WASR 8000 · Week 5
WASR 8000 · Environmental Tracers in Hydrology · Fall 2026

When Did This Water Last See the Sky?

An interactive primer on dating young groundwater — the decades-old water that supplies most wells and carries most contaminants. Three clocks that start the moment water loses contact with the atmosphere (³H/³He, the CFCs, SF₆), the corrections that set each clock to zero, the ways each clock breaks, and the lumped-parameter models that turn one concentration into a distribution of ages. Builds on Week 3; no tracer background needed.

Week 5 companion · read alongside the three readings ≈ 45–55 minutes 11 interactive demos · self-check quiz
1 · The opening question

A well is dated at 27 years. Fertilizer arrived 15 years ago. Is the nitrate there yet?

A water-supply well is sampled in 2017. Its dissolved SF₆ is 0.93 fmol L⁻¹ — the concentration that water in equilibrium with the 1990 atmosphere would carry, so the textbook reading is "recharged in 1990" — 27 years before the sample was drawn. The farmland upgradient switched to heavy nitrogen fertilizer 15 years ago, and nitrate travels with the water. (The situation is the worked example in Chambers et al.'s Fig. 4.) You are asked whether the well is at risk.

Chambers et al. run exactly this comparison. Give the well a single piston-flow age and its outflow contains no nitrate at all. Give it instead a broad (log-normal) age distribution with the same SF₆ concentration and the same mean — and more than 30% of its water is younger than 15 years, nitrate included. Same measurement, same "age," opposite management decisions. Their conclusion is the week's thesis: "it is the shape of the age distribution that will determine the 'breakthrough' of a contaminant."

That is Week 5 in one example. Week 3 argued in the abstract that a groundwater sample is a crowd of molecules with a distribution of ages. This week the argument gets teeth: the young-water tracers — ³H/³He, the chlorofluorocarbons, SF₆ — are the only clocks that can read the part of that distribution where contaminants live, and the lumped-parameter models (and the TracerLPM workbook that runs them) are how a handful of concentrations becomes a defensible statement about the shape.

The three readings divide the labour: Chambers et al. (2019) review the gas tracers and how their clocks are set and broken; Solomon & Cook (2000) do the same for tritium and its daughter helium-3; Jurgens, Böhlke & Eberts (2012) document the workbook that turns tracer concentrations into age distributions. This page is a map to all three, not a substitute for reading them.

2 · Young water, and three ways to date it

Why the under-100-year window gets its own week

Modern groundwater — recharged within the last ~50–100 years — is, by Chambers et al.'s reckoning, the most abundant and most accessible part of the active water cycle, roughly three times the volume of fresh surface water, and also the part most exposed to fertilizer, industry, land-use change and climate. It sits below the radiocarbon window (Week 3) and above the range where nothing happens fast enough to measure. Three clocks work here, and each starts the same way: the moment water stops exchanging with the atmosphere. Click each one.

Pick a clock to see how it keeps time, what it needs, and what it cannot see.

The Week 3 taxonomy still holds — decay clocks, accumulation clocks, event markers — but the young-water tracers are hybrids. ³H is a decaying tracer with a wildly transient input; ³He is its accumulating daughter; the CFCs and SF₆ are conservative gases whose "clock" is entirely the atmosphere's history, and whose "zero" has to be set by dissolving that history into water at the right temperature and pressure. Every difficulty this week lives at one of those two joints: the input history, or the setting of zero.
3 · The atmospheric calendar

Five curves the whole method hangs on

CFC-12 and CFC-113 production began in the early 1940s, CFC-11 and SF₆ in the 1950s (refrigeration and air-conditioning, semiconductor manufacture, electrical insulation). They leaked, mixed globally, and rose in the air until the Montreal Protocol bent the CFC curves over in the 1990s; SF₆, a greenhouse gas with no protocol, is still climbing. Tritium's history is a different animal: a cosmogenic background of a few TU, then thermonuclear tests from 1952 that pushed Northern-Hemisphere rain past 1,000 TU — a monthly mean above 5,000 TU at several stations, an annual mean of 3,278 TU at Ottawa in 1963 — followed by a long decay back toward background. Hover the chart.

Input histories, Northern Hemisphere

10 °C

CFCs and SF₆ in air

Left axis: CFCs · right axis: SF₆ (always ~100× less)

Tritium in precipitation, Ottawa

Log axis. Annual means (Fig. 13.1 plots monthly values); the pre-bomb level is the 8 TU TracerLPM assumes for Ottawa (Solomon & Cook: 3–6 TU in Europe and North America).

