Analysis of Soil Water Retention Data Using
Artificial Neural Networks
Sharad K. Jain1; Vijay P. Singh, F.ASCE2; and M. Th. van Genuchten3
Abstract: Many studies of water flow and solute transport in the vadose zone require estimates of the unsaturated soil hydraulicproperties, including the soil water retention curve 共WRC兲 describing the relationship between soil suction and water content. An artificial
neural network 共ANN兲 approach was developed to describe the WRC using observed data from several soils. The ANN approach was
found to produce equally or more accurate descriptions of the retention data as compared to several analytical retention functions
popularly used in the vadose zone hydrology literature. Given sufficient input data, the ANN approach was also found to closely describe
the hysteretic behavior of a soil, including observed scanning wetting and drying curves.
DOI: 10.1061/共ASCE兲1084-0699共2004兲9:5共415兲
CE Database subject headings: Neural networks; Soil water; Soil water storage; Hysteresis; Soil suction.

Introduction
The soil water retention curve 共WRC兲 describes the ability of asoil to store water at different suctions. This curve, also known as
the soil moisture characteristic, is one of the most basic water
properties of a soil relating soil suction 共or the matric or pressurehead兲 h, with the volumetric content ␪. As the suction increases,
progressively smaller pores lose their water and hence the watercontent decreases.
The WRC of a soil depends upon both soil texture and soil
structure. The amount of water retained at relatively low suctions
depends primarily upon capillary effects and the soil pore-size
distribution, and hence is strongly affected by the soil structure.
Water retention at higher suctions is increasingly due to adsorption, and hence is affected more by soil texture and the specificsurface area of a soil, as opposed to soil structure. Coarsetextured soils generally release their water much quicker thanfine-textured soils which often have a much broader pore-size
distribution so that when suction is increased, the water content
will decrease only gradually. Hence, a coarse-textured 共sandy兲
soil is able to retain less water as compared to a fine-textured
共clayey兲 soil.
One complication in the description of the soil hydraulic properties is the hysteretic nature of the water retention function, i.e.,the curve follows different paths depending upon the drying andwetting history of the soil. During wetting when the suction is
gradually lowered, the water content is generally higher than at
1

Scientist ‘F’, National Institute of Hydrology, Roorkee 247667,
India.
2
A. K. Barton Professor, Dept. of Civil and Environmental Engineering, Louisiana State Univ., Baton Rouge, LA 70803-6405. E-mail:cesing@lsu.edu
3
Soil Physicist, USDA-ARS Salinity Laboratory, 450 W. Big Springs
Rd., Riverside, CA 92507-4617.
Note. Discussion open until February 1, 2005. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on March 25, 2003; approved on January 7, 2004. This
paper is part of the Journal of Hydrologic Engineering, Vol. 9, No. 5,
September, 2004. ©ASCE, ISSN 1084-0699/2004/5-415– 420/$18.00.

the same suction during drying. Fig. 1 shows the hysteretic natureof the WRC for Caribou silt loam using data determined by Topp
共1971兲 as tabulated by Mualem 共1974兲 in his unsaturated soil
hydraulic property catalogue. Notice that in addition to the main
hysteretic loop, one also obtains secondary scanning curves asreflected by the smaller loops in Fig. 1.
Several theories have been advanced to explain the hysteretic
nature of soils, including the independent domain theory 共Everett
1955兲 and various modifications thereof 共e.g., Topp 1971兲. The
main causes of hysteresis 共Hillel 1971兲 are 共1兲 the ‘‘inkbottle’’effect due to geometric nonuniformity of individual pores; 共2兲 thecontact angle effect; 共3兲 aggregate impacts of such phenomena asswelling, shrinking, or aging; and 共4兲 the effect of entrapped air
pockets in soils that connect different size pores during wetting.
Water retention data are often conveniently described using analytical expressions. A large number of equations have been proposed for this purpose 共e.g., van Genuchten and Nielsen 1985;
Leij et al. 1997兲, the more popular ones being those by Brooks
and Corey 共1964兲 and van Genuchten 共1980兲. The Brooks andCorey 共1964兲 WRC model is given by␪⫽␪ r ⫹ 共 ␪ s ⫺␪ r 兲共 ␣h 兲 ⫺␭
⫽␪ s

