Improving the Predictive Capability of Popular SWCCs by Incorporating Maximum Possible Suction

doi:10.4236/ijg.2011.24049

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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

E-mail: [email protected]

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.





 (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



=ln

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.



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

eψa









(4)

 



ln 1

1ln 110

ψψ

Cψ ψ



  (5)



sr ea



 





(6)

where is the suction corresponding to the residu

water contental



. and other parameters ar

the v-G odel

tion of the permeability is important for

s in

ty of unsatu-

e same as in

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].









eey













aev

bsy

Keey











(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

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

itional fitting parameter Cr. Incorporating the

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

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

the literature, this study is intended to improv

B-C, v-G, and F-X

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

in Equation (8). To preserve the advantage of the B-

el, no add

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

ψa

Cψa

 













(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





is equal to zero at the

residual water content.









 (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

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

and compared with the B-C model for four different

comparison T

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.



sr a

 



quation (11) .





 (10)

The modified correction function is shown below in



0.5

max

















(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.









sr ea





he modified



 (12)

T correction function is shown below in

quation (13) . E



0.5

max















(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

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









 



aev

bsy

eey

Keey

















(14)

The function



is given by the following equation

quation (15) ).

 



Cψ

eψa







(15)

The modified correction factor





is given by

Equation (16) .



0.5

max



















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

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

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

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

[1] W. Gardner, “Mathematics of Isothermal Water Conduc-

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

Porous Media,” Hydrology Paper, Colorado State Uni-

versity, Fort Collins, Vol. 27, No. 3, 1964, pp. 22-27.

[3] M. Th. van Genuchten, “A Closed Form Eq

Predicting the Hydraulic Conductivity of Unsatura

Soils,” Soil Sc

No. 5, 1980, pp

[4] K. Kosugi, “The Parameter Lognormal Distribution

Model for Soil Water Retention,” Water Resource Re-

search, Vol. 30, No. 4, 1994, pp. 891-901.

S. H. KRISHNAPILLAI ET AL.

475

doi:10.1029/93WR02931

[5] D. G. Fredlund and A. Xing, “Equations for the Soil-Wa-

ter Characteristic Curve,” Canadian Geotechnical Jour-

nal, Vol. 31, No. 4, 1994, pp. 521-532.

doi:10.1139/t94-062

[6] D. G. Fredlund, A. Xing and S. Huang, “Predicting the

Permeability Function for Unsaturated Soils Using the

Soil-Water Characteristic Curve,” Canadian Geotechni-

cal Journal, Vol. 31, No. 4, 1994, pp. 533-546.

doi:10.1139/t94-062

[7] E. C. Leong, and H. Rahardjo, “Review of Soil-Water

Characteristic Curve Equations,” Journal of Geotechnical

and Geoenvironmental Engineering, Vol. 123, No. 12,

1997, pp. 1106-1117.

doi:10.1061/(ASCE)1090-0241(1997)123:12(1106)

. Pufahl and D. G. Fredlund, “The

.49.2.143

[8] S. K. Vanapalli, D. E

Influence of Soil Structure and Stress History on the

Soil-Water Characteristic of a Compacted Till,” Geo-

technique, Vol. 49, No. 2, 1999, pp. 143-159.

doi:10.1680/geot.1999

rop-

low Water

pp. 383-426.

[9] L. A. Richards, “Water Conducting and Retaining P

erties of Soils in Relation to Irrigation,” Proceedings of

International Symposium on Desert Research, Jerusalem,

1952, pp. 523-546.

[10] R. E. Moore, “Water Conduction from Shal

Tables,” Hilgardia, Vol. 1, 1939,