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Journal of Hydro-environment Research 5 (2011) 93e99www.elsevier.com/locate/jher
Research paper
Numerical analysis of virus transport through heterogeneous porous media
Pramod Kumar Sharma a,*, Rajesh Srivastava b
aDepartment of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, IndiabDepartment of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India
Received 13 August 2009; revised 13 April 2010; accepted 20 January 2011
Abstract
Virus transport through two-dimensional heterogeneous porous media at field scale is simulated using an advective dispersive virus transportequation with first-order adsorption and inactivation constant. An increasing exponential dispersivity function has been used to account forheterogeneity of the porous media. Implicit finite-difference numerical technique is used to get the solution of two-dimensional virus transportequation for virus concentration in suspension. The numerical model is used to investigate the effect of inactivation and mass transfer rateconstants on the relative concentration profile in two observation wells. The effect of correlation length and ln K variance on movement of virusthrough heterogeneous porous media has been studied. It is found that the higher values of mass transfer rate constant and inactivation constantlead to reduced virus concentration in both observations wells. An increase in the variance of conductivity field or its correlation length wasfound to result in an earlier arrival of the virus at the observation wells as well as a higher virus concentration.� 2011 International Association of Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rightsreserved.
Keywords: Virus transport; Heterogeneous porous media; Concentration profiles
1. Introduction
The transport of colloids (including viruses and bacteria aswell as abiotic colloids) in groundwater has long beenrecognized as a serious hazard to human health. It has beenidentified that most of the viruses in groundwater originatefrom human and animal sewage from nearby municipalwastewater discharges, septic tanks, sanitary landfills andagricultural practices. As wastewaters including bacteria andviruses are released into the subsurface environment, theyinfiltrate through the vadose zone, and upon reaching the watertable continue to migrate in the direction of groundwater flow.It is generally seen that the groundwater is often consumedwithout prior conventional water treatment. It is also seen thatpathogenic bacteria and viruses travel a large distance at fieldscale. Therefore it is necessary to understand the mechanismsgoverning the transport and fate of these viruses in subsurface
* Corresponding author. Tel.: þ91 1332 285684; fax: þ91 1332 275568.
E-mail address: [email protected] (P.K. Sharma).
1570-6443/$ - see front matter � 2011 International Association of Hydro-environment Engine
doi:10.1016/j.jher.2011.01.001
groundwater systems at field scale, so that health risk owing togroundwater pollution by viruses can be evaluated. It has beensuggested (Robertson et al., 1991) that the main source ofground water pollution is septic tank as shown in Fig. 1. Mostof the country’s population is using septic tanks for wastewaterdisposal and these represent the largest volumetric source ofcontaminant discharge to the ground water-zone. Previousfield observations have demonstrated that in some aquifersseptic tank viruses and bacteria can travel on the order ofseveral hundred meters. A field scale study (Masciopinto et al.,2008) suggests that for the case of municipal wastewaterinjection into fractured aquifers, the required most conserva-tive set back distance for drinking wells should be more than8000 m. There are several mathematical models available inliterature for virus transport in porous media, which describethe virus attachment onto the solid matrix and inactivationconstants (Vilker et al., 1978; Tim and Mostaghimi, 1991;Powelson et al., 1993; Redman et al., 2001). It is generallyrecognized that two processes occur during virus transportthrough porous media. The first process is the nonequilibriumreversible adsorption, which represents the rate of approach to
ering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Water table
Pumping Well
Septic-tank
Flow Direction
House
Fig. 1. Contamination of groundwater due to septic tank.
