A general memory function for modeling mass transfer in groundwater transport

Authors


Abstract

[1] A power-law extension of the gamma distribution is proposed as a general memory function for capturing rate limitations of retention in groundwater transport. Using moments, we show how the new memory function can be reduced to most other forms available in the literature, exactly or approximately. The proposed formulation is suitable for field scale or laboratory scale transport modeling. Rate limitation effects are illustrated for solute transport by considering the fractional mass release over a given transport scale. The equilibrium and no-retention cases set bounds for contaminant attenuation, between which the impact of rate limitations is clearly exposed.

1. Introduction

[2] Contaminant transport by groundwater takes places along pathways, starting from some recharge location at the surface or a source location in the subsurface, to a downstream discharge location. The main mechanism of groundwater transport is advection by spatially variable groundwater flow which is typically assumed to be at steady state [Dagan, 1984, 1989]. The next most important mechanism is retention due to mass transfer between mobile and immobile phases [Coats and Smith, 1964; Brusseau and Rao, 1989; Haggerty and Gorelick, 1995; Carrera et al., 1998]. Finally, transformation is important for many contaminants, in the form of decay, or more complex chemical reactions, and needs to be coupled with both variable advection and retention [Cvetkovic, 2012].

[3] The use of the memory function for modeling mass transport has origins in chromatography [Villermaux, 1974]. In statistical physics, a comparable notion is used to capture the complexity of trapping in particle transport, including a distribution of release rates. Particularly relevant is the “waiting time density” of which a number of forms have been introduced as part of the continuous-time random walk framework [Montroll and Scher, 1973]. Memory function formulations were used specifically for contaminant transport in the subsurface [Cvetkovic and Dagan, 1994], with spatial heterogeneity [Cvetkovic et al., 1998], or with multiple rates [Pedit and Miller, 1994; Haggerty and Gorelick, 1995; Haggerty et al., 2000] or both with spatial heterogeneity and multiple rates [Cvetkovic and Haggerty, 2002].

[4] The memory function concept has proven useful in field test applications [Haggerty et al., 2000, 2001], for interpreting tracer tests in crystalline rock [Cvetkovic et al., 2007; Cvetkovic and Cheng, 2011; Cvetkovic et al., 2010], as well as for transport modeling in the context of safety assessment [Painter et al., 2008]. Specifically “designed” forms of the memory function have been proposed for capturing microstructural heterogeneity of the rock adjacent to fractures [Cvetkovic, 2010a], demonstrating its conceptual flexibility. A particularly interesting form of the memory function was proposed recently as part of a fractional advection-dispersion equation formulation of groundwater transport [Baeumer and Meerschaert, 2010].

[5] In this paper we propose a new form of the memory function that generalizes earlier expressions extending primarily the works of Montroll and Scher [1973], Haggerty and Gorelick [1995]; Haggerty et al. [2000], and of Cvetkovic and Haggerty [2002]. The Lagrangian (or trajectory) based hydrological transport modeling framework [Dagan, 1984; Cvetkovic and Dagan, 1994; Cvetkovic et al., 1998] is first briefly summarized, followed by the new formulation and a comparison to existing models. The potential effect of mass transfer rate limitations on contaminant attenuation in groundwater is illustrated and the possibilities for model parameter inference using sample tests is discussed.

2. Solute Transport

[6] We consider solute transport by groundwater. The mean flow, which accounts for often complex boundary conditions and deterministic structures, is assumed resolved, e.g., by simulations or analytically as uniform mean flow in a limited section of an aquifer. Macroscopic dispersion is present due to spatial variability of hydraulic properties that cannot be deterministically resolved and is assumed random, whereas retention is present due to mass transfer of the solute between the mobile and immobile phases [Brusseau and Rao, 1989].

[7] Let L denote the transport scale in an aquifer that to some degree will act as a barrier to dissolved contaminants. Water and solute particles are traced from an arbitrary input location (in the subsurface or close to the surface in the unsaturated zone) to a discharge location of interest, such as a stream or a supply well. The basic assumptions of our analysis are: (i) Spatially variable tracer advection by groundwater flow across L is the dominant transport mechanism; (ii) the tracer is subject to exchange between a mobile and immobile phase; and (iii) groundwater is approximately at steady state and initially tracer free. With these assumptions, the main interest is to provide a general formulation of solute (tracer) transport with retention using a memory function.

