I I I II

I

SOILPROP

A Program for Estimating Unsaturated Soil Hydraulic Pr<)perties

and Their Uncertainty from Particle Size Distribution Data

Version ::.0

USE R' S G U IDE

}<Jnvironmental Systems & Technologies, Inc.

P. O. Box 10457, Blacksburg, VA 2-1062 U.S.A.

I

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(c) CDPyright Envir.onmental Systems & TechnDIDgies. Inc. 1988. 1989

P.O. BDX 10457. Blacksburg. VA 24061-0457. USA

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INTRODUCTION

SOILPROP is an interactive prol!:ram for estimatinl!: soil hydraulic properties

from particle size distribution data. Specific soil properties of interest are

the saturated hydraulic conductivity and parameters in the van Genuchten

(VG) model (van Genuchten. 1980) and/or the Brooks-Corey (BC) model (Brooks

and Corey. 1964) for water retention relations. SOILPROP also estimates the

error in individual parameters and an error covariance matrix to account for

correlation among parameters. This report describes prog"ram input

requirements and essentials of the parameter estimation methodolog"Y. Details

of the procedures used in SOILPROP are described by Mishra et a1. (1989) (see

Appendix A). Procedures for employing" parameter covariance data estimated

by SOILPROP to evaluate uncertainty in unsaturated flow predictions are

discussed by Mishra and Parker (1989) (see Appendix B). SOILPROP is based

on the premise that the soil-water retention function. B(h). reflects an

underlying" pore size distribution which can be deduced from the particle size

distribution (e.g" .• Arya and Paris. 1981).

SOIL HYDRAULIC PROPERTY MODELS

The two parametric hydraulic property models which are considered in this

prog"ram are those proposed by Brooks and Corey (1964) and van Genuchten

(1980). In the Brooks-Corey (BC) model. the soil-water retention function is

g"iven by the relation

(Ia)

Se = 1 (Ib)

where Se = Sj(1-Sr) = (B-Br )/(Bs-Br ). in which S is the deg"ree of saturation

of the pore space with water. Sr is the residual water saturation. B is the

volumetric water content. Br is the residual water content. Bs is the saturated

water content. h is capillary head. and hd and A are BC retention function

parameters. Note that S = B/Bs and Sr = B/Br The conductivity function for

the BC model is g"iven by

(2)

1

For the van Genuchten (VG) model the soil-water retention function is

described by

h '" a (3a)

Se = 1 h ,; a (3b)

and the conductivity function is

(4)

where IX and n are the VG retention parameters, m = l-l/n and other symbols

are as previously defined. Note that relative conductivity Kr = K / Ks.

INPUT OF PARTICLE SIZE DISTRIBUTION (PSD) DATA

Table 1 shows the various PSD classification schemes used in the prOl!ram.

The user is first asked to select a scheme, followinl! which SOIL PROP prompts

for the % mass fraction in each particle size class ranl!e. Alternatively, the

user can specify his/her own PSD classification scheme by providinl! the

number of classes, the mweimum particle diameter for each class and the

correspondinl! mass fraction.

cumulative distribution function

Input

(CDF)

mass fractions are

which is fitted

converted to a

to a 101!-normal

distribution model usinl! a nonlinear rel!ression procedure to calculate median

particle diameter, d 5 o. The information in Table 1 is also available as an

on-screen help menu. Note that SOILPROP treats fractions with particle

diameter> 12 mm (17.5 mm in the ASTM scheme) as 'inert' components - that

is, these fractions are assumed to occupy volume without affectinl! the pore

size distribution of the continuous finer material surroundinl! them.

INPUT AND/OR ESTIMATION OF BULK DENSITY AND SATURATED WATER CONTENT

SOILPROP requires the user to input the bulk density, "b, and/or saturated

water content, 8s ' If one of these is unknown, its value is estimated from a

correlation of porosity, bulk density and saturated water content, assuminl! a

particle density, Ps' of 2.65 I! cm-'. SOILPROP also prompts for the

uncertainty in as and pb as a percental!e of the input value (i.e .. the

2

coefficient of variation} if this is known. If 'inert' materials as described

above are present, the user should input as andlor pb values that are

representative of the whole soil. If this information is not available, the user

may input as and pb values for the fraction < 5 mm, which will be internally

corrected to account for the presence of coarser material by SOILPROP usinl'!

the procedure sUl'!l'!ested by Mehuys et a1. (1975).

Table 1. Classification schemes for PSD data used in SOILPROP.

d(nnn) USDA-l USDA-2 ASTM Limited Data

0.001 Clay Clay Clay

0.002 Fine Silt Fines

0.005 (clay Medium silt Silt and silt) Silt

0.02 Coarse silt

0.05 Very Fine Very Fine

0.08 Sand Sand 0.10 S

Fine Sand Fine Sand Fine 0.25 Sand A

Medium Sand Medium Sand 0.50 N

Coarse Sand Coarse Sand 1. 00 Medium D

Very Coarse Very Coarse Sand Sand Sand

2.00 Coarse Sand

5.00 Fine Gravel Fine Gravel Fine Gravel

12.0 Fine Gravel

17.5 Cobbles Cobbles Cobbles Cobbles

II II II II

3

CONVERSION OF PSD DATA TO SOIL-WATER RETENTION FUNCTION

The basic premise of this procedure is that the soil-water retention function.

9(h). reflects an underlyin.e: pore size distribution which can be deduced from

the particle size distribution (Arya and Paris. 1981). SOILPROP converts PSD

data to 9 (h) data usin.e: the al.e:orithm of Arya and Paris (AP) as modified by

Mishra et al. (Appendix A). Briefly. the AP method involves dividin.e: the

particle size ODF into a number of fractions. assi.e:nin.e: a pore volume and a

volumetric water content to each fraction. and then computin.e: a representative

pore radius and a correspondin.e: capillary pressure head. This results in a

complete 9 (h) functional relationship.

COMPUTATION OF SOIL-WATER RETENTION PARAMETERS

In principle. the AP model-.e:enerated 9(h) data can be fitted to either the van

Genuchten (VG) or Brooks-Oorey (BO) soil-water retention models usin.e:

nonlinear rel!ression methods in order to estimate soil-water retention

parameters. However. because of practical problems with automatic fittin.e: of

the BC model to 9(h) data. SOILPROP first fits the VG model to 9(h) data to

estimate IX. nand 9r •

parameters.

and then converts these to "equivalent" BO model

An option is provided for stipulatin.e: initial I!uesses for parameters used in

fittin.e: the VG model to AP model-.e:enerated 9 (h) data. The default startin.e:

values of " = 0.02 cm-'. n = l.80 and 9r = 0.001 used by SOILPROP will

normally be satisfactory. However. occasionally the optimization al.e:orithm may

not conver.e:e to the .e:lobal minimum of the objective function leadin.e: to

erroneous parameter values. If a low R' statistic ("0.85) and lar.e:e parameter

standard deviations (O.V."lOO%) are observed. it is recommended to rerun the

analysis with different startin.e: values to see if a local minimum has been

encountered in the inversion. Results yieldin.e: the hi.e:hest R' should be taken

as correct parameters. When BO model parameters are bein.e: estimated.

optional startin.e: values are specified in terms of equivalent BO parameters

which are internally converted to VG startin.e: values. Equivalent default

startin.e: values correspond to hd = 50 cm. A = 0.80 and 9r = 0.001.

4

The van Genuchten parameters. IX and n. are internally converted to the BC

model parameters. hd and A. usin.e: the method of Lenhard et' al. (1989) by

specifYin.e: the effective saturation at which the BC and VG 8(h} curves are

forced to cross. The default procedure selects the match point usin.e: an

empirical scheme proposed by Lenhard et al. Alternatively. the user may

stipulate a match point effective saturation between 0.5 and 0.95. For

problems in which better accuracy in the hi.e:h water saturation rerion is

desired. use of a hi.e:her match point effective saturation is recommended.

The residual water content. 8r • (or the residual saturation. SrI may be

estimated by SOILPROP. or specified by the user accordin.e: to one of the

followin.e: options.

1. 8r (or SrI may be fixed by the user at some pre-determined value.

A value of zero is recommended in cases in which initial attempts to

fit 8r yield lar.e:e parameter uncertainty (i.e .• C.V. for 8r ,'" 100%).

2. 8r may be estimated by SOILPROP from the AP model .e:enerated

equilibrium water retention curve simultaneously with the retention

parameters IX and n. (or hd and A).

3. For vertically intep;rated models. Quasi-static parameters are more

relevant because vertical redistribution under ,!!ravitational .e:radients

becomes ne.e:lirible when the hydraulic conductivity is very low. In

this case. SOILPROP will adjust 8r to represent 'field capacity'

defined at the user's discretion by one of the followin.e: options

water content at which conductivity equals specified minimum

flow rate (default 0.05 cm/ d)

water content at which capillary head equals specified critical

value (default 100 cm)

If the user-specified minimum flow rate results in a value of 8r for

which the effective saturation is ,!!reater than 0.8. the pro.e:ram

switches to a value of 8r correspondin.e: to a capillary head of 100

cm. This is done to make the parameter estimation more robust.