94 P.K. Sharma, R. Srivastava / Journal of Hydro-environment Research 5 (2011) 93e99
equilibrium between adsorbed and liquid phase virusconcentrations. The second is the filtration process, which isused as colloids for models describing virus transport throughporous media. Sim and Chrysikopoulos (1995) developedanalytical solution for virus transport in one-dimensionalhomogeneous, saturated porous media for the case of bothconstant flux as well as constant concentration boundarycondition. The effect of model parameters on virus transportwas investigated. Espinoza and Valocchi (1997) used pertur-bation approach to study the effect of chemical heterogeneityon one-dimensional transport of kinetically adsorbingpollutant in a porous media. Rehmann et al. (1999) usednumerical stochastic approach to study the effect of spatialvariability of hydraulic conductivity and virus transportparameters (attachment, detachment, and inactivation) on virustransport through heterogeneous porous media. Sim andChrysikopoulos (1999, 2000) developed analytical andnumerical models to examine the effect of soil moisturevariation on virus adsorption and inactivation in unsaturatedporous media. The model accounts for virus sorption onto theliquidesolid and aireliquid interfaces as well as inactivationof viruses suspended in the liquid phase and virus attached atboth interfaces. Jin and Flury (2002) reviewed the currentresearch work done on fate and transport of viruses in porousmedia which include (a) mechanisms and modeling of virussorption, (b) virus survival and factors affecting virus inacti-vation in the natural environment, and (c) mechanisms of virustransport in porous media and available modeling approaches.Bhattacharjee et al. (2002) used the numerical approach tosolve a two-dimensional mathematical model for virus trans-port in physically and geochemically heterogeneous porousmedia. They studied the effect of subsurface-layeredgeochemical and physical heterogeneity on the movement ofvirus through porous media. Bradford et al. (2003) carried outexperimental work for colloid transport in saturated porousmedia and predicted various breakthrough curves. Thesebreakthrough curves can be fitted by using attachment anddetachment models. This study is very useful for controllingdeposition mechanisms. Anders and Chrysikopoulos (2005)conducted field-scale experiment to investigate the fate andtransport of viruses during artificial recharge. Measured virus
concentrations were fitted using a mathematical model tosimulate virus transport in one-dimensional, homogeneoussaturated porous media accounting for virus sorption, virusinactivation, and time-dependent source concentration.According to Han et al. (2006) retention and transport ofcolloids and microorganisms are complex processes in thevadose due to more complicated water flow regime andadditional interfacial reactions involved. They also studied theretention and transport behavior of two bacteria phases MS-2and fX174 in homogeneous and chemically heterogeneousmedia under variable saturated conditions. Anders andChrysikopoulos (2006) conducted experiment to examine theeffect of temperature and the presence of sand on the inacti-vation of bacteriophase MS-2 and PRD1. The experimentaldata suggested that the inactivation process can be representedby a pseudo-first-order expression with time-dependent ratecoefficients. Further, Anders and Chrysikopoulos (2009)conducted experiment to investigate the factors that controlvirus inactivation as well as transport in saturated and unsat-urated porous media.
In this work we study the virus transport in heterogeneousporous media and evaluate the effect of rate constant andinactivation constant on subsurface virus transport. It isassumed that porous media is saturated and heterogeneous.The implicit finite-difference technique has been used to solvethe governing equations for the case of two-dimensional virustransport in heterogeneous porous media. The model is used tosimulate the field experimental data of spatial moments.
2. Governing equations
The two-dimensional virus transport in homogeneous,saturated porous media with first-order adsorption and inac-tivation is governed by the following equation (Sim andChrysikopoulos, 1995).
vC
vtþ r
q
vC�
vt¼ DL
v2C
vx2þDT
v2C
vy2�V
vC
vx� l��C� l�
r
qC� ð1Þ
where,C is the concentration of virus in suspension [M/L3],C*isthe mass of virus adsorbed on the solid matrix [M/M], DL is the
95P.K. Sharma, R. Srivastava / Journal of Hydro-environment Research 5 (2011) 93e99
longitudinal hydrodynamic dispersion coefficient [L2/T], DT isthe transverse hydrodynamic dispersion coefficient [L2/T], V isthe average interstitial velocity [L/T], r is the bulk density of thesolidmatrix [M/L3], l** is the inactivation constant of suspendedviruses [T�1], l* is the inactivation constant of adsorbed viruses[T�1], q is the porosity of soil medium [L3/L3] ,and t is time [T].
