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a Geological Institute, Univ. of Copenhagen, Øster Voldgade 10, 1350 Copenhagen K, Denmark
b Geological Survey of Denmark and Greenland, Øster Voldgade 10, 1350 Copenhagen K, Denmark
c Environment and Resources, Technical Univ. of Denmark, Building 204, 2800 Lyngby, Denmark
d USGS, 12201 Sunrise Valley Drive, Reston, VA 20192, USA
* Corresponding author (pe{at}geol.ku.dk)
Received 4 July 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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= 0.1–0.4) and unsaturated zones greater than approximately 20 m in thickness. In these cases the CFC distribution may still be from either the exponential or linear phase. In other cases, the use of numerical models, as described in our work and elsewhere, is an option.
Abbreviations: CFC, chlorofluorocarbon • GC, gas chromatography • GEUS, Geological Survey of Denmark and Greenland
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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For very shallow unsaturated zones of only a few meters thickness, diffusion and barometric pumping sufficiently mix the gases in the unsaturated zone so soil-gas CFC concentrations are similar to those in the troposphere. The input to the groundwater system is thus very close to the atmospheric changes in CFC concentrations. This simple approach often has been used for dating groundwater, where the concentrations measured in groundwater are used directly together with the atmospheric concentrations to estimate the time of recharge, thus neglecting any time lag of the CFCs in the unsaturated zone.
In deeper unsaturated zones the time lag can be important, and various processes affecting CFC transport become important. Gas diffusion and air–water exchange (solubility) are especially important in controlling the migration and attenuation rate. Both processes are a function of the water content and, thus, seasonal and year-to-year changes in infiltration and depth to the water table. Cook and Solomon (1995) investigated numerically the conditions under which the time lag of CFCs in the unsaturated zone is of importance by examining the relative effects of various soil parameters on the distribution of gases in the unsaturated zone. Busenberg and Plummer (2000) used the same model as Cook and Solomon (1995) to study lag times of SF6 in unsaturated zones and found that the lag times of SF6 are smaller than for CFCs because of its low solubility. Oster et al. (1996) measured 57 profiles of both CFC-11 and CFC-12 in a 4.5-m-thick unsaturated zone in a forest soil in Germany. A clear damping of the annual changes in CFC atmospheric concentrations were found with a relaxation time (time lag) of 30 d for 4 m. Weeks et al. (1982) used analytical and numerical transport models to simulate the observed distribution of CFC-11 and CFC-12 in a >50-m-deep unsaturated zone. The analytical model was a pure diffusion model, whereas the numerical model also included advective transport because of the downward movement of the water table caused by increased pumping in the aquifer below. Weeks et al. (1982) were able to demonstrate that the procedures for estimating tortuosity (and thus also gaseous diffusion in soils) developed from theoretical considerations or laboratory techniques apply for field conditions as well. Busenberg et al. (1993) found that soil gas concentrations from two multilevel piezometers in a 46-m-deep unsaturated zone of sedimentary and basalt deposits could be explained by diffusion theory. Using the same model as Weeks et al. (1982) they were able to fit the tortuosity coefficients for the sedimentary and basalt deposits to CFC-11 and CFC-12 profiles. Severinghaus et al. (1997) measured CFC-11, CFC-12, and CFC-113 profiles in relatively dry and deep (60 m) unsaturated zones in sand dunes. A diffusion transport model simulated the profiles very well. The reported mean age of air (time lag) was 10 yr, and diffusion was found to be much more important than advection. Recently, transport of CFC gases in deep unsaturated fractured systems was investigated to study recharge mechanisms in a regional aquifer (Plummer et al., 2000) and chemical and physical processes at Yucca Mountain, Nevada, USA (Thorstenson et al., 1998).
The main objective of this work has been to simulate the historical input of CFC-11, CFC-12, and CFC-113 to groundwater at the Rabis Creek field site in Denmark. Observations of gas profiles in the 16-m-thick unsaturated zone are used for investigating the important physical-chemical processes affecting transport. In contrast to previous investigations, we explore the effect of nonsteady flow. As an outcome of the work, we also present a new analytical solution for predicting the distribution of CFCs in the soil and the historical input of CFCs to groundwater. Hinsby et al. (unpublished data, 2004) used groundwater input functions to investigate CFC transport and degradation in the underlying aquifer.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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Field Investigations
Profiles of the CFC gases were from a single well. Sixteen 6-mm copper tube soil gas samplers with an approximately 50-mm screened section at the end were installed at intervals of 1 m from the surface to the water table. The copper tubes were attached to the outside of a 63-mm-diameter piezometer with a 2-m screen at the end. The piezometer with the copper tube soil gas samplers then was placed through a hollow auger stem that was augered down to just below the water table. Fine- to medium-grained quartz sand was used as gravel pack around the screened intervals, and a mixture of quartz sand and silt was used as low-permeability plugs to reduce the gas-permeability between the samplers. The copper tubes were equipped with fittings at the upper end for a convenient and air-tight connection to the CFC sampler. After installation, the copper tubes were purged simultaneously for 10 min by means of a small soil gas pump connected to a manifold. The installation then was left for one-half year before the first sampling was performed in September 1995.
