Simultaneous estimation of local-scale and flow path-scaledual-domain mass transfer parameters using geoelectrical monitoring
Martin A. Briggs,1 Frederick D. Day-Lewis,1 John B. T. Ong,1 Gary P. Curtis,2 and John W. Lane1
Received 21 December 2012; revised 24 June 2013; accepted 2 July 2013; published 13 September 2013.
[1] Anomalous solute transport, modeled as rate-limited mass transfer, has an observablegeoelectrical signature that can be exploited to infer the controlling parameters. Previousexperiments indicate the combination of time-lapse geoelectrical and fluid conductivitymeasurements collected during ionic tracer experiments provides valuable insight into theexchange of solute between mobile and immobile porosity. Here, we use geoelectricalmeasurements to monitor tracer experiments at a former uranium mill tailings site inNaturita, Colorado. We use nonlinear regression to calibrate dual-domain mass transfersolute-transport models to field data. This method differs from previous approaches bycalibrating the model simultaneously to observed fluid conductivity and geoelectrical tracersignals using two parameter scales: effective parameters for the flow path upgradient of themonitoring point and the parameters local to the monitoring point. We use regressionstatistics to rigorously evaluate the information content and sensitivity of fluid conductivityand geophysical data, demonstrating multiple scales of mass transfer parameters cansimultaneously be estimated. Our results show, for the first time, field-scale spatialvariability of mass transfer parameters (i.e., exchange-rate coefficient, porosity) betweenlocal and upgradient effective parameters; hence our approach provides insight into spatialvariability and scaling behavior. Additional synthetic modeling is used to evaluate the scopeof applicability of our approach, indicating greater range than earlier work using temporalmoments and a Lagrangian-based Damköhler number. The introduced Eulerian-basedDamköhler is useful for estimating tracer injection duration needed to evaluate masstransfer exchange rates that range over several orders of magnitude.
Citation: Briggs, M. A., F. D. Day-Lewis, J. B. T. Ong, G. P. Curtis, and J. W. Lane (2013), Simultaneous estimation of local-scaleand flow path-scale dual-domain mass transfer parameters using geoelectrical monitoring, Water Resour. Res., 49, 5615–5630,doi:10.1002/wrcr.20397.
1. Introduction
[2] Anomalous transport behavior, including solute con-centration rebound after pumping stops and long tailing,has been noted in numerous field and laboratory experi-ments [Feehley et al., 2000; Harvey and Gorelick, 2000].Such behavior is a principal control on the efficiency of aq-uifer remediation [Harvey et al., 1994] and water resourcemanagement using aquifer storage and recovery [Culkin etal., 2008]. Several mathematical models have been pro-posed to explain such behavior, including dual-domainmass transfer (DDMT) [van Genuchten and Wierenga,1976], multirate mass transfer [Haggerty and Gorelick,
1995], fractional dispersion [Benson et al., 2000], and con-tinuous time random walk [Berkowitz et al., 2006].Although these models can reproduce observed transportbehavior, identification of appropriate model parameters(e.g., mass transfer rate coefficient) is impeded by the lackof experimental methods that can directly interrogate theimmobile pore space. Commonly, model parameters areidentified by history matching and model calibration to mo-bile fluid tracer concentrations [Singha et al., 2007].Because the occurrence of anomalous-transport phenomenadepends strongly on experiment time scales [Haggerty etal., 2001], and the relative rates of advection and diffusion,parameters estimated based on a single flow path averagedexperiment may not prove effective under differentconditions.
[3] Recent work at pilot field [Singha et al., 2007] andlaboratory scales [Swanson et al., 2012] has demonstratedthat anomalous solute transport has an observable geoelec-trical signature that, in principle, can be exploited todirectly determine mass transfer parameters (i.e., rate coef-ficient, mobile porosity, and immobile porosity)[Day-Lewis and Singha, 2008]. Whereas, fluid samplingpreferentially draws from the mobile domain, electricalmeasurements are sensitive to solute tracer in both mobileand immobile domains. The relation between bulk and fluid
Additional supporting information may be found in the online version ofthis article.
1Office of Groundwater, Branch of Geophysics, U.S. Geological Sur-vey, Storrs, Connecticut, USA.
2Water Mission Area, National Research Program, U.S. Geological Sur-vey, Menlo Park, California, USA.
Corresponding author: M. A. Briggs, Office of Groundwater, Branch ofGeophysics, U.S. Geological Survey, 11 Sherman Place, Unit 5015, Storrs,CT 06269, USA. ([email protected])
WATER RESOURCES RESEARCH, VOL. 49, 5615–5630, doi:10.1002/wrcr.20397, 2013
conductivity appears hysteretic, or time dependent, in thepresence of rate-limited mass transfer. Hysteresis betweenfluid and bulk conductivity is observed because bulk con-ductivity (mobile and immobile domains) lags behind rela-tively quick changes in fluid conductivity (mobile domain)on both the rising and falling limbs of a solute break-through curve (BTC) due to rate-limited mass transferbetween mobile and immobile pore space. The electricalsignature of mass transfer was observed first in field-experimental data by Singha et al. [2007], but only limitedsensitivity analysis was performed for that data set, and rig-orous parameter estimation was not attempted. Swanson etal. [2012] recently inferred mass transfer parameters usingparametric sweeps for controlled homogeneous columnexperiments with independently evaluated immobile poros-ity. Forward simulations of transport were run for severalthousand combinations of mass transfer parameters, andthe fit to experimental data was evaluated. The previouswork detailed above relied on either (1) analysis based ontemporal moments [Day-Lewis and Singha, 2008], whichare subject to error in the presence of measurement noise or(2) parametric sweeps [Swanson et al., 2012], which pro-vide limited insight into parameter sensitivity or informa-tion content.
[4] Here, we extend the use of electrical measurementsto inference of field-scale mass transfer parameters at boththe effective flow path and local scales by adopting amodel-calibration strategy to match both fluid conductivityand geoelectrical tracer signatures simultaneously. Ourapproach facilitates rigorous sensitivity analysis to evaluatethe information content of the geophysical data, and pro-vides insight to describe the true variability in local masstransfer parameters. We apply our approach to conservativefield tracer data from a former uranium mill tailings sitenear Naturita, Colorado [e.g., Curtis et al., 2006; Brostenet al., 2011]. At this site, the fluid sample-geoelectricalmeasurement data exhibit the expected signature of masstransfer. We calibrate a series of one-dimensional DDMTmodels, for various sampling locations, to tracer andgeophysical data using nonlinear regression. For transportsimulation, we use the code Modular Three-DimensionalMultispecies model (MT3DMS) [Zheng and Wang, 1999].For model calibration, the transport model is linked to acomputer code for universal inverse modeling (UCODE_2005) [Poeter et al., 2005] nonlinear regression package,which also performs sensitivity analysis. Finally, we intro-duce an Eulerian-based Damköhler number useful for plan-ning experiments and evaluating whether tracer injectionduration was sufficient to observe hysteresis and determinelocal-scale mass transfer.
2. Field Experiment
[5] We conducted a forced-gradient tracer experiment ata radionuclide-contaminated site near Naturita, Colorado.The tracer test described here was conducted to comple-ment a companion tracer test that focused on uranium de-sorption from the sediments in this contaminated aquifer.The desorption experiments were performed by injectinguranium-free water with a relatively low ionic strength butunder forced-gradient conditions in order to minimize mix-ing with native groundwater. In the tracer tests described
here, the same forced-gradient conditions were used butwith a high ionic strength solution to permit monitoring byconventional fluid sampling and concurrent geoelectricalmeasurements. In this section, we detail the field site, ex-perimental setup, and data acquisition.
