Multiple-Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media with Pore-Scale...

WATER RESOURCES RESEARCH, VOL. 31, NO. 10, PAGES 2383-2400, OCTOBER 1995

Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity

Roy Haggerty and Steven M. Gorelick Department of Geological and Environmental Sciences, Stanford University, Stanford, California

Abstract. Mass transfer between immobile and mobil• zones is a consequence of simultaneous processes. We develop a "multirate" modeithat allows modeling of small- scale variation in rates and types of mass transfer by using a series of first-order equations to represent each of the mass transfer processes. The multirate model is incorporated into the advective-dispersive equation. First, we compare the multirate model to the standard first-order and diffusion models of mass transfer. The spherical, cylindrical, and layered diffusion models are all shown to be specific cases of the multirate model. Mixtures of diffusion from different geometries and first-order rate-limited mass transfer can be combined and represented exactly with the multirate model. Second, we develop solutions to the multirate .equations under conditions of no flow, fast flow, and radial flow to a pumping well. Third, using the multirate model, it is possible to acCUrately predict rates of mass transfer in a bulk sample of the Borden sand containing a mixture of different grain sizes and diffusion rates. Fourth, we investigate the effects on aquifer remediation of having a heterogeneous mixture of types and rates of mass transfer. Under some circumstances, even in a relatively homogeneous aquifer such as at Borden, the mass transfer process is best modeled by a mixture of diffusio n rates.

1. Introduction and Literature Review

The focus of this paper is on the impact of multiple and simultaneous rates of mass transfer. First, we review the commonly used first-order models and show how they are mathematically equivalent to each other. Second, we present the multiple rate, or "multirate," mass transfer model for use in representing multiple and simultaneous rates of mass transfer. Third, we show that the multirate model is a "superset" of other prevalent mass transfer models. In other words, in reduced form, the multirate model is mathematically equivalent to the spherical, cylindrical, or layered diffusion models, or the standard firSt-order mass transfer model. We show that the general multirate model can also be used to represent any mixture of these components. Fourth, we apply the multirate model to mass transfer in the Borden sand and to a hypothetical example where small-scale heterogeneity in mass transfer processes is particularly important to groundwater remediation. We give several solutions of the multirate model for different flow conditions.

Rates of mass transfer between zones of mobile and immo-

bile contaminant correspond in a general but often unlmown way to small-scale variations in aquifer properties. These properties are numerous and include at least the following: (I) the types of minerals and their spatial distribution [Pignatello, 1990; Wood et al., 1990; Ball and Roberts, 199 la, b; Barber et al., 1992]; (2) the geometry, chemistry, and mineralogy of coatings on the surfaces of aquifer particles [Weber et al., 1991; Barber et al., 1992]; (3) the volume, size, and geometry of macropo- rosity or microporosity in aquifer particles and aggregates of particles [Rao et al., 1980; Pignatello, 1990; Wood et al., 1990; Ball and Roberts; 1991a, b; Harmon et al., 1992; Harmon and Roberts, 1994]; (4) the external and internal geometry of small

Copyright 1995 by the American Geophysical Union.

Paper number 95WR01583. 0043-1397/95/95WR-01583505.00

clay lenses or other low-permeability material, and the proportions of this material [Shackelford, 1991; National Research Council, 1994; Wilson, 1995]; (5) variations in hydraulic conductivity within the aquifer [Valocchi, 1989; Cvetkovic and Sha- piro, 1990; Selroos and Cvetkovic, 1992; Quinodoz and Valocchi, 1993]; (6) the quantity and distribution of organic material [Karickhoff, 1984; Grathwohl, 1990; Barber et al., 1992]; and (7) the chemistry of the water and contaminant [Curtis et al., 1986; Brusseau and Rao, 1989a; Weber et al., 1991]. Clearly, there are many media properties, varying greatly through space, that influence rate-limited mass transfer. It is apparent that one should account for multiple rates of mass transfer at the small scale, as they affect larger-scale transport; yet the majority of existing models account for only one mass transfer process.

Given the multitude of processes and scales that contribute to rate-limited mass transfer, it is unlikely that a simple model such as the "one-site" or "two-site" first-order mass transfer

model [Coats and Smith, 1964; van Genuchten and Wierenga, 1976; Selim et al., 1976; Cameron and Klute, 1977] can accurately predict mass transfer in a complex, heterogeneous aquifer. Models that incorporate Fickian diffusion through assumed geometries [R•'61e• al., 1980; van Genuchten et al., 1984] may be adequate to predict mass transfer at the laboratory scale [Wu and GschWend, 1986; Ball and Roberts, 1991b], but may not be satisfactory to predict concentration histories in a heterogeneous medium. Natural soils may contain an extremely wide range of particle sizes, as well as aggregates of particles and lenses of clay or other low-permeability material. As noted by Wu and Gschwend [1988], Ball and Roberts [1991b], and Harmon et al. [1992], a model with a uniform aggregate size may not successfully model solute transport in natural porous media. Accurate representation of multiple mass transfer processes is essential to predict the tail concentrations during aquifer remediation and to assess the length of time needed for remediation.

2383

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2384 HAGGERTY AND GORELICK: MULTIPLE-RATE MASS TRANSFER MODEL

Some work, mostly in the chemical engineering and chromatography literature, evaluates the effects of mass transfer properties that vary within a single representative elementary volume. The effects on solute transport of variability in mass transfer properties at the small scale appear to be just as significant as larger-scale variability. According to Weber et al. [1992, p. 1962], "particle-scale heterogeneity and distributed reactivity can significantly affect contaminant transport processes at [the] field scale." In chromatographic systems, variability in the particle size and sorption properties has long been known to affect solute transport [Purnell, 1962; Doughany, 1972]. By comparing transfer functions, Villermaux [1981, 1987, 1990] maintains that the sophistication of explicitly accounting for transient diffusion into and out of particles of various sizes may be unnecessary and that the diffusion model can be sim- plified to a first-order model with multiple rates of mass transfer. Villermaux presents a first-order transfer function approximation of the variable particle size model with transient diffusion, which he calls transport with "retention sites in parallel." This model has been examined by Valocchi [1990] and by $ardin etal. [1991] for use with solute transport in groundwater. Valocchi [1990] noted that certain distributions of first- order mass transfer coefficients and diffusion from certain dis-

tributions of spheres could yield similar breakthrough curves. ElSewhere in the groundwater literature, Brusseau et al.

[1989] developed a multiprocess nonequilibrium model. This model consists of three first-order mass transfer reactions op- erating simultaneously, two in parallel, and the third in series with one of the first two. BruSseau et al.'s model allows greater flexibility than previous models in modeling mass transfer processes because of the greater number of possible reactions.

The multirate model is a mass transfer relationship with a distribution of rate coefficients coupled to an advective- dispersive solute transport model. Our work builds upon the theory developed by l•llermaux [1981, 1987, 1990], Brusseau et al. [1989], Valocchi [1990], and $ardin et al. [1991], who recognized the importance of accounting for more than one mass transfer process.

The main theoretical contribution of the present work is to show precisely how a mass transfer model with a distribution of first-order rate coefficients is mathematically equivalent to the diffusion models of mass transfer, and to provide a basis for comparing them. A result of this work, for example, shows that the spherical diffusion model is simply a first-order model with a specific distribution of rate coefficients. Many other distributions are possible that may be more useful for modeling rate- limited mabb U-ttnbt•. r'urtn•more, we are able to simultaneously model the effects of sorption and diffusion in different types of immobile zones, such as sorption sites, clay layers, and aggregates of particles. The multirate model can incorporate data from laboratory studies of sorption and diffusion, and field studies of the geologic environment, perhaps to improve predictions of concentration histories or contaminant remediation.

2. Overview of Common Mass Transfer Models

2.1. First-Order Models

In this section, we briefly review the most commonly used first-order mass transfer models and show how they are equivalent. This portion of the work extends the work of Nkedi-Kizza et al. [1984], who showed that two commonly used bicontinuum or "two-site" first-order models are equivalent to each other.

We show that both these models are mathematically equivalent to a simpler one-site model after a change of variables. The set of commonly used solute transport models may all be expressed, without loss of generality, in one dimension, as

OC m Oeim A -•-+ B •-•= D

02Cm OC m ax 2 - Q ax (1)

The first-order mass transfer models may be expressed as

OCim B --•-= o•(XCm- Cim) (2)

where A [dimensionless] is the parameter that relates concentration within the mobile zone to concentration within a unit

volume of the porous medium; B [dimensionless or M/L 3] is the parameter that relates concentration within the immobile zone to concentration within a unit volume of the porous medium; Cm [M/L 3] is the aqueous concentration in the mobile zone; Cim [M/L3 or M/M] is the sorbed concentration in the immobile zone for chemical models, or the aqueous concentration in the immobile zone for other models; D [L 2/T] is the hydrodynamic dispersion coefficient; Q [L/T] is the Darcy velocity; •0 [ T- 1 or ML - 3T- 1 ] is the first-order mass transfer rate coefficient; and • [dimensionless or L 3/M] is the distribution coefficient relating Cim to Cm at equilibrium. Some variables and parameters have two possible sets of units due to the different sets of units on immobile concentration.

The commonly used first-order mass transfer models can be classified as "chemical" or "physical," and each of these can be subdivided into one-site and two-site models. Mathematically, all are equivalent to each other and their solutions are inter- changeable after a change of variables. Their equivalence and how the models may be interchanged is shown in Figure 1. By exchanging the parameter groups in a given row with the constants and variables in (1) and (2), any model may be converted to any other model. Although not shown in the figure, initial and boundary conditions may also need transformation from one model to another. All parameters and variables are defined in the notation section. The multirate model, introduced at the bottom of Figure 1, is very similar in form to the standard first-order models, but its solutions are much different and it cannot be expressed as a standard first-order model. The mathematics of the multirate model will be introduced in sec-

tion 3.2.

