Stochastic modeling of colloid‐contaminant transport in physically and geochemically heterogeneous...

Stochastic modeling of colloid-contaminant transport in physically

and geochemically heterogeneous porous media

Hesham M. Bekhit1,2 and Ahmed E. Hassan1

Division of Hydrologic Sciences, Desert Research Institute, Las Vegas, Nevada, USA

Received 13 April 2004; revised 24 October 2004; accepted 11 November 2004; published 12 February 2005.

[1] A two-dimensional model is used to describe contaminant transport in the presence ofcolloids for heterogeneous porous media. The model accounts for both spatially varyingconductivity (physical heterogeneity) and the spatially varying distribution coefficient andcolloid attachment coefficient (chemical heterogeneity). The model is tested againstexperimental data, and the results are favorable. The model is implemented in a stochasticMonte Carlo fashion, and both absolute and relative dispersion frameworks are used forthe analysis. One finding in this study indicates that the presence of colloids reducesvariability in mass arrival times to a downstream control plane. The study also indicatesthat the effect of geochemical heterogeneity is important only if it is correlated to physicalheterogeneity. Spatial variability of the contaminant distribution coefficient or colloidattachment coefficient (i.e., geochemical heterogeneity) is not important in determiningthe mean plume shape and concentration values when it is not correlated to the hydraulicconductivity variability. Moreover, the comparison between absolute and relativedispersion shows that including the plume meandering in the overall ensemble (absolutemean) would reduce the peak concentration 20–30%. It is also found that the positivecorrelation structure of geochemical parameters and hydraulic conductivity reduces thedifference between absolute and relative dispersion results.

Citation: Bekhit, H. M., and A. E. Hassan (2005), Stochastic modeling of colloid-contaminant transport in physically and

geochemically heterogeneous porous media, Water Resour. Res., 41, W02010, doi:10.1029/2004WR003262.

1. Introduction

[2] The transport of contaminants in groundwater is pro-gressively acknowledged as a serious environmental prob-lem. Traditionally, contaminant transport in groundwater hasbeen treated as a two-phase system where contaminantscould partition between the mobile aqueous phase andimmobile solid phase. Consequently, contaminants havinga high affinity for the solid phase or a low solubility ingroundwater were considered as immobile contaminants(e.g., plutonium, americium, and europium). However, alarge body of field and laboratory evidence has shown thatsuch immobile contaminants can be mobilized via colloids,especially when the colloids are highly mobile in theaquifer [McCarthy and Zachara, 1989; Magee et al., 1991;Dunnivant et al., 1992; Smith and Degueldre, 1993]. Col-loids, which are defined as very small, suspended particlesranging in size from 1.0 nm to 10.0 mm [Russel et al., 1989]will provide a third phase for the contaminant and cansignificantly facilitate its migration. On the basis of thisevidence, understanding the fate and transport of colloidshas become a crucial issue for the hydrogeologic community.

[3] A significant effort has been devoted to studying howcolloids enhance contaminant migration rates in groundwa-ter systems. Most of these efforts have focused on devel-oping numerical models that were tested or verified withlaboratory data. Some of these models focused on colloidtransport and interaction with porous media while othersstudied the interaction among colloids, contaminants, andporous media. Examples of models that focused on under-standing colloid transport include the work of Mills et al.[1991], Abdel-Salam and Chrysikopoulos [1994], Wan andWilson [1994], Corapcioglu and Choi [1996], Corapciogluet al. [1999], James and Chrysikopoulos [1999, 2000],Lenhart and Saiers [2002], and Sirivithayapakorn andKeller [2003]. Examples of models that were developed tostudy contaminant transport in the presence of colloidsinclude Corapcioglu and Jiang [1993], Abdel-Salam andChrysikopoulos [1995a, 1995b], Ibaraki and Sudicky[1995a, 1995b], Knabner et al. [1996], Roy and Dzombak[1998], Saiers and Hornberger [1999], and Sen et al.[2002]. These studies presented the basic concepts ofcolloids and their transport and provided an understandingof the interactions among colloids, contaminants, andporous media.[4] Yet, most of the available models for contaminant

transport in the presence of colloids consider deterministichomogeneous conditions. Addressing the problem in real-istic heterogonous media has not been thoroughly investi-gated as compared to deterministic homogeneousconditions. We distinguish here between two types ofheterogeneity. The first type is physical heterogeneity at-

1Also at Irrigation and Hydraulics Department, Faculty of Engineering,Cairo University, Cairo, Egypt.

2Also at Hydrology Program, University of Nevada, Reno, Nevada,USA.

Copyright 2005 by the American Geophysical Union.0043-1397/05/2004WR003262

W02010

WATER RESOURCES RESEARCH, VOL. 41, W02010, doi:10.1029/2004WR003262, 2005

1 of 18

tributed to the spatial variability of hydraulic conductivity.The second type is geochemical heterogeneity that accountsfor spatial variability of both colloid deposition and con-taminant distribution coefficients.[5] Several attempts have been made to consider the

effects of physical or geochemical heterogeneity on colloidtransport. Saiers et al. [1994] conducted an experiment tostudy the behavior of colloidal silica in a heterogeneouslypacked column. The experimental results showed that silicatransport could be predicted using the advection-dispersionequation in two-dimensional, structured heterogeneousporous media by simulating the silica deposition with afirst-order kinetic process. Sirivithayapakorn and Keller[2003] conducted an experiment to study the effect ofcolloid exclusion from areas having small aperture sizes.They found that size exclusion phenomena occur when theratio of pore throat to colloid diameter is less than 1.5. Abdel-Salam and Chrysikopoulos [1995b] and Chrysikopoulos andAbdel-Salam [1997] developed a model to describe thetransport of colloids in a saturated fractured domain havinga spatially variable aperture. They assumed the aperture in thefracture plane to be a lognormally distributed random vari-able, which accounts for variability in the velocity field. Theyused the model to investigate the effect of colloid exclusionfrom areas of small aperture size on colloid transport.[6] Chrysikopoulos and James [2003] and James and

Chrysikopoulos [1999, 2000] added variability in colloidsize to the preceding model. These researchers used parti-cle-tracking simulation and assumed a lognormal distribu-tion for the colloid size. In their model, each colloidalparticle has its own molecular diffusion, which is a functionof particle diameter. Therefore the diffusion term in thetransport equation is considered variable and is changedwith the particle diameter (physical heterogeneity). Theyalso used the Smulochowski-Levich sorption relation toexpress the linear sorption coefficient as a function ofmolecular diffusion (particle diameter). Thus their modelaccounts for the variability in the sorption coefficient(geochemical heterogeneity) by considering the effect ofvariable particle diameter. Johnson et al. [1996] incorporatedboth patch-wise geochemical heterogeneity and randomsequential deposition dynamics in a one-dimensional, col-loid transport model. They also tested the model againstresults from column experiments where silica colloidstraveled through a column filled with geochemically het-erogeneous sand grains; the results were favorable. Sun etal. [2001a] developed a two-dimensional, colloid transportmodel for heterogeneous porous media where colloid trans-port equations were coupled with the fluid flow equationand the finite element method was used to numerically solvethese equations. In a subsequent study, Sun et al. [2001b]focused on the inverse problem of colloid transport ingeochemically heterogeneous porous media where theytried to use an inverse method to estimate colloid transportparameters. They found that the colloid deposition andrelease rate coefficients were interrelated.[7] A few studies have addressed the interactions among

colloids, contaminants, and porous media under heterogo-nous conditions. Saiers [2002] conducted six-columnexperiments to explore the effect of heterogeneity in chem-ical composition on colloid-facilitated transport. He useddifferent colloid types: either suspensions of homogenous

