## Tracer Model

Excel workbook for calculation and presentation of environmental tracer data for simple groundwater mixtures.

Böhlke, J.K., 2004, TRACERMODEL1. Excel workbook for calculation and presentation of environmental tracer data for simple groundwater mixtures. In: IAEA Guidebook on the Use of Chlorofluocarbons in Hydrology. International Atomic Energy Agency, Vienna, in press.

### In This Document

- Introduction
- Calculations
- Explanation of workbook TRACERMODEL1a
- Explanation of workbook TRACERMODEL1b
- Disclaimer
- References

### Introduction

Atmospheric environmental tracers commonly used to date groundwater on time scales of years to decades include CFC-11, CFC-12, CFC-113, SF6, 85Kr, 3H, and 3H/3H(0), where 3H(0) refers to initial tritium (3H + tritiogenic 3He) (Cook and Herczeg, 2000). Interpretation of age from environmental tracer data may be relatively simple for a water sample with a single age, but the interpretation is more complex for a sample that is a mixture of waters of varying ages. A mixture can be a natural result of convergence of flow lines to a discharge area such as a spring or stream, or it can be an artifact of sampling a long-screen well. TRACERMODEL1 contains a worksheet that can be used to determine hypothetical concentrations of atmospheric environmental tracers in water samples with several different age distributions. It is designed to permit plotting of ages and tracer concentrations in a variety of different combinations to facilitate interpretation of measurements. TRACERMODEL1 includes several different types of graphs that are linked to the calculations. The spreadsheet and accompanying graphs can be modified for specific applications. For example, the selection of atmospheric environmental tracers can be changed to reflect analytes of interest, the input tracer data can be modified to reflect local conditions or different time scales, and the analytes of interest can include other types of non-point-source contaminants such as nitrate (Böhlke, 2002). Previous versions of this workbook have been used to evaluate field data in studies of groundwater residence time and agricultural contamination (Böhlke and Denver, 1995; Focazio and others, 1998; Katz and others, 1999; Katz and others, 2001; Plummer and others, 2001; Böhlke and Krantz, 2003; Lindsey and others, 2003).

### Calculations

Calculations are based on equations given in Cook and Böhlke (2000), some of which were modified slightly from those of previous authors (Vogel, 1967; Zuber, 1986). Equations summarized here relate the concentration of a tracer in a sample collected at a specific time (Cs,ts) to the tracer concentrations in recharging waters of all ages represented in the sample. A water sample is considered to be a mixture of sub-parcels, each of which has a tracer concentration appropriate for the time it entered the system (Cin,ti), adjusted for first-order decay. The tracer concentration in the sample is given by the sum of the fractional contributions of all sub-parcels in the sample. The fractional contributions of the sub-parcels are given by various assumptions about their age-frequency distribution in the sample. Because the commonly used atmospheric environmental tracers are transient (their concentrations in air have changed over time), the tracer concentrations in any water sample will depend not only on the age distribution, but also on the date the sample was collected.

In the Piston-Flow Model (PFM), all the water in the sample has a single age (by definition equal to the mean age or residence time tPFM) (Chapter 3, this volume), and the time it entered the system (ti) is equal to ts-t. In this case, the concentration of a tracer in a sample collected at time ts is given by:

[1] Cs,ts = Cin,(ts-tPFM).

In the Binary Mixing Model (BMM), the sample is a mixture of two sub-parcels, one of which is young and contains measurable tracer and one of which is too old to have measurable tracer (Chapter 5, this volume). As currently formulated in the model worksheet, the young sub-parcel is assumed to have a discrete age. In this case, the concentration of a tracer in a sample collected at time ts is given by:

[2] Cs,ts = Xyoung · Cin,(ts-tyoung)

where Xyoung is the fraction of the sample consisting of the young sub-parcel and tyoung is the age of the young sub-parcel. Given a value of tyoung (which could be zero for dilution of modern water), the worksheet returns tracer concentrations for mixtures ranging from 0 to 100 % young, in 10 % increments.

In the Exponential Mixing Model (EMM), the sample consists of an infinite number of sub-parcels that have an exponential age distribution and an overall mean age of tEMM (Section 6.1, this volume). This could correspond to certain simple situations such as discharge from a spring or long-screen well in a homogeneous aquifer with evenly distributed recharge. In this case, tracer concentrations are calculated by stepping forward numerically through time and adding appropriate increments for each sub-parcel. At each time step:

[3] Cs,ti = Cs,(ti-1) + [1/tEMM] · [Cin,ti-Cs,ti-1],

where Cs,ti is the concentration of tracer in the mixture at time ti, Cs,(ti-1) is the concentration in the mixture at the previous time step, and Cin,ti is the concentration in the new recharge added to the mixture in the time step.

