Reactive solute transport in physically and chemically heterogeneous porous media with multimodal reactive mineral facies: The Lagrangian approach

Chemosphere 122 (2015) 235–244

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Reactive solute transport in physically and chemically heterogeneousporous media with multimodal reactive mineral facies: The Lagrangianapproach

http://dx.doi.org/10.1016/j.chemosphere.2014.11.0640045-6535/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 (937) 775 2201.E-mail addresses: [email protected] (M.R. Soltanian), robert.ritzi@

wright.edu (R.W. Ritzi), [email protected] (Z. Dai), [email protected](C.C. Huang).

1 Tel.: +1 (937) 775 2201.2 Tel.: +1 (505) 665 6387.3 Tel.: +1 (937) 775 2491.

Mohamad Reza Soltanian a,⇑, Robert W. Ritzi a,1, Zhenxue Dai b,2, Chao Cheng Huang c,3

a Department of Earth and Environmental Sciences, Wright State University, Dayton, OH 45435, United Statesb EES-16, Earth and Environmental Sciences Division, Los Alamos National Laboratory, Mailstop T003, Los Alamos, NM 87545, United Statesc Department of Mathematics and Statistic, Wright State University, 3640 Colonel Glenn Hwy, Dayton, OH 45435-0001, United States

h i g h l i g h t s

� Hierarchical reactive mineral facies is used to describe the distribution of reactive minerals.� Using Lagrangian theory, the analytical expression for reactive solute dispersion is derived.� Sensitivity analysis is performed to understand the effects of hydraulic and reactive attributes.

a r t i c l e i n f o

Article history:Received 4 September 2014Received in revised form 14 November 2014Accepted 26 November 2014Available online 19 December 2014

Handling Editor: I. Cousins

Keywords:Reactive transportReactive mineral faciesHierarchical porous mediaLagrangian-based theory

a b s t r a c t

Physical and chemical heterogeneities have a large impact on reactive transport in porous media. Exam-ples of heterogeneous attributes affecting reactive mass transport are the hydraulic conductivity (K), andthe equilibrium sorption distribution coefficient (Kd). This paper uses the Deng et al. (2013) conceptualmodel for multimodal reactive mineral facies and a Lagrangian-based stochastic theory in order to ana-lyze the reactive solute dispersion in three-dimensional anisotropic heterogeneous porous media withhierarchical organization of reactive minerals. An example based on real field data is used to illustratethe time evolution trends of reactive solute dispersion. The results show that the correlation betweenthe hydraulic conductivity and the equilibrium sorption distribution coefficient does have a significanteffect on reactive solute dispersion. The anisotropy ratio does not have a significant effect on reactive sol-ute dispersion. Furthermore, through a sensitivity analysis we investigate the impact of changing themean, variance, and integral scale of K and Kd on reactive solute dispersion.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Reactive transport in porous formations is controlled by heter-ogeneity in physical and chemical properties (Dagan, 1989; Bellinet al., 1993; Miralles-Wilhelm and Gelhar, 1996; Brusseau andSrivastava, 1997; Rajaram, 1997; Dai et al., 2009; Deng et al.,2010; Soltanian et al., 2014a). Examples of these properties arehydraulic conductivity (K), and the equilibrium sorption distribu-tion coefficient (Kd). It has been shown that these parameters are

scale-dependent (Allen-King et al., 1998, 2006; Davis et al., 2004;Ritzi et al., 2004, 2013; Dai et al., 2007; Ramanathan et al., 2010;Zhang et al., 2013). The spatial variations of physical and chemicalheterogeneity are known to be responsible for the scale-dependence of transport parameters such as the retardation factorand the macrodispersivity (Dai et al., 2009; Deng et al., 2013).

Different methods have been proposed for dealing with scale-dependent transport parameters. For example, it is common touse the upscaling process in order to incorporate the effect ofsmall-scale variability on solute transport (Rubin, 2003). Variousschemes have been suggested in the literature to upscale reactivetransport parameters, as reviewed by Dentz et al. (2011). Theseinclude volume averaging (e.g., Whitaker, 1999), stochastic averag-ing (e.g., Gelhar and Axness, 1983; Dagan, 1984), homogenization(e.g., Lunati et al., 2002), and renormalization (e.g., Zhang, 1998).For example, the time evolution of a conservative solute dispersion

236 M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244

has been investigated in detail for unimodal porous media (e.g.,Dagan, 1989; Rajaram and Gelhar, 1993; Rubin et al., 1999; Fioriand Dagan, 2002). The time- and scale-dependent effective retar-dation factor in a unimodal porous media was presented byRajaram (1997) and in a hierarchical porous media by Deng et al.(2013) using a Lagrangian-based theory. The time evolution ofreactive solute dispersion undergoing equilibrium sorption hasbeen investigated by Bellin et al. (1993) and Bellin and Rinaldo(1995) for unimodal porous media. Samper and Yang (2006) ana-lyzed the multicomponent cation exchange reactions in heteroge-neous porous media (see also Yang and Samper, 2009). In thispaper we use the Deng et al. (2013) conceptual model for porousmedia with hierarchical organization of reactive minerals (seeFig. 1) and develop a Lagrangian-based theory to analyze the reac-tive solute dispersion undergoing equilibrium sorption.

