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Modelling velocity and retardation factor of a nonlinearly sorbing solute plume.


Only limited developments, both theoretical and experimental, have been achieved so far for quantifying the transport parameters of reactive solutes. There is need to validate the theories that explain and predict the transport of such solutes. Heterogeneous geological formations as well as sorption processes of reactive solutes cause their transport under non-ideal conditions. Solute is adsorbed on soil grains by rate-limited chemical reactions called chemical sorption, and owing to concentration gradients, mass of solute is transferred between mobile and immobile fluid phases by Fickian type diffusion called physical sorption. The transport of reactive solutes is attributed to chemical (Roberts et al. 1986; Dagan and Cvetkovic 1993) or physical (Cushey and Rubin 1997) sorption, or to a combination of both processes (Miralles-Wilhelm and Gelhar 1996). However, to simplify the prediction of transport by applying equilibrium-based solute-transport models, sorption reactions are assumed sufficiently fast (Kabala and Sposito 1991; Chrysikopoulos et al. 1992). But evidence from laboratory and field investigations indicates significant kinetic effects and much slower sorption rates at the field scale than generally assumed (Nkedi-Kizza etal. 1983; Ptacek and Gillham 1992). An important problem in modelling sorption reactions is, therefore, to identify the actual processes governing the transport of reactive solutes.

The transport of reactive solutes in a heterogeneous porous medium occurs under the combined influence of velocity field and sorption reactions, both spatially variable; the former enhances spreading of the solutes while the latter retards their movement. The spatial variability of solute distribution coefficient imparts an additional variability in the local velocity of such solutes. The asymptotic stochastic-analytic theories of Garabedian (1987), Dagan (1989), and Kabala and Sposito (1991) predict much larger macrodispersion of reactive solutes than non-reactive ones. Mackay et al. (1986) and Sudicky (1986) noticed growing retardation factor of reactive solutes in time in a large-scale field test at Borden, Canada. Simple laboratory column experiments where sorption reactions are assumed to occur under linear equilibrium condition do not show this nature of retardation factor. Van der Zee and van Riemsdijk (1987), Bosma and van der Zee (1993), and Burr et al. (1994), in their numerical study, found growing retardation factor with increasing travel time and travel path of reactive solute plumes in steady-state flow field. Vereecken et al. (2002), using asymptotic approximation of large time (Jaekel et al. 1996), derived a power law function of travel time and displacement of reactive solutes for 1D transport in a homogeneous porous medium. For nonlinear sorption, they also derived a power law function of retardation factor and travel time of a solute plume at large time. Despite these developments, the variation of velocity and retardation factor of nonlinearly sorbing solutes at the field scale is yet not fully understood and predictable. This study therefore first identified the processes controlling the transport of an uranin (sodium fluorescein: [C.sub.20][H.sub.10][Na.sub.2][O.sub.5]) plume in a heterogeneous aquifer by evaluating field-measured breakthrough curves (BTCs) and then modeled the variation of velocity and retardation factor of the plume.

Materials and methods

Site description and tracer experiments

This study was conducted at the Krauthausen test site (size: 200 by 70m), 6km away from the research centre in Juelich, Germany, and located in the Lower Rhine Embayment. Detailed description of the test site and tracer experiments is found in Vereecken et al. (1999, 2000); only a summary is given here. The tracer experiments were conducted in the upper unconfined aquifer, which consists of quaternary sediments overlaid by loess. The base of the aquifer, 11-13 m below the ground surface, consists of thin layers of clay and silt. A 7-m-thick layer of gravels and sands (Rhine deposits) is on the base whose bottom 1-2 m is inter-layered with sand, gravel, and clay. The Rhine sediments are overlain by a 2.5-3.0-m-thick layer of dark brown gravel stemming from deposition of the river Rur. On the top of the Rur deposits, there is a 0.2-m-thick stony layer on which a 0.7-1.0-m surface soil developed from loess. The Rur sediments [93-95 m above mean sea level (MSL)], upper Rhine sediments (90-93 m above MSL), and lower Rhine sediments (88-90 m above MS L) are hereafter referred to Layer 1, Layer 2, and Layer 3, respectively.

