United States
Environmental Protection
Predicting Attenuation of
Viruses During
Percolation in Soils
1. Probabilistic Model

                                                            August 2002
Predicting Attenuation of Viruses During Percolation in Soils:
                   1. Probabilistic Model.
                      Barton R  Faulkner
          U S EPA Office of Research and Development
         National Risk Management Research Laboratory
         Subsurface Protection and Remediation Division
                     Ada  Oklahoma 74820
                       William G  Lyon
         ManTech Environmental  Research Services Corp
                     Ada  Oklahoma  78420
                       Famque A  Khan
                    U S  EPA  Headquarters
            Washington,  District of Columbia 20460
                     Sandip Chattopadhyay
                   Battelle Memorial  Institute
             Environmental Restoration Department
                     Columbus  Ohio 43230
                 Contract Number 68-C-98-138

                        Project Officer
                     Georgia A  Sampson

         National Risk Management Research Laboratory
              Ofhce of  Research and Development
             U S  Environmental  Protection Agency
                     Cincinnati OH 45268
                                                        /T~~y  Recycled/Recyclable
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                                                               processed chlorine free


The U S  Environmental Protection Agency through its Office of Research and De-
velopment funded and managed the research described here through m-house efforts
and under Contract 68-C-98-138 to ManTech Environmental Research Services Cor-
poration  It has been subjected to the Agency's peer and administrative review and
has been approved for publication as an EPA document  Use of trade names  or
commercial products does not constitute endorsement or recommendation for use

   All  research projects making conclusions or recommendations based on environ-
mental data  and funded by the U S Environmental Protection  Agency are required
to participate in the Agency Quality Assurance Program  This project was conduct-
ed under an  approved Quality Assurance Project Plan The procedures specified  in
this plan were used without exception   Information  on  the  plan and documenta-
tion of the quality assurance activities and results are available from the Principal

    Virulo and the user's guide have been subjected  to the Agency's peer arid ad-
ministrative  review and  have been approved for  publication as an EPA document
Virulo is made available on an as-is basis without guarantee or warranty of any kind,
express or implied   Neither the United  States Government (U S  EPA), ManTech
Environmental Research Services Corporation, Battelle Memorial  Institute,  Wash-
ington State Department of Ecology, nor any of the authors or reviewers accept any
liability resulting from the use of  Virulo, and interpretation  of the predictions of the
model are the sole responsibility of the user

The U  S  Environmental Protection Agency is charged by Congress with protecting
the Nation's land, air, and water resources  Under a  mandate of national environ-
mental laws, the Agency strives to formulate and implement actions leading to a
compatible balance  between human  activities and the ability of natural systems to
support and nurture life  To meet this mandate, EPA's research program is providing
data and technical support for solving environmental  problems today and building a
science knowledge base necessary to manage our ecological resources wisely, under-
stand how  pollutants affect our health  and  prevent or  reduce environmental risks
in the future

   The National Risk Management Research Laboratory (NRMRL) is the Agency's
center  for investigation of technological  and management approaches for prevent-
ing and reducing risks  from pollution that  threatens human health  and  the  envi-
ronment  The focus of the Laboratory's  research program is on methods and their
cost-effectiveness for prevention and control of pollution  to air  land, water,  and
subsurface  resources protection of water quality in  public water systems, remedi-
ation of contaminated  sites  sediments and ground  water, prevention and control
of  indoor air pollution, and restoration of ecosystems  NRMRL  collaborates with
both public and private sector  partners to foster technologies that reduce the cost
of compliance and to anticipate emerging problems  NRMRL's research provides so-
lutions to environmental problems by  developing and promoting  technologies that
protect and improve the environment advancing scientific and engineering informa-
tion to support regulatory and policy decisions, and  providing the  technical support
and information transfer to ensure  implementation of environmental regulations and
strategies at the national state, and community levels

    EPA's Office of Water is currently promulgating a Ground Water Rule to ensure
water supplies are safe from contamination by viruses  States may be required to
conduct hydrogeologic sensitivity assessments to predict whether a particular aquifer
is vulnerable to pathogens  This work presents the conceptual and  theoretical de-
velopment of a predictive screening model for virus attenuation above aquifers  It is
hoped this model will be a useful tool for State regulators utilities  and development
                             Stephen G  Schmellmg Acting Director
                             Subsurface Protection and Remediation Division
                             National Risk Management Research Laboratory

We present a probabilistic model  for predicting virus  attenuation   Monte Carlo
methods are used to generate  ensemble simulations of virus  attenuation  due  to
physical, biological   and  chemical  factors  The model generates a probability  of
failure to achieve a chosen degree of attenuation  We tabulated data from  related
studies to develop probability density functions for input parameters,  and utilized
a database  of soil hydraulic  parameters based on the  12  USDA  soil categories
Regulators can use the model  based  on limited information such  as  boring logs,
climate  data, and soil  survey reports for a particular site of interest  The model
may be  most useful as a tool  to aid in siting new septic systems

    Sensitivity analysis indicated the most important mam effects on probability of
failure to achieve 4-log (99 99%) attenuation in our model were mean  logarithm of
saturated hydraulic conductivity (+0 105) and the rate of microscopic mass transfer
of suspended viruses to the air-water interface (-0 099), where they are permanently
adsorbed and removed from suspension in the model  Using the  model, we predicted
the probability of failure of a 1-meter thick proposed hydrogeologic barrier to achieve
4-log attenuation Assuming a soil water content of 0 3  with the currently available
data  and  the associated  uncertainty, we predicted the following  probabilities  of
failure sand (p - 22 '5697), silt loam (p =- 6/2000000), and clay (p - 0/9000000)

    The model is extensible in the sense that probability density functions of param-
eters  can be modified as future studies  refine the uncertainty,  and  the lightweight
object-oriented design  of the  computer  model  (implemented in Java™) will facili-
tate reuse with modified classes, and implementation  in a geographic information


1  Introduction                                                       1

2  Abridged Literature Review                                         1

3  Mathematical Description                                          2

   3 1   Differential Equations                                           2

   3 2   Proposed Solution for the Differential  Equations                    4

4  Mass Transfer and Inactivation Rates of Viruses                     9

5  Modeling under Uncertainty                                       11

6  Sensitivity Analysis                                               14

7  Design of the Computer Model                                    19

8  Conclusions                                                      20

9  List of Symbols and Notation  Used                                24

10 References                                                       25

11 Internet  References                                               27

List of Figures

   1    Processes Considered in the Model.                              3

   2    Plot illustrating correlations                                       12

   3    Comparison of simulated and measured values                       13

   4    Frequency histogram of  values of - Iog10.4 for  poliovirus for
        Rosetta sands.                                                  15

