U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2009–5261

Prepared in cooperation with the Montana Department of Environmental Quality

Travel Times, Streamflow Velocities, and 

Dispersion Rates in the Yellowstone River, 

Montana
Cover. Looking downstream at the non-toxic dye plume from the Miles City Bridge over the Yellowstone River, near Miles City,

Montana. Photograph by Peter M. McCarthy, U.S. Geological Survey, taken September 23, 2008.
Travel Times, Streamflow Velocities, and 

Dispersion Rates in the Yellowstone River, 

Montana

By Peter M. McCarthy

Prepared in cooperation with the Montana Department of Environmental Quality

Scientific Investigations Report 2009–5261

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2009

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reproduce any copyrighted materials contained within this report.

Suggested citation:

McCarthy, P.M., 2009, Travel times, streamflow velocities, and dispersion rates in the Yellowstone River, Montana: 

U.S. Geological Survey Scientific Investigations Report 2009–5261, 25 p.
iii

Contents

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

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

Purpose and Scope ..............................................................................................................................2

Description of the Study Area ............................................................................................................2

Methods for Determining Travel Times, Streamflow Velocities, and Dispersion Rates ....................2

General Description of Dye Tracing...................................................................................................4

Dye Injections........................................................................................................................................5

Fluorometric Measurements ..............................................................................................................6

Development of Time-Concentration Curves .................................................................................10

Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River.....................10

Evaluation of the Transport-Model Estimates................................................................................16

Uncertainty in Estimating Mean Streamflow Velocities...............................................................17

Limitations of the Travel Times, Streamflow Velocities, and Longitudinal 

Dispersion Rates ...................................................................................................................23

Summary and Conclusions.........................................................................................................................23

Selected References...................................................................................................................................24

Appendixes ...................................................................................................................................................25

Appendix 1. Raw Data from the Self-Contained Underwater Fluorescence 

Apparatus for Measurement Sites along the Yellowstone River, 

Montana...................................................................................... in pocket at back of report

Appendix 2. Adjusted Data from the Self-Contained Underwater Fluorescence 

Apparatus for Measurement Sites along the Yellowstone River, 

Montana...................................................................................... in pocket at back of report

Appendix 3. Bench Analysis of Grab Samples Measured with Turner Designs 

Model 10 Fluorometer for Measurement Sites along the Yellowstone 

River, Montana........................................................................... in pocket at back of report

Appendix 4. Data and Equations for Three-Parameter Log-Normal Curves of the 

Self-Contained Underwater Fluorescence Apparatus and Grab-Sample 

Data for Measurement Sites along the Yellowstone River, 

Montana...................................................................................... in pocket at back of report

Figures

1. Map showing locations of the study area and sites used for dye injection or 

measuring, Yellowstone River, Montana ..................................................................................3

2. Graph showing a typical time-concentration curve for the movement of dye 

past a fixed measurement point downstream from a dye injection.....................................5

3–6. Maps showing:

3. Location of subreach for dye injection at Lockwood Bridge, Montana.....................7

4. Location of subreach for dye injection at Myers Bridge, Montana............................8

5. Location of subreach for dye injection at Cartersville Dam, Montana.......................8

6. Location of subreach for dye injection at Miles City Bridge, Montana......................9

7–18. Graphs showing:

7. Concentration of Rhodamine WT dye at Pompeys Bridge, Montana.......................11
iv

Tables

1. Injection and measurement sites along the Yellowstone River, Montana..........................4

2. Optimal mixing length for each injection site along the Yellowstone River, 

Montana .......................................................................................................................................10

3. Travel-time data and mean streamflow velocities for the Yellowstone River, 

Montana .......................................................................................................................................15

4. Longitudinal dispersion rates for the Yellowstone River, Montana....................................16

5. Calculated and estimated streamflow velocities for each subreach within 

the study area, Yellowstone River, Montana..........................................................................19

6. Estimated and measured peak Rhodamine WT dye concentrations for each 

subreach within the study area, Yellowstone River, Montana............................................20

8. Rhodamine WT dye time-concentration curve fit to data from the 

discrete samples collected at Forsyth Bridge after September 29 dye 

injection at Lockwood Bridge, Montana........................................................................12

9. Time-concentration curves for Rhodamine WT dye injected at 

Lockwood Bridge, Montana.............................................................................................12

10. Time-concentration curves for Rhodamine WT dye injected at 

Myers Bridge, Montana....................................................................................................13

11. Time-concentration curves for Rhodamine WT dye injected at 

Cartersville Dam, Montana ..............................................................................................13

12. Time-concentration curves for Rhodamine WT dye injected at 

Miles City Bridge, Montana .............................................................................................14

13. Relation between travel times of the peak concentration and 

durations of the dye plume for the Yellowstone River, Montana...............................17

14. Calculated and estimated velocity data for subreach upstream from 

each site..............................................................................................................................18

15. Calculated velocity of the centroid of the dye plume from Lockwood 

Bridge to Huntley Bridge and the velocity-streamflow relations 

developed from the transport model (McCarthy, 2006) and routine 

streamflow measurements for the Yellowstone River at Billings, 

Montana (gaging station 06214500)................................................................................21

16. Calculated velocity of the centroid of the dye plume from Forsyth 

Bridge to Cartersville Dam and the velocity-streamflow relations 

developed from the transport model (McCarthy, 2006) and routine 

streamflow measurements for the Yellowstone River at Forsyth, 

Montana (gaging station 06295000)................................................................................21

17. Calculated velocity of the centroid of the dye plume from Fort Keogh 

Bridge to Kinsey Bridge and the velocity-streamflow relations 

developed from the transport model (McCarthy, 2006) and routine 

streamflow measurements for the Yellowstone River at Miles City, 

Montana (gaging station 06309000)................................................................................22

18. Calculated velocity of the centroid of the dye plume from Fallon Bridge 

to Glendive Bridge and the velocity-streamflow relations developed 

from the transport model (McCarthy, 2006) and routine streamflow 

measurements for the Yellowstone River at Glendive, 

Montana (gaging station 06327500)................................................................................22
v

Conversion Factors, Acronyms, and Abbreviated Units

Multiply By To obtain

Length

mile (mi) 1.609 kilometer (km)

Volume

gallon (gal) 3.785 liter (L) 

Flow rate

foot per second (ft/s) 0.3048 meter per second (m/s)

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Mass

pound, avoirdupois (lb) 0.4536 kilogram (kg) 

Hydraulic gradient

foot per mile (ft/mi) 0.1894 meter per kilometer (m/km)

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

°F=(1.8×°C)+32

Horizontal coordinate information is referenced to both the North American Datum of 1927 

(NAD 27) and the North American Datum of 1983 (NAD 83).

Concentrations of chemical constituents in water are given in milligrams per liter (mg/L) and 

micrograms per liter (μg/L).

Acronyms used in this report

ADCP acoustic Doppler current profiler 

CD–ROM Compact Disk–Read-Only Memory

RWT Rhodamine WT

SCUFA Self-Contained Underwater Fluorescence Apparatus

USGS U.S. Geological Survey

Abbreviated units used in this report

ft/mi foot (feet) per mile

ft/s foot (feet) per second

ft3/s cubic foot (feet) per second

L liter

mg/L milligram(s) per liter

μg/L microgram(s) per liter

NTU nephelometric turbidity units

Travel Times, Streamflow Velocities, and Dispersion Rates 

in the Yellowstone River, Montana

By Peter M. McCarthy

Abstract

The Yellowstone River is a vital natural resource to the 

residents of southeastern Montana and is a primary source 

of water for irrigation and recreation and the primary source 

of municipal water for several cities. The Yellowstone River 

valley is the primary east–west transportation corridor through 

southern Montana. This complex of infrastructure makes the 

Yellowstone River especially vulnerable to accidental spills 

from various sources such as tanker cars and trucks. In 2008, 

the U.S. Geological Survey (USGS), in cooperation with the 

Montana Department of Environmental Quality, initiated a 

dye-tracer study to determine instream travel times, streamflow velocities, and dispersion rates for the Yellowstone River 

from Lockwood to Glendive, Montana. The purpose of this 

report is to describe the results of this study and summarize 

data collected at each of the measurement sites between 

Lockwood and Glendive. This report also compares the results 

of this study to estimated travel times from a transport model 

developed by the USGS for a previous study. For this study, 

Rhodamine WT dye was injected at four locations in late 

September and early October 2008 during reasonably steady 

streamflow conditions. Streamflows ranged from 3,490 to 

3,770 cubic feet per second upstream from the confluence of 

the Bighorn River and ranged from 6,520 to 7,570 cubic feet 

per second downstream from the confluence of the Bighorn 

River.

Mean velocities were calculated for each subreach 

between measurement sites for the leading edge, peak concentration, centroid, and trailing edge at 10 percent of the peak 

concentration. Calculated velocities for the centroid of the 

dye plume for subreaches that were completely laterally 

mixed ranged from 1.83 to 3.18 ft/s within the study reach 

from Lockwood Bridge to Glendive Bridge. The mean of the 

completely mixed centroid velocity for the entire study reach, 

excluding the subreach between Forsyth Bridge and Cartersville Dam, was 2.80 ft/s. Longitudinal dispersion rates of the 

dye plume for this study ranged from 0.06 ft/s for the subreach 

upstream from Forsyth Bridge to 2.25 ft/s for the subreach 

upstream from Calyspo Bridge for subreaches where the dye 

was completely laterally mixed. A relation was determined 

between travel time of the peak concentration and time for the 

dye plume to pass a site (duration). This relation can be used 

to estimate when the receding concentration of a potential 

contaminant reaches 10 percent of its peak concentration for 

accidental spills into the Yellowstone River.

