United States
Environmental Protection
Agency
Predict!
xxx
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EPA/600/R-02/051a
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
Faruque 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
Office of Research and Development
U.S. Environmental Protection Agency
-Cincinnati, OH 45268
Recycled/Recyclable
Printed with vegetable-based ink on
paper that contains a minimum of
50% post-consumer (iber content
processed chlorine free.
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Notice
The U.S, Environmental Protection Agency through its Office of Research and De-
velopment funded and managed the research described here through in-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
Investigator.
Virulo and the user's guide have been subjected to the Agency's peer and 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.
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Foreword
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
planners.
Stephen G. Schmelling, Acting Director
Subsurface Protection and Remediation Division
National Risk Management Research Laboratory
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Abstract
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 main 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
system.
IV
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Contents
1 Introduction
2 Abridged Literature Review
3 Mathematical Description
3.1 Differential Equations
3.2 Proposed Solution for the Differential Equations
4 Mass Transfer and Inactivation Rates of Viruses
5 Modeling under Uncertainty
6 Sensitivity Analysis
7 Design of the Computer Model
8 Conclusions
9 List of Symbols and Notation Used
10 References
11 Internet References
1
1
2
2
4
9
11
14
19
20
24
25
27
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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 -\ogwA for poliovirus for
Rosetta sands 15
5 Frequency histogram of values of —\ogwA for poliovirus for Rosetta
silt loams 16
6 Frequency histogram of values of -log10A for poliovirus for Rosetta
clays 17
7 Javadoc class documentation for Attenuator interface 22
VI
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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
VII
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Acknowledgments
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.
VIII
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1 Introduction
Impending regulations in U.S. 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
research.
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 Hassanizadeh (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
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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 in 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 interfacial 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:
d[OmC]
dt
dt
dt
d2C d[qC]
dz2
dz
- X*pC* -
0 (1)
where C = C(t, z) (ML 3) is the concentration of viruses in the mobile solution
phase, t is time, z (L) is the downward distance from the top of the proposed
hydrogeologic barrier, C*(t,z) (MM~1} is the adsorbed virus concentration at
the liquid-solid interface, CQ(t,z) (ML~3} is the adsorbed virus concentration at
the liquid-air interface, q (LT~l) is the specific discharge, Om (L3L~3) is the
moisture content, A (T~l) is the inactivation rate coefficient for the viruses in
the bulk solution, A* (T"1), the inactivation rate for the viruses that are sorbed
at the liquid-solid interface, and A° (T'1), the inactivation rate for the viruses
sorbed at the liquid-air interface, p (ML~3} is the soil bulk density, and Dz =
Qizq/0m + T>e (L?T~1} is the hydrodynamic dispersion coefficient, az (L} is the
vertical dispersivity, £>e = £>/r (L2T~l), where V (L2T~1} is the virus diffusivity
in water, and r (LL~l, greater than 1) is the tortuosity.
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C(M)
> inactivation
\ 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
geochemistry
• The soil does not induce preferential flow
Sorption and inactivation of viruses at the various interfaces is described by Sim
and Chrysikopoulos (2000) by
dC*
(2)
where k — KO,T, OT — 3(1 — 6s)/rp is the liquid-solid interfacial area in units
of (L2L~3}. The symbol fc (T~1} is the microscopic mass transfer rate and K
(LT~1} is called the mass transfer coefficient. In Eq. 2, Kd (L3M~l) is the
equilibrium partitioning coefficient, rp (L) is the average radius of soil particles,
and Bs (L3L~3} is the saturated water content. Analogously they derived the
change in concentration of viable viruses at the air-water interface,
_
dt
1.
— A.
(3)
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where fc° (T -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
K — K
(4)
where re* (L2L 3) 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 of viral
inactivation rates on their sorption state. A method for estimating a*n is discussed
below in the section on Mass Transfer and Inactivation Rates.
The Buckingham-Darcy flow equation for the downward water flux in the un-
saturated zone is (Jury et al., 1991, p. 88):
where h (L) is the capillary pressure head and z is positive downward.