   5    Frequency histogram of values of - Iog1().l for poliovirus for Rosetta
        silt  loams                                                        16

   6    Frequency histogram of values of   log|(,.4 for poliovirus for Rosetta
        clays                                                            17

   7    Javadoc class documentation for Attenuator interface.          22

List  of Tables

   1    Hydraulic Properties of Sand, Silt and Clay                         6

   2    Parameters Used  for Poliovirus                                10

   3    Main Effects on Probability of Failure                          19

   4    Classes Used in the Computer Model                          21


The authors would like to thank  Mohamed Hantush, U S  EPA, for his invaluable
suggestion to use the final value theorem modeling approach, and many other helpful
suggestions  We would also like to thank John Wilson of the U S EPA for providing
compiled data, Kathy Tynsky (Computer Sciences Corp )  for designing the graphics
on the cover,  and Joan Elliott (U S EPA) and Martha Williams (Computer Sciences
Corp ), for their advice in typesetting this document

1     Introduction

Impending regulations m U 5  EPA s forthcoming Ground Water Rule (EPA, 2000)
will require public water systems (PWS) to more closely monitor their ground-water
systems for contamination by pathogenic viruses  The Rule clarifies the conditions
that define risk to PWS from viruses  Regulators  can  use  the new definitions for
siting new septic systems  If it can be shown that the risk is low due to the presence
of a hydrogeologic barrier, a proposed site may be acceptable  The Rule defines a
hydrogeologic barrier as a subsurface region through which  viruses  must pass from
a source in order to reach PWS wells that provides at least a yet undetermined  but
specific degree of attenuation of active pathogenic viruses  The draft rule indicates
attenuation factors are  "physical,  biological, and chemical   acting  "singularly or in
combination "

    In instances where the ground-water system in question is connected to potential
virus sources by karst fractured rock gravel  or a  soil exhibiting preferential flow
the system will be classified as high  risk  In other cases the assessment process  will
benefit from prediction by mathematical  modeling  Therefore,  regulators and utility
operators  may benefit from simple, probabilistic quantitative models as tools in  the
context of responding to the Ground Water Rule (GWR) This document presents
the development of a proposed  model to evaluate attenuation  as viruses are carried
with percolating water in  an  unsaturated,  naturally existing soil layer   The model
itself is a computer application At  the time of this writing, a user's guide for this
model  is imminent in a companion document   Here we describe  the  conceptual
and mathematical development of the model, and  highlight areas of much needed
2   Abridged  Literature Review

Although several papers describing the modeling of virus transport in ground water
have recently been published, there is not yet a concensus on which factors have the
greatest impact on eliminating  active viruses as they pass through natural  porous
media   Keswick and  Gerba (1980) presented an  early review of factors affecting
viruses in ground  water  More recently, Schijven and  Hassamzadeh  (2000) wrote a
valuable review that is fairly comprehensive, and  Breidenbach et al  (in review) have
produced an environmental handbook and extensive bibliography on the subject  We
refer the reader to these works for a description  of the available data from field and
laboratory studies

    Current modeling approaches have been criticized  Yates (1995) demonstrated
by using a numerical dynamic model to  predict virus survival,  that field data  do  not
agree well with model predictions  In particular, their model-predicted attenuation
was dramatically greater than actual Yates and Jury (1995) have emphasized  the
sensitivity of a numerical dynamic model to input  parameters

    Modeling approaches themselves have  varied greatly depending on the scale of
the study and the specific interests of the investigators  Some treat virus transport as

a Fickian process, coupled with advection of ground water, others have incorporated
filtration theory, treating virus transport as a colloid filtration process

    Field studies in which viruses were released into the subsurface have documented
early arrival times, with arrivals at monitoring wells sometimes preceding those of
dissolved tracers  Viruses are  more likely to be attenuated during percolation m the
unsaturated zone than during transport through the same distance in the saturated
zone (Lance and Gerba, 1984)   Studies with unsaturated soils have shown that
hydrophobic colloids are adsorbed at the air-water  interface in greater proportion
than the mineral-water  interface  (Wan  and Wilson,  1994), and this is apparently
also the  case with viruses (Thompson  et  al ,  1998, Thompson and Yates.  1999)
Chu et al (2001) have shown that when sand column experiments were conducted
with reactive solids removed  (metals and metal oxides) the effect of the  air-water
interface was most pronounced, and suggested reactions at the solid-water interface
may be dominant when  reactive solids are  present  Sim and Chrysikopoulos  (2000)
developed a governing constitutive equation  for unsaturated-zone virus transport
that considers partitioning of viruses to the air-water interface   More recently Chu
et al   (2001)  expanded  this  model, more  closely considering  mterfacial  reactions
using  as  yet untabulated parameters
3    Mathematical  Description

3.1   Differential  Equations

Sim and Chrysikopoulos (2000) developed the following governing equation which
can describe the transport of viruses in a porous medium as depicted in  Figure 1
                                               A6>,,,r   A ,>("   A° #,„<"'   (1)
    where C  -  ( '(/.:) (ML  ^) is the concentration of viruses m the mobile solution
phase, t is time   : (L)  is the downward distance from the top of the proposed
hydrogeologic barrier, ('"(/.;) (MM   ') is  the  adsorbed virus concentration at
the liquid-solid interface C'"(/ :)  (ML ~ '') is the adsorbed  virus concentration at
the liquid-air interface  q  (LI  ') is the specific discharge,  H,n (Z,'/."')  is the
moisture content, A  (J  "') is the mactivation rate  coefficient for the viruses in
the bulk solution, A'  (/ "'),  the  mactivation rate for the viruses that  are sorbed
at the liquid-solid interface  and  A4' (7 ~'),  the  mactivation rate for the viruses
sorbed at the liquid-air  interface,  /> (.ML  '') is  the soil  bulk density, and  D
<>-<7/^iM  - A (L2T~l)  is  the hydrodynamic dispersion coefficient,  n   (L) is the
vertical dispersivity, P,  - P/r (L2T~ '), where T> (L2'l  ')  is the virus diffusivity
in water and r (LL~ '  greater than 1)  is the tortuosity


                 \  mass transfer

                Figure 1  Processes Considered in the  Model.

