Chemosphere 91 (2013) 248–257

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Chemosphere

journal homepage: www.elsevier .com/locate /chemosphere

Upscaling retardation factor in hierarchical porous media with multimodalreactive mineral facies

Hailin Deng a,⇑, Zhenxue Dai a,1, Andrew V. Wolfsberg a,2, Ming Ye b,3, Philip H. Stauffer a,4, Zhiming Lu a,5,Edward Kwicklis a,6

a Earth and Environmental Sciences Division, Los Alamos National Laboratory, EES-16, T003, Los Alamos, NM 87545, United Statesb Department of Scientific Computing and Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee, FL 3230, United States

h i g h l i g h t s

" A new concept, reactive mineralfacies to describe reactive mineraldistribution.

" Using Lagrangian approach, thescale-dependent analyticalexpressions are derived.

" Describe scaling effect of effectiveretardation factor in time and spacedomains.

0045-6535/$ - see front matter Published by Elsevierhttp://dx.doi.org/10.1016/j.chemosphere.2012.10.105

⇑ Corresponding author. Present address: CSIRO LanE-mail addresses: hailin@lanl.gov (H. Deng), daiz@l

(Z. Lu), kwicklis@lanl.gov (E. Kwicklis).1 Tel.: 505 665 6387.2 Tel.: 505 667 3599.3 Tel.: 850 644 4587.4 Tel.: 505 665 4638.5 Tel.: 505 665 2126.6 Tel.: 505 665 7408.

g r a p h i c a l a b s t r a c t

Concept model for reactive mineral facies at different spatial scales

.

a r t i c l e i n f o

Article history:Received 23 November 2011Received in revised form 29 October 2012Accepted 31 October 2012Available online 20 December 2012

Keywords:UpscalingRetardation factorReactive mineral faciesHierarchical porous media

a b s t r a c t

Aquifer heterogeneity controls spatial and temporal variability of reactive transport parameters and hassignificant impacts on subsurface modeling of flow, transport, and remediation. Upscaling (or homogeni-zation) is a process to replace a heterogeneous domain with a homogeneous one such that both repro-duce the same response. To make reliable and accurate predictions of reactive transport forcontaminant in chemically and physically heterogeneous porous media, subsurface reactive transportmodeling needs upscaled parameters such as effective retardation factor to perform field-scale simula-tions. This paper develops a conceptual model of multimodal reactive mineral facies for upscaling reac-tive transport parameters of hierarchical heterogeneous porous media. Based on the conceptual model,covariance of hydraulic conductivity, sorption coefficient, flow velocity, retardation factor, and cross-covariance between flow velocity and retardation factor are derived from geostatistical characterizationsof a three-dimensional unbounded aquifer system. Subsequently, using a Lagrangian approach the scale-dependent analytical expressions are derived to describe the scaling effect of effective retardation factors

Ltd.

d and Water, Private Bag No. 5, Wembley WA 6913, Australia. Tel.: +61 4 52196874.anl.go (Z. Dai), awolf@lanl.gov (A.V. Wolfsberg), mye@fsu.edu (M. Ye), stauffer@lanl.go (P.H. Stauffer), zhiming@lanl.gov

Nomenclature

a, b constants in linear correlation between K and Kd (i.e.lnKd = a lnK + b)

Cff ðnÞ covariance function of lnKCv1RðnÞ cross-covariance function of v1 and RCv1v1 ðnÞ covariance function of v1

CRRðnÞ covariance function of Rf perturbation of lnKIj(x) indicator variable of the j-th reactive mineral assem-

blageJ mean hydraulic gradient oriented in x1 directionk a three-dimension wave-number vector, k = (k1, k2, k3)T

K hydraulic conductivityKG geometric mean of lnK expressed as exp(MY)Kd Sorption coefficientKG

d geometric mean of lnKd

mj mean of the j-th reactive mineral assemblageMY composite mean of a random variablen porosity of mediumpj volumetric proportion of the j-th reactive mineral

assemblager perturbation of lnKd

R retardation factorR0 perturbation of retardation factorR(x) retardation factor as a function of spatial location~RðxÞ scale dependent effective retardation factor~RðtÞ time dependent effective retardation factorR mean retardation factor

SRR(k) spectrum density of retardation factorSrr (k) spectrum density of the fluctuations of lnKd

Sff(k) spectrum density of the fluctuations of lnKSfr(k) cross spectrum density of lnK and lnKd

Sv1v1 ðkÞ spectrum density of flow velocitySv1RðkÞ cross spectrum density of v1 and RT travel time for a sorbing soluteTp travel time for a conservative soluteu(x) Eulerian solute velocity~uðtÞ time dependent effective solute velocityv 01 perturbation of flow velocityv1 mean flow velocityv(x) Eulerian flow velocityx spatial coordinator vectorY(x) multimodal spatial random variable for K, Kd or RYj(x) random variable of the j-th reactive mineral assemblagen separation distance or lag distancer2

Y composite variance of a random variabler2

j variance of the j-th reactive mineral assemblager2

v1R cross-variance of flow velocity and retardation factorr2

v1variance of flow velocity

r2R variance of retardation factor

qb bulk density of mediumki correlation length of the i-th reactive mineral assem-

blagekI indicator correlation length

H. Deng et al. / Chemosphere 91 (2013) 248–257 249

in temporal and spatial domains. When time and space scales become sufficiently large, the effectiveretardation factors approximate their composite arithmetic mean. Correlation between the hydraulicconductivity and the sorption coefficient can significantly affect the values of the effective retardationfactor in temporal and spatial domains. When the temporal and spatial scales are relatively small, scalingeffect of the effective retardation factors is relatively large. This study provides a practical methodology todevelop effective transport parameters for field-scale modeling at which remediation and risk assess-ment is actually conducted. It does not only bridge the gap between bench-scale measurements tofield-scale modeling, but also provide new insights into the influence of hierarchical mineral distributionon effective retardation factor.

Published by Elsevier Ltd.

1. Introduction

Estimation of effective transport parameters in physically andchemically heterogeneous geological media has received growingamount of attention in the past decade (Rubin, 2003; Davis et al.,2004). Numerous investigations show that the transport parame-ters are spatially and/or temporally scale-dependent in both por-ous and fractured media (e.g. Rajaram, 1997; Liu et al., 2008; Daiet al., 2009). These studies indicate that the scale dependence isoriginated from physical and chemical heterogeneity of sedi-ments. For transport of a reactive solute with linear equilibriumadsorption, spatial variability of the sorption coefficients (Kd)has significant effect on retardation factors (R) because of the lin-ear relationship of R = 1 + (qb/n)Kd, where qb and n are bulk den-sity and porosity of the medium, respectively. Spatialdistributions of Kd for reactive solutes in sediments are heteroge-neous (Allen-King et al., 2006). The dilemma in reactive transportmodeling is that measurements of sorption coefficients are oftenobtained from small-scale column experiments (Cheng et al.,2007) while contaminant remediation and risk assessment areactually conducted at larger field scales. Therefore, to bridge the

scale gaps and to estimate effective transport parameters forfield-scale modeling, we need transfer of knowledge obtained atcolumn scale to a larger field scale through upscaling, which isone of the most important processes for field-scale transportmodeling.

The sorption coefficient and hydraulic conductivity are spa-tially correlated in sediments (Allen-King et al., 2006). Using asedimentary facies-based geostatistical approach, Allen-Kinget al. (1998) described the heterogeneous distributions of per-chloroethene (PCE) Kd and hydraulic conductivity (K) and theirspatial correlation at the Borden site; significantly positive corre-lations between Kd and K were found within and between differ-ent facies. Using a Lagrangian approach, Rajaram (1997) derivedanalytical expressions for scale-dependent effective retardationfactor in unimodal porous media by assuming various spatial cor-relations between Kd and K. Deng et al. (2010) applied dual-porosity model and concept of reactive mineral facies describingmultimodal fractured porous media and upscaled tortuosity andretardation factor for reactive transport. Attinger et al. (1999) re-ported time behavior of solute cloud in chemically heterogeneousporous media with a space-dependent retardation factor using

250 H. Deng et al. / Chemosphere 91 (2013) 248–257

stochastic method within Fourier transformation domain. Morestudies on the scaling of R in unimodal porous media can befound in Chrysikopoulos et al. (1990), Bellin et al. (1993), Cvetko-vic and Dagan (1994) and Fernàndez-Garcia et al. (2005).

Continuous time random walk (CTRW) has been proposed as anew theoretical framework for upscaling retardation factor andinvestigating anomalous transport (non-Fickian transport) in geo-logical porous media (Berkowitz et al., 2006). Using CTRW, Dentzet al. (2004) explored time behavior of effective solute transportin heterogeneous porous media, particularly the transition fromanomalous to normal transport. An interesting and significant re-sult given by Dentz and Castro (2009) based on CTRW formulation,displayed that in porous media with heterogeneously distributedretardation an (constant) effective retardation factor may not al-ways exist at practically relevant time and length scales. Dentzand Boster (2010) identified two different mechanisms causinganomalous transport in quenched random porous media due todifferent heterogeneities between broad disorder point distribu-tion and strong disorder correlations. Bolster and Dentz (2012) fur-ther studied anomalous dispersion in heterogeneous porous mediawith a spatially varying retardation factor and derived effectivetransport equation in a complex Fourier–Laplace space. It is alsoworthy to point out that the definition of retardation factor usedin CTRW framework is different from that in traditional advec-tion–dispersion transport equation.

Mineral facies is the physical and chemical basis for describingthe correlation between hydraulic conductivity and sorption coeffi-cient. Geological formations consist of various minerals and organicmaterials, which can be grouped into different types of facies. Forfluid flow and conservative solute transport, the sedimentary fa-cies-based characterizations can appropriately represent multi-modal conductivity distributions because variability of the sedi-mentary facies accurately reflects the physical controls of conduc-tivity and flow velocity (Ritzi et al. (2004), Dai et al., 2004). Fortransport of a reactive solute in porous media, the sedimentary fa-cies-based conceptual model does not always work well becausethe sedimentary facies does not always capture the chemical heter-ogeneity of the sorption coefficients (Allen-King et al., 1998). In por-ous media the values of sorption coefficients are mainly controlledby reactive minerals (Cheng et al., 2007) including organic carbon(Russo et al., 2010), which are distinguished as any mineral or min-eral facies controlling one or a group of specified sorption reactions(e.g., Allen-King et al., 2006). This renders a multimodal for thesorption coefficients. Therefore, a reactive mineral facies-basedconceptual model is conceived in this study for characterizing thephysical and chemical heterogeneity in porous media.

This study focuses on upscaling the retardation factor for reactivesolute transport in multimodal reactive mineral facies (RMFs) thatconsist of a porous medium. It is different from method used by Rus-so et al. (2010), which is based on nonlinear, rate-limited sorption/desorption with a continuous distribution function. A conceptualmodel of RMFs is first described for the heterogeneous porous mediawith hierarchical structures, followed by geostatistic characteriza-tion of random variables R and flow velocity. Then, the temporallyand spatially scale-dependent equations of effective retardation fac-tors are derived using a Lagrangian approach coupled with transitionprobability models, followed by discussions and conclusions.

2. Conceptual model for hierarchical multimodal porous media

Sorption reactions in porous media depend not only on aque-ous-phase chemical species and physiochemical conditions (e.g.,temperature, oxygen fugacity, pH, and Eh) but also on types ofreactive minerals (RMs) and their spatial distributions. From smallto large scales, the RMs constitute reactive mineral assemblages(RMAs), which in turn form reactive mineral facies (RMFs). A con-

ceptual model of the RMFs in quartz-feldspar sandstone is con-structed for reactive transport in heterogeneous porous mediawith multimodal Kd and K. As described in Fig. 1, the spatial distri-butions of uranium sorption coefficients are characterized at eachhierarchical level (For convenience of discussion, the species ura-nium is used as an example).

RM at the microform scale (10�6 to 10�2 m) refers to any min-erals that are sensitive to a specified geochemical reaction. Asshown by the four probability distributions of lnKd in Fig. 1C, dif-ferent RMs such as calcite, smectite, hematite and organic mattersusually have different sorption coefficients for uranium. Quartzand feldspar are not considered as RMs in the sandstone becausetheir sorption coefficients for the uranium are extremely low.However, these non-reactive minerals (NRMs) occupy a large vol-umetric proportion of the rock; the spatial distributions of NRMscan thus result in heterogeneous patterns of the RMs and in turnaffect the statistical properties (e.g. proportion and correlationlength) of the RMs.

A RMA at the mesoform scale (10�2 to 101 m) represents coex-isting of NRMs and RMs for a specified geochemical reaction.Because of different statistical distributions of the uranium sorp-tion coefficients among different RMs (Fig. 1A), three differentRMAs, i.e. Clay-Quartz-feldspar, Clay-Fe2O3-Quartz-Feldspar, andClay-Organic Mater-Quartz-Feldspar, may have different statisticaldistributions of uranium sorption coefficients (Fig. 1B). In otherwords, if different RMs have significantly different statistical distri-butions of the uranium sorption coefficients, the RMA composed ofthe RMs will have a multimodal structure for the uranium sorptioncoefficients (Fig. 1B and Table 1). Given that there can be one orseveral non-reactive mineral assemblages (NRMAs), the spatial dis-tributions of NRMAs among RMAs can also affect the spatial heter-ogeneity of RMA distributions in the sandstone.

A RMF is a body of rock that is characterized by an association ofRMAs (or RMAs and NRMAs) for a specified geochemical reaction.Due to temporal and spatial variability of sedimentation, weather-ing and oxidation of sandstones, the spatial distributions of RMAsare heterogeneous. Two types of RMFs, i.e. Calcite-Clay-OrganicMatter RFM (CCO-RMF) and Clay-Hematite RFMs (Fig. 1C), aresummarized in this study on intensity of oxidation and degreesto which the RMF influence uranium sorption in the sandstone.For the purpose of demonstration, only the CCO-RMF is used forupscaling from three RMAs in this study, and flow and transportare assumed to occur in CCO-RMF. Upscaling the reactive transportparameters from the RMF scale to a larger scale can be easily ex-tended, but is beyond the scope of this study.

This conceptual model of the hierarchical structure of the RMFsprovides a basis for upscaling the sorption coefficient. After allappropriate RMFs are identified and their hierarchical structure isestablished, multimodal cross-covariance functions are developedfor upscaling the retardation factor using indicator geostatisticsand a transition probability model (e.g., Dai et al., 2007a) to de-scribe the distribution patterns of RMAs.

3. Geostatistical characterization of random variables inmultimodal porous media

Although the conceptual model above contains the three hierar-chical levels of the K and Kd, for the purpose of demonstration, theupscaling equations developed below are only for two levels, i.e.from the RMA to the RMF. These equations can be easily extendedto any number of hierarchical levels as suggested in Dai et al.(2004). Consider a domain X filled with N RMA of mutually exclu-sive occurrences. Let Y(x) be multimodal spatial random variablesfor K, Kd or R at location x. It can be expressed via indicator geosta-tistics as:

-2 1

lnKd

Freq

uenc

yFr

eque

ncy

Freq

uenc

y

lnKd

lnKd

A. Reactive Mineral Facies

at scale of 101 ~103 m

B. Reactive Mineral

Assemblages at

scale of 10- ~101 m

C. Reactive Minerals

at scale of 10-6 ~10-2 mFr

eque

ncy

Fig. 1. Concept model for reactive mineral facies at different spatial scales.

H. Deng et al. / Chemosphere 91 (2013) 248–257 251

YðxÞ ¼XN

j¼1

IjðxÞYjðxÞ; ð1Þ

where Ij(x) is indicator variable within the domain X and Yj(x) arethe random variables of the j-th RMA. Following Ritzi et al.(2004), the composite mean MY and variance r2

Y of the random var-iable are computed via:

MY ¼XN

j¼1

pjmj; ð2Þ

r2Y ¼

XN

j¼1

pjr2j þ

12

XN

i¼1

XN

j¼1

pipjðmi �mjÞ2; ð3Þ

where pj, mj and r2j are volumetric proportion, mean and variance,

respectively. Consider a steady groundwater flow in a three dimen-sional unbounded porous medium with a mean hydraulic gradient(J) oriented in x1 direction. Similar to the method used by Rajaram(1997), we assume that the hydraulic conductivity K is also second-order stationary and log normal distributed. For reactive solutetransport in such a porous medium, the solute velocity u and retar-dation factor R are assumed to be second-order stationary, and thatthe sorption coefficient Kd follows a log normal distribution. Follow-ing Gelhar (1993) and Rajaram (1997), perturbations of R and v1 areobtained as:

R0 ¼ qb

nKG

d r; ð4Þ

v 01 ¼KGJn

1� k21

k2

!f ; ð5Þ

where the prime symbol indicates pertubations, superscript G de-notes geometric mean expressed as exp(MY), r and f are pertuba-

tions of lnKd and lnK, respectively. Therefore the spectrumdensities of the retardation factor and flow velocity are expressedas below:

SRRðkÞ ¼qb

nKG

d

� �2SrrðkÞ; ð6Þ

Sv1v1 ðkÞ ¼1n2 ðK

GJÞ2 1� k21

k2

!2

Sff ðkÞ; ð7Þ

where Sff and Srr are the spectral density of the fluctuations of lnKand lnKd, respectively, and k = (k1, k2, k3)T is a 3-D wave-numbervector. If K and Kd are perfectly correlated as ln Kd = a ln K + b, wherea and b are real constants, then the cross spectrum of lnK and lnKd

has a linear relationship with the spectrum of lnK, i.e. Sfr(k) = aSff(k)and the cross spectrum density of v1 and R is expressed as (Rajaram,1997):

Sv1RðkÞ ¼qb

n2 KGd KGJa 1� k2

1

k2

!Sff ðkÞ: ð8Þ

Using the relationship between the spectrum and the covari-ance function yields:

Sff ðkÞ ¼1

ð2pÞ3ZZZ 1

�1e�ik�nCff ðnÞdn; ð9Þ

where n is separation distance or lag distance. For the multimodalcovariance function of lnK, we can assume that it follows an expo-nential form (Dai et al., 2007b), i.e.

Table 1Synthetic parameters of three RMA used for model calculation and plotting.

RMF RMA Lj pj Parameter mj r2j MG

jkj kw Rj

Cc-Clay-OM RMF Cc-QF 50 0.6 lnK 1.5 0.6 4.48 10 6.67 2.39lnKd �2.2 0.22 0.11 12 7.5

Clay-Fe2O3-QF 23.5 0.15 lnK 0.5 0.3 1.65 6 4.62 4.76lnKd �1.2 0.12 0.3 8 5.71

Clay-OM-QF 26.7 0.25 lnK 0.05 0.15 1.05 9 6.21 10.26lnKd �0.3 0.1 0.74 7 5.19

Parameters lnK lnKd RStatistics MY r2

Y MGY

MY r2Y MG

YMY r2

Y MGY

Values 0.99 0.86 2.68 �1.58 0.84 0.21 4.93 5.6 3.59

Parameters �v1 r2v1R

kI n qb J

Values 0.21 0.2 20 0.2 2.5 0.01

Note: RMA = reactive mineral assemblage, Cc = calcite, Fe2O3 = iron oxides, QF = quartz and feldspar; For j-th RMA (j = 1, 2, 3), Lj = mean length (m), MGj = geometric mean,

kj = correlation length (m), kw = kjkI/(kj + kI), Rj = retardation factor; MY = global mean, r2Y = global variance, MG

Y = global geometric mean, �v1 = mean flow velocity (m d�1),r2

v1 R = cross covariance of flow velocity and retardation factor, kI = indicator correlation length (m), n = porosity, qb = bulk density of the porous media (g cm�3), J = averagehydraulic gradient, K = hydraulic conductivity (m d�1), Kd = sorption coefficient (cm3 g�1).

252 H. Deng et al. / Chemosphere 91 (2013) 248–257

Cff ðnÞ ¼XN

i¼1

p2i r

2i e�g1 þ

XN

i¼1

pið1� piÞr2i e�g2

þ 12

XN

i¼1

XN

j¼1

pipjðmi �mjÞ2e�g3 ; ð10Þ

where g1 ¼ ki=n, g2 ¼ kw=n; and g3 ¼ kI=n are the dimensionlessdistances. ki and kI are the correlation length of i-th RMA and theindicator correlation length for the whole domain, respectively;kw = kikI/(ki + kI), and N is the number of RMA in the multimodalporous medium. Substituting (10) into (9) and integrating (9), thespectrum corresponding to the multimodal covariance function isobtained as:

Sff ðkÞ ¼XN

i¼1

p2i

r2i k

3i

p2ð1þ k2k2i ÞþXN

i¼1

pið1� piÞr2

i k3w

p2ð1þ k2k2wÞ

(

þ12

XN

i¼1

XN

j¼1

pipjðmi �mjÞ2k3

I

p2ð1þ k2k2I Þ

): ð11Þ

Integration method used for Eq. (9) can be found in Appendix A.Then substituting (11) into (8), using the relationship between thespectrum and the covariance function and integrating (8) overwave number space, the cross-covariance of v1 and R for the mul-timodal isotropic porous medium is:

Cv1RðnÞ ¼Z Z 1

�1

Zeik�nSv1RðkÞdk

¼ qb

n2 KGd KGJa

XN

i¼1

p2i r

2i F1ðg1Þ þ

XN

i¼1

pið1� piÞr2i F1ðg2Þ

(

þ12

XN

i¼1

XN

j¼1

ðmi �mjÞ2pipjF1ðg3Þ); ð12Þ

where F1(g) = 4g3(1 � e�g) � 2ge�g � 4g2e�g. If the separation dis-tance approaches to zero, the cross-covariance of the v1 and R is:

r2v1R ¼

2qb

3n2 KGd KGJa

XN

i¼1

pir2i þ

12

XN

i¼1

XN

j¼1

pipjðmi �mjÞ2( )

; ð13Þ

Integration method used to attain Eqs. (12), (13) can be found inAppendix B. Similarly, using Eq. (7) and the relationship betweenthe spectrum and the covariance function, and then integratingproduces the covariance of v1, i.e.,

Cv1v1 ðnÞ ¼ðKGJÞ2

n2

XN

i¼1

p2i r

2i F2ðg1Þ

(þXN

i¼1

pið1

� piÞr2i F2ðg2Þþ

12

XN

i¼1

XN

j¼1

ðmi �mjÞ2pipjF2ðg3Þ); ð14Þ

where F2(g) = [8g2 + 48g3 + 96g4]e�g + [96g5 � 8g3](e�g � 1) .Thusthe variance of v1can be retrieved as the dimensionless distancesg ?1:

r2v1¼ 8

15ðKGJÞ2

n2

XN

i¼1

pir2i þ

12

XN

i¼1

XN

j¼1

pipjðmi �mjÞ2( )

; ð15Þ

In the same way, the multimodal covariance of R is obtainedaccording to Eq. (6), i.e.

CRRðnÞ ¼qb

nKG

d

� �2 XN

i¼1

p2i r

2rie�g1 þ

XN

i¼1

pið1� piÞr2rie�g2

(

þ 12

XN

i¼1

XN

j¼1

pipjðmri �mrjÞ2e�g3

); ð16Þ

where r2ri and mri are the variance and mean of lnKd of the i-th RMA,

respectively, kri is correlation length of lnKd for the i-th RMA. There-fore, the corresponding variance of retardation factors is

r2R ¼

qb

nKG

d

� �2 XN

i¼1

pir2ri þ

12

XN

i¼1

XN

j¼1

pipjðmri �mrjÞ2( )

; ð17Þ

The covariance and cross-covariance derived here will be ap-plied in the following development of time- and scale-dependentretardation factors for the multimodal porous media.

4. Time dependent effective retardation factor

For reactive solute transport with linear equilibrium sorption ina 3-D unbound saturated porous medium, the Eulerian solute veloc-ity is defined as u(x) = v(x)/R(x) with a spatially variable retarda-tion factor R(x). Using the assumptions made in Section 3 that u,v and R are all second order stationary, the effective retardation fac-tor in non-dimensional form is thus expressed as (Rajaram, 1997)

~RðtÞ ¼ �v1=~uðtÞ; ð18Þ

where upper bar indicates mean, and upper tilde denotes effectivevalue. Then Lagrangian mean velocity of a reactive solute is de-scribed as uiðXtÞ ¼ dXti=dt, where Xti means the component of posi-

H. Deng et al. / Chemosphere 91 (2013) 248–257 253

tion vector in the direction i-th (i = 1, 2, or 3). In the Lagrangianframework, Rajaram (1997) developed an equation for the timedependent effective retardation factor, ~RðtÞ, as:

R~RðtÞ¼ 1þr2

R

R2�

r2v1 R

�v1R

!�

~RðtÞR

1R2

r2R�CRRðn;0;0Þ

� �� 1

�v1R½r2

v1R�Cv1Rðn;0;0Þ�� �

;ð19Þ

where separation distance n ¼ �v1t=~RðtÞ. Eq. (19) shows that thebehavior of ~RðtÞ is determined by the covariance (CRR) and cross-covariance (Cv1R). Substituting the multimodal covariance (16) andcross-covariance (12) into (19), ~RðtÞ can be evaluated. For the multi-modal porous media, when the solute travel time (t) approaches toinfinity, R=~Rð1Þ ¼ 1, indicating that ~RðtÞ approaches the arithmeticaverage of R, which is similar to the result of Rajaram (1997) for uni-modal porous media. As t ? 0, the following equation holds:

R~RðtÞ¼ 1þ r2

R

R2� rv1R

�v1R

� : ð20Þ

5. Space dependent effective retardation factor

In the 3-D random velocity field, the uniform mean flow direc-tion is parallel to the x1-direction. When a solute particle startedtraveling at the origin in the 3-D field, the ensemble average ofthe travel time through the x2–x3 plane at a distance x from the ori-gin is expressed by T(x). According to Dagan et al. (1992) and Rajaram(1997), the differential equation of T(x) for a sorbing particle is:

dTdx¼ R

�v11þ 1

�v21

Cv1v1 ðx;0;0Þ �1

R�v1Cv1Rðx;0;0Þ

�ð21Þ

For a conservative solute particle, R is equal to 1. After integra-tion, Eq. (21) becomes:

Tp ¼x�v1þ ðK

GJÞ2�v3

1n2

XN

i¼1

p2i r

2i kiG1ðb1Þ

(þXN

i¼1

pið1� piÞr2i kwiG1ðb2Þ

þ 12

XN

i¼1

XN

j¼1

ðmi �mjÞ2pipjkIG1ðb3Þ)

ð22Þ

where G1ðbÞ ¼ 24b4ð1� e�bÞ � 24b3e�b � 4b2ð1þ 2e�bÞ þ 1; b1 ¼ki=x, b2 ¼ kw=x; and b3 ¼ kI=x are the dimensionless distances. Simi-larly, for a sorbing particle, substituting (12) and (14) into Eq. (21)and integrating over x obtain:

T ¼ RTp �qbKG

d KGJan2 �v2

1

XN

i¼1

p2i r

2i kG2ðb1Þ

(þXN

i¼1

pið1� piÞr2i kwiG2ðb2Þ

þ 12

XN

i¼1

XN

j¼1

ðmi �mjÞ2pipjkIG2ðb3Þ)

ð23Þ

where G2(b) = 2b2(e�b � 1) + 2be�b + 1. Integration method used forEqs. (22), (23) can be found in Appendix C. Finally, the scale depen-dent effective R for isotropic multimodal porous media is computedby:

~RðxÞ ¼ T=Tp; ð24Þ

6. Discussion and conclusions

In order to illustrate the scale dependence of retardation factorswith the above derived equations, we build a synthetic example inwhich the parameter values of the three RMAs are extracted from areal case (see Table 1). The time-dependent effective R for the RMFis plotted in Fig. 2A for three cases of correlations between K andKd: positively correlated (a = 1), uncorrelated (a = 0), and nega-tively correlated (a = �1). In all three cases, the effective Rincreases monotonically with time, but starting with different val-

ues and at different growth rates. Fig. 2A shows that the effective Rvalues of the two correlated are larger than that of uncorrelatedcase, in particular in the early time due to contribution from thecross-covariance between v1 and R (Eq. (12)). As expected, theeffective R in the three cases converges to the arithmetic mean ofR when time is sufficiently large.

Spatial variation of the effective R is plotted in Fig. 2B for thethree cases with the same parameters given in Table 1. In theuncorrelated case the effective R keeps constant, whereas in caseof the positive correlation the effective R rapidly increases withinthe distance of 100 m and then gradually increases with distance.In the case of negative correlation, the effective R rapidly decreasesat the first 100 m, and then gradually decreases with distance. Forboth positive and negative correlation cases, however, the effectiveR approaches to the arithmetic mean when the distance is suffi-ciently large.

Fig. 2C shows the time-dependent effective R changes with theindicator correlation length (kI) when the time is fixed at 1000 d. Inthe case of positive correlation, the effective R increases to a max-imum at about 15 m, and then gradually decreases until kI reaches1000 m, and finally decreases slowly with kI. In the case of negativecorrelation, the effective R reaches its maximum at kI = 200 m, andthen gradually decreases until it approximates to a minimum atkI > 10000 m. Under the situation of non-correlation, the effectiveR keeps constant until kI = 100 m, then gradually decreases withkI until reaching its minimum at about 50000 m. Although theeffective R decreases to a minimum for three cases, it reaches todifferent minimums. This reflects the different contribution fromthe cross-covariance function. Fig. 2D reveals the spatial scale-dependent effective R increases with kI in the case of negative cor-relation, but decreases with kI in the case of positive correlation. Ifthere is no correlation between K and Kd, the effective R does notchanges with kI.

Fig. 3 shows that the scale-dependent effective retardation fac-tors do not change much with variances of the hydraulic conduc-tivity. However, they do change greatly with variances of thesorption coefficients, especially when the variances are larger than0.1 for RMA 1 and great than 0.3 for RMA 2 (Fig. 4).

Our results above focus on methodology for upscaling retarda-tion factor, which is different from the study of Dentz and Castro(2009). They derived an analytical expression for effective trans-port equation based on CTRW formulation, which describes thebehavior of the ensemble averaged concentration of a sorping sol-ute. Dentz and Boster (2010) employed a three-step method toaverage two types of anomalous transport. This two studies didnot formulate effective retardation factor while they developedeffective transport equations. Direct comparison of our results withtheirs is beyond the scope of this paper. In future, using our equa-tions of effective retardation factor to model reactive transport canbe directly compared with the results of Dentz and Castro (2009)and Dentz and Boster (2010).

When collating our study with those by Dentz and Castro(2009) and Bolster and Dentz (2012), we noticed that our defini-tion of retardation factor is different from theirs in CTRW. Ours isderived from traditional advection–dispersion equation (ADE). Itis defined as R = 1 + (qb/n)Kd. Dentz and Castro (2009) defined theretardation factor (they called retardation coefficient) asR ¼ ½nþ ð1� nÞKd�=�n while Bolster and Dentz (2012) gave anotherdefinition for retardation factor, R = 1 + Kd. It is not clear that howdifferent modeling results can be caused by different definitions ofretardation factor.

In general, this study shows from a new theoretical angle thatthe correlation between K and Kd can significantly affect the effec-tive R in temporal and spatial domain. The temporal and spatialscale-dependent R originated from the physical and chemical het-erogeneity and the correlation of K and Kd. When the temporal and

A

4.0

4.2

4.4

4.6

4.8

5.0

5.2

0 500 1000 1500 2000 2500 3000

Time (day)

Eff

ecti

ve R

Positive correlationUncorrelatedNegative correlation

B

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

0 100 200 300 400 500

Distance (m)

Eff

ecti

ve R

C

4.0

4.2

4.4

4.6

5.0

5.2

1 10 100 1000 10000 100000

λ I (m)

Eff

ecti

ve R

D

4.2

4.4

4.6

4.8

5.0

5.2

5.4

5.6

1 10 100 1000 10000 100000λ I (m)

Eff

ecti

ve R

4.8

Fig. 2. Time-dependent (A) and scale-dependent (B) effective retardation factor for a porous medium with three RMFs. The time dependent effective retardation factors (C)change with indicator correlation length when time is fixed at 1000 d. The scale-dependent effective retardation factors (D) change with indicator correlation length when lagdistance is fixed at 1000 m.

4.85

4.90

4.95

5.00

0.01 0.1 1 10

σ12 or σ 2

2

Effe

ctiv

e R

correlation coefficient a=1 for RMA2correlation coefficient a=-1 for RMA2correlation coefficient a=1 for RMA1correlation coefficient a=-1 for RMA1

Fig. 3. Scale-dependent effective retardation factor changes with the variance ofthe hydraulic conductivity for RMA 1 and 2 when the lag distance is fixed at 1000 mand indicator correlation length fixed at 20 m. a = 1 and �1 denote a positive andnegative correlation, between the hydraulic conductivity and the sorption coeffi-cient, respectively.

4.5

5.0

5.5

6.0

0.001 0.01 0.1 1 10σ 1

2 or σ 22

Effe

ctiv

e R

a=1 for RMA2a=-1 for RMA2a=1 for RMA1a=-1 for RMA1

Fig. 4. Scale-dependent effective retardation factor changes with the variances ofthe sorption coefficients for RMA 1 and 2 when the lag distance is fixed at 1000 mand indicator correlation length fixed at 20 m. a = 1 and �1 denote a positive andnegative correlation, between the hydraulic conductivity and the sorption coeffi-cient, respectively.

254 H. Deng et al. / Chemosphere 91 (2013) 248–257

spatial scales are relatively small, scaling effect of the effectiveretardation factors is relatively large. The chemical heterogeneityhas greater influence on scale-dependent effective retardation fac-

tor than the physical heterogeneity does. The results from thisstudy provide a methodology to develop effective transport param-eters for field-scale modeling at which remediation and risk assess-ment is actually conducted.

H. Deng et al. / Chemosphere 91 (2013) 248–257 255

The conceptual model and its corresponding mathematicalmodel derived in this study can be tested with field data or byMonte Carlo simulations, as Deng et al. (2010) did for fracturedporous media. On the other hand, because the physical and chem-ical heterogeneities in aquifer have significant influences on uncer-tainty quantification of flow and transport (for instance, Ye andKhaleel, 2008; Deng et al., 2009), the conceptual model from thisstudy provides a new one for uncertainty assessment and modelselection that requires multiple different conceptual models (Yeet al., 2008) for heterogeneous porous media.

The methodology developed in this study can be applied to non-stationary porous media with hierarchical mineral distribution.Upscaling from RMA to RMF, even to assemblage of RMF is equiv-alent to classic stochastic upscaling approach from the local to theregional scale. Our upscaling procedure bridges lab bench scale tofield scale through quantitatively linking multimodal reactive min-eral distribution with scale-dependence of parameters. Therefore,our methodology provides new insights into the origin of the effec-tive retardation factor and has significant practical implications inreactive transport modeling at field scale.

Acknowledgements

The reported research was supported by Los Alamos NationalLaboratory’s Directed Research and Development Project (Number20070441ER). The fourth author is supported by the DOE projectDE-SC0002687 and ORAU/ORNL High Performance ComputingGrant. We are grateful to Harihar Rajaram for his constructive com-ments on this manuscript.

Appendix A. Derivation of spectrum density of fluctuations inln K

Considering a simple exponential function for autocovariance oflnK, that is:

Cff ðnÞ ¼ r2f e�

nk ðA1Þ

The spectrum density Sff of fluctuations in lnK is evaluated bytaking the Fourier transform of Eq. (9). Substituting A1 into Eq.(9), then it yields:

Sff ðkÞ ¼1

ð2pÞ3ZZZ 1

�1e�ik�nr2

f e�nkdn ðA2Þ

Integration of A2 is implemented in a spherical coordinate sys-tem. The scalar product k � n = kn cos h, where h is the angle be-tween wave number vector k and the separation vector n.Expressing the element volume dn in spherical coordinates (n, h,a), and integrating A2 results in:

Sff ðkÞ ¼1

ð2pÞ3Z 1

n¼0

Z p

h¼0

Z 2p

a¼0r2

f e�nke�ikn cos hn2 sin hdadhdn

¼r2

f k3

p2ð1þ k2k2ÞðA3Þ

A3 is the spectrum density for unimodal porous media. For themultimodal porous media, the multimodal covariance function oflnK (Eq. (10)) is easily integrated to obtain corresponding spectrumdensity (Eq. (11)) by using A3 three times for three exponentialterms.

Appendix B. Derivation of cross-covariance of v1 and R (Eq.(12))

Continue to consider the unimodal porous media from above; itscross-covariance can be expressed as below by substituting A3 intoEq. (8), then taking Fourier transform of (8) and integrating result in

Cv1RðnÞ ¼qb

n2 KGd KGJa

Z Z 1

�1

Zeik�n 1� k2

1

k2

!r2

f k3

p2ð1þ k2k2Þ2dk ðB1Þ

Set I is equal to the integration term of B1, then

I ¼ZZZ 1

�1eik�n 1� k2

1

k2

!r2

f k3

p2ð1þ k2k2Þ2dk ðB2Þ

Now changing to spherical coordinate system and defining:

k1 ¼ k cos b ðB3Þ

k1=k ¼ cos h cos xþ sin h sin x cos a ðB4Þ

dk1dk2dk3 ¼ k2 sin hdkdadh ðB5Þ

where x is the angle between the separation vector n and the direc-tion of mean flow k1, and h is the angle between n and k. Note thatthe x and n stand for coordinates of the autocovariance function,whereas k, h, and a are spherical coordinates in wave number space.Substitute B3, B4 and B5 into B2, it yields:

I¼r2

f

kp2

Z 1

k¼0

Z 2p

a¼0

Z p

h¼01�cos2 hcos2 x�sin2 hsin2 xcos2 ahn

�2coshcosxsinhsinxcosa� k2k4

ð1þk2k2Þ2eikncosh sinhdkdhda

)ðB6Þ

Let cos h = y and use the relation of eikny ¼ cos knyþ i sin kny, B6becomes B7 after several steps of operation:

I¼r2

f

kp2

Z 1

k¼0

Z 2p

a¼0

�Z 1

y¼�1ð1�y2 cos2 xÞ�ð1�y2Þsin2 xcos2 ah i k2k4

ð1þk2k2Þ2coskny

( )dkdyda

ðB7Þ

in which we have several integrations below:

Ia ¼Z 1

0

k2k4

ð1þ k2k2Þ2cos knydk ¼ p

4k 1� jnyj

k

� e�jnyjk ðB8Þ

Ib ¼Z 2p

0ð1� y2Þ sin2 x cos2 ada ¼ pð1� y2Þ sin2 x ðB9Þ

Ic ¼Z 2p

0ð1� y2 cos2 xÞda ¼ 2pð1� y2 cos2 xÞ

¼ pð2� 2y2 cos2 xÞ ðB10Þ

Of them, B8 can be found from Dwight (1961, p225: definiteintegral 859.013). Substituting B8, B9 and B10 into B7 yields:

I ¼r2

f

4

�Z 1

y¼�1½2� 2y2 cos2 x� ð1� y2Þ sin2 x� 1� jnyj

k

� e�jnyjk

� �dy

ðB11Þ

Expanding B11 results in

256 H. Deng et al. / Chemosphere 91 (2013) 248–257

I ¼r2

f

2

Z 1

0ð1þ cos2 xÞe�

nyk dyþ

Z 1

0ð1� 3 cos2 xÞy2e�

nyk dy

�

�Z 1

0

nkð1þ cos2 xÞye�

nyk dy�

Z 1

0

nkð1� 3 cos2 xÞy3e�

nyk dy�

¼r2

f

2ðI1 þ I2 � I3 � I4Þ ðB12Þ

In B12 the four integrations I1, I2, I3 and I4 are given below

I1 ¼Z 1

0ð1þ cos2 xÞe�

nyk dy ¼ � k

nð1þ cos2 xÞ e�

nk � 1

� �ðB13Þ

I2 ¼Z 1

0ð1� 3 cos2 xÞy2e�

nyk dy

¼ ð3 cos2 x� 1Þ kn

� e�

nk þ 2

kn

� 2

e�nk þ 2

kn

� 3

e�nk � 1

� �" #ðB14Þ

I3 ¼Z 1

0

nkð1þ cos2 xÞye�

nyk dy

¼ �ð1þ cos2 xÞ e�nk þ k

ne�

nk � 1

� � �ðB15Þ

I4 ¼Z 1

0

nk

1� 3 cos2 x�

y3e�nyk dy

¼ 3 cos2 x� 1�

e�nk þ 3

kn

� e�

nk þ 6

kn

� 2

e�nk þ 6

kn

� 3

ðe�nk � 1Þ

" #

ðB16Þ

Substituting B13, B14, B15, and B16 into B12, which in turn goesinto B1, finally turns out:

Cv1RðnÞ¼qb

2n2 KGd KGJar2

f ð1þcos2 xÞe�nkþð1�3cos2 xÞ

n

� e�nkþ2

kn

� e�

nkþ4

kn

� 2

e�nkþ4

kn

� 3

ðe�nk�1Þ

" #)ðB17Þ

When the separation vector is parallel to the flow direction,x = 0, cosx = 1, then:

Cv1RðnÞ ¼qb

n2 KGd KGJar2

f 4kn

� 3

ð1� e�nkÞ � 4

kn

� 2

e�nk � 2

kn

� e�

nk

" #

ðB18Þ

Let n ? 0, the cross-variance of v1 and R for unimodal porousmedia can be given below:

r2v1R ¼ Cv1Rð0Þ ¼

2qb

3n2 KGd KGJar2

f ðB19Þ

Using the same integration method for multimodal porousmedia, the corresponding multimodal cross-covariance function(Eq. (12)) can be obtained by using B18 three times for three expo-nential terms.

In a similar way the covariance function of flow velocity (v1) forunimodal porous media can be retrieved as following

Cv1v1 ðnÞ ¼r2

fðKGJÞ2

n2 8 kn

� �2þ 48 k

n

� �3þ 96 k

n

� �4 �

e�nk

�

þ 96ðkn Þ5 � 8ðkn Þ

3h i

ðe�nk � 1Þ

o ðB20Þ

The variance of flow velocity (v1) for unimodal porous media isobtained when separation distance n ? 0, that is:

r2v1¼ Cv1v1 ð0Þ ¼

815

r2f ðK

GJÞ2

n2 ðB21Þ

For multimodal porous media the covariance function of flowvelocity (Eq. (14)) can also be gained in a similar integration proce-dure used above.

Appendix C. Derivation of equation for space dependenteffective retardation factor

For a passive or conservative particle, retardation factor R isequal to 1. Eq. (21) is then reduced into one below:

dTp

dx¼ 1

�v11þ 1

�v21

Cv1v1 ðx;0;0Þ �

ðC1Þ

Substituting B20 into C1 and integrating C1, it brings out thetravel time of the passive particle in unimodal porous media as:

Tp ¼Z x

0

1�v1

1þ 1�v2

1

Cv1v1 ðx;0;0Þ �

dx ¼ x�v1þðKGJÞ2kr2

f

�v31n2

24kx

� 4(

� 1� e�xk

� � 24

kx

� 3

e�xk � 4

kx

� 2

1þ 2e�xk

� þ 1

)ðC2Þ

For a sobbing particle, substituting B18 and B20 into Eq. (21),and then integrating yields:

T ¼Z x

0

R�v1

1þ 1�v2

1

Cv1v1 ðx;0;0Þ �1

R�v1CRv1 ðx;0;0Þ

�dx

¼ RTp �qbKG

d KGJakr2f

n2 �v21

2kx

� 2

e�xk � 1

� þ 2

kx

� e�

xk þ 1

( )ðC3Þ

For multimodal porous media, similar integrations can be com-pleted as shown in Eqs. (22) and (23).

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