The water table in the aquifer fluctuated between 1.0 m in winter and 2.5 m in summer below ground surface with a fairly constant hydraulic gradient of 0.2% during the tracer experiments. The geometric mean of the hydraulic conductivity of the aquifer is 328.32 m/day with a variance of loge-transferred hydraulic conductivity of 1.3. Local velocities of groundwater, estimated by a radioactive tracer experiment, were lognormally distributed with a geometric mean of 0.47 m/day and a variance of 1.86.

As a reactive solute, 2 kg of uranin ([C.sub.20][H.sub.10][Na.sub.2][O.sub.5]), dissolved in 4500 L of groundwater, was injected at 6-7 m below the ground surface during a period of 5 h. Uranin is an alkaline salt of sodium fluorescein, which is non-toxic in low concentration, relatively inexpensive, slightly sorbed by most solids, and easily measurable in low concentrations. It has a bright yellowish-green colour in dilute solutions and was used as a good tracer by several investigators (Eiswirth and Hotzl 1994; Mikulla et al. 1997; Neretnieks 2002). As a non-reactive tracer, 135.9 kg of bromide, dissolved in 8000 L of groundwater, was also injected for 10.8 h at the same depth. The spatial and temporal evolution of the uranin and bromide plumes was monitored in 58 observation wells, the depth of which varied from 10 to 11 m. Each well was equipped with 24 multi-level samplers at a vertical spacing of 0.22, 0.30, or 0.35 m. The temporal evolution of uranin was monitored at all samplers for 449 days while that of bromide was monitored only in odd number of samplers for 398 days.

Solute-transport models

Two physical non-equilibrium based and one chemical non-equilibrium based solute-transport models were used to determine sorption processes of uranin with aquifer materials by evaluating their rate parameters. BTCs of uranin plume were analysed by the method of time moments to determine the velocity and retardation factor of the plume. In the physical non-equilibrium based models, the fluid inside the porous aggregate is assumed stagnant, and the liquid phase is partitioned into mobile and immobile zones. The governing solute-transport equation is given by (van Genuchten 1985):

[[theta].sub.m][R.sub.m] [[differential][C.sub.m]/[differential]t] + [[theta]][] [[differential][]/[differential]t] =[[theta].sub.m]D[[[differential].sup.2][C.sub.m]/[differential][x.sup.2]] - [[theta].sub.m]v [[differential][C.sub.m]/[differential]x] (1)

where the subscripts 'm' and 'im' refer to parameters for the mobile and immobile regions, respectively, C is the aqueous phase concentration, [theta] the porosity, R the retardation factor, D the dispersion coefficient, v the fluid velocity, x the distance, and t the time. The parameters [] and [] are expressed in terms of aggregate radius, a, local aqueous phase concentration inside the aggregate, [c.sub.a], and effective molecular diffusion coefficient within the aggregate, [D.sub.a]. Eqn 1 is reduced to a non-dimensional form by defining the following dimensionless variables (van Genuchten 1985) as:

T = [vt/L] [[[theta].sub.m]/[theta]] (2)

X = x/L (3)

P = vL/D (4)

[beta] = [[R.sub.m]/R][[[theta].sub.m]/[theta]] (5)

[gamma]' = [[D.sub.a]L/[a.sup.2]v][[[theta]]/[[theta].sub.m]] (6)

where T is a dimensionless time called pore volume, L the distance of transport, P the Peclet number, [theta] = [[theta].sub.m] + [[theta]] the total porosity, and R the total retardation factor. For linear adsorption of a solute, R is often expressed in terms of bulk density of the porous medium, [rho], and the equilibrium solute-distribution coefficient, [k.sub.d], as R = 1 + [rho] [k.sub.d]/[theta]. In Eqn 6, [gamma]' is the index of non-equilibrium mass transfer that represents mass balance inside the immobile zone of the porous medium. Dropping the subscript 'm', for simplicity, the dimensionless form of Eqn 1, called the diffusion physical non-equilibrium model, is:

[beta]R[[differential]C/[differential]T] + (1 - [beta])R[[differential][]/[differential]T = [1/P][[[differential].sup.2]C/[differential][X.sup.2]] - [differential]C/[differential]X (7)