5

ESTIMATION OF SATURATED CONDUCTIVITY

Saturated hydraulic conductivity. Ks. is estimated from the median particle

diameter. d S 00 saturated water content. 9 s. and the standard deviation of the

log-particle diameter. ''In(d). using a modified Kozeny-Carman equation. which

was developed from a data set of 250 soil samples for which measurements

and/or estimates of Ks. 9s • d so and <T}n(d) were available (Appendix A).

EVALUATION OF PARAMETER UNCERTAINTY

SOILPROP also evaluates the uncertainty associated with parameter estimates

usinl! a first-order error analysis procedure. Parameter standard deviations

are computed for Ks and 9s • and an error covariance matrix is calculated for

ex. nand 9 r • and/or hd. A and 9r .

IMPLEMENTATION NOTES

SOILPROP consists of two modules - (i) COMPUTE - a FORTRAN 77 routine

which performs the parameter estimation and uncertainty analysis tasks. and

(ti) GRAPllT - a BASIC routine which produces l!raphs of the h-9 and 9-Kr

functions on screen. A batch file called SOILPROP is provided to run both

modules interactively. System requirements are an IBM PC (or compatible)

with a math co-processor. 360KB RAM and an EGA/CGA adapter for l!raphics

display operating under DOS version 3.0 or later.

LITERATURE CITED

Ar;ya. L. M. and J. F. Paris. A physicoempirical model to predict soil moisture

characteristics from particle size distribution and bulk density data. Soil

Sci. Soc. Am. J .• 45. 1023-1030. 1981.

Brooks. R.H. and A. T. Corey. Hydraulic properties of porous media. Hydrolol!Y

paper no. 3. Colorado State U .• Fort Collins. CO. 1964.

Lenhard. R.J .• J. C. Parker and S. Mishra. On the correspondence between van

Genuchten and Brooks-Corey Models. J. Irr. Dr. Enl!. (in press). 1989.

6

Mehuys. G. R •• L. H. Stolzy. J. Letey and L. V. Weeks. Effect of stones on the

hydraulic conductivity of relativelY dry desert soils. Soil Sci. Soc. Am.

Proc .• 39. 37-42. 1975.

Mishra. S. and J. C. Parker. Effects of parameter uncertainty on predictions of

unsaturated flow. J. Hydrol. (in press). 1989.

Mishra. S .• J. C. Parker and N. Singhal. Estimation of soil hydraulic properties

and their uncertainty from particle size distribution data. J. Hydrol. (in

press). 1989.

van Genuchten. M. Th.. A closed-form equation for predictinl': the hydraulic

conductivity of unsaturated soils. Soil Sci. Soc. Am. J •• 44. 892-899. 1980.

7

APPENDIX A

ESTIMATION OF SOIL HYDRAULIC PROPERTIES AND THEm UNCERTAINTY

FROM PARTICLE SIZE DISTRIBUTION DATA

INTRODUCTION

Unsaturated soil hydraulic properties are commonly represented by

empirical models which define the relationships between wettinll: fluid

conductivity, saturation and capillary pressure. The problem of estimatinll: soil

hydraulic properties then reduces to estimatinll: parameters of the appropriate

constitutive model. Two such models that are widelY used are those sUll:ll:ested

by Brooks and Corey (1964) and van Genuchten (1980). In the Brooks-Corey

(BC) model, the soil water retention function, 8(h), and the hydraulic

conductivity function, K(8), are represented, respectively, by

Cla)

Se = 1 Clb)

K = (2)

while for the van Genuchten (VG) model, the functional forms are

h '" 0 (3a)

Se = 1 h if 0 (3b)

(4)

where Se = (8-8r )/(as-ar ) is effective saturation, h is capillary head. a is

volumetric water content, K is hydraulic conductivity, Ks is the saturated

hydraulic conductivity, ar is the residual water content, as is the saturated

water content, hd and A are BC model parameters. " and n are VG model

parameters and m = I-lin.

8

Several methods have been described in the literature for estimating

parameters in soil hydraulic property models. One common approach is to take

static B(h) measurements and fit them to the desired soil-water retention

model. (1) or (3). Once the retention function is estimated. the conductivity

relation. K(B). can be evaluated from (2) or (4) if the saturated conductivity.

Ks. is known. Another approach to model calibration is to conduct a dynamic

flow experiment (i.e .. infiltration. redistribution and/or drainage event). and

use the observed water content. pressure head and/or boundary flux data to

invert the governing initial-boundary value problem (Kool et aJ .• 1987).

Since soil textural information is more easily obtained than static or

dynamic hydraulic data. an appealinl'!: alternative for estimating soil properties

is from particle size distribution (PSD) data. Methods have been proposed for

computing soil-water retention relations from PSD data using rel'!:ression

equations (McCuen et aJ .. 1981; .Campbell. 1985; Rawls and Brakensiek. 1985) or

via models with quasi-physical bases (Arya and Paris. 1981). Methods for

estimating saturated conductivity, Kg, from PSD data are generally based on

use of the Kozeny-Carman equation or variations thereof (Dullien. 1979) which

involve a relationship between saturated conductivity. porosity and some

representative particle diameter.

Although the estimation of soil hydraulic properties from PSD data may

offer substantial savings in experimental effort over more direct calibration

methods. accuracy will generally be sacrificed. Without knowledl'!:e of the

confidence regions of estimated parameters. it is not feasible to evaluate their

utility in making predictions of fluid flow and transport in the unsaturated

zone within prescribed tolerance.

In this paper we describe a systematic methodology for estimating soil

hydraulic properties from particle size distribution data and for evaluating

parameter uncertainty. The procedure uses a modified form of the Arya and

Pads (AP) model to convert PSD data to an equivalent soil-water retention

function, which is then fitted to the VG and/or BC models. Saturated

conductivity is estimated by a modified Kozeny-Carman (KC) equation.

Procedures for quantifyinl'!: uncertainty in parameter estimates are presented

which are based on first-order error analysis methods.

9

EXPERTIMENTAL METHODS

Empirical relationships between particle size distribution data and soil

hydraulic properties were evaluated in this study from a data set consistinl!:

of 250 soil samples for which I!:rain size distribution, bulk density, and soil

hydraulic properties were available. These soil samples were obtained from a

wide ranl!:e of soil types from depths of 0-6 m at various locations within the

state of Virl!:inia, USA.

Particle size distribution was determined on all samples usinl!: the pipette

method for fractions L 0.05 mm and by wet sievinl!: for coarser fractions (Day,

1965). Bulk density and soil hydraulic parameters were determined on 54 mm

diameter by 40 mm lonl!: core samples taken with a thin wall samplinl!:

apparatus to minimize disturbance. Bulk densities of cores were determined

by strail!:htforward I!:ravimetric means. Saturated water contents were

determined after I!:radually brinl!:inl!: cores to zero capillary head. These water

contents do not I!:enerally correspond to true saturation but to a "field

saturation" pertinent to secondary imbibition, hereafter referred to simply as

saturated water content, 8s ' Hydraulic conductivities at field saturation, lis, were determined by fallinl!: head tests. Parameters IX, nand 8r in the VG

model were determined for the samples by one of two means: (1) static 8(h)

drainal!:e data were fitted by nonlinear rel!:ression to equation 1 for 48 cores,

and (2) data from one-step desorption tests were inverted numerically as

described by Kool et al. (1985) to obtain VG parameters for all other cores.

Parker et al. (1985) have discussed the relative merits of these two methods

and have illustrated their comparability. For our present purpose, we assume

that both methods yield parameters of approximately equal overall accuracy in

describinl!: soil K-8-h relations.

PARAMETER ESTTIMATION METHODOLOGY

Computation of statistics of PSD data

Particle size data are normally available in terms of mass fractions for

each of several size classes. This may be converted to a cumulative

distribution function (CDF) and then fitted to some theoretical CDF to compute

descriptive statistics. The 101!:-normal distribution provides a reasonable

10

representation of PSD data for a wide ranJJ;e of soils as will be subsequently

shown. The lolt-normal distribution has a CDF Itiven by

P(d) = I .J27rUIn(d)

d

J exp m

[- 1 (y - ~In(d)l' ] d 2 aIn(d) y

(5)

where P(d) is the cumulative frequency correspondinJJ; to particle diameter d.

and /.lln(d) and Uln(d). respectively. are the mean and standard deviation of

y = In(d). It is convenient to replace (5) with the polynomial approximation

Itiven by Abramowitz and Steltun (1965) (their equation 26.2.19)

P(x) = I (6)

where x = [In(d) - ~n(d)l / Uln(d) is the standard normal variate. and the

coefficients are c, = 0.0498673470. c, = 0.0211410061, C3 = 0.0032776263. C4 = 0.0000380036. cs = 0.0000488906 and c. = 0.0000053830. The theoretical model

Itiven by (6) is fitted to measured PSD data usinlt nonlinear reltression to

obtain/Lln(d) and aIn(d)' Median particle diameter. d so• is then computed as

d so = eXP[/LIn(d)]. For the entire data set of 250 samples. d. o ranlted from

0.0017 to 0.463 mm. The averalte correlation coefficient (R') in fittinlt the

10JJ;-normal distribution to PSD data was 0.938 for the entire data set. the best

and worst fit values beinlt 0.998 and 0.773. respectively.

