Geochernical Evolution of Water in the 

Madison Aquifer in Parts of Montana^ 

South DakotBj and Wyoming

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Geochemical Evolution of Water in the 

Madison Aquifer in Parts of Montana, 

South Dakota, and Wyoming

By JOHN F. BUSBY, L. NIEL PLUMMER, ROGER W. LEE, andEWJCE B. HANSHAW

GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE AND 

ASSOCIATED ROCKS IN PARTS OF MONTANA, NEBRASKA, NORTH DAKOTA, 

SOUTH DAKOTA, AND WYOMING

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1273-F

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1991
DEPARTMENT OF THE INTERIOR 

MANUEL LUJAN, JR., Secretary

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director

Any use of trade, product, or firm names in this publication

is for descriptive purposes only and does not imply endorsement

by the U.S. Government

Library of Congress Cataloging-in-Publication Data

Geochemical evolution of water in the Madison Aquifer in parts of Montana, South Dakota, and Wyoming : geology and 

hydrology of the Madison Limestone and associated rocks in parts of Montana, Nebraska, North Dakota, South 

Dakota, and Wyoming / by John F. Busby ... [et al.].

p. cm. (U.S. Geological Survey professional paper ; 1273 F) 

Includes bibliographical references.

1. Hydrogeology Madison Aquifer. 2. Water chemistry. I. Busby, John F. II. Series: Geological Survey 

professional paper : 1273 F. 

GB1019.G46 1991

551.48'09786 dc20 90-28405

CIP

For sale by the Books and Open-File Reports Section, U.S. Geological Survey, 

Federal Center, Box 25425, Denver, CO 80225
CONTENTS

Abstract --------------------------- Fl

Introduction-------------------------- 1

Acknowledgments--------------------- l

Description of project area ---------------- 3

Geologic setting- ----------------------- 3

Paleostructure ---------------------- 3

Stratigraphy ----------------------- 3

Depositional environment ----------------- 8

Mineralogy ------------------------ 9

Major minerals -------------------- 9

Relation of mineralogy to water chemistry ------ 9

Hydrology--------------------------- 9

Regional ground-water flow- --------------- 9

Leakage-------------------------- H

Estimation of regional hydraulic conductivity ------- n

Error analysis of regional hydraulic conductivity ----- 15

Geochemistry ------------------------- 16

Sample collection --------------------- 16

Sulfur isotopes sulfide phase ------------- 16

Sulfur isotopes sulfate phase ------------- 16

Field procedures --------------------- 16

Results of previous work ----------------- 17

Modeling strategy for mass-transfer models- ------- 17

Definition of flow paths for mass-transfer models ----- 17

Definition of recharge chemistry ------------- 17

Equilibrium-speciation calculations -------------- 18

Trends in water-quality data ----------------- 31

Analysis of sulfur-isotope data ---------------- 36

Sulfur-isotope dilution- ------------------ 37

Sulfur isotopes as a Rayleigh-distillation problem ----- 42

Mass-balance reaction models ----------------- 47

Equations ------------------------- 47

Mass-balance reaction models Continued

Preliminary mass-transfer results ------------- F55

Indications of uncertainties in the preliminary modeling

results------------------------ 55

Sources of error- --------------------- 58

Criteria used in reaction modeling ------------ 64

Selection of modeling parameters for each flow path - - - 66

Flow path l---------------------- 66

Flow path 2---------------------- 67

Flow path 3---------------------- 69

Flow path 4---------------------- 69

Flow path 5---------------------- 70

Flow path 6---------------------- 71

Flow path 7---------------------- 71

Flow path 8---------------------- 72

Summary of final mass-transfer results for each flow

path------------------------- 72

Flow path l---------------------- 73

Flow path 2---------------------- 73

Flow path 3---------------------- 73

Flow path 4---------------------- 74

Flow path 5---------------------- 74

Flow path 6---------------------- 74

Flow path 7---------------------- 74

Flow path 8---------------------- 75

Overview of chemical reactions in the Madison aquifer - - - - 75

Regional patterns in tJ 34Sanhydrjte --------------- 79

Carbon-14 ages ------------------------ 79

Flow velocities calculated from carbon-14 ages -------- 86

Rates of reactions- ---------------------- 86

Summary and conclusions- ------------------ 86

Selected references ---------------------- 87

ILLUSTRATIONS

FIGURES 1-3. Maps showing: Page

1. Location of project area and major structural features ---------------------------- F2

2. Relation between paleostructure and porosity of the Madison Limestone ------------------- 4

3. Relation between paleostructure, lithology, and thickness of the Madison Limestone ------------- 5

4. Photograph of solution karst features developed in Mission Canyon Limestone at Mission Canyon,

near Hayes, Montana ------- ---------------------------------------- 9

5. Map showing extent of halite and anhydrite in the Madison Limestone and in younger and older rocks -------- 10

6. Diagram showing relation between geologic units and units used in simulation model ----------------- 11

7-9. Maps showing:

7. Recharge rates, flow paths, flow directions, and discharge rates used in analysis of geochemical data - - - - 12

8. Relation between geologic structure and concentration of dissolved solids- ------------------ 13

9. Ground-water flow velocity based on adjusted carbon-14 ages, flow lines from recharge area to well, and

potentiometric surface- ------------------------------------------ 14

10-13. Graphs showing saturation indices as a function of dissolved-sulfate concentration for all wells and springs sampled:

10. Aragonite --------------------------------------------------- 24

11. Calcite----------------------------------------------------- 24

12. Dolomite --------------------------------------------------- 25

13. Gypsum ---------------------------------------------------- 26

III
IV CONTENTS

FIGURES 14-20. Graphs showing: p*s*

14. Log of ion activity product of calcite as a function of temperature for all wells and springs sampled - - - - F27

15. Log of ratio of the activity of calcium to the activity of magnesium as a function of temperature for all

wells and springs sampled- --------------------------------------- 27

16. Comparison of the temperature dependence of log K dolomite as given by the Van't Hoff equation and 

derived from the assumption of aragonite-dolomite equilibrium (eq 8) and calcite-dolomite equilibrium 

(eq 9) as a function of temperature for all wells and springs sampled ------------------ 28

17. Dolomite saturation index based on the Van't Hoff equation as a function of temperature for all wells and

springs sampled- -------------------------------------------- 28

18. Celestite saturation index as a function of dissolved-sulfate concentration for all wells and springs

sampled ------------------------------------------------- 29

19. Strontianite saturation index as a function of dissolved-sulfate concentration for all wells and springs

sampled ------------------------------------------------- 30

20. Dissolved-silica concentration as a function of temperature for all wells and springs sampled- ------- 31

21. Trilinear diagram showing chemical composition of water along all flow paths, in mole percent of cations and

anions ------------------------------------------------------ 33

22-29. Trilinear diagrams showing chemical composition of water along individual flow paths, in mole percent of cations 

and anions:

22. Flow path !-------------_----------------------------------- 34

23. Flow path 2------------------------------------------------- 35

24. Flow path 3------------------------------------------------- 36

25. Flow path 4------------------------------------------------- 37

26. Flow path 5------------------------------------------------- 38

27. Flow path 6------------------------------------------------- 39

28. Flow path 7------------------------------------------------- 40

29. Flow path 8------------------------------------------------- 41

30-33. Graphs showing trends in selected constituents or variation in selected properties as a function of dissolved-sulfate 

concentration along flow paths 1 to 8:

30. Trend in dissolved-calcium concentration ---------------------------------- 42

31. Trend in dissolved-magnesium concentration- ------------------------------- 43

32. Variation in pH----------------------------------------------- 44

33. Variation in partial pressure of carbon dioxide (atmospheres)- ----------------------- 45

34. Graph showing variation in pH as a function of temperature along flow paths 1 to 8 ---------------- 46

35. Graphs showing variation in log partial pressure of carbon dioxide (atmospheres) as a function of temperature

along flow paths l to 8---------------------------------------------- 47

36-39. Graphs showing variation in selected constituents as a function of dissolved-sulfate concentration along flow paths 

1 to 8:

36. Total alkalinity ----------------------------------------------- 48

37. Dissolved sodium ---------------------------------------------- 49

38. Dissolved potassium- -------------------------------------------- 50

39. Dissolved chloride- --------------------------------------------- 51

40-42. Graphs showing:

40. Trends in <5 34SSo4 with dissolved-sulfate concentration along flow paths lto8--------------- 54

41. Variation in 34A = d 34Ss04 - d 34SH s as a function of temperature for water in the Madison aquifer compared with waters from the Edwards and Floridan aquifers ---------------------- 57

42. Variation in the measured d 13C as a function of dissolved-sulfate concentration in the Madison aquifer

system ------------------------------------------------- 62

43. Map showing regional variation in anhydrite mass transfer (per kilogram of water) for all wells and springs

sampled ----------------------------------------------------- 76

44. Graph showing trend in the quantity of dolomite dissolved (per kilogram of water) as a function of anhydrite

dissolution along flow paths lto8---------------------------------------- 77

45. Graph showing trend in the quantity of calcite precipitated (per kilogram of water) as a function of anhydrite

dissolution along flow paths lto8---------------------------------------- 78

46-49. Maps showing regional variation in:

46. Millimoles of dolomite dissolved and of calcite transferred (per kilogram of water) for all wells and springs

sampled ------------------------------------------------- 80

47. Degree of ion exchange (divalent cation exchanged for sodium per kilogram of water) for all wells and

springs sampled- -------------------------------------------- 82

48. Millimoles of halite dissolved (per kilogram of water) for all wells and springs sampled- ---------- 83

49. Estimated sulfur-isotopic composition of dissolving anhydrite for all wells and springs sampled ------ 84

50. Graphs showing variation in measured carbon-14 content as a function of concentration of dissolved sulfate along

flow paths 1 to 8------------------------------------------------- 85
CONTENTS V

TABLES

TABLE 1. Generalized correlation chart of Paleozoic rocks -------------------------------------- F6

2. Wells and springs sampled and source of water- -------------------------------------- 7

3. Comparison of hydraulic-conductivity values calculated from ground-water flow velocities based on adjusted carbon-14

	ages with those calculated from digital simulation ----------------------------------- 15

4. Comparison of field- and laboratory-determined sulfide data -------------------------------- 17

5. Recharge waters selected for use in geochemical models of the Madison Limestone -------------------- 18

6. Summary of revised thermodynamic data ------------------------------------------ 19

7. Chemical and temperature data used in modeling (cations) --------------------------------- 20

8. Chemical data used in modeling (anions) ------------------------------------------ 21

9. Results of thermodynamic calculations ------------------------------------------- 22

10. Total concentrations of the major elements and redox state used in mass-balance models ----------------- 32

11. Isotopic data -------------------------------------------------------- 52

12. Initial estimate of sulfur-isotopic composition of dissolving anhydrite along flow paths ------------------- 55

13. Calculation of anhydrite and pyrite mass transfers using the isotope-dilution method- ------------------- 56

14. Calculation of anhydrite and pyrite mass transfers using the Rayleigh-distillation method ----------------- 56

15. Evaluation of uncertainties in anhydrite and pyrite mass transfers ----------------------------- 57

16. Selected phases for mass-balance modeling ----------------------------------------- 58

17. Preliminary results from mass-transfer model --------------------------------------- 59

18. Summary of average carbon-isotope values for recharge waters used for modeling carbon isotopes at downgradient

	wells ---------------------------------------------------------- 60

19. Summary of preliminary model parameters and carbon-isotope results --------------------------- 61

20. Summary of modeling alternatives applied to the Mysse Flowing Well (well No. 20 in Montana) -------------- 63

21. Summary of final mass-transfer model results --------------------------------------- 65

22. Summary of final model parameters and carbon-isotope results- ------------------------------ 66

23. Summary of selected modeling parameters evaluated at Sumatra (well No. 17 in Montana) on flow path 2- -------- 69

24. Summary of average net rates of reactions ----------------------------------------- 87

GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE AND ASSOCIATED ROCKS IN PARTS OF 

MONTANA, NEBRASKA, NORTH DAKOTA, SOUTH DAKOTA, AND WYOMING

GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER IN 

PARTS OF MONTANA, SOUTH DAKOTA, AND WYOMING

By JOHN F. BUSBY, L. NIEL PLUMMER, ROGER W. LEE, and BRUCE B. HANSHAW

ABSTRACT

The geochemical models BALANCE and WATEQF have been used 

to deduce the major chemical reactions controlling the ground-water 

chemistry in the Madison aquifer in parts of Montana, South Dakota, 

and Wyoming. Although the Madison aquifer extends into North Dakota, 

wells were not available in North Dakota from which an uncontaminated 

sample of water could be collected. The chemical system was assumed 

to be closed to mass transfer of carbon dioxide. The dominant reactions 

along each of the eight selected flow paths were found to be dedolomitization caused by the irreversible dissolution of anhydrite and the bacterially 

mediated reduction of sulfate. Some cation (calcium/sodium) exchange 

and halite dissolution were determined to occur along flow paths in central Montana.

After the geochemical models had been validated using carbon-13 data 

in an independent mass-balance calculation, the carbon-14 ages were 

adjusted for the contribution of sources and sinks of carbon. These 

carbon-14 ages were adjusted for incongruent dissolution by applying 

Rayleigh-distillation equations to the isotope fractionation. The adjusted 

carbon-14 ages were then used to calculate regional hydraulic conductivities, and these values agreed well with those obtained by a computerbased digital model of the ground-water flow simulation. The adjusted 

ages also were used to calculate apparent rates of reaction for dissolution of anhydrite, dolomite, and halite, reduction of sulfate, and calcite 

precipitation.

INTRODUCTION

This report describes the results of an investigation to 

determine the geochemical reactions controlling water 

chemistry in the Madison aquifer underlying the northern 

Great Plains in parts of Montana, South Dakota, and 

Wyoming (fig. 1). Although the Madison aquifer extends 

into North Dakota, little use is made of water from the 

aquifer in that State because of the large concentration 

of dissolved solids. For this reason, that part of the Madison aquifer underlying North Dakota was not investigated 

chemically and is excluded from this report. This investigation is part of a larger interdisciplinary study to 

describe the geohydrology of the Madison and associated 

aquifers.

In this report, the geohydrology of the Madison aquifer 

system is first summarized from information contained

in reports by Sando (1976b), Peterson (1978, 1981), 

Thayer (1981), Brown and others (1982), MacCary and 

others (1983), and Downey (1984) in order to set the scene 

for understanding the geochemistry. Following the geohydrologic section, revisions of the WATEQF data base 

and speciation calculations based on WATEQF are 

discussed. Geochemical reactions in the Madison aquifer 

are then characterized using chemical and isotopic massbalance techniques.

The mass-balance modeling was used to solve the inverse problem in geochemical modeling: determination of 

the chemical reactions controlling the water chemistry 

from the chemical and isotopic data. The program 

BALANCE (Parkhurst and others, 1982) was used to 

calculate the change, in moles, of selected minerals and 

gases entering and leaving the ground water between a 

point in the recharge area and a point downgradient along 

a flow path. If the determined mass transfer for a particular evolutionary set of reactions was thermodynamically consistent with the results of WATEQF and yielded 

calculated isotopic compositions in agreement with the 

field-measured values, that set of minerals and gases was 

considered to be a good representation of the geochemical 

reactions controlling the actual water chemistry.

After a viable model was derived, the results of this 

model were used to calculate an adjusted 14C (carbon-14) 

age, following the method of Wigley and others (1978, 

1979). The model-adjusted 14C age was then used to calculate flow rates, regional hydraulic conductivities, and 

relative rates of chemical reactions.

ACKNOWLEDGMENTS

Many personnel of the U.S. Geological Survey have 

assisted in the preparation of this report. In particular, 

the authors have had helpful and thoughtful discussions 

with William Back, Tyler B. Coplen, Ruth Deike, Donald 

W. Fisher, Blair F. Jones, Leonard F. Konikow, and 

Donald C. Thorstenson (Reston, Va.). Discussions with

Fl
F2 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

Robert 0. Rye (Lakewood, Colo.) helped with the understanding of sulfur-isotope geochemistry.

Especially helpful were the comments of Joe S. Downey 

and Briant A. Kimball (both at Lakewood, Colo.) and 

William J. Steinkampf (Tacoma, Wash.). Field assistance of Joel D. Johnson (Charleston, W. Va.), Carole 

L. Loskot (Lakewood, Colo.), and Michael R. Plummer 

(Fairfax, Va.) is gratefully acknowledged. The computer support of George Fleming and David Stanley 

(Reston, Va.) was invaluable, as was the typing of

Ana M. MacKay (Reston, Va.) and the editing and 

typing of Alicia A. Mitchell (Austin, Tex.). The drafting 

and redrafting of figures was well and cheerfully 

performed by Donald L. Leslie and John P. Maguire 

(both of Austin, Tex.). Graig C. Joy (Cheyenne, Wyo.) 

and Richard Feltis (Billings, Mont.) aided the work by 

providing copious amounts of well-completion data. 

Rochelle Rittmaster (Austin, Tex.) and Gary Mackey 

(Austin, Tex.) deserve credit for assistance in proofreading.

48 Sv r^\

j NORThTV.DAKOTA if 

I \ <

SOUTH J(DAKOTA/

^ / 

/

Base from U.S. Geological Survey 

1:7,500,000 map, 1970

0 50 100 150 200 MILES I P H H i ' '

0 50 100 150 200 KILOMETERS

Modified from Peterson (1981)

EXPLANATION

MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

BOUNDARY OF CENTRAL MONTANA 

TROUGH

LINEAMENT Probable extension 

of Nye Bowler fault zone

NUMBER USED TO IDENTIFY 

STRUCTURAL FEATURES IN 

OTHER FIGURES

 *-: *--.; "r..-,. PROJECT AREA BOUNDARY 

FIGURE 1. Location of project area and major structural features.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F3

DESCRIPTION OF PROJECT AREA

The climate of the project area is characterized as 

middle-latitude steppes occupying a transitional zone 

between arid lands to the west and more humid environments to the east. The annual precipitation of about 12 

inches per year occurs mostly in the spring as rain when 

warm moisture-laden air from the gulf coast meets colder 

arctic air masses, and in the winter as snow resulting from 

arctic air masses moving across the region. Because much 

of the recharge to the aquifer system occurs in the spring 

as the result of snowmelt, a ground-water temperature 

of about 15 °C (degrees Celsius) has been used as one of 

the criteria for a recharge water.

The major physiographic features in the area include 

the Bighorn Mountains, the Laramie Mountains, the Little 

Rocky Mountains, the Big Snowy Mountains, the Hartville uplift, and the Black Hills (fig. 1). These highland 

features serve as the principal recharge areas to the 

Madison aquifer (Downey, 1984). Extensive undulating 

plains dissected by streams lie between the uplift areas 

and the regional discharge area for the aquifer system 

in eastern North and South Dakota.

GEOLOGIC SETTING

During the time in which the Mississippian Madison 

Limestone was deposited, the study area was a part of 

the cordilleran platform bordered on the west by the cordilleran miogeosyncline, which trended approximately 

north-south in Idaho and western Montana. Most of the 

sediment in the synclinal trough came from the Antler 

erogenic belt, which probably was an island arc to the west 

that underwent intermittent tectonism during the 

Paleozoic. The transcontinental arch located to the south 

of the project area was emergent but low-lying, and it 

sporadically contributed minor quantities of sediment that 

were spread thinly by marine currents across the platform. In general, during Mississippian time the cordilleran 

platform was a shallow-water, shelf-and-basin system that 

received predominantly carbonate and evaporite sediments in alternating cycles of transgression and regression of a sea lying to the west (Sandberg, 1961; Sando,

1968).

PALEOSTRUCTURE

Uplifted areas (fig. 1) were not regionally significant 

tectonic elements until late Paleozoic or early Mesozoic 

time (Agnew and Tychsen, 1965; Peterson, 1981) and had 

little influence on Mississippian sedimentation. Structural 

deformation during Mississippian time has been explained 

in terms of horizontal compression (Sales, 1968; Stone,

1969), vertical tectonics (Stearns and others, 1975), and 

wrench-fault tectonics (D.L. Brown, Frio Oil Company, 

Denver, Colo., written commun., 1978). Although the

mechanics of deformation have little direct effect on the 

existing hydrologic system, there is general agreement 

that the current structure has been controlled by the 

preexisting structure of the Precambrian basement as 

modified by the Laramide orogeny. Several authors, including Sandberg (1962), Rose (1976), and Slack (1981), 

have suggested that the paleostructure in various parts 

of the study area has had a pronounced effect on the 

thickness and depositional environments of the sediments. 

The major porosity zones in the Madison aquifer occur 

along paleostructures that are predominantly faults (fig. 

2). The relation between the paleostructure and lithology 

is shown in figure 3. Although no trend in the type of 

lithofacies with paleostructure is apparent, changes in the 

type of lithofacies most commonly occur along structural 

elements.

STRATIGRAPHY

The Madison aquifer in this report is composed of the 

Madison Limestone or the Madison Group where divided, 

or stratigraphic equivalents. The Madison Group, from 

oldest to youngest, consists of the Lodgepole Limestone, 

the Mission Canyon Limestone, and the Charles Formation. The generalized correlation chart for Paleozoic rocks 

(table 1) shows the stratigraphic relation between the formations in the study area, and the data in table 2 relate 

to the stratigraphic intervals from which water samples 

were collected. The Lodgepole Limestone is a cyclic carbonate sequence consisting largely of fossiliferous to 

micritic dolomite and limestone units that are argillaceous 

and thinly bedded in most of the study area (Smith, 1972). 

The thickness of the unit ranges from 0 to more than 900 

ft (feet) in the study area and averages about 300 ft.

The Mission Canyon Limestone grades from a coarsely 

crystalline limestone at its base to finer crystalline limestone with evaporite minerals near the top. The sequence 

contains one evaporite cycle and shares a second with the 

lower part of the overlying Charles Formation. Bedded 

evaporites are absent in most of Wyoming and South 

Dakota but occur in southeastern Montana. The evaporite 

deposits thicken in the central Montana trough, reaching 

their maximum areal extent in the Williston basin of 

North Dakota. The thickness of the Mission Canyon Limestone ranges between 0 and about 650 ft and averages 

about 300 ft.

The Charles Formation is the uppermost unit of the 

Madison Group. It is a marine evaporite sequence consisting of anhydrite and halite with interbedded dolomite, 

limestone, and argillaceous units. The thickness of the 

Charles Formation ranges from 0 to more than 295 ft with 

an average thickness of about 98 ft in the study area. PreJurassic erosion has removed most of the Charles Formation in the western and southern parts of the study 

area.
F4 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

Because the sulfate and carbonate rocks of the Madison 

Limestone are relatively soluble in water, the development of karst (solution) features is common. These complex and interconnected solution features developed in 

limestones under weathering conditions when the rocks 

were exposed at the land surface (figure 4). Sando (1974)

describes ancient karst features, including enlarged joints, 

sinkholes, caves, and solution breccias, that developed in 

the Madison Limestone in north-central Wyoming. He further indicates that most of the open spaces were filled by 

sand and residual products reworked by a transgressive 

sea during Late Mississippian time. Large and extensive

- L~ ,.

Base from U.S. Geological Survey 

1:7,500,000 map, 1970

0 50 100 150 200 MILES I___ i ___ i i_____|

0 50 100 150 200 KILOMETERS

-.. 

Modified from Peterson (1981) 

and Downey (1984)

EXPLANATION

 ..£ MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

 . APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

 BOUNDARY OF CENTRAL MONTANA 

TROUGH

. FAULT, PROBABLE FAULT, OR LINEAMENT

 J _ ANTICLINE TREND OR DOME-Dashed line 

shows probable extension of feature

. ._. STRUCTURAL TREND, ARCH, OR UPLIFT 

~| MONOCLINE

. 91 LINE OF EQUAL THICKNESS OF ROCK WITH 

POROSITY EQUAL TO OR GREATER THAN 

10 PERCENT Interval 61 meters; hachures 

indicate area of closed low

 PROJECT AREA BOUNDARY

FIGURE 2. Relation between paleostructure and porosity of Madison Limestone.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F5

cave systems in outcrop areas of the Madison Limestone in 

the Bighorn Mountains and in the Black Hills are further 

evidence of the importance of the dissolution process in the 

development of secondary permeability in limestone units. 

Overlying the Charles Formation in parts of Montana 

and South Dakota are rocks of Late Mississippian age

belonging to the Big Snowy Group. The Big Snowy Group 

consists of, in ascending order, the Kibbey, Heath, and 

Otter Formations. These formations are composed mainly 

of shale and sandstone but also contain minor limestone 

beds. The Big Snowy Group is considered a confining unit 

of the underlying Madison aquifer (Downey, 1984).

48°

 i 

Base from U.S. Geological Survey 

1:7,500,000 map, 1970

0 50 100 150 200 MILES I i H H i ' '

0 50 100 150 200 KILOMETERS

Lincoln

«

__ 

Modified from Peterson (1981)

EXPLANATION

I-':'.;."--/.I MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

HIM DOLOMITE FACIES OF MADISON LIMESTONE

Y/X PRIMARY LIMESTONE AND SHALEY LIMESTONE FACIES OF MADISON LIMESTONE

 APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

 BOUNDARY OF CENTRAL MONTANA 

TROUGH

 FAULT, PROBABLE FAULT, OR LINEAMENT

 | ANTICLINE TREND OR DOME-Dashed line 

shows probable extension of feature

 ~ STRUCTURAL TREND, ARCH, OR UPLIFT

 | MONOCLINE

:_^-^-^ PROJECT AREA BOUNDARY

FIGURE 3. Relation between paleostructure, lithology, and thickness of Madison Limestone.
F6 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE 

TABLE I. Generalized correlation chart of Paleozoic rocks

SYSTEM

JURASSC

fPIASSIC

PERMIAN

PENNSYLVANIAN

MISSISSIPPIAN

DEVONIAN

SILURIAI\

VICIAN

B

ee 

O

1

ee to

t

SERIES

MIDDLE JURASSIC

UPPER PERMIAN

LOWER PERMIAN

UPPER 

PENNSYLVANIAN

MIDDLE 

PENNSYLVANIAN

LOWER 

PENNSYLVANIAN

UPPER 

MISSISSIPPIAN

LOWER 

MISSISSIPPIAN

UPPER 

DEVONIAN

MIDDLE 

DEVONIAN

LOWER 

DEVONIAN

UPPER 

SILURIAN

MIDDLE 

SILURIAN

LOWER 

SILURIAN

UPPER 

ORDOVICIAN

MIDDLE 

ORDOVICIAN

LOWER 

ORDOVICIAN

UPPER 

CAMBRIAN

MIDDLE 

CAMBRIAN

LOWER 

CAMBRIAN

PRECAMBRIAN

POWDER RIVER BASIN

Piper Formation

Goose Egg Formation

Sandstone ... , Minnelusa

Amsden 

Formation

Madison Limestone

Three Forks Formation

Jefferson Formation

y/lnterlake Formation

^> Stony Mountain

Bighorn < Red River 

Dolomite <^ Formation

<~ 

^.^ > Winnipeg 

'^" "~»i-v Formation

Snowy Range <f 

Formation or <^ 

Gallatin and ^ Deadwood 

Gros Venire ^ Formation 

Formations ^> 

or equivalents*^, , . ..... ..,.,.

^$££

m

SOUTH CENTRAL 

MONTANA

Piper Formation ' -*^^~**^^ *^^-^^ 

Chugwater Formation

i 1

Tensleep 

Sandstone

Amsden 

Formation

Charles 

-^Formation

Mission 

Madison c 

GrouP Limestone

Lodgepole 

Limestone

Three Forks Formation

Jefferson Formation

\

Bighorn Dolomite

^ ^ B^BBfll^F

Snowy Range <C 

Formation or <^ 

Gallatin and < Deadwood 

Gros Venire ^ Formation 

Formations > 

or equivalents^^ ...... i ... ... .. u . . m

^ :

WESTERN 

SOUTH DAKOTA

^ 

Gypsum Spring Formation 

-^~^_

Spearfish Formation

Minnekahte Formation

Opeche Formation

Minnelusa Formation

Madison Limestone 

or 

Pahasapa Limestone

Englewood Formetion

f

Red River Formation 

or 

Whitewood Dolomite

Winnipeg Formation

Deadwood 

Formation

P» m

WILLISTON BASIN

Piper Formation 

'-^-^-^~_^^^-^^_

Spearf sh Formation

Minnekahta Formation

Opeche Formation

,...;-..; , < ,,._..;-'. :

Minnelusa Formation

x^Amsden Group 

/ upper part)

Tyler Formation

Heath Formation 

Big Snowy Otter Formation 

Group Kibbey Formation

 ' .... - ;

Charles 

^Formation

Madison Mission 

Group CarV°n 

Limeatone

Lodgepole

Bakken Formation

Three Forks Formation

Birdbear Formation

Duperow Formation

Souris River Formation

Dawson Bay Formation

Prairie Formation

Winnipegosis Formation

t ;4

Interlake Formation

Stony Mountain Formation

Red River Formation

Winnipeg Formation

Deadwood Formation

CENTRAL MONTANA 

TROUGH

Piper Formation ^~^^~^~^-^ s~^^

Chugwater Formation

Tensleep Sandstong^- -""

Amsden Group 

(upper part)

^-- rvieT

<^~' Formation o 

^>Amsden Group

Heath Formation 

Big Snowy otter Formation 

GrouP Kibbey Formation

Charles 

-^Formation

Madison Mission Canyon 

Group Limestona

Lodgepole 

 Limestone 

Bakken For^tion^ZZ^=-

Three Forks Formation

Jefferson Formation

If Red River Formation

/ y Winnipeg Formation

"""""""*"*''"'""":

Emerson Formation

Flathead Sandstone

 H

NORTH CENTRAL 

MONTANA

Piper Formation

Madison Mi^ion Canyon 

Group Llmestone

Lodgepole 

Limestone

Bakken Formation "

Three Forks Formation

Birdbear Formation

Duperow Formation

Souris River Formation

Red River Formation^ J

Winnipeg Formation \-

Emerson Formation

Flathead Sandstone

*: ?|
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F7

TABLE 2. Wells and springs sampled and source of water

[The legal description (for example 02N 27E 35AAB01) is the location of the well based on the rules set forth in the ordinance of May 20,1785, "The establishment of the rectangular survey system"]

Well or

spring

number

Name

1

Legal description 2

Total

depth

(feet)

Interval

sampled

(feet)

Water-yielding unit

MONTANA

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

Gore Hill

Great Falls High School

Bozeman Fish Hatchery

Bough Ranch

McLeod Warm Spring

Big Timber Fish Hatchery

Hanover Flowing Well

Vanek Warm Spring

Lewistown Big Spring

Bluewater Spring

Landusky Spring

Lodgepole Warm Spring

HTH No. 3

Keg Coulee

Texaco C115X

Sumatra

20N03E28CDAD

20N 04E 07BDAC

01S06E34BCDA

22N 06E 09DDAB

03S13E34ABAB

01N 14E 15DAAA

16N16E22DCC

17N 18E 09BCAA

14N19E05ABCC

06S24E09BCAA

25N24E32DABC

26N25E24BCDB

02N 27E 35AAB01

11N32E24ADCD

11N32E15ABC

UN 32E 24ADCD

755

426

200

1,299

spring

spring

751

spring

spring

spring

spring

spring

7,175

.-

6,768

6,854

679-699

400-426

115-200

1,266-1,299

--

..

--

--

--

--

..

--

4,373-4,422

6,457-6,516

6,079-6,768

6,152-6,854

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Mission Canyon Limestone

Mission Canyon Limestone

Madison Limestone

Madison Limestone

Mission Canyon Limestone

Mission Canyon Limestone

Mission Canyon Limestone

Mission Canyon and Lodgepole

Mission Canyon Limestone

Limestones

Mission Canyon Limestone, Charles

Formation, and Lodgepole Limestone

18

19

20

21

22

23

24

26

27

Sleeping Buffalo

Sarpy Mine

Mysse Flowing Well

Colstrip

Moore

Ranch Creek

Belle Creek

Gas City

Buckhorn Exeter

32N32E35CDBD

01N37E26BDDD

12N 39E 09AACA

02N41E34BADA

ION 43E 21CDCA

09S 53E 22ABAC

08S54E21ADAD

14N55E27CDDB

11N31E36AA

3,199

-

5,102

9,337

flowing well

7,260

7,190

7,598

6,519

3,130-3,199

-

5,039-5,102

7,795-8,340

..

7,169-7,260

6,991-7,190

7,434-7,598

6,174-6,335

Mission Canyon Limestone

-

Mission Canyon Limestone

Mission Canyon and Lodgepole

Mission Canyon Limestone

Madison Limestone

Madison Limestone

Mission Canyon Limestone

Madison Limestone

Limestones

SOUTH DAKOTA

1

2

3

4

5

6

7

8

9

10

11

12

13

14

16

17

18

19

20

21

22

23

24

25

26

27

Kosken

McNenney

Provo

Rhoads Fork

Spearfish

Fuhs

Delzer No. 1

Delzer No. 2

Cascade Spring

Evans Plunge

Kaiser

Jones Spring

Black Hills Cemetery

Streeter Ranch

Cleghorn Spring

Lien

Ellsworth AFB

Philip

Dupree

Hamilton

Hilltop Ranch

Eagle Butte

Midland

Murdo

Prince

Bean

42N26W34ABCD

02N01E21BBC

10S02E03

02N02E15

06N02E10D

05N03E28BACBB

12N03E28BACCB

12N 03E 32ACBAC

08S05E20

07S 05E 24BAAA

06S05E23

04N05E23

05N07E08

06S 06E 15AACC

01N07E08

02N07E18BCA

02N08E13BDC

01N 20E 01ACDD

13N21E31BDDA

08N23W26CDA

05N24E17CBBD

12N24E17CBDD

01N 25E 06CAA

01S 26E 36ACA

05N27E22CD

09N03E20CDD

220

3,845

flowing well

879

551

4,557

5,453

spring

spring

..

spring

--

938

spring

3,921

4,436

4,009

4,501

3,760

..

4,324

3,320

3,314

2^38

3,510

2,661-2,936

144-220

..

--

650-879

548-551

..

..

..

--

689-781

--

-.

928-938

--

..

--

3,783-4,009

--

--

3,379-4,101

--

3,166-3,320

--

2,700-2,746

3,199-3,510

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Pahasapa Limestone

Madison Limestone

_

_

_

Madison Limestone

Pahasapa Limestone

-

Pahasapa Limestone

Pahasapa Limestone

Madison Limestone

Pahasapa Limestone

Pahasapa Limestone

Mission Canyon Limestone

Lodgepole Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Pahasapa Limestone

'
F8 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE 

TABLE 2. Wells and springs sampled and source of water Continued

Well or

spring Name

number 1

Legal description 2

Total

depth

(feet)

Interval

sampled

(feet)

Water-yielding unit

WYOMING

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Mock Ranch

Denius No. 1

Denius No. 2

Denius No. 3

Hole in the Wall

Storey Fish Hatchery

Mobil

Conoco No. 44

Shidler

MKM

Conoco No. 175

Barber Ranch Spring

Devils Tower

HTH No. 1

Upton

Coronado No. 2

Osage

JBJ

Seeley

Voss

Newcastle

Self

Martens Madison

Mallo Camp

Ranch A

57N87W21DBC01

58N87W32BA

58N87W32BAB

58N87W32BB

41N84W20BAC

53N 84W 13BDC

49N 83W 27DDA

41N81W09CDA

40N79W31BCA

39N78W26CDC

33N 75W 20AAC

32N74W03CBD

53N 65W 18BBD

57N 65W 15DA

48N65W25CC

46N 64W 13CCA

46N 63W 15BD

44N63W26CAC

46N 62W 18BDC

45N61W28AB

45N61W33AB01

45N 60W 07CA

45N60W04BDA

47N 60W 04BDA

50N 60W 1BC

1,594

--

--

--

432

764

1,112

--

6,155

7,178

..

1,467

479

4^41

3,159

4,521

3,071

6,880

2,677

2,738

2,637

3,596

.-

spring

--

--

942-3,143

--

--

..

..

2,680-2,880

5,141-6,155

6,391-7,178

8,845-9,154

-.

450-479

2,431-2,595

2,900-3,159

4,085-4,521

2,684-3,071

6,476-6,880

2,349-2,677

2,467-2,738

2,618-2,637

3,169-3,596

640-718

--

--

Madison Limestone

Lodgepole Limestone

Lodgepole Limestone

Lodgepole Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

_

Madison Limestone

Upper Mission Canyon Limestone

Lower Charles Formation

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

Madison Limestone

^Missing well numbers indicate that the well initially bearing that number was deleted after the report was in process. Wells were deleted when error 

found in the analysis, when it was learned that the well drew water from more than one source, or when the well was not completed in the formation thought.