3.2 Proposed Solution for the Differential Equations
Using the above governing equations, we would like to determine whether a proposed
hydrogeologic barrier is likely to produce a specified degree (e-log) of attenuation.
To solve the equations we will formulate an attenuation function f(t,z] for perco-
lating viruses, and focus our interest on the cumulative attenuation, the total mass
leached (M) at depth z from the bottom of the barrier (z = L). Considering a unit
area portion of the barrier bottom, the leached mass is (Hantush et al., 2000):
M(t,z = L] =
(6)
t—>oo
We will apply the final value theorem of operational mathematics. This theorem
states that given the Laplace transform M(s, z) of the cumulative attenuation, the
following identity holds (Hantush et al., 2000):
lim M(z] = \imsM(s,z)
t—+00 s—>0
(7)
In other words, we can determine the final value of the cumulative attenuation
function M by simply taking its Laplace transform, M, and evaluating it as the
Laplace domain variable s —> 0. Although this conceptualization is traditionally used
in the design of chemical reactors, Hantush et al. (2000) have shown the conditions
under which it may be valid for transport in natural environmental systems. They
discuss the differences between a complete mixing model and the more realistic
advective-dispersive model, applied to the final value of attenuation.
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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:
(8)
dC(ttoo)
dz
= 0
(9)
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:
dC_
dz
z=Q
(10)
Since it is assumed that the flow is due to gravity on\y(dh/dz = 0), then the
total head gradient is unity (dz/dz] and q = K(0m), To obtain K(0m),
Genuchten (1980) obtained the following:
van
= KS
1——
(11)
where Ks is the saturated hydraulic conductivity, Or is the residua! water content
(L3L~3), Os is the saturated water content (L3£~3), 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 Cma;cexp[-/3t]. Now we can write the attenuation
function:
~dz
qC
(12)
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 intial conditions (Eq. 8) yields
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Table 1: Hydraulic Properties of Sand, Silt, and Clay
Soil*
sand
silt loam
clay
Parameter
Or
&s
logio/0
logioa
logion
P
rp
az
T
Gr
os
l0gio#s
logiocv
logion
P
rP
az
T
6r
9S
logio/'G
logioa
logion
P
rp
a.z
T
N
308
308
991
308
308
1681
o§
It
1944*
330
330
751
330
330
1331
o§
1*
1944*
84
84
221
84
84
381
o§
1*
1944*
Mean
0.050
0.367
-0.691
0.5306
0.482
1.58 x 106
4.71 x 1Q-4
5.59 x 10~3
11.7
0.063
0.406
-2.160
-0.207
0.206
1.43 x 106
1.18 x 10~4
8.75 x 10~5
11.7
0.101
0.515
-2.085
0.276
0.114
1.29 x 10G
9.95 x 10"5
8.75 x Itr6
11.7
Standard
Deviation
0.003
0.032
0.218
0.034
0.077
1.42 x 105
1.60 x 10~5
0.00
7.38
0.013
0.050
-0.384
0.075
0.016
1.48 x 105
5.50 x 1(T5
0.00
7.38
0.011
0.085
0.0475
0.129
0.015
1.68 x 105
6.15 x 10~5
0.00
7.38
Units
i3!-3
L3£~3
log(m /ir"1)
logfm"1)
\og(dimensionless)
g m~3
m
m
0 Celsius
L3L~'A
&L'3
log(?n hr~l)
\og(m~l)
\Qg(dimensionless]
Q
g m
m
m
0 Celsius
LAL~6
L3L~3
logfm hr~1}
logfm"1)
\og(dimensionless)
g m~3
m
m
0 Celsius
* Generated with the Rosette program {Schaap et al. 1999). unless otherwise noted.
t Field lysimeter study by Poletika et al. (1995).
t Kaczmarek et al. (1997).
* Data from Remote Soil Temperature Network [I].
f From the UNSODA database (Leij et al. 1996).