␣h⭓1

␣h⬍1

(1)

where h⫽pressure head; ␪⫽volumetric water content;␪ s ⫽saturated water content; ␪ r ⫽residual water content;␣⫽parameter whose inverse (h a ⫽␣ ⫺1 ) is frequently referred toas the air entry value; and ␭ is sometimes named the pore-size
distribution index. van Genuchten 共1980兲 proposed an alternative
S-shaped model for the retention curve as follows:␪⫽␪ r ⫹ 共 ␪ s ⫺␪ r 兲关 1⫹ 兩 ␣h 兩 n 兴 ⫺m(2)

where ␣, n and m⫽empirical constants defining the shape of thecurve.
Recently, a two-parameter equation for water retention curve
was proposed by Assouline et al. 共1998兲
␪⫽␪ L ⫹ 共 ␪ s ⫺␪ L 兲 兵 1⫺exp关 ⫺␰ 共 h ⫺1 ⫺h L⫺1 兲 ␩ 兴 其

0⭐h⭐h L(3)

where h L ⫽suction head that corresponds to a very low watercontent ␪ L , where the hydraulic conductivity becomes negligible;

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Fig. 1. Hysteretic water retention curves for Caribou silt loam 共datafrom Topp 1971兲

and ␰ and ␩⫽empirical shape parameters. This model was tested
on water retention data sets of 12 soils representing a range of soil
textures from sand to clay. Agreement between the fitted curves
and the measured data was reported to be the same or better thanfor other models.
One disadvantage of analytical expressions, such as thosegiven by Eqs. 共1兲–共3兲, is that they are essentially empirical andincorporate certain assumptions about the shape of the WRC, including, for example the existence of a residual water contents in
Eqs. 共1兲 and 共2兲. For this purpose several investigators have usedalternative approaches to mathematically describe observed retention data, such as cubic splines 共Kastanek and Nielsen 2001兲 orother more flexible functions 共Prunty and Casey 2002兲. An emerging modeling technique that may be very well suited for thispurpose is the artificial neural networks 共ANNs兲. This empirical
technique is now being applied successfully to a wide range ofapplications in hydrology. It is this technique that we will explore
in this paper in an attempt to obtain improved descriptions of thewater retention curve.
The main purpose of this paper hence is to use ANNs to describe the water retention curve of soils using measured soil water
retention and suction data. Results will be compared with thoseusing Eqs. 共1兲, 共2兲, and 共3兲. We also apply the ANN model to acase with hysteresis involving drying and wetting curves andscanning curves. Below we first briefly review the use of ANNs inwater resources.

Artificial Neural Network Applications in Water
Resources
Artificial neural networks refer to computing systems whose central theme is borrowed from the analogy of biological neural networks. They represent highly simplified mathematical models of
biological neural networks. They include the ability to learn andgeneralize from examples to produce meaningful solutions toproblems even when input data contain errors or are incomplete,
and to adapt solutions over time to compensate for changing circumstances and to process information rapidly.
The ANN approach is faster compared to its conventionalcounterparts, robust in noisy environments, and flexible in therange of problems it can solve. An ANN has the ability to learn
from examples, to recognize a pattern in the data, to adapt solutions, and process information rapidly. Due to these advantages,
ANNs have been used in numerous real world applications, suchas image processing, speech processing, performing general mapping from input pattern to output pattern, and grouping similar
patterns. Applications of ANNs to hydrology are rapidly gaining
popularity due to their power and potential in mapping nonlinear
system data.

