U.S. Department of the Interior

U.S. Geological Survey

Techniques and Methods 4-F5

National Water Quality Assessment Project

DGMETA (Version 1): Dissolved Gas Modeling and 

Environmental Tracer Analysis Computer Program Chlorofluorocarbon-11, in parts per trillion by volume

100

50

0

150

200

250

300

1950 1960 1970 1980 1990 2000 2010 2020

Sample date

2015

Apparent

year of recharge

1980

Apparent

age

35 years

Atmospheric record

He(terr) / (He(s) - He(a)) 

Rs/Ra -air corrected ratio

0.2

0

0.4

0.6

0.8

1.2

1

1.4

1.6

1.8

2

Mantle He

10% Mantle + 90% Radiogenic He

Radiogenic He

accumulation

–0.2 –0.1 0.1 0.3 0.5 0.7 0.9

Pre-modern

aged

Decay of 

tritium

Decay of 

tritium

Air

Modern

aged

Mixed

age

2

4

6

8

10

0

2

4

6

8

10

0

10

20

30

40

0

10

20

30

40

0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Nitrogen, milligrams per liter

Xe to Ne ratio, cubic centimeters per gram of water

Unfractionated

excess air

Fractionated

excess air

Temperature

Excess air

Denitrification

Degassing

Henry solubility constant, moles per kilogram per atmosphere

0 10 20 30 40

Temperature, in degrees Celsius

0.0001

0.001

0.01

0.1

Noble gases

Tritium > 0.5TU

Tritium  0.5TU

Helium

Neon

Argon

Krypton

Xenon

Reactive gases

Nitrogen

Methane

Nitrous oxide

Oxygen

Carbon monoxide

Carbon dioxide

Hydrogen

Environmental tracers

CFC-11

CFC-12

CFC-13

SF6

Halon-1301
Cover Photos:

Solubility constant of gases 

(in freshwater) included in 

the program.

Plot of gases from samples and 

paths of dissolved gas models.

Analysis of helium isotope 

systematics.

Use of the environmental 

tracers for determining age 

of the water.
DGMETA (Version 1): Dissolved Gas 

Modeling and Environmental Tracer 

Analysis Computer Program

By Bryant C. Jurgens, J.K. Böhlke, Karl Haase, Eurybiades Busenberg, 

Andrew G. Hunt, and Jeffrey A. Hansen

National Water Quality Assessment Project

Techniques and Methods 4-F5

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

DAVID BERNHARDT, Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia: 2020

For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, 

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U.S. Government.

Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials 

as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner.

Suggested citation:

Jurgens, B.C., Böhlke, J., Haase, K., Busenberg, E., Hunt, A.G., and Hansen, J.A., 2020, DGMETA (version 1)—

Dissolved gas modeling and environmental tracer analysis computer program: U.S. Geological Survey Techniques and 

Methods 4-F5, 50 p., https://doi.org/10.3133/tm4F5.

ISSN 2328-7055 (online)
iii

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Methods...........................................................................................................................................................4

Gas Solubilities......................................................................................................................................4

Approximations for Local Pressure, Water Vapor Pressure, and Salinity .........................5

Forms of Solubility Equations .....................................................................................................6

Pressure Corrections ..................................................................................................................7

Real Gas Corrections ..................................................................................................................8

Unit Conversions ..........................................................................................................................8

Models of Excess Air............................................................................................................................9

Inverse Modeling of Dissolved Gases .............................................................................................10

Monte Carlo Simulations ...................................................................................................................11

Environmental Tracer Calculations ..................................................................................................11

Transient Atmospheric Gases ..................................................................................................12

Tritiogenic Helium-3 and Helium-4 ..........................................................................................12

Program Description ...................................................................................................................................15

DGMETA Worksheets .........................................................................................................................16

SuppInfo Worksheet ..................................................................................................................16

Database Worksheet .................................................................................................................16

Input_Gases Worksheet ...........................................................................................................19

Input_Tracers Worksheet .........................................................................................................20

DGMETA Operation .............................................................................................................................21

Database .....................................................................................................................................21

Graphing ......................................................................................................................................21

Gas Solubilities ..................................................................................................................22

_SolGph Worksheet ..........................................................................................................22

Gas-Gas Graphs ................................................................................................................22

_GasGph Worksheet ........................................................................................................24

Dissolved Gas Modeling ...........................................................................................................25

_ModOut Worksheet ........................................................................................................26

Environmental Tracer Concentrations ....................................................................................27

_TrcOut Worksheet ...........................................................................................................29

Formatting ...................................................................................................................................29

_AvgVal Worksheet ..........................................................................................................29

_ModFlt, _TrcFlt, and _AvgFlt Worksheets ...................................................................29

Examples .......................................................................................................................................................31

Example 1—Comparison of Noble Gas Solubility Data in Freshwater and Seawater ............31

Example 2—Inverse Modeling of Air-Equilibrated Freshwater and Seawater Samples .......34

Example 3—Dissolved Gas Modeling with Neon, Argon, and Nitrogen or Argon and 

Nitrogen ..................................................................................................................................35

Example 4—Tritiogenic Helium-3 Analysis in the Central Eastside San Joaquin 

Valley, California ....................................................................................................................40

Example 5—Degassed Groundwater ..............................................................................................44
iv

Installation Notes.........................................................................................................................................45

Disclaimer.....................................................................................................................................................45

References Cited..........................................................................................................................................46

Figures

1. Graphs showing air-water solubility equilibrium concentration of inert noble 

gases helium, neon, argon, krypton, and xenon in water that are commonly 

used for modeling air-water equilibrium conditions; air-water solubility 

equilibrium concentration of the biogeochemically reactive gases nitrogen, 

oxygen, carbon dioxide, methane, carbon monoxide, hydrogen, and nitrous 

oxide; air-water solubility equilibrium concentration of environmental 

tracers chlorofluorocarbon-11, -12, and -113, sulfur hexafluoride, and 

bromotrifluoromethane that are commonly used for determining residence 

times of water; and Henry solubility constant of noble gases, reactive gases, 

and environmental tracers ..........................................................................................................3

2. Graph of the excess air-corrected helium isotope ratio of the sample relative 

to air versus the concentration of terrigenic helium as a fraction of the total 

helium in the sample corrected for excess-air ......................................................................14

3. Image showing the Dissolved Gas Modeling and Environmental Tracer Analysis 

tab in the Excel ribbon ...............................................................................................................16

4. Image showing the SuppInfo worksheet, which contains supporting 

information on and references to program implementation; models of excess 

air; equations of local pressure, water density, and vapor pressure; conversion 

factors; gas solubility formulations used by the program; and a set of sample 

calculations of pressure, salinity, model gas concentrations, and mole fractions 

of environmental tracers ...........................................................................................................17

5. Image showing the Database worksheet ...............................................................................18

6. Image showing the Input_Gases worksheet and the Dissolved Gas Modeling 

and Environmental Tracer Analysis tab in the Excel ribbon ................................................19

7. Image showing the Input_Tracers worksheet and the DGMETA (Dissolved Gas 

Modeling and Environmental Tracer Analysis)tab in the Excel ribbon ..............................20

8. Image showing the form used for creating graphs of the solubility of a gas in 

water at equilibrium with the atmosphere .............................................................................22

9. Image showing the _SolGph worksheet created by the program that is used to 

compute and graph the solubility of a gas in water with different temperature, 

elevation (pressure), and salinity conditions .........................................................................23

10. Image showing the form used for creating graphs of gas concentrations from 

samples and dissolved gas models .........................................................................................24

11. Image showing an example _GasGph worksheet created by the program to 

compute and graph gas concentrations for samples and models .....................................25

12. Image showing the form for modeling dissolved gases.......................................................26

13. Image showing the _ModOut worksheet ...............................................................................27

14. Images showing the forms for selecting a model output worksheet for 

computing tracer concentrations and for selecting a tracer output worksheet 

for computing average tracer concentrations ......................................................................28
v

15. Image showing the _TrcOut worksheet, which contains calculations of 

atmospheric dry-air mole fractions and concentrations of tritium-helium-3 

components .................................................................................................................................28

16. Image showing the _AvgVal worksheet, which contains average values of 

tracer concentrations for each sample and date listed on the _TrcOut worksheet .......30

17. Image showing a _ModFlt worksheet, which is a single-row header version of 

a _ModOut, _TrcOut, or _AvgVal worksheet ..........................................................................31

18. Graphs showing neon (Ne) concentrations in freshwater and saltwater at 

equilibrium with air, based on solubility equations by Weiss (1971) that were 

expressed as the moist-air solubility (Ne_Weiss) and Bunsen solubility 

constant (Ne_WeissB), by Clever (1979a; Smith and Kennedy, 1983; Ne_Clvr), by 

Hamme and Emerson (2004; Ne_H&E), and by Jenkins and others (2019; Ne_Jnk) .......32

19. Graph showing percentage of deviation of neon solubility from measured data 

reported by Sano and Takahata (2005) in seawater for a range of temperatures ...........33

20. Image showing the Dissolved Gas Modeling form used to evaluate 

air-equilibrated samples prepared at the U.S. Geological Survey Noble 

Gas Laboratory in Denver, Colorado, and seawater samples from Sano and 

Takahata (2005)............................................................................................................................34

21. Graphs showing concentrations of nitrogen gas and argon and neon ratio, and 

nitrogen gas and argon only were measured in groundwater samples from the 

Texas coastal lowlands aquifer in 2013 ..................................................................................36

22. Image showing the form for Dissolved Gas Modeling for modeling recharge 

conditions and denitrification using neon, argon, and nitrogen gas concentrations .....37

23. Image showing the form for Dissolved Gas Modeling for modeling recharge 

conditions with denitrification using argon and nitrogen gases only ................................37

24. Images showing the helium isotope separation plot and table of helium isotope 

control variables on the TX_Final_TrcOut worksheet ..........................................................39

25. Graph showing tritium and sulfur hexafluoride concentrations in groundwater 

in the Texas coastal lowlands aquifer .....................................................................................39

26. Image showing the Dissolved Gas Modeling form used to generate 

unfractionated excess air models and closed-system equilibration models of 

dissolved gases on the basis of dissolved neon, argon, krypton, and xenon 

concentrations in groundwater in the central eastside of the San Joaquin 

Valley, California, in 2006 ...........................................................................................................41

27. Graph showing comparison of closed-system equilibration and unfractionated 

excess air model results for recharge temperatures among different age 

classifications based on tritium measurements of groundwater samples from 

the central eastside of the San Joaquin Valley, California, in 2006 ....................................42

28. Images showing graphs of sample concentrations of argon versus xenon to 

neon and krypton versus xenon to neon .................................................................................42

29. Images showing views of the modified Weise plot for samples that are tritiated 

and samples that are tritium-dead, and a table of helium isotope control 

variables on the Final_TrcOut worksheet ...............................................................................43

30. Images showing views of the modified Weise plot showing mixing of 

groundwater endmembers and a table of helium isotope control variables on 

the Final_TrcOut worksheet ......................................................................................................43

31. Graphs showing solubility concentrations for argon and krypton plotted versus 

the ratio of the solubility concentrations of xenon and neon for different 

recharge temperatures and excess air values ......................................................................45
vi

Conversion Factors

International System of Units to U.S. customary units

Multiply By To obtain

Length

centimeter (cm) 0.3937 inch (in.)

meter (m) 3.281 foot (ft)

Volume

cubic meter (m3) 6.290 barrel (petroleum, 1 barrel = 42 gal)

liter (L) 33.81402 ounce, fluid (fl. oz)

cubic meter (m3) 264.2 gallon (gal)

cubic meter (m3) 0.0002642 million gallons (Mgal)

cubic centimeter (cm3) 0.06102 cubic inch (in3)

liter (L) 61.02 cubic inch (in3)

cubic meter (m3) 35.31 cubic foot (ft3)

cubic meter (m3) 1.308 cubic yard (yd3)

cubic meter (m3) 0.0008107 acre-foot (acre-ft)

Mass

gram (g) 0.03527 ounce, avoirdupois (oz)

kilogram (kg) 2.205 pound avoirdupois (lb)

Pressure

kilopascal (kPa) 0.009869 atmosphere, standard (atm)

Energy

joule (J) 0.0000002 kilowatthour (kWh)

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows: 

°F = (1.8 × °C) + 32.

Datum

Elevation, as used in this report, refers to distance above the vertical datum.

Supplemental Information

Specific conductance (SC) is given in microsiemens per centimeter at 25 degrees 

Celsius (μS/cm).

Activities for radioactive constituents in water are given in picocuries per liter (pCi/L).
vii

Salinity (S) is given in parts per thousand (per mil).

3He and 4He. The element helium (He) has atoms with different masses: 3He has two protons and 

one neutron in its nucleus, and 4He has two protons and two neutrons in its nucleus.

The isotopic ratio of helium in a sample can be expressed as δ3He in units of percent:

( )

( )

3 4

3

3 4

/

1 100

/

sample

air

He He

He

He He

 

δ =   − ×

 

A tritium unit (TU) is equal to 3.19 picocuries per liter.

Concentrations of gases in water are given in several units:

pptv, volume of gas per volume of dry air, in parts per trillion.

mmol/L, millimoles of gas per liter of water.

mmol/kg, millimoles of gas per kilogram of water.

 μmol/L, micromoles of gas per liter of water.

nmol/kg, nanomoles of gas per kilogram of water.

pmol/kg, picomoles of gas per kilogram of water.

fmol/kg, femtomoles of gas per kilogram of water.

mg/L, milligrams of gas per liter of water.

cm3/g or cc/g, cubic centimeters of gas at standard temperature and pressure (STP) per 

gram of water.

cm3/kg, cubic centimeters of gas at standard temperature and pressure per kilogram 

of water.

Abbreviations

AEW Air-equilibrated distilled water

CE Closed-system equilibration

F Fractionation

ID Identification

PR Partial re-equilibration

T Temperature

UA Unfractionated excess air

USGS U.S. Geological Survey
viii

CHEMICAL NAMES

3H Tritium

3Hetrit Tritiogenic helium-3

4Herad Radiogenic helium-4

Ar Argon

CFC-11 Trichlorofluoromethane (chlorofluorocarbon-11)

CFC-113 1,1,2-Trichloro-1,2,2-trifluoroethane (chlorofluorocarbon-113)

CFC-12 Dichlorodifluoromethane (chlorofluorocarbon-12)

CH4 Methane

CO Carbon monoxide

CO2 Carbon dioxide

δ3He delta helium-3

H2 Hydrogen

Halon-1301 Bromotrifluoromethane

He Helium

Hea Helium originating from excess air

Heeq Helium in solubility equilibrium with the atmosphere

Hes Measured total dissolved helium

Heterr Terrigenic helium

Kr Krypton

N Nitrogen

N2 Nitrogen gas

N2O Nitrous oxide

NaCl Sodium chloride

Ne Neon

O2 Oxygen

Ra 3He/4He atom ratio of the atmosphere

Req Isotopic fractionation of helium during air-water solubility equilibrium

Rs Measured 3He/4He atom ratio of the sample

Rterr 3He/4He atom ratio of the terrigenic helium

SF6 Sulfur hexafluoride

Xe Xenon
DGMETA (Version 1): Dissolved Gas Modeling and 

Environmental Tracer Analysis Computer Program

By Bryant C. Jurgens1, J.K. Böhlke2, Karl Haase2, Eurybiades Busenberg2, Andrew G. Hunt3, and 

Jeffrey A. Hansen1

Abstract

DGMETA (Dissolved Gas Modeling and Environmental 

Tracer Analysis) is a Microsoft Excel-based computer program 

that is used for modeling air-water equilibrium conditions 

from measurements of dissolved gases and for computing 

concentrations of environmental tracers that rely on air-water 

equilibrium model results. DGMETA can solve for the 

temperature, salinity, excess air, fractionation of gases, or 

pressure/elevation of water when it is equilibrated with the 

atmosphere. Models are calibrated inversely using one or more 

measurements of dissolved gases such as helium, neon, argon, 

krypton, xenon, and nitrogen. Excess nitrogen gas, originating 

from denitrification or other sources, also can be included as a 

fitted parameter or as a separate calculation from the dissolved 

gas modeling results. DGMETA uses the air-water equilibrium 

models to separate measured concentrations of gases and 

isotopes of gases into components that are used for tracing 

water in the environment. DGMETA calculates atmospheric 

dry-air mole fractions (mixing ratios) for transient atmospheric 

gas tracers such as chlorofluorocarbons, sulfur hexafluoride, 

and bromotrifluoromethane (Halon-1301); and concentrations 

of tritiogenic helium-3 and radiogenic helium-4, which 

accumulate from the decay of tritium in water and the decay of 

uranium and thorium in rocks, respectively.

Sample data can be graphed to identify applicable 

models of excess air, samples that contain excess nitrogen 

gas, or samples that have partially degassed, for example. 

Monte Carlo analysis of errors associated with dissolved 

gas equilibrium model results can be carried through 

computations of environmental tracer concentrations to 

provide robust estimates of error. In addition, graphical 

routines for separating helium sources using helium isotopes 

are included to refine estimates of tritiogenic helium-3 when 

terrigenic helium from mantle or crustal sources is present 

1U.S. Geological Survey, Placer Hall, 6000 J St., Sacramento, CA 95819, 

bjurgens@usgs.gov

2U.S. Geological Survey, 431 National Center, Reston, VA 20192

3U.S. Geological Survey, Denver Federal Center, Lakewood, CO, 80225

in samples. Environmental tracer concentrations and their 

errors computed from DGMETA can be used with other 

programs, such as TracerLPM (Jurgens and others, 2012), 

to determine groundwater ages and biogeochemical reaction 

rates. DGMETA also produces output files in a format that 

meets the U.S. Geological Survey open data requirements for 

documentation of model inputs and outputs.