§ Generated with random deviates in soil textural triangle queried by USDA category,
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emsc + psc*
(13)
Likewise, the Laplace transform of Eq. 2 is
-A'pC"
(14)
Likewise, the Laplace transform of Eq. 3 is
(15)
Noting the boundary conditions given by Eq. 8, and solving for C* in Eq. 14,
we obtain
1
s , 1 , \*
(16)
We can solve for C® in Eq. 15 to obtain
(17)
Now we insert the substitutions into Eq. 13, which yields the following ordinary
differential equation, in terms of C:
(18)
where
7 = A +
££ j. ^L
fc "*" ^d
(19)
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To solve Eq, 18 we consider the homogeneous equation which has the character-
istic polynomial Dzr2 —VT —7. The general solution \sC(s,z) = tpiexp\Ti(s)z] +
1/33 exp[F2(5)2;], where, by the quadratic formula
ID,
(20)
The Laplace transform of Eq. 9 is dC(s,oo)/dz, which implies tp\ = 0 for a
physically reasonable solution, thus C(s,z) =
The Laplace transform of Eq. 12 is
dC_
dz
Substituting for C and its derivative, we have:
(21)
/(a, z) = -£»,0
The Laplace transform of Eq. 10 is
(22)
or
3=0
(23)
(7(5))
(24)
Thus, we find the value of tp% is
(25)
From Eq. 22:
From Eq. 19, we note
....
7 = hm 7 = A +
s—O '
\p
(26)
(27)
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We now apply the final value theorem:
T i f / r \ -\- atl \ ax-'
lim M(z - L) = lim M(s, z) =
t—too s—*0 ' p
(28)
The attenuation factor A is
A =
M(z = L)
M(z = 0)
To find M(z — 0), we integrate the input flux in time:
(29)
ft
M(z = 0) = qCmax /
Jo
(30)
thus
(31)
The average velocity of the percolating water is V — q/6m- Note the inactiva-
tion rate for the air-water interface drops out of the final expression. This is due to
the fact mass transfer to the air-water interface is irreversible; therefore, the rate of
decay is irrelevant after the virus moves to the air-water interface from the mobile
phase. We also note /? drops out, because in the limit t —> oo the exponentially
decaying input pulse appears as an instantaneous input of mass. It thus does not
behave differently from the Dirac-5 pulse which has been used to model pesticide
application (e.g., Hantush et al., 2000). The input concentration Cmax also drops
out because we are only interested in the attenuation factor.
We can now define a hydrogeologic barrier as a layer physically separating the
virus source and the ground-water supply under consideration which produces the
attenuation factor:
A=<10 £, or - log10A
(32)
For example, the current draft GWR frequently refers to a target value of e = 4,
meaning "4-log attenuation," or 99.99% attenuation of active viruses.
4 Mass Transfer and Inactivation Rates of Viruses
As viruses are carried with percolating water, their rate of inactivation depends on
many factors. Breidenbach et al. (2001) provided an overview and tabulation of
measured inactivation rates. Factors include the geochemical characteristics of the
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Table 2: Parameters Used for Poliovirus
Parameter*
logioA
log10A*
K
K°
rv
Kd (sand)
Kd (silt loam)
Kd (clay)
N
12
0*
It
It
0^
87
23
39
Mean
0.605
0.304
1.34 x ID"3
9.27xlO-3
1.375 x 10~8
2.43 x 1Q-4
3.77x UT4
7.20 x 10~4
Standard
Deviation
0.608
0.608
1.80 x 10~3
1.80 x 10~3
1.25 x 109
5.66x 10-4
7.16 x 10-4
9.74 x 10~4
Units
\og(hr~
\og(hr~
m hr~l
m hr~l
^ _ i
m g L
m3 g~l
m3 g~l
M
* Data complied by Breidenbacti et al. (2001) unless otherwise noted.
t From Gnu et al. (2001), see Appendix A for assumptions.
\ Yates and Ouyang (1992) assumed A* fa A/2.
§ 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 bacteriophages (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 geochemical
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 bacteriophage, 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 K, due
to Wilson and Geankoplis (1966), K = 1.09V* [r>/2rp0s]s 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
coefficient, ppn — 0.039, N = 23). The expression fails to account for major factors
that affect mass transfer of viruses at the molecular level, such as pH-dependent
electrostatic 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.
