Macroscopic behavior and random-walk particle tracking … · Macroscopic behavior and random-walk...

Macroscopic behavior and random-walk particle trackingof kinetically sorbing solutes

Anna M. Michalak and Peter K. KitanidisDepartment of Civil and Environmental Engineering, Stanford University, Stanford, California

Abstract. Analytical expressions are derived for the zeroth, first, and second spatialmoments of sorbing solutes that follow a linear reversible kinetic mass transfer model. Wedetermine phase-transition probabilities and closed-form expressions for the spatialmoments of a plume in both the sorbed and aqueous phases resulting from an arbitraryinitial distribution of solute between the phases. This allows for the evaluation of theeffective velocity and dispersion coefficient for a homogeneous domain without resortingto numerical modeling. The equations for the spatial moments and the phase-transitionprobabilities are used for the development of a new random-walk particle-trackingmethod. The method is tested against three alternate formulations and is found to becomputationally efficient without sacrificing accuracy. We apply the new random-walkmethod to investigate the possibility of a double peak in the aqueous solute concentrationresulting from kinetic sorption. The occurrence of a double peak is found to be dependenton the value of the Damkohler number, and the timing of its appearance is controlled bythe mass transfer rate and the retardation factor. Two ranges of the Damkohler numberleading to double peaking are identified. In the first range (DaI # 1), double peakingoccurs for all retardation factors, while in the second range (1 # DaI # 3), thisbehavior is most significant for R . 12.

1. Introduction

A thorough understanding of sorption processes is necessaryfor accurate groundwater transport predictions. Sorption inporous media is often described by assuming local equilibrium.The local-equilibrium assumption is not always valid, however,and sorption rates must be considered in some cases, especiallywhere the adsorption and desorption reactions are slow rela-tive to the seepage velocity [Brusseau and Rao, 1989]. Variousmultisite and multiprocess models have been proposed to de-scribe kinetic sorption, reviewed by Haggerty and Gorelick[1995]. However, a simple linear reversible mass transfermodel has been shown to be an accurate approximation tomore complicated nonequilibrium models in many cases[Nkedi-Kizza et al., 1984; van Genuchten, 1985; Parker andValocchi, 1986; Valocchi, 1990; Sardin et al., 1991].

One of the phenomena that may be attributable to kineticsorption is the formation of a transient double peak in theaqueous concentration profile. Such behavior has been ob-served in some reactive solute experiments, such as the onesconducted at the Borden [Mackay et al., 1986; Roberts et al.,1986] and Cape Cod [Garabedian et al., 1988; LeBlanc et al.,1991] field sites, although kinetic sorption has not been con-sidered as a possible causal mechanism for such observations.Quinodoz and Valocchi [1993] suggested that such doublepeaks appear only for very large retardation factors and dis-cussed no rate limitations on such behavior. Their claim wasbased on the evolution of the first spatial moment of the solutein the aqueous phase, which indicates that the mean velocitytemporarily drops below its asymptotic value for large retar-dation factors. However, this claim has not been verified nu-

merically. Bellin et al. [1991] investigated the formation ofdouble peaks in the breakthrough curves of column experi-ments involving solutes undergoing kinetic sorption. The con-centration profile was measured as a function of time at theoutflow of the column, and a necessary condition for doublepeaking was found to be

v/D .. k/hmv , (1)

where D is the dispersion coefficient, k is a mass transfercoefficient, hm is the mobile phase porosity, and v is theseepage velocity. In other words, most of the desorption had tooccur after the transport effects ceased to affect the break-through curve. Furthermore, double peaking occurred onlywhen (kL/hv) was “not too large,” where h is the total po-rosity and L is the length of the column. Sample runs wereperformed only for a Peclet number Pe 5 vL/D 5 10 and aretardation coefficient R 5 10, and only point sources wereexamined.

1.1. Previous Particle-Tracking Methods

A simple and efficient numerical method is needed to inves-tigate the behavior of kinetically sorbing solutes. The particle-tracking method [Ahlstrom and Foote, 1976; Prickett et al.,1981] is a Lagrangian approach in which a large number ofparticles is tracked in order to simulate conservative-tracertransport. In contrast to Eulerian simulation techniques, neg-ative concentrations cannot occur, concentration profile artifi-cial smoothing and oscillations are greatly reduced, and dis-persive fluxes are approximated accurately, provided that thenumber of particles is high enough. The random-walk methodis therefore particularly applicable to groundwater situations,where sharp concentration fronts are common and mixingrates are low relative to surface water and atmospheric appli-cations.

Copyright 2000 by the American Geophysical Union.

Paper number 2000WR900109.0043-1397/00/2000WR900109$09.00

WATER RESOURCES RESEARCH, VOL. 36, NO. 8, PAGES 2133–2146, AUGUST 2000

2133

The random-walk method has also been applied to sorptionproblems. Abulaban et al. [1998] used it to examine cases oflinear and nonlinear sorption isotherms, applying an algorithmdeveloped by Tompson [1993]. The experiments of Tompson[1993] focused on four hypothetical constituents, one beinginert and the other three independently obeying linear, Freun-dlich, and Langmuir partitioning relationships. Both Abulabanet al. [1998] and Tompson [1993] used constituents that arisefrom reversible, equilibrium sorption reactions. Nonequilib-rium sorption was not addressed.

Selroos and Cvetkovic [1992] proposed a method for cou-pling sorption kinetics and solute advection in particle-trackingmodels. However, their method is efficient only when sorptionrate coefficients can be assumed uniform. The solution for aninstantaneous injection was derived in the Laplace domain. Inthe case of uniform rate coefficients the inverse Laplace trans-formation yields an equation involving a modified first-orderBessel function of the first kind. The mass flux is dependentonly on the sorption coefficients and on the intrinsic traveltime. If the sorption coefficients are assumed uniform, themass flux at a given location along a streamline can be readilyobtained by evaluating the travel time of a particle from theorigin to the given location.

Valocchi and Quinodoz [1989] compared three methods fordealing with kinetic sorption assuming a linear driving force.The arbitrary time step method (ATSM) uses a three-stepprocess to determine the fraction of time that each particlespends in the aqueous phase during each time step. The con-tinuous time history method (CTHM) simulates the history ofphase changes for each particle during each time step. Thesmall time step method (STSM) is an operator-splitting ap-proach that was originally introduced by Kinzelbach [1987] andKinzelbach and Uffink [1991]. The method is based on thepremise that the probability of having more than one phasechange in a small time step is negligible, and a particle cantherefore be assumed to remain in its initial state for the entiretime step. Valocchi and Quinodoz [1989] concluded that theATSM is a more efficient and accurate method relative to theCTHM and STSM.