    We assumed the following

    •  Steady-state flow
    •  Gravity drainage only

    •  The soil is homogeneous in terms of
          hydraulic  properties
          virus properties

    •  The soil does not induce preferential  flow

    Sorption and inactivation of viruses at the various interfaces is described by Sim
and Chrysikopoulos (2000) by
    where /.  -  /-,«/  ";  -  .<(!   ^), ',  is the liquid-solid interfacial area in units
of (L'l.~ ()   The symbol /.  (1   ') \s the microscopic mass  transfer  rate and h
(LI   ')  is called  the  mass transfer coefficient   In  Eq   2  A,, (L^M  ') is  the
equilibrium partitioning coefficient /;, (I.) is the  average radius of  soil  particles
and H^  (L''L  '')  is the  saturated  water  content   Analogously they derived  the
change in concentration of viable viruses at the air-water interface,

    where /.c (1 ~') is the liquid to liquid-air interface mass transfer rate  The mass
transfer rate for the liquid to liquid-air interface is described by
                                 r   H-a;                               (4)

    where //  (/.-'/.. ~ ^) is the mass transfer coefficient and a'} is the estimated area
of the air-liquid interface as a function of the moisture content   Thompson et al
(1998) and Thompson and Yates (1999) have demonstrated  dependence ol viral
mactivation rates  on their sorption state  A method for estimating 
    Figure  1  is our conception of a proposed natural hydrogeologic barrier  The
output consists of remaining viable viruses  plus the amount destroyed due to the
attenuation factors of suspension and sorption coupled with virus-specific degrada-
tion  rates

    The initial and boundary conditions are

    Mass continuity through  the  upper boundary requires the additional boundary
condition which states the input equals the supplying concentration  subject to the
advection  and dispersion constraints of the porous medium
                                       DH,,,—      -,/<"                (10)
    Since  it is assumed that the flow is due to gravity only(t>/i '<):   0)  then the
total  head gradient  is unity  (/): <>:}  and <\  ---  K(f>,,,)   To obtain K(Hl:i)   van
Genuchten (1980) obtained the following
    where A _ is the saturated hydraulic conductivity, d, is the residual water content
(//'/.  !), W, is the saturated water content  (//'/,  !), and n is  a  well-tabulated
empirical curve fitting parameter (Table 1)

    We consider the case where the supplying concentration results from percolation
of water containing viruses  lasting for a  period  of time that is small compared to
the residence time in  a  proposed barrier   Such a situation would  result if  a septic
tank temporarily overflowed and was then pumped or otherwise corrected  thereby
stopping the virus source  Arrival of viruses at the input may be approximated as a
relatively sharp concentration front followed  by exponentially decreasing concentra-
tion of viruses such that the concentration of viruses  immediately above the upper
boundary region of the barrier is ( ',,,,,, <>\p   11   Now we can write the attenuation

    Having made the above assumptions, and  assuming dispersion and bulk den-
sity of the soil are  constant throughout the proposed barrier, taking the  Laplace
transform of Eq  1  and applying the mtial conditions (Eq  8) yields

Table 1  Hydraulic Properties of Sand, Silt, and Clay


silt loam



' ;'


0 367
-0 691
1 r>,X X 10"
1.71 x 10- '
r) :><) x 10 *
11 7
-2 160
-0 207
1 l:i x 10"
1 18 x 10 '
x 7") x 10 -"'
0 101
0 515
-2 085
0 276
0 114
1 2<) x 10"
-0 384
1 48 x 10"'
r> r)0 x If)' "'
7 38
0 129
1 (18 x 10 '
(>.r> x 10 "•
7 38

L*L- "^
L^L !
log(/// // r j
!og(/// )
log((/////f // S/(,'/l/( SS 1

                                                      -  AX"  - A  H,,,C   (13)

    Likewise, the Laplace transform of Eq 2 is

    Likewise  the Laplace transform of Eq 3 is

    Noting the boundary conditions given by Eq  8  and solving for (." in  Eq  14

we obtain
    We can solve for ('   in Eq  15 to obtain
    Now we insert the substitutions into Eq  13, which yields the following ordinary

differential equation, in terms of C
                                                  0                      (18)
                             a ~ -      ii :


   To solve Eq 18 we consider the homogeneous equation which has the character-
istic polynomial D  Y1 - VY - -  The general solution is f(.s. :) = ^\ rxpfFi (^)"-\ •
ri exp Fji ,s);j, where, by the quadratic formula
                                                 r   \;

   The Laplace transform of Eq  9 is dC( s x)/r/;, which implies
physically  reasonable solution thus r'(s. ;) _ ^j ('xp[r_>(.s):

   The Laplace transform of Eq 12 is

                                                                   -  0  for a
    Substituting for (" and its derivative we have
    The Laplace transform of Eq  10 is
                                        1 JT_,
    Thus, we find the value of - >  is
    From Eq  22
    We now apply the final value theorem

                 Inn  .We = /,) - Inn .V/(s. :)   'L^ll	            (28)
                / -- x              ^  -0                  ]

    The attenuation factor 1 \s
    To find -U(;  - (II,  we integrate the input flux in time
                                 .1   ,'-'-  '                             (31)

    The average velocity of the percolating water is \"   
                   Table 2  Parameters Used for Poliovirus
A',/ (sand)
A,/ (silt loam)
A,/ (clay)
1 :i4 x K)- <
 x 10""
2.4.5 x K)- '
,i 77 x K) '
7 20 x K)- '
0 M)8
0 (>OM
l.MO x K)- ^
1 MO x 10 ''
1.2r» < 10"
r, (if) x HP '
7 K) x 10 '
().7i x 10 '
in hi' '
;» /// - '

;;/5 ,/- '
//; '' (, '
/» < y '
* Data complied by Breidenbach et al  (2001) unless otherwise noted
1  From Chu et al  (2001), see Appendix A for assumptions
|  Yates and Ouyang (1992) assumed A* =: A _'
>t  Mazzone (1998) p  114
soil and water, such as temperature, pH, organic matter, and presence of metals, or
other ions

    Most of the  work related to natural hydrogeologic barriers has been  conducted
with bactenophages (viruses to bacteria), due to the restrictive conditions required
to obtain such  data  for human  pathogenic viruses without  posing  a risk to re-
searchers  The polio viruses are perhaps the most widely studied human pathogenic
viruses in this context  Table 2 lists the relevant properties of the viruses,  based
on Breidenbach  et al   (2001)  The data  cover a fairly wide range of geochem.cal
conditions, hence the high standard deviations  Measured mass transfer rate data
is largely lacking Vilker and Burge (1980) did early work that included some  mea-
surements for poliovirus  More recently, Chu et al  (2001) measured  mass transfer
parameters with  MS-2 bactenophage,  which has comparable size and  geometric
properties,  using an inverse modeling approach