[8] Tracer discharge from a source to a monitoring location at x = L is quantified using the tracer particle residence time density h written in the Laplace domain as [Cvetkovic et al., 1998]

display math

where the “hat” denotes Laplace transform (LT), s is the LT variable, math formula is the water travel time between the injection and detection locations, and angular brackets denote ensemble averaging. Equation (1) is derived from the mass balance equation system where pure advection is coupled with mass transfer; key steps of this derivation are summarized in Appendix A. The memory function math formula quantifies the mass transfer (retention) processes, and math formula is a spatially variable retention parameter vector following a trajectory as math formula. Averaging takes into account variability in all parameters. An useful simplification relevant in several important applications is to assume g in a factorized form, as math formula, where A is the only spatially variable retention parameter and P are approximately uniform (effective) values. In the factorized case (1) can be written as

display math

where averaging is over two random variables: the water travel time math formula and math formula. Equation (1) or (2) can also be expressed in a discretized form as a time domain random walk [Painter et al., 2008].

[9] If all retention parameters are assumed constant, the expression for tracer particle residence time h is

display math

where f is the water travel time density; the above expression was first proposed in chromatography [Villermaux, 1974] and subsequently extended to three-dimensional random flow paths [Cvetkovic and Dagan, 1994]. If solute particles are noninteractive (ideal) then math formula, or math formula, whereby math formula, i.e., the tracer and water particle residence time densities coincide.

[10] The choice of the water residence time math formula will reflect the nature of the macrodispersion. A general and convenient form of f is the tempered one-sided stable (TOSS) density [Cvetkovic and Haggerty, 2002; Cvetkovic, 2011a], summarized in Appendix B. The TOSS density can be reduced to most forms used for hydrological transport, including the solution of the advection-dispersion equation [Cvetkovic, 2011a]. For finite injection with a specified rate math formula [M/L2T] at the injection location, tracer discharge is computed by convolution as

display math

where for simplicity we consider the case with uniform retention parameters and the injection starts at t = 0. Parameter a is a cut-off rate, c is a scaling parameter, and α is an exponent in the range [0,1]. The density h is in effect a transfer function for subsurface transport.

[11] Equation (4) is general and can account for Fickian, non-Fickian, or anomalous transport, both in terms of advection (as quantified by f) and/or mass transfer (as quantified by g); in this paper we focus on the properties of g. The most common subsurface transport model is the solution of an advection-dispersion equation with equilibrium sorption and linear decay; it is obtained as a special case of (4) with math formula (Appendix B) and math formula, where Kd is a dimensionless partitioning coefficient, and math formula is the Dirac delta function.

3. A Generalized Memory Function

[12] In equations (1)(4), math formula is defined as math formula (Appendix A), where c is the mobile and math formula is the immobile solute concentration. In other words, g corresponds to a “partitioning coefficient” in the LT domain; it can also be viewed as a “transfer function” for a solute between mobile and immobile phases. Alternatively, normalized g may be interpreted as a probability density function (pdf) of tracer particle return time from the immobile to the mobile phase, once it has entered the immobile phase.

[13] Our basic requirement for a general form of g is that it can be expressed in the Laplace domain and be related to most existing models as special cases, exactly or approximately. Furthermore, we seek a parameterization that is simple, intuitive, and physically based to the extent possible, asymptotically capturing for instance cutoffs in the mass transfer which may eventually lead to an equilibrium exchange.

[14] Consistent with the above criteria, we propose here g that is defined in the Laplace domain as

display math

where A is dimensionless, T0 [T] is a characteristic retention time, and math formula and ν are exponents; we limit our present discussion to the range math formula. For math formula, normalized (5) is a gamma distribution; we therefore refer to (5) as a power-law extension of the gamma distribution. Equation (5) is factorized, hence spatial variability in A can be incorporated as in (2), assuming all other parameters of g as uniform (effective) values. In the following, our interest is the rate-limiting (kinetic) aspect of retention, hence our focus is set on the normalized form of g, math formula with math formula; for simplicity, the notation g may be used in the following as equivalent to the normalized form math formula.

[15] The exponent range in (5) is math formula and math formula. For math formula we recover the equilibrium model (model 1, Table 1), whereas for math formula, (5) reduces to the first-order kinetic model with the reverse rate equivalent to math formula (model 2, Table 1). Three specific forms used as “waiting time densities” by Scher and Lax [1973] and Montroll and Scher [1973], for instance, are obtained as math formula (“canonical form”), math formula, and math formula; multiple path waiting time density has been used in the form (5) with math formula and math formula by Caceres [2004].