Estimation of saturated hydraulic conductivity

Dullien (1979) summarizes available data on representinlt saturated

conductivity. Rs. by expressions of the form

where a is a proportionality factor. 8s is the saturated water content. and dp

is 'some representative particle diameter. We assume the simple functional

forms f,(8s )" 8s ' and f,(dp ) E (d. o )'. thus

Ks = a 8s (dso )' (8a)

The factor. a, was determined by fiUinlt (8a) to each of the 250 samples in the

data set described previously. This proportionality factor was found to

depend on the standard deviation of the PSD, Uln(d). as follows

11

o

Io~(a) = 7.445 - 0.642 uIn(d) (Bb)

where Ks is expressed in cm h- 1 and d SD is in cm. The standard error of

estimate for 101'((a) was 1.176. Fi/!ure 1 shows the relation between 101'((a) and

uln(d) as well as the rel'(ression line I'(iven by (8b) and one standard deviation

error intervals. Notice the wide confidence band associated with the

prediction of the factor, a, which translates to an equally larl'(e uncertainty in

predictinz:; Ks since the standard error in eSLimatinl'( 10dKs) is the same as

that in estimating 101'((a).

)0,--------------------------------,

8

c 6

l-' m 10 c ~

o ..J

4

DD D

D

D

D

D

2+-------.--------r-------.------~ o 2 3 4

STANDARD DEVIATION OF InCd)

Fig 1. Relation between 101'«(a) and Uln (d) for the 250 sample data set.

o W 10-1

f-< ..J => 10-2

U ..J < U 10-:3

10-3 10-2 10-1 loD 10 1 10 2 10 3

MEASURED Kc

Fil'( 2. Comparison of measured and predicted Ks for data set.

Fil'(ure 2 compares measured Ks values with those predicted by (8) for the

calibration data set. Also shown are the 1:1 line of perfect al'(reement and one

standard deviation error intervals. As evidenced from Fil'(ure 2, there is

roul'(hly an order of mal'(nitude uncertainty in estimatinl'( Ks from the simple

Kozeny-Carman type equation (8). We emphasize here the fact this equation is

intended to provide only a crude estimate of order of mal'(nitude accuracy for

saturated conductivity in the absence of actual measurements. Note that the

12

use of loda) in (8b) is necessitated by the lare;e rane;e of variation in

measured values of Kg (over 5 orders of mae;nitude).

Oorrelation of porosity and saturated water content

Since saturated water content is influenced by stress history of the soil

to a e;reater extent than it is by particle size distribution. we assume that

some independent information must be available to define as' In practice. as

may be known directly since it is readily measured. Alternatively. bulk

density may be known either via direct measurement or indirect means (e.e; ••

e;eophysical tests). To estimate as from bulk density we first note that

porosity. +. is e;iven by the expression

+ = 1 - Pb 1 Ps (9)

where pb is the measured bulk density and Ps is the particle density assumed

to be 2.65 e; cm-'. Furthermore. as will be normally less than + due to air

entrapment. A reduction factor. F = as/+. was calculated for the entire data

set and was found to rane;e from 0.60 to 0.99. No trend was observed between

the reduction factor. F. and the e;rain size distribution statistics. dso or

"1n(d)' Therefore the mean over the entire data set was chosen as a

representative estimate for F. This value was found to be F = 0.911 with a

the standard deviation beine; 0.073.

Oonversion of PSD data to soil-water retention function

The basic premise of the procedure is that the a(h) relationship reflects

an underlyine; pore size distribution which can be deduced from PSD data

usine; the model proposed by Arya and Paris (1981). Their method involves

dividine; the particle size ODF into a number of fractions. assie;nine; a pore

volume and a volumetric water content to each fraction. and then computine; a

representative pore radius and a correspondine; capillary head. Details of the

Arya and Paris (AP) model are explained below.

Arya and Paris first assume that when the particle size CDF is divided

into several fractions. the solid mass in each fraction can be assembled into a

discrete domain with a bulk density equal to that of the natural structure

sample. The pore volume associated with each size fraction is e;iven by

i=l.::z, ....• n (10)

13

where Vpi is the pore volume per unit sample mass and Wi is the solid mass

per unit sample mass in the i-th class. Ps is the particle density (taken to be

2.65 g cm-'). and e = ;/(1-;) is the void ratio. The quantity Wi is essentially

the frequency for each size class such that the sum of all Wi is unity. The

volumetric water content. which is computed by pro.e:ressively fillin.e: the pore

volumes .e:enerated by each of the size fractions. is .e:iven as

(11)

where 8i is the volumetric water content represented by a pore volume for

which the lar.e:est size pore corresponds to the upper limit of the i-th particle

size ran.e:e. and Pb is the sample bulk density.

Two further assumptions are required to formulate a relationship between

pore and particle radii. These are: (a) the solid volume in a size fraction can

be approximated as that of uniform spheres with radii equal to the mean

particle radius for that fraction. and (b) the volume of the resultin.e: pores

can be approximated as that of uniform cylindrical capillary tubes whose radii

are related to the mean particle radius for the fraction. From (a). we have

(12a)

and from assumption (b). we have

(12b)

Here Vsi is the total solid volume in the assembla.e:e. n.i is the number of

spherical particles. Ri is the mean particle radius. ri is the mean pore radius.

and Ii is the total pore len.e:th. A first approximation for the total pore

length is obtained by equatin.e: it to the number of particles that lie alon.e: the

total pore path times the len.e:th contributed by each particle. Since the

shape of actual soil particles is non-spherical. the len.e:th contributed by each

particle would be .e:reater than the diameter of an equivalent sph~re. The

total number of particles alon.e: the pore len.e:th can therefore be approximated

by niB. w~ere P is a tortuosity exponent. The total pore len.e:th thus becomes

Ii = 2RiniB. Now combining (12a) and (l2b). substituting for Ii and

rearranging. one . obtains an expression for the mean pore radius

[ (I-P)] 1/2 ri = 0.8165 Ri en~ (13)

14

The value of 1Ii can be obtained from (12a) , while the tort~osity exponent, 8,

is to be evaluated empirically by calibratinlt the model against known data.

After the pore radii are calculated for each of the size classes

corresponding to a particular volumetric water content, the equivalent

soil-water capillary pressure can be obtained from

(14)

where hi is the soil-water capillary pressure, 'Y is the surface tension of

water, Pw is the density of water and It is acceleration due to Itravity.

To calibrate the AP model against the present data set of 250 soil samples,

particle size data for each soil were first converted to volumetric water

contents using (10) and (11). For, each value of e, a corresponding value of

pressure head, hVG, was computed from the van Genuchten model. (3), using

known values of IX, n, er and es . The tortuosity exponent, 8, was then

calculated by a nonlinear regression procedure to minimize the function

(15)

where hAP, the pressure head predicted by the AP model, was obtained from

(12)-(14). The averalte root mean square error for In(h) calculated for the 250

soil samples was 2.41 with a standard deviation of 1.62.

Arya and Paris (1981) concluded that although 8 varied between 1.31 and

1.43, an average value of 1.38 yielded satisfactory results for the entire data

set used in their study. Recently, Schuh et al. (1988) applied the AP model to

a number of soils and found that 8 ranged from 0.8 to 2.0 with a composite

average of 1.36, and noted a dependence of 8 on soil textural class. For the

data set employed here, 8 was found to vary between 1.02 and 2.97 with an

average value of 1.41. We also observed a correlation between (3 and uln(d),

the' standard deviation of particle size CDF, which could be represented by

8 = exp r 0.183 uln(d) 1 (16)

The standard error in estimating 8 from this equation was 0.195. Figure 3

shows the variation of 8 with uln(d) for the entire data set, the relationship

given by (16) and one standard deviation error intervals.

15

3.0,------------------------------------------,

2.5

2.0 o ;.J w m

1.5

1.0

o o

o 0 o o

o &0

0.5 +----------.----------.---------.----------4 o 2 3 4

Standard OQviatian of lnCd)

Fig 3. Relation between (3 and "1n( d) for sample data set.

Estimation of VG parameters from retention data

The van Genuchten parametric model. (3), represents 8(h) as a function of

three unknown parameters (0:. n, 8r ), assuminJl: the saturated water content,

8s • is known independently. These unknown model parameters can be

estimated by a nonlinear reJl:ression scheme. which seeks to minimize

t = (17)

where 8(hi) are the water contents Jl:enerated by the AP model. and 6(hi:",n,8r )

are those predicted by the VG model. Minimization of the sum-of-squares

function defined in (17) is achieved by the LevenIJerJl:-Marquardt modification

of the Gauss-Newton minimization alJl:orithm (Beck and Arnold, 1977).

Conversion of VG retention parameters to equivalent BC parameters

It is possible, in principle at least, to fit 8(h) data derived from the AP

model directly to (1) and estimate Be model parameters. However. since there

are siJl:nificant practical problems with such an approach (Milly, 1987). we

16

adopt an alternative method which involves first fitting 9(h) data to the van

Genuchten model and then converting VG parameters to equivalent BC

parameters using an empirical procedure proposed by Lenhard et al. (1989).

To estimate the BC parameter A, Lenhard et al. su):!):!est eQuatin):! the

differential fluid capacities, aSe/ah, of the VG and BC models at Se = 0.5.