    Due  to the  difficulty of obtaining good experimental control, and  the  corre-
sponding sparsity of data,  a  popular semi-empirical correlation to estimate h  due
to W'lson and Geankophs (1966),  /, -  1 00\'   D'"2il,H,  ' is often employed  This
correlation can be used at low Reynolds numbers, however, comparison between this
expression  and measured values shows very poor correlation (Pearson's correlation
I) O.'i'i A" = 2ii)  The expression fails to account for major factors
that affect mass transfer of viruses  at the molecular level,  such as pH-dependent
e^ectrostatic interactions between the protein surfaces and soil  particles and /or the
air-water interface  Indeed the correlation was not developed for this purpose The
work of Chu et al  (2000) highlighted the enormous effect of oxides on sand grains
in their soil column experiments  Their work suggested the pH-dependent behavior
of oxide coatings has a stronger effect on mass transfer than  the air-water interface
Much  additional work is needed to develop realistic correlations to estimate mass
transfer of viruses The experimental control needed to conduct such studies can be
daunting  Most soil column studies must rely on plaque assay  methods that suffer
from virus aggregation effects and other sources of uncertainty

    As with h  a correlation has not been  established for t^ though we may now
rely on the results of Chu et al (2001)   These are listed in Table 2

    Rose and Bruce (1949) derived equations to estimate the air-water mterfacial
area by «-} - (/>,. t/liH,,,)/ rr In this expression, n (MLI ' ') is the  surface tension
of water, /;„ is  the density  of water, // is the acceleration due to gravity  Employing
the van Genuchten  (1980)  expression that relates «,„ to the  capillary pressure head
//  and expressing the effective saturation as S,  -  \H,,, -  H, ) CV,    0, ) we obtain
the following
    The benefit of using this expression is that it utilizes the well-tabulated fitting
parameters for which we have already developed multivanate distribution functions
which we will discuss in the next section

    Virus diffusivity P  (LJ7   ') is governed by Brownian motion and is described
by the Stokes-Emstem  equation
    in which /, i, is Boltzman's constant (MLI  '-}  I (  Celsius) is  temperature,
//    //( / ) is viscosity of water (.WZ.  '/   ')  and;,  is the equivalent  radius of the
5    Modeling  under Uncertainty
The draft Ground Water Rule states a specified degree of attenuation must occur in
order for a hydrogeologic medium to be considered a barrier  Using the mathematical
model with as much information about the geochemistry and other factors that can
help in  making  a  decision on the appropriate parameters  we operate under  the
premise that  it is possible to obtain a prediction of attenuation that is more useful
than a qualitative expression of confidence that the barrier can or cannot attenuate a
particular virus  It  is essential to develop a probabilistic expression of the confidence
that -•    log  attenuation will occur  encapsulating the possible sources of error in
the model parameters

    Virus diffusivity  P  is a physical parameter that can be calculated if the tem-
perature  /   and corresponding viscosity  of water are known  Unfortunately  the
tortuosity r,  is less easily measured since  it depends  not only on the modal particle
diameter, but also the pore geometry and connectedness of the pores  Tortuosity
for  unsaturated soils is often predicted with (Schaefer et al  1995)

           UNSODA sands

'1 9
1 Z



, ,*,* ,-,..

i_ _ _
log n
log n

*•• * -
0.15   0.05
Figure 2  Plot illustrating correlation of logmA\,  login//, #,, and logmn with
#,  for sands in the UNSODA soils database and a corresponding multivariate
normal ensemble simulation.
                                   ——   otherwise
    Although the data in Tables 1 and 2 list uncertainties that could introduce sig-
nificant error into the predicted attenuation, we needn't rely on these variabilities as
independent (orthogonal) sources of error  Many of the parameters, when measured
in controlled experiments, are correlated  Thus we can consider their space of vari-
ability as conditionally multivariate  normal and ;or lognormal  The five parameters
that display significant correlation (based on H(]  //,,,,   0  //i //,,,, ^ 0) are shown
in Figure 2

    We applied covariance- and histogram-honoring simulations using the  Monte
Carlo  approach  Details of the Cholesky decomposition approach we used are  de-
scribed by  other authors (e g  see Kitamdis, 1997,  Appendix  C3)  For  all  the
other  parameters, the Monte Carlo simulation used histograms from the parameters
independently  The advantage of the Monte Carlo method is that it produces a
histogram of attenuation factors as output and it allows us to assign a  probability
of failure to  achieve ~  - log attenuation  Figure 2 shows the space of variability for
the five (hydraulic) parameters which  were significantly correlated  The simulat-
ed values were generated by conditional  simulation  of multivariate normal density
functions parameterized  by the vanance-covariance  matrices of the parameters as
determined with the UNSODA database (Ley et al  1996)   These are listed in
Appendix B




;: •• o~ '.V;"'»'>4-' '•.•»-i';' ,7'.-. ''•'.

, :-, i^^r^^^S^^.

Kim etal. (1997) '
Anwar et al. (2000): !
25 mm,
iogKs =-0.165
>~! 50 mm,
log Ks = -0.082

log Ks = -0.029

    ,  Baked blasting sand
    i  Glass micro beads of 3 different mean diameters listed
    *  Generated by setting mean U,n " ti It), std  err (i _'

Figure 3  Comparison of values of 
    Clearly, this simple model  appears to underestimate u'-j when S,  is less than
about 0 6 (0,,,  less than about 0 25)  The measurements themselves, may be subject
to considerable error, and there is no data for finer soils and soils containing clays
At low water contents «/'  will apparently be  underestimated for sandy soils, tnus
the model should  not be used to test barriers which are proposed solely on account
of low water contents Other means of evaluation  should be used in those cases

    Figure 4 shows histograms  of attenuation factors of polio 1 virus using the data
in Tables 1  and  2,  for 1 meter thick soils  at water content equal to 0 30  "he
histograms were  generated  with Monte  Carlo simulations with  soil hydraulic data
from the UNSODA soils database (Ley  et al  , 1996) for categorical sand and silt
loam soils (Table  1)  The soil values themselves were generated in previous studies
using a  bootstrap method (Schaap et al , 1999)   The uncertainty associated with
hydraulic parameters is relatively well understood  The higher standard deviations
for clay soil are due to the various clay mineralogies that can be present having large
effects on water retention characteristics  On the other hand, the variation shown  in
Table 2  is largely  the result of  the various geochemical conditions under which the
measurements were made  Indeed,  State regulators may not have comprehensive
data available  for  a proposed hydrogeologic barrier  As more data become available,
the uncertainties  may be  reduced to primarily measurement error,  assuming the
investigator knows relevant details about the geochemical environment

    From the distribution of attenuations produced in the Monte Carlo simulations,
we simply compute the probability of failure to achieve a target attenuation factor
                   number of Monte Carlo runs that produced .1 < 10
       p (f a 11 u re I  =	
                          total number of valid Monte Carlo runs