When equilibrium distribution of solute is attained between the mobile and immobile zones, local equilibrium assumption (LEA) becomes valid and Eqn 7 reduces to:

R [[differential]C/[differential]T] = [1/P][[[differential].sup.2]C/[differential][X.sup.2]] - [differential]C/[differential]X (8)

Avoiding geometrical specifications of the stagnant region, the mass exchange between the mobile and immobile zones is also described by a first-order type rate equation (Coats and Smith 1964) as:

[[theta]][] [[differential][]/[differential]t = [alpha](C - []) (9)

with [alpha] an empirical mass-transfer coefficient. The dimensionless form of Eqn 9 is:

(1 - [beta])R [[differential][]/[differential]T] = [omega](C - [] (10)

where [omega] = [alpha]L/(v[[theta].sub.m]), a dimensionless mass-transfer parameter. A large [omega] indicates attainment of equilibrium distribution of solute between the mobile and immobile zones and therefore the validity of LEA. The models described in Eqns 7 and 10 are called the first-order physical non-equilibrium models.

When sorption is controlled by chemical non-equilibrium reactions, the governing equation for the transport model is given by:

[theta][[differential]C/[differential]t] + [rho][[differential]s/[differential]t] = D[theta][[[differential].sup.2]C/[differential][x.sup.2]] - [v.sub.[theta]][[differential]C/[differential]x (11)

with s the concentration of the adsorbed solute. The sorption rate, [differential]s/[differential]t, in Eqn 11 is described by a first-order linear kinetic expression as:

[differential]s/[differential]t = [k.sub.f]C - [k.sub.r]s (12)

where [k.sub.f] is the forwards rate coefficient and [k.sub.r] is the reverse rate coefficient for the kinetic sorption reactions. The dimensionless form of Eqn 11 is given by:

[differential]C/[differential]T + [[rho]/[theta]][[differential]s/[differential]T = [1/P][[[differential].sup.2]C/[differential][X.sup.2]] - [differential]C/[differential]X (13)

and that of Eqn 12 is given by:

[differential]s/[differential]T = F([k.sub.d]C - s) (14)

In Eqn 14, [k.sub.d] = [k.sub.f]/[k.sub.r], the equilibrium solute-distribution coefficient, and F= [k.sub.r]L/v, the dimensionless rate parameter. Like [omega] in Eqn 10, a large F indicates the attainment of equilibrium solute distribution between the mobile and immobile zones and supports the validity of LEA. Since all pore water is assumed mobile in this transport model, [[theta].sub.m] = [theta] and T= vt/L (Eqn 2). The solute-transport model given by Eqns 13 and 14 is called linear chemical non-equilibrium model. This model cannot describe the transport of solutes undergoing nonlinear sorption (Bosma and van der Zee 1993). Strong adsorption of reactive solutes at low concentrations results in a prolonged tail of the measured BTCs and this model is incapable to predict such a tail. Nonlinear sorption of reactive solutes is governed by Freundlich isotherm defined by:

s = [k.sub.F][C.sup.n] (15)

where n is the Freundlich exponent, [k.sub.F] is the Freundlich sorption parameter, and s and C are as defined above.

The method of time moments, being independent of solute-transport models, is a simple but powerful tool for characterising the behaviour of models for the transport of both non-reactive and reactive solutes. The first two time moments describe the mean breakthrough time of a solute and the degree of spreading of the response BTCs. The method is well established and documented in literature (e.g. Valocchi 1985). The first time moment, [tau], is calculated by:

[tau] = [[integral].sup.t.sub.0] tC(x, t)dt/[[integral].sup.t.sub.0] C(x,t)dt (16)

and the second central moment, [mu], is calculated by:

[mu] = [[integral].sup.t.sub.0][(t - [tau]).sup.2]C(x, t)dt/ [[integral].sup.t.sub.0] C(x, t)dt (17)

where C(x, t) is the aqueous phase concentration at distance x and time t. The dimensionless forms of the first time moment, denoted by [M.sub.1], and the second central moment, denoted by [M.sub.2], for the solute-transport models in Eqns 7, 8, 10, and 13 (adopted from Valocchi 1985) are depicted in Table 1. [M.sub.1] and [M.sub.2] are related to their corresponding dimensional forms [tau] and [mu] by:

[M.sub.1] = [tau][v/L][[[theta].sub.m]/[theta]] (18)

[M.sub.2] = [mu][([v/L][[[theta].sub.m]/[theta]]).sup.2] (19)

The expressions of Table 1 were used to determine the rate parameters (F, [k.sub.f], [k.sub.r], [k.sub.d], [alpha], [gamma]', and [omega]) of different solute-transport models to quantify the effect of rate limitations. This was necessary to know the underlying sorption process(es) of uranin with the Krauthausen aquifer materials. In addition to these expressions, the analytical solution of LEA-based model (Eqn 8) given by DeSmedt and Wierenga (1979) as:

C'(X, T) = [(PR/4[pi][T.sup.3]).sup.1/2] X exp {-PR/4T[(X - T/R).sup.2]} (20)

was also used. In Eqn 20, C'(X, T) is a relative concentration defined by the ratio of aqueous species concentration to the solute concentration, which would be attained if the total input mass of solute were uniformly distributed throughout the pore volume of the transport domain. Note that the models in Eqns 7, 8, 10, and 13 result in the same first time moment, [M.sub.1], in Table 1, showing that the travel time of a solute plume, when calculated by using the method of time moment, does not depend on a particular solute-transport model.

Calculation and analysis of average BTCs

The concentration of uranin in the monitoring wells differed considerably at different samplers. An appropriate averaging procedure to upscale the BTCs at different samplers to an average BTC for a well is therefore necessary. The average BTC was calculated for each monitoring well separately for each layer of the aquifer by averaging (i) the absolute concentrations, (ii) the normalised concentrations, and (iii) the velocity-weighted normalised concentrations measured at different samplers. The first procedure calculated average BTCs, which were dominated by a few very high concentrations in the time series. The other 2 procedures provided average BTCs, which, not being influenced considerably by such concentrations, resulted in comparable values of solute-transport parameters. Vanderborght and Vereecken (2001) also reported similar results for bromide. The third averaging procedure, considering it to be more objective, was used in this study.

The evolution of uranin and bromide, although monitored in 58 observation wells, was detected in 29 wells of which only 17 provided almost complete BTCs. The average BTCs of both plumes were calculated at these 17 wells for each layer and also for the entire depth of aquifer for the bromide plume. The first time moment of bromide, [tau]', and the first time moment, [tau], and second central moment, [mu], for uranin were calculated from the average BTCs using Eqns 16 and 17. Equating these time moments to the moment formulas in Table 1, through Eqns 18 and 19, the desired parameters of different solute-transport models were determined. Also fitted to the average BTCs of uranin was Eqn 20 to estimate the Peclet number, P, retardation factor, R, and dimensionless time, T. In this case, the velocity of the uranin plume was calculated from T by using Eqn 2, where [[theta]]/[theta] was taken from the results of moment analysis.