Assuming that the adsorption process consists of virusdiffusion to the outer layer of a solid particle by nonequilib-rium mass transfer, and virus immobilization onto the solidparticle while in equilibrium with the liquid phase virusconcentration in the outer layer, the sorption term in Eq. (1)can be written as (Sim and Chrysikopoulos, 1995):
r
q
vC�
vt¼ k
�C�Cg
�� r
ql�C� ð2Þ
where, k is the mass transfer rate constant and Cg is the liquidphase virus concentration in direct contact with solids. Liquidphase virus concentration can be evaluated from Langmuirisotherm suggowast;ested by Vilker (1981):
C� ¼ Q0bCg=�1þ bCg
�
where, Q0 is the Langmuir monolayer capacity, representativeof the solid phase concentration of adsorbed viruses, and b isa constant related to the bonding energy. It is assumed thatliquid phase virus concentration is very small ie. bCg� 1,then the modified expression of Cg can be written as:
Cg ¼ C�=Q0b ð3ÞAfter substituting the value of Cg from Eq. (3) into Eq. (2),
the following equation can be written:
r
q
vC�
vt¼ kC�
�k
Q0bþ r
ql��C� ð4Þ
The heterogeneity of the porous media represents thevariability of hydraulic conductivity. Yates (1990) usedhydrodynamic dispersion which is function of scale dependentdispersivity and developed analytical solution of transportequation. Yates (1992) used hydrodynamic dispersion which isan exponential function of position and developed analyticalsolution of transport equation. Gelhar (1993) used stochasticapproach to represent the heterogeneity of the porous media.Logan (1996) extended the work of Yates (1990, 1992) anddeveloped an analytical solution for transport in porous mediawith an exponential dispersion function and decay. Zou et al.(1996) used time-dependent dispersion function to study thesolute transport through porous media. Srivastava et al. (2002)used an exponential function for dispersion coefficient toconsider the heterogeneity of the porous media, which wasverified with previous numerical simulation results for reactivetransport through heterogeneous porous media.
In this study, for two-dimensional numerical analysis ofvirus transport through heterogeneous porous media thefollowing assumptions have been made:
1. Porous media is fully saturated and isotropic.2. Porosity and bulk density of soil media is constant.
3. The heterogeneity of the porous media may be accountedfor by using an exponential function for dispersion coef-ficient as follows:
DL ¼ D0 þDN
�1� e�tb
� ð5aÞ
DT ¼ D0 þ 0:1�DN
�1� e�tb
� ð5bÞ
where D0 is the molecular diffusion, DN is the macro-dispersion, and b defines the rate of increase of the dispersivitywith time. The parameters DN and b are related to theheterogeneity of the porous media and are given by DN ¼s2f lV and b¼ V/Rl, where V is the pore water velocity, R is theretardation factor (defined as the ratio of groundwater velocityto the contaminant velocity), s2f is the variance and l is thecorrelation length of log-normal hydraulic conductivity field(Gelhar, 1993).
3. Development of numerical model and validation
In this investigation, an implicit finite-difference numericaltechnique has been used to get the solution of virus transportequations for the case of two-dimensional virus transportthrough heterogeneous porous media (Eqs. (1) and (4)). Theimplicit finite-difference formulation of above equation can bewritten as:
Clþ1i;j �Cl
i;j
dtþ r
q
C� lþ1i;j �C� l
i;j
dt¼ DL
Clþ1iþ1;j � 2Clþ1
i;j þClþ1i�1:j
dx2
þDT
Clþ1i;jþ1 � 2Clþ1
i;j þClþ1i;j�1
dy2
�VClþ1
iþ1;j �Clþ1i�1;j
2 dx� l��Clþ1
i;j
� l�r
qC� lþ1
i;j
ð6Þ
r
q
C� lþ1i;j �C� l
i;j
dt¼ kClþ1
i;j ��
k
Q0bþ r
ql��C� lþ1
i;j ð7Þ
where l is known time; lþ 1 is unknown time; dx and dy aregrid size in x and y direction; and dt is the time step.
