U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigation Report 2004-5192

Prepared in cooperation with the Metropolitan Council, Minnesota Pollution Control Agency, 

Minnesota Department of Health, and the cities of St. Cloud, Minneapolis, and St. Paul, 

Minnesota

Estimation of Travel Times for Seven Tributaries of the 

Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003
Estimation of Travel Times for Seven 

Tributaries of the Mississippi River, St. 

Cloud to Minneapolis, Minnesota, 2003

By A.D. Arntson, D.L. Lorenz, and J.R. Stark

Prepared in cooperation with Metropolitan Council, Minnesota Pollution Control Agency, Minnesota Department of Health, and the cities of St. Cloud, Minneapolis, and St. Paul, Minnesota

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigation Report 2004-5192
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2004

For sale by U.S. Geological Survey, Information Services

Box 25286, Denver Federal Center

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reproduce any copyrighted materials contained within this report.

Suggested citation:

Arntson, A.D., Lorenz, D.L., and Stark, J.r., 2004, Estimation of travel times for seven tributaries of the Mississippi 

River, St. Cloud to Minneapolis, Minnesota, 2003: U.S.Geological Survey Scientific Investigation Report 2004-5192, 

17 p.
iii

Contents

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

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

Purpose and Scope ............................................................................................................................. 1

Hydrologic Setting ............................................................................................................................... 2

Methods ......................................................................................................................................................... 3

Estimated Travel Times of Seven Mississippi River Tributaries ............................................................ 7

Sauk River Time of Travel Study ............................................................................................................... 11

Acknowledgments ...................................................................................................................................... 17

Summary ...................................................................................................................................................... 17

References ................................................................................................................................................... 17

Figures

Figure 1. Locations of seven river watersheds tributary to the Upper Mississsippi River ............... 2

Figure 2. Lateral mixing and longitudinal dispersion of concentration of a water-soluble 

tracer downstream from a single, center, slug injection ...................................................... 4

Figure 3. Definition sketch for trace-response curves ........................................................................... 4

Figure 4. Downstream reach of Sauk River showing selected locations of estimated time of 

travel points. ................................................................................................................................. 5

Figure 5. Downstream reach of Elk River showing selected locations of estimated time of 

travel points. ................................................................................................................................. 6

Figure 6. Downstream reach of Crow River showing selected locations of estimated time 

of travel points. ............................................................................................................................. 7

Figure 7. Downstream reach of Rum river showing selected locations of estimated time 

of travel points. ............................................................................................................................. 8

Figure 8. Downstream reach of Elm Creek showing selected locations of estimated time 

of travel points. ............................................................................................................................. 9

Figure 9. Downstream reach of Coon Creek showing selected locaitons of estimated 

time of travel points. .................................................................................................................. 10

Figure 10. Downstream reach of Rice Creek showing selected locations of estimated 

time of travel points. .................................................................................................................. 11

Figure 11. Tracer-response curves at left, center, and right sampling sections on the 

Sauk River at County Road 121. ............................................................................................... 15

Figure 12. Tracer-response curves for three Sauk River sections. .................................................... 16

Tables

Table 1. Estimated low-flow travel times represented by 90 percent exceedance discharge ..... 12

Table 2. Estimated median-flow travel times represented by 50 percent exceedence 

discharge .................................................................................................................................... 13

Table 3. Estimated high-flow travel times represented by 10 percent exceedence discharge .... 14

Table 4. Comparison of estimated and measured travel times for reaches in the Sauk River ..... 16
iv

Conversion Factors, and Water-Quality Unit

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 2.590 square kilometer (km2) 

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)

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

 °C=(°F-32)/1.8

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

or micrograms per liter (μg/L).
Abstract

Travel times for seven streams tributary to the Mississippi River from St. Cloud to Minneapolis, Minnesota, were 

estimated for three flow conditions; low, median, and high. 

Travel times were estimated for Sauk, Elk, Crow, and Rum 

Rivers, and Elm, Coon, and Rice Creeks. Regression equations based on watershed characteristics of drainage area, river 

slope, mean annual discharge, and instantaneous discharge at 

the time of measurement from more than 900 streams across 

the nation were used to estimate travel times. Travel times 

were estimated for the leading edge, peak concentration, and 

trailing edge of tracer-response curves. To test the validity of 

these equations, a time of travel study, using a luminescent 

dye, was conducted on the Sauk River, from Rockville, to the 

confluence with the Mississippi River on June 16, 2003, at a 

discharge of 457 ft3

/s at Rockville. Dye was injected in the 

Sauk River at Rockville, and time and concentrations were 

measured at three sampling sections downstream; at County 

Road 121, Veterans Drive, and County Road 1 near the mouth. 