Table 1. Memory Function g in Real and LT Domainsa
(Model) Mass Transfer ProcessMemory Function (LT Domain) (–)Memory Function (Real Domain) [1 T−1]Limit of math formula and Parameters
  • a

    Diffusion in complex (disordered) structures has been considered, e.g., by Havlin and Ben-Avram [1987] and Giona [1992], where the power-law form of g seems to be most useful. Physical parameters are explained in the text. The parameter math formula for model 3 is a finite value parameter different from T0 in (5). Model 4 is a special case of model 3, and is consistent with forms found in the literature [e.g., Neretnieks, 1980; Cvetkovic et al., 1999].

(1) EquilibriumA math formula math formula
(2) First-order math formula math formula math formula, math formula
(3) Multirate math formula math formula math formula, math formula
(power-law dist)   math formula
(4) Diffusion-sorp math formula math formula math formula, math formula,
(unlimited)   math formula, math formula,
    math formula
(5) Diffusion-sorp (slabs) math formulaN/A math formula
    math formula
(6) Diffusion-sorp (spheres) math formulaN/A math formula
    math formula

[16] The first derivative of (5) with respect to s is

display math

For math formula, math formula as math formula is not defined and the retention process is anomalous, whereas for math formula, m1 is finite as are all other moments of g. Then m1 quantifies the first moment of tracer return times. In the following, cases with math formula and math formula will be treated separately. If the characteristic time of hydrodynamic transport is math formula, then a Damkohler number may be defined as math formula.

[17] Further generalization of (5) is possible by truncation with a fifth parameter, the cutoff rate d [1/T] as

display math

which secures finite moments. Our focus in this work will be on the four-parameter model with d = 0.

3.1. Gamma Distribution ( math formula = 1)

[18] With math formula, (5) reduces to

display math

where T0 [T] is a characteristic (cut-off) time, roughly the time when equilibrium is reached, and A is the partitioning coefficient between the mobile and immobile phases at equilibrium. Note that equilibrium is reached eventually only if T0 is finite.

[19] In the real domain, g is proportional to the gamma distribution, i.e.,

display math

[20] Current mass transfer models for groundwater transport available in the literature may be classified into two basic categories: (1) simplified geometric models (spheres, rectangles, etc.), and (2) multirate models of which the first-order kinetic model is a special case. A relationship between a number of models in the two categories has been established [Haggerty et al., 2000].

[21] The first two moments of g(8) are computed from (7) as

display math

[22] Consider now two classical geometric models: diffusion-sorption in rectangular slabs and diffusion-sorption in spheres. The memory functions for the two models are given in the LT domain in (1). The moments are

display math

for the spheres model where math formula [L] is the sphere radius and math formula is the apparent diffusivity [Haggerty et al., 2001], Dp is the pore diffusivity and math formula the retardation factor. For the slab model we have

display math

where math formula [L] is the extent (thickness) of the slabs.

[23] Based on moment expressions (9)(11), the parameters T0 and ν can be related to the parameters of the two geometric models. The simplest way is to equate the first moments; in 1 we summarize resulting relationships.

[24] The cumulative distribution functions (CDFs) and complementary cumulative distribution functions (CCDFs) for the two geometric models are compared with the model (8) in Figure 1, the parameters being related as given in Table 1. Indeed, the CDFs and CCDFs for the three models are very close, but not identical (blue, red, and green curves, Figure 1) since only the equivalence of the first moment is ensured. Note the ascending 1:2 slope of the CDF for math formula that is characteristic of mass transfer controlled by Fickian diffusion; it can be compared to the much steeper slope of 1:1 for the first-order kinetic model (solid black curve in Figure 1) obtained from (8) with math formula.

Figure 1.

Cumulative distribution function (CDF) and complementary cumulative distribution function (CCDF) for the dimensionless return time math formula compared between different mass transfer models. Characteristic time T0 in (8) with math formula is related to the parameters of the geometrical diffusion models by equating first moments of the return time.

3.2. Extended Gamma Distribution ( math formula)

[25] In this case, first and all higher moments of g are undefined. We explore the limits of the CDF math formula as math formula and math formula where math formula is defined by (5). Consider first the case math formula (or math formula); we have

display math

[26] Next, we explore the CCDF, i.e., math formula, in the limit math formula (or math formula); we get

display math

[27] In Figure 1, CDFs and CCDFs of the memory function (5) are illustrated for a range of math formula with math formula. As indicated by the above limits, the early time slopes of the CDF decrease for decreasing math formula, whereas the asymptotic slope of the CCDF for math formula becomes steeper with increasing math formula (thin solid lines in Figure 1). Retention in these cases is clearly anomalous with infinite moments. Note that in view of the above limit expressions, the case with math formula would result in identical limiting slopes of the CCDF for math formula, whereas the CDF for math formula would retain similar relative positioning of the ascending thin black lines but now positioned above the math formula slope (thicker black solid line in Figure 1) as indicated by math formula. The case math formula is also applicable with math formula discussed in the section 3.1.