Since analytical solutions for virus transport through two-dimensional heterogeneous porous media are not available, thenumerical model has been validated for the case of one-dimensional virus transport through homogeneous porousmedia. We used one-dimensional analytical solution forconstant concentration boundary condition (Sim andChrysikopoulos, 1995). The numerical and analytical resultsof spatial and temporal concentration profiles have been shownin Fig. 2(a and b) for various values of inactivation constant. Thetemporal concentration profile is predicted at a distance of 9 cmdown gradient of the contaminant source. The excellent agree-ment of the numerical results with the analytical solution vali-dated the model. As expected, larger values of inactivationconstant reduce the magnitude of virus in suspension.
0 100 200 300 400 500 6000.0
0.2
0.4
0.6
0.8
1.0
1.0
0.5
0.3
C/C
0
X (cm)
Analytical Numerical
0 3 6 9 12 150.0
0.2
0.4
0.6
0.8
1.0
1.00.5
0.3
C/C
0
Time (hr)
Analytical Numerical
a
b
Fig. 2. (a) Spatial distribution of virus concentration profile with various
inactivation constants (t¼ 240 h, V¼ 4 cm/h, D¼ 15 cm2/h, l*¼ 0 h�1,
k¼ 0.005 h�1). Numbers near the curve represent the value of l** in d�1). (b)
Temporal distribution of virus concentration profile with various inactivation
constants (V¼ 4 cm/h, D¼ 15 cm2/h, l*¼ 0 h�1, k¼ 0.005 h�1). Numbers
near the curve represent the value of l** in d�1).
0 5 10 15 200.0
0.2
0.4
0.6
0.8At distance, X=9 cm
C/C
0
Time (hr)
Pe=2 Pe=30 Pe=60 Pe=150
0 5 10 15 20 25 300.0
0.2
0.4
0.6
0.8
1.0
After time=30 hr
C/C
0
X (cm)
Pe=2 Pe=30 Pe=150
a
b
Fig. 3. (a) Temporal distribution of virus concentration profile for different
value of Peclet number. (l¼ 30 cm, l**¼ l*¼ 0.0 d�1, k¼ 0.05 h�1,
Kd¼ 0.0208 ml/mg). (b) Spatial distribution of virus concentration profile for
different value of Peclet number (l¼ 30 cm, l**¼ l*¼ 0.0 d�1, k¼ 0.05 h�1,
Kd¼ 0.0208 ml/mg).
96 P.K. Sharma, R. Srivastava / Journal of Hydro-environment Research 5 (2011) 93e99
Fig. 3a represents the temporal distribution of liquid phasevirus concentration at a distance of 9 cm down gradient fromthe source and Fig. 3b shows the spatial distribution of liquidphase virus concentration profile at a time of 30 h for differentvalues of column Peclet number (Pe¼ Vl/D where l isa reference length). The following boundary conditions havebeen used:
Cð0; tÞ ¼ C0 andvCðx; tÞ
vx
����x¼30
¼ 0
Peclet number represents the relative importance ofadvection and dispersion as transport processes. Figures 3aand 3b compare the numerical model with the analyticalsolution given by Sim and Chrysikopoulos (1995) for differentvalues of column Peclet number. It can be seen that thenumerical results match very well with the analytical solutionfor all the values of column Peclet number. To reduce thenumerical error, we kept value of both the grid Peclet numberand Courant number less than one. Grid size dx ¼ 0.1 cm and
time step dt ¼ 0.025 hr have been used in this simulation. Alarger Peclet number leads to reduced virus concentration atearly times as shown in Fig. 3a. However, for large times, thetemporal concentration profile depicts the complex interactionof advection, dispersion, and mass transfer processes andresults in a non-monotonic behavior. Similarly, as shown inFig. 3b, virus concentration decreases at a particular locationwith increase in Peclet number, except near the inlet where theboundary condition influences the profile.
4. Application of model
A two-dimensional rectangular area of porous media asshown in Fig. 4 is simulated in which two observation wellsare located at 50 m and 100 m downgradient from the centerof the source area. Virus concentration profiles have beenpredicted in these wells with various inactivation and mass
50 m 50 m
V
Well-1 Well-2 Source (5m x 5m)
10 m
150 m
15 m
Fig. 4. Source of contaminant plume and location of observation wells.