Andricevic and Foufoula-Georgiou [1990, 1991] developed amethod similar to the ATSM [Valocchi and Quinodoz, 1989].However, they proposed a different approximate approach fordetermining the fraction of time that each particle spends inthe aqueous phase during each time step, based on the first twomoments of the total time spent by each particle in the aque-ous phase.

Delay et al. [1996] developed a method for kinetic sorptionwith a linear isotherm assuming that for a given time step, theadvective-dispersive jump is instantaneous at the beginning ofthe time step. By analytically integrating the phase-transfer (ortransition) equation, the number of particles in the mobile andimmobile phases in a cell after a given time step are deter-mined based on the number of particles in the mobile andimmobile phases before that time step. This method is similarto that introduced by Kinzelbach [1987] and Kinzelbach andUffink [1991].

Mishra et al. [1999] developed a recursion formulation forthe transport of linearly sorbing solutes undergoing kineticsorption. The method is based on a Markov process model ofsorption-desorption. However, the method is based on dis-cretizing time and space as Dx 5 vDt , where Dx is the gridsize, Dt is the time step, and v is the seepage velocity, andeither advecting a solute particle by Dx or by 0 in a given time

step. Therefore the applicability of this method to heteroge-neous media is limited. Furthermore, the method does notincorporate local-scale dispersion, which can be important insome applications.

1.2. Scope of Paper

In this work, we apply the method of moments [Aris, 1956] tothe problem of kinetic sorption with a linear driving-forcekinetics model. We derive analytical expressions for the zeroth,first, and second spatial moments of the sorbed and aqueoussolute distributions for solute starting in either the aqueous orsorbed phase. The original method of moments was derived forconservative solutes [Aris, 1956]. Goltz [1986] and Goltz andRoberts [1987] derived the analytical solution for the zerothand first spatial moments of a kinetically sorbing solute in-jected into the aqueous phase. The second moment was de-rived only for a solute starting and ending in the aqueousphase. Solutions for a solute starting in the sorbed phase werenot presented.

On the basis of analytical moment expressions we deriveanalytical expressions for the first and second moments of thesolute distribution resulting from an arbitrary initial distribu-tion between the aqueous and sorbed phases. This allows for ageneral description of the solute distribution without resortingto numerical modeling, as would have been necessary in thepast.

Furthermore, spatial moments can be used to define advec-tive and dispersive steps for particle-tracking models [Kitanidis,1994]. We therefore develop a highly efficient random-walkmethod based on the spatial moment expressions. This methodis shown to be at least as accurate as, and more efficient than,any of the methods suggested by Valocchi and Quinodoz[1989].

Finally, we apply our new numerical method to the problemof double peaking in the aqueous concentration profile andshow that double peaking caused by kinetic sorption is con-trolled by the relative magnitude of the reaction rate as com-pared to the advection rate for the case of a linear driving-force sorption model and that such double peaking cantherefore occur for any retardation factor.

2. Mathematical Conceptualizationof the Problem2.1. General Development

The basic equations governing the transport of kineticallysorbing solutes are the advection-dispersion equation for theaqueous (or mobile) phase,

hCt 1 r

St 5 2hv i

C xi

1 h

xiSDij

C xj

D , (2)

and a phase-transition (or mass transfer) equation, which is,for a linear driving force,

St 5 k~KdC 2 S! , (3)

where C is the aqueous concentration [M/Lsolution3 ], h is the

porosity, S is the contaminant mass sorbed per mass of aquifersolids [M/M], r is the bulk solids density (mass of aquifersolids per total volume) [M/L3], v is the velocity [L/T], D isthe dispersion coefficient [L2/T], t is the time variable [T], xis the space variable [L], Kd is the distribution coefficient

MICHALAK AND KITANIDIS: BEHAVIOR AND TRACKING OF SORBING SOLUTES2134

[L3/M], k is a mass transfer coefficient [1/T], and the re-peated subscript designates summation. The dispersion coef-ficient will be assumed to be a scalar, for which case thetransport equation (equation (2)) simplifies to

hCt 1 r

St 5 2hv i

C xi

1 hD2C

xi xi. (4)

The dimensionless retardation coefficient is defined as

R 5 1 1rKd

h. (5)

We derive analytical solutions for the zeroth, first, and sec-ond spatial moments based on the advection-dispersion equa-tion for the aqueous phase (equation (4)), and the phase-transition equation (equation (3)). All analytical expressionsare derived for spatially uniform and temporally constant co-efficients. Each type of moment is derived for four separatebase cases: (1) aqueous distribution for solute originating inthe aqueous phase; (2) aqueous distribution for solute origi-nating in the sorbed phase; (3) sorbed distribution for soluteoriginating in the aqueous phase; and (4) sorbed distributionfor solute originating in the sorbed phase.

Since the governing equations (3) and (4) are linear, super-position is applicable, and all cases are covered by the analyticmoment solutions for the four base cases. Therefore momentscan be evaluated for any initial distribution between the aque-ous and sorbed phases by linear combination of the four basecases.

In random-walk models, solute concentrations are equiva-lent to particle densities, when a sufficient number of particlesis used. Furthermore, owing to the linearity of the problem, theprobability distribution of a single particle in both space andphase is independent of other particles. When the spatial mo-ments of concentrations are interpreted as those of particles,the spatial-moment behavior of any particle is known as afunction of time, regardless of its original or final phase. Thephase-transition probabilities, which are a function of the spe-cific transport parameters as well as the chosen time step, arederived based on the analytical solutions for the zeroth mo-ments.

For the particle-tracking model developed in this work, theinitial and final phase of each particle is determined with aBernoulli trial on the appropriate phase-transition probabilityat each time step. The particle is advected according to thedisplacement mandated by the corresponding analytical first-moment equation, and the random dispersive redistribution ofparticles is based on the corresponding analytical second-moment equation.

2.2. Development of Moment Equations

Although the spatial moments of the aqueous and sorbeddistributions (C and S , respectively) are generally defined inthree dimensions [Aris, 1956], they are presented here in onedimension for simplicity:

mn 5 E2`

`

xnhC~t , x! dx nn 5 E2`

`

xnrS~t , x! dx , (6)

where mn and nn are the nth aqueous and sorbed phase mo-ments, respectively. This formulation results in the zeroth mo-ment being equal to the total mass of solute in each of thephases.