Results and discussion

Velocity of bromide plume and groundwater flow

Figure 1 illustrates the variation of travel path, x, with travel time, [tau]', of the bromide plume for each of the 3 layers and entire depth of the aquifer. Layer 1 shows slightly lower and layer 2 slightly higher velocity than the average velocity of the aquifer. No remarkable difference in travel path v. travel time relationship is, however, visualised when considering the 3 layers and the entire depth of the aquifer. A linear relationship is therefore assumed (coefficient of determination = 0.89) between the travel path and travel time without significant error (Fig. 1). A similar relationship was also reported for bromide plume by Vereecken et al. (2000). Since bromide is a non-reactive tracer, the slope of the regression line, 0.64 m/day, provides the velocity of groundwater flow in the aquifer. The immobile water in the aquifer consisting of relatively coarse materials was neglected in the estimation of this velocity. The constant velocity is an evidence of a spatially and temporally uniform velocity field and hence steady-state flow during the tracer experiments as was also reported for the Borden aquifer in Canada (Brusseau 1998). Several different velocities of bromide plume were, however, reported for the Krauthausen aquifer. After 110 days of injection, Vereecken et al. (2000) obtained a mean average velocity of 0.5 m/day, which was consistent with the effective velocity of bromide plume, 0.41-0.84 m/day, estimated by the method of time moment from BTCs (Vanderborght and Vereecken 2001). But this velocity was lower than that of groundwater flow, 0.7-1.2 m/day, estimated from a radioactive tracer (eosin) experiment (Doring 1997). Also, the mean average velocity was considerably lower than the estimates from local velocity of groundwater (1.7m/day) and grain size data analysis (0.72 0.92 m/day) by Vereecken et al. (1999). The effective velocity at a reference plane of Vanderborght and Vereecken (2001) first decreased with distance from the plane of injection and then increased giving rise to smaller travel times of bromide plume at higher distance than at lower distance. This result could not be explained physically since, for example, a bromide plume needed to arrive in a monitoring well at 122.8 m earlier than in a monitoring well at 55.1 m from the injection plane. As a reason for this non-realistic observation, Vanderborght and Vereecken (2001) thought that monitoring breakthrough data at limited number of locations (monitoring wells) on a reference plane might not represent the overall breakthrough of bromide particles on the plane.


Transport processes of uranin

Table 2 lists the non-dimensional ratio, [beta], immobile water fraction, [[theta]], and rate parameters of the solute-transport models (Eqns 7, 8, 10, and 13), such as F, [k.sub.f], [k.sub.r], [k.sub.d], [alpha], [gamma]', and [omega] along with their standard deviations and coefficients of variation. The 1D physical non-equilibrium model (Eqn 7) estimates 0.03 [m.sup.3]/[m.sup.3] of immobile water compared with 0.26 [m.sup.3]/[m.sup.3] of total water content in the aquifer. The small mass-transfer coefficient, [alpha] = 0.03/day (Eqn 9), being the result of rate-limited mass transfer between the mobile and immobile water phases, and the large equilibrium distribution coefficient, [k.sub.d] = 0.32 (Eqn 14), of uranin demonstrate the presence of kinetic sorption reactions. The average values of [gamma]', [omega], and F in Table 2 are not large enough to ensure attainment of chemical equilibrium (Valocchi 1985). The rate parameters might also be confounded with retardation factor, R, as expected from Table 1, in attaining such equilibrium. Laboratory batch experiments of Doring (1997) showed nonlinear adsorption of uranin with the sediments of the Krauthausen aquifer that was described by Freundlich isotherm (Eqn 15). Vereecken et al. (1999, 2000), in their sorption-desorption study with sediment of this site, identified 3 different sorption processes for uranin: Freundlich equilibrium sorption with hysteresis, linear nonequilibrium sorption, and nonlinear nonequilibrium sorption, with the last sorption process dominating. Following these lines of evidence, it is assumed that chemical non-equilibrium sorption processes governed the transport of uranin in the Krauthausen aquifer. The evidence of non-equilibrium transport of reactive solutes was also reported at Borden (Roberts et al. 1986) and Cape Cod (Wood et al. 1990) test sites.

Variation of velocity with travel path of uranin plume

With growing concern of the effect of nonlinear sorption on the velocity of reactive solutes, it is a necessity to identify the relationship between local velocity and travel path of such solutes; the local velocity is the velocity at any point on the travel path. As illustrated in Fig. 2, the travel time of the uranin plume increases with increasing travel path of the plume at an increasing rate due to nonlinear sorption of uranin with the aquifer materials. Consequently, the slope of the plot that is equal to the local velocity decreases with increasing travel time. The local velocity is therefore predicted by the derivative of the relationship between the travel path and travel time of the uranin plume. The plot of travel path, x, v. travel time, [tau], of the plume, being linear on log-log scale (Fig. 3), provides power law relation given by:

x = k[[tau].sup.[gamma]] (21)