10
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As with K, a correlation has not been established for K® 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 interfacial
area by a^ — (pwgh&m)/cr. In this expression, a (MLT~2) is the surface tension
of water, pw is the density of water, g is the acceleration due to gravity. Employing
the van Genuchten (1980) expression that relates 9m to the capillary pressure head
h, and expressing the effective saturation as Se = (9m — 9r)/(9s — Or] we obtain
the following:
Q!fT
l/n
(33)
The benefit of using this expression is that it utilizes the well-tabulated fitting
parameters for which we have already developed multivariate distribution functions
which we will discuss in the next section.
Virus diffusivity T> (L?T~1} is governed by Brownian motion and is described
by the Stokes-Einstein equation:
kbT
(34)
in which kb is Boltzman's constant (MLT 2), T (° Celsius) is temperature,
= (j,(T] is viscosity of water (ML~lT~l), and rv is the equivalent radius of the
virus.
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 e — log attenuation will occur, encapsulating the possible sources of error in
the model parameters.
Virus diffusivity, T>, is a physical parameter that can be calculated if the tem-
perature, T, 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):
11
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UNSODA sands
simulated
0.05
1 A-
1*»
•1O
1 Z
4.
n
X
X X
X
X
x logK
* log n
log a.
X
X
X
X
•«•***»*»* • w
+ **» **•*»•» + +*•
0.10
0.15 0.05
0.15
Figure 2: Plot illustrating correlation of \ogioKai Iogi0n, 8a, and Iogi0a with
9r for sands in the UNSODA soils database and a corresponding multivariate
normal ensemble simulation.
r(0m) =
if 0™ < 0.2
(35)
,7/3
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 HQ; ppn — 0, HI. ppn •£ 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 Kitanidis, 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 e - 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 variance-covariance matrices of the parameters, as
determined with the UNSODA database (Leij et al., 1996), These are listed in
Appendix B.
12
-------
0.8 1.0
A Kim etal. (1997)
Anwar et al. (2000):*
• 25 mm,
logKs =-0.165
• 50 mm,
log Ks = -0.082
* 75 mm,
log Ks = -0.029
simulated
I Baked blasting sand.
| Glass micro-beads of 3 different mean diameters listed.
* Generated by setting mean 6m — 0.16, std. err. 0.2.
Figure 3: Comparison of values of a^ used in model simulation for Rosetta
sands with measured values.
Recently, workers have measured air-water interfacial areas for partially saturat-
ed sand (Kim et al., 1997) and for variously-sized glass microbeads (Anwar et al.,
2000). A comparison between these measurements and a conditional simulation is
presented in Figure 3. The simulated values were obtained by using the multivariate
probability density functions with the means for sand as obtained from the Rosetta
program (Schaap et al., 1999).
13
-------
Clearly, this simple model appears to underestimate a^ when Se is less than
about 0,6 (6m 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 a^ will apparently be underestimated for sandy soils, thus
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. The
histograms were generated with Monte Carlo simulations with soil hydraulic data
from the UNSODA soils database (Leij 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:
p(failure)
number of Monte Carlo runs that produced A < 10'
total number of valid Monte Carlo runs
The probability of failure for poliovirus to achieve 4-log attenuation was p=
22/5697 for the sands. Model users would more likely be interested in 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= 6/2000000. For this particular set of data, the Rosetta clays produced a
probability of failure p= 0/9000000 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 (1646) 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
14
-------
Histogram
D fieuln and Accumulate
Right Tftlncated Histogram
Vertical Axis is Count
Hits -Rijns
22 :5637
Figure 4: Frequency histogram of values of —log1QJ4 for poliovirus for Rosetta
sands.
15
-------
Figure 5: Frequency histogram of values of —log10A for poliovirus for Rosetta
silt loams.
16
-------
280-
2$0-
240-
220-
200-
140-
80 -
Right TrlinEated Histogram
Vertical Aids-IS'Cdunt
Figure 6: Frequency histogram of values of —\ogwA for poliovirus for Rosetta
clays.