Aquifer architecture is often conceptualized as a hierarchy withfacies types defined at each scale comprising assemblages of faciestypes defined at a smaller scale, across any number of hierarchicallevels (e.g. Bridge, 2006). Using the information about facies typesdefined at different scales can considerably simplify the task ofcharacterizing subsurface heterogeneity. It has been shown thatfacies types at different scale control the scale-dependence of Kand Kd (Dai et al., 2004, 2005; Allen-King et al., 2006; Ritzi andAllen-King, 2007; Zhou et al., 2014; Gershenzon et al., 2014a,b).Importantly, Kd is known to vary with sedimentary facies types(Allen-King et al., 1998, 2006; Ritzi et al., 2013; Soltanian andRitzi, 2014) and reactive mineral facies (Cheng et al., 2007; Denget al., 2013). Information from sedimentary facies types and/orreactive mineral facies at different scales could be used in develop-ing models for understanding reactive transport processes.

Facies classifications are not unique (Dai et al., 2005; Soltanianand Ritzi, 2014). What is important is that the classification shouldbe useful, and usefulness depends upon context. Reactive mineralscan be used to define facies for characterizing heterogeneity both

Fig. 1. Conceptual model for reactive mineral facies at different spatial scales, and correDeng et al. (2013).

in K and in Kd. The Deng et al. (2013) conceptual model is usefulfor geologic architecture within bedrock in which the type of reac-tive minerals and their spatial distributions exert the strongestcontrol on the attributes of interest. Using this classification mightnot be appropriate for deposits where K does not co-vary with min-eralogic facies controlling reactivity. We accept that the Deng et al.(2013) classification is useful for geologic architecture within cer-tain settings and our goal is to derive a Lagrangian-based theoryusing their classification for hierarchical organization of reactiveminerals. To our knowledge, there is no theory available for analyz-ing reactive solute dispersion in hierarchical porous media withhierarchical organization of reactive minerals. Note that the theorydeveloped in this paper can be extended to any type of system clas-sification. In developing the theory we follow Rajaram (1997)which is different in part from the Lagrangian-based model pre-sented by Bellin et al. (1993). Here the theory is developed in a for-mation with multiple K and Kd modes and hierarchicalorganization across scales. Note that while Bellin et al. (1993) rep-resented three-dimensional isotropic formations with one scale ofspatial variability, in this study we derived a model for three-dimensional anisotropic formations.

In Section 2 we briefly review the Deng et al. (2013) conceptualmodel for hierarchical multimodal porous media with reactiveminerals. Section 3 presents the derivation of a Lagrangian-basedtheory for reactive solute dispersion. In Section 4 an example isused to illustrate the utility of the developed theory.

2. Conceptual model for hierarchical multimodal porous mediaand geostatistical characterization

Aside from aqueous-phase chemical species and physiochemi-cal conditions such as temperature and pH, the sorption reactionsin porous media depends on types of reactive minerals and theirspatial distributions (Deng et al., 2010, 2013). Mineral reactivity

sponding modes for the log sorption distribution coefficient, ln (Kd). Modified from

M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244 237

is defined in terms of sorption/desorption process. Deng et al.(2013) presented a conceptual model of reactive minerals inquartz-feldspar sandstone with multimodal ln(Kd) and ln(K) sub-populations. In their conceptual model mineral facies have a hier-archical organization and there is a corresponding hierarchy ofln(Kd) subpopulations (see Fig. 1). In the hierarchy, the reactiveminerals (RMs) constitute reactive mineral assemblages (RMAs),which, in turn, form reactive mineral facies (RMFs). Here we brieflyreview the hierarchy of reactive minerals.

The base of the hierarchical organization of reactive mineralsis the microform scale (10�6 to 10�2 m). The microform scale isassociated with RMs. This scale is related to the scale of mineralgrains in a rock. RMs are minerals that are sensitive to a specifiedgeochemical reaction. There is ln(Kd) subpopulations in each RM.Examples of RMs are calcite, smectite, and hematite which havedifferent sorption coefficients (see also Zavarin et al., 2004).There are also non-reactive minerals (NRMs) with a large volumeproportions in quartz-feldspar sandstone. These NRMs (e.g.,quartz and feldspar) have low sorption capabilities. The secondhierarchical level is the mesoform scale (10�2 to 101 m). Themesoform scale has RMAs with occurrences of both NRMs andRMs. Examples of RMAs are Clay–Quartz–Feldspar, Clay–Fe2O3–Quartz–Feldspar, and Clay–Organic Mater–Quartz–Feldspar. TheRMA composed of RMs has a multimodal structure for uraniumsorption coefficients. Also, there can be one or several non-reac-tive mineral assemblages (NRMAs). The third hierarchical level isthe macroform scale (101 to 103 m) with reactive mineral facies(RMF). These are a body of rock characterized by an associationof RMAs (or RMAs and NRMAs). Two types of RMFs are Cal-cite–Clay–Organic Matter (CCO–RMF) and Clay–Hematite couldbe found in sandstone. Similar to Deng et al. (2013), for the pur-pose of demonstration, only the CCO–RMF with three RMAs isused in this study, and thus flow and transport are assumed tooccur within a CCO–RMF. The analysis can be easily extendedto also include the RM scale. However, it is not in the scope ofthis study.

