5.4 Calculation of Tracer Age
As a transient tracer, the general concept of [latex]{SF}_{6}[/latex] dating involves comparing a measured concentration in a water sample with historical concentrations in the atmosphere. Henry’s Law is used to relate the measured aqueous concentration of a sample to the equilibrium gas‑phase concentration. Because the Henry Coefficient is temperature dependent, and because the partial pressure of [latex]{SF}_{6}[/latex] in the atmosphere depends on total pressure, both the recharge temperature and pressure (i.e., elevation) are needed for age dating. Furthermore, if groundwater contains excess air, and therefore some excess SF6 above equilibrium solubility, the amount of excess air is also required.
The temperature and salinity dependence of the Henry Coefficient was investigated by Bullister and others (2002), who provided an empirical equation for calculating the equilibrium constant ([latex]K_{h}[/latex]) for the solubility form of Henry’s Law as shown by Equation (23).
| [latex]C = K_{h}(T,S)\left( P - e(T) \right)[/latex] | (23) |
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where:
| C | = | aqueous concentration (molM-1, typically in moles/kg) |
|---|---|---|
| Kh | = | temperature- and salinity-dependent equilibrium constant (molM‑2LT2 typically in moles kg‑1 atm‑1) |
| x | = | dry air mole fraction (dimensionless) |
| P | = | atmospheric pressure (ML-1T-2, typically in atm) |
| e | = | temperature dependent vapor pressure of water (ML-1T-2, typically in atm) |
Values for [latex]K_{h}[/latex] and e at various temperatures for fresh water are shown in Table 6. If the recharge temperature (T), salinity (S), and elevation (and hence P) are known, Equation (23) can be solved for the dry air mole fraction (x) and then x can be computed for a measured aqueous concentration. The dry air mole fraction can then be compared to the history of [latex]{SF}_{6}[/latex] in the atmosphere to obtain a recharge date as illustrated in Figure 21.
Table 6 ‑ Values of Kh for SF6 and water vapor (e) at various temperatures for fresh water (salinity = 0).
| Temperature (°C) | Kh [mole kg-1 atm-1] | e [atm] |
|---|---|---|
| 0 | 0.0006216 | 0.0060 |
| 10 | 0.0003969 | 0.0121 |
| 20 | 0.0002739 | 0.0230 |
| 30 | 0.0002023 | 0.0418 |
| 40 | 0.0001587 | 0.0726 |
Kh from Bullister and others (2002)
e from Antoine (1888) and Section 2.3 of this book
Equation (23) represents equilibrium conditions and does not include the effects of excess air. Solomon and others (2010) derived an expression for the dry-air mole fraction—which is essentially equivalent to the mixing ratio—that includes excess air as given by the CE model presented in Section 2.2 and Equation (24).
| [latex]x = \frac{C}{K_{h}(P - e) + \frac{(1 - F)A}{1 + \frac{FA}{K_{h}(P - e)}}}[/latex] | (24) |
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where:
| x | = | dry air mole fraction (dimensionless) |
|---|---|---|
| C | = | aqueous concentration (molM-1, typically in mole/kg) |
| Kh | = | temperature‑ and salinity‑dependent equilibrium constant (molM‑2LT2 typically in moles kg‑1atm‑1) |
| P | = | atmospheric pressure (ML-1T-2, typically in atm) |
| e | = | temperature‑dependent vapor pressure of water (ML-1T-2, typically in atm) |
| A | = | volume of gas trapped in the porous media per unit mass of water (molM-1, typically in moles/kg) |
| F | = | fractionation factor (dimensionless) |
As discussed in Section 2, when gas bubbles completely dissolve, the fractionation factor (F) is 0 and the CE model reverts to the UA model; such that Equation (20) becomes Equation (25).
| [latex]x = \frac{C}{K_{h}(P - e) + A'}[/latex] | (25) |
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where:
| A’ | = | amount of excess air dissolved per unit mass of water (molM-1, typically in moles/kg) |
|---|
As discussed in Section 2.6, A in Equation (24) and A’ in Equation (25) are not generally equivalent except when air bubbles are fully dissolved as represented by F = 0.
A general procedure for calculating a tracer age using [latex]{SF}_{6}[/latex] is as follows:
- Measure the aqueous concentration (C) of [latex]{SF}_{6}[/latex] in a groundwater sample.
- Estimate the recharge elevation (Z) of the sample.
- Compute the recharge pressure (P) using Equation (5): [latex]P = P_{0}\exp\left( - \frac{Z}{Z_{s}} \right)[/latex] where [latex]P_{0}[/latex] is the pressure at Z[latex]=[/latex]0 (1 atm) and [latex]Z_{s}[/latex] is a scaling factor ([latex]Z_{s}[/latex] = 8,300 m for Z < 1,800 m).
- Estimate the recharge temperature for the sample. This could utilize noble gas thermometry, if available, or could be taken as the average annual temperature of the recharge zone for the sample.
- Estimate the recharge salinity for the sample. In many cases this is assumed to be zero for fresh water in the recharge area.
- Compute the equilibrium constant ([latex]K_{h}[/latex]) for the temperature and salinity of the recharge area. This could be computed using the equations of Bullister and others (2002) or could be approximated from the values given in Table 6.
- Compute the vapor pressure of water (e) at the recharge temperature. This could be computed using empirical equations such as Antoine (1888; Section 2.3) or could be interpolated from the values given in Table 6.
- Estimate the amount of excess air in the sample. If noble gas data are available, the values of A and F for the CE model or A’ for the UA model can be obtained by fitting theoretical values to measured values as discussed in Section 2.6. The units for A or A’ must match those of C. If a noble gas model provides A or A’ in units of ccSTP/g, these can be converted to mole/g by dividing by the molar volume of an ideal gas (22414 ml/mole).
- Compute the mole fraction (x) of [latex]{SF}_{6}[/latex] in the atmosphere that would have resulted in the measured aqueous concentration using (24) for the CE model or Equation (25) for the UA model.
- Compare the computed mole fraction—which is essentially equivalent to the mixing ratio—to the atmospheric history (Figure 20) to derive the estimated year the sample recharged.
- Compute the age as sample date minus recharge year.
Exercise 7⮧ provides an opportunity to practice calculating tracer age using [latex]{SF}_{6}[/latex].