clay colloids or mixtures of clay colloids and dissolvedorganic matter. In addition, Saiers [2002] developed aone-dimensional model that accounts for the transport ofcolloids having varying sorption capacities. The resultspresented in his work indicated that assuming homoge-nous colloid suspensions could not produce quantitativedescriptions of colloid-facilitated transport in the field.Marseguerra et al. [2001a, 2001b] developed a stochasticmodel based on the Kolmogorov-Dmitriev theory ofbranching stochastic processes to account for contaminanttransport in the presence of colloids. The model accountedfor the interaction between colloids and contaminant bymeans of a transition rate instead of using the classicaladvection-dispersion equation. They used the model tosimulate the transport of plutonium�239 in the presence ofcolloids at Mol (Belgium) by simulating the field as aone-dimensional, homogenous domain.[8] In the current work, the stochastic analysis and

uncertainty assessment of contaminant transport in thepresence of colloids are the focus. One of the mainobjectives in this work is to seek better understanding ofthe cotransport of contaminant and colloids under bothphysical and geochemical heterogeneity in saturated po-rous media. We approach this problem by developing atwo-dimensional, stochastic numerical model. The physicalheterogeneity is represented by using a random hydraulicconductivity field. We also incorporate geochemical hetero-geneity by assuming a random colloid deposition rate andrandom contaminant distribution coefficient. We focus onstudying the impact of these heterogeneities on meanbehavior (i.e., mean concentration and mean spreading)and uncertainty of the total mobile contaminant plume. Wealso evaluate Monte Carlo results using both absolutedispersion and relative dispersion frameworks. In theabsolute dispersion, the ensemble mean plume and asso-ciated variance (measure of uncertainty considered here)are impacted by velocity variations at all scales (i.e., scaleslarger, smaller, and equivalent to plume size). In therelative dispersion framework, however, only variationsin velocity at scales smaller than or equal to the plume sizeare considered. The term ‘‘ensemble mean plume’’ is usedhere to refer to a mean concentration plume at a certain time,which thus implies a certain location in the simulationdomain.[9] The remaining sections of the paper are organized as

follows. Section 2 presents the mathematical model, the setof partial differential equations governing contaminanttransport in presence of colloids, and numerical solutionsfor these equations. In section 3, we discuss some tests andevaluations that verify the model and compare its results toexperimental data. Section 4 presents the results and dis-cusses the impact of different correlations between physicaland geochemical parameters on contaminant concentrationmean and variance. This impact is studied under both theabsolute and relative dispersion frameworks. Section 5summarizes the study and presents the main conclusionsdrawn from the results.

2. Model Formulation

[10] Different models have been developed to describethe interactions among colloids, contaminants, and porous

2 of 18

W02010 BEKHIT AND HASSAN: STOCHASTIC MODELING OF COLLOID-CONTAMINANT TRANSPORT W02010

media. Here we adapt the model developed by Corapciogluand Jiang [1993] and subsequently used by Corapciogluand Kim [1995], Corapcioglu and Jiang [1996], andKim and Corapcioglu [1996]. However, we assume thegeneral case of nonequilibrium, yet linear, reactions forcolloids with porous media, contaminant with porous me-dia, and contaminant with colloids.

2.1. Mathematical Model

[11] Adapting Corapcioglu and Jiang’s [1993] model andassuming the general case of nonequilibrium reactions, sixcoupled partial differential equations governing colloid-facilitated contaminant transport can be written. Theseequations are expressed as

@ Cmc

@ t¼ @

@ xiDij

@ Cmc

@ xj

� �� Vi

@ Cmc

@ xi� K1C

mc þ K2

qCimc ; ð1Þ

@ Cimc

@ t¼ K1q Cm

c � K2 Cimc ; ð2Þ

@ Smc Cmc

� �@ t

¼ @

@ xiDij

@ Smc Cmc

� �@ xj

� �� Vi

@ Smc Cmc

� �@ xi

þ Kma Cd

� Kmr S

mc C

mc � K1S

mc C

mc þ K2

qCimc Simc ; ð3Þ

@ Simc Cimc

� �@ t

¼ Kima qCd � Kim

r Simc Cimc þ K1 q Smc Cm

c � K2 Simc Cim

c ;

ð4Þ

@ Cd

@ tþ 1

q@ Ss@ t

¼ @

@ xiDij

@ Cd

@ xj

� �� Vi

@ Cd

@ xi� Km

a þ Kima

� �Cd

þ Kmr S

mc C

mc þ Kim

r

qSimc Cim

c ; ð5Þ

@ Ss@ t

¼ KrKd q Cd � Kr Ss: ð6Þ

In equations (1)–(6), Ccm represents the concentration of

mobile colloids (mass of mobile colloids per unit aqueousvolume [ML�3]); Cc

im is the mass of captured colloids(immobile colloids) per unit total volume of porous media[ML�3]; Sc

m is the mass of contaminant adhered to mobilecolloids per unit mass of mobile colloids [MM�1]; Sc

im is themass of contaminant adhered to immobile colloids per unitmass of captured colloids [MM�1]; Cd is the mass ofdissolved contaminant per unit aqueous volume [ML�3]; Ssis the mass of contaminant sorbed on solid matrix per unittotal volume of porous media [ML�3], q is the porosity, Dij

is the ij component of the dispersion tensor (assumed to bethe same for colloid and contaminant dispersion); Vi is thevelocity component in the ith direction [LT�1]; K1 is thedeposition rate coefficient of mobile colloids [T�1]; K2 isthe release rate coefficient of captured colloids [T�1]; Ka

m

and Kaim are the attachment rates of contaminant to mobile

and immobile colloids [T�1]; respectively; Krm and Kr

im arethe release (or detachment) rates of contaminant frommobile and immobile colloids [T�1]; respectively; Kr is the

reaction rate coefficient [T�1] governing the sorption ofcontaminant directly onto solid matrix; and Kd is thedistribution coefficient defining the apportioning of con-taminant between the aqueous and solid phases. Inequations (1)–(6), repeated indices imply summation.

2.2. Numerical Model

[12] We consider a two-dimensional, heterogeneous po-rous medium. The physical heterogeneity of the domain isrepresented by a statistically homogeneous, second-orderstationary, random log conductivity distribution that can beexpressed as Y = log K = logK + f, where logK is theconstant mean of the log conductivity field and f is the zero-mean fluctuating log conductivity. The log conductivity ismodeled as a correlated random field with a Gaussiancovariance structure of the form CY(u) = sf

2e�u2/l2

, whereCY is the correlation function; sf

2 is the fluctuating log-Kvariance; u is the spatial lag; and l is the integral scale ofthe fluctuating log conductivity.[13] The simulation domain is assumed rectangular with

dimensions 51.2l � 25.6l. A mean constant head gradient,J, is imposed on the system along x1 direction. The flowequation is solved by employing a block-centered, finitedifference scheme. The domain is discretized into 256 �128 blocks with 5 blocks per integral scale. For the flowproblem, no-flow boundaries are employed along the baseand the top of the model, and constant-head boundaries areassumed along the left and right edges of the model domain.These conditions result in a unidirectional flow in thepositive x1 direction, and with the heterogeneous conditions,both positive (upward) and negative (downward) verticalvelocity components exist in x2 direction.[14] For the transport problem, an instantaneous pulse of

contaminant is released into the simulation domain at timet = 0. A colloid plume is assumed to coincide with thedissolved contaminant plume as an initial condition. Thiscase represents leakage from a storage tank or a landfill(e.g., resulting from a heavy rainstorm) of both dissolvedcontaminants and fine colloidal particles that eventuallyreach the saturated zone as two coincidental plumes. Theinitial and boundary conditions of the transport problem aresummarized as

C x1; x2; 0ð Þ ¼ Co; ð7Þ

C 0; x2; tð Þ ¼ 0 or background concentrationð Þ; ð8Þ

@C

@x1Lx1 ; x2; tð Þ ¼ 0; ð9Þ

@C

@x2x1; 0; tð Þ ¼ @C

@x2x1;Wx2 ; tð Þ ¼ 0; ð10Þ

where C represents any of the three mobile concentrations(Cc

m, Cd, and Scm); C0 is the initial concentration distribution;

Lx1 is the domain length in x1 direction; and Wx2is the

domain width in x2 direction. The three immobileconcentrations are initially set equal to zero.[15] In addition to the physical heterogeneity, we assume

geochemical heterogeneity by using spatially random and

W02010 BEKHIT AND HASSAN: STOCHASTIC MODELING OF COLLOID-CONTAMINANT TRANSPORT

3 of 18

W02010

correlated colloid-deposition coefficient, K1, and contami-nant distribution coefficient, Kd. These two random spacefunctions can be decomposed into their means and fluctua-tions about the means:

logK1 ¼ logK1 þ k1 ð11Þ

logKd ¼ logKd þ kd ; ð12Þ

which, under the assumption of second-order stationarity,implies that logK1 and logKd are the stationary, space-invariant means of the log K1 and log Kd fields, respectively,and that k1 and kd are the zero-mean fluctuations aroundthese means.[16] The assumption of random colloid deposition rate

could simulate one or more of the following conditions:spatially variable solid matrix or surface properties[e.g., Sun et al., 2001a; Song et al., 1994], multiple particlesizes [e.g., Chrysikopoulos and James, 2003; James andChrysikopoulos, 1999, 2000]. In addition, one could attri-bute the heterogonous contaminant distribution coefficientto spatially variable solid matrix material, solid surfaceroughness, and different surface charges.[17] Several studies reported that physical and geochem-

ical properties affecting contaminant transport are spatiallyvariable and that the model parameters describingthese processes might be correlated [e.g., Allen-King et al.,1998; Ball and Roberts, 1991; Barber, 1994; Barber et al.,1992; Fiori and Bellin, 1999; Hu et al., 1995; Huang andHu, 2001; Robin et al., 1991; Sun et al., 2001a, 2001b]. Forexample, Barber et al. [1992] reported a negative correlationbetween the sorption coefficient of chlorobenzene andhydraulic conductivity, K. Conversely, Ball and Roberts[1991] obtained the opposite trend for sorption of tetrachloro-ethene (PCE) in a sandy aquifer. However, both positiveand negative correlations in addition to the no-correlationcondition could be expected in the subsurface. Thus weassume the hydraulic conductivity field,K, colloid depositionrate, K1, and contaminant distribution coefficient, Kd, to becorrelated. Hence three possible correlations betweeneither K1 or Kd and the hydraulic conductivity field, K, areconsidered: (1) perfect positive correlation, (2) perfect neg-ative correlation, and (3) no correlation. In the case of perfectcorrelation, the geochemical parameters are generated as

Kj ¼ KGj e�f ; ð13Þ

where KjG represents the geometric mean of the colloid

deposition coefficient, K1G, or the contaminant distribution

coefficient, KdG, and f is the zero-mean perturbation of the

log conductivity. The minus sign in equation (13) gives riseto the perfect negative correlation case, and the plus signgives the perfect positive correlation case. For the no-correlation condition, the same isotropic Gaussian correla-tion structure used for the hydraulic conductivity is adaptedfor K1 and Kd. In this case, K1 or Kd is generated as

Kj ¼ KGj ew; ð14Þ

where w is a normally distributed random space functionwith a zero mean and variance sw

2 assumed to be equal to

sf2. Therefore the value and sign of the covariance of K1 (or

Kd) and K, f k1 (or f kd) reflect the nature of the assumedcorrelation. That is, f k1 < 0 indicates the negativecorrelation case, f k1 > 0 indicates the positive correlationcase, and f k1 = 0 indicates the no-correlation case.[18] Individual realizations of the two-dimensional distri-

bution of K are generated using a Fast-Fourier-Transform-based algorithm as described in detail by Hassan et al.[1997]. For the correlation cases, K1 or Kd is generated fromthe conductivity realizations using equation (13). Figure 1presents an example of a single realization of K1 showing itsspatial distribution compared to the spatial distribution of Kin a negative correlation case.[19] After solving the flow problem and generating the

transport parameters, the colloid-contaminant cotransportproblem is solved by discretizing the transport equations,equations (1)–(6), using finite differences with a third-ordertotal variation-diminishing (TVD) algorithm for the advec-tion terms. The TVD algorithm relies on the UniversalLimiter for Transient Interpolation Modeling of AdvectiveTransport Equations (ULTIMATE) algorithm [cf. Leonardand Niknafs, 1991; Zheng and Wang, 1999].[20] Once transport equations (1)–(6) are solved, the total

contaminant concentration is obtained as the summation ofCd and Sc

mCcm. A control plane is selected at a distance of

about 8 l from the initial source location and used to obtainthe cumulative mass arrival and mass flux breakthroughcurves for the total mobile contaminant. The values of themodel parameters used to obtain the results discussed insection 4 are listed in Table 1.

3. Model Verification

[21] Under simplified assumptions of homogeneous con-ditions, the model was verified using mass balance tests,comparisons with simple analytical solutions, and compar-isons with experimental data. This partial verification ispresented in detail by H. M. Bekhit and A. E. Hassan (Two-dimensional modeling of contaminant transport in porousmedia in the presence of colloids, submitted to Advances inWater Resources, 2004, hereinafter referred to as Bekhit andHassan, submitted manuscript, 2004). As discussed in thatstudy, these tests showed very good accuracy for thedeveloped numerical model under homogeneous conditions.The tests, however, are not sufficient to test the modelperformance under heterogeneous conditions.[22] Owing to the absence of analytical solutions to the

coupled equations of colloid-contaminant transport, espe-cially under heterogeneous conditions, we verify the modelby simulating the transport of colloidal silica throughstructured heterogeneous porous media. The model resultsare compared with the experimental data of Saiers et al.[1994] under heterogeneous conditions.[23] In their experiment, Saiers et al. [1994] examined

the transport of colloidal silica through laboratory columnscontaining a single structured heterogeneity. They used aglass chromatography column having an internal diameterof 4.8 cm, and to generate the structured heterogeneity, athin-walled glass tube (internal diameter of 1.6 cm) wasplaced into the center of the column. Fine sand was thenadded to the column around the thin glass tube, which wasfilled with coarse sand and slowly withdrawn from thecolumn. Colloidal suspensions of amorphous silica were

4 of 18


injected as pulse injections, and the injection was terminatedwhen the colloid concentration in the effluent reached 95%of the input concentration. The experiment was conductedusing an influent colloid concentration of 1000 mg/L.[24] To simulate the experiment we use the same porosity,

flow velocity, dispersivity, and pulse duration values as usedor predicted experimentally [i.e., Saiers et al., 1994, Tables 1and 3]. The deposition rate coefficient of mobile colloids,K1, and release rate coefficient of captured colloids, K2, inboth layers are used as fitting parameters. For K1, the bestfitting values for coarse and fine sand are found to be 1.6and 0.1 h�1, respectively. The best fitting values of K2 forcoarse and fine sand are 1.8 and 0.818 h�1, respectively.Figure 2 shows a comparison between the effluent colloidconcentration, Cc

m, normalized with the initial concentration,Cc0

m, observed experimentally and the model results. As canbe seen from Figure 2, a very good match is obtained wherethe modeled breakthrough time and overall trend of thecurve agree well with the experimental data.

[25] Owing to the lack of experimental data describingcolloid-contaminant transport in heterogonous porous me-dia, we could not fully verify the model. However, given thegood match presented in Figure 2 and the verificationsdescribed by Bekhit and Hassan (submitted manuscript,2004) the model seems to be functioning properly and canbe used to investigate the different interactions amongcolloids, contaminant, and porous media under heteroge-neous conditions.

4. Results and Discussion

[26] To organize the analysis and explanation of the results,we divide the following discussion into three main topics.First, we investigate the effect of physical heterogeneitywhere we study contaminant migration in heterogeneousporous media under two levels of heterogeneity, sf

2 = 0.4and sf

2 = 1.2, with and without colloids. We compare theseresults to the transport results for homogenous conditions.