A system may be nonlinear and multivariate, and the variables
involved may have complex inter-relationships. Artificial neuralnetworks are capable of adapting their complexity and their accuracy increases as more and more input data are made available
to them. They are capable of extracting the relation between the
input and output of a process without any knowledge of the underlying principles. Because of the generalizing capabilities of theactivation function, one need not make any assumption about therelationship 共linear or nonlinear兲 between input and output. All
these properties make ANNs an attractive tool for water resources
practitioners.
In the field of water resources, ANNs have been used for flow
predictions, flow/pollution simulation, parameter identification,and to model complex nonlinear input–output time series. Hsu
et al. 共1995兲 have shown that the ANN approach provides a better
representation of the rainfall–runoff relationship of a medium
sized basin than does the ARMAX approach or the Sacramento
soil moisture model. Raman and Sunilkumar 共1995兲 investigated
the use of ANNs for synthetic inflow generation and compared
the model performance with that of a multivariate time series
autoregressive moving average model. Minns and Hall 共1996兲
applied an ANN to rainfall–runoff modeling. Dawson and Wilby
共1998兲 used an ANN for river flow forecasting. Artificial neuralnetworks were used for reservoir inflow prediction by Jain et al.
共1999兲. Birikundavyi et al. 共2002兲 found that an ANN can achieve
accuracy superior to that of ARMAX and deterministic models
for 7-day ahead forecasting. Kumar et al. 共2002兲 concluded that
the ANN can predict reference crop evapotranspiration for an area
better than the Penman–Monteith method.
A set of two papers published by the ASCE task committee on
application of ANNs in hydrology 共ASCE 2000a, 2000b兲 contains
a detailed review of the theory and applications of ANNs in water
resources. Govindaraju and Rao 共2000兲 have described many applications of ANNs to water resources.
Artificial neural networks have also recently found widespread
application to predictions of the water retention curve from soil
texture and related data using pedotransfer functions 共PTFs兲 共e.g.,
Pachepsky et al. 1996; Schaap et al. 1998; Koekkoek and Booltink 1999; Minasny and McBratney 2002兲. The basic premise of
these approaches is to assume the applicability of a certain retention model 关e.g., Eqs. 共1兲 or 共2兲兴, and then to use ANNs to correlate several or all of the unknown model parameters to more
readily available or more easily measured data, such as soil texture, bulk density, organic matter content, and/or soil structure. As
an example, Schaap et al. 共1998兲 used ANNs to estimate the parameters in Eq. 共2兲, as well as the saturated hydraulic conductivity. To facilitate the practical use of PTFs, they designed a hierarchical structure to allow input of both limited and more
extended sets of predictors. They combined their ANNs with thebootstrap method 共Efron and Tibshirani 1993兲 to additionally obtain an estimate of the uncertainty in the PTF predictions 共Schaap
et al. 1998兲. The PTFs were calibrated on some 2,100 soil hydraulic data sets, and further tested on more than 47,000 records
from the Natural Resources Conservation Services soil characterization database. Schaap and Leij 共2000兲 subsequently expanded
the hierarchical approach to include unsaturated hydraulic conductivities. Their work resulted in a windows-based software
package, Rosetta 具http://www.ussl.ars.usda.gov/models/rosetta/rosetta.htm典 that incorporates the different PTFs.
An advantage of the PTF approach is that estimates of thecomplete retention function are obtained. Still, the predictions
always remain somewhat approximate since they are based on
generic, not site-specific data. If site-specific data are available,

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Table 1. Soil Data Sets and Mualem 共1974兲 Curve Types Used in This
Study
Soil type
Beit Netofa clay soil
Rideau clay loam
Touchet silt loam
Pachappa loam
Rubicon sandy loam
Pachappa fine sandy clay
Sable de riviereFig. 2. Three-layer feed forward artificial neural network with one
input node 共for matric potential兲, four nodes in hidden layer, and one
output node 共representing soil moisture content兲

several approaches may be used to obtain accurate descriptions of
the retention data. This includes: 共1兲 fitting the data with one of
the empirical retention functions mentioned earlier; 共2兲 using
cubic splines 共Kastanek and Nielsen 2001兲 or other methods forinterpolation between measured retention data; and 共3兲 using
ANNs for the interpolation. It is the latter approach that is followed in this paper.

Development of Artificial Neural Network Model andAnalysis of Data
A three-layer feed forward ANN was used in this study. According to Hsu et al. 共1995兲, three-layer feed forward ANNs can beused to model real-world functional relationships that may be of
unknown or poorly defined form and complexity. Such an ANN isshown in Fig. 2. The input to the network is received by theneurons in the input layer. The data passing through the connections from one neuron to another are manipulated by weights
which control the strength of a passing signal. When theseweights are modified, the data transferred through the networkchanges and the network output alters. The neurons in a layer
share the same input and output connections, but do not interconnect among themselves. Each layer performs specific functions.
All the nodes within a layer act synchronously, meaning at any
point of time they will be at the same stage of processing. The
level of activity generated at the output node共s兲 is the network’s
response to the inputs presented to it.
In this study, we applied the ANN results to the same data sets
as used in the recent study by Assouline et al. 共1998兲. The samedata were also previously used by van Genuchten and Nielsen
共1985兲 in their comparisons of the relative accuracy of the Brooks
and Corey 共1964兲 and van Genuchten 共1980兲 retention models.
All data sets were taken from the hydraulic property catalogue of
Mualem 共1974兲. Details of the data sets are given in Table 1. van
Genuchten and Nielsen 共1985兲 and Assouline et al. 共1998兲 bothfitted the WRC model parameters to the data using an iterative
nonlinear regression procedure based on the Marquardt–
Levenburg algorithm 共Marquardt 1963兲. Following Assouline
et al. 共1998兲, we compared the results of different approaches
using the norm index 共NI兲 defined as