    The probability of  failure for  poliovirus to achieve 4-!og attenuation was p=
22/.r>i»07 for the sands  Model users would more likely be interested m tighter soils
more  likely to be proposed as hydrogeologic  barriers   Figure 5 shows the results
for  poliovirus for silt loams  Probability of  failure for the  1-meter thick barrier
was p— (> '2000000  For this particular set of data, the Rosetta clays produced a
probability of failure p- 0 ''1000000 for the data shown in Tables 1 and 2
6    Sensitivity Analysis

Beres and Hawkins (2001)  listed the advantages of applying the Plackett-Burman
(Plackett  and Burman  1946) method of sensitivity  analysis   These include the
ability to measure  two-way  interaction effects among  parameters and the freedom
to apply any desired  domain of plausiblility to input parameters  The reader  is
referred to Beres and Hawkins  (2001) for a detailed description of how to apply
the method  and Plackett and Burman (1946) gave the statistical foundation and
the optimum cyclic permutations  (pattern of parameter variation) of parameter
combinations that  may be used  for a given number of variables

    We  considered the 17  input parameters that  are used  in the model solution
Table 3 lists each  parameter and the plausible domains used, based on the means

 File Edit  Run Help   J> Start Simulation     ^ Stop   Threshold Attenuation 
 File  Edit Run  Help    |> Start Simulation     |) Stop   Threshold Attenuation <£):  4  (-loglG)

 jJJgl Flow Parameters >  ,','. Virus Parameters    Jj| Histogram    7^ Probability

                                if  Retain and Accumulate
             Right Truncated Histogram
             Vertical Axis is Count
Figure 5   Frequency histogram of values of  log!0-4 for poliovirus for Rosetta
silt loams.

 File Edit  Run Help   p» Start Simulation     ^ Stop   Threshold Attenuation <£):  4  (  loglO)

  yjsJK Flow Parameters    ',",  Virus Parameters   jjjjj Histogram   ZJ^ Probability

                                 v Retain and Accumulate

              Right Truncated Histogram
              Vertical Asir i? Count
Figure 6   Frequency histogram of values of   logm.4 for poliovirus for Rosetta

and standard deviations shown  in Tables 1 and  2   We used the following  cyclic
permutation of the parameters, which was derived by Plackett and Burman (L946)
for a 20 parameter model
    Where the values in Table 3 are greater than zero, the effect  is proportional to
the probability of failure, and parameters which produce effects less than zero are
inversely proportional to the  probability of failure

    We may interpret the effect as the expected amount and direction of change in
the response (probability of failure to achieve 4-log attenuation)  that results from
changing  the particular parameter by the + and - values given  for the plausible
domains  It  is a resolution IV factorial design, on account of the foldover, the lower
half of the permutation matrix (Beres and  Hawkins,  2001)   It is not unexpected
that the probability  of failure to  achieve 4-log attenuation is  sensitive to parame-
ters that affect water flow  for the given  plausible domains  It should be noted the
effects of the correlated hydraulic parameters (logi()A\, Ws  login//, M< • and logi()n)
listed in Table 3 are listed  only to show  the effect of a change in the mean value
A  special  vanance-covanance matrix  was calculated for all  the parameters of the
LINSODA database (see Appendix B)  This matrix was used in the sensitivity anal-
ysis However, it was not adjusted for the effects in  means resulting from the cyclic
permutations of the  parameter mean  values

    For the  given plausible domains, logarithm of saturated hydraulic conductivity
(logioAs) was  the  most important parameter   In  light  of  experimental evidence
that preferential flow plays an  important role  in unsaturated  mass transfer, users
should do their best to obtain improved  estimates of the effective logioA\ if there
is  evidence of soils that could produce preferential  flow  The second most impor-
tant parameter was  KC  the rate  of microscopic mass transfer from the suspended
phase  to  the air-water interface, where the viruses are  adsorbed  and  effectively
removed from the system  This result highlights the importance  of estimating the

               Table 3  Main Effects on  Probability of Failure
logio "
logio A
logio A'
6i m

logio n
/ ,
-2 12
0 5
6 233 x 10 "'
-305 ; 10 4
0 0647
0 15
0 12
1 250 , 10 s
0 0
u 8 75 x 10 "'
Domain' ^
0 023
1 997 > 10 '
1 213
0 912
1 20 * 10 ^
19 08
0 28
1 500 x 10 s
3 14 x 10 <
5 59 < 10 '
0 376
+0 105
-0 099
+0 020
-0 017
+0 005
-0 004
air-water mterfacial area, a difficult endeavor  and an important subject of research
for unsaturated-zone  contaminant transport

   Two-way interaction effects were also measured  These are listed completely in
Appendix C  Most notable  among these include /. ><  logniA  (-0 072) which is not
surprising due to the  increased residence time experienced by the viruses  in thicker
soil layers. /, x  A,,  (+0054), and logn>/\. * /,  (-0095  the parameters which
exhibit important (and opposing) mam effects
7   Design of the  Computer Model
The Java programming language (Gosling et al   2000) was used to implement the
model   This language allows object- oriented design to  be relatively easily  imple-
mented   Table 4  lists the classes and  interfaces used   Java class documentation
will be available on-line and will be described in a forthcoming  user's guide for the
model  Some of the classes use the Matrix class of JAMA  a Java matrix package

   We consider these classes to be "lightweight,1  in this  context,  meaning that
they are high-level  easily implemented with other applications abstracted from the
computer hardware and its operating system (i e ,  they are portable) they have a
small footprint  and they require few memory and computational resources

   The model ("Virulo") can be implemented as an applet or application depending
on the launcher used (ViruloApplet java or Virulo java)  The Swing graphical com-
ponents of the Java Foundation Classes were used in the Graphical User  Interface
(GUI)  The model and its source code can be found at


   The classes were written using the Javadoc tool [5]  The Javadoc tool uses
the philosophy  of programming created by Professor  Donald Knuth  at  Stanford
University  Knuth advocates that  programs should be written  to be read  not only
by machines, but also  by humans (Knuth,  1992)   He  designed  a programming
language called WEB Programs written in this language are parsed by an  engine
that generates Pascal source code  as well as source code that  can  be typeset with
Knuth's T[=X typesetting program

   The Javadoc commenting system follows a similar  philosophy  It is natural  to
make  classes of an  object-oriented program easier to read  by other programmers
so they can  be implemented in other programs  This tool allows special comment
tags, embedded in the body of the source code, but ignored  by the compiler, tc  be
parsed by the Javadoc application  It thus generates formated class documentation
that describes the structure and function of the class  Programmers can  use it  to
implement classes without having  to resort to reading  the Java source code  The
Javadoc system generates HTML code in a  common  style   Much of the  HTML
document comes from interpreting the Java source code, but the system also allows
commenting by the author  Figure 7 shows the Javadoc documentation for the
Attenuator class of Virulo
8   Conclusions
We developed a  probabilistic model to predict the effectiveness of a hydrogeologic
barrier to pathogenic viruses in  the unsaturated zone  It is based on physics, and
we can conclude that the following assumptions must be employed