In the Exponential-Piston Model (EPM), the sample consists of an infinite number of sub-parcels that have an exponential age distribution (EMM), but the whole assemblage is shifted in time (Section 6.3, this volume). This could correspond to a situation in which an aquifer receives distributed recharge in an upgradient unconfined area, then continues beneath a downgradient area that does not receive recharge (e.g., confined area). The EMM age distribution is established in the recharge area, then all sub-parcels in the mixture age together as the water moves laterally beneath the are without recharge. The same result would be obtained for a perfectly soluble tracer in the reverse situation in which the sample represents an aquifer receiving distributed recharge following piston flow through an unsaturated zone. That is, the EPM model and its reverse (PEM) would be indistinguishable for a tracer like 3H. The procedure used in the model worksheet for the EPM is as follows: (1) Assign an age to the PFM part of the flow system (tPFM) and a ratio of tPFM/tEMM and use these values to calculate the total mean age (tEPM) and the mean age of the EMM part of the system (tEMM). (2) Locate the EMM tracer concentration result corresponding to tEMM for a displaced sample date of ts-tPFM. (3) Add first-order decay to the displaced EMM result for the time represented by tPFM (decay was already accounted for in the EMM calculation).

### Explanation of workbook TRACERMODEL1a

General: TRACERMODEL1 consists of two worksheets and several charts. The first worksheet ("models") contains the tracer calculations for hypothetical age distributions and the second worksheet ("samples") contains measurements for representative samples for comparison with the models (real data can be substituted here). In the models worksheet, bold italic entries in shaded boxes are entries made by the user. All other entries should be left alone except when modifying the calculations. The models worksheet is arranged as follows:

Column A: Times (in years) for tracer input values. The time interval for the input data (Cell A6, currently 2 rows per year) is used in the decay equation and elsewhere. This can be changed to accommodate other input data sets (e.g., annual, monthly, etc.) as long as the table is filled at the corresponding time scale for all tracers of interest and the entry in Cell A6 is correct.

Columns C-G: Tracer input values. The current entries (CFC-11, CFC-12, CFC-113, SF6, 3H) represent atmospheric abundances in the mid-Atlantic region of North America. The current entries for the gases are atmospheric mixing ratios in parts per trillion by volume (pptv) from Table 13.2-1 (this volume). Model output in these units is generalized and can be compared to measurements from water samples if the measurements have been converted to equilibrium mixing ratios at appropriate temperatures and pressures. Alternatively, the input tracer values can be replaced with equilibrium aqueous concentrations for a specific set of conditions. The entries for 3H are from the Washington DC record (IAEA GNIP; R.L. Michel, written communication, 2004). Concentrations of 3H before 1959 and after 2000 are estimated. Cells C6:G6 contain half-lives for the tracers, which can be altered to evaluate effects of first-order decay. An alternative tracer (e.g., 85Kr or nitrate) can be substituted for a current one simply by replacing the entries in one of these columns.

Columns J-N: Interactive exponential mixing model. The mean age can be changed here (Cell J6) for a quick comparison of different tracers in a sample collected at any time. This section is independent of the larger calculation blocks to the right.

Columns S-Z: Comparison of PFM, EM, EPM, and BMM results for a single tracer (currently CFC-11). For PFM (Columns S and T), EMM (Columns U and V), and EPM (Columns W and X), mean ages are tabulated with corresponding tracer concentrations. For BMM (Columns Y and Z), results for mixtures of varying proportions of old and young water are given. Important entries here are the sample date (Cell S6, applies to all models), the ratio of tPFM/tEMM (Cell W6, for the EPM model), and the age of the young fraction (Cell Z6, BMM model). The values of mean age considered by the EMM calculations (Cells AD6-BL6) can be altered if necessary (entries in this section are used for all tracers in the worksheet).

Columns AA-BL: The worksheet block for EMM calculations for various mean ages and collection times. These data are used to fill in the summaries in Columns S-Z and they also are used directly in some types of graphs (see below).

Columns BO-IV: Four more blocks of data and calculations identical to those in Columns S-BL, but for different tracers. The calculations in these other blocks refer to the input data and half-lives in appropriate columns from C-G.

Graphs: Representative plot types are shown to illustrate different approaches to data evaluation. Hypothetical tracer concentrations from any of the models can be plotted against sample dates, mean ages, and each other. Measurements added to these and other related plots can be evaluated for concordance with different model assumptions, for concordance among different tracers in a sample, and for evidence of degradation or contamination. Measured values are shown in the tracer-tracer plots for 3 hypothetical samples that yield concordant results for different age distributions. All tracer measurements in each of these samples are consistent if interpreted as follows: Sample #1 has a single discrete age (tPFM) of 14 years; sample #2 is a binary mixture in which the young sub-parcel is 60 % of the sample and has an age (tyoung) of 14 years; sample #3 is an exponential mixture with a mean age (tEMM) of 14 years. Other graphics programs may provide additional options for the presentation of the data.