DGMETA is a versatile and adaptable program that 

allows users to add solubility data for new gases, modify the 

existing set of gas solubility data, modify the default set of 

gases used for modeling, choose calculations based on real 

(non-ideal) gas behavior, and select various concentration 

units for data entry and results to match laboratory reports and 

study objectives. DGMETA comes with a set of gases widely 

used in hydrology and oceanography and many gases include 

multiple solubilities from previous work. Seventeen dissolved 

gases are included in the default version of the program: noble 

gases (helium, neon, argon, krypton, and xenon), reactive 

gases (nitrogen, oxygen, methane, carbon dioxide, carbon 

monoxide, hydrogen, and nitrous oxide), and environmental 

tracers (chlorofluorocarbon-11, chlorofluorocarbon-12, 

chlorofluorocarbon-113, sulfur hexafluoride, and Halon-1301).

Introduction

Analysis of dissolved gases in water is common in 

oceanographic and terrestrial hydrologic studies. Various 

combinations of dissolved gases can provide records of 

environmental change and a deeper understanding of 

groundwater flow systems (Heaton and Vogel, 1981; Heaton 

and others, 1983; Stute and others, 1995; Aeschbach-Hertig 

and others, 1999; Ballentine and Hall, 1999; Cey and others, 

2008; Cartwright and others, 2017), groundwater discharges 

to surface water (Heilweil and others, 2015; Sanford and 

others, 2015; Gilmore and others, 2016), ocean circulation 

patterns and mixing (Schlosser and Winckler, 2002; Stanley 

and Jenkins, 2013; Loose and Jenkins, 2014), and emission 

rates of greenhouse gases from parts of the hydrosphere 

(Matthews and Fung, 1987; Dalal and Allen, 2008; Jeffrey and 

others, 2018).
2 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Dissolved gases with constant atmospheric abundances 

like the inert noble gases helium (He), neon (Ne), argon 

(Ar), krypton (Kr), xenon (Xe; fig. 1A) or the reactive gas 

nitrogen (N2; fig. 1B) are commonly used to evaluate the 

conditions (temperature or excess air, for example) of water 

when it is equilibrated with the atmosphere. Reactive gases 

such as N2, oxygen (O2), carbon dioxide (CO2), methane 

(CH4), carbon monoxide (CO), hydrogen (H2), and nitrous 

oxide (N2O; fig. 1B) can be used to evaluate biogeochemical 

processes. Relatively stable, but transient (time-varying) 

atmospheric trace gases like sulfur hexafluoride (SF6), 

chlorofluorocarbon-11 (CFC-11), chlorofluorocarbon-12 

(CFC-12), chlorofluorocarbon-113 (CFC-113), and 

bromotrifluoromethane (Halon-1301) are commonly used to 

trace water in the environment (fig. 1C). Transient atmospheric 

gases and the isotopes of helium (helium-3/helium-4, 3He/4He; 

or the delta helium-3, δ3He)—tritiogenic helium-3 (3Hetrit) 

and radiogenic helium-4 (4Herad)—are commonly known as 

“environmental tracers” and are frequently used to evaluate 

residence times of water and groundwater. Tritiogenic 

helium-3 can accumulate in water from the decay of tritium 

(3H) in water and 4Herad can accumulate in water from 

uranium and thorium in subsurface rocks.

The measured concentrations of reactive gases, transient 

atmospheric gases, and He with He isotopes are not typically 

meaningful because the concentration that was gained or 

lost from biogeochemical reactions or the concentration 

used for age-dating water, like the dry-air mole fraction 

or the amount of He produced from the decay of tritium 

(3Hetrit), are calculated quantities that are dependent on the 

air-water equilibration conditions when the water was last 

in contact with the atmosphere. The air-water equilibration 

conditions such as temperature and excess air contribution 

may be estimated or, more frequently, they are determined 

by inverse modeling of noble gases or Ar-N2. Consequently, 

samples for the noble gases (with He isotopes), reactive gases, 

and transient atmospheric tracers are increasingly collected 

together to gain a more complete analysis of gases in water.

Different sets of dissolved gases commonly are 

sampled in separate containers, analyzed at different 

laboratories, and may be interpreted separately using software 

programs available publicly, in literature, or provided by 

the laboratories. Noble90 (Aeschbach-Hertig and others, 

1999, 2000; Peeters and others, 2003), NOBLEBOOK 

(Aeschbach-Hertig and others, 1999, 2000), and PANGA 

(Jung and Aeschbach-Hertig, 2018) are programs available 

for modeling air-water equilibrium conditions from noble 

gases. Noble90 also can be used to calculate He components 

from model results. These programs rely on a limited set of 

gas solubility equations that are widely used but cannot be 

changed or updated without a substantial alteration of the 

program. In addition, no single program is available that can 

compute a wide range of commonly sampled environmental 

tracers, such as CFCs, SF6, and Halon-1301, or provide a way 

to interactively compute He components (3Hetrit, 4Herad) in 

the context of local conditions through graphical inspection 

and manipulation of dependent terms. Because environmental 

tracer interpretations are dependent on dissolved gas model 

results, and different combinations of gases may be included 

in various studies, a single program that is capable of 

simultaneously and consistently interpreting the full range of 

dissolved gases and environmental tracers is desirable.

The purpose of this report is to document the program 

DGMETA: Dissolved Gas Modeling and Environmental 

Tracer Analysis is a Microsoft Excel-based program that is 

compatible with Excel Office versions 2007 or later. DGMETA 

can be used for inverse modeling of air-water equilibrium 

conditions (temperature, excess air, gas fractionation, and 

pressure/elevation) from stable atmospheric dissolved gases 

(He, Ne, Ar, Kr, Xe, and N2); for evaluating environmental 

tracers of relatively stable, but transient atmospheric trace 

gases (such as the CFCs, SF6, and Halon-1301); and for 

evaluating environmental tracers produced within the 

subsurface from the decay of tritium in water, like 3Hetrit, 

or from the decay of uranium and thorium in rocks such as 

4Herad. DGMETA can be used to model air-water equilibrium 

conditions assuming that excess air is unfractionated (UA; 

Heaton and Vogel, 1981), fractionated following the partial 

re-equilibration model (PR; Stute and others, 1995) of excess 

air, or fractionated following the closed-system equilibration 

model (CE; Aeschbach-Hertig and others, 2000) of entrapped 

air. Gases that are transient or biologically reactive also 

are included. These gases can be used to identify O2 and 

nitrate reduction, CH4 production, and other biogeochemical 

processes. Finally, DGMETA has the added capability for 

users to modify or add to the set of gas solubility equations 

included in DGMETA’s solubility database, including 

atmospheric gas tracers. These modifications allow DGMETA 

to grow as new tracers are identified or as solubility equations 

are revised.

This report contains three sections: Methods, Program 

Description, and Examples. The “Methods” section describes 

the theoretical and empirical basis for the dissolved gas 

and tracer models and the computational methods used by 

DGMETA. The “Program Description” section describes 

the contents of the worksheets in DGMETA, the input data 

required for operation of the software, and how to use the 

functions provided in the DGMETA Excel ribbon. The 

“Examples” section includes five examples of using DGMETA 

to process dissolved gas data of different types.
Introduction 3

A

1.0E−08

1.0E−07

1.0E−06

1.0E−05

1.0E−04

1.0E−03

1.0E−02

1.0E−01

1.0E+00

C D

1.0E+00

0 10 20 30 40

Millimoles per kilogram Millimoles per kilogram

B

1.0E−08

1.0E−07

1.0E−06

1.0E−05

1.0E−04

1.0E−03

1.0E−02

1.0E−01

1.0E+00

Millimoles per kilogram Moles per kilogram per atmosphere

Temperature, in degrees Celsius

Temperature, in degrees Celsius

0 10 20 30 40

Temperature, in degrees Celsius

0 10 20 30 40

Temperature, in degrees Celsius

1.0E−12

1.0E−11

1.0E−10

1.0E−09

1.0E−08

1.0E−07

1.0E−06

1.0E−05

1.0E−04

1.0E−03

1.0E−02

1.0E−01

0 10 20 30 40

EXPLANATION EXPLANATION

He_Weiss

Ne_Weiss

Ar_Weiss

Kr_Weiss

Xe_Clvr

EXPLANATION

CFC11_Warner

SF6_Bull

CFC12_Warner

H1301_B

CFC113_Bu

N2_Weiss

CH4_Wies

N20_Weiss_Nonldeal

O2_B&K

CO_Wies

CO2_Weiss_Nonldeal

H2_Wies

EXPLANATION

Nitrogen

Methane

Nitrous oxide

Oxygen

Carbon monoxide

Carbon dioxide

Hydrogen

Helium

Neon

Argon

Krypton

Xenon

CFC-11

CFC-12

CFC-113

SF6

Halon-1301

0.0001

0.001

0.01

0.1

Figure 1. A, air-water solubility equilibrium concentration of inert (non-reactive) noble gases helium (He), neon (Ne), argon (Ar), 

krypton (Kr), and xenon (Xe) in water that are commonly used for modeling air-water equilibrium conditions; B, air-water solubility 

equilibrium concentration of the biogeochemically reactive gases nitrogen (N2), oxygen (O2), carbon dioxide (CO2), methane (CH4), 

carbon monoxide (CO), hydrogen (H2), and nitrous oxide (N2O); C, air-water solubility equilibrium concentration of environmental tracers 

chlorofluorocarbon-11, -12, -113 (CFC-11, CFC-12, CFC-113), sulfur hexafluoride (SF6), and bromotrifluoromethane (halon-1301) that 

are commonly used for determining residence times of water; and D, Henry solubility constant of noble gases, reactive gases, and 

environmental tracers. Solubility concentration of the gases were determined for freshwater (salinity = 0 per mil) over the temperature 

range of 0–40 degrees Celsius at sea level (atmospheric pressure = 1 standard atmosphere). The names of the lines consist of the 

name of the gas and a descriptor of the reference for the sources of the gas solubility data; and they are unique identifiers for the gas 

solubility data used by Dissolved Gas Modeling and Environmental Tracer Analysis (DGMETA; see the Database worksheet of DGMETA).
4 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Methods

The dissolved concentration of a gas, Ct, in water 

includes three major components: (1) the concentration of 

dissolved gas attributable to its solubility equilibrium with the 

atmosphere, Ceq; (2) an amount of gas in excess of solubility 

equilibrium resulting from the dissolution of air bubbles, Ca; 

and (3) an amount of gas that is produced or consumed in situ, 

either in an ocean, aquifer, surface water, or streambed, Cis.

Ct = Ceq + Ca + Cis (1)

For some gases, the in situ production in water is zero 

or negligible. For other gases, the in situ production may be 

from biological processes (N2; Heaton and others, 1983), 

radioactive decay of aquifer materials (4Herad; Andrews and 

Lee, 1979), the decay of tritium in water (3Hetrit; Tolstikhin 

and Kamensky, 1969), or upwards migration of deep crustal 

or mantle fluids (Craig and others, 1978). Some gases can be 

consumed by biological processes that result in negative in situ 

concentration (for example, O2 reduction). Because the noble 

gases (He, Ne, Ar, Kr, and Xe) are chemically non-reactive, 

have stable concentrations in the atmosphere, and (except for 

He) have in situ production rates that are often negligible, 

they provide the best tool for determining the temperature and 

excess air component of water when it equilibrated with the 

atmosphere (Kipfer and others, 2002).

Relatively stable, transient atmospheric trace gases, 

such as CFCs, SF6, and Halon-1301, are often used to 

determine the residence time of reservoirs of water because 

their concentrations in the atmosphere have changed since 

the 1930s (Gammon and others, 1982; Bullister and Weiss, 

1983; Bullister, 1984; Bullister and Weiss, 1988; Busenberg 

and Plummer, 1992; Bu and Warner, 1995; Busenberg and 

Plummer, 2000; Bullister and others, 2002; Deeds, 2008; 

Beyer and others, 2014, 2015, 2017). To determine residence 

time, it is necessary to compute the atmospheric mole fraction 

(of dry air) of the gas when the water equilibrated with the 

atmosphere. Because the air-water equilibrium conditions are 

often unknown, it is common to measure noble gases (and 

nitrogen; N) along with CFCs, SF6, or Halon-1301 to provide 

the best estimate of air-water equilibrium conditions. The 

dry-air mole fraction of a CFC, SF6, or Halon-1301 in water 

can be compared to the time-varying record of dry-air mole 

fractions in the atmospheric record to determine the year 

or range of years in which the water could have entered a 

reservoir of water. These methods also can apply to other more 

chemically reactive gases or transient gases to understand 

reaction rates and climate change records or deviations from 

non-equilibrium conditions.

Gas Solubilities

The solubility equilibrium concentration of a gas in 

water, Ceq, is governed by Henry’s law, which says that the 

concentration of a gas dissolved in water is proportional 

to the partial pressure of the gas, p, above the water; this 

proportional relationship is defined as the Henry solubility 

constant: Kcp = 

C_eq

p (commonly referred to as “K”). Henry’s 

law also is frequently defined as a volatility constant, 

Kpc (commonly known as “KH”), which is the inverse of 

Kcp or Kpc = 1 / Kcp; as this name implies, it is often used 

for expressing the volatility of sparsely soluble organic 

compounds from water. Henry’s law also can be defined as 

a dimensionless molar ratio, H, that represents the molar 

concentration of the gas in water to the molar concentration 

of the gas in air. The Henry solubility constant, volatility 

constant, and dimensionless Henry constant are related by 

H = RTKcp = RT/Kpc, where R is the universal gas constant 

[82.056 cubic centimeters (cm3)-atm per mol per Kelvin] and 

T is the temperature in Kelvin. For solubility studies of the 

atmospheric gases, the Henry solubility constant, Kcp, is the 

most commonly used form of Henry’s law, and this form is 

adopted by DGMETA.

The concentration of a dissolved gas in equilibrium with 

the atmosphere is calculated using the following formula:

 Ceq = x(Ph −ew  G(T,S) × correction factors (

where

Ceq is the solubility concentration in equilibrium 

with the atmosphere and may be computed 

in volumetric (cm3/g), mass (milligrams 

per liter; mg/L), molar (millimole per L; 

mmol/L), or molal (mmol/kilogram; kg) 

concentration units,

x is the dry-air mole fraction of the gas in the 

atmosphere in mole fraction,

Ph is the local total atmospheric pressure in atm,

ew is the water vapor pressure (or saturation 

vapor pressure of water) in atm,

G(T,S) is the generalized Henry solubility constant 

of a gas in water and is a function of 

temperature, T, and salinity, S, and 

may be expressed in any concentration 

units per atm,

T is the temperature of water in Kelvin,

S is the salinity of water in parts per thousand 

(per mil) or g per kg (equivalent to 

1 part dissolved solute per 1,000 parts of 

water), and

correction factors (for example, Pc, Rc, UC, are defined 

in equations 7–12) are used as needed, 

depending on the form and units of G(S,T).
Methods 5

Equation 2 is the general equation used by the program for computing the solubility of a gas in water and it is a function 

of the partial pressure of a gas [ x(Ph −ew)] and the solubility function G(T,S). As written, G(T,S) is equal to the Henry’s la

solubility constant, Kcp, and is a function of temperature and salinity in any concentration units per atm. Equation 2 has been 

generalized to account for different definitions, formulations, and units of the solubility function, G(T,S), that was derived in the 

original solubility study of a gas. DGMETA builds specific forms of equation 2 in a modular fashion, such that the dry-air mole 

fraction, correction factors, and conversion terms can be added or removed based on the form of the solubility function, G(T,S). 

By default, DGMETA converts all gas concentrations to cm3/g of water—see the “Unit Conversions” section below. For gases 

that have transient dry-air mole fractions (x), the equilibrium concentration of the gas will depend on the year in which the water 

was last in contact with the atmosphere. For such gases, the dry-air mole fractions can be manually adjusted from the Database 

worksheet by entering a different mole fraction from the atmospheric record for a particular year of interest.

DGMETA includes solubility equations for several gases: He, Ne, Ar, Kr, Xe, N2, CO2, H2, O2, CH4, N2O, CO, CFC-11, 

CFC-12, CFC-113, SF6, and Halon-1301. This list can be expanded to suit user needs by entering gas solubility data and 

properties into the Database worksheet. The solubility equations were primarily derived to compute gas concentrations in fresh 

water and seawater over the salinity range of 0–35 per mil (‰ or g/kg), temperature range of 0–40 degrees Celsius (°C), and 

pressure range of 0.5–1.0 atm, which corresponds to altitudes between sea level (1 atm) and 6,000 meters (m) above sea level, 

approximately.

Approximations for Local Pressure, Water Vapor Pressure, and Salinity

DGMETA uses the U.S. Standard Atmosphere (National Aeronautics and Space Administration , 1976) and barometric 

formula to approximate local atmospheric pressure based on an elevation specified by the user. The elevation may be based 

on the site location or the elevation where water enters an aquifer as recharge. DGMETA calculates pressure for the base level 

U.S. Standard Atmosphere only. The following calculation is valid for elevations less than 11,000 m:

( )

0

*

0

a

b

g M

R L

b

h o

b b

T P P

T Lhh

  =   + −  

(3)

where

Ph is the local atmospheric pressure in atm,

Po is the sea level standard atmospheric pressure in atm (1 atm = 1013.25 millibars),

Tb is the standard temperature, 288.15 Kelvin,

Lb is the standard lapse rate, –0.0065 Kelvin per m,

h is the elevation in m above sea level,

h0 is the elevation of the bottom layer, 0 m above sea level,

g0 is the gravitational acceleration, 9.80665 m per second squared,

Ma is the molar mass of air, 0.0289644 kg per mol, and

R* is the universal gas constant, defined as 8.3144598 joules (J) per mol per Kelvin by the U.S. Standard 

Atmosphere.

DGMETA approximates the water vapor pressure using the equation from Weiss and Price (1980):

 

6745.09 exp 24.4543 4.8489 0.000544

100 w

T

e ln S

T

    = −− −        

(4)

where

ew is the water vapor pressure (or saturation vapor pressure of water) in atm,

T is the temperature of water in Kelvin, and

S is salinity in per mil or g/kg.