冑兺
n

NI⫽

i⫽1

共 ␪ oi ⫺␪ ci 兲 2(4)Catalog number

Curve type

1006
3101
3304
3403
3501
3503
4118

First drying
Main drying
First drying
Main drying
Main drying
Main drying
Main drying

where ␪ oi and ␪ ci ⫽observed and computed values of the watercontent for the ith observation, respectively.
A three-layer feed-forward ANN was trained for each data set.
The input layer had one neuron that received matric potential
values as input. The signal from the only neuron in the output
layer represented the corresponding moisture content. The number of neurons in the hidden layer was determined by trial anderror; five neurons gave the best results for the data used in thisstudy. The sigmoid transfer function was used. This function isy j ⫽1/关 1⫹exp共 ⫺z i 兲兴

(5)where z i ⫽⌺w i j x i ; w i j ⫽weight of the connection from the ith
neuron in the previous layer to the current neuron; and x i ⫽input
to ith neuron in the previous layer. The sigmoid function is continuously differentiable. Its derivative is y ⬘ ⫽y(1⫺y).
The weights of the ANN were estimated using an error backpropagation method. In this algorithm, a set of inputs and outputs
was selected from the training set and the network calculated theoutput based on the inputs. The actual output was compared withthe target output to find the output-layer errors. The weights of all
neurons were adjusted based on the strength of the signal in theconnection and the total measure of the error. The total error at
the output layer was then reduced by redistributing this error
backwards through the hidden layers until the input layer was
reached. This process continued for a number of prescribed iterations or until a prescribed error tolerance was reached.
The mean square error 共MSE兲 over the training samples is atypical objective function and can be expressed as
N

E⫽

m

兺 兺 共 T pn ⫺O pn 兲 2p⫽1 n⫽1(6)

where T pn ⫽target 共or observed兲 value of the nth neuron for thepth pattern; O pn ⫽output value of the nth neuron for the pth pattern; N⫽total number of training patterns 共sets of matric potential
and soil moisture content values兲; and m⫽total number of output
neurons. About 10,000 iterations were performed to train the
ANN for each data set.
The number of water retention ␪ – h data pairs for the main
drying curves of the soil used in this study ranged from 14 to 34.
Before training, the ␪ values were normalized by using the following formula:␪ N ⫽ 共 ␪⫺␪ min兲 /⌬␪

(7)where ␪ min⫽value slightly less than the minimum of the ␪ values;and ⌬␪⫽difference between the minimum and the maximum values. A different formula will be more suitable for a variable thatvaries within a certain range. Minns and Hall 共1996兲 have rightly
emphasized the importance of the correct standardization.
The suction values were normalized by dividing by the maximum h value. This, however, caused a problem in the ANN train-

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Table 2. Comparison of Norm Index for Various Soils
Norm index obtained by

Soil type

CatalognumberBrooks andCorey
共1964兲
model

van
Genuchten
共1980兲
variable m, nvan
Genuchten
共1980兲
m⫽1⫺1/nAssouline
et al.
共1998兲

This study
共artificial neural network兲

Beit Netofa clay soil
Rideau clay loam
Touchet silt loam
Pachappa loam
Rubicon sandy loam
Pachappa fine sandy clay
Sable de riviere

1006
3101
3304
3403
3501
3503
4118

0.044
0.02
0.015
0.055
0.015
0.034
—

0.02450.019
0.01050.051
0.01480.03
0.00118

0.034
0.033
0.045
0.06530.05
0.03460.00118

0.017
0.028
0.034
0.030
0.038
0.041
0.042

0.012
0.016
0.019
0.058
0.025
0.036
0.0074ing for soils for which the difference between the maximum andthe minimum values of suction was extremely large. For instance,
the maximum and minimum values of the matric potential for the
Pachappa loam soil were 319⫻104 cm and 10 cm of water. If thesuction values were normalized by merely dividing by the maximum value for such data, most values in the lower suction range
clustered close to zero and this resulted in poor training. To overcome this problem, the suction values for such soils were first log
transformed and then standardized to fall within the range of 0–1.
All available pairs of ␪ – h values were used to train the ANNs.
These days, a number of software packages are available, commercially as well as in public domain, that can be used to train an
ANN.