Usin):! (1) and (3), this leads to

A = m [ 1 - 0.5,/m 1 (18)

(l-m)

where m is related to the VG parameter, n, by m = I-lin. The BC parameter,

hd' which represents the air-entry capillary head, can be obtained by

equating the functions at some match-point effective wettin):! fluid saturation,

= - 'IA Sy [ - -'1m Sx - 1

IX

l,-m (19)

where '" is a VG model parameter, and Sx is the match-point effective

saturation ):!iven by the followinl! empirical expression

Sx = O. 72 - O. 35 e}:p r -n 41 (20)

Equation (20) was developed by minimizin):! deviations between Se-In(h) curves

predicted by VG and BC models usin):! a wide ran):!e of soil properties. Via

(18)-(20) the VG model parameters, '" and n, may thus be converted to BC

model parameters, A and hd'

The estimation of VG andlor BC retention parameters from PSD data is

based on the assumption that the retention function predicted by the AP

model reasonably approximates true soil behavior. When soil structure

deviates from the simple physical model postulated by Arya and Paris (e.):!.,

due to a):!gregation induced by clay fractions Dr or):!anic matter) these

procedures may no lon):!er apply.

EVALUATION OF PARAMETER UNCERTAINTY

There are several sources of error which produce imprecision in parameter

estimates determined by the procedure described in the previous section.

17

Estimates of saturated conductivity, Ks, are imprecise due to error ·in

estimating' or measuring' the saturated water content, 8s ' uncertainty in the

estimated value of ds 0' and uncertainty in the KozenY-Carman parameter, a,

Errors in the estimated VG parameters, IX, nand 8r , may arise due to

uncertainty in AP model-g'enerated 8(h) data associated with uncertainty in the

tortuosity exponent, f1, as well as imperfect correspondence between AP

model-g'enerated 8(h) data and the fitted VG model. Uncertainty may also

arise due to inherent inability of the assumed parametric K-8-h model to

accurately describe true soil properties. When BC parameters are estimated,

errors in VG parameters are propai!ated durini! their conversion to equivalent

BC model parameters. Methods of quantifying' uncertainty in parameter

estimates due to each of these sources are addressed next.

General methodolo.<rY of uncertainty analysis

The uncertainty associated with a process due to error in its parameters

is . commonly evaluated usini! either Monte-Carlo simulation or first-order error

analysis (e."., Ben.iamin and Cornell, 1970). Monte-Carlo simulation involves

forming' random vectors of input parameters from prescribed probability

distributions, repeatedly simulatini! the process, and computing' summary

statistics (i.e .. mean and variance) of process performance. First-order error

analysis is based on a Taylor expansion around the mean values of parameters

assuming' small parameter perturbations and neg'lig'ible hig'her-order terms.

Summary statistics can be estimated if mean, variance and/or covariance of the

input parameters are known. This work uses the first-order error analysis

approach to evaluate parameter uncertainty as outlined in the following'.

Consider a quantity, f, which depends on the parameter vector, x. A

first-order Taylor expansion then g'ives

f(x) + 1: i

where x is the vector of estimated parameters.

operator on both sides of this expression, we obtain

E[f] + 1: i

(21)

Usini! the expected value

(22)

Assuming small parameter perturbations around tbe mean values. we get

(23)

18

The variance of f is defined as

Var[f] = Uf2 = E [ ( f - E[fl )2 ] (24)

and can be calculated by substitutin~ (21) and (23) in (24). which leads to

Vadfl " E E af af

E [ (x--;;')(x .-;;.) ] ----i .i aXi ax' ~ ~ .1 .1

.1

Vadf1 " E E af af

Cov[x'x'l (25) ----i .i aXi ax' ~ .1

.1

The covariance of two random variables. Y, and Y2 • where both are functions

of the parameter vector. x. can be obtained in a similar manner as

Cov(Y,Y2l = l: l: i .i

Cov[x'x'1 ~ .1 (26)

These expressions. i.e. (25) for the variance and (26) for the covariance. will

be used to estimate parameter uncertainties due to input parameter error.

Error in estimatim! saturated conductivity

The variance of Ks. assumin~ independence of the parameters a. 8 sand

d sD• can be derived from (8a). using (25). as

Vadlo~(Ks) J Vadlo~(a)l + + (27)

As before. we specify the variance in lo.e:(Kg). since saturated conductivity. Kg.

may ran.e:e over several orders of ma.e:nitude. The first term on the ri~ht

hand side was previously determined to be 1.383 (i.e. 1.1762). The estimated

value and variance of d SD are obtained when fittin~ the "lo~-normal

distribution to a specific soil particle size CDF. When the value of 8s is

known. Var[8s 1 can be calculated from the uncertainty in 8s • if any.

Otherwise. 8s is calculated from porosity and bulk density data usin~ (9). and

the variance of 8s is given by

Vad8s 1 8

2

s = [

VarfF1 F2 +

18

F2 Varf Ph 1 ] 28 2

Ps s (28)

Substitution of (28) in (27) results in an estimate of the uncertainty in Irs. In

.£(eneral, the coefficient of variation for either Bs or dso is not expected to be

.£(reater than 10%. Thus, both the second and third terms in the ri.£(ht hand

side of (27) will be roughly ~0.01 and hence the uncertainty in predictin.£( Ks

will be dominated by the first term in (27). i.e .. Varno.£«a)l. This is an

artifact of the attempt to model permeability in a broad variety of soils usin.£(

the simple expression given in (8). and underscores the need for makin.£(

independent physical measurements of Ks'

Uncertainty in estimatinJ! VG model parameters

Information concerning uncertainty in the parameter estimates obtained by

solvin.£( (17). i.e. in fiUin.£( the VG model to B (h) data, is contained in the

parameter covariance matrix, defined by

C = E [ (b - b) (b - b) T 1 (29)

where b is the vector of estimated parameters. b is the vector of true

parameters, and E denotes statistical expectation. For nonlinear reg;ression

problems. a first-order approximation to the covariance matrix. C. is g;iven by

(Beck and Arno1 i, 1977)

c = (30)

where s' is the least squares error. J is the parameter sensitivity matrix. M is

the number of observations and P is the number of unknown parameters. The

elements of C are computed during; the nonlinear estimation of a. nand Br

from B(h) data .£(enerated with the AP model. The elements of C are the

individual parameter variances. Var[al. Var[n] and VarrBrl. and the

covariances. Cov[anl. Cov[nBrl and Cov[aBrl.

The error covariance matrix associated with fittin.£( the van Genuchten

model to AP model-predicted B(h) data is assumed to represent uncertainty

due to the inherent inability of the VG parametric model to accurately

represent true soil K-B-h relations. This surrog;ate for estimating; inherent

model error is taken as a reasonable approximation in view of the lack of

complete measurements of K(B) and B(h) for the calibration data set which

would enable more direct assessment.

19

covariance matrix associated with the BC parameters can then be estimated

from the correspondin!': matrix for the VG parameters usin!': (25) and (26).

These are !':iven by the followin!,: expressions

Uhd

2 " [ ~ r U 2 + [ ~ r U 2 + 2~~ Uo:n ao: 0: an n acx. an

[ 2

U)! " aA

] un2

an

uAhd " ~~ Uo:n + aA ahd U 2 (34) an ao: an an n

u9 r hd " ahd

u0:9 r + ahd un9r aex an

The variance of 9r is unchan!,:ed durin!': this process. The sensitivity

coefficieni.s. ahdlan. ahdlaex and aA/an are computed numerically from (18) and

(19) usin(; a forward difference approximation similar to that used in (33).

EXAMPLE APPLICATION_S

The parameter estimation and uncertainty analysis met"odolo!,:y described

in the previous sections has been implemented in an interactive FORTRAN code

SOILPROP (Mishra and Parker. 1989a). To test the methodolo!':y. soil-water

retention functions and their uncertainty were evaluated for three soils not in

the· calibration data set. and for which direct measurements of the 9-h

relations were available. Particle size distribution and bulk density data for

these soils are riven in Table 1. Soil 111 is the Caribou silt loam described by

Topp (1971). Soil 112 is the well-!':raded sandy sample referred to as Soil 1 in

Lenhard and Parker (1988). Soil 113 is the sandy clay loam horizon Btl at site

1 of the Norfolk sand in Blackville. South Carolina. USA. reported by

Quisenberry et al. (1987).

21

Table 1. Particle size distribution data, and bulk density data for example problems.

Bulk

Particle Dia (mm)

1. 000 - 2.00 0.500 - 1. 00 0.250 - 0.50 0.100 - 0.25 0.050 - 0.10 0.002 - 0.05

< 0.002

Density (g cm-' )

Soil III Soil lI2 Soil lI3

B.7 5.0 2.B 5.B 7.1

54.4 16.2

1.11

Percent mass in each size class

3.3 2.0 40.3 11.1 37.0 20.1 14.0 17.4 2.6 4.0 O.B 15.7 1.9 29.7

1.65 1.58

The PSD data of Table 1 were used to compute the van Genuchten

retention parameters (oc. n. 9r lo the Brooks-Corey retention parameters

(hd. A. 9r ) and the correspondinl! error covariance matrices usinl! the methods

described in the previous section. Results of this analysis are presented for

the VG model in Table 2. and for the BC model in Table 3.

Table 2. Estimated VG retention parameters and error covariance matrix for example problems.

Estimated Covariance Matrix Soil Value IX n 9r

oc (cm- 1 ) .llle-Ol .602e-04 lI1 n .1l7e+01 -.24Be-03 .71le-02

9r .375e-06 -.386e-03 .614e-02 .692e-02

IX (cm- I ) .41Se-01 .303e-03 lI2 n .21ge+Ol .352e-02 .44Ie-Ol

9r .554e-02 .104e-04 .153e-03 .202e-05

oc (cm-') .768e-Ol .106e-02 lI3 n .120e+Ol -.396e-03 .25Se-02

9r .64ge-01 -.336e-03 .769e-03 .392e-03

22

Table 3. Estimated BC retention parameters and error covariance matrix for example problems.

Estimated Covariance Matrix Soil Value hd A 8r

hd (cm) .810e+02 .274e+04 #1 A .16ge+00 .968e+00 .654e-02

8r .375e-06 .20ge+Ol .58ge-02 .692e-02

hd (cm) .150e+02 .34ge+02 #2 A .856e+00 -.684e+00 .145e-01

8r .554e-02 -.352e-02 .874e-04 .202e-05

hd (cm) .1l3e+02 .207e+02 #3 A .197e+00 .17ge-01 .224e-02

8r .64ge-01 .371e-01 .717e-03 .392e-03

The retention function. 8(h). was calculated for each soil from the retention

parameters usinl'< (Ia) for the Be model and (3a) for the VG model. The error

in predictinl'< these functions was evaluated by expressions similar to (25) from

the estimated covariance matrices and numerically computed sensitivity

coefficients. Measured retention data are compared with VG/BC model

predictions in Fil'<ures 4-6. Also shown as dashed lines are one standard

deviation error intervals associated with the predicted functions. For Soil #l.

the VG model underpredicts capillary heads at hil'<h water contents. and

overpredicts heads at low water contents. The Be model consistently

overpredicts capillary heads for this soil. The al'<reement between measured

and predicted capillary heads is much better for Soil #2. althoul'<h there is

some underprediction with cboth VG and Be models for near-saturated

conditions. On the other hand. capillary heads are underpredicted at low

water contents for Soil #3 with both VG and Be models. In I'<eneral. the

al'<reement between measured and predicted 8 (h) data is reasonably I'<ood

considerinl'< that the predictions were based on PSD data only. Moreover. the

measured retention functions are found to lie within the one standard

deviation error interval bounds for all three soils analyzed.

23

FiJ'! 4. Soil n.

" 10:

E U

v

lD'

0 < W I lD Z

>-0: < -l 1 D 1 -l ~

n.. < U JOe

0.30

" 1 0 ~ E U

v

JO' 0 < W I

lD' >-0: < -l

lD 1 -l ~

n.. < U

JOe

0.30

SOIL N I VC modQ] -"'- ... ---... - .. .....:. Co .. _

- ... ~ C C -----_!:_ .. c "-,,_ .. -"'__ c ........

_... .. ..... 0,

-"'-.. \0 , , " \" \'

e pr",dictQd mClolllurl:ld

D.3S 0.40

WATER CONTENT

SOIL ~ I

, \

0.45

c~-______________ ~~_m_C_dO_'_

--~ C C - ___ 0 c ::: __ -;.-___ = -------':...__ C

e prQdictgd mClo&urc:ad

0.35

-----.EL __ c

0.40

WATER CONTENT

O. 45

Coma prison of measured and predicted retention data for Dashed lines show one standard deviation error intervals.

24

~

E U

v

0 <: l1J I

r a: <: ...J ...J ~

CL <: u

~

Fig 5. Soil 112.

E U

v

0 <: l1J I

r a: <: ...J ...J ~

CL <: U

]0'

]0.

]0'

]OC

0.05

10'

10·

10 '

IOc

0.05

-'-

C

-, -, -, -,

SOIL iI 2 VG modgl

-'-__ Do ........ -___ c~

prgd1ctgd mlOlcs:ur-gd

'_ C .......... \\0 "-,

O. 10 O. 15 D. 20 O. 25 D. 30 0.35 D. 40

c

WATER CONTENT

prgd1ctgd ITIQCGur""gd

SOIL ~ 2 Be modQl

-'­'-,

-------~

D. 10 D. 15 D. 20 O. 25 O. 3D O. 35 O. 40

WATER CONTENT

Comparison of measured and predicted retention data for Dashed lines show one standard deviation error intervals.

25

FiJ! 6. Soil #13.

1"'\ 10.t E U

v

o < UJ I

10 • >­a:: < ...J ..J 10 1 ~

Q.

< U

o pz-gd1etQd mg",.,urgd

10 D +---,----,-----,----4>----1 0.20 0.25 0.30 0.35 0.40 0.45

WATER CONTENT

A 10· E U

V SOlL N ::3

10' Be medAl

D

0 '0

< , . UJ ;?i- ........ I

10 • c ........ >- -,-a:: --, < - -...J '-p

...J 10 ' -"'- ... -~

prcdlctgd Q. 0 < mQc .. urgd

U IO

D

0.20 0.25 0.30 0.35 0.40 0.45

WATER CONTENT

,

Comaprison of measured and predicted retention data for Dashed lines show one standard deviation error intervals.

26

Retention parameters derived from PSD data were also used to predict the

relative conductivity function. Kr = K(B)/Ks ' for Soil #1. which was the only

soil for which measurements of Kr were available. Fi~ure 7 shows the

comparison between measured relative conductivity data and those predicted

usin/! the VG model (4) and the BC model (2). alon/! with tID" error intervals.

The VG model consistently underpredicts the relative conductivity function.

whereas the BC model consistently overpredicts this function. However. it is

not possible to make any ~eneral conclusions as to the choice of any

particular model (i.e .• VG or BC) for predictin/! unsaturated conductivity from

retention parameters based on just one sample.

H

> -l H 10 I-U ::J o Z o u w > H

I-- 10-" < --1 W e::: 10-.!!

r 100

I-H

> H

I-

O. 25

~ 10-1

o Z o U

W 10-2

> H

I­< --1 W e::: 10-2

O.2S

SOIL II l' VG modgl

0.30 0.35 0.40

WATER CONTENT

0.30 0.35 0.40

WATER CONTENT

0.45 0.50

0.45 0.50

Fi/! 7. Comparison of measured and predicted conductivity data for Soil 111. Dashed lines are tID" error intervals.

27

Topp (1971) reports a saturated conductivity value of 0.598 cm hr- I for

Soil til. whereas equation (8) predicts a value of 2.04 cm h- I with one

standard deviation error intervals being 0.135 cm h- ' ,f Ks ,f 30.96 cm h- I •

For Soil 113. Ks was estimated to be 1.27 cm h- ' by extrapolating measured K-8

data to 8=8s • while the predicted value using (8) was found to be 6.216 cm h- 1

with one standard deviation error intervals of 0.41 cm h- 1 ,f Ks ,f 94.1 cm h- 1 •

No saturated hydraulic conductivity measurement was available for Soil 112.

although Ks was predicted to be 566.4 cm h- 1 with an error interval of 37.44

cm h- 1 ,f Ks ,f 8568 cm h- 1•

the argument for making

The magnitude of these uncertainties reinforces

independent physical measurements of Ks. and

su,,;,Etests that the use of (8) as a predictive tool is appropriate for providins:

order of mas:nitude estimates only.

S~Y AND CONCLUSIONS

Although methods for estimating soil hydraulic properties from PSD data

have been reported in the literature previously (e.g .• McCuen ·et m.. 1981.

Arya and Paris. 1981, Campbell. 1985: Rawls and Brakensiek. 1985). we believe

this IS the first work to provide a methodolos:y for Quantifyins: the

uncertainty in these parameter estimates. A unified approach to parameter

estimation. which provides a mean value for soil properties as well as the

associated error. is of fundamental importance in assessing the reliability of

unsaturated flow model predictions. Elsewhere we present applications of the

approach developed in this work to examine the uncertainty in field-scale

simulations of unsaturated flow (Mishra and Parker. 1989b).

Although PSD data provide an easy source for estimatins: soil properties.

it is important to remember that these parameter estimates. particularly Ks.

may· be associated with large uncertainty. Another consideration in the use of

PSD data and parameters derived from them is that they represent very small

sampling volumes. Proper scaling of such small-scale parameters to effective

parameters at the grid-block scale of a numerical model is as yet a lars:ely

unresolved problem. especially for unsaturated flow - although some recent

studies have begun to address this issue (Mantoglou and Gelhar. 1987).

28

REFERENCES

Abramowitz. M. and 1. A. Steg:un. 1965. Handbook of Mathematical Functions.

Dover Publications Inc .• New York. 1045 PP.

Arya L. M. and J. F. Paris. 1981. A physico-empirical model to predict soil

moisture characteristics from particle-size distribution and bulk density

data. Soil Sci. Soc. Am. J .• 45:1023-1030.

Beck. J. V. and K. J. Arnold. 1977. Parameter Estimation in Enl!ineering: and

Science. John Wiley and Sons. New York, 393 pp.

Benjamin, J. R. and C. A. Cornell. 1970. Probability. Statistics. and Decision for

Civil Engineers. McGraw Hill Book Co •• New York. 684 pp.

Brooks. R. H. and A. T. Corey. 1964. Hydraulic Properties of Porous Media.

Hydrology Paper No.3. Colorado State U .• Fort Collins.

Campbell. G. S. 1985. Soil Physics with BASIC, Transport Models for Soil-Plant

Systems. Elsevier Science Publishing: Co. Inc .• New York. 150 pp.