   •  Viruses reach the top of the proposed barrier following release in an overlying
      source, and following their arrival  the  input concentration decays exponen-
      tially  This condition corresponds  to an accidental release, such  as from an
      overflowing septic tank, which is subsequently corrected

   •  Water flow in the proposed barrier is due to gravity only

   •  The virus of interest is approximately spherical in shape

   •  The proposed barrier does not contain significant numbers of predatory mi-
      croorganisms (the model estimate  is conservative in this sense)

   •  The percolating water does not contain significant  amounts of surface active
      agents, such as detergents that could change the hydraulic properties, decay
      rates,  or adsorption

                  Table 4  Classes  Used in the Computer Model
  FlowComboPanel extends JF'anel
  FlowPanel extends  JPanel
  implements Observer

  Gossiper implements Observer

  HistoPanel extends JPanel



  Medium implements Cloneable
  Normal implements Cloneable
  Operandum implements Cloneable
  OutputPanel extends JTextPane


  StnngAsChars  implements Cloneable


  Virulo (or ViruloApplet)
  ViruloFrame extends  JFrame
  VirusComboPanel extends JF'anel

No  of
i  0
!  i





           Computes the water flux and  1
           Interface for sorting callback (due to Eckel 1998)
           Subclass of Compare for sorting callback
           Combo box for selecting soil type

           Panel to display flow parameter text boxes
           Java translation of the popular C program gasdev.c
           (Press et al   1989)
           Observer object (Gamma et al  1995) that notifies
           subscribing objects of actions without violating
           object-orientation bv creating unwanted dependencies
           Polvmorphable class for generating  histograms
           Uses Histogram to create a Bufferedlmage for
           Panel for displaying histogram image
           Has useful static methods for oft-used  math
           Conducts and manages  the Monte Carlo Simulations
           for Virulo
           Polvmorphable image canvas of Geary  (1999)
           Allows image files to be retrieved on the flv from
           a Java Archive (jar)  file
           Cloneable data structure for soil parameters
           Generate a realization of a  multivariate normal
           distribution   Behaves like the SAS' v  macro
           mvn. sas [3]
           Cloneable data structure for a parameter in Virulo
           Data  structure for parameter text fields
           Cloneable data structure for virus parameters
           Prints dipboardable text output
           Polvmorphable random deviate generator due to
           Java Numerical  Toolkit  [4]
           Holds necessarv parameters for each soil type
           selectable with FlowComboPanel
           Sorts  a  Java  Vector  object, due to  Eckel (1998)
           Cloneable data structure for a soil or virus name
           Stores vanance-covariance matrices as  computed with
           Rosetta for each soil type
           Useful oft-used static methods for use  with Java
           Vector objects
           Launcher for  the application (or applet)
           GUI frame for Virulo
           The analogv  of FlowComboPanel  but  for
           virus parameters
           Analogy of SoilStack  but for virus parameters
           by virus name
i  Names in italics represent Java interfaces
i  Names in boldface represent classes that are part of the Java  language

 File  Edtt  View  Seated  Go  i<:ulrn*'Ki  T3i*   Help

  V  ,   ."       '**->! |, litt ^ijp e-td't' -r;/H-.'3;"urrtjii u^'-tteni-a'^" i ltr(                »|  > Search   •*  -
  da p3'> EESIXtee Deprecated Index Help
  •FV :, a4;; NE- T c LA;;                                     IJKAMM> NO t RAM[s»  AJ o^
Interface Attenuator
i:iiDl.' ;.i'-r;-t"- Atlenuaior
 Method Summary

 Method Detail

get Medium

      Figure 7  Javadoc class documentation for  Attenuator interface.

   •  The proposed barrier matrix consists of one of the 12 USDA soil types

   •  Provided distribution  functions for saturated  hydraulic conductivity  do not
      account for preferential flow, thus it must be assumed preferential flow will
      not occur in the site of interest
    We found  that for a  1-meter thick proposed hydrogeologic barrier with a volu-
metric water content  of 0 3, only soils classified as clays did not fail to produce the
4-log attenuation in the  these simulations  User s  may have additional information
that could change the outcome of the probabilistic model

    This study revealed several areas of much needed research  These include
    •  Table 3 lists in order of magnitude  the parameters that most strongly affect
      the results of this model  It suggests the parameters which should receive the
      most research  attention through experiments

    •  The  issue of accurate  estimation of the air-water interfacial  area is  an  im-
      portant one, not only for modeling transport of contaminants subject to hy-
      drophobicity effects but also for unsaturated-zone virus transport modeling

    •  More experimentation  is needed with  real  proteins or the ammo-acids that
      have surfaces that behave like viruses,  rather than artificial  or inorganic col-

    •  Geochemical effects can produce profound changes in the sorption and survival
      of the viruses  and more work is needed to  identify the causes

    •  Although plaque assays are appropriate for testing of natural water for  the
      presence of viruses,  the associated uncertainty when large numbers of viruses
      are used  lead to lack of experimental control at the level of accuracy needed
      to study viruses in unsaturated soil columns For these types of studies more
      accurate assay methods are needed

    •  Correlations need to be developed to predict mass transfer coefficients specifi-
      cally for viruses which sorb at both the solid-water and the air-water interface
      Current correlations are not relevant for viruses

    •  Out of about 36 soil column studies in the  literature,  only those of Jm et  al
      (2000) and Chu et  al  (2001)  were done with unsaturated columns  More
      unsaturated column studies are needed
    Because of the large uncertainty in parameters needed to predict virus transport
in  the  unsaturated zone  probabilistic models that encapsulate and  propagate the
uncertainty in those parameters in the predictions should be used