where [gamma] is an exponent and k is a proportionality constant. The value of [gamma] and k found to be 0.614 and 1.699, respectively raises 2 notable points. First, [gamma] closely agrees with the mean value of n (= 0.69) (Eqn 15) determined by asymptotic analysis of column BTCs of uranin (Vereecken et al. 1999). Second, Eqn 21 is of the form similar to that derived between the travel path of a reactive solute front and its travel time (Jaekel et al. 1996, their eqn 23) with [gamma] as the n exponent. The power law behaviour of the movement of the uranin plume with its travel time (Eqn 21) is consistent with the finding of Jaekel et al. (1996), who reported variation of the travel path of a nonlinearly sorbing solute front in time with Freundlich exponent. Eqn 21 is also consistent with the finding of Vereecken et al. (2002), who showed a power law relationship between x and [tau] at large time with the exponent equal to n of Freundlich isotherm. Since nonlinear sorption, described by Eqn 15, controlled the transport of uranin in the Krauthausen aquifer, it is intuitively understood that the exponent, [gamma] (Eqn 21), governs the nonlinearity between x and [tau], and is equivalent to the exponent of Freundlich isotherm (Eqn 15). Applying asymptotic theory of Jaekel et al. (1996) in 72 BTCs of uranin in 3 observation wells, Vereecken et al. (1999) found an average n to be 0.58 with a variance of 0.006 for the whole aquifer. Laboratory batch experiments of Vereecken et al. (2000) with materials of Layers 2 and 3 together resulted in n that varied between 0.71 and 0.95. These values are higher than that obtained in this study (0.614) and indicate moderate to weak nonlinearity. Nevertheless, our result is in good agreement with the results obtained from the field measured BTCs by Vereecken et al. (1999). The observed difference in n between the field and laboratory estimates might be due to the different sorption reactions in the field and laboratory. In the batch experiments, measurements were done when sorption reactions reached equilibrium condition, while the transport of uranin occurred in the field under non-equilibrium condition. Ball (1989) and Quinodoz and Valocchi (1993) also found significantly lower kinetic rate parameters from field tests than in laboratory measurements for Borden aquifer. The constant k (Eqn 21) depends on the properties of aquifer materials (mainly bulk density and porosity), velocity of flow, type of solute, and mass of injected solute (Jaekel et al. 1996, their eqn 23). For linear sorption process, [gamma] becomes unity and k represents the constant velocity of the solute plume.


Figure 4 compares the performance of equilibrium based model (Eqn 8) in estimating local velocity, [v.sub.s], and travel time, [tau], of the uranin plume by least square fitting of Eqn 20 to BTCs with that of time moment method. Since the method of time moment estimates the same travel time for all solute-transport models (Table 1), the relationship between the travel path and travel time of the plume is regarded as a reliable one. In comparison to the method of moment, Eqn 8 underestimates travel time of uranin plume over small distance (<24m from injection plane) and overestimates it over large distance (Fig. 4). The reason for the early time underestimation of travel time is not intuitive. The overestimation over large distance might be due to the fact that the transport of uranin occurred under non-equilibrium condition and the local equilibrium-based model (Eqn 8) predicted response BTCs, which occurred too late (Valocchi 1985).


The derivative of Eqn 21 is the time-dependent local velocity, [v.sub.s], of the uranin plume also called retarded velocity that is expressed by:

[v.sub.s](= [differential]/x/[differential]/[tau]) = k[gamma][[tau].sup.[gamma]-1] (22)

Equation 22 shows that [v.sub.s] decreases asymptotically with increasing travel time, [tau], as was also reported by Vereecken et al. (2002) for reactive solute plumes. Figure 5 depicts a fundamental difference between the velocity of the uranin plume calculated by power law relation (Eqn 22) and that determined by the ratio of travel path to the first time moment, [tau] (Eqn 16). Overlooking this difference misinterprets the local velocity, [v.sub.s], of the plume with a consequent erroneous prediction of uranin-transport parameters from BTCs. The velocity of the uranin plume calculated by using [tau] provides an average velocity, [v.sub.a], over travel path, and is considerably higher than [v.sub.s] (Eqn 22). This is due to fact that the velocity of reactive solutes decreases with increasing time and the method of moment considers the past history of solute movement (mainly the kinetically controlled sorption reactions).