Results and Discussion
The values of the norm index obtained by Assouline et al. 共1998兲
and in this study are given in Table 2. Also included in Table 2 arenorm values obtained by applying the van Genuchten 共1980兲
model assuming both independent m and n values, or the often
assumed relationship m⫽1⫺1/n 共van Genuchten and Nielsen
1985兲. Notice that the NI obtained using ANNs is significantly
smaller 共except for one soil兲 than those obtained by Assouline
et al. 共1998兲, with the results of van Genuchten being somewhat
better than those by Assouline et al. 共1998兲, especially when mand n are kept independent. These results are consistent withmany other studies in which ANNs have been found to model the
behavior of real-world data better than the empirical/regression
models.
Fig. 3 graphs observed values of soil suction versus the moisture content for Rideau clay loam. Also shown in the figure areplots of the WRC curve obtained by using Eq. 共3兲 共the parameters
as given by Assouline et al. 1998 were used兲 and the ANN-based
curve. Note that the input to the ANN model was only the suction
head. It can be seen that the ANN curve is closer to the measured
data points than the curve obtained using Eq. 共3兲. The results ofTable 2 also show that the performance of ANN is much better
than the model of Assouline et al. 共1998兲 and is comparable to themodel of van Genuchten 共1980兲 and Brooks and Corey 共1964兲.
Note that application of ANN requires that the data over the entire
water retention range is available. Also, the Assouline et al.
共1998兲 model cannot be extrapolated to lower pressure heads
共higher suctions兲 without risking physically unrealistic retention
values.

Modeling of Hysteresis
The second part of this study is concerned with describing thehysteretic behavior of soils using ANNs. This problem is more

complex than modeling WRC because the shape of drying andwetting limbs of the curve is not the same. Additionally, an ANN
has to be provided with enough data so that it can unambiguously
decide which curve to follow.
As before, the soil moisture data were normalized using Eq.
共6兲, while the suction data were normalized by dividing by the
maximum value. The root mean square error 共RMSE兲 for the
training/testing was calculated as

冑兺
n

RMSE⫽

1
共 ␪ ⫺␪ mi 兲 2
n i⫽1 ai(8)

where n⫽number of data points, and ␪ ai and ␪ mi ⫽actual andmodeled values of the moisture content, respectively.
Initially, the ANN model was provided with two input values(h i and h i⫺1 , i.e., the observed value of suction at the current andprevious points兲 by reasoning that these would provide enoughinformation for the model to determine which 共main wetting or
drying兲 branch to follow, and then correctly estimate the soil
moisture. Note that in this case, the ANN training set consisted of
all the available pairs of h – ␪ values for drying as well as wetting
curves. The RMSE was quite small for the Rideau clay loam but
was quite large for the Caribou silt loam. This experiment was
followed up by including one more input in the ANN for h i⫺2 .
This additional input should provide the ANN with the definite
information whether the data is on the wetting limb or on thedrying limb. As expected, inclusion of this additional input resulted in a large improvement in training results as reflected by a
substantial reduction in the RMSE for both soils. The reduction in

Fig. 3. Measured water retention data and curves obtained using
artificial neural networks and empirical model of Assouline et al.
共1998兲

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Table 3. Results of Artificial Neural Network Modeling: Root Mean

Square Error 共RMSE兲 for Soils for Different Input Vectors
RMSE
Rideau clay loam
共No. 3101兲

InputsCaribou silt loam
共No. 3301兲

I. Main drying and wetting data

Fig. 4. Observed and artificial neural network modeled water
retention curve for Rideau clay loam