Data from this dye-tracer study were used to evaluate 

velocity and concentration estimates from a transport model 

developed as part of an earlier USGS study. Comparison of 

the estimated and calculated velocities for the study reach 

indicate that the transport model estimates the velocities of 

the Yellowstone River between Huntley Bridge and Glendive Bridge with reasonable accuracy. Velocities of the peak 

concentration of the dye plume calculated for this study averaged 10 percent faster than the most probable velocities and 

averaged 12 percent slower than the maximum probable velocities estimated from the transport model. Peak Rhodamine WT 

dye concentrations were consistently lower than the transport 

model estimates except for the most upstream subreach of 

each dye injection. The most upstream subreach of each dye 

injection is expected to have a higher concentration because of 

incomplete lateral mixing. Lower measured peak concentrations for all other sites were expected because Rhodamine WT 

dye deteriorates when exposed to sunlight and will sorb onto 

the streambanks and stream bottom.

Velocity-streamflow relations developed by using routine 

streamflow measurements at USGS gaging stations and the 

transport model can be used to estimate mean streamflow 

velocities throughout a range of streamflows. The variation in 

these velocity-streamflow relations emphasizes the uncertainty 

in estimating the mean streamflow velocity for streamflows 

outside of the streamflows measured in this study.

Introduction

The Yellowstone River is a vital natural resource to the 

residents of southeastern Montana. The River originates in 

Yellowstone National Park and flows about 545 river miles 

to the Montana–North Dakota State line (fig. 1). The Yellowstone River is important to the economy of this region as a 

primary source of water for irrigation and recreation, and is 

the primary source of municipal water for the cities of Laurel, 

Billings, Lockwood, Forsyth, Miles City, and Glendive. 

Other communities along the Yellowstone River obtain their 
2 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

municipal water from shallow alluvial aquifers adjacent to the 

Yellowstone River and its tributaries.

The Yellowstone River valley is the primary east-west 

transportation corridor through southern Montana. Transportation infrastructure within this corridor includes Interstate 

Highways I-90 and I-94; Montana Rail Link and Burlington 

Northern Santa Fe Railroads; and many state highways, county 

roads, and city streets. Numerous roadways and railways cross 

over the Yellowstone River or are adjacent to the Yellowstone 

River. This complex of infrastructure makes the Yellowstone 

River especially vulnerable to accidental spills from various 

sources such as tanker cars and trucks. Knowledge of instream 

travel times, streamflow velocities, dispersion, and dilution 

of contaminant concentrations is a requirement for effective 

emergency response to a contaminant spill into the Yellowstone River. Additionally, travel times, streamflow velocites, 

and longitudinal dispersion rates are needed to calibrate 

hydraulic and water-quality models of the Yellowstone River.

McCarthy (2006) developed a transport model to 

provide estimates of instream travel times and concentrations of a soluble compound in the Yellowstone River. This 

model utilizes regression equations developed by Jobson 

(1999), which were based on empirical time-of-travel data 

from numerous dye-tracer studies conducted throughout the 

United States. The model developed by McCarthy (2006) was 

not calibrated because travel-time data were not available. 

However, precipitation-driven flood-wave velocities were 

estimated using hydrographs, and these velocities were used 

to constrain the model in lieu of measured travel-time data. In 

2008, the U.S. Geological Survey (USGS), in cooperation with 

the Montana Department of Environmental Quality, initiated a 

dye-tracer study to determine travel times, streamflow velocities, and longitudinal dispersion rates for the Yellowstone 

River in order to validate the transport model and to calibrate a 

water-quality model for the river.

Purpose and Scope

The purpose of this report is to describe the results of a 

study to determine travel times, streamflow velocities, and 

longitudinal dispersion rates for a reach of the Yellowstone 

River for one streamflow condition. For this study, Rhodamine WT (RWT) dye was injected at four locations in late 

September and early October 2008. Dye concentrations were 

measured at an average of four sites downstream from each 

dye injection. These data were used to determine travel times, 

streamflow velocities, and longitudinal dispersion rates for 

a 266-mi reach of the Yellowstone River from Lockwood to 

Glendive. Included in this report are the time-of-travel data 

collected during the dye tracer study, analysis of the data 

to determine streamflow velocities and longitudinal dispersion rates, and comparisons of the data to estimates from the 

transport model developed by McCarthy (2006). The traveltime data collected and analyzed for this study are in appendixes 1–4 [Compact Disk-Read Only Memory (CD–ROM) 

located in the inside back cover].

Description of the Study Area

The upstream end of the study reach is downstream 

from Billings, Montana, where U.S. Highway 87 crosses the 

Yellowstone River (Lockwood Bridge) at river mile 360.6 

(table 1). The study reach extends approximately 266 river 

miles downstream to the West Bell Street Bridge (Glendive 

Bridge), river mile 94.6, in Glendive, Montana (fig.1) [river 

miles derived from the Montana Department of Natural 

Resources and Conservation (1976) river mile index of the 

Yellowstone River and Google, Inc. (2008)]. Within the study 

reach, three major tributaries join the Yellowstone River: 

Bighorn River, Tongue River, and Powder River. Mean annual 

streamflow for the Yellowstone River at Billings, based on 

73 years of record from 1928 to 2002, is 6,970 ft3/s. Mean 

annual streamflow for the Yellowstone River at Glendive, 

based on 10 years of record from 1897 to 1934, is 13,300 ft3/s. 

Mean annual streamflow for the Bighorn River is 3,860 ft3/s, 

whereas the mean annual streamflows for the Tongue River 

and Powder River are 409 ft3/s, and 576 ft3/s, respectively 

(McCarthy, 2005). The channel slope of the Yellowstone River 

within the study reach ranged from 10.6 ft/mi at Billings to 

3.2 ft/mi at Glendive (McCarthy, 2006). The channel substrate 

changes from large cobbles mixed with large gravel and small 

boulders at Billings to gravel and sand at Glendive.

Throughout the study reach, the Yellowstone River is 

used for municipal-water supply, irrigation supply, and recreation. Forsyth, Miles City, and Glendive are three communities within this study reach that obtain their municipal-water 

supply from the Yellowstone River. Lockwood and Billings 

obtain their municipal-water supply from the Yellowstone 

River upstream from the study reach. Three low-head dams 

located upstream from Huntley, downstream from Pompeys 

Bridge, and at Forsyth are used to divert water for irrigation 

in the river valley. Pumps also are used at other locations 

along the river to pump water from the Yellowstone River for 

irrigation. The scenic beauty and natural conditions along the 

river support a major riparian corridor for numerous wildlife 

species, recreational fishing, hunting, and paddling.

Methods for Determining Travel Times, 

Streamflow Velocities, and Dispersion 

Rates

Dye tracers are commonly used for time-of-travel studies 

because the dyes are water soluble and behave in the same 

manner as the water particles. A measure of the movement of 

the dye tracer will, in effect, be a measure of the movement 

of the fluid in the stream and of its dispersion characteristics (Kilpatrick and Wilson, 1989). Dispersion and mixing 

of the injected dye tracer occurs in all three dimensions of 

the channel: vertical, lateral, and longitudinal. The speed 

at which mixing of the dye occurs depends upon hydraulic 
Figure 1. Locations of the study area and sites used for dye injection or measuring, Yellowstone River, Montana.

River

River

Powder

Yellowstone

River

Bighorn

River

Tongue

Jordan

108° 107° 106° 105°

47°

46°

Glendive

Terry

Miles 

City

Forsyth

Hysham Locate

Bighorn

Fallon

Marsh

Billings

Lockwood

Huntley Pompeys 

Pillar

Kinsey

Custer Rosebud

Hardin

Colstrip

Ashland

Broadus Pryor

06295000

EXPLANATION

Cartersville Dam

Bridge used for dye-tracer study

U.S. Geological Survey 

 streamflow-gaging station 

 and number

Figure 6 area

Figure 5 

area

Figure 4 

Figure 3 area area

0 20 40 60 MILES

0 20 40 60 KILOMETERS

Base from Digital Chart of the World 

digital data, 1993, 1:1,000,000

Albers Equal-area Conic projection

Standard parallels 29°30’N and 45°30’N,

central meridian 106°00’W

North American Datum of 1983 (NAD 83) 

06327500

06295000

06214500

116° 104°

0 30 60 MILES

0 30 60 KILOMETERS

Base modified from U.S. 

Geological Survey, 1965 

North American Datum of 1927 

(NAD 27)

CANADA

UNITED STATES

NORTH DAKOTA DAKOTA

SOUTH 

IDAHO

Yellowstone

National

Park

49°

45° MONTANA

WYOMING

Sidney

Glendive

Terry

Miles City

Forsyth

Hysham

Billings

Bighorn

Laurel

06214500

06327500

06295000

110°

River

Yellowstone

06309000

06038500

Forsyth

Miles 

City

06038500

06309000

Methods for Determining Travel Times, Streamflow Velocities, and Dispersion Rates 3
Table 1. Injection and measurement sites along the Yellowstone River, Montana.

[River mileages from Montana Department of Natural Resources and Conservation (1976) and Google, Inc. (2008). Abbreviations: NAD 83, North American 

Datum of 1983; mi, miles]

Site names

Site location

(NAD 83) Site description

River 

mileage 

(mi)

Used for dye 

injection

Used for dye 

measuring Latitude Longitude

Lockwood Bridge N 45°47′47′′ W 108°28′12′′ U.S. Route 87 bridge near Lockwood 360.6 Yes No.

Huntley Bridge N 45°54′14′′ W 108°19′03′′ U.S. Route 312 bridge near Huntley 348.6 No Yes.

Pompeys Bridge N 45°59′48′′ W 108°00′32′′ Bundy Road bridge near Pompeys Pillar 328.6 No Yes.

Custer Bridge N 46°08′32′′ W 107°32′56′′ State Route 310 bridge near Custer 300.1 No Yes.

Myers Bridge N 46°15′16′′ W 107°20′44′′ State Route 311 bridge near Myers 283.3 Yes Yes.