The physical model of mobile-immobile mass transfer may be rewritten in a general form that will be used for the remain- der of this paper:

OCm ot +0-w=v. Vcm -Vcm

- -- (e - -- (3)

where D [L2/T] is the hydrodynamic dispersion tensor; v [L/T] is the pore water velocity vector; R m [dimensionless] is the retardation factor of the mobile zone; q [T-1] is a source or sink of water; and • [M/L 3] is the concentration within the source or sink. Hereafter the right-hand side will be designated by the operator •(*), where the asterisk represents the dependent variable, Cm in this case. The term/3 [dimensionless] is the "capacity ratio" and is •he ratio of total contaminant mass in the immobile zone to the mass in the mobile zone at equilibrium:

HAGGERTY AND GORELICK: MULTIPLE-RATE MASS TRANSFER MODEL 2385

Solute Transport Equation: •C m 3Cim 32Cm 3C m A 3t +B 3t =D 3x 2 -Q 3x

First-Order Mass Transfer Equation: aCim OkCm Cim) B 3t = co -

•.• Constants & '••riables A B Cm Cim f0 • D Q Auxiliary Definitions:

Model • One-Site (•) 0 p c E Pbk2 K d 0cti•v 0v

(5) _ (5) _ Two_Site (a) o+ (1-P)p b c c 1 Pbk2 K d 0CtLV 0V ca=KdC {5) f'PbKd

Mobile/ Immobile (3) Om elm Cm ½im OimOt '(6) 1 OmOtLV OmV Mobile/ Immobile (4) Om + 0im + with Equil. Sorp. fPbr d (l_f)Pbr d Cm Cim 0imOt '(6) 1 0mCti•v 0mY •=Kd[fCm+(1-f)Cim]

Multi-Rate i fmPbKd, m i (f•,)jpb0Cd•,)j C m (C'an)j (Oim)j• j' 1 mGtLV OmV U= fmKdmCm +Z (fimt(rd,imt(Cimt j=l

Figure 1. Mathematical equivalence of commonly used first-order mass transfer models. The multirate model, shown at the bottom, contains N first-order equations and is described fully in section 3. Definitions of the constants, parameters, and variables are given in the notation section. For rim and f the mobile zone may contain some fraction of sorbed mass in sorption equilibrium with the mobile zone; although this fraction of mass is not truly mobile, we consider it to be part of the mobile zone. The sum of fm and all (fim)i is 1. The footnotes are as follows: (1) Various authors, including Lapidus and Amundson [1952], Oddson et al. [1970], and Bahr and Rubin [1987]. (2) Selim et al. [1976], Cameron and Klute [1977], and Nkedi-Kizza et al. [1984]. (3) Coats and Smith [1964]. (4) Van Genuchten and Wierenga [1976] and Nkedi-gizza et al. [1984]. (5) The term (1 - f) appears before K d in Nkedi-Kizza et al. [1984] due to a slightly different definition of the immobile concentration. Our definition of the immobile concentration for this model is the mass per unit mass of the specific sorption site available, whereas Nkedi-Kizza et al. defined the corresponding immobile concentration as mass per unit mass of total sorption sites available. (6) The porosity is commonly lumped together with a' [i.e., Nkedi-Kizza et al., 1984]. For the sake of consistency with diffusion models and with the multirate model, we separate them.

Oim -}- ( ] -- f) pbKd gimOim [• : Om _}_ f[jbgd '- gmOm (4)

where Pb [M/L 3] is the bulk density of the porous medium; f [dimensionless] is the fraction of the sorbed phase in sorption equilibrium with the mobile phase; and Rim [dimensionless] is the retardation factor associated with the immobile zones. The

retardation factors are defined as

Rrn: i -t- [fj•bg'd/0rn] (5)

Rim-- ] + [(1--f)pbKd/Oim] (6)

The mass transfer equation (2) becomes

O--T = "(Cm -- C,m) (7)

where a - a'/Rim and where a' is the first-order mass transfer rate coefficient for a physical model with no sorption.

2.2. Diffusion Models

One of the most commonly used models of rate-limited mass transfer is the diffusion model, whereby diffusion into and out

of an immobile zone is explicitly modeled using Fick's law [Rao et al., 1980; van Genuchten et al., 1984]. In addition, solute sorbs at equilibrium to surfaces in both the mobile and immobile zones. The solute transport equation may be written

a½ m a•im

where/3 is the total capacity ratio for the diffusion model. The immobile concentration •im [M/L3] is the immobile concentration averaged over the immobile zone, and is given by

v0a •im : •-• r(V-1)c r dr (9)

where c r [M/L 3] is the actual immobile zone concentration, varying with distance r [L] from the center of the immobile zone; a [L] is the distance from the center to the edge of the immobile zone; and •, is the dimensionality of the immobile zone. If •, = 1, the immobile zone is composed of layers; if •, - 2, the immobile zone is composed of cylinders; if •, = 3, the immobile zone is composed of spheres. The concentration


within the immobile zone is determined from the diffusion

equation:

ocr a0[ ocr] 0-•-= r (v-l) Or r(V-1) Or ] (10) where D a [L 2/T] is the apparent pore diffusion coefficient and is equal to the effective pore diffusion Dp [L2/T] divided by the immobile zone retardation factor Rim [ Wu and Gschwend, 1986; Harmon et al., 1989]. The boundary conditions for (10) are

OC r --= 0 at r = 0 (11) Or

C r: C m at r = a (12)

Analytic solutions for this set of equations for no-flow and various initial conditions are given by Crank [1975]. The semianalytic solution of these equations for radial flow can be found by taking the Laplace transform of (8) to (10) and solving with the technique outlined for the multirate model in section 5.

model, in which everything in Figure 2a is modeled by a single set of multirate equations. On the right of the figure, we show that diffusion in and out of grains or grain aggregates is modeled by spherical diffusion. Surface sorption reactions are modeled by a linear first-order equation. On the left, we show that diffusion in and out of clay layers and pods is modeled as diffusion in ideal layers or in cylinders. All these processes are modeled simultaneously. The multirate model could also represent diffusion in some other geometry or combination of geometries if its size, shape, and diffusion rate were known.

In this paper we apply the multirate model to mass transfer in saturated porous media only. Although it is beyond the scope of the present paper to explore other applications, we see the model as having potential use in unsaturated as well as saturated systems. In unsaturated systems it may be necessary to add extra terms to describe mass transfer between water and

air.

3.2. Mathematical Model

The multirate solute transport equation can be written as follows:

3. The Multirate Model

In this section we describe the multirate model. We provide a description of the conceptual model, and then we show how this conceptual model can be framed mathematically.

3.1. Conceptual Model The multirate model is different from other mass transfer

models in that it allows numerous types and rates of mass transfer to occur simultaneously. A similar model, first presented by Villermaux [1981], allows for a distribution of resi- dence times, which are inversely proportional to first-order mass transfer coefficients. The model describes mass transfer

between a mobile zone and any number of immobile zones of varying properties, and is coupled with the advective-dispersive transport model for solute transport. We extend the model for use in modeling subsurface contaminant transport and show how it is mathematically equivalent in reduced form to diffusion models of mass transfer. The mobile-immobile mass trans-

fer may be diffusion into/from spheres, cylinders, layers, or some other geometry, or the mass transfer may be a first-order • surface reaction. Total mass transfer in the system is allowed to be any combination or probability density function of these.

Figure 2a is a schematic representation of multiple mass transfer processes that occur in a porous medium. As discussed above, several different properties of a porous medium result in rate-limited mass transfer. At the grain scale, shown on the right of the figure, a dissolved compound may be held in small stagnant zones of water or "side pockets," or stagnant zones around the outside of grains. Intraparticle pores and fractures, and larger aggregates of grains, are also the cause of slow diffusion. Additionally, the compound may be adsorbed to any of the internal or external surfaces of minerals and mineral

coatings. At a slightly larger scale, shown on the left of Figure 2a, contaminant may diffuse into and out of clay layers or "pods." This larger scale is approximately the scale of a grid cell in many numerical models. Each immobile zone or phase results in a different rate of uptake from or release to the mobile zone, as shown by the length of the arrows.

Figure 2b is a representation of the multirate mass transfer

N

OCm O(Cim)j •(Cm ) (13) Ot + • 13j Ot - j=l

The mass transfer equations for the multirate model are

a(Cim)j Ot -- Olj[C m -- (Cim)j], j = 1, 2,..., N (14)

where

Olj ---- ol;/(Rim)j (15)

The ratio of total contaminant mass in each of the immobile

zones to the total mass in the mobile zone at equilibrium is

N N (Rim)j (aim) j E •j-- E Rrnarn j=l j=l

=/3 (16)

where all variables are exactly the same as those in the standard first-order mass transfer model (3) to (7), except that there are now N distinct immobile phases, each designated by an indexj. All variables associated with an immobile zone have this index. The variable (½im)j [M/L3] is the concentration in the jth immobile zone; a i [T -1] is the apparent first-order mass transfer coefficient for the j th immobile zone, and so forth. The capacity ratios/3i [dimensionless] are the capacity ratios of each of the immobile zones, and/3 is the total capacity ratio of all immobile zones added together, as it is in the standard first-order mass transfer and diffusion models. The

retardation factor for the mobile zone is

Pbgd,mf m Rm-- 1 + (17)

while the retardation factors for the immobile zones are

Pb(gd,im)j(fim)j (Rim)j = 1 + (aim) j (18)

where gd, m [L 3/M] is the distribution coefficient of the mobile zone; fm [dimensionless] is the mass fraction of the sorbed phase in sorption equilibrium with the mobile zone; (Kd,im)i [L 3/M] is the distribution coefficient of the jth immobile zone;


(b) Multi-rate model

Figure 2. (a) Cartoon illustration of a porous medium with heterogeneous mass transfer processes indicated by the arrows. Mass transfer limitations in the right of the figure are due to various "side pockets," intraparticle porosity, and sorption. At the larger scale, shown at the left of the figure, mass transfer limitations are due to zones of low permeability such as clay layers, where slow diffusion may be the dominant transport process. (b) Illustration of how a multirate mass transfer model would simultaneously describe the various mass transfer processes. Diffusion in the grains and grain aggregates is modeled as spherical diffusion, while diffusion at the scale of day layers is modeled as layered and cylindrical diffusion. Surface reactions are modeled as first-order linear reactions. Dark areas refer to high concentration, and light areas refer to low concentration.

and (fim)j [dimensionless] is the mass fraction of the sorbed phase in sorption equilibrium with the j th immobile zone. The sum of fm and all (fim)j is 1.

Let us examine the mass transfer parameters in the multirate model. The two most important parameters are the apparent mass transfer coefficient a i and the capacity ratio /3•. These parameters vary locally with changes in the mass transfer properties in the aquifer. The capacity ratio should be thought of as a weighting function: For everyj (i.e., for each type and rate of local mass transfer), a weight/3• is assigned to each associated rate coefficient a•. The sum of the weights must add up to the total capacity ratio /3. Although the functions a• and /3• are made up of various other functions and variables, such as (Kd,im)•, (f•m)•, and (0•m)•, in practice it is very unlikely that each of these could be estimated independently. Therefore a• and/3• are the basic parameters of the model, and any estima- tion procedure would try to find values for these rather than their constituent parts.

The parameters a• and /3• should be considered a single

distribution rather than two sets of N independent parameters. In one case, this distribution might be statistically based, such as a uniform distribution, a lognormal distribution, or some other probability density function characterized by a mean and a variance. Alternatively, the distribution may be physically based, and directly related to diffusion. We show in the next section how a specific distribution of a• transforms the multirate model into a model of diffusional mass transfer by diffusion into/ from spheres, cylinders, or layers. In a third case the values of a• and/3• may be determined by a statistical distribution of diffusion rates. In any case, the values of a• and/3• are determined by the parameters of a distribution, whether that distribution has a physical interpretation or a statistical interpretation.

