Experimental Visualization of Solute Transport and Mass Transfer

Tailing is not accounted for in groundwater solute transport models that ... Table 1. Experiments on Artificial Heterogeneous Porous Media ... 7.6 and...
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Environ. Sci. Technol. 2004, 38, 3916-3926

Experimental Visualization of Solute Transport and Mass Transfer Processes in Two-Dimensional Conductivity Fields with Connected Regions of High Conductivity BRENDAN ZINN,† LUCY C. MEIGS,‡ C H A R L E S F . H A R V E Y , * ,† R O Y H A G G E R T Y , § WILLIAM J. PEPLINSKI,‡ AND CLAUDIUS FREIHERR VON SCHWERIN§ Ralph M. Parsons Laboratory, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, Flow Visualization Laboratory, Geohydrology Department, Sandia National Laboratories, Albuquerque, New Mexico 87185, and Department of Geosciences, 104 Wilkinson Hall, Oregon State University, Corvallis, Oregon 97331

Solute transport displaying mass transfer behavior (i.e., tailing) occurs in many aquifers and soils. Spatial patterns of hydraulic conductivity may play a role because of both advection and diffusion through isolated low conductivity areas. We demonstrated such processes in laboratory experiments designed to visualize solute transport through a thin chamber (40 cm × 20 cm × 0.64 cm thick) packed with glass beads and containing circular emplacements of smaller glass beads with lower conductivity. The experiments used three different contrasts of conductivity between the large-bead matrix and the emplacements, targeting three different regimes of solute transport: low contrast, targeting macrodispersion; intermediate contrast, targeting advection-dominated mass transfer between the high-conductivity regions and the emplacements; and high contrast, targeting diffusion-dominated mass transfer. Use of a strong light source, a high-resolution CCD camera, and a colorimetric dye produced images with a spatial resolution of about 400 µm and a concentration range of approximately 2 orders of magnitude. These images confirm the existence of the three different regimes, and we observed tailing driven by both advection and diffusion. Outflow concentration measured by spectrophotometer achieved 3 orders of magnitude in concentration range and showed good agreement with known models in the case of dispersion and diffusive mass transfer, with estimated parameters close to a priori predictions. Existing models for diffusive mass transfer did not fit the breakthrough curves from the intermediate-contrast chamber, but a model of slow advection through cylinders did. Thus, both breakthrough curves and chamber images confirm that different contrasts in small-scale K lead to different regimes of

* Corresponding author phone: (617)258-0392; fax: (617)258-8850; e-mail: [email protected]. † Massachusetts Institute of Technology. ‡ Sandia National Laboratories. § Oregon State University. 3916

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solute transport and thus require different models of upscaled solute transport.

Introduction Aquifers and soils often possess very complex spatial patterns of hydraulic conductivity (K), leading to extremely complex solute transport behavior. Late breakthrough of dilute concentrations, or tailing, of solute plumes is a particularly vexing phenomenon because the breakthrough of moderate and low contaminant concentrations over long periods of time is difficult to predict and adds significant time and cost to aquifer remediation. Tailing is not accounted for in groundwater solute transport models that describe solute spreading through heterogeneous aquifers as Fickian (i.e., macrodispersive or advective-dispersive formulations). Tailing is often explained as a result of a mass transfer process (e.g., ref 1), a conceptualization that assumes that solute moves between an immobile and a mobile state. The transfer may be between chemical states, such as desorption from a surface or dissolution from a nonaqueous phase, but can also occur by solute movement between regions of high and low K. Recent work by Zinn and Harvey (2) focusing on the structure of aquifer heterogeneity required to create this tailing suggests that a mass transfer model might more accurately describe transport through conductivity patterns with well-connected high K values than a purely Fickian model and that advection of solute in to and out of low K regions can contribute significantly to tailing, in addition to diffusion. The importance of advection in low K regions has also been discussed at short spatial scales in two-region model studies (3, 4) and at longer spatial scales with numerical modeling (5, 6). These results suggest that it may be necessary to examine both diffusion and advection in low K regions to effectively characterize tailing. We investigate solute tailing in spatially heterogeneous porous media through a series of experiments conducted at the Flow Visualization Laboratory at Sandia National Laboratories. We constructed a quasi-2D (i.e., thin) porous medium of packed glass beads, consisting of a high K matrix surrounding cylindrical low K emplacements and used highresolution digital photography to image solute transport through it. The resulting “movies” (see Supporting Information) provide insight into the dynamics of the physical processes that control solute transport through porous media as well as a novel pedagogical tool for demonstrating these processes. The high K matrix surrounding lower K regions provides a connected structure, which previous work suggested is important for tailing (2). We examined both smallscale spatial variability from images of the medium and the upscaled effects of that variability from outflow concentration. Solute Transport Visualization in Porous Media. Visualization of flow and transport in rocks and soils is inherently difficult because it requires imaging tracers in an opaque medium. Traditional methods of imaging geologic media have used high-energy methods (e.g., X-rays) that can penetrate the medium, producing depth-averaged 2D images (e.g., refs 7 and 8). Modern 3D techniques, such as MRI (e.g., ref 9), show promise in allowing more detailed study of flow and transport in rock. Porous media can also be studied by visible light transmission (e.g., refs 7 and 10-12) if the medium is transparent (or relatively translucent), and packed glass beads (13, 14) are capable of creating such a medium. High-quality (i.e., high-clarity (transparency) and strong 10.1021/es034958g CCC: $27.50