At the time of sampling, the copper tubes were connected to the CFC sampler and the soil gases then were pumped through the borosilicate sampling ampoules by means of a small soil gas pump. At least three (3–10) copper tube volumes were flushed through the sample ampoule before the soil gas sample was taken by fusing the ampoule with a small torch for an air-tight seal. The ampoules were taken to the laboratory at the Geological Survey of Denmark and Greenland (GEUS) and transferred to the gas chromatograph and analyzed for CFC-11, CFC-12, and CFC-113 as described by Busenberg and Plummer (1992). A second set of samples was taken in June 1996 and analyzed at the USGS Chlorofluorocarbon Laboratory in Reston, VA. The CFCs were measured using purge-and-trap gas chromatography (GC). The CFCs were trapped at –30°C on a short Porasil C-Porapak T trap. The CFCs were released by heating the trap at 100°C and separated with a 3 by 3.13 mm o.d. stainless-steel column packed with Porasil C. The N2 carrier gas flow was 40 cm3 min–2. The GC oven temperature was maintained at 70°C, and the CFCs were quantified with an electron capture detector held at 290°C (Busenberg and Plummer, 1992). During the second sampling, the soil gas samplers and sampling ampoules were purged exactly three times with a 50-mL syringe before sample collection.
An approximately 14-m-long sediment core was analyzed for its organic content (foc), water content (
w), and total porosity (). The core was subdivided into 56 subsamples. Each subsample was classified according to grain size. Parts of each subsample from the same classification (five in total) then were pooled and foc was measured, thus yielding average values of the organic content for each classification. The water content and porosity were measured for each of the 56 subsamples giving a vertical spatial distribution of the porosity and, at the time of sampling, of the water content.
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THEORY |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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The 16-m-thick unsaturated zone at Rabis Creek is a homogeneous porous, nonfractured media. The primary mechanisms affecting transport of CFCs in an aerobic unsaturated zone is shown in Fig. 1 (Plummer and Busenberg, 1999). Gas transport in a variably saturated soil takes place as a combination of advective and diffusive–dispersive processes in both air and water. The water content changes over the season and diffusion in air and advective transport in both phases are sensitive to the water content (Cook and Solomon, 1995). The simulated average yearly infiltration at the site ranges from 266 to 623 mm yr–1 (see Water Content in Results below). A high water content, which may occur close to the soil surface under periods of infiltration and close to the water table, leads to reduced transport by gas diffusion by decreasing the effective area for diffusive flow and connectivity of gas-filled pores. However, at the same time, advective–dispersive transport in the water phase increases. Advective transport in the air phase also may occur as a result of variations in barometric pressure, wind blowing across the surface, temperature gradients, water infiltration and redistribution, and a moving water table. Weeks et al. (1982) and Kimball and Lemon (1972) found that many of these mechanisms contributed little to advective transport at depths of more than 1 m below the soil surface. On the other hand, the analysis by Massmann and Farrier (1992) suggested that gas can migrate several meters into the unsaturated zone by advective transport caused by atmospheric pressure fluctuations.
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Chlorofluorocarbon sorption–desorption may be controlled by kinetics with a time scale much longer than for air–water exchange. Little is known about CFC sorption. In the following we assume that sorption is an equilibrium process. Generally, it can be assumed that degradation of CFCs only occur in anaerobic environments (Plummer and Busenberg, 1999). There are only a few small-scale studies that demonstrate the onset of degradation with a change in redox environment. For example, Cook et al. (1995) found a significant decrease in CFC-11 concentration coincided with a decrease in O2 concentration to below 0.5 mg L–1. They estimated first-order degradation rates to 0.2 to 1.6 yr–1. Happell et al. (2003) observed that CFC-11, CFC-12, and CFC-113 disappeared with a significant increase in methane concentration, which they attributed to the activity of methanogenic bacterial degradation of the CFC gases. The decrease or increase in solute concentration of different redox environment indicators may therefore point to the presence of degradation. Anaerobic microenvironments where degradation occurs may exist in the unsaturated zone, although the bulk part will remain aerobic without any degradation. First-order degradation is included in the model because detectable methane concentrations in the lower part of the unsaturated zone were observed.
Transport in the Water Phase
Transport of CFCs in the water phase of a variably saturated porous media can be described by
 | [1] |
w is the volumetric water content, 
w is the water density, cw is the concentration of gas dissolved in water, ca is the concentration of gas in the air, Dw is the dispersion coefficient, qw is the Darcy flux, t is time, and z is depth oriented positive downwards. The dispersion coefficient is given by Dw = 
Lvw, where L is the longitudinal dispersivity and vw is the pore water velocity given as qw/w. In Eq. [1], it is assumed that the CFC can be affected by linear sorption (R) and first-order decay (
). Sorption is modeled by the retardation factor R = 1 + b/(ww)Kd, where Kd is the solid–water distribution coefficient and b is the bulk density of the media. First-order degradation may be specified as a function of depth. The last term on the right-hand side of Eq. [1] accounts for the mass exchange between the water and air phases. The solution to Eq. [1] was based on steady or nonsteady simulations of the flow field. A numerical code was used to solve the Richards equation for one-dimensional nonsteady flow based on local climatic information. The same unsaturated flow code was used by Engesgaard et al. (1996) to predict the average recharge from 1961 to 1970 at the same field site.
Transport in the Air Phase
Advective transport due to volumetric displacement, for example, from infiltration, is the only advective gas transport process included in the numerical code. Advective transport in air is approximated, as a complementary part of unsaturated water flow as the water- and gas-filled pore volumes must equal the total porosity. Assuming that air is incompressible and that the water table constitutes an impermeable boundary, advective transport in the numerical cell located just above the water table is calculated from the volumetric changes in water and air from one time step to the next. If the air volume increases, gas is transported into the cell from the cell above and vice versa for a decreasing air volume. Advective transport then can be calculated from the water table to the soil surface by keeping track of the volumetric changes and gas transport from the underlying cell. Using this procedure, a potentially important mechanism of advective gas transport is considered and a CFC mass balance over the two phases in the system is obtained. Variations in barometric pressure were not included in the simulations, but this mechanism was assumed to cause complete mixing of soil air in the upper 1 m. The CFC soil transport simulations were thus performed from 1 m below the soil surface giving a total length of 15 m unsaturated soil column.
The net transport of gas can be described by
 | [2] |
a is the volumetric air content; Da = DCFC is the effective gas diffusion coefficient, where is the tortuosity of the porous media and DCFC is the self-diffusion coefficient for the CFC gas into atmospheric air; and qa is the advective flux of air as discussed above. The last term on the right-hand side accounts for the mass exchange between the air and water phases. No sorption occurs in the air phase as all the solids are assumed to be fully surrounded by water. No degradation is assumed to take place in the air phase.
Air–Water Exchange
Assuming equilibrium exchange of CFC between the air and water phases, the relation between the air and water CFC concentrations can be expressed by Henry's Law as
 | [3] |
Numerical and Analytical Solutions
Equations [1] and [2] are coupled through the water–air exchange term that can be computed from Eq. [3]. Both [1] and [2] were solved using a finite difference discretization and the coupled system of equations was solved with an Iterative Operator Splitting technique (Steefel and MacQuarrie, 1996).
There were no meteorological data available for the site before 1961. To estimate initial 1961 soil CFC concentrations, the mean water content and mean water flux for the period 1961 to 1996 was calculated and used as input in a CFC transport simulation based on steady flow from the start of atmospheric CFC release (1940–1950) to 1961. The upper boundary condition is that of specified concentration with the concentration in the air phase equal to the atmospheric concentration. The upper boundary condition for the concentration in water then is cw (z = 1m,t) = caKwa. The lower boundary condition, at the water table, is assumed to be in the form of a zero-gradient in concentration, or (
ca/z)(z = L,t) = (cw/z)(z = L,t) = 0, where L is the total length of the soil profile. For the air phase, this implies a zero-flux boundary (impermeable), and for the water phase a pure advective flux boundary (no dispersion).
A spatial discretization step of 0.1 m was used in all simulations, resulting in 151 node points over the 15-m-deep profile. The simple procedure to calculate the advective component in air, qa, was affected by numerical oscillations if a time step too large was used. A time step of 1 d was used, matching the frequency with which climatic input data were available. No oscillations were found using this time step, and simulations using a time step of 1 h gave almost identical results. If the advective component was not included, small oscillations were seen simply because mass was not conserved during a time step when the simulated nodal water content changed.
If degradation is neglected (typical in unsaturated zones), then transport of CFC for steady water flow can be described by adding Eq. [1] and [2], averaging all parameters, using Eq. [3] and the definition of the retardation coefficient, yielding
 | [4] |
*, D*, and q* are the average volumetric air content, bulk diffusion coefficient, and advective flux of air, respectively, defined by
 | [5] |
 | [6] |
 | [7] |
w, 
w, and a are the mean water content, mean Darcy flux, and mean advective flux of air. Equations [4] and [5] through [7] are identical to the ones solved by Cook and Solomon (1995) in their steady flow analysis with no degradation and 
a = 0.
