F-X model was
ms
predict finite suctio n values at very low degree of satura-
nd residual water content
t in these modified models. The pre-
the modified models in low suction
uation for
ted
ience Society of America Journal, Vol. 44,
. 892-898.
doi:10.2136/sssaj1980.03615995004400050002x
odified as part of this research. The modified equation
tions. Both maximum suction a
can be used as inpu
dictive capability of
range could not be verified because there is no experi-
mental moisture-suction data available at low degree of
saturations. However, the flexibility of these modified
models has been improved by the introduction of maxi-
mum suction as one of the model parameters.
5. References
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tion in Unsaturated Soils,” Highway Research Board Spe-
cial Report 40, International Symposuim on Physico-
Chemical Phenomenon in Soils, Washington DC, 1956,
pp. 78-87.
[2] R. H. Brooks and A. T. Corey, “Hydraulic Properties of
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[3] M. Th. van Genuchten, “A Closed Form Eq
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Soils,” Soil Sc
No. 5, 1980, pp
[4] K. Kosugi, “The Parameter Lognormal Distribution
Model for Soil Water Retention,” Water Resource Re-
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S. H. KRISHNAPILLAI ET AL.
475
doi:10.1029/93WR02931
[5] D. G. Fredlund and A. Xing, “Equations for the Soil-Wa-
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[7] E. C. Leong, and H. Rahardjo, “Review of Soil-Water
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[8] S. K. Vanapalli, D. E
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rop-
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[9] L. A. Richards, “Water Conducting and Retaining P
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