### Explanation of workbook TRACERMODEL1b

General: TRACERMODEL1b is similar to TRACERMODEL1a except that TRACERMODEL1b includes provision for evaluating 3H-3He data. This is done by treating 3H both as a stable species (3H(0)) and as a radioactive species (3H) and reserving one block of calculations for the ratio 3H/3H(0). The remaining two calculation blocks can be used for any other tracers (currently CFC-12 and SF6).

Columns S-BL: This block is different from all the others because the results are calculated directly by dividing the model results for radioactive 3H (3H) by the model results for "stable" 3H (3H(0)). Results of these calculations can be compared to 3H-3He measurements by assuming 3H(0) = 3H + tritiogenic 3He (see "samples" worksheet).

### Disclaimer

The current version of TRACERMODEL1 contains several modifications from previous versions that have not been tested fully. Please report errors or suggestions to the author. Also, because the worksheets are not protected from accidental changes, users are advised to keep unaltered backup copies.

### References

Böhlke, J.K., 2002, Groundwater recharge and agricultural contamination: Hydrogeology Journal, v. 10, p. 153-179 [Erratum, 2002, Hydrogeology Journal, v. 10, p. 438-439].

Böhlke, J.K., and Denver, J.M., 1995, Combined use of groundwater dating, chemical, and isotopic analyses to resolve the history and fate of nitrate contamination in two agricultural watersheds, Atlantic coastal plain, Maryland: Water Resources Research, v. 31, p. 2319-2339.

Böhlke, J.K., and Krantz, D.E., 2003, Isotope geochemistry and chronology of offshore ground water beneath Indian River Bay, Delaware: U.S. Geological Survey Water-Resources Investigations Report 03-4192, 37 p, http://pubs.water.usgs.gov/wri03-4192.

Cook, P.G., and Böhlke, J.K., 2000, Determining timescales for groundwater flow and solute transport, in Cook, P. G., and Herczeg, A., eds., Environmental tracers in subsurface hydrology: Kluwer Academic Publishers, Boston, p. 1-30.

Cook, P.G., and Herczeg, A.L., eds., 2000, Environmental tracers in subsurface hydrology: Kluwer Academic Publishers, Boston, 529 p.

Focazio, M.J., Plummer, L.N., Böhlke, J.K., Busenberg, E., Bachman, L.J., and Powars, D.S., 1998, Preliminary estimates of residence times and apparent ages of ground water in the Chesapeake Bay watershed, and water-quality data from a survey of springs: U.S. Geological Survey Water Resources Investigations Report 97-4225, 75 p.

Katz, B.G., Böhlke, J.K., and Hornsby, H.D., 2001, Timescales for nitrate contamination of spring waters, northern Florida, USA: Chemical Geology, v. 179, p. 167-186.

Katz, B.G., Hornsby, H.D., Böhlke, J.K., and Mokray, M.F., 1999, Sources and chronology of nitrate contamination in spring waters, Suwanee River basin, Florida: U.S. Geological Survey Water Resources Investigations Report 99-4252, 54 p.

Lindsey, B.D., Phillips, S.W., Donnelly, C.A., Speiran, G.K., Plummer, L.N., Böhlke, J.K, Focazio, M.J., Burton, W.C., and Busenberg, E., eds., 2003, Residence times and nitrate transport in ground water discharging to streams in the Chesapeake Bay Watershed: U.S. Geological Survey Water-Resources Investigations Report 03-4035, 201 p, http://pa.water.usgs.gov/reports/wrir03-4035.pdf.

Plummer, L.N., Busenberg, E., Böhlke, J.K., Nelms, D.L., Michel, R.L., and Schlosser, P., 2001, Groundwater residence times in Shenandoah National Park, Blue Ridge Mountains, Virginia, USA: a multi-tracer approach: Chemical Geology, v. 179, p. 93-111.

Vogel, J.C., 1967, Investigation of groundwater flow with radiocarbon: Isotopes in Hydrology, International Atomic Energy Agency, Vienna, p. 355-368.

Zuber, A., 1986, Mathematical models for the interpretation of environmental radioisotopes in groundwater systems., in Fritz, P., and Fontes, J. C., eds., Handbook of Environmental Geochemistry, Volume 2, The Terrestrial Environment B: Elsevier, p. 1-59.