Forsyth Bridge N 46°15′51′′ W 106°41′45′′ U.S. Route 12 bridge at Forsyth 238.8 No Yes.

Cartersville Dam N 46°16′32′′ W 106°40′47′′ Diversion dam at Forsyth 237.7 Yes Yes.

Rosebud Bridge N 46°16′29′′ W 106°27′53′′ State Route 446 bridge near Rosebud 224.2 No Yes.

Fort Keogh Bridge N 46°23′54′′ W 105°53′45′′ Fort Keogh Road bridge near Miles City 186.2 No Yes.

Miles City Bridge N 46°25′15′′ W 105°51′36′′ State Route 59 bridge at Miles City 183.7 Yes No.

Kinsey Bridge N 46°31′53′′ W 105°42′51′′ Kinsey Road bridge near Kinsey 171.9 No Yes.

Calypso Bridge N 46°46′47′′ W 105°24′40′′ Milwaukee Road bridge near Calypso 144.9 No Yes.

Fallon Bridge N 46°51′20′′ W 105°06′59′′ CB Route Road bridge near Fallon 126.9 No Yes.

Glendive Bridge N 47°06′21′′ W 104°43′09′′ West Bell Street pedestrian bridge at 

Glendive

94.6 No Yes.

4 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

characteristics of the stream, velocity variations, and how 

the dye is injected into the stream. Typically, vertical mixing 

throughout the water column requires the shortest distance 

from the point of injection. Lateral mixing across the width 

of the channel requires a greater distance than vertical mixing 

and until the dye is mixed laterally, its movement does not 

represent that of the total flow. Lateral mixing of the dye will 

occur more quickly if the dye is injected at multiple points 

across the channel. Longitudinal mixing or dispersion, the 

dispersion component of primary interest, occurs indefinitely 

(Kilpatrick and Wilson, 1989).

General Description of Dye Tracing

Dye-tracer concentrations measured continuously at 

points downstream from an instantaneous dye-injection can 

be plotted as a function of time to create a time-concentration 

curve. The time-concentration curves are the basis for 

determining time-of-travel and dispersion characteristics 

of streams. Theoretical characteristics of a time-concentration curve for a stream are shown in figure 2. This timeconcentration curve can be used to describe dye movement 

past a fixed measurement point located downstream from a 

dye injection (Kilpatrick and Wilson, 1989).

The dye concentration and movement characteristics pertinent 

to time-of-travel measurements are the following:

	 Tl, travel time of leading edge of dye plume;

	 Tp, travel time of peak concentration of dye 

plume;

	 Tc, travel time of centroid of dye plume;

	 T10p, travel time of trailing edge of dye plume 

where dye concentration is reduced to 

10 percent of the peak concentration; and

	 Tt, travel time of trailing edge of dye plume.

The mean travel time (t

c

) for the flow along the stream line is 

the difference in elapsed time of the centroids of the timeconcentration curves defined upstream and downstream on the 

same stream line:

t

c

=Tc(n+1)-Tc(n), (1)

where n is the number of the sampling site, and all other terms 

are as previously defined. The time required for the centroid 

of a dye plume to travel through a river reach (t

c

) is used to 

calculate the mean velocity of the centroid of the dye plume, 

which represents the mean streamflow velocity for the reach.
Figure 2. A typical time-concentration curve for the movement of dye past a fixed measurement point 

downstream from a dye injection (modified from Kilpatrick and Wilson, 1989).

Cp, peak concentration of dye plume

Tl ,travel time of leading edge of dye plume

Tp, travel time of peak concentration of dye plume 

Tc, travel time of centroid of dye plume

T10p, travel time of trailing edge of dye plume where 

dye concentration is reduced to 10 percent of 

the peak concentration 

Tt, travel time of trailing edge of dye plume

Cp

Tl Tp Tc T10p Tt

TIME AFTER INJECTION

CONCENTRATION

Methods for Determining Travel Times, Streamflow Velocities, and Dispersion Rates 5

The mean travel times of the leading edge (t

l

), peak 

concentration (t

p

), 10 percent of the peak concentration 

(t

10p

), and trailing edge (t

t

) along a given stream line are, 

respectively,

t

l

=Tl(n+1)-Tl(n), (2)

t

p

=Tp(n+1)-Tp(n), (3)

t

10p

=T10p(n+1)-T10p(n); and (4)

t

t

=Tt(n+1)-Tt(n), (5)

where all terms are as previously defined. These travel times 

are used to calculate the mean velocity of the travel-time 

components through a river reach. These mean velocities are 

then used to estimate longitudinal dispersion rates and also are 

used for management decisions for source-water intakes when 

a contaminant is present in the river.

The time, Td, necessary for the dye plume to completely 

pass a measurement site in a section is:

Td=Tt(n)-Tl(n). (6)

The time, T10d, necessary for the dye plume to pass a measurement site until the concentration of the dye plume on the trailing edge is 10 percent of the peak concentration (duration) is:

T10d=T10p(n)-Tl(n). (7)

Dye Injections

For this study, RWT dye was injected into the Yellowstone River at four sites: Lockwood Bridge, Myers Bridge, 

Cartersville Dam, and Miles City Bridge. Four separate 

instantaneous dye injections were made to keep peak dye 

concentrations below 10 μg/L at municipal water intakes 

while maintaining measureable concentrations of the dye. 

The maximum allowable peak concentration of RWT dye at 

municipal water intakes was determined by the U.S. Environmental Protection Agency to be 10 μg/L (Office of the Federal 

Register, 1998). Thirteen bridges and one low-head dam were 

used for injecting and measuring dye. Bridges were selected 

for injecting and measuring because they provided the safest 

access to the centroid of the river. Dye that was injected at 
6 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

Lockwood Bridge was measured downstream at Huntley 

Bridge, Pompeys Bridge, Custer Bridge, Myers Bridge, and 

Forsyth Bridge (fig. 3). Dye that was injected at Myers Bridge 

was measured downstream at Forsyth Bridge, Cartersville 

Dam, and Rosebud Bridge (fig. 4). Dye that was injected 

at Cartersville Dam was measured downstream at Rosebud 

Bridge, Fort Keogh Bridge, and Kinsey Bridge (fig. 5). Dye 

that was injected at Miles City Bridge was measured downstream at Kinsey Bridge, Calypso Bridge, Fallon Bridge, and 

Glendive Bridge (fig. 6).

RWT dye at 20-percent stock solution was used as the 

dye tracer for this time-of-travel study because it is a nearly 

conservative, water-soluble, nontoxic dye that can be detected 

at low concentrations using a fluorometer. The volume of 

RWT dye injected at Lockwood Bridge, Myers Bridge, Cartersville Dam, and Miles City Bridge was 68.0, 21.0, 33.0, and 

51.5 L, respectively. The resulting instantaneous concentration 

of dye in the Yellowstone River at each of the injection sites 

was 160, 25, 38, and 55 mg/L, respectively. Multiple lateralinjection points were used at each injection site to accelerate lateral mixing because travel times of the dye, once it is 

completely mixed laterally, more accurately represent mean 

streamflow velocity of the Yellowstone River. The distance 

required for complete lateral mixing (optimal mixing length, 

table 2) for each injection site was calculated based on flow 

and channel characteristics just downstream from each injection site using equation 7 from Kilpatrick and Wilson (1989). 

Three lateral-injection points (across the channel) were used 

for injections at Miles City Bridge, Cartersville Dam, and 

Myers Bridge. At each of these three injections sites, the 

total volume of dye injected was split equally and injected at 

points of 25, 50, and 75 percent of total streamflow measured 

from the left streambank to the right streambank. Similarly, 

four lateral-injection points were used for the injection at 

Lockwood Bridge because the streamflow was split equally 

into two channels. Thus, dye was injected at points of 40 and 

60 percent of streamflow in each channel.

Until the dye is completely mixed in the lateral dimension, its movement does not represent that of the total streamflow. The dye is considered completely mixed when the 

time-concentration curves for laterally separate points across 

the stream at the same location are virtually equal. Traveltime data represent the mean travel time of a river reach only 

when the dye is completely laterally mixed throughout the 

entire river reach between measurement sites. For each of the 

four injections (Lockwood Bridge, Myers Bridge, Cartersville Dam, and Miles City), travel times that were calculated 

from the injection site to the first measurement site downstream (Huntley Bridge, Forsyth Bridge, Rosebud Bridge, and 

Kinsey Bridge, respectively) do not accurately represent the 

mean travel time of those subreaches because the dye was not 

completely laterally mixed for the entire subreach. Therefore, 

each of these subreaches were overlapped by subsequent dye 

injections such that representative travel times could be calculated for the entire study reach from Huntley Bridge to Glendive Bridge. Only the unmixed travel times could be measured 

for the subreach from Lockwood Bridge (farthest upstream 

injection site) to the Huntley Bridge (farthest upstream 

measurement site). These travel times probably are not representative of the mean streamflow travel times because the dye 

was not completely laterally mixed for the entire subreach. For 

the purpose of this report, calculated velocities for subreaches 

where the dye is completely laterally mixed will be referred to 

as completely mixed velocities. In contrast, calculated velocities for subreaches where the dye is not completely laterally 

mixed will be referred to as unmixed velocities.

Fluorometric Measurements

At each measurement site downstream from a dyeinjection site, a Self-Contained Underwater Fluorescence 

Apparatus (SCUFA) was used to measure the fluorescence 

of the dye (Turner Designs, 2004). The SCUFA fluorometer 

converts the measured fluorescence of the dye to a direct 

concentration when it is calibrated to known dye concentrations. Prior to each of the dye injections, the SCUFAs were 

calibrated against known dye concentrations of 0 μg/L, 5 μg/L, 

15 μg/L, and 25 μg/L depending on the peak dye concentration 

expected at the measurement site. The SCUFAs were deployed 

from bridges at each measurement site and were suspended 

in the water column in the centroid of flow. The centroid of 

flow was determined by making an instantaneous streamflow measurement with an acoustic Doppler current profiler 

(ADCP) at the measurement site. Instantaneous streamflow 

measurements were made in accordance with the procedures 

described by Oberg and others (2005). At measurement sites 

where streamflow measurements could not be made, streamflow was estimated from upstream and downstream streamflow measurements.