[28] In order to further grasp the impact of exponents math formula on the exchange between the mobile and immobile phase, fractional return times are plotted in Figure 2, as the 1-percentile (Figure 2a), 50-percentile (or median) (Figure 2b), and 99-percentile (Figure 2c), for the ranges math formula and math formula. The 1- and 50-percentile surfaces have a similar form, except that the values for the 1-percentile are significantly lower, where the impact of math formula is mostly exhibited for low ν (Figures 2a and 2b). By contrast, the 99-percentile return time math formula is significantly more sensitive to the value of the exponent math formula for higher ν (Figure 2c); the general trend for the late tracer return to the mobile phase is that math formula increases significantly for decreasing math formula. With math formula, the 1- and 50-percentile return times are more sensitive to the exponent ν than is math formula (Figures 2a2c).

Figure 2.

The normalized return times of different percentiles that indicate the early, medium, and late return times and their dependence on exponents math formula and ν for the general model (5).

4. Impact on Groundwater Transport

[29] An important question for applications is how the rate limitations of mass transfer (or retention) influence contaminant transport by groundwater. The Lagrangian framework summarized in section 2 provides arguably the most compact means of exploring this impact.

[30] Consider an aquifer on some scale for which the mean water travel time is math formula. In crystalline rock for instance, a mean water travel time of 1 year would be typical on the scale of one to a few hundred meters [Frampton and Cvetkovic, 2011]; in an unconsolidated and more permeable aquifer, the mean water travel time on a comparable scale could be a few months. Assume further that the macrodispersion on the same scale of interest is quantified by the coefficient of variation of the water travel time math formula; as noted in Appendix B, math formula, where math formula is the longitudinal macrodispersivity and L is the scale of the transport problem. The value of math formula of the domain length is relatively large but not uncommon for heterogeneous porous media, including fractured rock [Neuman, 1990]; we shall assume this value for illustration, whereby math formula.

[31] With math formula and math formula specified, we need to assume the nature of the hydrodynamic transport with the choice of the exponent α, in order to use the transport expression (4); in this example Fickian macrodispersion is assumed with math formula in (4). Equation (4) can now be used for exploring the impact of retention on groundwater contaminant transport by inserting, e.g., (5) (or any other form of g) into (3). Equation (4) also requires the input function J0, as the total injected mass distributed over time.

[32] Rather than consider full breakthrough curves (4), we shall focus here on tracer attenuation, as quantified by the released mass fraction at the transport scale L over which math formula is specified, math formula and math formula in (4). The injection function J0 is assumed as a pulse of unit mass, i.e., math formula, whereby J = h. Furthermore, the tracer will be subject to some type of loss approximated as linear decay; the mass fraction released over scale L for math formula (denoted by math formula), is then obtained from (4) (or (3)) by substituting math formula (see Appendix A) as

display math

where math formula [1/T] is a decay rate. A useful measure of math formula is an order-of-magnitude count math formula, referred to as the “attenuation index” [Cvetkovic, 2011b]:

display math

[33] The quantity math formula summarizes in a compact and intuitive manner all dominant transport mechanisms: hydrodynamic transport, retention, and transformation, and is therefore suitable for exploring relative effects.

[34] To specify retention by (5), we require the characteristic retention time as quantified by the Damkohler number math formula, the exponents ν and math formula, and A. Defining math formula, we get from (13)

display math

after utilizing (5) and (B11). A convenient representation is to normalize math formula by math formula, i.e.,

display math

where

display math

is obtained from (14) by setting math formula. Our focus in the following is on the impact of math formula, A, ν, and math formula on tracer transport as quantified by the relative attenuation index math formula(15).

[35] In Figure 3a, math formula(16) is first illustrated as a function of math formula for five values of A [0,1,10,100,1000]; math formula in Figure 3a quantifies attenuation in case the retention process is assumed at equilibrium, including the case without retention (A = 0). Clearly math formula strongly depends on A.

Figure 3.

The attenuation indices (16) and (15). (a) math formula as a function of math formula for different A; (b) math formula as a function of ν for different math formula and math formula with math formula; (c) math formula as a function of ν for different math formula and math formula with math formula; and (d) math formula as a function of ν for different math formula and math formula with math formula.

[36] In Figures 3b3d, math formula(15) is illustrated as a function of ν for math formula [1/10,1/2,1], math formula [0.001,1,1000], and math formula [1,10,100]. In all cases, the curves are monotonically decreasing, with the maximum value being 1. Thus for a given math formula, Figures 3b3d provide the entire range of kinetic effects for different values of ν, math formula, and math formula. The thick curves are for the special case with finite moments ( math formula); it is seen that this case sets bounds, implying that for quantifying math formula the three parameter model ((5) with math formula, and math formula) may be substituted with the simpler two-parameter model ((8) with ν and math formula), irrespective of math formula. Special case of interest is for math formula and math formula, consistent with the classical diffusion models as discussed earlier (see Table 1). Rate limitations in this case are significant and the effect on attenuation notable, in particular for low math formula and increasing math formula (compare solid thick red curves in Figures 3b, 3c, and 3d at math formula).

[37] As an application example, consider solute transport in a fractured crystalline rock [Neretnieks, 1980; Cvetkovic et al., 1999] with effective matrix block size math formula. It is known that in this case [Cvetkovic et al., 1999]

display math

where math formula is the effective diffusivity, math formula is the pore diffusivity, R is the retardation factor, and sf [1/L] is the active specific surface area; in the above expression math formula, where Kd is the sorption coefficient, math formula is the rock density, and math formula is the matrix porosity.

[38] Comparing parameters A and T0 of (7) ( math formula), with parameter groups in (17), we have (consistent with expressions in Table 1 for model 5)

display math

which can now be used in (14).

[39] Take a rock domain for which math formula yr and math formula (which corresponds to the dispersivity being math formula/2 = 12.5% of the domain length). With a typical value math formula m2 yr−1, let us consider a tracer corresponding to plutonium with roughly math formula = 15,000 and math formula 1 yr−1. The parameter sf is difficult to estimate in the field; for illustration we assume a value of 10,000 1 m−1, which falls within the range inferred from tracer tests and obtained by numerical simulation [Cvetkovic and Frampton, 2010, 2012].

[40] In Table 2 we summarize math formula computed from (14) for a range of exponents α (characterizing macrodispersion) and ν (characterizing retention processes); math formula indicates possible deviations from Fickianity, and math formula indicates deviation from the mass transfer controlled by Fickian diffusion. The equilibrium model ( math formula) assumes that all the sorption sites are instantaneously available to sorption, resulting in strong attenuation; note that the first calculations for assessing feasibility of geological disposal of nuclear waste were based on the equilibrium assumption [de Marsily et al., 1977]. The common retention model ( math formula) implies that the sorption sites are being accessed as a rate limiting process by Fickian diffusion, which results in significantly lower attenuation compared to the equilibrium model; deviation from Fickian diffusion can affect attenuation, depending on whether ν is smaller or larger than 1/2. The nature of macrodispersion (choice of α) can strongly affect attenuation, in particular for the lower range of ν. Values math formula imply earlier arrival compared to the advection-dispersion equation model ( math formula), and hence lower attenuation, and vice versa for math formula. Deviations of ν from 1/2 and α from 1/2 have been explored [Cvetkovic et al., 2007; Cvetkovic and Frampton, 2012], however, more field data is required for further assessment of these deviations under different field conditions.

Table 2. The Attenuation Index (14) for Different Values of Exponents α and ν
 α = 1/3α = 2/5α = 1/2α = 3/5
math formula6697179348
math formula16.7202636
math formula7.589.411
math formula2.72.833

5. Discussion

[41] Rate limitations in groundwater transport are a rule rather than an exception [Brusseau and Rao, 1989], due to often complex microstructure of the secondary porosity where groundwater is essentially stagnant (immobile) [Wood et al., 1990; Cvetkovic, 2010a]. A memory function, in particular its Laplace transform, is an effective means of accounting for rate limitations in groundwater transport modeling using (4). A variety of memory function formulations have been considered in the literature for groundwater transport, starting from the classical cases of first-order kinetics and diffusion-controlled mass transfer with simple geometries, to multirate models with different distributions of exchange rates [Haggerty et al., 2000].

[42] The key to accurate quantification of groundwater transport is to effectively combine macroscopic heterogeneity effects (as quantified by the water travel time density f) with effects of mass transfer (quantified by the memory function g); this coupling is captured by the Lagrangian mass balance expressions (1) and (2) or the simplified form for macroscopically uniform retention parameters (3); in (1)(3), g plays a central role as a physically based representation of the dominant underlying retention mechanisms. Field-scale experiments [Haggerty et al., 2001; Cvetkovic et al., 2007], laboratory tests [Gouze et al., 2008a, 2008b], or their combination [Cvetkovic, 2010b] can be used for a specific “design” of g, including interpretations of measured temporal moments [Luo et al., 2008]. Reproducing flow heterogeneity effects on breakthrough tailing using a memory function has been explored by Willmann et al. [2008]. A recent study based on generic groundwater transport simulations [Fiori et al., 2011] is a good example of possibilities in designing a geometric-type g (spheres, Table 1); the work of Fiori et al. [2011] exposes very clearly the combined effect of advection and mass transfer, by providing for the first time a physically based parameterization of g that explicitly incorporates both heterogeneous hydraulic and microscale diffusion/dispersion properties.