17
-------
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 (1946)
for a 20 parameter model:
i
2
3
4
5
6
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
_ _ _ _ + +
-- + + + + -
_ _ + - -
+ -- + + ---
- + -- + + --
-- + -- + + -
---- + -
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 (log10A's, &s, logical, &r, and logioa)
listed in Table 3 are listed only to show the effect of a change in the mean value.
A special variance-covariance matrix was calculated for all the parameters of the
UNSODA 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
(logio/^s) 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 \ogioKs if there
is evidence of soils that could produce preferential flow. The second most impor-
tant parameter was «°, 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
18
-------
Table 3: Main Effects on Probability of Failure
Parameter
logio Ks
K°
L
logio n
rp
logio A
logio A*
Kd
Or
6m
T
P
logio <*
rv
K
az
os
Plausible
-2.12
0.0
0.5
0.23
6.233 x 10~5
0.0
0.0
-3.05 x 10'4
0.0647
0.15
4.32
975333
0.12
1.250 x 10"8
0.0
8.75 x 10~5
0.433
Domain*
0.42
0.023
20
0.31
1.997 x 10~4
1.213
0.912
1.20 x 10'3
0.0753
0.40
19.08
1891333
0.28
1.500 x 10~8
3.14 x 10^3
5.59 x 10~3
0.376
Effect
+0.105
-0.099
-0.039
+0.030
-0.026
-0.025
+0.020
-0.017
+0.015
+0.012
+0.007
+0.006
+0.005
-0,004
+0.002
-0.001
+0.001
ComP1
i it th.
listed
± the mean of the
tandard deviation for the dryii
n the UNSODA database.
air-water interfacial 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 L x log10A* (-0.072), which is not
surprising due to the increased residence time experienced by the viruses in thicker
soil layers, rv x Kd (+0.054), and \ogwKs x K* (-0.095, the parameters which
exhibit important (and opposing) main 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
[2]-
We consider these classes to be "lightweight," 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.
19
-------
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
http : //www . epa . gov/ada/
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^< 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, to 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.
20
-------
Table 4: Classes Used in the Computer Model
Class
No. of
Public
Methods
Description
Attenuator
Compare*
DoubleCompare
FlowComboPanel extends JPanel*
FlowPanel extends JPanel
implements Observer
GasDev
Gossiper implements Observer
Histogram
HistoPlot
HistoPanel extends JPanel
HspBasicMath
HspMonteCarlo
ImageCanvas
JarLoadable
Medium implements Cloneable
Mvn
Normal implements Cloneable
NormalF
Qperandum implements Cloneable
OutputPanel extends JTextPane
Random
SoilStack
SortVector
StringAsChars implements Cloneable
VarCov
VectorParser
Virulo (or ViruloApplet)
ViruloFrame extends JFrame
VirusComboPanel extends JPanel
VirusStack
t Names in italics represent Java interfaces.
$ Names in boldface represent classes that are part of the Java language
Computes the water flux and A.
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 by creating unwanted dependencies.
Polymorphable class for generating histograms.
Uses Histogram to create a Buffered I mage for
display.
Panel for displaying histogram image.
Has useful static methods for oft-used math
operations.
Conducts and manages the Monte Carlo Simulations
for Virulo.
Polymorphable image canvas of Geary (1999).
Allows image files to be retrieved on the fly from
a Java Archive (jar) file.
Cloneable data structure for soil parameters.
Generate a realization of a multivariate normal
distribution. Behaves like the SASrM 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 clipboardable text output.
Polymorphable random deviate generator due to
Java Numerical Toolkit [4].
Holds necessary 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 variance-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 analogy of FlowComboPanel, but for
virus parameters.
Analogy of SoilStack, but for virus parameters
by virus name.
21
-------
EBURS NO FRAMES
DETAIL: FIELD | CON8TH 1 METHOD
Package HMTr** D»Pr«caltd |nd«x
PfiEV CLASS NEK? CLASS
SUMMAfiV: NESTED I FIELD I CQN6TH (METHOD
Interface Attenuator
public Interfax Attenuator
This Is UK Interfax that generalizes attenuation of an Operandtan (e,g., a solute, colloid, heat) in & Medium (e.g,, aiafce, stream, porous
medium, aoi I).