SCUFAs were deployed at each measurement site to 

record the dye concentrations from the arrival to the trailing 

edge at 10 percent of the peak concentration of the dye plume. 

The SCUFAs were programmed to collect a reading every 

30 seconds for most sites. For two sites, the collection rate was 

increased to every 10 seconds in an attempt to record more 

data before algae accumulation affected the measurements. 

Because of time constraints, the concentration of the trailing 

edge was measured until it decreased to less than 10 percent of 

the peak concentration.

Fluorescence measurements are commonly affected by 

temperature and turbidity. The SCUFA is ideal for RWT dye 

studies because the instruments are temperature compensated and also measure turbidity. Temperature compensation 

eliminates substantial errors that can arise from fluctuating 

water temperatures. Although the SCUFA fluorometer is very 

effective at limiting interference from turbidity, highly turbid 

waters may still affect measurements. Turbidity measurements 

and RWT dye concentrations were recorded simultaneously to 

evaluate turbidity interference (Turner Designs, 2004). Turbidity measurements can be affected by stream turbulence, which 

was experienced during this study when algal blooms along 
310

370

340

350

360

290

300

260 270

280

320

330

108°30’ 108°00’ 107°30’ 107°00’

45°45’46°00’46°15’

0 10 20 MILES0 10 20 KILOMETERS

Base map from U.S. Geological Survey National Elevation Dataset (NED) raster digital data, 1999, 1:24,000, and U.S. Census Bureau TIGER digital data, 1990 and 2000, 1:100,000Lambert Conformal Conic projectionStandard parallels 45°00'N, and 49°00'N, central meridian 109°30'WNorth American Datum of 1983 (NAD 83)

Site used for dye inject

310 River miles measured f

 (Montana Departmen Conservation, 1976; G

EXPLA

Myers

Pompeys Pillar

Lockwood

Huntley

Yellowstone

Hysham

Custer

Billings

87

Myers Bridge 283.3

Custer Bridge 300.1

Pompeys Bridge 328.6

Huntley Bridge 348.6

Lockwood Bridgeinjection point 360.6

B I G H O R N

C O U N T Y

Y E L L O W S T O N E

C O U N T Y

M U S S E L S H E L L

C O U N T Y

T R E A S U R E

C O U N T Y

B I G H O R N

C O U N T Y

Y E L L O W S T O N E

C O U N T Y

M U S S E L S H E L L

C O U N T Y

T R E A S U R E

C O U N T Y

Bighorn River

Myers

Lockwood

Huntley

Hysham

Custer

Billings

Bighorn River

94

94

90

47

3

12

River

300.1

Figure 3. Location of subreach for dye injection at Lockwood Bridge, Montana.
106°30’ 106°00’

46°15’46°30’

0 

0 5 

240

220

Miles City

Cartersville Daminjection point 237.7

Kinsey Bridge 171.9

Rosebud Bridge 224.2

Forsyth Bridge 238.8

Miles City Bridge 183.7

Fort Keogh Bridge 186.2 R O S E B U D C O U N T Y

C U S T E R C O U N T Y

Tongu

e

R

i

v

e

r

Yellowstone

Tongu

e

R

i

v

e

r

Yellowstone

230 210

200

190

180

170

Fort Keogh

94

12

59

94

River

Rosebud Forsyth

Miles City

Fort Keogh

Myers

Hysham 

Forsyth

Cartersville Dam 237.7

Myers Bridgeinjection point 283.3

Forsyth Bridge 238.8

R O S E B U D C O 

T R E A S U R E C O U N T Y

R O S E B U D C O U N T Y

C U S T E R C O U N T Y

Yellowstone River Yellowstone River

280

260

250

270

240

230 

107°00’ 

46°12’46°18’

0 0 5 

94

94

12

39

94

Base maps frElevation Da1:24,000, and data, 1990 anLambert ConStandard paracentral meridNorth Americ

Site u mea

180

River  Yell Dep Con

183.7

Figure 4. Location of subreach for dye injection at Myers Bridge, Montana.Figure 5. Location of subreach for dye injection at Cartersville Dam, Montana.
Figure 6. Location of subreach for dye injection at Miles City Bridge, Montana.

Kinsey

Fallon

160

To

ngu

e

R

i

v

e

r

Calypso Terry

Fallon

Glendive

Miles City

C U S T E R C O U N T Y

M C C O U N T Y N E

M C C O N E

C O

D AW S O N C O U N T Y

P R A I R I E C O U N T Y

F A L L O N C O U N T Y

Fallon Bridge126.9

Kinsey Bridge171.9

Calypso Bridge144.9

Glendive Bridge94.6

Miles City Bridgeinjection point183.7

Fort Keogh Bridge186.2

Powder

River

90

190

170

180

140

150

130

110

120

100

105°30’ 105°00’ 104°30’

46°15’46°30’46°45’47°00’

0 5 0 5 10 KILO

Base map from U.S. Elevation Dataset (N1:24,000, and U.S. C1990 and 2000, 1:100Lambert Conformal CStandard parallels 45central meridian 109°North American Datu

W I B A U X 

C O U N T Y

C U S T E R C O U N T Y

D AW S O N C O U N T Y

P R A I R I E C O U N T Y

F A L L O N C O U N T Y

Site used for  river mile

90

River miles m Yellowstone  of Natural R 1976; Goog

EXPL

Yellowstone River

Kinsey

To

ngu

e

R

i

v

e

r

Calypso

Glendive

Miles City

Powder

River

Yellowstone River

94

12

200S

94

94.6
Table 2. Optimal mixing length for each injection site along the Yellowstone River, Montana.

[Time reported in military time as hours and minutes. Abbreviations: ft3/s, cubic feet per second; mi, miles]

Injection site Date of injection Time of injection

Instantaneous 

streamflow 

(ft3/s)

Number of 

lateral injection 

points

Optimal mixing 

length 

(mi)

Lockwood Bridge October 6, 2008 1410 3,500 4 1.9

Myers Bridge September 29, 2008 1700 6,750 3 4.1

Cartersville Dam September 26, 2008 1000 16,860 3 4.1

Miles City Bridge September 23, 2008 1003 7,420 3 6.5

1

 Instantaneous streamflow estimated from upstream and downstream streamflow measurements where discharge could not be measured.

10 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

the Yellowstone River caused buildup of algae on the deployed 

SCUFAs. When algae accumulated on the SCUFAs, the turbulence increased, which caused spikes in turbidity and fluorescence. To reduce the interference of algae on the fluorescence 

readings, the SCUFAs were frequently cleaned.

In addition to deploying the SCUFAs, grab samples 

also were collected at measurement sites at 15-, 30-, 45-, 

60-, or 90-minute intervals depending on the expected timeof-passage of the dye. These grab samples were collected 

at a single point near the centroid of flow where water was 

assumed to be well mixed.

Development of Time-Concentration Curves

SCUFA data (appendix 1) for each site were reviewed, 

and outliers from the time-concentration curve were removed 

from the data set. Outliers were defined as data values with 

an associated spike in turbidity or as values that substantially 

deviated from the time-concentration curve. Outliers were 

selected using best professional judgement. After outliers 

were removed from the data set, all the data were shifted up or 

down such that concentration values preceding the arrival of 

the dye (that is, background values) were zero (appendix 2). 

The SCUFA data may have background concentrations different than zero because the SCUFA measures fluorescence and 

converts this value to a concentration based on the calibration 

performed in the field. When background fluorescence values 

of the measurement site are different than the calibration 

standard of 0 μg/L, the calculated concentration of dye may be 

greater than or less than zero. Raw SCUFA data and adjusted 

SCUFA data collected at Pompeys Bridge are presented in 

figure 7 as an example of the adjustments made to the data set.

The discrete grab samples were analyzed with a bench 

fluorometer for RWT dye concentrations by the USGS 

Montana Water Science Center, Helena, Montana, and the 

resulting data were used to develop time-concentration curves. 

A Turner Designs Model 10 bench fluorometer was calibrated 

using standards with concentrations of 0 μg/L and 5 μg/L 

of RWT dye created in the laboratory of the Montana Water 

Science Center. RWT dye standards with concentrations of 

0.1 μg/L, 0.5 μg/L, and 15 μg/L were used to verify the calibration of the fluorometer. The discrete samples were analyzed 

at a temperature of 20°C. Data from the samples are presented 

in appendix 3.

Time-concentration curves were developed by fitting 

smooth curves to the SCUFA and discrete sample data 

(Kilpatrick and Wilson, 1989). Smooth curves were fit to the 

data by using a three-parameter log-normal equation. The 

three-parameter log-normal equation, parameters, and data for 

these curves are presented in appendix 4. In instances where 

the SCUFA data and discrete data were substantially different, 

the three-parameter log-normal curves were adjusted to fit the 

discrete data rather than the SCUFA data because the SCUFA 

data were affected by field conditions that were eliminated 

when analyzing the discrete data. The final time-concentration 

curves fit to the SCUFA and discrete data (for example Forsyth 

Bridge, fig. 8) were then used to determine the travel times for 

the arrival of the leading edge, peak concentration, centroid, 

trailing edge at 10 percent of the peak concentration, and the 

trailing edge of the plume for each measurement site. Timeconcentration curves for each of the four dye injections are 

shown in figures 9–12.

Travel Times, Streamflow Velocities, 

and Dispersion Rates in the 

Yellowstone River

The travel times, streamflow velocities, and dispersion 

rates of the subreaches were calculated by using the timeconcentration curves developed from data collected at the 

measurement sites. Mean velocities between measurement 

sites were calculated for the leading edge, peak concentration, 

centroid, and trailing edge at 10 percent of the peak concentration for each subreach (table 3). The mean velocity for the 

centroid of the dye plume most accurately represents the mean 

streamflow velocity of the river, whereas the mean velocities 
Figure 7. Concentration of Rhodamine WT dye at Pompeys Bridge, Montana. A, Raw data from SelfContained Underwater Fluorescence Apparatus (SCUFA); B, Adjusted SCUFA data.

-5

0

5

10

15

20

30

40

50

10 12 14 16 18 20 22 -5

RHODAMINE WT DYE CONCENTRATION, IN MICROGRAMS PER LITER

0

5

10

15

20

TIME AFTER DYE INJECTION, IN HOURS

A

B

Raw SCUFA data

SCUFA data without outliers, 

adjusted for background values 

Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River 11

of the other dye plume components typically are used for 

emergency response to contaminants spilled into the river.

As mentioned in the section “Dye Injections,” travel-time 

data and mean streamflow velocities between the dye-injection 

site and first measurement site downstream do not accurately 

represent the mean streamflow velocity because of incomplete lateral mixing. Similarly, when large tributaries join 

the Yellowstone River, the dye will not be completely mixed 

laterally across the channel below the confluence and therefore 

the mean velocity of the dye may not accurately represent the 

mean velocity of the streamflow until the dye is completely 

laterally mixed. The Bighorn River joins the Yellowstone 

River within the study area between Custer Bridge and 

Myers Bridge at river mile 295.3, which, during this study, 

contributed approximately 42 percent of the streamflow 

(2,890 ft3/s measured at mouth of Bighorn River, October 7, 

2008). Calculated velocities of the dye plume at Myers Bridge 

might have been affected by incomplete lateral mixing of the 

Bighorn River with the Yellowstone River.

Completely mixed velocities (calculated velocities for 

subreaches where the dye was completely laterally mixed for 

the entire subreach) for the centroid of the dye plume most 

accurately represents the mean streamflow velocities of the 

river. The completely mixed velocities of the centroid (table 3) 

ranged from 1.83 ft/s for the subreach upstream from Cartersville Dam to 3.18 ft/s for the subreach upstream from Calypso 

Bridge. Excluding the short subreach affected by backwater 

upstream from Cartersville Dam, the mean of the completely 
RHODAMINE WT DYE CONCENTRATION, IN MICROGRAMS PER LITER

TIME AFTER DYE INJECTION, IN HOURS

48 54 60 66 72 78 84 90 96 -0.5

0.0

0.5

1.0

1.5

2.0

2.5

Trailing edge

(Tt) 

92.7 hours 

Trailing edge at 10 percent of 

Cp (T10p) 78.6 hours 

Peak concentration (Cp

)

2.05 micrograms per liter

Leading edge

(Tl)

57.7 hours 

Centroid (Tc) 

67.2 hours

Peak (Tp) 

65.7 hours

SCUFA data

Discrete sample

Time-concentration

 curve

Figure 8. Rhodamine WT dye time-concentration curve fit to data from the discrete samples collected at 

Forsyth Bridge after September 29 dye injection at Lockwood Bridge, Montana.

Figure 9. Time-concentration curves for Rhodamine WT dye injected at Lockwood Bridge, Montana.

TIME AFTER DYE INJECTION, IN HOURS

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96

RHODAMINE WT DYE CONCENTRATION, IN MICROGRAMS PER LITER

0

2

4

6

8

10

20

30

40

Huntley Bridge

Pompeys Bridge

Custer Bridge

Myers Bridge

Forsyth Bridge

River miles downstream 

 from injection point 

(12.0)

(12.0)

(32.0)

(60.5)

(77.3) (121.8)

12 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana
Figure 10. Time-concentration curves for Rhodamine WT dye injected at Myers Bridge, Montana.

TIME AFTER DYE INJECTION, IN HOURS

18 24 30 36 42 48

RHODAMINE WT DYE CONCENTRATION, IN MICROGRAMS PER LITER

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

River miles downstream 

 from injection point

Forsyth Bridge

Cartersville Dam

Rosebud Bridge

(59.1)

(44.5)

(45.6)

(59.1)

Figure 11. Time-concentration curves for Rhodamine WT dye injected at Cartersville Dam, Montana.

TIME AFTER DYE INJECTION, IN HOURS

12 18 24 30 36 42 48 54

RHODAMINE WT DYE CONCENTRATION, IN MICROGRAMS PER LITER

0

2

4

6

8

10

River miles downstream 

 from injection point

Rosebud Bridge

Fort Keogh Bridge

Kinsey Bridge

0 6

(13.5) (13.5)

(51.5) (65.8)

Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River 13
Figure 12. Time-concentration curves for Rhodamine WT dye injected at Miles City Bridge, Montana.

TIME AFTER DYE INJECTION, IN HOURS

12 18 24 30 36 42 48 54 60 66

RHODAMINE WT DYE CONCENTRATION, IN MICROGRAMS PER LITER

0

1

2

3

4

5

10

15

20

25

30

0 6

(11.8)

(38.8)

(56.8)

(89.1)

River miles downstream

 from injection point

Kinsey Bridge

Calypso Bridge

Fallon Bridge

Glendive Bridge

(11.8)

14 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

mixed centroid velocities for the entire study reach was 

2.80 ft/s, with a standard deviation of 0.24 ft/s, for the streamflow conditions during this study. The fastest mean streamflow 

velocities generally occurred downstream from Fort Keogh 

Bridge, near Miles City. The slowest mean streamflow velocities generally occurred upstream from Fort Keogh Bridge.

Completely mixed velocities (calculated velocities for 

subreaches where the dye was completely laterally mixed for 

the entire subreach) for other points on the time-concentration 

curve typically are used by water resource managers for 

emergency reponse to contaminant spills into the river. The 

velocities of the leading edge, peak concentration, and trailing 

edge at 10 percent of the dye plume provide water resource 

managers information on when and for how long to stop withdrawals from the Yellowstone River for municipal supply. The 

completely mixed velocities of the leading edge of the dye 

plume ranged from 2.07 ft/s for the subreach upstream from 

Cartersville Dam to 3.77 ft/s for the subreach upstream from 

Calypso Bridge. Excluding the short subreach affected by 

backwater upstream from Cartersville Dam, the mean of the 

completely mixed velocities of the leading edge for the entire 

study reach was 3.27 ft/s with a standard deviation of 0.31 ft/s. 

The completely mixed velocities of the peak concentration of 

the dye plume ranged from 2.10 ft/s for the subreach upstream 

from Cartersville Dam to 3.39 ft/s for the subreach upstream 

from Calypso Bridge. Excluding the short subreach affected 

by backwater upstream from Cartersville Dam, the mean of 

the completely mixed velocities of the peak concentration for 

the entire study reach was 2.88 ft/s with a standard deviation 

of 0.27 ft/s. The completely mixed velocities of the trailing 

edge at 10 percent of the peak concentration of the dye plume 

ranged from 1.21 ft/s for the subreach upstream from Cartersville Dam to 2.74 ft/s for the subreach upstream from Forsyth 

Bridge. Excluding the short subreach affected by backwater 

upstream from Cartersville Dam, the mean of the completely 

mixed velocities of the trailing edge at 10 percent of the peak 

concentration for the entire study reach was 2.37 ft/s with a 

standard deviation of 0.27 ft/s.

Completely mixed velocities of the leading edge and peak 

concentration of the dye plume averaged 16.8 and 2.9 (respectively) percent faster than the completely mixed velocities of 

the centroid of the dye plume. Completely mixed velocities of 

the trailing edge at 10 percent of the peak concentration averaged 15.4 percent slower than the velocities of the centoid of 

the dye.

Longitudinal dispersion of the dye plume, having no 

boundaries, continues indefinitely and is the dispersion 

component of primary interest in dye-tracer studies (Kilpatrick and Wilson, 1989). Longitudinal dispersion occurs when 

the leading edge of a dye plume travels in the downstream 

direction faster than the trailing edge of the dye plume. 

The longitudinal dispersion rate (table 4) was estimated for 
Table 3. Travel-time data and mean streamflow velocities for the Yellowstone River, Montana.

[Time reported in military time as hours and minutes. Abbreviations: ft3/s, cubic feet per second; hr, hours; μg/L; micrograms per liter; ft/s, feet per second; L, liters. Symbols: %, percent; --, no data]

Site

River 

miles 

downstream 

from dye 

injection 

Instantaneous 

streamflow 

(ft3/s)

Travel time after dye injection to measurement site (hr)

Time for 

dye plume 

to pass site 

(hr),

Td

Measured 

peak dye 

concentration 

(μg/L),

Cp

Mean velocity for each component of the dye 

plume for subreach upstream from the site (ft/s)

Leading 

edge, 

Tl

Peak 

concentration, 

Tp

Centroid, 

Tc

Trailing 

edge at 

10% peak 

concentration,

T10p

Trailing 

edge,

Tt

Leading 

edge

Peak 

concentration Centroid

Trailing 

edge at 

10% peak 

concentration

Slug injection of dye (68 L) at 1410 on October 6, 2008, at Lockwood Bridge 

Lockwood Bridge 0.0 3,500 0.00 0.00 0.00 0.00 0.00 -- -- -- -- -- --Huntley Bridge 12.0 3,490 4.59 5.39 5.69 7.35 12.5 7.90 30.4 13.83 13.27 13.09 12.39Pompeys Bridge 32.0 23,600 13.4 15.7 16.7 22.1 37.3 23.9 8.90 3.33 2.85 2.66 1.99Custer Bridge 60.5 3,770 27.6 32.6 34.2 43.2 61.0 33.4 4.63 2.93 2.47 2.40 1.98Myers Bridge 77.3 6,520 35.5 42.0 43.8 54.8 70.0 34.5 2.27 33.14 32.62 32.57 32.13Forsyth Bridge 121.8 6,600 57.7 65.7 67.2 78.6 92.7 35.0 2.05 2.94 2.76 2.79 2.74