2.1. Site Description
[6] The field experiments were conducted at a formeruranium mill site located along the San Miguel River insouthwestern Colorado, approximately 3 km northwest ofthe town of Naturita (Figure 1). The aquifer is recharged bythe river at the southeast end of the site, and dischargesback into the river along the northern end [Curtis et al.,2006], creating long flow paths of groundwater/surfacewater exchange. The aquifer is recharged during snowmeltand runoff in the spring and water elevations are generallyconstant from August until March with an average satu-rated 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 ofsand, gravel, and cobbles, with mineralogy consisting ofquartz 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 ofapproximately 1 m minimizes local recharge from the sur-face at the test site. The aquifer is underlain by the BrushyBasin Member, a fine-grained shale approximately 30 mthick. The tracer tests were conducted downgradient of themaximum observed uranium concentrations observed at thesite. The depth to water and saturated thickness of the aqui-fer varies seasonally, and was approximately 3.4 and 1.6 m,respectively, at the time of the field investigation. Addi-tional details of the overall site characteristics are presentedelsewhere [Davis et al., 2004, 2006; Kohler et al., 2004;Brosten et al., 2011], as is a description of an earlier experi-ment on uranium transport that was conducted at the tracertest site [Curtis and Davis, 2006]. The results of the currentwork provide independently determined mass transfermodel parameters which will be used to simulate observeduranium desorption.
[7] The uranium mill was in operation from the 1930’suntil its closure in 1963. Remediation efforts included re-moval of contaminated soil from the southern two thirds ofthe site between 1996 and 1998. The site has been the focusof U.S. Geological Survey research on reactive transport ofuranium (VI) for over 10 years [Curtis et al., 2004, 2006;Davis et al., 2004; Kohler et al., 2004]. Several well clus-ters were installed for previous research efforts at the site.For our experiments, a new cluster (Figure 1) was installedwith wells designed for dual-purpose fluid sampling andgeoelectrical monitoring.
2.2. Tracer Injection
[8] Ten polyvinyl chloride (PVC) fluid/geoelectricalmonitoring wells and an injection gallery were installedusing a Geoprobe direct-push rig. Each monitoring wellconsists of fifteen 2.5 cm wide, 4.1 cm diameter stainlesssteel electrodes spaced 22 cm apart, and four levels of 0.64cm diameter sampling ports spaced 30.5 cm apart, installedto a total depth of approximately 5.5 m below ground sur-face. Wells were installed a meter apart in the east-westdirection and spaced 2 m apart in the north-south direction
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(Figure 1). The injection gallery consists of nine 0.02 m di-ameter wells, installed to depths of approximately 4.5 m,with a 0.6 m screen at the bottom of each well ; injectionwells are distributed in two rows, spaced 0.5 m apart.
[9] Figure 2a shows the conceptual injection experimentas a 2-D cross section and how the experiment is reducedto a 1-D transport problem between the injection galleryand co-located fluid conductivity (�f) and bulk conductivity(�b) observation locations distributed along vertical arrays.Forced injection, at a rate of 0.9 6 0.1 L/min, was con-ducted for 27 days from 24 October 2011 to 19 November2011. A NaCl tracer solution at approximately 3.74 g NaClL�1 with mean conductivity of 6900 (SD 271) �S cm�1
was injected for 15 days and followed directly by a fresh-water flush injection of 292 (SD 4) �S cm�1 for 12 days.Although the goal of the tracer injection was a constantconcentration/rate, the reality showed some variability,which was addressed by the models in a manner describedin section 3.1. Groundwater fluid parameters (specific elec-trical conductivity, pH, and temperature) were measuredonce per day using an Orion five-Star meter; conductivitychange resulting from the tracer (Cl�) addition wasassumed to be conservative [Davis et al., 1998]. The meterwas calibrated daily for electrical conductivity and pH.Prior to sampling, sampling ports and tubing were purgedusing a peristaltic pump. All fluid conductivity measure-ments, including the injection tank, were converted to a ref-erence temperature equal to the mean groundwater sample
temperature (14.5�C), to make fluid measurements compa-rable to the geoelectrically measured bulk conductivitymeasurements.
2.3. Geoelectrical Measurements
[10] Single-borehole four-electrode resistivity profileswere collected twice daily for every monitoring well verti-cal profile. Wenner-style measurements [e.g., Telford et al.,1990] were collected using a SuperSting R8 IP (AdvancesGeosciences, Inc.) eight-channel resistivity meter. In theWenner measurement configuration, four adjacent electro-des are involved, with the exterior electrodes A and B serv-ing as current source and sink, respectively, and interiorelectrodes M and N (spaced 22 cm apart) serving as poten-tial electrodes (Figure 2b). Given the current I appliedbetween A and B and the measured voltage, DV, betweenelectrodes M and N, apparent resistivity is calculated as
�a ¼ KDV
I; ð1Þ
where K is the appropriate geometric factor:
K ¼ 4�=1
rAMþ 1
r0AM� 1
rBM� 1
r0BM� 1
rAN� 1
r0ANþ 1
rBNþ 1
r0BN
� �;
ð2Þ
where rXY is the distance separating electrodes X and Y, andr0XY is the distance separating the electrode in the symmetrical
mirror image used to account for the no-flow boundary
Figure 1. Map of the U.S. Department of Energy Uranium Mill Tailings Remedial Action site nearNaturita, Colorado, showing the locations of the former mill, ore tailings, and monitoring wells. The lay-out of the tracer site used for this study is depicted in the inset.
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condition at the Earth’s surface. Figure 2b illustrates how arepresentative volume of geoelectrical measurement betweentwo potential electrodes may encompass both mobile andimmobile pore space, and is colocated with a fluid samplingport.
[11] For comparison of fluid sampling and geoelectricalmeasurements, a petrophysical model is required. Com-monly, Archie’s law is used to relate bulk and fluid conduc-tivity [Archie, 1942], here formulated for a dual-domainmedium under the assumption of domains acting as con-ductors in parallel as [Singha et al., 2007]:
where nmob is the porosity of the mobile domain, nimmob isthe porosity of the immobile domain, �mob is the fluid con-ductivity in the mobile domain, �immob is the fluid conduc-tivity in the immobile domain, and m is the cementationexponent, which is a function of the connectivity of thepore space, assumed to be identical between the domains[Singha et al., 2007]. We determined m experimentally asthe slope of the best fit �b-�f line for background (preinjec-tion) and plateau (equilibrium) data (supporting informa-tion Figure S1). The fit was forced through the origin tohonor Archie’s law, which changed the estimated value of1.3 by a negligible amount, indicating surface conductionmay not be an important factor. The cementation exponentis within the expected range [Telford et al., 1990] and con-sistent with unconsolidated sediment. In developing theregression for m, data from sampling location E14.6 (i.e.,well E1, port at 4.6 m depth) were neglected because thisport showed anomalously high �b plateau (>2500 �Scm�1) compared to other locations’ values (mean �1000�S cm�1), and thus exerted strong leverage on the regres-sion. The use of an ‘‘average’’ m is a potential source oferror to the modeling effort, and the sensitivity of simula-
tions to the m parameter is evaluated through sensitivityanalysis described below. It was decided that the field-measured average m was preferable to determining m inde-pendently for each well location, as this might impair rela-tive comparisons of mass transfer parameters, or toestimating m through inverse modeling because of the highcorrelation with other model parameters.
3. Data Analysis Approach
[12] Our analysis approach involves model calibrationapplied simultaneously to fluid conductivity data reflectingtracer breakthrough, and geoelectrical measurements alsosensitive to the tracer, to estimate mass transfer parametersat varied scales. In this section, we detail the transport sim-ulation, model calibration, and sensitivity analysis.
3.1. Transport Simulation
[13] Groundwater flow and solute transport were simu-lated using the finite-difference model MODFLOW-2000[Harbaugh et al., 2000] combined with particle tracking inMT3DMS [Zheng and Wang, 1999] to solve the 1-D advec-tive-dispersive transport equation with DDMT:
nmob@cmob
@tþ nimmob
@cimmob
@t¼ @
@xnmobD
@cmob
@x
� �
� @
@xnmobvcmobð Þ; ð4aÞ
and
nimmob@cimmob
@t¼ � cmob � cimmobð Þ; ð4bÞ
where cmob is the mobile-domain concentration [M/L3] ;cimmob is the immobile-domain concentration [M/L3]; D is
Figure 2. Schematic diagram of (a) the field-scale tracer injection which was observed in full at a sub-set of the original monitoring points (not to scale); (b) the local-scale colocated fluid (�f) and bulk (�b)conductivity sampling and associated hysteresis resulting from mass transfer ; and (c) the 1-D numericalmodel framework partitioned into effective upgradient flow path and local-scale parameter sets.
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the hydrodynamic dispersion coefficient [L2/T] ; v is thepore water velocity [L/T] ; � is the mass transfer rate coeffi-cient [T�1]; and t is time [T]. Optimized model values of �were related to a diffusive length scale using the open waterdiffusion coefficient (D
�) for chloride (1.7556 cm2 d�1)
[Culkin et al. 2008]:
�
nimmob� D�
rl2: ð5Þ
[14] The model domain was discretized into 0.01 mcubic grid cells. The Euclidean distance between the injec-tion zone and observation locations was taken as the effec-tive flow path length for each simulation. Modelparameters for forward simulation and calibration were par-titioned into (1) effective parameters for the flow pathupgradient of the observation location (nmob1, nimmob1, �1,D) and (2) local parameters at the observation location(nmob2, nimmob2, �2) for each modeled aquifer location; thisapproach is shown in Figure 2c. Therefore, in addition tothe flow path–scale mass transfer parameters, a second setof mass transfer parameters at the observation point modelnode and the following nine grid cells was included to sim-ulate the dynamics of the geoelectrical representative vol-ume. Observed deformation of the input tracer signalthrough fluid sampling (�f) primarily results from transportand mass transfer processes along the flow path upgradientof the sampling port, and is therefore commonly summar-ized with ‘‘average’’ parameter sets that describe a Lagran-gian process of change through space and time. The �b
measurements describe change in one space (representativevolume) through time: an Eulerian process that is sensitiveprimarily to processes within the local measurement vol-ume when variability of the incoming tracer BTC has beenaccounted for. Therefore, an additional set of model param-eters local to the observation model node is included. Thisapproach allows the variability in local mass transfer prop-erties to be investigated. This variability represents an im-portant complexity that in the absence of geoelectrical datawould be obscured at the effective flow path scale. When �f
and �b tracer BTCs indicate differing mass transfer proper-ties, both observation sets can be fit with one model andreveal two scales of mass transfer.
[15] The model gradient used to simulate the forced-gradient tracer injection test was determined using Darcy’slaw to reproduce the observed median tracer transporttimes observed at the 5.0 m depth fluid port at the E1 wellassuming the hydraulic conductivity (8.64 m d�1) andeffective porosity (0.2) values determined for the area byCurtis et al. [2006]. This assumption was necessary to setthe fundamental gradient needed for the MODFLOWmodel and allowed relative differences in nmob1 to beassessed. The local nmob2 is not strongly affected by thisassumption, as it is primarily informed by the �b data set asdiscussed below. The mass transfer parameters (nimmob and�) were also less affected by this assumption, and are ofmost interest to this facet of the larger uranium transportstudy because they control much of the long-term storageand release of contaminants.
[16] The 15 day tracer injection and subsequent 12 dayflush were simulated by adjusting a constant concentrationboundary condition in MT3DMS. The goal for the experi-
mental tracer injection was constant concentration/rate, butthe immediate downstream observation point (RD) and ev-ery �f plateau indicate a variable input. This pattern oftracer input appears highly consistent across the well field.We approximate the observed pattern by representing theinput boundary condition as varying over four stress peri-ods, 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’’ scenar-ios to illustrate the BTC and hysteresis response we mightexpect from (1) moderate mass transfer at the effectiveflow path and local scales ; (2) moderate mass transfer atthe effective flow path scale, but negligible at the localscale; and (3) negligible mass transfer at the effective flowpath scale, but considerable at the local scale. One meterdomain forward models were run with a 15 day constantrate injection (7000 �S cm�1) followed by a 12 day flush(300 �S cm�1) to approximate the field experiment. For allscenarios, the mobile porosity and mass transfer rate coeffi-cients were held constant at all scales, while the immobile-domain porosity was adjusted at varied scales to createmore/less mass transfer (Table 1).
3.2. Model Calibration
[18] The mass transfer parameters (and D) were optimizedthrough nonlinear regression with UCODE_2005 [Poeter etal., 2005; Hill and Tiedeman, 2007] using MATLAB (Math-works, Inc., Natick, MA) codes in an intermediate step togenerate the necessary input text files from forward modeloutput. The �f and �b observations from each location weretypically given a coefficient of variation (CV) weighting of3%, or the general estimated error in conductivity observa-tions based on inspection of repeat and reciprocal errors.Changes in �f and �b through time were assumed to resultfrom the break through of the conservative NaCl tracer, hy-pothetical BTCs and resulting hysteresis are shown in Figure2b. It should be noted that we fit the �b and �f observationsdirectly, not a hysteresis pattern between them, because errorpropagation is compounded when considering ratios ofmeasurements; hence, an optimal fit to a hysteresis patternwould likely result in a poorer fit to the underlying BTCdata. With UCODE_2005, a sum of squared weighted resid-uals (SOSWR) objective function is minimized through theperturbation of parameter values providing the ‘‘best fit’’between weighted observations and their simulated
Table 1. Parameter Values for Three Synthetic Examples of Spa-tially Variable Mass Transfer Parametersa
aScenario 1 has moderate mass transfer at the effective flow path andlocal scales; Scenario 2 has moderate mass transfer at the effective flowpath scale, but negligible mass transfer at the local scale; and Scenario 3has negligible mass transfer at the effective flow path scale, but consider-able mass transfer at the local scale.
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equivalents [Poeter et al., 2005]. Several different startingvalues were used to help insure the models converged on atrue global minimum. Before model parameters were opti-mized, UCODE_2005 was run in ‘‘sensitivity analysis’’
mode to assess possible parameter correlation. Parameterswith correlation greater than 0.95 were estimated independ-ently during initial optimization runs, then typically com-bined later for simultaneous convergence. The criterion formodel convergence was set at the UCODE_2005 default, orless than 1% change in parameter values between iterations[Poeter et al., 2005]. A nonlinear confidence interval (CI)analysis was performed for the optimized parameter setsusing the UCODE_2005 sensitivity analysis mode.
3.3. Sensitivity Analysis
[19] To assess relative model fits, the SOSWR was deter-mined with equal observational weighting (CV¼ 0.03) andwas normalized to the total number of observations, orn¼ 46 for profile E1 and n¼ 50 for profile E2. Compositescaled sensitivities (CSS) of the optimized parameters tothe observational data (�f and �b) were estimated with aforward or central perturbation technique. Parameters thathave a CSS of less than 1.0 and (or) are more than 2 ordersof magnitude less sensitive than the most sensitive parame-ter in the simulation may be difficult to determine [Hill andTiedeman, 2007]. The dimensionless scaled sensitivities(DSS) were also determined with UCODE_2005 to assessthe importance of each observation to the optimized modelparameters.
4. Results
[20] Example synthetic forward modeling was used todefine three type scenarios of effective and local masstransfer variability. Experimental field data, model calibra-tion using nonlinear regression, and sensitivity analysis arepresented for aquifer locations where full tracer BTCs andapparent electrical conductivity time histories wererecorded. Observations are closely matched, and the cali-bration procedures yield spatially variable mass transfer pa-rameters. Sensitivity analysis is used to show parametersensitivity to both �b and �f observations.
4.1. Synthetic Modeling Scenarios
[21] Synthetic modeling of three type scenarios providesinsight into how different combinations of mass transfer pa-rameters for the effective upgradient flow path and localsupport volume manifest in observed tracer and geoelectri-cal measurements. In Scenario 1, the base-case scenario,mass transfer parameters are identical for the upgradientand local scales; in this scenario, the �f signal showedstrong tailing (Figure 3a), the �b signal had a moderate pla-teau value of 1700 �S cm�1 (Figure 3b), and hysteresisseparation between �b and �f was observed (Figure 3c).Scenario 2 was analogous to Scenario 1, except mass trans-fer at the local scale was essentially eliminated by reducingnimmob2 to 0.01 (Table 1). The resulting �f signal was nearlyidentical to Scenario 1 (Figure 3a), but the �b BTC plateauwas of much lower magnitude at 860 �S cm�1 (Figure 3b).In the absence of local mass transfer, a linear relationshipwas observed between �b and �f (Figure 3c). In Scenario 3,effective flow path–scale mass transfer was eliminated bydecreasing nimmob1 to 0.01, while local exchange wasenhanced greatly by increasing nimmob2 to 0.35. The ab-sence of upgradient mass transfer in Scenario 3 results innegligible tailing in the �f signal (Figure 3a), but enhancedlocal porosity and mass transfer results in the highest �b
Figure 3. Synthetic examples showing the tracer/geoelec-trical response to spatially variable mass transfer parametersunder three scenarios. (a) Fluid conductivity breakthroughcurves have tailing under moderate effective flow path–scalemass transfer (Scenarios 1 and 2), while the input tracer sig-nal shows little deformation in the absence of flow path–scale mass transfer (Scenario 3). (b) Although all bulk con-ductivity signals show tailing, plateau levels are governed bydiffering total porosity. (c) Hysteresis does not occur withnegligible local mass transfer in accordance with Archie’slaw (Scenario 2), but hysteresis separation increases as theimmobile domain becomes a more dominant fraction of totalporosity at the local scale. Note the fluid conductivity of theflush injectate is lower than that of the background pore fluidso the hysteresis does not ‘‘close,’’ consistent with the fieldexperiment.
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BTC plateau of 3100 �S cm�1 (Figure 3b) and strongesthysteresis separation (Figure 3c).
4.2. Tracer Test Fluid and Bulk Conductivity
[22] The injected tracer was observed at all well loca-tions, except for the �f sampling ports that did not functionat the 4.6 and 5.2 m depths along profiles E2 and E6, andthe total suite along E7. The magnitude of tracer responsewas variable, and many locations were not seemingly in thedirect path of the injected plume. The tracer time series(both �b and �f) at wells E15.2, E3, E4, and E8 showedstrongly truncated tailing due to relatively slow loading andsubsequent transition to flush; therefore, the 27 day experi-mental data collection did not run long enough to capturethe necessary data at these locations. The tracer tail com-monly contains the majority of the information regardingeffective flow path–scale mass transfer, and is needed toinvestigate full hysteresis patterns, therefore these locationsalso were not modeled. The locations along well E5 werebetter aligned with the direct path of the inject plume andshowed both the tracer loading and flush dynamics, but thetiming of tracer appearance was inexplicably truncated by
several days. This may have been caused by preferentialinfiltration from the surface along the well casing during arain event, but as the cause was inconclusive, it was judgedbetter not to force the model inputs to account for this dif-ference, and the well locations were not simulated. There-fore, the remaining five locations, which showed strongtracer and flush response (E14.6, E14.8, E15.0, E24.8, andE25.0) at a Euclidean distance of 2 m from the injection,were modeled using the effective and local mass transferframework discussed above.
[23] Tracer plateau �f at all modeled downgradient loca-tions was similar, ranging from 6682 to 6976 �S cm�1, andwas comparable to the mean preflush injection tank �f of6900 �S cm�1, indicating that equilibrium concentrationwas reached in the mobile domains, and there was negligi-ble dilution by unlabeled groundwater (Figure 4). All �f
BTCs showed strong visual evidence of anomalous trans-port, or a shift in mass from early to late time, except E14.6,which had very little tailing (Figures 4 and 5). Tailing dur-ing the flush (except E14.6) was fit with exponential curveshaving r2> 0.94, confirming the appropriateness of theDDMT model. The �b signals at each location showed
Figure 4. Fluid and bulk conductivity observations and optimized simulations are shown in respectivecolumns, with the lettered rows corresponding to profile locations: (a) E14.6, (b) E14.8, and (c) E15.0.
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minimal change after day 15 (Figures 4 and 5), indicatingequilibrium had been reached or very slow mass transfer.All �b tailing during flush was exponential, even at theE14.6 location, and fit with models with r2> 0.95. Unlike�f, �b values near the end of the injection phase differedconsiderably, ranging from 700 to 2500 �S cm�1, indicat-ing fundamental differences in total porosity of the respec-tive support volumes.
[24] Experimental �b was plotted versus �f to identifythe hysteresis diagnostic of DDMT (Figure 6). The conduc-tivity of the flush water (�292 �S cm�1) was lower than
that of the background, native pore water (�1500 �Scm�1), thus affecting steady state �f and �b and causingsmall differences in pre- and postexperiment conditions;this difference manifests as accentuated hysteresis in thepresence of mass transfer and nonclosure of the hysteresisloop. Hysteresis separation was notable at the E14.6 andE15.0 locations, but not at E14.8 (not shown), which was in-termediate to these locations in the vertical. Location E14.6
in particular, which had the highest plateau �b, showedextreme hysteresis, with a separation between the risingand falling limbs of approximately 900 �S cm�1. Similar
Figure 5. Fluid and bulk conductivity observations and optimized simulations are shown in respectivecolumns, with the lettered rows corresponding to profile locations: (a) E24.8 and (b) E25.0.
Figure 6. Hysteresis patterns between �b and �f, simulated and observed, along the (a) E1 profile thatshowed 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 explicitlyfit during parameter optimization their structure was well simulated in all cases.
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to E14.8, the relationship of �b to �f was essentially linear atboth locations along the E2 profile.
4.3. Model Calibration
[25] The calibrated model closely matches the observed�b and �f characteristics at all five sites; the normalizedSOSWR ranges from 4.2 �S cm�1 at E15.0 to 20.1 �S cm�1
at E14.6, and the accuracy of fits did not seem to show biasto the degree of �b or �f tailing (Figures 4 and 5 and Table2). We note that the �b��f hysteresis patterns were notdirectly fit during the parameter optimization procedure,rather, the calibration target was to fit the time histories(BTCs), not the hysteresis pattern; however, the fitting pro-cess to the combined �f and �b observations effectivelyreproduces 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 wasprimarily related to the variable tracer input concentration(supporting information Figure S2), and these generalshapes were also well matched by the model, which usedthe same variable input signal for all simulations, as dis-cussed in section 2.2. Specific mass transfer parameter val-ues varied strongly in space, and between the effective andlocal scales (Table 2). The 95% parameter confidence inter-val width for each parameter was inversely correlated toparameter sensitivity, which is discussed in section 3.3.The precision of parameter estimates as evaluated by theUCODE 95% confidence interval was highest for the flowpath scale nimmob1 and �1 when there was notable �f tailing
and highest for the local scale nimmob2 and �2 when hystere-sis was observed.
[26] The optimized nmob for all locations (0.10–0.31)reflected the observed differences in median tracer trans-port times, which ranged from 1.0 to 3.1 days (Table 2).The effective nimmob1 ranged from 0.03 at E14.6, where little�f tailing was observed, to 0.32 at E14.8 and E15.0, whichshowed a large tracer shift to late time (tailing). The adja-cent aquifer locations of E14.6 and E14.8 displayed nearlyopposite effective and local mass transfer relations. Despitethe lack of mass transfer observed in the �f signal at E14.6,the corresponding �b created a strong hysteresis separationand resulted in the largest nimmob2 estimate of 0.37, whichwas accompanied by a relatively small nmob2 of 0.11. Con-versely, although there was substantial modeled effectivemass transfer for E14.8, hysteresis separation was notobserved, and the local mass transfer coefficient was verysmall (0.002 d�1) and imprecise. Additionally, nmob2 wasprecisely estimated at 0.19, similar to the effective value.
[27] The simulation for location E15.0 shows strong masstransfer in the �f signal due to high effective nimmob1 and�1, but the precisely estimated local parameters differmarkedly, particularly the low nmob2 (0.06) and moderate�2. This unique parameter combination resulted in hystere-sis, which was subdued compared to E14.6. The two adja-cent locations along the E2 profile had statisticallyidentical local and effective mass transfer parameters,except for nmob2, which was 60% higher at E24.0. Masstransfer parameters for all models were related to an openwater diffusive length scale using equation (5). The lengthscale estimates are reported in Table 3 and range from 1.2to 2.4 cm for effective parameters, and 2.0 to 9.4 cm forlocal parameters.
4.4. Sensitivity Analysis
[28] The CSS of all model parameters to the paired �b
and �f observations is in the desired range of being within 2orders of magnitude of the most sensitive parameter in re-spective sets, and greater than 1.0 (Table 2), with theexception of D at E14.6, so that term was neglected. In addi-tion, although m was not estimated using UCODE_2005, itwas found to be one of the most sensitive parameters ineach simulation. The ordinal ranking of parameter sensitiv-ity varied by location and the degree of both effective and
Table 2. Optimized Effective and Local Mass Transfer Parameters With 95% Confidence Intervals in Italics, the Combined Sum ofSquared Weighted Residuals (SOSWR) to the �b and �f Model Simulations Normalized by Number of Observations, and EstimatedEffective and Local Diffusive Length Scales Based on Respective Mass Transfer Parameters Location
Table 3. The Composite Scaled Sensitivities for all EstimatedModel Parametersa
Effective Mass Transfer Local Mass Transfer
DLocation nmob1 nimmob1 �1 nmob2 nimmob2 �2
E14.6 39.2 7.8 4.1 14.9 43.1�
12.7 n/aE14.8 9.0 18.9 6.5 20.7
�7.8 2.8 1.1
E15.0 9.3 18.6 5.4 10.8 22.4�
6.0 1.4E24.8 79.9
�20.5 5.5 23.7 5.8 2.1 2.6
E25.0 81.8�
20.1 5.9 29.7 8.5 2.5 2.1
aLocations that showed appreciable hysteresis separation are shown inbold, asterisks are used to designate the most sensitive parameter in thesimulation.
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local mass transfer, but generally, the porosity parameterswere more sensitive than the exchange coefficients and D.The location which had little �f BTC tailing, E14.6, showsthe lowest sensitivity to effective mass transfer parameters;whereas locations that show only weak hysteretic relationsbetween �f and �b (E14.8, E24.8, E25.0) have the lowest sen-
sitivity to nimmob2 and �2. As reflected in the respectiveeffective mass transfer parameter estimates, locations thatshare the same �f sampling port have similar sensitivities tothese parameters. Interestingly, in cases of little hysteresis,nmob2 is highly sensitive to �b. Conversely, the two loca-tions that showed stronger hysteresis (E14.6, E15.0) have
Figure 7. The absolute value of dimmensionless scaled sensitivity (DSS) for each mass transfer param-eter to both �b and �f obervations in the hysteresis pattern recorded at E15.0. The nmob1 plot displays theelapsed time in days that each sample was taken. The size of each dot indicates sensitivity to �b observa-tions, while the color indicates senstivity to �f observations; note the larger color scale for all effectiveparameters in row 1 due to much higher sensitivies to �f.
Figure 8. (a) Simulated mobile and immobile breakthrough curves of a synthetic 100 day injection atE25.0 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 suc-cessively longer injections times using the mass transfer parameters optimized to the 15 day field injec-tion. The widening of hysteresis separation with injection time illustrates how very slow exchange rateswith the immobile domain may be better characterized by longer injection, effectively increasing the‘‘window of detection’’ of the method.
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nimmob2 as the most sensitive parameter in their respectivesets, with corresponding �’s the most sensitive of localtransfer coefficients.
[29] As expected, in general, effective parameters (nmob1,nimmob1, �1, D) are more sensitive to the informational con-tent of �f whereas local parameters (nmob2, nimmob2, �2, D)are most sensitive to �b. However, there is some overlap,e.g., effective parameters showed some sensitivity to �b,particularly nmob1 and nimmob1. The observed hysteresis atthe most effectively modeled location of E15.0 (based onSOSWR) is used to illustrate the sensitivity of effectiveand local mass transfer parameters to specific temporalobservations (Figure 7). In this analysis, the effectivemass transfer parameters are sensitive to the tailing of thesignal, expressed as ‘‘hot’’ colors at later times in the hys-teresis pattern. In particular, nimmob1 is most sensitive tothe late-time �f, but also shows sensitivity to the informa-tional content of the �b tail, shown here as larger diameterpoints. In addition to the �f tail, nimob1 is most sensitive toearly time arrival of both the �b and �f BTCs. The �1 isonly sensitive to �f, with hot spots at the initial and late-time falling BTC limb. None of the effective parametersshowed sensitivity to the plateau concentrations of either�b or �f.
[30] The local mass transfer parameters at E15.0 are pri-marily informed by �b (Figure 7). The nimmob2 is the mostsensitive parameter, effective or local, and shows a rela-tively even distribution of sensitivity to �b along the entirehysteresis, including at plateau. Additionally, this localimmobile porosity is sensitive to the late-time �f, in a simi-lar but subdued manner compared to its effective upgra-dient counterpart. The nimmob2 also is sensitive (although tolesser degree) to the �b plateau, but shows essentially nosensitivity to the �f data. Finally, �2 was informed by theentire �b signal, though overall sensitivity was low, withsome minimal sensitivity to the �f tail in a similar bimodalpattern as �1.
[31] To investigate how long it would take to achieve �b
plateau (full hysteresis) based on locations with very smallmodeled �2 (e.g., 0.003 d�1), the E25.0 simulation wasrerun using 20, 40, 60, and 100 day injection times (Figure8). The longer simulation results indicate that the very slowestimated exchange processes would necessitate an approx-imate 100 day injection to reach tracer plateau at �b. Toinvestigate the lower bounds of what �2 would be neededto achieve tracer �b plateau within the confines the fieldinjection, the original 0.003 d�1 value was iterativelyincreased until equilibrium was reached within 15 days;the result was almost one full order of magnitude greater at0.019 d�1.
5. Discussion
[32] The discussion proceeds from the implications ofthe synthetic modeling scenario results to a comparison andevaluation of the mass transfer parameters determinedacross the Naturita site. Finally, we discuss the broaderimplications of this novel modeling effort and introduce anew Eulerian-based Damköhler number to help plan exper-imental tracer injection durations, which maximize theapplicability of the method to investigating local-scaleexchange processes.
5.1. Synthetic Modeling Scenarios
[33] Forward modeling of the three type scenarios illus-trates how variation in mass transfer at both the effectiveflow path and local scales results in characteristic BTC andhysteresis shapes (Figure 3). Furthermore, these scenariosshow how tailing in �f is primarily controlled by effectiveflow path–scale mass transfer processes upgradient of theobservation point, while the �b plateau and hysteresis sepa-ration result from local mass transfer processes at the ob-servation point, or within the geoelectrical representativevolume. For Scenario 2 with negligible nimmob2, the �b pla-teau is low as there is less total porosity (0.21) to load withthe conductive tracer, compared to Scenario 3, where totalporosity is large (0.55) due to a sizeable immobile domain.Also for Scenario 2, a linear relationship was observedbetween �b and �f as would be predicted by Archie’s law(equation (3)) in the absence of mass transfer, even thoughthe moderate effective flow path–scale mass transfer resultsin a heavy �f tail. With negligible local mass transfer, bulkconductivity tracks linearly with fluid conductivity (Sce-nario 2), but hysteresis results from local mass transferwith the immobile domain (Scenarios 1 and 3) as has beenshown previously [Singha et al., 2007; Day-Lewis and Sin-gha, 2008]. The tracer flush will amplify hysteresis separa-tion if mass transfer occurs, but the flush alone will notcreate hysteresis in the absence of mass transfer (Figure3c). In actual field-scale transport, all mass transfer param-eter values may vary simultaneously, creating more com-plicated patterns than those presented by these three typescenarios, but the basic characteristic shapes of effectiveand local mass transfer still apply.
5.2. Dual-Domain Mass Transfer at Naturita
[34] The exponential decline in conductivity during theflush observed in �f and �b, combined with accurate simu-lations of both sets of observations, indicates the DDMTmodel with first-order exchange coefficient is an appropri-ate description of nonreactive solute transport for theNaturita site (Figures 4 and 5). This finding is further sup-ported by the reasonable simulated fits to observed hystere-sis between �f and �b, patterns that were not explicitly fitduring 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 ofthese, but this is not surprising given the history of tracertests at the site [Curtis and Davis, 2006]. This spatial tracerdistribution indicates complex flow path dynamics in thesubsurface, some of which we were able to describe withboth effective-flow path and local-scale mass transferparameters.
[35] It is clear from multiple lines of evidence collectedduring this study that mass transfer parameters vary spa-tially 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 �f signals,variable �b ‘‘plateau’’ values, and disparate �b versus �f
hysteresis patterns. Contrasting patterns occurred even atadjacent locations along the same vertical profiles. Forexample, the �f signal observed at E14.6 showed little defor-mation from the input tracer signal similar to the syntheticScenario 3, while strong tailing was observed at E14.8 andE15.0 along the same profile (Figure 4). The differences in
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flow path–scale transport of the solute tracer translated todisparate optimizations of effective transport parameters,which were shown to be particularly sensitive to the infor-mational content of �f (Figure 7). Effective immobile po-rosity was only 0.03 for the subdued tailing at the E14.6
location, but estimated to be 0.32, with much higher re-spective rate coefficient, at the adjacent profile locationswith strong �f tailing. This relationship of tailing to strongmass transfer is consistent with the synthetic examples andthe laboratory findings of Swanson et al., [2012] in thepresence of known immobile pore space.
[36] One of the more striking aspects of the E1 profileparameter estimates is the differences in local-scale proc-esses between observation locations and their contrast withrespective effective upstream transport. The �b BTC atE14.6 reached a value of 2500 �S cm�1, much higher thanthe 700–1160 �S cm�1 observed at E15.0 and E14.8, respec-tively. Assuming negligible surface conductance and tortu-osity differences and relatively constant m (determinedexperimentally at 1.3), the differences in �b maximum andbackground equilibrium values correspond directly to var-ied total porosity, with greater �b during the late-time tracerinjection, indicating greater local porosity. Furthermore,the hysteresis at E14.6 showed the largest separationbetween injection and flush of any modeled location. Thismeans that at E14.6 there was high total porosity (estimatedat 0.48), and much of this was immobile (77%), as shownby observed hysteresis and the optimized parameter values.This type of porosity distribution is indicative of a clay de-posit [McWhorter and Sunada, 1977], and supports previ-ous observation of clay-size particle pockets. An alternateexplanation for the relatively large immobile porosity atlocation E14.6 is formation disturbance during well installa-tion and poor collapse. Interestingly, if this porosity reflectsthe true undisturbed well field condition, the most dominantlocal-scale immobile zone size was found along a flow pathwith negligible effective mass transfer, again similar tosynthetic Scenario 3.
[37] The opposite pattern was observed at E14.8, whichshowed little hysteresis and subdued local mass transfer af-ter a strong effective mass transfer signal (Figures 4 and 5and Table 2). Here, the wide confidence intervals aroundthe local parameters and lower sensitivities indicate therewas less immobile zone information contained in the obser-vations and that local porosity was dominated by the pre-cise estimate of nmob2 (0.19). The vertical variability inlocal mass transfer along the E1 profile is apparent, becausethe E14.8 location separates E14.6 and E15.0 in space, andthe latter together showed the largest hysteresis separationsof any modeled locations (Figure 6). The optimized esti-mates of nmob2 at E14.6 and E15.0 were relatively preciseand relatively low (0.06–0.11) (Table 2), serving to par-tially balance the corresponding higher nimmob2 (0.12–0.37), and as discussed above may indicate the presence ofclay. The optimized local mass transfer rate coefficientsspanned 3 orders of magnitude between the three E1 loca-tions, with �2 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 �2 at E14.8 of 0.002 d�1 isassumed, immobile residence time at the local scale wouldbe >100 days, suggesting there may be pockets of substan-tial solute retention and slow release at the Naturita site.
[38] Similar to E14.8, locations E24.8 and E25.0 showedminimal hysteresis separation between the tracer injectionand flush (Figure 6). Optimized effective parameter esti-mates were identical along this profile, showing relativelybalanced porosities and substantial mass transfer. And in asimilar fashion to synthetic Scenario 2 (Figure 3), moderatemass transfer at the effective scale contrasts with ‘‘negligi-ble’’ mass transfer at the local scale, where both field loca-tions had a low nimmob2 (0.06/0.07) and modest �2 (0.003d�1). These values were again quite similar to E14.8 ; thus,there is commonality among some local mass transfer pa-rameters at the site, both in the vertical and horizontal. Thestark difference between the E2 local parameters is innmob2, which is 60% greater at E24.8. Both locations hadhighly sensitive nmob2 to �b (Table 3), resulting in preciseparameter estimates, the difference here stemming fromvaried maximum �b (Figure 5), which as discussed aboveindicates a difference in total local porosity.
[39] The optimized � and nimmob parameter estimatesindicated effective flow path–scale mass transfer occurredover shorter diffusive lengths (as expressed in open water)compared to local mass transfer processes. During the in-stallation of the wells, we observed median stratified sedi-ment textures ranging from fine (<2 mm) to cobbles,characteristic of mixed fluvial deposits. Effective flow pathlengths (1.2–2.4 cm) may indicate mobile-immobileexchange over length scales corresponding to individualgrains, whereas the larger local length scales (2.0–9.4 cm)may indicate more macroscale exchange with pockets offine-grained heterogeneity, which is very slow and notobservable in �f data.
[40] In summary, all effective parameters, based onstrong �f tailing during flush, indicated mass transfer wasan important process at the flow path scale, except at E14.6,where little tailing was observed. Many local parametersshowed significant variation both from their effectiveupgradient counterparts and from adjacent locations in thevertical, with rate coefficients spanning 3 orders of magni-tude. As discussed below in section 5.4, a longer tracerinjection could serve to confirm or rule out the very slowlocal mass transfer processes (e.g., �2¼ 0.003 d�1) esti-mated for three locations.
5.3. Method for Effective and Local ParameterEstimation
[41] Commonly, mass transfer parameters are calibratedto �f alone [e.g., Feehley et al., 2000], which permits theestimation of averaged flow path–scale processes. Becauseconventional �f data are not directly sensitive to the immo-bile zone at the point of observation, this representativeupgradient flow path length, which is balanced by advec-tive velocity, must be sufficient to allow enough exchangebetween the two domains to inform immobile parameterestimation (optimal Damköhler number) [Bahr and Rubin,1987]. At this flow path scale, the dominant mass transfermechanisms expressed in the �f BTC may not well repre-sent the true variability in mass transfer parameters. Thismay lead to poor predictions of contaminant retention,transport, and remediation response, particularly if rela-tively large immobile porosities and slow exchange ratesare missed. In this study, a parameter sensitivity analysiswas used to show that mass transfer parameters at both the
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effective flow path and local scale could be simultaneouslydetermined when �b observations are included into the pa-rameter optimization process. For the first time, a sampleof the true local variability in field-scale solute transportmass transfer properties has been described.
[42] As detailed above, the resulting description of masstransfer at the Naturita site was heterogeneous. In fact, vari-ability along some single flow paths was so high (E14.6,E14.8, E15.0), indicated by disparate effective and local pa-rameter estimates, that the models would not converge tothe combined �b and �f observations when just one set ofeffective mass transfer parameters was considered; i.e.,there was no combination of parameters that could reason-ably match observed �b and �f patterns simultaneously,because they describe different-scale processes. When thelocal parameters were included in the optimization process,both sets of conductivity observations were effectivelysimulated, resulting in reasonable reproduction of theobserved hysteresis patterns. Synthetic modeling exampleswere used to show how multiscale variability in mass trans-fer could explain the seemingly contrasting �b and �f pat-terns. If E14.6 were modeled conventionally, matching onlythe observations during the optimization process, we wouldassume negligible mass transfer along that flow path, yetthe physical explanation, based on previous column experi-ments [Swanson et al., 2012], for the strong hysteresisbetween �b and �f at that location is the existence of a sub-stantial immobile zone. The optimized 12� increase in nim-
mob from the flow path to local scale, combined with anorder of magnitude increase in rate coefficient, directlyillustrates the dominant local immobile processes thatwould have gone undetected using traditional DDMT char-acterization methods.
[43] The ‘‘local’’ scale for this experiment refers to asample volume approximately 22 cm long, which is gov-erned by the spacing of the inner electrodes (M and N).The distance across this volume would be too short to con-duct a traditional tracer analysis using �f data alone basedon a Damköhler number assessment (equation (6)) and isapproximately one ninth the length of the effective flowpaths used here. The elongated representative volume isoriented in the vertical, roughly normal to the effectiveflow path direction (Figure 2b); the intersection of the twois where the local heterogeneity was included into the 1-Dmodel by independently estimating mass transfer parame-ters at that specific node. However, heterogeneity may existwithin this 22 cm long sample volume that was obscuredthrough this technique. Future experiments with relativelyshort effective flow path lengths can be designed withcloser electrode spacing, so the ‘‘local’’ scale is more dis-parate from the flow path scale, and local heterogeneitiesmore precisely described.
[44] For two pairs of aquifer locations (E14.8, E15.0, andE24.8, E25.0), the same �f sampling port was used because itwas located in the overlap of their electrical representativevolumes for �b (Figure 2). Hypothetically, as effective pa-rameters are most sensitive to �f, each set of pairs shouldhave analogous effective parameter estimates, even thoughthe �b observations at each location were quite different.We found this rule to hold for all effective parameters,illustrated by the overlap of 95% confidence intervalsbetween analogous parameters, indicating they were statis-
tically identical (Table 2). Similarity was also borne out inthe sensitivity analysis, which found comparable sensitiv-ities for analogous parameters (Table 3). The identicaleffective parameter results for locations that shared �f
observations instill confidence in the parameter optimiza-tion and sensitivity analysis using UCODE_2005 and indi-cate that the inclusion of �b observations is not somehowbiasing the estimation of effective parameters.
[45] The sensitivity analysis also indicated that the param-eter m, which was determined as the average value from thefield data, was consistently one of the most sensitive parame-ters in each simulation. Care must therefore be exercisedwhen determining this exponent, as the estimated value mayaffect optimized estimates of mass transfer parameters(equation (3)). Alternative averaging procedures could beused in place of arithmetic averaging. Although the bicontin-uum model has been used successfully in several studies,more work is required to fully evaluate the range of applic-ability of the model to DDMT. Depending on the internalconnectivities of the mobile and immobile domain, otheraveraging procedures of m (e.g., geometric, harmonic) couldprove more effective. Given the high information content ofgeoelectrical data for inference of DDMT, it may also bepossible to infer the form of the petrophysical model.
[46] We found that when hysteresis between �b and �f isobserved, there is more information regarding the localimmobile and exchange parameters within the data set,which supports previous work [Day-Lewis and Singha,2008]. Lack of hysteresis can result from two physical rea-sons: (1) there is no substantial immobile porosity at thelocal scale, or (2) mass transfer processes at the local scaleare outside the limit of detection of the experiment, due toinappropriate mobile advective velocities (very slow) or atracer injection that was too short to capture very slowexchange rates. Recent work has shown that hysteresis canresult 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 thathysteresis between �b and �f can be observed over approxi-mately 2 orders of magnitude in �2, which is a somewhatlarger range than was predicted by Day-Lewis and Singha[2008]. These differences may result from the fact that weallowed local parameters, including nimmob2, to vary simul-taneously using UCODE_2005. When the ratio of nimmob2
to nmob2 is high, hysteresis may be observed with an �2
ranging from 5.0 to 0.01 d�1 during an injection of 15 daysunder these conditions. As this definition suggests, and hasbeen observed elsewhere [Haggerty et al., 2001; Haggerty,2004; Day-Lewis and Singha, 2008], estimates of �2 aresensitive in part to the time scale of the injection. Assumingreasonably fast flow path–scale pore water velocities, suchas the 1.0 m d�1 found for location E15.0, �2> 5.0 d�1
results in rapid loading of the immobile zone and thereforelittle offset in cm and cim. The consequence is that there isessentially no ‘‘immobile zone’’ relative to solute transportas the loading, and subsequent unloading, converges on ad-vective transport, and the system essentially functions as asingle domain [Zheng and Wang, 1999]. But as we show inFigure 7, hysteresis may not be initially observed at verysmall �2, because the extremely slow loading of the immo-bile zone looks analogous to a lack of immobile zone at‘‘early time.’’
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[47] The parameter optimization reveals the presence ofsmall immobile zones and slow mass transfer exchange atlocations that showed little (but not zero) hysteresis. Threelocations (E24.8, E24.8, and E25.0) did not appear to achieve�b equilibrium. We attribute this behavior to small �2 atthose locations, i.e., locally slow mass transfer. The opti-mized parameters for location E25.0 were used to determinewhat injection length would be necessary to achieve �b equi-librium at that location, and thus produce a ‘‘full’’ hysteresisloop over the course of the experiment. Additionally, theoptimized �2 was iteratively increased to determinethe threshold �2 that would facilitate �b equilibrium duringthe 15 day tracer injection implemented in this study.
[48] Figure 8a illustrates that if we assume the �2 of 3 �10�3 for the E25.0 location is accurate, it would have takenapproximately 100 days of injection to fully load the corre-sponding immobile zone. Furthermore, during our 15 dayinjection, we could only expect �b to reach equilibrium pla-teau for �2¼ 2 � 10�2 d�1 or higher. Therefore, the win-dow of detection for � can be effectively increased on thelow end by running longer tracer injections. It does notseem necessary to achieve full �b plateau, however, asthere will be an exponential decrease in hysteresis separa-tion with time as Cim approaches Cm. Figure 8b depicts thatby 40 days, two thirds of the hysteresis that would resultfrom �2¼ 3 � 10�3 d�1 is achieved, providing abundantinformation to estimate �2 more precisely. Furthermore, ifno hysteresis is observed at these longer time scales, masstransfer on the low end of � effectively can be ruled out.
5.4. Local Damköhler Number
[49] Various design criteria must be considered wheninvestigating effective- and local-scale mass transfer simul-taneously; particularly as previous work has indicated thatslow exchange processes may keep the immobile domain atdisequilibrium with the mobile domain even at very largetimes during a constant rate solute injection [Haggertyet al. 2001], directly affecting local hysteresis patterns. Theeffective flow path scale fundamentally depends on aLagrangian-viewpoint Damköhler number (DaIE), as iscommonly presented in transport literature [Bahr andRubin, 1987; Haggerty, 2004];
DaIE ¼ �1tad 1þ nimmob1
nmob1
� �; ð6Þ
where cumulative exchange with the immobile zone alonga 1-D flow path under a given set of mass transfer parame-ters depends on the time scale of advection (tad) or the ratioof flow path length to flow path velocity. If we shift to anEulerian viewpoint, we can use the local Damköhler num-ber (DaIL) concept to evaluate whether our injection timescale (duration) was appropriate to investigate local masstransfer processes, where exchange with the local immobilezone under a given set of mass transfer parameters dependson the time scale of mobile zone tracer loading (tld) as
DaIL ¼ �2tld 1þ nmob2
nimmob2
� �: ð7Þ
[50] For simplicity, we can take tld to be the time overwhich �f BTC plateau is maintained, assuming that local
mobile tracer plateau is reached in a relatively shortamount of time relative to the loading of the local immobilezone. Note, here we use the inverse of the original DaIE po-rosity ratio to scale the DaIL by the rate of change in Cim-
mob, similar to the transport-governing equation ((4b)), notthe total tracer mass that enters nimmob as presented in equa-tion (5). Therefore, a faster rate of change in Cimmob resultsin a lower time to equilibrium (full hysteresis) and a higherDaIL for a given set of conditions. It should also be clarifiedthat the DaIL is designed for use with the MT3DMS versionof �, which differs from some other related mass transfercoefficients 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 injectionwas not long enough to load the local immobile zone to-ward equilibrium, and hysteresis will not be observed.Conversely, DaIL> 1 indicates the tracer injection wasunnecessarily long to investigate local mass transfer,and the ‘‘local equilibrium assumption’’ [Miller et al.,1990], or full hysteresis, will have been satisfied forsome time.
[52] If we return to the example of E25.0, where we didnot observe considerable hysteresis during our 15 dayinjection, the resulting local DaIL 0.14 (Figure 8a), while ifwe ran the injection for 40 days to achieve considerablehysteresis, the resulting DaIL¼ 0.38. If the experiment wasrun 100 days to equilibrium between the two domains (Fig-ure 8a), the local DaIL¼ 0.95, which is similar to value ofDaIL¼ 0.94 that is determined for a 15 day injection using�2¼ 2 � 10�2 d�1, or the minimum exchange coefficientneed to achieve equilibrium at E25.0 under the current ex-perimental conditions. At location E24.8, which has analo-gous mass transfer to E25.0 but much larger mobileporosity, we observed slightly greater hysteresis for thesame length (15 day) injection (Figure 6), and this isreflected in the larger DaIL¼ 0.2. The comparison betweenthese two locations illustrates the utility in inverting theoriginal DaIE porosity ratio to scale DaIL, as tracer concen-tration will rise faster is the local immobile zone when mo-bile porosity is relatively larger. For comparison, the DaIL
for E15.0, where �b equilibrium was just achieved by 15days, is 0.94; while DaIL for E14.6, where �b equilibriumwas reached in <<15 days, is 4.5. The new Eulerian-basedlocal DaIL should be useful to future research that utilizeschemical/geoelectrical tracer signals to resolve mass trans-fer and potentially in the stream environment when usingtracers to define exchange between the main channel andspecific zones of surface transient storage [Briggs et al.,2009], such as those found downstream of restorationstructures.
[53] When hysteresis is not observed under study condi-tions, no matter what the explanation, the sensitivity analy-sis indicated important information regarding nmob2 andflow path–scale porosities can still be contained in �b
observations. This informational content is due to the sensi-tivity of bulk conductivity to porosity, as predicted byArchie’s law, and the sensitivity of flow path parameters tothe timing of the �b BTC. This is an advantage of ourmodel-calibration approach over analysis of temporalmoments [Day-Lewis and Singha, 2008]. We found a largecontrast in �b between the aquifer locations that did notshow hysteresis separation (E14.8, E24.8, E25.0), which
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could be explained almost entirely by differences in nmob2,as the sensitivity of nmob2 to �b was particularly high incases of little hysteresis. Therefore, although the character-istic hysteresis of mass transfer was not observed, impor-tant information regarding local-scale transport and masstransfer parameters can be gleaned from the timing, shape,and plateau of �b. Although nmob2 does not describe immo-bile zone solute retention, varied nmob2 will have a strongeffect on local transport ; for example, the enhanced nmob2
at E15.0 increased local mobile pore water velocities by60% compared to E14.8, which may be particularly impor-tant to biogeochemically relevant parameters such as con-tact time.
6. Conclusions
[54] The variability between effective-upgradient flowpath and local-scale DDMT parameters has been describedusing field data for the first time. The major findings of thisstudy are: (1) both �b and �f observations were closelymatched by regression modeling using UCODE_2005 anda multiscale DDMT model, with generally high parametersensitivity; (2) strong variability in mass transfer wasobserved throughout the well field, including along thesame flow path, with � spanning 2 orders of magnitude;(3) hysteresis patterns are a local process, and hysteresisstructure was reproduced well in all cases, even thoughthese patterns were not explicitly fit during the parameteroptimization process; (4) as indicated in previous work,greater hysteresis separation between the injection andflush limbs of BTCs yielded greater local-scale mass trans-fer and higher sensitivity of the related parameters, but �b
observations were still useful for determining local porosityand optimizing effective parameters in cases of little hys-teresis; and (5) the method’s ‘‘window of detection’’ oflocal-scale mass transfer can be increased using longertracer injections to include very slow local mass transferprocesses, which may influence long-term solute retentionand release, and experimental design can be evaluated witha Eulerian-based local Damköhler number.
[55] Given the combination of tracer and geoelectricaldata collected during an ionic tracer test, it is possible todiscriminate between the effects of upgradient-flow pathand local mass transfer. The ability to characterize spa-tially variable mass transfer parameters at the field scale isa substantial advance, particularly with respect to assess-ing the true variability in local-scale mass transfer param-eters, which are now identifiable using solute tracertechniques for the first time because a representativelength (e.g., as determined with DaIE) is no longer needed.Improved estimation of immobile porosity distributionshould facilitate more accurate estimates of the volume ofimmobile contaminant present in the field and, combinedwith local mass transfer rates, the time it may take toremediate. Additionally, our approach provides informa-tion for parameterization of field-scale multirate masstransfer models where the rate distribution is directlyinformed by measurements of the true local-scale masstransfer. Detailed characterization of mobile-immobilemass transfer should provide improved understanding ofthe long-term fate of uranium at contaminated sites suchas Naturita.
[56] Acknowledgments. The authors gratefully acknowledge field as-sistance from Emily Voytek, Eric White, Peter Joesten, and Cian Dawson.This work was supported by U.S. Department of Energy Subsurface Bio-geochemical Research Program grants DE-SC0003681 and DE-SC0001773 and the U.S. Geological Survey Toxic Substances HydrologyProgram. Any use of trade, firm, or product names is for descriptive pur-poses only and does not imply endorsement by the U.S. government.
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