The central spatial moments are

m9n 5 E2`

`

~ x 2 m1!nhC~t , x! dx (7a)

n9n 5 E2`

`

~ x 2 n1!nrS~t , x! dx . (7b)

Replacing C and S with their Fourier transforms

C~t , x! 5 E2`

`

C~t , s! exp ~ j2pxs! ds (8a)

S~t , x! 5 E2`

`

S~t , s! exp ~ j2pxs! ds , (8b)

where j 5 =21, and applying the auxiliary conditions

limx36`

C 5 0 limx36`

S 5 0, (9)

limx36`

C x 5 0 lim

x36`

S x 5 0 (10)

yields

hCt 1 r

St 1 2hvjpsC 1 4hDp2s2C 5 0, (11)

where C and S are the Fourier transforms of the aqueous andsorbed concentrations, respectively. The same approach forthe phase-transition equation yields

St 2 k~KdC 2 S! 5 0. (12)

The spatial moments of the concentration in the real domaincan be evaluated from the concentration Fourier coefficientsby [Bracewell, 1986; Goltz and Roberts, 1987]

mn 5h

~22pj!n

nCsn U

s50

nn 5r

~22pj!n

nSsnU

s50

. (13)

All spatial moments will be presented for the four possiblephase combinations: solute starting and ending in the aqueousphase mn

(C), solute starting in the sorbed phase and ending inthe aqueous phase mn

(S), solute starting in the aqueous phaseand ending in the sorbed phase nn

(C), and solute starting andending in the sorbed phase nn

(S).Additionally, we define spatial moments normalized by their

respective zeroth moments:

m*1 5 m1/m0 n*1 5 n1/n0 (14)

m*2 5 m92/m0 n*2 5 n92/n0. (15)

2.2.1. Zeroth moments. The equations for the zeroth mo-ments in Fourier space are (from (13))

m0 5 hC u s50 n0 5 rS u s50. (16)

Substituting (16) into (12) and setting s 5 0 yields

m0 51

~R 2 1! S 1k

n0

t 1 n0D . (17)

2135MICHALAK AND KITANIDIS: BEHAVIOR AND TRACKING OF SORBING SOLUTES

Substituting (16) into (11) and setting s 5 0 yields

m0

t 1n0

t 5 0. (18)

Letting the initial conditions for the zeroth moment be

m0~0! 5 M1 n0~0! 5 M2, (19)

where M1 and M2 are the initial mass in the aqueous andsorbed phases, respectively, and the total mass is M 5 M1 1M2, the solutions to (17) and (18) are

m0~t! 5 M1

~1 1 bA!

~b 1 1!1 M2

~1 2 A!

~b 1 1!(20)

n0~t! 5 M1

b~1 2 A!

~b 1 1!1 M2

~b 1 A!

~b 1 1!, (21)

where A 5 exp (2Rkt) and b 5 R 2 1.The examination of the spatial moments of solute distribu-

tions allows for the identification of characteristic timescales.According to the definition of the term A , 1/kR is the char-acteristic time for the zeroth moments approaching theirsteady state values. This characteristic time will also apply for

the first- and second-moment expressions. The simplified ze-roth-moment equations resulting from the initial solute sourcebeing entirely in a single phase are presented in Table 1, andthe zeroth-moment behavior is illustrated in Figure 1 for thecase of R 5 1.5 for the four possible phase combinations.These moments will also be interpreted, in section 3.3, asphase-transfer probabilities for particles starting and ending inthe specified phases. For the cases for which analytical expres-sions were already available, the zeroth-moment equations de-rived in this work match those presented in the literature[Goltz, 1986; Goltz and Roberts, 1987]. The same is true forfirst- and second-moment expressions.

2.2.2. First moments. The derivation of the first mo-ments is similar to that of the zeroth moments and is outlinedin Appendix A. The first-moment expressions normalized bytheir corresponding zeroth moments are presented in Table 1.First-moment behavior and rates of change are presented inFigure 2 for the case of R 5 1.5 for the four possible phasecombinations. The values of the first moments can be inter-preted as the total displacement of the centroid of fractions ofa solute plume starting and ending in the specified phases. The

Table 1. Longitudinal Moment Equations for Kinetically Sorbing Solutes

Zeroth Moment Normalized First Moment Normalized Second Central Moment

Aqueous phase originatingin aqueous phase(m0 5 M, n0 5 0)

m0~t! 5M~1 1 bA!

~b 1 1!m*1~t! 5 t

vb 1 1 1 tA

bv~b 2 1!

~b 1 1!~1 1 bA!

12vb~1 2 A!

k~b 1 1!2~1 1 bA!

m*2~t! 5 t2Av2b~b 2 1!2

~b 1 1!2~1 1 bA!2 1 t S 2Db 1 1 1

2v2b

k~b 1 1!3D1 tAS4bv2~2b2A2b2 2b11!

k~11bA!2~b11!3 12Db~b21!

~b11!~11bA!D1

2bv2~1 2 A!~3~b2A 2 1! 2 b~A 1 1!!

k2~1 1 bA!2~b 1 1!4

14Db~1 2 A!

k~1 1 bA!~b 1 1!2

Aqueous phase originatingin sorbed phase(m0 5 0, n0 5 M)

m0~t! 5M~1 2 A!

~b 1 1!m*1~t! 5 t

vb 1 1 1 tA

v~1 2 b!

~b 1 1!~1 2 A!

1v~b 2 1!

k~b 1 1!2

m*2~t! 5 2t2Av2~b 2 1!2

~1 2 A!2~b 1 1!2 1 t S 2Db 1 1 1

2v2b

k~b 1 1!3D1 tAS 2D~1 2 b!

~b 1 1!~1 2 A!1

4v2b

k~b 1 1!3~1 2 A!D1

v2~b2 2 6b 1 1!

k2~b 1 1!4 12D~b 2 1!

k~b 1 1!2

Sorbed phase originatingin sorbed phase(m0 5 0, n0 5 M)

n0~t! 5M~A 1 b!

~b 1 1!n*1~t! 5 t

vb 1 1 1 tA

v~b 2 1!

~A 1 b!~b 1 1!

12vb~A 2 1!

k~A 1 b!~b 1 1!2

n*2~t! 5 t2Av2b~1 1 b2 2 2b!

~A 1 b!2~b 1 1!2 1 t S 2Db 1 1 1

2v2b

k~b 1 1!3D1 tAS 2D~b 2 1!

~A 1 b!~b 1 1!1

4v2b~b2 2 b 2 A 2 1!

k~A 1 b!2~b 1 1!3 D1

2v2b~A 2 1!~3~b2 2 A! 1 b~A 2 1!!

k2~A 1 b!2~b 1 1!4

14Db~A 2 1!

k~A 1 b!~b 1 1!2

Sorbed phase originatingin aqueous phase(m0 5 M, n0 5 0)

n0~t! 5Mb~1 2 A!

~b 1 1!n*1~t! 5 t

vb 1 1 1 tA

v~1 2 b!

~b 1 1!~1 2 A!

1v~b 2 1!

k~b 1 1!2

n*2~t! 5 2t2Av2~b 2 1!2

~1 2 A!2~b 1 1!2 1 t S 2Db 1 1 1

2v2b

k~b 1 1!3D1 tAS 2D~1 2 b!

