spatial moments for colloid-enhanced radionuclide transport in heterogeneous aquifers

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Advances in Water Resources 30 (2007) 101–112

Spatial moments for colloid-enhanced radionuclidetransport in heterogeneous aquifers

Gerardo Severino a,*, Vladimir Cvetkovic b, Antonio Coppola c

a Division of Water Resources Management, Department of Agricultural Engineering, Naples University,

via Universita’ 100, 80055 Portici, Naples, Italyb Division of Water Resources Engineering, KTH, Stockholm, Sweden

c DITEC Department, University of Basilicata, Potenza, Italy

Received 30 June 2005; received in revised form 2 March 2006; accepted 3 March 2006Available online 19 April 2006

Abstract

We consider colloid facilitated radionuclide transport by steady groundwater flow in a heterogeneous porous formation. Radionuclidebinding on colloids and soil-matrix is assumed to be kinetically/equilibrium controlled. All reactive parameters are regarded as uniform,whereas the hydraulic log-conductivity is modelled as a stationary random space function (RSF). Colloid-enhanced radionuclide transportis studied by means of spatial moments pertaining to both the dissolved and colloid-bounded concentration. The general expressions ofspatial moments for a colloid-bounded plume are presented for the first time, and are discussed in order to show the combined impact ofsorption processes as well as aquifer heterogeneity upon the plume migration. For the general case, spatial moments are defined by the aidof two characteristic reaction functions which cannot be expressed analytically. By adopting the approximation for the longitudinal fluidtrajectory covariance valid for a flow parallel to the formation bedding suggested by Dagan and Cvetkovic [Dagan G, Cvetkovic V. SpatialMoments of Kinetically Sorbing Plume in a Heterogeneous Aquifers. Water Resour Res 1993;29:4053], we obtain closed form solutions.

For illustrative purposes, we consider the case when sorption/desorption between solution and moving colloids is a linear non-equi-librium process, whereas sorption onto the soil-matrix is a linear equilibrium process. Based on the flow and transport parameters per-taining to the alluvial aquifer at the Yucca Mountain Site (Nevada), we investigate the potential enhancing role of colloidal particles bycomparing radionuclide spatial moments with and without colloids, and mainly investigate the sensitivity to the reverse rate parameter.The most potentially significant effects are obtained when radionuclide attachment to colloidal particles is irreversible. The simplicity ofour results makes them suitable for quick assessments of the potential impact of colloids on contaminant transport in heterogeneousaquifers.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Colloids; Radionuclide transport; Heterogeneity; Stochastic modelling; Spatial moments

1. Introduction

Most radionuclides strongly sorb onto the soil-matrix ofan aquifer whereby their mobility and their potential envi-ronmental impact are significantly reduced. However, non-soluble particles in form of colloids (such as clay particles,large molecules, etc.), normally found in groundwater, maybind radionuclides enhancing de facto their mobility (e.g.

0309-1708/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2006.03.001

* Corresponding author. Tel.: +39 081 2539426; fax: +39 081 2539412.E-mail address: [email protected] (G. Severino).

[31]). The potential role of colloids in enhancing radionu-clide migration in the subsurface has been discussed inseveral studies (e.g. [10,21,26,20,15,23,40,2]). The mainevidence on colloid-enhanced radionuclide transport atfield scale is by Kersting et al. [24] who observed very sig-nificant effects (although the colloidal concentration levelwas quite low) in an aquifer at the Nevada Test Site.

Several models have been developed to account for col-loids in enhancing solute transport in porous media. Mostof them have assumed equilibrium interactions betweencolloidal particles and dissolved contaminants (e.g.

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102 G. Severino et al. / Advances in Water Resources 30 (2007) 101–112

[1,30,34,16]). However, there are indications that desorp-tion from colloidal phases may be slow (e.g. [3,40]), imply-ing that for realistic predictions of colloid facilitated solutetransport kinetic effects must be taken into account.

Many studies have incorporated the possibility ofkinetic limitations in the exchange between colloidal anddissolved phases. For example, Smith and Degueldre [45]showed that when irreversible sorption of contaminantson colloids is considered, the enhancement of transport dis-tances through fractured porous media could be signifi-cant; this model assumed constant colloid concentrationsand equilibrium interactions with fracture walls. The modelpresented by Corapcioglu and Jiang [6] accounts for equi-librium exchange between colloids and dissolved phase aswell as for rate limited contaminant exchanges, but withseparate equations for the equilibrium and non-equilibriumprocesses. This approach was modified by Saiers andHornberger [44] to account for equilibrium and rate limitedtransfer with the colloidal particles by adopting the two-sites approach where sorption is assumed to occur on afraction of sites in local equilibrium with dissolved phase,and with rate limited exchange on the remaining sites.Based on the laboratory data of Lu et al. [28,27], Painteret al. [39] developed a fairly robust procedure to estimatethe linear partitioning coefficient as well as the forward ratefor radionuclide sorption on inorganic colloids.

It is well known that subsurface systems are physicallyheterogeneous (see e.g. [19,12]). Among the first attemptsin considering the effects of the formation heterogeneityon colloid-enhanced solute transport there are studiesby Abdel-Salam and Chrysikopoulos [1] and Chrysikopo-ulos and Abdel-Salam [7]. The same problem in heteroge-neous aquifers was first considered by Cvetkovic [9] withthe focus on tracer discharge. By assuming that the log-conductivity of the porous formation is a stationaryRSF, Cvetkovic [9] proposed a Lagrangian frameworkusing a two-sites sorption model. In a recent paper,Cvetkovic et al. [10] used a simplified (steady state) ver-sion of such a Lagrangian formulation to study modeland parametric sensitivities for a hypothetical release ofradionuclide in the alluvial aquifer near the Yucca Moun-tain in Nevada.