CFC-11 CFC-12 CFC-113 SF₆
year probed
hover either chart
CFC-11 / CFC-12 / CFC-113
pptv in air
SF₆
pptv in air
³H in rain
TU, that year's precipitation

Reconstructed histories: smoothed annual curves anchored to the USGS Reston Groundwater Dating Laboratory air curves (the data behind Chambers et al.'s Fig. 1) and to the Ottawa record as summarized by Solomon & Cook (Fig. 13.1). The Southern Hemisphere lags by 1–2 years for the gases, about one year for the tritium peak, and its peak was ~100× smaller (37.7 TU at Kaitoke, New Zealand, in 1964). For real work use the stored histories in TracerLPM.

What the shapes give you

  • A rising limb is a one-to-one calendar. Pre-1990 CFC concentrations, and SF₆ to this day, map a concentration to a single year.
  • Different shapes are a mixing detector. CFC-12 flattened while SF₆ kept climbing; a mixture of old and young water lands off the piston-flow curve in a CFC-12 vs SF₆ plot (Section 11).
  • A spike is a marker. The 1963 tritium peak is a dated horizon you can find in a profile (Section 8).

What the shapes take away

  • The Montreal bend. After ~1990 a single CFC concentration matches two years (Section 5).
  • A fading input. Tritium in Ottawa rain fell from 3,278 TU (1963) to 18.5 TU (1993); the Southern Hemisphere never had much. A lone ³H value rarely dates anything now.
  • Tiny numbers. Atmospheric equilibrium means picomoles per litre (CFCs) and femtomoles per litre (SF₆): the detection limits Chambers et al. quote are 0.01 pmol L⁻¹ and 0.1 fmol L⁻¹, and the same smallness makes contamination trivially easy (Section 6).
4 · Setting the clock to zero

Henry's law, and the four numbers you must know about the recharge zone

A gas tracer in water tells you nothing until you convert it back to the air it came from. That conversion is Henry's law — dissolved concentration = solubility × partial pressure — and the solubility depends on temperature (colder water holds more gas), the partial pressure on elevation (thinner air holds less), a little on salinity, and the whole budget on excess air: bubbles trapped and forced into solution as the water table rises, adding gas in atmospheric proportions on top of equilibrium. Chambers et al. call this setting the groundwater-dating "clock" to zero, and every age you read afterward inherits whatever you assumed here.

The clock-setter

Choose a tracer and the recharge conditions. The chart shows the concentration water would carry if it recharged in each year — the curve you match a sample against. Then set a measured concentration and read the recharge year(s) it implies. Move the temperature 2 °C and watch the year move.

10 °C
0 cm³ L⁻¹
0 m

Dissolved CFC-12 expected in water recharged each year

solubility at this T (Henry's)
mol kg⁻¹ atm⁻¹, freshwater
equilibrium share of the sample
the excess-air "correction factor" of Chambers et al. Fig. 3
implied recharge year(s)
piston flow, no mixing
if T were 2 °C warmer
the age shifts by…
Cw = KH(T, S) · x · P(z) + (EA / Vm) · x equilibrium + excess air; x = mixing ratio in air, Vm = 22,414 cm³ mol⁻¹

Temperature and elevation

Recharge temperature is best fixed from other dissolved gases — N₂/Ar or the noble gases (Week 7) — and in a pinch from shallow groundwater temperature. Chambers et al.: an uncertainty of ±2 °C typically shifts the age of a pre-1990 sample by less than 3 years. Elevation lowers pressure exponentially, but the lapse rate (6.5 °C km⁻¹) cools the recharge and raises solubility, so the two corrections tend to cancel rather than compound. Salinity at infiltration is usually near zero; even 1,000 µS cm⁻¹ water (<1‰) changes the answer by under 1%.

Excess air and its opposite

A few cm³ L⁻¹ of excess air is typical of sedimentary aquifers, but it varies, and it matters most for the least soluble gases — SF₆ and ³He — where a small bubble is a large fraction of a tiny equilibrium amount (compare the CFC-12 and SF₆ curves of Fig. 3 in the clock-setter above). The reverse problem, degassing, is increasingly common: methane from anoxic organic matter or N₂ from denitrification forms a gas phase below the water table that strips dissolved gases — SF₆ can be lost entirely. Both are diagnosed with the noble gases, which is why Chambers et al. recommend them as the standing companion measurement.