~b 1 1!~1 2 A!1

4v2b

k~b 1 1!3~1 2 A!D1

v2~b2 2 6b 1 1!

k2~b 1 1!4 12D~b 2 1!

k~b 1 1!2


rates of change of the first moments correspond to the instan-taneous advective velocity of fractions of a solute plume start-ing and ending in the specified phases. Note that the evolutionof the first moment is the same for all solute ending in a phaseopposite from its initial phase. Whether a fraction of solutestarts in the aqueous phase and is known to end in the sorbedphase or vice versa, the mean amount of time spent in eachphase is the same, and the evolution of the first moment istherefore identical. The same will be seen to hold true for theevolution of the second moments. The weighted mean pre-sented in Figure 2a corresponds to the overall first-momentbehavior resulting from an initial solute distribution in whichthe total solute mass is distributed between the phases accord-ing to the equilibrium distribution, that is, a 1/R fraction in theaqueous phase and a (R 2 1)/R fraction in the sorbed phase.The plume is then left to evolve according to the sorptionkinetics specified.

Since there is no advective transport in the transverse direc-tion(s) in a homogeneous field, the transverse first momentsequal zero for both the aqueous and sorbed phases. The trans-verse moments are presented in Table 2.

2.2.3. Second moments. An outline of the derivation ofthe second moments is given in Appendix B. Longitudinalsecond central moment expressions normalized by the corre-sponding zeroth moments are presented in Table 1. Centralsecond-moment behavior and rates of change are presented inFigure 3 for the case of R 5 1.5 for the four possible phasecombinations. The values of the second moments can be in-terpreted as the mean spread about the centroid of fractions ofa solute plume starting and ending in the specified phases. Therates of change of the second moments represent 2 times theinstantaneous dispersion coefficient of fractions of a soluteplume starting and ending in the specified phases.

The normalized second central moments are defined as, forthe aqueous and sorbed moments, respectively,

m*2 5m2 2 m1

2

m0n*2 5

n2 2 n12

n0, (22)

where the asterisk denotes the normalized second central mo-ments.

The weighted mean presented in Figure 3 corresponds to theoverall second-moment behavior resulting from an initial sol-ute distribution in which the total solute mass is distributedbetween the phases according to the equilibrium distribution,that is, a 1/R fraction in the aqueous phase and a (R 2 1)/Rfraction in the sorbed phase. The plume is then left to evolveaccording to the sorption kinetics specified. Note that the over-all normalized second central moment grows faster than themoments of the four individual phase combinations. In otherwords, the overall plume is spread around its center of massmore than the individual distributions resulting from the fourpossible phase combinations are spread around their individ-ual centers of mass, as would be expected since the individualplume fractions have different centroids.

For the transverse second moments the derivation is iden-tical to that for the longitudinal case, with the exception thatthe transverse aqueous phase velocities are zero, thereby sim-

Figure 1. Zeroth moments as a function of time for R 5 1.5with k 5 0.1 [1/T], M as mass injected in aqueous (C) orsorbed (S) phase, and m0 and n0 as solute mass in the aqueousand sorbed phases, respectively.

Figure 2. Behavior of normalized first moments for R 5 1.5with v 5 1 [L/T], k 5 0.1 [1/T], and m*1 and n*1 as normal-ized first spatial moments for solute in the aqueous and sorbedphases, respectively. (a) Normalized first moments as a func-tion of time. (b) Rates of change of normalized first moments.


plifying the final moment equations. These expressions arepresented in Table 2.

2.3. Phase Transition Probabilities

On the basis of the analytical solutions for the zeroth spatialmoments we derive the particle phase-transition probabilities.The total mass in the system is assumed to be M . Assumingthat at time t 5 0, all the solute is in the aqueous phase(m0(0) 5 M and n0(0) 5 0), (20) and (21) simplify to

m0~C! 5 M

~1 1 bA!

~b 1 1!, (23)

n0~C! 5 M

b~1 2 A!

~b 1 1!, (24)

where the superscript (C) represents solute originating in theaqueous phase. Conversely, assuming that at time t 5 0, allthe solute is in the sorbed phase (m0(0) 5 0 and n0(0) 5 M),(20) and (21) are

m0~S! 5 M

~1 2 A!

~b 1 1!, (25)

n0~S! 5 M

~b 1 A!

~b 1 1!, (26)

where the superscript (S) represents solute originating in thesorbed phase.

Normalizing (23) to (26) by the total mass gives the proba-bilities that a particle starting in a given phase is in a certainphase after time t:

PC3S 51M n0

~C! PC3C 51M m0

~C! (27)

PS3C 51M m0

~S! PS3S 51M n0

~S!, (28)

where the subscript P1 3 P2 refers to a particle starting inphase P1 and ending in phase P2, with S designating thesorbed phase and C designating the aqueous phase. The tran-sition probabilities are plotted in Figure 1. Valocchi and Qui-

Figure 3. Behavior of normalized second central momentsfor R 5 1.5 with v 5 1 [L/T], k 5 0.1 [1/T], D 5 0, andm*2 and n*2 as normalized second central spatial moments forsolute in the aqueous and sorbed phases, respectively. (a) Nor-malized second central moments as a function of time. (b)Rates of change of normalized second central moments.

Table 2. Transverse Moment Equations for Kinetically Sorbing Solutes

Normalized FirstMoment Normalized Second Central Moment

Aqueous phase originatingin aqueous phase(m0 5 M, n0 5 0)

m*1~t! 5 0 m*2~t! 5 t2D

b 1 1 1 tA2Db~b 2 1!

~b 1 1!~1 1 bA!1

4Db~1 2 A!

k~1 1 bA!~b 1 1!2

Aqueous phase originatingin sorbed phase(m0 5 0, n0 5 M)

m*1~t! 5 0 m*2~t! 5 t2D

b 1 1 1 tA2D~1 2 b!

~b 1 1!~1 2 A!1

2D~b 2 1!

k~b 1 1!2

Sorbed phase originatingin sorbed phase(m0 5 0, n0 5 M)

n*1~t! 5 0 n*2~t! 5 t2D

b 1 1 1 tA2D~b 2 1!

~A 1 b!~b 1 1!1

4Db~A 2 1!

k~A 1 b!~b 1 1!2

Sorbed phase originatingin aqueous phase(m0 5 M, n0 5 0)

n*1~t! 5 0 n*2~t! 5 t2D

b 1 1 1 tA2D~1 2 b!

~b 1 1!~1 2 A!1

2D~b 2 1!

k~b 1 1!2


nodoz [1989] derived these expressions by relying on resultsobtained by Parzen [1962] and on Kolmogorov differentialequations. Conversely, the current derivation is obtained di-rectly from expressions for the zeroth moments of the aqueousand sorbed phase distributions.

3. Overall Moment Behavior: Analytical ResultsWe use the moment equations (Tables 1 and 2) and the

transition probabilities (equations (27) and (28)) in order todefine general properties of kinetically sorbing solute trans-port. Since the two equations governing the problem (equa-tions (3) and (4)) are linear, their moments can be superposed.Therefore, regardless of the initial distribution of the solutebetween the two phases, the location of the center of mass andthe degree of spreading of the solute can now be determinedanalytically, without resorting to numerical modeling. Eventhough the first and second moments are explicitly derived onlyfor the cases of all the solute being in either the aqueous orsorbed phase at time t 5 0, the solute distribution can bedescribed for any initial distribution by applying the properweighting function to the analytical moment equations.

For an initial mass fraction c in the aqueous phase and 1 2c in the sorbed phase, the first moment and second centralmoment are

m*1 5 cPC3Cm*1~C! 1 ~1 2 c! PS3Sn*1

~S! 1 cPC3Sn*1~C!

1 ~1 2 c! PS3Cm*1~S! (29)

m*2 5 cPC3C@m*2~C! 1 ~m*1

~C! 2 m*1!2#

1 ~1 2 c! PS3S@n*2~S! 1 ~n*1

~S! 2 m*1!2#

1 cPC3S@n*2~C! 1 ~n*1

~C! 2 m*1!2#

1 ~1 2 c! PS3C@m*2~S! 1 ~m*1

~S! 2 m*1!2# (30)

where m*1 and m*2 are the moments of the overall solutedistribution (sorbed and aqueous fractions) and the super-scripts (C) and (S) represent solute originating in the aqueousand sorbed phases, respectively. The aqueous and sorbed mo-ments are

m*1~C,S! 5

1jC

@cPC3Cm*1~C! 1 ~1 2 c! PS3Cm*1

~S!# (31)

n*1~C,S! 5

1jS

$@~1 2 c! PS3Sn*1~S! 1 cPC3Sn*1

~C!#% (32)

m*2~C,S! 5

1jC

$cPC3C@m*2~C! 1 ~m*1

~C! 2 m1~C,S!!2#

1 ~1 2 c! PS3C@m*2~S! 1 ~m*1

~S! 2 m1~C,S!!2#% (33)

n*2~C,S! 5

1jS

$~1 2 c! PS3S@n*2~S! 1 ~n*1

~S! 2 m1~C,S!!2#

1 cPC3S@n*2~C! 1 ~n*1

~C! 2 m1~C,S!!2#% , (34)

where the superscript (C , S) represents solute starting ineither phase, and the solute fractions present in each phase aredefined as

jC 5 cPC3C 1 ~1 2 c! PS3C, (35)

jS 5 ~1 2 c! PS3S 1 cPC3S, (36)

where jC and jS are the fractions in the aqueous and sorbedphases, respectively.

On the basis of these analytical relations the first and secondmoments of any instantaneous point source can be deter-mined. For nonpoint sources the first moment and secondcentral moment can be determined by simply adding the initialmoment value to the point source moment value. For nonin-stantaneous sources a convolution integral must be performedfor each moment in order to determine the overall momentbehavior.

As an example, the moment behavior for the case of a steadystate solute distribution between the aqueous and sorbedphases will be used to develop transport parameters. A steadystate is defined herein as one for which the total solute mass isdistributed between the phases according to the equilibriumdistribution but for which local equilibrium is not satisfied atevery point in the medium. In other words, overall, the frac-tions of solute in the aqueous and sorbed phases are those thatwould have been predicted by the local-equilibrium assump-tion, that is, 1/R and (R 2 1)/R , respectively. At steady statea 1/R fraction of the total solute mass will be in the aqueousphase. Therefore the cumulative rate of change of the overallfirst moment can be defined as

veff 51Dt F PC3C

~b 1 1!m*1

~C! 1bPS3S

~b 1 1!n*1

~S!

1PC3S

~b 1 1!n*1

~C! 1bPS3C

~b 1 1!m*1

~S!G5

1Dt ~m*1! 5

vR ,

(37)

where all moments are calculated at time t 5 Dt . This resultis illustrated by the weighted mean in Figure 2b.

The effective dispersion coefficient Deff can be defined in asimilar manner by taking a weighted mean of the average ratesof change of the individual plume components (each compo-nent corresponding to a given initial and final phase):

Deff 51

2Dt H PC3C

~b 1 1!@m*2

~C! 1 ~m*1~C! 2 m*1!2#

1bPS3S

~b 1 1!@n*2

~S! 1 ~n*1~S! 2 m*1!2#

1PC3S

~b 1 1!@n*2

~C! 1 ~n*1~C! 2 m*1!2#

1bPS3C

~b 1 1!@m*2

~S! 1 ~m*1~S! 2 m*1!2#J , (38)

Deff~t3`! 5D

b 1 1 1v2b

k~b 1 1!3 , (39)

where all moments are calculated at time t 5 Dt and theasterisks indicate that the normalized first moments and sec-ond central moments are being used.

4. Particle-Tracking ImplementationOn the basis of the analytical solution for the spatial mo-

ments and phase-transition probabilities we develop a particle-tracking model, referred to as the semianalytical momentmethod (SAMM). We compare our approach to the three


models presented by Valocchi and Quinodoz [1989]. A shortdescription of the three alternate methods is presented alongwith the implementation details of the proposed method.

4.1. Semianalytical Moment Method (SAMM)

The implementation of the SAMM is based on the analyticalexpressions for the spatial moments and the phase-transitionprobabilities. The values of the four first and second moments(Tables 1 and 2) as well as the four phase-transition probabil-ities (equations (27) and (28)) are calculated for t 5 Dt .

For each time step the final phase of each particle is deter-mined by a Bernoulli trial based on its initial phase and thecorresponding phase-transition probability. Each particle isthen displaced by the value of the first moment at t 5 Dtcorresponding to the particle’s initial and final phase. Thesemoment expressions are plotted in Figure 2a. To account fordispersion, each particle is displaced by a distance proportionalto the square root of the value of the normalized secondcentral moment corresponding to each particle’s initial andfinal phase. These distances are weighted using normally dis-tributed random numbers with a mean of zero and a varianceof one. The second-moment expressions are plotted in Figure3a. A flow chart for the method, applied to a single particle, ispresented in Figure 4.

Since the analytical results for the zeroth, first, and secondmoments are applied, and therefore enforced, directly, themethod is guaranteed to reproduce these values precisely andaccurately if a sufficient number of particles is used.

Although only the lower moments are directly specified, themethod reproduces the transient skewness (or tailing) charac-teristic of solute subjected to kinetic sorption. The accuracywith which the skewness is represented is a function of the timestep used. However, although this effect was not quantified,

time step limitations were not observed for the examined pa-rameters.

4.2. Continuous Time History Method (CTHM)

The CTHM simulates the history of phase changes for eachparticle during each time step by generating a sequence ofexponentially distributed phase residence times. The processstops when the total time elapsed for a given particle is equalto the specified time step. At that point the waiting times foreach state are summed, and the particle is advected and dis-persed using a time step equal to the time spent in the aqueousphase [Valocchi and Quinodoz, 1989].

The exponential phase residence times can be expressed as

tC 52ln ~PC,C!

k~R 2 1!tS 5

2ln ~PS,S!

k , (40)

where tC and tS are the exponential phase residence times fora particle in the aqueous and sorbed phases, respectively. PC ,C

and PS ,S can be randomly generated from a uniform distribu-tion in the range [0 1]. Note that the probabilities denoted asPC3C and PS3S in (27) and (28) represent the probability thata particle is in its original phase after a given time, whereas theprobabilities represented as PC ,C and PS ,S represent the prob-ability that a particle remains in its original phase throughoutan examined period. In short, PC ,C and PS ,S are the zerophase-transition probabilities.

4.3. Small Time Step Method (STSM)

The STSM [Kinzelbach, 1987; Kinzelbach and Uffink, 1991] isbased on the fact that for small enough time steps, the transi-tion probabilities become

PC3S 5 kbDt PS3C 5 kDt . (41)

Figure 4. Semianalytical moment method flow chart for a single particle.


Since the number of phase transitions can be considered aPoisson process, the probability of having more than one phasechange in a small time step is negligible. Therefore a particle isassumed to remain in its initial state for the entire time step,and its final state is adjusted by a Bernoulli trial, where a valuerandomly generated from a uniform distribution in the range[0 1] is compared to the appropriate transition probability in(41).

4.4. Arbitrary Time Step Method (ATSM)

The ATSM was introduced by Valocchi and Quinodoz[1989]. This method uses a three-step procedure to determinethe fraction of each time step spent by each particle in theaqueous phase. The first step involves a Bernoulli trial todetermine whether a phase change occurs according to thezero phase-transition probabilities:

PC,C 5 exp ~2k~R 2 1!Dt! , (42)

PS,S 5 exp ~2kDt! . (43)

If a phase transfer occurs, the final phase is determined usingthe transition probabilities (equations (27) and (28)), adjustedto eliminate the possibility of zero transfers. Finally, the frac-tion of time spent by each particle in the aqueous phase isdetermined from the probability density functions of the frac-tion of time spent in the aqueous phase for each of the initial/final phase combinations [Keller and Giddings, 1960]:

fC3C~g! 5 Î abg

1 2 gexp @2a~1 2 g! 2 bg#

z I1~ Î4abg~1 2 g!! (44)

fC3S~g! 5 a exp @2a~1 2 g! 2 bg#I0~ Î4abg~1 2 g!! (45)

fS3C~g! 5ab fC,S~g! (46)

fS3S~g! 51 2 g

gfC,C~g! , (47)

where fC3C, fC3S, fS3C, and fS3S are the probability dis-tribution functions of the fraction of time g spent by a particle

in the aqueous phase, a 5 kDt , and b 5 kbDt . The initial andfinal phases of the particle are indicated by the subscripts Cand S , corresponding to the aqueous and sorbed phases, re-spectively. I0 and I1 are modified Bessel functions of the firstkind of order 0 and 1, respectively.

These probability distribution functions are numerically in-tegrated over the range 0 # g # 1 and normalized by their totalintegral in order to obtain cumulative probability distributionfunctions of the fraction of time spent in the aqueous phase.This integration is performed once, before the start of timestepping. For each time step and particle the random numberused to determine the fraction of time spent in the aqueousphase is compared to the probabilities of each fraction g, andlinear interpolation is performed between the two closest val-ues to obtain an estimate of g for the given particle.

4.5. Model Comparison

The four available models were tested using a method sim-ilar to that of Valocchi and Quinodoz [1989]. Each model wasrun 10 times for each set of parameters in order to obtain anestimate of the mean error in the aqueous solute distributionresulting from each method, relative to the available analyticalsolutions. In this manner the standard deviation of the errorwas also estimated. The simulations were performed with thereaction rates and number of particles presented in Table 3.All particles were initially in the aqueous phase. The velocity vwas unity (in arbitrary length over time units), the local dis-persion D was set to zero, and the retardation factor was R 51.5. The numerical integration required for the ATSM wasperformed with 100 and 10 discretization points using an adap-tive recursive Simpson’s rule [Forsythe et al., 1977].

Each method was implemented in Matlab using both a vec-torized algorithm, and a loop-structured algorithm. The algo-rithm that minimized CPU time was selected individually foreach application. A vectorized algorithm is one that uses mul-tiplication by matrices or vectors containing logical operatorsinstead of using a looped if/else structure, where required.Vectorized algorithms were found to be more efficient for allcases except the fine integration discretization runs of theATSM model, for which a loop-structured algorithm was

Table 3. Mean Relative Error, Standard Deviation of Error, and Operation Cost of Particle Positions at T 5 100

MethodTimeStep

Relative Error RelativeCPUTime

Flops

Value, % Standard Deviation, % Absolute Relative

Slow reaction rate withkr 5 0.1 andN 5 100,000

CTHM 1 0.35 0.14 2.5 1.1 3 109 5.7STSM 0.1 0.77 0.54 3.7 5.0 3 108 2.5ATSMa 1 0.54 0.45 11.3 7.7 3 107 0.4ATSMb 1 0.39 0.20 11.0 2.2 3 109 11.0SAMM 1 0.33 0.33 1.0 2.0 3 108 1.0

Moderate reaction rate withkr 5 1 andN 5 100,000


Fast reaction rate withkr 5 10 andN 5 10,000


aRuns performed using a 100-point discretization for probability density function integration.bRuns performed using a 10-point discretization for probability density function integration.


found to be more efficient (although, as seen in Table 3, thevectorized algorithm results in significantly higher floating-point operation counts).

The error in the values of the mean position were very smallin general, and only results for the aqueous concentrationvariance are discussed. These results are presented in Table 3.

4.6. Results and Discussion

The examined methods offer different advantages and dis-advantages. Although the CTHM is computationally simple, itrequires the generation of a large number of waiting times,especially when reaction rates are high, which makes it lessefficient at high reaction rates. The STSM is also simple toimplement. However, this method requires that the time stepbe chosen to ensure that PC3S and PS3C are much smallerthan unity. Otherwise, the method overestimates the secondmoment of the aqueous article distribution. This requirementmakes the STSM inefficient for high reaction rates. As men-tioned in section 4.4, the ATSM requires numerical integrationof the probability density functions and interpolation betweenthe integrated values for each particle for each time step. Thisrequirement makes the method inefficient when the integra-tion is performed with a fine discretization.

Overall, results of comparable quality were obtained withthe CTHM, the ATSM (with fine discretization), and theSAMM, while the STSM produced larger errors. As expected,accuracy was found to decrease for all models when the num-ber of particles was reduced.