Table 1. Model Parameters for the Base Case

Description Parameter Value

Initial colloid concentration, kg/m3 Cc0

m 100Initial contaminant concentration, kg/m3 Cd0

100Porosity q 0.3Geometric mean of hydraulic conductivity, m/day KG 1Longitudinal and transversal dispersivities, m aL, aT 0.1, 0.05Contaminant and colloid reaction rates with porous media, day�1 Kr = K2 0.5Geometric mean of contaminant distribution coefficient Kd

G 2.0Geometric mean of colloid deposition rate, day�1 K1

G 1.0Contaminant attachment rate to mobile and immobile colloid, day�1 Ka

m = Kaim 10.0

Release rate coefficient of contaminant from mobile colloids, day�1 Krm 1.0

Release rate coefficient of contaminant from immobile colloids, day�1 Krim 5.0

Figure 1. Spatial distribution of K and K1 in a single realization with perfect negative correlationbetween the two parameters.


5 of 18

W02010

[27] Second, we study the impact of physical and geo-chemical heterogeneity on colloid-facilitated transport usingdifferent correlations between physical and chemical param-eters. For this analysis, we first establish two referencecases. The first reference case is the contaminant transportin heterogeneous, spatially correlated K fields with a deter-ministic uniform distribution coefficient, Kd, in the absenceof colloids (i.e., ff 6¼ 0 but kdkd = 0). This case is referred toas case 1 in the subsequent discussion. The second referencecase, case 2, represents contaminant-colloid transport inheterogeneous K fields with deterministic uniform K1 andKd (i.e., ff 6¼ 0 but kdkd ¼ k1k1 = 0). We then present eightcases that are compared to each other and to the tworeference cases. The details of these eight cases with thetwo reference cases are listed in Table 2.[28] Third, we evaluate the results of Monte Carlo sim-

ulations in the absolute and relative dispersion frameworks.Comparisons are made between the two frameworks interms of mean, variance and spatial moments of the totalcontaminant plume. Before presenting the three sets of

results, we establish in the following subsection how MonteCarlo results converge on a local and global basis.

4.1. Monte Carlo Convergence

[29] The mean squared error averaged over the entiredomain is used as a criterion to ensure the global conver-gence of Monte Carlo results. The convergence criterion isdefined as [e.g., Bellin et al., 1994; Hassan et al., 1998]

e E;MCð Þ ¼ 1

N �M

XNi¼1

XMj¼1

E MCð Þ i; jð Þ � E MC�1ð Þ i; jð Þ� �2

; ð15Þ

where MC is the Monte Carlo realization number; N is thetotal number of rows in the domain; M is total number ofcolumns; and E(MC)(i, j) is the statistic or ensemble quantityE computed at cell (i, j) based on MC realizations.Convergence is attained when e vanishes [Bellin et al.,1994; Hassan et al., 1998].[30] In addition, we check local convergence using the

squared error estimate evaluated at single points. Five

Table 2. Different Correlation Cases Studied

Case K1 Kd

Reference case 1 contaminant transport without colloids and with a deterministic Kd

Reference case 2 contaminant transport with colloids and with a deterministic K1 and Kd

Case 3 negatively correlated with K ( f k1 < 0) Kd = KdG

Case 4 positively correlated with K ( f k1 > 0) Kd = KdG

Case 5 K1 = K1G negatively correlated with K ( f kd < 0)

Case 6 K1 = K1G positively correlated with K ( f kd > 0)

Case 7 negatively correlated with K ( f k1 < 0) negatively correlated with K ( f kd < 0)Case 8 positively correlated with K ( f k1 > 0) positively correlated with K ( f kd > 0)Case 9 random with no correlation with K ( f k1 = 0) Kd = Kd

G

Case 10 K1 = K1G random with no correlation with K ( f kd = 0)

Figure 2. Comparison between the breakthrough curves as predicted from the model and theexperimental data [Saiers et al., 1994].

6 of 18


points are selected to evaluate the local convergence.These points are the center of mass of the contaminantplume (CM), two points at distance 10l from the CM inthe upstream and downstream directions, and two points atdistance 5l above and below the CM in x2 direction. Thepreceding convergence criterion is adapted and used tocheck the local convergence where we calculate e valuefor each point as

e SE lð Þ;MC�

¼ SElð ÞMCð Þ � SE

lð ÞMC�1ð Þ

h i2; l ¼ 1; 2 . . . 5; ð16Þ

where SE is the single-point ensemble mean or variance ofthe concentration.[31] Figures 3a and 3b show the global convergence of

the Monte Carlo ensemble moments, where e(E(l), MC), l =1, 2, 3, 4 are plotted against the number of realizations (upto 6000 realizations). In these plots, E(1) and E(2) are theensemble mean of the total mobile contaminant concentra-tion (Cd + Sc

mCcm) 30 and 80 days after releasing the plume,

respectively, whereas E(3) and E(4) are the correspondingensemble variances. Figure 3c displays e(SE(l), MC), l = 1,2, . . ., 5 as a function of the number of Monte Carlorealizations, where SE is the single-point mean contaminantconcentration. In addition, Figure 3d shows the results forthe concentration variance at these five points. It is evidentfrom the results in Figure 3 that all e values are very smallafter 300 realizations, and they decay quickly as the numberof realizations increases. However, to ensure proper con-vergence, we use 3000 Monte Carlo realizations for all theresults presented here unless stated otherwise.

4.2. Effect of Physical Heterogeneity

[32] The effect of colloids on facilitating contaminanttransport has mostly been studied in homogenous systems.

The impact is usually shown by earlier mass arrival and anaccelerated breakthrough curve. In heterogeneous systems,stochastic analysis of contaminant transport in the absenceof colloids has provided a mechanism for quantifying theuncertainty associated with mass arrival times and break-through curves. Therefore it is of interest to examinewhether the uncertainty stemming from conductivity spatialvariability overshadows the colloid effect on transportpredicted for homogeneous conditions. Two scenariosare investigated for this purpose. In the first scenario, weselect values for reaction rates and distribution coefficients(Table 1), so that colloids can significantly facilitate con-taminant transport. We then solve the transport equationsassuming a homogenous porous media having a uniformflow velocity of 0.5 m/day in x1 direction. In the secondscenario, we solve the reactive transport problem in theabsence of colloids assuming a heterogeneous porousmedium. The parameters of the flow equation are chosen,so that the macroscale mean velocity in x1 direction is equalto 0.5 m/day.[33] For the Monte Carlo realizations, we compute the

prediction quantiles (PQs) by ranking the 3000 realizationsbased on the 50% mass arrival times. For a 95% confidenceinterval, we use the realization ranked 75 as an upper bound(97.5th quantile), realization ranked 2925 as a lower bound(2.5th quantile), and the average of realizations 1500 and1501 as the median (50th quantile). The comparison be-tween the two scenarios introduced in the preceding para-graph is shown in Figure 4. It is important to note that allgeochemical parameters are kept uniform and deterministicin these two scenarios.[34] Figure 4a presents a comparison between the mass

flux breakthrough curves (BTCs) of the total mobile con-taminant for the first scenario and the upper PQs for the

Figure 3. Global and local convergence of computations for the mean concentration and the variance.


7 of 18

W02010

second scenario with sf2 = 0.4 and sf

2 = 1.2. The resultsshow that the effect of colloids is very significant andcannot be accounted for using uncertainty bounds. Eventhough the upper bounds of the 95% PQs for both sf

2 = 0.4and sf

2 = 1.2 reasonably approximate the peak arrival timeof the homogeneous, colloid-facilitated transport case, theseupper bounds cannot produce the peak mass flux value. It isalso clear from the cumulative mass arrival curves inFigure 4b that the upper 95% PQs in the heterogonous casegive earlier first arrival than the colloid-facilitated transportcase, but the effect of colloids becomes more significant atlater times. For example, the total plume passes the controlplane after 51 days in the homogeneous case with colloids,

whereas it took more than 80 days for the upper PQrealization without colloids.[35] A second aspect of this investigation addresses how

uncertainty in mass arrival times and BTCs in colloid-facilitated transport problems compares with reactive trans-port in the absence of colloids. Figure 5 shows a comparisonbetween variability in contaminant cumulative mass arrivalin the absence (Figure 5a) and in the presence (Figure 5b) ofcolloids. The 95% PQs shown in this figure are calculatedusing the same approach described in the previous para-graph. Again, the purpose of this comparison is to investi-gate the effect of colloids on the uncertainty bounds ofBTCs.

Figure 4. Comparison between reactive contaminant transport in a homogeneous domain with colloidsand in a heterogonous domain without colloids: (a) mass flux breakthrough curves and (b) cumulativemass arrival curves (PQ is the 95% prediction quantile).

8 of 18


[36] One observes from Figure 5 that the presence ofcolloids reduces the uncertainty range in the contaminantBTCs represented by the two PQs. Results indicate thatusing the 95% confidence interval, the range of the 50%mass arrival time is about 73 days for sf

2 = 1.2 and 33 daysfor sf

2 = 0.4, whereas in the presence of colloids theseranges are about 45 and 19 days, respectively. To betterquantify these results, we choose the coefficient of variation(CV) as a quantitative measure to assess the variability inthe plume arrival time with and without colloids. Wecompute the mean and standard deviation of the 50% massarrival time across all realizations and obtain the CV. TheCV value is found to be about 30% for the colloid-facilitated transport case and about 39% for the case without

colloids. This result is attributed to the fact that colloidssignificantly enhance the transport of slow plumes (plumesexperiencing slow-velocity realizations), whereas theyslightly enhance the transport of fast plumes (plumesexperiencing fast-velocity realizations). Thus the presenceof colloids reduces the variability of mass arrival timesaround the ensemble mean arrival time.[37] In addition, we use statistical hypothesis testing to

verify whether the variance of the 50% mass arrival time inthe presence of colloids s250%

��w= colloids

equals the variance inthe absence of colloids s250%

��w=o colloids

. In order to usestandard hypothesis testing, we first verify that the 50%mass arrival time is normally distributed in the two cases(i.e., with and without colloids). The distribution is found to

Figure 5. Comparison between the contaminant cumulative mass flux (a) in the absence of colloids and(b) in the presence of colloids.


9 of 18

W02010

be normal in both cases using the chi square and Kolmo-gorov-Smirnov goodness-of-fit tests. Then, we randomlydraw two independent samples from the two populations.Each sample consisted of 30 realizations (note that we didnot consider the entire 3000 realizations because for such alarge number any difference in the variance will be statis-tically significant). Consequently, the null hypothesis,H0: s250%

��w= colloids

= s250%��w=o colloids

, and the alternative

hypothesis, H1: s250%��w=o colloids

> s250%��w= colloids

, are tested

using a standard F test with a critical rejection regionF ¼ ðS250%

��w=o colloids

=S250%��w= colloids

Þ Fa(n � 1, m � 1).

Here the significance level a is taken as 5%, n = m = 30,(n � 1) and (m � 1) are the degrees of freedom for the twosamples, and S2 is the sample variance. The hypothesistesting is repeated using four sets of two samples and thenull hypothesis H0 is rejected in all tests, indicating that thepresence of colloids decreases the uncertainty in mass arrivaltimes.[38] The results in Figure 5 also indicate that colloids

have an increasing effect on contaminant plumes withlonger residence times (slow-velocity realizations). Slow-velocity realizations imply that residence times for contam-inants are longer allowing more of the slow kinetic sorptionprocess to occur. When colloids are present, they carry thecontaminant leading to a decreased sorption effect. However,when flow velocity is fast, the contaminant residence time isshort and does not allow significant sorption to occur (i.e.,retardation locally is not large). The presence of colloidsthus slightly facilitates the contaminant movement in thefast-velocity realizations (i.e., by decreasing the retardationfactor that already is not very large).

4.3. Effect of Geochemical Heterogeneity

[39] The ten cases previously described and summarizedin Table 2 are analyzed using a log conductivity variance,sf2, of 1.2. Figure 6 shows the ensemble mean concentration

for the total mobile contaminant after 80 days underdifferent physical and geochemical heterogeneities. Thedifferent correlation assumptions affect the shape and dis-persion pattern of the mean concentration plume. Forexample, in addition to increasing the longitudinal disper-sion, the negative correlation assumption (cases 3, 5, and 7)strongly reduces solute desorption from the immobile phaseand results in a heavy tail. Therefore the plume exhibitsskewness toward the upstream. On the other hand, thepositive correlation cases (cases 4, 6, and 8) result in almostsymmetric plumes around the center of mass.[40] The skewness of the plume toward the upstream

direction is observed in all cases having negatively corre-lated physical and chemical parameters. On the other hand,all positive correlation cases lead to symmetric plumesbecause positive correlation implies smaller sorption rateswith the low conductivity, and higher sorption rates withhigh conductivity. Positive correlation increases mobility inthe slow portion of the plume (thus decreasing the tail) andreduces mobility in the front of the plume (fast portion ofthe plume) leading to generally symmetrical plumes.[41] The cases with random K1 not correlated to K (case 9)

and random Kd uncorrelated to K (case 10) yield meanplumes that lie between the positive and negative correla-tion cases. The mean plume has more skewness than plumesin positive correlation cases but less than in negative

correlation cases. Interestingly, the mean plumes in cases 9and 10 are very similar to the reference case 2 withdeterministic uniform K1 and Kd. This observation indicatesthat the spatial variability of K1 or Kd is not important indetermining the mean plume shape and concentration valuesas long as it is not correlated to the hydraulic conductivityvariability. In other words, to be an important factorinfluencing colloid-facilitated contaminant transport, geo-chemical heterogeneity has to be correlated to physicalheterogeneity. Similar results were obtained experimentallyby Chen et al. [2001]. Chen et al. [2001] showed colloiddeposition rate and transport behavior to be independent ofthe spatial distribution of chemical heterogeneity in porousmedia. They added that the mean value of chemical hetero-geneity rather than its distribution governs the colloidtransport behavior in packed columns.[42] In order to examine the variability associated with

different correlation cases, we compute the ensemble vari-ance of the total mobile contaminant concentration. Thespatial distribution of the variance is plotted as a snapshotafter 80 days from the release of the plume (Figure 7). Thefigure shows that, under all examined correlations, theplume exhibits more variability (i.e., higher variance) atthe tail than at the front. This could occur because theconcentration range at the plume tail is higher than at theplume front. This difference is induced by plumes havingvery slow or very fast velocities (compared to the averageadvection velocity of the domain). Figure 8 displays aschematic diagram for the computation of an ensemblemean plume resulting from two independent realizations.The slow plume P1 (i.e., plume experiencing a slow-velocity realization) is not strongly influenced by mixingand dispersion, and therefore, concentrations within P1remain relatively high. Accordingly, P1 yields high concen-tration values at the ensemble plume tail and negligiblevalues at its front (see Figure 8). On the other hand, the fastplume P2 (i.e., plume experiencing a fast-velocity realiza-tion) experiences significant dispersion and mixing, andtherefore, concentrations within P2 are much lower than P1.As a result, P2 yields small concentration values at theensemble plume front and negligible values at the tail.Consequently, a much wider concentration range results atthe tail of the ensemble plume as compared to the front,leading to a much higher variance at the tail.

4.4. Relative Versus Absolute Dispersion

[43] As previously mentioned, absolute dispersionaccounts for velocity variations at all scales, whereasrelative dispersion only considers variability in subplume-scale velocity. As a result, absolute dispersion accounts forplume meandering, but it artificially smears the ensemblemean plume. Consequently, absolute dispersion predictsmore diluted concentration than concentration predictedby a single realization. On the other hand, relative disper-sion accounts for the heterogeneity on the plume scale andremoves the meandering effect. Therefore relative disper-sion provides better estimates of the second moment (mea-sure of dispersion) and concentration that are directlycomparable to the real single-realization field conditions.[44] Relative dispersion is obtained by shifting the

plumes before averaging to remove the meandering effect.For each realization, the center of mass (CM) of the totalmobile plume is computed and the ensemble mean CM is

10 of 18


obtained by averaging over all realizations. The individualCM locations are then set to coincide with the ensemblemean CM. A new ensemble is formed where all realizationsare set to have the same CM. They only differ in thespreading pattern caused by subplume velocity fluctuations.

Concentration statistics for the relative dispersion case canthus be obtained based on the new ensemble.[45] Figure 9 shows the relative mean concentration for

the total mobile contaminant plume after 80 days for thedifferent correlation cases. Figure 9 reveals several interest-

Figure 6. Ensemble absolute mean of total mobile contaminant concentration (Cd + ScmCc

m) at t =80 days for different correlation cases between geochemical and physical parameters. Concentrations arenormalized with respect to the initial contaminant concentration, and the contour interval is 0.004.


11 of 18

W02010

ing issues. Considering a positive correlation between Kd

and K (or K1 and K) leads to a higher peak concentrationthan the reference case (case 2). For example, the total peakcontaminant concentration in case 8, having positive corre-

lation between K and Kd and between K and K1, isapproximately 17% higher than the peak concentration inthe reference case of only random K (case 2). The peak isalso 18% higher than the peak for the random Kd case

Figure 7. Ensemble absolute variance of total mobile contaminant concentration (Cd + ScmCc


12 of 18


without correlation to K (case 10). In negative correlationcases, the opposite effect occurs; for example in case 7(having negative correlation between Kd, K1, and K), thepeak mean concentration is 34% smaller than in case 2where only K is heterogeneous. The randomness of K1 or Kd

under the assumption that they are not correlated withhydraulic conductivity does not significantly change theconcentration values. This is similar to the absolute disper-sion results shown in Figure 6.[46] Comparing snapshots of the relative dispersion

plume displayed in Figure 9 and absolute dispersion plume(Figure 6), we conclude that including plume meandering inthe overall ensemble (absolute mean) would reduce the peakand lead to artificial smoothing attributed to velocity varia-tions at scales larger that the plume size. Generally, theabsolute mean concentrations are about 20–30% lower thanthe relative mean concentrations.[47] Owing to the large-scale heterogeneity (larger than

the plume scale) which is typical of geologic environ-ments, we cannot precisely estimate where the plume as awhole will migrate. Consequently, a simple first-orderquantity such as the location of the plume’s center ofmass is associated with a large degree of uncertainty. Itshould be realized that absolute mean and relative meanare both important, and depending on the application, onemay be more suitable than the other. For example, plumemeandering and fluctuations of the centroid are importantinformation in designing groundwater monitoring net-works. On the other hand, the more realistic estimate ofconcentration mean and variance obtained by using therelative dispersion framework may be more suitable forrisk assessment purposes.[48] Figure 10 presents the spatial distribution of the

relative concentration variance as a snapshot at t = 80 days.The results show that the concentration variance in therelative dispersion case is dramatically smaller than thevariance in the absolute dispersion analysis. Again, thisoccurs because relative dispersion only accounts for vari-ability in the subplume-scale velocity, whereas absolutedispersion accounts for variability at both smaller and largerscales than plume size. Unlike absolute variance, relative

variance shows only one peak (close to the CM), becausethe effect of slow or fast plumes is removed in the relativedispersion. Therefore the second variance peak, (i.e., cases3, 5, 7, 9, and 10 (Figure 7)) which occurred at the plumetail in the absolute ensemble, is eliminated in the relativedispersion framework. To quantitatively compare the abso-lute and relative dispersion frameworks, we calculate thespatial moments of the mean plume and discuss them next.

4.5. Spatial Moments for Absolute andRelative Dispersion

[49] Spatial moments are calculated to investigate theeffect of geochemical heterogeneity on plume dispersionand travel distance under both absolute and relative disper-sion frameworks. These spatial moments are computed as

Xi ¼

ZR2

qC x1; x2ð Þxi dx1 dx2

ZR2

qC x1; x2ð Þ dx1 dx2i ¼ 1; 2 ð17Þ

Xii ¼

ZR2

qC x1; x2ð Þ xi � Xið Þ2 dx1 dx2

RR2

qC x1; x2ð Þ dx1 dx2i ¼ 1; 2; ð18Þ

where Xi is the first spatial moment in the xi direction and Xii

is the second spatial moment in the xi direction.[50] Figure 11 displays the spatial moments for the

relative mean plume under different correlation combina-tions. It is clear that the plume travels faster under negativecorrelation conditions between K1, Kd, or both and K(cases 3, 5, and 7). For example, when f k1 < 0 and f kd <0 (case 7), X1 is about 12.8 l at t = 40 days, whereas it is10.6 l for case 8 with f k1 > 0 and f kd > 0. The randomnessof K1 and/or Kd under no correlation with K slightlyincreases the travel distance compared to the case consid-ering only physical heterogeneity. This is consistent with theresults of Huang and Hu [2001] in their study of reactivecontaminant transport.[51] Figures 11c–11d show the effect of geochemical

heterogeneity and the correlation to physical heterogeneityon the relative second longitudinal moment. The momentincreases in the presence of colloids, which means thatcolloids increase dispersion in the longitudinal direction.That is attributed to the fact that the plume moves faster inthe presence of colloids, and therefore, the plume experi-ences greater heterogeneity leading to more dispersion. Onecan observe also that ignoring geochemical heterogeneity(case 2) underestimates plume dispersion except in the caseswhere positive correlation between Kd (or K1) and hydraulicconductivity is assumed. However, the geochemical hetero-geneity effect is more apparent at later times (i.e., (vt/l) > 5)where the plume has traveled enough correlation scales toexperience the geochemical heterogeneity. At earlier times,the plume does not experience enough heterogeneity, andthus the effect is not significant at these times.[52] It can also be seen from Figure 11 that negative

correlation between Kd (or K1) and K increases longitudinaldispersion of the plume compared to physical heterogeneity

Figure 8. Schematic of ensemble averaging leading tohigh variance at the ensemble plume tail.


13 of 18

W02010

Figure 9. Ensemble relative mean of total mobile contaminant concentration (Cd + ScmCc

m) at t = 80 daysfor different correlation cases between geochemical and physical parameters. Concentrations arenormalized with respect to the initial contaminant concentration, and the contour interval is 0.004.

14 of 18


only (case 2), whereas positive correlation decreases theplume dispersion in the longitudinal direction. For the caseswhere the negative correlation exists, low hydraulic con-ductivity is associated with high K1 (or Kd) values and vice

versa. This implies that if a portion of the plume (whetherthe dissolved contaminant, Cd, or contaminant carried oncolloids, Sc

mCcm) encounters a low-conductivity zone (and

thus low velocities), it becomes somewhat trapped in this

Figure 10. Ensemble relative variance of total mobile contaminant concentration (Cd + ScmCc



15 of 18

W02010

zone due to the high sorption coefficient that corresponds tothe low K value. At the same time, the portion of the plumethat travels through a high-conductivity zone is mobilizedmore because of a low sorption coefficient value. Thus thenegative correlation increases the mobility of the fastest partof the plume and decreases the mobility of the slowest part(leading to increased dispersion). The opposite effect isobtained in the cases of positive correlation where the low Kis associated with a low sorption coefficient, and the high Kis associated with a high sorption coefficient. This leads toreducing mobility in the fastest part of the plume and

increasing mobility in the slowest part, and hence plumedispersion decreases.[53] Figures 11e–11f show a similar influence of the

geochemical heterogeneity on the transverse secondmoment. However, the effect in the transverse direction isnot as strong as in the longitudinal direction (note thedifference in y axis scale between longitudinal and transver-sal directions). This is because the mean velocity is orientedalong the longitudinal direction, which increases the impactof geochemical heterogeneity in this direction. However,both positive and negative vertical velocity components exist

Figure 11. Effect of the geochemical heterogeneity and its correlation to physical heterogeneity on(a–b) the first longitudinal moment, (c–d) the second longitudinal moment, and (e–f) the secondtransverse moment.