The estimated travel times for the leading edge, peak concentration, and trailing edge at County Road 1 were 10.6 hrs, 11.9 

hrs, and 14.6 hrs, respectively. The measured travel times for 

the leading edge, peak concentration, and trailing edge were 

13.4 hrs, 15.5 hrs, and 20.5 hrs, respectively for the 15.7 mile 

reach.

Introduction

The cities of St. Cloud, Minneapolis, and St. Paul, Minnesota obtain most of their drinking water from the Mississippi River. Spills or discharges of contaminants into the 

Mississippi River, or its tributaries upstream from the city 

water intakes could threaten water supplies (Minnesota Pollution Control Agency, 2001). If a contaminant spill or discharge 

occurs, the managers of the water supplies need to know when 

to stop pumping water from the river and how long to wait 

before again pumping water from the river to protect their 

water supplies. Water managers need a reliable estimate of the 

travel time of the contaminant from a spill to water intakes and 

an estimate of the dispersion of the contaminant in the river. 

Although many excellent models are available to estimate 

travel time and dispersion of contaminants, none can be used 

with confidence without the calibration and verification along 

a particular river reach (Jobson, 1996). The availability of 

measured field data is usually the most difficult and expensive 

kind of data needed to predict the rate of movement, dilution, 

and mixing of contaminants in rivers and streams.

Previous studies of travel times on the Mississippi 

River were conducted by the U.S. Army Corps of Engineers 

(USACOE) (1997) and the U.S. Geological Survey (USGS). 

The USACOE determined travel times as part of the River 

Defense Network Program. The travel times from reservoir 

pools to the Twin Cities (Minneapolis and St. Paul) were 

needed to adequately plan for water for use in river navigation 

and for providing water supplies in times of low flow. The 

USGS measured travel times on the Mississippi River from 

Anoka to Hastings during the low-flow years of 1976–77. The 

study, using dye trace methods, was done for the Metropolitan 

Waste Control Commission to understand wastewater treatment effluent flow and dispersion in time of low flow (U.S. 

Geological Survey, written commun., 1978). Travel times 

along tributary streams were not measured or calculated. 

To address the need for reliable estimates of travel time, the 

USGS, in cooperation with the cities of St. Cloud, Minneapolis, and St. Paul, the Minnesota Pollution Control Agency, the 

Minnesota Department of Health, and the Metropolitan Council conducted a study to aid in implementation of source-water 

protection efforts in the Upper Mississippi River Basin.

Purpose and Scope

The purpose of this report is to present the results of the 

cooperative study, which includes: (1) estimates of travel times 

in seven tributaries of the Upper Mississippi River in Minnesota: (2) measurements of travel times in the Sauk River, at 

median flow conditions, using a dye tracer; and (3) comparison of estimated and measured travel times in the Sauk River 

to evaluate the estimation technique. Results are based on 

conservative, water-soluble materials.

Estimation of Travel Times for Seven Tributaries of the 

Mississippi River, St. Cloud to Minneapolis, Minnesota, 

2003

By A.D. Arntson, D.L. Lorenz, and J.R. Stark
2 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

Hydrologic Setting

The Mississippi River and confluence points for seven 

of its tributaries are located in the east-central part of the state 

(fig. 1). The Mississippi River drainage area at St. Cloud, 

downstream from the confluence with the Sauk River, is 

13,300 mi2, and at Minneapolis is 19,100 mi2. The distance 

from St. Cloud to Minneapolis along the Mississippi River is 

approximately 75 miles. The topography of the seven tributary 

watersheds consists of gently rolling hills west of the Mississippi River and generally flat to the east. The watersheds 

receive about 29 inches of precipitation a year with about 25 

inches as rainfall. The average temperatures are 68o F in the 

summer months and 12o F in the winter months.

The Sauk River watershed covers an area of 1,043 mi2. 

The topography generally is gently rolling hills with a predominately agricultural land use. The St. Cloud metropolitan 

area, with a population of 66,000 (U.S. Department of Commerce, 2000), is located at the confluence of the Sauk River 

with the Mississippi River. The upper watershed is an area of 

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Figure 1. Locations of seven river watersheds tributary to the Upper Mississsippi River
many lakes that retain precipitation from snowmelt and rainfall. Lakes, ponds, and wetlands (storage) in the Sauk River 

watershed covers about 18 percent of the watershed, with 

about 4 percent as lakes. Discharge and stage increase more 

rapidly in the St. Cloud area following rainfall and snowmelt 

events primarily from urban runoff and shorter overland and 

tributary travel distances. Stage can be sustained for extended 

periods of time due to the slower release of runoff amounts 

from lake and wetland storage in the upper watershed.