The choice of rate coefficient distributions will be made

based on the amount of information that one has for a given medium. If detailed laboratory studies are available it may be possible to use a distribution that describes diffusional mass transfer; examples of this are given in section 5. In other cases where such detailed information on diffusion and sorption is


unavailable, a probability density function may be chosen a priori (i.e., lognormal). The parameters of the distribution (mean, variance) would be fit to available data.

Distinguishing between equilibrium and nonequilibrium sites, or between mobile and immobile zones, is a matter of determining which mass transfer rates are fast enough relative to solute transport to be lumped together as equilibrium. Much work has been done on this problem for the standard first- order model [i.e., Valocchi, 1986; Bahr and Rubin, 1987]. This work is directly applicable to the multirate model for each individual mass transfer relationship and may be used to dis- tinguish equilibrium from nonequilibrium. In the case of one- dimensional flow and transport, each mass transfer reaction has a Damkohler I number Da/i associated with it:

Da/j = [aj(/3• + 1)RmL]/v x (19)

where L [L] is the length scale of the problem and v x [L/T] is the pore water velocity. For a column experiment, L would be the distance from the input to the point of measurement. This Damkohler I number reflects the rate of mass transfer

relative to the rate of advective solute transport, for the j th mass transfer relationship. If Da/i is greater than approximately 100, the mass transfer relationship is effectively at equilibrium [Bahr and Rubin, 1987]. Any sorbed mass associated with the j th site would effectively be part of the mobile zone or domain. Additionally, any sites with Da/• greater than 100 could be lumped together and modeled with a single equilibrium mass transfer relationship.

4. Equivalence of Multirate Model to Diffusion Models

In this section we show that the multirate model is a gener- alization of the various different models of diffusion-limited

mass transfer. We show the sets of parameters that make the multirate model mathematically equivalent to diffusion into/ from spheres, cylinders, or layers. We also show how the multirate model can be used to represent a heterogeneous mixture of mass transfer processes that occur in aquifer particles or clay layers and pods, or that are otherwise described by first-order surface reactions.

One way of comparing the multirate model to the diffusion models is to compare the concentrations generated by each model to one another. We will perform this comparison on the Laplace transform of the solute transport models because the Laplace-domain models are much simpler than the time- domain models. The complete development of the Laplace- domain equations is given in Appendix A. By transforming (13) and (14) and combining, the Laplace-domain solution for the multirate concentrations is

SCm- Ct q- (S•m- (Cim)j) -- •(•--•m) : 0 (20) m '= Sq- Olj

where s is the Laplace parameter; c• is the initial mobile concentration; and (c •m)j is the initial immobile concentration in the j th immobile zone. Overbars above the concentrations indicate the Laplace-domain concentrations. By transforming (8)-(10) and combining, the Laplace-domain solution for the spherical diffusion model is

SC m -- C • n q- 3/3(S•mm- •Jm) S

'[coth(••s) - ••sl-] - 3•(•mm) = 0 (21)

where •Jm [M/L3] is the uniform initial concentration within the sphere. For the purposes of this paper we will examine only cases where the initial solute concentration of the sphere is uniform, so (c Jm)j = • Jm over all j. If we put the solution for the multirate model on the left-hand side, put the solution for the spherical diffusion model on the right-hand side, and can- cel like terms we get the following equation:

• /3•a• ] j [coth (•D•2•s) - •5s1-1 •__•11s+a•l =3/3 • 1 (22)

If a solution to (22) can be found for % and/3• (j = 1, 2, -.., •) that holds true for all/3 and D a/a 2, and over all s, which is complex valued, then the multirate model is mathematically equivalent to the spherical diffusion model. This equation is analogous to a nonlinear integral equation, and it would be extremely difficult to solve without any information about % and/3i. However, using our knowledge of the spherical diffusion model, we are able to find a "trial solution":

otj = j2•'2(Da/a2 ) j = 1, 2, 3, ..., • (23)

/3j = (6/j2• 2)/3 j = 1, 2, 3, ..., • (24)

This trial solution is based on knowledge of diffusion into or out of a sphere that is placed within a well-stirred infinite solution, and is completely developed in Appendix B. Inserting this trial solution into (22) and rearranging, we have

Da : -- coth s 2 •- j2•.2 S j=l S q- •-

Dal a 2 s

(25)

Equation (25) is true over all 13, Da/a 2, and s. Although this solution to (22) may not be mathematically unique, we believe it is the only solution that has positive, real values for all % and /3•, and is therefore the only physically meaningful solution. We numerically evaluated the equality over a very wide range of/3, Da/a 2, and s, and found it to be accurate to at least 10 digits. With greater numerical precision, we expect that the equality would hold to a greater number of digits. An alternative development of the mathematical equivalence of the multirate and diffusion models, using temporal moments, is given in Appendix C. In addition to solving (22) and the equations shown in Appendix C, the parameters in (23) and (24) were tested in the solutions for diffusion with various boundary conditions as given by Crank [1975].

The same development is possible for layered or cylindrical diffusion, or for some other model of mass transfer. The solution to equation (22), which we term the "multirate series", is shown with the corresponding solutions for cylinders and layers in Table 1.

When the series in Table 1 are put into the multirate model, the multirate model becomes a diffusion model. It is not an

approximation. The multirate series for layered diffusion and spherical diffusion are very simple; the multirate series for cylindrical diffusion is slightly more complicated, involving the zeros of the zero-order Bessel function of the first kind


[Abramowitz and Stegun, 1972]. It is also possible to find the multirate series that transforms the multirate model into dif-

fusion models of other geometries, but this is reserved for future work.

The multirate series for spherical and layered diffusion are graphed in Figure 3. The figure shows the j th capacity coefficient normalized by the total capacity coefficient, versus the jth mass transfer rate coefficient scaled by the ratio Da/a 2. When the values of a i and /3i shown in Figure 3 are put into the multirate model, the model is mathematically equivalent to a diffusion model. Figure 3 also shows the approximations to layered and spherical diffusion first suggested by Valocchi [1985] for use in a standard first-order mass transfer model. In the figure, we see that the first few mass transfer rate coefficients in the multirate series dominate the series, and that larger coefficients have very small weights. This is why the first few terms of the multirate series do a very good job of approx- imating the entire series. In the figure we also see that the Valocchi [1985] approximation for the standard first-order model lies very close to the first term of the multirate series. In fact, the standard first-order approximation is equal to the harmonic mean of the multirate series:

a s = •- • (26)

where a• is the first-order mass transfer coefficient for the approximations to the diffusion models. Van Genuchten [1985] and Wu and Gschwend [1988] have suggested slightly larger values of a• to approximate the diffusion models. Such larger values do a better job at early times, but cause a poorer approximation for late times. Figure 3 suggests that if we are interested in late-time concentrations and we want to use a

standard first-order model, it would be best to use the first •erm of the multirate series. For spherical diffusion, this value is af = 'n'2Da/a2; for cylindrical diffusion, a,r • 5.78Da/a2; and for layered diffusion, ai = ,n'2/4(Da/a2).

For practical use, the multirate series must be truncated at some point, N. Fortunately, these are well-behaved series, and if they are truncated properly, errors are very small even when using only the first few terms of the series. In our experience, a satisfactory way of truncating the series is to use the first (N - 1) terms of the multirate series just as they are written

Table 1. First-Order Coefficients and Capacity Coefficients That Transform Multirate Model Into Diffusion Models of

Different Geometries and Into the Standard First-Order

Rate Transfer Model

Layered diffusion (2j -- 1)2'I1 '2 Da 8 4 a 2 (2j- 1)2'I1 '2

Da 4

Cylindrical diffusion u• a•- •/3 D a 6

Spherical diffusion j2'n'2 a2 •/3 j2.tr2 Standard first-order model a, j = 1 /3, j = 1

The value u i is thejth solution OfJo(ui) = 0 andJ o is the zero-order Bessel function of the first kind. For the diffusion models, j = 1, 2, ..., o•, where j is the index for the multirate series.

Approximation to Layered

Diffusion N•

Relative

Capacity Coefficient

-]

0.1

0.01

lO -3

10 4

Approximation to Spherical

Diffusion

I I I

1 10 100 10 3

Relative Mass a 2 Tr•fer Coefficient % • [-]

10 4

Figure 3. The terms of the multirate series, starting fromj = 1. These multirate series exactly transform the multirate model into the spherical and layered diffusion models. The parameters that approximate these diffusion models with a standard first-order model are also shown at the top of the figure. Note that/3• is the capacity coefficient for the j th immobile zone in the multirate model, while/3 is the total capacity coefficient for the diffusion model.

in Table 1. For the Nth term, we us•e %=N as it is written in the table. However, we take all the remaining/3•, from j = (N + 1) to j = o•, and add them into /3•=N. The truncated terms correspond to fast'mass transfer rates and equilibrium behavior, and therefore only their weights need to be accounted for to provide a good approximation. Take, for example, the case of spherical diffusion in a batch reactor, which will be discussed in the next section. With N = 100, the error has a maximum of 1%, and this is at the extremely early time of 10-6(Da/a2)t. Error falls to 10-4% at 10-1(Da/a2)t. With N = 25, the maximum error is still only 1% at 10-4(Da/a2)t, and falls to 10-2% at 10- l(Da/a2)t. With N = 5, the error is less than 1% after 10-2(Da/a2)t. In contrast, the error for the standard first-order model, N - 1, approaches 100%. In practice, the value of N should be large enough that all truncated terms represent fast mass transfer and therefore can be represented by instantaneous equilibrium. For a one-dimensional advective-dispersive-reactive system, N would be chosen large enough that Da/N from (19) is greater than 100.

An important feature of the multirate model is that it is very easy to combine different diffusion geometries and surface reactions. To do this, we simply combine the various multirate series in the appropriate proportions. For example, if we had a laboratory sample with 30% small spheres and 70% large spheres, we would use the values of % given in Table 1 for each sphere, and multiply the corresponding/3• values by 0.30 and 0.70, respectively. If we decide to truncate the multirate series at N = 25, we would need 25 % and/3• values for each of the two sphere sizes, for a total of 50 % and 50 /3• values. These 100 numbers, however, are completely determined by the D a/a 2 and /3 values of the two spheres and should not be thought of as independent parameters. Note that the sum of the/3• is not necessarily 1.0, but may be any value, depending on the ultimate amount of contaminant contained in the im-

mobile zones at equilibrium. Given a set of experimental data, one would be faced with

the task of choosing the multirate series % and /3• (j - 1, 2, ..., N). The first step in doing this would be to choose the

2390 HAGGERTY AND GORELICK: MULTIPLE-RATE MASS T •RANsFER MODEL

type of distribution to use. Based on knowledge of the data set, one would choose either a distribution based on one or several

diffuõion models, or a distribution based on a probability density function. For a diffusion model, the entire multirate series is determined by Da/a 2 and/3 according to Table 1 (i.e., two parameters). For a model based on a probability density function, the multirate series would be determined by the mean, variance, and total/3 (i.e., three parameters).

5. Solutions to the Multirate Equations and Consequences for Remediation

In this section, we solve the mUltirate equations, (13) and (14), for different flow conditions. The purpose of this section is fourfold: (1) to give solutions for the multirate model that others may be able to apply; (2) to examine the various properties of the multirate model; (3) to compare the model to other mass transfer models; and (4) to investigate the effects of multiple and simultaneous mass transfer processes on remediation. Conditions of no flow are the easiest conditions under

which to compare the multirate model to other mass transfer models, so this solution is presented first. Solutions under advection and dispersion are necessary to investigate effects on remediation. Since the easiest solution of equations (13) and (14) is obtained for conditions of fast flow, we present this next. Third, we give the semianalytiC solution for the case of radial flow to a pumping well.

For each solution, we show two examples. The first example deals with data from the Borden sand collected and examined

by Ball and Roberts [1991a, b], and the second set of examples deal with a hypothetical field case. In all cases we chose the distributions of multirate parameters based on diffusion and surface reaction models, as given by Table 1.

5.1. Batch Reactor (No Regional Flow)

For the case of no advection or dispersion, and no sources or sinks, •(Cm) is 0. This represents a batch reactor, and is also the approximate solution to any field or laboratory case where advection and dispersion rates a. re slow with respect to mass transfer rates. Equations (13) and (14) can be written in matrix notation as

dC

A •-= BC (27) where C is the vector of mobile and all immobile concentra-

tions, A is a sparse upper triangular matrix, and B is a sparse lower triangular matrix. Both A and B are of dimension (N + 1) by (N + 1). The vector C is defined as

C---[Cm, (Cim)l , (tim)2, øøø , (tim)N] T (28) where the superscript T indicates the transpose of a vector. The solution to (27) is

C = exp [ (A-1B)t]C0 (29) where Co is the vector of initial concentrations. The matrix A-•B is

N

j=l

•1131

Ot2• 2

OI N• N

Ol 1 Ol 2 ß . ß Ol N

-aN]

(3O)

Table 2. Grain Sizes, Diffusion Parameters, and Capacity Coefficients Measured by Ball and ROberts [1991a, b] for PCE Desorption From the Borden Sand

Size Range Class ( = 2a), mm Da/a 2, S- 1 /3

1 0.85-1.7 3.1 x 10 -8 0.0406 2 0.42-0.85 9.2 x 10 -8 0.1699 3 0.25-0.42 2.3 X 10 -7 0.2731 4 0.18-0.25 2.7 X 10 -7 0.2592 5 0.125-0.18 9.4 X 10 -7 0.1548 6 0.075-0.125 1.7 X 10 -6 0.0620 7 <0.075 1.4 x 10 -6 0.0404

bulk (all sizes) 2.8 X 10 -7 1.0

The seven classes together make up the bulk sample. The capacity coefficients/3 associated with each of the seven sizes are shown in the right-hand column.

All elements not shown in the matrix are zero. Evaluation of the matrix exponential in (29) was accomplished by using a Pad6 9pproximation in combination with "scaling and squar- ing." The fundamentals of this algorithm are given by Golub and Van Loan [1989].

Borden sand: Laboratory study in a batch reactor. Here we show how the multirate model predicts solute Uptake into a composite set of grain sizes from the Borden sand, and compare this prediction to that of a spherical diffuSioh model. Ball [1989] and Ball and Roberts [1991a, b] studied the long-term sorption of perchloroethylene, or PCE, in the Borden sand, from Borden, Canada. Recognizing that multiple timescales for diffusion would be very important, they looked at the uptake of PCE into different size fractions and bulk samples of the sand, and modeled this uptake using the spherical diffusion model. By matching the model to their data, they were able to estimate values of D•/a 2 for each size fraction and for the bulk sample. The size fractions, their average values for D•/a 2, and average capacity coefficients are shown in Table 2. We give each size fraction a "class number" for easy identification. The capacity coefficients for each individual size fraction were determined from Ball and Roberts' [1991b] "ultimate fractional uptake," F:

13 = F/(1 - F) (31)

To obtain the capacity coefficient for a size fraction as part of a bulk sample, we first scaled the capacity coefficients by their mass fraction. Then we renormalized by the sum of the capacity coefficients so that the capacity coefficient of the bulk sample would be the same as the total capacity coefficient of the bulk sample. These are the capacity ratios shown in Table 2. We assumed that there is no instantaneously sorbing fraction, and that the PCE isotherm is linear.

Ball and Roberts [1991b] modeled the "apparent" capacity coefficient, ]3 app, scaled by the "ultimate" capacity coefficient, /3, for PCE in the Borden sand. The apparent capacity coefficient is the ratio of contaminant mass in the immobile zones to

the mass in the mobile zone at a given time. Their fit to the data from the bulk sample is labeled as "single sphere model" in Figure 4a. We modeled the same data set using a "composite model" composed of the measured amounts of each of seven size fractions. Our model of their data is also shown in

Figure 4a, and is labeled, "seven sphere sizes: composite model." As seen in Figure 4a, both models fit the data well. The key

difference is that the composite model, based on the parame-


ters from the individual size fractions, is a prediction of the data, whereas the single sphere model is a curve fit to the data. The multirate model with seven sphere sizes is a 7% closer approximation to the data (mean square error, 0.014) than the single sphere model (mean square error, 0.015).

Figure 4b is a graph of normalized concentration versus time, plotted in log-log space: 1.0 on the vertical axis indicates initial conditions, while 0.0 would indicate equilibrium. The figure shows how the two models of the same data differ in their prediction of the approach to equilibrium. Both models give essentially the same prediction at times less than 3 days and far from equilibrium. However, the single sphere model

(a)

(b)

Normalized Concentration

Cm, eq- C_m [_] Cm, eq- Cm

0.1 1 10 100

Time[d]

0.1 -

0.01 -

10 -3

lOO

(c)

0.1

Capacity 0.01 Coefficient

•j[-] 10 '•

1 0 4

1 0 's 1 0 '7 1 0 '6 1 0 's 1 0 '4 1 0-3 O. 01 O. 1

Mass Transfer

Rate Coefficient ctj Is 'l] Figure 4. (a) Apparent capacity coetficient normalized by the total capacity coefficient versus time for the Ball and Rob- erts [1991b] data. Two models of this are shown, the composite model using the seven sphere sizes, and the single sphere model of Ball and Roberts. (b) Normalized concentration through time for the composite and bulk models. The value of 1.0 indicates initial conditions, and 0.0 indicates equilibrium. (c) The set of multirate series that make up the composite model.

Table 3. Parameters Associated With Different Immobile Zones and Surface Reactions

Class Immobile Zone D ,da 2, s - • /3

1 porous grains (spheres) 2.8 x 10 -? 0.35 2 grain aggregates (spheres) 1.75 x 10 -8 0.20 3 clay layers 1.43 X 10 -9 0.15 4 clay pods (cylinders) 1.00 X 10 -9 0.15

Class Immobile Zone a, s -• /3 :

5 surface reaction 2.76 X 10 -6 0.05 6 surface reaction 4.42 X 10 -5 0.10

For the immobile zones, the apparent diffusion coefficients divided by the square of the radius or half width of the immobile zones for the hypothetical field case. For the surface reactions, the first-order mass transfer coefficients are shown. The capacity coefficients/3 associated with each immobile zone are shown in the right-hand column.

predicts that 99.9% of equilibrium is achieved approximately five times sooner than the composite model. As recognized and predicted by Ball and Roberts, this is due to the inability of the single sphere model to account for the small amount of very slow diffusion caused by larger grain sizes. This may have important consequences on predictions of remediation times, depending on other parameters such as pumping rate.

Figure 4c is a plot of the parameters used in the multirate formulation to model the seven sphere sizes. The diagonal lines indicate the seven different multirate series, composed of 50 coefficients each, that were added together to come up with the composite model. Each line is labeled with the class size from Table 2, and the label is placed as close as possible to the first point of the multirate series. Where dots do not fall on the lines, capacity coefficients were added together because they corresponded to the same mass transfer coefficient. The dots that are circled represent the "truncation control" discussed in section 4. The capacity coefficients at these points include the sum of all other capacity coefficients from later in the series.

Hypothetical aquifer with a mixture of mass transfer processes: No advection. In this section we show how a very heterogeneous porous medium, containing diffusion into and out of grains, spherical aggregates, clay layers and pods, and surface reactions could be modeled under conditions of no

advection. Table 3 lists a set of immobile zones that might cause diffusion-limited and sorption-limited mass transfer in a field setting. Similar to the porous medium in Figure 2, this medium contains small-scale grains and aggregates of grains. On some surfaces, relatively fast surface reactions occur. At a larger scale, contaminant can diffuse into and out of clay layers or clay pods. Table 3 indicates how each immobile zone is associated with a capacity coe•cient, and with a class number for later reference. Values ofD•/a 2 in Table 3 are chosen to be representative of possible field values, and to demonstrate the use of the multirate model. Values of D•/a 2 for spheres are chosen in the range calculated by Ball and Roberts [199!b] for PCE desorption from the Borden sand. Values of D•/a 2 for clay are chosen in the range of possible values for volatile organic compounds diffusing through thin clay layers [Myrand et al., 1989] (as reported by Shackelford [1991]). Values for rates of surface reactions are chosen from the range reported by Brusseau and Rao [1989b].

Figure 5a shows normalized concentration in the hypothetical field site, labeled the "composite model," as concen•tra-


(a) 1

Normalized 0.1 - Concentration

Cm•q_•_•- Cm [_] Cm, eq- C•n 0.01 -

10 -3

0.01

(b) I 0.1 -

Capacity 0.01- Coefficient

•j [-] 10 -3 _

10 -4 _

10 -5 10

(1) (6)

Mass Transfer [s-l] Rate Coefficient

0.01

Figure 5. The normalized concentrations through time for the composite model of the hypothetical field site. Parameters for each immobile zone are from Table 3. The composite medium, shown in bold, is composed of the given fractions of each of the immobile zones. The other curves represent the concentrations that would occur if the entire field site were

composed of a single type of immobile zone. (b) The set of multirate series that make up the composite model. The points represented by a cross represent the first-order surface reactions.

tions approach equilibrium through time. A value of 1.0 on the vertical axis represents the initial conditions, and 0.0 represents equilibrium. Also shown in this graph are the normalized concentrations that may occur in a homogeneous medium. Each of these six curves illustrates how concentrations may behave through time if the hypothetical field site contained only a single type of immobile zone. For the composite model, we see that about 80% of equilibrium is attained within 10 days, but that it takes 500 days to reach 90% of equilibrium, and 10,000 days to approach equilibrium. In addition, we see that each type of immobile zone, if modeled by itself, may give a very poor prediction of concentration through time. The first-order surface reactions and the spheres, on their own, would predict that equilibrium would be attained within a small number of days. On the other hand, the clay pods and layers would predict that it would take several thousands of days to obtain 80% of equilibrium, drastically overestimating the time.