9     List  of  Symbols and  Notation   Used
                     Water retention curve fitting parameter
                     Vertical hydrodvnamic dtspersivitv
                     Coefficient of exponential decav of virus concentration
                     Soil water content
                     Residual soil water content
                     Saturated soit water content
                     Measured water content equivalent to (),,
                     Suspended to solid sorbed virus mass transfer coefficient
                     Suspended to air-sorbed virus mass transfer coefficient
                     Suspended phase virus mactivation rate
                     Solid sorbed phase virus mactivation  rate
                     Air sorbed phase virus mactivation rate
                     Viscosity of water
                     General Gaussian random variable
                     Soil bulk density
                     Pearson s correlation coefficient
                     Density of water
                     Surface tension of water
                     Variance operator
                     Unsaturated soil water tortuosity
                     Dummy variable of integration
                     The predicted attenuation factor ((',,(', , , }
                     Concentration of viruses in suspended phase
                     Concentration of viruses in the solid-sorbed phase
                     Concentration of viruses in the air-sorbed phase
                     Maximum (initial) concentration of viruses entering, top of proposed hvdrogeologic barrier
                     Concentration  of viable suspended viruses exiting the proposed hydrogeologic barnei
                     Molecular  virus diffusivitv
                     Effective molecular virus diffusi^ity
                     Vertical hydrodvnamic dispersion coefficient
                     Dahmkohler number for mass transfer  suspended to solid water  inlerface
                     Dahmkohler number for mass transfer  suspended to air-water interface
                     Cumulative virus attenuation function
                     Laplace transform of cumulative virus attenuation function
                     Measured flux  of percolating water equivalent to <(
                     Unsaturated hydraulic conductivity
                     Equilibrium distribution coefficient (solid-suspended)
                     Saturated  hydraulic conductivity
                     Thickness of proposed hydrogeologic barrier
                     Number of observations
                     Effective soil water saturation
                     Mean percolation velocity
                      Measured velocity of percolating water
                      Measured velocity of percolating water
                     Air water mterfacial area
                     Solid-water mterfacial area (soil specific surface area)
                     Virus attenuation function
                     Acceleration due to gravity
                     Soil capillary pressure head
                     Suspended  to solid sorbed  virus mass transfer rate
                     Suspended  to air-sorbed virus mass transfer rate
                      Mass transfer rate equivalent to A
                      Mass transfer rate of equivalent to A
                      Boltzmann s constant
                     Water retention curve fitting parameter
                      Flux of percolating water
                     Vector of calculated mean soM particle diameters
                      Mean soil particle radius
                      Virus radius
                      Laplace domain variable
                      General Gaussian random variable
                      General Gaussian random variable
                      Distance  downward from top of proposed hydrogeologic barrier

10    References
Anwar, A H M F ,  Bettahar, M , Matsubayashi, U 2000  A method for determining
air-water mterfacial area in variably saturated porous media  J  Contain  Hydro/
43 129-146

Beres, D L ,  Hawkins, D M  2001  Plackett-Burman technique for sensitivity anal-
ysis of many-parametered models  Ecol  Modell 141 171-183

Boas  M 1983  Mathematical Methods in the Physical Sciences, Second Edition
John Wiley & Sons  New York  793 p

Breidenbach  P Chattopadhyay, S  Lyon, W G Survival and Transport of Viruses
in  the Subsurface, An Environmental Handbook  U S EPA Document in  prepara-

Chu  Y  . Jm  Y Flury  M   Yates, M V  2001  Mechanisms of virus removal during
transport in unsaturated porous media  Water Resour  Res  37(2) 253-263

Eckel, B 1998  Thinking in Java Prentice Hall PTR,  Upper Saddle River  New
Jersey  1098 p

EPA  2000   National Primary Drinking Water Regulations   Ground Water Rule
Fed  Regis  65(91) 30193-30274

Gamma,  E ,  Helm  R  Johnson,  R  Vlissides  J 1995  Design Patterns, Elements
of Reusable  Object-Oriented Software  Addison-Wesley  Reading, Massachusetts
395 p

Geary D M  1999   Graphic Java' w 2, Mastering the JFC,  3rd Edition, Volume
II,  Swing Sun Microsystems Press Palo Alto   1622 p

Gosling, J  Joy B  Steele, G   Bracha,  G  2000   The Java1 w Language Spec-
ification, Second Edition Addison-Wesley, Boston   505 p

Hantush  M  M  Marino M A   Islam  M R  2000   Models for leaching of pesti-
cides in  soils and groundwater J  Hydro  227  66-83

Jm, Y  Chu, Y   Yunsheng, L   2000  Virus  removal and  transport in saturated
and unsaturated sand columns  J  Contam  Hydrol 43 111-128

Jury W A  Gardner W R , Gardner  W H  1991  Soil Physics, Fifth  Edition John
Wiley & Sons  Inc  New York  328 p

Kaczmarek  M , Hueckel, T , Chawla V  Imperiali P  1997  Transport through
a clay barrier with the contaminant concentration  dependent permeability  Trans-
port in Porous Media 29 159-178

Keswick, B H ,  Gerba,  C P  1980  Viruses  in groundwater   Environ   Sci   Tech-
nol  14 1290-1297

Kim,  H  , Rao, PS C ,  Annable,  M D  1997   Determination  of effective air-water
mterfacial area in  partially saturated porous media using surfactant adsorption  Wa-
ter Resour Res  33(12)2705-2711

Kitanidis, P K  1997   Introduction to Geostatistics  Applications in  Hydrogeolo-
gy  Cambridge University Press, Cambridge  U K 249 p

Knuth, D E  1992  Literate Programming, CSLI  Lecture Notes Number 27  Center
for the Study of Language and Information, Leland Stanford  Junior University  S-
tanford, California 368 p

Lance  J C  Gerba,  C  P  1984   Effect of ionic  composition of suspending solu-
tion  on virus adsorption by ad soil column  Environ  Sci  Technol  14  1290-1297

Ley,  F J  Alves WJ ,  van  Genuchten R  1996   The UNSODA Unsaturated Hy-
draulic Database, User's Manual  Version 1 0 EPA • 600; R-96/095

Lyon, W G ,  Faulkner,  B F ,  Khan  F   Chattopadhyay,  S ,  Cruz, J  2002   Pre-
dicting Attenuation of Viruses in Unsaturated  Natural Barriers  2  User's Guide to
Virulo (in preparation)

Mazzone, H M  1998  CRC Handbook of Viruses, CRC Press, Boca Raton,  FL

Plackett, R L , Burman, J P 1946  Design of optimum multifactonal experiments
Biometnka 33 305-325

Poletika, N N , Jury, W A ,  Yates, M V  1995   Transport of bromide,  simaz^ne
and  MS-2 coliphage in  a lysimeter containing undisturbed, unsaturated soil  Water
Resour  Res  31(4) 801-810

Press W H   Teukolsky, S A ,  Vetterlmg, W T   Flannery, B P  1992   Numerical
Recipes in C, The Art of Scientific Computing, Second Edition  Cambridge Univer-
sity Press Cambridge  U K  994 p

Rose, W  Bruce  W A  L949  Evaluation of capillary character in petroleum reser-
voir  rock  Trans  Am  Inst   Mm Metall  Eng  186 127-142