5 · When one number gives two answers

The Montreal bend, and why you measure three gases

Chambers et al.'s Fig. 1 makes the point with a single red line: a CFC-11 concentration a little below the peak matches the rising limb in the late 1980s and the falling limb in 2005. One measurement, two recharge years. Their fix is not a better measurement but a second tracer with a different shape — SF₆, still rising — whose value picks out the right branch (their dashed line, drawn for a recharge year of 2005). (All of this still assumes piston flow; mixing is Section 9's problem.)

Which branch?

not measured

CFC-11 (left) and SF₆ (right), dissolved at 10 °C, sampled 2017

CFC-11 alone says
candidate recharge years
SF₆ says
SF₆ has never bent
together
agreement, or a warning
Notice what the second tracer buys you when the two disagree: not a date, but a diagnosis. A CFC-11 year that no SF₆ year can match means one of the two gases has been contaminated, degraded, or degassed — Section 6 — or the sample is a mixture — Section 9. Chambers et al.'s recommendation is to measure CFC-11, CFC-12 and SF₆ together every time, precisely so the disagreements can do their work.
6 · What breaks the calendar clocks

Non-correctable, but diagnosable

Temperature, pressure and salinity are corrections; Chambers et al. reserve a harsher word — non-correctable — for the processes that add or remove tracer after recharge. You cannot fix them with arithmetic. You can only detect them, usually because three gases with different chemistries stop agreeing. Learn the signatures, then try the diagnosis board.

addsContamination

Equilibrium concentrations are so small (pmol L⁻¹) that "only minor contamination is required to render the CFCs somewhat meaningless in terms of tracers of groundwater age." Chambers et al.'s scale-setter: one-tenth of the CFC-12 in a refrigerator of older design could push a moderately sized aquifer above ten times modern atmospheric equilibrium. Sources are industrial activity and leaking landfills, from the air or from below — worst in urban and peri-urban aquifers where fractured horizons carry them fast. The sample reads "younger than today," which is impossible, or the CFCs disagree wildly with each other. SF₆ is far less prone (high-voltage switchgear, Mg and Al smelting, landfills), but has its own problem, next.

addsTerrigenic SF₆

Rocks make SF₆. Anomalously high concentrations occur in some sedimentary aquifers and in aquifers with fluorite or metallic-sulphide mineralization — "a far greater concern" for SF₆ than industrial leaks. The signature is SF₆ above any possible atmospheric equilibrium while the CFCs look sane, or an SF₆ "age" much younger than the CFC ages. (Old, tracer-free groundwater with measurable SF₆ is the giveaway.)

removesMicrobial degradation

Under anoxic conditions the CFCs are consumed, CFC-11 typically faster than CFC-12. Under strongly reducing conditions they vanish below detection. So the field parameters — dissolved oxygen, redox potential — belong on the sampling sheet, and a CFC-11 age older than the CFC-12 age of the same sample is the classic tell. A detectable DO does not clear you: redox can change along the path, and a mixed sample can carry oxic water from one flow line and stripped water from another. SF₆ does not degrade.

removesSorption, degassing, the vadose zone

Sorption is minor except in organic-rich matrices, and increases in the order CFC-12 < CFC-11 < CFC-113. Degassing (methanogenesis, denitrification) strips the least soluble gas first — SF₆ can go to zero — and is caught by the noble gases. A thick unsaturated zone delays the arrival of the atmospheric signal at the water table, by an amount set by each gas's diffusion coefficient, solubility and the soil moisture; gas-phase diffusion is fast enough that in most practical situations the lag is negligible, but "most" is doing work in that sentence. Thermal effects (CFC-12 preferentially lost) have been seen at two springs, in Belgium and the UK, and nowhere else.

The diagnosis board

Each row is a pattern in the three-gas data from one well. Pick the most likely explanation. (One process per row is the best fit; the feedback explains the runner-up.)

Chambers et al. close their theory section with a line worth keeping: contamination "may be useful for fingerprinting water sources." A CFC-12 concentration ten times atmospheric is useless as a clock and excellent as a label — it says this water passed under that town. The failure of one question is often the answer to another.
7 · The self-starting clock

³H/³He: a clock that carries its own zero

Tritium is hydrogen — it is part of the water molecule, so it goes exactly where the water goes and reacts with nothing. It decays by beta emission to ³He with a half-life of 12.32 years (Lucas & Unterweger, 2000; Solomon & Cook's chapter uses the older 12.43). The trouble with tritium alone is the input: to turn a concentration into an age you must know what the rain contained, and the rain's record is a decaying spike. Tolstikhin & Kamensky's 1969 idea sidesteps the whole problem. Measure the daughter too. If the ³He produced by decay stays in the water, the ratio of daughter to surviving parent depends only on elapsed time:

t = (1/λ) · ln( 1 + ³He* / ³H ) Solomon & Cook eq. 13.2; Chambers et al. eq. 2 — with ³He* the tritiogenic ³He, in the same units as ³H

The catch is in the asterisk. Groundwater holds ³He from three other sources — atmospheric solubility, excess air, and nuclear reactions in the rock (plus mantle helium in volcanic terrain) — and each must be subtracted before the ratio means anything. One tritium unit is one ³H¹HO molecule per 10¹⁸ molecules of water; when a TU of tritium has fully decayed it has produced 2.487 pcm³ (STP) of ³He per kilogram of water — against an atmospheric-equilibrium background of about 63.7 pcm³ kg⁻¹ at 10 °C, and up to ~100 pcm³ kg⁻¹ more from excess air. Small signal, large background: the accounting is the method.

The ³H/³He clock

Set the tritium the water carried when it crossed the water table and let it age. The bars show what is in the water; the readouts show the apparent age the ratio returns — and how much an unrecognized cm³ of excess air would corrupt it. Solomon & Cook's worked example (20 TU, 2 years vs 30 years) is the first preset.

20 TU
2 yr
0 cm³ kg⁻¹

What the water holds (TU-equivalents)

Parent, daughter and their sum through time

³H remaining
TU
³He* produced
TU-equiv · pcm³ kg⁻¹
³H/³He apparent age
eq. 13.2
age error from the excess air
if its ³He were mistaken for ³He*

The helium ledger (eqs. 13.3–13.6)

³Hetotal = ³Heatm + ³He* + ³Henuc + ³Heman, and ³Heatm = ³Hesol + ³Hee. The solubility term comes from temperature (helium solubility changes by only about 0.5% per °C near 10 °C, and the dissolved ³He/⁴He ratio, 1.36 × 10⁻⁶, sits just below air's 1.384 × 10⁻⁶). The excess-air term is fixed from neon, which has no source but the atmosphere. The nucleogenic term rides on radiogenic ⁴He: crustal helium has ³He/⁴He of order 10⁻⁸, a hundred times below air, so it matters only when ⁴Herad is large — easy to see, hard to convert. With no radiogenic helium and no excess air the whole ledger collapses to ³He* = ⁴Hem(R₀ − Rsol): the measured ³He/⁴He ratio minus the equilibrium one, times the helium you measured.

Where it is sensitive, and where it is not

  • Young water is fragile. Two-year-old 20-TU water holds 5.25 × 10⁻¹² cm³ kg⁻¹ of ³He* (5.29 with the 12.32-year half-life the demo uses) — about 10% of the 6.4 × 10⁻¹¹ dissolved from the air. One cm³ kg⁻¹ of excess air adds 7.2 × 10⁻¹², as much as the signal.
  • Bomb-era water is robust — because of what it started with. At 30 years the same water's ³He* has grown to 4.0 × 10⁻¹¹, comparable to the solubility background, so "the sensitivity of ³H/³He ages to determinations of atmospheric ³He decreases the older the water becomes." The error from an unrecognized bubble, though, scales with 1/³H₀: Cook & Solomon's figures — roughly −5 years per cm³ kg⁻¹ of excess air for very young water, falling to −0.25 by 25 years — fall because water recharged in the early 1970s carried ten times the tritium of today's rain.
  • The clock does not start until the water table. In the vadose zone, helium exchanges freely with soil gas, so ³He* is lost; Solomon & Cook's Fig. 13.11 shows ³H/³He ages offset from total travel time by exactly the unsaturated-zone transit.
  • The clock can leak. Below the water table, ³He can diffuse back up and out. With an effective ³He diffusion coefficient of 1.3 × 10⁻⁴ m² day⁻¹, loss is under 20% if the vertical velocity exceeds ~0.1 m yr⁻¹ (≈30 mm yr⁻¹ recharge at 30% porosity), under 1% above 0.5 m yr⁻¹; below ~0.01 m yr⁻¹ diffusion dominates and little travel-time information survives (Fig. 13.9). Slow recharge, no clock.
  • Mantle helium lies. Its ³He/⁴He is ~10× atmospheric, indistinguishable from decay. Rare in young groundwater; a real problem near young volcanic rock and in geothermal water.
8 · The marker horizon

Watching the bomb peak sink

The 1963–64 rain is a dated layer. Smith et al. found it 4 m down in the English Chalk in October 1968 — a 600-TU spike, four to five years after the peak, hence 0.9 m yr⁻¹ downward. Andersen & Sevel tracked it through 22 m of Danish outwash at 4.5 m yr⁻¹. At Sturgeon Falls, Ontario, Solomon et al. (1993) found the ³H and ³He peaks together at 11 m in 1991 — with the ³H peak shrinking between the 1986 and 1991 samplings while the ³He peak grew. The elevator below animates the idea: profile shape from the input history, decay in the parent, ingrowth in the daughter, and a ³H/³He age line whose slope is the velocity.

The bomb-peak elevator

One-dimensional downward flow at velocity v below a water table, driven by the Ottawa input; the unsaturated-zone lag shifts the ³H/³He clock. (The Smith et al. Chalk profile was in the unsaturated zone; here its 0.9 m yr⁻¹ is run below a water table.) Presets transcribe the reported velocities; dispersivities and lags are illustrative. Drag the sampling year forward and watch the parent fade, the daughter grow, and their sum — which "effectively corrects for radioactive decay" (Schlosser et al., 1988) — keep the peak readable for decades longer than tritium alone.

0.45 m yr⁻¹
1991
0.20 m
3 yr

Concentration vs depth below the water table

Age vs depth

³H ³He* (TU-equiv) ³H + ³He* (initial tritium) total travel time ³H/³He age
depth of the 1963 layer
v × years since (after the lag)
peak ³H today vs 1963 rain
what decay alone has done
age gradient
1/v — the recharge rate is v × porosity

Three ways tritium alone has been read

  • Peak displacement (Fig. 13.2–13.3): depth of the marker ÷ time = velocity; × water content = recharge. Fails where vapour transport or root uptake moves the tracer without moving the water, which is why Tyler & Walker insist on velocities measured below the root zone, and why arid profiles often carry their maximum tritium right at the surface.
  • Mass balance (eq. 13.1): total tritium stored in the profile ÷ decay-corrected tritium that fell = the fraction of rain that recharged. Akrotiri, Cyprus: 75.2 TU m in the ground against 849 TU m fallen → 36 mm yr⁻¹, versus 48–63 mm yr⁻¹ from peak displacement — "reasonably in agreement," which is also a statement about the method's precision. No flow assumption needed, but the local input must be known accurately — Atakan et al. put the error in estimating local fallout at a German site near 20%.
  • Discharge models (Fig. 13.6): for a spring or open borehole, a well-mixed-reservoir model relates tritium to recharge/volume — Polda Basin, 1.0–4.9 TU → R/V of 0.005–0.02 yr⁻¹ → ~30 mm yr⁻¹. Time series beat single values: Siegenthaler's spring falling from 290 to 230 TU over 3 years fit an exponential model with a 13-year mean residence time. A single value is non-unique (the Wairau River's decay line cuts the input curve twice, Fig. 13.5).

What ³H/³He added

  • Sturgeon Falls. The 1963 peak at 11 m plus a 2-D flow model gave recharge of 0.15 m yr⁻¹; the ³H/³He age gradient alone gave a vertical velocity of ~0.45 m yr⁻¹, which at porosity 0.35 is 0.16 m yr⁻¹. Same answer — but the second needed only a few samples near the water table, not a profile deep enough to find the peak.
  • Szigetköz, Hungary. Bank infiltration from the Danube makes flow one-dimensional and horizontal; apparent ³H/³He age vs distance gives ~530 m yr⁻¹ (Fig. 13.12), from ordinary long-screened wells.
  • Cape Cod. A contaminant plume marks the flow path; ages along it give 90 ± 20 m yr⁻¹, date the spill (1975 ± 3), and locate its source to ±250 m — neither the time nor the place of the release was known beforehand (Fig. 13.13). Portniaguine & Solomon then inverted ages and heads together: heads constrain recharge/conductivity, ages constrain recharge/porosity; jointly they pin conductivity to ~50%, far better than pumping tests.
  • Dispersion is the caveat. ³H and ³He diffuse at rates differing by ~4×, so hydrodynamic dispersion separates apparent age from travel time — mostly at and below the bomb peak (Fig. 13.10): minimal on Long Island sand, ~35% near the peak in Ontario silty sand.
9 · From a number to a distribution

The lumped-parameter zoo

Every age in Sections 4–8 assumed piston flow: one parcel, one path, one date. Real samples come from screens metres long, from springs where flow lines converge, from aquifers that disperse. Lumped-parameter models (LPMs) keep the arithmetic simple and drop the fiction: the aquifer is a black box whose only property is its exit-age distribution g(t) — the fraction of the sample that is each age. Maloszewski & Zuber gave the field its classic shapes in 1982; TracerLPM ships five, and lets any two be blended. Click a model to see the aquifer it stands for; then push its parameters.

Age-distribution explorer

The distribution g(t) for the selected model, on a fixed 0–150-year axis so that every slider visibly reshapes it. The readouts answer the Section 1 question directly: how much of this water is younger than 15 years?

27 yr

Exit-age distribution

younger than 15 yr
the nitrate question
younger than 50 yr
inside the young-tracer window
median age
half the water is older
youngest water present
the delayed-response clue
Cout(t) = ∫−∞t Cin(t′) · g(t − t′) · e−λ(t − t′) dt′ the convolution: Chambers et al. eq. 1; Jurgens et al. eq. 1; Cook & Böhlke eq. 1.6 (Week 3)
Two models, one distribution. The EPM and the PEM are drawn for different plumbing — a recharge area followed by a confined stretch, versus a well that skips the top of the aquifer — but Jurgens et al. show they produce identical g(t) whenever the mean ages are equal and EPM ratio = ln(PEM ratio + 1) (their eq. 12). Tracer data cannot tell them apart; only your knowledge of the site can. That is the general lesson of this zoo: the tracers choose the parameters, the hydrogeology chooses the model.
10 · The convolution machine

How a distribution becomes a concentration — and a concentration becomes a forecast

The convolution integral is the hinge of the week. Read it right to left: for every parcel in the sample, look up what the atmosphere (or the rain) held the year that parcel recharged, decay it for the years since, weight it by how much of the sample is that old, and add. Chambers et al.'s Fig. 4 draws it as a box with an input curve going in and a number coming out. The machine below lets you turn the crank.

Turn the crank

Pick a tracer, a model and a sampling year. The upper panel shows the input history; the shaded curve is the age distribution laid on the calendar (the sample's parcels, by recharge year); the lower panel is their product — the pieces that add up to the measured concentration.

20 yr
2017

Input history and the sample's parcels

Contribution of each recharge year to the sample

modelled concentration in the sample
Cout
the same tracer, piston flow at the same τ
what "one age" would predict
share of the sample younger than 15 yr
from g(t) alone

Mixtures and apparent ages

Blend water of one age with tracer-free old water, or with water of a second age, and ask each clock what it reads. The ³H/³He clock is a ratio, so diluting with tritium-free water does not move it; the calendar clocks read the diluted concentration as older. Neither is wrong about the water it can see.

50%
10 yr
tracer-free
true mean age of the blend
f·t₁ + (1−f)·t₂
³H/³He apparent age
sampled 2026, Ottawa-type input
CFC-12 piston-flow age
10 °C, no excess air
SF₆ piston-flow age
10 °C, no excess air
11 · Tracer–tracer plots

Reading mixing without a model of the aquifer

Because the input histories have different shapes, every model traces a different path through a plot of one tracer against another, with age ticking along each path. Chambers et al.'s Fig. 5 (CFC-12 vs SF₆) shows the piston-flow "bow," the binary-mixing straight line from tracer-free water to modern recharge, and the exponential family between them. A sample lands somewhere on this map, and where it lands is a statement about its age distribution — obtained, in Chambers et al.'s words (citing Gooddy et al., 2006), "without any prior knowledge of the physical properties of the aquifer system." This is the tracer-tracer method that TracerLPM's first workgroup is built around.

Tracer–tracer explorer

Choose the axes and the sampling year. Curves are the model outputs for mean ages 0–200 years (dots every 10 years, labels every 20). The dashed line is binary mixing between tracer-free water and water recharged in the sampling year, in 20% steps. Set a sample with the sliders and read the nearest model's mean age — TracerLPM's "lookup mean ages," by eye.

2017
1.00

CFC-12 vs SF₆, dissolved at 10 °C

PFM EMM EPM DM binary mixing (tracer-free + modern)
nearest model point
by relative distance, as the workbook's lookup does
binary-mixing reading
fraction of modern water, if on the dashed line
piston-flow reading
one age per axis — do they agree?
The bow is the whole argument in one curve. Along it the two piston-flow ages agree; off it they cannot both be right, and the direction of the miss says why — toward the origin along a straight line means dilution by old water, into the exponential family means mixing across ages, above the SF₆ axis means terrigenic SF₆ or contamination. But Chambers et al. are blunt about the limit: "the inversion of tracer concentrations to derive age distributions is a non-unique process, even when several tracers are measured simultaneously." A tracer–tracer plot narrows the family of possible distributions; it never picks one. Jurgens et al. make the same point as a rule of thumb — you need at least p + 1 tracers for a model with p parameters, and even then several local minima may fit.
12 · The workbook

TracerLPM: the argument, operationalized

Jurgens, Böhlke & Eberts wrote TracerLPM for the USGS's "Transport of Anthropogenic and Natural Contaminants to supply wells" studies, after finding that piston-flow ages of public-supply wells "are often misleading," that contaminant forecasts built on distributions differ substantially from those built on single ages, and that LPMs calibrated to tracers can reproduce particle-tracking distributions from full flow models at a fraction of the cost. It is an Excel workbook with a compiled add-in that evaluates the convolution monthly for each model. Its worksheets encode a method, and the method is the point.