Figure 5b shows the combination of multirate series needed to model this hypothetical field situation. The three diagonal lines represent the diffusion models of two sets of spheres, one set of cylinders, and one set of layers. The two points labeled with a cross represent the two surface reactions. Again, dots not on the lines represent capacity coefficients that are added

together because they correspond to the same mass transfer coefficient, or they represent truncation control.

5.2. Fast Flow

From the multirate model in (13) and (14), it is possible to give a simple estimate for the mass fraction remaining in a small contaminant plume subject to multirate mass transfer. Suppose that pumping rates are large enough that clean water from outside the plume is moved rapidly through the plume to the well. In this case C•n is very small, and (14) becomes

a(Cim)i at = --Otj(Cim)j j = 1, 2,'", N (32)

Solving these equations for the ratio of total contaminant mass remaining in the system to the initial contaminant mass in the system, the mass fraction remaining, gives

N

• /3j(ci%)j exp (-a•t) M j=l

- (33) M0 N

' • ' c m + 13(C,m) j=l

where (C•m)j [M/L 3] is the initial concentration of solute in the jth immobile zone.

Borden sand: Column experiment. In this section we predict how PCE sorbed to the Borden sand would be flushed in

a laboratory column. We assume, for the purpose of this study, that the rates of release from the grains are the same as the rates of uptake into the grains. We use the same parameters shown in Table 2 and discussed in section 5.1.

Figure 6a shows the predl.'cted mass fraction removed versus time for PCE in the Borden sand. The initial condition is

equilibrium, with 50% of the PCE in the mobile domain, and 50% in the immobile domain. The mass fraction remaining, as predicted by the single sphere model, is shown with a dashed line ("single sphere model"), and the mass fraction remaining, as predicted by a mixture of the seven sphere sizes and apparent diffusion coefficients of Ball and Roberts, is shown by the solid line ("composite model"). We see that the composite model's prediction of cleanup time is approximately four times longer than the single sphere model's. Although both models fit well to the data in Figure 4a, they predict very different equilibration times and times to reduce the mass fraction remaining. The differences highlighted in Figures 4b and 6a are due to the fraction of grains with very slow diffusion rates. This fraction makes little difference at early times and large concentrations, as highlighted in Figure 4a, because the composite model contains a large fraction of grains with the same diffusion properties as the single sphere model. However, the difference becomes very apparent at small concentrations, where the small fraction of grains with slow diffusion properties dom- inates the composite model.

Hypothetical aquifer with a mixture of mass transfer processes: Fast pumping rates. In this section we show how concentrations in a rapidly pumped, heterogeneous aquifer might respond over time. We use the same parameters shown in Table 3 and discussed in section 5.1.

Figure 6b shows predictions of the mass fraction of contaminant remaining in an aquifer undergoing very rapid flushing. The curve in bold, labeled "composite model," shows the mass fraction remaining over time in the heterogeneous aquifer.


(a)

0,1 - Mass

Fraction

Remaining [4

0.01 -

10 -o ,

i i i i i

0 20 80 100 4o 6o

Time[d]

(b)

Mass 0.1 -• ,.•',, •'•,,,.•2••,,,,•'\ \. • • •

Re•_•g •'

0.01 0.1 1 10 100 10 3 10 4 10 s

Time [d]

Figure 6, (a) The mass fraction remaining over time for rapid flushing of the Borden sand. The •o models of the rapid flushing are the same as in Figure 4. (b) The mass fraction remaining over time for rapid flushing of the hypothetical field site. The models are the same as in Figure 5.

The other curves show how the mass fraction remaining would change over time if the aquifer were modeled using only a single component and the same total capacity ratio. Time is graphed on a log scale because of the large spread in time of the curves. The labeled percentages indicate the fraction of contaminant in the given immobile zones relative to that in the mobile zone prior to flushing. We see in this figure that about 80% of the contaminant is removed from the heterogeneous medium within 10 days of very rapid flushing. It takes another 200 days to remove 10% more of the total mass, and then an additional 30 years to remove most of the last 10% of the contaminant. Similar to the case of the batch reactor, none of the homogeneous mass transfer models would do a good job of predicting the overall remediation time. Smaller, faster immobile zones may dramatically underestimate remediation time, while the larger, slower immobile zones may dramatically over- estimate the time needed to remove most of the contaminant.

If the aquifer were modeled using only clay pods, the model would predict 1000 days to remove 80% of the contaminant mass, when it could actually be done in 10 days.

5.3. Radial Flow to a Pumping Well

For the case of radial flow to a pumping well, which is a reasonable model for many pump-and-treat scenarios, the multirate model is written nondimensionally as follows:

O--• + O-• - 2 p -b-•p 2 + P > p w (34) /=1

oSj or = a?(13•C- S•) j = 1, 2,..., N (35)

with the boundary conditions

C (p --> oo, z) = 0 (36)

a C(p = p0, ,) = 0 (37)

Op

Several nondimensional groups are defined below and labeled with their nondimensional interpretation.

Concentration in the mobile zone

c = cdc' (38) tot

Concentration in the immobile zone

Sj--' ½j(Cim)j/C' tot

Position p = r/aL

Time z = ant

Mass transfer coefficient

Pumping rate

(39)

(40)

(41)

0 = Q*/•rbRmOmøt2øtn (43)

Well radius p0 = rw/aL (44)

where c lot [M/L 3] is the initial total (mobile and all immobile) concentration at the well; r [L ] is the radius from the center of the well; an [ l/T] is the apparent first-order mass transfer rate coefficient attributed to a specified immobile zone; Q* [L3/T] is the pumping rate; b [L] is the constant aquifer thickness; and rw [L] is the radius of the well. The purpose of an is to have a timescale of reference, 1/an. Although the particular value of an is not important, the harmonic mean is a good choice with which to scale the problem.

The equations for radial flow and transport used here assume spatially homogeneous hydraulic conductivity, porosity, and aquifer thickness. They also assume that the mobile zone retardation factor, each of the immobile zone retardation factors, and each of the first-order mass transfer coefficients are homogeneous in space.

The problem is solved by taking the Laplace transform of (34) to (37), determining the Green's function solution for the transformed equations, and then performing the back- transform numerically. The Laplace transform collapses the (N + 1) coupled equations, (34) and (35), to a single equation. This equation is the inhomogeneous Airy equation [Abramow- itz and Stegun, 1972]. Chen and Woodside [1988] solved this differential equation for equilibrium mass transfer using a Green's function approach, and Harvey et al. [1994] extended the method to include first-order mass transfer with a single mass transfer equation. The semianalytic solution is detailed in Appendix D.

Borden sand: Remediation of a homogeneous aquifer. Once the concentrations C and S i are obtained, the cleanup history for an aquifer can be predicted. In this section we model the removal of PCE from the Borden aquifer under realistic pumping rates. We assume, for the purpose of dem- onstration, that rate-limited mass transfer is adequately de-

*--' Olj/Ol n (42) O/j


(a)

Mass

Fraction

Remaining

(b)

Mass

Fraction

Remaining [-1

0.1

0.01

10 -3

10 -4 0 100 200 300 400 500

Time [d]

1

o.1

O.Ol

ß ß

10 100 10 3 10 4

Time [d]

Figure 7. (a) The mass fraction remaining over time for pump-and-treat remediation of the Borden sand. The two mass transfer models are the same as in Figures 4 and 6a. (b) The mass fraction remaining over time for remediation of the hypothetical field site. The mass transfer models are the same as in Figures 5 and 6b.

scribed by the seven grain sizes detailed by Ball and Roberts [1991a, b] and shown in Table 2.

We use the following specifications: Q* - 1 L/s; b = 20 m, r w - 0.1 m, a L = 2 m, and Om -- 0.25. The contaminant plume is initially at equilibrium with concentrations proportional to exp [- (r - rw)2/50 m 2] yielding concentrations at 15 m equal to 1% of the concentration at the well.

Figure 7a is a graph of mass fraction remaining versus time for the bulk and the composite models of the Ball and Roberts [1991a, b] data set. Unlike the case of rapid flushing, there is no significant difference between the predicted remediation times for the two models. For the given pumping rates, this system is at local equilibrium, and therefore the particular mass transfer model makes no difference, provided that the total capacity coefficients are the same. According to the evaluation criterion given by Valocchi [1986] for local equilibrium in a convergent flow field, the system would be at local equilibrium for the single sphere model, and would be very close to local equilibrium for the composite model. Slower pumping rates would bring the system even closer to equilibrium, but would also slow the remediation, while faster pumping rates would take the system away from local equilibrium. Extremely fast pumping rates would yield the same solution as shown in Figure 6a.

Hypothetical aquifer with a mixture of mass transfer processes: Remediation of a heterogeneous aquifer. In this section, we model the removal of a contaminant from an aquifer with multiple and simultaneous mass transfer processes, under realistic pumping rates. We use the set of mass transfer pa-

rameters in Table 3 and discussed in section 5.1. We assume

that the mass transfer properties are locally heterogeneous, but that they have the same distribution at all points in space. The specifications for the pumping rates and other aquifer simulation parameters are the same as for the previous example.

Figure 7b is a graph of mass fraction remaining versus time, in log space. The curve in bold is for the composite model of mass transfer, and the other curves are for models that assume the aquifer is homogeneous and has one specific type of mass transfer. As in the case with rapid flushing, the composite model predicts a very different remediation history than do the other models. Again, none of the homogeneous models pro- duce a satisfactory approximation to the remediation history in a heterogeneous aquifer. Relative to the composite model, all other models either underpredict remediation time by one to two orders of magnitude or seriously overpredict the time needed to remove the majority of the contaminant.

As in the case for the Borden sand in Figure 7a, the immobile zones with rapid mass transfer properties (small spheres, and slow and fast surface reactions) are at local equilibrium for the given pumping rate of 1 L/s. These are also the types of immobile zones that are most easily investigated and quanti- fied in the laboratory. If we assumed that these were the only immobile zones present in the aquifer, the resulting model would predict local equilibrium conditions and seriously underestimate the cleanup time.

Another interesting feature of Figure 7b is that the late-time results for the composite model and the other slow mass transfer models are remarkably similar to those in Figures 5a and 6b. At these late times for the slowest of immobile zones, the advection and dispersion of solute make very little difference to the model. The time it takes for remediation of the heterogeneous aquifer is virtually the same, regardless of pumping rates, because the system is dominated by diffusion-limited mass transfer from the slowest of the immobile zones.

6. Discussion and Conclusions

6.1. The Multirate Model

The multirate model presented here may be useful in modeling simultaneous mass transfer of different types and rates. Among the advantages of the model are (1) different types and rates of mass transfer can be examined by determining their multirate series or sets of first-order mass transfer coefficients

and adding the parameters together in the correct proportions; (2) the first-order nature of the model allows incorporation of diffusion-limited mass transfer into existing numerical models with relative ease and efficiency; (3) the multirate formulation, because of its flexibility, can model types of mass transfer that could not otherwise be easily modeled; and (4) the multirate model is a direct method for comparing the various models of mass transfer. Among the disadvantages of the model are (1) the formulation of the multirate model for use with diffusion is

not as intuitive as the diffusion model itself; (2) some computer simulations using the multirate model are slower than diffusion and standard first-order models; and (3) the model is more complicated than other models of mass transfer.

Use of the multirate model to examine the Ball and Roberts

[1991a, b] data set shows how it is possible to use individually measured diffusion coefficients from samples with the same grain size to predict the solute uptake history in the heterogeneous mixture. Similarly, it may be possible to measure or estimate individual diffusion coefficients for larger immobile


zones, such as clay layers, and then to predict the remediation history of a contaminant in a heterogeneous field site.

6.2. Implications for Groundwater Remediation

With the multirate model, we can examine the effect on groundwater remediation of mixtures of multiple and simultaneous mass transfer processes. From Figures 6 and 7 we see that a small amount of aquifer material with very slow diffusion properties may cause strikingly long cleanup times. Although this small amount of material may not be apparent from a standard short-term laboratory or field experiment conducted on bulk samples, it may control remediation time.

In a heterogeneous aquifer, a model that assumes the aquifer has a homogeneous diffusion-limited mass transfer rate may underpredict remediation time by several orders of magnitude. If the aquifer has clay lenses or other large immobile zones, and if the contaminant has been in the aquifer long enough to diffuse into these zones, the clay lenses will be the dominant factors in remediation time. Even if these zones of

low hydraulic conductivity do not severely affect flow, they dramatically lengthen groundwater remediation. On the other hand, it would be a poor practice to assume the "worst case scenario," that the aquifer contains only the slowest types of immobile zones. This would result in greatly underpredicting the amount of contaminant that can be rapidly removed from the aquifer. In locations where these large, slow immobile zones exist, the best management of the site may be to remove as much contaminant as quickly as possible, and then move to a long-term containment strategy.

For aquifers with very heterogeneous mass transfer properties, characterizing the heterogeneity may be the single most important factor in design of the remediation system. Infor- mation on large, slow immobile zones is not likely to be found by performing a tracer test unless the test is run over a very long period. The short-duration tracer tests that are commonly used are unlikely to be sensitive enough to long-term diffusion processes, even though these are a critical factor in aquifer remediation. Laboratory and field work is also needed to de- termine diffusion coefficients for larger immobile zones such as clay lenses.

6.3. Conclusions

The main conclusions of this work are the following: 1. All standard first-order models are fundamentally the

same, even beyond that recognized by Nkedi-Kizza et al. [1984]. Figure 1 shows that in addition to first-order models of "physical" and "chemical" mass transfer being the same, "one-site" and "two-site" models are also mathematically equivalent after a change of variables.

2. The multirate model, a model describing mass transfer by using a set of first-order relationships between the mobile and immobile phases, is precisely equivalent in reduced form to models of diffusion from layers, cylinders, or spheres.

3. The spherical, cylindrical, and layered-diffusion models of mass transfer may each be thought of as first-order mass transfer models with a specific probability density function for the mass transfer rate coefficient. The probability density function is given in Table 1 for each model.

4. The multirate formulation is able to describe mass trans-

fer in heterogeneous mixtures collectively from layers, cylinders, and spheres, and first-order surface reactions.

5. Measuring the faster and slower mass transfer processes individually appears likely to be the key to making accurate

predictions of aquifer remediation times. To obtain reliable estimates of bulk mass transfer rates, individual processes need to be identified and measured. Measurements of mass transfer

rates in a heterogeneous bulk sample in the field or laboratory will probably yield a poor prediction of aquifer remediation times unless very precise measurements are made at late times.

Appendix A: Development of Laplace-Domain Equations

A1. Multirate Model

The Laplace transform of (13) is

N

SCm- C•n q- Z •j[S(Cim)j- (Clm)j] : '•(•mm) /=1

(A1)

where s is the Laplace variable; (C•m)j [M/L 3] is the initial concentration in the jth immobile zone; and overbars above the variables indicate Laplace domain. Taking the Laplace transform of (14) and rearranging,

aj 1 ( C im ) j = S q- Ol• C--• q- -- ( C ;m ) j j = 1, 2,'", N (A2) S q- OZj

Substituting (A2) into (A1) and rearranging, we arrive at (20)'

SC m -- C• q- (S•mm- (C im)j) -- '•(•mm) = 0 ß = s+ otj

(A3)

A2. Spherical Diffusion Model

The following derivation is after Harvey and Gorelick [1995]. The Laplace transform of (8), combined with the Laplace transform of (9), is

SC m -- C•n q- •S • r2•rr dr- /3c} = •(•mm) (A4)

where c 'r [M/L 3] is the initial concentration within the sphere and overbars above the variables indicate Laplace domain. For the purposes of this paper we are starting with uniform initial concentrations, so C'r is independent of r and equal to •t tm•

where •tim [M/L 3] is the average initial concentration within the sphere. The Laplace transform of (10) is

[raV] SC•--C; = r2 dr dr ] (A5) The solution of (•) for the bounda• conditions (11) and (12) is

(•) a sinh(r s•) c' c•= •- 7sinh(a •s/Da) + • (A6) s

Noting that c } is equal to • Jm, we combine (A6) with (A4) and integrate, to yield the following, which is (21):

SCm- C•n q- 3/3(S•mm- •m) D• 1 'Icøth(•D-•2aS)- •aD•3]-•(•mm' :0 (A7)


Appendix B: Finding the Multirate Series Equation (25) is analogous to a nonlinear integral equation.

Although this equation would be difficult to solve directly, the equation can be used to show that a given series converts the multirate model into a diffusion model of mass transfer. Sim-

ilarly, equation (C8) cannot be solved directly, but it also can be used to show that a given series converts the multirate model into a diffusion model of mass transfer. To find a possible solution, we search for a set of boundary conditions that put the multirate equations and the diffusion equations into forms that are easily compared. We find this set of boundary conditions in the model of a well-mixed, infinite bath, or infi- nitely fast flow of clean water.

For the case of a fast flow to a pumping well, discussed in section 5.2, the boundary conditions on the immobile region are that C m = 0 after one pore volume has been removed and replaced by clean water. The multirate model is then given by (32). If we allow N to be infinity, and initial concentrations to be the same everywhere in the immobile zone, the solution to (32) for the ratio of mass in the immobile zone to the initial mass in the immobile zone is

concentration histories to each other. According to the "braid- ing theory" [Smith, 1993] any two cumulative distributions cross each other at least n times if the probability density functions have the same n first moments. In other words, if two concentration histories share the same first three moments, the two cumulative mass arrival curves have the same values and

cross each other at least three times. The more moments in

common, the more the two curves cross. Consequently, if two concentration histories have an infinite number of equal moments, then the two histories must be the same.

Using the method developed by Harvey and Gorelick [1995], the nth temporal moment of the multirate model, m,•, is given by the solution to the following moment-generating equation forn- > 1:

n N 2 ' mi_ 1 flj flj(C im)j

-nmn_l-n! i•• (i_ 1), • (otj)n_ i n! ': ' '= j:l (01j)n = •(mn) (C1)

If n = 0, the equation is

Mi m o• Mim,O = • -•- exp ( - 090 (B 1)

where /3 is the total capacity coefficient for the system. For diffusion, consider the case of diffusion into or from a sphere with initial concentration the same everywhere in the sphere. In this case, the expression for the ratio of mass in the sphere to the initial mass in the sphere is (modified from Crank [1975, p. 91])

Mira ø• 6 ( j2,rr2 D ) Mimo = • j-7-•2 exp - • t (B2) ' j=l

From (B1) and (B2) it is easy to see that two solutions are exactly the same if

Otj = j27r2 Da •-• j = 1, 2, 3, ..-, • (B3) and

6

/3j = j-2--•2/3 j = 1, 2, 3,-.., o• (B4) Together, the two give us a possible multirate series for spherical diffusion. This series can be substituted into (25), and we see that the spherical and multirate models must be mathematically equivalent. Similarly, we can substitute this series into (C8), and we see that the spherical and multirate models then have the same temporal moments. In addition, we can test this series in the multirate solution for a batch reactor and

compare the solution to that given by Crank [1975, p. 94], to see that they are the same. Exactly the same process can be used to derive the multirate series for diffusion from layers, cylinders (shown in Table 1), or some other geometry. We could also derive the multirate series for other processes, such as combinations of reaction in series and reactions in parallel, as in Brusseau et al.'s [1989] multiprocess nonequilibrium.

Appendix C: Alternative Development of Multirate Series

Another way of comparing the multirate model to the diffusion models is to compare the temporal moments of their

N

--C •n -- E f•j(C i%)j • • (mo) j=l

(C2)

For the diffusion models, the corresponding equation (n -> 1) is given by [after Harvey and Gorelick, 1995]

n mi_lA(n_l)(a•a)n_ i --rtmn_ 1 -- n!f• E (i- 1)! i=1

-- n !A n f}•' i'm

= •C•(mn) (C3)

where A,• is the n th coefficient for a given geometry as defined by Harvey and Gorelick's [1995] Table 2. The initial concentration, Cr', is the same everywhere in the immobile zone. For n = 0,

-c•- /3•'jm = •(m0) (ca)

where c• and •m [M/L3] are the initial conditions for the mobile and immobile zones, respectively.

Comparing (C1) and (C3), and (C2) and (C4), we see that the form of the expressions for the multirate and diffusion models is the same. Given that the coefficients in •( ) are the same in the two models, and the initial mobile mass is fixed, the zeroth moments are the same if

N

/=1

(C5)

where (C•m)j is the initial concentration in the jth site of the multirate model. The terms on the left are for the multirate

model, and the terms on the right are for the diffusion models. This equation states that the total initial mass in the immobile zone is the same for both models. Comparing like terms in (C1) and (C3), the first moments (n = 1) are the same if

N

j=l

(C6)

and


Z •j(½ im)j __ •Zll • ;m (C7) j=l O•j

Since we are trying to see when the models are the same, we are only interested in cases where the concentrations are the same, (c •m)i = •' •'m over all j. Using the same logic, we find equations that make (C1) and (C3) equivalent over all values of n. Therefore all temporal moments are the same at every point, and the two models are mathematically equivalent provided that the following set of conditions is met:

ß = a• Ak /3 k=0,1,2,- ,0: (C8) These equations are an infinite series of infinite order polyno- mials, with an infinite number of unknowns. Equation (C8) cannot be solved directly, but it can be used to check the validity of a potential solution. The solutions shown in Table 1 were tested in (C8) and found to match exactly.

Appendix D: Semianalytic Solution for Radial Flow

The Laplace transform of (34)is

•v 2p (d2C ) dC

sC - c' + [sS- s;] = ¾ + j=l

(D1)

and the transform of (35) is -- ! --

sSj- Sj= a•(13•C- S) j= 1, 2, ...,N (D2)

where s is the Laplace parameter, C' is the initial nondimensional mobile concentration, and Sj is the initial nondimensional immobile concentration in the j th immobile zone. Over- bars above the concentrations indicate the Laplace-domain concentrations. Equation (D2) can be rearranged to

a•3• -- 1 Sj= C+ ' j=l 2,--- N (D3) s+7 ' '

Substituting (D3) into (D1) and rearranging collapses the (N + 1) equations into the following single ordinary differential equation for •"

4, c' + j=l

d2C dC 2ps a•18• -- =dp 2 +dp qb 1 + C (D4)

/=1

By defining the parameter groups

P=-•- 1+ s+a /=1

(D5)

= c' + ' ß = s + a• S• (D6)

we can write (D4) as

d2C dC

-•(p)p = dp• + •pp- pPC (D7) which is the inhomogeneous Airy equation. The boundary conditions in the Laplace domain are

C(p --> •) = 0 (D8)

dC(p = P0) = 0 (D9) dp

We employ the solution presented by Chen and Woodside [1988], and used by Harvey et al. [1994], in which a Green's function accounts for the inhomogeneous term on the left- hand side of (D7). This solution determines the concentrations at a finite number of nodes and interpolates the concentrations between nodes. Integrations are done numerically with Gaus- sian quadrature. Airy functions are evaluated with Interna- tional Mathematics and Statistics Libraries, Inc. [1991] routines. After the Laplace-domain mobile concentrations are found, the Laplace-domain immobile concentrations are given by (D3). The results for both •' and • are back-transformed using the Stehfest [1970] algorithm to provide the time-domain concentrations C and Si at any time during the pumping period.

Notation

Several variables for concentration also have a Laplace- domain form, and these are indicated in the notation below. Unless otherwise noted, an overbar above the variable indicates Laplace-domain concentration in the text of the paper.

a distance from the center to the edge of an immobile zone for a diffusion model [L ].

A parameters that relate concentration within mobile zone to concentration within unit volume

of porous medium [dimensionless]. A sparse upper triangular matrix.

A n n th coefficient for a given diffusion geometry from Harvey and Gorelick's [1995], Table 2 [dimensionless].

b aquifer thickness [L]. B parameters that relate concentration within

immobile zone to concentration within unit

volume of porous medium [dimensionless or M/ L3].

B sparse lower triangular matrix. •' aqueous concentration within the source or sink

[M/L3]. • total sorbed concentration [M/M]. C nondimensional aqueous concentration (also has

Laplace-domain form) [dimensionless]. C' initial nondimensional aqueous concentration

[dimensionless]. C vector of concentrations [M/L3].

C ira sorbed concentration in immobile zone for chemical models, and aqueous concentration in immobile zone for other models [M/M or M/L 3].

Cim aqueous cqncentration of immobile zone [M/L3]. •im aqueous concentration averaged over the

immobile zone in diffusion models (also has Laplace-domain form) [M/L3].

-' initial aqueous concentration averaged over the C im

immobile zone in diffusion models [M/L 3].


(ci,,,)i aqueous concentration within the jth immobile zone (also has Laplace-domain form) [M/L3].

(c •m)i initial concentration within the j th immobile zone [M/L3].

Cm aqueous concentration in mobile zone [M/L3]. C m aqueous concentration of mobile zone (also has

LaPlace-domain form) [M/L3]. c• initial mobile concentration [M/L3]. C r actual aqueous concentration at a particular point

Within an immobile zone, for a diffusion model (also has Laplace-domain form) [M/L3].

Clot initial total (mobile and all immobile) concentration at a pumping well [M/L3].

Co vector of initial concentrations [M/L3]. •11 SOrbed concentration within slow sorption sites

[m/m]. c-• sorbed concentration within instantaneous

sorption sites [M/M]. D hydrodynamic dispersion [L 2/r]. D hydrodynamic dispersion tensor [L2/T].

D a apparent pore diffusion coefficient [L2/T]. Da/i Damkohler I number associated with j th

immobile zone [dimensionless]. Dp effective pore diffusion coefficient [L2/T].

f fraction of the sorbed mass in sorption equilibrium with the mobile zone [dimensionless].

f' fraction of sorbed mass that instantly achieves equilibrium [dimensionless].

F Ball and Roberts' [1991b] ultimate fractional uptake [dimensionless].

fm fraction of the sorbed mass in sorption equilibrium with the mobile zone [dimensionless].

•(p) parameter group dependent upon radial position in inhomogeneous Airy equation [dimensionless].

K a distribution coefficient [L3/M]. (Ka,im): distribution coefficient for jth immobile zone [L3/

MI. Ka, m distribution coefficient for mobile zone [L 3/MI.

k 2 first-order mass transfer rate coefficient for chemical models [ T- 1].

L length scale of a one-dimensional problem [L]. •( ) operator representing the advective, dispersive,

and source/sink terms in the solute transport equation.

M mass of contaminant remaining in an aquifer [M]. Mim mass of contaminant remaining in all immobile

zones within an aquifer [M]. Mim,O initial mass of contaminant .in all immobile zones

within an aquifer [MI. m n n th temporal moment of a solute breakthrough

curve at a point. Mo initial mass of contaminant in an aquifer [MI.

P parameter group in inhomogeneous Airy equation [dimensionless].

q source or sink of water in solute transport equation [ T- 1 ].

Q Darcy velocity [L/T]. Q* pumping rate [L3/T].

r radial or spherical coordinate Rim retardation factor of the immobile zone

[dimensionless]. (Rim)j retardation factor of the jth immobile zone

' [dimensionless].

R m retardation factor of the mobile zone [dimensionless].

s Laplace parameter. S i nondimensional immobile concentration (also has

Laplace-domain form) [dimensionless]. S•j initial nondimensional immobile concentration

[dimensionless]. t time [r]. v pore water velocity vector [L/T].

Vx pore water velocity in one-dimensional problem

x Cartesian coordinate [L]. a apparent first-order mass transfer rate coefficient

for physical models [ T- 1]. a' first-order mass t•ansfer rate coefficient for

physical models and the case of no sorption [T-l]. "•

aœ first-order mass transfer coefficient for approximation to diffusion models [ T- 1].

a• apparent first-order mass transfer rate coefficient, associated with the j th immobile zone, for physical models [ T- 1].

aj first-order mass transfer rate coefficient, associated with the j th immobile zone, for physical models and the case of no sorption [T-1].

a• nondimensional first-order mass transfer rate coefficient, associated with the j th immobile zone [dimensionless].

aL longitudinal dispersivity [L]. a n first-order mass transfer scaling coefficient for

dimensionless problem [ T- 1]. /• capacity ratio, that is, the ratio of total

contaminant mass in the immobile zone to the

mass in the mobile zone at equilibrium [dimensionless].

•app apparent capacity ratio at any given time before equilibrium is reached [dimensionless].

/•i capacity ratio associated with the j th immobile zone [dimen•gi0nless].

0 porosity [dimensionless]. Oim porosity of immobile zone [dimensionless].

(Oim)• porosity of jth immobile zone [dimensionless]. 0,,, porosi• of mobile zone [dimension!ess].

X distribution coefficient relating Cim to C,,, at equilibrium [dimensionless or L 3/M].

u dimensionality of the immobile zone for diffusion models (i.e., 1, 2, or 3).

p nondimensional position in radial coordinates [dimensionless].

Pb bulk density of porous medium [M/L3]. Po nondimensional well radius [dimensionless]. ß nondimensional time [dimensionless]. & nondimensional pumping rate [dimensionless]. •o first-order mass transfer rate coefficient

[ML-3T -1 or T-i].

Acknowledgments. Funding for this study was provided by the Of- fice of ReSearch and Development, U.S. Environmental Protection Agency, under agreement R'819751 through the Western Region Haz- ardous SUbstanc9 Research Center. The content of this study does not necessarily represent the views of the agency. Additionally, we are grateful to the National Sciences and Engineering Research Council of Canada for graduate student SUpport. Computer facilities were provided by a grant from the Hewlett-Packard Company, with additional


support from the National Science Foundation BCS 8957186, for which we are most grateful. We would like to thank Charles Harvey for the many ideas and much help from beginning to end. We are grateful to Bill Ball, Jeff Cunningham, Mark Goltz, Tom Harmon, Paul Roberts, Tom Soerens, A1 Valocchi, and the anonymous reviewers for their very helpful comments and discussion.

References

Abramowitz, M., and I. A. Stegun (Eds.), Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.

Bahr, J. M., and J. Rubin, Direct comparison of kinetic and local equilibrium formulations for solute transport affected by surface reactions, Water Resour. Res., 23(3), 438-452, 1987.

Ball, W. P., Equilibrium sorption and diffusion rate studies with halogenated organic chemicals and sandy aquifer material, Ph.D. disser- tation, Stanford Univ., Stanford, Calif., 1989.

Ball, W. P., and P. V. Roberts, Long-term sorption of halogenated organic chemicals by aquifer material, 1, Equilibrium, Environ. Sci. Technol., 25, 1223-1237, 1991a.

Ball, W. P., and P. V. Roberts, Long-term sorption of halogenated organic chemicals by aquifer material, 2, Intraparticle diffusion, En- viron. Sci. TeChnol., 25, 1237-1249, 1991b.

Barber, L. B., II, E. M. Thurman, and D. D. RunnelIs, Geochemical heterogeneity in a sand and gravel aquifer: Effect of sediment mineralogy and particle size on the sorption of chlorobenzenes, J. Con- tam. Hydrol., 9, 35-54, 1992.

