Environ. Sci. Technol. 2004, 38, 2089-2096
Measurement of Mixing-Controlled Reactive Transport in Homogeneous Porous Media and Its Prediction from Conservative Tracer Test Data SURABHIN C. JOSE AND OLAF A. CIRPKA* Institut fu ¨ r Wasserbau, Universita¨t Stuttgart, Pfaffenwaldring 61, 70550 Stuttgart, Germany
Numerical and theoretical studies have indicated that pore-scale mixing can be the limiting process for reactions among dissolved compounds in porous media. It has been claimed that multicomponent reactions in porous media could be accurately estimated using mixing coefficients obtained from point-like measurements of conservative tracer concentrations. In this study, we verify these concepts experimentally by tracer tests in a homogeneously packed saturated sand column. Fiber-optic fluorometry was applied to detect point-related concentrations of fluorescein, which was used as both the conservative and the reactive tracer. In the reactive tracer experiment, an acidic solution containing the tracer was displaced by an alkaline solution without tracer. Since the fluorescence of fluorescein is quenched at low pH, the fluorescence intensity measured in the reactive breakthrough curve indicated the mixing of the two solutions. The measured reactive breakthrough curves were compared to predictions based on the conservative breakthrough curves. Predictions and measurements agreed well. Our results imply that incomplete mixing on the pore scale is of minor significance for field-scale applications. On this scale, however, even weak sorption might influence mixing significantly.
Introduction Since the beginning of the past decade, reactive mixing of dissolved compounds in porous media has gained much attention (1-6). A better understanding of mixing in reactive transport is necessary to predict effectively the fate and behavior of contaminants in groundwater (e.g., in the context of engineered and intrinsic bioremediation). Two compounds initially occupying adjacent but separate regions in an aquifer can react with each other only when they are mixed. In the absence of sorption, pore-scale dispersion is the major mechanism leading to the overlapping of the compounds at the contact interface. We denote this overlapping of adjacent plumes as mixing. On the field scale, the solute plume also undergoes deformation of its original shape. We refer to this mechanism as spreading. It is caused by spatial variability of advection due to aquifer heterogeneities. Without pore-scale dispersion, two nonsorbing compounds that initially did not occupy the same space will remain in their own specific packages of groundwater and cannot interact with each other. Pore-scale * Corresponding author phone: +49(711)685-8218; fax: +49(711)685-7020; e-mail:
[email protected]. 10.1021/es034586b CCC: $27.50 Published on Web 02/24/2004
2004 American Chemical Society
dispersion is required to make the two plumes overlap. In essence, diffusion is the actual process that leads to mixing. Spreading only enhances the mixing by increasing the contact area. Commonly, field-scale dispersion is quantified by socalled macrodispersion coefficients, defined as half the rate of change of the second central spatial moments of very large conservative plumes (7). Until the early 1990s, macrodispersion coefficients were also used in field-scale simulations of mixing-controlled reactive transport. The macrodispersion coefficients, however, quantify the cumulative effects of both spreading and mixing. Therefore, reactive transport simulations applying these coefficients, while keeping the reaction terms unchanged, overestimated the mixing of reactants and the overall reaction rates (1-3, 6, 8). Some numerical and theoretical studies have indicated that effective dispersion coefficients, calculated from the width of point-related breakthrough curves, may accurately quantify dispersive mixing of reacting compounds (6, 9-11). These concepts have so far rarely been tested by experiments. Raje and Kapoor (4) performed an experiment in which solutions of dissolved aniline and 1,2-naphthoquinone-4sulfonic acid replaced each other in a 18 cm long column. The two reactants and the reaction product, 1,2-naphthoquinone-4-aminobenzene, were measured in the outflow of the column in a flow-through spectrophotometer cuvette. To fit a model to the measured breakthrough curves of conservative tracers and of the reaction product, the authors needed to account for pore-scale segregation of the reactants. Since concentrations were measured only in the total outflow of the domain, the authors could not test whether pointrelated concentrations of the conservative tracers had been more suitable to predict the reactive behavior. In the experiment of Gramling et al. (5), such a comparison was possible. They used sodium EDTA and copper sulfate as reactants in a transparent, quasi-two-dimensional device with dimensions of 36 cm × 5.5 cm. The reaction product, CuEDTA, was detected throughout the domain by light transmission imaging. The authors found that pore-scale dispersion coefficients, derived from conservative tracer studies and applied to an advection-dispersion reaction model, overpredicted the formation of the reaction product. The experiment, however, was restricted to a travel distance of 36 cm. Toward the end of the domain, the increase of the reaction product mass was predicted by the model quite well. As we will discuss in the following, we believe that the lack of pore-scale mixing that has been found by both Raje and Kapoor (4) and Gramling et al. (5) is of minor significance in larger scale applications. The objective of the present study is to test the concepts mentioned in refs 6, 9, and 10 by conservative and reactive tracer experiments in a homogeneously packed sand column. From breakthrough curves of a conservative tracer, measured at single points within the porous medium, we derive mixingrelated dispersion coefficients. These are used to predict the breakthrough curves at the same points for a compound that would be formed by a mixing-controlled multicomponent reaction. We compare the predicted breakthrough curves to measurements in reactive tracer tests. As primary criterion for the goodness of the prediction, we consider the area underneath the reactive breakthrough curve, which is proportional to the total mass of the reaction product passing the measurement device. In principle, we consider an instantaneous reaction of the following type: VOL. 38, NO. 7, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
2089
+ ) (1) A B C nonfluorescent nonfluorescent fluorescent in which reactants A and B are nonfluorescent educts and component C is a fluorescent reaction product. Hence, we are able to monitor the mixing-controlled reaction by in-situ fluorometry.
Theory The difference between actual mixing and common dispersion has been discussed in subsurface hydrology primarily for field-scale transport (3, 6, 9-11). We may briefly review some results of stochastic theory, derived for heterogeneous aquifers, and subsequently transfer these results to porescale processes. An upper limit for the longitudinal macrodispersion coefficient (Dl,∞) in an isotropic heterogeneous aquifer is given by Dagan’s (12) well-known result:
Dl,∞ ) vj λσ2Y
(2)
in which vj is the mean seepage velocity, λ is the integral length scale of the aquifer heterogeneities, and σ2Y is the variance of the log conductivity. The macrodispersion coefficient Dl/(t) increases with time until it reaches its asymptotic value Dl,∞. The time dependence can be described approximately by an exponential function with a characteristic time of about 2τa ) 2λ/vj in which τa is the advective time scale related to the length scale of the heterogeneities:
D/l (t) ≈ Dl,∞(1 - exp(t/(2τa)))
(3)
While macrodispersion describes the combined effects of spreading and mixing, the effects of mixing alone can be parametrized by the so-called effective dispersion coefficients, Del (t), describing the expected spreading rate of a point source rather than a very large plume (6, 11). Since reactions occur at single points, it makes intuitively sense that point-related dispersion coefficients are better suited to parametrize reactive mixing than coefficients describing the behavior of extended plumes (6). At the large-time limit, both types of dispersion coefficients become identical. At early times, however, effective dispersion is considerably smaller than macrodispersion. The catching-up of mixing with spreading is controlled by transverse pore-scale dispersion. Dentz et al. (11) derived the following approximation for intermediate times:
t Del (t) ) D/l (t) (π/8)τDt + t
(4)
in which τDt ) λ2/Dt is the time scale for transverse porescale dispersion, and Dt is the transverse pore-scale dispersion coefficient. Typically, τDt . τa, that is, macrodispersion reaches much faster its asymptotic value than effective dispersion. On the pore scale, we can observe similar effects. Velocity fluctuations within and between single pores lead to spreading of solute fronts. Front irregularities are smoothed out by transverse diffusion leading to actual longitudinal mixing. Using Darcy’s law for flow already implies averaging over pore-scale fluctuations so that in the related transport considerations a distinction between spreading and mixing in single pores is not possible. Thus, the typical pore-scale dispersion coefficients applied to transport in homogeneous porous media reflect “macrodispersion”, as defined above, applied to pore-scale heterogeneities. True mixing within the pores would be described by corresponding effective dispersion coefficients. The experimental studies of Raje and Kapoor (4) and Gramling et al. (5) indicated lack of pore2090
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 7, 2004
TABLE 1. Characteristic Times for the Pore and Column Scales in This Study as Compared to Other Published Works ref 4
our expt mean grain diameter, d50 (m) length of experimental setup, L (m) velocity,a v (m/s) molecular diffusion coeff,b D (m2/s) hydrodynamic dispersion coeff, D* (m2/s) grain-scale advective time, τa,g ) d50/v (s) grain-scale diffusive time, τd,g ) d502/D (s) column-scale advective time, τa,c ) L/v (s) column-scale dispersive time, τa,c ) L2/D* (s) grain Peclet no., τd,g/τa,g mixed-scale Peclet no., τd,g/τa,c column-scale Peclet no., τd,c/τa,c
ref 5
1.5 × 10-3
1.5 × 10-3 1.3 × 10-3
2
0.18
0.36
1.75 × 10-4 7 × 10-4 10-9 10-9
1.21 × 10-4 10-9
5 × 10-7
2.3 × 10-6 1.75 × 10-7
8.57
2.15
10.74
2250
2250
1690
11429
257.14
2975
8×
1.4 ×
7.4 × 105
106
104
262 0.20
1050 8.75
157 0.57
700
55
25
a Only velocities that are comparable to our experiments are considered. b Assumed.
scale mixing in a column of 18 cm and a two-dimensional setup of 36 cm length, respectively. Decisive for the question of whether the typical porescale dispersion coefficients are applicable to mixing-controlled reactive transport is the time scale of pore-scale transverse diffusion in comparison to the time scale of advective transport from the injection point to the point of observation. We may denote this quantity as the mixed-scale Peclet number. Table 1 lists the computed time scales for the experiments of Raje and Kapoor (4) and Gramling et al. (5). In the following brief estimation, we may apply a set of parameters in the range of these studies. The mean grain diameter (d50) is the characteristic length λ of the pore-scale heterogeneities. Here, we use a value of 1.5 × 10-3 m. As mean seepage velocity (vj ) we take a value of 1.5 × 10-4 m/s, which is typical for laboratory experiments, although it is rather high for field-scale applications. On the pore scale, the transverse exchange is by molecular diffusion only, with typical coefficients of ≈10-9m2/s. Finally, as characteristic column-scale length (L), we use a value of 30 cm. According to eq 3, the typical pore-scale dispersion coefficients are approached exponentially with a characteristic time of 2τa ) 2d50/vj ≈ 20 s. Actual mixing lags behind. According to eq 4, the characteristic time for pore-scale mixing to catch up with pore-scale dispersion is (π/8)d502/D ≈ 880 s. For comparison, the column-scale characteristic time of advection is L/vj ) 2000 s. That is, when the tracer front reaches the end of the domain, the typical pore-scale dispersion coefficients have already become applicable to describe mixing. In mixing-controlled reactive transport, the effective dispersion coefficient determines the rate of change of the reaction product. The actual total mass of the product is the rate of change integrated over the travel time and thus reflects mixing over the entire travel distance. Even if we have already reached the regime in which “macrodispersion” coefficients are applicable to estimate the rate of change of a reaction product, the total mass will be affected by the lack of mixing at earlier times. A lack in the product mass will be observed at all times. However, the lack will no more grow whereas the mass itself does. In fact, Gramling et al. (5) show a time curve of the measured total product mass as compared to a prediction based on a constant pore-scale dispersion coefficient (their Figure 6). The prediction overestimated the
FIGURE 2. General setup of conservative and reactive tracer experiments. The crosses mark point-like probes for in-situ fluorometry.
FIGURE 1. (a) Position of probes along the column length. (b) Radial arrangement of probes at each measurement level. total mass at all times, but the difference did no more increase toward the end of the experiment where about twice the transverse diffusive time scale (τDt) had passed. In most field-scale applications, we are interested in transport over at least several meters. Also, typical groundwater velocities are smaller than 1 m/d. We conclude that on these scales the lack of pore-scale mixing is of minor significance. Nonetheless, there will be a discrepancy between macrodispersion and actual mixing because of aquifer heterogeneities on larger scales.
Experimental Section We performed conservative and reactive tracer experiments in a 2 m long PVC column with 10 cm i.d. The column was packed relatively homogeneously with quartz sand (grain size ) 1-2.5 mm; hydraulic conductivity ) 1.69 × 10-2 m/s). We computed a porosity of 0.4 from the specific discharge and the mean arrival time of the tracer. Thus, the pore volume of the column was approximately 6.3 L. We used fiber-optic fluorometry to measure concentrations within the porous medium (13). The excitation light, originating from blue LEDs and filtered by an optical filter, is coupled into an optic fiber ending directly in the porous medium. The excitation light penetrates a few millimeters into the porous medium. A dissolved fluorophore within that volume may be excited and emits fluorescence light of a longer wavelength. The emission light is partially sampled by a second optic fiber parallel to that for the excitation light, filtered, and detected by a photomultiplier. The fiber-optic probes were produced from standard Duplex polymer fibers
(Mitsubishi Rayon Co. Ltd. Grade GH 4002-P, Duplex POF1000/2,2*4,4PE-LWL). The measurement tips were stripped of the insulation and re-sealed by heat-shrinkable tubes, leading to a reduced total diameter of 2.5 mm. The sampling volume of the probe may be approximated by a sphere with 3 mm diameter leading to a volume of ≈14.1 µL. A comparison with the grain diameters ranging between 1 and 2.5 mm shows that the tracer concentrations detected by the fiber-optic probes are not concentrations in single pores but averages over a few pores each. The fluorescence intensity of the tracer was measured at seven measurement levels along the column length, which were equally spaced at a distance of 0.25 m each (Figure 1). The first and the last measurement levels were at 0.25 and 1.75 m from the inlet of the column. At each measurement level, there were eight fiber-optic probes arranged at equal distances along the radius of the column (Figure 1). In addition, a probe was each placed at the inlet and at the outlet of the column, constituting a total of 58 probes, out of which 56 were within the porous medium. Three 19channel fluorometers (Hermes Messtechnik, Germany) were used in the experiment. Two types of experimentssa conservative tracer test and a reactive tracer testswere carried out (Figure 2). The specific discharge in both tests was 7 × 10-5 m/s. We used fluorescein as tracer for both the conservative and the reactive tracer tests. Fluorescein has been reported a good conservative tracer for silica sand (14). We could use fluorescein also as the reactive tracer because its fluorescence depends on pH. The fluorescence intensity of fluorescein is essentially zero under acidic conditions (pH < 4) and maximal under alkaline conditions (pH > 8) (Figure 3). Thus, in our experiments, “reactant A” was an acidic solution (pH 4) containing fluorescein, and “reactant B” an alkaline solution (pH 10) without fluorescein. Where the two solutions mixed, the pH was about neutral and some, now fluorescent, fluorescein was present. This system has been used previously to study mixing in turbulent flows (15). In comparison to the above-mentioned turbulent-mixing studies, the solutions used in the present study had to be modified because fluorescein sorbs to quartz at low pH (14, VOL. 38, NO. 7, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
2091
FIGURE 3. Relative fluorescence intensity of fluorescein as function of pH as measured by fiber-optic fluorometry. 16). To keep fluorescein in solution, we added a nonionic surfactant above the critical micelle concentration (cmc) to the deionized water used in the experiment. The surfactant was Lutensol GD 70 (BASF, Germany), an alkyl polyglucoside. The mean molecular weight of the surfactant is 600 g/mol, the cmc is 0.5 g/L, and the density of the pure surfactant is 1170 kg/m3. In our study, density effects were not relevant as the injected solution and the displaced solution had the same surfactant concentration of 2% (vol). To prevent biomass decay of the surfactant, sodium azide was added to all alkaline solutions at a concentration of 0.1 mmol/L. Conservative Tracer Test. In the conservative tracer test, an alkaline surfactant solution (pH 10) was displaced by another alkaline surfactant solution containing fluorescein (pH 10, tracer concentration ) 100 µg/L). The alkaline solutions were prepared by adding a concentrated NaOH solution (2 mol/L) to the surfactant solution. The tracer solution was introduced continuously into the column. Breakthrough curves (BTCs) of the fluorescence intensity were measured continuously at all measurement points and stored on a PC. The fluorescence intensity BTCs obtained from the conservative test were normalized with respect to the maximum and the minimum intensities of each BTC:
X(t) )
I(t) - Imin Imax - Imin
(5)
in which I(t) is the measured intensity at a specific point and time. Imin and Imax are the minimum and maximum intensities measured at that point. Since the fluorescence intensity obtained in the conservative test is a linear function of the tracer concentration alone (17), the normalized fluorescence intensity is identical to the normalized tracer concentration. The normalized tracer concentration X(t) may also be interpreted as mixing ratio, that is, the volumetric ratio of the injected to the total solution within the measurement volume of a given measurement device. Reactive Tracer Test. In the reactive test, an alkaline surfactant solution (pH 10) without tracer was injected into the column, displacing an acidic surfactant solution containing fluorescein (pH 4, tracer concentration ) 100 µg/L) that originally occupied the porous medium. The acidic solution was prepared by adding concentrated nitric acid (65%) to the surfactant solution. The measured fluorescence intensity BTCs resulted from the tracer at about neutral pH, indicating the mixing of the two solutions within the porous medium. The fluorescence intensity BTCs obtained in the reactive tracer tests were a function of both the tracer concentration and the pH of the solution. The reactive BTC at each measurement location was normalized according to the 2092
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 7, 2004
FIGURE 4. pH value as function of the fraction Xbase of the alkaline surfactant solution in the mixture with the acidic surfactant solution. maximum and the minimum fluorescence intensities of the corresponding conservative BTC. Additional Batch Tests. To compare predicted to measured reactive BTCs, we had to determine the relative fluorescence intensity of fluorescein for a given mixing ratio of the two solutions. Assuming that the impact of the soil matrix on pH and fluorescein behavior was negligible, we performed two additional tests in batch studies without sand: (i) a titration test of the two surfactant solutions, yielding the pH as function of the mixing ratio X, and (ii) a test on the relative fluorescence intensity (Irel) of fluorescein as function of the pH. In the titration test, 100 mL of the pH 4 solution (surfactant concentration ) 2% (vol); fluorescein concentration ) 100 µg/L) were transferred into a brown bottle and stirred continuously by a magnetic stirrer. An alkaline solution at pH 10 (surfactant concentration ) 2% (vol); sodium azide concentration ) 0.1 mmol/L) was added in varying increments to the acidic solution. The pH was measured by an electrode, and its value was noted after 10 s, the maximum time required for the pH to stabilize in the sampled solution. For the test on relative fluorescence intensity, an acidic (pH 3) and an alkaline (pH 11) surfactant solution both containing fluorescein were prepared (surfactant concentration ) 2% (vol); tracer concentration ) 100 µg/L). A total of 50 mL of the acidic solution was transferred into a brown bottle, where the pH was measured by an electrode and the fluorescence intensity was measured by a fiber-optic probe identical to those used in the column. The solution was stirred continuously by a magnetic stirrer. The alkaline surfactant solution was added in increments, and the resultant pH and fluorescence intensity were measured. The background fluorescence intensity occurring in the setup was measured using surfactant solutions without tracer. The background fluorescence was found independent of pH; it was subtracted from all measurements with tracer. The measured fluorescence intensity was normalized from zero to one, yielding a relative fluorescence intensity (Irel(pH)).
Results and Discussion Constitutive Relationships. Figure 4 shows the pH measured in the titration test of the two surfactant solutions. The titration curve indicates buffering of the surfactant solution at slightly alkaline pH values. We fitted a standard buffer equation with two buffering compounds to the titration curve, using the Levenberg-Marquardt algorithm. For estimated equilibrium constants Ka1 and Ka2 of the buffer reactions and estimated total buffer concentrations [Buf1,tot] and [Buf2,tot], we could calculate the proton concentration from the cumulative charge of all counterions, that is, all ions except for the buffer, hydronium, and hydroxonium ions, by iterative
solution of
(x
[H+] )
(
Ka1[H+][Buf1,tot]
charge2 + 4 10-14 +
1 2
+
[H+] + Ka1
)
Ka2[H+][Buf2,tot] [H+] + Ka2
- charge
)
(6)
in which the charge of the counterions for a certain mixing ratio is given by linear interpolation of the charges in the acidic and alkaline end point solutions. The fitted concentration of the first buffer was 5.32 × 10-4 mol/L ( 6 × 10-6 mol/L with a fitted pKa of 8.541 ( 0.006. The fitted concentration of the second buffer was 5.27 × 10-5 mol/L ( 1.5 × 10-6 mol/L with a pKa of 6.416 ( 0.047. The simulated titration curve is included in Figure 4. Although the second pKa agrees with pKa values of fluorescein reported in the literature, we do not believe that fluorescein is the major buffering compound in this pH range because the fitted buffer concentration is about 2 orders of magnitude higher than the fluorescein concentration added to the acidic solution. Since the exact chemical structure of the surfactant mixture is not published by the supplier, we cannot relate the fitted pKa values and buffer concentrations to specific functional groups of the surfactant molecules. Figure 3 shows the measured relative fluorescence intensity of fluorescein as detected by our fiber-optic fluorometer setup. Minimizing the root mean square error (RMSE), the following empirical equation was fitted to the measured data:
(
)
1.135 ,1 pH9.269 + 3.179 × 107
Irel(pH) ) max pH9.269 ×
(7)
The empirical fit is included in Figure 3. We tested whether the relative fluorescence intensity as function of pH could be explained by the relative concentration of the deprotonated fluorescein according to the pKa of the compound. The simulated curve deviated from the measured one in the pH range of 4-6.5. The deviation may be caused by an unaccounted pH dependence of another component in the measurement chain. In the following analysis, we used therefore the empirical fit given in eq 7. Apparent Dispersion Coefficients. The normalized conservative breakthrough curves were analyzed by the method of temporal moments. The kth noncentral temporal moment (mk) of a tracer introduced as Dirac pulse is defined by
mk )
∫
∞ k
0
t cδ(t) dt
(8)
in which cδ(t) is the normalized concentration resulting from a pulse injection. The corresponding second central moment is
m2c )
∫
∞
0
(t - m1)2cδ(t) dt ) m2 - m12
(9)
In the experiments, however, we introduced the conservative tracer continuously rather than by a pulse. We analyzed the corresponding normalized concentrations, denoted by cH(t), based on their truncated temporal moments. The kth truncated moment is
Tk(cH,te) )
∫
te k
0
t cH(t) dt
(10)
in which te is the truncation time that needs to be chosen such that the maximum concentration has already been reached. For the normalized concentrations, we calculated the pulse-related temporal moments from the truncated moments of the BTCs originating from continuous injection by
mk ) tke - kTk-1(X,te)
(11)
The resulting first temporal moment (m1) calculated from each BTC is the mean arrival time of the tracer at the measurement point. We can use it to determine the apparent seepage velocity (vapp):
vapp )
x m1(x)
(12)
The second noncentral temporal moments (m2) were calculated for each BTC. At a given measurement level, we computed two apparent dispersion coefficients, a macrodispersive one (D/app(x)) and an effective one (Deapp(x)): /
Dapp (x) )
Deapp(x) )
2 x2 〈m2〉 - 〈m1〉 2 〈m 〉3 1
〈
(13)
〉
2 x2 m2 - m1 2 m13
(14)
in which 〈 〉 denotes averaging over all probes in the same measurement level x, to which we refer as cross-sectional average in the following. In the given context, the apparent macrodispersion coefficient describes the solute spreading and mixing by analyzing the moments of the concentration BTCs averaged over all probes at that level. The effective dispersion coefficient, by contrast, is evaluated for each probe individually and may be averaged over the cross-section subsequently. That is, the effective dispersion coefficient is related to the spread observed in BTCs of single probes, whereas the macrodispersion coefficient includes the variability in the mean breakthrough time among the probes within a measurement level. The coefficients are denoted as apparent ones because they do not describe the rate of change of moments. Instead, they interpret the moments of the BTCs as if caused by one-dimensional transport with constant coefficients (9). The moment-derived apparent transport parameters were sensitive to small fluctuations in the measured concentrations. As an alternative, we fitted the analytical solution of the advection-dispersion equation with uniform coefficients to the locally obtained BTCs by adjusting the parameters v and D (18):
(
)
x - vt 1 Xsim(t) ) erfc 2 2xDt
(15)
Strictly speaking, eq 15 is the analytical solution for the residence concentration in a one-dimensional infinite domain with Heaviside initial condition. The correct analytical expression of the residence concentration for stepwise injection into the flux of a semi-infinite domain contains additional correction terms that become very small for column-scale Peclet numbers: Pe ) xv/D > 10 (18). The dispersion coefficient calculated from a point-related tracer concentration is the apparent effective dispersion coefficient (Deapp) at that point, whereas the dispersion coefficients calculated from the concentration averaged over all probes in the measurement level is the apparent macrodispersion coefficient (D/app(x)) for that level. VOL. 38, NO. 7, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
2093
FIGURE 5. Conservative tracer breakthrough curves of two probes at travel distances of 0.5 and 1.75 m. Comparison between measured data and fit based on temporal moments according to eq 15. The measured data are plotted as markers.
FIGURE 6. Apparent velocity as function of travel distance calculated from the first temporal moments of the conservative tracer breakthrough curves. Markers: apparent velocities at individual probes; solid line: average over all probes in the same measurement level. Figure 5 shows a comparison of measured and simulated breakthrough curves for two measurement points at x ) 0.5 and 1.75 m. As expected for homogeneous porous media, the parametric model of eq 15 agreed very well with the measurements. Figure 6 shows the apparent velocities determined from the individual BTCs and those determined from the BTCs averaged over all probes at a given level. The small variations in the local velocities confirm that the column was packed relatively homogeneously; a few outliers indicate remaining heterogeneity. Figure 7 shows the apparent effective and macrodispersion coefficients determined from the conservative tracer BTCs within the column as a function of travel distance. The apparent effective dispersion coefficients (Deapp) calculated in the column were mostly about 5 × 10-7 m2/s with a few outliers at x ) 0.75 and 1.25 m and consistently higher values at the last measurement level, x ) 1.75 m. The value Deapp ) 5 × 10-7 m2/s corresponds to a characteristic effective dispersivity (Reff ) Deapp/vapp) of 2.5 mm, which is on the order of the maximum grain size. At most measurement levels, 〈Deapp(x)〉 and D/app(x) showed little difference, with the exception of the values at the travel distances of 1.25 and 1.75 m. The similarity in effective dispersion and macrodispersion is expected for homogeneous packing. The differences at x ) 1.25 and 1.75 m may be due to heterogeneities introduced during the packing of the column. Figure 8 shows the conservative BTCs obtained at all measurement locations in level 7 (x ) 1.75 m), where the maximum dispersion coefficients were found. The cross2094
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 7, 2004
FIGURE 7. Apparent effective dispersion coefficients Deapp and macrodispersion coefficients D/app(x) as function of travel distance calculated from the temporal moments of the conservative tracer breakthrough curves. The calculated effective dispersion coefficients at all measurement points are plotted as markers.
FIGURE 8. Locally measured and cross-sectionally averaged conservative BTCs at measurement level 7 (distance ) 1.75 m). sectionally averaged BTC is also plotted in the same figure. The average velocity observed in the level was 1.86 × 10-4 m/s ((7.09 × 10-6 m/s). The cross-sectionally averaged effective dispersion coefficient in level 7 is 1.71 × 10-6 m2/s ((4.55 × 10-7 m2/s) while the macrodispersion coefficient calculated from the cross-sectionally averaged conservative BTC is 1.95 × 10-6 m2/s.
Prediction of Reactive Mixing from Conservative Test In the reactive tracer test, the major processes assumed to lead to the measured fluorescence intensity were as follows: (i) the dispersive mixing of the two surfactant solutions at the displacement front, (ii) the acid-base reaction of the two surfactant solutions, and (iii) the pH dependence of specific fluorescence intensity of fluorescein. We predicted the total tracer concentration (fluorescent plus nonfluorescent fluorescein) in the reactive tracer test at each measurement point for all times from the conservative tracer test. Since fluorescein was displaced in the reactive tracer test, its normalized concentration was predicted by 1 - X(t) of the conservative tracer test. The pH of the mixed solution was estimated by the buffer model derived from the titration test (eq 6) and the relative fluorescence intensity by the empirical relationship Irel(pH) derived from the batch test with fluorescein at 100 µg/L at differing pH (eq 7). Combining these relations leads to the following prediction of the normalized fluorescence intensity:
Ipred(t) ) (1 - X(t)) × Irel(pH(X(t)))
(16)
Figure 9 shows the observed and predicted reactive BTCs for the same probes as used in Figure 5. Two types of
FIGURE 9. Comparison between predicted and observed reactive breakthrough curves at two measurement levels. The observed data are plotted as markers. predictions, applying eq 16, are included: (i) predictions based on the actual conservative BTCs at the measurement points and (ii) predictions based on the parametric model (eq 15) with parameters derived from the actual conservative BTCs. In general, the observed and predicted reactive BTCs agreed well with regard to shape and arrival time at regions near the inlet but showed differences at distances of more than 0.5 m from the inlet. As seen in Figure 3, the relative fluorescence intensity does no more increase with pH if the pH exceeds a value of 8. According to Figure 4 and the buffer fit using eq 6, the pH value of 8 relates to a volume fraction of the alkaline solution Xbase of 0.32. In the reactive tracer test, the normalized concentration of tracer at this point is 0.68. Hence, also the predicted normalized intensity is 0.68. It is reasonable to expect this value as the maximum intensity. For Xbase < 0.32, the pH and thus the relative fluorescence intensity are smaller; whereas for Xbase > 0.32, the tracer concentration is smaller. For a travel distance of 0.5 m, Figure 9 shows indeed a maximum intensity in the reactive test of about 0.68. For the travel distance of 1.75 m, the observed maximum is considerably smaller (namely, 0.55), meaning that the pH 8 was reached at Xbase of about 0.45 instead of 0.32. We also see a breakthrough later than predicted. This implies that the process of pH increase with incoming alkaline solution is retarded at large distances. If there was no sorption of the tracer under acidic conditions, the declining part of the observed reactive BTC would perfectly match with that of the predicted BTC since the tail of the peak depends not on pH. However, the observed tails are slightly shifted toward later times, indicating weak sorption of the tracer at low pH values. We may approximate the retardation factor of the increasing and declining fronts by the ratio of the observed to the predicted breakthrough time (t50) of half the peak concentration. For the BTC at 1.75 m shown in Figure 9, the resulting retardation factors were 1.09 and 1.05 for the injected and displaced front, respectively. The same analysis revealed no retardation for distances less than 0.5 m, neither for the injected nor the displaced front. Our model for the prediction of the reactive BTC is based on the assumption that the grain surface is totally inert with respect to the compounds considered in the multicomponent reaction. Even though adding the surfactant reduced sorption of fluorescein to an extent that it could not be determined in batch tests, there might have been residual sorption (e.g., by an exchange between free-floating micelles containing fluorescein and hemi-micelles attached to the grain surface). At the same time, acid-base reactions may have occurred on the quartz surface, so that both the tracer wash-out and the penetration of the alkalinity front were slightly retarded. As these mass-transfer processes include attachment and
FIGURE 10. Observed and predicted zeroth temporal moment of the reaction product as function of distance in the column. detachment of micelles, it is reasonable to assume slow kinetics. As a consequence, the early reactive BTC at x ) 0.5 m would hardly be affected by kinetic mass transfer, whereas the late BTC at x ) 1.75 m showed some impact. We tried to quantify the sorption of fluorescein in the acidic surfactant solution onto quartz grains by independent batch tests. In this test, we measured the concentration decrease in the surfactant solution after contact with the sand. The accuracy in the concentration measurement, however, was insufficient to quantify sorption leading to a retardation factor in the range of 1.1 in the column. Measuring the retardation of fluorescein in the surfactant solution by conservative tracer tests under acidic conditions was not possible since the fluorescence of the tracer is quenched under these conditions. As a quantitative comparison