 2004 American Chemical Society Published on Web 06/16/2004

TABLE 1. Experiments on Artificial Heterogeneous Porous Media ref

porous medium dimensions (cm) length width depth inclusion dimensions (cm) length width depth σ2Y (approx) outflow concentration resolution (C/C0) interior concentration resolution (C/C0) interior spatial resolution (cm) length width depth Pe (approx)i Da (approx)i

4a

3a

19a

5.0-24 b b

4.4-30 2.7-14 g

28.2 19 g

89 45 10

1 1.3 g 0.9 ∼0.1 none

1 1.3 g 1.4 ∼0.1 none

1.3 3 3.9 0.15 ∼0.005 ∼0.01

na na na 18.2 0.055

na na na

1 0.234 0.234 2500 0.005

9

17

this worka

20

18

1000 120 6

160 67 9.5

560 100 80

40 20 0.64

c c 10 1.1 d c

∼51 ∼5 6 1.2 none ∼0.01

∼3 ∼3 9.5 0.5 none ∼0.01

∼120 ∼20 ∼20 1.4 none f

g 2.54 0.64 0.7, 7.2, 12.4 ∼0.001 ∼0.01

∼10 ∼5

d 10

3 3

0.04 0.04

na na

10.5 0.135

20 0.34

50 25 10 14.7 0.009

h h

a Multiple experiments/media. b Cylindrical column, diameter not specified. c 1 “fractal” pattern; 1 random field, l ) 6 cm by 3 cm. d Measured heads/effective K; no transport measured. e Two transects of measurements, 7.6 and 8.8 m downstream. f Resolution not given. g Cylindrical column/inclusion; diameter given in width column. h See Tables 3 and 4. i See eqs 13 and 14.

TABLE 2. Bead Sizes and Light Filters Used in the Three Chambersa chamber

dominant behavior

large bead diameter (mm)

small bead diameter (µm)

K contrast

filter (%)

low contrast intermediate contrast high contrast

advection-dispersion advection-dominated tailing diffusion-dominated tailing

2.1 2.1 2.1

900 135 57

6 300 1800

50 12.50 6.25

a Mean bead diameter outside the emplacements indicated by large bead size and inside the emplacements by small bead size. K contrast represents approximate K outside the emplacements divided by approximate K inside. Filter denotes the percentage of light allowed to enter the large bead area (100% entered the emplacements).

physical integrity) packed beads possess many useful characteristics, including good light transmission, predictable K and porosity, low reactivity, and low intragranular porosity. However, an artificial porous medium does not fully reproduce the physical characteristics of geologic media, such as depositional patterns, diagenesis, or consolidation. Previous studies (e.g., refs 15 and 16) have demonstrated that the dye FD&C Blue 1 works well for solute visualization using visible light transmission. Most experimental studies of flow and transport in artificial heterogeneous porous media have used materials with two conductivity values, although some combined materials with multiple discrete values to mimic more continuous distributions of K. These experiments have varied in their focus, including measurements of internal distributions of solute, outflow concentrations, and distribution of head values (summarized in Table 1; refs 3, 4, 9 and 17-20). Our experiments extend the results of earlier work in several ways. Two of our chambers contained materials with much higher contrasts in K, an essential characteristic to produce tailing (e.g., refs 2 and 6)sprevious constructed laboratory experiments have ln(K) variances of less than 2 (with the exception of fracture flow; e.g., ref 21). Furthermore, our experimental design also allows us to study the effects of diffusion on solute tailing processes, to differentiate between tailing caused by advective and diffusive processes (via chamber images), and to study solute behavior in a domain significantly larger than the scale of the heterogeneity.

Experimental Methods Our experiments simultaneously measure internal concentration and the outflow concentration of the medium. Internal measurements possess sub-millimeter spatial accuracy over

a concentration range of more than 2 orders of magnitude. Outflow concentration measurements have an accuracy range of 3 orders of magnitude. The measurements allow detailed comparisons of small-scale and upscaled behavior in the chamber. We constructed three experimental chambers (see Table 2), each with a different K contrast in order to study a different dominant solute transport process: low contrast, studying advection-dispersion; intermediate contrast, studying advection-dominated mass transfer; and high contrast, studying diffusion-dominated mass transfer. The experiments required several pieces of equipment in addition to the specially constructed chambers (see Figure 1), which we briefly detail in this section. A significantly expanded version of this section that details design, construction, and validation is included in the Supporting Information. Chamber Construction and Other Equipment. The K contrasts necessary for the processes of interest require glass beads of significantly different sizes. Table 2 lists the sizes and the conductivity ratio created (K of packed large beads divided by that of packed small beads). Three chambers were built, one for each of the conductivity ratios, named according to relative contrast in K. Conductivities were estimated using the Carmen-Kozeny-Bear equation (22) for packed spheres and incorporating edge effects (23). Measuring the effective conductivity of the high-contrast chamber yielded a K of the packed large beads of 4.1 × 10-4 m/s versus a predicted conductivity of 2.4 × 10-4 m/s. We suspect that discrepancies in the predicted and measured K are primarily due to the walls of the emplacements acting as an additional, unaccounted for edge effect. To keep the two bead packs separated without inhibiting flow between the two regions, we constructed “emplacements” that consisted of thin stainless steel meshes wrapped VOL. 38, NO. 14, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 3. A Priori Estimates of Transport Parameters chamber low contrast high contrast intermediate contrast

FIGURE 1. Completed experimental setup from two perspectives. Panel A shows the view of the system as seen by the camera (the lightbox is seated behind the chamber). Panel B shows a side view of the path that light follows from the light box to the camera. See the Supporting Information for more details. and welded into a circular shape, similar to thin slices of cylindrical porous pipe. Emplacements were 2.54 cm in diameter and 5.5 mm high, topped by a 1 mm thick O-ring. The smallest mesh in the emplacements was 30 µm, fine enough to contain the smallest beads used, and the total volume of the screen wall was less than 4% of the total volume of each emplacement. Fifty-three emplacements were placed in the chamber in a computer-generated random pattern, with emplacements no closer to each other than 6 mm and no closer to the edges of the chamber than 4 mm (emplacements therefore constitute 33.5% of the total tank volume). The emplacements were glued to a sheet of plastic to ensure stability. Light transmission intensities through the two bead packs were equalized using a neutral density filter (a thin translucent plastic sheet) between the light source and the chamber (without filters, light transmission through large bead areas could be much greater than in small bead areas). Filter strength for each chamber is noted in Table 2. We cut 1.9 cm diameter circular holes in the filter at emplacement locations, so that the filter blocked light entering the large bead areas without significantly reducing transmission through small bead areas. Tests showed that minimum and maximum light intensities in both regions differed by no more than a factor of 2. Figure 1a shows the experimental setup from the perspective of the camera, while Figure 1b shows a side view of the light path. Figures in the Supporting Information provide additional details and perspectives. Each chamber consisted of several pieces stacked in a “sandwich” and held together 3918

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vin vx flow rate (mL/min) (cm/min) (cm/min) 1.32 4.1 0.66 1.32 0.66 1.32

0.04 0.12 0.00006 0.00013 0.00038 0.00076

0.2 0.59 0.11 0.23 0.11 0.23

Pe

Da

285 886 0.47 0.95 2.9 5.7

0.022 0.007 0.037 0.018 0.037 0.019

along the sides by a clamping mechanism. The neutraldensity filter and plastic sheet with the emplacements were sandwiched between two thick sheets of glass. Anodized aluminum side rails ran lengthwise between the plastic sheet and the glass to form the side boundaries of the flow area (flow area was 40 cm long, 20 cm wide, approximately 0.64 cm thick). Steel bars and screw plates clamped the sandwich together along the side rails. Flow at the top and bottom of the chamber was controlled by specially designed manifoldss the bottom (inflow) manifold was designed to produce an even flow along the width of the chamber and the top (outflow) manifold was designed to create even outflow with minimal residence time and mixing (see Supporting Information for more details). Water was pumped into the chamber using a computercontrolled ISCO model 500D syringe pump. Tests found that the pump produced accurate and consistent flow rates to well below 0.01 mL/min. Hydraulic gradients ranged from 0.03 for the slowest (0.66 mL/min) experiments to 0.12 for the fastest (4.1 mL/min) experiments (see Table 3 for experimental flow rates). Pumping rates were held fixed during all experiments. Outflow dye concentration was measured with a Varian spectrophotometer. Our tracer, FD&C Blue 1, possesses a distinct absorbance peak with strong linearity (15, 16). All components of the chamber were tested to ensure that there was no detectable mass loss due to sorption of dye. The photometer was computer-controlled and programmed to target the 630- and 409-nm wavelengths of FD&C Blue 1. The 630-nm peak was effective at measuring concentrations from 0.03 to approximately 20 mg/L. The 409-nm peak was effective from 0.1 mg/L to well above our maximum concentration of 30 mg/L. In the experiments run on the low-contrast chamber, outflow was collected in small sample bottles and then poured into a 1-cm path cuvette for scanning. Outflow for the other two chambers was passed into an 80-µL flow-through cell placed in the spectrophotometer, allowing high measurement frequency and reducing errors that occur in manual samples. Narrow tubing and the small outflow manifold volume produced measurement lag times at the photometer of less than 3 min at the slowest flow rate. A 14-bit liquid-cooled CCD (charge-coupled device) camera captured a series of images of the chamber as the light absorbance of the solution within it altered during transport. A combination of camera lens filters created a 600-700-nm band-pass filter for optimal light absorbance. Data were recorded in terms of pixel intensities (0-4095) and analyzed using the Scanalytics, Inc. software program IPLab. Each pixel resolved an area of the chamber approximately 400 × 400 µm. The chamber was mounted in front of a diffused light source composed of a bank of high-frequency (60 MHz), high-output fluorescent lights controlled through feedback circuitry and fan cooling to maintain a constant temperature. A stepped density wedge (photographic gray scale step tablet) was imaged along with the chamber for calibration purposes (see Figure 1).

Experimental Procedure. We initially saturated the chamber with de-aired clean (i.e., no dye) water and took an image of the chamber to be used as the baseline “clean” image (see eq 1). Porosity (volume of pore space in the bead pack (pore volume) divided by total volume) was estimated from the mass of water needed to saturate the chamber, the mass that drained from the large bead areas under gravity, and the known total volumes of both bead packs. We estimated porosity between 0.4 and 0.45 for all chambers in both the large and small bead areas. After drying the chamber, it was saturated with de-aired water with a FD&C Blue 1 concentration of 30 mg/L. We let the chamber sit for at least 24 h to ensure that any small concentration fluctuations equilibrated. The lower (inflow) manifold was flushed with clean water, leaving the inflow tubing and manifold reservoir clean of dye. The experiment then commenced with the pump, camera, and spectrophotometer set at their desired rates. We adjusted measurement settings and took hand samples as needed during the experiment. These experimental conditions simulate the flushing of an aquifer that has been exposed to a contaminant for a large period of time (thus giving solute time to reach highconcentration values in low K regions). Although many field scenarios possess similar conditions, it should be noted that this experiment is not analogous to all (or even most) field situations. Imaging the Spatial Patterns of Concentration. Chamber images were processed into concentration maps through a series of steps performed separately for each experiment. First, we compared the light intensity of parts of the stepped density wedge in each image to their corresponding intensities in a baseline image. The difference was used to recalibrate the light intensity values of all pixels in the picture (e.g., refs 7 and 24). We then used the wedge (a fixed point in space) to adjust for spatial drift in the camera at both full- and partial-pixel scale (7, 24). These steps created images that were normalized to identical position and background light intensity. Concentrations were calculated at each pixel in each image by comparing that pixel’s intensity in the clean water and dye-saturated chamber images using the Beer-Lambert law:

ln(Ii) - ln(Ii,w) Ci ) C0 ln(Ii,0) - ln(Ii,w)

(1)

where Ci is the dye concentration at pixel i, C0 is the dyesaturated concentration (30 mg/L), Ii is the measured light intensity of pixel i, Ii,w is the light intensity of that pixel in the clean water (0 mg/L) image, and Ii,0 is the light intensity in the dye-saturated (30 mg/L) image of pixel i. Breakthrough Curve Calculation. Standard curves of FD&C Blue 1 photometer absorbance (10 values spanning 3 orders of magnitude) were run through the flow cell before and after each experiment. All standard curves were fit using least-squares linear regression, and this line was used to transform photometer measurements into outflow concentrations. All standards for both the 630-nm absorbance peak and the 409-nm peak possessed r 2 values in excess of 0.999. Outflow concentrations were calculated from spectrophotometer measurements at both wavelengths. A cutoff concentration value where both measurements agreed with each other was picked (near 10 mg/L). Concentrations calculated by the 409-nm calibration line were used above the cutoff value, while the 630-nm line was used below the cutoff value. This method did not introduce discontinuities in the data because of the strong agreement between the two different estimates within this overlap range.

FIGURE 2. Diagram of the cylindrical advection model. Flow is discretized into tubes that spread longitudinally but do not mix laterally. Breakthrough curve parameters were estimated using the computer program STAMMT-L (25). The program modeled advective-dispersive transport coupled with mass transfer in and out of cylindrical immobile regions (e.g., ref 26):

(

)

∂Cm ∂2Cm ∂Cm ∂Cim +β ) vx RL 2 ∂t ∂t ∂x ∂x

(2a)

[ ]

∂Ca D* ∂ ∂Ca r ) ∂t r ∂r ∂r Cim )

2 R2

∫

R

0

(2b)

Car dr

(2c)

where Cm is the mobile domain concentration, Cim is the average concentration in the immobile domain, vx is the mean velocity of the plume in the high K areas, RL is the dispersivity, D* is the pore diffusion coefficient, and R is the radius of the cylinder. β is the ratio of immobile domain pore space to mobile domain pore space (thus larger values of β indicate a larger percentage of the porous medium is immobile/low velocity). The model assumes a one-dimensional mobile domain in which advection and dispersion (with diffusion neglected) occur, coupled with solute diffusing in and out of immobile cylinders (eq 2a). The distribution of concentration in the cylinders (Ca) is governed by Fick’s law (eq 2b), with volume-averaged solute concentration in the cylinder given by eq 2c. Analytical solutions have been developed for this scenario (see in particular Appendix 1 of ref 26), which STAMMT-L used in order to estimate vx, RL, β, and D*. However, STAMMT-L does not possess an algorithm for solving advection through a cylinder; therefore, we developed a model for this process using previous analytical work. Approximate Model of Advection through a Cylinder with Longitudinal (Neglecting Transverse) Dispersion/ Diffusion. Advection through cylinders has been studied analytically by others (e.g., refs 27 and 28). In particular, we exploit the results of Dagan and Lessoff (27) that flow through the cylinders occurs at an approximately uniform velocity. At time ) t0, the cylinder is dye-saturated and the outer boundary around the cylinder changes from dye-saturated to clean. This assumption requires that flow outside the cylinder is significantly faster than inside. Flow in the cylinder is assumed from left to right, and clean water enters the cylinder from the left semicircle boundary and exits the right semicircle (see Figure 2). Images from the tank suggest that these assumptions are approximately valid. We approximate transport through the cylinder by dividing the flow into a series of horizontal stream tubes, each carrying an equal volumetric fluid flux that does not mix VOL. 38, NO. 14, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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with its neighbors. With N tubes, total dye mass flux out of the cylinder is N

m(t′) ) 2θwthvin

∑C (t′)

(3)

i

i)1

where t′ is equal to t - t0, vin is the (uniform) velocity inside the cylinder, and Ci(t) is the concentration in each tube as a function of time, C0 is the initial concentration of solute in the cylinder, wt is the width of each tube (equal to the radius divided by the number of tubes), h is the thickness of the chamber, and θ is the porosity inside the cylinder. Assuming longitudinal Fickian spreading only, the analytic solution for concentration from each tube is

(

Ci(t′) ) C0 1 -

(

xi 1 erfc 2 2xDint′

))

(4)

where Din is the dispersion or diffusion coefficient in the tube and xi is the length of each stream tube. Using our requirement that we are dividing the outflow into equally sized tubes, the lengths will then be

xi ) 2xR 2

r 2i

(5)

where ri (the distance from the center of the cylinder to the center of the stream tube) is

(

ri ) i -

1 R 2N

)

i ) 1, 2, ..., N

(6)

which leaves a mass release from the cylinder of

m(t′) )

2θRhvinC0 N

N

∑ i)1

(

1-

1 2

erfc

( )) xi

2xDint′

(7)

with xi given by eqs 5 and 6. We chose a value of 100 for N; values of R, θ, h, and C0 are properties of the chamber or the experiment. Incorporating this description of mass transfer into a model to fit chamber breakthrough curves required additional steps. We first made the assumption that the fraction of the total fluid flux through the cylinders is approximated by

Fcyl ≈ Ncyl

2R vin w vx

(8)

with Ncyl as the number of cylinders, vx as the mean velocity in the high K region, and w as the chamber width (20 cm). This assumes that flow lines do not pass through multiple emplacements, a reasonable approximation if the velocity difference between the two regions is large. It also assumes identical porosities in the two regions. Thus, the concentration as a function of time is

C(t) ≈ (1 - Fcyl)AD + Fcyl

m(t) M0

(9)

where AD is the breakthrough concentration for 1D advection-dispersion and M0 is mass from the cylinder at time t0. The outflow from the cylinder(s) is assumed to be M0 until time t0, which we approximate by the average travel time in the high-velocity regions:

L t0 ≈ vx

(10)

with L as the length of the tank (40 cm). 3920

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Validation of Imaged Concentrations. We ran several experiments designed to ensure that our chamber imaging method gave consistent and accurate results (see Supporting Information for full details). We flooded a chamber with several fixed concentrations of dye and compared the mean and variance of estimates of the concentration value using our imaging methods to the actual concentration used. The tests showed that eq 1 slightly overestimated actual concentrations, but that the correct concentration-intensity relation was also log-linear and could be accounted for by minor modification to concentrations calculated using eq 1 (see eq SI1, Supporting Information). The coefficient of variation in these concentration estimates showed a significant increase in variability between concentrations of 0.2 and 0.1 mg/L. We concluded from these tests that, by compensating for the minor Beer-Lambert deviation of light transmission intensity, concentration calculations within the chamber were accurate and reliable down to approximately 0.2 mg/L (about 0.0066 of our maximum concentration, C0, of 30 mg/L).

Results A Priori Estimates of Transport Parameters. We estimated relevant transport parameters prior to running the experiments. These relate the approximate time scales of advection and diffusion in the emplacements to the time scale of advection through the chamber in the large bead matrix (summarized in Table 3). The interior velocity of the emplacements was approximated using the ratio of K inside and outside the emplacements, along with the porosity (θ), height (h), and width (w) of the chamber:

vin )

Kinside Q × Koutside hwθ

(11)

where Q is the flow rate of the experiment (Table 3). Exterior velocity was approximated as

(

vx ) 1 -

)

Kinside Q × Koutside hwθ

(12)

(i.e., the velocity driven by the fluid flux that does not enter the emplacements (under our assumed calculation of vin)). The Peclet number (Pe) is the ratio of the time scale of diffusion through an emplacement to the time scale of advection across an emplacement:

Pe )

vin R 2 vinR ) R D* D*

(13)

where D* is the diffusion coefficient of the dye within the bead pack. The aqueous diffusion coefficient of FD&C Blue 1 was measured by Bowman et al. (29) in the lab to be 3.4 × 10-4 cm2/min (using the Taylor dispersion technique). We assumed a pore diffusion coefficient of half that value, thus D*) 1.7 × 10-4 cm2/min (Bear (22), for example, suggests a pore diffusion coefficient between 0.4 and 0.7 of the aqueous diffusion coefficient, depending on packing and grain size distribution). The Damko¨hler number (Da) is the ratio of the time scale of advection across the chamber to the time scale of diffusion through an emplacement:

Da )

LD* vx R 2

(14)

The numerical values for the complete set of estimates are given in Table 3. Please note that we subsequently estimate

TABLE 4. Parameter Estimates from Model Fits to Experimental Breakthrough Curvesa chamber low contrast high contrast intermediate contrast

flow rate (mL/min)

vx (cm/min)

dispersivity, rL (cm)

1.32 4.1 0.66 1.32 0.66 1.32

0.2 0.57 0.15 0.3 0.13 0.28

0.46 1.22 0.68 0.55 0.6 0.7

diff coeff, D* (cm2/min) 0.0058 0.004 0.00021 0.00021 na na

β

vin (cm/min)

Dispin (cm)

0.04 0.01 0.48 0.46 na na

na na na na 0.0016 0.0025

na na na na 0.19 0.28

a Mobile domain velocity (v ) and dispersivity (R ) estimated for all experiments. Diffusion coefficient (D*) and capacity coefficient (β) estimated x L using the cylindrical diffusion model for the low- and high-contrast experiments. Interior velocity (vin) and longitudinal dispersivity ( Dispin) estimated using the model of cylindrical advection for the intermediate-contrast experiments.

vx, vin, and D* from the breakthrough curves of each experimentsthe a priori estimates (Table 3) differ from the a posteriori values calculated by fitting models to the breakthrough curves (Table 4). Spatial Concentration Images. Concentration images accurate over about 2 orders of magnitude (relative to initial) were recorded with high-frequency (intervals between images varied from every 2 min for the fastest experiments to every hour during late times in chambers with tailing behavior). The images from each chamber (Figure 3) illustrate observed transport behaviors, but the experimental results are best understood by viewing a complete data set from each chamber in the form of Quicktime movies (see Supporting Information). Images show the full chamber area (40 cm long × 20 cm wide), with each chamber subjected to the same flushing rate (1.32 mL/min). Although experimental flow was from bottom to top, we have rotated the images 90° counter-clockwise for easier viewing. Thus, flow is from right to left in these images (and in the movies), with initial saturation at 30 mg/L (red), followed by flushing with clean water at a constant flow rate. The images from the low-contrast chamber (first column, Figure 3) resemble typical macrodispersive spreading, with a front propagating through the entire chamber and velocity variation leading to spreading. Flow in emplacements is slower, but it appears that solute does not lag significantly behind the main front. At this observation scale, the emplacements seem to produce Fickian spreading and minimal tailing. Images from the high-contrast chamber (next two columns, Figure 3) significantly differ from the low-contrast chambersflow through emplacements in this chamber is much slower than in high K areas. Flow diverts around the emplacements, leaving them with high dye concentration as clean water passes by in the high K areas. As time passes, solute slowly diffuses out of the emplacements. After 1 d, there is still significant solute left in the emplacements, while the low-contrast chamber was clean after 4 h. Images of an individual emplacement (column following chamber images) show classic behavior for diffusion out of a cylinderscircularly symmetric solute concentrations with highest values in the middle of the emplacement. However, it should be noted (this is especially visible in the movie) that the circularly symmetric solute distribution appears to displace slightly in the downgradient direction, suggesting that there is also some advection occurring, albeit very slow. The images of the high-contrast chamber display minor problems caused by beads settling in the emplacements. Because beads were packed in a different orientation than the final chamber position, they sometimes settled enough that small gaps formed at the top of the emplacements. These open spaces had no discernible effects on flow but resulted in imaging difficulties because the gaps were not completely covered by the light filter. We fixed this problem by taping small slivers of filter over these areas, resulting in resolution