Brusseau, M. L., and P.S. C. Rao, Sorption nonideality during organic contaminant transport in porous media, Crit. Rev. Environ. Control, 19(1), 33-99, 1989a.

Brusseau, M. L., and P.S. C. Rao, The influence of sorbate-organic matter interactions on sorption nonequilibrium, Chemosphere, 18(9/ 10), 1691-1706, 1989b.

Brusseau, M. L., R. E. Jessup, and P.S. C. Rao, Modeling the transport of solutes influenced by multiprocess nonequilibrium, Water Resour. Res., 25(9), 1971-1988, 1989.

Cameron, D. R., and A. Klute, Convective-dispersive solute transport with a combined equilibrium and kinetic adsorption model, Water Resour. Res., 13(1), 183-188, 1977.

Chen, C.-S., and G. D. Woodside, Analytical solution for aquifer decontamination by pumping, Water Resour. Res., 24(8), 1329-1338, 1988.

Coats, K. H., and B. D. Smith, Dead-end pore volume and dispersion in porous media, J. Soc. Pet. Eng., 4, 73-84, 1964.

Crank, J., The Mathematics of Diffusion, 2nd ed., Oxford Univ. Press, New York, 1975.

Curtis, G. P., P. V. Roberts, and M. Reinhard, A natural gradient experiment on solute transport in a sand aquifer, 4, Sorption of organic solute and its influence on mobility, Water Resour. Res., 22(13), 2059-2067, 1986.

Cvetkovic, V. D., and A.M. Shapiro, Mass arrival of sorptive solute in heterogeneous porous media, Water Resour. Res., 26(9), 2057-2067, 1990.

Dougharty, N. A., Effect of adsorbent particle-size distribution in gas- solid chromatography, AIChE J., 18(3), 657-659, 1972.

Golub, G. H., and C. F. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins Univ. Press, Baltimore, Md., 1989.

Grathwohl, P., Influence of organic matter from soils and sediments of various origins on the sorption of some chlorinated aliphatic hydro- carbons: Implications on Koc correlations, Environ. Sci. Technol., 24, 1687-1693, 1990.

Harmon, T. C., and P. V. Roberts, Comparison of intraparticle sorption and desorption rates for a halogenated alkene in a sandy aquifer material, Environ. Sci. Technol., 28, 1650-1660, 1994.

Harmon, T. C., L. Semprini, and P. V. Roberts, Nonequilibrium transport of organic contaminants in groundwater, in Reactions and Movement of Organic Chemicals in Soils, edited by B. L. Sawhney and K. Brown, pp. 405-437, Soil Sci. Soc. of Am., Madison, Wis., 1989.

Harmon, T. C., L. Semprini, and P. V. Roberts, Simulating groundwater solute transport using laboratory determined sorption parameters, J. En v. Eng., 118(5), 666-689, 1992.

Harvey, C. F., and S. M. Gorelick, Temporal moment-generating equations: Modeling transport and mass transfer in heterogeneous aquifers, Water Resour. Res., 31(8), 1895-1911, 1995.

Harvey, C. F., R. Haggerty, and S. M. Gorelick, Aquifer remediation: A method for estimating mass transfer rate coefficients and an evaluation of pulsed pumping, Water Resour. Res., 30(7), 1979-1991, 1994.

International Mathematics and Statistics Libraries, Inc., IMSL MATH/ LIBRARY Special Functions User's Manual, version 2.0, Houston, Tex., 1991.

Karickhoff, S. W., Organic pollutant sorption in aquatic systems, J. Hydraul. Eng., 110(6), 707-735, 1984.

Lapidus, L., and N. R. Amundson, Mathematics of adsorption beds, 6, The effects of longitudinal diffusion in ion exchange and chromatographic columns, J. Phys. Chem., 56, 984-988, 1952.

Myrand, D., R. W. Gillham, J. A. Cherry, and R. Johnson, Diffusion of volatile organic compounds in natural clay deposits, report, Dep. of Earth Sci., Univ. of Waterloo, Waterloo, Ont., Canada, 1989.

National Research Council, Alternatives for Groundwater Cleanup, National Academy Press, Washington, D.C., 1994.

Nkedi-Kizza, P., J. W. Biggar, H. M. Selim, M. T. van Genuchten, P. J. Wierenga, J. M. Davidson, and D. R. Nielsen, On the equivalence of two conceptual models for describing ion exchange during transport through an aggregated oxisol, Water Resour. Res., 20(8), 1123-1130, 1984.

Oddson, J. K., J. Letey, and L. V. Weeks, Predicted distribution of organic chemicals in solution and adsorbed as a function of position and time for various chemical and soil properties, Soil Sci. Soc. Am. Proc., 34, 412-417, 1970.

Pignatello, J. J., Slowly reversible sorption of aliphatic halocarbons in soils, 2, Mechanistic aspects, Environ. Toxicol. 'Chem., 9, 1117-1126, 1990.

Purnell, H., Gas Chromatography, John Wiley, New York, 1962. Quinodoz, H. A.M., and A. J. Valocchi, Stochastic analysis of the

transport of kinetically sorbing solutes in aquifers with randomly heterogeneous hydraulic conductivity, Water Resour. Res., 29(9), 3227-3240, 1993.

Rao, P.S. C., D. E. Rolston, R. E. Jessup, and J. M. Davidson, Solute transport in aggregated porous media: Theoretical and experimental evaluation, Soil Sci. Soc. Am. J., 44, 1139-1146, 1980.

Sardin, M., D. Schweich, F. J. Leij, and M. T. van Genuchten, Mod- eling the nonequilibrium transport of linearly interacting solutes in porous media: A review, Water Resour. Res., 27(9), 2287-2307, 1991.

Selim, H. M., J. M. Davidson, and R. S. Mansell, Evaluation of a two-site adsorption-desorption model for describing solute transport in soil, in Proceedings of the Computer Simulation Conference, pp. 444-448, Am. Inst. of Chem. Eng., Washington, D.C., 1976.

Selroos, J.-O., and V. Cvetkovic, Modeling solute advection coupled with sorption kinetics in heterogeneous formations, Water Resour. Res., 28(5), 1271-1278, 1992.

Shackelford, C. D., Laboratory diffusion testing for waste disposal--A review, J. Contam. Hydrol., 7, 177-217, 1991.

Smith, J. E., Moment methods for decision analysis, Manage. Sci., 39(3), 340-358, 1993.

Stehfest, H., Numerical inversion of Laplace transforms, Commun. ACM, 13, 47-49, 1970.

Valocchi, A. J., Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils, Water Resour. Res., 21(6), 808-820, 1985.

Valocchi, A. J., Effect of radial flow on deviations from local equilibrium during sorbing solute transport through homogeneous soils, Water Resour. Res., 22(12), 1693-1701, 1986.

Valocchi, A., Spatial moment analysis of the transport of kinetically adsorbing solutes through stratified aquifers, Water Resour. Res., 25(2), 273-279, 1989.

Valocchi, A. J., Use of temporal moment analysis to study reactive solute transport in aggregated porous media, Geoderma, 46, 233- 247, 1990.

van Genuchten, M. T., A general approach for modeling solute transport in structured soils, in Proceedings of the 17th International Con- gress, International Association of Hydrogeologists, Tucson, Ariz.: Hy- drogeology of Rocks of Low Permeability, pp. 513-526, UAH Press, Huntsville, Ala., 1985.

van Genuchten, M. T., and P. J. Wierenga, Mass transfer studies in sorbing porous media, I, Analytical solutions, Soil Sci. Soc. Am. J., 40, 473-480, 1976.

van Genuchten, M. T., D. H. Tang, and R. Guennelon, Some exact solutions for solute transport through soils containing large cylindrical macropores, WaterResour. Res., 20(3), 335-346, 1984.

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https://www.researchgate.net/publication/222148397_Laboratory_diffusion_testing_for_waste_disposal_-_A_review?el=1_x_8&enrichId=rgreq-adfe4de1-0637-4b13-8ac8-a4977d5281b9&enrichSource=Y292ZXJQYWdlOzIyOTQyODkzMTtBUzoxMDM4NTM4NjUzMDgxNjNAMTQwMTc3MjA5OTM0NQ==

https://www.researchgate.net/publication/222148397_Laboratory_diffusion_testing_for_waste_disposal_-_A_review?el=1_x_8&enrichId=rgreq-adfe4de1-0637-4b13-8ac8-a4977d5281b9&enrichSource=Y292ZXJQYWdlOzIyOTQyODkzMTtBUzoxMDM4NTM4NjUzMDgxNjNAMTQwMTc3MjA5OTM0NQ==

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Villermaux, J., Theory of linear chromatography, in Percolation Pro- cesses, Theory and Applications, edited by A. E. Rodrigues and D. Tondeur, NATO ASI Ser., Ser. E, 33, 83-140, 1981.

Villermaux, J., Chemical engineering approach to dynamic modelling of linear chromatography: A flexible method for representing complex phenomena from simple concepts, J. Chrornatogr., 406, 11-26, 1987.

Vil!ermaux, J., Dynamics of linear interactions in heterogeneous media: A systems approach, J. Pet. Sci. Eng., 4(1), 21-30,.1990.

Weber, W. J., Jr., P.M. McGinley, and L. E. Katz, Sorption phenomena in subsurface systems: Concepts, models and effects on contaminant fate and transport, Water Resour. Res., 25(5), 499'528, 1991.

Weber, W. J., Jr., P.M. McGinley, and L. E. Katz, A distributed reacti•vi•y model for•Sorption by soils and sediments, 1, Conceptual basis and equilibrium assessments, Environ. Sci. Technol., 26, 1955- 1962, 1992.

Wilson, J., Removal of aqueous phase dissolved contamination: Non- chemically enhanced pump and treat, in Subsurface Restoration, ed-

ited by C. H. Ward, J. Cherry, and M. R. Scalf, Lewis, Boca Raton, Fla., in press, 1995.

Wood, W. W., T. F. Kraemer, and P. P. Hearn Jr., Intragranular diffusion: An importan t mechanism influencing solute transport in clastic aquifers?, Science, 247, 1569-1572, 1990.

Wu, S., and P.M. Gschwend, Sorption kinetics of hydrophobic organic compounds to natural sediments and soils, Environ. Sci. Technol., 20, 717-725, 1986.

Wu, S., and P.M. Gschwend, Numerical modeling of sorption kinetics of organic compounds to søil and sediment particles, Water Resour. Res., 24(8), 1373-1383, 1988.

S. M. Gorelick and R. Haggerty, Department of Geological and Environmental Sciences, 138 Mitchell Building, Stanford University, Stanford, CA 94305-2115.

(Receive d November 25, 1994; revised May 9, 1995; accepted May 23, 1995.)

Multiple-Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media with Pore-Scale...

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