The present paper aims at quantifying the potential effectsof colloidal particles on spatial development of a radionu-clide plume in a heterogeneous aquifer. The work expandson the Lagrangian framework, and provides, for the firsttime, the representation of colloid-enhanced radionuclidetransport by means of spatial moments. Our study is orga-nized as follows. In Section 2, we formulate Eulerian equa-tions for colloids (Section 2.1) as well as aqueous/non-aqueous radionuclide concentrations (Section 2.2).These equations represent the starting point to obtain theLagrangian version of the same set of equations (Section2.3). A general methodology for deriving spatial momentsin a simple manner for both the dissolved and colloidal-bounded concentration is presented in Section 3. InSection 4 we show possible scenarios in which the presence

of colloidal particles may lead to enhancement of radionu-clide transport. Concluding remarks are reported in Section 5.

2. Problem formulation

We consider a dissolved radionuclide that is releasedinto an aquifer, and limit our analysis to a steady velocityfield V which satisfies the continuity equation $ Æ (#V) = 0,and is related to the hydraulic conductivity K, to the poros-ity # (assumed constant in the present paper), and to thehydraulic head U through Darcy’s law V ¼ � K

#rU. Due

to its erratic variations and owing to the scarcity of avail-able data, we regard the log-conductivity Y = ln(K) as astationary RSF with mean hYi (the operator h i denotesthe ensemble average), and covariance CY(r) (r = x � y

represents the relative separation distance between twoarbitrary points x � (x1,x2,x3) and y � (y1,y2,y3) belong-ing to the flow domain). The covariance function CY(r) ishypothesized of exponential type, i.e.

CY ðrÞ ¼ r2Y

� exp �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix1� y1

I

� �2

þ x2� y2

I

� �2

þ x3 � y3

Iv

� �2s24 35;

ð1Þwhere r2

Y ¼ CY ð0Þ represents the variance, and I and Iv arethe horizontal and vertical integral scales of Y, respectively.As a consequence, V is also a RSF whose statistics willdepend upon those of Y, and the boundary conditionsimposed on U. In particular, we consider the case of anaverage uniform flow in an unbounded domain, andassume without loss of generality that the mean groundwa-ter velocity U is parallel to the x1-coordinate, i.e.hVi � (U, 0,0). Statistics of such a flow field are given byDagan [11], and will be used herein to derive spatialmoments of solute plumes.

2.1. Transport of colloids

A detailed description of the dynamics of colloids atfield scale is not an easy task. The reason is that, whereasthe main mechanisms which determine the transport of col-loids are well established (see e.g. [44]), their characteriza-tion is very uncertain because of the extreme difficulty tomeasure the controlling parameters (e.g. [37]). In view ofsuch restrictions, and following previous studies [10,39,9],we shall introduce some simplifying assumptions that per-mit to catch the impact of colloids upon radionuclidetransport.

Hence, we consider a groundwater containing a homo-geneous collection of colloidal particles. It is assumed thatthe properties and mass of colloids are not influenced bysorption of radionuclides [36], and that colloids areadvected with the groundwater velocity V (i.e. we considersufficiently small particles as to be dynamically inert). Gen-erally, the advection velocity of the colloidal particles is

G. Severino et al. / Advances in Water Resources 30 (2007) 101–112 103

somewhat larger than V due to the exclusion phenomena;however, this effect is negligible at column scale (see e.g.[29,44]), and it is even less relevant at aquifer scale[10,9,32]. We also neglect the effect of hydrodynamic dis-persion since it is of minor importance when dealing withplume moments [18,17]. Finally, we assume quasi steady-state colloidal concentration. This assumption is reason-able since the temporal stability of colloids depends mainlyon the hydraulic conditions (steady in our case) prevailingin the aquifer [32,33]. Nevertheless, in recent laboratoryexperiments [29] designed to mimic field conditions, radio-nuclides were injected in the column only after recoveringsteady-state conditions in the colloidal concentration.

With these assumptions, the governing equation for thecolloidal concentration K (defined as colloidal mass perunit volume of fluid) writes as

V ioKoxi¼ ��Kþ x: ð2Þ

The filtration (i.e. removal) rate � and the generationparameter x are not simple to quantify. A commonassumption is that � is proportional to the mean groundwa-ter flow [37,22], and therefore the filtration rate may beconsidered approximately constant. The generation of col-loids is more difficult to estimate, both at field and labora-tory scales. It is evident, however, that generation isstrongly related to the removal [43].

On these grounds, we assume that in the mean the filtra-tion is compensated by the generation, i.e. x � �K, andtherefore we deduce from (2) that K can be approximatelyregarded as uniform. Such an assumption is acceptable alsoin view of other sources of uncertainties mainly related todata measurements (e.g. [42]), and it has already beenadopted to analyze colloid facilitated radionuclide trans-port both at laboratory [39], and field scale [10,9]. Finally,it is worth to emphasize here that even in the case in whichthe removal of colloids is prevailing upon the generation,assuming that colloids are uniformly distributed in thespace leads to overestimation of potential effects due tothe presence of colloidal particles, therefore providing anupper bound of the associated environmental risk.

2.2. Eulerian formulation of radionuclide transport

The Eulerian formulation of colloid facilitated radionu-clide transport is based on two mass balance equations:one pertaining to the radionuclides confined in the liquidphase, and the second one accounting for the radionuclidespresent in the non-aqueous (i.e. colloidal) phase. Further-more, these equations are supplemented by relationshipsdescribing sorption/desorption processes involving the dif-ferent phases. We note that the term sorption is used in abroad (phenomenological) sense such that several differentphysical and/or chemical processes which affect the fate ofradionuclides in aquifers can be accounted for. A phenom-enological approach to sorption is common in treating col-loid facilitated transport according to simplified models

(e.g. [10,9,47,46,6]). Thus, we shall assume that sorptiontakes place on two types of sites: fast sites where sorptionis relatively rapid, and slow sites where sorption takes placeover a relatively long time. The multiple sites approach hasbeen recently adopted for interpreting laboratory (e.g.[39,36,41,5]) as well as field scale (e.g. [10]) data.

Let C and Cc define the liquid and colloidal concentra-tion as mass of radionuclide dissolved and bound tocolloids per volume of liquid, respectively. We denote byN (defined per unit bulk volume) the concentration ofradionuclide immobilized on the porous matrix. With thesedefinitions, the system of governing equations is written as

RoCotþ RcV � rC ¼ wcðC;CcÞ þ wðC;NÞ � RkC; ð3aÞ

Rc

oCc

otþ RcV � rCc ¼ �wcðC;CcÞ � RckcCc; ð3bÞ

oNot¼ �wðC;NÞ; ð3cÞ

where the constants R = Rc + Kd, and Rc = 1 + Kc are re-ferred as retardation factors. The dimensionless coefficientsKd and Kc are the linear equilibrium coefficients on the por-ous matrix and colloidal particles, respectively. Kineticsorption on colloids is accounted for by the exchange termwc = wc(C,Cc), and similarly kinetic sorption on the porousmatrix is described by w = w(C,N). The functional depen-dence of wc and w upon the various concentrations C, Cc,and N will be specified later on the bases of the assumedexchange model. We also consider linear degradation inthe dissolved and colloidal phase through the coefficientsk and kc, respectively.

It is worth to note that in such a formulation, we canclearly distinguish kinetics from equilibrium. Thus, pureequilibrium in the colloidal phase corresponds to wc = 0,while pure equilibrium on the matrix implies w = 0. Colloidfacilitated radionuclide transport under equilibrium condi-tions has been extensively studied both at laboratory [39],and at field [9] scale.

2.3. Lagrangian formulation of radionuclide transport

The Lagrangian version of Eqs. (3a)–(3c) is based on thetrajectories X(t;a) � (X1,X2,X3) of a marked fluid particlestarting from a point a � (a1,a2,a3) belonging to a givenarea A0 on the injecting plane (Fig. 1). More specifically,the relationship between the Eulerian and Lagrangian con-centrations is obtained by adopting the methodology pro-posed by Cvetkovic and Dagan [8]. Indeed, we replacethe independent variable x � (x1,x2,x3) with n � (n1,n2,n3)defined along a streamline (Fig. 1).

n1 ¼ sðx1; aÞ; n2 ¼ x2 � gðx1; aÞ; n3 ¼ x3 � fðx1; aÞ;ð4Þ

where s = s(x1;a) is the travel time of a particle departingfrom a on the injecting plane, and reaching a control planelocated at x1. The travel time is obtained by solving

mean flow

injection area accessible environment

AQUIFER

colloidal plume front

dissolved plume front

A0

Fig. 2. Configuration sketch of the colloid facilitated solute transport ataquifer scale.

MEAN FLOW

Injecting plane

Control plane

x1

x2

x3

( η ζ),,11 xV

A0

( )a1V

ad

Fig. 1. Definition sketch of a particle path starting from the injectingplane and arriving at the control plane.


x1 = X1(t;a) with respect to time t, whereas g(x1;a) = X2

(s;a) and f(x1;a) = X3(s;a). Thus, g(x1;a) and f(x1;a)define the equations of a streamline originating at theinjecting plane (see Fig. 1) and intersecting the controlplane at t = s. The functions s, g, and f previously defineddepend upon the velocity field through the following kine-matical relationships [8]

dsdx1

¼ 1

V 1ðnÞ;

dgdx1

¼ V 2ðnÞV 1ðnÞ

;dfdx1

¼ V 3ðnÞV 1ðnÞ

: ð5Þ

With the details reported in Appendix A, Eqs. (3a)–(3c) canbe recast in terms of streamlines as

R0oCotþ oC

os¼ 1

Rc

½wcðC;CcÞ þ wðC;NÞ� � R0kC; ð6aÞ

oCc

otþ oCc

os¼ � 1

Rc

wcðC;CcÞ � kcCc; ð6bÞ

oNot¼ �wðC;NÞ ð6cÞ

(with R0 ¼ RRc

). In view of the computation of spatial mo-ments, it is convenient [8] to write C and Cc as

Cðs; t; aÞ ¼ q0ðaÞV 1ðaÞ

cðs; tÞ; Ccðs; t; aÞ ¼ q0ðaÞV 1ðaÞ

ccðs; tÞ;

ð7Þ

where c = c(s, t) and cc = cc(s, t) are termed as characteris-

tic reaction functions, and are explicitly calculated inAppendix B (q0(a) represents the injected radionuclidemass per unit net area).

Summarizing, we have transformed the three-dimen-sional Eulerian system (3a)–(3c) into the one-dimensionalsystem of equations (6a)–(6c). Thus, the original three-dimensional nature of the problem has been encapsulatedin that of the travel time s. The characteristic functions cand cc will constitute the starting point for calculating spa-tial moments.