reduction and light-smearing errors (visible in the images) in areas that were filtered twice or covered with tape. We believe this was primarily responsible for the larger mass balance errors in this chamber (see Supporting Information). Results from the intermediate-contrast chamber (last three columns, Figure 3) are similar to the high-contrast chamber at early timessthe advective front moves past emplacements quickly, leaving solute behind in the emplacements. However, the late-time behavior is much differentssolute predominantly advects across emplacements instead of diffusing out of them. In the single emplacement images, the high-contrast chamber shows a more radial release of solute, but the intermediate-contrast case is more akin to plug flow through the emplacement; clean water is transported in at the upstream end and dyed solute flows out at the downstream end. This advection occurs concurrently with a significant amount of diffusion, causing deviation from a simplistic plug flow characterization. Chamber images suggest that slow advection occurs faster than diffusion, as overall concentration in the advective mass transfer chamber at 1035 min is already lower than the diffusive mass transfer chamber at 1710 min. Breakthrough Curves. Breakthrough curves were fit with either the cylindrical diffusion or cylindrical advection model (Table 4; vx and vin differ from the a priori estimates in Table 3). A log-log plot of the complete set of breakthrough curves is graphed as a function of pore volume flushed in Figure 4. Concentrations below 0.001 C0 were below the reliability of our measurements; thus, the lower boundary of the graph is the approximate minimum measurable concentration. Breakthrough curves are shown as function of absolute time (minutes) in Figure 5. The breakthrough curves of the low-contrast chamber (Figure 5a) show no tailing. A pinhole leak in the top right corner of the chamber caused air to entrain in one of the outflow hoses, causing measurement problems in the flow cell. Therefore, all data for these breakthrough curves were collected by hand sample. The curves are well fit by an advective-dispersive model (i.e., parameter estimates from STAMMT-L give very low estimated values of β), meaning that only velocity and dispersivity are required to characterize the curve (and from Figure 5a, the advective-dispersive model with minimal mass transfer clearly fits the data well). Breakthrough curves for the high-contrast chamber (Figure 5b) are clearly different, with a tail that is matched by the cylindrical diffusion model (28). The flow cell allowed high data frequency (over 1000 data points in each curve), and the best fits nearly perfectly overlap the data. Parameter estimates for these experiments agree well with the physical parameters of this system. The emplacement volume (34% of total chamber volume) suggests a β estimate of 0.5, close to the estimates from the breakthrough curves (Table 4). Pore diffusion coefficients (D*) were estimated at 2.1 × 10-4 cm2/min for both experiments (Table 4), close to our a priori prediction of 1.7 × 10-4 cm2/min. D* estimates for the two VOL. 38, NO. 14, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 3. Series of color images representing solute concentration as a function of time for each of the three chambers; all three experiments run at 1.32 mL/min. Concentration is represented as fraction of initial, as noted on color bar to the right. Close-up of a single emplacement is shown for the two mass transfer cases, illustrating one dominated by diffusion and the other by slow advection.

FIGURE 4. Breakthrough curves for the six experiments performed, shown as a function of pore volume flushed into the chamber. The experiments in the intermediate- and high-contrast chambers are composed of more than 500 data points each (thus appearing like lines, although they are actually discrete points). Differences between the tails of the high- and intermediate-contrast experiments are greater at the faster flow rate (1.32 mL/min) than the slower flow rate (0.66 mL/min). 0.001 C0 represents the lower limit of outflow measurement accuracy.

FIGURE 5. Experimental breakthrough curves and their best fits with cylindrical diffusion and/or advection models. The shape of the curves for the low-contrast chamber (a) indicates that little mass transfer (tailing) is occurring. The high-contrast experiments (b) show tailing, well fit by the cylindrical diffusion model. The faster intermediate-contrast experiment (c) is more accurately fit by a cylindrical advection model than by a cylindrical diffusion model. The slower experiment (d) is not particularly well fit by either the advection or the diffusion model. experiments differed by less than 2%, suggesting that diffusion (unaffected by the head gradient) is the driving force of tailing. Dispersivity estimates (Table 4) in the three chambers are similar with values ranging from 0.46 to 0.7 cm, a length of about half of the emplacement radius except for one outlying experiment, the low-contrast chamber with a flow rate of 4.1 mL/min, which has a dispersivity estimate that is twice as high as any other experiment. We attribute this large

difference to air entrapment in one of the outflow tubes causing periodic flow blockage, leading to increased spreading as solute moved laterally to the other two outflow ports. Increasing the relative importance of diffusion (by reducing the flow rate) does not appear to increase dispersivity in emplacements. This agrees with earlier work (2) that suggested that dispersion and diffusion are not simply additive Fickian processes and that in some cases diffusion may reduce VOL. 38, NO. 14, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 6. Graphical representation of flow regimes observed in our experiments (location of experiments denoted by yellow circles, using Pe and Da estimates from Table 3, with dashed lines indicating change in hydraulic gradient with same conductivity contrast). Three regimes of solute spreading occursone with Fickian spreading and two possessing tailing (i.e., mass transfer)swith either advection or diffusion dominating the tailing. Several experiments from previous works are indicated by light-blue squares. dispersion through processes similar to those described by the Taylor-Aris model of dispersion. Breakthrough curves from the intermediate-contrast chamber (Figure 5c,d) display different tailing than the highcontrast experiments. Comparison of the intermediatecontrast experiments to their high-contrast counterparts show a flatter early part of the tail, followed by a sharper downturn in concentration. For the faster flow experiment (Figure 5c), the breakthrough curve is poorly fit by the cylindrical diffusion model and is better represented by our cylindrical advection model, indicating that slow advection drives much of the tailing. For the slower experiment (Figure 5d), both models reproduce parts of the breakthrough curve well, suggesting that both diffusion and advection may play roles in transporting solute from emplacements. Furthermore, for the slower flow rate (0.66 mL/min), the breakthrough curve for the intermediate-contrast chamber is more similar in shape to its corresponding high-contrast (diffusiondominated) experiment, also suggesting that diffusion plays a greater role in flushing solute from the emplacements. It should be noted that best fits using the cylindrical diffusion model yield unrealistically high values of β(≈0.65) for the intermediate-contrast experiments. Dispersivity estimates inside the emplacements (Dispin) are much larger than the average bead diameter in the emplacements, and dispersion coefficient values (vin × Dispin) are of the same order of magnitude as D*, suggesting that diffusion plays a large role in longitudinal spreading in the emplacements. The observed solute transport behaviors are the result of interaction between three competing time scalesstime to 3924