When direct measurements of salinity or total dissolved solids are not available, salinity, S, can be approximated from 

measurements of specific conductance (SC) using an equation taken from Pickering (1981):

S(permilorg / kg) = 5.572 × 10−4(SC) + 2.02 × 10−9 (SC)2 (5)

where

S is salinity in per mil or g/kg, and

SC is the specific conductance in microsiemens per centimeter (μS/cm).
6 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Salinity is dependent on the ionic composition of water; therefore, relations between S and SC can vary for different 

environmental settings. Equation 5 was derived from more complex equations published in Standard Methods (American Public 

Health Association, 2005) with the assumption that the dissolved ions in water are identical to those in seawater. Differences 

between the predictions of equation 5 and the equations from Standard Methods typically result in less than a 0.1 percent 

change in the salinity factor over a wide range of temperature (0–40 °C) and specific conductance (0–67,000 μS/cm) values 

(U.S. Geological Survey, 2011). Consequently, this equation provides a reasonable approximation for most study objectives and 

applications.

Forms of Solubility Equations

As written in equation 2, G(T,S) is equal to the Henry’s law solubility constant, Kcp, and is a function of temperature and 

pressure and is usually expressed in volumetric or gravimetric concentrations per atm. Equation 2 has been generalized to 

account for different definitions, formulations, and units of the solubility function, G(T,S), that was derived in the original 

solubility study of a gas. In general, two forms of G(T,S) are most often used for reporting solubility results. In one form, G(T,S) is 

a function for the solubility constant (Kcp), Bunsen solubility constant (β; Weiss, 1970), mole fraction (X; Clever, 1979a; Smith 

and Kennedy, 1983), solubility constants that require a modified form of equation 2 for real gas behavior, such as Ko (Weiss, 

1974; Weiss and Price, 1980), or that approximate real gas behavior at pressures near 1 atm, such as K’ (Warner and Weiss, 

1985; Bu and Warner, 1995). In these cases, the corrections applied to equation 2 may only require conversion to different 

units—see the “Unit Conversions” section or correction for real gas behavior. The other form of G(T,S) that is frequently used is 

the moist-air solubility, which is the volume or concentration of a gas in water at equilibrium with water saturated air at a total 

pressure of 1 atm. In other words, this is the volume or concentration of a gas expected to be dissolved in seawater or freshwater 

at sea level. This form is noted as C* (Weiss, 1970) and it includes the partial pressure of the gas at sea level in the solubility 

expression [x(P0 −ew)]. It is, therefore, necessary to remove the pressure term from equation 2 to compute the solubilit

concentration at other elevations (or pressures)—see the “Correction” sections below. Similar to C*, Weiss and Price (1980) and 

Benson and Krause (1984) derived equations that included the effects of real gas behavior for moist-air solubility concentrations 

in seawater or freshwater at sea level. To use their equations in equation 2, the pressure terms and non-ideality terms at 1 

atm must be corrected in order to compute the solubility concentration at other elevations (pressure)—see the “Correction” 

sections below.

The solubility function, G(T,S), is usually a two- or three-part polynomial with coefficients that express the temperature 

and salinity dependence of the gas in water at 1 atm. Most often the solubility equation is based on the integrated van ’t Hoff 

equation and the Setchénow relation for salinity. The coefficients of the polynomial are found by fitting experimental solubility 

data to the function. The concentration units of G(T,S) are, therefore, dependent on the concentrations used in the experiment. 

DGMETA accepts most concentration units expressed as volumetric, gravimetric, molar, or molal units per atm. DGMETA 

will convert the concentrations internally to a common unit of cm3/g and can output concentrations in either cm3/g, mmol/L, 

mmol/kg, or mg/L—see the “Unit Conversions” section.

Below is a list of solubility equations that can be used to compute dissolved gas concentrations in water:

ln C*or ln β = A1 + A2(100 _

Τ ) + A3 ln(

_Τ

100) + A4(

_Τ

100) + S[B1 + B2(

_Τ

100) + B3 (

_Τ

100)

2

] (G1)

ln Ko,lnK′, orln F* = A1 + A2(100 _

T ) + A3 ln(

_T

100) + A4 (

_T

100)

2

+ S[B1 + B2(

_T

100) + B3 (

_T

100)

2

] (G2)

ln X = A1 + A2(100 _

Τ ) + A3 ln(

_Τ

100) − NaCl[B1 + B2(100 _

Τ ) + B3 ln (

_T

100)

2

] (G3)

ln C*= A1 + 

A_2

Τ + 

_

A3

Τ2 + 

_

A4

Τ3 + 

_

A5

Τ4 + S[B1 + 

B_2

Τ + 

_

B3

Τ2] (G4)

ln C* = A1 + 

_

A2

Ts

 + 

_

A3

Ts

2 + 

_

A4

Ts

3 + 

_

A5

Ts

4 + 

_

A6

Ts

5 + S[B1 + 

B_2

Ts

 + 

_

B3

Ts

2 + 

_

B4

Ts

3] + C1 S2 (G5)

ln C*= A1 + A2(100 _

Τ ) + A3 ln(

_Τ

100) + A4(

_Τ

100) + S[B1 + B2(

_Τ

100) + B3 (

_Τ

100)

2

] + C1 S2 (G6)
Methods 7

where

ln is the natural logarithm, a logarithm to base e

(a mathematical constant approximated as 

2.71828),

C* is the moist-air solubility concentration and 

is the volume or concentration of a gas in 

water at equilibrium with water saturated 

air at a total pressure of 1 atm,

β is the Bunsen coefficient and is the volume of 

gas at standard temperature and pressure 

(STP) that partitions into the aqueous 

phase per unit volume of water at the 

temperature of the measurement when 

the partial pressure of the gas is 1 atm in 

milliliters (mL) of gas per L of water,

Aj is the gas temperature coefficient for the j-th 

term (1–6),

Bj is the gas salinity coefficient for the j-th 

term (1–4),

C1 is the gas salinity coefficient for certain 

formulations,

S is the salinity of the water in per mil or g/kg,

T is the water temperature in Kelvin for 

equations G1, G2, G3, and G6,

Ts is the scaled temperature of water, unitless, 

for equations G4 and G5, calculated 

as ln(298.15 _− t 273.15 + t

), where t is temperature in °C,

Ko is the fugacity solubility constant in mol per 

kg per atm (Weiss and Price, 1980; Warner 

and Weiss, 1985),

K′ is the partial pressure equilibrium solubility 

constant in mol per kg of water per atm 

(Warner and Weiss, 1985),

F* is moist-air solubility concentration adjusted 

for real gas behavior (Weiss and Price, 

1980; Warner and Weiss, 1985),

X is the mole fraction solubility concentration 

(Clever 1979a, b, 1980; Smith and 

Kennedy, 1983), and

NaCl is the sodium chloride concentration in 

mol per L and is calculated following 

Aeschbach-Hertig and others (1999) as 

equation 6 below.

 NaCl = _S

MNaCl ρ(t,S) / 1000 (6)

where

MNaCl is the molar mass of sodium chloride, 

58.443 g/mol, and

ρ(t,S) is the density of water as function of 

temperature and salinity in kg per cubic 

meter (m3)—see the “Unit Conversions” 

section for the calculation.

Equation G1 was used by Weiss (1970, 1971) to compute 

solubilities for dissolved N, O2, and Ar (Weiss, 1970), for He 

and Ne (Weiss, 1971), and for CH4, CO, and H2 (Wiesenburg 

and Guinasso, 1979) in fresh water and seawater. This 

equation is formulated with the integrated van ’t Hoff equation 

for temperature and the Setchénow relation for salinity and 

can be used to compute gas solubilities from Bunsen constants 

(β) or as moist air solubilities (C*). Weiss and Price (1980) 

used equation G2 to compute the fugacity solubility coefficient 

(K0) and fugacity adjusted moist-air solubility (F*) for N2O 

and CO2. Equation G3 is from Clever (1979a, b) and Smith 

and Kennedy (1983) and was used to compute solubilities of 

the noble gases in fresh water and seawater. Equation G4 was 

used by Benson and Krause (1984) to compute dissolved O2

concentrations in fresh water and seawater. Equation G5 was 

used by Garcia and Gordon (1992, 1993) to compute dissolved 

O2 solubility and later was used by Hamme and Emerson 

(2004) for Ne, Ar, and N in fresh water and seawater. Equation 

G6 was used by Jenkins and others (2019) to fit noble gas 

solubility data in water and seawater.

Pressure Corrections

Some expressions of gas solubility were derived for 

moist air at sea level (total pressure is 1 atm, adjusted for 

water saturated air) to facilitate use in oceanographic studies. 

These solubility forms are often noted as C* or F* (eqs. G1, 

G4, G5, and G6), and these forms need pressure corrections 

to be used at other elevations (pressures; Benson and Krause, 

1984; Kipfer and others, 2002). DGMETA will apply the 

following correction term when the value of the “Use pressure 

correction, Pc” field is “Yes” on the Database worksheet:

Pc = _1

Po −ew (

where

Pc is an optional pressure correction for water 

vapor when the solubility constant G(S,T) is 

normalized to moist air at 1 atm,

Po is the sea level standard atmospheric pressure 

in atm (1 atm = 1013.25 millibars), and

ew is the water vapor pressure (or saturation 

vapor pressure of water) in atm.
8 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Real Gas Corrections

Two methods for dealing with non-ideal behavior of real 

gases are included in the program. DGMETA will compute 

additional gas corrections for non-ideal behavior when the 

value in the “Treat non-ideal” field is “Yes” on the Database 

worksheet. Some gases can deviate from ideal behavior, 

and these deviations may limit the accuracy of the solubility 

computations. For many gases this deviation is small at total 

pressures near 1 atm in comparison to solubility measurements 

(Weiss and Price, 1980; Warner and Weiss, 1985). Benson and 

others (1979) and Benson and Krause (1984) used the second 

virial coefficient to correct O2 for real gas behavior. Similarly, 

Weiss and Price (1980) derived a relationship for the second 

virial and cross virial coefficient to correct N2O and CO2 for 

real gas behavior:

Benson and others (1979) and Benson and Krause (1984):

Rc = 

1_− θ Ph

1 − θ (8)

where

Rc is the local real gas correction, in atm,

θ is a polynomial function for the second virial 

coefficient between oxygen and air (eq. 22

in Benson and others, 1979), and

Ph is the local atmospheric pressure in atm.

Weiss and Price (1980):

2 1 * 1

exp exp h h

c h h

BP P R P v PB v

RT RT RT

      + −   − = + =+                

 (9)

where

Rc is the real gas correction,

Ph is the local atmospheric pressure in atm,

B* is a polynomial function (eq. 11 in Weiss and 

Price, 1980) relating the quantity 

B_+ 2δ

RT , 

where B is the second virial coefficient and 

δ is the cross-virial coefficient between the 

trace gas and air,

 v ̄ is the partial molal volume in cm3 per mol,

R is the gas constant (82.056 cm3-atm per mol 

per Kelvin), and

T is the temperature in Kelvin.

For CO2, Weiss (1974) did not define the expansion 

series, B*, like was done for N2O (Weiss and Price, 1980). 

To facilitate the use of equation 9 with CO2, the function 

B* (eq. 11 in Weiss and Price, 1980) was fitted to the 

quantity B_+ 2δ

RT based on equations for the second virial 

coefficient and cross-virial coefficient given by Weiss (1974). 

For formulations in moist air at sea level, F*, the real gas 

correction at sea level must be removed to apply it at other 

elevations. This is done by dividing the real gas correction 

(eq. 9) by the real gas correction at sea level, exp(B*).

Unit Conversions

Solubilities can be reported in gravimetric, volumetric, 

molar, or molal concentration units or mole fractions, 

X. By default, DGMETA automatically converts all gas 

solubilities internally from the units specified in the “Units of 

solubility constants” field on the Database worksheet to cubic 

centimeters of gas at an STP per gram of water, which is the 

most common unit of measurement of the noble gases. Most 

units can be converted using the density of water, molar gas 

volume, and molar mass of each gas. In general, conversion to 

and from cubic centimeters per gram use the molar volumes 

of each gas defined on the Database worksheet rather than 

the molar volume of an ideal gas (M

I = RT ≈ 22.414). Th

default molar volumes are computed by dividing the molar 

mass by the molar density. It is possible that the original 

solubility study used MI for conversions, and therefore, 

in some cases the MI may be more appropriate than the 

individual molar gas volumes used by default. Solubilities as 

mole fractions (Clever, 1979a, b, 1980; Smith and Kennedy, 

1983) are converted to cubic centimeters per gram following 

Aeschbach-Hertig and others (1999). All dissolved gas 

modeling results report gas concentrations of each model 

component in cm3/g, mmol/L, and mg/L:mmol/L to cm3/g:

Uc = _V

ρ(t,S) (10)

mg/L to cm3/g:

Uc = _V

M ρ(t,S) (11)

mole fraction to cm3/g:

Uc = _V

MH2O(

_X

1 − X) 

ρ

_(t,S=0)

ρ(t,S) (12)

Bunsen solubility (cm3/mL) to cm3/g:

Uc = _1

ρ(t,S) (13)

where

V is the molar volume of a gas in mols per cm3

(molar mass in g/mol divided by molar 

density in g/cm3),

ρ(t,S) is the density of water as function of 

temperature and salinity in kg/m3—see 

the “Unit Conversions” section for 

the calculation,

M is the molar mass of a gas in g per mol,

X is the mole fraction of a gas in water, in mol 

of gas per mol of water, and

MH2O is the molar mass of water (18.016 g per mol).
Methods 9

The density of water, ρ(t,S), in kg per cubic meter (m3), is computed from the equations derived by Bigg (1967a, b) and 

Millero and Poisson (1981). It is computed in two parts:

ρ(t,S) = ρ(t) + AS + B S3_

2 + CS (14)

where

 ρ(t) = 999.842594 + 6.793952 × 10−2 t − 9.09529 ×10−  t2 + 1.001685   10−  t3 − 1.120083 ×1−6  t4 + 6.53633 ×  1−9

Bigg (1967a, b):

A = 0.824493 − 4.0899 ×10−3 t + 7.648 ×  0−5  t2 − 8.2467 ×10−7  t3 + 5.875 × 1

 B = − 5.72466 ×10−3 + 1.027 ×  10−4 t − 1.66546 ×1

 C = 4.8314 × 10−4

S is the salinity of the water in per mil or g/kg,

t is the temperature in °C.

Models of Excess Air

DGMETA can be used to model air-water equilibrium conditions under different theories of the presence and partitioning 

of excess air, represented as Ca in equation 1. Excess air (A) is a phenomenon in which dissolved gas concentrations exceed 

the concentrations attributable to equilibrium in the atmosphere (Heaton and Vogel, 1981). The composition of the excess air is 

similar to the composition of the atmosphere, hence the name “excess air.” This extra amount of air is from air-bubbles entrained 

in water that may be partially or completely dissolved into solution. Excess air may form in groundwater from water-table 

fluctuations during recharge or in ocean and surface-water bodies from wave action.

Dissolved gas concentrations of excess air can be unfractionated or fractionated relative to the atmospheric composition. 

Excess air will be unfractionated (UA) when air bubbles are completely dissolved into a solution (Heaton and Vogel, 1981; 

eq. 15). Fractionated excess air can occur from the diffusion of gases across a water table or by incomplete dissolution of 

entrapped air bubbles (Ae). Stute and others (1995) proposed a partial re-equilibration (PR) model of fractionated excess air 

that describes a process where water loses its excess air component through progressive re-equilibration with the atmosphere 

via diffusion of gases across a water table or water surface (eq. 16). Because lighter gases will diffuse more quickly than 

heavier ones, the water is depleted in lighter gases relative to the heavier gases. The fractionation parameter, F, used in the 

PR model is the degree to which the gases have fractionated and may range from zero (unfractionated amount of excess air) 

to infinity (zero excess air). Excess air also can be fractionated if entrapped air-bubbles are only partially dissolved into a 

solution. Aeschbach-Hertig and others (2000) proposed a closed-system equilibration (CE) model of fractionated excess air 

that results from a gas phase (entrapped air-bubble), Ae, that is partially dissolved into a solution due to increased hydrostatic 

pressure (eq. 17). The fractionation of the gases dissolved into a solution is governed by the solubilities of the gases, which 

again tends to fractionate the lighter gases with lower solubilities relative to the heavier gases with higher solubilities (fig. 1D). 

Thus, water following this model will be enriched in heavier gases as compared to the lighter gases. For the CE model, the F

parameter relates the final and initial gas volumes of the entrapped air, such that F = 

V_b

Va

; where Va and Vb are the initial a

final gas volumes. Commonly, the final volume of the gas phase is smaller than the initial volume such that the F parameter 

typically ranges from zero (complete dissolution of entrapped air into solution and excess air is unfractionated) to 1 (no excess 

air formed). A unique feature of this model is that it also can describe degassing or the formation of a gas phase (air bubble) in 

water. In this case, a drop in the total gas pressure could lead to the formation of a bubble such that the final volume is greater 

than the initial volume. In this case, the F parameter is greater than 1—see example 5.
10 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

DGMETA uses the following equations for the excess air 

concentration in water:

UA model: Ca = xA (15)

PR model: exp a

Ne

D C xA F

D

  = −   

(16)

CE model: Ca = 

x Ae _(1 − F)

1 + 

_x AeF

Ceq 

 (17)

where

Ca is the dissolved concentration of a gas in 

excess of air-water equilibrium from the 

entrainment of air in water in cm3 (STP) 

per g of water,

x is the dry-air mole fraction of the gas in the 

atm in mole fraction,

A is the amount of air dissolved in water in cm3

(STP) per g of water,

D is the molecular diffusion coefficie

of the gas,

DNe is the molecular diffusion coefficient of N

Ae is the amount of entrapped air as a gas phase 

in water in cm3 (STP) of air per g of water,

F is the fractionation parameter that describes 

the degree of diffusive fractionation when

used with the PR model or the final to

initial volume of entrapped air (Ae) when 

used with the CE model, and

Ceq is the solubility concentration in equilibrium 

with the atmosphere in water in cm3 (STP) 

per g of water.

Diffusion coefficients (D) for gases are computed from 

the Arrhenius equation:

 exp

Ea D B

RT

  =    

(18)

where

B is the diffusion coefficient when t

temperature goes to infinity, in square

meters per second (m2/s),

Ea is the activation energy in J per mol,

R is the gas constant (8.3143 J per mol per 

Kelvin), and

T is the temperature in Kelvin.

Diffusion coefficients were compiled for He, Ne,

Kr, Xe, CO2, and CH4 from Jähne and others (1987); for 

Ar from Holocher and others (2002); for N2 and O2 from 

Ferrell and Himmelblau (1967); for chlorofluorocarbons-11,

-12 from Zheng and others (1998); and SF6 from King and 

Saltzman (1995).

Inverse Modeling of Dissolved Gases

DGMETA can compute best-fit parameter estimates

for temperature, salinity, elevation (pressure), excess air 

(or entrapped air), fractionation of gases, and excess N2 for 

different models of excess air and solubility equations. Best-fit

parameters are determined from inverse methods that solve 

equation 1 repeatedly until the error between modeled and 

measured concentrations, relative to the measurement error, 

is minimized.

DGMETA uses an inverse modeling method similar 

to that described by Ballentine and Hall (1999) and

Aeschbach-Hertig and others (1999). More specifically,

DGMETA uses a weighted, nonlinear least-squares 

parameter-estimation routine that was originally developed 

from the Gauss-Newton method (with scaling parameter)

described by Johnson and Faunt (1992) and modified to allow

for constrained parameters. The Jacobian matrix is numerically 

estimated using the five-point Lagrange differentiation.

Standard errors of the fitted parameters are from the diagonal

terms of the variance-covariance matrix.

Model fits are computed by minimizing the

chi-square test statistic between measured and modeled gas 

concentrations relative to the gas concentration error:

χ2 = ∑i=1 n (Oi _

 − Mi)2

σi

2 (19)

where

n is the number of modeled gases,

Oi is the observed (measured) concentration of 

the i-th gas,

Mi is the modeled total gas concentration (eq. 1) 

of the i-th gas, and

σi is the standard error of the observed 

(measured) concentration of the i-th gas.

The chi-square probability is calculated using the 

chi-square distribution and the degrees of freedom (number of 

gases minus the number of model-fit parameters). Chi-square

probabilities are only calculated for models having a degree of 

freedom greater than zero.

Temperature is constrained to the range of 0–100 °C.

Excess air/entrapped air is constrained to the range of 

0–250 cm3/kg. Salinity is constrained to the range of 

0–50 per mil (or g/kg). Pressure is constrained by elevation

and can range from –2,000 to 10,000 m above sea level,

which corresponds to about 1.2–0.26 atm. Fractionation is

constrained to the range of 0–5. Excess N2 is constrained to 

the range of 0–500 mg/L as N.
Methods 11

Monte Carlo Simulations

Monte Carlo simulations are included in this program 

to provide better estimates of the parameter uncertainty for 

dissolved gas models. For some models, like the CE model, 

parameter uncertainties can be larger than the computed 

parameter estimates. This can occur when the F parameter in 

the CE model approaches the inflection point at 1 (0.8 less 

than F less than 1.2) or when gas samples appear to have 

more than one model solution or have large entrapped air 

values (less than 50 cm3/kg)—see Jung and Aeschbach-Hertig 

(2018) for a more complete description. In cases where the 

parameter uncertainty is as large or larger than the parameter 

estimates, the results of the CE model are ambiguous. These 

large uncertainties become problematic with computing 

tracer concentrations because the large uncertainties in model 

parameters produce large uncertainties in computed tracer 

concentrations. This situation can be less of an issue with more 

precise gas measurements that have uncertainties of less than 

0.5 percent. Uncertainties of gas measurements determined 

from existing methods are typically between 1 and 5 percent.

One method to assess the error around the best-fit 

parameters is to compute Monte Carlo simulations. PANGA 

(Jung and Aeschbach-Hertig, 2018) provides an interactive 

Monte Carlo method for exploring the individual cases listed 

above and potentially refining the model estimates in cases 

where there are multiple solutions. DGMETA also can be 

used to generate Monte Carlo simulations and store the results 

in a worksheet, but DGMETA does not provide automated 

graphical tools to explore the distribution of simulations 

for patterns. As an alternative, DGMETA can compute 

parameter uncertainties more locally around the best-fit 

parameter estimates computed from the inverse modeling 

method. Calculations of tracer concentrations from these error 

estimates have lower tracer uncertainties (see example 4).

Monte Carlo simulations are performed by generating 

random sets of gas concentrations from a normal distribution 

using the measured gas and error concentration for a sample 

and then solving for the best-fit model parameters. This 

process is repeated many times in order to produce a large 

population of simulations that can provide estimates of the 

possible solution space and parameter uncertainty in the 

model. The number of simulations is typically repeated until 

the variance of the parameter values asymptotically approach 

a stable value. The number of Monte Carlo simulations 

needed to give uncertainty estimates is typically greater 

than 500, but less than 10,000. By default, DGMETA uses 

5,000 simulations. The random gas samples and Monte Carlo 

simulation results can be output to a separate worksheet 

if needed.

Random sets of gas samples can be generated by one 

of two options: (1) by assuming the gases are positively 

correlated and not independent or (2) by assuming complete 

independence. The default used by DGMETA is to compute 

random gas concentrations using option 1. For this option, gas 

concentrations are computed using a single random number to 

generate the entire set of gas concentrations. This will produce 

sets of gases with concentrations that are higher or lower than 

the original set of concentrations, but within the error of each 

gas. Alternatively, completely independent sets of gases can 

be generated. This approach computes a random number for 

each gas, such that the set of gases produced from this method 

are truly random. Unfortunately, that approach can generate 

parameter errors that are as large as the parameters themselves 

and hence do not improve the model confidence. In these 

cases, Jung and Aeschbach-Hertig (2018) recommend a more 

detailed analysis of the Monte Carlo results using PANGA. 

Alternatively, it is possible to compute parameter uncertainties 

that are more local to the best-fit parameters using sets of gas 

concentrations that increase and decrease in unison (option 1). 

Parameter estimates and their uncertainties computed using 

these dependent sets of gases tend to more strongly adhere to 

the best-fit parameter values based on the original gas data in 

comparison to results obtained from completely independent 

sets of gas concentrations. Users can choose either Monte 

Carlo simulation option.

Environmental Tracer Calculations

Environmental tracers such as CFCs, SF6, Halon-1301, 

3Hetrit, and 4Herad can be used to determine the time elapsed 

since the water was last in contact with the atmosphere. The 

measured concentrations of these tracers are not typically 

meaningful because the concentration used for computing 

the age of water; like the dry-air mole fraction, the amount 

of He produced from the decay of tritium in water (3Hetrit), 

or from the decay of uranium and thorium in rocks (4Herad); 

are calculated quantities that are dependent on the air-water 

equilibration conditions when the water was last in contact 

with the atmosphere. Because these conditions (temperature 

and excess air contribution) are often unknown, they may be 

estimated or, more frequently, they are determined by inverse 

modeling of noble gases or Ar-N2.

DGMETA can compute atmospheric dry-air mole 

fractions for transient atmospheric gases and concentrations 

of 3Hetrit and 4Herad using air-water equilibration conditions 

determined from inverse modeling of dissolved gases. 

Computations of air-water equilibration conditions and 

environmental tracer concentrations using DGMETA provides 

a consistent treatment of data that may have been analyzed 

at different laboratories and reported in different units. 

In addition, the concentrations of tracers and their errors 

computed by DGMETA can be used in newer versions of 

TracerLPM (Jurgens and others, 2012) to compute water ages 

using the same error-weighted inverse modeling technique 

described in this report.
12 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Transient Atmospheric Gases

Trace atmospheric gases, such as CFCs, SF6, and Halon-1301, have dry-air mole fractions that have increased in the 

atmosphere over the last 100 years—though mole fractions for the CFCs have been declining since the 1990s. The dry-air mole 

fraction is computed by rearranging equation 1 and substituting the total concentration for the measured concentration of the 

gas and ignoring the in situ production term. Equations for the dry-air mole fraction depend on the model of excess air and are 

computed using the equations below:

UA model: x = _Cs

G(T,S)(Ph −ew) 

A_

V (20)

PR model: x = ________________ Cs G(T,S)(Ph −

ew) 

A[exp(− F 

_D _DNe)] V

(21)

CE model: x = _________________ Cs G(T,S)(Ph −

ew)   A

V _(1 − F)

(1 + 

_F Ae

G(T,S)(Ph −ew)

(22)

where

x is the dry-air mole fraction of the gas in the atmosphere in mole fraction,

Cs is the sample concentration (measured value) of a gas tracer (CFC, SF6, or Halon-1301) in mols per kg,

G(T,S) is the solubility function for the Henry solubility coefficient in mols per kg,

Ph is the local atmospheric pressure in atm,

ew is the water vapor pressure in atm,

A is the excess air component in cm3 (standard temperature pressure) per g of water,

V is the molar volume of a gas in mol per cm3,

F is the fractionation parameter that describes the degree of diffusive fractionation when used with the PR model 

or the final to initial volume of entrapped air (Ae) when used with the CE model,

D is the molecular diffusion coefficient of the gas,

DNe is the molecular diffusion coefficient of Ne, and

Ae is the entrapped air volume (volume of trapped air per volume of water) in cm3 (standard temperature pressure) 

per g of water,

Tritiogenic Helium-3 and Helium-4

Tritiogenic helium-3 and terrigenic He components are computed from the mass balance of 3He and 4He and are described 

in more detail in Schlosser and others (1989), Solomon (2000), and Solomon and Cook (2000). Air-water equilibrium model 

results are used to determine the concentration of He in solubility equilibrium with the atmosphere (Heeq) and the concentration 

of He originating from excess air (Hea). The amount of terrigenic helium (Heterr) is calculated by subtracting the amount of Heeq

and Hea from the measured total dissolved helium (Hes) in each sample. Terrigenic helium includes He that accumulates in water 

from 4Herad emitted from radioactive decay of uranium and thorium in aquifer materials and He from the Earth’s mantle (Heman) 

that has migrated upwards through faults or is released by weathering of volcanic sediments. Terrigenic helium that is dominated 

by radiogenic sources in the Earth’s crust generally has a low 3He/4He ratio (Rterr) that is typically less than 5 × 10–8, whereas 

Heterr that is dominated by mantle He generally has high Rterr that is typically greater than 5 × 10–6.
Methods 13

DGMETA determines Hea from dissolved gas modeling results so the calculation of 3Hetrit (Schlosser and others, 1989) is 

in the equations below:

3Hetrit = H es Rs −

(  es −   eterr  Ra +   ee  Ra(1 − α) −   eter  Rterr (2

where

3Hetrit is the helium concentration from the decay of tritium in cm3 (STP) per g of water,

Hes is the helium concentration of the sample in cm3 (STP) per g of water,

Rs is the 3He/4He ratio of the sample corrected for the decay of tritium (half-life is equal to 12.32 years; Lucas and 

Unterweger, 2000) for the time between the dates of sample collection and analysis,

Heterr is the helium concentration of terrigenic origin in cm3 (STP) per g of water,

Ra is the 3He/4He ratio of air (Ra = 1.384 × 10–6; Clarke and others, 1976),

Heeq is the helium concentration in equilibrium with the atmosphere in cm3 (STP) per g of water,

α is the isotopic fractionation factor of He during air-water solubility equilibrium, and

Rterr is the 3He/4He ratio of terrigenic helium.

 H eterr = H es − H eeq − H ea (24)

where

Heterr is the helium concentration of terrigenic origin in cm3 (STP) per g of water,

Hes is the helium concentration of the sample in cm3 (STP) per g of water,

Heeq is the helium concentration in equilibrium with the atmosphere in cm3 (STP) per g of water, and

Hea is the helium concentration of excess air in cm3 (STP) per g of water.

 α = exp[− (− 0.0299645 +

19.8715 _

T + 

− 1833.92 _

T2 )(1 + 0.000464S)] (25)

where

α is the isotopic fractionation factor of He during air-water solubility equilibrium,

T is the temperature of water in Kelvin, and

S is salinity in per mil or g/kg.

Samples that have Hes less than Heeq usually have experienced gas loss or the gases are fractionated. When Hes is less 

than Heeq, calculations of 3Hetrit using equation 23 will underestimate 3Hetrit concentrations in the sample. In these cases, 

DGMETA will compute 3Hetrit as the excess of solubility equilibrium, which is usually the largest component of He in samples 

without Heterr:

3Hetrit = H es(Rs −

Req) (2

where

3Hetrit is the helium concentration from the decay of tritium in cm3 (STP) per g of water,

Hes is the helium concentration of the sample in cm3 (STP) per g of water,

Rs is the 3He/4He ratio of the sample corrected for the decay of tritium (half-life is equal to 12.32 years; Lucas and 

Unterweger, 2000) for the time between the dates of sample collection and analysis, and

Req is the 3He/4He ratio of helium in equilibrium with the atmosphere and is equal to the isotopic fractionation 

factor times the 3He/4He ratio of air (αRa).
14 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

For many calculations of 3Hetrit, the presence of Heterr

in samples requires the Rterr value to be known or to be 

determined from an analysis of He isotope systematics. 

In these situations, the samples where 3H is low or absent 

(tritium-dead) can be used to identify the source and 

appropriate Rterr value to apply to samples with 3H. The Rterr

can often be identified using a modified Weise plot (Weise 

and Moser, 1987; Stute and others, 1992) where the ratio 

of Heterr normalized to the excess-air-corrected total He 

concentration, Heterr/(Hes–Hea), is plotted against the ratio of 

the excess-air-corrected Rs to the Ra.

Figure 2 shows an example of the modified Weise plot 

with several lines that illustrate possible evolutionary paths for 

the accumulation of He in water from different sources of He. 

Water that enters a water body such as an aquifer, for example, 

will tend to follow a temporal process that starts on the left 

near the atmospheric equilibrium ratio of water, Rs/Ra, (0.983) 

and will generally evolve to the right as the water ages and 

accumulates He from terrigenic sources. In most groundwater 

systems, water will gain Heterr from the decay of uranium 

and thorium in sediments causing the Rs to evolve along the 

lower dashed line showing a radiogenic source of He (dashed 

line in fig. 2). In some cases, for example, in basaltic rock 

aquifers or in areas of seismic activity that allow deep mantle 

gases to migrate upwards along faults (Craig and others, 

1978; Plummer and others, 2000), the He in groundwater may 

be substantially influenced by a mantle-derived He source 

(Heman) and cause Rs/Ra end-member values greater than 5 

(dotted line in fig. 2). In other cases, mantle He may be only a 

minor component caused by the weathering of small amounts 

of volcanic sediments in an aquifer (Jurgens and others, 2016), 

and the groundwater may show a mixed signature of mantle 

and radiogenic sources of He (dashed-dotted line in fig. 2). 

Tritiogenic helium-3 is another source of He. For water that 

entered a system after 1950, terrigenic sources of He may not 

be a substantial component of the total He, so the resulting Rs/

Ra will mainly reflect the contribution of 3He from the decay 

of tritium. Samples that exemplify this process will contain 3H 

and will plot along the red arrow in the upper left of figure 2. 

Samples with a mixture of post-1950s water and water older 

than 1,000 years will tend to have multiple sources of He. 

Samples that contain measurable levels of 3H and substantial 

amounts of Heterr with a radiogenic source may plot above the 

dashed line on figure 2.

Heterr/(Hes–Hea)

Rs/Ra -air corrected, ratio

0.2

0

0.4

0.6

0.8

1.2

1

1.4

1.6

1.8

2

Decay of

tritium

Decay of

tritium

Air 10% Heman + 90% Herad

Radiogenic He

accumulation

Mantle He

–0.1 0.1 0.3 0.5 0.7 0.9 Zero terrigenic helium

accumulation

EXPLANATION

Best-fit

Radiogenic

Mantle

Mix: rad+man

Tritiated

Tritium dead

Figure 2. Excess air-corrected helium isotope ratio of the sample relative to air versus the concentration of 

terrigenic helium as a fraction of the total helium in the sample corrected for excess-air.
Program Description 15

To facilitate the He isotope analysis, samples are 

categorized as either “tritiated” or “tritium dead” using a 

tritium threshold value that is specified by the user (default 

is 0.5 tritium unit, TU). DGMETA automatically creates 

a graph like figure 2 and computes a best-fit line through 

all tritium-dead samples (blue circles). Those samples are 

expected to contain the largest proportion of Heterr and will 

usually indicate the Rs/Ra value of the terrigenic source. For 

the blue samples on figure 2, a radiogenic He source appears 

to be the main source of Heterr and, therefore, the best-fit 

Rterr value is 2 × 10–8. The best-fit line and Rterr value are 

determined from the equation below:

 Y = (1 − x)

(Req −

Rterr 

_Tb

H eeq) + Rterr (27)

where

Y is the computed R/Ra value,

x is the excess air corrected value of terrigenic 

helium (Heterr/Hes – Hea),

Req is the 3He/4He ratio of helium in equilibrium 

with the atmosphere and is equal to the 

isotopic fractionation factor times the 

3He/4He ratio of air (αRa),

Rterr is the 3He/4He atom ratio of the 

terrigenic helium,

Tb is the background 3Hetrit in water, and

Heeq is helium in solubility equilibrium 

with the atm.

The He isotope graph on the tracer calculation worksheet 

(“_TrcOut”; fig. 2) is controlled by variables listed in a table 

to the right of the figure. The table lists the best-fit Rterr value 

computed by the program and this value is used as the default 

value to calculate 3Hetrit concentrations using equation 23—

see example 4. Changes to the Rterr value will affect the graph 

and 3Hetrit calculations. The best-fit Rterr can be changed 

manually based on user-defined fits and also can be specified 

for each sample individually in the 3He/4Heterr column of 

the Tracer Output worksheet. Users also can change the 

percentage of mantle He value and the 3He/4He ratio of mantle 

helium (Rman) to explore different mixing lines of radiogenic 

and mantle He components on the graph (dash-dot-dot line). 

The default in the DGMETA program is set to 10 percent 

mantle He.

DGMETA is programmed to only report 3Hetrit

concentrations for samples where the percentage of Heterr is 

less than 70 percent of the total He. Uncertainty in the 3He/4He 

ratio of the Heterr can lead to unreliable estimates of 3Hetrit for 

samples with large amounts of Heterr. DGMETA computes 

3Hetrit for all samples, but does not report it for samples with 

Heterr greater than 70 percent of the total He; instead, the 

output includes an additional column with a recommended 

3Hetrit value to report. Tritiogenic helium is not reported for 

samples with Heterr greater than 70 percent of the total He. 

In cases where the computed 3Hetrit is small or negative, but 

the 3H concentration indicates that 3Hetrit should be present, 

DGMETA reports a value of less than 1 TU. In other cases, the 

concentration may not be reliably quantifiable.

Program Description

DGMETA includes an Excel ribbon tab that contains 

buttons to operate the program and four worksheets for 

entering sample data or gas solubility data. The DGMETA 

tab is in the Excel ribbon at the top of the Excel window 

(fig. 3). The worksheet called SuppInfo contains supporting 

information on and references to program implementation; 

models of excess air; equations of local pressure, water 

density, and vapor pressure; conversion factors; gas solubility 

formulations used by the program; and a set of sample 

calculations of pressure, salinity, model gas concentrations, 

and mole fractions of environmental tracers (fig. 4). The 

worksheet called Database is used for entering new or 

modifying existing gas solubility data and properties (fig. 5). 

The worksheet called Input_Gases is used for entering 

dissolved gas concentrations (fig. 6), and the worksheet 

called Input_Tracers is used for entering measurements of 

environmental tracers (fig. 7). The program is operated from 

buttons in the DGMETA ribbon menu that can be used to 

update gas solubility equations, create graphs, model dissolved 

gases, and calculate tracer concentrations. Some of the 

subroutines produce worksheets with graphs, model results, or 

tracer concentrations. Graphs of dissolved gas concentrations 

are output to a worksheet named by the user with the suffix 

“_Graphs” appended to the worksheet name. Dissolved gas 

modeling results are output to a worksheet with the suffix 

“_ModOut.” Environmental tracer calculations are output 

to a worksheet with the suffix “_TrcOut.” The “Collapse 

Tracer Calculation” button will compute average tracer 

concentrations and results are output to a worksheet with the 

suffix “_AvgVal.” Flat files with a single-row header can be 

created for dissolved gas model output files (“_ModOut”), 

calculated tracer concentrations (“_TrcOut”). or average tracer 

concentrations (“_AvgVal”).
16 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Figure 3. The Dissolved Gas Modeling and Environmental Tracer Analysis tab in the 

Excel ribbon.

DGMETA Worksheets

The following sections describe the function and 

purpose of four worksheets included in DGMETA: SuppInfo, 

Database, Input_Gases, and Input_Tracers. Worksheets 

colored gray (SuppInfo and Database) are mainly for 

reference, whereas worksheets colored blue are used for 

entering sample data.

SuppInfo Worksheet

The SuppInfo worksheet contains supporting information 

on and references to program implementation; models 

of excess air; equations of local pressure, water density, 

and vapor pressure; conversion factors; and gas solubility 

formulations used by the program (fig. 4). Two important 

pieces of information are codes used by the program to 

call up different models of excess air and for specifying the 

solubility equation of a gas. Worksheet functions are available 

to calculate total gas concentrations (eq. 1) and mole fractions 

of gases. These functions use the two-character short name of 

the excess air model to compute concentrations—see example 

3. Examples of how to call up and use the worksheet functions 

are given at the bottom of the SuppInfo worksheet. When 

specifying new gas solubility data on the Database worksheet, 

the solubility equation used by the program is specified 

by an integer number that refers to one of the equations 

below the Dissolved Gas Solubility Formulations heading 

on the SuppInfo worksheet. Note that addition of new gas 

solubility data and other customizations of DGMETA do not 

require changes to the SuppInfo worksheet. A calculator for 

converting between pressure and elevation is provided at the 

bottom of the worksheet for convenience.

Database Worksheet

The Database worksheet contains a table of solubility 

coefficients, form of solubility equations, dry-air mole 

fractions, solubility units, molar volumes, diffusivities, 

and non-ideal behavior used to calculate the solubility 

concentration and diffusivity of gases in water. The table can 

be used for entering new or modifying existing gas solubilities 

and their formulations (fig. 5). References to source data are 

provided where applicable. There are comments associated 

with many fields at the top of the table that contain specific 

information on the values that can be entered in those fields.

The solubility table is loaded into memory when the 

program is first opened or when the button “Update Gas 

Database” is clicked from the DGMETA tab in the Excel 

ribbon. Users can change the primary solubility equations 

used for different gases to explore the effects of using different 

solubility equations on dissolved gas model results—see 

examples 1 and 2. A value of “Yes” next to the gas (column A) 

instructs the program to use that formulation for dissolved gas 

modeling and tracer concentration calculations. The default 

set of equations are the Weiss solubilities for He, Ne, Ar, and 

Kr and the Clever solubility for Xe. The program can reset to 

the default set of gas solubilities using button “Reset Gases to 

Defaults” from the DGMETA tab in the Excel ribbon (fig. 3).

A unique name for each solubility equation must be 

specified when entering new gas solubility data on the 

Database worksheet. Worksheet functions are available to call 

up gas solubility concentrations for different solubility data. 

These worksheet functions use the unique name of the gas to 

call up the solubility formula listed in the solubility table—see 

example 1.
Program Description 17

Figure 4. The SuppInfo worksheet, which contains supporting information on and references to program implementation; models 

of excess air; equations of local pressure, water density, and vapor pressure; conversion factors; gas solubility formulations 

used by the program; and a set of sample calculations of pressure, salinity, model gas concentrations, and mole fractions of 

environmental tracers.
18 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Figure 5. The Database worksheet.
Program Description 19

Input_Gases Worksheet

The Input_Gases worksheet is used to enter sample 

information, initial or fixed model values, and dissolved 

gas concentration data for water samples (fig. 6). Entries 

for Sample identification (ID), Sample Date, and values for 

the initial or fixed air-water model parameters are required 

fields for each sample. Values for the model parameters 

may be estimated initially and solved for later—see the 

“Dissolved Gas Modeling” section. Values for salinity and 

elevation (commonly referred to as the recharge elevation 

in groundwater studies), and excess air or entrapped air are 

needed for all models (UA, PR, and CE). The CE and PR 

models require a value for F. Models that include N2 as a 

model constraint require a value for excess N2, which is most 

often set to zero unless there is evidence of denitrification or 

some other contribution of excess N2 in the sample. Parameter 

values are treated as initial starting values when those 

parameters are optimized during dissolved gas modeling. 

Parameter values are fixed if the model parameter is not 

chosen to be optimized. Commonly, salinity and elevation 

are not optimized so these values are often fixed using site 

estimates. Salinity can be estimated from measurements 

of total dissolved solids or from SC using equation 5. The 

elevation is used to calculate the atmospheric pressure during 

air-water equilibrium. In groundwater studies, the elevation 

of air-water equilibrium occurs at the phreatic surface where 

recharge occurs. The elevation of the well or spring where the 

dissolved gas sample was collected is often used when the 

actual recharge elevation is unknown.

Figure 6. The Input_Gases worksheet and the Dissolved Gas Modeling and Environmental Tracer Analysis tab in the Excel 

ribbon.
20 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Dissolved gas concentrations and uncertainty for each 

sample are entered below the section “Select and Enter Gases 

Below to Model Air-Water Equilibrium Conditions or Graph.” 

Each gas requires an estimate of the measurement uncertainty 

(usually 1 standard error) in order to find the best-fit solution 

to the data. Measurement uncertainty for dissolved gases can 

vary by gas and analytical method, but are typically between 

1 and 5 percent of the measured value.

The list of gases in columns below the section “Select 

and Enter Gases Below to Model Air-Water Equilibrium 

Conditions or Graph” can be changed to different gases and 

different concentration units by the user. Worksheet cells 

that have blue text contain a drop-down list of gases or units 

(fig. 6). Row 2 contains a list of gases and row 3 contains a 

list of units that the program will accept. The list of gases in 

the drop-down menu is controlled by the list of gases entered 

on the Database worksheet. Adding new gases to the Database 

worksheet will expand the gases listed in the drop-down 

menus on the Input_Gases worksheet.

Input_Tracers Worksheet

The Input_Tracers worksheet is used to enter laboratory 

results for 3H, He isotopes, dissolved gas data, and transient 

atmospheric gas tracers (fig. 7). The program uses the tracer 

information on this worksheet to calculate environmental 

tracer concentrations of dry-air mole fractions and He 

components from dissolved gas model results listed in a 

dissolved gas modeling output worksheet, which is identified 

by the suffix “_ModOut.” Dissolved gas model output is 

based on information entered on the Input_Gases worksheet 

so samples entered on the Input_Tracers worksheet should 

correspond to samples entered on the Input_Gases worksheet.

Figure 7. The Input_Tracers worksheet and the DGMETA (Dissolved Gas Modeling and Environmental Tracer Analysis) tab in the 

Excel ribbon.
Program Description 21

The first several columns of the Input_Tracers worksheet 

pertain to sample information. The Sample ID and Sample 

Date are used to pair environmental tracer measurements on 

the Input_Tracers worksheet with samples listed in a dissolved 

gas modeling output worksheet. As such, multiple dissolved 

gas models and multiple rows of tracer measurements 

will produce multiple rows of calculated environmental 

tracer concentrations on the Environmental Tracer Output 

worksheet (“_TrcOut”).

Measurements of 3H, isotopes of He (3He/4He), He, 

and Ne are used to compute the concentration of He derived 

from the decay of tritium (3Hetrit) and the amount of Heterr in 

a sample. Measured concentrations of these constituents are 

entered below the column “Tritium-Helium analysis.” Helium 

isotope concentrations and errors can be entered as δ3He or Rs. 

If the analysis date of the He isotope measurement is provided 

by the user, the isotopes will be corrected for the decay of 

tritium between the sample date and the analysis date. If the 

analysis date is unknown, the program will use a presumed 

analysis date that is 6 months after the sample date. The 

measurement uncertainty must also be entered or estimated for 

each measured value.

Atmospheric Gas Tracers are listed to the right of 

Tritium-Helium analysis (not shown in fig. 7). For each gas 

listed in the “Atmospheric Gas Tracers” section, the program 

will compute the dry-air mole fraction in the Environmental 

Tracer Output worksheet (_TrcOut). The list of gases can 

be modified by the user for cells with blue text. Worksheet 

cells that have blue text contain a drop-down list of gases or 

units. Row 2 contains a list of gases and row 3 contains a list 

of units that the program will accept. The list of gases in the 

drop-down menu is controlled by the list of gases entered in 

the Database worksheet. Adding new gases to the Database 

worksheet will expand the gases listed in the drop-down 

menus on the Input_Gases and Input_Tracers worksheets.

DGMETA Operation

The DGMETA tab is used to operate the program 

(fig. 3). The buttons on the tab are organized into four areas 

of program functionality: Database, Graphing, Modeling and 

Tracer Calculations, and Formatting.

Database

There are two buttons that cause the database, stored 

in memory, to change. The first button called “Update Gas 

Database” is used when changes to the database table are 

made. Additional gases can be added to the database by 

following the formatting and conventions provided on the 

Database worksheet. The field comments at the top of the 

database table can contain specific information on the values 

that can be entered in those fields. The button “Update Gas 

Database” also will need to be clicked when changes to the 

existing gases are made. For example, many reactive gases, 

like CO2, have transient atmospheric mole fractions (mixing 

ratios) not just the environmental trace gases. As such, it may 

be necessary to adjust the mole fractions of these gases for 

calculating solubility and modeling biogeochemical reactions. 

In addition, the reactive gases can be included as tracers 

and their mole fractions can be derived from dissolved gas 

modeling results.

A common use of the Update Gas Database function 

is to test the fit of various solubility data to sample gas 

concentrations during dissolved gas modeling. Dissolved gas 

modeling fits are performed with the set of solubility data 

indicated with a “Yes” next to the gas name. To change the 

default set of solubility data, replace the “Yes” with a “No” 

for updated solubility data and replace the “No” with a “Yes” 

for the desired solubility data. For each gas, only one set of 

solubility data can be chosen as the primary solubility data. 

Example 2 tests the fit of different sets of solubility data to 

air-equilibrated and seawater samples. The second button, 

“Reset Gases to Defaults,” will reset the primary solubility 

data values to the defaults. The set defaults are the Weiss 

solubilities for He, Ne, Ar, and Kr and the Clever solubility 

(with salting coefficients from Smith and Kennedy, 1983) 

for Xe. These solubilities are the most commonly used for 

groundwater studies of past recharge temperatures and age 

dating (Aeschbach-Hertig and others, 1999; Kipfer and others, 

2002; Aeschbach-Hertig and Solomon, 2013).

Graphing

Graphs are a simple way to aid the evaluation of 

dissolved gas models to fit sample data and to evaluate the 

behavior of gases under different conditions. DGMETA 

can create two types of interactive graphs: (1) graphs of the 

solubility of gases in water at equilibrium with the atm and 

(2) graphs of gas concentrations of samples with dissolved 

gas model output. Users are encouraged to explore the Graphs 

worksheet to see how built-in functions can be used to create 

custom graphs and functionality.
22 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Gas Solubilities

The solubility of a gas in water can be graphed using 

the “Gas Solubilities” button on the DGMETA tab (fig. 3). 

This button will display a form (fig. 8) that can be used to 

select gases and graph their solubility in water over a range 

of temperature, elevation (pressure), and salinity conditions 

(fig. 8). The list of gases displays the unique name entered 

in the database table. The user can specify which conditions 

to vary (temperature, elevation, or salinity values), the range 

of parameter values, the step increment, the values of fixed 

parameters (temperature, elevation, or salinity), and the units 

of the concentrations. Example 1 uses this routine to examine 

the solubility of Ne in water and seawater for different sets of 

Ne solubility data. Once the form is complete, the “CREATE 

GRAPH” button will create a new worksheet with a name 

specified by the user and a suffix of “_SolGph,” and this new 

worksheet will contain a graph of the solubility concentration 

in water (fig. 9).

_SolGph Worksheet

The _SolGph worksheet is organized into three sections: 

(1) the graphical display of charts, (2) a set of control 

variables that can be adjusted to change the graph, and (3) a 

section below the control variables that contains formulas 

for calculating the dissolved gas concentration (fig. 9). The 

worksheet has a set of control variables that are initially 

populated by the program based on the values specified on the 

form by the user (fig. 8). The values on the worksheet can be 

changed to see how the solubility of a gas is affected.

Gas-Gas Graphs

Graphs of gas concentrations from samples and from 

model output can be created to evaluate different models of 

dissolved gas concentrations (UA and CE) that might explain 

measured concentrations in samples. The “Gas-Gas Plots” 

button will display a form named “Create Graphs” that can 

be used to select samples to graph from the Input_Gases 

worksheet or from the Output of Dissolved Gas Models 

(_ModOut) worksheet (fig. 10)—see the “Dissolved Gas 

Modeling” section. Up to four graphs can be created, and 

graphs can include selected ratios of gases in addition to 

the gas concentrations. Samples can be plotted as groups 

according to their labels entered in the Study Area column 

on the Input_Gases or _ModOut worksheets; although these 

worksheets should be sorted by Study Area before plotting the 

samples as groups. Figure 8. The form used for creating graphs of the solubility 

of a gas in water at equilibrium with the atmosphere.
Program Description 23

Figure 9. The _SolGph worksheet created by the program that is used to compute and graph the solubility of a gas in water with 

different temperature, elevation (pressure), and salinity conditions.
24 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Figure 10. The form used for creating graphs of gas concentrations from samples and dissolved gas models.

_GasGph Worksheet

The Create Graphs program will create a new worksheet 

with a suffix of “_GasGph” that contains interactive graphs 

with a set of control variables above each graph, which are 

used for exploring sample gas concentrations in relation to 

modeled dissolved gas concentrations from the UA and CE 

models (fig. 11). Each graph created on a _GasGph worksheet 

will have a set of black lines that correspond to a range in 

temperatures (0–40 °C) and amounts of UA (0–10 cm3/kg). 

The black lines will generally bound most well-behaved 

samples. Samples that plot outside the black lines can indicate 

degassed samples (see example 5), extreme F of gases, and 

samples that have undergone in situ production of gases, such 

as radiogenic helium or denitrification (see examples 3 and 4).

Each graph also will include a green line of the UA 

model that spans across the bounded range of black lines and 

a red-dotted line of fractionated excess air (CE model) for 

various amounts of entrapped air (0–200 cm3/kg; fig. 11). 

The position of the UA model (green line) and CE model will 

move based on changes to the temperature value set in the 

control variables above each graph. The CE model also will 

change shape based on changes to the F value. Graphs that 

include N2 gas concentrations will have an orange line that 

shows UA for various amounts of excess N2 gas (0–10 mg/L 

as N) that could be contributed by denitrification or deep 

fluids. The origin of the excess N2 gas line will intersect the 

UA model line by the value set in the “Init. excess air (UA), 

cm3/kg” field.
Program Description 25

Figure 11. An example _GasGph worksheet created by the program to compute and graph gas concentrations for samples and 

models.

Above each graph is a set of variables that control the 

lines on each graph (fig. 11). The values are initially set by 

the program based on averages of the initial values on the 

Input_Gases worksheet or best-fit values from a _ModOut 

worksheet. The values for salinity and elevation affect the 

gas concentrations for all lines displayed. The value for 

temperature affects the lines for the UA model (green line), 

CE model (red-dotted line), and excess N2 (orange line). The 

“Init. excess air (UA), cm3/kg” field affects the position of the 

excess N2 line and the F value affects the CE model.