The workflow, worksheet by worksheet

Samplestracers, dates, corrected concentrations TracerInputinput histories, UZ lag, scaling …Outputup to 4 models × 1,000 mean ages …Graphstracer–tracer or time-series, by eye …Fitsbest-fit within bounds you set SavedModelAgesthe record LPM_AgeDistribution · Forecastingthe payoff
  • Corrections come first. The Samples sheet expects concentrations already corrected for recharge temperature, elevation, excess air, terrigenic helium and degradation — Sections 4 and 7 are prerequisites, not options. Gases are entered as the atmospheric mixing ratio (pptv) they were in equilibrium with, which strips local temperature and elevation out of the comparison.
  • Conceptualize, then look, then fit. Start with PFM and EMM — one parameter each — and see where the sample falls. Off both curves, try EPM, PEM or DM; off all of them, suspect a binary mixture or a broken tracer. Only then run the fit, with bounds no wider than ~20 years on age and ~0.2 on a shape parameter, because "multiple local minima in the residual errors … can exist."
  • Count your unknowns. At least p + 1 tracers for p parameters: PFM/EMM need 2, EPM/PEM/DM need 3, a BMM-DM-DM needs 6 — usually impossible at one well, hence the habit of borrowing constraints from nearby wells.
  • Tritium is four tracers. Enter ³H once and the workbook derives ³Hetrit (eq. 16), initial tritium ³H₀ = ³H + ³Hetrit, and the ratio ³H/³H₀ — each with a different sensitivity to the distribution.
  • Total mean age = saturated-zone mean + UZ travel time (eq. 22), with the UZ lag settable per tracer, because soluble ³H rides the water down while the CFCs can outrun it through soil air.

Forecasting: why the shape decides the future

A nitrate history of the Modesto type — rising loads from the 1950s, held constant from 2000, then cut to zero — pushed through two calibrated distributions. Jurgens et al.'s two examples had opposite tempers: the Modesto supply well (a PEM with mean age ~65 years and no water younger than the screen's depth allows) responds late and keeps rising after inputs stabilize; the upper Missouri River (a BMM-EMM-PFM: 84% groundwater with a 4.3-year mean, 16% prompt runoff) responds at once and then trails. Drag the cessation year and compare.

2020
0 yr⁻¹

Nitrate at the water table (dashed) and in the discharge (solid)

peak in discharge
year and fraction of the input maximum
still above half the input maximum until
the legacy after cessation
first arrival above 5% of input
set by the youngest water present
13 · Reading the three sources

Three readings, three blocks, one method

Chambers et al. supply the gas clocks and the discipline of setting them; Solomon & Cook supply the tritium clock and the field cases that made it credible; Jurgens et al. supply the instrument that turns concentrations into distributions. The guides below are maps, not substitutes: what each asks, where its load-bearing figures live, which equations to recognize, which numbers to be able to quote, and where a careful reader might press. The notes and the critique are yours to write.

Block 1 · Core reading

Chambers, L. A., Gooddy, D. C., & Binley, A. M. (2019). Use and application of CFC-11, CFC-12, CFC-113 and SF₆ as environmental tracers of groundwater residence time: A review. Geoscience Frontiers, 10(5), 1643–1652.

Ten open-access pages (CC BY-NC-ND) that walk from sampling to the meaning of "age," recommending complementary techniques at every turn. doi:10.1016/j.gsf.2018.02.017

Block 2 · Supplementary reading

Solomon, D. K., & Cook, P. G. (2000). ³H and ³He. Ch. 13 of Cook & Herczeg (eds.), Environmental Tracers in Subsurface Hydrology. Kluwer, pp. 397–424.

The canonical method chapter: sources and units, sampling, four ways to read tritium alone, then the ³H/³He method with its helium ledger, its limitations, and five field applications. doi:10.1007/978-1-4615-4557-6_13

Block 3 · Tool (hands-on demonstration)

Jurgens, B. C., Böhlke, J. K., & Eberts, S. M. (2012). TracerLPM (Version 1): An Excel® workbook for interpreting groundwater age distributions from environmental tracer data. U.S. Geological Survey Techniques and Methods 4-F3, 60 p.