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diffuse out of an emplacement, time to advect across an emplacement, and time to advect through the entire chamber. We summarize these in Figure 6, using our formulation of Peclet and Damko¨hler number to create a grid that delineates the three time scales of interest in these experiments. When either of the processes within the emplacements is faster than advection across the chamber (Da > 1 or Da > Pe-1), solute spreading at the large scale may be described as Fickian and fully described by a dispersive model. When both advection and diffusion through emplacements are slower than advection across the chamber (Da < 1 and Da < Pe-1), the breakthrough curve will show tailing. The tailing will be dominated by whichever emplacement process is faster (Pe < 1 for diffusion, Pe > 1 for advection). Our a priori predictions of Pe and Da, in the context of Figure 6, match up with the observed behavior in both the breakthrough curves and the chamber images. In general, the parameters used to estimate Pe and Da match up well with subsequent a posteriori estimates, although vin values in the intermediate-contrast tank (Table 4) are about a factor of 3 higher than a priori estimates (Table 3), possibly due to higher head gradients in the emplacements that were not accounted for in our estimation. The estimated Pe and Da values for some of the other experiments (Table 1) also provide reasonable predictions of solute behavior. Experimental results generally agree with our prediction (Figure 6)sthe Barth et al. (17) and Silliman and Zheng (20) experiments showed mostly Fickian spreading, while the Li et al. (4) and Hoteit et al. (18) experiments displayed some tailing. Our large conductivity contrasts create

the substantially smaller values of Pe in the intermediateand high-contrast experiments. Identifying which process drives tailing is important to accurately predicting solute transport at different spatial and temporal scales and hence to making decisions about remediation and waste disposal. The rate of mass transfer driven by slow advection will be altered by changes in hydraulic gradient, but diffusion-driven mass transfer may not be. Additionally, the processes controlling mass transfer may be altered by changing the gradientsin Figure 6, changing the gradient shifts position in the regime map along a diagonal line. If Pe-1 is greater than the Damko¨hler number, decreasing the gradient may shift the behavior from advective mass transfer, to diffusive mass transfer, to Fickian spreading. If Pe-1 is less than Da tailing will not occur, regardless of the hydraulic gradient (as was the case in our low-contrast chamber). Site assessment should account for possible shifts in mass transfer rates and processes under different hydraulic gradients (e.g., pump tests and natural conditions); prediction of transport under a different gradient will be difficult if the mechanism that drives tailing is not identified. In summary, the experiments in this work successfully visualized small-scale solute transport patterns while simultaneously measuring outflow concentration from an artificially constructed porous medium with conductivity contrasts much larger than previously reported experiments. Different K contrasts displayed three behaviors (macrodispersion, advection-dominated tailing, and diffusion-dominated tailing) both inside the porous medium chamber and in the outflow, demonstrating fundamentally different smallscale behaviors driving discernibly different upscaled behaviors. The resulting dynamic images provide a novel view into solute transport through porous media. Breakthrough curve parameter estimates give values of diffusion coefficients, capacity coefficients, and dispersivities that are reasonable, confirming our observations of small-scale behavior in the chambers. Parameter estimates confirmed that no significant tailing occurs in the low-contrast chamber, and diffusion coefficient estimates confirmed that the tailing in the high-contrast chamber was diffusion driven. For the intermediate-contrast chamber, we developed a model to describe the flow, incorporating slow advection through the emplacements, and applying this model to breakthrough curves suggested that the tailing in this chamber was primarily advective. With reduction of the hydraulic gradient in the intermediate-contrast chamber, breakthrough began to look more like the analogous diffusion-dominated experiments. This work suggests that aquifers may produce solute tailing driven by both diffusion and advection, depending on the pattern and characteristics of hydraulic conductivity heterogeneity in the aquifer and the conditions of the flow field. Many aquifers have low K regions possessing a distribution of geometries and sizes, adding additional complexity. An understanding of the mechanism causing solute tailing may be critical for predicting transport at length and time scales that differ from those used experimentally, particularly if the hydraulic gradient changes.

Acknowledgments We thank Robert Glass, James Brainard, and Vincent Tidwell of Sandia National Laboratory for their assistance with lab space, equipment, and helpful suggestions. We would also like to thank Patricia Culligan, as well as the Journal reviewers and editors for their useful edits and suggestions. This work was funded by the National Science Foundation (EAR9875995) and the Department of Energy, Basic Energy Sciences Division (DE-FG02-ooER15029 and DE-FG02ooER15030). Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed

Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. Financial support does not constitute an endorsement by DOE of the views expressed in the paper.

Supporting Information Available Three movies, additional text, and figures. This material is available free of charge via the Internet at http://pubs.acs.org.

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Received for review September 1, 2003. Revised manuscript received April 8, 2004. Accepted April 19, 2004. ES034958G