of the predicted and observed reactive BTCs, we computed the zeroth temporal moment (m0) of the normalized intensity curves, which is the area underneath the BTC. The zeroth moment (m0) is proportional to the total mass of the reaction product passing the probes. Figure 10 shows the observed and predicted zeroth moment, averaged over the cross-section, with increasing distance from the inlet. At small travel distances, the model underpredicted the product mass, whereas at larger distances, it led to a slight overprediction. The overprediction of product mass at small distances, despite fitting the peak time and height, is because the predicted curve was less spread than the observed one. Particularly, the observed tailing was stronger than the predicted one. Again, such behavior could be caused by slow sorption kinetics. At large distances, the effects due to retardation are more prominent, resulting in a lower peak concentration for the observed product. The better agreement in the product mass may thus be coincidental. We also compared the first and second central moments of the reaction product in the measurements and predictions. The mean arrival time was predicted almost exactly for x ) 0.25 m; the observed mean breakthrough time was about 3% later than the predicted one for x ) 0.50 m; and the relative difference increased to 4.5% until x ) 1.75 m. The observed temporal variance of the product BTC was, throughout the column, about 50% larger in the observations than in the predictions. Table 1 gives a comparison of the characteristic times for pore scales and column scales computed for the present experiments and for the experiments mentioned in refs 4 and 5. Most of the BTCs discussed in both papers are at observation time intervals less than the pore-scale diffusion time (τd). This is the major reason for the incomplete mixing observed in their experiments. Although the concentrations were measured in our experiments at time intervals considerably larger than the grain-scale diffusive time, a conclusive result on whether VOL. 38, NO. 7, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
2095
complete pore-scale mixing could be assumed in reactive transport predictions could not be drawn due to kinetic sorption observed in the system. It becomes clear, however, that for larger travel distances even weak sorption is important for mixing. This experimental finding of our study, although unintended, is in full agreement with theoretical studies indicating that sorption differences, no matter how weak, become dominating in the large time mixing of solutes (19). While we consider 20% accuracy in the mass balance an excellent value for the prediction of reactive mixing, we hope yet to improve the accuracy of prediction, particularly with respect to the BTC shape, by completely avoiding tracer sorption and incorporating the pH behavior in the presence of sand into our predictive model. While we admit that quasitwo-dimensional visualization studies benefit from a higher density of information, they cannot be used to investigate reactive mixing in three-dimensional settings. This is possible with the fiber-optic probes used in our study. Since the real significance of effective dispersion coefficients are in predicting reactive mixing in heterogeneous porous media, experiments of the same type will be conducted in a heterogeneously packed large-scale sandbox soon.
Acknowledgments This work was funded by the Emmy Noether Program of the Deutsche Forschungsgemeinschaft (DFG) under Grant Ci 26/3-2. We are grateful for the valuable comments of three anonymous reviewers.
Literature Cited (1) Molz, F. J.; Widdowson, M. A. Water Resour. Res. 1988, 24 (4), 615-619. (2) Ginn, T. R.; Simmons, C. S.; Wood, B. D. Water Resour. Res. 1995, 31 (11), 2689-2700.
2096
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 7, 2004
(3) Kapoor, V.; Gelhar, L. W.; Miralles-Wilhelm, F. Water Resour. Res. 1997, 33 (4), 527-536. (4) Raje, D. S.; Kapoor, V. Environ. Sci. Technol. 2000, 34 (7), 12341239. (5) Gramling, C. M.; Harvey, C. F.; Meigs, L. C. Environ. Sci. Technol. 2002, 36 (11), 2508-2514. (6) Cirpka, O. A. J. Contam. Hydrol. 2002, 58 (3-4), 261-282. (7) Gelhar, L. W. Stochastic Subsurface Hydrology; Prentice-Hall: Englewood Cliffs, NJ, 1993. (8) Kemblowski, M. W.; Chang, C. M.; Kamil, I. Stochastic Hydrol. Hydraul. 1997, 11 (3), 255-266. (9) Cirpka, O. A.; Kitanidis, P. K. Water Resour. Res. 2000, 36 (5), 1221-1236. (10) Cirpka, O. A.; Kitanidis, P. K. Water Resour. Res. 2000, 36 (5), 1209-1220. (11) Dentz, M.; Kinzelbach, H.; Attinger, S.; Kinzelbach, W. Water Resour. Res. 2000, 36 (12), 3591-3604. (12) Dagan, G. J. Fluid Mech. 1984, 145, 151-177. (13) Nielsen, J. M.; Pinder, G. F.; Kulp, T. J.; Angel, S. M. Water Resour. Res. 1991, 27 (10), 2743-2749. (14) Kasnavia, T.; Vu, D.; Sabatini; D. A. Ground Water 1999, 37 (3), 376-381. (15) Koochesfahani, M. M.; Dimotakis, P. E. J. Fluid Mech. 1986, 170, 83-112. (16) Sabatini, D. A.; Alaustin, T. Ground Water 1991, 29 (3), 341345. (17) Guilbault, G. G Practical Fluorescence; Marcel Dekker Inc.: New York, 1990. (18) Kreft, A.; Zuber, A. Chem. Eng. Sci. 1978, 33 (11), 1471-1480. (19) Oya, S.; Valocchi, A. J. Water Resour. Res. 1998, 34 (12), 33233334.
Received for review June 11, 2003. Revised manuscript received January 15, 2004. Accepted January 21, 2004. ES034586B