[43] The extended gamma distribution (5) proposed in this work expands further modeling possibilities with a memory function, and can reproduce exactly or approximately a number memory functions used in the literature, most common of which are summarized in Table 1. In the following, we wish to relate (5) to a few forms of g not included in Table 1.

5.1. Incomplete Gamma Function

[44] An interesting formulation of g that relates to the fractional diffusion equation for groundwater transport has been recently proposed as an incomplete gamma function [Baeumer and Meerschaert, 2010]

display math

where the parameters T0, A, and ν have a similar meaning as in (8). In the real domain, the expression reads

display math

where math formula is the incomplete gamma function.

[45] The first two moments of g are easily obtained from (19) as

display math

Comparing (21) with (9) we see that for m1 in (21) and (9) to be equivalent, we require T0[(20)] = 2 T0[(8)]. With this constraint, it can be shown that (20) and (8) in fact yield very close responses. Thus (20) can be related to classical models of (1), similarly as was done with (8) in Table 1.

5.2. Multirate Models

[46] A multirate model was originally introduced for describing trapping in a variety of physical systems [Schmidlin, 1977; Noolandi, 1977a, 1977b], and more recently for transport in groundwater [Haggerty and Gorelick, 1995; Haggerty et al., 2000, 2001; Cvetkovic and Haggerty, 2002]. In the Laplace domain, the multirate g (discrete form) reads

display math

where math formula is the probability of the rate ki, and math formula if the summation is over all available sites.

[47] The special case of power-law math formula is of interest. It has been shown that power-law math formula yields math formula [Haggerty et al., 2000]. As noted in Table 1, a power-law g can be recovered from (8) by assuming math formula while math formula is finite.

[48] Various distributions have been considered for math formula, the most common being lognormal and the gamma distribution [Haggerty et al., 2000]. In both cases it can be shown that

display math

In other words, the CDF of return times always has 1:1 slope in the limit math formula, as the first-order model with a single rate. This can be set in contrast to the memory functions for diffusion controlled exchange where the corresponding slope is 1:2.

[49] A mass transfer model with multiple rates that follows a gamma distribution was proposed by Culver et al. [1997], and further elaborated by Haggerty et al. [2000]. Relatively simple closed form expressions in the LT and real domains are respectively

display math

[50] The asymptotic slope of (23) for math formula (or math formula) is controlled by ν, and in fact (23) has no finite moments. Closer inspection of (23) indicates that in the limit math formula (corresponding to math formula), math formula with (23) behaves like t for math formula, i.e., like a first-order model as deduced from (8) with math formula. This implies that the model (23) is not suitable for capturing the Fickian diffusion-type mass transfer (common in the subsurface), for which the ascending slope is 1/2, i.e., math formula as math formula.

[51] An instructive comparison of the memory functions (23) and (5) is by their impact on contaminant transport, in the present case quantified by math formula(12), or math formula(14). In Figure 4 the attenuation index math formula(14) is shown as a function of math formula, over a six orders-of-magnitude range, for three values of the exponent ν in (23), [0.1,0.5,1]. The red dashed line in Figure 4 is obtained from math formula(14) with g(5) and setting math formula since only in that case the short time slope of g(23) is identical with that of g(5), as has been shown above.

Figure 4.

Comparison between three memory functions (5), (23), and (22) with math formula lognormal. (a) Attenuation index math formula(14) as a function of the Damkohler number math formula; in this computation math formula, A = 10, and math formula. (b) math formula as a function of s (real part).

[52] When math formula in (23), math formula obtained by the two models essentially coincide over the entire range of math formula (compare the red and blue dashed curve in Figure 4). With decreasing ν in (23), the two mass transfer models yield diverging math formula only when the transport conditions converge to equilibrium with say math formula. Note however that T0 (and hence math formula) for the two mass transfer models have a different meaning since (8) has finite moments and the first moment is proportional to T0(9), whereas moments of (23) are undefined. Because all curves in Figure 4 increase monotonically, any math formula value associated with the blue curves (for different ν in (23)), can in principle be reproduced by a suitable choice (or interpretation) of the characteristic retention time T0.