Day, P. R. 1965. Particle fractionation and particle size analysis. In C. A. Black

et ai. (ed). Methods of Soil Analysis. Monol!raph 9. American Society of

Al!ronomy. Madison. 1:545-567.

Dullien. F. A. L. 1979. Porous Media Fluid Transport and Pore Structure.

Academic Press. Inc .• New York. 396 pp ..

Kool. J. B .• J. C. Parker and M. Th. van Genuchten. 1985. Determining: soil

hydraulic properties from one-step outflow experiments by parameter

estimation: I. Theory and numerical studies. Soil Sci. Soc. Am. J .•

49:1348-1354.

Kool. J. B .• J. C. Parker and M. Th. van Genuchten. 1987. Parameter

estimation for unsaturated flow and transport models. A Review. J.

Hydrol.. 91:255-293.

Lenhard. R. J. and J. C. Parker. 1988. Experimental validation of the theory of

extending: two-phase saturation-pressure relations to three-phase systems <

for monotonic draing:age paths. Water Resour. Res .• 24:373-380.

Lenhard. R. J.. J. C. Parker and S. Mishra. 1989. On the correspondence

between Brooks-Corey and van Genuchten models. ASCE J. Irr. Dr. Eng:.

(in press).

Mantol!lou. A. and L. W. Gelhar. 1987. Stochastic analysis of larl!e-scale

transient unsaturated flow systems. Water Resour. Res .. 23:37-46.

29

McCuen. R. H •• W. J. Rawls and D. L. Brakensiek. 1981. Statistical analysis of

the Brooks-Corey and Green-Ampt parameters across soil textures. Water

Resour. Res .• 17:1005-1013.

Milly. P. C. D. 1987. Estimation of Brooks-Corey parameters from water

retention data. Water Resour. Res .• 23:1085-1089.

Mishra. S. and J. C. Parker. 1989a. User's Guide to SOILPROP. Environmental

Systems and Technolo.e:ies. Blacksbur.e:. 7pp.

Mishra. S. and J. C. Parker. 1989b. Effects of parameter uncertainty on

predictions of unsaturated flow. J. Hydrol. (in press).

Parker. J. C .• J. B. Kool and M. Th. van Genuchten. 1985. Determinin.e: soil

hydraulic properties from one-step outflow experiments by parameter

estimation: II. Experimental studies. Soil Sci. Soc. Am. J .• 49:1354-1359.

Quisenberry, V. L., D. K. Cassel. J. H. Dane and J. C. Parker. 1987. Physical

Characteristics of Soils in the Southern Re.e:ion. SCS Bulletin 263. South

Carolina A.e:. Expt. Station. Clemson U .• 307 pp.

Rawls, W. J. and D. L Brakensiek. 1985. Prediction of water properties for

hydrolo.e:ic modellin.e:. Proc. Symposium on Watershed Mana.e:ement. ASCE,

Denver, CO. 293-299.

Schuh. W. M .• R. L. Cline and M. D. Sweeney. 1988. Comparison of a laboratory

procedure and a textural model for predictin.e: in situ soil water

retention. Soil Sci. Soc. Am. J .• 52:1218-1227.

Topp, G. E. 1971. Soil water hysteresis in silt loam and clay loam soils. Water

Resour. Res .• 7:914-920.

van Genuchten. M. Th. 1980.

hydraulic conductivity of

44:892-899.

A closed-form equation

unsaturated Boils. Soil

30

for predictin.e: the

Sci. Soc. Am. J.,

Two common methods for evaluatinlt, the uncertainty as socia

process due to error in its parameters are Monte Carlo simu Iti

first-order error analysis (e.lt.. Benjamin and Cornell. 1970). Iv

simulation involves forminlt random vectors of input paramE er

prescribed probability distributions. repeatedly simulatinlt the PI

computinlt summary statistics (i.e •• mean and variance) of process p 'fe

First-order error analysis is based on a first-order Taylor expam

the mean values of parameters assuminlt small parameter pel UI

Summary statistics can be estimated if mean. variance and/or covari

input parameters are known.

The first-order error analysis procedure has proven to be me P.

than Monte Carlo simulation as it entails considerably less computal

for problems with small numbers of parameters while often provid lit

of comparable accuracY. It has been used by several invest

analyzinlt the uncertainty in predictions of subsurface flow models RI

(1977) used published data on mean and variance of the soil micro'

properties to estimate the mean and variance of soil water flux in

profile for Panoche soil usinlt both first-order error analysis and

simulation and noted ltood agreement between results from both ro

Tanlt and Pinder (1977. 1979) used a first-order error analysi~ D

based on perturbation theory to simulate ltroundwater flow and ma~

under parameter uncertainty. Dettinlter and Wilson (1981) applied [it

analysis of uncertainty usinlt Taylor expansion of vector quan'

derived expressions relatinlt the mean and variance of

predictions to the statistics of aquifer parameters. Wal':ner and Gol

used a numerical model of saturated flow and transport wit

regression to estimate model parameters and their uncertainty. Th

error covariance matrix was used to quantify the error in simulatE

heads and concentrations usinl': first-order error analysis. which .u,

with nonlinear stochastic optimization for I':roundwater quality mana _.

The objective of this paper is to apply first-order er~ r I techniques to assess the reliability of unsaturated flow model

I subject to parameter uncertainty for two specific cases. The fi It I estimatinl': parameters and their uncertainty from particle size

(PSD) data for a layered field soil and comparinl': predictions of su id

water content distributions durinl': a drainalte experiment ,,'

32 I . ,

grel

flui

sub

com

defi

of I

be (

thei

unc~

the

over

para

medi

accu

para

para

unsa

measurements. The second case concerns a hypothetical homo/teneous soil with

parameters estimated by numerical inversion of a transient flow experiment.

Predictions of water content distributions are compared with 'actual' and

estimated parameters for a simulated rainfall-redistribution sequence. In both

cases, the nature of uncertainty in model predictions caused by imprecision in

model parameter estimates is examined.

THEORY

Governinl'f Equation for Unsaturated Flow

The governin", equation for one-dimensional vertical transient flow in a

non-deformable porous medium is "iven by

where h is pressure head (L), x is depth (L) and t is time (T); C = dS/dh is

the water capacity with S the volumetric water content, and K is the hydraulic

conductivity (LT-l). Tbe nonlinear unsaturated hydraulic properties, K(S) and

9(h), are assumed to be described by van Genuchten's (1980) parametric model

as modified by Kool and Parker (1987) to allow for hysteresis and air

entrapment in the S(h) relation. Details of the model are "iven by Kool and

Parker (1987) and only the most salient features are reviewed here. The main

draina",e branch of the S(h) relation is described by

-m S = { 9r + (Ssd - 9r ) [1 + (-~dh}nl

Ssd

and the main wetting branch by

-m

S = { Sr + (SsW - Sr) [1 + (-~Wh}nl

Ssw

33

h " 0

(2) h'" 0

h " 0 (3)

h '" 0

where 8r (VL-') is the residual water content, 8s d and 8sw (V r;-') are

satiated water contents correspondinJt to zero pressure head on the main

drainaJte and main wettinJt branches, respectively, cP and exW (L- 1 ) are curve

shape parameters for drainaJte and wettinJt branches, n (LO) is a parameter

assumed independent of saturation path and m = I-lin. ScanninJt curves in

the 8(h) function are determined usinJt an empirical scheme proposed by Scott

et al. (1983) with satiated water contents of primary weitinJt scanninJt curves

assiJtned using a method due to Land (1968). This procedure introduces no

additional parameters in the constitutive relation model.

The conductivity function, K(8), assumed to be non-hysteretic, is

(4)

where Ks = Ks(8 s d ) (LT-l) is the saturated conductivity and Se = (8-8 r) I (8 sd-8 r) is the effective saturation. For monotonic saturation paths. or

when hysteresis and air entrapment are neJtliJtible. the parametric model

described by (2)-(4) reduces to van Genuchten's ori)!inal model.

Nethodolol[Y of First-order Error Analysis

Consider a Quantity, f, which depends on the parameter vector, x. A

first-order Taylor expansion then gives

f(x) = f(x) (5)

where x is the vector of estimated parameters. UsinJt the expected value

operator on both sides of this expression, we obtain

Erfl = f(x) + af ~

1: -- Erx'-x'l . ax' ). 1. ). ).

(6)

Assuming small parameter perturbations around the mean values, we get

Erfl = f(x) (7)

The variance of f is defined as

Varifl = af' = E [ (f - Erfl), ] (8)

34

which can be calculated by substituting (5) and (7) in (8), leading to

Var[f]

Varff1

E [ (x·-x·)(x·-x·) ] ~~ .J.J

covrx-x'l ~ .J (9)

Thus, the expected value (first moment) is the same as that obtained with the

estimated parameters, while the variance (second moment) depends on the

variance-covariance relation of the input parameters as well as the sensitivity

of the process to these parameters. Although previous workers have sometimes

used a second-order approximation for the expected value (e.g. Dettinger and

Wilson, 1981), only first-order approximations as given in (7) and (9) will be

considered in this work.