SeeAlio:
Kediun,Opotsndu*
This method returns the mean alwction rate supplied by the Medium,
This method returns the attenuation factor which la by definition the quantity of Dperandum remaining divided by the
initial input quantity of Operandum.
void aspJaa(>C*diu» udiim, Opetandun opaiandm)
A replace is used to replete the data for purposes of a time-series or a Monte Carlo Implementation.
getMedtum
public Hodiu*
get Operandum
public OpBCOndun g»fcDper«B(fct>0
afe Nft SB I Pa
Figure 7: Javadoc class documentation for Attenuator interface.
22
-------
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 amino-acids that
have surfaces that behave like viruses, rather than artificial or inorganic col-
loids.
• Geochemica! 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 Jin 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.
23
-------
9 List of Symbols and Notation Used
Symbol
Description
A
C
c"
D,
Dai
AT,
L
N
S,
T
V
/(*,*)
9
h
k
k"
Water retention curve fitting parameter
Vertical hydrodynamic dispersivity
Coefficient of exponential decay of virus concentration
Soil water content
Residual soil water content
Saturated soil water content
Measured water content equivalent to Om
Suspended to solid sorbed virus mass transfer coefficient
Suspended to air-sorbed virus mass transfer coefficient
Suspended phase virus inactivation rate
Solid-sorbed phase virus inactivation rate
Air-sorbed phase virus inactivation 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 (Cf/C,,ia;i.)
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 hydrogeologic barrier
Concentration of viable suspended viruses exiting the proposed hydrogeologic barrier
Molecular virus diffusivity
Effective molecular virus diffusivity
Vertical hydrodynamic dispersion coefficient
Dahmkohler number for mass transfer, suspended to solid-water interface
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 q
Unsaturated hydraulic conductivity
Equilibrium distribution coefficient (solid-suspended)
Saturated hydraulic conductivity
Thickness of proposed hydrogeologic barrier
Number of observations
Effective soil water saturation
Temperature
Mean percolation velocity
Measured velocity of percolating water
Measured velocity of percolating water
Air-water interfacial area
Solid-water interfacial 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 k
Mass transfer rate of equivalent to fc°
Boltzmann's constant
Water retention curve fitting parameter
Probability
Flux of percolating water
Vector of calculated mean soil particle diameters
Mean soil particle radius
Virus radius
Laplace domain variable
Time
General Gaussian random variable
General Gaussian random variable
Distance downward from top of proposed hydrogeologic barrier
24
-------
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[5] JAVADOC TOOL HOME PAGE,
http://j ava.sun.com/j2se/javadoc/index.html
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Appendix A
Back-Calculation of Mass Transfer Coefficients of Chu et al. (2001)
1. Assume Chu et al. (2001) ki is our k and their £3 is our fc*, based on Chu
et al. Figure 1. Also their Jw is our q, and their Ov is our Om.
2. From Chu et al. Table 1 calculate the velocities as V0x~removed = Jw/@v =
4.86/0.209 = 23.3cm//ir and Vwater-washed = 1.14/0.24 = 4.75 cm/hr =
0.0475 m/hr
3. Experiment 1 obtained Day, — 2.92 at Vox_removed — 23.3 cm/hr, thus
&MS-2 = Vox-removedDa3/L = (23.3cm//ir)(2.92)/(10cro) - 6.80/w--1
Experiment 2 obtained Da\ — 12.1 at Vwater-washed — 4.74 cm/hr, thus
4. Let' refer to values obtained from the sand centroid of the soil triangle. Now
= 3(1-6,,) = 3(1-0-37) = 42 % l = 4295 l
«T rfi/ 0.044cm I±^-^'J t-/fi -izao in
=> KMS_2 = 5.75/42.95 - 0.134 cm/hr=0.00134 m/hr
5.