Consider a domain X filled with N number of RMA of mutuallyexclusive occurrences. Let Y(x) be multimodal spatial random vari-ables for ln(K) or ln(Kd) at location x. It can be expressed using indi-cator geostatistics as:

YðxÞ ¼XN

j¼1

IjðxÞYjðxÞ ð1Þ

where Ij(x) is indictor variable within the domain X and Yj(x) arevariables of the j-th RMA. Following Ritzi et al. (2004), the compos-ite mean MY and variance r2

Y of Yj(x) are computed as (see alsoHuang and Dai, 2008):

MY ¼XN

j¼1

pjmj ð2Þ

r2Y ¼

XN

j¼1

pjr2j þ

12

XN

i¼1

XN

j¼1

pjpiðmi �mjÞ2 ð3Þ

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

respectively. The multimodal covariance function of ln(K) and ln(Kd)has been presented from previous studies (see Dai et al., 2004;Soltanian et al., 2014a):

CYðnÞ ¼XN

j¼1

p2j r

2j e� n

kj þXN

i¼1

pjð1� pjÞr2j e�

nku

þ 12

XN

i¼1

XN

j¼1

pipjðmi �mjÞ2e�nkI ð4Þ

where kj and kI are the integral scale of the j-th RMA unit and theindicator integral scale of the RMAs, respectively; ku ¼ kjkI=ðkj þ kIÞ.

3. The Lagrangian-based theory

Spatial variability of velocity experienced by reactive solutes isthe first step in characterizing reactive solute spreading (Bellinet al., 1993; Rajaram, 1997). The Lagrangian velocity for reactivesolutes is as follows:

uðxÞ ¼ vðxÞRðxÞ ð5Þ

where u is the reactive solute velocity, v is the groundwater veloc-ity, and R is the retardation factor with a constant mean R, variancer2

R and a stationary spatial covariance CRR(n) (Bellin et al., 1993;Rajaram, 1997). For transport of a reactive solute with the linearequilibrium sorption assumption, spatial variability of R is relatedto the spatial variability of Kd by the relationship R(x) = 1 + (qb/n)Kd(x) where qb and n are the bulk density and porosity of themedium, respectively. The perturbation of the reactive solute veloc-ity, u0i, is found by linearizing Eq. (5) as:

u0i ¼v 0iR�

�v iR0

R2ð6Þ

where R0 is the R perturbation, and v 0i is the perturbation of thegroundwater velocity (Rajaram, 1997). Using Eq. (6), and the spec-tra of the flow velocity and the retardation factor (Sv iv j

and SRR) andtheir cross-spectral density, Sv iR; the spectral density of the reactivesolute velocity is given by:

SuiujðkÞ ¼ 1

R2Sv iv jðkÞ þ

�v i �v j

R4SRRðkÞ �

�v i

R3Sv jRðkÞ �

�v j

R3Sv iRðkÞ ð7Þ

where k = (k1, k2, k3)T is a 3-D wave-number vector (Rajaram, 1997).Thus, the following relationship defines the covariance of reactivesolute velocity:

CuiujðnÞ ¼ 1

R2Cv iv jðnÞ þ

�v i �v j

R4CRRðnÞ �

�v i

R3Cv jRðnÞ �

�v j

R3Cv iRðnÞ ð8Þ

Following Dagan (1984, 1989) and Bellin et al. (1993) and usingthe Lagrangian-based theory the time-dependent dispersion tensorfor a reactive solute, eDR

ijðtÞ, is found as:

eDRijðtÞ ¼

Z t

0CuiujðsÞds ð9Þ

We assume that the velocity covariance depends on the meanparticle trajectory instead of the actual one. Therefore, n can beapproximated by �v1t=R: This assumption has been successfullyused for analyzing the dispersion of nonreactive and reactive sol-utes (Dagan, 1989; Bellin et al., 1993; Bellin and Rinaldo, 1995;Dai et al., 2004).

Bellin et al. (1993) showed that the transverse retarded velocityis independent of the variability of R. Our focus is on analyzing thelongitudinal dispersion of a reactive plume as has been observed infield experiments (Roberts et al., 1986; Garabedian et al., 1988).Therefore, we assume that the velocity field is uniform in average,and that the mean velocity is aligned with the x1 axis such thatv = (v1, 0, 0).

Thus,

eDR11ðtÞ ¼

Z t

0Cu1u1

v1

Rs

� �ds

¼ 1R2

Z t

0Cv1v1

�v1

Rs

� �ds ð10aÞ

238 M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244

þ�v2

1

R4

Z t

0CRR

�v1

Rs

� �ds ð10bÞ

�2�v1

R3

Z t

0Cv1R

v1

Rs

� �ds ð10cÞ

Bellin et al. (1993) used both the cross-covariance of ln(K) andR, and the cross-covariance of the hydraulic head and R by per-forming first-order approximation in order to derive Cv iRðnÞ: Herewe follow Rajaram (1997) in which Cv iRðnÞ is derived directly fromits spectral density, Sv iRðkÞ; using the cross-correlation betweenln(K) and ln(Kd).

The retardation factor can also be expressed as R(x) = 1 + (qb/n)ew(x) where w is ln(Kd). Using stochastic theory w(x) can bereplaced by �wþw0 where �w and w0 are the mean and the perturba-tion of w. Soltanian et al. (2014b) derived the following expressionfor the perturbation of R:

R0 ¼ R� R ¼ qb

nKG

d ew0 � er2

w2

h i !ð11Þ

where KGd is the geometric mean of Kd(x), and r2

w is the variance ofw. Using the Taylor series expansion for ew0 , we calculate CRR

approximately as:

CRRðnÞ ¼qb

nKG

d

� �2e½r

2w �ðeCwðnÞ � 1Þ ð12Þ

Therefore, the corresponding variance of the retardation factoris:

r2R ¼

qb

nKG

d

� �2e½r

2w �ðe½r2

w � � 1Þ ð13Þ

Following Gelhar (1993), Rajaram (1997), and Deng et al. (2013)the perturbation of v is obtained as:

v 0 ¼ KGJn

1� k21

k2

!f ð14Þ

where f is the perturbation of ln(K), J is the average hydraulic gradi-ent, and KG is the geometric mean of K(x).