2The legal description (for example, 02N 27E 35AAB01) is the location of the well based on the rules set forth in the ordinance of May 20, 1785, "The establis 

of the rectangular survey system."

DEPOSITIONAL ENVIRONMENT

According to Sando (1976b), the deposition of the 

Mississippian System occurred in a set of three cycles of 

major transgression and regression. Each of the three 

cycles is divided into phases of minor transgression and 

regression. The Madison Limestone or Group was 

deposited during cycle 1 as described by Sando (1976b). 

Cycle 1 deposition was followed in cycle 2 by a period of 

epeirogenic uplift and erosion that marked the end of 

Madison deposition.

During much of Mississippian time most of the 

study area was covered by a alternately transgressiveregressive warm shallow sea (Sando, 1976b), termed the 

"knee-deep" environment by Hanshaw (B.B. Hanshaw, 

U.S. Geological Survey, written commun., 1985). This 

environment was conducive to prolific biologic productivity, sediment accumulation, and the development and 

preservation of shoals and reefs. Many of these small reefs

and associated oolite- and crinoid-bank shoals formed 

lagoons that had restricted inflow from the sea.

Extensive dolomitization occurred on the shelf environment, and the lagoonal system served as centers for 

anhydrite precipitation. As the seas regressed, precipitation of halite occurred (Kinsman, 1969; Hanshaw and 

others, 1971).

Evidence of more restricted areas or areas with a more 

rapid rate of evaporation, probably during regressive 

phases of the cycles, is the accumulations of bedded 

evaporites found in the Williston basin and the Central 

Montana trough (fig. 5). These shallow-water depositions! 

environments grade both laterally and vertically from 

shallow-marine carbonate and evaporite facies to deepwater facies. The Big Snowy Group was deposited during 

subsidence of the Central Montana trough and Williston 

basin. Regression of the sea from these restricted basins 

resulted in the deposition of extensive anhydrite and some 

halite.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F9

FIGURE 4. Solution karst features developed in Mission Canyon Limestone at Mission Canyon, near Hayes, Montana.

MINERALOGY

MAJOR MINERALS

The major minerals in the Madison Limestone are 

calcite (CaCOs), dolomite [CaMgtfCOs^], and anhydrite 

(CaS04) (Thayer, 1981; R.G. Deike, U.S. Geological 

Survey, written commun., 1984). Minor quantities of 

goethite (FeOOH), hematite ^6263), and quartz (SiC>2) 

also are common.

R.G. Deike (written commun., 1984) also reported the 

presence of talc [Mg3Si40io(OH)2], fluorite (CaF2), and 

chalcedony (SiC>2) in the Osagean interval penetrated by 

well HTH No. 1 and amphibole in the Meramecian interval penetrated by the same well. Thayer (1981) reported 

one example of dedolomite (calcite pseudomorphic after 

dolomite) in well HTH No. 1. The most common clay 

minerals in the system are kaolinite [Al2 81265 (OH)4] 

and illite [K0.6Mg0.23Al2.3Si3.50io(OH)2 ], for example, 

some of which is in a mixed layer of illite-smectite. Smectites, such as a calcium montmorillonite [Cao.i7Al2.33 

Sis.670io(OH)2], for example, also occur but in quantities 

subordinate to those of other clay minerals.

RELATION OF MINERALOGY TO WATER CHEMISTRY

The most common mechanism operating in the Madison 

aquifer is the incongruent dissolution of calcite caused by 

the dissolution of anhydrite and dolomite. The paucity of 

dedolomite reported by Thayer (1981) probably is due to 

a kinetic control preventing its formation. Although 

secondary calcite commonly is observed, it is not pseudomorphous after dolomite (dedolomite). Because the 

stoichiometry of the incongruent dissolution of calcite 

caused by the dissolution of anhydrite and dolomite is consistent with the formation of dedolomite, the term 

"dedolomitization" has been applied to this reaction, even 

in the absence of mineral phase "dedolomite," for the sake 

of brevity.

HYDROLOGY 

REGIONAL GROUND-WATER FLOW

Downey (1984) developed a digital simulation model of 

the hydrologic system with two major aquifers and two 

confining units as shown in figure 6. The results of his 

simulation are shown in figure 7.
F10 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

98°

46°

46°

, SOUTH \ DAKOTA 

\ \ /

- - _ _ '"~ ^O --^^

44°

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

Modified from Peterson (1981)

50 100 150 200 MILES

0 50 100 150 200 KILOMETERS

EXPLANATION

F^jji/n MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

 APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

 BOUNDARY OF CENTRAL MONTANA 

TROUGH

 FAULT, PROBABLE FAULT, OR LINEAMENT

 | ANTICLINE TREND OR DOME-Dashed line 

shows probable extension of feature

_. .__ STRUCTURAL TREND, ARCH, OR UPLIFT

 i MONOCLINE

LIMIT OF DUNHAM SALT OF INFORMAL 

SUBSURFACE USAGE, JURASSIC

LIMIT OF PINE SALT OF INFORMAL 

SUBSURFACE USAGE, TRIASSIC

LIMIT OF OPACHE SALT OF INFORMAL 

SUBSURFACE USAGE, PERMIAN

____. LIMIT OF CHARLES SALT OF INFORMAL 

SUBSURFACE USAGE, MISSISSIPPIAN

 LIMIT OF ANHYDRITE IN THE MIDDLE 

PART OF THE MADISON LIMESTONE, 

MISSISSIPPIAN

 LIMIT OF PRAIRIE SALT OF INFORMAL 

SUBSURFACE USAGE, DEVONIAN

.,^_«^M PROJECT AREA BOUNDARY

FIGURE 5. Extent of halite and anhydrite in Madison Limestone and in younger and older rocks.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER Fll

Downey (1984) found that the Madison aquifer functioned as a confined aquifer system with recharge in the 

western highlands and a general direction of flow toward 

the northeast. Major discharge areas for the aquifer 

system are along the subcrop of the Madison Limestone 

in east-central South Dakota and eastern North Dakota, 

and in the saline lakes in northern North Dakota and 

saline springs and seeps in the Canadian province of 

Alberta (Grossman, 1968; Downey, 1984).

Recharge entering the ground-water flow system in the 

Black Hills follows two general paths. One component 

flows north along the front of the Black Hills and then 

to the east in response to structural and hydrologic control. The other component flows south along the front of 

the Black Hills and then to the east, also in response to 

structural and hydrologic control. Very little water 

recharged from the Black Hills enters the Madison aquifer 

system in the Powder River basin of Wyoming.

The same general situation occurs for recharge waters 

entering the Madison aquifer from the Bighorn Mountains. Waters entering the northern end of the Bighorn 

Mountains flow to the east, whereas waters entering 

along the flanks of the mountains are diverted to the north 

by structural features along the eastern front of the mountain range.

Lower Cretaceous

Pennsylvanian, Permian, 

Triassic, Jurassic, and 

Charles Formation of 

Mississippian age

Lodgepole and Mission 

Canyon Limestones of 

the Madison Group

Silurian-Devonian formations 

and Bakken Formation 

of Mississippian age

Red River, Stonewall, 

Stony Mountain, Winnipeg, and 

upper Deadwood Formations of 

Cambrian to Ordovician age

Precambrian

Upper boundary

Confining layer

Madison aquifer

Confining layer

Cambrian-Ordovician 

aquifer

Lower boundary

FIGURE 6. Relation between geologic units and units used in simulation model.

As shown in figure 7, little recharge to the ground-water 

flow system occurs from the Laramie Mountains, and the 

recharge that does enter the system in this region moves 

toward the center of the Powder River basin.

A map (fig. 8) relating concentration of dissolved solids 

to geologic structure indicates a significant control of the 

dissolved-solids distribution by the geologic structure. The 

area in east-central Montana bounded by the Cat Creek 

fault and Porcupine dome shows a large dissolved-solids 

concentration, indicating a possible retardation of flow 

by geologic structure emanating from the Big Snowy 

Mountains. The high dissolved-solids concentration could 

also be the result of the availability of soluble material 

in the aquifer material. In eastern Wyoming the set of 

anticlines and minor faults along the western front of the 

Black Hills diverts recharge north and south around the 

Black Hills and away from the Powder River basin of 

Wyoming. In southern Wyoming the Casper Mountain 

fault and Casper arch appear to retard flow north from 

the Laramie Range.

LEAKAGE

The possibility of significant crossformational leakage 

into the Madison aquifer from underlying CambrianOrdovician strata was considered during formulation of 

the geochemical models and rejected. This conclusion was 

based on the hydrologic modeling results of Downey 

(1984,1986), which show that vertical hydraulic conductivity between the Cambrian-Ordovician and Madison 

aquifers is less than 10 ~ 6 ft/d throughout the study area. 

This compares to horizontal hydraulic conductivities 

within the Madison aquifer ranging from approximately 

1 to 10 ft/d. This comparison indicates that on a regional 

scale crossformational leakage between the Madison and 

Cambrian-Ordovician aquifers can be disregarded.

ESTIMATION OF REGIONAL HYDRAULIC CONDUCTIVITY

The regional hydraulic conductivity (Ki4C ), in feet per 

second, was estimated from the 14C data after adjustment of the data for reaction as explained in the section 

"Carbon-14 Ages."

The flow paths used in this analysis (fig. 9) were selected 

to be orthogonal to the potentiometric surface, with the 

exception of those in southern Wyoming, where the flow 

lines were selected to follow the dissolved-solids gradient.

The calculation of regional hydraulic conductivity 

(K"i4C ) based on the work of Hanshaw and others (1964) 

uses the Darcy equation in the form

vB (1)
F12 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

where

K is the hydraulic conductivity, in feet per second;

v is the average linear flow velocity, in feet per 

second;

6 is the porosity, dimensionless; 

Lh is the change in hydraulic head, in feet; and 

LI is the length of the flow path, in feet.

The value of .Khyd* (the regional hydraulic conductivity) 

was calculated by dividing the transmissivity, obtained 

by either aquifer tests or model simulation, by the aquifer 

thickness. Both the thicknesses and transmissivities were 

distance-weighted averages along the length of the flow 

path. 

The values for the hydraulic conductivities

 """" fci mull...

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

0 50 100 150 200 MILES I i H H i ' '

0 50 100 150 200 KILOMETERS

Modified from Downey, J. S., 

U.S. Geological Survey, 

written communication (1987)

EXPLANATION

OR OLDER ROCKSExposed at land surface

lrl5.0dJAREA OF GROUND-WATER DISCHARGENumber is discharge, in cubic feet per second

|+13.98| AREA OF GROUND-WATER RECHARGENumber is recharge, in cubic feet per second

-4J OUTLINE OF FLOW PATH-Arrow shows 

principal direction of flow. Numbers in 

box are those of flow path

22

WELL REPRESENTS CONTROL POINTS

USED IN GEOCHEMICAL MODEL-Number 

is identification number of well

; PROJECT AREA BOUNDARY

NOTE: Wells not within flow path were used 

for recharge

FIGURE 7. Recharge rates, flow paths, flow directions, and discharge rates used in analysis of geochemical data.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F13

derived from the calculated flow velocity on the basis of 

14C age and an assumed porosity of 3 percent and for the 

hydraulic conductivities (Khyd) derived from analysis of 

the transmissivity maps (fig. 29) developed by Downey

using a computation-based digital model are shown in 

table 3 (Downey, 1984). The differences between the 

values derived from the two different methods are within 

the range of values typical of carbonate aquifer systems

48

46

IDAHO ,

i WYOMING

Cheyenne

_^jLJ

COLORADO i i

 - L___ 

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

0 50

i i

100 150 200 MILES

Lincoln

i

__ 

Modified from Downey (1984)

I i H H i ' '

0 50 100 150 200 KILOMETERS

EXPLANATION

MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

BOUNDARY OF CENTRAL MONTANA 

TROUGH

FAULT

ANTICLINE TREND OR DOME-Dashed line 

shows probable extension of feature

SYNCLINE

LINEAMENT

1000- LINE OF EQUAL DISSOLVED-SOLIDS

CONCENTRATION-Interval, in milligrams 

per liter, is variable

i360. CONCENTRATION OF DISSOLVED SOLIDSNumber is concentration in milligrams per 

liter. Well number shown in figure 7

; PROJECT AREA BOUNDARY

NOTE: Numbers beside structural features refer to 

feature names shown on figure 1

FIGURE 8. Relation between geologic structure and concentration of dissolved solids.
F14 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

and show a remarkable consistency between the two sets 

of data, with the worst case being within a factor of 8. 

The KUc values range from 1.12 x 10 ~ 6 to 29.87 x 10 ~ 6 

ft/s (feet per second), a factor of about 30 and a reasonable 

range for carbonate aquifers. The larger values at Dupree 

(well No. 20), Eagle Butte (well No. 23), Kosken (well No. 

1), and Prince (well No. 26) in South Dakota along flow

48°

path 8 are measured from near a zone of greater permeability postulated by Downey (1986).

The smallest values of regional hydraulic conductivity 

(Ki4C) calculated on the basis of 14C age are found along 

flow path 5 in Wyoming, an area expected to have received detrital material from the transcontinental arch 

during deposition (Sando, 1976b).

 ~ - 

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

50 .00 150 200 MILES i i i i_____|

0 50 100 150 200 KILOMETERS

 -i 

Modified from Downey (1986)

EXPLANATION

|::V-:i:.::.-| MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

 APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

 BOUNDARY OF CENTRAL MONTANA 

TROUGH

 2000- POTENTIOMETRIC CONTOURS-Shows

altitude of potentiometric surface. Dashed 

where inferred. Contour interval 200 feet. 

Datum is sea level

7 2, WELL USED FOR CONTROL-Number is

flow velocity in feet per year, calculated 

from adjusted carbon-14 data. Well number 

shown in figure 7

 FLOW LINE FROM RECHARGE AREA 

TO WELL

i_-_r. PROJECT AREA BOUNDARY

FIGURE 9. Ground-water flow velocity based on adjusted carbon-14 ages, flow lines from recharge area to well, and potentiometric surface.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F15

TABLE B. Comparison of hydraulic-conductivity values calculated from, groundswaterflow velocities based on adjusted carbon-14 

ages with those calculated from digital simulation (Downey, 1984)

[ft, feet; mi, miles; ft/yr, feet per year; ft/s, feet per second; porosity of 3 percent assumed]

Well

Keg Coulee

Sarpy Mine

Mysse Flowing

Well

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 2

Conoco No. 175

MKM

Shidler

Conoco No. 44

Upton

Evans Plunge

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Well 

No.

15

19

20

14

23

24

8

11

10

9

8

15

10

1

19

24

25

22

26

21

23

20

State

Mont.

Mont.

Mont.

Wyo.

Mont.

Mont.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow 

path

2

3

3

4

4

4

4

5

5

5

5

6

7

8

8

8

8

8

8

8

8

8

Change 

in 

hydraulic 

head 

(ft)

3,002

3,002

3,097

1,260

1,260

1,260

1,499

1,631

1,001

1,001

1,001

1,125

1,499

2,539

2,241

2,342

2,500

2,500

2,552

2,434

2,500

2,402

Horizontal 

flow 

length 

(mi)

143

174

99

40

68

68

28

43

62

50

31

25

31

174

106

124

130

106

124

106

149

124

Hydraulic 

gradient 

(xlO-3)

3.98

3.27

5.92

5.97

3.51

3.51

10.14

7.02

3.06

3.79

6.12

8.52

9.16

2.76

4.00

3.58

3.64

4.47

3.90

4.35

3.18

3.67

Carbon-14 

age 

(x!03yr)

23.0

11.2

22.5

7.3

9.2

9.1

5.6

7.6

21.0

11.6

22.6

8.3

2.4

10.6

16.8

16.3

14.5

10.0

7.8

13.3

11.2

10.8

Hydraulic conductivity 

(in ft/s x 10-6) 

based on:

Velocity 

(ft/yr)

32.8

82.0

23.2

28.9

39.0

39.4

26.4

29.9

15.6

22.8

7.2

15.9

68.2

86.7

33.3

40.2

47.3

56.0

83.9

42.1

70.2

60.6

Carbon-14 

age

7.85

23.86

3.73

4.61

10.58

10.69

2.48

4.05

4.85

5.71

1.12

1.78

7.08

29.87

7.92

10.67

12.38

11.91

20.47

9.20

21.0

15.71

Digital 

simulation

6.07

3.29

3.48

7.81

7.09

7.45

6.96

7.14

4.26

4.36

4.69

6.20

7.15

5.45

3.54

6.23

4.56

5.18

4.26

3.54

5.74

4.00

ERROR ANALYSIS OF REGIONAL HYDRAULIC CONDUCTIVITY

The major sources of error in hydraulic-conductivity 

values derived from 14C data, as suggested by L.F. 

Konikow (U.S. Geological Survey, written commun., 

1984), were in the flow-path lengths, hydraulic-head 

changes, and porosities used.

To these suspected sources of error should also be added 

the error in the measurement of the unadjusted 14C age 

of the water [ ± 500 yr (years)]; errors in the geochemical 

model, including the choice of plausible phases; inadequacies in the thermodynamic data; and assumptions 

regarding the temperature dependence of the equilibrium 

constant. Using the well at Keg Coulee (well No. 15 in 

Montana) as an example and assuming the validity of all 

chemical data, the effect of other errors can be calculated 

if the range of the errors is known.

For the purposes of this exercise, the error in hydraulichead change (A/0 is assumed to be + 20 percent, error in 

length of flow path (AZ) to be ± 20 percent, error in porosity (0) to be ±70 percent, and error in the unadjusted 

14C age to be ± 500 yr.

Let A/i = 3,002 ft - 600 ft = 2,402 ft; 

AJ = 143 mi + 29 mi = 172 mi; 

0 = 0.03 + 0.021 = 0.051; and 

14C = 23,000 yr - 500 yr = 22,500 yr before 

present.

The hydraulic gradient becomes = 2.64xlO~3 , the A£

velocity becomes 40.4 ft/yr, and the calculated hydraulic 

conductivity is 14.5 x 10 ~ 6 ft/s, a net relative error of 

about 86 percent compared with the original estimate of 

7.85x10 ~ 6 ft/s. 

Using the set of minimum-value assumptions,

A/i = 3,602 ft, 

AU114mi, 

0 = 0.009, and 

14C = 23,500 yr before present,

the hydraulic gradient is 5.98 x 10~ 3, the velocity is 25.61 

ft/yr, and the calculated hydraulic conductivity is 4.07 

x 10 ~ 6 ft/s, a net relative error of 48.2 percent compared
F16 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

with the original estimate of 7.85 x 10 ~ 6 ft/s. Although 

the error in the calculation of the hydraulic conductivities 

depends on the error ranges chosen for the porosity, 

length of flow path, hydraulic gradient, and .age, this exercise does indicate the relation between the errors in the 

parameters.

The effect of differences in the choice of mass-transfer 

models may be investigated by comparing the results of 

Back and others (1983), which are based on a model using 

only the incongruent dissolution of calcite for the evolution of water chemistry from the recharge area to the well 

at Midland (well No. 24 in South Dakota), with the model 

(along the same flow path) incorporating both the incongruent dissolution of calcite and sulfate reduction. Using 

the values of AZ and A/?, from this study and 0 = 0.03, the 

calculated KI^ from the data of Back and others (1983) 

for the well at Midland is 5.64 x 10~ 6 ft/s, a relative error 

of 89 percent compared with the value of 10.67 x 10 ~ 6 

ft/s calculated using the reaction model described in this 

report.

GEOCHEMISTRY

SAMPLE COLLECTION

The selection criteria used in choosing a well or spring 

for sampling were (1) a record of the well or spring had 

to be available, (2) the sample had to represent a point 

source in some member of the Madison Group, and (3) the 

well or spring had to be free of contamination.

Samples were collected for measurement of major and 

minor elements using the methods described in Brown and 

others (1970). The analytical results for all data collected 

by the Madison aquifer study are reported in Busby and 

others (1983). Only the subset of those data judged useful 

in the geochemical model is presented in this report. Field 

determinations were made for pH, alkalinity, temperature, and specific conductance following the methods of 

Wood (1976).

Samples were collected for the analysis of the radioactive isotopes 14C and tritium and the stable isotope ratios 

d 13C, 6D, and d 180 using the methods described in Busby 

and others (1983). The collection of samples for the 

analysis of d 34S was changed from that used by Busby 

and others (1983) and is described below.

SULFUR ISOTOPES SULFIDE PHASE

Although the odor of hydrogen sulfide (£[28) was 

detected in many of the wells sampled by Busby and 

others (1983), d 34S was measurable in the sulfide phases 

in only one sample. Possible reasons for this are (1) coprecipitation of carbonate could have resulted in an inordinately large sample bulk on the filter, leading the field

personnel to the false conclusion that an adequate quantity of sulfide had been collected; (2) analytical error; or 

(3) oxidation of sulfide by dissolved oxygen introduced 

during transfer of the sample from the filter plate to the 

sample bottle.

It was judged that the likelihood of such a large number 

of analytical errors was remote, and a careful review of 

the sampling procedure revealed no obvious error. The 

possibility of a judgment error in estimating the quantity 

of the sulfide phase collected or of oxidation by occluded 

water was not amenable to testing.

In an attempt to eliminate suspected causes, L. Niel 

Plummer and Eurybiades Busenberg (U.S. Geological 

Survey, written commun., 1982) performed a set of experiments on sulfide stabilities. The investigation resulted 

in the following procedure:

1. The field-determined concentration of sulfide 

was used to calculate the quantity of cadmium acetate 

(Cd (CH3C02)2) required to precipitate the sulfide present, and then a large excess of cadmium acetate was 

added to a 10-L (liter) carboy.

2. The carboy was filled with water, and concentrated 

hydrochloric acid (HC1) was added to decrease the pH to 

about 4.0.

3. The carboy was capped, shaken well, and allowed to 

stand until precipitation appeared complete.

4. After precipitation, the carboy was again shaken and 

its contents transferred into a stainless-steel kettle and 

filtered under nitrogen pressure through a 0.1-jum (micrometer) millipore filter. After filtration, the filter was 

removed and placed in an actinic glass bottle, which was 

then sparged with nitrogen.

5. The bottle was wrapped with foil, retightened, and 

sealed with paraffin prior to shipment to the laboratory.

Although this procedure was a marked improvement, 

problems remain. Successful determinations of d 34S of 

dissolved sulfide were made for only 70 percent of the 

samples, indicating the need for further improvement of 

the sampling procedure.

SULFUR ISOTOPES SULFATE PHASE

The sampling procedure for d 34S of dissolved sulfate 

was modified for convenience. The new procedure involved precipitation of the sulfate from an acidified (HC1) 

200-mL sample of water with barium chloride (BaCl2) and 

filtration through a 0.45-/um filter using a syringe pump.

FIELD PROCEDURES

In addition to the field procedures reported by Busby 

and others (1983), dissolved sulfide was also determined 

in the field using the methylene-blue method with a Each 

model DR-2 spectrophotometer. The procedure followed 

was that described in the Each Chemical Company
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER FIT

manual for analysis of water and waste water (Hach 

Chemical Co., 1982). The results determined in the field 

and from duplicate samples submitted to the U.S. 

Geological Survey laboratory in Arvada, Colorado, are 

shown in table 4. The sulfide concentrations determined 

by the methylene-blue method are consistently less than 

those determined using the iodometric method used by 

the U.S. Geological Survey laboratory.

RESULTS OF PREVIOUS WORK

The major conclusions in the report by Busby and others 

(1983) were the following:

1. Deuterium and oxygen-isotope ratios are consistent 

with present-day meteoric waters, and no unconnected 

14C age greater than 30,000 yr was measured.

2. The major geochemical mechanisms controlling the 

water quality are (a) dissolution of anhydrite, (6) incongruent dissolution of dolomite, and (c) bacterially mediated sulfate reduction.

3. The precipitation of calcite in the well-bore caused 

by the large loss of C(>2 in wells with temperatures 

greater than 60 °C may result in diminished well capacity. This statement is not supported by the present study, 

however.

4. There is significant leakage from the overlying Pennsylvanian and Permian sedimentary rocks in the vicinity 

of Midwest, Wyoming. This leakage was thought to be 

caused by withdrawal of water from the Madison aquifer 

for secondary recovery of petroleum from sedimentary 

rocks of Cretaceous age. This statement is not supported 

by the present study.

MODELING STRATEGY FOR MASS-TRANSFER MODELS

Because the chemical data generally met the criteria 

for the saturation-sufficient case of Plummer and others 

(1983), mass-balance models of hypothesized reactions 

were used to predict the quantity of material transferred 

to or from the solid phase. Because sulfate reduction was 

evident, the mass balance on d 34S for the sulfide and 

sulfate phases was included in the system of reactions. 

The direction of the predicted mass transfers was compared for consistency with the results from WATEQF 

(Plummer and others, 1976); if, for example, the masstransfer model predicted the dissolution of calcite, but 

results from WATEQF indicated the system to be oversaturated with respect to calcite, the mass-balance model 

was not consistent with the results of WATEQF. Therefore, the concept of dissolution of calcite was eliminated.

The model results were then partially validated using 

the possible mass-balance models to predict the fractionation of 6 13C. If a model shows a substantial lack of agreement between the observed and predicted values of 6 13C, 

other reactants and (or) products need to be considered.

TABLE 4. Comparison of field- and laboratory-determined sulfide data

[Field Data determined in the field using a Hach model DR-2 spectrophotometer as a detector for methylene-blue determination of sulfide. Laboratory Data determined by the U.S. 

Geological Survey laboratory in Arvada, Colorado, using an iodometric titration; mg/L, 

milligrams per liter]

Concentration of dissolved 

sulfide determined in:

Well Well 

number

Flow 

path

Field 

(mg/L)

Laboratory 

(mg/L)

Sleeping Buffalo 18 1 0.04 0.3

Delzer No. 1 74 .04 .2

Delzer No. 2 84 .17 1.8

Evans Plunge 10 7 .03 .3

Murdo 25 8 .14 .7

Streeter Ranch 14 -- .02 .3

Dupree 20 8 1.0 19

Hilltop Ranch 22 8 .07 .9

Midalnd 24 8 .02 .3

Kosken 1 8 .13 .7

Philip 19 8 .01 .3

Bean 27 8 .04 .2

Fuhs 6 8 .05 .5

Black Hills Cemetery 13 8 .02 .2

Kaiser 11 8 .03 .1

Prince 26 8 .12 .6

Thus, through this elimination process, reaction models 

(that is, particular sets of reactants and products) may 

be identified that are consistent with all the chemical and 

isotopic data, and those that are not are shown to be 

invalid.

DEFINITION OF FLOW PATHS FOR MASS-TRANSFER MODELS

The ground-water flow paths used in this report generally were along flow lines selected to agree with the 

direction of flow as determined from the potentiometric 

surface of the Madision aquifer (fig. 7). Flow path 5 in 

southern Wyoming is an exception to this generality. The 

potentiometric gradient is orthogonal to the gradient in 

dissolved solids. Because pressure heads in this area were 

determined from drill-stem test data with rather large 

error ranges, the direction of increasing concentration of 

dissolved solids was chosen to indicate the direction of 

flow.

DEFINITION OF RECHARGE CHEMISTRY

The chemical composition of rainfall and snowmelt were 

not used to define the recharge water; instead the chemistry of waters near the beginning of the flow path, waters 

that had passed through the soil zone but had not interacted significantly with the rock matrix, were used to 

define the chemistry of the recharge water.

With one exception, all waters chosen to represent 

recharge met at least four of the following criteria: (1)
F18 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 5. Recharge waters selected for use in geochemical models of the Madison Limestone

[°C, degrees Celsius; mg/L, milligrams per liter; TU, tritium units]

Spring 

or well

Lewistown Big Spring

Bozeman Fish Hatchery

Big Timber Fish Hatchery

Mock Ranch

Denius No. 1

Mock Ranch

Storey Fish Hatchery

Mobil

Denius No. 1

Hole in the Wall

Barber Ranch Spring

Mallo Camp

Rhoads Fork

Rhoads Fork

Kaiser

Jones Spring

Cleghorn Spring

Spring 

or well 

number

10

4

7

1

2

1

6

7

2

5

12

24

4

4

11

12

16

State

Mont.

Mont.

Mont.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow 

path

1

2

2

3

3

4

4

4

4

5

5

6

6

7

8

8

8

Temperature (°C)

10.6

8.6

10.7

10.8

9.0

10.8

8.0

7.1

9.0

8.2

15.7

7.7

5.7

5.7

15.0

12.9

11.6

Sulfate 

(mg/L)

140.0

23.0

27.0

8.1

22

8.1

3.2

7.7

22.0

6.2

34

1.8

3.1

3.1

14

5.8

23

Carbon-14 

(percent 

modern)

35.8

84.1

103.9

57.7

8.4

57.7

88.8

62.2

8.4

87.4

83.7

92.9

92.9

92.9

81.1

100.0

91.6

Tritium 

(TU)

0.8

115

124

35.4

19.8

35.4

107

56.7

19.8

.5

53.0

-

62.2

62.2

27.7

276

182

The well or spring had to be located in a recharge area; 

(2) temperature, measured at the wellhead or spring 

outlet, had to be less than or equal to the mean annual 

air temperature of 15 °C; (3) sulfate concentration had 

to be less than 100 mg/L; (4) the unadjusted 14C value had 

to be at least 50 percent modern; and (5) the tritium value 

had to be greater than 10 TU (tritium units).

Criteria for the recharge water selected for the modeling effort are shown in table 5. The Lewistown Big Spring 

(well No. 10 in Montana) failed to meet all but the 

temperature criterion but was retained because it was the 

only site along flow path 1 that at least met one of the 

criteria.

EQUILIBRIUM-SPECIATION CALCULATIONS

In order to investigate thermodynamic controls on the 

water composition, equilibrium-speciation calculations 

were made using WATEQF (Plummer and others, 1976). 

These calculations provide a saturation index (SI) of 

minerals that may be reacting in the system. The SI of 

a particular mineral is defined as

SI=\ogIAP/KT, (2)

where

IAP is the ion activity product of the mineral; and 

KT is the thermodynamic equilibrium constant adjusted to the temperature of the given analysis.

The SI is greater than zero for oversaturation and less 

than zero for undersaturation. Before the speciation and 

SI calculations could be made, several revisions and updates were made to the thermodynamic data base of 

WATEQF, as outlined in table 6. These changes were 

necessary because the previous thermodynamic data for 

the carbonate system were reliable to 50 °C, whereas the 

waters in many wells completed in the Madison aquifer 

have temperatures of 80 °C and higher.

The chemical data presented in tables 7 and 8 are from 

Busby and others (1983). The results from WATEQF 

calculations are summarized in table 9. These include Si's 

of calcite, aragonite (CaCOa), dolomite, gypsum, anhydrite, siderite (FeCOs), strontianite (SrCOs), celestite 

(SrS04), and barite (BaS(>4), as well as the calculated in 

situ log Pc02 (Partial pressure of carbon dioxide). In 

calculating the saturation state of siderite, it was assumed 

that the total dissolved iron was in the ferrous state.

Uncertainties in Si's occur because of uncertainties in 

mineral and aqueous thermodynamic data used in the 

model, different degrees of analytical accuracy, and errors 

in pH caused by the more rapid outgassing of carbon dioxide from the higher temperature waters (table 9). As a 

result, some judgment is required in interpreting the Si's. 

Generally a tolerance of ±0.1 can be assigned to the SI 

of calcite, aragonite, gypsum, and anhydrite; ± 0.2 for the 

iS7 of dolomite, celestite, and strontianite; and much larger 

uncertainties on the order of ± 0.5 to the SI of barite and 

siderite. Because the solubilities of barite and, to some
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F19

TABLE 6. Summary of revised thermodynamic data

[Aff ° standard enthalpy of reaction in kilocalories per mole at 298.15 kelvins and 1 atmosphere pressure; K, equilibrium constant; T, temperature in kelvins]

log* Analytical expression Reference

Reaction (species'): 

Ca2++HCO: = CaHCO:

o a

= CaCO°

C02(g) = C02(aq)

C02(aq) + H20 = H+ + HCO"

HCO: =

HCO: =SrHCO*

O O

=SrCO°

o a

=SrSO°

Reaction (minerals):

4.11

3.556

-4.776 

2.177 

3.561 

6.05 

5.22 

1.6

1.095 logXc ,ICO+=1209.120 + 0.31294r-34765.06/r-478.7821ogr (1) °^ 3

3.224 \ogKCaCQ° = -1228.732 -0.299444T+35512.75/T -I- 485.818 log T (1)

-1.468 log Kg = 108.3865 + 0.01985076T - 6919.53/T- 40.45154 log T + 669365/5^ (1)

-6.352 logK^ = -356.3094-0.06091964T+ 21834.37/T + 126.8339logT- 1684915/T2 (1) 

-10.329 log K2 = -107.8871 - 0.03252849T + 5151.79/7V 38.92561 log T- 563713.90/T2 (1)

(2)

(2)

(3)

1.18 log#SrHCO+ =-3.248+ 0.014867T

3

2.81 logtfg^QO =-1.019-1-0.0128267

3

2.55 

CaSO = Ca + SO 4 4

CaCO, =Ca2+ + C02~

3 3

BaSO, = Ba2+ -i- SO2"

4 4

3~ 3

arso4 = a* + sojCaMg(C03)2 = Ca2+ + Mg2* + 2C02-

CaSO . 2H,0 = Ca2+ + SO 2" + 2H.O 42 42

FeCO, = Fe2+ + CO2"

o o

3 ~ 3

Anhydrite

Aragonite

Barite

Calcite

Celestite

Dolomite

Gypsum

Siderite

Strontianite

-4.3

-2.589

6.141

-2.297

.228

-9.436

-.028

-6.14

-.40

-4.384

-8.336

-9.978

-8.480

-6.578

-17.09

-4.602

-10.57

-9.271

--

log*A = -171.9773 - 0.077993T -I- 2903.293/T + 71.595 log T

logKc = -171.9065 - 0.077993T + 2839.319/T+ 71.595 log T

log#Cel = 73.415 - 3603.34LT- 29.8115 log T

log#G = 82.090 - 3853.936/T- 29.8115 log T

log Ka = 155.0305 - 7239.594/T - 56.58638 log T

0 otront

(4)

(1)

(5)

(4)

(6)

(7)

(8)

(9)

(2)

(1) Plummer and Busenberg (1982).

(2) Busenberg and others (1984).

(3) Smith and Martell (1976).

(4) Log K of Harvie and Weare (1980) adjusted to be consistent with log KQ AH° from Parker and others (1976).

(5) Parker and others (1976).

(6) Calculated from the data of Gallo (1935) for SrSO precipitated as given by Linke and Seidell (1965) and consistent with SrSO° data of Smith and Martell (1976).

(7) Robie and others (1978).

(8) Calculated from the data of Marshall and Slusher (1966) using the aqueous model of WATEQF.

(9) Log K of Smith (1918) recalculated using aqueous model of WATEQF at 30 °C. Adjusted to 25 °C using AH° from Parker and others (1976).

extent, celestite and gypsum depend on the crystallinity 

of the solid phase, it is sometimes possible to interpret 

seemingly constant levels of undersaturation or oversaturation as uncertainties in the thermodynamic data, 

rather than as actual indications of nonequilibrium.

The SI values of aragonite, calcite, dolomite, and gypsum as a function of dissolved-sulfate concentration are 

shown in figures 10-13. As dissolved sulfate .increases, 

the waters approach gypsum saturation (fig. 13). Although 

gypsum is not known to occur in the Madison Limestone, 

thermodynamic calculations based on gypsum are more

reliable that those for anhydrite as a result of the availability of high-quality solubility data (table 6).

The slight oversaturation of many waters with calcite 

(fig. 11) could be due to pH error (Pearson and others, 

1978), with the increase in pH at higher temperatures due 

to the increased outgassing of carbon dioxide (C02) during field measurement of pH (table 7). Significant pH 

error is, however, discounted, as the pH measurements 

were made in a closed flow cell to minimize the effects 

of C02 outgassing, and the effect of the expected increase 

in pH with temperature is not apparent in figure 14.
F20 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 7. Chemical and temperature data used in modeling (cations)

[°C, degrees Celsius; all constituent concentrations are in milligrams per liter; letter in front of flow-path number indicates whether the spring or well was used as recharge (R) 

or on the flow path (F); Ca, calcium; Mg, magnesium; Na, sodium; K, potassium; Ba, barium; Fe, iron; Li, lithium; Mn, manganese; Sr, strontium]

Spring 

Spring or well State 

or well number

Lewistown Big Spring

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

Bozeman Fish Hatchery

Big Timber Fish Hatchery

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Mock Ranch

Denius No. 1

Colstrip

Sarpy Mine

Gas City

Bluewater Spring

Moore

Mysse Flowing Well

Storey Fish Hatchery

Mobil

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Hole in the Wall

Barber Ranch Spring

Conoco No. 175

MKM

Shidler

Conoco No. 44

Mallo Camp

Rhoads Fork

Seeley

Coronado No. 2

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

Jones Spring

Kaiser

Cleghorn Spring

Lien

McNenney

Ellsworth AFB

Fuhs

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

10

8

9

12

13

18

4

7

6

17

15

16

1

2

21

19

26

11

22

20

6

7

14

23

24

7

8

5

12

11

10

9

8

24

4

19

16

21

17

15

13

20

22

18

10

9

12

11

16

17

2

18

6

1

19

24

25

22

26

21

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow Temperature pH 

path (-C)

Rl

Fl

Fl

Fl

Fl

Fl

R2

R2

F2

F2

F2

F2

R3,

R3,

F3

F3

F3

F3

F3

F3

R4

R4

F4

F4

F4

F4

F4

R5

R5

F5

F5

F5

F5

R6.R7

R6.R7

F6

F6

F6

F6

F6

F6

F7

F7

F7

F7

F7

R8

R8

R8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

10.6

20.4

19.6

20.4

31.6

40.9

8.6

10.7

24.6

83.7

61.7

84.7

10.8

9.0

97.5

83.0

91.5

14.3

86.9

63.0

8.0

7.1

46.2

52.7

56.2

22.8

55.6

8.2

15.7

65.0

88.1

56.0

32.2

7.7

5.7

13.0

39.8

25.0

23.4

25.7

17.1

26.1

29.5

45.1

30.5

20.0

12.9

15.0

11.6

11.9

11.5

49.0

10.7

63.5

68.0

71.0

59.1

66.0

57.0

58.5

7.58

7.63

7.40

7.24

7.08

7.00

7.84

7.64

7.40

6.61

6.50

6.40

7.50

7.60

6.52

6.70

6.61

7.29

6.67

6.61

7.30

7.31

6.99

6.94

7.01

7.51

6.77

7.25

7.46

7.07

6.71

7.08

7.17

7.60

7.35

7.30

7.16

7.20

7.20

7.12

7.20

7.30

7.30

6.70

6.90

6.89

7.20

7.40

7.41

7.41

7.18

7.01

7.21

6.68

6.65

6.69

6.68

6.70

6.67

6.66

Ca

75

84

130

260

260

510

50

51

71

220

350

320

56

40

220

320

370

530

380

450

52

32

180

190

200

510

550

35

56

110

290

300

240

57

66

66

56

64

70

88

110

63

73

73

210

540

71

47

42

46

87

91

88

240

220

270

300

360

410

420

Mg

28

29

40

99

96

120

17

16

23

34

53

52

20

29

28

51

61

67

68

110

15

19

44

46

48

110

110

32

19

25

41

49

61

24

23

27

27

29

27

41

38

29

31

22

41

83

22

18

19

25

24

33

63

48

58

65

65

77

83

90

Na

2.4

2.8

3.6

41

79

310

1

9.2

1.5

1,800

1,500

1,700

.4

.6

140

48

1,400

71

1,700

730

1.4

2.2

36

38

38

36

45

1.6

8.3

77

760

490

380

1.3

.9

1.3

1.8

2.6

2.1

2.6

3.7

2.6

2.5

11

86

27

1.5

9.7

4.9

2.4

2.1

4.9

5.8

27

18

25

46

85

72

34

K

0.9

1.2

1.3

9.2

12

26

.6

2.9

1.4

130

120

150

.6

1.1

67

49

110

2.7

130

99

.5

1.0

7.6

8.0

8.0

35

12

.6

2.6

9.8

69

35

25

.7

.6

1.2

1.7

1.7

1.4

2.2

1.5

1.7

2.1

4.8

11

5.2

.6

2.9

2.7

1.8

1.4

3.7

3.6

11

7.3

9.7

13

13

20

15

Ba

0.0

.0

.0

.0

.0

.0

.0

.0

.0

.1

.1

.0

.2

..

.0

.2

.2

.2

.2

.2

.0

.0

.2

.0

.2

.1

.1

.0

.0

.1

.1

.0

.1

--

--

.1

.0

--

--

.0

.0

.0

.0

.0

--

..

--

.1

--

.1

..

--

.0

--

--

__

0.0

.0

.2

.0

Fe

0.04

.01

.03

.03

.02

.44

.10

.01

.00

.02

.39

.05

.02

.09

.34

.56

.1

.01

.02

.02

_

.09

5.8

.02

.05

7.9

5.7

.03

.02

2.4

6.9

2.4

.87

--

--

.02

--

--

--

--

.03

--

.02

--

--

..

.00

.08

--

.01

..

--

.01

--

--

..

1.4

.86

.42

3.0

Li

0.02

.02

.02

.08

.14

.34

.00

.00

.00

3.80

3.50

4.4

.00

.00

.57

.33

1.5

.05

2.3

2.1

.01

.00

.04

.04

.04

.09

.06

.01

.01

.15

.87

.39

.36

.00

--

.00

.00

--

-.

.00

.01

.00

.01

.03

--

..

--

.01

--

.00

..

--

.01

--

--

_.

0.07

.17

.12

.02

Mn

0.00

.00

.00

.00

.01

.02

.00

.00

.00

.14

.08

.05

.00

.02

.01

.02

.02

.01

.01

.01

.00

.07

.16

.02

.01

.15

.06

.00

.01

.30

.15

.24

.02

.00

--

.00

.00

--

--

.00

.00

.00

.02

.22

--

..

--

.03

--

.00

..

.-

.00

--

--

..

0.05

.11

.02

.05

Sr

0.98

1.6

2.5

4.9

5.6

11

.14

.40

.49

12

13

13

.20

.17

9.0

11

15

5.6

15

11

.07

.09

4.5

4.8

4.7

12

11

.10

.4

1.6

6.0

4.6

3.4

-

.07

.25

.38

.38

.32

2.9

.39

.61

6

2.9

6.2

.12

.24

.10

.14

.89

.94

.50

4.6

4.5

5.5

6.6

8.0

9

9.2
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER 

TABLE 7. Chemical and temperature data used in modeling (cations) Continued

F21

Spring 

or well

Eagle Butte

Dupree

Gore Hill

Great Falls High School

Bough Ranch

HTH No. 3

Buckhorn Exeter

Provo

Spearfish

Black Hills Cemetery

Streeter Ranch

Bean

Denius No. 2

Denius No. 3

Martens Madison

Ranch A

Spring 

or well State 

number

23 S.Dak.

20 S.Dak.

2 Mont.

3 Mont.

5 Mont.

14 Mont.

27 Mont.

3 S.Dak.

5 S.Dak.

13 S.Dak.

14 S.Dak.

27 S.Dak.

3 Wyo.

4 Wyo.

23 Wyo.

25 Wyo.

Flow 

path

F8

F8

--

--

--

..

--

--

--

--

__

--

--

--

--

--

Temperature pH

54.7

56.5

14.0

13.1

17.9

51.8

80.0

60.0

13.2

12.4

42.2

41.5

8.3

8.5

15.0

12.2

6.84

6.82

7.13

7.40

7.06

6.80

6.40

6.97

7.39

7.28

7.61

7.10

7.55

7.69

7.50

7.26

Ca

390

390

170

96

320

490

380

110

52

55

40

160

55

74

61

120

Mg

110

94

76

37

200

100

66

30

23

23

15

51

16

30

26

30

Na

60

110

130

32

150

95

1,600

200

2.1

2.3

9.2

14

.6

1.9

2.7

2.6

K

28

49

13

3.9

20

39

120

18

1.1

1.3

2.3

5.0

.5

.0

2.0

1.4

Ba

.0

.0

.0

.0

.0

.0

..

..

..

.3

.0

.0

.0

.0

..

.0

Fe

3.0

.30

1.0

.08

.18

.97

.5

..

.02

.02

.04

.01

.01

..

--

Li

.11

.23

.33

.06

.30

.40

--

..

-.

.00

.01

.02

.00

.01

.00

.00

Mn

.09

.04

.02

.02

.02

.46

.04

-.

-.

.00

.00

.00

.00

.02

.01

.00

Sr

11

12

3.2

1.1

8.6

9.5

-

2.2

0.15

.33

..28

3.3

.11

.55

--

1.1

TABLE 8. Chemical data used in modeling (onions)

[Letter in front of flow-path number indicates whether the spring or well was used as recharge (R) or on the flow path (F); Cl, chloride; S04 , sulfate; HC03 , bicarbonate; H2S,

hydrogen sulfide; B, boron; Br, bromide; F, fluoride; P04 , orthophosphate; H4Si04 , silicic acid]

Spring 

Spring or well State 

or well number

Lewistown Big Spring

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

Bozeman Fish Hatchery

Big Timber Fish Hatchery

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Mock Ranch

Denius No. 1

Colstrip

Sarpy Mine

Gas City

Bluewater Spring

Moore

Mysse Flowing Well

Storey Fish Hatchery

Mobil

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Hole in the Wall

Barber Ranch Spring

Conoco No. 175

MKM

Shidler

Conoco No. 44

Mallo Camp

Rhoads Fork

10

8

9

12

13

18

4

7

6

17

15

16

1

2

21

19

26

11

22

20

6

7

14

23

24

7

8

5

12

11

10

9

8

24

4

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

Flow 

path

Rl

Fl

Fl

Fl

Fl

Fl

R2

R2

F2

F2

F2

F2

R3.R4

R3.R4

F3

F3

F3

F3

F3

F3

R4

R4

F4

F4

F4

F4

F4

R5

R5

F5

F5

F5

F5

R6, R7

R6.R7

a

1.8

1.5

2.4

19

67

190

1.1

2.8

1.1

2^00

2,000

2,100

.5

.7

96

21

1,900

2.4

2,400

630

1.4

.9

54

56

57

25

67

1.3

4.4

70

1,200

620

490

1.0

.4

S04

140

170

330

970

990

2,000

23

27

120

1,200

1,300

1,400

8.1

22

740

960

1,300

1,500

1,500

1,900

3.2

7.7

460

480

520

1,600

1,700

6.2

34

340

920

970

840

1.8

3.1

HC03

190

200

200

210

200

150

220

230

190

330

340

360

250

240

140

140

390

220

300

320

220

190

210

210

210

110

170

260

230

120

130

110

220

300

310

H2S

B Br F

(field) (laboratory)

..

..

..

.-

--

0.04 0.3

..

..

..

46

57

132

..

..

4

2.9

54

..

19

8.8

..

..

..

..

--

.04 .2

.17 1.8

..

..

.08

.07

.12

.30

..

..

0.03

.03

.06

.21

.41

1.0

.01

.04

.01

6.8

6.1

7.9

.23

.01

.67

.34

4.7

.17

3.3

3.2

.01

.01

.09

.09

.09

.14

.17

.02

.02

.20

.89

.43

.36

.00

--

0.0

.0

.0

.1

.2

.6

.0

.0

.0

12

11

12

.0

.0

.6

.1

.1

.0

6.9

2.4

.0

.2

.1

.2

.1

.2

.0

.1

.4

3.5

1.9

4.3

--

--

1.1

1.4

1.5

1.6

1.7

3.3

.2

3.9

.6

4.9

5.2

5.4

.2

.7

4.7

4.5

4.7

1.4

5.4

4.2

.1

.2

2.8

3.3

3.6

3.1

3.6

.2

.3

1.4

4.4

3.5

3.0

.1

.1

P04

0.00

.00

.00

.00

.00

.00

.00

.00

.00

.00

.00

.01

.00

.00

.00

.03

.06

.06

.00

.00

.03

.00

.03

.09

.00

.00

.00

.03

.06

.00

.00

.00

.03

.00

--

H4SiO4

6.7

8.9

9.3

12

16

18

4.8

16

11

9.1

55

58

6.3

7.8

69

49

58

13

56

36

6.8

9.3

30

33

32

17

29

6.7

19

19

59

32

24

12

8.8
F22 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 8. Chemical data used in modeling (anions) Continued

Spring 

or well

Seeley

Coronado No. 2

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

Jones Spring

Kaiser

Cleghorn Spring

Lien

McNenney

Ellsworth AFB

Fuhs

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Gore Hill

Great Falls High School

Bough Ranch

HTH No. 3

Buckhorn Exeter

Provo

Spearfish

Black Hills Cemetery

Streeter Ranch

Bean

Denius No. 2

Denius No. 3

Martens Madison

Ranch A

Spring 

or well 

number

19

16

21

17

15

13

20

22

18

10

9

12

11

16

17

2

18

6

1

19

24

25

22

26

21

23

20

2

3

5

14

27

3

5

13

14

27

3

4

23

25

State

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Mont.

Mont.

Mont.

Mont.

Mont.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Flow 

path

F6

F6

F6

F6

F6

F6

F7

F7

F7

F7

F7

R8

R8

R8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

--

--

--

..

-

-

--

--

..

--

--

--

--

--

Cl

.8

.8

1.2

.7

0.8

2.6

1.7

2.1

7.6

110

31

.5

3.7

2.2

1.5

.6

1

3.9

40

22

28

61

160

110

48

63

120

110

14

64

39

2,500

270

.6

1.8

2.5

17

.3

.6

1.4

1.4

S04

17

28

47

50

170

210

43

86

200

550

1,500

5.8

14

23

14

100

200

200

680

640

800

860

1,200

1,200

1,300

1,300

1,400

570

230

1,600

1,800

1,300

310

8

11

9

450

3

43

11

200

HC03

300

280

290

300

280

270

310

280

100

230

240

340

230

200

240

260

200

340

130

160

150

150

130

130

150

180

160

340

260

260

160

310

180

270

270

200

190

240

310

310

270

H2S

(field) (laboratory)

..

-.

-.

--

..

..

..

..

.03 .3

..

..

.03 .1

..

--

.-

..

.05 .5

.13 .7

.01 .3

0.02 0.3

.14 .7

.07 .9

.12 .6

1.4

.9

1.0 19

.9

..

8.2

4.5

34

..

..

.02 .2

.02 .3

.04 .2

..

..

..

--

B

.01

.01

-.

..

--

.00

.01

.01

.04

--

..

.03

.03

..

.02

..

..

.02

.00 

--

0.10

.14

.18

.13

.25

.49

.51

.09

.33

.37

--

-

-

.01

..

--

.01

.01

.03

.03

Br

.0

--

--

-.

--

..

--

.0

--

--

..

--

.1

..

.1

..

..

.0

.2

--

..

0.3

4.2

.5

.2

.2

.4

.7

.1

.3

.5

--

--

-.

.0

.0

.4

.0

.0

.0

.1

F

.3

.2

.3

.4

1.2

.5

.3

.6

1.4

1.1

1.6

.1

.5

.0

.2

.3

.6

.2

1.7

2.6

2.8

2.6

2.9

3.4

3.4

3.7

3.6

2.4

.9

3.2

.7

4.9

1.3

.2

.3

.4

1.7

.2

.3

.3

.3

P04

.03

.02

-.

-.

--

.06

.06

.00

.00

--

..

.00

.00

--

.00

..

..

.00

--

--

..

--

--

0.00

.03

..

--

.00

.00

0.00

.01

--

-.

--

.03

..

--

--

.00

.00

.00

1148104

12

14

12

11

12

11

12

13

4.5

23

15

8.4

10

11

11

12

21

12

39

33

35

33

36

29

31

30

34

15

12

9.7

7.0

58

30

10

10

17

20

5.2

8.9

11

8

TABLE 9. Results of thermodynamic calculations

[Letter in front of flow-path number indicates whether the spring or well was used as recharge (R) or on the flow path (F)]

Spring

or well

Lewistown Big Spring

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

Bozeman Fish Hatchery

Big Timber Fish Hatchery

Spring

or well

number

10

8

9

12

13

18

4

7

Saturation index

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Flow

path

Rl

Fl

Fl

Fl

Fl

Fl

R2

R2

Log

PC02

-2.40

-2.38

-2.17

-2.01

-1.80

-1.81

-2.60

-2.37

Calcite

0.01

.25

.14

.15

.11

.18

.19

.05

Aragonite

-0.15

.11

-.01

.00

-.03

.04

.04

-.11

Dolomite

-0.27

.35

.06

.20

.22

.20

.01

-.27

Anhydrite

-1.80

-1.60

-1.24

-.70

-.61

-.15

-2.68

-2.59

Siderite

-1.46

-1.75

-1.58

-1.85

-1.98

-.81

-.74

-1.86

Strontianite

-1.37

-1.03

-1.15

-1.18

-1.22

-1.21

-1.82

-1.53

Celestite

-1.68

-1.45

-1.09

-.60

-.58

-.22

-3.17

-2.66

Barite

..

--

--

--

--

..

--

--
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER 

TABLE 9. Results of thermodynamic calculations Continued

F23

Spring 

or well

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Mock Ranch

Denius No. 1

Colstrip

Sarpy Mine

Gas City

Bluewater Spring

Moore

Mysse Flowing Well

Storey Fish Hatchery

Mobil

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Hole in the Wall

Barber Ranch Spring

Conoco No. 175

MKM

Shidler

Conoco No. 44

Mallo Camp

Rhoads Fork

Seeley

Coronado No. 2

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

Jones Spring

Kaiser

Cleghorn Spring

Lien

McNenney

Ellsworth AFB

Fuhs

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Gore Hill

Great Falls High School

Bough Ranch

Spring 

or well 

number

6

17

15

16

1

2

21

19

23

11

22

20

6

7

14

23

24

7

8

5

12

11

10

9

8

24

4

19

16

21

17

15

13

20

22

18

10

9

12

11

16

17

2

18

6

1

19

24

25

22

26

21

23

20

2

3

5

Saturation index

State

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Mont.

Mont.

Mont.

Flow 

path

F2

F2

F2

F2

R3,R4

R3,R4

F3

F3

F3

F3

F3

F3

R4

R4

F4

F4

F4

F4

F4

R5

R5

F5

F5

F5

F5

R6

R6.R7

F6

F6

F6

F6

F6

F6

F7

F7

F7

F7

F7

R8

R8

R8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

..

.-

--

Log

pco2

-2.14

-.83

-.88

-.68

-2.20

-2.32

-.89

-1.20

-.70

-2.08

-.88

-.94

-2.06

-2.14

-1.57

-1.47

-1.52

-2.57

-1.41

-1.94

-2.17

-1.74

-1.21

-1.90

-1.85

-2.23

-1.98

-1.91

-1.63

-1.76

-1.75

-1.69

-1.84

-1.82

-1.85

-1.58

-1.55

-1.61

-1.75

-2.11

-2.19

-2.12

-1.86

-1.57

-1.79

-1.34

-1.18

-1.24

-1.32

-1.36

-1.40

-1.32

-1.45

-1.47

-1.71

-2.08

-1.78

Calcite

.02

.18

.05

.07

-.01

-.11

.12

.27

.56

.37

.48

.29

-.32

-.59

.18

.23

.35

.37

.25

-.48

-.02

.07

.20

.18

.21

.12

-.09

-.05

.08

-.02

.01

-.03

.00

.12

.16

-.71

-.03

.10

-.06

-.15

-.30

-.19

-.17

.03

-.09

-.05

.03

0.13

.01

.06

-.01

.05

.23

.17

.04

.05

.01

Aragonite

-.13

.07

-.07

-.04

-.16

-.26

.02

.16

.46

.22

.37

.17

-.48

-.75

.05

.10

.22

.23

.13

-.64

-.17

-.04

.10

.05

.07

-.04

-.25

-.20

-.05

-.16

-.13

-.17

-.15

-.02

.02

-.84

-.17

-.05

-.21

-.30

-.45

-.34

-.32

-.10

-.25

-.17

-.09

0.01

-.11

-.05

-.13

-.07

.11

.04

-.11

-.10

-.14

Dolomite

-.10

-.16

-.26

-.38

-.33

-.25

-.60

-.00

.52

.09

.45

.42

-1.11

-1.35

.25

.33

.56

.43

.27

-.92

-.29

-.04

-.22

.05

.26

-.06

-.60

-.31

.33

-.02

-.06

-.02

-.21

.27

.36

-1.44

-.34

-.30

-.45

-.51

-.79

-.48

-.74

.12

-.18

-.36

-.09

0.04

-.17

-.12

-.25

-.10

.40

.18

-.05

-.11

.11

Anhydrite

-1.74

-.48

-.38

-.28

-3.07

-2.80

-.30

-.18

-.20

-.38

-.20

-.11

-3.51

-3.32

-.82

-.75

-.68

-.32

-.04

-3.41

-2.41

-.94

-.32

-.43

-.72

-3.76

-3.50

-2.70

-2.31

-2.17

-2.13

-1.57

-1.48

-2.21

-1.85

-1.37

-.84

-.33

-3.13

-2.86

-2.71

-2.91

-1.88

-1.29

-1.68

-.48

-.51

-0.36

-.38

-.20

-.21

-.17

-.24

-.20

-1.08

-1.54

-.56

Siderite

..

-1.76

-.77

-1.64

-1.65

-.98

-.68

-.41

-.94

-2.48

-1.71

-1.93

..

-1.39

.79

-1.63

-1.13

.52

.38

-1.78

-1.61

.54

.65

.20

-.20

--

--

-1.73

--

--

--

--

-1.68

--

-1.39

--

--

..

--

-1.07

--

-2.03

..

--

-2.22

--

--

..

-0.19

-.44

-.88

.03

.24

-.83

-.34

-1.18

-1.33

Strontianite

-1.72

-.88

-1.14

-1.13

-1.93

-1.94

-1.10

-1.00

-.66

-1.19

-.74

-1.10

-2.64

-2.59

-1.12

-1.09

-1.01

-.89

-1.21

-2.48

-1.72

-1.50

-1.27

-1.37

-1.29

-

-2.49

-1.96

-1.71

-1.81

-1.89

-1.10

..

-1.66

-1.51

-1.46

-1.53

-1.46

-2.32

-1.94

-2.40

-2.18

-1.65

-1.64

-1.84

-1.52

-1.42

-1.34

-1.39

-1.36

-1.42

-1.36

-1.06

-1.10

-1.24

-1.41

-1.17

Celestite

-2.07

-.37

-.34

-.30

-3.49

-3.12

-.36

-.27

-.25

-.47

-.24

-.28

-4.31

-3.78

-.84

-.80

-.80

-.17

-.25

-3.88

-2.65

-1.30

-.63

-.74

-.86

~

-4.36

-3.12

-2.79

-2.56

-2.61

-1.24

..

-2.59

-2.14

-.83

-.97

-.46

-3.90

-3.17

-3.32

-3.41

-1.86

-1.69

-1.92

-.73

-.76

-0.63

-.56

-.42

-.38

-.36

-.29

-.23

-.88

-1.52

-.29

Barite

__

.09

.41

--

.06

--

--

.60

.32

1.70

.39

.92

..

--

.88

..

.77

1.26

.76

--

--

.28

.05

--

.90

--

--

-.12

--

--

--

--

.41

--

--

--

--

..

--

-.18

-

-.14

..

--

.58

--

--

..

--

--

0.96

-.43

..

--

--

-

--
F24 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 9. Results of thermodynamic calculations Continued

Spring 

or well

HTH No. 3

Buckhorn Exeter

Provo

Spearfish

Black Hills Cemetery

Streeter Ranch

Bean

Denius No. 2

Denius No. 3

Martens Madison

Ranch A

Spring 

or well 

number

14

27

3

5

13

14

27

3

4

23

25

Saturation index

State

Mont.

Mont.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Flow Log 

I«A P<X>2

-1.50

-.66

-1.51

-2.04

-1.93

-2.20

-1.75

-2.27

-2.31

-2.08

-1.93

Calcite

.13

.16

.07

-.08

-.18

.30

.15

-.01

.32

.17

.03

Aragonite

.01

.05

-.05

-.24

-.34

.17

.02

-.17

.16

.01

-.13

Dolomite

.06

-.11

.06

-.34

-.58

.68

.28

-.48

.34

.17

-.37

Anhydrite

-.09

-.27

-1.06

-3.08

-2.94

-2.83

-.90

-3.55

-2.32

-2.89

-1.51

Siderite

-.46

-.64

--

--

-1.80

-.91

-1.37

-1.99

-1.78

~

~

Strontianite

-1.33

-

-1.35

-2.11

-1.89

-1.48

-1.22

-2.16

-1.27

--

-1.52

Celestite

-.29

~

-1.25

-3.62

-3.15

-3.30

-.96

-4.18

-2.39

-

-1.56

Barite

..

--

--

--

.21

..

--

--

--

--

.58

1 .00

&Q

z 0.50

z

o

scc

K 0.00

< 

W

LU

z

8 -0.50

<

cc

<

-1 .00

 1 1 1 1 I 1 1 1 \ 1 

m

m

m (DO Ojm rn m m rn m __ rn _- m UJ

rfll pi rfij rrim^ffl ^iDtl! m C

g njrjm|P * CP t]] (T] D m QC]t]] a ^Equilibrium

W m m

. cP

m

m

m

i i i i i i i i i i

-1.00 1.00 3.00 5.00 7.00 9.00 11.00 13.00 15.00 17.00 19.00 21.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 10. Saturation indices of aragonite as a function of dissolved-sulfate concentration for all wells and springs sampled.

1.00

x

0 0.50

0.00

-o.5o

-1.00

m

Q

m m

m m

m m 

n 

n S a

Equilibrium

-1-00 1.00 3.00 5.00 7.00 9.00 11.00 13.00 15.00 17.00 19.00 21.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 11. Saturation indices of calcite as a function of dissolved-sulfate concentration for all wells and springs sampled.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F25

1.00

0.50

i 0.00

p? -0.50 

w

t

5

g -1.00

o 

o

-1.50

-2.00

m m

m

n

m 

m

m m

m 

m__

Equilibrium

o 

m

m 

m

-1.00 1.00 3.00 5.00 7.00 9.00 11.00 13.00 15.00 17.00 19.00 21.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 12. Saturation indices of dolomite as a function of dissolved-sulfate concentration for all wells and springs sampled.

Alternatively, laboratory studies have shown that the 

presence of dissolved magnesium and higher temperatures favor precipitation of aragonite rather than calcite 

(Bischoff, 1968). Thus, under the in situ conditions, aragonite may be controlling the CaCOs solubility product 

rather than calcite. The X-ray diffraction analysis of 

samples of scale and travertine from the outlets of the 

flowing wells of Mysse (well No. 20 in Montana), Moore 

(well No. 22 in Montana), and Philip (well No. 19 in South 

Dakota) showed the presence of both calcite and aragonite, though aragonite was not noted in the subsurface by 

either Thayer (1981) or R.S. Deike, (U.S. Geological 

Survey, written commun., 1984).

The data in figure 12 show that, except for some waters 

with very small sulfate concentrations, waters in the 

Madison aquifer are saturated with dolomite. Because the 

solubility of dolomite as a function of temperature is not 

well known, the dolomite SI has been calculated using the 

thermodynamic data in table 6 by assuming a constant 

change in enthalpy (AH£) for the dissolution of dolomite 

throughout the entire range of temperature determined 

for the Madison aquifer and adjusting the equilibrium constant for the effects of temperature by using the Van't 

Hoff equation:

log

#298.15 T 298.15

(3)

where

KDT is the equilibrium constant for dolomite dissolution;

#298.15 is the equilibrium constant for dolomite dissolution at 298.15 K;

Aff o is the standard enthalpy of the dissolution

reaction at 1 atm pressure and 298.15 K;

R is the gas constant (0.08205 L-atm/mol K);

and 

T is the final temperature, in kelvins.

If the waters from the Madison aquifer are in equilibrium 

with aragonite (or calcite) and dolomite, it is possible to 

derive more accurate expressions for the temperature 

dependence of the dolomite equilibrium constant from the 

field data.

The data in figure 15 show an increase in log(aCa2+/ 

aMg2+ ) with temperature corresponding to the following relation:

1

aCa2+ \ 

 =1.907 + 0.0072 T, aMg2+ |

where

T is temperature, in kelvins; and 

a is the activity of the specified ion.

(4)
F26 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

1.00

0.50

0.00

-0.50

-1.00

-1.50

-2.00

-2.50

-3.00

-3.50

-4.00

'Equilibrium

 to

D

§

§

m 

n

'TT

D

D Q]

D n D (3

m

-1.00 1.00 3.00 5.00 7.00 9.00 11.00 13.00 15.00 17.00 19.00 21.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 13. Saturation indices of gypsum as a function of dissolved-sulfate concentration for all wells and springs sampled.

For water in equilibrium with aragonite (or calcite) and 

dolomite, equation 5 may be written from the following 

reactions:

2\og KA 

- log KD

2CaC03

Ca2+ + Mg2+ + 2CO§- = CaMg(C03)2

2CaC03 + Mg2

for which the equilibrium constant is 

aCa2+ \

2+ 

KA -log KD, (5)

(6)

where the subscripts A and D denote aragonite and 

dolomite, respectively. Assuming that the dolomite is in 

equilibrium with aragonite and using the approximately 

linear relation for log (aCa2+/aMg2+ ) from figure 15 

with the aragonite equilibrium-constant expression from 

table 6, the temperature dependence of the dolomite 

equilibrium constant is approximately

log KD = 2 log KA -log (7)

log KD = -342.05 - 0.163186T + 5806.586/T

+ 143.191og T. (8)
LOG OF RATIO OF THE ACTIVITY OF CALCIUM 

TO THE ACTIVITY OF MAGNESIUM

LOG ION ACTIVITY PRODUCT OF CALC

3 

O

M>

3 

"8

B

B

B

B

I

I

g*

z o

o ~ 

x

m m 

V) o>

O P 

m 0

W o 

CO

B
F28 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

Alternatively, if the waters from the Madison aquifer are 

in equilibrium with calcite rather than aragonite,4he equation becomes

log KD = -341.91 - 0.163186 T+ 5678.638/T7 

+143.19 log T. (9)

Application of the Van't Hoff equation to the data in table 

6 leads to

log KD = -24.01 + 2062.035/r. (10)

The data in figure 16 compare log KD calculated from 

equations 8 to 10 and show that the Van't Hoff equation 

used in this report for calculation of the dolomite equilibrium constant probably is accurate to ±0.1 log K at 

60 °C, but that it may overestimate the value by as much 

as 0.4 log K units at 80 °C. If equations 8 or 9 more 

accurately describe the temperature dependence of the 

dolomite equilibrium constant, then the computed dolomite SI for waters at 80 °C, for example, should be increased by 0.2 to 0.4 SI units. The data in figure 17 show 

a trend toward undersaturation with respect to dolomite

FIGURE 16. Comparison of temperature dependence of log K dolomite 

as given by Van't Hoff equation and derived from assumption of 

aragonite-dolomite equilibrium (eq 8) and calcite-dolomite equilibrium 

(eq 9) as a function of temperature for all wells and springs sampled.

-16.00

-16.50

-17.00

-17.50

-18.00

-18.50

-19.00

-19.50

-20.00

EQUATION 8

VAN'T HOFF

EQUATION 9

0.00 20.00 40.00 60.00 80.00 100.00 

TEMPERATURE, IN DEGREES CELSIUS

1.00

0.50

X

LLJ

o

? 0.00

-0.50

-1.00

-1.50

-2.00

m m

m

m

m

m 

a

m

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

TEMPERATURE, IN DEGREES CELSIUS

FIGURE 17. Dolomite saturation index based on Van't Hoff equation as a function of temperature for all wells and springs

sampled.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F29

at higher temperatures based on the Van't Hoff approximation to log KD (table 6). From the results of figure 16, 

this trend probably reflects the inappropriateness in using 

a constant Afl£ at higher temperatures rather than undersaturation with respect to dolomite.

The data in figure 18 show that with increasing sulfate 

concentration the celestite SI approaches a constant value 

near -0.2 between 14 and 21 mmol/L (millimoles per 

liter) sulfate. This indicates saturation with a celestite 

phase slightly less soluble than that of the laboratoryprecipitated celestite (Gallo, 1935) given in table 6.

An application of the Gibbs phase rule to the CaCOsCaS04-SrS04 system at constant temperature and pressure indicates that waters in equilibrium with calcite (or 

aragonite), gypsum, and celestite cannot also be in equilibrium with strontianite. For example, assume that a 

water is saturated with aragonite, gypsum, and celestite:

log (aCa2+ aCO§- ) = log KA, 

log (aCa2+ aSO|- ) = log KG , and 

log(aSr2 +

where the subscripts A, G, and Cel represent aragonite, 

gypsum (assuming unit activity of water), and celestite, 

respectively. The SI of strontianite is then dependent on

i.oo

0.50

0.00

-0.50

-1.00

p -1.50 

<

tr3

(A

-2.00

-2.50

-3.00

ft

-3.50 9 

3 

D

-4.00

-4.50

'Equilibrium

G3

m

ED

rrjD

m

m

n

m

G3

m

D

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 18. Celestite saturation index as a function of dissolved-sulfate concentration for all wells and springs sampled.
F30 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

the equilibrium constants of aragonite, gypsum, and 

celestite according to

= log

KGKst

(11)

where St represents strontiamte, and the other terms are 

as defined above.

A constant level of undersaturation with respect to 

strontianite near -1.0 would be expected if the data in 

table 6 at 25 °C are substituted into equation 11. The data 

in figure 19 show that the strontianite SI, as a function 

of sulfate concentration, is near -1.2 + 0.2 throughout the 

Madison aquifer, in agreement with equation 11. Based 

on this determination, if strontianite were present in the 

Madison aquifer, it would dissolve irreversibly, causing 

precipitation of celestite, calcite, and further dissolution 

of gypsum.

The results of the thermodynamic calculations in table 

9 show considerable variation for barite and siderite, partly reflecting larger uncertainties in the analytical data for 

dissolved barium and iron. The calculations indicate,

however, that most waters of the Madison aquifer are 

probably saturated or oversaturated with barite and 

undersaturated with siderite.

The data in figure 20 show that most of the waters are 

oversaturated with respect to quartz but either saturated 

or slightly undersaturated with respect to chalcedony 

throughout the temperature range.

In addition to providing SI information, the equilibriumspeciation calculations may yield information on other 

parameters of use during the modeling process. These include Pco2 (table 9), the redox state of the water (RS) 

(Parkhurst and others, 1980), the total concentration of 

inorganic carbon (molal units), and a conversion of the 

total concentrations of the other analyzed elements to the 

molal scale. The molal-scale concentrations (mmol/kg of 

water) of the elements and redox state are summarized 

in table 10 for the average recharge water of each flow 

path and for each subsequent well on the flow path. These 

values were used in a mass-balance analysis of the reactions and are discussed after an examination of trends in 

the water-quality data.

1.00

0.50

0.00

-0.50

-1.00

-1.50

-2.00

-2.501

-3.00

'Equilibrium

B

m m

H

° JD m

itD

a

m 0 m mm 

L a

a

a

a

a

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 19. Strontianite saturation index as a function of dissolved-sulfate concentration for all wells and springs sampled.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F31

90.00

80.00

70.00

tc 60.00

HI 

QL

V) 

5 

<

0 50.00

< 40.00

3

V)

a

LLJ

30.00

20.00

10.00

0.00

Chalcedony saturation.

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

TEMPERATURE, IN DEGREES CELSIUS

FIGURE 20. Dissolved-silica concentration as a function of temperature for all wells and springs sampled.

TRENDS IN WATER-QUALITY DATA

Trends in the water-quality data as a function of dissolved sulfate are an indication of further reactions in the 

system. These trends also can indicate other reactions that 

need to be considered; for example, calcite and dolomite 

may be either dissolving or precipitating as a function of 

reaction progress, responding almost reversibly to the 

dissolution of anhydrite (a partial-equilibrium system).

In the preceding section, thermodynamic calculations 

were used to determine, from the saturation states, which 

minerals may react reversibly and which may react irreversibly in the Madison aquifer. Based on these calculations, it is likely that anhydrite dissolves irreversibly in 

most of the system, whereas the water remains at or near 

saturation with calcite and dolomite. It is appropriate then 

to choose total dissolved sulfate as a reaction-progress 

variable in examining trends in the water-quality data.

Changes in predominant cations and anions are depicted 

(on a mole percent basis) on the trilinear diagram in figure 

21. No single flow path shows the entire evolutionary 

trend, but the combined eight flow paths have different 

segments of the overall reaction path. The evolutionary 

path for cations proceeds from a predominance of calcium 

and magnesium to sodium predominance.

The anions proceed from a predominance of bicarbonate 

to sulfate to chloride. This trend indicates that the majorelement chemistry is initially determined by dissolution 

of calcite and dolomite followed by dissolution of anhydrite, and along several flow paths halite dissolution may 

be important.

There are distinct differences in the extent of reaction 

progress along each flow path as shown in separate trilinear diagrams (figs. 22-29). Flow paths 1, 4, 7, and 8 

(figs. 22, 25, 28, and 29) begin as calcium-magnesium 

bicarbonate waters but are significantly affected by
F32 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 10. Total concentrations of the major elements and redox state used in mass-balance models

[All constituent concentrations are in millimoles per kilogram of water; letter in front of flow-path number indicates whether the spring or well was used as recharge (R) or on the flow path 

(F); Ca, calcium; Mg, magnesium; Na, sodium; K, potassium; Cl, chloride; C, total dissolved inorganic carbon; S, total dissolved sulfide; S04 , sulfate; Fe, total dissolved iron]

Spring 

or well

Recharge No. 1

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

Recharge No. 2

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Recharge No. 3

Colstrip

Sarpy Mine

Gas City

Bluewater Spring

Moore

Mysse Flowing Well

Recharge No. 4

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Recharge No. 5

Conoco No. 175

MKM

Shidler

Conoco No. 44

Recharge No. 6

Seeley

Coronado No. 2

Newcastle

Osage

Upton

Devils Tower

Recharge No. 7

Voss

Self

JBJ

Evans Plunge

Cascade Spring

Recharge No. 8

Lien

McNenney

Ellsworth AFB

Fuhs

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Spring 

or well 

number

..

8

9

12

13

18

-.

6

17

15

16

--

21

19

26

11

22

20

--

14

23

24

7

8

--

11

10

9

8

--

19

16

21

17

15

13

--

20

22

18

10

9

--

17

2

18

6

1

19

24

25

22

26

21

23

20

State

Mont.

Mont.

Mont.

Mont.

Mont.

.-

Mont.

Mont.

Mont.

Mont.

-.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

--

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

--

Wyo.

Wyo.

Wyo.

Wyo.

--

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

--

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

--

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow 

path

Rl

Fl

Fl

Fl

Fl

Fl

R2

F2

F2

F2

F2

R3

F3

F3

F3

F3

F3

F3

R4

F4

F4

F4

F4

F4

R5

F5

F5

F5

F5

R6

F6

F6

F6

F6

F6

F6

R7

F7

F7

F7

F7

F7

R8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

F8

Ca

1.87

2.10

3.25

6.50

6.50

12.77

1.26

1.77

5.52

8.78

8.03

1.20

5.50

8.00

9.28

13.26

9.54

11.28

1.12

4.50

4.75

5.00

12.76

13.76

1.14

2.75

7.26

7.50

6.00

1.54

1.65

1.40

1.60

1.75

2.20

2.75

1.65

1.57

1.82

1.82

5.25

13.51

1.33

1.15

2.17

2.27

2.20

6.00

5.50

6.75

7.50

9.00

10.25

10.50

9.75

9.75

Mg

1.15

1.19

1.65

4.08

3.96

4.95

.68

.95

1.41

2.19

2.15

1.01

1.15

2.10

2.52

2.76

2.82

4.54

.85

1.81

1.89

1.98

4.54

4.54

1.05

1.03

1.69

2.02

2.51

.97

1.11

1.11

1.19

1.11

1.69

1.56

.95

1.19

1.28

.91

1.69

3.42

.81

1.03

.99

1.36

2.59

1.98

2.39

2.72

2.68

3.17

3.42

3.71

4.53

3.88

Na

0.10

.12

.16

1.79

3.44

13.53

.22

.07

78.78

65.63

74.42

.02

6.10

2.09

61.24

3.10

74.44

31.89

.05

1.57

1.65

1.65

1.57

1.96

.22

3.35

33.17

21.37

16.57

.05

.06

.08

.11

.09

.11

.16

.04

.11

.11

.48

3.75

1.18

.23

.10

.09

.21

.25

1.18

.78

1.09

2.00

3.70

3.14

1.48

2.62

4.80

K

0.02

.03

.03

.24

.31

.67

.04

.04

3.35

3.09

3.86

.02

1.72

1.26

2.83

.07

3.35

2.54

.02

.19

.20

.20

.90

.31

.04

.25

1.77

.90

.64

.02

.03

.04

.04

.04

.06

.04

.02

.04

.05

.12

.28

.13

.05

.05

.04

.09

.09

.28

.19

.25

.33

.33

.51

.38

.72

1.26

a

0.05

.04

.07

.54

1.89

5.38

.05

.03

65.27

56.74

59.61

.02

2.71

.59

53.90

.07

68.14

17.85

.02

1.52

1.58

1.61

.71

1.89

.08

1.98

33.97

17.53

13.85

.02

.02

.02

.03

.02

.02

.07

.01

.05

.06

.21

3.11

.88

.06

.04

.02

.03

.11

1.13

.62

.79

1.72

4.52

3.11

1.36

1.78

3.39

C

3.31

3.42

3.53

3.81

3.72

2.81

3.85

3.35

6.31

6.78

6.37

4.30

3.58

2.99

7.54

3.97

6.02

6.87

4.05

3.99

4.05

3.95

1.88

3.45

4.45

2.23

2.81

2.00

3.98

5.46

5.51

5.13

5.34

5.54

5.25

5.03

5.73

5.56

5.00

2.20

4.59

4.87

4.67

4.31

4.96

3.80

6.41

2.83

3.57

3.26

3.24

2.77

2.80

3.22

3.54

3.17

S

..

--

--

--

--

0.001

--

-.

1.358

1.682

3.898

..

.118

.085

1.594

..

.561

.259

--

--

..

.001

.005

--

.002

.002

.004

.009

--

..

--

--

--

--

..

--

--

--

--

.001

-.

.001

--

--

..

.002

.004

.000

.001

.004

.002

.004

.041

.027

.029

S04

1.46

1.77

3.44

10.11

10.32

20.89

.26

1.25

12.57

13.61

14.67

.16

7.72

10.01

13.61

15.65

15.72

19.86

.11

4.79

5.00

5.42

16.70

17.74

.21

3.54

9.61

10.12

a?e

.03

.18

.29

.49

.52

1.77

2.19

.03

.45

.90

2.08

5.73

15.65

.15

.15

1.04

2.08

2.08

7.09

6.67

8.34

8.97

12.52

12.52

13.56

13.56

14.61

Fe

0.0007

.0002

.0005

.0005

.0004

.0079

.0010

.0000

.0004

.0070

.0009

.0010

.0061

.0100

.0018

.0002

.0004

.0004

.0012

.1040

.0004

.0009

.1418

.1023

.0004

.0430

.1240

.0431

.0156

.0000

.0004

--

--

--

--

.0005

.0000

--

.0004

--

-.

--

.0007

.0002

--

..

.0002

--

--

--

.0251

.0154

.0075

.0538

.0538

.0054

Redox 

state

21.9

24.2

34.7

75.9

76.8

136.5

16.9

20.9

97.9

105.4

105.6

18.1

60.3

71.8

108.6

109.8

117.2

146.1

16.8

44.9

46.2

48.3

107.9

120.4

19.0

30.2

69.1

68.8

68.5

21.9

23.0

22.2

24.2

25.2

31.6

33.2

23.1

24.9

25.3

21.2

52.7

113.4

19.5

18.1

26.0

27.7

38.1

53.8

54.2

63.0

66.7

86.2

86.3

94.2

95.6

100.2
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F33

100

wvw\

100 60 80 100

Cl

FIGURE 21. Trilinear diagram showing chemical composition of water along all flow paths, in mole percent of

cations and anions.

anhydrite dissolution. Flow paths 2, 3, and 5 (figs. 23,24, 

and 26) are significantly affected by halite dissolution. 

Flow path 6 (fig. 27) is controlled predominantly by carbonate mineral reactions. Flow path 4 (fig. 25) represents 

a relatively mature water in which dissolution of anhydrite 

dominates over dissolution of the carbonate minerals. 

Although these waters are dominated by halite dissolution, they also show evidence of earlier evolution in the 

calcite-dolomite-anhydrite reaction system. For example, 

flow paths 3 (fig. 24) and 5 (fig. 26) begin with waters 

similar in composition to those at the end of flow paths

1 (fig. 22) and 8 (fig. 29), and the initial well on flow path

2 (fig. 23) predominantly is calcium-magnesium bicarbonate. It is expected that better well control along the 

beginnings of flow paths 2, 3, and 5 would show the 

full reaction path from predominantly Ca>Mg:HC03 

to Ca>Mg:S04 >HC03 to Na>Ca>Mg:Cl>S04>HC03 

waters.

Because the trilinear diagrams only indicate changes 

in the relative proportions of cations and anions, more 

reaction information may be gained by examining actual 

changes in concentration as a function of reaction progress (sulfate concentration).

Systematic increases in both calcium and magnesium 

as sulfate increases are shown in figures 30 and 31. An 

increase in dissolved sulfate also corresponds with a 

general decrease in pH and an increase in PCOZ (^Ss- 32 

and 33).

These trends result from the dedolomitization reaction 

as discussed by Back and others (1983). Beginning with 

recharge water near saturation with calcite and dolomite, 

dissolution of anhydrite adds calcium to the ground water, 

causing precipitation of calcite. Calcite precipitation 

causes pH to decrease (fig. 32) (as a result of H+ released 

from HCOjj" in incorporation of C0§~ in calcite). The 

decrease in pH increases the proportion of carbonic acid
F34 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

100

100 100

FIGURE 22. Trilinear diagram showing chemical composition of water along flow path 1, in mole percent of cations

and anions.

(H2C03) in solution and thus increases the calculated 

PCO£ (fig- 33). The shift in the carbonate system with 

lesser pH causes the water to be undersaturated with 

respect to dolomite, which in turn leads to dolomite 

dissolution and an increase in dissolved magnesium (fig. 

31). Apparently the dissolution of anhydrite and dolomite 

exceeds the precipitation of calcite, resulting in a net increase in total dissolved calcium (fig. 30). The data in 

figures 34 and 35 show that pH and PCQ^ are verv sfr°ng 

functions of temperature. This relation is not well understood but is in part a function of the strong temperature 

dependence of the Hemy's-law constant for C0£ gas 

(Plummer and Busenberg, 1982).

Two trends are evident in total alkalinity as a function 

of dissolved sulfate (fig. 36). First, most waters show a 

decrease in alkalinity with increasing sulfate. Second, the 

waters associated primarily with flow paths 2 and 3 

display an opposite trend, having some of the maximum

alkalinity concentrations (>300 mg/L as HCOg ) with 

large concentrations of dissolved sulfate. The decrease in 

alkalinity is to be expected for evolutionary paths following the dedolomitization reaction, but large alkalinity 

concentrations are not expected with large sulfate concentrations. Based on this determination, additional reactions beyond those among calcite, dolomite, and anhydrite 

can be expected along flow paths 2 and 3.

To help understand what these reactions are, we display 

the variations in sodium, potassium, and chloride with 

sulfate in figures 37-39. Generally waters with large concentrations of sodium also have large concentrations of 

potassium and chloride, indicating evaporite sources 

(halite and sylvite) for these ions. Sodium and chloride concentrations are small and nearly independent of sulfate 

along flow paths 1, 4, 6, 7, and 8. Flow paths 2, 3, and 

5 have much greater concentrations of sodium, potassium, 

and chloride (figs. 37-39). Because the concentrations of
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F35

100

100 100

FIGURE 23. Trilinear diagram showing chemical composition of water along flow path 2, in mole percent of cations

and anions.

sodium plus potassium are similar to that of chloride for 

flow path 5, it is likely that evaporite minerals are the 

primary source for these ions along flow path 5. However, 

the concentrations of sodium plus potassium are nearly 

10 mmol/kg of water greater than chloride for flow paths 

2 and 3 (figs. 37-39). These are the same waters that contain unusually large concentrations of bicarbonate (fig. 

36). Thus, there is a tendency to form sodium bicarbonate 

waters along flow paths 2 and 3.

Several reactions could lead to the formation of sodium 

bicarbonate waters, including (1) the dissolution of 

nahcolite (NaHCOs), (2) the dissolution of carbonate 

minerals accompanying calcium and magnesium exchange 

for sodium on clay minerals, and (3) a reaction similar to 

case 2 above, but with the added possibility of incorporation of magnesium (and possibly calcium) in secondary clay 

minerals. Because nahcolite is not known or expected in 

the Madison aquifer, the possibility of its dissolution may

be discounted. However, the data need to be examined 

more closely with mass-balance models in conjunction with 

the carbon-isotopic data in order to determine the reactions responsible for the excess sodium bicarbonate along 

flow paths 2 and 3.

Whereas flow paths 1, 4, 6, 7, and 8 show little variation in sodium and chloride, there is an indication of a 

progressive increase in potassium with increasing sulfate 

(fig. 38). This possibly is due to dissolution of trace quantities of detrital potassium silicate, such as orthoclase 

(KAlSisOg), in the system.

Based on analysis of the chemical trends, the reactions 

along each flow path are summarized as follows. Flow 

paths 1, 4, 7, and 8 are examples of the dedolomitization 

reaction caused irreversibly by anhydrite dissolution at 

calcite-dolomite saturation. Flow paths 2 and 3, which 

combine the dedolomitization reaction with dissolution of 

halite and sylvite and possibly exchange (Ca2+ and
F36 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

100

100 80 20 40 60 80 100

FIGURE 24. Trilinear diagram showing chemical composition of water along flow path 3, in mole percent of cations

and anions.

Mg2 "1" for Na+ ), are the most complex chemically. Flow 

path 5 also shows evidence for the dedolomitization reaction accompanied by halite dissolution. Flow path 6 is controlled primarily by CCVcarbonate mineral reactions with 

only small quantities of anhydrite dissolution.

ANALYSIS OF SULFUR-ISOTOPE DATA

Most of the waters downgradient from recharge areas 

contain traces of hydrogen sulfide, and, at least along flow 

path 2, H2S concentrations larger than 100 mg/L have 

been determined (table 8). Therefore, in addition to the 

dissolution and precipitation reactions discussed above, 

the possibility of redox reactions (especially sulfate reduction) must also be considered.

In order to quantify redox reactions, it is necessary to 

evaluate mass-balance models, incorporating changes in 

the redox state of the waters in conjunction with the

sulfur-isotope data as discussed by Plummer and others 

(1983).

Because the only likely sources of sulfate are gypsum 

(CaS04'2H20) or anhydrite, dissolved sulfate was 

selected as the reaction-progress variable for the analysis 

of the chemical trends. Because the sulfate reduction 

process occurring in much of the ground-water system 

removes sulfate ion from solution, the quantity of 

anhydrite dissolved may be significantly larger than the 

quantity of sulfate in solution. Additionally, the quantity 

of hydrogen sulfide in solution is not always a good indication of the extent of sulfate reduction because of the 

negligible solubility of iron sulfides such as pyrite:

8SOf ' + 4FeOOH + 15CH20^4FeS2

15HC03-+9H20 (12)
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F37

100

100 100

FIGURE 25. Trilinear diagram showing chemical composition of water along flow path 4, in mole percent of cations

and anions.

If a source of iron is available, it is possible for considerable sulfate reduction to occur with only traces of 

dissolved-sulfide species remaining in solution. In cases 

of abundant hydrogen sulfide, it can be concluded that sufficient sources of dissolved iron are not available to 

remove all the dissolved sulfide:

SOf- + 2CH20-*H2S + 2HC03 (13)

Therefore, unless it is known that sulfate reduction is 

not occurring, it is not possible to calculate the anhydrite 

mass transfer directly from the change in sulfate concentration. Two methods of estimating the quantities of 

anhydrite dissolved and of pyrite precipitated have been 

considered. Each involves the sulfur-isotope compositions 

of dissolved-sulfur species (table 11). The first method is 

similar to that used by Plummer and others (1983) and

assumes that the degree of sulfate reduction is sufficiently 

small so that the process can be treated as a problem in 

isotope dilution. The second method is more rigorous and 

treats the problem as one of Rayleigh distillation. This 

method is more accurate in cases of extensive sulfate 

reduction.

SULFUR-ISOTOPE DILUTION

The required information in treating the problem of the 

anhydrite and pyrite mass transfers as a case of isotope 

dilution included the sulfur-isotopic compositions of the 

dissolving anhydrite (<J 34Sanhydrite) and of total dissolved 

sulfate and sulfide (<J 34Sr). These data were then used 

to solve simultaneously a mass-balance equation for sulfur 

and a mass-balance equation for the sulfur isotope. In 

using a mass-balance equation for the sulfur isotope, the 

assumption was made that the degree of sulfate reduc-
F38 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

100

100 40 60 

Cl

80 100

FIGURE 26. Trilinear diagram showing chemical composition of water along flow path 5, in mole percent of cations

and anions.

tion was small enough that the effects of Rayleigh distillation could be ignored. The sulfur mass-balance equation is

Sr(initial) + "anhydrite + 2«pyrite = Sr(final), (14)

where S^^tiai) and Sr(finai) are the total molalities of 

sulfur (sulfate plus total sulfide) in the initial (recharge) 

water and the final (downgradient) water, respectively. 

The mass transfers of anhydrite (aanhydrite) and pyrite 

(ffpyrite) have positive signs for dissolution and negative 

signs for precipitation. 

The sulfur-isotope mass-balance equation is

Sr(initial) <* 34S(initial) + ^anhydrite <* 34Sanhydrite 

+ 2«pyrite <* 34Spyrite = Sr(final)

(15)

where d^S^tiai), d^Sanhydrite, d34Spyrite, and d 34S(final) 

denote the isotopic compositions of sulfur in the initial

water, anhydrite, pyrite, and the final water, respectively. It is assumed that d 34Spyrite is that of the dissolved 

sulfide (ignoring the small fractionation between dissolved 

sulfide and pyrite). If both sulfate and sulfide species are 

present, then we have

d 34 dS

(final) =

d 34SSO 4 + mH2sd34SH2s,

(16)

where mgo4 and mn2S are the molalities of sulfate and 

hydrogen sulfide, respectively; and <J 34Sso4 an(* ^ 34^H2S 

are the isotopic compositions of sulfate and hydrogen 

sulfide, respectively.

Only limited data on the sulfur-isotopic composition of 

anhydrite in the Madison aquifer were available. R.G. 

Deike (U.S. Geological Survey, oral commun., 1984) 

reported two values of d^Samhydrite °f 12.8 and 13.9 per 

mil from the HTH No. 1 well and six values at the HTH
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F39

100

100 100

FIGURE 27. Trilinear diagram showing chemical composition of water along flow path 6, in mole percent of cations

and anions.

No. 2 well between 14.6 and 22.6 per mil. Most of these 

values are considerably lighter than that expected for 

open marine evaporites of Mississippian age (Claypool and 

others, 1980), indicating a terrigenous source, such as 

pyrite, for some of the sulfur. This source was subsequently oxidized, precipitated as gypsum, and diagenetically 

altered to the anhydrite, as now found in the Madison 

Limestone.

In the absence of abundant isotopic data for anhydrite, 

the isotopic composition of dissolved sulfate as a function 

of the sulfate concentration for each flow path has been 

examined. Assuming a nearly constant (average) isotopic 

composition of the anhydrite dissolving along a flow path, 

no change in <J 34Sso4 would be observed if no sulfate 

reduction were occurring. Alternatively, if there was 

sulfate reduction, a trend to heavier <J 34Sso4 with increased concentration of sulfate (reaction progress) would 

be determined as a result of the large sulfur isotope fractionation involved in the bacterially mediated reduction 

of sulfate. In either case, the <J 34Sso4 *ren(^ should extrapolate back to the isotopic composition of anhydrite 

when no sulfate is present in solution. This method was 

used to approximate d 34Sanhydrite f°r eacn flow path. The 

<* 34Sanhydrite derived by this method is reported to the 

nearest 0.1 per mil.

Although all eight flow paths show significant trends 

to heavier <5 34Sso4 with increasing concentrations of 

dissolved sulfate (fig. 40A-#), it is clear that there are 

distinct differences between flow paths: For example, all 

<5 34Sso4 values along flow path 1 (fig. 40A) are two to 

three times heavier than those along flow path 5 (fig. 

40#). Because the data for several of the flow paths 

(notably 3, 4, 5, 6, and 8) indicate that <J 34Sso4 increases 

linearly with sulfate concentration, the assumption has 

been made that all trends of figure 40 may be approximated with a linear relation.
F40 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

100

80

100 20 40 60 

Cl

80 100

FIGURE 28. Trilinear diagram showing chemical composition of water along flow path 7, in mole percent of cations

and anions.

Several flow paths (such as 1, 2, and 7) contain some 

values of <J 34Sso4 that deviate significantly from the 

selected linear trend (fig. 40). These may be due to 

regional variation in the anhydrite isotopic composition, 

or result from differences in the extent of sulfate reduction along the flow paths, or indicate that these wells 

do not lie entirely on the given flow path. In the preliminary mass-balance modeling discussed in a later 

section of this report, it was assumed that deviations from 

the selected linear trends of figure 40 are due to variations in the extent of sulfate reduction along the flow 

path.

The estimated initial isotopic compositions of anhydrite 

for flow paths 1 to 8, as indicated by the linear trends 

in figure 40, are summarized in table 12. The derived 

<* 34Sanhydrite values appear to be mappable, with lightest 

values (7.9 to 10.0 per mil) along flow paths 5, 6, and 7 

in northeastern Wyoming and the adjacent southwestern 

corner of South Dakota. Moving northeast from these

light values into South Dakota, and north and northwest 

into Montana, a progressive increase to heavier estimated 

values of the initial isotopic composition of dissolving 

anhydrite is found, with a maximum value of 17.1 per mil 

reached in north-central Montana. This pattern in estimated anhydrite isotopic composition may reflect the 

depositional environment of the Madison Limestone and 

surrounding exposed topography during the time of its 

emplacement (Sando, 1976b), indicating more abundant 

terrestrial sources of sediment, enriched in sulfide, in 

northeastern Wyoming and the area surrounding the 

Black Hills than in the rest of the area.

Presumably, the environment of anhydrite deposition 

in the Madison Limestone was affected by both marine 

and terrestrial sources of sulfate, with a greater abundance of terrigenous sulfate being introduced in northeastern Wyoming.

Returning to equations 14 and 15, we can use the data 

for <J 34Sanhydrite (table 12) with the concentrations and
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F41

100

100 80 60 40 20 40 60 80 100

FIGURE 29. Trilinear diagram showing chemical composition of water along flow path 8, in mole percent of cations

and anions.

isotopic compositions of dissolved sulfate and total sulfide 

to estimate the anhydrite and pyrite mass transfers at the 

12 wells in the Madison aquifer (table 13) for which sufficient analytical and isotopic data are available. 

From equations (14) and (15), we have

"anhydrite -

34 (d ^anhydrite -

pyrite

(Sy d 34S)final - (ST

(17)

(18)

(19)

AS T = - S ^(initi

The average concentration and isotopic composition of 

sulfate in the recharge water associated with each well 

(tables 10 and 11) were used in calculating the pyrite and 

anhydrite mass transfers in table 13. As discussed by 

Plummer and others (1983), although the computed mass 

transfer is not very sensitive to uncertainties of several 

parts per thousand in the isotopic composition of dissolved 

sulfide, it does depend significantly on the selected value 

for the isotopic composition of dissolving anhydrite. For 

example, if we assign an uncertainty of ± 1 per mil to the 

isotopic composition of dissolving anhydrite along flow 

path 2, the anhydrite and pyrite mass transfers along the 

flow path to the Sumatra well are 14.22 ±0.83 and 

-0.22 ± 0.41 mmol/kg of water, respectively. Variations 

of several parts per thousand in (J 34Sanhydrite °^ ^dividual 

anhydrite samples are likely along each flow path, but 

these variations are moderated somewhat because the 

values of (J 34Sanhydrite derived in figure 40 can be interpreted as average values for the flow path.
F42 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

SULFUR ISOTOPES AS A RAYLEIGH-DISTILLATION PROBLEM

The results in table 13 show that for at least several 

wells, such as Dupree, relatively substantial sulfate reduction may be occurring. Therefore, the estimation of the 

anhydrite and pyrite mass transfers via the isotopedilution equation may not be justified. A more precise 

method of estimating the anhydrite and pyrite mass 

transfers is to account for the fact that the sulfur-isotopic 

compositions of dissolved sulfate, dissolved sulfide, and 

pyrite vary with the extent of sulfate reduction. This is

accounted for by treating the sulfur-isotope data as a problem in Rayleigh distillation.

In this particular case sulfate reduction involves the 

dissolution of anhydrite with an assumed constant isotopic composition, the reduction of dissolved-sulfate to 

dissolved-sulfide species with a fractionation that averages about 30 per mil (table 13), and the precipitation of 

a proportion of this dissolved sulfide as pyrite, involving 

only a small (but unknown) fractionation. By assuming 

that the sulfur-isotopic composition of pyrite is equal to 

that of the dissolved-sulfide species, the sulfate-reduction

15.00

13.75

12.50

£ 11.25 tI

LL.

o

I 10.00

CC 

O

£ 8.75

a.

V)

LU 

O

I 7.50

Q 6.25

5.00

< 3.75

2.50

1.25

0.00

- Flow path 1

- Flow path 2

- Flow path 3

- Flow path 4

- Flow path 5

- Flow path 6

- Flow path 7

- Flow path 8

(3D

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 30. Trend in dissolved-calcium concentration as a function of dissolved-sulfate concentration along flow paths 1 to 8.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F43

process can be treated as one input (anhydrite) and one 

fractionating output (dissolved-sulfide species and pyrite) 

of the Rayleigh-distillation problem. The analytical solution to the Rayleigh-distillation problem in terms of the 

sulfur-isotopic composition of dissolved sulfate is adopted 

from Wigley and others (1978, 1979) and is

(20)

ft <* 34S°so4 -<* 34Sso4 + i- I o +C* ^S04~ _

r /L wso4J r

Equation 20 describes the isotopic composition of dissolved sulfate, <S 34Sso4 » as a function of reaction progress. The notation of equation 20 is defined as follows:

d 34Sjgo4 is the isotopic composition of sulfur in dissolved sulfate of the starting water;

d 34S>so4 *s t*16 isotopic composition of sulfur in anhydrite; 

£ps is the additive fractionation factor - 34A =

P is \dlld0\, the absolute value of the ratio of 

moles of incoming sulfur to outgoing sulfur 

(as dissolved-sulfide species and pyrite);

mso4 is the molality of sulfate in the starting 

water;

mso4 is the molality of sulfate in the final water;

an(* tn 

B is the parameter defined as (1 + -^ ).

1,000 T

5.50

5.00

4.50

CJ 4.00 

O

3.50

| 3.00

z

| 2.50 

cc

K

LU 

O

§ 2.00 

o

^ 

2 

w

Z 1.50

C3

<

1.00

o

01 o

*

r

O - Flow path 1

A - Flow path 2

V - Flow path 3

O - Flow path 4

+ - Flow path 5

+ - Flow path 6

X - Flow path 7

if - Flow path 8

o

0

O

**

O

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

20.00 22.00

FIGURE 31. Trend in dissolved-magnesium concentration as a function of dissolved-sulfate concentration along flow paths 1 to 8.
F44 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

Equation 20 has been used to solve iteratively for T (the 

sulfur input to output ratio). Here we take £ps as the 

measured value, e^ = - ^A = d34SH2S - <* 34Sso4 » which is 

summarized in table 14 along with the rest of the 

measured parameters required by equation 20. The values 

of f calculated using equation 20 are also presented in 

table 14.

The calculated value of f is then combined with the 

mass-balance equation for sulfur and analytical data for 

dissolved sulfide to determine the anhydrite and pyrite 

mass transfers. The following two equations were used:

r = aanhydrite

and

 ^anhydrite ~ ^apynte 

(21)

(22)

In equations 21 and 22, the mass transfers of anhydrite 

and pyrite, o-anhydrite and a-pyrite» are both treated as 

positive numbers because T is defined as the absolute

value of the ratio of moles. Solving equations 21 and 22 

for the anhydrite mass transfer, we obtain

^anhydrite = r

(H2 Sr-ASr)

d-o '

^anhydrite ffpyrite -

(23)

(24)

where I^Sy is the total concentration of dissolved-sulfide 

species, and ASy is the change in the total concentration 

of dissolved-sulfur species (sulfate plus sulfide species). 

The anhydrite and pyrite mass transfers are in millimoles 

per kilogram of water, the unit of concentration.

Comparison of the calculated anhydrite and pyrite mass 

transfers of table 14 with those calculated by the isotopedilution method (table 13) shows only minor differences 

between the results from the two methods (table 15). For 

example, the anhydrite mass transfers differ by less than 

4 percent between the two methods, which is probably 

better than the overall accuracy of the data. Therefore,

8.00

7.75

7.50

7.25

a 7.00

6.75

6.50

6.25

6.00

0

0

**

0

 KD

*"%

* * 

ft * ^

0 - Flow path 1 

A - Flow path 2 

V - Flow path 3 

^ - Flow path 4 

+ - Flow path 5 

* - Flow path 6 

X - Flow path 7 

W - Flow path 8

0

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 32. Variation in pH as a function of dissolved-sulfate concentration along flow paths 1 to 8.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F45

we conclude that negligible uncertainty is introduced in 

the anhydrite and pyrite mass transfers by treating 

sulfate reduction as a simple isotope-dilution problem. 

The results in table 13 show that the quantity of anhydrite dissolved exceeds the sulfate concentration by values 

that range between 0.81 mmol/kg of water (at Sarpy 

Mine, well No. 19 in Montana) and 4.8 mmol/kg of water 

(at Dupree, well No. 20 in South Dakota). Wells in central Montana, which contain relatively large quantities of 

dissolved-sulfide species, indicate only minor pyrite 

precipitation, whereas wells in South Dakota, which show 

relatively small quantities of dissolved-sulfide species, indicate relatively substantial pyrite precipitation. Based 

on this determination and the discussion in the section 

"Depositional Environment," it is likely that sources of 

dissolved iron, such as detrital FeOOH, are more abundant in the Madison aquifer in South Dakota than in central Montana. Detrital FeOOH would be expected to be 

more prevalent near the source area of the sediments. 

This further supports the earlier hypothesis regarding the 

source of terrigenous sulfur in the vicinity of South 

Dakota and northeastern Wyoming.

Although the sulfur-isotopic composition of dissolved 

sulfate is known for nearly all the wells or springs sam0.00

-0.25

-0.50

-0.75

-1.00

o 

x

g -1.25

-1.75

-2.00

-2.25

-2.50

-2.75

O - Flow path 1

A - Flow path 2

V - Flow path 3

^ - Flow path 4

 f - Flow path 5

+ - Flow path 6

X - Flow path 7

 jfr - Flow path 8

* * *

* *

CD

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

20.00 22.00

FIGURE 33. Variation in log partial pressure of carbon dioxide (atmospheres) as a function of dissolved-sulfate concentration

along flow paths 1 to 8.
F46 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

pled in the Madison aquifer, the sulfur-isotopic composition of dissolved sulfide was obtained for only 12 wells 

(table 11). As a result, it was necessary to use an estimation procedure to calculate the anhydrite and pyrite mass 

transfers elsewhere in the Madison aquifer. This procedure was based on the parameter 34A (Pearson and 

Rightmire, 1980), where

(25)

An approximately linear correlation with temperature 

among 34A data from the Madison aquifer (table 13) and 

the Edwards aquifer in the Cretaceous Edwards Limestone of Texas (F.J. Pearson, Jr. and P.L. Rettman, U.S. 

Geological Survey, unpublished data) is shown in figure 

41. As shown by Rye and others (1981) for the limestones 

of the Floridan aquifer of Florida, 34A approaches that expected for isotopic equilibrium [based on experimental 

data of Ohmoto and Rye (1979) at 200 to 350 °C].

Values of 34A in the Edwards and Madison aquifers 

probably result from kinetic fractionation during biologically mediated sulfate reduction. The kinetic fractionation in the Edwards and Madison aquifers appears to be 

a linear function of temperature (fig. 41) and is approximated by

34A = 54-0.40 i, (26)

where t is the temperature, in degrees Celsius.

Based on the scatter of values in figure 41, values of 

^A estimated from equation 26 have uncertainties of ± 10 

per mil. In the absence of additional data on the isotopic 

composition of dissolved sulfide in the Madison aquifer, 

the measured temperature, the isotopic composition of 

dissolved sulfate, and the correlation of equation 26 have 

been used to estimate <J 34SH2s» which equals <J 34Spyrite , 

in the remaining wells sampled in the Madison aquifer. 

These estimates have then been used in the mass-balance 

models to calculate the anhydrite and pyrite mass 

transfers elsewhere in the Madison aquifer.

As a test of the magnitude of potential uncertainties 

due to the use of equation 26 (in lieu of actual data on the

7.80

7.60

7.40

7.20

7.00

6.80

6.60

6.40

6.20

if o

* ** * «,

0

0

X *

**

O - Flow path 1 

A - Flow path 2 

V - Flow path 3 

$ - Flow path 4 

+ - Flow path 5 

+ - Flow path 6 

X - Flow path 7 

K- Flow path 8

A

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

TEMPERATURE, IN DEGREES CELSIUS

FIGURE 34. Variation in pH as a function of temperature along flow paths 1 to 8.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F47

sulfur-isotopic composition of dissolved sulfide), the 

isotope-dilution calculation of the anhydrite and pyrite 

mass transfers shown in table 13 was repeated using 

values of d 34SH2S (which equals d 34Spyrite) derived from 

equation 26. These mass-transfer values are compared 

with those derived from the measured d 34SH2S data via 

the isotope-dilution and Rayleigh-distillation methods in 

table 15. The largest deviations of values derived from 

equation 26, from the measured values (±10 per mil 

uncertainty in d 34SH2s) lead to a 9 percent underestimation of the anhydrite mass transfer at the Dupree well 

and an 8 percent overestimation of the anhydrite mass 

transfer at the Moore well. All other anhydrite masstransfer values calculated using equation 26 agree 

significantly more closely with the measured values (table 

15). Uncertainties of less than 1.0 mmol/kg of water in 

the pyrite mass-transfer result from the approximation 

of equation 26.

MASS-BALANCE REACTION MODELS

EQUATIONS

Mass-balance reaction models are used to help define 

the net masses of minerals dissolved or precipitated along 

each flow path between the recharge area and each downgradient well. This type of modeling has been considered 

in detail by Plummer and others (1983). As shown by these 

authors, the chemical evolution of the water along the flow 

path is constrained by relations of conservation of mass 

and electrons that are represented by the equations

p-i

and

p

I = &RS.

(27)

(28)

-0.50

-0.75

-1.00

9 -1.25

9 -1.50 

X

o

Q

O

£ -1.75

-2.00

r= -2.25

-2.50

-2.75

O - Flow path 1

A - Flow path 2

V - Flow path 3

O - Flow path 4

+ - Flow path 5

+ - Flow path 6

X - Flow path 7

Tfr - Flow path 8

o

o

o

o o

**

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

TEMPERATURE, IN DEGREES CELSIUS

FIGURE 35. Variation in log partial pressure of carbon dioxide (atmospheres) as a function of temperature along flow

paths 1 to 8.
F48 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

Equation 27 requires that the change in total moles of 

the kth element in solution along the flow path, Am^fc 

(final water minus initial water), be equal to the sum of 

the net moles of that element entering or leaving solution as a result of dissolution, precipitation, biological 

degradation, gas transfer, and so forth of P phases along 

the flow path. Here the mass-transfer coefficient ap is the

number of moles of the pth mineral entering (positive) 

or leaving (negative) the solution, and bpi ]c is the stoichiometric coefficient of the kth element in the pth mineral. 

There are j equations of the form of equation 27, one for 

each of the j elements (excluding hydrogen and oxygen) 

required to define the composition of the P phases 

(minerals) considered in the model. Thus a reaction model

400.00

375.00

350.00

325.00

300.00

275.00

± 250.00

225.00

200.00

175.00

150.00 -

125.00 -

100.00 -

75.00

O - Flow path 1

A - Flow path 2

V - Flow path 3

$ - Flow path 4

+ - Flow path 5

+ - Flow path 6

X - Flow path 7

X - Flow path 8

Q* O

O 

O

if if o

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 36. Variation in total alkalinity as a function of dissolved-sulfate concentration along flow paths 1 to 8.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F49

is a particular choice of phases that react to give the actual 

final-water composition.

If the reaction also includes changes in the oxidation 

state, equation 28 provides an additional constraint on the 

reaction. Equation 28 follows from the fact that hydrated 

electrons do not exist in aqueous solution. If electron 

transfer does take place (that is, there is a redox reaction), the electrons transferred are conserved among the 

dissolved species and the masses (ap) of the P phases in 

the model. Equation 28 relates the sum of operational 

valences of the constituents in the pth plausible phase 

(up) and the stoichiometric coefficient of the pth plausible phase in the mass-balance calculation to A/25 (the 

difference in redox state between the initial and final 

waters).

In equation 28 the change in the redox state of the solution (A/25) is calculated from the analytical data, as is 

Amy, and includes the (operational) valence v$ of the ith 

.species in solution. The operational valence is defined by

Plummer and others (1983) and in some cases differs by 

convention from the recognized oxidation state. Using 

those conventions, the term A/25 is defined as

A/25 = J. Vimi (final)- J. (initial), (29)

where w^ is the molality of the ith species in solution. 

(The initial condition refers to the recharge area, and the 

final condition indicates the downgradient well). The 

change in the redox state is then set equal to the number 

of electrons transferred among the P phases.

The data required to solve a mass-balance reaction 

model are (1) the set of phases thought to be reacting 

along the flow path, and (2) the concentrations of the 

elements in the initial and final waters that correspond 

to the composition of the chosen phases. The analytical

90.00

80.00

70.00

60.00

50.00

40.00

30.00

20.00

10.00

0.00

O - Flow path 1

A - Flow path 2

V - Flow path 3

O - Flow path 4

+ - Flow path 5

+ - Flow path 6

2? - Flow path 7

1C - Flow path 8

O

.* O

&, X. * +.

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER 

FIGURE 37. Variation in dissolved sodium as a function of dissolved-sulfate concentration along flow paths 1 to 8.
F50 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

data and redox states used in mass-balance modeling of 

the Madison aquifer are summarized in table 10.

The most obvious phases to include in the mass-balance 

modeling, based on the analysis of the chemical trends, 

the calculated values of SI, and the mineralogy, are summarized in table 16. Organic matter (CH^O) is included 

as the most likely electron donor for bacterially mediated 

sulfate reduction. The formula CH^O is used only to 

denote carbon of valence zero. Sulfate reduction is known 

to occur because of the presence of dissolved-sulfide 

species in many wells (table 8) and of the occurrence of 

sulfate-reducing bacteria as facultative thermophiles in 

one of the test wells drilled during the Madison Limestone 

Project (Olson and others, 1982; Blankennagel and others, 

1979). As discussed in the previous section, if sufficient 

sources of dissolved iron are available for precipitation 

of pyrite, sulfate reduction need not be associated with 

large concentrations of dissolved-sulfide species. Therefore, ferric hydroxide and pyrite have been included in 

the mass-balance models. Dolomite, calcite, and anhydrite

are included to quantify the dedolomitization reaction. 

Dissolution of halite was indicated by the earlier analysis 

of trends, and excess sodium is attributed to sodiumcalcium ion exchange according to the reaction

Ca2+ = CaX + 2Na (30)

Carbon dioxide gas is included to account for the possibility of the exchange of CC>2 between the waters from 

the Madison aquifer and other reservoirs. Near-zero 

values for the C(>2 gas mass transfer indicate that 

ground-water conditions approximate a closed system. 

The deep ground-water system, with the exceptions of a 

small area along flow path 2 in Montana (the area of extensive faulting associated with the Little Rocky and Big 

Snowy Mountains) and the extreme eastern edge of the 

project area at the subcrop of the Madison aquifer, is 

expected to be virtually closed. Nonzero values of C(>2 

gas mass transfer in other areas probably indicate errors 

in the analytical data or in the choice of mineral phases.

4.50

a 4.00 

I

3.50

3.00

=! 2.50

2.00

8 1-50

w

£ 1.00

0.50

0.00

<D - Flow path 1

A - Flow path 2

V - Flow path 3

$ - Flow path 4

+ - Flow path 5

+ - Flow path 6

X - Flow path 7

 £ - Flow path 8

 ,* ** ** 0°

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 38. Variation in dissolved potassium as a function of dissolved-sulfate concentration along flow paths 1 to 8.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F51

As discussed in the previous section, the anhydrite and 

pyrite mass transfers probably can be estimated accurately by the sulfur mass-balance equation (eq 27 for sulfur) 

and an isotope-dilution equation for A 34Sy. In the form 

of equations 27 and 28, the sulfur-isotope balance equation is

p=l

where

(31)

s

6p> s is the stoichiometric coefficient of sulfur of the 

pth phase;

d 34Sp is the sulfur-isotopic composition, in parts per 

thousand, of the pth phase;

ST and d 34Sr denote the total molality of sulfur in 

solution (sulfate plus sulfide species) and the average isotopic composition, in parts per thousand, of 

total dissolved sulfur, respectively; and

ap is the mass transfer of the pth phase.

Values of d 34S for sulfate and sulfide are required in 

the solution of the mass transfers of anhydrite and pyrite. 

Measured values have been used where available (12 wells, 

table 11). For the other wells, A 34Sr was estimated using 

equation 26 in conjunction with the measured temperature 

and d^S of the dissolved sulfate. Apparent values of d^S 

for anhydrite are summarized in table 12 for each flow 

path.

For the 10 plausible phases listed in table 16, 10 equations of the form of equations 27, 28, and 31 are needed 

to calculate the mass-transfer coefficients (ap). These 

equations are

1. Mass balance (eq 27):

"calcite + 2adolomite + "CH20 + "C02 (gas) = 

"anhydrite + 2"pyrite = 

"calcite + "dolomite + "anhydrite - "exchange - 

"dolomite =

, c> (32) 

,S, (33) 

* (34) 

, Mg» (35)

80.00

70.00

< 60.00 

oc

50.oo

40.00

o

jZ 30.00

20.00

10.00

0.00

0.00

O - Flow path 1 

A - Flow path 2 

V - Flow path 3

* - Flow path 4 

+ - Flow path 5

* - Flow path 6 

X - Flow path 7 

X- Flow path 8

 t ~e L

O

*.** «P

2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 39. Variation in dissolved chloride as a function of dissolved-sulfate concentration along flow paths 1 to 8.
F52 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 11. Iso topic data

[Letter in front of flow-path number indicates whether the spring or well was used as recharge (R) or on the flow path (F); TU, tritium units]

Spring 

or well

Lewistown Big Spring

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

Bozeman Fish Hatchery

Big Timber Fish Hatchery

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Mock Ranch

Denius No. 1

Colstrip

Sarpy Mine

Gas City

Bluewater Spring

Moore

Mysse Flowing Well

Storey Fish Hatchery

Mobil

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Hole in the Wall

Barber Ranch Spring

Conoco No. 175

MKM

Shidler

Conoco No. 44

Mallo Camp

Rhoads Fork

Seeley

Coronado No. 2

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

Jones Spring

Kaiser

Cleghorn Spring

Lien

McNenney

Ellsworth AFB

Fuhs

Kosken

Philip

Spring 

or well 

number

10

8

9

12

13

18

4

7

6

17

15

16

1

2

21

19

26

11

22

20

6

7

14

23

24

7

8

5

12

11

10

9

8

24

4

19

16

21

17

15

13

20

22

18

10

9

12

11

16

17

2

18

6

1

19

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Wyo.

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Plow 

path

Rl

Fl

Fl

Fl

Fl

Fl

R2

R2

F2

F2

F2

F2

R3,R4

R3,R4

F3

F3

F3

F3

F3

F3

R4

R4

F4

F4

F4

F4

F4

R5

R5

F5

F5

F5

F5

R6

R6.R7

F6

F6

F6

F6

F6

F6

F7

F7

F7

F7

F7

R8

R8

R8

F8

F8

F8

F8

F8

F8

Tritium 

(TU)

0.8

.0

5.3

21.9

31.8

..

115.0

124.0

45.5

--

..

--

35.4

19.8

--

..

--

.5

--

--

107.0

56.7

5.4

--

--

.0

.0

.5

53.0

.8

.2

1.2

.0

--

62.2

.8

.0

.1

.5

--

1.5

2.3

.0

.3

--

..

276.0

27.7

182.0

16.6

11.4

--

65.6

.1

--

8l»0 

(per mil)

-18.25

-18.25

-18.35

-18.25

-17.80

-18.40

-18.90

-18.60

-18.50

-16.90

-16.80

..

-18.35

-18.50

-19.05

-19.80

--

-19.35

-18.00

-18.25

-18.35

-17.45

--

-18.10

-18.25

-19.66

-18.13

-18.30

-18.20

-19.25

-19.70

-20.15

-20.10

--

-17.22

..

-17.60

-17.66

-18.15

-18.18

-17.85

-17.40

-17.60

-17.95

-16.71

-15.48

-14.61

-12.13

-13.23

-14.19

-17.43

-14.13

-16.13

-16.75

-17.55

SD 

(per mil)

-139.20

-140.45

-139.65

-138.75

-134.75

-138.35

-143.70

-142.65

-140.70

-136.15

-136.40

--

-138.00

-137.85

-146.05

-151.40

--

-146.90

-141.75

-141.50

-138.95

-132.00

--

-137.60

-137.90

..

--

-139.10

-137.65

-145.50

-152.05

-153.65

-154.05

--

-125.00

-133.50

-133.30

-130.00

-135.00

-133.00

-139.95

-130.65

-131.80

-130.65

-121.00

-118.00

-110.00

--

-103.00

-

-127.00

-107.00

--

-126.60

-125.00

8^8 

(sulfate) 

(per mil)

14.93

17.48

17.78

21.33

23.86

21.95

3.34

-4.46

17.95

17.01

17.75

18.69

--

9.73

14.67

13.79

--

--

17.06

16.30

..

11.51

11.60

11.75

11.71

15.20

14.67

--

7.75

8.23

8.52

8.64

8.41

--

6.22

9.94

--

9.84

10.44

12.16

11.68

10.09

10.49

11.68

11.62

12.50

6.67

9.00

5.75

-3.20

11.36

--

4.50

12.93

14.52

8% 

(sulfide) 

(per mil)

..

--

--

--

--

..

--

--

--

-2.60

-.22

--

-.

--

-9.20

--

--

-7.73

-22.09

..

..

--

..

--

..

-17.37

--

-

--

..

--

--

--

--

..

--

--

--

--

..

--

--

--

--

..

--

--

--

--

..

--

--

--

--

813C 

(per mil)

-4.97

-5.32

-5.18

-7.46

-7.04

-3.22

-8.98

-14.11

-7.57

-3.61

-1.68

..

-7.10

-6.88

-2.67

-2.33

--

-9.51

-2.40

-2.34

-7.82

-9.75

-6.63

-6.17

-6.02

-4.60

-2.61

-10.77

-11.74

-6.23

-4.66

--

-5.14

-8.00

-11.00

-7.82

-7.51

-10.40

-10.00

-8.20

-6.80

-7.26

-6.60

-4.24

-9.70

-9.10

-11.60

-8.36

-9.60

-8.03

-11.50

-9.10

-9.96

-6.20

-7.20

Carbon-14 

(percent 

modern)

35.8

25.4

29.3

--

28.0

4.2

84.1

103.9

52.5

-

1.0

..

57.7

8.4

--

3.3

--

--

1.6

.8

88.8

62.2

12.7

10.0

9.5

4.6

2.8

87.4

83.7

13.9

2.6

6.2

1.8

92.9

92.9

61.4

36.0

46.2

54.7

14.7

59.0

44.5

31.2

4.8

28.5

19.4

100.0

81.1

91.6

68.5

79.6

5.8

53.8

7.8

2.8
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER 

TABLE 11. Isotopic data Continued

F53

Spring 

or well

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Gore Hill

Great Falls High School

Bough Ranch

HTH No. 3

Buckhorn Exeter

Provo

Spearfish

Black Hills Cemetery

Streeter Ranch

Bean

Denius No. 2

Denius No. 3

Martens Madison

Ranch A

Spring 

or well 

number

24

25

22

26

21

23

20

2

3

5

14

27

3

5

33

14

27

3

4

23

25

State

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Mont.

Mont.

Mont.

Mont.

Mont.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Flow 

path

F8

F8

F8

F8

F8

F8

F8

--

.-

--

..

..

..

--

..

--

--

--

--

--

Tritium 

(TU)

1.3

.0

..

--

.6

10.2

1.0

16.1

.0

.1

..

..

87.5

--

10.6

..

-.

--

--

41.6

8«0 

(per mil)

-17.62

-17.35

-17.60

-17.87

-17.74

-18.25

-19.00

-19.35

-19.00

-18.15

..

..

-17.09

-17.06

-15.04

-12.05

..

..

--

-17.45

-17.49

8D

(per mil)

-128.00

-131.45

-132.40

..

-

-137.10

-143.65

-148.75

-146.20

-141.25

..

..

-131.00

-126.00

-

-88.40

..

.-

--

-130.95

--

834S 

(sulfate) 

(per mil)

14.96

14.56

12.96

15.58

15.22

15.99

16.31

22.91

7.71

19.99

__

17.38

10.89

3.19

-5.10

..

11.50

--

--

--

11.20

834s 

(sulfide) 

(per mil)

-19.12

--

..

-12.22

-7.21

-4.77

--

--

-26.21

__

-4.23

--

..

--

..

.-

--

--

--

--

8>3C 

(per mil)

-6.20

-5.50

-6.63

-4.72

-3.50

-2.40

-2.05

-3.18

-7.20

-3.80

-3.68

..

-9.40

-10.20

-9.88

-6.65

-6.73

-

~

-9.56

-10.81

Carbon- 14 

(percent 

modern)

2.4

3.2

4.6

4.0

2.4

2.2

2.8

3.6

24.2

1.3

..

--

7.3

74.3

69.1

52.2

3.3

-

--

61.9

74.4

"halite + 2aexchange 

"sylvite

"halite + "sylvite 

"FeOOH + "pyrite 

2. Conservation of electrons (eq 28):

, Na> (36) 

',K, (37) 

,Cb (38) 

= Amy, Fe- (39)

4«calcite + + $<* anhydrite + 4«C02 (gas)

(40)

Note that terms for CH20 and pyrite do not appear in 

equation 40 because their operational valence is zero.

3. Sulfur-isotope balance (eq 31):

"anhydrite <* ^anhydrite + 2apyrite ^Spyrite = (41)

Equations 32 to 41 provide the 10 independent equations required to solve for the unknown mass-transfer 

coefficients (ap). The following are algebraic solutions to 

equations 32 to 41, which define the mass transfers of 

anhydrite, pyrite, sylvite, halite, Ca2+/Na+ ion exchange, 

dolomite, goethite, calcite, CC>2 gas, and CH20.

"anhydrite ~

0 ^anhydrite

(42)

"pyrite = (Am^s ~ "anhydrite )/2, (43) 

"sylvite = Awr,K, (44) 

"halite = Am^ ci - "sylvite , (45) 

"exchange = (Aw^ Na ~ "halite )/2, (46) 

"dolomite =Amr( Mg, (47) 

"FeOOH = Amy, Fe ~ "pyrite, (48) 

"calcite = ^^T, Ca + "exchange - "anhydrite - "dolomite» (49) 

"CH20 = Amrt c - «C02 (gas) ~ 2a<iolomite ~ "calcite, (50)

(51) 

ite ~ 8gdolomite ~ 6»anhydrite)

"C02 (gas) = 

(ARSTo facilitate the calculations, equations 32 to 41 were 

solved using the computer program BALANCE (Parkhurst and others, 1982).
F54 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

26.00

20.00

18.00

16.00

Flow path 1

0.00 4.00 8.00 12.00 16.00 20.00 24.00

19.00

16.00

Flow path 2

0.00 4.00 8.00 12.00 16.00 20.00 24.00

20.00

Flow path 3

18.00

16.00

12.00

" 10.00

DC 0.00 4.00 8.00 12.00 16.00 20.00 24.00

Flow path 5

20.00

18.00

16.00

12.00

Flow path 4

0.00 4.00 8.00 12.00 16.00 20.00 24.00

13.00

Flow path 6

0.00 2.00 4.00 6.00 8.00 10.00 12.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00

13.00

Flow path 7

8.00

Flow path 8

6.00 4.00 8.00 12.00 16.00 20.00 24.00 0 00 4.00 8.00 12.00 16 00 20.00 24.00

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F55

TABLE 12. Initial estimate ofsuljur-isotopic composition 

of dissolving anhydrite along flow paths

Flow 

path

1

2

3

4

5

6

7

8

State

Mont.

Mont.

Mont.

Wyo.

Wyo.

Wyo.

Wyo.

S. Dak.

S34o 

anhydrite 

(per mil)

17.1

14.7

11.8

10.3

7.9

9.5

10.0

11.0

PRELIMINARY MASS-TRANSFER RESULTS

Preliminary mass-transfer coefficients, in millimoles per 

kilogram of water, calculated for 42 waters along the 8 

selected flow paths are summarized in table 17. Waters 

not included either do not fall on any of the eight flow 

paths, do not have a d 34S value for dissolved sulfate, or 

were judged to be from multiple sources.

The computed mass transfer in table 17 represents the 

net millimoles of each phase entering (positive sign) or 

leaving (negative sign) a kilogram of water by dissolution 

or precipitation as that kilogram of water flows from the 

recharge area to the downgradient well. For example, according to the preliminary results in table 17, a kilogram 

of water moving along flow path 8 from the recharge area 

in the Black Hills eastward into South Dakota, some 106 

miles to the Hamilton well, evolves according to the 

following net reaction:

recharge water + 2.90 CaMg(C03)2 + 15.90 CaS04 

+ 4.68 CH20 + 1.28 FeOOH + 0.14 Na2* 

+ 0.97 NaCl + 0.33 KC1 = 9.49 CaC03 (52) 

+ 1.22 FeS2 + 0.14 CaZ+ 2.45 C02 

+ Hamilton well water,

where the mass-transfer coefficients are in millimoles per 

kilogram of water. Similar net reactions can be written 

for each downgradient well using the preliminary results 

from table 17.

FIGURE 40. Trends in (J 34SSo4 with dissolved-sulfate concentration 

along flow paths 1 to 8.

INDICATIONS OF UNCERTAINTIES IN THE 

PRELIMINARY MODELING RESULTS

The validity of the preliminary mass-transfer results 

given in table 17 depends on many factors, including the 

choice of plausible phases, the accuracy of the correlation 

of 34A with temperature described by equation 26, the 

assumed constant sulfur-isotopic composition of anhydrite 

along each flow path (table 12), and the accuracy of the 

analytical data. One particularly uncertain aspect of the 

preliminary modeling results in table 17 is the required 

outgassing of significant quantities of C02 gas from many 

of the wells. About one-half of the wells modeled indicate 

C02 outgassing of 1.0 to 4.0 mmol/kg of water.

One way of checking the modeled mass transfer is to 

use the preliminary mass-transfer results in table 17 and 

the Rayleigh-distillation equations of Wigley and others 

(1978, 1979) to predict the measured values of d 13C 

(d 13Cmeas) of each downgradient water. A detailed example of how these calculations are made is given by Plummer and others (1983). If d 13Cmeas cannot be predicted 

within reasonable uncertainties of d 13C values of the carbon sources (CH20 and dolomite), then this is a strong 

indication of modeling error. In making such a calculation, we have used the average d 13Cmeas value for each 

recharge area of each flow path (table 11) as the starting 

point.

In the carbon-isotope modeling, computed positive 

values of C02 gas mass transfer (table 17) were interpreted as dissolution of soil gas C02 associated with the 

recharge process. The values of d 13C (table 18) were 

calculated from the recharge waters and assumed to be 

representative of the average soil gas. The 14C content 

of the downgradient well also was calculated, assuming 

no radioactive decay. The prenuclear-detonation 14C content of the recharge waters, shown in table 18, was computed from mass-balance calculations assuming a 100 

percent modern source for soil gas C02 and 0 percent 

modern sources of dissolving calcite and dolomite. Along 

each flow path, the 13C content of reacting dolomite 

(<* 13Cdolomite) and of CH20 (d 13CcH2o) was taken as 2.0 

and -20 per mil, respectively. The ^C content of calcite 

and of CH20 was assumed to be 0 percent modern. The 

d 13C and 14C content of the downgradient well, shown 

in table 19, was computed using the Rayleigh-distillation 

equations of Wigley and others (1978, 1979).

Cases involving possible C02 outgassing were treated 

with the one input (dolomite, CH20) and two fractionating output (C02 gas, calcite) equations. The case of apparent ingassing of C02 was solved with the one input 

(dolomite, CH20, C02 gas) and one fractionating output 

(calcite) equations. Carbon-isotope fractionation factors 

were those used by Plummer and others (1983), taking 

into account variation in temperature along the flow 

path.
F56 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 13. Calculation of anhydrite and pyrite mass transfers using the isotope-dilution method

[°C, degrees Celsius; mmol/kg, millimoles per kilogram of water]

Sulfate Total sulfide Mass transfer1

S34SS04-

Spring 

or well

Bough Ranch^

Sumatra

Buckhorn Exeter*

Texaco C115X

Sarpy Mine

Moore

Mysee Flowing Well

Delzer No. 2

Murdo

Hamilton

Eagle Butte

Dupree

Spring 

or well 

number

5

17

27

16

19

22

20

8

25

21

23

20

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

S.Dak

S.Dak

S.Dak.

S.Dak.

S.Dak.

Flow 

path

2

..

2

3

3

3

4

8

8

8

8

8*8 

anhydrite 

(per mil)

17.1

14.7

14.7

14.7

11.8

11.8

11.8

10.3

11.0

11.0

11.0

11.0

Temperature

17.9

83.7

80.0

84.7

83.0

86.9

63.0

55.6

59.1

58.5

54.7

56.5

mmol/kg

16.67

12.57

13.54

14.67

10.01

15.72

19.86

17.74

8.97

13.56

13.56

14.61

(per mil)

19.99

17.01

17.38

18.69

13.79

17.06

16.30

14.67

14.56

15.22

15.99

16.31

mmol/kg

0.24

1.36

1.00

3.90

.09

.56

.26

.01

.004

.04

.03

.03

(per mil)

-26.21

-2.60

-4.23

-.22

-9.20

-7.73

-22.09

-17.37

-19.12

-12.22

-7.21

-4.77

534SH2S 

(per mil)

46.20

19.61

21.61

18.91

22.99

24.79

38.39

32.04

33.68

27.44

23.20

21.08

Anhydrite

17.78

14.22

15.41

18.59

10.82

19.81

22.35

20.44

9.89

15.90

17.16

19.41

Pyrite

-0.44

-.22

-.56

-.15

-.44

-1.84

-1.19

-1.40

-.54

-1.22

-1.86

-2.46

Sulfate 

(reduced)

1.11

1.65

1.87

3.92

.81

4.09

2.49

2.70

.92

2.34

3.60

4.80

^Positive for dissolution, negative for precipitation.

'Assuming zero sulfate in the recharge water and that the dissolving anhydrite is similar in isotopic composition to that of flow path 1.

3Assumed to be on flow path 2.

TABLE ^. Calculation of anhydrite and pyrite mass transfers using the Rayleigh-distillation method

[mmol/kg, millimoles per kilogram of water]

Spring

or well

Bough Ranch

Sumatra

Buckhorn Exeter

Texaco C115X

Sarpy Mine

Moore

Mysee Flowing Well

Delzer No. 2

Murdo

Hamilton

Eagle Butte

Dupree

Spring

or well

number

5

17

27

16

19

22

20

8

25

21

23

20

6^SO4 Dissolved 

sulfate 

(mmol/kg)

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow

path

..

2

--

2

3

3

3

4

8

8

8

8

CaSO4

17.1

14.7

14.7

14.7

11.8

11.8

11.8

10.3

11.0

11.0

11.0

11.0

Starting

water

3.34

3.34

3.34

3.34

9.73

9.73

9.73

10.62

7.14

7.14

7.14

7.14

Final

water

19.99

17.01

17.38

18.69

13.79

17.06

16.30

14.67

14.56

15.22

15.99

16.31

Starting

water

0.26

.26

.26

.26

.16

.16

.16

.11

.15

.15

.15

.15

Final

water

16.67

12.57

13.54

14.67

10.01

15.72

19.86

17.74

8.97

13.56

13.56

14.61

eps

(per mil)

-46.20

-19.61

-21.61

-18.91

-22.99

-24.79

-38.39

-32.04

-33.68

-27.44

-23.20

-21.08

Calculated

r

15.25

8.05

7.75

4.72

11.45

4.77

8.62

7.42

9.40

5.83

5.26

4.02

Dissolved 

concentration 

(mmol/kg)

Sulfide

species

0.24

1.36

1.00

3.90

.09

.56

.26

.01

.004

.04

.03

.03

ASr

16.65

13.67

14.28

18.31

9.94

16.12

19.96

17.64

8.82

13.45

13.44

14.49

Mass transfer 

(mmol/kg)

Anhydrite

17.56

14.06

15.25

18.28

10.79

19.69

22.29

20.38

9.87

16.19

16.56

19.25

Pyrite

-0.46

-.19

-.48

.01

-.43

-1.78

-1.16

-1.37

-.52

-1.37

-1.56

-2.38

Sulfate

reduced

1.15

1.75

1.97

3.87

.94

4.13

2.59

2.75

1.05

2.78

3.15

4.79

The data in figure 42 show that there is a systematic 

variation to heavier values in the measured d 13C of 

waters in the Madison aquifer with increasing concentration of dissolved sulfate. The preliminary modeling results 

in table 19 show that, typically, the calculated d 13C is 5 

to 10 per mil lighter than the measured value. Six samples 

of Madison dolomite had an average d 13C value of 

1.8 ± 1.5 per mil (R.G. Deike, U.S. Geological Survey, written commun., 1982). The value of 2.0 per mil for d 13C of

dolomite has been used in modeling the calculated d 13C 

of the water. Organic-carbon sources are expected to be 

light isotopically and are typically near - 25 per mil. The 

modeling results of table 19 were obtained using a heavier 

d 13C for organic matter (-20 per mil), but even in this 

case, the calculated values of d 13C are systematically too 

light. Sensitivity analysis showed that no reasonable 

values of d 13C for dolomite or organic matter were 

capable of predicting values of d 13C similar to the
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F57

TABLE 15. Evaluation of uncertainties in anhydrite and pyrite mass transfers

[°C, degrees Celsius; mmol/kg, millimoles per kilogram of water]

834SH2S

Anhydrite mass transfer 

(mmol/kg)

(per mil)

Spring

or well

Bough Ranch

Sumatra

Buckhorn Exeter

Texaco C115X

Sarpy Mine

Moore

Mysee Flowing Well

Delzer No. 2

Murdo

Hamilton

Eagle Butte

Dupree

Spring

or well

number

5

17

27

16

19

22

20

8

25

21

23

20

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow

path

..

2

..

2

3

3

3

4

8

8

8

8

Temperature (°C)

17.9

83.7

80.0

84.7

83.0

86.9

63.0

55.6

59.1

58.5

54.7

56.5

Measured

-26.21

-2.60

-4.23

-.22

-9.20

-7.73

-22.09

-17.37

-19.12

-12.22

-7.21

-4.77

Equation

26

-26.85

-3.51

-4.62

-1.43

-7.01

-2.18

-12.50

-17.09

-15.80

-15.38

-16.81

-15.09

Isotope

dilution

17.78

14.22

15.41

18.59

10.82

19.81

22.35

20.44

9.89

15.90

17.16

19.41

Rayleigh

distillation

17.56

14.06

15.25

18.28

10.79

19.69

22.29

20.38

9.87

16.19

16.56

19.25

Isotope

dilution

and

equation 26

17.61

14.19

15.38

18.58

10.92

21.28

23.29

20.46

10.03

15.61

15.54

17.47

Pyrite mass transfer 

(mmol/kg)

Isotope

dilution

-0.44

-.22

-.56

-.15

-.44

-1.84

-1.19

-1.40

-.54

-1.22

-1.86

-2.46

Rayleigh

distillation

-0.46

-.19

-.48

.01

-.43

-1.78

-1.16

-1.37

-.52

-1.37

-1.56

-2.38

Isotope

dilution

and

equation 26

-0.48

-.26

-.55

-.13

-.49

-2.58

-1.66

-1.41

-.60

-1.08

-1.05

-1.49

90.00

80.00

70.00

60.00

50.00

40.00

30.00

20.00

10.00

0.00

O Madison system

X Edwards Limestone aquifer

+ Range for Floridan aquifer

1 Line 1 represents equilibrium Extrapolated from 

Ohmoto and Rye (1979)

2 Line 2 represents 34 A = 54.0-0.40f where 34 A = 534SSO< - &*SHf

t is temperature, in degrees Celsius (Equation 26 in text)

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

TEMPERATURE, IN DEGREES CELSIUS

FIGURE 41. Variation in 34A = d 34Sgo4 - d 34SH § as a function of temperature for water in Madison aquifer compared

with waters from Edwards and Floridan aquifers.
F58 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 16. Selected phases for mass-balance modeling

Phase

Calcite (aragonite)

Dolomite

Anhydrite

Organic matter

Carbon dioxide gas

Ferric hydroxide

(goethite)

Pyrite

Cation exchange

Halite

Sylvite

Composition

CaCOs

CaMg(C03)2

CaSO4

CH2O

CO2

FeOOH

FeS2

(Ca-Na2)X

NaCl

KC1

Redox state

(Up)

4.0

8.0

6.0

.0

4.0

3.0

.0

.0

.0

.0

measured values for many of the wells or springs. For 

example, if 6 13C for dolomite and organic matter were 

2.0 and -20.0 per mil, respectively, the calculated d 13C 

at Dupree (well No. 20 in South Dakota) is -11.58 per 

mil. Even if 6 13C for organic matter is increased to -15 

per mil and d 13C for dolomite to +4.0 per mil, the 

calculated d 13C at Dupree is still only -7.85 per mil, 

which compares with the measured d 13C value of -2.05 

per mil.

Another indicator of modeling error is based on the 

calculated 14C activity. This activity is corrected for reaction effects only, and because radioactive decay is not 

applied to the calculated value, the calculated 14C is expected to be larger than the measured. The age of the 

water is a function of the ratio of calculated to measured 

14C. Modern samples will have ratios of about 1.0 with 

the ratio increasing with age. Ratios less than 1.0 are impossible and indicate errors in the modeling process. Many 

of the calculated 14C activities for wells downgradient are 

about equal to or less than the measured 14C activities, 

indicating very young to impossibly young waters at 

points where, based on ground-water flow modeling, the 

ages of the waters are expected to be in excess of 10,000 

to 20,000 yr. For example, the calculated 14C activity at 

Dupree is 1.07 percent modern, which is less than the 

measured 2.80 percent modern. As the water at Dupree 

is expected to be in excess of 10,000 yr in age, the 

calculated 14C activity is expected to be greater than 11 

percent modern. Again, a serious flaw is indicated in the 

modeling process.

SOURCES OF ERROR

Many sources of error have been considered. These 

include (1) uncertainties in the analytical data, (2) uncertainties in defining the chemical and isotopic compositions 

of the recharge waters, (3) uncertainties in the 13C fractionation factors as a function of temperature, (4) uncertainties in the 13C content of dolomite and organic 

matter, (5) uncertainties in the estimation of d 34S of 

dissolved sulfide (pyrite), (6) uncertainties in the estimation of the sulfur-isotopic composition of dissolving 

anhydrite, (7) the possibility of other reactions such as 

magnesium-sodium ion exchange and the in situ generation of methane (methanogenesis), and (8) uncertainties 

in determining the flow paths. Certainly all of these factors contribute to the modeling uncertainties, but only a 

few affect the results significantly.

From inspection of equations 42 to 51, it can be seen 

how errors in the analytical data propagate through the 

computed mass transfer, with the C02 gas and CH20 

mass transfers most sensitive to combined analytical 

error. Because most of the waters balance in charge 

within 1 percent or less, it is unlikely that there are large 

errors in the analytical data. Uncertainties in defining the 

chemical and isotopic compositions of recharge areas are 

real and probably contribute to modeling errors of ground 

waters near recharge areas which have not been affected 

by significant chemical reaction. Test calculations show, 

however, that differences in recharge-water composition 

have little effect on modeling results. Uncertainties in the 

13C fractionation factors have only minor effect on the 

modeling. For example, assuming that there is no fractionation between the bicarbonate ion and calcite, the 

calculated value of d 13C at Dupree (well No. 20 in South 

Dakota) was changed by only 0.5 per mil. The calculated 

14C content (in percent modern) is even less dependent 

on uncertainties in the fractionation factors (Wigley and 

others, 1978,1979). It already has been shown that even 

for unrealistic choices of d 13C for organic matter and 

dolomite, large modeling errors remain.

Uncertainties in the compositions of the sulfur isotopes 

are likely and contribute to modeling errors. For example, the correlation described by equation 26 has been used 

to estimate d 34SH2s» using the measured temperature 

and d34Sso4 > where no value of d 34Sn2s was available. 

Deviations of ±10 per mil in the calculated d 34Sn2s 

values are likely (fig. 42). The Dupree well is used as an 

example. Here the measured d 34§H2S is -4.77 per mil. 

Using this value, the preliminary modeling results of table 

17 indicate oxidation of 9.29 millimoles of CH20 per 

kilogram of water, outgassing of 4.08 millimoles of C02 

per kilogram of water, and a calculated d 13C of -11.58 

per mil (table 19). Using the calculated d 34SH2S of 15.09 

per mil (eq 26) and otherwise identical modeling conditions, the quantity of CH20 oxidized is decreased to 5.8 

mmol/kg of water, the quantity of C02 outgassed is 

lessened to 2.4 mmol/kg of water, and the calculated 

d 13C is -9.6 per mil. Although a 10 per mil difference 

in d 34SH2S can significantly alter the modeled mass 

transfer, in this case the shift is far from sufficient to
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F59

TABLE 17. Preliminary resuttsfrom mass-transfer model

[All mineral and gas mass transfers are in millimoles per kilogram of water. Negative for precipitation, positive for dissolution]

Spring 

or well

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Colstrip

Sarpy Mine

Moore

Mysse Flowing Well

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Conoco No. 175

MKM

Shidler

Conoco No. 44

Seeley

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

McNenney

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Spring 

or well State 

number

8 Mont.

9 Mont.

12 Mont.

13 Mont.

18 Mont.

6 Mont.

17 Mont.

15 Mont.

16 Mont.

21 Mont.

19 Mont.

22 Mont.

20 Mont.

14 Wyo.

23 Mont.

24 Mont.

7 S.Dak.

8 S.Dak.

11 Wyo.

10 Wyo.

9 Wyo.

8 Wyo.

19 Wyo.

21 Wyo.

17 Wyo.

15 Wyo.

13 Wyo.

20 Wyo.

22 Wyo.

18 Wyo.

10 S.Dak.

9 S.Dak.

2 S.Dak.

1 S.Dak.

19 S.Dak.

24 S.Dak.

25 S.Dak.

22 S.Dak.

26 S.Dak.

21 S.Dak.

23 S.Dak.

20 S.Dak.

Flow 

path

1

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

4

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

Dolomite

0.04

.49

2.93

2.80

3.80

.27

.73

1.51

1.47

.14

1.09

1.81

3.54

.96

1.04

1.12

3.68

3.68

-.02

.64

.97

1.47

.14

.23

.14

.72

.60

.25

.33

-.04

.74

2.48

.18

1.17

1.58

1.91

1.87

2.36

2.61

2.90

3.72

3.07

Calcite

-0.20

-1.20

-7.36

-8.26

-11.15

-1.01

-2.36

-3.19

-4.07

-2.72

-3.75

-8.47

-7.53

-2.36

-2.43

-2.72

-9.83

-11.32

-1.02

-3.85

-2.48

-3.68

-.18

-.60

-.41

-1.87

-1.62

-.71

-1.00

-1.77

-2.64

-6.95

-.31

-3.92

-4.91

-6.08

-5.40

-8.40

-8.08

-9.49

-11.80

-12.84

Anhydrite

0.40

2.10

9.76

10.97

22.62

1.18

14.22

15.08

18.59

9.41

10.82

19.81

22.35

4.87

5.12

5.56

18.64

20.44

3.38

9.73

10.16

8.67

.15

.47

.51

1.86

2.27

.42

.88

2.16

5.93

16.52

.91

7.47

7.55

9.74

9.89

13.35

14.54

15.90

17.16

19.41

Organic 

matter

0.16

.23

2.07

3.95

5.98

.37

3.75

3.46

8.34

3.49

1.82

8.04

4.99

.36

.43

.47

3.87

5.28

.09

.65

.47

.21

.01

.01

.02

.22

.20

.01

.02

.20

.44

1.69

.03

1.00

1.93

2.91

2.03

1.84

4.08

4.68

7.04

9.29

Goethite

0.04

.06

.55

1.05

1.60

.10

.22

.03

.14

.87

.45

1.84

1.19

.19

.11

.13

1.16

1.50

.06

.29

.16

.07

.00

.00

.01

.06

.05

.00

.01

.05

.12

.45

.01

.26

.51

.77

.56

.50

1.09

1.28

1.91

2.47

Pyrite

-0.04

-.06

-.55

-1.05

-1.59

-.10

-.22

-.03

-.15

-.87

-.44

-1.84

-1.19

-.09

-.12

-.13

-1.02

-1.40

-.02

-.16

-.12

-.05

.00

.00

-.01

-.06

-.05

.00

-.01

-.05

-.12

-.45

-.01

-.27

-.52

-.78

-.54

-.49

-1.09

-1.22

-1.86

-2.46

Ion 

exchange

0.02

.02

.70

.89

4.37

-.07

8.32

5.88

9.23

2.54

1.36

4.81

8.28

.09

.12

.10

.86

.16

.72

.40

2.28

1.59

.01

.04

.03

.05

.04

.03

.03

.17

.44

.19

-.06

.05

.06

.16

.19

-.36

.16

.14

.66

1.22

Halite

-0.02

.01

.27

1.56

4.68

-.02

61.92

53.65

55.74

1.00

-.66

6480

15.31

1.33

1.37

1.40

-.20

1.58

1.69

32.16

16.60

13.17

-.01

-.01

-.02

-.04

.03

.01

.01

.10

2.83

.75

-.03

.84

.43

.54

1.38

4.18

2.59

.97

1.06

2.13

Sylvite

0.01

.01

.21

.28

.64

-.01

3.30

3.04

3.82

1.69

1.23

3.32

2.52

.17

.18

.18

.88

.29

.21

1.73

.86

.60

.01

.03

.02

.04

.02

.03

.04

.11

.27

.12

-.02

.23

.13

.19

.28

.28

.46

.33

.66

1.20

Carbon 

dioxide 

gas

0.06

.21

-.07

-.89

-2.93

-.39

-.38

-.36

-4.70

-1.78

-1.57

-1.47

-1.97

.01

-.09

-.11

-3.58

-1.93

-1.26

.26

-2.38

.06

-.07

.01

.17

.00

-.21

.04

-.41

-1.88

-.42

-.56

.20

-1.26

-1.29

-2.07

-1.79

-.07

-3.10

-2.45

-3.82

-4.08

account for the lack of agreement between calculated 

(-9.6 per mil) and measured (-2.05 per mil) <J 13C at 

Dupree.

In summary, the main indicators of modeling errors in 

the preliminary results (tables 17 and 19) are (1) the 

tendency for unreasonable quantities of C02 outgassing, 

(2) calculated 6 13C values that are significantly lighter

than tiie measured values, and (3) calculated 14C activities 

that are about equal to or less than the measured values 

when they are expected to be significantly larger than 

these values. Factors or processes that, if applied uniformly to the modeling exercise, could systematically improve 

the discrepancies in calculated 6 13C, 14C, and C02 outgassing were examined and are discussed below.
F60 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 18. Summary of average carbon-isotope values for 

recharge waters used for modeling carbon isotopes at downgradient wells

Recharge 1 

Recharge 2 

Recharge 3 

Recharge 4 

Recharge 5 

Recharge 6 

Recharge 7 

Recharge 8

«13c(water) 

(per mil)1

-4.97 

-11.55 

-6.99 

-7.89 

-11.26 

-9.50 

-11.00 

-9.85

»13cC02(gas) 

(per mil)2

-9.44 

-20.49 

-13.36 

-14.64 

-20.24 

-17.39 

-19.90 

-17.16

"c

(percent 

modern)3

52.68 

56.37 

52.34 

53.88 

55.63 

54.64 

55.29 

57.39

1 Average of wells defined as recharge water for each flow path.

2Estimated, assuming recharge water evolved from reaction of 

rain with soil carbon dioxide, calcite, and dolomite where del carbon13 of calcite and dolomite is taken as 0.

3Estimated, prenuclear-detonation carbon-14 content of 

recharge waters calculated assuming a 100 percent modern source of 

carbon dioxide gas and 0 percent modern carbonate mineral sources.

As the preliminary model results show (tables 17 and 

19), only two sources of carbon are expected: a light 

source (CH^O), and a heavy source (dolomite). The modeling discrepancies in calculated d 13C values indicate a need 

to either decrease the quantity of CH20 reacting, increase the quantity of dolomite dissolving, or find a sink 

for light <J 13C. Three modeling alternatives that can improve the results are (1) a mineral sink for magnesium 

such as Mg2+/Na+ ion exchange or formation of a 

magnesium-silicate phase, (2) methanogenesis, and (3) 

variation in <J 34SanhydriteAn added sink for magnesium will increase the quantity of dolomite that can dissolve, leading to heavier values 

of calculated d 13C. Likely sinks for magnesium include 

ion exchange of Mg2+ for Na+ and the formation of 

authigenic magnesium-silicate minerals such as sepiolite 

[Mg2Si307.5(OH)-3H20] and stevensite [Nao.67Mg2.6?Si4 

Oio(OH)2], which are significant weathering products of 

pyroclastic rocks (Jones and Weir, 1983). Calculations 

using various additional sinks for magnesium show that 

heavier d 13C values can be calculated. However, the mass 

transfer is confined to the calcite-dolomite phases, with 

more calcite precipitated as more dolomite dissolves. The 

C02 outgassing term is not a function of noncarbonate 

magnesium sinks. Furthermore, by increasing the quantity of dolomite dissolved and calcite precipitated, the 

calculated 14C content is significantly decreased, as required by incorporation of 14C in precipitated calcite.

To demonstrate this point and other modeling options, 

an example using several modeling alternatives at the 

Mysse Flowing Well (well No. 20 in Montana) was made. 

The data in table 20 show, for the modeling conditions of 

Ca2+/Na+ exchange, no methanogenesis, <J 34Sanhydrite

equal to that derived for flow path 3 (table 12), 

<J 13Cdoiomite equal to 2.0 per mil, and <J 13CcH20 equal to 

-25.0 per mil, that the calculated value of <Ji3C (-9.39 

per mil) is substantially different from the measured value 

(-2.34 per mil) and that a significant quantity of C02 

outgassing (1.97 mmol/kg of water) is indicated (case 1, 

table 20). By providing a sink for magnesium, for example, as a pure Mg2+/Na+ exchange reaction in place of 

pure Ca2+/Na+ exchange, the dolomite and calcite mass 

transfers are almost tripled (case 2, table 20), but there 

is no change in the large calculated quantity of C02 outgassed. Although the calculated value of <J 13C (-4.31 per 

mil) is now closer to the measured value (-2.34 per mil), 

the added magnesium sink still leads to a calculated 14C 

value less than the measured value (case 2, table 20).

Analysis of the chemical trends indicates that excess 

bicarbonate is found primarily along flow paths 2 and 3, 

indicating significant ion exchange. The preliminary masstransfer results confirm this trend because values of about 

zero for the ion-exchange mass-transfer coefficient were 

determined almost everywhere in the Madison aquifer 

system (table 17) except along flow paths 2 and 3. Thus, 

Mg2+/Na+ exchange or combined (Ca2+ + Mg2+)/Na+ 

exchange cannot be invoked to account for the 13C discrepancy throughout the Madison aquifer. Furthermore, 

by adding the magnesium sink, the CCVoutgassing problem is not affected and the 14C discrepancy is even larger.

Another modeling alternative is the possibility of 

methanogenesis. A large biological fractionation accompanies methanogenesis, resulting in the formation of 

CH4, which is typically 40 ± 20 per mil lighter in 13C than 

the CH20 from which it was derived. Methane as light 

as -80 per mil is not uncommon (Hoefs, 1973). The 

C02-CH4 equilibrium fractionation factor is about 70 per 

mil at 25 °C and decreases to 50 per mil at about 90 °C 

(Bottinga, 1969).

Only traces of dissolved CH4 were found in the dissolved gases of 12 wells completed in the Madison aquifer 

(Busby and others, 1983). The maximum concentrations 

(0.45 to 0.87 mg/L) were in water from wells along flow 

paths 2 and 3. The modeling of the mass transfer along 

these flow paths indicates that such concentrations of 

methane are insufficient to significantly affect the 

calculated value of d13C. Larger concentrations of CH4 

are required to increase the calculated value of d 13C, 

which would necessitate a system open to both C02 and 

CH4 outgassing.

Returning to the modeling example of the Mysse Flowing Well, the mass transfer to this well was calculated 

assuming the formation of a gas containing 20 percent of 

CH4 and 80 percent of C02- The CH4 produced was 

assumed to be 60 per mil lighter than the d 13C of the 

aqueous solution. The remainder of the modeling parameters were as before: Ca2+/Na+ exchange, <J 34Sanhydrite
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F61

TABLE 19. Summary of preliminary model parameters and carbon-isotope results

[Calculated carbon-14 assumes no radioactive decay]

Spring or well

Hanover Plowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Colstrip

Sarpy Mine

Moore

Mysse Flowing Well

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Conoco No. 175

MKM

Shidler

Conoco No. 44

Seeley

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

Lien

McNenney

Fuhs

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Spring 

or well 

number

8

9

12

13

18

6

17

15

16

21

19

22

20

14

23

24

7

8

11

10

9

8

19

21

17

15

13

20

22

18

10

9

17

2

6

1

19

24

25

22

26

21

23

20

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow 

path

1

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

4

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

8

8

&*s

anhydrite 

(per mil)

17.10

17.10

17.10

17.10

17.10

14.70

14.70

14.70

14.70

11.80

11.80

11.80

11.80

10.30

10.30

10.30

10.30

10.30

7.90

7.90

7.90

7.90

9.50

9.50

9.50

9.50

9.50

10.00

10.00

10.00

10.00

10.00

11.00

11.00

11.00

11.00

11.00

11.00

11.00

11.00

11.00

11.00

11.00

11.00

8l3C 

(per mil)

Organic 

matter

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

Dolomite

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

813C, water 

(per mil)

Calculated

-5.62

-4.68

-4.85

-7.55

-7.52

-10.01

-13.49

-10.84

-12.89

-13.81

-7.84

-13.59

-7.55

-6.02

-5.85

-5.78

-4.08

-8.21

-10.10

-11.59

-4.16

-6.23

-8.86

-8.76

-9.31

-7.67

-7.62

-10.17

-9.26

-9.65

-8.84

-6.63

-8.20

-9.47

-5.59

-7.28

-8.22

-8.55

-7.14

-7.78

-8.05

-8.86

-8.99

-11.58

Measured

-5.32

-5.18

-7.46

-7.04

-3.22

-7.57

-3.61

-1.68

-

-2.67

-2.33

-2.40

-2.34

-6.63

-6.17

-6.02

-4.60

-2.61

-6.23

-4.66

-5.14

-7.82

-10.40

-10.00

-8.20

-6.80

-7.26

-6.60

-4.24

-9.70

-9.10

-8.03

-11.50

-9.96

-6.20

-7.20

-6.20

-5.50

-6.63

-4.72

-3.50

-2.40

-2.05

Carbon-14, 

water 

(percent modern)

Calculated

49.09

37.38

5.66

3.46

.61

43.82

19.81

16.10

5.90

20.02

17.22

5.35

5.79

30.59

28.70

27.28

1.02

1.84

54.73

33.08

25.32

26.56

51.80

50.14

51.68

40.09

41.85

50.62

48.68

53.57

38.04

15.75

52.62

53.02

34.80

23.15

16.55

10.32

13.14

9.44

4.50

3.91

1.65

1.07

Measured

25.40

29.30

--

28.00

4.20

52.50

--

1.00

--

--

3.30

1.60

.80

12.70

10.00

9.50

4.60

2.80

13.90

2.60

6.20

1.80

61.40

46.20

54.70

14.70

59.00

44.50

31.20

4.80

28.50

19.40

68.50

79.60

53.80

7.80

2.80

2.40

3.20

4.60

4.00

2.40

2.20

2.80

Apparent 

age 

(years)

5,446

2,013

--

modern

modern

modern

--

22,970

-.

--

13,659

9,986

16,368

7,266

8,716

8,720

modern

modern

11^29

21,026

11,631

22,252

modern

677

modern

8,294

modern

1,065

3,677

19,942

2,386

modern

modern

modern

modern

8,995

14,689

12,059

11,679

5,944

981

4,026

modern

modern

equal to the average flow-path value (11.8 per mil), and 

<J 13CcH20 and d 13Cdoiomite of -25.0 and 2.0 per mil, 

respectively.

The mass-transfer and carbon-isotope calculations are 

summarized in table 20 (case 3). Inclusion of methanogenesis significantly increases the calculated quantity of

outgassing. As expected, the quantity of organic matter 

oxidized is increased and there is only minor improvement 

in the calculated value of d 13C compared with case 1. 

The combined effect of pure Mg2+/Na+ exchange and 

methanogenesis (case 4, table 20) differs little from that 

of Mg2+ /Na+ exchange alone (case 2).
F62 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

Substantial methanogenesis is not known to occur when 

dissolved sulfate is present (Fenchel and Blackburn, 1979). 

Because many of the waters are almost saturated with 

respect to anhydrite, methanogenesis is not expected to 

be an important process in the Madison aquifer.

One final and potentially more likely modeling alternative considered was the possibility that the sulfurisotopic composition of dissolving anhydrite may differ 

from that derived for each flow path (table 12) and may

vary along the flow path. If, for example, 

were heavier than previously estimated (table 12), less 

sulfate reduction would be calculated, the quantity of 

isotopically light CH20 reacting would decrease, and the 

value of d 13C would be heavier. In addition, the decrease 

in CH£0 mass transfer would decrease the quantity of 

C02 outgassed, and thus it may be possible to find a 

"reasonable" value of d^Sanhydrite that indicates a system closed to C02 gas. Finally, decreased CH20 mass

-13.00

-12.00

-11.00

-10.00

-9.00

-8.00

£ -7.00

o.

-6.00

-5.00

-4.00

-3.00

-2.00

-1.00

o 

o

* *

o

o

*

 t

0 - Flow path 1

A - Flow path 2

V - Flow path 3

^ - Flow path 4

+ - Flow path 5

+ - Flow path 6

X - Flow path 7

 ft - Flow path 8

O

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 42. Variation in measured <J 13C as a function of dissolved-sulfate concentration in Madison aquifer system.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F63

TABLE 20. Summary of modeling alternatives applied to the Mysse Flowing Well (well No. 20 in

Montana)

[mmol/kg, millimoles per kilogram of water] 

Modeling parameters

Case

0*43 

(per mil)^

Ion

exchange 

XMg2

Proportion 

of carbon 

dioxide in

Solutionmethane

fractionation 

(per mil)

513C 

(per mil)

Organic matter Dolomite

11.8

11.8

11.8

0.0 

1.0 

.0

1.0 

1.0 

.8 -60.

-25.0

-25.0

-25.0

+2.0 

+2.0 

+2.0

4

5

6

11.8

15.5

15.5

1.0

.0

.0

.8

1.0

1.0

-60.

-

~

-25.0

-25.0

-25.0

+2.0

+2.0

+4.0

Calculated results

Case

1

2

3

4

5

6

Mass transfer 

(mmol/kg)

Dolomite

3.54

11.82

3.54

11.82

3.54

3.54

Calcite

-7.53

-24.09

-7.53

-24.09

-7.53

-7.53

Anhydrite

22.35

22.35

22.35

22.35

20.15

20.15

Organic 

matter

5.00

5.00

6.31

6.31

0.87

0.87

Gas

-1.97

-1.97

-3.28

-3.28

-0.04

-0.04

5"c 

(per mil)

Calculated

-9.39

-4.31

-7.90

-3.78

-3.57

-2.20

Measured

-2.34

-2.34

-2.34

-2.34

-2.34

-2.34

Carbon-14 

activity 

(percent modern)

Calculated

5.79

.28

4.56

.22

12.30

12.30

Measured

0.80

.80

.80

.80

.80

.80

^Dissolving anhydrite.

2XMg is the fraction of magnesium/sodium ion exchange;

0.0 = pure calcium/sodium ion exchange,

1.0 = pure magnesium/sodium ion exchange. 

^Proportion of carbon dioxide gas in a carbon dioxide-methane mixture;

1.0 = pure carbon dioxide gas,

0.0 = pure methane gas. 

4Content corrected for reaction effects but not radioactive decay.

5730 [~ carbon-14 (calculated)

Calculated age =

In 2

IT ca 

In -

L c:arbon-14 (measured)

, in years.

transfer, caused by increasing d^Sanhydrite* would result 

in less dilution of 14C by 14C-depleted carbon, and thus 

older ages would be calculated.

The data in table 20 show that for the Mysse Flowing 

Well (case 5), the calculated C02 outgassing is about 0.0 

mmol/kg of water if cJ 34Sanhydrite is 15-5 Per mil rather 

than the extrapolated value of 11.8 per mil (table 12), 

which resulted in C02 outgassing of 1.97 mmol/kg of 

water. If we use this heavier value of d^Sanhydrite* the 

CH^O mass transfer is decreased from 5.00 to 0.87 

mmol/kg of water. The quantity of anhydrite dissolved is 

decreased from 22.35 to 20.15 mmol/kg of water, and the 

calculated value of d 13C is only 1.23 per mil lighter than 

the measured value.

Minor variations in many of the other modeling parameters can lead to almost identical calculated and measured 

values of d 13C. For example, if d 13C of the dissolving 

dolomite is about 4.0 per mil rather than 2.0 per mil (case 

6, table 20), the calculated d 13C value is similar to the 

measured value. Other factors, which separately or combined would result in similar calculated and measured 

values of d 13C, are (1) lighter d 34SH2s> (2) variation 

in d 13Cdolonute, (3) variation in d 13CCH20, (4) (Ca2+ + 

Mg2+)/Na+ ion exchange, and (5) minor uncertainties in 

the analytical data. It is apparent that an almost unlimited 

number of minor variations in reaction parameters can 

lead to close agreement in calculated and measured values 

of d 13C. But, through all of these changes, the overall
F64 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

mass-transfer results are changed only slightly and are 

similar to those of case 5 (table 20).

Of the three modeling alternatives investigated (Mg2+/ 

Na+ ion exchange, methanogenesis, and variation in 

<* 34Sanhydrite), only by increasing the value of d^Sanhydrite 

can all major model discrepancies be resolved. That is, by 

increasing <J 34Sanhydrite> the calculated CQz outgassing is 

minimized, indicating the expected closed system; the 

calculated <J 13C value is heavier and similar to the 

measured values; and the calculated 14C activity, corrected for reaction effects, is now significantly larger than 

the measured value, which results in more reasonable age 

estimates.

For example, at the Mysse Flowing Well (case 5, table 

20) the calculated 14C activity of 12.30 percent modern, 

when used with the measured 14C activity (0.80 percent 

modern), indicates a water age in excess of 22,000 yr, 

which is within the reasonable range as determined from 

ground-water flow models (Downey, 1984). None of the 

other modeling alternatives considered (Mg2+/Na"f ion 

exchange and methanogenesis) can yield the same 

reasonable results.

Based on this analysis of modeling uncertainties, the 

single most significant factor contributing to modeling 

errors in the preliminary results (tables 17 and 19) is 

uncertainty in the sulfur-isotopic composition of dissolving anhydrite. Unfortunately, almost no data are available 

for <J 34Sanhydrite m the Madison aquifer. As mentioned 

earlier, R.G. Deike (U.S. Geological Survey, written commun., 1984) reported values of rf^Sanhydrite m HTH No. 

1 (well No. 14 in Wyoming) on flow path 4 of 12.8 and 

13.9 per mil, and six values at HTH No. 2 were between 

14.6 and 22.6 per mil. Claypool and others (1980) reported 

marine evaporites of Mississippian age with <J 34Sanhydrite 

in the range of 16 per mil (Upper Mississippian) to 25 per 

mil (Lower Mississippian). Because the <J 34Sso4 of gypsum or anhydrite formed by oxidation of a terrigenous 

sulfide with subsequent precipitation of anhydrite would 

be lighter than that from a marine evaporite, the 

measured values are reasonable.

CRITERIA USED IN REACTION MODELING

In preparing the final mass-transfer models, the initial 

set of plausible reactants and products was virtually maintained (table 16), whereas the isotopic composition of 

dissolving anhydrite was varied. The following requirements were met in preparing the final mass-transfer 

models:

1. It was assumed that different values of d 34S of 

anhydrite may be appropriate for different wells or 

springs on the same flow path.

2. The value of d 34Sanhydrite needs to be equal to or less 

than the value of <J 34Ss04 for each well or spring. That

is, if there were no sulfate reduction, <J 34Sso4 would be 

equal to <J 34Sanhydrite3. The value of <J 34Sanhydrite was varied (usually increased) in order to minimize (and decrease to zero, if 

possible) the CC>2 gas mass transfer.

4. Only wells or springs with measured values of 

<J 34Ss04 were modeled. The value of <J 34SH2s usgd was 

either the measured value or, if missing, the calculated 

value using equation 26.

5. Pyrite always was maintained as a product and 

goethite as a reactant. For example, in some cases in 

order to decrease the <*c02(gas) term to zero, the value 

of <J 34Sanhydrite was increased to the point that massbalance calculations showed pyrite as a reactant and 

goethite as a product. In these cases the value of 

<* 34Sanhydrite was decreased until pyrite appeared as a 

product and FeOOH as a reactant. Pyrite is a product 

because of the presence of H2S.

6. For wells or springs where the CC>2 gas and pyrite 

mass transfers are not sensitive to <J 34Sanhydrite> the 

selected criterion for <J 34Sanhydrite was that the measured 

and calculated values of d 13C agree in the final solution.

7. The value of <J 13CcH20 was varied between -20 to 

- 27 per mil but usually chosen to be - 25 per mil.

8. The value of 6 13Cd0iomite was varied between 0.0 and 

6.7 per mil but usually chosen as 2.0 per mil.

9. Magnesium/sodium ion exchange was found 

necessary for wells or springs along flow paths 2 and 3, 

where ion exchange is an important reaction, based on 

the analysis of the chemical trends. A variable (Ca2 "1" + 

Mg^/Na* ion-exchange reaction was considered in 

which the fraction of magnesium exchange, XMg, was 

adjusted in the range 0<XMg<1.0. The value of XMg 

equal to 1.0 corresponds to pure Mg2+ /Na"f exchange, 

and the value of JfMg equal to 0.0 represents pure 

Ca2+ /Na"f exchange. Cases of extensive Mg2+/Na"f ion 

exchange indicate formation of magnesium-enriched 

minerals such as sepiolite or stevensite.

10. In a few cases, the magnitude of the ion-exchange 

term was not sufficient to increase the calculated value 

of d 13C to a value similar to the measured value, and for 

these cases, the possibility of methanogenesis was investigated.

The data in table 21 are the best estimates of the mass 

transfers of dolomite, calcite, anhydrite, CH20, goethite, 

pyrite, ion exchange, halite, sylvite, and C02 gas, in 

mmol/kg of water, for 42 wells or springs in the Madison 

aquifer. The carbon- and sulfur-isotopic values used and 

the nature of the ion-exchange reaction considered are 

summarized in table 22. The calculated and measured 

d 13C values also are compared in table 22. The few examples of dissimilarity between calculated and measured 

values of <J 13C, such as the Voss well (well No. 20 in 

Wyoming), usually correspond to wells near recharge
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F65

TABLE 21. Summary of fined mass-transfer model results

[All mineral and gas mass transfers are in millimoles per kilogram of water. Negative for precipitation, positive for dissolution]

Spring or well

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Colstrip

Sarpy Mine

Moore

Mysse Flowing Well

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Conoco No. 175

MKM

Shidler

Conoco No. 44

Seeley

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

McNenney

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Spring 

or well 

number

8

9

12

13

18

6

17

15

16

21

19

22

20

14

23

24

7

8

11

10

9

8

19

21

17

15

13

20

22

18

10

9

2

1

19

24

25

22

26

21

23

20

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow 

path

1

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

4

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

Dolomite

0.04

.49

2.93

2.80

3.80

.27

5.72

7.39

10.70

2.68

2.18

6.61

3.54

.96

1.04

1.12

3.68

3.68

.70

.64

.97

1.47

.14

.23

.14

.72

.60

.25

.33

-.04

.74

2.48

.18

1.17

1.58

1.91

1.87

2.36

2.61

2.90

3.72

3.07

Calcite

-0.20

-1.20

-7.36

-7.41

-8.26

-.95

-12.10

-14.95

-22.52

-6.35

-5.35

-16.07

-5.33

-2.36

-2.33

-2.61

-8.81

-9.14

-2.47

-3.85

-2.48

-3.59

-.18

-.60

-.41

-1.87

-1.62

-.71

-1.00

-1.77

-2.64

-7.33

-.31

-3.35

-4.34

-5.00

-4.70

-7.75

-6.47

-7.15

-8.22

-8.17

Anhydrite

0.40

2.10

9.76

10.13

19.73

1.12

13.97

15.08

18.59

7.97

10.23

17.81

20.15

4.87

5.03

5.46

17.62

18.26

3.38

9.73

10.16

8.58

.15

.47

.51

1.86

2.27

.42

.88

2.16

5.93

16.91

.91

7.10

6.98

8.66

9.19

12.70

12.94

13.56

13.58

14.74

Organic 

matter

0.16

.23

2.07

2.36

.56

.25

3.29

3.46

8.34

.78

.72

4.28

.87

.36

.25

.27

1.97

1.19

.09

.65

.47

.05

.01

.01

.02

.22

.20

.01

.02

.20

.44

2.42

.03

.30

.86

.88

.71

.63

1.07

.30

.33

.53

Goethite

0.04

.06

.55

.63

.15

.07

.15

.03

.14

.15

.16

.84

.09

.19

.07

.07

.66

.41

.06

.29

.16

.02

.00

.00

.01

.06

.05

.00

.01

.05

.12

.65

.01

.08

.23

.23

.21

.18

.29

.11

.12

.13

Pyrite

-0.04

-.06

-.55

-.63

-.15

-.07

-.15

-.03

-.15

-.15

-.15

-.84

-.09

-.09

-.07

-.07

-.52

-.31

-.02

-.16

-.12

-.01

.00

.00

-.01

-.06

-.05

.00

-.01

-.05

-.12

-.65

-.01

-.08

-.23

-.24

-.19

-.17

-.28

-.06

-.07

-.13

Ion 

exchange

0.02

.02

.70

.89

4.37

-.07

8.32

5.88

9.23

2.54

1.36

4.81

8.28

.09

.12

.10

.86

.16

.72

.40

2.28

1.59

.01

.04

.03

.05

.04

.03

.03

.17

.44

.19

-.06

.05

.06

.16

.19

-.36

.16

.14

.66

1.22

Halite

-0.02

.01

.27

1.56

4.68

-.02

61.92

53.65

55.74

1.00

-.66

64.80

15.31

1.33

1.37

1.40

-.20

1.58

1.69

32.16

16.60

13.17

-.01

-.01

-.02

-.04

.03

.01

.01

.10

2.83

.75

-.03

.84

.43

.54

1.38

4.18

2.59

.97

1.06

2.13

Sylvite

0.01

.01

.21

.28

.64

-.01

3.30

3.04

3.82

1.69

1.23

3.32

2.52

.17

.18

.18

.88

.29

.21

1.73

.86

.60

.01

.03

.02

.04

.02

.03

.04

.11

.27

.12

-.02

.23

.13

.19

.28

.28

.46

.33

.66

1.20

Carbon 

dioxide

0.06

.21

-.07

-.14

-.40

-.33

-.17

-.36

-4.70

-.52

-1.06

.29

-.04

.01

-.01

-.02

-2.70

-.02

-1.26

.26

-2.38

.14

-.07

.01

.17

.00

-.21

.04

-.41

-1.88

-.42

-.90

.20

-.93

-.79

-1.12

-1.18

.49

-1.70

-.40

-.69

.00

areas where modeling is particularly sensitive to the starting d 13C conditions. As previously discussed, the similarity in calculated and measured values of d 13C was a 

modeling requirement and dependent on the choice of 

modeling parameters.

Although methanogenesis was investigated for many 

of the wells or springs, none of the final modeling results 

included this process. The calculated (adjusted for reaction) 14C content and measured 14C content (percent

modern) also are included in table 22 along with an age 

estimate calculated by the following equation:

14CCalc A, 573°1 Lt = In 

In 2

14Cr

(53)

where bt is the travel time, in years, since the ground 

water became isolated from the soil 14C reservoir (Wigley 

and Muller, 1981), and 14Ccaic and 14Cmeas are in percent 

modern (table 22).
F66 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

TABLE 22. Summary of final model parameters and carbon-isotope results

[Calculated carbon-14 assumes no radioactive decay]

Spring 

or well

Hanover Flowing Well

Vanek Warm Spring

Landusky Spring

Lodgepole Warm Spring

Sleeping Buffalo

McLeod Warm Spring

Sumatra

Keg Coulee

Texaco C115X

Colstrip

Sarpy Mine

Moore

Mysse Flowing Well

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 1

Delzer No. 2

Conoco No. 175

MKM

Shidler

Conoco No. 44

Seeley

Newcastle

Osage

Upton

Devils Tower

Voss

Self

JBJ

Evans Plunge

Cascade Spring

McNenney

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Spring 

or well 

number

8

9

12

13

18

6

17

15

16

21

19

22

20

14

23

24

7

8

11

10

9

8

19

21

17

15

13

20

22

18

10

9

2

1

19

24

25

22

26

21

23

20

State

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Mont.

Wyo.

Mont.

Mont.

S.Dak.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow 

path

1

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

4

5

5

5

5

6

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

8

8

8

8

8%

anhydrite 

(per mil)

17.10

17.10

17.10

20.00

21.90

17.00

15.00

14.70

14.70

14.00

13.00

14.00

15.50

10.30

10.90

10.90

12.60

13.60

7.90

7.90

7.90

8.30

9.50

9.50

9.50

9.50

9.50

10.00

10.00

10.00

10.00

9.00

11.00

12,40

12.90

13.70

13.30

12.30

14.30

15.00

15.80

16.00

(per mil)

Organic 

matter

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-20.00

-20.00

-20.00

-20.00

-20.00

-20.00

-25.00

-25.00

-25.00

-25.00

-27.00

-22.00

-25.00

-20.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

-25.00

Dolomite

2.00

1.00

.00

2.00

.00

2.00

4.00

5.00

5.00

4.00

4.00

6.70

4.00

1.20

.90

1.00

.00

4.00

2.50

2.00

2.00

4.00

2.00

2.00

2.00

.50

2.00

2.00

2.00

2.00

.00

1.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

4.90

Ion 

exchange 

(Xjjg)

0.0

.0

.0

.0

.0

.0

.6

1.0

1.0

1.0

.8

1.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

813C 

(per mil)

Calculated

-5.84

-5.19

-7.33

-7.05

-3.22

-10.15

-3.72

-1.85

-3.19

-2.64

-2.41

-2.47

-2.21

-6.66

-6.18

-6.01

-4.58

-2.60

-6.29

-11.59

-4.69

-5.10

-8.86

-8.77

-9.33

-8.19

-7.79

-10.17

-9.28

-9.92

-9.67

-9.08

-9.51

-6.14

-7.20

-6.18

-5.52

-6.72

-4.73

-3.59

-2.35

-2.07

Measured

-5.32

-5.18

-7.46

-7.04

-3.22

-7.57

-3.61

-1.68

-

-2.67

-2.33

-2.40

-2.34

-6.63

-6.17

-6.02

-4.60

-2.61

-6.23

-4.66

-5.14

-7.82

-10.40

-10.00

-8.20

-6.80

-7.26

-6.60

-4.24

-9.70

-9.10

-11.50

-6.20

-7.20

-6.20

-5.50

-6.63

-4.72

-3.50

-2.40

-2.05

Carbon-14 

(percent modern)

Calculated

49.09

37.38

5.66

5.45

3.63

45.29

2.92

1.66

.15

10.97

12.75

2.43

12.30

30.59

30.61

28.72

1.99

5.49

34.84

33.08

25.32

27.79

51.80

50.14

51.68

40.09

41.85

50.62

48.68

53.57

38.04

13.72

53.02

28.01

21.49

17.31

18.40

15.35

10.26

12.01

8.53

10.29

Measured

25.40

29.30

-

28.00

4.20

52.50

..

1.00

-

-

3.30

1.60

.80

12.70

10.00

9.50

4.60

2.80

13.90

2.60

6.20

1.80

61.40

46.20

54.70

14.70

59.00

44.50

31.20

4.80

28.50

19.40

79.60

7.80

2.80

2.40

3.20

4.60

4.00

2.40

2.20

2.80

Adjusted 

age 

(years)

5,446

2,013

-

modern

modern

modern

HISS

-

~

11,173

3,448

22,588

7,266

9,247

9,147

modern

5,560

7,597

21,026

11,631

22,624

modern

677

modern

8,294

modern

1,065

3,677

19,942

2,386

modern

modern

10,568

16,847

16,332

14,461

9,960

7,784

13,310

11,203

10,763

*An adjusted carbon-14 age of 23,000 years is found if H2S is introduced by crossfonnational leakage (see text).

SELECTION OF MODELING PARAMETERS 

FOR EACH FLOW PATH

FLOW PATH 1

As modeling progressed, it became evident that the five 

wells and springs selected for flow path 1 do not approximate a single flow path but are instead all relatively

young waters affected primarily by varying quantities of 

anhydrite dissolution and local recharge conditions. The 

Hanover Flowing Well (well No. 8 in Montana) appears 

to be nearest to recharge-water composition, with only 

0.04 mmol of anhydrite dissolved per kilogram of water. 

The C02 outgassing is about zero and is insensitive to 

uncertainties in d^S- As the dolomite and
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F67

mass transfers are about zero (table 21) at the Hanover 

Flowing Well, the calculated value of d 13C is insensitive 

to values selected for d 13Cdoiomite and d 13CcH20. The 

small differences in calculated and measured d 13C probably are due to differences in the d 13C value of the 

recharge area.

The Vanek Warm Spring water (spring No. 9 in Montana) is similar to water from the Hanover Flowing Well 

except that more anhydrite has dissolved and the 

dedolomitization reaction has begun. The C02 outgassing 

is again small and insensitive to d 34Sanhydrite- The 

previously defined value of d 34Sanhy{jrite (17.10 per mil) 

was used (table 12) for both the Hanover Flowing Well 

and Vanek Warm Spring. The calculated and measured 

d 13C agree if d 13C of the reacting dolomite is 1.00 per 

mil rather than 2.00 per mil.

Landusky Spring (spring No. 12 in Montana) is approximately half-saturated with anhydrite, and the masstransfer calculations indicate a moderate dedolomitization 

reaction (table 21). The C02 outgassing term is about zero 

if 17.10 per mil is again used for d^Sanhydrite- The 

calculated value of d 13C agrees with the measured value 

if d 13Cdoiomite is about 0.0. The spring water is affected 

by moderate ion exchange (0.70 mmol/kg of water) and 

significant sulfate reduction (2.07 mmol of CH20 oxidized 

per kilogram of water).

The two remaining waters along flow path 1, Lodgepole 

Warm Spring (spring No. 13 in Montana) and Sleeping 

Buffalo (well No. 18 in Montana), probably dissolve 

isotopically heavier anhydrite. Using 17.10 per mil for 

<* 34Sanhydrite» the preliminary modeling indicated 0.9 and 

2.9 mmol of C02 outgassing per kilogram of water for 

Lodgepole Warm Spring and Sleeping Buffalo well, 

respectively, which is unlikely. Furthermore, the calculated value of d 13C was too light at Sleeping Buffalo well, 

consistent with excessive calculated sulfate reduction and 

CH20 oxidation. The adjustment procedure was as 

follows: First d^Sanhydrite was increased to minimize the 

C02 outgassing term. Estimated values of 20.0 and 21.9 

per mil were used for d 34Sanhydrite at Lodgepole Warm 

Spring and Sleeping Buffalo well, respectively.

At Lodgepole Warm Spring, 62 outgassing changes 

to ingassing of 0.09 mmol/kg of water if d 34Sanhydrite is 

21.0 per mil rather than 20.0 per mil (corresponding to 

0.14 mmol of C02 outgassing per kilogram of water). 

Considering the magnitude of the uncertainties in total 

inorganic carbon in the recharge water, further refinement of the d 34Sanhydrite parameter is not warranted at 

Lodgepole Warm Spring. The calculated value of d 13C is 

similar to the measured value at Lodgepole Warm Spring 

if dolomite with d 13C of 2.0 per mil and CH20 with 

d 13C of -25.0 per mil are reacting.

The C02 outgassing at Sleeping Buffalo well cannot be 

decreased to zero without using d 34Sanhydrite values

greater than d34SSo4 (21.95 per mil). The value of 21.90 

per mil for d 34Sanhydrite was selected, resulting in 0.40 

mmol of C02 outgassing per kilogram of water, which 

again could be due to uncertainties in total inorganic 

carbon of the recharge water. Using 21.9 per mil for 

<* 34Sanhydrite> ^e measured d 13C at Sleeping Buffalo well 

is then matched if d 13CcH20 is -25.0 per mil and 

<* 13Cdolomite is 0.0 per mil.

The 14C age estimates are summarized in table 22. All 

waters along flow path 1 appear to be younger than about 

6,000 yr. The Hanover Flowing Well and Vanek Warm 

Spring water ages are relatively well defined because of 

the smaller mass transfer involved in calculating the 

adjusted 14C content. No 14C data were available for 

Landusky Spring. Water from Lodgepole Warm Spring 

is likely contaminated with a modern source, because the 

measured 14C value is significantly larger than the 

calculated value. This indication of a modern component 

in water from Lodgepole Warm Spring is further supported by a tritium content of 31.8 TU (table 11).

The importance of reaction corrections in 14C age 

dating is demonstrated by water from the Sleeping 

Buffalo well, which has a relatively small 14C content 

(4.2 percent modern) that is equivalent to an unadjusted 

14C age (assuming a recharge water of 50 percent modern 

14C) in excess of 20,000 yr. After correction for extensive dedolomitization, the age is, within the uncertainties 

of the calculation, virtually modern. The short residence 

time along flow path 1 in Montana, caused by a local cell 

of recharge-discharge, results in water of virtually modern 

age at the Sleeping Buffalo well (Downey, 1984).

FLOW PATH 2

The McLeod Warm Spring (spring No. 6 in Montana) 

is located nearest the recharge area of flow path 2, and 

the water composition indicates that only minor reactions 

have occurred. Water from this spring is similar in composition to the average recharge water along flow path 

2, except that an additional 1.1 mmol of anhydrite per 

kilogram of water have dissolved. The calculated mass 

transfer is not sensitive to d 34Sanhydrite or reasonable 

estimates of d 13C for CH20 and dolomite. For example, 

using 14.7 per mil for d 34Sanhydrite, -25-0 for d 13CcH20> 

and 2.0 per mil for d 13Cdoiomite> the calculated value of 

d 13C is -10.47 per mil. Increasing d^Sanwirite to 17.0 

per mil increases the calculated value of d 13C to -10.15 

per mil. If d 34Sanhydrite is increased to 17.5 per mil (approximately that of dissolved S0l~ at McLeod Warm 

Spring) and d 13Cdoiomite is increased to 4.0 per mil, the 

calculated value of d 13C is increased to -9.81 per mil. 

The difference in calculated and measured d 13C probably 

is due to differences in d 13C of the recharge water, as the 

measured d 13C is -7.57 per mil.
F68 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

The recharge waters used to define the average recharge of flow path 2 (Bozeman Fish Hatchery, well No. 

4 in Montana; Big Timber Fish Hatchery, spring No. 7 

in Montana) have a considerable variation in d 13C of 

-8.98 and -14.11 per mil, respectively. The calculations 

along flow path 2 included the average d 13C of -11.55 

per mil for the recharge area. The final values used 

at McLeod Warm Spring were 17.00 per mil for 

d 34Sanhydrite> -25.0 per mil for d 13CcH20, and 2.0 per 

mil for 6 13Cd0iomite The calculated 14C is not a function 

of uncertainty in the d 34S and 6 13C values and indicates 

that water from McLeod Warm Spring is virtually 

modern.

The remaining three wells along flow path 2 Sumatra 

(well No. 17 in Montana), Keg Coulee (well No. 15 in Montana), and Texaco C115X (well No. 16 in Montana) pose 

far more difficult problems in interpretation of chemical 

reactions. Because of the presence of significant concentrations of H2S in these waters (as much as 132 mg/L at 

Texaco C115X), pyrite would be expected to precipitate 

as a product if it was included in the model. This places 

a severe constraint on modeling because the d^Sanhydrite 

cannot be much heavier than the estimated flow-path 

value of 14.70 per mil without resulting in a calculated 

dissolution of pyrite. But if a value of d^Sanhydrite of 

14.70 per mil is used, the calculated values of <5^3C are 

very light (ranging from -12 to -15 per mil) when the 

measured values are -1.68 and -3.61 per mil, respectively, at Keg Coulee and Sumatra. No carbon-isotope 

data are available for Texaco C115X, but because of its 

close proximity to Sumatra and Keg Coulee, similar values 

are expected.

In order to calculate heavy 6 13C values at these three 

wells, it was necessary to assume an additional sink for 

magnesium so that additional dissolution of dolomite 

would increase the calculated 6 13C values. For modeling 

purposes, this was done in the form of Mg2+/Na+ exchange, although, as discussed earlier, other authigenic 

magnesium-silicate minerals may be forming along flow 

path 2. Methanogenesis also was considered along flow 

path 2, but it had only a minor effect on the calculated 

value of d 13C. As an example, some of the modeling 

parameters investigated for Sumatra are shown in table 

23 along with the resulting calculated value of d 13C.

Only by including substantial Mg2+ /Na+ exchange 

(X"Mg = 0.6) and relatively heavy sources of 13C (-20.00 

per mil for C^O and 4.00 per mil for dolomite) was the 

calculated 6 13C value at Sumatra similar to the measured 

value. Even more extensive magnesium ion exchange 

(X"Mg = 1.0) and a d 13C value for dolomite of 5.00 per mil 

are required to calculate heavy values of <5 13C at Keg 

Coulee and Texaco C115X (table 22). The calculated 

dolomite mass transfer was significantly increased in 

obtaining the heavy values. For example, without the

added magnesium sink, only 1.51 mmol of dolomite is 

dissolved per kilogram of water at Keg Coulee (table 17), 

but by adding Mg2+/Na+ ion exchange, dissolution of 

7.39 mmol of dolomite per kilogram of water is indicated 

(table 21).

Although the increased dolomite dissolution resulted in 

reasonable agreement between calculated and measured 

values of <5 13C, the calculated 14C content decreased 

significantly. Without Mg2+/Na+ ion exchange, the calculated 14C age at Keg Coulee (table 19) is about 23,000 

yr (but there is little agreement with the measured value 

of 6 13C). With the magnesium sink, the additional dolomite dissolution at Keg Coulee decreased the calculated 

14C age to about 4,000 yr (table 22), but the calculated 

and measured values of 6 13C showed agreement.

Based on distance from the recharge area in central 

Montana, the waters are expected to be old along flow 

path 2, and the 23,000-yr age for water at Keg Coulee 

is not unreasonable. Other possibilities that can result in 

relatively old water with a heavy calculated value of 

6 13C were considered. One is that the large concentrations of dissolved H2S along flow path 2 were introduced 

by leakage from other formations and are not the product 

of in situ sulfate reduction. To investigate this possibility, the mass-balance calculations were repeated, but 

HaS was treated as a separate phase. Significant improvements in the modeling results were found, but an 

adjustment was required for the total inorganic carbon 

of the recharge water along flow path 2. If (1) the water 

recharging flow path 2 contains about 1.5 mmol of C0£ 

per kilogram of water more than that at Big Timber Fish 

Hatchery and Bozeman Fish Hatchery (average total 

inorganic carbon of 3.8 mmol/kg of water); (2) the H2S 

found along flow path 2 formed in other formations 

and migrated into the Madison aquifer; and (3) the 

d 13C<Iolomite value along flow path 2 is 4.0 per mil, then 

there is similarity between the calculated and measured 

6 13C values of the water. The calculated value of 14C is 

then 16.5 percent modern, giving an age of 23,000 yr at 

Keg Coulee.

Measured 14C data at the Sumatra and Texaco C115X 

wells are not available because organic-corrosionprevention additives were present in the wells. Calculated 

values are 27.1 and 9.0 percent modern, respectively, for 

the same assumptions as at Keg Coulee. In addition to 

resulting in more reasonable age estimates along flow 

path 2, the calculations indicate that rf 34Sanhydrite varies 

between 18.0 and 18.6 per mil for Sumatra, Keg Coulee, 

and Texaco C115X; these values are similar to the 

reported values for Mississippian evaporite deposits 

(Claypool and others, 1980). The hypothesis of an extraformational source for the H2S along flow path 2 results in reasonably good agreement with the analytical 

data.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F69

TABLE 23. Summary of selected modeling parameters evaluated at Sumatra (well No. 17 in Montana) onflow path 2

[Xco , fraction of carbon dioxide gas in carbon dioxide-methane mixture; Xytg, fraction of magnesium ion exchange]

anhydrite 

(per mil)

17.0

14.7

16.0

15.0

15.0

15.0

15.0

15.0

15.0

Ion 

exchange 

<XMg>

0.0

.0

.0

.0

1.0

.5

.5

1.0

.6

Gas 

compositior

1.0

1.0

1.0

1.0

1.0

1.0

.9

1.0

1.0

i

Methane Organic 

matter

-25.0

-25.0

-25.0

-25.0

-25.0

-25.0

-80.0 -25.0

-20.0

-20.0

(per mil)

Dolomite

2.0

2.0

2.0

2.0

2.0

4.0

4.0

4.0

4.0

Calculated

-11.5

-15.8

-13.4

-15.3

-4.3

-5.6

-5.6

-1.9

-3.7

Measured

-3.61

-3.61

-3.61

-3.61

-3.61

-3.61

-3.61

-3.61

-3.61

Pyrite 

dissolution (+) 

or 

precipitation (-)

+

-

+

_

-

-

.

-

-

FLOW PATH 3

Problems in modeling reactions along flow path 3 are 

similar to those of flow path 2 except that concentrations 

of H2S in most waters are less. As along flow path 2, it 

was necessary to resort to sinks for magnesium (treated 

as Mg2+ /Na+ exchange), to isotopically heavy dolomite 

(<* 13Cdolomite of 4.0 per mil), and to relatively heavy 

organic matter (d 13CcH20 of -20 per mil) in order to obtain the calculated values of d 13C. In minimizing the 

C02 mass transfer, values of d 34Sanhydrite of 13.0 to 15.5 

per mil were used rather than the estimated value of 11.8 

per mil (table 12).

The measured value of d 13C (-2.67 per mil) at Colstrip 

(well No. 21 in Montana) is almost matched if d 34Sanhydrite 

is 14.0 per mil, pure Mg2+ /Na+ exchange is included, 

<* 13CcH20 ig -20.0 per mil, and d 13Cdoiomite is 4-0 per mil. 

No data are available for the measured 14C, but the adjusted 14C is 10.97 percent modern. If the measured 

14C of Colstrip is similar to that at the nearby Sarpy Mine 

well (well No. 19 in Montana) (3.30 percent modern), the 

water at Colstrip is at least 10,000 yr old.

Modeling of the Sarpy Mine well yields results similar 

to those at Colstrip. The measured value of d 13C (-2.33 

per mil) is almost matched if d 34Sanhydrite is 13 per mil, 

predominantly Mg2+/Na+ ion exchange (XMg=0.8) is 

included, d 13CcH20 is -20 per mil, and d 13Cdolomite is 4.0 

per mil. The calculated 14C age is about 11,000 yr.

The Moore Flowing Well (well No. 22 in Montana) 

posed the most difficult modeling problems along flow 

path 3. The C02 mass transfer was minimized using a 

d 34Sanhydrite value of 14.0 per mil. Taking this value, and 

including pure Mg2+/Na+ exchange, a d 13CcH20 value 

of - 20.0 per mil, and a d 13Cd0iomite value of 4.0 per mil, 

the calculated value of d 13C was - 4.40 per mil, which is

similar to the measured value of - 2.40 per mil. By varying the d 13C of dolomite, but holding the other parameters constant, a value of 6.70 per mil for 6 13Cdoiomite 

is required to match the measured value of d 13C. This 

probably is unrealistic and indicates further uncertainties 

in the modeling exercise. It is noted that the Moore Flowing Well has a relatively large concentration of dissolved 

H2S (table 8) and may be affected by leakage of H2S from 

other formations, as was suggested earlier for waters 

along flow path 2. A further indication of problems in 

modeling the Moore Flowing Well is a calculated 14C 

age of about 3,000 yr in an area where the age of the 

water likely ranges from 10,000 to 20,000 yr old.

Contrary to the previous three wells modeled along flow 

path 3, the Mysse Flowing Well (well No. 20 in Montana) 

is relatively less complicated to model and was treated 

in the same fashion as wells along flow path 1. The C02 

mass transfer is minimized at Mysse if d 34Sanhydrite is 

near 15.5 per mil, and a calculated d 13C value similar 

to the measured value is determined if d 13Cd0iomite is 

4.0 per mil and d 13CcH20 is -25.0 per mil. No Mg2 "1"/ 

Na+ exchange was required. The calculated 14C age 

is about 23,000 yr, similar to that indicated for flow 

path 2.

FLOW PATH 4

Most of the wells along flow path 4 were modeled using 

the dedolomitization reaction and by making small adjustments to values of d34S of reacting anhydrite. The 

estimated flow-path value of d 34Sanhydrite along flow path 

4 used in the preliminary modeling was 10.3 per mil. 

Values between 10.3 and 13.6 per mil were found to 

minimize the C02 outgassing term. Ion exchange is not 

an important reaction along flow path 4.
F70 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

The CC>2 outgassing term is not sensitive to 

at HTH No. 1 (well No. 14 in Wyoming), and similar 

calculated and measured <5 13C values were determined by 

changing 6 13Cd0iomite to 1.2 per mil. Calculated and 

measured values of <5 13C at Ranch Creek (well No. 23 in 

Montana) were matched with a <J 34Sanhydrite °f 10-9 per 

mil, a <J 13CcH£0 °f -25.00 per mil, and a <J 13Cd0iomite of 

0.90 per mil. Belle Creek (well No. 24 in Montana) is 

located within 13 miles of Ranch Creek, and very similar 

modeling results were obtained there. The values for 

<* 34Sanhydrite, <* 13CcH20, and 6 13Cdoiomite were 10.9, 

-25.0, and 1.0 per mil, respectively.

The calculated 14C ages of Belle Creek and Ranch 

Creek differ by only 100 yr (9,200 versus 9,100 yr) even 

though the measured and adjusted 14C contents differ 

somewhat.

Delzer No. 1 (well No. 7 in South Dakota) and Delzer 

No. 2 (well No. 8 in South Dakota) are very near each 

other, yet in this case the models do not agree. Reasonable 

results were found for Delzer No. 2 if d^Sanhydrite is 13.6 

per mil, <5 13CCH20 is -22.0 per mil, and (J l3Cdoiomite is 

4.0 per mil, resulting in a 14C age of 5,560 yr.

The data for Delzer No. 2 are considered more reliable 

than those for Delzer No. 1. The unusually low flow at 

Delzer No. 1 apparently resulted in collection of a less 

reliable sample, as evidenced by a 20 °C temperature difference between the reported bottom-hole and wellhead 

temperatures and the observed evolution of C0£ gas. A 

comparison of the analytical data for Delzer No. 1 and 

No. 2 shows a significant difference in total inorganic carbon for the two wells, a result consistent with the field 

observations. The modeling results also were consistent 

with the field observations; a large transfer of C0£ gas 

from the system was required regardless of the value 

chosen for <J 34Sanhydrite- The modeling results in tables 

21 and 22 are based on the reported analytical data and 

show what modeling parameters were required to match 

the measured <5 13C value. These results indicate a modern 

age for the waters at Delzer No. 1, a result inconsistent 

with the conceptual model of the flow system. Results in 

better agreement with the conceptual model of the flow 

system are obtained for Delzer No. 1 if a value for total 

inorganic carbon similar to the value measured for Delzer 

No. 2 is used.

FLOW PATH 5

Waters along flow path 5 are unusual in that they have 

the lightest values of <5 34Ss04 in the Madison system (8.2 

to 8.6 per mil with an extrapolated lower limit of 7.90 

per mil). Each well on flow path 5 posed special modeling 

problems, particularly in determining modeling alternatives that match the measured, relatively heavy values 

of d 13C.

The preliminary results in table 17 are based on a model 

that does not incorporate Mg2+/Na+ exchange. These 

results show no net dissolution along the flow path to 

Conoco No. 175 (well No. 11 in Wyoming). Because there 

is no mass transfer of dolomite, the relatively heavy <5 13C 

value of - 6.23 per mil cannot be calculated even with a 

<5 13CcH20 value of -20.00. Thus, it was necessary to 

propose either a magnesium sink or methanogenesis in 

modeling Conoco No. 175. In addition, it was determined 

that the sulfur-isotopic composition of dissolving 

anhydrite could not be heavier than 8.2 per mil or else 

dissolution of pyrite would be predicted. By increasing the 

<* 34Sanhydrite value from 7.9 to 8.2 per mil, the quantity 

of CC>2 outgassed is decreased by only 0.03 mmol/kg of 

water. Thus, Conoco No. 175 poses an additional problem 

in the carbon balance. Differences in total inorganic carbon in the recharge area are suspected. Using the average 

flow-path value of <J 34Sanhydrite (7.9 per mil), Conoco No. 

175 has been modeled two ways: (1) by assuming methanogenesis, and (2) by assuming a magnesium sink.

The model incorporating methanogenesis includes 

Ca2+/Na+ exchange, dedolomitization, and outgassing of 

a gas containing 14 percent of CH4 (d 13CcH4 of -80.00 

per mil). If, in addition, <J 13CcH20 and <* 13Cdolomite have 

values of - 25.00 and 2.00 per mil respectively, then the 

measured <5 13C value is almost matched. Using the model 

incorporating methanogenesis, the calculated 14C age is 

10,300 yr. Conoco No. 175 also can be modeled by assuming there that there is a sink for magnesium. Treating this 

as pure Mg2+/Na+ exchange, the measured <5 13C value 

at Conoco No. 175 is almost matched if <J 13CcH20 is 

-25.0 per mil and <J 13Cd0iomite is 2.5 per mil. The 

calculated dissolution of an additional 0.70 mmol/kg of 

dolomite per kilogram of water dilutes the 14C reservoir 

somewhat, leading to a slightly younger calculated age 

(7,600 yr).

Of the two modeling alternatives, the choice incorporating pure magnesium ion exchange is preferred. 

Methanogenesis is not thought to be an important 

mechanism in the presence of dissolved sulfate (3.47 mmol 

of sulfate per kilogram of water), and small concentrations of CH4 (less than 1.88xlO~ 5 mmol/kg of water) 

have been measured in the Madison aquifer system (Busby 

and others, 1983).

Even more difficult problems are encountered in modeling the MKM well (well No. 10 in Wyoming) on flow path 

5. Using the value of 7.9 per mil for <J 34Sanhydrite> dissolution of 0.26 mmol of C0£ gas per kilogram of water was 

computed. This can be decreased to the expected zero 

value (closed system) if the <J 34Sanhydrite value is 7.4 per 

mil. It is, however, more likely that the carbon imbalance 

is due to the difference in total inorganic carbon in the 

recharge water for the MKM well. Using - 25.0 per mil 

for <5 13CcH20> 2.0 per mil for <5 13Cd0iomite> and 7.9 per
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F71

mil for d^Sanhydrite , the calculated value of d 13C (-12.27 

per mil) was significantly lighter than the measured value 

(-4.66 per mil). Furthermore, only 0.4 mmol of ion exchange per kilogram of water are calculated, so that even 

with pure Mg2+ /Na+ ion exchange, and using heavy 

CH20 and dolomite (-20.0 and 4.0 per mil, respectively), the calculated value of d 13C only increased to -9.17 

per mil compared with the value of -11.59 per mil in the 

preliminary model results (table 19). Thus, either there 

are serious errors in the analytical data for MKM or else 

the process of methanogenesis needs to be considered as 

a cause of the measured heavy value of 6 13C. But in order 

to compute significant methanogenesis, the carbon 

balance needs to be corrected in order for the mass 

balance to calculate a net loss of gas. Because the 

magnitude of the required adjustments to the carbon 

balance are unknown, the mass transfer to the MKM well 

cannot be finalized, and thus the preliminary results have 

been retained in the final mass-transfer results (table 21). 

Because most of the mass transfer is accounted for by the 

dedolomitization reaction, there probably are no large 

errors in the calculated 14C age, which, based on the 

preliminary results, is about 21,000 yr.

A measured d 13C value is not available for the Shidler 

well (well No. 9 in Wyoming), but it is assumed to be in 

the range of - 4.0 to - 6.0 per mil, similar to that for the 

other wells along flow path 5. The large, calculated 

C(>2 outgassing is probably due to uncertainties in the 

composition of recharge water. Using 7.9 per mil for 

<* 34Sanhydrite, -25.0 per mil for d 13CcH20> 2.0 per mil for 

<* 13Cdoiomite> ^^ Ca2+ /Na+ exchange, a reasonable value 

of d 13C (-4.69 per mil) was calculated for the Shidler 

well.

The Conoco No. 44 well (well No. 8 in Wyoming) was 

modeled using 8.30 per mil for d 34Sanhydrite, Ca2+/Na+ 

ion exchange, and -25.0 and 4.0 per mil for d 13CcH20 

and d 13Cdoiomite» respectively. Some C(>2 dissolution was 

calculated but probably is due to differences in recharge 

waters. If the carbon balance is adjusted to obtain a 

system closed to C(>2, lighter values of d 13Cdoiomite can 

be included in the model. Carbon-dioxide dissolution was 

treated as input of soil gas using the average calculated 

recharge-area soil gas d 13C value (table 18) of -20.24 per 

mil for flow path 5. If the carbon balance were adjusted 

to calculate zero C(>2 mass transfer, no net input of 

-20.24 per mil soil C(>2 gas would be included in the 

13C calculation and a lighter d 13Cdoiomite would be 

indicated.

FLOW PATH 6

The water along flow path 6 has the least quantities of 

mass transfer of the entire Madison aquifer. This can 

probably be attributed to rapid ground-water flow on the

west side of the Black Hills, which has removed most of 

the reactants, CH20, and anhydrite, thereby limiting the 

driving force for reaction. If this is the case, a relatively 

younger age would be expected for water along flow path 

6.

Using the value of 9.5 per mil for d 34Sanhydrite (table 

12), all wells along flow path 6 are found to be virtually 

closed to C(>2 gas. Ion exchange does not occur and minor 

sulfate reduction is calculated only for the wells at Upton (well No. 15 in Wyoming) and Devils Tower (well No. 

13 in Wyoming). Because the dolomite mass transfer 

also is very small, particularly at Seeley (well No. 19 in 

Wyoming), Newcastle (well No. 21 in Wyoming), and 

Osage (well No. 17 in Wyoming), there is only a minor 

carbon source, and thus the calculated d 13C value 

depends significantly on the starting (recharge water) 

d 13C value. Calculated and measured d 13C values differ 

by about 1.00 per mil, (except at Upton), but this probably 

is not an indication of mass-transfer errors.

The calculated 14C age at Upton is about 8,000 yr, and 

its water is the oldest along flow path 6. Three other 

waters (Seeley, Osage, and Devils Tower) probably are 

less than 1,000 yr old but cannot be dated more closely 

because the measured 14C value is slightly larger than the 

calculated value. Tritium concentrations range from 0.0 

to 0.8 TU along flow path 6, so a large modern component is not expected. The Newcastle well has 0.10 TU, 

and the corrected 14C age there is about 700 yr.

FLOW PATH 7

Reactions along flow path 7 are similar to those along 

flow path 6, but with somewhat more reaction progress. 

The calculated d 13C values at three of the wells (Voss, 

well No. 20 in Wyoming; Self, well No. 22 in Wyoming; 

and JBJ, well No. 18 in Wyoming) could not be matched 

closely to the measured d 13C values, probably because of 

relatively light 13C in the recharge area. Rhoads Fork 

(well No. 4 in South Dakota) has been used as the recharge 

water for flow path 7 and has a measured d 13C value of 

-11.0 per mil. A recharge water with a d 13C value of 

about -8.0 per mil is indicated by the modeling results 

of table 21. Waters at Voss and Self evolved by relatively slight dedolomitization (0.42 and 0.88 mmol of anhydrite 

dissolved per kilogram of water). Neither water is 

sensitive to variation in d 34Sanhydrite> and both appear, 

within reasonable uncertainties in the recharge-water 

composition, to be closed to C(>2. A large calculated C(>2 

outgassing at the JBJ well is not sensitive to d^Sanhydrite 

and probably indicates uncertainties in the recharge 

composition.

Significantly larger quantities of anhydrite have dissolved in the Evans Plunge and particularly the Cascade 

Spring waters. The C(>2 mass transfer at Evans Plunge
F72 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

is minimized to -0.20 mmol/kg of water if d 34Sanhydrite 

is 11.6 per mil, but in this case the measured d 13C value 

cannot be exactly matched. Using -25.0 per mil and 0.0 

per mil for d 13CcH20, and d 13Cdoiomite, respectively, 

the calculated value of d 13C at Evans Plunge is about 

1.00 per mil heavier than the measured value (-9.70 per 

mil). In order to obtain the best match between calculated 

and measured d 13C values, a smaller average value for 

<* 34Sanhydrite (10-0 per mil) has been used, resulting in 

more sulfate reduction and lighter calculated values of 

d 13C.

A similar determination was made for the Cascade 

Spring water. The measured d 13C value (-9.10 per mil) 

is very light, particularly for a water almost saturated 

with anhydrite and dolomite, which likely is the result of 

substantial calcite precipitation. For example, using 11.5 

per mil for d 34Sanhydrite> the C02 mass transfer is 

decreased to zero at Cascade Spring, but even values 

of -25.0 per mil for d 13CcH20 and 0.0 per mil for 

<* 13Cdolomite resu& m a calculated 13C value that is too 

heavy (-7.00 per mil). In order to match the measured 

d 13C value, an isotopically lighter anhydrite source is 

required. Calculated and measured values of d 13C are 

almost matched if d 34Sanhydrite ig 9.0 per mil, d 13CcH20 

is -25.0 per mil, and d 13Cdoiomite is 1-0 per mil. The computed CC>2 gas mass transfer (-0.90 mmol/kg of water) 

may be due to differences in recharge composition.

Because these waters flow along a path consisting of 

highly transmissive rocks, it is not surprising that all of 

these waters (table 22) are less than 4,000 yr old, except 

for the JBJ well. Water from the latter well, which is 

located on the western side of the flow-controlling structure on the western side of the Black Hills in Wyoming 

(Downey, 1984), has an age of about 20,000 yr before 

present.

FLOW PATH 8

From the preliminary modeling, several serious problems were evident along flow path 8. In addition to difficulties in matching the measured d 13C values and 

unreasonably large calculated C02 outgassing, the calculated 14C ages increase with increasing reaction progress. 

Waters with the most extensive reaction progress were 

found to be impossibly young (calculated 14C content less 

than the measured content). Varying the sulfur-isotopic 

composition of anhydrite, however, generally improved 

model results.

As with many of the previously modeled wells that are 

associated with recharge areas, calculated and measured 

d 13C values at the McNenney well (well No. 2 in South 

Dakota) could not be exactly matched; the calculated 

d 13C value was several parts per thousand heavier than 

the measured value and insensitive to variation in

<J 34Sanhydrite, <* 13CcH20, and d 13Cdoiomite The data in 

table 11 show that the McNenney well contains tritium 

(11.4 TU), and the calculated 14C content was found to 

be less than the measured content. It was concluded, 

therefore, that the McNenney well is not completely 

isolated from the recharge process. The remaining wells 

along flow path 8 have anhydrite mass transfers of 6.98 

to 14.74 mmol/kg of water. For these, the modeling 

results were improved by increasing d 34Sanhydrite fr°m 

the estimated value of 11.0 per mil.

The largest value of d 34Sanhydrite tested for the Kosken 

well (well 1 in South Dakota) was 12.9 per mil; this 

minimized the quantity of CH20 reacted. The CC>2 outgassing could not be decreased to zero by further increasing d 34Sanhydrite without leading to impossible reactions 

such as the formation of CH20. Using 12.9 per mil for

<J 34Sanhydrite, - 25-00 Per mil for <* 13CcH20, and 0.30 

per mil for d 13Cd0iomite> tne calculated d 13C value almost 

matches the measured value. If d 13Cd0iomite were 2.0 per 

mil, then the measured d 13C value would almost be 

matched if d 34Sanhydrite were 12.4 per mil. This latter 

result for Kosken has been retained in tables 21 and 22.

Modeling of the remaining wells along flow path 8 

followed a similar pattern. The sulfur-isotopic composition of dissolving anhydrite used in the modeling process 

was varied from 12.3 per mil at Hilltop Ranch (well 

No. 22 in South Dakota) to 16.0 per mil at Dupree 

(well No. 20 in South Dakota). By varying d 34Sanhydrite> 

all but one of the wells along flow path 8 could be modeled 

using -25.0 per mil for d 13CcH20 and 2.0 per mil for 

d 13Cdolomite- Dupree required a heavier dolomite (4.9 per 

mil) to obtain a match between calculated and measured 

values of d 13C.

Most of the wells along flow path 8 indicated an outgassing of less than 1.0 mmol of CC>2 per kilogram of water, 

probably indicating a smaller dissolved CC>2 concentration 

in the recharge area than was used in the calculations.

The calculated 14C ages indicate that the modeling 

results in tables 21 and 22 are correct. Unlike the 

preliminary modeling results (table 19), the final modeling results (table 22) are relatively constant with respect 

to the ages of the waters. Because most of the wells are 

located along a north-trending line normal to the direction of flow along flow path 8, generally similar ages are 

expected. Excluding McNenney, which has been shown 

to be associated with the recharge process, the average 

age of waters along flow path 8 is 12,300 ± 2,900 yr.

SUMMARY OF FINAL MASS-TRANSFER RESULTS 

FOR EACH FLOW PATH

The mass-transfer results of the modeling excercises for 

each flow path are summarized below and detailed in 

tables 21 and 22.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F73

FLOW PATH 1

The evolution of waters at five wells and springs in 

north-central Montana was modeled using the composition of Lewistown Big Spring (spring No. 10 in Montana) 

as the starting point. The mass-transfer results in table 

21 show that water at sites nearest the recharge area, 

the Hanover Flowing Well (well No. 8 in Montana) and 

Vanek Warm Spring (spring No. 9 in Montana), is affected 

by some anhydrite dissolution accompanying the dissolution of dolomite and the precipitation of calcite. No other 

reactions are important for these two waters. At Landusky Spring (spring No. 12 in Montana), Lodgepole 

Warm Spring (spring No. 13 in Montana), and Sleeping 

Buffalo (well No. 18 in Montana), further progress in the 

anhydrite-driven dedolomitization reaction is determined 

with more than 19 mmol of anhydrite dissolved per 

kilogram of water at Sleeping Buffalo. Accompanying the 

extensive dedolomitization reaction is substantial sulfate 

reduction (about 0.56 mmol CH^O oxidized per kilogram 

of water), pyrite formation balanced by dissolution of 

FeOOH (as much as 0.15 mmol/kg of water), Ca2+/Na+ 

exchange (4.37 mmol/kg of water), and dissolution of 

halite and sylvite (4.68 and 0.64 mmol/kg of water, 

respectively).

All of the waters along flow path 1 appear to be young 

(about 5,000 yr to modern), and it is likely that the water 

from the five wells and springs selected is primarily affected by local recharge conditions and probably is not 

part of a regional flow system.

FLOW PATH 2

Data from wells at Bozeman Fish Hatchery (well 

No. 4 in Montana) and Big Timber Fish Hatchery (well 

No. 7 in Montana) were averaged to define the recharge 

water for flow path 2. Water at McLeod Warm Spring 

(spring No. 6 in Montana), nearest the recharge area, is 

affected by dedolomitization (with 1.12 mmol of anhydrite 

dissolved per kilogram of water) and sulfate reduction 

(0.25 mmol CH20 oxidized per kilogram of water). Proceeding downgradient some 124 miles to oilfield waterflood wells at Sumatra (well No. 17 in Montana), Keg 

Coulee (well No. 15 in Montana), and Texaco C115X (well 

No. 16 in Montana), which are all within 23 miles of each 

other, all reactions except pyrite formation and goethite 

dissolution are significant. Anhydrite mass transfer varies 

between 13.97 and 18.59 mmol/kg of water between these 

three wells and the recharge area.

Large modeling uncertainties remain along flow path 

2 concerning primarily the calcite and dolomite mass 

transfers. If there is a major sink for magnesium (such 

as Mg2+/Na+ exchange), then substantial dolomite dissolution (as much as 10.70 mmol/kg of water) and calcite 

precipitation (to 22.5 mmol/kg of water) are indicated.

Although this model results in agreement between 

calculated and measured <J 13C values, the calculated 14C 

activity indicates very young ages (about 4,000 yr) for 

waters with ages likely ranging from 10,000 to 20,000 yr.

Based on the indications of leakage and the extensive 

faulting in the area around flow path 2 (fig. 9), it is likely 

that the H2S found along flow path 2 has migrated into 

the Madison aquifer from the underlying CambrianOrdovician aquifer and is not really a product of in situ 

sulfate reduction. The hypothesis of an external source 

of hydrogen sulfide gives reasonable model results. If this 

is the case, much of the calcium from anhydrite dissolution is removed from the water by ion exchange, with 5.88 

to 9.23 mmol of calcium exchanged per kilogram of water. 

The removal of calcium results in decreased calcite 

precipitation (2.36 to 4.07 mmol/kg of water) and dolomite 

dissolution (0.73 to 1.51 mmol/kg of water). This modeling alternative almost results in a match between 

calculated and measured d 13C values and 14C ages of 

about 23,000 yr.

Regardless of the source of H2S, as much as 61.92 

mmol of halite and 3.82 mmol sylvite dissolve in each 

kilogram of water along flow path 2. Exchange of Ca2+ 

for Na+ , producing excess sodium bicarbonate, is more 

significant along flow path 2 than for any other flow path 

in the Madison aquifer.

FLOW PATH 3

The wells used to define the composition of the recharge 

water for flow path 3 were at Mock Ranch (well No. 1 

in Wyoming) and Denius No. 1 (well No. 2 in Wyoming), 

some 31 miles south of flow path 3 in north-central Wyoming. No d 34S data were obtained for HTH No. 3 (well 

No. 14 in Montana) and Gas City (well No. 26 in Montana), 

which precluded mass-transfer modeling of water from 

these wells. Water from the remaining four wells, Colstrip (well No. 21 in Montana), Sarpy Mine (well No. 19 

in Montana), Moore (well No. 22 in Montana), and Mysse 

Flowing Well (well No. 20 in Montana), is dominated by 

a considerable range of reactions, including extensive 

dedolomitization driven by dissolution of more than 20 

mmol of anhydrite per kilogram of water. Halite dissolution is significant at the Moore and Gas City wells (the 

latter was calculated independent of the lack of sulfurisotope data) and of little importance at the Colstrip and 

Sarpy Mine wells. Varying degrees of ion exchange are 

present, with between 1.36 and 8.28 mmol of calcium exchanged per kilogram of water along flow path 3.

Ages along flow path 3 ranged from about 3,000 yr at 

the Moore well to about 23,000 yr at the Mysse Flowing 

Well. The latter value probably is more reasonable for 

flow path 3, and the younger ages probably are due to 

uncertainties in the mass transfer used to correct the 

calculated 14C value.
F74 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

FLOW PATH 4

The recharge area of flow path 4 is well defined by four 

wells in north-eastern Wyoming: Storey Fish Hatchery 

(well No. 6 in Wyoming), Mobil (well No. 7 in Wyoming), 

Mock Ranch (well No. 1 in Wyoming), and Denius No. 1 

(well No. 2 in Wyoming). Five wells were modeled along 

flow path 4, which has a length of more than 155 miles 

through northeast Wyoming, southeast Montana, and 

northwest South Dakota; the wells modeled were HTH 

No. 1 (well No. 14 in Wyoming), Ranch Creek (well No. 

23 in Montana), Belle Creek (well No. 24 in Montana), 

Delzer No. 1 (well No. 7 in South Dakota), and Delzer No. 

2 (well No. 8 in South Dakota). Very similar mass 

transfers were found for the HTH No. 1, Ranch Creek, 

and Belle Creek wells, which are within 31 miles of each 

other and midway along the flow path. The mean 

anhydrite mass transfer at this point is 5.12 mmol/kg of 

water, with an average of 2.43 mmol of calcite 

precipitated and an average of 1.04 mmol of dolomite 

dissolved per kilogram of water. Only minor sulfate reduction has occurred (0.30 mmol CH£0 oxidized per kilogram 

of water) with formation of traces of pyrite. Halite dissolution and ion exchange are not significant, at least to 

the midpoint of flow path 4. Farther downgradient at the 

Delzer No. 1 and No. 2 wells, which are less than 0.62 

miles apart, more extensive dedolomitization caused by 

dissolution of 17.94 mmol of anhydrite per kilogram of 

water was determimed. Sulfate reduction is more significant at this farthest point sampled along flow path 4 (1.19 

to 1.97 mmol of CH£0 oxidized per kilogram of water) 

with acompanying formation of 0.31 to 0.52 mmol of 

pyrite per kilogram of water. Neither ion exchange nor 

halite dissolution contribute significantly to the mass 

transfer throughout the entire 155-mile distance sampled 

along flow path 4.