For a system controlled by diffusion (neglecting advective flow of gas and water, qa = qw = 0) Weeks et al. (1982) and Cook and Solomon (1995) adapted an analytical solution from Carslaw and Jaeger (1959) to Eq. [4] using an upper boundary condition of exponentially increasing atmospheric concentration. Cook and Solomon (1995) used the analytical solution to define the time lag as the time it takes a CFC gas to move through the unsaturated zone. The time lag to an arbitrary depth is thus found from the relation
 | [8] |
 | [9] |
Equation [9] only is valid after a certain time has elapsed since initial conditions (terms dropped in original solution).
The atmospheric input function has a linear increase since the early 1970s to approximately the early 1990s for CFC-11 and CFC-12, and a much shorter period with a linear increase from 1985 to 1990 for CFC-113 (see below). Equation [9] is not necessarily valid for CFC profiles measured during the mid 1990s, such as in this study, and an analytical solution must, therefore, be based on a linear increase at the top boundary. Carslaw and Jaeger (1959) provided an analytical solution for the case of linear increase in atmospheric concentrations and zero initial concentrations as
 | [10] |
 | [11] |
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 | [12] |
 | [13] |
The time lag at the water table (z = L) is
 | [14] |
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DATA |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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The three atmospheric CFC functions at the two sites are shown in Fig. 2. The increases in CFC-11 and CFC-12 concentrations are roughly exponential from the time of initial release to the early to mid 1970s. For CFC-113, the exponential growth period continues to the mid 1980s. Following the exponential period is a period of an approximately linear increase until the late 1980s to early 1990s, where CFC-11 and CFC-113 concentrations level off and recently decline. The atmospheric concentrations measured at the two locations were almost identical for CFC-12 and CFC-113. However, some differences for CFC-11 are found, especially for post-1985 concentrations. Although the CFC concentrations measured at unpolluted sites show little variation between monitoring stations on the northern hemisphere, Oster et al. (1996) found significant differences in local atmospheric CFC concentrations at four sites in geographically diverse regions in Germany and Switzerland. They showed that the concentrations of CFC-11 and CFC-12 increased with closer proximity to urban (industrialized) regions. At the Rabis Creek site no significant local sources were expected.
Only a few observations of atmospheric CFC concentrations are available at the Rabis Creek field site and in Denmark in general. The atmospheric CFC curves are compared with the few atmosphere measurements at Rabis Creek in Fig. 2. The measurements by GEUS consistently are higher than those by USGS, for CFC-11, with up to a 10% difference. Even when disregarding these analytical differences, it is not possible from these few measurements to decide which input function is the most appropriate to use at Rabis Creek. To quantify the sensitivity to the choice of atmospheric input function, the CFC distributions in the unsaturated zone were simulated with both input functions and compared with the observations from the multilevel samplers. The results showed that the simulations based on the Niwot Ridge curves gave the best fit to the observed data. Therefore the Niwot Ridge curves were used as input functions in the following analyses of CFC transport in the subsurface.
Climatic Data
Daily values of precipitation, potential evapotranspiration, and temperature collected at a nearby meteorological station were available. The same data were used by Engesgaard et al. (1996) to study tritium transport in the unsaturated zone at Rabis Creek.
Physical and Chemical Parameters for Soil and CFC Gases
Some of the main physicochemical parameters for the CFC gases are given in Table 1. DCFC values for CFC-11 and CFC-12 were taken from Weeks et al. (1982) and corrected to the ambient groundwater temperature of 8°C using the approximation D1/D2 = (T1/T2)m, where Di and Ti are the diffusion coefficient and temperature, respectively, and m was taken as 1.75 (Thomas, 1982).
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w, and flux of water, qw, in Eq. [1] were obtained by solving the Richards equation, as explained above. Hydraulic properties for the sandy soil at the study site were not measured. Instead, measured properties for similar Danish coarse sand were specified. The soil was chosen so that the numerical simulations gave approximately the same mean water content as observed. A cubic spline function was fitted to measured retention data in the range from wilting point to porosity. Changes in unsaturated hydraulic conductivity with w were predicted using a power function. The flow code was used to predict the variations in w and qw for 1961 through 1996.
