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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45
Beck, J. V. and K. J. Arnold. 1977. Parameter Estimation in Enl!ineerinl! and
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47
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