To the left of each graph is a set of controls for setting 

the units of gas concentrations (fig. 11). These can be changed 

to display gases in units of (1) cm3/g, (2) mmol/L, (3) mg/L, 

or (4) mmol/kg. Gas concentrations are converted to different 

units using equations 10–11.

The _GasGph worksheets also contain calculations 

for the different gas models and the sample concentrations 

converted to different units. These calculations call up custom 

Excel worksheet functions that can be manipulated and used to 

create custom calculations and graphs to suit user needs.

Dissolved Gas Modeling

DGMETA is used most often to compute dissolved 

gas models and environmental tracer concentrations; these 

tasks are completed in three steps: (1) enter dissolved 

gas measurements on the Input_Gases worksheet and 

environmental tracer measurements on the Input_Tracers 

worksheet, (2) compute dissolved gas models for samples 

listed on the Input_Gases worksheet, and (3) compute 

environmental tracer concentrations for samples listed on 

the Input_Tracers worksheet using the dissolved gas models 

computed from step 2. Dissolved gas models can be computed 

using the “Dissolved Gas Modeling” button in the DGMETA 

tab (fig. 3). This program will compute best-fit estimates of 

temperature, excess air, and other model parameters and will 

report the results in a program created output worksheet with 

the suffix “_ModOut.” Environmental tracer concentrations 

can be computed using the “Env. Tracer Calcs” button 

in the DGMETA tab. For samples listed in the _ModOut 

worksheet, environmental tracer concentrations will be 

computed for every corresponding Sample ID and Date listed 

on the Input_Tracers worksheet. Tracer calculations will be 

reported on a worksheet created by the program with the 

suffix “_TrcOut.”
26 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

The program for modeling dissolved gases is initiated 

from the “Dissolved Gas Modeling” button in the DGMETA 

tab in the Excel ribbon (fig. 3). This button will display the 

Dissolved Gas Modeling form, which allows users to select 

one or more samples to model, gases to include in the model, 

parameters to optimize, and one or more models to evaluate 

(fig. 12). The user also has several model options that can be 

included. The first option enables the program to automatically 

rerun models with poor fits by removing the gas with the 

poorest fit in an attempt to look for an acceptable model 

solution for the sample. The program defines a poor fit as a 

model with a chi-square probability less than 1 percent or 

0.01—see example 4. The second option enables the program 

to calculate excess N2 for samples where dissolved N was 

measured, but is not included as a model-fit parameter such 

that the dissolved gas model is based on measured gases other 

than N. For example, if noble gases and N were measured in 

a sample, a user could select Ne, Ar, Kr, and Xe for modeling 

the temperature and excess air, and the program would 

subsequently calculate the amount of excess N2 using the 

model results and the dissolved N measurement. Alternatively, 

the user could include excess N2 as a model-fit parameter. 

The third option will run Monte Carlo simulations for models 

with acceptable probabilities in order to more accurately 

define the parameter errors. The user can choose the number 

of simulations and to output all the simulation results in a 

separate worksheet. The fourth option, Excess N2 for Ar/N2

only, allows users to solve for excess N2 gas for a range of 

excess air values (see example 3) when only Ar and N2 were 

measured. This analysis computes a set of dissolved gas 

models by setting excess air to a constant value incrementally 

over a range of values and computing the best-fit recharge 

temperature and excess N2 value. If the N2 values are greater 

than or equal to zero, the model is accepted and the average 

and standard deviation of the set of accepted models are 

returned as the best-fit parameter and parameter error values.

_ModOut Worksheet

Results of dissolved gas modeling are output to a 

worksheet named by the user with the suffix “_ModOut” 

appended to the name (fig. 13). All sample information, 

initial model parameter values, dissolved gas modeling 

results, measured concentrations, modeled concentrations, 

and Monte Carlo results are written to the output worksheet. 

Results of dissolved gas modeling include the model-fit and 

optimal parameters and their uncertainty. If a parameter was 

not included in the best-fit parameter, but was required by the 

model, then the initial value was used (“init. val.”). Only the 

CE and PR models require a value for F, and only samples 

with dissolved N can be used to estimate excess N2; all other 

parameters are required. The gas with the highest contribution 

to the chi-square error, deviations between measured and 

modeled solubility concentrations, and generic comments 

about the gas concentrations and model results are provided 

to help evaluate the model results. Modeled gas solubility, 

excess air, and total gas concentrations are provided in cm3 at 

STP per g of water, mmol/L, and mg/L. If the CE model was 

used to find the best-fit parameter of entrapped air, then the 

equivalent excess air concentration for each gas is calculated 

and included in the output.

Figure 12. The form for modeling dissolved gases.
Program Description 27

Figure 13. The _ModOut worksheet. There are more than 250 columns of values that are produced in the output, organized into 

groups that relate fields to sample information, initial model values, model results, measured concentrations and computations of 

model components and Monte Carlo results.

Monte Carlo simulation results are provided in the last 

set of columns in the worksheet and contain the results of 

the average and standard deviations of modeled simulations. 

When computing tracer concentrations from dissolved gas 

model results where Monte Carlo simulations were computed, 

the Monte Carlo parameter errors from these simulations will 

be automatically used for tracer calculations; these values are 

red in the _TrcOut worksheet—see example 4.

There are more than 250 columns of values that are 

produced in the output from the _ModOut worksheet. 

Many of the fields are used for documenting the measured 

concentrations and internal calculations of model components 

so that results satisfy U.S. Geological Survey (USGS) open 

data requirements for models.

Environmental Tracer Concentrations

Environmental tracer concentrations can be calculated 

for every dissolved gas model listed in a _ModOut worksheet. 

This requires that samples have been modeled using the 

dissolved gas modeling routines listed in the previous section. 

The tracer calculations are initiated by the “Env. Tracer 

Calcs” button in the DGMETA tab in the Excel ribbon (fig. 3). 

The user will be prompted to select a _ModOut worksheet 

(fig. 14A) to calculate tracer concentrations for each dissolved 

gas model that is listed in the worksheet. The program will 

match tracer measurements for each corresponding Sample 

ID and Sample Date listed in the _ModOut and Input_Tracers 

worksheets. Calculated tracer concentrations are output to 

a worksheet named by the user with the suffix “_TrcOut” 

appended to it (fig. 15).
28 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

A B

Figure 14. The forms A, for selecting a model output worksheet 

for computing tracer concentrations and B, for selecting a tracer 

output worksheet for computing average tracer concentrations 

(single row per Sample and Date).

Figure 15. The _TrcOut worksheet, which contains calculations of atmospheric dry-air mole fractions and concentrations of 

tritium-helium-3 components.
Program Description 29

_TrcOut Worksheet

The TrcOut worksheet (fig. 15) contains the calculated 

dry-air mole fractions (mixing ratios) of atmospheric transient 

gas tracers and concentrations of 3H-He components for 

every model listed in the corresponding _ModOut worksheet. 

Parameter errors that are red are errors computed from Monte 

Carlo simulations reported in the _ModOut worksheet—see 

example 4. These errors can be preferred when parameter 

errors determined from dissolved gas model fits are of the 

same magnitude as the fitted parameters. In particular, the CE 

model can have large parameter errors when the F parameter 

approaches 1.

Details of the methods used for the calculations of tracer 

concentrations were given in the “Environmental Tracers” 

section. The 3Hetrit calculations are dependent on the Rterr

and the _TrcOut worksheet, which provides a graphical 

approach for evaluating different sources of Heterr and their 

effect on 3Hetrit concentrations. The calculated value of 

Rterr is to the right of the δ3He graph and is blue; that value 

should be evaluated using the graph (fig. 2) on the _TrcOut 

worksheet. The Rterr can be manually changed by entering 

a new value in the cell on the _TrcOut worksheet. These 

parts of the worksheet and functionality are demonstrated in 

examples 3 and 4.

Formatting

There are two routines in the DGMETA tab that can 

be used to reformat worksheets created by DGMETA. The 

first routine, “Collapse Tracer Calcs” can collapse multiple 

tracer calculations for a single Sample ID and Sample Date, 

which can be useful for finalizing the reporting of results 

and transferring them to other programs. The second routine 

reformats the _ModOut, _TrcOut, and _AvgVal worksheets 

into a new worksheet with a single-row header that is useful 

for exporting the results into another program. This output 

worksheet is in a format that meets USGS model open data 

requirements for documentation of model inputs and outputs.

If there are multiple dissolved gas models in a 

_ModOut worksheet or multiple tracer measurements in 

the Input_Tracers worksheet for a single Sample ID and 

Sample Date, the _TrcOut worksheet can contain multiple 

rows of tracer results for the same Sample ID and Sample 

Date. Multiple models and tracer measurements may provide 

additional estimates of uncertainty for the calculated tracer 

concentrations or identify samples that may be contaminated, 

degraded, or were collected improperly. The “Collapse Tracer 

Calcs” button in the DGMETA tab can be used to collapse the 

multiple rows into a single row for each unique Sample ID and 

Sample Date. The program will prompt the user (fig. 14B) to 

select a _TrcOut worksheet and create a new worksheet with 

a suffix “_AvgVal” appended to the name. The worksheet is 

organized and formatted the same as the _TrcOut worksheet, 

but the program will compute the average tracer concentration 

for multiple tracer results.

_AvgVal Worksheet

This worksheet contains average dissolved gas model 

results and tracer concentrations for every unique Sample 

ID and Sample Date reported in the _TrcOut worksheet 

(fig. 16). The _AvgVal worksheet can be useful for collapsing 

multiple tracer measurements from different laboratories or for 

comparing the effect of different dissolved gas model results 

on tracer concentration calculations. The _AvgVal worksheet 

should not be used, however, as a sole source of results when 

reporting the model outcomes. In cases where the dissolved 

gas models or computed tracer concentrations are very 

different, the averaging of the results could be undesirable, 

so users are advised to look closely at the _TrcOut worksheet 

before averaging the results.

_ModFlt, _TrcFlt, and _AvgFlt Worksheets

Worksheets with the suffix _ModFlt, _TrcFlt, and 

_AvgFlt are reformatted versions of the _ModOut, _TrcOut, 

and _AvgVal worksheets (fig. 17). These worksheets are 

identical to the original worksheets, but contain a single-row 

header that enable the worksheets and results to be portable to 

other programs and graphing software.
30 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Figure 16. The _AvgVal worksheet, which contains average values of tracer concentrations for each sample and date listed on 

the _TrcOut worksheet.
Examples 31

Figure 17. A _ModFlt worksheet, which is a single-row header version of a _ModOut, _TrcOut, or _AvgVal worksheet.

Examples

The following examples provide guidance for entering 

and comparing different solubility equations, strategies 

for modeling dissolved gas concentrations with a limited 

set of measured gases, the identification of mantle He in 

groundwater, and samples that have been degassed. This is not 

a comprehensive set of examples to show the full capability 

of the program. All figures included in the examples were 

created using the DGMETA workbooks that are included in 

the DGMETA installation package. Users are encouraged to 

use them while following the written examples below.

Example 1—Comparison of Noble Gas Solubility 

Data in Freshwater and Seawater

DGMETA includes multiple solubility data for each noble 

gas—see the Database worksheet for the list of gases. For 

most noble gases, the solubilities from different studies vary 

less than 1 percent over temperatures between 0 and 40 °C and 

salinities between 0 and 35 per mil. However, the solubility 

of Ne among studies frequently varies by more than 1 percent 

over this range of temperature and salinity (fig. 18).
32 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

A B

C D

0 10 20 30 40 50

6.0E-06

6.5E-06

7.0E-06

7.5E-06

8.0E-06

8.5E-06

9.0E-06

9.5E-06

1.0E-05

1.1E-05

0 10 20 30 40 50

–2

–1

0

1

2

3

4

5

6

7

8

Deviation from Weiss (1971), in percent

6.5E-06

7.0E-06

7.5E-06

8.0E-06

8.5E-06

9.0E-06

0 5 10 15 20 25 30 35

Salinity, in per million or in parts per thousand

0 5 10 15 20 25 30 35

–0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Salinity, in per million or in parts per thousand

Deviation from Weiss (1971), in percent

Millimoles per kilogram Millimoles per kilogram

Temperature, in degrees Celsius Temperature, in degrees Celsius

EXPLANATION

Ne_Weiss Ne_WeissB Ne_Clvr Ne_H&E Ne_Jenk

Figure 18. Neon (Ne) concentrations in freshwater and saltwater at equilibrium with air, based on solubility equations by Weiss 

(1971) that were expressed as the moist-air solubility (Ne_Weiss) and Bunsen solubility constant (Ne_WeissB), by Clever (1979a; Smith 

and Kennedy, 1983; Ne_Clvr), by Hamme and Emerson (2004; Ne_H&E), and by Jenkins and others (2019; Ne_Jenk). A, variation in 

neon solubility concentrations in freshwater (salinity = 0 per mil) over the temperature range of 0–50 °C; B, deviation of the solubility 

concentrations derived from all four equations compared to those derived from the gravimetric equation of Weiss (1971) as a function 

of temperature; C, variation of neon solubility concentrations at a temperature of 15 °C over the salinity range of 0–35 per mil; and 

D, deviation of the solubility equations from those of the gravimetric equation of Weiss (1971) as a function of salinity. The names of the 

lines consist of the name of the gas and a descriptor of the reference for the source of the gas solubility data and are unique identifiers 

for the gas solubility data used by DGMETA (see the Database worksheet of DGMETA).

DGMETA includes solubility data for Ne in freshwater 

and seawater from five studies. These solubility data were 

graphed using the “Gas Solubilities” routine in the DGMETA 

tab (fig. 3) and selecting the Ne solubilities listed in the Gas 

Solubilities form (fig. 8). In general, these studies measured 

the solubility of Ne in freshwater and seawater over a range of 

temperatures and fit polynomial equations to the experimental 

solubility data. The derived expressions provide a reasonable 

approximation of the solubility of the gas in water as a 

function of temperature and salinity for the conditions used in 

each experiment. Many of the studies used different methods 

and instruments to measure the solubility. The purpose of 

the analysis presented here is to compare the differences 

in solubility to evaluate their applicability over a range of 

environmental conditions rather than a detailed examination 

and comparison of the methods used to develop the solubility 

functions or to identify a “correct” set of solubility equations.
Examples 33

Weiss (1971) provided two equations for the solubility of 

Ne: (1) one equation based on the moist-air solubility at 1 atm 

(Ne_Weiss) and (2) another equation for the Bunsen solubility 

(Ne_WeissB). Both equations used the van ’t Hoff equation 

and Setchénow relation for solubility and salinity corrections 

(G1). Clever (1979a) determined the solubility of Ne in water 

in terms of the mole fraction solubility at standard pressure (1 

atm) from a compilation of several solubility studies, and these 

solubility equations were combined with salting coefficients 

for solutions of NaCl (0.0–5.2 mol/L) computed by Smith and 

Kennedy (1983; G3). Hamme and Emerson (2004) reexamined 

the solubility of Ne (G5) using the equations of Garcia and 

Gordon (1992; G5). Most recently, Jenkins and others (2019) 

performed solubility experiments for Ne in freshwater and 

seawater and reported their solubility using a modified version 

of the Weiss (1970) equation to include an additional salinity 

term introduced by Garcia and Gordon (1992; G6). These five 

solubility equations can be easily compared in DGMETA by 

clicking the “Gas Solubility” button on the DGMETA tab (see 

the “Graphing” section).

The Weiss solubility equations are the default or 

primary solubility equations used by the program because 

the experiments used to derive the equations were done over 

a wide range of temperatures and salinities using the same 

or similar methods for each gas. The Weiss solubility for 

Ne was reported in terms of volumetric Bunsen coefficients 

(Ne_WeissB) and gravimetric moist air solubilities 

(Ne_Weiss). Deviations between the two forms are 

less than 0.04 percent over the temperature range of 

0–40 °C (figs. 18A, B). The solubilities derived from 

the mole-fraction-based equations of Clever (1979a) for 

freshwater (Ne_Clvr) deviate from the Weiss solubilities 

usually less than 2 percent, whereas solubilities from 

Hamme and Emerson (2004; Ne_H&E) are more than 

0.5 percent higher than the Weiss solubilities over the 

entire temperature range (figs. 18A, B). The Jenkins (2019) 

solubilities for Ne in freshwater were more than 1 percent 

over the entire temperature range. At temperatures greater 

than 30 °C, Hamme and Emerson (2004) solubilities grow 

to more than 2 percent because their experiments were 

done for the temperature range of 0–25 °C and seawater 

experiments for the temperature range of 0–30 °C. Solubility 

equations determined by Clever (1979a, b, 1980) were fit 

to several reported sets of measured solubility equations 

over the temperature range of 5–66 °C. For temperatures 

less than 5 °C, this may partly explain deviations between 

the Clever and Weiss solubilities. It should be noted that 

the Clever solubilities for He, Ar, and Kr were partly based 

on solubility data from Weiss; and therefore, the Clever 

solubilities can be similar to the Weiss solubilities.

These differences highlight the importance of choosing 

solubility equations best suited to the environmental 

conditions being studied. For freshwater terrestrial 

environments, the Weiss solubilities combined with Xe 

solubility from Clever (1979b) and Smith and Kennedy (1983) 

are commonly used. Newer solubility studies by Hamme and 

Emerson (2004) and Jenkins (2019) may be more common in 

oceanographic or saltwater studies.

Hamme and Emerson (2004) noted that small 

inaccuracies of the solubility equations can lead to large 

differences in biological fluxes of O2 production. In their 

study, they showed that the most commonly used solubility 

data for Ne in freshwater and seawater was most often lower, 

about 1 percent or more, than the solubility of Ne they 

determined (Ne_H&E; figs. 18B, D). In particular, the Weiss 

solubilities had the highest deviations. Sano and Takahata 

(2005) measured equilibrium concentrations of noble gases 

in seawater for nine temperatures ranging from 1 to 29.6 °C. 

The measured Ne data were compared to Ne concentrations 

predicted from the Ne_Weiss, Ne_Clvr, Ne_H&E, and Ne_

Jenk solubilities. Figure 19 shows the percentage of deviation 

from the measured data reported by Sano and Takahata (2005). 

The Weiss solubility of Ne were lower than the measured data 

for the entire temperature range and were more than 1 percent 

lower for many temperatures. Almost all of the predicted Ne 

were lower than the measured Ne for temperatures greater 

than 15 °C. Neon from Ne_H&E, Ne_Clvr, and Ne_Jenk were 

most similar to the Ne data from Sano and Takahata (2005) 

and indicate that those solubilities may be better suited for 

oceanographic studies.

Temperature, in degrees Celsius

–5

–4

–3

–2

–1

0

1

2

0 5 10 15 20 25 30 35

Deviation from Sano and Takahata (2005), 

in percent

Sano and Takahata Ne_Weiss Ne_Clvr Ne_H&E Ne_Jenk

EXPLANATION

Figure 19. Percentage of deviation of neon (Ne) solubility from 

measured data reported by Sano and Takahata (2005) in seawater 

(salinity equals 34.2 per mil) for a range of temperatures. The 

names of the points consist of the name of the gas and a descriptor 

of the reference for the source of the gas solubility data and are 

unique identifiers for the gas solubility data used by Dissolved Gas 

Modeling and Environmental Tracer Analysis (DGMETA; see the 

Database worksheet of DGMETA).
34 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Despite the differences in Ne solubility, overall 

differences in gas solubilities are usually less than 1 percent 

among all types of water. Solubility equations can be 

chosen to maximize precision for specific temperature and 

salinity conditions. Most solubility equations are valid for 

temperatures between 0 and 40 °C and salinity from 0 to 

35 per mil, although individual solubility equations may have 

broader or shorter temperature and salinity ranges based on the 

experimental data used to derive the solubility equation.

Example 2—Inverse Modeling of 

Air-Equilibrated Freshwater and 

Seawater Samples

A common application of measuring noble gases in 

water is to determine the temperature of the water when 

it equilibrated with the atmosphere. The temperature is 

determined by inverse modeling of the measured noble gas 

concentrations with modeled concentrations predicted by 

solubility. The noble gas concentrations reported by Sano and 

Takahata (2005) were from nine seawater samples equilibrated 

with air at different temperatures. Inverse modeling of 

the measured concentrations should, in theory, predict the 

air-water equilibration temperature of each sample in the 

experiment (1, 5.5, 7.5, 10, 14.4, 17, 21.3, 24.6, and 29.6 

°C). Similarly, the USGS Noble Gas Laboratory in Denver, 

Colorado, routinely prepares air-equilibrated distilled-water 

(AEW) samples for laboratory quality-assurance purposes 

(Hunt, 2015). Between April 2016 and August 2017, the 

USGS Noble Gas Laboratory prepared 51 AEW samples that 

were equilibrated with room air at about 20 °C.

The temperatures of these air-equilibrated water and 

seawater samples were modeled using the full suite of 

measured noble gases using different sets of solubility. The 

default set used the Weiss solubilities for He, Ne, Ar, and Kr, 

and the solubility of Xe from Clever (1979b) and Smith and 

Kennedy (1983). The second set used the solubility equations 

from Clever (1979a, b, 1980) and Smith and Kennedy (1983). 

The third set used the solubility of Ar and Ne from Hamme 

and Emerson (2004), He from Weiss (1971), Kr from Weiss 

and Kyser (1978), and Xe from Clever (1979b) and Smith and 

Kennedy (1983). The fourth set used the solubility of noble 

gases reported by Jenkins and others (2019). The Dissolved 

Gas Modeling form was configured to find the temperature of 

AEW and seawater samples at air-water equilibrium for each 

set of noble gas solubilities (fig. 20). For the USGS Noble 

Gas Laboratory AEW samples, He was not modeled because 

a known source of excess atmospheric He at the USGS Noble 

Gas Laboratory contaminated the AEW samples.

Figure 20. The Dissolved Gas Modeling form used to evaluate air-equilibrated samples prepared 

at the U.S. Geological Survey (USGS) Noble Gas Laboratory in Denver, Colorado, and seawater 

samples from Sano and Takahata (2005). Helium was not included in dissolved gas model fits for 

air-equilibrated, distilled-water samples from the USGS Denver Noble Gas Laboratory.
Examples 35

Results of the dissolved gas modeling for seawater 

samples showed that none of the four sets of solubility data 

fit the seawater sample data well. It also was counterintuitive 

that the modeled temperature agreed more with the reported 

temperature for the sets of solubilities with poorer fits than 

for sets of solubilities with better fits. The median chi-square 

probabilities were 3.2, 12, 18.4, and 14.5 percent for the 

Weiss, Clever, Hamme, and Jenkins solubilities, respectively. 

Temperature agreements, measured by the median of the 

percent deviation from the reported temperature, were –0.24, 

0.04, –1.5, and –4.4 percent for the Weiss, Clever, Hamme, 

and Jenkins solubilities, respectively. Despite some poor fits 

to the data, the modeled temperatures were generally within 

0.5 °C of the reported equilibrated temperature.

Results of the dissolved gas modeling for AEW samples 

prepared by the USGS Noble Gas Laboratory showed that 

all sets of solubilities fit the measured data well. Unlike the 

seawater samples, the sets of solubilities with better fits also 

had better temperature agreement. The median chi-square 

probabilities were 80.8, 77.8, 76.5, and 70.7 percent for the 

Weiss, Clever, Hamme, and Jenkins solubilities, respectively. 

Temperature agreements, measured by the median of the 

percent deviation from the equilibration temperature of 

20 °C, were –0.73, –0.76, –1.1, and –2.9 percent for the 

Weiss, Clever, Hamme, and Jenkins solubilities, respectively. 

Temperature disagreements were within 0.5 °C for most of 

the fits.

Overall, the solubility comparisons for AEW and 

seawater samples indicate that the Weiss and Clever 

solubilities may perform better than other solubilities when 

estimating air-water equilibration temperatures. These 

solubilities gave better fits to the measured freshwater 

AEW samples and had better temperature agreements 

than other solubilities. Although the Weiss and Clever 

solubilities gave poorer fits to the seawater samples than 

other solubilities, the Weiss and Clever solubilities gave 

better temperature agreement. This interesting result could 

reflect the complexities and difficulties in the preparation and 

measurement of gases in concentrated salt solutions. Although 

the analysis of modeled fits to AEW samples and seawater 

support the use of the Weiss data and the Clever data for Xe 

for most applications, the data used in this comparison were 

not rigorously assessed, and comparisons to other data or 

to data collected over a different range of temperatures may 

support other sets of solubility equations. Therefore, users are 

given the option to substitute different solubility equations and 

parameters to explore dissolved gas model fits and results for 

temperature and excess air.

Example 3—Dissolved Gas Modeling 

with Neon, Argon, and Nitrogen or Argon 

and Nitrogen

This example shows how data from different labs can be 

combined to model recharge temperatures and excess air in an 

aquifer where denitrification complicates the interpretation. 

Samples where denitrification has increased the amount of 

dissolved N2 gas can make it difficult to determine temperature 

and excess air from Ar and N2 concentrations alone or 

from Ar, N2, and Ne concentrations. In this situation, the 

temperature, excess air, and excess N2 need to be determined; 

but in one case only two gases are available and in the other 

case Ne Ar, and N2 are available. This situation is best solved 

by working with samples that have Ar, N2, and Ne data first 

to provide ranges of possible temperatures and excess air and 

then to use those estimates to inform the modeling of samples 

with Ar and N2 only.

Water samples for analysis of N2 and Ar were collected 

from 26 wells in the Texas coastal lowlands aquifer in August 

and September 2013. These samples were measured at the 

USGS Groundwater Dating Laboratory (Reston, Virginia) 

using gas chromatography (U.S. Geological Survey, 2020) 

and were reported in mg of gas per L of water. Samples from 

a subset of 19 wells were analyzed for He, Ne, and δ3He at 

the Lamont-Doherty Earth Observatory (Ekwurzel and others, 

1994; Ludin and others, 1998). Five of the 19 samples were 

analyzed at the USGS Noble Gas Laboratory in Denver, 

Colorado (Hunt, 2015). Concentrations of He and Ne were 

reported in units of cm3 of gas at an STP per g of water. 

DGMETA can handle gases reported in different units and can 

convert them to common units for graphical display of the 

data and for inverse modeling of recharge temperature, excess 

air, and excess N2.

Graphs of N2 concentration versus the Ar-to-Ne ratio and 

N2 versus Ar concentrations were made to identify patterns 

in temperature, excess air, and excess N2 that could be 

caused by denitrification in the aquifer (fig. 21). These graphs 

show that groundwater likely was recharged at temperatures 

between 10 and 20 °C and had excess air values between 

1 and 3 cm3/kg. Samples that may have been affected by 

denitrification tend to plot above samples unaffected or less 

affected by denitrification. Samples affected by denitrification 

are orange and green on figure 21; samples that are green have 

excess N2 between 0.5 and 2 mg/L as N, and samples that are 

orange have excess N2 greater than 2 mg/L as N. Samples 

that are blue on figure 21 have little to no excess N2 and plot 

near lower excess air concentrations, typically between 2 and 

4 cm3/kg. The temperature ranges and trends of samples are 

more consistent on the Ar-N2 graph than the Ar/Ne-N2 graph 

because the concentrations were measured from the sample 

bottle and used the same analytical method.
36 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

0

2

4

6

8

10

0

2

4

6

8

10 0

5

10

15

20

25

30 35 40

0

5

10

15

20

25

30 35 40

0

2

4

6

8

10

0

2

4

6

8

10

0

5

10

15

20

25

30 35 40

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Temperature,

in degrees Celsius

Excess air, in cubic centimeters

per gram of water

Nitrogen gas, in milligrams per liter

Argon, in milligrams per liter

0

5

10

15

20

25

30

35

1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000

Temperature, in degrees Celsius

Excess air, in cubic

centimeters per 

gram of water

Nitrogen gas, in milligrams per liter

Argon to neon ratio, in milligrams per liter

A B

Nitrogen gas greater than 2

Nitrogen gas 0.5–2

Nitrogen gas less than 0.5

Closed-system equilibration model

Unfractionated excess air model, plus excess nitrogen gas

Unfractionated excess air model

Temperature or excess air boundary line

EXPLANATION

Figure 21. Concentrations of A, Nitrogen gas (N2) and argon and neon ratio, and B, N2 and argon only were measured in groundwater 

samples from the Texas coastal lowlands aquifer in 2013. Samples are color coded for the amount of excess N2 (presumably from 

denitrification; milligrams per liter as nitrogen) determined from graphical analysis and model fits to the data.

In order to assess the extent of potential denitrification, 

19 sites with measured values of Ne, Ar, and N2 were modeled 

first using the UA model with fit parameters of temperature, 

excess air, and excess N2 (fig. 22). Model results indicate 

that most samples had temperatures around 16 °C (median of 

samples), excess air of about 2.5 cm3/kg, and denitrification of 

about 0.6 mg/L. In some cases, modeled recharge temperatures 

were too low (8 °C) or too high (42 °C) because the Ne did not 

agree well with measured Ar and N2 results. These samples 

and other samples with near-zero excess air were reevaluated 

using Ar and N2 only. Recharge conditions from models that 

include Ne may differ from Ar-N2 recharge conditions because 

of different sample collection techniques that lead to different 

chemistry of samples, or from denitrification, or biogenic-CH4

production that causes partial gas loss.

As a next step, all the samples were modeled again 

using Ar and N2 only, with fit parameters of temperature and 

excess nitrogen gas. The option “Excess N2 for Ar/N2 only” 

will generate multiple models for a range of excess air values 

specified on the user form (fig. 23). For each model generated, 

the best-fit temperature and excess N2 values are computed 

and the average temperature, excess N2, and excess air value 

are reported. The median excess air concentration using Ne, 

Ar, and N2 was 2.5 cm3/kg and most models with no excess or 

a minimal amount of excess N2 had excess air between 1.5 and 

3.5 cm3/kg. Consequently, a range of 1.5–3.5 cm3/kg was used 

as an estimate of excess air in groundwater that likely had 

undergone denitrification.

The option “Excess N2 for Ar/N2 only” method works 

best when all samples are from the same aquifer and were 

recharged under similar conditions. If there are large 

differences in recharge conditions in an aquifer, this method 

may overestimate or underestimate the contribution of 

denitrification to samples.
Examples 37

Figure 22. The form for Dissolved Gas Modeling for modeling recharge conditions and 

denitrification using neon (Ne), argon (Ar), and nitrogen gas (N2) concentrations. In this example, 

all samples will be modeled using available Ne, Ar, and N2. Temperature, excess air, and excess 

N2 are the parameters to be used to fit the dissolved gas models to the sample data. Elevation and 

salinity will be specified from the Input_Gases worksheet.

Figure 23. The form for Dissolved Gas Modeling for modeling recharge conditions with 

denitrification using argon (Ar) and nitrogen gases (N2) only. The option “Excess N2 for Ar/N2

only” will compile a list of models that have excess air values between the range specified by the 

user, 1.5–3.5 cubic meters per kilogram (cm3/kg) in this case, and will report the corresponding 

concentrations of excess N2 (presumably from denitrification).
38 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

For samples with multiple models of dissolved gas 

results, the output from each model should be evaluated 

against one another to select a single dissolved gas model that 

best describes the set of measured gas concentrations. For 

this example, selected models are included in the example file 

and each sample was categorized in the study area column 

according to the excess N2 concentration of the chosen model. 

In general, models with Ne, Ar, and N2 were preferred over 

models that used only Ar and N2 when the excess air was 

greater than 1 or less than or equal to 6 mg/L as N. In a few 

instances, the Ne was not useful for determining temperatures 

and excess air. The final set of models listed in the worksheet 

called TX_Final_ModOut (in the example workbook) was 

used to make graphs on figure 21. In this example, the 

dissolved gas models that were derived based on Ne, Ar, 

and N2, provided the range of likely recharge temperatures, 

excess air, and extent of denitrification in the aquifer. This 

information was then used to inform samples with Ar and N2

data only. Samples with larger excess air values than the range 

typical of samples with Ne, Ar, and N2 were likely to have 

excess N2 present. In these cases, the UA model was run with 

a different amount of excess N2 (entered on the Input_Gases 

worksheet) until the excess air or temperature ranges were 

similar to those determined using Ne, Ar, and N2.

Combining data from different sample collection types 

and analytical methods can highlight some of the sources of 

uncertainty in the laboratory results and the assumptions that 

underlie the models. Böhlke and others (2007) compared 

model results using Ar and N2 (only) with model results 

using Ne, Ar, and N2 for an aquifer in Nebraska receiving 

irrigation recharge. The modeling approach that included Ne 

data indicated variation in recharge temperature and excess 

air (assumed to be unfractionated, using the “UA” model). 

After accounting for those variations, oxic samples had a 

mean calculated excess N2 concentration of –6 plus or minus 

9 micromoles per L (μmol/L) as N (–0.1 mg/L), indicating 

a minor bias that might have resulted from combining data 

from different sample collection types and analytical methods. 

Similar comparisons of calculated excess air in oxic samples 

for an aquifer in Long Island, New York (Böhlke and others, 

2009), yielded a mean excess N2 value of –14 plus or minus 

11 μmol/L as N (0.2 mg/L) when calculated using all three 

gases. These negative values were interpreted as an indication 

of uncertainty in the estimation of excess N2 from such data.

Tracer concentrations for the Texas samples were 

computed for the final set of dissolved gas models listed in the 

TX_Final_ModOut worksheet. In this example, concentrations 

of SF6, 3H, and 3Hetrit were calculated using the “Env. Tracer 

Calcs” button in the toolbar. As part of the 3Hetrit computation, 

the 3He/4He ratio of Heterr was determined automatically by 

DGMETA. The calculated best-fit line explains the δ3He 

evolution of tritium-dead samples. In this example, the best-fit 

Rterr value was 4.43 × 10–8 and is listed in the blue cell in 

the table to the right of the graph (fig. 24). This value is very 

similar to the radiogenic helium line (2 × 10–8; black dashed 

line) but is slightly elevated probably because the amount 

of He in some samples appears to be overestimated from 

dissolved gas models that do not fit the He data well. In this 

case, the authors decided to replace the Rterr value in the blue 

cell to the right of the graph with 2 × 10–8 and the best-fit 

value was ignored.

Average concentrations of the tracers and their errors can 

be computed as a final step using the “Collapse Tracer Calcs” 

button in the toolbar. In this study area, SF6 concentrations 

above 1.0 volume of gas per volume of dry air, in parts per 

trillion (pptv), in samples with low 3H (less than 0.5 TU) 

indicate that SF6 was contaminated, likely by terrigenic 

sources or perhaps from atmospheric contamination during 

sample collection, storage, or shipping. The contamination 

bias can be identified from a graph of 3H and SF6 (fig. 25). 

Because SF6 is a tracer of water that entered an aquifer after 

about 1975 or later, groundwater recharged after that date 

should contain SF6 concentrations between 0.5 and 9 pptv and 

have detectable levels of 3H (greater than 0.5 TU). However, 

3H concentrations in most samples were less than 0.1 TU, 

which indicates most of the groundwater was recharged before 

about 1950, such that all SF6 concentrations of environmental 

origin should be below about 0.5 pptv (gray-shaded area 

in fig. 25). Assuming the laboratory analyses were reliable 

and sample collection procedures were followed correctly, 

groundwater samples with SF6 concentrations above 

0.5 pptv and 3H concentrations less than 0.1 TU indicate 

the introduction of SF6 from a terrigenic source of SF6—

diffused from igneous or volcanic rocks (Busenberg and 

Plummer, 2000; red-shaded area in fig. 25). On the other 

hand, contamination from exposure to atmospheric sources 

of SF6 during sample collection, storage, or shipping could 

be another explanation. By comparison, the atmospheric 

mole fraction concentration of SF6 is about 10 pptv. 