16 of 18


in the transverse direction leading to a less significant impactof geochemical heterogeneity in the transverse direction.[54] Figures 11c–11f also display the absolute dispersion

results for the second longitudinal and transverse momentsat two times (t = 30 and 80 days). The triangular symbols inthe figure indicate that significant dispersion results fromthe meandering effect inherent in the absolute dispersionframework.[55] Comparisons between the absolute and relative dis-

persion in terms of the second moment are summarized inTable 3. The ratio of the absolute second moment to therelative second moment decreases with time because theplume experiences more heterogeneity. As time progresses,the nonergodic conditions begin to diminish as the plumegrows to a scale comparable to the macroscale of velocityvariation. Therefore the plume trajectory converges gradu-ally to the large-scale mean velocity trajectory leading toless plume meandering and thus smaller differences be-tween absolute and relative dispersion.[56] Another interesting outcome is that the presence of

colloids decreases the difference between the absolute andrelative second spatial moment, especially at later times. Thisresult stems from the fact that in the presence of colloids,the contaminant plume moves faster, experiencing greaterheterogeneity, and its size increases (i.e., colloids increaserelative dispersion). This leads to a reduced differencebetween absolute and relative dispersion. The results alsoreveal that the positive correlation assumption decreases thedifference between absolute and relative second spatialmoments. To explain this result, recall that the positivecorrelation between K and Kd or between K1 and K reducesthe mobility of short residence time realizations and increasesmobility of long residence time realizations. Therefore thepositive correlation decreases the effect of plumemeanderingand thus decreases absolute dispersion.

5. Summary and Conclusions

[57] In this study, we present a stochastic, two-dimen-sional numerical analysis for contaminant transport inheterogeneous porous media in the presence of colloids.The presented model is based on simulating the cotransportof contaminants and colloids as a three-phase system(aqueous, solid matrix, and colloids) with six constituentsrepresenting concentrations of colloids and contaminant.The model accounts for both physical and geochemicalheterogeneity. Hydraulic conductivity, K; contaminant dis-

tribution coefficient, Kd; and colloid deposition rate, K1, areassumed to be random variables and correlated. Threepossible correlations are considered (pure negative correla-tion, pure positive correlation, and no correlation). Theanalysis is performed within a stochastic framework usinga Monte Carlo simulation technique. The mean squarederror averaged over the entire domain is adapted as acriterion to ensure the convergence of Monte Carlo results.Both global and local convergence tests indicate that theresults are stable, and the number of realizations is sufficientfor convergence.[58] The model is used to understand interactions be-

tween colloids and contaminants under physical and geo-chemical heterogeneity. In addition, the model is used toevaluate results of Monte Carlo simulations in absolute andrelative dispersion frameworks. We present a comparisonbetween the two frameworks in terms of the ensemblemean, ensemble variance, and spatial moments of the totalcontaminant plume.[59] Results show that colloids reduce the uncertainty

ranges in contaminant arrival times as compared to contam-inant transport in the absence of colloids. It is also foundthat the wide range of uncertainty resulting from spatialvariability cannot compensate for the colloid effect oncontaminant transport. Analyses of the effects of geochem-ical heterogeneity and its correlation to physical heteroge-neity on colloid-facilitated transport yield the followingconclusions.[60] 1. Negative correlations between Kd (or K1) and

hydraulic conductivity increase total mobile contaminantplume (Cd + Sc

mCcm) dispersion, whereas positive correla-

tions decrease plume dispersion.[61] 2. Negative correlations between Kd (or K1) and

hydraulic conductivity increase variance in the plume tail,whereas positive correlations decrease variance as com-pared to the case where only physical heterogeneity isconsidered.[62] 3. First moment of the contaminant plume in

negative correlations is slightly larger than in positivecorrelations.[63] 4. Spatial variability of K1 or Kd is not important in

determining mean plume shape and concentration values ifit is not correlated to hydraulic conductivity variability.[64] 5. Inclusion of plume meandering in the overall

ensemble (absolute mean) reduces the peak and leads toartificial smoothing.[65] 6. Absolute mean concentrations are approximately

20–30% lower than relative mean concentrations.[66] 7. Concentration variance in the relative dispersion

analysis is approximately 40–60% of the variance in theabsolute dispersion case.[67] 8. Ratio between the absolute second spatial moment

to the relative second spatial moment decreases with time(i.e., nonergodic conditions diminish with time).[68] 9. Largest difference between absolute and relative

dispersion is obtained under the negative correlationassumption, whereas the smallest difference results whenthe positive correlation assumption is employed.

ReferencesAbdel-Salam, A., and C. V. Chrysikopoulos (1994), Analytical solutions forone-dimensional colloid transport in saturated fractures, Adv. WaterResour., 17, 283–296.

Table 3. Ratio Between Absolute Second Longitudinal Moment

to Relative Second Moment at t = 30 and 80 Days

Case

X11 Absolute Dispersionð ÞX11 Relative Dispersionð Þ

t = 30 days t = 80 days

Case 1 1.59 1.46Case 2 1.63 1.35Case 3 1.62 1.36Case 4 1.47 1.31Case 5 1.63 1.37Case 6 1.47 1.31Case 7 1.64 1.36Case 8 1.38 1.27Case 9 1.57 1.35Case 10 1.57 1.35


17 of 18

W02010

Abdel-Salam, A., and C. V. Chrysikopoulos (1995a), Analysis of a modelfor contaminant transport in fractured media in the presence of colloids,J. Hydrol., 165, 261–281.

Abdel-Salam, A., and C. V. Chrysikopoulos (1995b), Modeling of colloidand colloid-facilitated contaminant transport in a two-dimensional frac-ture with spatially variable aperture, Transp. Porous Media, 20, 197–221.

Allen-King, R. M., R. M. Halket, and D. R. Gaylord (1998), Characterizingthe heterogeneity and correlation of perchloroethene sorption and hy-draulic conductivity using a facies-based approach, Water Resour. Res.,34, 385–396.

Ball, W. P., and P. V. Roberts (1991), Long-term sorption of halogenatedorganic chemicals by aquifer material: 1. Equilibrium, Environ. Sci.Technol., 25, 1223–1236.

Barber, L. B. (1994), Sorption of chlorobenzenes to Cape Cod aquifersediments, Environ. Sci. Technol., 28, 890–897.

Barber, L. B., E. M. Thurman, and D. D. Runnells (1992), Geochemicalheterogeneity in a sand and gravel aquifer: Effect of sediment mineralogyand particle size on the sorption of chlorobenzenes, J. Contam. Hydrol.,9, 35–54.

Bellin, A., Y. Rubin, and A. Rinaldo (1994), Eulerian-Lagrangian approachfor modeling of flow and transport in heterogeneous geological forma-tions, Water Resour. Res., 30, 2913–2924.

Chen, J. Y., C. H. Ko, S. Bhattacharjee, and M. Elimelech (2001), Role ofspatial distribution of porous medium geochemical heterogeneity in col-loid transport, Colloids Surf. A, Physicochem. Eng. Aspects, 191, 3–16.

Chrysikopoulos, C. V., and A. Abdel-Salam (1997), Modeling colloid trans-port and deposition in saturated fractures, Colloids Surf. A, Physicochem.Eng. Aspects, 121, 189–202.

Chrysikopoulos, C. V., and S. C. James (2003), Transport of neutrallybuoyant and dense variably sized colloids in a two-dimensional fracturewith anisotropic aperture, Transp. Porous Media, 51, 191–210.

Corapcioglu, M. Y., and H. Choi (1996), Modeling colloid transport inunsaturated porous media and validation with laboratory column data,Water Resour. Res., 32, 3437–3449.

Corapcioglu, M. Y., and S. Jiang (1993), Colloid-facilitated groundwatercontaminant transport, Water Resour. Res., 29, 2215–2226.

Corapcioglu, M. Y., and S. Jiang (1996), Dimensional analysis of colloid-facilitated ground-water contaminant transport, J. Hydrol. Eng., 1, 139–142.