The Elk River watershed covers an area of 611 mi2 and 

converges from the north with the Mississippi River at the city 

of Elk River, Minnesota. Elk River flows through Orono Lake, 

a backwater formed from the power generating dam near the 

mouth at Elk River. Land use in the Elk River watershed is 

primarily agricultural with urban development increasing near 

the mouth.

The Crow River watershed covers an area of 2,760 mi2

and converges from the southwest with the Mississippi River 

at Dayton, Minnesota. Land use in the Crow River watershed 

is primarily agricultural with some scattered urban development.

The Rum River watershed covers an area of 1,560 mi2

and converges from the north with the Mississippi River at 

Anoka, Minnesota. Lake Mille Lacs forms the headwaters 

of the Rum River. Land use in the Rum River watershed in 

primarily agricultural with urbanization occurring in the lower 

reach of the watershed. A power generating dam is located on 

the Rum River in Anoka.

The Elm Creek watershed covers an area of 106 mi2

and converges from the southwest with the Mississippi River 

in Champlin, Minnesota. Elm Creek flows through Hayden 

Lake a short distance upstream from the confluence with the 

Mississippi River. Land use in the Elm Creek watershed has 

steadily changed from agricultural to urban in recent years. A 

wier-crested dam at East River Road provides storage for Mill 

Pond.

The Coon Creek watershed covers an area of 109 mi2

and converges from the northeast with the Mississippi River 

and downstream from the Coon Rapids Dam in Coon Rapids, 

Minnesota. The upper reaches of the Coon Creek watershed 

contain wetlands while the lower reaches continue to become 

more urbanized.

The Rice Creek watershed covers an area of 180 mi2

 and 

converges from the east with the Mississippi River in Fridley, 

Minnesota. The upper reaches of the Rice Creek watershed 

form a chain of lakes while the lower reaches are fully urbanized.

Methods

The U.S. Geological Survey (Jobson, 1996) developed 

a method to estimate travel times in rivers based on extensive 

sets of readily available stream and watershed-characteristic 

data collected from rivers throughout the nation. The method, 

based on measured dye trace studies, provides estimates of (1) 

the rate of movement of a conservative contaminant (hereinafter, contaminant) through a river reach, (2) the rate of 

attenuation of the peak concentration of a contaminant with 

time, and (3) the length of time required for the contaminant to 

pass a point in the river. Data used to develop these regression 

equations include information about drainage area, river slope, 

mean annual discharge, and instantaneous discharge at the 

time of the measurement. All additional references to Jobson 

in this report are from Jobson (1996).

The accuracy of the estimated time of travel for the Sauk 

River was tested by measuring the time of travel for the river 

with an injected water-soluble tracer dye. The river reach 

was divided into three reaches of similar channel slope, and 

sampling locations were established near the points where 

the channel slope changed. Rhodamine WT 20 tracer dye was 

injected into the Sauk River just downstream from County 

Road 139 in Rockville, Minnesota. Water samples were collected at three sample locations along the study reach. Dye 

concentrations were measured and tracer-response curves were 

developed at these sampling locations to determine times of 

peak concentrations and travel times between the sampling 

locations.

Time-of-travel studies are conducted to quantify travel 

time and dispersion for rivers. A known quantity of a watersoluble tracer dye is instantaneously injected into a stream 

and concentrations are measured as it moves downstream. The 

theoretical distribution of tracer concentrations resulting from 

the injection is shown in figure 2. The tracer-response curves 

shown in figure 2 are a function of longitudinal distance, and 

not as a function of time. The results of a water soluble tracer 

injection are plotted as concentration varies with time (tracer 

response curve) at one or more cross sections along the study 

reach downstream from the injection point, as shown in figure 

3. The tracer-response curve is the basis for determining time 

of travel in streams when referenced to injection or spill times. 

Travel times, peak velocities, and unit-concentrations 

from selected locations to the mouth for seven tributaries were 

determined using Jobson’s equations. The tributaries are the 

Sauk (fig. 4), Elk (fig. 5), Crow (fig. 6), and Rum (fig. 7) Rivers; and Elm (fig. 8), