Simultaneous estimation of local-scale and flow path-scale dual-domain mass transfer parameters using geoelectrical monitoring



[1] Anomalous solute transport, modeled as rate-limited mass transfer, has an observable geoelectrical signature that can be exploited to infer the controlling parameters. Previous experiments indicate the combination of time-lapse geoelectrical and fluid conductivity measurements collected during ionic tracer experiments provides valuable insight into the exchange of solute between mobile and immobile porosity. Here, we use geoelectrical measurements to monitor tracer experiments at a former uranium mill tailings site in Naturita, Colorado. We use nonlinear regression to calibrate dual-domain mass transfer solute-transport models to field data. This method differs from previous approaches by calibrating the model simultaneously to observed fluid conductivity and geoelectrical tracer signals using two parameter scales: effective parameters for the flow path upgradient of the monitoring point and the parameters local to the monitoring point. We use regression statistics to rigorously evaluate the information content and sensitivity of fluid conductivity and geophysical data, demonstrating multiple scales of mass transfer parameters can simultaneously be estimated. Our results show, for the first time, field-scale spatial variability of mass transfer parameters (i.e., exchange-rate coefficient, porosity) between local and upgradient effective parameters; hence our approach provides insight into spatial variability and scaling behavior. Additional synthetic modeling is used to evaluate the scope of applicability of our approach, indicating greater range than earlier work using temporal moments and a Lagrangian-based Damköhler number. The introduced Eulerian-based Damköhler is useful for estimating tracer injection duration needed to evaluate mass transfer exchange rates that range over several orders of magnitude.

1. Introduction

[2] Anomalous transport behavior, including solute concentration rebound after pumping stops and long tailing, has been noted in numerous field and laboratory experiments [Feehley et al., 2000; Harvey and Gorelick, 2000]. Such behavior is a principal control on the efficiency of aquifer remediation [Harvey et al., 1994] and water resource management using aquifer storage and recovery [Culkin et al., 2008]. Several mathematical models have been proposed to explain such behavior, including dual-domain mass transfer (DDMT) [van Genuchten and Wierenga, 1976], multirate mass transfer [Haggerty and Gorelick, 1995], fractional dispersion [Benson et al., 2000], and continuous time random walk [Berkowitz et al., 2006]. Although these models can reproduce observed transport behavior, identification of appropriate model parameters (e.g., mass transfer rate coefficient) is impeded by the lack of experimental methods that can directly interrogate the immobile pore space. Commonly, model parameters are identified by history matching and model calibration to mobile fluid tracer concentrations [Singha et al., 2007]. Because the occurrence of anomalous-transport phenomena depends strongly on experiment time scales [Haggerty et al., 2001], and the relative rates of advection and diffusion, parameters estimated based on a single flow path averaged experiment may not prove effective under different conditions.

[3] Recent work at pilot field [Singha et al., 2007] and laboratory scales [Swanson et al., 2012] has demonstrated that anomalous solute transport has an observable geoelectrical signature that, in principle, can be exploited to directly determine mass transfer parameters (i.e., rate coefficient, mobile porosity, and immobile porosity) [Day-Lewis and Singha, 2008]. Whereas, fluid sampling preferentially draws from the mobile domain, electrical measurements are sensitive to solute tracer in both mobile and immobile domains. The relation between bulk and fluid conductivity appears hysteretic, or time dependent, in the presence of rate-limited mass transfer. Hysteresis between fluid and bulk conductivity is observed because bulk conductivity (mobile and immobile domains) lags behind relatively quick changes in fluid conductivity (mobile domain) on both the rising and falling limbs of a solute breakthrough curve (BTC) due to rate-limited mass transfer between mobile and immobile pore space. The electrical signature of mass transfer was observed first in field-experimental data by Singha et al. [2007], but only limited sensitivity analysis was performed for that data set, and rigorous parameter estimation was not attempted. Swanson et al. [2012] recently inferred mass transfer parameters using parametric sweeps for controlled homogeneous column experiments with independently evaluated immobile porosity. Forward simulations of transport were run for several thousand combinations of mass transfer parameters, and the fit to experimental data was evaluated. The previous work detailed above relied on either (1) analysis based on temporal moments [Day-Lewis and Singha, 2008], which are subject to error in the presence of measurement noise or (2) parametric sweeps [Swanson et al., 2012], which provide limited insight into parameter sensitivity or information content.

[4] Here, we extend the use of electrical measurements to inference of field-scale mass transfer parameters at both the effective flow path and local scales by adopting a model-calibration strategy to match both fluid conductivity and geoelectrical tracer signatures simultaneously. Our approach facilitates rigorous sensitivity analysis to evaluate the information content of the geophysical data, and provides insight to describe the true variability in local mass transfer parameters. We apply our approach to conservative field tracer data from a former uranium mill tailings site near Naturita, Colorado [e.g., Curtis et al., 2006; Brosten et al., 2011]. At this site, the fluid sample-geoelectrical measurement data exhibit the expected signature of mass transfer. We calibrate a series of one-dimensional DDMT models, for various sampling locations, to tracer and geophysical data using nonlinear regression. For transport simulation, we use the code Modular Three-Dimensional Multispecies model (MT3DMS) [Zheng and Wang, 1999]. For model calibration, the transport model is linked to a computer code for universal inverse modeling (UCODE_2005) [Poeter et al., 2005] nonlinear regression package, which also performs sensitivity analysis. Finally, we introduce an Eulerian-based Damköhler number useful for planning experiments and evaluating whether tracer injection duration was sufficient to observe hysteresis and determine local-scale mass transfer.

2. Field Experiment

[5] We conducted a forced-gradient tracer experiment at a radionuclide-contaminated site near Naturita, Colorado. The tracer test described here was conducted to complement a companion tracer test that focused on uranium desorption from the sediments in this contaminated aquifer. The desorption experiments were performed by injecting uranium-free water with a relatively low ionic strength but under forced-gradient conditions in order to minimize mixing with native groundwater. In the tracer tests described here, the same forced-gradient conditions were used but with a high ionic strength solution to permit monitoring by conventional fluid sampling and concurrent geoelectrical measurements. In this section, we detail the field site, experimental setup, and data acquisition.

2.1. Site Description

[6] The field experiments were conducted at a former uranium mill site located along the San Miguel River in southwestern Colorado, approximately 3 km northwest of the town of Naturita (Figure 1). The aquifer is recharged by the river at the southeast end of the site, and discharges back into the river along the northern end [Curtis et al., 2006], creating long flow paths of groundwater/surface water exchange. The aquifer is recharged during snowmelt and runoff in the spring and water elevations are generally constant from August until March with an average saturated thickness of approximately 3–4 m [Davis et al., 2004]. Uranium contamination occurs in the shallow, unconfined aquifer at the site; the aquifer is composed of sand, gravel, and cobbles, with mineralogy consisting of quartz and lesser amounts of detrital feldspars, carbonates, magnetite, and fine clay-size materials [Davis et al., 2004; Curtis et al., 2006]. A caliche layer present at a depth of approximately 1 m minimizes local recharge from the surface at the test site. The aquifer is underlain by the Brushy Basin Member, a fine-grained shale approximately 30 m thick. The tracer tests were conducted downgradient of the maximum observed uranium concentrations observed at the site. The depth to water and saturated thickness of the aquifer varies seasonally, and was approximately 3.4 and 1.6 m, respectively, at the time of the field investigation. Additional details of the overall site characteristics are presented elsewhere [Davis et al., 2004, 2006; Kohler et al., 2004; Brosten et al., 2011], as is a description of an earlier experiment on uranium transport that was conducted at the tracer test site [Curtis and Davis, 2006]. The results of the current work provide independently determined mass transfer model parameters which will be used to simulate observed uranium desorption.

Figure 1.

Map of the U.S. Department of Energy Uranium Mill Tailings Remedial Action site near Naturita, Colorado, showing the locations of the former mill, ore tailings, and monitoring wells. The layout of the tracer site used for this study is depicted in the inset.

[7] The uranium mill was in operation from the 1930's until its closure in 1963. Remediation efforts included removal of contaminated soil from the southern two thirds of the site between 1996 and 1998. The site has been the focus of U.S. Geological Survey research on reactive transport of uranium (VI) for over 10 years [Curtis et al., 2004, 2006; Davis et al., 2004; Kohler et al., 2004]. Several well clusters were installed for previous research efforts at the site. For our experiments, a new cluster (Figure 1) was installed with wells designed for dual-purpose fluid sampling and geoelectrical monitoring.

2.2. Tracer Injection

[8] Ten polyvinyl chloride (PVC) fluid/geoelectrical monitoring wells and an injection gallery were installed using a Geoprobe direct-push rig. Each monitoring well consists of fifteen 2.5 cm wide, 4.1 cm diameter stainless steel electrodes spaced 22 cm apart, and four levels of 0.64 cm diameter sampling ports spaced 30.5 cm apart, installed to a total depth of approximately 5.5 m below ground surface. Wells were installed a meter apart in the east-west direction and spaced 2 m apart in the north-south direction (Figure 1). The injection gallery consists of nine 0.02 m diameter wells, installed to depths of approximately 4.5 m, with a 0.6 m screen at the bottom of each well; injection wells are distributed in two rows, spaced 0.5 m apart.

[9] Figure 2a shows the conceptual injection experiment as a 2-D cross section and how the experiment is reduced to a 1-D transport problem between the injection gallery and co-located fluid conductivity (σf) and bulk conductivity (σb) observation locations distributed along vertical arrays. Forced injection, at a rate of 0.9 ± 0.1 L/min, was conducted for 27 days from 24 October 2011 to 19 November 2011. A NaCl tracer solution at approximately 3.74 g NaCl L−1 with mean conductivity of 6900 (SD 271) μS cm−1 was injected for 15 days and followed directly by a freshwater flush injection of 292 (SD 4) μS cm−1 for 12 days. Although the goal of the tracer injection was a constant concentration/rate, the reality showed some variability, which was addressed by the models in a manner described in section 3.1. Groundwater fluid parameters (specific electrical conductivity, pH, and temperature) were measured once per day using an Orion five-Star meter; conductivity change resulting from the tracer (Cl) addition was assumed to be conservative [Davis et al., 1998]. The meter was calibrated daily for electrical conductivity and pH. Prior to sampling, sampling ports and tubing were purged using a peristaltic pump. All fluid conductivity measurements, including the injection tank, were converted to a reference temperature equal to the mean groundwater sample temperature (14.5°C), to make fluid measurements comparable to the geoelectrically measured bulk conductivity measurements.

Figure 2.

Schematic diagram of (a) the field-scale tracer injection which was observed in full at a subset of the original monitoring points (not to scale); (b) the local-scale colocated fluid (σf) and bulk (σb) conductivity sampling and associated hysteresis resulting from mass transfer; and (c) the 1-D numerical model framework partitioned into effective upgradient flow path and local-scale parameter sets.

2.3. Geoelectrical Measurements

[10] Single-borehole four-electrode resistivity profiles were collected twice daily for every monitoring well vertical profile. Wenner-style measurements [e.g., Telford et al., 1990] were collected using a SuperSting R8 IP (Advances Geosciences, Inc.) eight-channel resistivity meter. In the Wenner measurement configuration, four adjacent electrodes are involved, with the exterior electrodes A and B serving as current source and sink, respectively, and interior electrodes M and N (spaced 22 cm apart) serving as potential electrodes (Figure 2b). Given the current I applied between A and B and the measured voltage, ΔV, between electrodes M and N, apparent resistivity is calculated as

display math(1)

where K is the appropriate geometric factor:

display math(2)

where rXY is the distance separating electrodes X and Y, and inline image is the distance separating the electrode in the symmetrical mirror image used to account for the no-flow boundary condition at the Earth's surface. Figure 2b illustrates how a representative volume of geoelectrical measurement between two potential electrodes may encompass both mobile and immobile pore space, and is colocated with a fluid sampling port.

[11] For comparison of fluid sampling and geoelectrical measurements, a petrophysical model is required. Commonly, Archie's law is used to relate bulk and fluid conductivity [Archie, 1942], here formulated for a dual-domain medium under the assumption of domains acting as conductors in parallel as [Singha et al., 2007]:

display math(3)

where nmob is the porosity of the mobile domain, nimmob is the porosity of the immobile domain, σmob is the fluid conductivity in the mobile domain, σimmob is the fluid conductivity in the immobile domain, and m is the cementation exponent, which is a function of the connectivity of the pore space, assumed to be identical between the domains [Singha et al., 2007]. We determined m experimentally as the slope of the best fit σb-σf line for background (preinjection) and plateau (equilibrium) data (supporting information Figure S1). The fit was forced through the origin to honor Archie's law, which changed the estimated value of 1.3 by a negligible amount, indicating surface conduction may not be an important factor. The cementation exponent is within the expected range [Telford et al., 1990] and consistent with unconsolidated sediment. In developing the regression for m, data from sampling location E14.6 (i.e., well E1, port at 4.6 m depth) were neglected because this port showed anomalously high σb plateau (>2500 μS cm−1) compared to other locations' values (mean ∼1000 μS cm−1), and thus exerted strong leverage on the regression. The use of an “average” m is a potential source of error to the modeling effort, and the sensitivity of simulations to the m parameter is evaluated through sensitivity analysis described below. It was decided that the field-measured average m was preferable to determining m independently for each well location, as this might impair relative comparisons of mass transfer parameters, or to estimating m through inverse modeling because of the high correlation with other model parameters.

3. Data Analysis Approach

[12] Our analysis approach involves model calibration applied simultaneously to fluid conductivity data reflecting tracer breakthrough, and geoelectrical measurements also sensitive to the tracer, to estimate mass transfer parameters at varied scales. In this section, we detail the transport simulation, model calibration, and sensitivity analysis.

3.1. Transport Simulation

[13] Groundwater flow and solute transport were simulated using the finite-difference model MODFLOW-2000 [Harbaugh et al., 2000] combined with particle tracking in MT3DMS [Zheng and Wang, 1999] to solve the 1-D advective-dispersive transport equation with DDMT:

display math(4a)


display math(4b)

where cmob is the mobile-domain concentration [M/L3]; cimmob is the immobile-domain concentration [M/L3]; D is the hydrodynamic dispersion coefficient [L2/T]; v is the pore water velocity [L/T]; α is the mass transfer rate coefficient [T−1]; and t is time [T]. Optimized model values of α were related to a diffusive length scale using the open water diffusion coefficient (D*) for chloride (1.7556 cm2 d−1) [Culkin et al. 2008]:

display math(5)

[14] The model domain was discretized into 0.01 m cubic grid cells. The Euclidean distance between the injection zone and observation locations was taken as the effective flow path length for each simulation. Model parameters for forward simulation and calibration were partitioned into (1) effective parameters for the flow path upgradient of the observation location (nmob1, nimmob1, α1, D) and (2) local parameters at the observation location (nmob2, nimmob2, α2) for each modeled aquifer location; this approach is shown in Figure 2c. Therefore, in addition to the flow path–scale mass transfer parameters, a second set of mass transfer parameters at the observation point model node and the following nine grid cells was included to simulate the dynamics of the geoelectrical representative volume. Observed deformation of the input tracer signal through fluid sampling (σf) primarily results from transport and mass transfer processes along the flow path upgradient of the sampling port, and is therefore commonly summarized with “average” parameter sets that describe a Lagrangian process of change through space and time. The σb measurements describe change in one space (representative volume) through time: an Eulerian process that is sensitive primarily to processes within the local measurement volume when variability of the incoming tracer BTC has been accounted for. Therefore, an additional set of model parameters local to the observation model node is included. This approach allows the variability in local mass transfer properties to be investigated. This variability represents an important complexity that in the absence of geoelectrical data would be obscured at the effective flow path scale. When σf and σb tracer BTCs indicate differing mass transfer properties, both observation sets can be fit with one model and reveal two scales of mass transfer.

[15] The model gradient used to simulate the forced-gradient tracer injection test was determined using Darcy's law to reproduce the observed median tracer transport times observed at the 5.0 m depth fluid port at the E1 well assuming the hydraulic conductivity (8.64 m d−1) and effective porosity (0.2) values determined for the area by Curtis et al. [2006]. This assumption was necessary to set the fundamental gradient needed for the MODFLOW model and allowed relative differences in nmob1 to be assessed. The local nmob2 is not strongly affected by this assumption, as it is primarily informed by the σb data set as discussed below. The mass transfer parameters (nimmob and α) were also less affected by this assumption, and are of most interest to this facet of the larger uranium transport study because they control much of the long-term storage and release of contaminants.

[16] The 15 day tracer injection and subsequent 12 day flush were simulated by adjusting a constant concentration boundary condition in MT3DMS. The goal for the experimental tracer injection was constant concentration/rate, but the immediate downstream observation point (RD) and every σf plateau indicate a variable input. This pattern of tracer input appears highly consistent across the well field. We approximate the observed pattern by representing the input boundary condition as varying over four stress periods, with a fifth stress period to simulate the tracer flush, and the same injection pattern was used for all simulations (supporting information Figure S2).

[17] In addition to simulation of the field experiment, three synthetic forward models were run as “type” scenarios to illustrate the BTC and hysteresis response we might expect from (1) moderate mass transfer at the effective flow path and local scales; (2) moderate mass transfer at the effective flow path scale, but negligible at the local scale; and (3) negligible mass transfer at the effective flow path scale, but considerable at the local scale. One meter domain forward models were run with a 15 day constant rate injection (7000 μS cm−1) followed by a 12 day flush (300 μS cm−1) to approximate the field experiment. For all scenarios, the mobile porosity and mass transfer rate coefficients were held constant at all scales, while the immobile-domain porosity was adjusted at varied scales to create more/less mass transfer (Table 1).

Table 1. Parameter Values for Three Synthetic Examples of Spatially Variable Mass Transfer Parametersa
ScenarioEffective ParametersLocal Mass Parameters
nmob1nimmob1α1 (d−1)nmob2nimmob2α2 (d−1)
  1. a

    Scenario 1 has moderate mass transfer at the effective flow path and local scales; Scenario 2 has moderate mass transfer at the effective flow path scale, but negligible mass transfer at the local scale; and Scenario 3 has negligible mass transfer at the effective flow path scale, but considerable mass transfer at the local scale.

3.2. Model Calibration

[18] The mass transfer parameters (and D) were optimized through nonlinear regression with UCODE_2005 [Poeter et al., 2005; Hill and Tiedeman, 2007] using MATLAB (Mathworks, Inc., Natick, MA) codes in an intermediate step to generate the necessary input text files from forward model output. The σf and σb observations from each location were typically given a coefficient of variation (CV) weighting of 3%, or the general estimated error in conductivity observations based on inspection of repeat and reciprocal errors. Changes in σf and σb through time were assumed to result from the break through of the conservative NaCl tracer, hypothetical BTCs and resulting hysteresis are shown in Figure 2b. It should be noted that we fit the σb and σf observations directly, not a hysteresis pattern between them, because error propagation is compounded when considering ratios of measurements; hence, an optimal fit to a hysteresis pattern would likely result in a poorer fit to the underlying BTC data. With UCODE_2005, a sum of squared weighted residuals (SOSWR) objective function is minimized through the perturbation of parameter values providing the ‘‘best fit'' between weighted observations and their simulated equivalents [Poeter et al., 2005]. Several different starting values were used to help insure the models converged on a true global minimum. Before model parameters were optimized, UCODE_2005 was run in “sensitivity analysis” mode to assess possible parameter correlation. Parameters with correlation greater than 0.95 were estimated independently during initial optimization runs, then typically combined later for simultaneous convergence. The criterion for model convergence was set at the UCODE_2005 default, or less than 1% change in parameter values between iterations [Poeter et al., 2005]. A nonlinear confidence interval (CI) analysis was performed for the optimized parameter sets using the UCODE_2005 sensitivity analysis mode.

3.3. Sensitivity Analysis

[19] To assess relative model fits, the SOSWR was determined with equal observational weighting (CV = 0.03) and was normalized to the total number of observations, or n = 46 for profile E1 and n = 50 for profile E2. Composite scaled sensitivities (CSS) of the optimized parameters to the observational data (σf and σb) were estimated with a forward or central perturbation technique. Parameters that have a CSS of less than 1.0 and (or) are more than 2 orders of magnitude less sensitive than the most sensitive parameter in the simulation may be difficult to determine [Hill and Tiedeman, 2007]. The dimensionless scaled sensitivities (DSS) were also determined with UCODE_2005 to assess the importance of each observation to the optimized model parameters.

4. Results

[20] Example synthetic forward modeling was used to define three type scenarios of effective and local mass transfer variability. Experimental field data, model calibration using nonlinear regression, and sensitivity analysis are presented for aquifer locations where full tracer BTCs and apparent electrical conductivity time histories were recorded. Observations are closely matched, and the calibration procedures yield spatially variable mass transfer parameters. Sensitivity analysis is used to show parameter sensitivity to both σb and σf observations.

4.1. Synthetic Modeling Scenarios

[21] Synthetic modeling of three type scenarios provides insight into how different combinations of mass transfer parameters for the effective upgradient flow path and local support volume manifest in observed tracer and geoelectrical measurements. In Scenario 1, the base-case scenario, mass transfer parameters are identical for the upgradient and local scales; in this scenario, the σf signal showed strong tailing (Figure 3a), the σb signal had a moderate plateau value of 1700 μS cm−1 (Figure 3b), and hysteresis separation between σb and σf was observed (Figure 3c). Scenario 2 was analogous to Scenario 1, except mass transfer at the local scale was essentially eliminated by reducing nimmob2 to 0.01 (Table 1). The resulting σf signal was nearly identical to Scenario 1 (Figure 3a), but the σb BTC plateau was of much lower magnitude at 860 μS cm−1 (Figure 3b). In the absence of local mass transfer, a linear relationship was observed between σb and σf (Figure 3c). In Scenario 3, effective flow path–scale mass transfer was eliminated by decreasing nimmob1 to 0.01, while local exchange was enhanced greatly by increasing nimmob2 to 0.35. The absence of upgradient mass transfer in Scenario 3 results in negligible tailing in the σf signal (Figure 3a), but enhanced local porosity and mass transfer results in the highest σb BTC plateau of 3100 μS cm−1 (Figure 3b) and strongest hysteresis separation (Figure 3c).

Figure 3.

Synthetic examples showing the tracer/geoelectrical response to spatially variable mass transfer parameters under three scenarios. (a) Fluid conductivity breakthrough curves have tailing under moderate effective flow path–scale mass transfer (Scenarios 1 and 2), while the input tracer signal shows little deformation in the absence of flow path–scale mass transfer (Scenario 3). (b) Although all bulk conductivity signals show tailing, plateau levels are governed by differing total porosity. (c) Hysteresis does not occur with negligible local mass transfer in accordance with Archie's law (Scenario 2), but hysteresis separation increases as the immobile domain becomes a more dominant fraction of total porosity at the local scale. Note the fluid conductivity of the flush injectate is lower than that of the background pore fluid so the hysteresis does not “close,” consistent with the field experiment.

4.2. Tracer Test Fluid and Bulk Conductivity

[22] The injected tracer was observed at all well locations, except for the σf sampling ports that did not function at the 4.6 and 5.2 m depths along profiles E2 and E6, and the total suite along E7. The magnitude of tracer response was variable, and many locations were not seemingly in the direct path of the injected plume. The tracer time series (both σb and σf) at wells E15.2, E3, E4, and E8 showed strongly truncated tailing due to relatively slow loading and subsequent transition to flush; therefore, the 27 day experimental data collection did not run long enough to capture the necessary data at these locations. The tracer tail commonly contains the majority of the information regarding effective flow path–scale mass transfer, and is needed to investigate full hysteresis patterns, therefore these locations also were not modeled. The locations along well E5 were better aligned with the direct path of the inject plume and showed both the tracer loading and flush dynamics, but the timing of tracer appearance was inexplicably truncated by several days. This may have been caused by preferential infiltration from the surface along the well casing during a rain event, but as the cause was inconclusive, it was judged better not to force the model inputs to account for this difference, and the well locations were not simulated. Therefore, the remaining five locations, which showed strong tracer and flush response (E14.6, E14.8, E15.0, E24.8, and E25.0) at a Euclidean distance of 2 m from the injection, were modeled using the effective and local mass transfer framework discussed above.

[23] Tracer plateau σf at all modeled downgradient locations was similar, ranging from 6682 to 6976 μS cm−1, and was comparable to the mean preflush injection tank σf of 6900 μS cm−1, indicating that equilibrium concentration was reached in the mobile domains, and there was negligible dilution by unlabeled groundwater (Figure 4). All σf BTCs showed strong visual evidence of anomalous transport, or a shift in mass from early to late time, except E14.6, which had very little tailing (Figures 4 and 5). Tailing during the flush (except E14.6) was fit with exponential curves having r2 > 0.94, confirming the appropriateness of the DDMT model. The σb signals at each location showed minimal change after day 15 (Figures 4 and 5), indicating equilibrium had been reached or very slow mass transfer. All σb tailing during flush was exponential, even at the E14.6 location, and fit with models with r2 > 0.95. Unlike σf, σb values near the end of the injection phase differed considerably, ranging from 700 to 2500 μS cm−1, indicating fundamental differences in total porosity of the respective support volumes.

Figure 4.

Fluid and bulk conductivity observations and optimized simulations are shown in respective columns, with the lettered rows corresponding to profile locations: (a) E14.6, (b) E14.8, and (c) E15.0.

Figure 5.

Fluid and bulk conductivity observations and optimized simulations are shown in respective columns, with the lettered rows corresponding to profile locations: (a) E24.8 and (b) E25.0.

[24] Experimental σb was plotted versus σf to identify the hysteresis diagnostic of DDMT (Figure 6). The conductivity of the flush water (∼292 μS cm−1) was lower than that of the background, native pore water (∼1500 μS cm−1), thus affecting steady state σf and σb and causing small differences in pre- and postexperiment conditions; this difference manifests as accentuated hysteresis in the presence of mass transfer and nonclosure of the hysteresis loop. Hysteresis separation was notable at the E14.6 and E15.0 locations, but not at E14.8 (not shown), which was intermediate to these locations in the vertical. Location E14.6 in particular, which had the highest plateau σb, showed extreme hysteresis, with a separation between the rising and falling limbs of approximately 900 μS cm−1. Similar to E14.8, the relationship of σb to σf was essentially linear at both locations along the E2 profile.

Figure 6.

Hysteresis patterns between σb and σf, simulated and observed, along the (a) E1 profile that showed prominent hysteresis separation (E14.8, which did not have strong hysteresis, not shown) and (b) E2 profile, which showed minimal hysteresis separation. Although hysteresis patterns were not explicitly fit during parameter optimization their structure was well simulated in all cases.

4.3. Model Calibration

[25] The calibrated model closely matches the observed σb and σf characteristics at all five sites; the normalized SOSWR ranges from 4.2 μS cm−1 at E15.0 to 20.1 μS cm−1 at E14.6, and the accuracy of fits did not seem to show bias to the degree of σb or σf tailing (Figures 4 and 5 and Table 2). We note that the σb − σf hysteresis patterns were not directly fit during the parameter optimization procedure, rather, the calibration target was to fit the time histories (BTCs), not the hysteresis pattern; however, the fitting process to the combined σf and σb observations effectively reproduces the general shapes and orientation of hysteresis (Figures 4-6). The notable hysteresis at E14.6 and E15.0 showed some structure during the injection limb that was primarily related to the variable tracer input concentration (supporting information Figure S2), and these general shapes were also well matched by the model, which used the same variable input signal for all simulations, as discussed in section 2.2. Specific mass transfer parameter values varied strongly in space, and between the effective and local scales (Table 2). The 95% parameter confidence interval width for each parameter was inversely correlated to parameter sensitivity, which is discussed in section 3.3. The precision of parameter estimates as evaluated by the UCODE 95% confidence interval was highest for the flow path scale nimmob1 and α1 when there was notable σf tailing and highest for the local scale nimmob2 and α2 when hysteresis was observed.

Table 2. Optimized Effective and Local Mass Transfer Parameters With 95% Confidence Intervals in Italics, the Combined Sum of Squared Weighted Residuals (SOSWR) to the σb and σf Model Simulations Normalized by Number of Observations, and Estimated Effective and Local Diffusive Length Scales Based on Respective Mass Transfer Parameters Location
 Effective Mass TransferLocal Mass TransferD (m2 d−1)SOSWRD* Length Scale
nmob1nimmob1α1 (d−1)nmob2nimmob2α2 (d−1)Effective (cm)Local. (cm)

[26] The optimized nmob for all locations (0.10–0.31) reflected the observed differences in median tracer transport times, which ranged from 1.0 to 3.1 days (Table 2). The effective nimmob1 ranged from 0.03 at E14.6, where little σf tailing was observed, to 0.32 at E14.8 and E15.0, which showed a large tracer shift to late time (tailing). The adjacent aquifer locations of E14.6 and E14.8 displayed nearly opposite effective and local mass transfer relations. Despite the lack of mass transfer observed in the σf signal at E14.6, the corresponding σb created a strong hysteresis separation and resulted in the largest nimmob2 estimate of 0.37, which was accompanied by a relatively small nmob2 of 0.11. Conversely, although there was substantial modeled effective mass transfer for E14.8, hysteresis separation was not observed, and the local mass transfer coefficient was very small (0.002 d−1) and imprecise. Additionally, nmob2 was precisely estimated at 0.19, similar to the effective value.

[27] The simulation for location E15.0 shows strong mass transfer in the σf signal due to high effective nimmob1 and α1, but the precisely estimated local parameters differ markedly, particularly the low nmob2 (0.06) and moderate α2. This unique parameter combination resulted in hysteresis, which was subdued compared to E14.6. The two adjacent locations along the E2 profile had statistically identical local and effective mass transfer parameters, except for nmob2, which was 60% higher at E24.0. Mass transfer parameters for all models were related to an open water diffusive length scale using equation (5). The length scale estimates are reported in Table 3 and range from 1.2 to 2.4 cm for effective parameters, and 2.0 to 9.4 cm for local parameters.

Table 3. The Composite Scaled Sensitivities for all Estimated Model Parametersa
 Effective Mass TransferLocal Mass TransferD
  1. a

    Locations that showed appreciable hysteresis separation are shown in bold, asterisks are used to designate the most sensitive parameter in the simulation.


4.4. Sensitivity Analysis

[28] The CSS of all model parameters to the paired σb and σf observations is in the desired range of being within 2 orders of magnitude of the most sensitive parameter in respective sets, and greater than 1.0 (Table 2), with the exception of D at E14.6, so that term was neglected. In addition, although m was not estimated using UCODE_2005, it was found to be one of the most sensitive parameters in each simulation. The ordinal ranking of parameter sensitivity varied by location and the degree of both effective and local mass transfer, but generally, the porosity parameters were more sensitive than the exchange coefficients and D. The location which had little σf BTC tailing, E14.6, shows the lowest sensitivity to effective mass transfer parameters; whereas locations that show only weak hysteretic relations between σf and σb (E14.8, E24.8, E25.0) have the lowest sensitivity to nimmob2 and α2. As reflected in the respective effective mass transfer parameter estimates, locations that share the same σf sampling port have similar sensitivities to these parameters. Interestingly, in cases of little hysteresis, nmob2 is highly sensitive to σb. Conversely, the two locations that showed stronger hysteresis (E14.6, E15.0) have nimmob2 as the most sensitive parameter in their respective sets, with corresponding α's the most sensitive of local transfer coefficients.

[29] As expected, in general, effective parameters (nmob1, nimmob1, α1, D) are more sensitive to the informational content of σf whereas local parameters (nmob2, nimmob2, α2, D) are most sensitive to σb. However, there is some overlap, e.g., effective parameters showed some sensitivity to σb, particularly nmob1 and nimmob1. The observed hysteresis at the most effectively modeled location of E15.0 (based on SOSWR) is used to illustrate the sensitivity of effective and local mass transfer parameters to specific temporal observations (Figure 7). In this analysis, the effective mass transfer parameters are sensitive to the tailing of the signal, expressed as “hot” colors at later times in the hysteresis pattern. In particular, nimmob1 is most sensitive to the late-time σf, but also shows sensitivity to the informational content of the σb tail, shown here as larger diameter points. In addition to the σf tail, nimob1 is most sensitive to early time arrival of both the σb and σf BTCs. The α1 is only sensitive to σf, with hot spots at the initial and late-time falling BTC limb. None of the effective parameters showed sensitivity to the plateau concentrations of either σb or σf.

Figure 7.

The absolute value of dimmensionless scaled sensitivity (DSS) for each mass transfer parameter to both σb and σf obervations in the hysteresis pattern recorded at E15.0. The nmob1 plot displays the elapsed time in days that each sample was taken. The size of each dot indicates sensitivity to σb observations, while the color indicates senstivity to σf observations; note the larger color scale for all effective parameters in row 1 due to much higher sensitivies to σf.

[30] The local mass transfer parameters at E15.0 are primarily informed by σb (Figure 7). The nimmob2 is the most sensitive parameter, effective or local, and shows a relatively even distribution of sensitivity to σb along the entire hysteresis, including at plateau. Additionally, this local immobile porosity is sensitive to the late-time σf, in a similar but subdued manner compared to its effective upgradient counterpart. The nimmob2 also is sensitive (although to lesser degree) to the σb plateau, but shows essentially no sensitivity to the σf data. Finally, α2 was informed by the entire σb signal, though overall sensitivity was low, with some minimal sensitivity to the σf tail in a similar bimodal pattern as α1.

[31] To investigate how long it would take to achieve σb plateau (full hysteresis) based on locations with very small modeled α2 (e.g., 0.003 d−1), the E25.0 simulation was rerun using 20, 40, 60, and 100 day injection times (Figure 8). The longer simulation results indicate that the very slow estimated exchange processes would necessitate an approximate 100 day injection to reach tracer plateau at σb. To investigate the lower bounds of what α2 would be needed to achieve tracer σb plateau within the confines the field injection, the original 0.003 d−1 value was iteratively increased until equilibrium was reached within 15 days; the result was almost one full order of magnitude greater at 0.019 d−1.

Figure 8.

(a) Simulated mobile and immobile breakthrough curves of a synthetic 100 day injection at E25.0 where tracer equilibrium is reached, and a comparison of the resulting local Damköhler numbers (DaIL) with the original 15 day injection duration. (b) Simulations of observed hysteresis at E25.0 at successively longer injections times using the mass transfer parameters optimized to the 15 day field injection. The widening of hysteresis separation with injection time illustrates how very slow exchange rates with the immobile domain may be better characterized by longer injection, effectively increasing the “window of detection” of the method.

5. Discussion

[32] The discussion proceeds from the implications of the synthetic modeling scenario results to a comparison and evaluation of the mass transfer parameters determined across the Naturita site. Finally, we discuss the broader implications of this novel modeling effort and introduce a new Eulerian-based Damköhler number to help plan experimental tracer injection durations, which maximize the applicability of the method to investigating local-scale exchange processes.

5.1. Synthetic Modeling Scenarios

[33] Forward modeling of the three type scenarios illustrates how variation in mass transfer at both the effective flow path and local scales results in characteristic BTC and hysteresis shapes (Figure 3). Furthermore, these scenarios show how tailing in σf is primarily controlled by effective flow path–scale mass transfer processes upgradient of the observation point, while the σb plateau and hysteresis separation result from local mass transfer processes at the observation point, or within the geoelectrical representative volume. For Scenario 2 with negligible nimmob2, the σb plateau is low as there is less total porosity (0.21) to load with the conductive tracer, compared to Scenario 3, where total porosity is large (0.55) due to a sizeable immobile domain. Also for Scenario 2, a linear relationship was observed between σb and σf as would be predicted by Archie's law (equation (3)) in the absence of mass transfer, even though the moderate effective flow path–scale mass transfer results in a heavy σf tail. With negligible local mass transfer, bulk conductivity tracks linearly with fluid conductivity (Scenario 2), but hysteresis results from local mass transfer with the immobile domain (Scenarios 1 and 3) as has been shown previously [Singha et al., 2007; Day-Lewis and Singha, 2008]. The tracer flush will amplify hysteresis separation if mass transfer occurs, but the flush alone will not create hysteresis in the absence of mass transfer (Figure 3c). In actual field-scale transport, all mass transfer parameter values may vary simultaneously, creating more complicated patterns than those presented by these three type scenarios, but the basic characteristic shapes of effective and local mass transfer still apply.

5.2. Dual-Domain Mass Transfer at Naturita

[34] The exponential decline in conductivity during the flush observed in σf and σb, combined with accurate simulations of both sets of observations, indicates the DDMT model with first-order exchange coefficient is an appropriate description of nonreactive solute transport for the Naturita site (Figures 4 and 5). This finding is further supported by the reasonable simulated fits to observed hysteresis between σf and σb, patterns that were not explicitly fit during the UCODE_2005 optimization process (Figure 6). Although tracer was observed at all monitoring locations, the injection BTC was only fully captured at a subset of these, but this is not surprising given the history of tracer tests at the site [Curtis and Davis, 2006]. This spatial tracer distribution indicates complex flow path dynamics in the subsurface, some of which we were able to describe with both effective-flow path and local-scale mass transfer parameters.

[35] It is clear from multiple lines of evidence collected during this study that mass transfer parameters vary spatially across this site. Even before the model was applied, there were strong physical indications of this variability, which manifested as differences in tailing of the σf signals, variable σb “plateau” values, and disparate σb versus σf hysteresis patterns. Contrasting patterns occurred even at adjacent locations along the same vertical profiles. For example, the σf signal observed at E14.6 showed little deformation from the input tracer signal similar to the synthetic Scenario 3, while strong tailing was observed at E14.8 and E15.0 along the same profile (Figure 4). The differences in flow path–scale transport of the solute tracer translated to disparate optimizations of effective transport parameters, which were shown to be particularly sensitive to the informational content of σf (Figure 7). Effective immobile porosity was only 0.03 for the subdued tailing at the E14.6 location, but estimated to be 0.32, with much higher respective rate coefficient, at the adjacent profile locations with strong σf tailing. This relationship of tailing to strong mass transfer is consistent with the synthetic examples and the laboratory findings of Swanson et al., [2012] in the presence of known immobile pore space.

[36] One of the more striking aspects of the E1 profile parameter estimates is the differences in local-scale processes between observation locations and their contrast with respective effective upstream transport. The σb BTC at E14.6 reached a value of 2500 μS cm−1, much higher than the 700–1160 μS cm−1 observed at E15.0 and E14.8, respectively. Assuming negligible surface conductance and tortuosity differences and relatively constant m (determined experimentally at 1.3), the differences in σb maximum and background equilibrium values correspond directly to varied total porosity, with greater σb during the late-time tracer injection, indicating greater local porosity. Furthermore, the hysteresis at E14.6 showed the largest separation between injection and flush of any modeled location. This means that at E14.6 there was high total porosity (estimated at 0.48), and much of this was immobile (77%), as shown by observed hysteresis and the optimized parameter values. This type of porosity distribution is indicative of a clay deposit [McWhorter and Sunada, 1977], and supports previous observation of clay-size particle pockets. An alternate explanation for the relatively large immobile porosity at location E14.6 is formation disturbance during well installation and poor collapse. Interestingly, if this porosity reflects the true undisturbed well field condition, the most dominant local-scale immobile zone size was found along a flow path with negligible effective mass transfer, again similar to synthetic Scenario 3.

[37] The opposite pattern was observed at E14.8, which showed little hysteresis and subdued local mass transfer after a strong effective mass transfer signal (Figures 4 and 5 and Table 2). Here, the wide confidence intervals around the local parameters and lower sensitivities indicate there was less immobile zone information contained in the observations and that local porosity was dominated by the precise estimate of nmob2 (0.19). The vertical variability in local mass transfer along the E1 profile is apparent, because the E14.8 location separates E14.6 and E15.0 in space, and the latter together showed the largest hysteresis separations of any modeled locations (Figure 6). The optimized estimates of nmob2 at E14.6 and E15.0 were relatively precise and relatively low (0.06–0.11) (Table 2), serving to partially balance the corresponding higher nimmob2 (0.12–0.37), and as discussed above may indicate the presence of clay. The optimized local mass transfer rate coefficients spanned 3 orders of magnitude between the three E1 locations, with α2 at E14.6 and E15.0 at 0.166 and 0.030 d−1, respectively, but likely not greater than 0.004 d−1 at E14.8 (Table 2). If the mean optimized α2 at E14.8 of 0.002 d−1 is assumed, immobile residence time at the local scale would be >100 days, suggesting there may be pockets of substantial solute retention and slow release at the Naturita site.

[38] Similar to E14.8, locations E24.8 and E25.0 showed minimal hysteresis separation between the tracer injection and flush (Figure 6). Optimized effective parameter estimates were identical along this profile, showing relatively balanced porosities and substantial mass transfer. And in a similar fashion to synthetic Scenario 2 (Figure 3), moderate mass transfer at the effective scale contrasts with “negligible” mass transfer at the local scale, where both field locations had a low nimmob2 (0.06/0.07) and modest α2 (0.003 d−1). These values were again quite similar to E14.8; thus, there is commonality among some local mass transfer parameters at the site, both in the vertical and horizontal. The stark difference between the E2 local parameters is in nmob2, which is 60% greater at E24.8. Both locations had highly sensitive nmob2 to σb (Table 3), resulting in precise parameter estimates, the difference here stemming from varied maximum σb (Figure 5), which as discussed above indicates a difference in total local porosity.

[39] The optimized α and nimmob parameter estimates indicated effective flow path–scale mass transfer occurred over shorter diffusive lengths (as expressed in open water) compared to local mass transfer processes. During the installation of the wells, we observed median stratified sediment textures ranging from fine (<2 mm) to cobbles, characteristic of mixed fluvial deposits. Effective flow path lengths (1.2–2.4 cm) may indicate mobile-immobile exchange over length scales corresponding to individual grains, whereas the larger local length scales (2.0–9.4 cm) may indicate more macroscale exchange with pockets of fine-grained heterogeneity, which is very slow and not observable in σf data.

[40] In summary, all effective parameters, based on strong σf tailing during flush, indicated mass transfer was an important process at the flow path scale, except at E14.6, where little tailing was observed. Many local parameters showed significant variation both from their effective upgradient counterparts and from adjacent locations in the vertical, with rate coefficients spanning 3 orders of magnitude. As discussed below in section 5.4, a longer tracer injection could serve to confirm or rule out the very slow local mass transfer processes (e.g., α2 = 0.003 d−1) estimated for three locations.

5.3. Method for Effective and Local Parameter Estimation

[41] Commonly, mass transfer parameters are calibrated to σf alone [e.g., Feehley et al., 2000], which permits the estimation of averaged flow path–scale processes. Because conventional σf data are not directly sensitive to the immobile zone at the point of observation, this representative upgradient flow path length, which is balanced by advective velocity, must be sufficient to allow enough exchange between the two domains to inform immobile parameter estimation (optimal Damköhler number) [Bahr and Rubin, 1987]. At this flow path scale, the dominant mass transfer mechanisms expressed in the σf BTC may not well represent the true variability in mass transfer parameters. This may lead to poor predictions of contaminant retention, transport, and remediation response, particularly if relatively large immobile porosities and slow exchange rates are missed. In this study, a parameter sensitivity analysis was used to show that mass transfer parameters at both the effective flow path and local scale could be simultaneously determined when σb observations are included into the parameter optimization process. For the first time, a sample of the true local variability in field-scale solute transport mass transfer properties has been described.

[42] As detailed above, the resulting description of mass transfer at the Naturita site was heterogeneous. In fact, variability along some single flow paths was so high (E14.6, E14.8, E15.0), indicated by disparate effective and local parameter estimates, that the models would not converge to the combined σb and σf observations when just one set of effective mass transfer parameters was considered; i.e., there was no combination of parameters that could reasonably match observed σb and σf patterns simultaneously, because they describe different-scale processes. When the local parameters were included in the optimization process, both sets of conductivity observations were effectively simulated, resulting in reasonable reproduction of the observed hysteresis patterns. Synthetic modeling examples were used to show how multiscale variability in mass transfer could explain the seemingly contrasting σb and σf patterns. If E14.6 were modeled conventionally, matching only the observations during the optimization process, we would assume negligible mass transfer along that flow path, yet the physical explanation, based on previous column experiments [Swanson et al., 2012], for the strong hysteresis between σb and σf at that location is the existence of a substantial immobile zone. The optimized 12× increase in nimmob from the flow path to local scale, combined with an order of magnitude increase in rate coefficient, directly illustrates the dominant local immobile processes that would have gone undetected using traditional DDMT characterization methods.

[43] The “local” scale for this experiment refers to a sample volume approximately 22 cm long, which is governed by the spacing of the inner electrodes (M and N). The distance across this volume would be too short to conduct a traditional tracer analysis using σf data alone based on a Damköhler number assessment (equation (6)) and is approximately one ninth the length of the effective flow paths used here. The elongated representative volume is oriented in the vertical, roughly normal to the effective flow path direction (Figure 2b); the intersection of the two is where the local heterogeneity was included into the 1-D model by independently estimating mass transfer parameters at that specific node. However, heterogeneity may exist within this 22 cm long sample volume that was obscured through this technique. Future experiments with relatively short effective flow path lengths can be designed with closer electrode spacing, so the “local” scale is more disparate from the flow path scale, and local heterogeneities more precisely described.

[44] For two pairs of aquifer locations (E14.8, E15.0, and E24.8, E25.0), the same σf sampling port was used because it was located in the overlap of their electrical representative volumes for σb (Figure 2). Hypothetically, as effective parameters are most sensitive to σf, each set of pairs should have analogous effective parameter estimates, even though the σb observations at each location were quite different. We found this rule to hold for all effective parameters, illustrated by the overlap of 95% confidence intervals between analogous parameters, indicating they were statistically identical (Table 2). Similarity was also borne out in the sensitivity analysis, which found comparable sensitivities for analogous parameters (Table 3). The identical effective parameter results for locations that shared σf observations instill confidence in the parameter optimization and sensitivity analysis using UCODE_2005 and indicate that the inclusion of σb observations is not somehow biasing the estimation of effective parameters.

[45] The sensitivity analysis also indicated that the parameter m, which was determined as the average value from the field data, was consistently one of the most sensitive parameters in each simulation. Care must therefore be exercised when determining this exponent, as the estimated value may affect optimized estimates of mass transfer parameters (equation (3)). Alternative averaging procedures could be used in place of arithmetic averaging. Although the bicontinuum model has been used successfully in several studies, more work is required to fully evaluate the range of applicability of the model to DDMT. Depending on the internal connectivities of the mobile and immobile domain, other averaging procedures of m (e.g., geometric, harmonic) could prove more effective. Given the high information content of geoelectrical data for inference of DDMT, it may also be possible to infer the form of the petrophysical model.

[46] We found that when hysteresis between σb and σf is observed, there is more information regarding the local immobile and exchange parameters within the data set, which supports previous work [Day-Lewis and Singha, 2008]. Lack of hysteresis can result from two physical reasons: (1) there is no substantial immobile porosity at the local scale, or (2) mass transfer processes at the local scale are outside the limit of detection of the experiment, due to inappropriate mobile advective velocities (very slow) or a tracer injection that was too short to capture very slow exchange rates. Recent work has shown that hysteresis can result within a certain window of mass transfer conditions [Singha et al., 2007, 2008; Day-Lewis and Singha, 2008; Swanson et al., 2012]. The Naturita data set indicates that hysteresis between σb and σf can be observed over approximately 2 orders of magnitude in α2, which is a somewhat larger range than was predicted by Day-Lewis and Singha [2008]. These differences may result from the fact that we allowed local parameters, including nimmob2, to vary simultaneously using UCODE_2005. When the ratio of nimmob2 to nmob2 is high, hysteresis may be observed with an α2 ranging from 5.0 to 0.01 d−1 during an injection of 15 days under these conditions. As this definition suggests, and has been observed elsewhere [Haggerty et al., 2001; Haggerty, 2004; Day-Lewis and Singha, 2008], estimates of α2 are sensitive in part to the time scale of the injection. Assuming reasonably fast flow path–scale pore water velocities, such as the 1.0 m d−1 found for location E15.0, α2 > 5.0 d−1 results in rapid loading of the immobile zone and therefore little offset in cm and cim. The consequence is that there is essentially no “immobile zone” relative to solute transport as the loading, and subsequent unloading, converges on advective transport, and the system essentially functions as a single domain [Zheng and Wang, 1999]. But as we show in Figure 7, hysteresis may not be initially observed at very small α2, because the extremely slow loading of the immobile zone looks analogous to a lack of immobile zone at “early time.”

[47] The parameter optimization reveals the presence of small immobile zones and slow mass transfer exchange at locations that showed little (but not zero) hysteresis. Three locations (E24.8, E24.8, and E25.0) did not appear to achieve σb equilibrium. We attribute this behavior to small α2 at those locations, i.e., locally slow mass transfer. The optimized parameters for location E25.0 were used to determine what injection length would be necessary to achieve σb equilibrium at that location, and thus produce a “full” hysteresis loop over the course of the experiment. Additionally, the optimized α2 was iteratively increased to determine the threshold α2 that would facilitate σb equilibrium during the 15 day tracer injection implemented in this study.

[48] Figure 8a illustrates that if we assume the α2 of 3 × 10−3 for the E25.0 location is accurate, it would have taken approximately 100 days of injection to fully load the corresponding immobile zone. Furthermore, during our 15 day injection, we could only expect σb to reach equilibrium plateau for α2 = 2 × 10−2 d−1 or higher. Therefore, the window of detection for α can be effectively increased on the low end by running longer tracer injections. It does not seem necessary to achieve full σb plateau, however, as there will be an exponential decrease in hysteresis separation with time as Cim approaches Cm. Figure 8b depicts that by 40 days, two thirds of the hysteresis that would result from α2 = 3 × 10−3 d−1 is achieved, providing abundant information to estimate α2 more precisely. Furthermore, if no hysteresis is observed at these longer time scales, mass transfer on the low end of α effectively can be ruled out.

5.4. Local Damköhler Number

[49] Various design criteria must be considered when investigating effective- and local-scale mass transfer simultaneously; particularly as previous work has indicated that slow exchange processes may keep the immobile domain at disequilibrium with the mobile domain even at very large times during a constant rate solute injection [Haggerty et al. 2001], directly affecting local hysteresis patterns. The effective flow path scale fundamentally depends on a Lagrangian-viewpoint Damköhler number (DaIE), as is commonly presented in transport literature [Bahr and Rubin, 1987; Haggerty, 2004];

display math(6)

where cumulative exchange with the immobile zone along a 1-D flow path under a given set of mass transfer parameters depends on the time scale of advection (tad) or the ratio of flow path length to flow path velocity. If we shift to an Eulerian viewpoint, we can use the local Damköhler number (DaIL) concept to evaluate whether our injection time scale (duration) was appropriate to investigate local mass transfer processes, where exchange with the local immobile zone under a given set of mass transfer parameters depends on the time scale of mobile zone tracer loading (tld) as

display math(7)

[50] For simplicity, we can take tld to be the time over which σf BTC plateau is maintained, assuming that local mobile tracer plateau is reached in a relatively short amount of time relative to the loading of the local immobile zone. Note, here we use the inverse of the original DaIE porosity ratio to scale the DaIL by the rate of change in Cimmob, similar to the transport-governing equation ((4b)), not the total tracer mass that enters nimmob as presented in equation (5). Therefore, a faster rate of change in Cimmob results in a lower time to equilibrium (full hysteresis) and a higher DaIL for a given set of conditions. It should also be clarified that the DaIL is designed for use with the MT3DMS version of α, which differs from some other related mass transfer coefficients by a factor of nimmob−1 [Ma and Zheng, 2011].

[51] Assuming mass transfer is occurring at the local scale, values of DaIL << 1 indicate the tracer injection was not long enough to load the local immobile zone toward equilibrium, and hysteresis will not be observed. Conversely, DaIL > 1 indicates the tracer injection was unnecessarily long to investigate local mass transfer, and the “local equilibrium assumption” [Miller et al., 1990], or full hysteresis, will have been satisfied for some time.

[52] If we return to the example of E25.0, where we did not observe considerable hysteresis during our 15 day injection, the resulting local DaIL 0.14 (Figure 8a), while if we ran the injection for 40 days to achieve considerable hysteresis, the resulting DaIL = 0.38. If the experiment was run 100 days to equilibrium between the two domains (Figure 8a), the local DaIL = 0.95, which is similar to value of DaIL = 0.94 that is determined for a 15 day injection using α2 = 2 × 10−2 d−1, or the minimum exchange coefficient need to achieve equilibrium at E25.0 under the current experimental conditions. At location E24.8, which has analogous mass transfer to E25.0 but much larger mobile porosity, we observed slightly greater hysteresis for the same length (15 day) injection (Figure 6), and this is reflected in the larger DaIL = 0.2. The comparison between these two locations illustrates the utility in inverting the original DaIE porosity ratio to scale DaIL, as tracer concentration will rise faster is the local immobile zone when mobile porosity is relatively larger. For comparison, the DaIL for E15.0, where σb equilibrium was just achieved by 15 days, is 0.94; while DaIL for E14.6, where σb equilibrium was reached in <<15 days, is 4.5. The new Eulerian-based local DaIL should be useful to future research that utilizes chemical/geoelectrical tracer signals to resolve mass transfer and potentially in the stream environment when using tracers to define exchange between the main channel and specific zones of surface transient storage [Briggs et al., 2009], such as those found downstream of restoration structures.

[53] When hysteresis is not observed under study conditions, no matter what the explanation, the sensitivity analysis indicated important information regarding nmob2 and flow path–scale porosities can still be contained in σb observations. This informational content is due to the sensitivity of bulk conductivity to porosity, as predicted by Archie's law, and the sensitivity of flow path parameters to the timing of the σb BTC. This is an advantage of our model-calibration approach over analysis of temporal moments [Day-Lewis and Singha, 2008]. We found a large contrast in σb between the aquifer locations that did not show hysteresis separation (E14.8, E24.8, E25.0), which could be explained almost entirely by differences in nmob2, as the sensitivity of nmob2 to σb was particularly high in cases of little hysteresis. Therefore, although the characteristic hysteresis of mass transfer was not observed, important information regarding local-scale transport and mass transfer parameters can be gleaned from the timing, shape, and plateau of σb. Although nmob2 does not describe immobile zone solute retention, varied nmob2 will have a strong effect on local transport; for example, the enhanced nmob2 at E15.0 increased local mobile pore water velocities by 60% compared to E14.8, which may be particularly important to biogeochemically relevant parameters such as contact time.

6. Conclusions

[54] The variability between effective-upgradient flow path and local-scale DDMT parameters has been described using field data for the first time. The major findings of this study are: (1) both σb and σf observations were closely matched by regression modeling using UCODE_2005 and a multiscale DDMT model, with generally high parameter sensitivity; (2) strong variability in mass transfer was observed throughout the well field, including along the same flow path, with α spanning 2 orders of magnitude; (3) hysteresis patterns are a local process, and hysteresis structure was reproduced well in all cases, even though these patterns were not explicitly fit during the parameter optimization process; (4) as indicated in previous work, greater hysteresis separation between the injection and flush limbs of BTCs yielded greater local-scale mass transfer and higher sensitivity of the related parameters, but σb observations were still useful for determining local porosity and optimizing effective parameters in cases of little hysteresis; and (5) the method's “window of detection” of local-scale mass transfer can be increased using longer tracer injections to include very slow local mass transfer processes, which may influence long-term solute retention and release, and experimental design can be evaluated with a Eulerian-based local Damköhler number.

[55] Given the combination of tracer and geoelectrical data collected during an ionic tracer test, it is possible to discriminate between the effects of upgradient-flow path and local mass transfer. The ability to characterize spatially variable mass transfer parameters at the field scale is a substantial advance, particularly with respect to assessing the true variability in local-scale mass transfer parameters, which are now identifiable using solute tracer techniques for the first time because a representative length (e.g., as determined with DaIE) is no longer needed. Improved estimation of immobile porosity distribution should facilitate more accurate estimates of the volume of immobile contaminant present in the field and, combined with local mass transfer rates, the time it may take to remediate. Additionally, our approach provides information for parameterization of field-scale multirate mass transfer models where the rate distribution is directly informed by measurements of the true local-scale mass transfer. Detailed characterization of mobile-immobile mass transfer should provide improved understanding of the long-term fate of uranium at contaminated sites such as Naturita.


[56] The authors gratefully acknowledge field assistance from Emily Voytek, Eric White, Peter Joesten, and Cian Dawson. This work was supported by U.S. Department of Energy Subsurface Biogeochemical Research Program grants DE-SC0003681 and DE-SC0001773 and the U.S. Geological Survey Toxic Substances Hydrology Program. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. government.