Corapcioglu, M. Y., and S. Kim (1995), Modeling facilitated contaminanttransport by mobile bacteria, Water Resour. Res., 31, 2639–2647.

Corapcioglu, M. Y., S. Jiang, and S.-H. Kim (1999), Comparison of kineticand hybrid-equilibrium models simulating colloid-facilitated contaminanttransport in porous media, Transp. Porous Media, 36, 373–390.

Dunnivant, F. M., P. M. Jardine, D. L. Taylor, and J. F. McCarthy (1992),Cotransport of cadmium and hexachlorobiphenyl by dissolved organiccarbon through columns containing aquifer material, Environ. Sci. Tech-nol., 26, 360–368.

Fiori, A., and A. Bellin (1999), Non-ergodic transport of kinetically sorbingsolutes, J. Contam. Hydrol., 40, 201–219.

Hassan, A. E., J. H. Cushman, and J. W. Delleur (1997), Monte Carlostudies of flow and transport in fractal conductivity fields: Comparisonwith stochastic perturbation theory, Water Resour. Res., 33, 2519–2534.

Hassan, A. E., J. H. Cushman, and J. W. Delleur (1998), A Monte Carloassessment of Eulerian flow and transport perturbation models, WaterResour. Res., 34, 1143–1163.

Hu, B. X., F. W. Deng, and J. Cushman (1995), Nonlocal reactive transportwith physical and chemical heterogeneity: Linear nonequilibrium sorp-tion with random Kd, Water Resour. Res., 31, 2239–2252.

Huang, H., and B. X. Hu (2001), Nonlocal reactive transport in heteroge-neous dual-porosity media with rate-limited sorption and interregionalmass diffusion, Water Resour. Res., 37, 639–647.

Ibaraki, M., and E. A. Sudicky (1995a), Colloid-facilitated contaminanttransport in discretely fractured porous media: 1. Numerical formulationand sensitivity analysis, Water Resour. Res., 31, 2945–2960.

Ibaraki, M., and E. A. Sudicky (1995b), Colloid-facilitated contaminanttransport in discretely fractured porous media: 2. Fracture network ex-amples, Water Resour. Res., 31, 2961–2969.

James, S. C., and C. V. Chrysikopoulos (1999), Transport of polydispersecolloid suspensions in a single fracture,Water Resour. Res., 35, 707–718.

James, S. C., and C. V. Chrysikopoulos (2000), Transport of polydispersecolloids in a saturated fracture with spatially variable aperture, WaterResour. Res., 36, 1457–1465.

Johnson, P. R., N. Sun, and M. Elimelech (1996), Colloid transport ingeochemically heterogeneous porous media: Modeling and measure-ments, Environ. Sci. Technol., 30, 3284–3293.

Kim, S., and M. Y. Corapcioglu (1996), A kinetic approach to modelingmobile bacteria-facilitated groundwater contaminant transport, WaterResour. Res., 32, 321–331.

Knabner, P., K. U. Totsche, and I. Kogel-Knabner (1996), The modeling ofreactive solute transport with sorption to mobile and immobile sorbents:1. Experimental evidence and model development, Water Resour. Res.,32, 1611–1622.

Lenhart, J. J., and J. E. Saiers (2002), Transport of silica colloids throughunsaturated porous media: Experimental results and model comparisons,Environ. Sci. Technol., 36, 769–777.

Leonard, B. P., and H. S. Niknafs (1991), Sharp monotonic resolution ofdiscontinuities without clipping of narrow extrema, Comput. Fluids, 19,141–154.

Magee, B. R., L. W. Lion, and A. T. Lemley (1991), Transport of dissolvedorganic macromolecules and their effect on the transport of phenanthrenein porous media, Environ. Sci. Technol., 25, 323–331.

Marseguerra, M., E. Patelli, and E. Zio (2001a), Groundwater contaminanttransport in presence of colloids: I. A stochastic nonlinear model andparameter identification, Ann. Nucl. Energy, 28, 777–803.

Marseguerra, M., E. Patelli, and E. Zio (2001b), Groundwater contaminanttransport in presence of colloids: II. Sensitivity and uncertainty analysison literature case studies, Ann. Nucl. Energy, 28, 1799–1807.

McCarthy, J. F., and J. M. Zachara (1989), Subsurface transport of con-taminants, Environ. Sci. Technol., 23, 496–502.

Mills, W. B., S. Liu, and F. K. Fong (1991), Literature review and model(COMET) for colloid/metals transport in porous media, Ground Water,29, 199–208.

Robin, M. J., A. Sudicky, W. Gillham, and R. G. Kachanoski (1991),Spatial variability of strontium distribution coefficients and their correla-tion with hydraulic conductivity in the Canadian forces base Bordenaquifer, Water Resour. Res., 27, 2619–2632.

Roy, S. B., and D. A. Dzombak (1998), Sorption nonequilibrium effects oncolloid-enhanced transport of hydrophobic organic compounds in porousmedia, J. Contam. Hydrol., 30, 1–2.

Russel, W. B., D. A. Saville, and W. R. Schowalter (1989), ColloidalDispersions, Cambridge Univ. Press., New York.

Saiers, J. E. (2002), Laboratory observations and mathematical modelingof colloid-facilitated contaminant transport in chemically heteroge-neous systems, Water Resour. Res., 38(4), 1032, doi:10.1029/2001WR000320.

Saiers, J. E., and G. M. Hornberger (1999), The influence of ionic strengthon the facilitated transport of cesium by kaolinite colloids, Water Resour.Res., 35, 1713–1727.

Saiers, J., G. M. Hornberger, and C. F. Harvey (1994), Colloidal silicatransport through structured, heterogeneous media, J. Hydrol., 163,271–288.

Sen, T. M., S. P. Mahajan, and K. C. Khilar (2002), Colloid-associatedcontaminant transport in porous media: 1. Experimental studies, AIChEJ., 48, 2366–2374.

Sirivithayapakorn, S., and A. Keller (2003), Transport of colloids in satu-rated porous media: A pore-scale observation of the size exclusion effectand colloid acceleration, Water Resour. Res., 39(4), 1108, doi:10.1029/2002WR001583.

Smith, P. A., and C. Degueldre (1993), Colloid-facilitated transport of radio-nuclides through fractured media, J. Contam. Hydrol., 13, 143–146.

Song, L., P. R. Johnson, and M. Elimelech (1994), Kinetic of colloiddeposition onto heterogeneously charged surface in porous media, En-viron. Sci. Technol., 28, 1164–1171.

Sun, N., M. Elimelech, N. Z. Sun, and J. N. Ryan (2001a), A novel two-dimensional model for colloid transport in physically and geochemicallyheterogeneous porous media, J. Contam. Hydrol., 49, 3–4.

Sun, N., N. Z. Sun, M. Elimelech, and J. N. Ryan (2001b), Sensitivityanalysis and parameter identifiability for colloid transport in geochemi-cally heterogeneous porous media, Water Resour. Res., 37, 209–222.

Wan, J., and J. L. Wilson (1994), Colloid transport in unsaturated porousmedia, Water Resour. Res., 30, 847–864.

Zheng, C., and P. P. Wang (1999), MT3DMS: A modular three-dimensionalmultispecies transport model for simulation of advection, dispersion, andchemical reactions of contaminants in groundwater systems, documenta-tion and user’s guide, Contract Rep. SERDP-99-1, 202 pp., U.S. ArmyEng. Res. Dev. Cent., Vicksburg, Miss.

��H. M. Bekhit and A. E. Hassan, Division of Hydrologic Sciences,

Desert Research Institute, 755 East Flamingo Road, Las Vegas, NV 89119,USA. ([email protected]; [email protected])

18 of 18


Stochastic modeling of colloid‐contaminant transport in physically and geochemically heterogeneous...

Documents

Transcript of Stochastic modeling of colloid‐contaminant transport in physically and geochemically heterogeneous...