Table 1
Parameter values used for the simulations.
Parameter Value Parameter Value
b 1.05� 10�11 mL/virus DN 1.152 m2/d
Q0 1.89� 108 sites/mg b 0.192 d�1
r 1.5 g/cm3 Correlation length 5 and 10 m
q 0.25 D0 0.005 m2/d
V 0.96 m/d ln K variance 0.24 and 0.5
Initial plume size 5 m� 5 m
0 500 1000 1500 20000.00
0.04
0.08
0.12
0.16
0.20
1.2
0.12
1.2
0.12
C/C
0
Time (days)
Well-1 Well-2
Fig. 5. Virus concentration profile at the observation wells with different
values of mass transfer rate constants. Numbers near the curve represents the
value of k in d�1 (other parameter values were kept as l*¼ 0.003 d�1,
l**¼ 0.006 d�1).
97P.K. Sharma, R. Srivastava / Journal of Hydro-environment Research 5 (2011) 93e99
transfer rate constants and contour plots of concentration isshown for different rate constants.
Following initial and boundary conditions have been used:
Cðx;y;0Þ¼0Cðx;y;tÞ¼C0 for 10�x�15 and 5�y�10 ð8aÞ
vCðx; y; tÞvx
����x¼0
¼ vðx; y; tÞvx
����x¼150
¼ 0 andvCðx; y; tÞ
vy
����y¼0
¼ vðx; y; tÞvy
����y¼15
¼ 0 ð8bÞ
where C0 is the source concentration.The numerical model was used with the input parameters
taken from Sim and Chrysikopoulos (1995) and listed in Table1, to simulate virus concentration in the observation wells.Fig. 5 shows the virus concentration profile for the wells with
0 30 600.0
7.5
15.0
Y (
m)
0 30 600.0
7.5
15.0
Y (
m)
Fig. 6. Concentration contour at t¼ 300 d, with different values of mass transfer con
values were kept constant as l*¼ 0.003 d�1 and l**¼ 0.006).
different values of mass transfer rate constant with a varianceof 0.24 for the log hydraulic conductivity. As expected, thevirus concentration in the first well, which is closer to thecontamination source, is higher than that in the second well.For both wells, the virus concentration initially increases withtime, but at higher time it becomes constant. For large timesand small value of mass transfer rate constant, the predictedmagnitude of virus concentration in the first well is 14% of thesource concentration as compared to 2% in the second well.Higher values of mass transfer rate constant lead to reducedmagnitude of virus in suspension, as seen in Fig. 5. The meancontour plot of virus concentration is shown in Fig. 6 with twodifferent values of the mass transfer rate constants.
The virus concentration profiles in the wells with differentvalues of inactivation constant of the suspended virus areshown in Fig. 7. The magnitude of relative virus concentrationagain approaches an asymptotic value at large times. Thisasymptotic value for small value of inactivation constant is15% for the first well and approximately 2% for the second
90 120 150
X (m)
90 120 150
X (m)
stant: k¼ 0.12 d�1 for upper and k¼ 1.2 d�1 for lower figure (other parameter
0 500 1000 1500 20000.00
0.04
0.08
0.12
0.16
0.20
0.01
0.001
0.01
0.001
C/C
0
Time (days)
Well-1 Well-2
Fig. 7. Virus concentration profile in the observation wells with different
values of inactivation constants of suspended viruses. Numbers near the curve
represent the value of l** in d�1 (k¼ 0.24 d�1, l*¼ 0.003 d�1).
0 100 200 300 400 5000.0
0.2
0.4
0.6
0.8
1.0
2
Well-1
C/C
0
Time (days)
Fig. 9. Relative virus concentration profile in the observation wells
(k¼ 0.0 d�1, l**¼ l*¼ 0.0 d�1).
98 P.K. Sharma, R. Srivastava / Journal of Hydro-environment Research 5 (2011) 93e99
well. Large value of inactivation rate constant lead to reducedasymptotic concentration of virus in both wells.