[53] To better understand Figure 4 for large math formula (or small T0), it is instructive to compare plots of math formula for models (8) and (22) with a lognormal and gamma math formula; g(22) with math formula as the gamma distribution is given in (23). In all cases, math formula or math formula. Clearly math formula is monotonically decreasing but at different rates. Model (8) with math formula 1/2 decreases more slowly and with significantly larger values in the tail, typical for diffusion controlled mass transfer. Since math formula directly affects math formula(13), it follows that the limit math formula (or math formula) is most important when quantifying contaminant transport by groundwater, e.g., for risk assessment.

[54] Finally, it is noted that a truncated form of (23) has also been discussed in the literature as a “waiting time density” within the continuous-time random walk formulation of transport by groundwater [Dentz et al., 2004]; in such a case s in (23) is replaced by s + d, where d [1/T] is a truncation parameter (or math formula the truncation time). The CDF slope of a truncated (23) for math formula is still 1:1 as for (23), making it unsuitable for quantifying diffusion controlled mass transfer.

6. Conclusions

[55] In this work we have proposed a simple form of the memory function for quantifying solute retention in groundwater (5), and have analyzed its properties. Based on the results, we draw the following main conclusions.

[56] 1. A memory function g is an expression of the mass transfer kinetics for systems where exchange takes place between a mobile and immobile fluid. In its normalized form it can be interpreted as a transfer function between the mobile and immobile fluid, or as a probability density function of return times form the immobile to the mobile phase.

[57] 2. Equation (5) is a general and convenient formulation of the memory function that can be reduced exactly or approximately to most of the forms that have been considered in the literature for groundwater transport [Haggerty et al., 2000], but also for studying exchange in more general physical settings [Scher and Lax, 1973; Scher and Montroll, 1975]; the proposed form is particularly suitable as part of the Lagrangian modeling framework [Dagan, 1984; Cvetkovic and Dagan, 1994; Cvetkovic et al., 1998] for coupling complex retention with Fickian or non-Fickian hydrodynamic transport using, e.g., (3).

[58] 3. In its most general truncated form, (6) depends on five parameters, two exponents math formula and ν, A, and two characteristic times T0 and math formula; a three parameter form (8) reduces to most of the models used in the literature (Table 1) and may be sufficient for most applications. The fact that g is factorized with A, makes it suitable for incorporating at least part of the spatial variability in the retention using B in (2).

[59] 4. The probability of short return times from the immobile to the mobile phase is critical for early tracer arrival and in particular for contaminant attenuation in groundwater. CDF of short return times (or high exchange rates) in the limit math formula behaves for (5) as math formula; this can be compared to the first-order kinetic case with math formula and Fickian diffusion case with math formula for math formula.

[60] 5. In the asymptotic limit math formula, CCDF of (5) behaves like math formula which can be compared to the first-order kinetics and diffusion-controlled exchange where the CDF converges to 0; if needed, this behavior can be reproduced using the truncated form (6) by a suitable choice of the cutoff rate d; thus (6) retains the same behavior of g(5) for short times but asymptotically converges to 0. The overall behavior of the return time distribution (5) is well summarized in Figure 2.

[61] 6. The impact of rate limitations on groundwater transport is suitably quantified by the attenuation index (13) and (14), as an intuitive and practically relevant measure. The equilibrium case with instantaneous accessibility of exchange sites and infinitely fast return times on the one hand, and the case without retention (g = 0) on the other hand, set the bounding values for tracer attenuation (Figure 3). All cases in between these limits in Figure 3 can in principle be reproduced by the two-parameter model (8).

[62] 7. In most applications, the three-parameter model (8) will be sufficient to capture key features of retention for field-scale groundwater transport. The value of around math formula is particularly relevant, as it is consistent with exchange controlled by Fickian diffusion in different geometrical forms. Although math formula may be quite robust in crystalline rock [Cvetkovic et al., 2007, 2008], possible deviations need to be further explored for other type of geological media, in particular given the effect deviations of ν from 1/2 may have on attenuation (Table 2).

Appendix A:: Mass Balance Formulation

[63] We consider the transport of a dissolved tracer in a spatially variable three-dimensional, steady state velocity field math formula with a mean drift set parallel to the x axis, i.e., math formula, and math formula, where math formula. In a heterogeneous aquifer, or a fractured rock formation, the flow is driven by a mean gradient. In an aquifer, the variability of flow velocity is primarily due to variability in the hydraulic conductivity K, whereas in a fractured rock formation the flow velocity is usually considered nonzero only in fractures of a three-dimensional network with random hydraulic properties.

[64] As a base case, consider a point source (i.e., on a scale of the same order or less than the support scale for V), with a pulse injection of tracer mass m0. The injected tracer is subject to advection by the random velocity field V, dispersion due to fluctuations on a scale smaller than the support scale of V and retention due to an exchange process with the immobile phase.