The only unknowns in Eq. [1] through [3] are , Kd, , and L. Both tortuosity and sorption will have a retarding effect on CFC gas transport, and various combinations of the two can fit a CFC profile (Weeks et al., 1982). Both are regarded as fitting parameters. Based on measured values for foc, a theoretical distribution coefficient can be calculated from the relation Kd = focKocw, where Koc is the water–organic C partition coefficient (Table 1). This relation is derived for sorbing of nonpolar chemicals to aquifer material with foc > 0.1%. Although the CFCs are not strictly nonpolar chemicals, they have been observed to act as such with respect to sorption (Ciccioli et al., 1980). For the soil at this site, Nielsen and Ørbeck (1995) measured a value of foc in the range of 0.005 to 0.014% below depths of 0.09 m and 0.129% for the top 0.09 m, representing an upper organic-rich layer. The depth-averaged foc value for the profile below 0.09 m is 0.008%. Theoretical Kd values of 0.008, 0.019, and 0.014 g g–1 for CFC-11, CFC-12, and CFC-113, respectively, can be calculated, although foc is <0.1% and the above relation is, thus, not strictly applicable. Given the low organic C content we anticipate little retardation and a starting assumption is Kd = 0 for all CFC gases.
Small amounts of methane (0.015% [v/v]) were detected in the lower 4 m of the profile, indicating that degradation in microenvironments is a possibility. Specifying a first-order rate is not easy because of the lack data and because the model uses a bulk rate (i.e., the effective rate should model degradation inside a numerical cell volume with primarily aerobic zones and only small volumes of microenvironments). The only source of information we could find is the study by Cook et al. (1995), who estimated rates of 5 to 40 x 10–4 d–1 in a groundwater aquifer with sudden depletion of O2.
Engesgaard et al. (1996) found a longitudinal dispersivity of approximately L = 1 m by matching an analytical solution to three observed tritium profiles in the unsaturated zone at the Rabis Creek study site. The analytical solution assumed that flow was steady over the considered time period of 1961 through 1970. The dispersive effects from nonsteady flow were incorporated in the calibrated value of L. The L value of 1 m must be an upper bound, and a lower L is expected.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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5% for CFC-11 and CFC-12 and 20% for CFC-113. The uniform concentration profiles indicate that an efficient mixing takes place in the unsaturated zone. However, a noticeable peak is present in the concentration profiles for CFC-11 in the upper 7 m of the profile, indicating instead that the rather uniform unsaturated zone profiles are a result of the recent stagnation and turnover of the atmospheric input functions (Fig. 2).
A comparison between numerically calibrated (based on nonsteady flow and thus variable w) and observed CFC profiles is shown in Fig. 5. Transport of CFC-12 was simulated first because it was expected that CFC-12 would be less affected by sorption and degradation. With no degradation and sorption for CFC-12 a tortuosity coefficient of = 0.10 was found. This value was used for CFC-11 and CFC-113 as well. For all gases, the best match to the observations was obtained for L = 0.05 m. This value is less than the 1 m found by Engesgaard et al. (1996), which partly can be explained by inclusion of nonsteady flow. However, the simulations showed a lack of sensitivity to changes in L because of the nature of the atmospheric input function, with a linear increase in concentrations up to the early 1990s.
Figure 5a shows the simulated profile for CFC-11 assuming no sorption and with and without degradation in the bottom 4 m (where methane was observed). A good match to the observations below a depth of 5 m was obtained with a of 0.0004 d–1, while assuming no degradation predicts concentrations that are too high. This rate is a factor of 1 to 10 less than that found by Cook et al. (1995). The difference may be explained by the coexistence of aerobic and anaerobic microenvironments. The rate is equivalent to a half-life of about 5 yr. Notice that the simulated profile is affected over the whole thickness of the unsaturated zone, even though degradation only occurs in the bottom 4 m. This is because degradation only takes place in the aqueous phase, which causes a transfer of CFC-11 from the air to the aqueous phase in the bottom 4 m and, consequently, then an increased diffusion from top to bottom. An almost similar fit can be obtained by neglecting degradation and instead allowing CFC-11 to sorb to the soil matrix with a Kd = 0.06. This Kd value is higher than the theoretical Kd value. Given our observations, especially with the small changes in CFC-11 concentrations, we are not able to conclusively say if sorption or degradation affects CFC-11 transport. The simulated concentrations are higher than observed in the upper 5 m. This difference can be explained by the atmospheric input function (Fig. 2a), where the leveling off and decline in concentration began around 1992 to 1996. The simulation, therefore, is sensitive to the choice of input function. The input function from Adrigole/Macehead, Ireland levels off too early, whereas the input function from Colorado, USA, used in the simulation, levels off too late.
For CFC-12, a good match to the observed profile was obtained assuming no sorption or degradation (Fig. 5b). The fit between the simulation and observation for CFC-12 is better than that for CFC-11 mainly because of less uncertainty in the input function (Fig. 2b). However, the observations in the top 3 m indicate that the leveling off in atmospheric CFC-12 concentrations started earlier than used in the input function. The observation at z = 15 m is higher than predicted by the model. The Kd value for CFC-113 also was chosen to be zero.