Figure 6 illustrates the variation of local velocity, [v.sub.s], and average velocity, [v.sub.a], of the uranin plume along with the average velocity of the bromide plume, [v.sub.b], over the travel path of both plumes. The average velocity of the uranin plume is always higher than its local velocity, which decreases asymptotically over travel path and travel time of the plume (Fig. 5). After 370 days of injection (Fig. 5) or equivalently after a movement of 64 m of the plume (Fig. 6), the local velocity decreases to about 0.1 m/day from an initial asymptotic large value. The decrease in [v.sub.s] over travel path relative to the constant velocity of the bromide plume reveals the existence of nonlinear sorption reactions of uranin with the aquifer materials. Ball (1989) and Wood et al. (1990) also reported a similar reason for decreasing velocity of reactive solute plumes over travel path. Initially, the transport process of uranin was similar to that of bromide (Fig. 6) since most ions of uranin, like bromide ions, were in the aqueous phase. But with increasing travel time, kinetic sorption reactions of uranin with the aquifer materials might result in characteristic nonlinearities in the velocity of the uranin plume. An asymptotic high velocity ofuranin, higher than the constant velocity of the bromide plume, is noticed in Fig. 6 at early time within the first 3.5 m from the plane of injection. This is regarded as a classical behaviour of power law relation at early asymptotic limit. Nonetheless, a large quantity of uranin (4500 L) and bromide (8000 L) solutions, injected in relatively short period of time (5 h for uranin and 10.8 h for bromide), might cause transient flow with large hydraulic gradients as well as high velocity in the vicinity of the plane of injection.


Variation of retardation factor with travel path of uranin plume

The retarding effect of sorption of solutes with porous media is characterised by a coefficient called retardation factor. This factor, widely used in chemical engineering, is defined in several ways, which have quite different meanings for the transport of reactive solutes. In one definition, retardation factor, [R.sub.1], is expressed by the ratio of average velocity of a non-reactive solute plume, v, to local velocity of a reactive solute plume, [v.sub.s], as:

[R.sub.1] = v/[v.sub.s] (23)

This, after substitution of [v.sub.s] from Eqn 22, becomes:

[R.sub.1] = v/k[gamma] [[tau].sup.1-[gamma]] (24)

In another definition, retardation factor, [R.sub.2], is expressed by the ratio of average velocity of a non-reactive solute plume to that of a reactive solute plume, [v.sub.a], for a given travel path of both plumes. [R.sub.2] is also defined by the ratio of travel time of a reactive solute plume, [tau], to that of a non-reactive solute plume, [tau]', for a given travel path of the plumes. Thus:

[R.sub.2] = v/[v.sub.a] = [tau]/[tau]' (25)

[R.sub.1] (Eqn 24) differs from [R.sub.2] (Eqn 25) exactly in the way the local velocity of a reactive solute plume differs from its average velocity. Hereafter, [R.sub.1] and [R.sub.2] is referred to as the local and average retardation factor, respectively. The second definition of retardation factor is consistent with that used in different solute-transport models, and is obtained from Eqn 20 and Table 1 (R = [R.sub.2]). The retardation factor of the uranin plume, when calculated from its first time moment, is equivalent to [R.sub.2]. Figure 7 illustrates the difference between [R.sub.1] and [R.sub.2], and their variation over travel path of the plume. [R.sub.1] is always larger than [R.sub.2] by a factor, f which is a function of both travel path, x, and travel time, [tau], of the uranin plume and increases with increasing travel path of the plume as:

f = 1/k[gamma] x/[[tau].sup.[gamma]] (26)