RMSE was much larger for Caribou silt loam as compared to
Rideau clay loam, with both soils now showing the same order ofmagnitude for RMSE.
Many other input combinations were tried. Another useful way
to guide the ANN to the correct branch was found by including
h init as input. This is the suction at the initial point of the h – ␪curve. This will be the maximum suction value for the wetting
curve and the minimum value for the drying curve. We found that
including this information also led to a large reduction in the
RMSE value. The observed and ANN-modeled curves for the twosoils for this combination of inputs are given in Figs. 4 and 5.
These figures show a good match between the observed and computed curves, except near saturation. Thus, a properly trained
ANN which has four neurons in the input layer representing h i ,h i⫺1 , h i⫺2 , and h init can satisfactorily reproduce the hysteretic
behavior of a soil. The RMSE values for the various input combinations for the two soils are given in Table 3, Part I.
An attempt was also made to model the behavior of the scanning curves using ANN. In this case, the training set consisted of
all the available pairs of h – ␪ values for main drying and wetting
curves and data for one dry scan and one wet scan curve. The
same combination of inputs, as earlier, was tried using the data ofmain drying and wetting as well as one drying scanning curve and
one wet scanning curve. The results are given in Part II of Table
3. These results also show that the training RMSE was quite small
when the inputs were h i , h i⫺1 , h i⫺2 , and h init .
Since data for more than one dry and wet scan curves for eachsoil were available, the goodness of the ANN was tested using thedata of a dry scan and wet scan that was not used in the training.
The results are given in Parts III and IV of Table 3 and Figs. 6 and7. Evidently, the match between observed and computed dry andwet scan curves was good for Soil 3101 while it was not that goodfor the wet scan curve of Soil 3301. However, the shape of a scancurve is significantly influenced by the drying and wetting history

Fig. 5. Observed and artificial neural network modeled water
retention curve for Caribou silt loam

h i , h i⫺1
h i , h i⫺1 , h i⫺2
h i , h i⫺1 , h init
h i , h i⫺1 , h i⫺2 , h init

0.006073
0.004783
0.008849
0.002466

0.015291
0.004679
0.002032
0.002431

II. Main drying, wetting, and dry/wet scan data
h i , h i⫺1 , h init
h i , h i⫺1 , h i⫺2 , h init

0.013179
0.007314

0.009246
0.003045

III. Test—dry scan curve
h i , h i⫺1 , h init
h i , h i⫺1 , h i⫺2 , h init

0.008189
0.006781

0.023480.00608

IV. Test—wet scan curve
h i , h i⫺1 , h init
h i , h i⫺1 , h i⫺2 , h init

0.004979
0.0084850.046360.02063of the soil and when seen with this perspective, coupled with data
errors, etc., the results appear to be very good and useful.
Limited numerical experiments to determine the impact of
missing data and errors in data on ANN training were also made.
In the first case, three or four successive data points were deleted
from a segment of WRC and the ANN was trained. We found thatfor our data sets RMSE changed by less than 0.5%. Introductionof 10% error in some data points of the training set also led toabout the same change in RMSE. As the ANN tries to generalize
the data behavior, it assumes the behavior shown by the erroneous
data set to be the true behavior of WRC and therefore, RMSE
does not change significantly.
It may be added that when the values of norm index and
RMSE are small, these indices may not be adequate to differentiate among the models and additional criteria, such as model
reliability need to be used.

Conclusions
The WRC of a soil was modeled by an ANN using the measured
data of soil moisture content and suction. The ANN used in thisFig. 6. Observed and artificial neural network modeled scan curves
for Rideau clay loam

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Fig. 7. Observed and artificial neural network modeled scan curves
for Caribou silt loam

study was a three-layer feed-forward architecture. It was foundthat the ANN with only suction head data as input was able todescribe WRC better than did the empirical model of Assouline
et al. 共1998兲 and the performance was comparable to the van
Genuchten 共1980兲 model. More importantly an ANN, whose inputs were the current and previous values of suction heads, wasfound to reproduce the hysteretic behavior and scan curves in asatisfactory manner. This application of ANN as a fitting tool
should be useful in soil-moisture modeling.

Notation
The following symbols are used in this paper:h ⫽ soil suction 共or matric or pressure head兲;h L ⫽ soil suction corresponding to very low water content;␪ ⫽ volumetric water content of soil;␪ L ⫽ very low water content of soil at which hydraulicconductivity becomes negligible;␪ r ⫽ residual 共volumetric兲 water content of soil; and␪ s ⫽ saturated 共volumetric兲 water content of soil.

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