The computational costs differ among the methods. For aslow reaction rate the relative number of floating-point oper-ations was lowest for the ATSM (fine discretization), followedby the SAMM, the STSM, and the CTHM. For higher reactionrates the CTHM and STSM require much greater computa-tional effort since the number of phase changes in the CTHMand the number of required time steps in the STSM increasedramatically. For the fast reaction rate both these methodsrequire approximately 25 times more floating-point operationsthan the SAMM. The number of floating-point operationsrequired for the ATSM (fine discretization), however, is belowthat required for the SAMM for all simulations.

The actual CPU time gives additional insight into the mod-els’ performance because the methods differ significantly in thenumber of relational and logical operations required. TheSAMM was faster than all other models for all parametersettings. For the CTHM the ratios of the computational timesrelative to the SAMM are about double the ratios of the flopcounts. For the ATSM, however, the computational time ratiosvary from approximately 11 to 55. The difference between theflops and computational time ratios is due to the fact that sincethe ATSM requires numerical integration, random numbersgenerated for each particle for each time step must be com-pared to n numbers, where n is the number of discretizationpoints used for the numerical integration of the cumulativeprobability distributions.

In order to decrease the CPU time requirements of theATSM method, the runs were repeated with a coarser discreti-zation of the probability density functions. The relative CPUtime decreased but remained an order of magnitude greaterthan that of the SAMM. Furthermore, for high reaction rates,where the cumulative probability density functions exhibitsharp fronts, the ATSM relative error in the second momentincreased significantly.

Therefore the SAMM was found to be at least as accurate as

all other available methods and significantly more computa-tionally efficient than these methods. It is important to notethat the computational efficiency of the examined methodscould potentially be improved. Specifically, for the ATSM abinary search technique could be used for comparing the ran-dom number generated for determining g to the values of thediscretized cumulative density function. Alternatively, an ap-proximate analytical expression for the density functions couldbe found. However, such extensions are beyond the scope ofthis work.

5. Investigation of Double-Peak BehaviorDouble peaks in the aqueous concentration distribution are

sometimes observed in reactive solute experiments such as theones conducted at the Borden [Mackay et al., 1986; Roberts etal., 1986] and Cape Cod [Garabedian et al., 1988; LeBlanc et al.,1991] field sites. Although double peaking can occur as a resultof physical heterogeneity [Freyberg, 1986], it has been sug-gested that double peaks may also be caused by kinetic sorp-tion [Quinodoz and Valocchi, 1993]. Therefore the SAMMmodel was used to investigate the conditions under whichdouble peaks can occur as a result of kinetic sorption.

A large range of parameters was used, but the initial condi-tion always involved all the mass being in the aqueous phase.Double-peaking behavior was found to occur for any value ofthe retardation factor, contrary to the suggestion [Quinodozand Valocchi, 1963] that double peaking can occur only for verylarge retardation factors. The current study suggests that dou-ble peaking is a function of the Damkohler number of the firstkind defined as

DaI 5 klxR/v , (48)

where lx is the initial length of the plume. The Damkohlernumber is a ratio of the advection timescale to the reactiontimescale. Therefore a low Damkohler number indicates thatthe advection rate is fast relative to the reaction rate. Doublepeaking occurred for all retardation factors for DaI # 1 andwas very pronounced for DaI , 1/3. In essence, double peak-ing occurred when the aqueous plume had time to pass aheadof the sorbed plume before the sorbed fraction desorbed. Thiscan be accomplished by a high mean velocity v/R , a slowreaction rate k , or a short length of the plume lx.

In cases where double peaking occurred, the time at whichthe double peak is most pronounced was found to be a functionof the reaction rate and the retardation factor. A dimension-less time was defined as

t* 5 k~R 2 1!t (49)

since double peaking can only occur for R . 1, regardless ofthe value of the Damkohler number. Where double peaking inthe aqueous phase occurs, it starts approximately at time t* ;2, is most pronounced for 3 # t* # 5, and dies out by t* ;10. Maximum double peaking was defined as the time at whichthe concentrations of the two peaks were equal (see Figure 5a).However, if the local dispersion D is sufficiently large or if thetime after which double peaking would occur is sufficientlylong (such as in cases where the retardation factor is extremelyclose to unity), the dispersion-induced smearing of the concen-tration profile can eliminate the double-peaking effect, as wasalso suggested by Bellin et al. [1991].

Results for DaI 5 0.1, 0.33, and 1.0 are presented inFigure 5. The Peclet number, defined as


Pe 5 vlx/D , (50)

was kept constant at a value of 100. The retardation factor waskept constant at R 5 2, demonstrating that double peakingcan occur for low retardation factors. For comparison, anaqueous concentration distribution for R 5 20 and DaI 5 4.0is presented in Figure 6c. No double peaking occurs in thiscase, despite the fact that the retardation factor is very large(R .. 10).

Double-peak formation can also occur in the range 1 #

DaI # 3. However, in this range, the occurrence of the doublepeaks is retardation factor dependent. Double-peak formationfor the higher Damkohler number case is related to the be-havior of the first aqueous moment for solute starting in theaqueous phase. For all retardation factors greater than 2, thevelocity of the aqueous center of mass of solute starting in theaqueous phase (i.e., the rate of change of the first spatialmoment) temporarily drops below its asymptotic value. Thisbehavior has been reported for “very high” retardation factors[Quinodoz and Valocchi, 1993], but a threshold had not yetbeen identified. For retardation factors greater than 12, in fact,the rate of change of this first moment drops below zero. Thisbehavior is illustrated in Figure 7 for R 5 20. This drop in themean velocity can be explained by the small fraction of soluteremaining in the aqueous phase downgradient of the source, asa majority of solute entering uncontaminated zones is sorbed.The sorbed solute remains immobile until the aqueous plumehas passed and is then released. If the retardation factor is highenough, the concentration of desorbing solute is larger thanthe concentration of downgradient aqueous solute, and thevelocity of the centroid is thereby slowed to below its asymp-totic value and for R . 12 is slowed to below zero.

Consequently, double-peak formation in this higherDamkohler number range is insignificant for R , 2 and ismost pronounced for R . 12. Furthermore, the appearance ofthe peaks is different from the DaI # 1 case. For the 1 #

DaI # 3 case the two peaks have approximately the samewidth, whereas for the case of an arbitrary retardation factorand DaI # 1, the upgradient peak is wide relative to thedowngradient peak. Examples of double peaking in the higherDamkohler number range are presented in Figures 6a and 6bfor R 5 20 and DaI 5 1.0 and 2.0, respectively.