The applications considered here involve the use of numerical models of

unsaturated flow. Hence it is necessary to obtain sensitivity coefficients by

parameter perturbations. The sensitivity of a system attribute, f, (e.g. water

content, e) to anY arbitrary parameter, xi, is evaluated approximately as

f(Xj+AXj) f(Xj)

where AXi is taken to be O.Olxi'

Back.<!round

ERROR ANALYSIS FOR LAYERED FIELD SOIL

WITH PARAMETERS ESTTiMATED FROM PSD DATA

(10)

Data for the field soil considered here were reported by Quisenberry et

al. (1987). The ;oil belongs to the Norfolk series in Blackville, South Carolina,

and is described as fine-loamy and siliceous with five identifiable soil

horizons. Experiments were conducted in the field and in the laboratory to

collect information on (a) texture and density, (b) soil water retention

characteristics of undisturbed cores, (c) in-situ soil water tensions and soil

water contents during drainage, and (d) hydraulic conductivity as a function

of water content for each layer also determined from in situ drainage data.

35

'we used particle size distribution and bulk density data to predict the

retention function, e(hl. and compared it with measured retention data. The

field drainal<e experiment was then simulated usinl< soil properties estimated

from PSD data to predict in situ tensions and water contents, which were also

compared with measured data. The uncertainty in these predictions was

quantified from the uncertainty in parameter estimates usinl< first-order error

analysis procedures described in the previous section.

Data Collection

Procedures for data collection in the field and laboratory are discussed in

detail by Quisenberry et al. (1987), and are only summarized here. In the

field, a plot 3.06 m by 3.06 m was prepared and instrumented with

tensiometers to a depth of 152.4 cm at rel<ular intervals. Water was ponded at

the surface until no sil<nificant chanl<e in pressure heads were noted. This

was followed by a 30 day drainal<e period with the soil surface protected

al<ainst evaporation and rainfall. In situ water contents and capillary tensions

were measured periodically. Unsaturated hydraulic conductivity was computed

as a function of water content by the instantaneous profile method. For

laboratory analyses, 7.5 cm lonl< by 7.5 cm diameter cores were taken in

triplicate at each tensiometer depth. Particle size data were obtained by usin~

the pipete method (Day, 1965) for clay, with sand determined by wet sievin~

and silt by difference. Soil water retention data were obtained by desorption,

followinl< which bulk density was estimated I<ravimetrically.

Table 1. Particle size distribution and bulk density data for layered soil.

% mass in size classes (nnn ) Bulk Layer Depth 2.0- 1. 0- 0.5- 0.25- 0.1- 0.05- < density

ern 1.0 0.5 0.25 0.10 0.05 0.002 0.002 I< ern-3

1 15.2 1.27 14.03 28.93 26.76 6.20 18.37 4.47 1. 79 2 30.5 1.53 9.10 21.43 21.10 5.07 17.87 23.87 1.66 3 45.7 1.97 11.13 20.13 17 .. 37 4.00 15.67 29.70 1.58 4 91. 7 2.50 12.10 20.53 17.76 3.97 12.90 30.30 1.53 5 121.9 3.00 14.33 20.37 16.97 3.83 12.20 29.27 1.66

36

Estimation of retention function from PSD data

The parameters of the van Genuchten model (2)-(4). assumin",

non-hysteretic constitutive relations. were estimated from PSD data (Jl:iven in

Table 1) usinJl: the model of Arya and Paris (1981) as implemented by Mishra et

al. (1989) with the parameter error covariance matrix estimated usinJl:

first-order error analysis. This procedure was carried out for each of the five

soil layers. with results Jl:iven in Table 2.

Table 2. Hydraulic properties estimated from PSD data for layered soil.

Layer Property Estimated Covariance Matrix Value ex n Sr

1 ex (em-I) 0.029 0.345e-03 n 1.503 0.293e-02 0.277e-Ol Sr 0.017 0.287e-04 0.353e-03 0.1l5e-04 Ks (em h-1 ) 0.16711

2 IX (em-I) 0.049 0.686e-03 n 1.250 0.122e-02 0.681e-02 Sr 0.053 -0. 340e-04 0.694e-03 0.181e-03 Ks (em h- 1 ) 1. 33311

3 ex (em-I) 0.074 0.240e-02 n 1.187 0.176e-03 0.200e-02 Sr 0.063 -0. 744e-03 0.441e-03 0.491e-03 Ks (em h-1 15.37111

4 ex (em-I) 0.066 0.109e-02 n 1.230 0.622e-03 0.568e-02 Sr 0.083 -0.145e-03 0.805e-03 0.258e-03 Ks (em h-1 ) 13.46511

5 ex (em-I) 0.070 0.145e-02 n 1.250 0.877e-03 0.498e-02 Sr 0.080 -0.179e-03 0.473e-03 0.166e-03 Kg (em h-1 ) 5.27111

#I - Ks obtained by fittin~ measured K(S) data to the VG model.

37

10' ~

E 0

v

0 10 " < W I

W n:

10' ::J UJ 1I}

w n: [L

10"

0.00

10' ~

E 0

v

0 10" < W I

W n: ::J 10' UJ 1I} W n: [L

10"

D. )S

10' \ , • , , ,

~ , • , '\ LAYER E , ,

LI\YER 2 , • 0 ,

• , , , , , v , , , , - • -,

................... Ctiibo 0 10 " -'- -, -

......... -.. -~- < , , W -'- ,

'\ I , ,

................ , , '\ - W , ':

, '\ n: \ ,

10' '\ , ::J , \ 1I} \ , prgd1ctAd 1I} prAdJctCld ,

I 0 1n-situ dClt= W " in-situ dota \ , • cc;lrCl data n: • cor-I:! doto \1 [L , ,

10" ,

0.05 O. 10 O. 15 0.20 0.25 0.30 D. ]0 0.15 0.20 0.25 0.30 0.35

WATER CONTENT . WATER CONTENT

'\ 10' • '\ , , ~

,. '\ • , , , E LAYER :3 '\ ,~ , •

, 0 , , , , LAYER 4

" •

, , v " , , , , , , , ,~

, , , , , , 10"

, , , 0 , , , < , , '"" , , ,

W -, ~O~ , -, '\ I " . , , , , , '\ , \ ,

"\ , • W , '\

, n: , , \ 10' '\ ,

'\ , ::J , \ prAd1ctcd ,

\ 1I} '\ , \ 1I} prcrd1ctgd in-lPltu dote , ,

cerCI dote! \ w " in-situ dota \ n: • corl:! dota , '

0.25 0.35

WATER CONTENT

)0 ' ~

E 0

v

0 )0 " < W I

W n:

10' ::J 1I} 1I}

w n: [L

10"

O. 10

Fi" 1. Comparison data for field soil.

, [L

10" 0.45

D. 15 0.25 0.35

WATER CONTENT

, '\ • '\ , '\ , •

'\ , LAYER 5 , , • '\ ,

" \, , , ,

" '" , , '''' , '''' , , "01 , , .. , , .' , .b, '\ , , \

'\ , prl:ldlctQd ,

\ " in-situ date. \

\1 • corQ data , , , 0.20 0.30 0.40

WATER CONTENT

of predicted and laboratory measured retention Dashed lines represent "10" error intervals.

38

, \ \ , , \ \ , , ,

0.40

0.35

z: .!.I

0.30 z: :J J 0.25'

r .!.I 0.20 . <:

" o. 15

·0. 10

0

Soil water retention curves calculated from the predicted van Genuchten

parameters are compared in Filrure 1 with measured retention data from

undisturbed cores. The agreement between measured and predicted data is

relatively Irood for the top three horizons. but is poor for the bottom two

layers. In general. the measured values fall within one standard deviation

intervals around the predicted values (shown as dashed lines in Filrure 1).

Simulation of drainage experiment

The drainage experiment was simulated usinlr a linear finite element code

(van Genuchten. 1982). The soil profile was assumed to be initiallY saturated.

Drainage was modeled by usinlr a zero pressure Irradient condition at the

lower boundary (assumed to be located at 160 em). Saturated conductivity

values were estimated by fittinlr the unsaturated hydraulic conductivity

function obtained with the instantaneous profile method to the van Genuchten

model (4). This approach was adopted because preliminary simulations with

saturated conductivity values derived from particle size· data via a modified

Kozeny-Carman type equation (Mishra et al .• 1989) did not yield Irood results.

Other hydraulic properties were taken to those obtained from PSD data.

--------;,. .. _---..! 1.- - _- ---, ,----

/ / . , " / I , I ,

I . , I o

I prCldictCld

• Clb":ClrVgd

. 3D 60 90 120 150

DEPTH (em)

O.'O~---------------------------------'

I- 0.35 Z W f-Z 0.30 o U

ffi 0.25

f-< 3: 0.20

.. - .. -::=.:::. ... _--/-----: .. --_. --/' .--, r--

/ o' I ,

I o

I .0 I ,

I •

PrQd1ctcd ObslOIrvCld

0.15~----~r_----_r--~--._-----T----__1 o 3D 60 90 120 15[

DEPTH

Filr 2. Comparison of predicted and measured water content profiles for field soil at 25 hr (left) and 361 hr (rilrht).

39

L

Soil water content profiles were calculated at 25. hand 361 h. and are

compared with measured data in FiJture 2. The aJtreement between both data

sets is surprisinJtly good considerinJt the assumptions made in estimatinJt soil

properties. The uncertainty in predictions of water contents was estimated by

the first-order error analysis procedure described previously. The numerical

code was modified to enable computation of the sensitivity coeffcients. as well

as the overall variance in model predictions usinJt (9). One standard deviation

error intervals associated with each predicted value are shown in FiJture 2. In

both cases. the measured data fall within the estimated error intervals.