Some of the more reliable 14C age dates are calculated 

along flow path 4, particularly in the vicinity of the HTH 

No. 1, Ranch Creek, and Belle Creek wells; the ages for 

the latter three wells range between about 7,300 and 

9,200 yr.

FLOW PATH 5

Waters at the Hole in the Wall well (well No. 5 in 

Wyoming) and Barber Ranch Spring (spring No. 12 in 

Wyoming) were averaged to define the recharge composition for flow path 5. Four wells were modeled along a 

62-mile flow path in northeast Wyoming: Conoco No. 175 

(well No. 11 in Wyoming), MKM (well 10 in Wyoming), 

Shidler (well No. 9 in Wyoming), and Conoco No. 44 (well 

No. 8 in Wyoming). In spite of difficulties experienced in 

modeling waters along flow path 5 (see previous section), 

a similar pattern in mass transfer resulted for all four 

wells. Waters from these wells are affected by moderately

extensive dedolomitization, with as much as 10.16 mmol 

of anhydrite dissolved per kilogram of water, and only 

minor sulfate reduction and pyrite formation. Halite 

dissolution is a major contributor to the water quality 

along flow path 5 (as much as 32.16 mmol dissolved per 

kilogram of water), and moderate ion exchange was 

detected (as much as 2.28 mmol of calcium exchanged per 

kilogram of water). Carbon-14 ages of about 8,000 to 

about 23,000 yr were calculated.

FLOW PATH 6

The Mallo Camp (well No. 24 in Wyoming) and Rhoads 

Fork (well No. 4 in South Dakota) wells were selected to 

define the average recharge-water composition of flow 

path 6. Five wells in northeastern Wyoming were modeled 

along flow path 6, a distance of about 50 miles. These are 

the Seeley (well No. 19 in Wyoming), Newcastle (well No. 

21 in Wyoming), and Osage (well No. 17 in Wyoming) 

wells clustered within 19 miles of each other on the west 

side of the Black Hills, and the Upton (well No. 15 in 

Wyoming) and Devils Tower (well No. 13 in Wyoming) 

wells at the end of the flow path. The mass transfers listed 

in table 21 show that very little, if any, reaction takes 

place along flow path 6.

The three wells nearest the recharge area (Seeley, 

Newcastle, and Osage) averaged only 0.38 mmol of 

anhydrite dissolved per kilogram of water, with some 

dolomite dissolution and calcite precipitation. No other 

reactions were detected at these wells. At the Upton and 

Devils Tower wells, more extensive dedolomitization was 

evidenced by as much as 2.27 mmol of anhydrite dissolved 

per kilogram of water and small increases in the dolomite 

and calcite mass transfers.

Some sulfate reduction probably occurs near the end 

of flow path 6, as about 0.20 mmol of CH£0 was oxidized 

per kilogram of water. Ion exchange and halite dissolution are not significant, and all waters appear to be 

modern to 1,000 yr old except water from the Upton well, 

which is about 8,000 yr old.

FLOW PATH 7

Water from the Rhoads Fork well (well No. 4 in South 

Dakota) was selected as representative of the recharge 

composition of flow path 7 on the southwest side of the 

Black Hills. Reactions were modeled for water from 

4 wells and 1 spring along flow path 7, a distance of 

more than 50 miles. Waters from the three wells nearest 

the recharge area, Voss (well No. 20 in Wyoming), Self 

(well No. 22 in Wyoming), and JBJ (well No. 18 in 

Wyoming), are effected almost exclusively by some 

dedolomitization. Water at JBJ well indicated some 

amount of dolomite precipitation (-0.04 mmol/kg of
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F75

water), which probably indicates either that there are 

errors in the water analysis or that water from the Rhoads 

Fork well is not representative of recharge for the JBJ 

well. No other reactions are significant near the recharge 

area of flow path 7.

Farther downgradient along flow path 7 to Evans 

Plunge (well No. 10 in South Dakota) and Cascade Springs 

(spring No. 9 in South Dakota) the calculations show 

significant increases in the extent of dedolomitization 

(16.91 mmol of anhydrite dissolved per kilogram of water 

at Cascade Spring). In addition, sulfate reduction was inferred based on 2.42 mmol of CH^O oxidized per kilogram 

of water with accompanying pyrite precipitation at 

Cascade Spring. Some ion exchange and halite dissolution were indicated near the end of flow path 7. Excluding 

water from well JBJ, which may not be part of the flow 

system, the waters along flow path 7 are young, ranging 

between modern and 4,000 yr old.

FLOW PATH 8

The average recharge for flow path 8 was defined using 

the composition of waters at Jones Spring (spring No. 12 

in South Dakota), the Kaiser well (well No. 11 in South 

Dakota), and Cleghorn Spring (spring No. 16 in South 

Dakota) on the east side of the Black Hills. From here, 

reactions were modeled to 10 wells east and northeast of 

the Black Hills for a distance of about 124 miles. Water 

from the McNenney well (well No. 2 in South Dakota) near 

the recharge area is slightly affected by dedolomitization, 

with 0.91 mmol of anhydrite dissolved per kilogram of 

water. No other reactions were important at this site.

The remaining nine wells are all located along the farthest part of the flow path in western South Dakota. 

Modeling of waters from those wells showed a progressive 

increase in mass transfer along a line normal to the direction of flow extending from the Kosken well (well No. 1 

in South Dakota) in south-central South Dakota to the 

Dupree well (well No. 20 in South Dakota) about 106 miles 

to the north-northwest in northwestern South Dakota. 

Along this line, progressive increases in the quantities of 

anhydrite dissolved (increasing from 7.10 to 14.74 

mmol/kg of water), calcite precipitated (increasing from 

3.35 to 8.22 mmol/kg of water), and dolomite dissolved 

(increasing from 1.17 to 3.72 mmol/kg of water) were 

determined. This trend in mass transfer probably is due 

to regional variations in the abundance of anhydrite in 

the Madison aquifer system, with availability of anhydrite 

for reaction increasing northward in western South 

Dakota toward the Williston basin. No trends in any other 

reactions were evident. The quantity of CH20 oxidized 

ranged between 0.30 and 1.07 mmol/kg of water, and 

pyrite formation and ion exchange generally were of 

minor significance.

Ion exchange was more significant at the Dupree well 

and the Eagle Butte well (well No. 23 in South Dakota) 

(0.66 and 1.22 mmol Ca2+ exchanged per kilogram of 

water, respectively). Some halite dissolution (0.97 to 4.18 

mmol dissolved per kilogram of water) was found at the 

Murdo (well No. 25 in South Dakota), Hilltop Ranch (well 

No. 22 in South Dakota), Prince (well No. 26 in South 

Dakota), Hamilton (well No. 21 in South Dakota), Eagle 

Butte, and Dupree wells.

Trends in 14C age were neither expected nor found for 

the nine wells near the end of flow path 8. Calculated ages 

vary between about 8,000 and about 17,000 yr and 

average 12,400 ±3,000 yr.

OVERVIEW OF CHEMICAL REACTIONS 

IN THE MADISON AQUIFER

Complete reaction models were developed at wells for 

which carbon- and sulfur-isotope data were available (table 

21). Partial reaction information for the mass transfers 

of anhydrite, dolomite, halite, and ion exchange was obtained from the chemical analyses in cases of limited 

isotopic data. In addition, results from some wells in the 

vicinity of the Black Hills not previously modeled have 

been added to the mass-transfer maps (figs. 43 and 47-50) 

to improve control.

Some estimated values of anhydrite mass transfer are 

displayed in figure 43. In the absence of sulfur-isotope 

data, anhydrite mass transfer was estimated from the 

difference between recharge and final dissolved sulfate 

content, ignoring sulfate reduction. The total amount of 

anhydrite dissolved will be larger than the change in 

dissolved sulfate content by the amount reduced to 

hydrogen sulfide. The extent of sulfate reduction has been 

found to be relatively small in the Madison aquifer system 

where sulfur-isotope data have permitted a more complete 

evaluation (table 21).

The mass transfer of dolomite is defined (eq 47) by the 

change in magnesium content independently of the sulfurand carbon-isotope data and may, therefore, be compared 

with all other modeled results. The mass transfers of 

sylvite, halite, and ion exchange are also defined independently of the carbon- and sulfur-isotope data 

through mass-balance relations involving changes in 

dissolved sodium, potassium, and chloride (eqs 44-46).

The data in table 21 indicate that dedolomitization 

that is, dissolution of anhydrite and dolomite accompanying precipitation of calcite is the predominant 

reaction throughout the entire Madison aquifer. The extent to which this reaction proceeds appears to be both 

a function of flow distance (age) and availability of 

anhydrite for reaction. The anhydrite mass transfer is 

about 20 mmol dissolved per kilogram of water along flow 

paths 1, 2, 3, and 4. Along flow path 8 the quantity of
F76 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

anhydrite dissolved increases northward normal to the 

eastward direction of flow, indicating that the increased 

availability of anhydrite northward toward the Williston 

basin is a more dominant cause of reaction progress than 

distance down the flow path. Other waters show little or 

no anhydrite dissolution (flow path 6), indicating little 

abundance of the mineral there and more rapid flow 

velocities. The importance of mineral availability for reactions is further demonstrated by the fact that some of the 

largest quantities of anhydrite dissolution are found in 

some of the youngest (flow paths 1 and 7) and oldest (flow 

paths 2 and 3) waters.

There are systematic variations in the regional pattern 

of anhydrite dissolution in the Madison aquifer based on 

mass-transfer calculations (fig. 44). The quantity of 

anhydrite dissolved is affected by distance from recharge

48

r enne i 

JM J

COLORADO

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

50 100 150 200 MILES

0 50 100 150 200 KILOMETERS

EXPLANATION

^vijij MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

- APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

-10 LINE OF EQUAL ANHYDRITE MASS TRANSFERNumber is millimoles anhydrite removed from ( ) 

or added to ( + ) the solution phase per kilogram 

of water. Interval variable. Dashed where inferred

« ANHYDRITE MASS TRANSFER-ln mass-transfer 

units. E is estimated value. Well number shown 

in figure 7

PROJECT AREA BOUNDARY

FIGURE 43. Regional variation in anhydrite mass transfer (per kilogram of water) for all wells and springs sampled.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F77

areas; progressive increases in anhydrite dissolution occur with distance downgradient. This is most noticeable 

for the Madison aquifer in the area surrounding the Black 

Hills.

As expected for dedolomitization, the dolomite and 

calcite mass-transfer patterns are similar to the anhydrite 

mass-transfer results. The data in figures 44 and 45 indicate nearly linear relations between the mass transfers 

of anhydrite and dolomite, and between anhydrite and 

calcite. The slopes of points in figures 44 and 45 indicate 

that for flow paths 1 to 8, about 0.2 mmol/kg of water 

of dolomite dissolves for every millimole per kilogram of 

water of anhydrite dissolved, causing the precipitation of

approximately 0.5 mmol/kg of water of calcite. These 

mass transfers are similar to those calculated from thermodynamic simulations of hypothetical dedolomitization 

using the geochemical mass-transfer program PHREEQE 

(Parkhurst and others, 1980).

In a simulation of the dedolomitization reaction, 14.74 

mmol/kg of water of anhydrite was dissolved in a hypothetical recharge water saturated with calcite and 

dolomite at 10~ 2 atmPc02 aTi& 12 °C- Temperature was 

increased in the calculation to 56.5 °C while maintaining 

equilibrium with calcite and dolomite. The calculated mass 

transfer showed that for each millimole of anhydrite 

dissolved, an average of 0.26 mmol/kg of water of

11.00

10.00

9.00

8.00

I 7 '°°

6.00

5.00

4.00

O 3.00

2.00

1.00

0.00

O - Flow path 1 

A - Flow path 2 

V - Flow path 3

* - Flow path 4 

^ - Flow path 5

* - Flow path 6 

X - Flow path 7 

« - Flow path 8

O

O

* *

X

O

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

ANHYDRITE DISSOLVED, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 44. Trend in quantity of dolomite dissolved (per kilogram of water) as a function of anhydrite dissolution along flow

paths 1 to 8.
F78 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

dolomite dissolved and 0.58 mmol/kg of water of calcite 

precipitated. These hypothetical mass transfers are 

similar to the mass-balance results (table 21) for the 

Dupree well (well No. 20 in South Dakota). The only 

significant exceptions to these determinations in the 

calcite and dolomite mass transfers are several wells along 

flow paths 2 and 3 where substantial ion exchange was 

indicated. Here the dissolved-dolomite and precipitatedcalcite mass transfers were larger than the trend shown 

for the rest of the Madison aquifer.

Regionally, the dolomite mass-transfer pattern (fig. 

46A) generally parallels the anhydrite dissolution pattern 

(fig. 43). The quantity of dolomite dissolved also is found 

to increase normal to the direction of flow along flow path 

8, in response to the irreversible dissolution of anhydrite. 

Similar variations exist for the calcite mass transfer (fig. 

465).

Occurring with the dedolomitization reaction generally is some lesser organic-matter oxidation, dissolution of 

ferric hydroxide, and pyrite formation. Several flow paths,

24.00

22.00

20.00

18.00

16.00

14.00

§ 12.00

10.00

8.00

6.00

4.00

2.00 -

0.00

O - Flow path 1 

A - Flow path 2 

V - Flow path 3 

^ - Flow path 4 

+ - Flow path 5 

+ - Flow path 6 

X - Flow path 7 

if - Flow path 8

* * O

*

if

O

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 22.00 

ANHYDRITE DISSOLVED, IN MILLIMOLES PER KILOGRAM OF WATER

FIGURE 45. Trend in quantity of calcite precipitated (per kilogram of water) as a function of anhydrite dissolution along flow paths

Ito8.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F79

notably 5, 6 and 7, show little or no evidence of sulfate 

reduction, whereas water from wells in the Central Montana trough (flow paths 2 and 3) and points along flow 

paths 4 and 8 indicate greater sulfate reduction. For example, in water from wells in the Central Montana trough, 

as much as 4 to 8 mmol of organic matter is oxidized per 

kilogram of water. The mass-balance calculations indicate 

that less than 0.50 mmol of organic matter oxidized is 

typical of much of the Madison aquifer.

Accompanying the relatively substantial sulfate reduction along flow path 2 generally is some lesser pyrite 

precipitation and dissolution of FeOOH. As discussed 

earlier, this probably reflects lesser quantities of detrital 

FeOOH deposited in the Central Montana trough. Here 

most of the reduced sulfur species remain in solution. 

As a result of the modeling exercise it appears that the 

H£S found in wells along flow path 2 may have been 

introduced from other formations. Elsewhere in the 

Madison aquifer, sulfate reduction accompanies proportionately larger quantities of FeOOH and greater pyrite 

mass transfer.

Halite dissolution and ion exchange are significant only 

in limited parts of Montana and Wyoming, presumably 

reflecting availability of halite and clay minerals there.

Ion exchange is important along flow paths 2 and 3 and 

begins to contribute significantly to the mass transfer at 

points along flow paths 1, 5, and 8, but except for flow 

paths 2 and 3, ion exchange is not important throughout 

most of the Madison aquifer. There are systematic variations in the degree of ion exchange (as calculated from 

eq 46), increasing significantly northeastward from central Montana (fig. 47). Dissolution of halite is very important along flow paths 2, 3, and 5 and minor to absent 

throughout the rest of the Madison aquifer. Halite dissolution generally contributes 1 mmol of chloride per kilogram 

of water or less to most of the waters of the Madison 

aquifer, except in east-central Montana where, very 

abruptly, waters from most wells have more than 50 

millimoles of halite dissolved per kilogram of water (fig. 

48). The dissolution of a sylvitelike phase always accompanies halite dissolution but at mass-transfer levels of only 

5 percent that of halite.

As discussed earlier, the C02 gas mass transfer is expected to be near zero, and the magnitude of this term 

has been used as a criterion in refining the mass-balance 

models. For most of the wells modeled, the C02 gas mass 

transfer is 0.0 ± 0.5 mmol/kg of water, which, taking into 

account the many modeling assumptions, reasonably supports the conclusion of a closed system. Other wells show 

larger C02 gas mass transfers that are attributed to 

uncertainties in the composition of recharge waters or 

other errors in the modeling exercise.

The thermodynamic calculations in table 9 show that 

most of the waters are undersaturated with anhydrite,

asymptotically approaching anhydrite saturation farther 

downgradient. For those few waters in equilibrium with 

anhydrite, dissolution of anhydrite can take place in 

response to the exchange of Ca2+ for Na+ and sulfate 

reduction; these reactions decrease the anhydrite saturation state. The calcite, dolomite, pyrite, and FeOOH mass 

transfers are directly linked thermodynamically to the 

irreversible dissolution of anhydrite and sulfate reduction. 

Halite dissolution also responds irreversibly, but its mass 

transfer is a function of halite availability. The ionexchange reaction probably proceeds reversibly, but the 

degree of the reaction probably is related to the abundance of sites available for exchange.

REGIONAL PATTERNS IN (J 34Sanhydrite

The best estimates of the sulfur-isotopic composition of 

dissolving anhydrite in the Madison aquifer are given in 

table 22. The lightest values (about 8 per mil) are found 

in northeast Wyoming and eastward through the Black 

Hills (9 to 10 per mil). As shown in figure 49, these modelderived values of <J 34Sanhydrite increase progressively to 

heavier values northward into east-central Montana and 

northward through western South Dakota. The lighter 

sulfur-isotopic values in northeastern Wyoming previously 

have been interpreted as indicating leakage of waters 

from overlying Pennsylvanian and Permian rocks in 

response to extensive pumpage from the Madison aquifer 

(Busby and others, 1983). The values in northeast Wyoming (fig. 49) are not necessarily anomalous; they may 

instead be part of a regional pattern in the Madison 

aquifer.

If the sulfur-isotope variation in figure 49 reflects that 

of the in situ anhydrite, then the pattern indicates a 

terrigenous source of sulfur in northeast Wyoming and 

southwest South Dakota. According to Sando (1976b), 

land masses associated with the transcontinental arch 

were emergent in southeast Wyoming and northeast 

Nebraska throughout the Mississippian and could have 

contributed light sulfur (presumably from pyrite) to 

deposits in northeastern Wyoming and southwestern 

South Dakota. The inferred trend to heavier sulfurisotopic composition of anhydrite northward through 

western South Dakota and through east-central Montana 

probably corresponds to a decrease in deposition of terrigenous sulfur and the increasing effect of a marine 

evaporite facies.

CARBON-14 AGES

Carbon-14 ages are reaction-model dependent. The 

calculation of 14C age requires a detailed knowledge both 

of all mass transfers of carbon into and out of solution 

and of the 14C content of the incoming carbon. It is like-
F80 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

ly that all sources of carbon (dolomite, CH20) contain no 

14C, so the primary concern in 14C dating is the accuracy 

of the derived mass transfers. The 14C ages in table 22 

may be used with varying degrees of confidence. The most 

reliable ages were determined from waters that have 

undergone only the dedolomitization reaction. The tritium

data (table 11) indicate that some waters, particularly 

those near recharge areas, may be partially contaminated 

with modern 14C.

As discussed earlier, the 14C ages are particularly sensitive to the sulfur-isotopic composition of dissolving 

anhydrite and to the extent to which a magnesium sink,

48

46°

NORTH DAKOTA

Bismarck

-4QUTrt>C DAKOTA

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

50 100 150 200 MILES

0 50 100 150 200 KILOMETERS

EXPLANATION

MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

LINE OF EQUAL DOLOMITE MASS TRANSFERNumber is mass-transfer units of dolomite. Interval variable. Dashed where inferred

s DOLOMITE MASS TRANSFER-ln mass-transfer 

units. Well number shown in figure 7

PROJECT AREA BOUNDARY

NOTE: Mass-transfer units are millimoles of substance 

added to the aqueous phase per kilogram of water

FIGURE 46. Regional variation in millimoles of (A) dolomite dissolved and of (B) calcite transferred (per kilogram of water) for all wells

and springs sampled.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F81

such as Mg2+ /Na+ exchange, is present. Through the 

modeling process, lighter values of d 34Sanhydrite cause 

calculation of more sulfate reduction and thus oxidation 

of more (14C-depleted) organic matter. This additional 

dilution of the 14C leads to smaller calculated (reactioncorrected) 14C values and younger ages. Similarly, an

additional sink for magnesium causes calculation of more 

extensive dissolution of 14C-depleted dolomite, which 

again results in younger ages. Consequently, some of the 

more uncertain 14C ages are those along flow paths 2 and 

3, where extensive sulfate reduction and Mg2+/Na+ exchange have been included in the reaction models.

B

112°

48°

46°

42°

NORTH DAKOTA 

Bismarck

44°

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

50 100 150 200 MILES

0 50 100 150 200 KILOMETERS

EXPLANATION

:::;: .-{ MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

 APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

-5 LINE OF EQUAL CALCITE MASS TRANSFERNumber is mass-transfer units of calcite. 

Interval variable. Dashed where inferred

5 0 CALCITE MASSTRANSFER-ln mass-transfer 

units. Well number shown in figure 7

PROJECT AREA BOUNDARY

NOTE: Mass-transfer units are millimoles of substance 

removed from ( ) the aqueous phase per 

kilogram of water

FIGURE 46. Continued.
F82 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

The importance of the reaction-model corrections to 

14C dating of the waters, however, cannot be overlooked. 

The variation in the measured 14C content of all wells and 

springs along flow paths 1 to 8 as a function of the computed molal scale sulfate concentration is shown in figure

50. An abrupt and systematic decrease is seen in the 14C 

content as anhydrite dissolves. This decrease is in part 

due to radioactive decay because the older waters generally have greater concentrations of dissolved anhydrite. The 

measured 14C values are further decreased by dilution

48°

46°

ro>f \ 

|__

44°

 - J 

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

50 100 150 200 MILES I I I

0 50 100 150 200 KILOMETERS

EXPLANATION

f-^jvj MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

- APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

 2 LINE OF EQUAL ION EXCHANGE-Number is 

mass-transfer units of exchange. Interval 

variable. Dashed where inferred

ION EXCHANGE-ln mass-transfer units. Well 

number shown in figure 7

PROJECT AREA BOUNDARY

NOTE: Mass-transfer units are millimoles of substance 

added to the aqueous phase per kilogram of water

FIGURE 47. Regional variation in degree of ion exchange (divalent cation exchanged for sodium per kilogram of water) for all wells and

springs sampled.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F83

from dissolution of dolomite and oxidation of CE^O as 

well as by incorporation of 14C in secondary calcite 

precipitated via the dedolomitization reaction.

Taking Sleeping Buffalo (well No. 18 in Montana) as an 

example, the measured 14C content of 4.20 percent 

modern is equivalent to an age of 26,000 yr if no corrections for reaction are made; that is, the unadjusted age 

is equal to (5730/ln 2) (100/measured percent modern). 

Correcting for an assumed congruent dissolution of carbonate minerals in an original CCVHgO solution, the 

model of Pearson and White (1967) results in an age for 

water from the Sleeping Buffalo well of about 17,000 yr

48

42°

NORTH DAKOTA

 

Bismarck

44°

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

50 100 150 200 MILES

i_____i____i____i

0 50 100 150 200 KILOMETERS

EXPLANATION

:::/j MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

 APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

 1 - LINE OF EQUAL HALITE MASS TRANSFERNumber is mass-transfer units of halite. Interval 

variable. Dashed where inferred

!:'# HALITE MASS TRANSFER-ln mass-transfer

units. E is estimated value. Well number shown 

in figure 7

PROJECT AREA BOUNDARY

NOTE: Mass-transfer units are millimoles of 

substance added to the aqueous phase 

per kilogram of water

FIGURE 48. Regional variation in millimoles of halite dissolved (per kilogram of water) for all wells and springs sampled.
F84 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

(assuming <J 13Cdoiomite is 0 per mil and <J 13Cco2 gas is 

- 9.44 per mil; see table 18). When the data are corrected 

for incongruent dissolution of carbonate minerals and 

sulfate reduction using the Rayleigh-distillation equations 

of Wigley and others (1978, 1979)-that is, the final 

calculated mass transfer (table 21) the water at Sleeping Buffalo is found to be approximately modern (probably 

less than 5,000 yr old). Not all examples are as extreme 

as the water from the Sleeping Buffalo well. For example, the Mysse Flowing Well (well No. 20 in Montana) has 

an unadjusted 14C age of 40,000 yr. By correcting for 

assumed congruent reaction (Pearson and White, 1967),

Base from U.S. Geological Survey, 

1:7,500,000 map, 1970

50 100 150 200 MILES

0 50 100 150 200 KILOMETERS

EXPLANATION

^ MISSISSIPPIAN OR OLDER ROCKSExposed at land surface

 APPROXIMATE EASTERN LIMIT OF 

MADISON LIMESTONE

 10- LINE OF EQUAL 534SANHYDRITE -Number is 

isotopic fractionation in per mil. Interval 

2 per mil. Dashed where inferred

^34SANHYDRITE Isotopic fractionation in per mil. 

Well number shown in figure 7

PROJECT AREA BOUNDARY

NOTE: Short dashes across mountains (stipple 

pattern) indicate that the uplift was not present 

at the time of depression

FIGURE 49. Regional variation in estimated sulfur-isotopic composition of dissolving anhydrite for all wells and springs sampled.
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F85

the 14C age is decreased to 25,500 yr, which compares 

favorably with 22,600 yr (table 22) based on all corrections for reactions.

In an earlier publication (Back and others, 1983), several 

waters from the Madison aquifer in the vicinity of the 

Black Hills were 14C-dated by correcting for incongruent 

dissolution only (the dedolomitization reaction). This led 

to a 14C age at the Philip well (well No. 19 in South 

Dakota), for example, of 20,000 yr. When both the 

dedolomitization reaction and sulfate reduction were corrected, the water from the Philip well was found to be 

younger, about 16,800 yr old (table 22).

The curve in figure 50 shows the calculated 14C content along a hypothetical flow path to the Dupree well 

(well No. 20 in South Dakota). The 14C content was 

calculated using the computed mass transfer to the 

Dupree well and by assuming constant relative rates 

of reaction in proportion to the computed mass transfer. Although the curve in figure 50 is based on the

mass transfer to the Dupree well, similar variation is 

expected for other waters that are affected predominantly by the dedolomitization reaction. Modern waters 

will have measured 14C contents similar to those along 

the curve as a function of anhydrite dissolution. Older 

waters have 14C values that plot below the curve. The 

ratio of the calculated (corrected) and measured 14C 

at the Dupree well results in an age of about 10,800 yr. 

Waters with large anhydrite mass transfers and relatively 

large values of measured 14C, such as Cascade Spring and 

the Evans Plunge Spring (fig. 50), are interpreted to be 

either virtually modern or possibly mixtures of older 

waters that have been contaminated in part with modern 

soil C02 gas. Unfortunately, no tritium data are available 

for these two waters to check for contamination. Water 

from Lodgepole Warm Spring (fig. 50) is contaminated 

with tritium (31.8 TU; see table 11), indicating either a 

modern water or an older water mixed with a modern 

source.

90.00

80.00

70.00

60.00

~ 50.00

o

^ 40.00

O m 

tr 

<

0 30.00 

o

20.00

10.00

0.00

 \

Calculated reaction path to Dupree

O

O

O

O - Flow path 1

A - Flow path 2

V - Flow path 3

O - Flow path 4

+ - Flow path 5

+ - Flow path 6

X - Flow path 7

 £ - Flow path 8

+ 

, V

^Dupr O

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00 

SULFATE CONCENTRATION, IN MILLIMOLES PER KILOGRAM OF WATER

22.00

FIGURE 50. Variation in measured carbon-14 content as a function of concentration of dissolved sulfate along flow paths 1 to 8.
F86 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

FLOW VELOCITIES CALCULATED 

FROM CARBON-14 AGES

Calculated 14C ages for selected waters from the 

Madison aquifer, as well as estimates of the length of the 

assumed horizontal, straight flow path from the recharge 

area to the well or spring, calculated flow velocities, and 

regional hydraulic-conductivity values are summarized in 

table 3. The flow velocities vary between 7.2 and 86.7 

ft/yr, the average being about 43.3 ft/yr. The slowest 

velocities are found along flow path 5 in northeast Wyoming, where the average is 18.9 ft/yr. Waters along flow 

paths 2 and 3 have velocities ranging between 23.2 and 

82.0 ft/yr. Flow velocities were calculated along flow path 

4 to the Belle Creek and Ranch Creek wells and averaged 

about 39 ft/yr. The velocity to Evans Plunge on flow path 

7 (68.2 ft/yr) is high, similar to that at the Sarpy Mine 

well (82.0 ft/yr), whereas the flow velocity at the Upton 

well (15.9 ft/yr) on flow path 6 indicates a retardation in 

flow similar to that suggested by L.F. Konikow (U.S. 

Geological Survey, written commun., 1984). A greater 

range of velocities is calculated along flow path 8 (33.3 

to 86.7 ft/yr). An average velocity of 57.8 ft/yr probably 

is most representative of waters along flow path 8.

RATES OF REACTIONS

Average net rates of reaction (table 24) were calculated 

for the same set of wells or springs for which average flow 

velocities were calculated (table 3). The average net rate 

of reaction was calculated using the net mass transfer 

given in table 21 and the 14C age (table 3). The unit of 

the rate of the reaction is millimoles per kilogram of water 

per year and has not been corrected to unit mineralsurface area, as the mineral-surface area in contact with 

a kilogram of water is not known for the Madison aquifer.

The reactions involving dissolution of anhydrite and 

halite and organic-matter oxidation are regarded as irreversible in the Madison aquifer, with the exception of a 

few waters that appear to be anhydrite saturated (fig. 11). 

The dissolution of dolomite and precipitation of calcite, 

in contrast, are reversible and are linked to the dissolution of anhydrite. The rates of anhydrite dissolution vary 

from 2.2 to 32.6 x 10~ 4 mmol/kg of water per year. The 

rate variability probably is a function of the availability 

of anhydrite for dissolution. From laboratory studies, the 

rate of dissolution of gypsum (Liu and Nancollas, 1971) 

and presumably anhydrite is expected to be rapid. In the 

laboratory, stirred suspensions of gypsum crystals reach 

equilibrium within hours to days depending on the solid 

surface area/solution ratio. Because waters throughout 

the Madison aquifer are not in equilibrium with anhydrite, 

it is evident that mineral abundance is rate limiting. 

Several waters that indicate very rapid anhydrite dissolution, such as Delzer No. 2 (well No. 8 in South Dakota) 

and Evans Plunge (spring No. 10 in South Dakota) probably flow in part through bedded evaporites or are receiving waters from other formations. Waters with very slow 

rates of anhydrite dissolution probably flow through areas 

where much of the anhydrite has already been dissolved. 

The slow rates probably represent the dissolution of 

nodules or pockets of anhydrite within the dolostone that, 

through dolomite dissolution, occasionally come into 

contact with the ground water. Once the anhydrite is 

exposed, dissolution probably is very rapid. From a 

regional perspective the average effect is one of slow 

anhydrite dissolution.

As shown in figures 44 and 45, the dolomite and calcite 

mass transfers are approximately linear functions of the 

anhydrite mass transfer, and thus a significant correlation among calcite, dolomite, and anhydrite rates is 

expected. The calculated rates of calcite precipitation vary 

between 1.6 and 16.3 x 10 ~ 4 mmol/kg of water per year, 

whereas the dolomite dissolution rate varies from 0.3 to 

6.6xlO~ 4 mmol/kg of water per year.

SUMMARY AND CONCLUSIONS

In interpreting the water chemistry of the Madison 

aquifer, extensive use has been made of thermodynamicspeciation calculations, mass-balance calculations, and 

carbon- and sulfur-isotopic data. The major conclusions 

are as follows:

1. The predominant ground-water reaction in the 

Madison aquifer is dedolomitization (calcite or aragonite 

precipitation accompanying dolomite dissolution), which 

is driven by the irreversible dissolution of anhydrite. 

Sulfate reduction, [Ca2+ + Mg2+]/Na+ exchange, and 

halite dissolution are locally important, particularly in central Montana.

2. The carbon-isotope data, coupled with the massbalance calculations, indicate that incorporation of magnesium in a secondary mineral, such as through ion 

exchange or formation of authigenic magnesium-silicate 

minerals, is an important reaction in parts of central Montana and northeast Wyoming. The indicated loss of 

magnesium causes additional dissolution of dolomite in 

accounting for the measured heavy d 13C values (about 

-2.0 per mil).

3. With the possible exception of flow path 2 and the 

beginning of flow path 3, the ground-water system was 

assumed to be closed to CC>2 gas.

4. The sulfur-isotopic composition of dissolving 

anhydrite in northeastern Wyoming and western South 

Dakota, estimated through the modeling process, appears 

to be significantly lighter than that of Mississippian 

marine evaporites, indicating, in part, a terrigenous 

source of sulfur. The lighter sulfur-isotopic values in
GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F87

TABLE 24. Summary of average net rates of reactions

Rate 

(millimoles per kilogram of water per year)

Spring 

or well

Keg Coulee

Sarpy Mine

Mysse Flowing Well

HTH No. 1

Ranch Creek

Belle Creek

Delzer No. 2

Conoco No. 175

MKM

Shidler

Conoco No. 44

Upton

Evans Plunge

Kosken

Philip

Midland

Murdo

Hilltop Ranch

Prince

Hamilton

Eagle Butte

Dupree

Spring 

or well 

number

15

19

20

14

23

24

8

11

10

9

8

15

10

1

19

24

25

22

26

21

23

20

State

Mont.

Mont.

Mont.

Wyo.

Mont.

Mont.

S.Dak.

Wyo.

Wyo.

Wyo.

Wyo.

Wyo.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

S.Dak.

Flow 

path

2

3

3

4

4

4

4

5

5

5

5

6

7

8

8

8

8

8

8

8

8

8

Anhydrite 

dissolution do-4)

6.6

9.1

9.0

6.7

5.5

6.0

32.6

4.5

4.6

8.8

3.8

2.2

24.7

6.7

4.2

5.3

6.3

12.7

16.6

10.2

12.1

13.7

Calcite 

precipitation do-4)

6.5

4.8

2.4

3.2

2.5

2.9

16.3

3.3

1.8

2.1

1.6

2.3

11.0

3.4

2.6

3.1

3.2

7.8

8.3

5.4

7.4

7.6

Dolomite 

dissolution do-4)

3.2

2.0

1.6

1.3

1.1

1.2

6.6

.9

.3

.8

.7

.9

3.1

1.1

.9

1.2

1.3

2.4

3.4

2.2

3.3

2.8

CH20 

oxidation do-4)

15.0

6.4

4.4

4.9

2.7

3.0

21.3

1.2

3.1

4.1

.2

2.7

18.3

2.8

5.1

5.4

4.9

6.3

13.7

2.3

3.0

4.9

Halite 

dissolution do-4)

23.3

--

6.8

1.8

1.5

1.5

2.8

2.2

15.3

14.3

5.8

--

11.8

.8

.3

.3

1.0

4.2

3.3

.7

.9

1.9

northeastern Wyoming are thought to be part of a 

regional depositional pattern for the Madison aquifer, 

rather than the result of a withdrawal-induced leakage 

from the Pennsylvanian and Permian Minnelusa 

Formation.

5. Ground-water ages vary from virtually modern to 

about 23,000 yr. The 14C ages indicate flow velocities of 

between 7 to 87 ft/yr. Hydraulic conductivities based on 

average carbon-14 flow velocities are similar to those 

based on digital simulation of the flow system (Downey, 

1984).

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Bischoff, J.L., 1968, Kinetics of calcite nucleation Magnesium ion 

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Bottinga, Y., 1969, Calculated fractionation factors for carbon and 

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Brown, D.L., Blakennagel, R.K., MacCary, L.M., Peterson, J.A., 1982, 

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Busenberg, E., Plummer, L.N., and Parker, V.B., 1984, The solubility 

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F88 GEOLOGY AND HYDROLOGY OF THE MADISON LIMESTONE

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Downey, J.S., 1982, Machine-readable data files from the Madison 

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GEOCHEMICAL EVOLUTION OF WATER IN THE MADISON AQUIFER F89

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