Periodically wetting and drying of the soil leads to variable air- and water-filled pore space fractions and, thus, to varying pore water velocities. The effect of changes in the water content will be greatest for the more soluble CFC-11 gas. Therefore, simulated CFC-11 concentration profiles based on steady and nonsteady flow, respectively, may give different results. A simulation with steady flow using the long-term mean infiltration of qw = 439 mm yr–1 and 
w = 0.1 is shown in Fig. 5a. Sorption was not included, but CFC-11 was allowed to degrade with the same rate as in the simulation with nonsteady flow. The maximum difference in concentration (at the water table) is <2 pptv.
Based on the results described above, it seems appropriate to test the simple analytical models for diffusion in soil air. For CFC-11, the simulated concentration in the gas phase at the water table in September 1995 was 260 pptv (Fig. 5a). This concentration corresponds to an atmospheric concentration in January 1989. Thus the soil gas appears to be 6.5 yr old if dated with CFC-11. Cook and Solomon (1995) defined these 6.5 yr as the "apparent time-lag" for the unsaturated zone. The period of linear increase in atmospheric concentrations starts in 1975 (with a start concentration of c0 = 115 pptv) and ends in approximately 1991 (Fig. 2a). The rate of linear increase is estimated to be kl = 9.3 pptv yr–1, which gives a (synthetic) initial time of linear increase from zero concentration at approximately t0 = 1962. The numerically computed time lag of 6.5 yr indicates that the observations closest to the water table are from the end of the linear growth phase. The ratio De = D*/* in Eq. [11] thus is calibrated to match the bottom of the profile (1–2 data points closest to the water table). The results of the analytical solution also are shown in Fig. 5a, using De = 19 m2 yr–1. This estimate can be compared with the results from the numerical analysis using Eq. [5] and [6], if qa = qw = 0 is assumed. The flow solution gave w = 0.10, and using b = 1.8, = 0.10, Kd = 0.0, and data from Table 1, a value of De = 21.7 m2 yr–1 is obtained, close to the result of the analytical calibration. Using Eq. [14] and De = 19 m2 yr–1, tlL = 5.2 yr is obtained. The time lag is lower than the apparent time lag of 6.5 yr, which is because degradation or sorption is not included in the estimation. If we choose to use a Kd of 0.06, then De = 18.50 and tlL = 6.1 yr.
The analytical solution overestimates the observed concentrations of CFC-11 (Fig. 5a). The leveling off in the CFC-11 atmospheric input function and even the decrease are therefore seen in the observed profile. The numerically simulated profile better matches the observed profile, but the actual decrease in concentrations may have happened slightly earlier than simulated.
For CFC-12, a reasonable fit of the analytical solution to the lower half of the unsaturated zone results (Fig. 5b). The analytical solution was fitted to a larger part of the observed profile, because the trend with a linear increase in atmospheric concentrations continues 2 yr longer than for CFC-11, and the leveling off is much less pronounced. Also, the onset of the linear phase is estimated from Fig. 2b to be in 1971 (at a concentration of 143 pptv). The start of the linear increase in the model is set at t0 = 1961 using a linear rate of increase of 16.0 pptv yr–1. The fitted De of 29 m2 yr–1 is similar to the 28 m2 yr–1 calculated on the basis of Eq. [5] and [6]. The analytically estimated time lag is 3.9 yr, whereas the apparent time lag based on the observed concentration at the water table is approximately 4.3 yr.
The simulated CFC-113 profile is shown in Fig. 5c. Clearly, the high concentrations at depths of 3 to 9 m cannot be simulated with the model. We suspect that this may be caused by contamination during sampling or by an unknown local source. Thus, analytical modeling has not been pursued.
Input of CFC to Groundwater
The atmospheric and numerically simulated recharge input functions in the air phase at the groundwater table for all three gases are shown in Fig. 6
. The smaller apparent time lag for CFC-12 is a result of the higher rate of increase in the atmospheric input function (the slope of linear growth being 16 pptv yr–1 compared with 9.3 pptv yr–1 for CFC-11) leading to a higher concentration gradient. CFC-113 concentrations differ from the two other gases during the exponential growth period, which continues to 1985. The input function, therefore, has a linear growth period of only 5 yr (to 1990) before the growth rate decreases.
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Sensitivity Study
Figure 7
shows the results of changing the infiltration rate on the CFC-11 time lag for two water contents (0.1 and 0.15). All other parameters are the same as before. The infiltration rate, qw, was varied between 0 and 600 mm yr–1. Thus, it was assumed that the water content was the same at all infiltration rates. The time lag was calculated for the 1983 atmospheric concentration. For reference, the time lags calculated with the fully dynamic model and the analytical solution [14] also are shown. The case of w = 0.10, which corresponds to the Rabis Creek conditions, is first discussed. Ideally, the numerical model simulation should match the analytical solution for qw = 0. However, there is a difference of 0.1 yr, which is explained from differences in the upper boundary condition in the two solutions. The numerical model uses the observed atmospheric input function, whereas the analytical model uses an exact linear input. The difference of 0.1 yr represents the uncertainty in assuming a linear increase and estimating kl from the atmospheric curve. For qw = 439 mm yr–1 (long-term average), the difference is 0.05 yr between the computed time lags using steady and nonsteady flow. This result was also shown in Fig. 5a. The infiltration rate was higher (500 mm yr–1) than the average in the period 1980 to 1995, Fig. 4. If 500 mm yr–1 is used, there is no difference between the calculated time lag using steady or nonsteady flow simulations. In the range 0 to 600 mm yr–1, the difference in time lag always is less than half a year; thus, flow has had little effect on the historical input of CFC-11 to groundwater. Also, the analytical estimate of time lag is adequate for displacing the atmospheric input function because the difference between the two is only 0.5 yr. If the unsaturated zone is thicker than 15 m, then the decrease in time lag by including water flow still may be a small fraction of the total time lag. However, this decrease may be so large that it must be taken into account when predicting the historical input to groundwater. The effects discussed here are less for CFC-12 and CFC-113 because these gases are less soluble than CFC-11. Water flow, however, becomes more important at water content of 0.15. Zero infiltration gives a time lag of 6.4 yr. Infiltration rates need to be >600 mm yr–1 to reduce the time lag by more than 1 yr.
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= 0.10 and 0.36), and the depth of the unsaturated zone (5–50 m). = 0.10 is equal to the calibrated value for Rabis Creek, whereas = 0.36 is calculated from the expression proposed by (Millington, 1959), = a7/3/2, where is the total porosity (0.39) and a is calculated as – w with w = 0.1. Kd = 0.06 was assumed for CFC-11, whereas CFC-12 was nonsorptive. No degradation was included. The analytically computed time lags for three depths (5, 15, and 50 m), with two different tortuosity coefficients during the phases of linear and exponential increase in atmospheric concentrations are shown in Fig. 8 . If the unsaturated zone is shallow, L = 5 m, then the time lag is very small (<1 yr) for both gases. Because the linear phase ended in approximately 1991 and 1993 for CFC-11 and CFC-12, respectively, this means that data must be from before approximately 1992 or 1994 for the analytical solutions to apply.
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w = 0.10, and = 0.10 corresponds to the conditions at the Rabis Creek site (assuming qa = qw = 0), which shows time lags >1 yr. The time lags in the linear phase for CFC-11 and CFC-12 were calculated to 5.2 to 6.1 and 3.9 yr, respectively (Table 2). Sampling after 1998, therefore, would not detect this period, and the analytical solution would not apply.
The time lag is important for larger depths (L 
50 m). In the case of = 0.1 and a water content of 0.15, the time lag for CFCs after the start of the linear phase (early to mid 1970s) is about 50 (CFC-12) and 80 (CFC-11) years. CFCs from after the mid 1970s can likely be found in 50-m-deep unsaturated zones. In the exponential phase the time lags are 20 to 30 yr. The time lags in the two different phases now are different, about 30 to 50 yr. In reality, there will be a much smoother transition in the time lag. So although the calculated time lags of the exponential phase indicate that sampling today would not detect this phase, it is likely that a 50-m-deep unsaturated zone would be a mixture of CFC gases from both the linear and exponential phases. This would make it difficult to use the analytical approach. For the case of = 0.36, sampling today only would detect the linear phase for CFC-11 because the time lags for the other cases are about 10 yr and the linear phase ended more than 10 yr ago.