Equation 24 demonstrates an asymptotic increase of local retardation factor with increasing travel time. Although nonlinear sorption, intraparticle diffusion and field-scale mixing processes caused by aquifer heterogeneities may cause the time dependence of retardation factor (Burr et al. 1994), nonlinear sorption is assumed the most dominant cause of asymptotic behaviour of velocity and retardation factor of the uranin plume in the Krauthausen aquifer. Closed-form analytical expressions for the velocity and retardation factor of nonlinearly sorbing reactive solutes (Vereecken et al. 2002) also show power law behaviour with time. Retardation factor of such a solute depends on its concentration, which, in turn, depends on travel path and travel time of the solute plume. The local velocity of a reactive solute plume, [v.sub.s], is therefore a function of both time and distance. The travel path of a linearly sorbing solute plume becomes a linear function of its travel time ([gamma] = 1 in Eqn 21), giving rise to a constant velocity of the plume (= k). Although cumulative retardation effects of the aquifer materials are invoked in [R.sub.2], [R.sub.1] does not encounter such effects. The concentration of uranin in the mobile phase decreases with increasing travel path due to continuous dispersion and spreading of the plume. Consequently, the desorption rate of uranin decreases asymptotically according to the classical behaviour of Freundlich isotherm (Eqn 15). The slow desorption rate causes a consequent increase in the retardation factor, [R.sub.1], and might be responsible, as a major cause, for the time-dependent behaviour of the retardation factor.


The average velocity of a bromide plume, 0.64m/day, provides the velocity of groundwater flow in the Krauthausen aquifer, Germany. The constant velocity of flow implies a spatially and temporally uniform velocity field where groundwater flows at steady-state condition.

A local equilibrium-based ID solute-transport model underestimates travel time of an uranin plume in comparison to the method of time moment. The rate parameter, F, of the chemical non-equilibrium model and equilibrium distribution coefficient, [k.sub.d], of uranin indicate chemical non-equilibrium transport process. The travel time of the uranin plume increases asymptotically with increasing travel path of the plume following power law. The agreement of the exponent of power law with the exponent of Freundlich isotherm indicates that uranin was adsorbed nonlinearly with the aquifer materials. The derivative of the relationship between displacement of the uranin plume and its travel time provides local velocity/retarded velocity of the plume that decreases asymptotically with increasing travel time. The local retardation factor derived from the local velocity is considerably larger than the average retardation factor derived from travel time. The travel time of the uranin plume takes into account rate-limited sorption reactions and consequently, the retardation factor determined from travel time is larger than that determined from the local velocity of the plume.


During this work, the principal author held a postdoctoral research fellowship offered by the Alexander von Humboldt Foundation, Germany, and was on deputation from the Bangladesh Agricultural University (BAU), Mymensingh. The authors gratefully acknowledge the assistances of both the Alexander von Humboldt Foundation and BAU authority to carry out this research.

Manuscript received 15 July 2004, accepted 12 May 2005


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M. A. Mojid (A,C) and H. Vereecken (B)

(A) Department of Irrigation and Water Management, Bangladesh Agricultural University, Mymensingh--2202, Bangladesh.

(B) Institute of Chemistry and Dynamics of the Geosphere, ICG--IV: Agrosphere, Forschungszentrum Juelich GmbH, D-52425 Juelich, Germany.

(C) Corresponding author. Email:
Table 1. Time moment formulas for equilibrium and non-equilibrium
solute-transport models for Dirac input of solute

Moment Local Physical non-equilibrium model
 equilibrium Diffusion controlled

[M.sub.1] XR XR

[M.sub.2] 2X[R.sup.2]/P 2X[R.sup.2]/P + [2/15]
 [X[(1 - [beta]).sup.2]

 Physical non-equilibrium Linear chemical
Moment Local model non-equilibrium
 equilibrium First-order model

[M.sub.1] XR XR XR

[M.sub.2] 2X[R.sup.2]/P 2X[R.sup.2]/P + 2X[R.sup.2]/P +
 2X[(1 - [beta]).sup.2] 2X(R - 1)/F

Table 2. Means, standard deviations, and coefficients of
variation of the rate parameters for uranin estimated by
different solute-transport models

Rate parameters [beta] [[theta]] F [k.sub.f]

Mean 0.89 0.03 195 1.16
s.d. 0.23 0.03 222 2.37
CV (%) 26 87 114 205

Rate parameters [k.sub.r] [k.sub.d] [alpha] [gamma]' [omega]

Mean 3.04 0.32 0.03 3.02 6.98
s.d. 3.14 0.19 0.01 0.76 1.35
CV (%) 103 60 41 25 19
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Author:Mojid, M.A.; Vereecken, H.
Publication:Australian Journal of Soil Research
Geographic Code:8AUST
Date:Nov 1, 2005
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