The concentration profile for virus in suspension withdifferent values of inactivation constants of adsorbed virus isshown in Fig. 8. During early time the magnitude of relativeconcentration remains same for both values of inactivationconstant of adsorbed viruses. It indicates that there is no effectof inactivation constant of adsorbed virus for small times.However, for large times the virus concentration in suspensionattains a constant value. An increase in the inactivationconstant results in smaller virus concentration in both obser-vation wells. For large time and smaller values of inactivationconstant of adsorbed viruses, the virus concentration obtainedin first well is 3.9% of the source concentration, while insecond well virus concentration is very small.
Fig. 9 shows the virus concentration profile in the observationwells for conservative virus transport, i.e., without any
0 500 1000 1500 20000.00
0.02
0.04
0.010.03
0.01
C/C
0
Time (days)
Well-1 Well-2
Fig. 8. Virus concentration profile of virus in the observation wells with
different values of inactivation constants of adsorbed viruses. Numbers near
the curve represent the value of l* in d�1 (k¼ 0.12 d�1, l**¼ 0.003 d�1).
inactivation constant and mass transfer rate constant. However,due to the heterogeneity of the porous media the relative virusconcentration profile in both observationwells deviates from thesharp front expected for a media with low dispersion. The twoprofiles clearly indicate an increase in dispersivitywith distance,as expected in a heterogeneous porous medium. At large time,the magnitude of virus concentration obtained in first well is69% of the source concentration and, in the second well, it is55%. Fig. 10 represents the virus concentration profiles withdifferent values of the hydraulic conductivity variance inheterogeneous porous media. Larger conductivity varianceleads to an early arrival of virus at the observationwell as well asan increase in the asymptotic magnitude of virus concentration.The asymptotic value of the virus concentration obtained in firstwell is 55%of the source concentration, while in the secondwellit is 37%. Similar inferences about the effect of correlation
0 500 1000 1500 20000.0
0.2
0.4
0.6
0.8
0.5
0.5
0.1
0.1
C/C
0
Time (days)
Well-1 Well-2
Fig. 10. Relative concentration profile of virus in suspension with time at two
well for different values of ln K variance. Numbers near the curve represent
the value of ln K variance (k¼ 0.24 d�1, l**¼ 0.003 d�1, l**¼ 0.006 d�1,
correlation length¼ 5 m).
0 500 1000 1500 20000.0
0.2
0.4
0.6
0.8
10
10
1
1
C/C
0
Time (days)
Well-1 Well-2
Fig. 11. Relative concentration profile of virus in suspension with time at two
well for different values of correlation length (m). Numbers near the curve
represent the value of correlation length in m (k¼ 0.2 d�1, l**¼ 0.003 d�1,
l**¼ 0.006 d�1, ln K variance¼ 0.24).
99P.K. Sharma, R. Srivastava / Journal of Hydro-environment Research 5 (2011) 93e99
length on virus concentration in both observation wells may bedrawn from Fig. 11.
5. Summary
An implicit finite-difference technique has been used to getthe solution of virus transport model for the case of two-dimensional viruses transport through heterogeneous porousmedia at field scale. An exponential dispersivity function hasbeen used to represent the heterogeneity of the porous media.The effect of mass transfer rate and inactivation rate constantson the virus concentration in two observation wells has beenstudied and various figures are shown quantifying this effect.The results show that the higher values of mass transfer rateconstant and inactivation constant lead to reduced virusconcentration in both observation wells. An increase in thevariance of conductivity field or its correlation length wasfound to result in an earlier arrival of the virus at the obser-vation wells as well as a higher virus concentration. Theseresults would be helpful in designing a soil remediationprogram or placement of monitoring wells near septic tanks.
The effect of Peclet number on temporal and spatial virusconcentration in one-dimensional homogeneous porous mediawas also studied. Higher value of Peclet number leads toreduced virus concentration during small transport times butshowed an increase at large times. It is also observed that forhigher value of Peclet number the virus concentration isreduced with increase in travel distance.
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