Slug injection of dye (21 L) at 1700 on September 29, 2008, at Myers Bridge

Myers Bridge 0.0 6,750 0.00 0.00 0.00 0.00 0.00 -- -- -- -- -- --Forsyth Bridge 44.5 6,890 19.1 22.3 22.9 27.6 33.2 14.1 1.52 13.42 12.93 12.85 12.36Cartersville Dam 45.6 26,890 19.9 23.1 23.8 29.0 35.3 15.4 1.43 2.07 2.10 1.83 1.21Rosebud Bridge 59.1 26,890 26.8 30.0 31.0 37.1 45.0 18.2 1.22 2.86 2.83 2.75 2.44

Slug injection of dye (33 L) at 1000 on September 26, 2008, at Cartersville Dam

Cartersville Dam 0.0 26,860 0.00 0.0 0.00 0.00 0.00 -- -- -- -- -- --Rosebud Bridge 13.5 26,860 5.58 5.90 6.32 7.87 16.9 11.3 9.27 13.55 13.36 13.13 12.52Fort Keogh Bridge 51.5 27,320 22.2 25.5 26.2 31.4 38.8 16.6 2.08 3.36 2.85 2.80 2.36Kinsey Bridge 65.8 27,350 28.5 32.2 33.1 39.1 47.8 19.3 1.88 3.29 3.14 3.07 2.73

Slug injection of dye (51.5 L) at 1003 on September 23, 2008, at Miles City Bridge

Miles City Bridge 0.0 7,420 0.00 0.00 0.00 0.00 0.00 -- -- -- -- -- --Kinsey Bridge 11.8 27,470 4.61 4.98 5.11 5.90 7.23 2.62 26.2 13.75 13.48 13.39 12.93Calypso Bridge 38.8 7,570 15.1 16.7 17.6 21.8 33.2 18.1 4.18 3.77 3.39 3.18 2.49Fallon Bridge 56.8 7,380 22.2 25.2 26.3 32.2 43.2 21.0 2.82 3.70 3.08 3.02 2.54Glendive Bridge 89.1 7,480 36.4 42.4 43.6 52.5 63.2 26.8 1.63 3.35 2.76 2.74 2.33

1 Mean velocity of dye plume was affected by incomplete lateral mixing of dye (unmixed velocity).2 Instantaneous streamflow was estimated from upstream and downstream streamflow measurements where discharge could not be measured.3 Mean velocity of dye plume might have been affected by incomplete lateral mixing downstream from confluence with Bighorn River.

Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River 15
Table 4. Longitudinal dispersion rates for the Yellowstone River, 

Montana.

[Abbreviation: ft/s, feet per second]

Site Longitudinal dispersion rate for subreach 

upstream from the site (ft/s)

Huntley Bridge 1,22.43 

Pompeys Bridge 22.15

Custer Bridge 21.18

Myers Bridge 2.38

Forsyth Bridge 2.06

Cartersville Dam 21.25

Rosebud Bridge 2.83

Fort Keogh Bridge 2.81

Kinsey Bridge 21.00

Calypso Bridge 22.25

Fallon Bridge 21.08

Glendive Bridge 2.97

1

 Longitudinal dispersion rate of dye plume was affected by incomplete 

lateral mixing of dye.

2

 Velocity of the trailing edge of the dye plume was determined from fitting 

a three-parameter log-normal curve to the travel-time data.

16 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

subreaches by subtracting the velocity of the trailing edge of 

the dye plume from the velocity of the leading edge of the dye 

plume. Because longitudinal dispersion continues indefinitely, 

the trailing edge of the dye plume was estimated from the 

three-parameter log-normal curves that were fit to the travel 

time data collected during this study (figs. 8–12). The threeparameter log-normal curves fit may not accurately represent 

when the trailing edge of the dye plume passes a measurement point, which may affect the estimated dispersion rates in 

table 4.

Longitudinal dispersion rates are dependant on the stream 

characteristics of the reach. Some stream characteristics that 

affect longitudinal dispersion rates are streamflow, the presence of split channels and pools, and the sinuosity of the river. 

Estimated longitudinal dispersion rates of the dye plume for 

this study ranged from 0.06 ft/s for the subreach upstream 

from Forsyth Bridge to 2.25 ft/s for the subreach upstream 

from Calyspo Bridge (table 4) for subreaches where the dye 

was completely laterally mixed.

Kilpatrick and Wilson (1989) found the time, T10d, 

necessary for the dye plume to pass a sampling point until 

the concentration of the dye plume on the trailing edge 

is 10 percent of the peak concentration (duration) can be 

estimated from the travel time of the peak concentration. 

A relation was determined between the travel time of the 

peak concentration and duration using data from this study 

for reaches where the dye was completely laterally mixed 

(fig. 13). This relation can be used to estimate when the 

receding concentration of a potential contaminant reaches 

10 percent of its peak concentration for accidental contaminant spills into the Yellowstone River. The relation between 

travel time of the peak concentration and duration determined 

from this study closely resembles the relation determined by 

Kilpatrick and Wilson (1989); however, the relation developed 

for this study may be affected by changes in streamflow in the 

Yellowstone River.

Evaluation of the Transport-Model Estimates

Data from this dye-tracer study were used to evaluate 

velocity and concentration estimates from the transport model 

developed by McCarthy (2006). The transport model uses 

equations developed by Jobson (1999) based on data from 

numerous dye-tracer studies conducted throughout the United 

States. The model estimates travel times based on estimates 

of both the most probable velocity and the maximum probable velocity. Estimates of the most probable velocity and the 

maximum probable velocity were derived by Jobson based on 

measured velocities from previous dye-tracer studies.

Travel-time data and completely mixed velocities (calculated velocites for subreaches where the dye was completely 

laterally mixed for the entire subreach) calculated for this 

study were compared with estimated travel-time data and 

streamflow velocities from the model. Figure 14 shows the 

difference between the estimated and calculated velocities 

for the leading edge, peak concentration, and trailing edge 

at 10 percent of the peak concentration. Comparison of the 

estimated and calculated velocities for the study reach (fig. 14) 

indicate that the transport model estimates the velocities of 

the Yellowstone River between Huntley Bridge and Glendive Bridge with reasonable accuracy. However, differences 

between estimated and measured velocities are evident.

The transport model uses several variables to estimate 

streamflow velocity, one of which is the stream slope. The 

slope used in the transport model (McCarthy, 2006) was an 

average slope for long reaches of the Yellowstone River. For 

example, the transport model uses an average slope (about 

4.7 ft/mi) for the subreach between Lockwood and the confluence with the Bighorn River even though the stream slope 

varies from 7.1 ft/mi at the upstream end of this subreach to 

4.1 ft/mi at the downstream end of this subreach. The variability of the stream slope in the subreach from Lockwood 

Bridge to Forsyth Bridge would partly account for the some of 

the calculated velocities being faster than the maximum probable velocities in the upper end of this subreach and some of 

the calculated velocities being slower than the most probable 

velocities at the lower end of this subreach (fig. 14). Similarly, 

the transport model uses average stream slopes between the 

following sites: confluence of the Bighorn River and Forsyth 

Bridge, Forsyth Bridge and Miles City Bridge, Miles City 

Bridge and confluence of the Powder River, confluence of 

the Powder River and Glendive Bridge. The variability of the 

stream slope in these reaches would partly account for differences between estimated velocities and calculated velocities 
Figure 13. Relation between travel times of the peak concentration and durations of the dye plume for 

the Yellowstone River, Montana.

TRAVEL TIME OF THE PEAK CONCENTRATION (Tp), IN HOURS

10 100

DURATION OF THE DYE PLUME (T10d), IN HOURS

1

10

100

T10d = 0.77Tp

0.80

R2 = 0.84 

T10d

 = 0.7Tp

0.86 (Kilpatrick and Wilson, 1989)

Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River 17

determined during this study. Velocities of the peak concentration of the dye plume calculated for this study averaged 

10 percent faster than the most probable velocities and averaged 12 percent slower than the maximum probable velocities 

estimated from the transport model (table 5).

Estimated peak concentrations from the transport model 

were compared with the measured peak dye concentrations for 

each measurement site (table 6). The transport model estimates 

a most probable and maximum probable peak concentration 

calculated from the most probable and maximum probable 

streamflow velocities.With the exception of the subreaches 

affected by incomplete lateral mixing, the measured peak 

concentrations for this study were consistently smaller than 

the transport model estimates of most probable and maximum 

probable peak concentrations. Measured peak concentrations 

are expected to be smaller than estimates because, even though 

RWT dye is considered to behave as a conservative constituent, it may not completely behave in a conservative manner. 

Solar degradation and sorption of the dye onto the streambanks, stream bottom, and algae are expected at a limited rate 

(Wilson and others, 1986) and may partly account for smaller 

measured peak concentrations. The mass of dye passing each 

measurement site was calculated from the time-concentration 

curves and ranged from 88 to 100 percent of the mass of the 

dye injected. The average mass recovery for the most downstream site of each dye injection was 96 percent.