Note that we used a nonlinear expansion for ln(Kd), Eq. (11), andfirst-order for ln(K), Eq. (14). Bellin et al. (1993) and Bellin andRinaldo (1995) have used the same inconsistent expansion in orderto analyze the time-dependent dispersion of reactive solutes (seeEqs. (10a) and (17) in Bellin et al., 1993). Their results were testedagainst numerical simulations and validated by Bosma et al. (1993)for relatively small variances (<1.6). Here we assume similarlysmall variances. Thus, the developed theory is valid for aquiferswith mild heterogeneity contrast (r2

f ; r2w < 1).

The Sv1RðkÞ is derived by using Eqs. (11) and (14). The Sv1RðkÞ isfound as:

Sv1RðkÞ ¼qb

n2 KGKGd J 1� k2

1

k2

!f ew0 � e½

r2w2 �

� �ð15Þ

Using the Taylor series expansion for ew0 and considering thepoint that the odd moments of a log normal distribution are zero,Sv1RðkÞ is found as:

Sv1RðkÞ ¼qb

n2 KGKGd J 1� k2

1

k2

!sinhðrwÞ

rwSfwðkÞ ð16Þ

where Sfw(k) is the spectral density of the fluctuations of f � w, andsinh is hyperbolic sine function. The K and Kd are assumed to be per-fectly correlated as ln(Kd) = a ln(K) + b, where a and b are real con-stants. Then the cross-spectral density of ln(K) and ln(Kd) has a

linear relationship given by ln(K), i.e. Sfw(k) = aSff(k). Therefore,Sv1RðkÞ is expressed as:

Sv1RðkÞ ¼qb

n2 KGKGd Ja

sinhðrwÞrw

1� k21

k2

!Sff ðkÞ ð17Þ

Using Eq. (14) Sv1v1 ðkÞ is easily found as follows:

Sv1v1 ðkÞ ¼KGJn

!2

1� k21

k2

!2

Sff ðkÞ ð18Þ

The relationship between the spectral density and the covari-ance function is expressed as follows:

SðkÞ ¼ 1

ð2pÞ3ZZZ 1

�1e�ik:nCðnÞdn ð19Þ

The Cv1v1 ðnÞ and Cv1RðnÞ are used in form of their spectral density tofind the longitudinal dispersion of a reactive solute. Therefore, Eq.(10) can be rewritten as follows:

eDR11ðtÞ ¼

Z t

0Cu1u1

v1

Rs

� �ds

¼ 1R2

Z t

0

ZZX

Ze�ik

�v1R

sSv1v1 ðkÞdsdk ð20aÞ

þ�v2

1

R4

Z t

0CRR

�v1

Rs

� �ds ð20bÞ

�2�v1

R3

Z t

0

ZZX

Ze�ik

�v1R

sSv1RðkÞdsdk ð20cÞ

By substituting Eqs. (12), (17) and (18) into Eq. (20) and usingEq. (4) to represent the multimodal covariance function of ln(K)and ln(Kd), the final expression for longitudinal dispersivity of areactive solute in multimodal porous media is found as:

aR11ðtÞ¼

eDR11ðtÞR�v1

¼XN

j¼1

p2j r

2fikiF1ðkiÞþ

XN

j¼1

pjð1�pjÞr2fikuF1ðkuÞ

þ12

XN

i¼1

XN

j¼1

pipjðmfi�mfjÞ2kIF1ðkIÞþ�v1

R3

qb

nKG

d

� �2

�e½r2w �

Z t

0e

XN

j¼1

p2jr2

wje�

�v1Rki

s

þXN

i¼1

pjð1�pj Þr2wj

e�

�v1Rku

s

þ12

XN

i¼1

XN

j¼1

pi pj ðmwi�mwjÞ2 e�

�v1RkI

s

�1

0BBB@1CCCAds

0BBBB@1CCCCA

�2Rqb

n2 KGKGd Ja

sinhðrwÞrw

XN

j¼1

p2j r

2fikiF2ðkiÞþ

XN

j¼1

pjð1�pjÞr2fikuF2ðkuÞ

(

þ12

XN

i¼1

XN

j¼1

pipjðmfi�mfjÞ2kIF2ðkIÞ)

ð21Þ

where

F1ðkÞ ¼ 1� e�s � eZ 1

02rJ1ðbÞ

2u32 � erðv þ 2uÞ

v2u32

(

þ ð2� b2ÞJ1ðbÞ � bJ0ðbÞrs2

" #e3r3ðv þ 4uÞ þ u

32ð5v � 4uÞ

v3u32

" #dr

);

F2ðkÞ ¼ 1� e�s � 2e2Z 1

0r2 1

2vu32þ 1

v2u12� 1

v2er

� �J1ðbÞdr

� ;

b ¼ rs;

u ¼ 1þ r2;

v ¼ 1þ r2 � e2r2;

Table 1Synthetic parameters used for model calculation and plotting. Modified from Deng et al. (2013).

RMF RMA Lj Pj Parameters mj r2j MG

jkj ku Rj

Cc–Clay–OM RMF Cc–QF 50.0 0.6 ln(K) 1.5 0.6 4.48 10 6.67 2.39ln(Kd) �2.2 0.22 0.11 12 7.5

Clay–Fe2O3–QF 23.5 0.15 ln(K) 0.5 0.3 1.65 6 4.62 4.76ln(Kd) �1.2 0.12 0.3 8 5.71

Clay–OM–QF 26.7 0.25 ln(K) 0.05 0.15 1.05 9 6.21 10.26ln(Kd) �0.3 0.1 0.74 7 5.19

Parameters ln(K) ln(Kd) 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 20.25 3.59Parameters �v1 rv1R kI n qb JValues 0.21 0.228 20.0 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), ku ¼ kikI=ðki þ kIÞ, Rj = retardation factor; MY = global mean, r2Y = global variance, MG

Y = global geometric mean, v1 = mean flow velocity (m d�1),rv1R = 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).

M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244 239

s ¼�v1tRk

;

Also, e is the anisotropy ratio defined as the vertical integralscale of the hydraulic conductivity to the horizontal component.The J0 and J1 are the zero and first order Bessel functions, respec-tively. The integration method to find Eq. (21) can be found inAppendix A, B, C, and D. The expressions in Eq. (21) cannot be inte-grated in closed form. We use the quadrature method in order tonumerically integrate and evaluate Eq. (21) at a number of pointsin the parameter space, and present the results in Section 4.

4. Results and discussion

In order to analyze the reactive solute dispersion using thedeveloped Lagrangian-based model, we use an example presentedby Deng et al. (2013) as summarized in Table 1. In this example theparameter values of the three RMAs are extracted from a real case.

The aR11ðtÞ is plotted in Fig. 2 for three cases of correlation

between ln(K) and ln(Kd): positively correlated (a = 1), uncorre-lated (a = 0), and negatively correlated (a = �1). The aR

11ðtÞincreases monotonically with time for all cases. Fig. 2 shows thatthe values of aR

11ðtÞ for the negatively correlated case are largerthan those of the uncorrelated and positively correlated cases. Inall cases the value of aR

11ðtÞ converges to a constant value when

Fig. 2. The longitudinal dispersivity aR11ðtÞ for a porous medium with three RMFs. Three

correlation (a = �1) are shown. The influence of the anisotropy ratio, e; for all of the thr

time is sufficiently large. Fig. 2 also shows the influence of anisot-ropy ratio, e; on aR

11ðtÞ. It is observed that e has a relatively smallimpact upon dispersion of reactive solutes in this example. TheaR

11ðtÞ is slightly enhanced at early times by a smaller e due to lat-eral mass transfer between streamlines with different velocitieslocated adjacent to each other (Rubin, 2003). Furthermore, asshown in Section 4, heterogeneity in ln(Kd) has a relatively largerimpact on reactive solute transport than ln(K) heterogeneity orits anisotropy ratio.

Fig. 3 shows aR11ðtÞ changes with the indicator correlation length

(kI) when the time is fixed at 1000 d. In all cases aR11ðtÞ increases to

a maximum at about kI ¼ 300 m, and then it stays constant.Although aR

11ðtÞ reaches a maximum for three cases, it reaches dif-ferent values. This reflects the contribution from the cross-correla-tion between R and v1, represented by Eq. (20c).

In order to better understand the impact of heterogeneity inboth ln(K) and ln(Kd) on reactive solute dispersion we study theeffect of changing the mean, variance, and integral scale of ln(K)and ln(Kd) for each RMA below.

4.1. The impact of ln(K) heterogeneity

The influence of changing the mean of log hydraulic conductivity(mj) for each RMA is shown in Fig. 4A. We set mj between �2.5 and

different cases with positive correlation (a = 1), no correlation (a = 0), and negativeee cases is presented.

Fig. 3. The longitudinal dispersivity aR11ðtÞ changes with indicator correlation length when time is fixed at 1000 d. Three different cases with positive correlation (a = 1), no

correlation (a = 0), and negative correlation (a = �1) are shown. The anisotropy ratio, e; is 1.

Fig. 4. The longitudinal dispersivity aR11ðtÞ changes with (A) the mean hydraulic conductivity; (B) the variance of the hydraulic conductivity; (C) the integral scale of hydraulic

conductivity; (D) the mean of the sorption distribution coefficient; (E) the variance of the sorption distribution coefficient; (F) the integral scale of the sorption distributioncoefficient. Two different cases with positive correlation (a = 1) and negative correlation (a = �1) are shown for each RMA. Time at 1000 d. Anisotropy ratio, e; is 1.

240 M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244

M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244 241

2.5 and time is fixed at 1000 d. Fig. 4A shows that any changes in val-ues of mean of hydraulic conductivity can affect aR

11ðtÞ: However, itdepends on the type of RMA. For example, decreasing the meanhydraulic conductivity for RMA1 does not significantly affect aR

11ðtÞfor both positive and negative correlation, whereas aR

11ðtÞ canincrease if the mean of hydraulic conductivity for RMA 1 increases.For RMA 2, by decreasing the mean of hydraulic conductivityaR

11ðtÞ increases. The same is true for RMA 3. Note that RMA 1 hasthe largest variance and mean of hydraulic conductivity and thesmallest mean sorption. In contrast, RMA 3 has the smallest varianceand mean of hydraulic conductivity and the largest mean sorptioncoefficient.

Fig. 4B illustrates how changes in variances of log hydraulicconductivity for RMAs can affect aR

11ðtÞ when the time is fixed at1000 d. Increases in variance of hydraulic conductivity for all RMAslead to a linear increase in aR

11ðtÞ. The dispersivity is more sensitiveto changes in variance of log hydraulic conductivity for RMA1 thanother two RMAs which can be attributed to the fact that RMA1 hasthe largest volume proportion. The aR

11ðtÞ is more sensitive tochanges in variance of log hydraulic conductivity for RMA 3 thanRMA 2 because the volume proportion of RMA 3 is slightly largerthan RMA 2.