— 1
l/n
if Q-m = 6v(Chu) an^ other values taken from sand centroid, then
a£ - 7.333 cm"1 = 733.3 m"1
and
_2 = 6.80/7.333 - 0.927 cm/hr-0.00927 m/hr
6. To compute the propagation of error let &2{} be the variance operator. Then
according to Boas (1983) p. 734, consider a function v of 2 normally dis-
tributed variables, x and y. We use overlines to denote the mean. If they are
uncorrelated, then
x=x, y=y
From the UNSODA database, we estimated mean particle radii, for each soil
classified as sand, from sieve data as follows:
a
sieve
size
ai
0-2
as
an
b
particle
fraction
bi
62
63
bn
c
weighted
component
((ai — ao)/2 + a(j) x &i
((a2 — a-i)/2 + ai) x 62
((as — aa)/2 + a-}} x 63
ffa _a i)/2 + a i) x 6
1 V * 7
j» — i \ n .
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With this we obtained an overall mean, rp — 1.73 x 10 4 m, and a standard
deviation cr{rp} = 9.1 x 10~5. These are the values listed in Table 1.
We then computed Pearson's correlation coefficient (ppn) between these com-
puted rp values and the values of 6S for the UNSODA database which were
fitted with the least squares computer program RETC. We obtained the fol-
lowing;
ppn = -0.137
number of observations: 93
Under HQ : ppn = 0,
we obtained p = 0.191, thus we cannot reject the null hypothesis, and con-
clude 9a and rP are uncorrelated for soils classified as sand.
Thus we may proceed using the propagation of error formula listed above:
Noting that
And evaluating at the means, we obtain:
Var[aT]=3.33x 107
In a likewise manner, we obtain
Var[«]= 3.235 x lO"6
Due to the lack of data on K° we assume the only significant contributions to
the error in K° are the lack of fit error in the Dahmkohler number as computed
by Chu et al. plus the error in a£ is °f similar magnitude as that of a^. We
also note (T2{lack of fit} -C
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Appendix B.
Variance-Covariance Matrices of Correlated Hydraulic Parameters
sand
Or
os
logioa
logion
logiQ-ftTs
silt loam
Or
Qs
logioa
logion
logio-Ks
clay
Or
Os
logioa
logion
logio^
everything*
Or
Os
logioa
logion
logio^s
Or
+0.00001
+0.00003
-0.00009
+0.00012
+0.00042
+0.00016
+0.00049
-0.00015
+0.00000
-0.00050
+0.00011
+0.00090
+0.00110
-0.00006
+0.00469
+0.00034
+0.00103
-0.00262
-0.00099
-0.00430
os
+0.00003
+0.00103
+0.00021
-0/00038
+0.00191
+0.00049
+0.00251
-0.00146
+0.00030
+0.01017
+0.00090
+0.00727
+0.00871
-0.00038
+0.03863
+0.00103
+0.00469
-0.00718
-0.00293
+0.00016
logioa
-0.00009
+0.00021
+0.00113
-0.00185
-0.00446
-0.00015
-0.00146
+0.00560
-0.00114
-0.01506
+0.00110
+0.00871
+0.01676
-0.00152
+0.04797
-0.00262
-0.00718
+0.09467
+0.01776
+0.12027
logion
+0.00012
-0.00038
-0.00185
+0.00593
+0.01506
+0.00000
+0.00030
-0.00114
+0.00026
+0.00425
-0.00006
-0.00038
-0.00152
+0.00023
-0.00179
-0.00099
-0.00293
+0.01776
+0.02035
+0.08733
logio/G
+0.00042
+0.00191
-0.00446
+0.01506
+0.04731
-0.00050
+0.01017
-0.01506
+0.00425
+0.14744
+0.00469
+0.03863
+0.04797
-0.00179
+0.22576
-0.00430
+0.00016
+0.12027
+0.08733
+0.52026
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^f CM ^" ^? CM O ^^ CN rH CJ CO CO
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^ dddddddddddo
o. . . .ppppppppppppp
bi ddddddddddddd
_O + + +II + II++III
«. . .pppppppppppppp
13:1 dddddddddddddd
+ i i 1 + 1 i 1 + 1 i 1 + 1
£. .ooppppppppppppp
^ dddddddddoooooo
1 + 1 1 + 1 1 + 1 i++i+ +
&- oooopppppppppppp
* ??????????????¥?
bo bO bO
hfl
« a
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