[65] Following the Lagrangian methodology (see Figure 1 in Cvetkovic and Dagan [1994] for the conceptualization), let math formula math formula denote the trajectory staring at math formula. The water travel time along the trajectory is defined by the integral equation math formula, where math formula is the x component of the velocity along a trajectory, and backward flow is neglected. This constraint can be easily bypassed if the intrinsic length along a trajectory is used. The starting location is at a in the plane at x = 0 and the end location is b at the plane x = L, where math formula.

[66] Let c [M/L3] denote tracer concentration at x and t in the mobile phase, and let math formula [M/L3] denote the concentration in the immobile phase, both per volume of fluid. The mass balance equations are written as

display math
display math

where q [M/L2T] is the mass flux, and S [M/L3T] is a sink-source term. The above equations can be solved in a general way based on the two following assumptions.

[67] 1. Exchange processes are linear in a generalized sense [Villermaux, 1974], which in the Laplace domain implies math formula with g being a “memory function” independent of solute concentration.

[68] 2. Fluctuations on a scale smaller than the support scale for V are negligible, i.e., advection by V and retention are dominant macroscopic transport mechanisms.

[69] Under above assumptions, the system (A1) and (A2) reduces to [Cvetkovic and Dagan, 1994]

display math

with the star denoting convolution, and math formula [T] the water travel time. The solution of the system (A3) is

display math

where n is the flow porosity (assumed constant), math formula and math formula are the transverse coordinates of the trajectory crossing location b in the control plane at x, and V1 is the x component of the velocity here transformed onto the trajectory; thus the independent variable is math formula, or the position of the control plane x. The function math formula is defined by

display math

[70] From the concentration, the tracer mass flux is computed as

display math

[71] Integration of q1 over the control plane yields solute discharge as

display math

[72] Typically, the interest is to compute at least the expected value of the mass discharge J, i.e., math formula for an arbitrary input that is given as a function of time and space (for a spatially distributed source), math formula. Note that for a unit pulse math formula, implying that math formula is in effect a transfer function for a single trajectory. Averaging of math formula(A5) yields h(3).

[73] Let a total amount of solute mass M0 be released over a given area math formula at t = 0. We have

display math

[74] If the injected solute over an area math formula is discharged at a given control plane (or point, e.g., a well), then the normalized solute discharge is

display math

where L denotes the distance between the injection and detection surfaces. In the general case, h may also depend on a. For a tracer subject to first-order decay, the LT solution can be easily modified by substituting in (A5), math formula.

Appendix B:: Macrodispersion

[75] The spreading of tracer particles due to flow variability results in macrodispersion. The basic model for macrodispersion of noninteracting tracers is the solution of the advection-dispersion equation. In this study, we use the tempered (or truncated) one-sided stable (TOSS) density as a general model for macrodispersion, that can capture Fickian, non-Fickian as well as anomalous transport [Cvetkovic and Haggerty, 2002; Cvetkovic, 2011].

[76] Consider transport along independent segments. The probability density function of math formula along each segment is described by the TOSS density in the Laplace domain as

display math

where s is the LT variable, and the parameters ai and ci are related to the first and second moments by

display math

Thus ai is a cut-off rate, math formula is the mean, and math formula is the coefficient of variation of math formula, and math formula is an exponent.

[77] Computation along the entire trajectory composed of n segments is obtained by convolution:

display math

where math formula and math formula ensure that the density is infinitely divisible; in other words, the cut-off rate a is the same for segments and for the entire trajectory, whereas c scales as a sum over all segments. These conditions also imply that the mean and variance for segments and for the entire trajectory, scale in the same manner, i.e., math formula, where math formula is the segment variance, and math formula the trajectory variance with math formula.

[78] The above equations imply that the TOSS model is consistent with a time domain random walk approach to particle transport. For the specific value math formula, the TOSS density is a solution of the advection-dispersion equation with injection and detection in the flux [Kreft and Zuber, 1978]. Assuming in (B1) and (B2) math formula and math formula, reduces (B1) to an exponential distribution with math formula as the parameter [Cvetkovic, 2011b], which is the “canonical form” of the “waiting time density” in the continuous-time random walk approach [Montroll and Scher, 1973].

Acknowledgments

[79] Critical comments by Aldo Fiori (University of Rome, Italy) and Jesus Gomez (NM Tech, USA) have helped to improve the original version of the manuscript; their effort is sincerely appreciated. The authors gratefully acknowledge the financial support by the Swedish Nuclear Fuel and Waste Management Co. (SKB).