Fig. 4C shows aR11ðtÞ changes with integral scales of log hydraulic

conductivity when time is fixed at 1000 d. The changes in integralscales of log hydraulic conductivity for RMA 2 and 3 only producea very small change in aR

11ðtÞ: In contrast, for RMA1 it produces alarge change in aR

11ðtÞ with a maximum at about kj ¼ 300 m, andthen it remains constant. Therefore, aR

11ðtÞ is more sensitive to theintegral scale of log hydraulic conductivity for RMA 1 because ithas the largest volume proportion and also the largest integral scale.

4.2. The impact of ln(Kd) heterogeneity

Fig. 4D shows how aR11ðtÞ changes with the mean log sorption

distribution coefficient when time is fixed at 1000 d. In general,increases from �2.5 to 0 results in slight decreases in aR

11ðtÞ: How-ever, increases from 0 to 2.5 lead to increases aR

11ðtÞ; more for RMA2 and 3.

Fig. 4E illustrates how the changes in the variance of the logsorption distribution coefficient for RMAs affect aR

11ðtÞ when thetime is fixed at 1000 d. Increases in variances of the sorption distri-bution coefficient for all RMAs lead to increases in aR

11ðtÞ: In con-trast to changing the variance of ln(K), increasing the variance ofln(Kd) leads to nonlinear increases in aR

11ðtÞ. By comparing Fig. 4Eto B it is clear that aR

11ðtÞ is more sensitive to changes in the vari-ance of the log sorption distribution coefficient for RMAs thanthe variance of the log hydraulic conductivity for RMAs.

Fig. 4F shows aR11ðtÞ changes with the integral scales of the log

sorption distribution coefficient when time is fixed at 1000 d.The aR

11ðtÞ is sensitive to changes in the integral scale of the logsorption distribution coefficient for RMA 1 which has the largestvolume proportion and also the largest integral scale.

5. Conclusions

We developed a Lagrangian-based theory for analyzing thetime-dependent reactive solute dispersion in hierarchical porousmedia with multimodal reactive mineral facies. This study demon-strates that the cross-correlation between K and Kd has a largeimpact on the longitudinal dispersivity aR

11ðtÞ: The scale-depen-dence of reactive solute dispersion originated from heterogeneityin both K and Kd and their cross-correlation.

We considered three types of cross-correlation: perfectly posi-tive cross-correlation, perfectly negative cross-correlation, and nocross-correlation. The positive cross-correlation reduces the

reactive solute dispersivity, since heterogeneities in K and Kd coun-teract each other, which means that low K regions occurs with lowretardation or high K regions occurs with high retardation. Thenegative cross-correlation causes an opposite effect. The resultsalso indicate that anisotropy ratio does not significantly affectthe transport of reactive solutes undergoing equilibrium sorptionin the example used.

Furthermore, reactive solute dispersion is scale-dependent butnot a linear function of the indicator correlation scale. We also per-formed analyses of sensitivity to changes in the integral scales ofln(K) and ln(Kd), and in the means and variances of ln(K) and ln(Kd)for each reactive mineral assemblage. The results show that heter-ogeneity in both ln(K) and ln(Kd) can significantly affect aR

11ðtÞ. TheaR

11ðtÞ is very sensitive to changes in mean and variance of bothln(K) and ln(Kd). The aR

11ðtÞ is most sensitive to changes of integralscales of ln(K) and ln(Kd) for the reactive mineral assemblage witha larger volume proportion.

Appendix A. Derivation of spectral density of fluctuations inln(K)

Considering an exponential covariance function of ln(K):

Cff ðnÞ ¼ r2f eð�j

nljÞ ¼ r2

f eð�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1l1

� �2

þ n2l2

� �2

þ n3l3

� �2r

ÞðA1Þ

where li are integral scale. The spectral density of ln(K) is evaluatedby taking the Fourier transform of Eq. (19). Substituting (A1) into(19), then it yields:

SðkÞ ¼ 1

ð2pÞ3ZZZ 1

�1e�ik:nr2

f eð�jnljÞdn ðA2Þ

Integrating (A2) leads to the following expression for spectraldensity of ln(K):

Sff ðkÞ ¼r2

f l1l2l3

p2

1

ð1þ ðlkÞ2Þ2 ðA3Þ

where (lk)2 = (l1k1)2 + (l2k2)2 + (l3k3)2. Note that (A3) is also usedfor spectral density of ln(Kd).

Appendix B. Derivation of Eq. (20a)

Eq. (20a) contains the following integral:

ð20aÞ ¼ 1R2

Z t

0Cv1v1

�v1

Rs;0;0

� �ds

Based on Eq. (18) we can write the following:

Sv1v1 ðkÞ ¼KGJn

!2

1� k21

k2

!2

Sff ðkÞ ¼ �v21 1� k2

1

k2

!2

Sff ðkÞ

¼�v2

1r2f l1l2l3

p2 1� k21

k2

!21

ð1þ ðlkÞ2Þ2

Therefore,

Cv1v1 ðnÞ ¼�v2

1r2f l1l2l3

p2

Z 1

�1

Z 1

�1

Z 1

�11� k1

k

� �2 !2

1

ð1þ ðlkÞ2Þ2

� cosðk:nÞdk

Hence,

242 M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244

Cv1v1

�v1

Rs;0;0

� �¼

�v21r2

f l1l2l3

p2

Z 1

�1

Z 1

�1

Z 1

�11� k1

k

� �2 !2

� 1

ð1þ ðlkÞ2Þ2 cosðk1

�v1

RsÞdk ¼

k0i¼liki

�v21r2

f

p2

Z 1

�1

�Z 1

�1

Z 1

�1

l�22 k2

2 þ l�23 k2

3

l�21 k2

1 þ l�22 k2

2 þ l�23 k2

3

!2

� 1

ð1þ k2Þ2 cosðk1

�v1

RsÞdk

Changing variables s0 ¼ l�11 R�1 �v1s and s ¼ �v1t=l1R gives:

ð20aÞ ¼�v1

R

r2f l1

p2

Z s

0

Z 1

�1

Z 1

�1

Z 1

�1

l�22 k2

2 þ l�23 k2

3

l�21 k2

1 þ l�22 k2

2 þ l�23 k2

3

!2

� 1

ð1þ k2Þ2 cosðk1s0Þdkds0

Now we have l1 ¼ l2 ¼ k and l3 ¼ ek which e is the anisotropyratio. Thus,

ð20aÞ ¼�v1

R

r2f k

p2

Z s

0

Z 1

�1

Z 1

�1

Z 1

�11� k2

1

k21 þ k2

2 þ e�2k23

!21

ð1þ k2Þ2

� cosðk1sÞdkds

Now changing to spherical coordinate system and defining:

k1 ¼ r cos b; k2 ¼ r sin b; k3 ¼ k3

dk1k2k3 ¼ rdrdhdk3

Therefore,

ð20aÞ ¼�v1

R

r2f k

p2

Z s

0dsZ 1

0rdrZ 2p

0dhZ 1

�11� r2 cos2 h

r2 þ e�2k23

!2

� cosðsr cos hÞ

ð1þ r2 þ k23Þ

2 dk3 ðB1Þ

In order to derive the above integral the following general inte-grals are used:Z 1

�1

1

ða2 þ x2Þ2dx ¼ p

2a3 ðB2aÞ

Z 1

�1

1

ða2 þ x2Þ2ðb2 þ x2Þdx ¼ p

2ðb2 � a2Þa3

� p

ðb2 � a2Þ2

1a� 1

b

� �ðB2bÞ

Z 1

�1

1

ða2 þ x2Þ2ðb2 þ x2Þ2 dx ¼ p

2ðb2 � a2Þ2

1a3 þ

1

b3

� �

� p

ðb2 � a2Þ3

1a� 1

b

� �ðB2cÞ

In case of integral in (B1) the following parameters are used in(B2a)–(B2c):

a ¼ 1þ r2 ¼ u;

b ¼ er;

a2 � b2 ¼ 1þ r2 � e2r2;

Thus, one can derive (B1) as follows:

ð20aÞ ¼�v1r2

f k

R

Z s

0dsZ 1

0f r

2u32

AðsrÞdr

þ 2e2r3 1

2vu32þ 1

v2u32� 1

v2er

� �BðsrÞ

þ e4r5 1

2v2u32þ 1

2v2ðerÞ3þ 2

v3u12� 2

v3er

" #CðsrÞgdr

where

AðsrÞ ¼ 2JoðsrÞ ðB3aÞ

BðsrÞ ¼ 2JoðsrÞsr � 2J1ðsrÞsr

ðB3bÞ

CðsrÞ ¼ 2ðsrÞ3JoðsrÞ � 6JoðsrÞðsrÞ þ 12J1ðsrÞ � 4J1ðsrÞðsrÞ2

ðsrÞ3ðB3cÞ

The Jo and J1 are the zero and first order Bessel functions,respectively. Now we change the variable b = rs. Thus,

ð20aÞ ¼ 1Rf�v1r2

f kZ s

0dsZ 1

0

r

2u32

AðsrÞdr

þ �v1r2f kZ 1

04e2r3 1

2vu32þ 1

v2u32� 1

v2er

� �J1ðbÞdr

þ �v1r2f kZ 1

0e4r5 1

2v2u32þ 1

2v2ðerÞ3þ 2

v3u12� 2

v3er

" #

� 2J1ðbÞb2 � 4J1ðbÞ þ 2JoðbÞbb2

" #drg

In above integral one can use the following general integral:Z s

0dsZ 1

0

r

2u32

AðsrÞdr ¼Z 1

0

sJoðbÞ1þ r2 þ r

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2p dr ¼ 1� e�s ðB4Þ

Finally, (20a) is found as follows:

ð20aÞ ¼�v1

Rr2

f kf1� e�s � eZ 1

0½2rJ1ðbÞ

2u32 � erðv þ 2uÞ

v2u32

þ F1ðrÞ�dr ðB8Þ

where

F1ðrÞ ¼ð2� b2ÞJ1ðbÞ � bJoðbÞ

rs2

" #e3r3ðv þ 4uÞ þ u

32ð5v � 4uÞ

v3u32

" #;

u ¼ 1þ r2;

b ¼ rs;

v ¼ 1þ r2 � e2r2;

Appendix C. Derivation of Eq. (20c)

Eq. (20c) contains the following integral:

ð20cÞ ¼ 2�v1

R3

Z t

0Cv1R

�v1

Rs;0;0

� �ds

Here we use Eq. (17), the cross spectrum between v1 and R, in orderto find Cv1RðnÞ:

Cv1RðnÞ ¼qb

n2 KGKGd Ja

sinhðrwÞrw

r2f l1l2l3

p2

Z 1

�1

Z 1

�1

�Z 1

�11� k1

k

� �2 !2

1

ð1þ ðlkÞ2Þ2 cosðk:nÞdk

M.R. Soltanian et al. / Chemosphere 122 (2015) 235–244 243

Therefore,

Cv1R�v1

Rs;0;0

� �¼ qb

n2 KGKGd Ja

sinhðrwÞrw

r2f l1l2l3

p2

Z 1

�1

Z 1

�1

�Z 1

�11� k1

k

� �2 !2

1

ð1þ ðlkÞ2Þ2 cosðk1

��v1

RsÞdk ¼

k0i¼liki

qb

n2 KGKGd Ja

sinhðrwÞrw

r2f l1l2l3

p2

�Z 1

�1

Z 1

�1

Z 1

�1

l�22 k2

2 þ l�23 k2

3

l�21 k2

1 þ l�22 k2

2 þ l�23 k2

3

!2

� 1

ð1þ k2Þ2 cosðk1

�v1

l1RsÞdk

We change the variable as s0 ¼ l�11 R�1 �v1s and s ¼ �v1t=l1R and

(20c) changes to:

ð20cÞ ¼ 2R2

qb

n2 KGKGd Ja

sinhðrwÞrw

r2f l1

p2

Z s

0

Z 1

�1

Z 1

�1

�Z 1

�1

l�22 k2

2 þ l�23 k2

3

l�21 k2

1 þ l�22 k2

2 þ l�23 k2

3

!21

ð1þ k2Þ2 cosðk1s0Þdkds

Now we have l1 ¼ l2 ¼ k and l3 ¼ ek. Thus

ð20cÞ ¼ 2R2

qb

n2 KGKGd Ja

sinhðrwÞrw

r2f l1

p2

Z s

0

Z 1

�1

Z 1

�1

�Z 1

�11� k2

1

k21 þ k2

2 þ e�2k23

!21

ð1þ k2Þ2 cosðk1sÞdkds

Now changing to spherical coordinate system and defining:

k1 ¼ r cos b; k2 ¼ r sin b; k3 ¼ k3

dk1k2k3 ¼ rdrdhdk3

Therefore we get:

ð20cÞ ¼ 2R2

qb

n2 KGKGd Ja

sinhðrwÞrw

r2f k

p2

Z s

0dsZ 1

0rdr

�Z 2p

0dhZ 1

�11� r2 cos2 h

r2 þ e�2k23

!cosðsr cos hÞð1þ r2 þ k2

3Þdk3 ðC1Þ

In order to derive the integral in Eq. (C1) we use the generalintegrals in(B2a) and (B2b). Furthermore, we use (B3a) and (B3b)along with (B4). The final form of Eq. (20c) is presented as follows:

ð20cÞ ¼ 2R2

qb

n2 KGKGd Ja

sinhðrwÞrw

r2f kf1� e�s þ 2e2

�Z 1

0r2 1

2vu32þ 1

v2u12� 1

v2er

� �J1ðbÞdrg ðC2Þ

Appendix D. Derivation of Eq. (20b)

Eq. (20b) contains the following integral:

ð20bÞ ¼�v2

1

R2

Z t

0CRR

�v1

Rs; 0;0

� �ds

CRR(n) is as follows(see Eq. (12)):

CRRðnÞ ¼qb

nKG

d

� �2e½r

2w �ðeCwwðnÞ � 1Þ

where

CwwðnÞ ¼ r2weð�j

nljÞ ¼ r2

we�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1l1

� �2

þ n2l2

� �2

þ n3l3

� �2r !

We consider the mean flow direction along x axis. Thus,

Cwwðn;0;0Þ ¼ r2weð�j

nljÞ ¼ r2

weð�n1l1Þ

We choose l1 ¼ k so that we can calculate (20b) as:

ð20bÞ ¼�v2

1

R2

Z t

0CRR

�v1

Rs;0;0

� �ds

¼�v2

1

R2

qb

nKG

d

� �2e½r

2w �Z t

0er2

weð�

�v1Rk

sÞ

—1

!ds ðD1Þ

Integral in (D1) can be easily derived as explained below. Interms of multimodal porous media the derivation is not straightforward and a numerical integration is performed. In order toderive (D1) for unimodal porous media we use the following singu-lar integrals.

EiðxÞ ¼ �Z 1

�x

e�s

sds ðD2Þ

(D2) is known as exponential integral.

EinðxÞ ¼ �Z �x

0

1� e�s

sds ¼ cþ lnðxÞ þ EiðxÞ

¼ �Xn

i¼1

xn

nn!for x > 0 ðD3Þ

(D3) is known as entire function, and c = 0.577. . . is the Euler’sconstant. Using the above integrals we can derive the followingintegral:Z t

0eAe�as

ds ¼X1n¼0

Z t

0

Ane�nas

n!ds

¼X1n¼1

1�na

An

n!ðe�nat � 1Þ þ t

¼ �1aX1n¼1

Ane�nat

nn!þ 1

aX1n¼1

An

nn!þ t

¼ 1a½hðAÞ � hðAe�atÞ� þ t ¼ 1

a½EiðAÞ � EiðAe�atÞ� ðD4Þ

Therefore,Z t

0eAe�as

ds ¼ 1a½EiðAÞ � EiðAe�atÞ� ðD5Þ

Therefore, in case of unimodal distribution of Cww(n) the resultof (20b) is:

ð20bÞ ¼�v1

Rqb

nKG

d

� �2e½r

2w �k½Eiðr2

wÞ � Eiðr2we�

�vRk

tÞ� ðD6Þ

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