Uncertainty in Estimating Mean Streamflow 

Velocities

The scope of this dye-tracer study was to determine travel 

times, streamflow velocities, and longitudinal dispersion rates 

for one streamflow condition along the Yellowstone River 

from Lockwood to Glendive. Determination of travel times, 

streamflow velocities, and longitudinal dispersion rates for 

other streamflows would require additional dye-tracer studies; 

however, streamflow velocities are commonly estimated 

using other methods. Velocity-streamflow relations developed 

from routine streamflow measurments made at USGS gaging 

stations are commonly used to estimate mean streamflow 

velocities throughout a range of streamflows. Routine streamflow measurements typically are from ideal locations where 

streamflow is uniform and well distributed across the channel; 

thus, velocity-streamflow relations developed by using 

these measurements may not represent the mean streamflow 

conditions for subreaches upstream or downstream from the 

gage. The transport model developed by McCarthy (2006) 

also could be used to estimate mean streamflow velocities 
Bighorn River

Miles City Bridge

Powder River

Bighorn River

Miles City Bridge

Powder River

Bighorn River

Miles City Bridge

Powder River

EXPLANATION

Range between estimated

Maximum probable velocity

Most probable velocity

Calculated velocity

Pompeys Bridge

Custer Bridge

Myers Bridge

Forsyth Bridge

Cartersville Dam

Rosebud Bridge

Fort Keogh Bridge

Kinsey Bridge

Calypso Bridge

Fallon Bridge

Glendive Bridge

0

1

2

3

4

0

1

2

3

4

B

C

VELOCITY, IN FEET PER SECOND

0

1

2

3

4

5

A

Figure 14. Calculated and estimated velocity data for subreach upstream from each site. A, Velocity data for leading edge of 

dye plume. B, Velocity data for peak of the dye plume. C, Velocity data for trailing edge at 10 percent of the peak concentration 

of the dye plume. Streamflows ranged from 3,490 to 3,770 cubic feet per second upstream from confluence with Bighorn River. 

Streamflows ranged from 6,520 to 7,570 cubic feet per second downstream from the confluence with Bighorn River.

18 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana
Table 5. Calculated and estimated streamflow velocities for each subreach within the study area, Yellowstone River, Montana.

 [Time reported in military time as hours and minutes. Estimated values were rounded to two significant figures in the transport model (McCarthy, 2006). Abbreviations: ft/s, feet per second; L, liter; hr, hours]

Site

Instantaneous 

streamflow 

(ft3/s)

Mean velocity for the leading edge 

of dye plume for reach upstream 

from the site 

(ft/s)

Mean velocity for the peak concentration 

of dye plume for reach upstream 

from the site 

(ft/s)

Mean velocity for the trailing edge at 

10 percent of the peak concentration of 

dye plume for reach upstream from the site 

(ft/s)

Estimated 

most 

probable 

velocity

Measured 

velocity

Estimated 

maximum 

probable 

velocity

Estimated 

most 

probable 

velocity

Measured 

velocity

Estimated 

maximum 

probable 

velocity

Estimated 

most 

probable 

velocity

Measured 

velocity

Estimated 

maximum 

probable 

velocity

Slug injection of dye (68 L) at 1410 on October 6, 2008, at Lockwood Bridge

Huntley Bridge 3,490 2.3 13.83 2.7 2.0 13.27 2.4 1.6 12.39 1.8

Pompeys Bridge 23,600 2.3 3.33 2.7 2.1 2.85 2.4 1.6 1.99 2.0Custer Bridge 3,770 2.3 2.93 2.8 2.1 2.47 2.6 1.7 1.98 2.0Myers Bridge 6,520 2.8 33.14 3.4 2.5 32.62 3.0 1.9 32.13 2.2Forsyth Bridge 6,600 3.0 2.94 3.6 2.6 2.76 3.3 2.2 2.74 2.7

Slug injection of dye (21 L) at 1700 on September 29, 2008, at Myers Bridge

Forsyth Bridge 6,890 3.1 13.42 3.8 2.7 12.93 3.4 2.3 12.36 2.7

Cartersville Dam 26,890 3.0 2.07 3.8 2.6 2.10 3.3 1.6 1.21 2.0Rosebud Bridge 26,890 2.9 2.86 3.6 2.6 2.83 3.2 2.0 2.44 2.4

Slug injection of dye (33 L) at 1000 on September 26, 2008, at Cartersville Dam

Rosebud Bridge 26,860 2.9 13.55 3.6 2.6 13.36 3.2 2.0 12.52 2.4Fort Keogh Bridge 27,320 2.9 3.36 3.7 2.7 2.85 3.3 2.1 2.36 2.7Kinsey Bridge 27,350 3.0 3.29 3.7 2.6 3.14 3.3 2.1 2.73 2.5

Slug injection of dye (51.5 L) at 1003 on September 23, 2008, at Miles City Bridge

Kinsey Bridge 27,470 2.9 13.75 3.7 2.6 13.48 3.3 2.0 12.93 2.4

Calypso Bridge 7,570 3.0 3.77 3.6 2.6 3.39 3.3 2.1 2.49 2.6Fallon Bridge 7,380 3.0 3.70 3.7 2.6 3.08 3.3 2.0 2.54 2.6Glendive Bridge 7,480 3.0 3.35 3.6 2.6 2.76 3.4 2.1 2.33 2.6

1 Mean velocity of dye plume was affected by incomplete lateral mixing of dye.2 Instantaneous streamflow was estimated from upstream and downstream streamflow measurements where discharge could not be measured.3 Mean velocity of dye plume might have been affected by incomplete lateral mixing downstream from confluence with Bighorn River.

Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River 19
Table 6. Estimated and measured peak Rhodamine WT dye concentrations for each subreach within the study area, 

Yellowstone River, Montana.

[Time reported in military time as hours and minutes. Estimated values were rounded to two significant figures in the transport model (McCarthy, 

2006). Abbreviations: ft3/s, cubic feet per second; μg/L, micrograms per liter; L, liter; hr, hours]

Site

Instantaneous 

streamflow 

(ft3/s)

Peak concentrations of Rhodamine WT dye 

(μg/L)

Estimated probable Measured Estimated maximum

Slug injection of dye (68 L) at 1410 on October 6, 2008, at Lockwood Bridge

Huntley Bridge 3,490 24 130.4 28

Pompeys Bridge 23,600 11 8.90 13

Custer Bridge 3,770 6.7 4.63 7.9

Myers Bridge 6,520 3.2 2.27 3.7

Forsyth Bridge 6,600 2.4 2.05 2.8

Slug injection of dye (21 L) at 1700 on September 29, 2008, at Myers Bridge

Forsyth Bridge 6,890 1.8 11.52 2.1

Cartersville Dam 26,890 1.8 1.43 2.1

Rosebud Bridge 26,890 1.4 1.22 1.7

Slug injection of dye (33 L) at 1000 on September 26, 2008, at Cartersville Dam

Rosebud Bridge 26,860 6.9 19.27 8.2

Fort Keogh Bridge 27,320 2.3 2.08 2.8

Kinsey Bridge 27,350 1.9 1.88 2.3

Slug injection of dye (51.5 L) at 1003 on September 23, 2008, at Miles City Bridge

Kinsey Bridge 27,470 11 126.2 13

Calypso Bridge 7,570 4.3 4.18 5.2

Fallon Bridge 7,380 3.2 2.82 3.8

Glendive Bridge 7,480 2.2 1.63 2.7

1

 Measured peak concentration was affected by incomplete lateral mixing of dye.

2

 Instantaneous streamflow was estimated from upstream and downstream streamflow measurments where discharge could not be measured.

20 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

throughout a range of streamflows. The transport model uses 

regression equations developed from numerous dye-tracer 

studies conducted nationwide, but velocity-streamflow relations developed using this model may not accurately estimate 

mean streamflow velocities throughout a range of streamflows 

in the Yellowstone River.

Velocity-streamflow relations were developed by using 

routine streamflow measurement data from the USGS gaging 

stations at Billings, Forsyth, Miles City, and Glendive (water 

years 2005–08, http://waterdata.usgs.gov/mt/nwis). Velocitystreamflow relations also were developed for the most 

probable velocity and maximum probable velocity using the 

transport model for subreaches at or near each of the gaging 

stations (Lockwood Bridge to Huntley Bridge, Forsyth Bridge 

to Cartersville Dam, Fort Keogh Bridge to Kinsey Bridge, and 

Fallon Bridge to Glendive Bridge, respectively). The calculated mean streamflow velocities for the centroid of the dye 

plume determined for this study were plotted with the velocity-streamflow relations developed from the USGS gaging 

stations and the transport model (figs. 15–18). The variation in 

these velocity-streamflow relations emphasizes the uncertainty 

in estimating the mean streamflow velocity for streamflows 

outside of the range of streamflows measured in this study. For 

example, at a streamflow of 10,000 ft3/s, the mean streamflow velocity can range from 3.26–4.95 ft/s, 2.18–4.03 ft/s, 

2.68–3.81 ft/s, and 2.96–3.95 ft/s for the subreaches upstream 

from Huntley Bridge, Cartersville Dam, Kinsey Bridge, and 

Glendive Bridge, respectively (figs. 15–18).
Figure 15. Calculated velocity of the centroid of the dye plume from Lockwood Bridge to Huntley Bridge 

and the velocity-streamflow relations developed from the transport model (McCarthy, 2006) and routine 

streamflow measurements for the Yellowstone River at Billings, Montana (gaging station 06214500).

STREAMFLOW (Q), IN CUBIC FEET PER SECOND (ft3/s)

VELOCITY (v), IN FEET PER SECOND

1,000 10,000 100,000

v = 0.13Q0.3952

0

2

4

6

8

10

12

Measured streamflow and average velocity, water years 2005–08

 (where Q is greater than 2,000 ft

3

/s)

Calculated velocity of centroid of dye plume from Lockwood Bridge

 to Huntley Bridge

Most probable velocity (McCarthy, 2006)

Maximum probable velocity (McCarthy, 2006)

Figure 16. Calculated velocity of the centroid of the dye plume from Forsyth Bridge to Cartersville Dam 

and the velocity-streamflow relations developed from the transport model (McCarthy, 2006) and routine 

streamflow measurements for the Yellowstone River at Forsyth, Montana (gaging station 06295000).