3. Spatial moments

Let a dissolved radionuclide be injected into the ground-water at time t = 0 over an area A0. At t > 0, two plumesdevelop and migrate toward the accessible environment(Fig. 2). We wish to quantify the evolution of both the dis-solved and colloid-bounded radionuclide plumes by meansof spatial moments. Toward this aim, we first consider thefluid DC and colloidal DCc concentration associated to aparcel moving in the aqueous and non-aqueous phase,respectively. Thus, starting from the characteristic reactionfunctions c and cc previously defined, it has been shown [8]that the aforementioned particle concentrations are calcu-lated as

DC ¼ #q0ðaÞV 1ðx1; g; fÞ

cðs; tÞdðx2 � gÞdðx3 � fÞ; ð8aÞ

DCc ¼#q0ðaÞ

V 1ðx1; g; fÞccðs; tÞdðx2 � gÞdðx3 � fÞ: ð8bÞ

With the parcel concentrations given by (8a) and (8b), thecorresponding field concentrations C and Cc are simply ob-tained by integrating over A0, i.e.

Cðx; tÞ ¼ #Z

A0

daq0ðaÞV 1ðx1; g; fÞ

cðs; tÞdðx2 � gÞdðx3 � fÞ; ð9aÞ

Ccðx; tÞ ¼ #Z

A0

daq0ðaÞV 1ðx1; g; fÞ

ccðs; tÞdðx2 � gÞdðx3 � fÞ: ð9bÞ

For such plumes, spatial moments are defined as

lðmnrÞðtÞ ¼Z

xm1 xn

2xr3Cðx; tÞdx

¼Z

A0

q0ðaÞZ 1

a1

xm1 gnfr

V 1ðx1; g; fÞcðs; tÞdx1 da; ð10aÞ

lðmnrÞc ðtÞ ¼

Zxm

1 xn2xr

3Ccðx; tÞdx

¼Z

A0

q0ðaÞZ 1

a1

xm1 gnfr

V 1ðx1; g; fÞccðs; tÞdx1 da; ð10bÞ


where m, n, r are non-negative integers. We exchange thevariable x1 with s by the aid of V 1 ¼ dx1

ds (first of (5)) tocome up with the final expression for spatial moments:

lðmnrÞðtÞ ¼Z

A0

Z 1

a1

X m1 gnfrcðs; tÞdsda;

lðmnrÞc ðtÞ ¼

ZA0

Z 1

a1

X m1 gnfrccðs; tÞdsda: ð11Þ

Due to the dependence of s, g, and f upon the RSF V, alsomoments (11) will result RSFs.

By assuming that both the plumes are ergodic [12], weconsider the ensemble average of (11), and wish to deriveexpressions for the first three spatial moments. In orderto do this, it is convenient to define the following auxiliaryfunctions:

CðpÞðtÞ ¼Z 1

0

spcðs; tÞds;

CðpÞc ðtÞ ¼Z 1

0

spccðs; tÞds ðp ¼ 0; 1; 2; . . .Þ: ð12Þ

The zero order moments (i.e. the masses) M(t) and Mc(t)pertaining to the fluid and colloidal phase are obtainedby setting m = n = r = 0 in (11) (with a1 = 0), and ensembleaveraging, i.e.

MðtÞ ¼Z

A0

Z 1

0

cðs; tÞdsda ¼ M0Cð0ÞðtÞ;

M cðtÞ ¼Z

A0

Z 1

0

ccðs; tÞdsda ¼ M0Cð0Þc ðtÞ ð13Þ

being M0 ¼ #R

A0daq0ðaÞ.

The first moments are obtained by setting m = 1,n = r = 0 in (11), taking the expectation and normalizingby the respective masses (13) to yield:

hR1ðtÞi ¼ R1ð0Þ þ UCð1ÞðtÞCð0ÞðtÞ

;

hR1cðtÞi ¼ R1cð0Þ þ UCð1Þc ðtÞCð0Þc ðtÞ

ð14Þ

(for simplicity we have set hRii = hRici = 0 for i = 2,3). In asimilar manner, the second longitudinal central momentsare evaluated as

hS11ðtÞi ¼ S11ð0Þ þ U 2 Cð2ÞðtÞCð0ÞðtÞ

� UCð1ÞðtÞCð0ÞðtÞ

� �2

þ 1

Cð0ÞðtÞ

Z 1

0

dsX 11ðsÞcðs; tÞ; ð15aÞ

hS11cðtÞi ¼ S11cð0Þ þ U 2 Cð2Þc ðtÞCð0Þc ðtÞ

� UCð1Þc ðtÞCð0Þc ðtÞ

" #2

þ 1

Cð0Þc ðtÞ

Z 1

0

dsX 11ðsÞccðs; tÞ: ð15bÞ

Higher-order moments can be evaluated similarly. In deriv-ing (14), (15a) and (15b) we have used the result X = a +hVit + X 0 [11], where X 0 = X � hXi is the fluctuation of

the particle displacement X, whereas X 11 ¼ hX 021 i representsthe variance of X1.

It is seen from (13) and (14) that the zero and first ordermoments are completely defined through the auxiliaryfunctions (12) which are evaluated as follows:

CðpÞðtÞ ¼ ð�1Þp dp

dspcðs; tÞ

s¼0

; CðpÞc ðtÞ ¼ ð�1Þp dp

dspccðs; tÞ

s¼0

;

ð16Þwhere c and cc represent the Laplace transforms over s of cand cc, i.e.

cðs; tÞ ¼Z 1

0

ds expð�ssÞcðs; tÞ;

ccðs; tÞ ¼Z 1

0

ds expð�ssÞccðs; tÞ: ð17Þ

Thus, like contaminant transport without colloids (see [8])the masses and the first order moments are not affected bythe heterogeneity, and depend only upon the reaction pro-cesses via the C-functions (12). On the contrary, the com-bined effect of sorption/desorption processes and physicalheterogeneity is clearly shown in the second order centralmoments (15a) and (15b). In particular, the second andthird term in (15a) and (15b) exist for deterministic anduniform velocity fields, and represent the spread due tomass transfer between dissolved and colloid-bounded con-centration. The last term appearing in (15a) and (15b) is ofmajor interest since it couples the effect of velocity random-ness and sorption processes (the quantities S11(0) andS11c(0) represent the initial solute spread).

In order to evaluate the second order central longitudi-nal moments (15a) and (15b) explicit expressions for cand cc are also required. Generally speaking, the system(6a)–(6c) can be solved only numerically. Alternatively,one can (in some cases) analytically solve such a systemin the Laplace domain (e.g. [9]) and subsequently, by anumerical inversion of the Laplace transform, computethe reaction functions c and cc (e.g. [10]).

To avoid any numerical step in the solution, we proposea different approach which leads to simple results. Inessence, we shall use the approximate expression for X11

for flow parallel to the bedding in three-dimensional aniso-tropic heterogeneous porous media in the form [13]:

X 11ðtÞ ¼2

bðrY IÞ2½expð��btÞ þ �bt � 1�; ð18Þ

where

�b ¼ bth

; b ¼ 1þ ð19� 10e2Þe2

16ð1� e2Þ2�

eð13� 4e2Þarcsinffiffiffiffiffiffiffiffiffiffiffiffi1� e2p� �

16ð1� e2Þ3=2:

ð19Þ

The anisotropic ratio e appearing in (19) is defined ase ¼ Iv

I , and th ¼ IU is a characteristic heterogeneity time.

Expression (18) is appropriate for many applications as itprovides sufficient accuracy for e 6 1 compared to the ex-act analytical expression [13].


Inserting (18) into the integrals appearing on the righthand side of (15a) and (15b) yields:Z 1

0

dsX 11ðsÞcðs; tÞ

¼ 2r2Y I UCð1ÞðtÞ� I

bCð0ÞðtÞ�

Z 1

0

dsexpð��bsÞcðs; tÞ� � �

;

ð20ÞZ 1

0

dsX 11ðsÞccðs; tÞ

¼ 2r2Y I UCð1Þc ðtÞ�

Ib

Cð0Þc ðtÞ�Z 1

0

dsexpð��bsÞccðs; tÞ� � �

:

ð21Þ

We calculate the integrals on the right hand side of (20) byusing the following property:Z 1

0

ds exp½�ðsþ �bÞs�f ðs; tÞ ¼ f ðsþ �b; tÞ; ð22Þ

which after taking the limit s! 0 on both sides, yields:Z 1

0

ds expð��bsÞf ðs; tÞ ¼ f ð�b; tÞ: ð23Þ

Thus, the longitudinal central second moments (15a) and(15b) write as

hS11ðtÞi ¼ S11ð0Þ þ U 2 Cð2ÞðtÞCð0ÞðtÞ

� UCð1ÞðtÞCð0ÞðtÞ

� �2

þ 2r2Y I U

Cð1ÞðtÞCð0ÞðtÞ

� Ib

1� cð�b; tÞCð0ÞðtÞ

� � �; ð24aÞ

hS11cðtÞi ¼ S11cð0Þ þ U 2 Cð2Þc ðtÞCð0Þc ðtÞ

� UCð1Þc ðtÞCð0Þc ðtÞ

" #2

þ 2r2Y I U

Cð1Þc ðtÞCð0Þc ðtÞ

� Ib

1� ccð�b; tÞCð0Þc ðtÞ

" #( ): ð24bÞ

Of course, the suggested approach does not permit deriva-tion of transverse spatial moments hS22i and hS33i unlessexpressions similar to (18) are provided for X22 and X33.However, in most applications the spread of the plume isprevailing in the mean flow direction [12], and longitudinalmoments are of prime interest.

So far, we have derived solute spatial moments pertain-ing to the fluid and colloidal phase. In particular, we haveadopted the approximation (18) to remove the computa-tional effort related to the solution of the system (6a)–(6c). The result is that in order to evaluate spatial moments,it is sufficient to calculate the Laplace transform over thetravel time s of the reaction functions c and cc.

4. Illustration of results

For illustration purposes, we consider radionuclidetransport in a three-dimensional statistically anisotropicheterogeneous porous medium, and we wish to demon-strate the role of the colloidal particles. In particular, we

want to identify those conditions under which colloidsmight produce significant effects upon groundwater con-taminations. Specifically, we deal with a hypotheticalradionuclide release into the alluvial aquifer near YuccaMountain (Nevada) for which basic parameter estimatesare available [38,35].

4.1. Input parameters

The log-conductivity Y and the groundwater velocity V

are RSFs with th � 0.2 y, and r2Y � 1. We would like to

emphasize that the present analysis relies on a small-pertur-bation analysis following the linear theory of Dagan [11]. Itis generally accepted that the linear theory captures theforemost features of transport phenomena whenever r2

Y issufficiently small. However, numerical simulations [4] aswell as recent theoretical results [14] concerning transportin strongly heterogeneous formations have shown abroader range of applicability (say till to r2

Y � 1:5) of thelinear theory (because of the mutual cancellation effectsof the errors induced by the linearization procedure). Thisenables us to use a relatively large value for the variance ofthe hydraulic log-conductivity.

Although we do not have specific data on anisotropicproperties, we shall assume some statistical stratification(in view of the alluvial nature of the aquifer) having ananisotropy ratio e � 0.1. We consider a radionuclide (likePlutonium) with a relatively large retardation R = 40,000[9]. Colloid concentration K � 0.65 mg/l for the alluvialaquifer is available from Painter et al. [38], and Kingstonand Whitbeck [25]. Detailed data upon Rc are quite difficultto obtain. A distribution of possible values of Rc has beenrecently provided by Wolfsberg and Reimus [48] using fil-tration theory. The median value for Rc was estimated tobe 20, and it is the value we shall adopt in this work. Weassume that degradation in the liquid and colloidal phaseis regulated by the same coefficient k ¼ kc ¼ ln 2

t1=2, and we

set the half-life time t1/2 equal to 2.5 · 104 y (which is typ-ical of a Plutonium-like radionuclide).

For illustrative purposes, we assume that the exchangebetween the fluid phase and colloids is described by the firstorder linear model

1

Rc

wcðC;CcÞ ¼ �aC þ bCc: ð25Þ

A more general model for wc = wc(C,Cc) would be the so-called bilinear model which accounts for a limited numberof colloidal binding sites. However, Cvetkovic et al. [10]found that the model (25) is often sufficient for field scaleapplications where both C and Cc are low. In (25) the sorp-tion rate (a 0) and the desorption rate (b 0) are scaled by theretardation factor Rc, i.e. a ¼ a0

Rcand b ¼ b0

Rc, to account for

the fact that at any given time, only the solute mass presentin the aqueous phase is available for exchanges with col-loids. For simplicity, we consider only linear equilibriumsorption to the matrix, i.e. w = 0 in (6a) and (6c).


Based on experimental data of Lu et al. [28,27] for differ-ent minerals, and based on the estimates of Cvetkovic et al.[10] and Painter et al. [39], we have set the value of the for-ward sorption rate a 0 = 0.148 y�1. Currently, we have nosite-specific data concerning the intrinsic desorption rateb 0, and we consider it as a sensitivity parameter. However,it seems reasonable to assume for b ¼ b0

Rca range of variabil-

ity similar to that which has been established for a (see [9]).Such a working assumption has been already adopted inthe past [10]. Thus, given the characteristic heterogeneitytime th, and accounting for the estimates of a providedby Painter et al. [39], we consider the normalized desorp-tion rate varying in a relatively wide range, i.e.10�6

6 bth 6 10�3.We calculate the first three spatial moments Mc(t),

hR1c(t)i and hS11c(t)i by the aid of (13), (14) and (15b),and compare them with those (say eM ; heR1i; and heS 11i)pertaining to the same aquifer in which the effect of colloidsis neglected. In particular, we are interested into assessingpossible ranges of the backward dimensionless rate bth

which lead to potentially significant effects of colloids onradionuclide transport. Toward this goal, we have com-puted the following quantities (see Appendix B):

lðtÞ ¼ M cðtÞeM ðtÞ ; qðtÞ ¼ hR1cðtÞiheR1ðtÞi

; vðtÞ ¼ hS11cðtÞiheS 11ðtÞi

ð26Þ

and illustrated their dependence upon time for several val-ues of bth.

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

10 100 10

t/

μ/αth

Fig. 3. Relative zero order moment l normalized by ath versus th

4.2. Discussion

With the input data as summarized in the previous sec-tion, we have computed relative spatial moments (26).The first of the relative measures in (26) represents themost important quantity as it gives insights into the massthat is associated with colloids as compared with thesame mass that one would recover in the liquid phasewhen colloids are not accounted for. For the sake of gen-erality, we found more convenient to depict in Fig. 3 thescaled relative mass l

athversus dimensionless time t

th. Due

to the initial conditions, at early times almost all the massis in the fluid phase. Thus, there is no significant desorp-tion, and l

athis practically insensitive to b. At the other

extreme (say t P 105th), when significant sorption hastaken place, the impact of different values of the desorp-tion rates is more prominent. By this time, for bth P 10�4

the partitioning between the fluid and colloidal phaseshas reached the equilibrium, and therefore the corre-sponding curves level on. On the other hand, forbth 6 10�4 kinetic effects are still present, and thisexplains why the mass attached to colloids is still increas-ing. In order to reach the equilibrium, a longer periodhas to be considered.

To render our general analysis more quantitative as wellas to provide a simple example showing the profoundpotential of Fig. 3, let us consider one possible value forthe forward rate, i.e. ath = 10�3. At t = 105th, and consid-ering a relatively fast reverse rate (i.e. bth = 10�4) werecover from Fig. 3

00 10000 100000

th

10-4

10-5

10-6

βth=10-3

e dimensionless time tth

, and for bth = 10�6, 10�5, 10�4, 10�3.


lð105thÞath

� 104 ) lð105thÞ � 10410�3

) M cð105thÞ ¼ 10 eM ð105thÞ: ð27ÞThus, at t = 105th and for the particular choice of the for-ward/backward rates the radionuclide mass attached tocolloids is roughly 10 times the one that is recovered inthe liquid phase in absence of colloids. Of course, sucha scenario becomes much more dramatic if we deal withlesser values of the dimensionless reverse rate. For in-stance, taking ath = 10�3 and bth = 10�6 (which in ourcase represents the most critical situation) the radionu-clide mass which is ‘‘colloid advected’’ is 100 times theone we would expect if colloids are not considered. It isclear that the most interesting (in view of the risk assess-ment problem) insight from the large time behavior of l

ath

is related to relatively low values of b. As the quantity laccounts for the pattern of radionuclide mass, l

athis not

affected by the formation heterogeneity (see also [8]),and therefore it provides very robust predictions uponthe environmental risk due to radionuclide migration.For this reason, we believe that the general insights con-tained in Fig. 3 represent one of the most important re-sult of the present paper.

The impact of the parameter b upon the relative firstmoment q is shown in Fig. 4. Unlike the previous case,for illustrative purposes it has been necessary to specifysomehow a realistic value for the forward rate. Indeed,we decided to choose ath on the base of the Yucca Moun-tain data-set, i.e. ath ¼ a0

Rcth ¼ 0:148

20y�1

� 0:2 y ’ 10�3.

Although at early times hR1ci is at least four orders of mag-nitude greater than heR1i, a simple comparison with Fig. 3shows that at these times the radionuclide mass which actu-ally is moving with colloids is very small. As time increases

1.0E+02

1.0E+03

1.0E+04

1.0E+05

10 100 1000

t/th

ρ

Fig. 4. Relative first moment q versus tth

, and for ath

(say for t P 103th) the curve corresponding to the fastdesorption starts to decrease. This is due to the fact thatmost of the radionuclide mass is bound to the immobilephase, and therefore the effect of colloidal particles as‘‘radionuclide carriers’’ is drastically reduced. The largetime regime shows the most interesting features. The valuesof q range from 300 to 2000, for bth ranging from 10�3 to10�6, respectively. At a first glance, we can argue that thepresence of colloids into radionuclide transport is impor-tant when there are relatively slow backward kinetics.Although the values attained by q clearly indicate the pos-sibility that at large time radionuclides may travel distanceslarger than those without colloids, we have to account forthe effective amount of mass which is attached onto colloi-dal sites. Inspection of Fig. 3 suggests that the radionuclidemass adsorbed to the colloidal particles is potentially sig-nificant for b < a.

The second order relative moment v is illustrated inFig. 5. Like the previous cases, we do not analyze the pat-tern of v at early times as the colloidal radionuclide masswhich would realistically spread is very low (see Fig. 3).To the contrary, we are mainly interested in the long timeregime as practical applications aim to predict the fate ofradionuclides on large time scales (e.g. [9]). Thus, fort = 105th we can see that the highest dispersion is attainedfor the lowest values of bth. This behavior is attributed tothe effect of kinetics, which is still important at this time.On the other hand, when bth P 10�4 sorption is at equilib-rium and this leads to a lower dispersion. Thus, at interme-diate times (say t ’ 103th) dispersion in the colloidal phasehas reached its maximum for those cases in which the equi-librium is attained, i.e. for bth = 10�4; whereas for lowerdesorption rates dispersion in the colloidal concentrationfurther increases.

10000 100000

10-4

10-510-6

βth=10-3

= 10�3. The values of bth are the same as Fig. 3.

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

10 100 1000 10000 100000

t/th

10-4

10-510-6

βth=10-3

χ

Fig. 5. Relative second moment v versus tth

, for r2Y ¼ 1 and e = 0.1. The remaining parameter values are the same as Fig. 4.


Finally, let us observe that in the case of no-sorption onthe soil-matrix, radionuclide (and more generally solute)spatial moments are still different from those resultingwhen solute particles are considered totally inert. This isdue to the fact that sorption on moving colloids acts as aresistance which delays (although to a lesser extent sincecolloids are advected) the solute migration [8].

5. Summary and conclusions

General expressions for expected spatial moments of atracer carried by colloids and dissolved in the fluid, havebeen derived for quantifying radionuclide transport inheterogeneous aquifers. As spatial moments are littleaffected by pore-scale dispersion, this has been neglected.Furthermore, throughout the present study the generalsorption model considers linear non-equilibrium sorptionon moving colloids and equilibrium sorption onto porousmatrix. In order to remove the computational difficulties,the approximation for the fluid particles longitudinal var-iance X11 proposed by Dagan and Cvetkovic [13] hasbeen adopted. Such approximation permits to calculatespatial moments using only the Laplace transforms overthe travel time s (that can be calculated in closed analyt-ical form) of the characteristic reaction functions c andcc.

We have illustrated some potential effects of colloidsupon radionuclide transport by comparing radionuclidetransport with and without colloids. The main flow andtransport parameters have been based on the alluvial aqui-fer at the Yucca Mountain (Nevada). When the desorptionrate is relatively high the equilibrium between the variousphases is reached, and the mobile radionuclide mass carriedby colloids may result quite low. On the other hand, for

low desorption rates kinetics are still important. In partic-ular, when a a� b the radionuclide mass attached to thecolloidal particles may result much greater than the onein the aqueous phase when colloids are not accountedfor. This implies that with the current sorption model,the most significant effect of colloids on transport isexpected when b ’ 0.

The analysis of the first and second order relativemoments strictly depends upon the zero order moments,as the evolution of the former would result of minor impor-tance for pollution risk assessment if a very small quantityof mass is attached to colloids. At large times, it is seen thatthe first order moment of the colloid-bounded plume (i.e.hR1ci) and the longitudinal dispersion (i.e. hS11ci) aregreater than the ones pertaining to the same aquifer with-out colloids.

Any specific evaluation of the potential impact of col-loids on contaminant transport risk assessment wouldclearly require an analysis with a broader range of param-eter and model sensitivity than those presented in thiswork. However, if the exchange processes can be assumedlinear, then the proposed methodology provides a relativelysimple and robust tool for carrying out such analyses. Weunderline that a systematic comparison between the linearand non-linear sorption models for colloid facilitatedtransport indicate that the linear exchange model providesa reasonable (upper bound) approximation even in caseswhere non-linear sorption effects (like Langmuir or bilineartype) are significant [10]. The current study may beextended in many directions, like accounting for severaloverlapping plumes pertaining to different populations ofcolloids, or considering more complex interactions betweeneach phases. Most of these extensions are topics of ongoingresearch.


Acknowledgments

This study was supported by grant ‘‘Programma discambi internazionali per mobilita di breve durata’’ (#048058, U.P.I.M.D.S.), Naples University (Italy).

We are indebted to Dr. Gabriella Romagnoli for review-ing the manuscript. Comments and suggestions from fiveanonymous reviewers have been deeply appreciated, andhave significantly improved the early version of the manu-script. The first author wishes to thank Prof. GiancarloBarbieri (Naples University) for promoting his visit at theRoyal Institute of Technology in Stockholm, and all thepeople at the Division of Water Resources Engineeringwho have helped him during his stay in Sweden.

Appendix A. Derivation of Eqs. (6a)–(6c)

In order to derive the Lagrangian version of (3a)–(3c),we calculate the spatial derivatives o

oxiaccording to the

chain rule, and make use of Eqs. (4) and (5) to have

o

ox1

¼ 1

V 1ðnÞo

on1

;

o

oxi¼ 1

V iðxÞV iðxÞ � V iðnÞ

V 1ðxÞV 1ðnÞ

� �o

oniði ¼ 2; 3Þ: ðA:1Þ

In this way the system (3a)–(3c) becomes

RRc

oCotþ V 1ðxÞ

V 1ðnÞoCon1

þ V iðxÞ � V iðnÞV 1ðxÞV 1ðnÞ

� �oConi

¼ 1

Rc

½wcðC;CcÞ þ wðC;NÞ� � RRc

kC; ðA:2Þ

oCc

otþ V 1ðxÞ

V 1ðnÞoCc

on1

þ V iðxÞ � V iðnÞV 1ðxÞV 1ðnÞ

� �oCc

oni

¼ � 1

Rc

wcðC;CcÞ � kcCc; ðA:3Þ

oNot¼ �wðC;NÞ ðA:4Þ

(summation for i = 2, 3 is performed in (A.2) and (A.3)).For x2 = g and x3 = f, i.e. along the stream line ni = 0(see Eqs. (4)), we have V(x) � V(n) and Eqs. (A.2)–(A.4) re-duce to (6a)–(6c).

Appendix B. Derivation of l, q, and v

To calculate the relative indicators (26), we first derive cc

which is needed to calculate the CðpÞc -functions appearing inthe spatial moments pertaining to the colloidal plume (seeEqs. (13), (14) and (15b)), and then recapitulate spatialmoments assuming that colloids are not present, i.e.eM ; eR1; and eS 11.

The reaction function cc(s, t):To derive the reaction function cc, we consider a radio-

nuclide pulse injection in the liquid phase, i.e c(0, t) = d(t)

and cc(0, t) = 0, assuming that the aquifer is initially radio-nuclide free, i.e. c(s, 0) = cc(s, 0) = 0. We consider1

Rcwcðc; ccÞ ¼ �acþ bcc as model to account for linear

non-equilibrium exchange between the fluid and colloidalphase, and set w = 0 so that the system (6a) and (6b)(rewritten after substituting Eq. (7)) becomes

R0ocotþ oc

os¼ �ðaþ R0kÞcþ bcc; ðB:1Þ

occ

otþ occ

os¼ ac� ðbþ kcÞcc: ðB:2Þ

To solve the system (B.1) and (B.2), we apply the Laplacetransform (denoted with hat symbol) over s (the corre-sponding complex variable is s) to obtain the following ini-tial value problem:

dcd�t ¼ �ðsþ aþ kÞcþ bcc þ dð�tÞ

R0 ;

1R0

dcc

dt ¼ ac� ðsþ bþ kcÞcc;

(cðs; 0Þ ¼ ccðs; 0Þ ¼ 0;

ðB:3Þ

where retardation due to R 0 has been encapsulated in thescaled-time �t ¼ t

R0. Again, we apply Laplace transform (de-noted with tilde symbol) over �t (the corresponding complexvariable is q) to (B.3) yielding

~ccðs; qÞ ¼a

P 2ðs; qÞðB:4Þ

(for the sake of brevity, we do not explicitly report theexpression of ~c) in which we have set P2(s,q) = (q + s + k + a)[q + R0(s + kc + b)nY]� R0ab. After observing that

~ccðs; qÞ ¼a

q1 � q2

1

q� q1

� 1

q� q2

� �; ðB:5Þ

where q1,2 are the roots of P2, i.e.

q1;2 ¼1

2�½R0ðsþ bþ kcÞ þ sþ aþ k�f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½R0ðsþ bþ kcÞ � ðsþ aþ kÞ�2 þ 4abR0

q �ðB:6Þ

inversion of q-Laplace transforms gives

ccðs; tÞ ¼a

q1 � q2

expq1tR0� �

1� exp �ðq1 � q2Þt

R0h in o

:

ðB:7ÞInsertion of (B.7) into (16) provides the CðpÞc -functionsneeded to compute Mc(t), hR1c(t)i, and hS11c(t)i.

Spatial moments without colloids:The Lagrangian governing equation of solute transport

by a groundwater in absence of colloids writes as

R0o

otcðs; tÞ þ o

oscðs; tÞ ¼ �kcðs; tÞ; ðB:8Þ

which, solved under the boundary/initial conditionsc(0, t) = d(t) and c(s, 0) = 0, yields

cðs; tÞ ¼ expð�ksÞdðt � R0sÞ: ðB:9Þ


By introducing (B.9) in the first of (13) and (14), and into(15a) gives

eM ðtÞ ¼ M0

R0exp � kt

R0

� �; ðB:10Þ

heR1ðtÞi ¼ eR1ð0Þ þUtR0; ðB:11Þ

heS11ðtÞi ¼ eS 11ð0Þ þ2

bðrY IÞ2 exp �

�btR0

� �þ

�btR0� 1

� �ðB:12Þ

with the various parameters appearing in (B.10)–(B.12)already defined in Section 3.

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