The time lags for CFC-11 are higher and more variable than for CFC-12 (and CFC-113) because of CFC-11's higher solubility. One of the greatest uncertainties in estimating time lag is associated with the tortuosity coefficient. For the linear phase, the time lag scales linearly with the inverse of the tortuosity coefficient (see Eq. [14]). In our study and the studies by Weeks et al. (1982) and Busenberg et al. (1993), the tortuosity coefficient has been a fitting parameter. It is difficult to estimate this parameter independently except for well-defined conditions, such as in the sand dunes investigated by Severinghaus et al. (1997). These dry sand dunes were 70 m deep, receiving about 50 mm yr–1 precipitation. They found that diffusion model results with a tortuosity coefficient of 0.6 for a dry soil (uncalibrated; see Millington [1959] and Weeks et al. [1982]) fitted observed CFC profiles. Another case where a relatively high tortuosity coefficient was found was reported by Oster et al. (1996). The forest soil was described as well aerated, which could explain the high effective diffusion coefficient (we estimate = 0.44 on the basis of the effective diffusion coefficient). In the general case, the tortuosity coefficient will be a complex function of time-dependent changes in water content, pore connectivity, and other physical properties of the soil.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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= 0.1 was fitted to the observed profiles. CFC-11 was shown to be slightly affected by either degradation or sorption, whereas CFC-12 and CFC-113 were found to move conservatively through the unsaturated zone. Degradation of CFC-11, with a first-order rate of 0.0004 d–1, in the bottom 4 m is likely because of the presence of methane at these depths. The degradation rate corresponds to a half-life of 5 yr in the bottom 4 m. In comparison, the time lag in air is 5.2 to 6.1 yr for the 15-m-thick unsaturated zone. Sorption of CFC-11, however, was modeled with a high Kd value of 0.06, which was not expected due to the very low organic content. Analytical models for diffusion and time lags were adapted from Cook and Solomon (1995) for the case of exponential increase and from Carslaw and Jaeger (1959) for linear increase in atmospheric CFC concentrations. The model for linear increase was fitted to CFC-11 and CFC-12 profiles in the lower part of the profile, providing an estimate of the effective diffusion coefficient. The analytical solutions agreed well with the numerical solution.
For the conditions at the Rabis Creek field site, the results show that it is possible to neglect water flow when quantifying transport of the three gases through the unsaturated zone. For the most soluble gas, CFC-11, the error in the calculated time lag is on the order of 1 yr, when neglecting water flow. For CFC-12 and CFC-113 the error is less than for CFC-11. The analytically derived time lags will suffice to displace the atmospheric input function by fixed time lags to predict the historical input to groundwater. However, water flow becomes important at a mean water content of 0.15 and infiltration rates >400 mm yr–1.
A sensitivity analysis using the analytical models demonstrates the importance of the depth of the unsaturated zone, the water content, and tortuosity coefficient on the estimated time lag. The most critical parameter is the tortuosity coefficient. For the model with a linear increase, the time lag scales inversely with tortuosity coefficient. High tortuosity coefficients ( = 0.36) mean that only the linear phase can be found today in deep (>20 m) unsaturated zones. The analytical solution presented in this work (Eq. [14]) for the linear phase then may apply. Low tortuosity coefficients ( = 0.10) result in a large difference in time lags in the linear and exponential phases (30–50 yr), which is not realistic. It is, therefore, difficult to predict if, for example, gas concentrations from the exponential phase are no longer present in the a deep unsaturated zone or whether a gas profile is a mixture of concentrations from both the linear and exponential phases. The use of analytical methods is therefore questionable. Numerical simulations may be an option, as described in our work, but only for conditions where, for example, barometric pumping of air through fractures or gross heterogeneity in transport properties are not present.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSTHEORYDATARESULTSCONCLUSIONSAPPENDIXREFERENCES |
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ca, concentration of gas in air (moles per cubic centimeter of gas)
c0, initial concentration of gas in air (moles per cubic centimeter of gas)
Dw, dispersion coefficient (square meters per year)
Da, = DCFC, CFC gas diffusion coefficient (square meters per year)
D*, bulk diffusion coefficient (square meters per year)
De, effective diffusion coefficient (square meters per year)
DCFC, self-diffusion coefficient for a CFC gas into atmospheric air (square meters per year)
foc, soil organic content (grams of C per gram of solid)
ke, rate of exponential increase in the atmospheric CFC function (per year)
kl, rate of linear increase in the atmospheric CFC function (pptv per year)
Kd, solid–water distribution coefficient (grams of water per gram of solid)
Kwa, water–air partitioning coefficient (cubic centimeters gas per gram water)
Koc, octanol–water distribution coefficient (grams of water per gram of C)
L, depth of unsaturated zone (meters)
R, = 1 + b/(ww)Kd (no unit)
qw, Darcy flux (cubic meters of water per year per square meter)
w, mean Darcy velocity (cubic meters of water per year per square meter)
qa, advective flux of air (cubic meters of gas per year per square meter)
a, mean advective flux of air (cubic meter of gas per year per square meter)
q*, bulk advective flux of air (cubic meters of gas per year per square meter)
t, time (years)
teL, time lag for the exponential phase (years)
tlL, time lag for the linear phase (years)
t0, initial time for start of CFC release (years)
vw, pore water velocity (cubic meters of water per year per square meter)
z, depth oriented positive downwards (meters)
L, longitudinal dispersivity (meters)
w, water density (grams per cubic centimeter of water)
b, bulk density of the media (grams per cubic centimeter)
, total porosity (cubic centimeters of porous medium per cubic centimeter)
w, volumetric water content (cubic centimeters of water per cubic centimeter)
w, mean volumetric water content (cubic centimeters of water per cubic centimeter)
a, volumetric air content (cubic centimeters of gas per cubic centimeter)