Sulfur hexafluoride concentrations substantially above the 

atmospheric levels would indicate a terrigenic source was 

more likely. Unfortunately, the presence of terrigenic SF6

can limit the utility of SF6 for dating water because it may 

not be possible to tell if the SF6 was derived from terrigenic 

or atmospheric sources even when the concentrations are at 

expected environmental levels.
Examples 39

Heterr/(Hes–Hea)

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Rs/Ra -air corrected, ratio

A B

EXPLANATION

Best-fit

Radiogenic

Mantle

Mix: rad+man

Tritiated

Tritium dead

Best-fit Rterr

Figure 24. The A, helium isotope separation plot and B, table of helium isotope control variables on the TX_Final_TrcOut worksheet. 

Dissolved Gas Modeling and Environmental Tracer Analysis automatically calculates a best-fit line for samples with tritium less than 

0.5 tritium unit in order to compute the helium-3/helium-4 of terrigenic helium (Rterr), which is output in the blue cell to the right of the 

graph on the _TrcOut worksheet. Sulfur hexafluoride, in parts per trillion by volume 

0

1

2

3

4

5

6

7

8

9

10

-0.5 0 0.5 1 1.5 2 2.5 3

Tritium, in tritium units

Terrigenic

contamination

Environmental

concentrations

Non-detect

Figure 25. Tritium and sulfur hexafluoride (SF6) 

concentrations in groundwater in the Texas coastal 

lowlands aquifer. Sulfur hexafluoride concentrations below 

the tritium cutoff line at 0.5 tritium unit (TU) are indicative 

of terrigenic sources whereas SF6 concentrations greater 

than 0.5 TU generally reflect expected environmental 

concentrations. The presence of terrigenic SF6 may limit 

the utility of SF6 for determining ages of water.
40 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Example 4—Tritiogenic Helium-3 Analysis in the 

Central Eastside San Joaquin Valley, California

The following example shows how dissolved gas 

modeling and environmental tracer concentrations were 

determined for a set of samples used for determining age 

distributions of groundwater in wells in the central eastside 

of the San Joaquin Valley, California (Green and others, 

2016; Jurgens and others, 2016). Even though measurements 

of noble gases typically provide the best way to determine 

air-water equilibrium conditions (for example, the temperature 

and excess air of recharge) in an aquifer, the data from this 

example do not provide strong evidence that one dissolved 

gas model is the best choice for determining recharge 

conditions. Example 4 also demonstrates the importance of 

Monte Carlo simulation methods for determining parameter 

uncertainty from CE models. Samples of dissolved gases 

were modeled using Ne, Ar, Kr, and Xe concentrations and 

the UA (fig. 26A) and CE models (fig. 26B). To evaluate the 

results of the dissolved gas models, samples were assigned 

age classifications of Modern, PreModern, or Mixed based on 

their 3H activity in relation to the 3H in the precipitation record 

(Michel and others, 2018) corrected for decay from the time of 

precipitation to the sample date. This age classification system 

is described in Lindsey and others (2019). Samples with 3H 

less than 0.34 TU were categorized as PreModern because 

activities below this value would indicate that groundwater 

was recharged before 1950. Samples with 3H greater than 2.6 

TU were categorized as Modern because activities greater 

than this value would indicate that groundwater was recharged 

after 1950. Samples with activities between 0.34 and 2.6 were 

categorized as Mixed and indicate that groundwater was a 

mixture of water recharged before and after 1950.

Comparison of CE to UA model results showed that the 

CE model generally predicted higher recharge temperatures 

than the UA model for the same sample (fig. 27), but gave 

better fits and He balance (Heterr = Hes – Heeq – Hea ≈ 0) 

for most of the Modern samples (23 of 40). The CE model 

may be more relevant to samples of Modern groundwater in 

this region because recharge is primarily from agricultural 

irrigation and can produce seasonal water-table fluctuations 

of around 1 m annually (Phillips and others, 2007; Jurgens 

and others, 2008). These fluctuations may trap large volumes 

of air in water that enters the aquifer as recharge. The 

UA model indicated that median excess air values were 

2.6 cm3/kg for Modern samples, 2.5 cm3/kg for Mixed 

samples, and 1.41 cm3/kg for PreModern samples, which 

indicates that Modern water has a larger amount of excess air 

than PreModern water.

Final dissolved gas models were chosen on a 

case-by-case basis by comparing model-fit parameters and 

statistics. Generally, samples with larger amounts of excess 

air or concentrations of gases indicative of F were fit better by 

the CE model than by the UA model (fig. 28). Final dissolved 

gas models are listed in the Final_ModOut worksheet in the 

example 4 workbook.

Chlorofluorocarbon and 3Hetrit concentrations were 

calculated from CFC and δ3He samples and the dissolved 

gas models listed in the Final_ModOut worksheet. Tracer 

concentrations are computed in the Final_TrcOut worksheet 

and values with a red font are parameter errors taken from the 

Monte Carlo analysis. The Rterr value was determined from a 

best-fit line for samples with 3H activities less than 0.29 TU, 

which is lower than the threshold used to define PreModern 

groundwater above, because it isolates the samples with the 

lowest 3H concentrations. These samples had a Rterr value 

of about 5.0 × 10–7, which indicates a mixture of radiogenic 

(2 × 10–8 cm3/g) and mantle (1 × 10–5 cm3/g) He (fig. 29). 

The amount of mantle He in the mixture is small, about 

5 percent, and is probably derived from weathering of volcanic 

sediments in the aquifer (Page and Balding, 1973).

Tritiogenic helium concentrations were calculated 

from the He mass balance (eqs. 23–25). Calculated 3Hetrit

concentrations, for samples with Heterr concentrations of 

greater than 70 percent of the measured He concentration, 

are not reported because the amount of Heterr in such samples 

can make it difficult to accurately determine the amount of 

He derived from tritium decay. Samples that plot below the 

best-fit line shown on figure 29 will have negative or zero 

concentrations of 3Hetrit even though some of the samples 

have measurable 3H. Those samples, and perhaps others, 

could be from sites where volcanic sediments are absent in the 

subsurface and have a purely radiogenic source of Heterr. For 

those sites, the Rterr values could be individually set to 2 × 10–8

on the _TrcOut worksheet, but the concentrations of 3Hetrit will 

be small. The best-fit Rterr value (5 × 10–7) gives lower, more 

conservative estimates of 3Hetrit than a radiogenic Rterr, which 

may be preferable for a regional dataset. Figure 30 shows how 

the δ3He graph can be manually adjusted to identify the 3Hetrit

content of samples with mixtures of Heterr. For that graph, a 

concentration of 30 TU was set for the 3Hetrit value in the table 

to the right of the graph (fig. 30B). The graph shows that many 

samples with the Heterr plot near and along this mixing line, 

which indicates those samples have 3Hetrit values near 30 TU. 

Overall, 36 samples had calculated concentrations of 3Hetrit

greater than 1 TU, 24 samples were not quantifiable, and 

15 samples had calculated concentrations of less than 1 TU.
Examples 41

A

B

Figure 26. The Dissolved Gas Modeling form used to generate A, unfractionated excess air models 

(UA) and B, closed-system equilibration models (CE) of dissolved gases on the basis of dissolved neon 

(Ne), argon (Ar), krypton (Kr), and xenon (Xe) concentrations in groundwater in the central eastside of 

the San Joaquin Valley, California, in 2006.
42 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

10

15

5

20

25

5 10 15 20 25

Closed-system equilibration model

recharge temperature, degrees Celsius

Unfractionated excess air model recharge temperature,

degrees Celsius

1 to 1

Mixed

Modern

PreModern

EXPLANATION

Figure 27. Comparison of closed-system equilibration and 

unfractionated excess air model results for recharge temperatures 

among different age classifications based on tritium measurements 

of groundwater samples from the central eastside of the San 

Joaquin Valley, California, in 2006.

0 2 4 6 8 10

0 2 4 6 8 10

0

5

10

15

20

25

30

35 40

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

0 2 4 6

8

10

0

5

10

15

20

25

30 35 40

0

5

10

15

20

25

30

35

40

0.0001

0

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

Argon, in cubic centimeters at standard temperature 

and pressure per gram of water

Krypton, in cubic centimeters at standard temperature 

and pressure per gram of water

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Xenon to neon, ratio Xenon to neon, ratio

0.0E+00

2.0E-08

4.0E-08

6.0E-08

8.0E-08

1.0E-06

1.2E-07

1.4E-07

1.6E-07

1.8E-07

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

A B

Unfractionated excess air model

Closed-system equilibration model

Closed-system equilibration model

Unfractionated excess air model

Temperature or excess air boundary line

EXPLANATION

Figure 28. Graphs of sample concentrations of A, argon versus xenon to neon and B, krypton versus xenon to neon. Samples are color 

coded based on the final dissolved gas model for each sample. Samples where the unfractionated excess air (UA) model was used 

are orange and samples where the closed-system equilibration (CE) model was used are blue. Most samples that follow a UA model 

(orange circles) had excess air values between 1 and 4 cubic centimeters per kilogram (cm3/kg). Many samples also followed a CE 

model with excess air values between 3 and 7 cm3/kg. The lines for the UA (green-solid line) and CE (red-dotted line) were plotted using 

a temperature of 17 degrees Celsius and the fractionation parameter of the CE model was set to 0.5.
Examples 43

Heterr/(Hes–Hea)

–0.3 –0.1 0.1 0.3 0.5 0.7 0.9

Rs/Ra - air corrected, ratio

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 A B

EXPLANATION

Best-fit

Radiogenic

Mantle

Mix: rad+man

Tritiated

Tritium dead

Figure 29. Views of A, the modified Weise plot (Weise and Moser, 1987) for samples that are tritiated (tritium concentration greater 

than 0.29 tritium unit; TU) and samples that are tritium-dead (less than 0.29 TU) and B, a table of helium isotope control variables on the 

Final_TrcOut worksheet. The best-fit line (solid blue) through tritium-dead samples has a helium-3/helium-4 of terrigenic helium (Rterr) of 

about 5.0 × 10–7 and indicates a mixture of radiogenic and mantle helium.

Heterr/(Hes–Hea)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

–0.3 –0.1 0.1 0.3 0.5 0.7 0.9

Rs/Ra -air corrected, ratio

A B

EXPLANATION

Best-fit

Radiogenic

Mantle

Mix: rad+man

Tritiated

Tritium dead

Figure 30. Views of A, the modified Weise plot showing mixing of groundwater endmembers and B, a table of helium isotope control 

variables on the Final_TrcOut worksheet. The blue line illustrates a mixture of water from one source that has no terrigenic helium 

(Heterr) and 30 tritium units (TU) of tritiogenic helium-3 (3Hetrit), and another water with 0 TU of 3Hetrit and is almost entirely composed of 

terrigenic helium that has a helium-3/helium-4 ratio (Rterr) of 5.0 × 10–7.
44 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

Finally, the example workbook that accompanies 

DGMETA (DGMETA_Example_4.xlsm) has an additional 

_ModOut worksheet with results from the samples where 

the CE model was used. These results have Monte Carlo 

simulations that were computed assuming the gases are 

completely independent. These results can be compared 

to the Final_ModOut version that included Monte Carlo 

simulations that were done assuming that the gases were 

not independent and were randomized as a group. The gas 

samples generated, assuming complete independence, are truly 

random and provide a more accurate picture about the true 

range of possible solutions to the measured gas data. Thus, 

the parameter error estimates from this method can often be 

large when the range of possible solutions is large. In addition, 

the average results of the ensemble may not be similar to the 

best-fit results computed from the measured gas data. In this 

example, the temperature, entrapped air, and F results from 

the Monte Carlo results deviate from the best-fit results by 

about –2.6, –87, and 9.2 percent, respectively. Overall, the 

completely independent results produced warmer temperatures 

with larger amounts of entrapped air than indicated by the fit 

computed from the measured data. These results also could 

be used to compute tracer concentrations, but the atmospheric 

mole fractions have errors as large as the computed mole 

fraction, which makes the tracer results ineffective for 

determining ages of water.

In contrast to the completely independent set of Monte 

Carlo simulations, the gas samples generated as a randomized 

group provided a more local estimate of the range of possible 

solutions and, therefore, had smaller parameter errors. In 

addition, the Monte Carlo simulations more strongly agreed 

with best-fit results computed from the measured data. Here, 

the temperature, entrapped air, and F values of the Monte 

Carlo simulations deviate from the best-fit parameter estimates 

by about 0.3, –4.7, and 0.17 percent, respectively. The 

dependent set of Monte Carlo errors also give lower computed 

errors associated with tracer concentrations. Consequently, this 

approach to parameter uncertainty can provide a reasonable 

substitute for the large uncertainty surrounding some CE 

model results.

Although this approach for dealing with large parameter 

uncertainties is advantageous for reporting CE model results 

and for computing tracer concentrations from those models, it 

may sidestep a more detailed analysis of Monte Carlo results 

recommended by Jung and Aeschbach-Hertig (2018). But 

more importantly, it also may mask important yet unexplained 

physical controls that might better explain peculiar sets of 

noble gas data. A more detailed analysis of the noble gas 

data in the context of new or existing physical processes 

and mixing of water with different recharge histories in the 

subsurface could therefore reveal a better model of the noble 

gas data.

Example 5—Degassed Groundwater

Aeschbach-Hertig and others (2008) showed that the CE 

model can explain some situations in which gases exsolve 

from a solution into a gas or bubble phase. Degassing can 

cause the gases to fractionate according to their solubility 

such that the lighter noble gases with lower solubilities are 

preferentially depleted from the solution relative to the heavier 

noble gases. The CE model simulates this behavior when F 

is greater than 1, which physically can be interpreted as the 

creation of bubbles from the water in response to a drop in 

hydraulic pressure. Bubble creation may occur as groundwater 

is extracted from deeper depths or moves to an area of lower 

pressure or warmer temperature. This phenomenon also 

can occur in groundwater where oversaturation of N2 from 

denitrification or CH4 from methanogenesis cause other 

gases to exolve from a solution or to be undersaturated in 

the subsurface (Aeschbach-Hertig and others, 2008; Cey and 

others, 2009). In the mathematical formulation of the CE 

model, the F parameter includes the ratio of the final volume 

of entrapped gas to the initial volume of entrapped gas. When 

F is greater than 1, the final volume of entrapped gas is greater 

than the initial volume indicating the enlargement or creation 

of a trapped volume of gas.

DGMETA can be used to graphically identify cases in 

which degassing likely occurred and estimate dissolved gas 

model parameters in which the CE model describes degassing 

in samples. A convenient way to identify degassed samples 

is to make graphs of the Xe/Ne ratio because degassing tends 

to cause Ne (a lighter gas) to be depleted relative to Xe (a 

heavier gas) and leads to higher Xe/Ne ratios in comparison 

to samples that are not degassed. Figure 31 shows samples 

from Aeschbach-Hertig and others (2008) in relation to the 

CE model with a F parameter value set to 2 and a recharge 

temperature set to 9 °C on the _GasGph worksheet. The 

samples plot far to the right of normal samples with dissolved 

excess air concentrations and higher Xe/Ne ratios.

In order to numerically solve for the best-fit parameters 

of degassed samples, the initial value of the F parameter 

should be set to a value of greater than 1 on the Input_Gases 

worksheet because the excess air component of the CE 

model vanishes at F=1. For this reason, it is recommended 

that degassed samples first be identified through graphical 

inspection of gas-gas graphs. Setting the initial F factor to 

a value greater than 1 ensures the numerical solution will 

identify the proper best-fit values of degassed samples. For 

the degassed samples of Aeschbach-Hertig and others (2008), 

the initial F value was set to 1.5 and best-fit solutions were 

computed. Results computed using DGMETA were nearly 

the same as best-fit values reported by Aeschbach-Hertig and 

others (2008).
Disclaimer 45

0 2 4 6 8 10

0 2 4 6 8 10

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

0 2 4

6

8

10

0

5

10

15

20

25

30 35 40

0

5

10

15

20

25

30

35

40

Temperature boundary line 1 Temperature boundary line 2

EXPLANATION

Excess air boundary line 1 Excess air boundary line 2

Unfractionated excess air model Closed-system equilibration model

Degassed

2.0E-08

0.0E+00

4.0E-08

6.0E-08

8.0E-08

1.0E-07

1.2E-07

1.4E-07

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

Krypton, in cubic centimeters at standard 

temperature and pressure per gram of water

0

Argon, in cubic centimeters at standard 

temperature and pressure per gram of water

0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 0.12

Xenon to neon, ratio Xenon to neon, ratio

A B

Figure 31. Solubility concentrations for A, argon and B, krypton plotted versus the ratio of the solubility concentrations of xenon and 

neon, in cubic centimeters per gram of water (cm3/g), for different recharge temperatures and excess air values. Degassed samples 

from Aeschbach-Hertig and others (2008) are plotted to show the area where degassed samples can be identified. Dissolved gas 

model output for the unfractionated excess air model is shown by the green line and the closed-system equilibration (CE) model is 

the red-dotted line. The samples are consistent with the CE dissolved gas model with a recharge temperature of 9 degrees Celsius 

and a fractionation value of about 2. The salinity and elevation of the degassed samples were approximately 1.3 per mil and 88 meters 

(pressure = 0.99 atm), respectively. The red dots give CE dissolved gas model concentrations and ratios for different amounts of 

entrapped excess air.

Installation Notes

The program was written in Visual Basic for Applications 

within the Microsoft Excel environment. As such, the program 

requires Excel macros to be enabled in order to work. An 

Excel warning may appear in the toolbar when the program 

is first opened, indicating that macros have been detected 

and are disabled. Users must enable Excel macros for the 

program to work.

Disclaimer

This software has been approved for release by the 

U.S. Geological Survey (USGS). Although the software has 

been subjected to rigorous review, the USGS reserves the 

right to update the software as needed pursuant to further 

analysis and review. No warranty, expressed or implied, 

is made by the USGS or the U.S. Government as to the 

functionality of the software and related material nor shall 

the fact of release constitute any such warranty. Furthermore, 

the software is released on condition that neither the USGS 

nor the U.S. Government shall be held liable for any damages 

resulting from its authorized or unauthorized use.
46 DGMETA (v.1): Dissolved Gas Modeling and Environmental Tracer Analysis Computer Program

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