Double peaking in the higher Damkohler number range isprobably the behavior that was reported by Quinodoz andValocchi [1993], although the authors discussed no limitationson the reaction rates that lead to the double-peak formation,whereas current results suggest a rate limitation based on thedependence on the Damkohler number. Another indication ofthe dependence on reaction rates is that as the reaction ratesincrease, the local equilibrium model becomes asymptoticallyvalid, which never exhibits double peaking. Therefore it is clearthat double-peak formation is rate-dependent.

It should be noted that we have performed this investigationassuming a linear driving-force model for sorption, and it isunclear if a different kinetics model for sorption would yieldidentical results. Specifically, double peaking in the higherDamkohler number range may be an artifact of assuming lin-ear driving-force kinetics, since it appears to be a result of thevelocity of the center of mass of the aqueous phase, for solutestarting in the aqueous phase, temporarily dropping below itsasymptotic value for R . 2. Such behavior has neither beenconfirmed nor refuted for other nonequilibrium sorption models.

Figure 5. Maximum peaking as a function of Damkohlernumber with R 5 2, v 5 1.0 [L/T], lx 5 1.0 [L], and0.05 , k , 1.0 [1/T]; 102,400 particles used. (a) Concen-tration profile, DaI 5 0.1, t* 5 3.6. (b) Concentrationprofile, DaI 5 0.33, t* 5 3.2. (c) Concentration profile,DaI 5 1.0, t* 5 3.0.


6. Conclusions and Future DirectionsWe have presented zeroth, first, and second spatial moments

for kinetically sorbing solutes following a linear reversible masstransfer model for both initial and both final phases. Owing tothe linearity of the problem, moments resulting from all pos-sible initial distributions between the phases can be obtainedby superposition. Moments resulting from nonpoint and non-instantaneous sources can also be determined by summationand convolution, respectively. Therefore a general analyticaldescription of solute transport is available for any source interms of effective velocities and dispersion coefficients.

The general expression relating the Fourier coefficients tothe spatial moments of the solute (equation (13)) applies alsoto the third and higher moments. Hence extending our analysisto these moments would be straightforward. However, expres-sions become increasingly long and complicated for highermoments, and since these moments were not necessary for thepresented applications, their development is beyond the scopeof the current study.

The analytical moment expressions derived in this paperallow for the implementation of a highly efficient and accuraterandom-walk method. The approach automatically simplifiesto the local-equilibrium case for very large reaction rates,whereas other methods, such as the continuous time stepmethod and the small time step (operator split) method, be-come inefficient and/or inaccurate at high reaction rates, wherethe local-equilibrium assumption becomes more applicable.Therefore the implementation of these methods requires oneto determine, a priori, whether the local-equilibrium assump-tion is valid. No such limitations are encountered with the newmethod, as the entire range of reaction rates is covered effi-ciently and accurately.

Using the new technique, we found that, contrary to pasthypotheses, double concentration peak behavior resultingfrom kinetic sorption can occur even for low retardation fac-tors, but it is controlled by the Damkohler number of the firstkind, with double peaks occurring for DaI # 1 for all retar-dation factors. For higher retardation factors, double peakingcan also occur for higher Damkohler numbers but is limited to

Figure 6. Maximum peaking as a function of Damkohlernumber with R 5 20, v 5 1.0 [L/T], lx 5 1.0 [L], and0.05 , k , 1.0 [1/T]; 102,400 particles used. (a) Concen-tration profile, DaI 5 1.0, t* 5 4.2. (b) Concentrationprofile, DaI 5 2.0, t* 5 4.2. (c) Concentration profile,DaI 5 4.0, t* 5 4.0.

Figure 7. Rates of change of normalized first moments withR 5 20, v 5 1 [L/T], k 5 0.1 [1/T], D 5 0, and m*1 andn*1 as normalized first spatial moments for solute in the aque-ous and sorbed phases, respectively.


the range DaI # 3. Double peaking in this higher Damkohlernumber range, however, may be an artifact of assuming lineardriving-force kinetics, as other nonequilibrium sorption mod-els may not exhibit a transient aqueous center of mass velocitydrop below its asymptotic value for solute starting in the aque-ous phase.

Although the analysis presented in this paper was limited tospatially and temporally constant coefficients, extensions toheterogeneous fields are possible. The analytical results foundin this paper could be extended to spatially varying velocityfields with negligible local dispersion and constant sorptionproperties. These analytical expressions could be derived byusing the stochastic-convective approach [Simmons, 1982; Sel-roos and Cvetkovic, 1992]. For cases where the sorption param-eters are not constant or where the local dispersion is notnegligible, analytical expressions cannot yet be readily ob-tained. However, the random-walk method developed in thiswork can be applied directly to gradually varying heteroge-neous media without significant increases in the computationalcost, although the potential limitations of such an applicationrequire further study.

Appendix A: Derivation of First MomentEquations

The equations for the first moments in Fourier space are(from (13))

m1 5h

~22pj!CsU

s50

n1 5r

~22pj!SsU

s50

. (A1)

Taking the first derivative of (11) and (12) with respect to s ,substituting (16) and (A1), and setting s 5 0 yields

m1 51

~R 2 1! S 1k

n1

t 1 n1D (A2)

vm0 5m1

t 1n1

t . (A3)

Letting the initial conditions for the first moments be

n1~t 5 0! 5 0 m1~t 5 0! 5 0, (A4)

the solutions to (A2) and (A3) are developed after substitutingthe transient solution for the zeroth moment m0.

Appendix B: Derivation of Second MomentEquations

The equations for the second moments in Fourier space are(from (13))

m2 5 2h

4p2

2Cs2 U

s50

n2 5 2r

4p2

2Ss2U

s50

. (B1)

Taking the second derivative of (11) and (12) with respect to s ,substituting (16), (A1), and (B1), and setting s 5 0 yields

m2 51

~R 2 1! S 1k

n2

t 1 n2D (B2)

vm1 1 Dm0 5m2

t 1n2

t . (B3)

Letting the initial conditions for the first moments be

n2~t 5 0! 5 0 m2~t 5 0! 5 0, (B4)

the solutions to (B2) and (B3) are developed after substitutingthe transient solutions for the zeroth moment m0 and the firstmoment m1.

Acknowledgments. Funding for this study was provided by a Stan-ford Graduate Fellowship for Anna M. Michalak and by the NationalScience Foundation under project EAR-9522651, “Factors AffectingSolute Dilution in Heterogeneous Formations.” The views presentedin this paper do not necessarily reflect those of Stanford University orthose of the National Science Foundation. The authors are indebted toO. Cirpka, J. Cunningham, T. Erickson, and M. MacWilliams Jr. fortheir technical input and reviews.

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P. K. Kitanidis and A. M. Michalak, Department of Civil and En-vironmental Engineering, Stanford University, Terman EngineeringCenter, Stanford, CA 94305-4020. ([email protected];[email protected])

(Received July 26, 1999; revised April 10, 2000;accepted April 12, 2000.)


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