Public domain. Model theory (pp. 3–9), tracer notes (pp. 10–14), a worksheet-by-worksheet manual (pp. 14–35), and three worked examples (pp. 35–48) whose numbers are the ones to know. doi:10.3133/tm4F3

Cross-paper synthesis, in one line: Chambers et al. tell you how to set a clock and when to distrust it, Solomon & Cook show a clock that sets itself and still needs a ledger, and Jurgens et al. insist that no clock reads a single time — only a distribution, chosen by hydrogeology and constrained by tracers. The question that carries forward: for the water your proposal cares about, which of this week's clocks can see it, and what would you have to measure alongside the clock to believe its reading?
14 · Practice

Four situations to reason through

Each scenario gives the kind of evidence a Week 5 reader should now be able to interrogate. Choose the reading you would defend first, then compare with the debrief.

Habit to keep: before quoting a young-water age, say aloud which corrections were applied to set the zero, which of the three gases (or which helium terms) were measured to check it, and which distribution the "age" is the mean of. An age that cannot survive those three questions is a concentration wearing a costume.
15 · Self-check

Ten questions before the meeting

Immediate feedback, no grade, no record. If you miss one, the linked section is the fix.

16 · Vocabulary

Glossary

The Week 5 working vocabulary, with a few terms you will meet again in Weeks 6, 7 and 11. Search or browse.

17 · Where to go next

The Week 5 readings — and two optional companions

Papers are not posted for this course: retrieving them from the citation and DOI — via the UGA Libraries, GALILEO, or the publisher — is part of the training. Two of this week's three are open access, and the workbook itself is a free download from the USGS.

CORE

Chambers, L. A., Gooddy, D. C., & Binley, A. M. (2019). Use and application of CFC-11, CFC-12, CFC-113 and SF₆ as environmental tracers of groundwater residence time: A review. Geoscience Frontiers, 10(5), 1643–1652.

Sampling and analysis; setting the clock (temperature, excess air, pressure, salinity); the non-correctables (contamination, terrigenic SF₆, degradation); three concepts of age; LPMs and tracer–tracer plots; links to flow models and hydrochemistry. Open access. doi:10.1016/j.gsf.2018.02.017

METHOD

Solomon, D. K., & Cook, P. G. (2000). ³H and ³He. In Cook & Herczeg (eds.), Environmental Tracers in Subsurface Hydrology (pp. 397–424). Kluwer.

Tritium sources and units; profiles, mass balance and discharge models; the ³H/³He age equation and helium ledger; excess air, dispersion and ³He confinement; Sturgeon Falls, the Danube, Cape Cod. doi:10.1007/978-1-4615-4557-6_13

TOOL

Jurgens, B. C., Böhlke, J. K., & Eberts, S. M. (2012). TracerLPM (Version 1): An Excel® workbook for interpreting groundwater age distributions from environmental tracer data. USGS Techniques and Methods 4-F3.

Five LPMs plus binary mixtures, tracer notes, the worksheet manual, and the Modesto, Albuquerque and Missouri River examples. Public domain; workbook and example files download with the report. doi:10.3133/tm4F3

OPTIONAL

Gilmore, T. E., et al. (2021). The ³H/³He groundwater age-dating method and applications. · Busenberg, E., & Plummer, L. N. (2000). Dating young groundwater with sulfur hexafluoride: natural and anthropogenic sources of sulfur hexafluoride. Water Resour. Res., 36(10), 3011–3030.

Gilmore and colleagues update the ³H/³He method two decades on, with applications; Busenberg & Plummer is the foundational SF₆ paper — solubility, the atmospheric record, and the terrigenic sources that Section 6 warns about. Not discussed in a block.

Next week — contaminant source identification I: nitrate and stable isotopes

The nitrate that this week's forecasts moved around gets its own fingerprint: δ¹⁵N and δ¹⁸O of nitrate, the dual-isotope cross-plot that apportions fertilizer, manure and soil sources, and the predictable slope along which denitrification drags a sample. Everything this week said about age distributions now applies to when a nitrate signal arrived; next week is about where it came from.

Carry-forward question, for your proposal one-pager (due Week 6): if your question needs the age of water younger than ~60 years, which of this week's clocks can see it — and what would you measure alongside it (noble gases? dissolved oxygen? a second gas? a time series?) so that a skeptic could not dismiss the reading in one sentence?