Schaap   M G ,  Ley,  F  J , van  Genuchten, M  T  1999  Bootstrap-neural network
approach to predict soil hydraulic parameters  (In)  M T van  Genuchten, F  J  Leij,
L  Wu (Eds )  Characterization and Measurement  of the Hydraulic Properties, of
Unsaturated Porous Media,  Part 2  Proceedings of the  International  Workshop
University of California Riverside pp  1237-1250

Schaefer  C E , Arands, R R ,  van der Sloot  H A , Kosson, D S  1995   Predic-
tion  and experimental validation  of liquid-phase  diffusion resistance in unsaturated
soils  J Contam  Hydrol 20 145-166

Schijven, J F  Hassanizadeh  S M   2000   Removal of viruses by soil passage
overview of modeling, processes, and parameters  Critical Reviews of Environmental
Science and Technology 30(1) 49-127

Sim  Y  Chrysikopoulos,  C V 2000  Virus  transport  in unsaturated  porous me-
dia  Water Resour Res 36(1)173-179

Thompson, S S ,  Flury,  M , Yates, M V ,  Jury, W A 1998  Role of the air-water-
solid interface in  bactenophage  sorption  experiments   Appl  Environ  Microbiol
64(1)  304-309

Thompson  S S ,  Yates, M V  1999  Bactenophage mactivation at the air-water-
solid interface in  dynamic: batch systems  Appl Environ  Microbiol  65(3) 1186-

van Genuchten  M Th  1980  A closed-form equation for  predicting the hydraulic
conductivity of unsaturated soils  So//So Soc  Am  J  44892-898

van Genuchten  M Th  Ley,  F J  Yates  S R 1992  The RETC Code for Quanti-
fying the Hydraulic Functions of Unsaturated Soils  EPA 600 52-91 065  10 p

Vilker  V L,  Burge W D  1980   Adsorption mass transfer model for  virus trans-
port in soils  Water Res  14 793-790

Wan  J , Wilson   J L  1994  Visualization  of the role  of the gas-water interface
on the fate and transport  of colloids  in porous media  J Appl Environ  Microbiol
60 509-516

Wilson  E J   Geankoplis,  C J  1966  Liquid mass transfer  at  very low Reynolds
numbers in packed beds  I&EC Fundam   5(1) 9-14

Yates,  M V  1995  Field  evaluation  of the GWDR s natural disinfection criteria
J Am  Water Works Assoc 87 76-84

Yates   M V  and W A  Jury   1995  On  the use of virus transport modeling for
determining compliance J Environ  Qual 24 1051-1055

Yates   M V   Ouyang, Y  1992   VIRTUS  a model of  virus  transport  in unsatu-
rated  soils  Appl  Environ  Microbiol 58 1609-1616
11    Internet References

[1] Mount H R 2000  Remote Soil Temperature Network
http://www statlab  iastate.edu/soils/nssc/temperature/rstnl.htm
[2] JAMA  A Java Matrix Package
http  //math.nist gov/javanumerics/jama/

[3] SAS Institute 2000  MVN macro Generating multivariate normal data,

[4] NIST  1998  Java Numerical Toolkit


                               Appendix A

    Back-Calculation of Mass Transfer Coefficients of Chu et al  (2001)

1  Assume Chu et  al  (2001) A i is our /,•  and their A <, is our  /,  , based on  Chu
   et  al  Figure 1   Also their  ./„ is our  (in In
   ().()47.r) //, In

3  Experiment  1 obtained  DH <, -  2 !)2 at \,,,.,, ,„,,,,,/  -   2.5.4 cm hr   thus
   /M^->  -  V,,, ,,,„„,,,//)(/i//,  = 12.4 .->(•/;/  //; 1(2.02) 'il(lr/;M -= (> M)///   '
   Experiment  2 obtained Z9o i     121  at I ,,„,,,  „„./,,,/     1 7 1  cm, hr,  thus
   /, Ws  j    ~>  7i/u  '

4  Let  refer to values obtained from the sand centroid of the soil triangle   Now
   -• /, Ws  _,  --- ."i 7".  12.()r. = (I  1.44 cm hr^O 00134  m  hr
   if ft1,,,  - ft1, ,r,<,.,  and other values taken from sand centroid  then
   a] ----  7 .4:4.5 ( m  '  -- 7.5.5 .4 /;/  '
             (>Mi 7:5.4.4   (I 1)27 cm hr=-0 00927m  hr
6  To compute the propagation of error let u1 {} be the variance operator Then
   according to Boas  (1983) p  734, consider a function c of 2 normally dis-
   tributed variables  i and //  We use overlmes to denote the mean  If they are
   uncorrelated  then
   From  the  UNSODA database, we estimated mean particle radii  for each soil
   classified as sand  from sieve data as follows

With this we obtained an overall mean, rp = 1 73 x 10  ' ni, and a standard
deviation ,,„  - 0,
we obtained f> - 0 101, thus we cannot reject the null hypothesis, and con-
clude f^ and /,, are uncorrelated for soils classified as sand
Thus we may proceed using  the propagation of error formula listed above

Var[«T]-  (^

Noting that
And evaluating at the means, we obtain
Varl/;/]^  3 33 x K)7

In a likewise manner, we obtain
      = 3 2:5") x l()-fl
Due to the lack of data on h' we assume the only significant contributions to
the error in i^ are the lack of fit error in the Dahmkohler number as computed
by Chu  et al   plus the error  in «c; is of similar magnitude as that of 
                       Appendix B.

Vanance-Covanance Matrices of Correlated Hydraulic Parameters
silt loam

-0 00001
+ 000003
-0 00009
^-0 00042

-0 00049
log,,," 1 -000015
logm A" ..
>- 000000
-0 00050

-0 00011
, 0 00090
. 0 00110
-0 00006
- 0 00469

. 0 00034
-0 00099
logi,,A\ -000430

• 0 00003
• 0 00103
-0 00038

- 0 00049
• 0 00030
t 0 01017

- 0 00090
. 0 00727
-0 00038

- 0 00469
-0 00718
• 0 00016

-0 00009
- 0 00113
-0 00446

-0 00146
-0 00114

- 0 00110
- 0 01676
-0 04797

-0 00262
-0 09467
• 0 12027

-0 00038
-0 00185
i 0 01506

* 0 00000
- 0 00030
+-0 00026

-0 00006
-0 00038
-0 00152
-0 00179

-0 00099
-0 00293
-0 01776
. 0 08733

- 0 00042
- 0 00191
-0 00446
-0 01506

-0 00050
- 0 00425
,-0 14744

-0 00469
-0 03863
- 0 04797
-0 00179
0 22576

-0 00430
-0 12027
-0 52026

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