STREAMFLOW (Q), IN CUBIC FEET PER SECOND (ft3/s)

VELOCITY (v), IN FEET PER SECOND

v = 0.0022Q0.7491

Measured streamflow and average velocity, water years 2005–08

Calculated velocity of centroid of dye plume from Forsyth Bridge

 to Cartersville Dam

Most probable velocity (McCarthy, 2006)

Maximum probable velocity (McCarthy, 2006)

0

2

4

6

8

10

12

1,000 10,000 100,000

Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River 21
STREAMFLOW (Q), IN CUBIC FEET PER SECOND (ft3

/s)

VELOCITY (v), IN FEET PER SECOND

v = 0.0163Q0.5541

0

2

4

6

8

10

12

Measured streamflow and average velocity, water years 2005−08

 (where Q is greater than 5,300 ft

3

/s) 

Calculated velocity of centroid of dye plume from Fort Keogh Bridge

 to Kinsey Bridge

Most probable velocity (McCarthy, 2006)

Maximum probable velocity (McCarthy, 2006)

1,000 10,000 100,000

Figure 17. Calculated velocity of the centroid of the dye plume from Fort Keogh Bridge to Kinsey Bridge 

and the velocity-streamflow relations developed from the transport model (McCarthy, 2006) and routine 

streamflow measurements for the Yellowstone River at Miles City, Montana (gaging station 06309000).

Figure 18. Calculated velocity of the centroid of the dye plume from Fallon Bridge to Glendive Bridge 

and the velocity-streamflow relations developed from the transport model (McCarthy, 2006) and routine 

streamflow measurements for the Yellowstone River at Glendive, Montana (gaging station 06327500). 

2

4

6

8

10

12

VELOCITY (v), IN FEET PER SECOND

STREAMFLOW (Q), IN CUBIC FEET PER SECOND (ft3/s)

0

v = 0.0212Q0.5392

Measured streamflow and average velocity, water years 2005−08 

Calculated velocity of centroid of dye plume from Fallon Bridge to

 Glendive Bridge 

Most probable velocity (McCarthy, 2006)

Maximum probable velocity (McCarthy, 2006)

1,000 10,000 100,000

22 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana
Summary and Conclusions 23

Limitations of the Travel Times, Streamflow 

Velocities, and Longitudinal Dispersion Rates

Data collected as part of this study reflect limited hydrologic conditions (reasonably steady streamflow) over a limited 

period (late September and early October 2008); therefore, the 

travel times, streamflow velocities, and longitudinal dispersion 

rates determined in this study are limited to these streamflow 

conditions. Streamflows ranged from 3,490 to 3,770 ft3/s 

upstream from the confluence with the Bighorn River and 

ranged from 6,520 to 7,570 ft3/s downstream from the confluence with the Bighorn River. Care should be exercised when 

estimating travel times, streamflow velocities, and longitudinal 

dispersion rates for streamflows other than those observed 

during this study. Determination of travel times, streamflow 

velocities, and longitudinal dispersion rates for other streamflow conditions would require additional dye-tracer studies 

under a variety of streamflow conditions.

reach was 2.80 ft/s. Similarly, the mean of the completely 

mixed velocities of the leading edge, peak concentration, and 

trailing edge at 10 percent of the peak concentration were 

3.27 ft/s, 2.88 ft/s, and 2.37 ft/s, respectively.

Longitudinal dispersion of the dye plume, having no 

boundaries, continues indefinitely and is dependant on the 

stream characteristics of the reach. The longitudinal dispersion 

rate was estimated for subreaches by subtracting the velocity of the trailing edge of the dye plume from the velocity of 

the leading edge of the dye plume. Longitudinal dispersion 

rates of the dye plume for this study ranged from 0.06 ft/s for 

the subreach upstream from Forsyth Bridge to 2.25 ft/s for 

the subreach upstream from Calyspo Bridge for subreaches 

where the dye was completely laterally mixed. A relation was 

determined between the travel time of the peak concentration 

and the time for the dye plume to pass a site (duration of the 

dye plume). This relation can be used to estimate when the 

receding concentration of a potential contaminant reaches 

10 percent of its peak concentration for accidental spills into 

the Yellowstone River.

Data from this dye-tracer study were used to evaluate 

velocity and concentration estimates from a transport model 

developed as part of an earlier USGS study. Comparison of the 

estimated and calculated velocities for the study reach indicate 

that the transport model estimates the velocities of the Yellowstone River between Huntley Bridge and Glendive Bridge with 

reasonable accuracy. However, differences between estimated 

and calculated velocities are evident. The transport model uses 

an average slope (about 4.7 ft/mi) for the subreach between 

Lockwood and the confluence with the Bighorn River even 

though the stream slope varies from 7.1 ft/mi at the upstream 

end of this subreach to 4.1 ft/mi at the downstream end of this 

subreach. The variability of the stream slope in these reaches 

would partly account for differences between estimated 

velocities and calculated velocities determined during this 

study. Velocities of the peak concentration of the dye plume 

calculated for this study averaged 10 percent faster than the 

most probable velocities and averaged 12 percent slower than 

the maximum probable velocities estimated from the transport 

model. Measured peak RWT dye concentrations were consistently lower than the transport model estimates except for 

the most upstream subreach of each dye injection. The most 

upstream subreach of each dye injection is expected to have 

a higher concentration because of incomplete lateral mixing. 

Lower measured peak concentrations for all other sites were 

expected because RWT dye deteriorates when exposed to 

sunlight and will sorb onto the streambed and stream bottom.

Velocity-streamflow relations developed from routine 

streamflow measurements at USGS gaging stations and the 

transport model can be used to estimate mean streamflow 

velocities throughout a range of streamflows. The variation in 

these velocity-streamflow relations emphasizes the uncertainty 

in estimating the mean streamflow velocity for streamflows 

outside of the streamflows measured in this study.

Summary and Conclusions

In 2008, the U.S. Geological Survey (USGS), in cooperation with the Montana Department of Environmental Quality, 

initiated a dye-tracer study to measure instream travel times 

in the Yellowstone River. Data collected from late September 

and early October 2008 were used to determine the travel 

times, streamflow velocities, and longitudinal dispersion rates 

for a 266-mile reach of the Yellowstone River. The upstream 

end of the study reach is downstream from Billings where 

U.S. Highway 87 (Lockwood Bridge) crosses the Yellowstone 

River at about river mile 360.6 and extended approximately 

266 mi downstream to the West Bell Street Bridge (Glendive Bridge), river mile 94.6, in Glendive. Within the study 

reach, three major tributaries join the Yellowstone River: 

Bighorn River, Tongue River, and Powder River. Streamflows 

were reasonably steady and ranged from 3,490 to 3,770 ft3/s 

upstream from the confluence with the Bighorn River and 

ranged from 6,520 to 7,570 ft3/s downstream from the confluence with the Bighorn River. For this study, Rhodamine WT 

(RWT) dye was injected at four locations: Lockwood Bridge, 

Myers Bridge, Cartersville Dam, and Miles City Bridge.

Mean velocities were calculated for each subreach 

between measurement sites for the leading edge, peak concentration, centroid, and trailing edge at 10 percent of the peak 

concentration. Calculated velocities for the centroid of the dye 

plume for subreaches that were completely laterally mixed 

ranged from 1.83 to 3.18 ft/s within the study reach from 

Lockwood Bridge to Glendive Bridge. Excluding the subreach 

between Forsyth Bridge and Cartersville Dam, which is a 

backwater area created by the Cartersville Dam, the mean of 

the completely mixed centroid velocity for the entire study 
24 Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana

Selected References

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traveltime prediction equation for streams in Arkansas: 

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google.com/intl/en/userguide/v5/ug_measuring.html.

Jobson, H.E., 1999, Contaminants in rivers and streams—

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of time of travel in streams by dye tracing: U.S. Geological Survey Techniques of Water-Resources Investigations, 

book 3, chap. A9, 27 p.

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water-quality model for low-flow conditions on the Red 

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2002: U.S. Geological Survey Scientific Investigations 

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McCarthy, P.M., 2006, A computer program for estimating 

instream travel times and concentrations of a potential 

contaminant in the Yellowstone River, Montana: U.S. Geological Survey Scientific Investigations Report 2006–5057, 

16 p.

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Appendixes

The raw SCUFA data (appendix 1), adjusted SCUFA data (appendix 2), bench fluorometer data 

(appendix 3), and three-parameter log-normal data and equation parameters (appendix 4) were 

compiled and organized into four Excel spreadsheets (files) that each contain two worksheets. 

The first worksheet of each appendix contains descriptions of the headings used in each appendix. Appendixes 1–4 are located in the CD–ROM on the inside back cover of this report and can 

be downloaded as Excel files from http://pubs.usgs.gov/sir/2009/5261/.

Appendix 1. Raw Data from the Self-Contained Underwater Fluorescence Apparatus for 

Measurement Sites along the Yellowstone River, Montana.

Appendix 2. Adjusted Data from the Self-Contained Underwater Fluorescence Apparatus for 

Measurement Sites along the Yellowstone River, Montana.

Appendix 3. Bench Analysis of Grab Samples Measured with Turner Designs Model 10 Fluorometer 

for Measurement Sites along the Yellowstone River, Montana.

Appendix 4. Data and Equations for Three-Parameter Log-Normal Curves of the Self-Contained 

Underwater Fluorescence Apparatus and Grab-Sample Data for Measurement Sites along the 

Yellowstone River, Montana.
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For more information concerning this publication, contact:

Director, USGS Montana Water Science Center

3162 Bozeman Ave.

Helena, MT 59601

(406) 457–5900

Or visit the Montana Water Science Center Web site at:

http://mt.water.usgs.gov

McCarthy—Travel Times, Streamflow Velocities, and Dispersion Rates in the Yellowstone River, Montana—Scientific Investigations Report 2009–5261

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