Back.!!round

ERROR ANALYSIS FOR HYPOTHETICAL SYSTEM;

WITH P~TERS ESTTIMATED FROM FLOW ~RSION

Here we consider a homo"eneous soil column extendin" to a water table at

200 cm. The soil water retention function is assumed to be hysteretic.

Hydraulic properties and other pertinent information are presented in Table 3.

Table 3. Assumed parameters for hypothetical system.

Old (em-') "'w (em-') n 9r 9s d

9sw Ks (em h-')

0.040 0.075 1.600 0.080 0.430 0.380 1.000

A ponded infiltration/redistribution sequence was simulated in this system

to "enerate a data set that was to be used in conjunction with a nonlinear

reJtression approach to estimate soil hydraulic properties. Be.e:inninJt from

equilibrium conditions. the surface was ponded for 12 hours. followed by a

redistribution period lastin.e: 3 days. This wettin.e:/dryin.e: sequence was

40

re-peated one more time. Water content measurements were assumed to be taken

at depths of 5. 15. 25. 35. 50. 70 and 90 cm depths. and -pressure head

measurements were taken at 15 cm. These measurements were taken once

durinlt each -pondinlt/redistribution period at 5. 36. 90 and 144 hours. The

data were then -perturbed by addinlt a normally distributed random error

term. with on = 2 cm and as = 0.02. The noisy data were input to a proltram

for estimatinlt the van Genuchten parameters. " and n. and saturated

conductivity. Kg. from transient flow events (Kool and Parker. 1988) usinlt a

simulation-optimization method.

Parameter estimation from inversion of flow data

In the inversion. hydraulic properties were assumed to be non-hysteretic.

The -position of the lower boundary was assumed to be unknown. and hence

'sam-pled' water contents from 90 cm were iriternally converted to -Pressure

heads and used as an a-p-proximate first-type (head s-pecified) boundary

condition. The residual water content. Sr. was assumed to be 0.072. which is

the averalte value of Sr for all soil types from a number of samples as

documented by Carsel and Parrish (1988). The saturated water content. 8 s • was

fixed at the averalte of water content measurements within the weLted zone at

the end of the first -pondinlt -period. Table 4 shows the -parameter estimates

and error covariance matrix obtained from the inversion -proe:ram.

Table 4. Parameters estimated from flow inversion for hypothetical homo~eneous soil.

Property

" (em-I) n Ks (em h-1 ) 8r 8s

Estimated Value

0.022 1.641 0.229 0.072# 0.420#

"

0.102e--04 0.11le--03 0.166e--03

Covariance Matrix n

0.120e--Ol 0.246e--02

# - parameters assumed as described in the text.

41

0.399e--02

Parameters were estimated by < minimizinlt a sum-of-sQuares objective

function with the Levenberlt-MarQuardt modification of the Gauss-Newton

minimization all':orithm (Beck and Arnold, 1977), and the error covariance

matrix was approximated by first-order Taylor expansion, The actual

hysteretic constitutive relations are compared with the non-hysteretic relations

I<enerated with estimated parameters in Filture 3. Also shown are one standard

deviation error intervals around predictions made with the estimated

parameters. There appears to be some overprediction of the e(h) function, and

underprediction of the K(e) relation, but the overall al!reement is reasonable.

10 •

" E U

v 10' "

0 < 10' W I

W 10' 0:: :::J UJ 0 UJ 10'

0 0

W f'1ttCld ncn-hygt 0

0:: 0 0 input wQtting

[L " input dr-ring

0

100+--. __ ,-_--.-_-, __ ~---.-~._.-.l___i D. as O. 10 D. 15 D. 20 O. 25 D. 30 O. 35 O. 40 D. 45

WATER CONTENT

" 10" I -, o ,9< L 10

o _,

E 10 -2 D3-~ .. ~

1:1,..-9--.. _" u c.--" .. -v -, p.-- .. -10 p'" ",' , ,--. , -" ,,-' >- 10 .d ,-f- It.

, , H 10 -s , , / > I. , / H -,

I. / f- lO , , U 10-7 I I :::J , 0 I f'fttCld ngn-hYlilt

Z 10 -8 ,

0 input

0 I

U 10 -(I

0.050.100.150.200.250.300.350.400.45

WATER CONTENT

Fil! 3. Comparison of retention and conductivity functions I!enerated with actual and estimated oarameters for hypothetical soil. Dashed lines show one standard deviation error intervals.

42

Simulation of hypothetical rainfall/runoff event

A rainfall/runoff event was then simulated usinl': the actual hysteretic soil

properties. as well as the estimated non-hysteretic parameters to further

examine the validity of the estimated parameters. and to demonstrate the

applicability of the first-order error analysis procedure. A rainfall flux of 0.1

cm h- 1 was applied at the surface for 50 hours. followed by a redistribution

period of 150 hours. The water content profile in the system was evaluated at

50 and 200 hours. The error covariance matrix estimated from the inversion

was used to calculate the uncertainty in model predictions made with the

estimated parameters usinl': (9). FiJ,;ure 4 shows the predictions made with

actual and estimated parameters and one standard deviation error intervals

associated with the latter case. The al':reement between both sets of

predictions is reasonable. althoul':h there is a consistent overprediction when

using the estimated parameters. The error intervals are larl':er than those

estimated for the draina"e experiment. The "reatest uncertainty appears to

coincide with the location of the wettin" front at 30 cm in Fil':ure 4a and 70

em in Fi"ure 4b. This is possibly due to the inadequacy of the first-order

error analysis to model the lar"e saturation/pressure I':adients associated with

the wettin" front.

A Monte Carlo simulation was performed to determine the variability of

water content predictions due to parameter uncertainty. and thereby examine

the accuracy of the first-order error analysis procedure for this particular

problem. Assuminl': I':aussian error distribution for ex. n and Kg. the joint

normal PDF for these variables was sampled to produce 100 random parameter

vectors. The rainfall-runoff event described previously was simulated for each

of these realizations. Predicted water contents at each spatial and temporal

location was then averal':ed to compute means and standard deviations. These

are also shown for comparison in Fi"ure 4. In I':eneral. the mean water

contents predicted· with the Monte Carlo and first-order methods al':ree well.

whereas the standard deviation is slil':htly overpredicted by the first-order

procedure particularly in the vicinity of the wettinl': front where sharp

saturation "radients occur. These results attest to the I':eneral utility of

first-order error analysis for predictin" the uncertainty in model predictions

due to parameter uncertainty for typical situations.

43

0.50

I- O. 40 Z W I-Z 0.30 0 U

fY W

0.20

I-« :;. 0.10

0.00

0

0.50

I- 0.40 Z W I-Z 0.30 0 U

fY W

0.20

I-« :;. 0.10

0.00

0

*\ " \\ " \ \ . . , ' . \ \ • ----:=--::=--=-\ 'b-----;.-=-.:.. ____ -

\ ,/'

25

\ .............

-•

50

DEPTH

FJ,..gt-e>rdg,.. Me>rn.a-Ce>rl e 'trua' peremgtarg

75 100

• '-------=-====,. !;:::;==.- .... _"=".::::::- • .. -. . .

25

-... ::: .... -.. -h-----~ - ,/

A--o<

•

50

DEPTH

"-... _--f"J,..gt-e,..dgr Menta-eerle 'trua' pe",cmgtg,..g

75 100

Fil'! 4. Comparison of water content profiles simulated with actual and estimated parameters for hypothetical soil at 50 hr (top) and 200 hr (bottom). Dashed lines are '1 standard dev error intervals.

44

SUMMARY AND CONCLUSIONS

In this study, we have examined the error in predictions of unsaturated

flow due to parameter uncertainty when parameters are estimated by two

different procedures. In the first case, soil hydraulic properties and their

uncertainty as represented by an error covariance matrix were estimated from

particle size distribution data, and in the second case from the inversion of

transient flow data. Error intervals on model predictions evaluated by a

first-order error analysis procedure were found to reasonably bracket the

true behavior of the system as obtained from actual measurements for the

first case, or from simulations using true parameters for the second case.

Thus, this work also provides an indirect verification of the utility of soil

hydraulic properties derived from particle size distribution data and from the

inversion of transient flow experiment data. Results of the first-order error

analysis agreed well with results obtained from Monte Carlo simulations

performed for the second problem.

The computational requirements of the first-order error analysis procedure

are quite reasonable. The total number of model runs is N+l, where N is the

number of uncertain parameters. For a resonable number of parameters, this

compares very favorably with Monte Carlo simulation which may require more

than 103 runs, and even with Latin-Hypercube sampling which commonly

requires about 102 model runs. There is thus greater than an order of

magnitude reduction in computational effort for the first-order method if N~lo.

For distributed parameter fields, the first-order analysis will require a

greater number of model evaluations to compute the sensitivity coefficients

and hence will involve comparable computational effort vis-a-vis Monte Carlo

type methods.

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moisture characteristics from particle-size distribution and bulk density

data, Soil Sci. Soc. Am. J .. 45:1023-1030.

45

Beck, J. V. and K. J. Arnold. 1977. Parameter Estimation in Enl!ineerinl! and

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46

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47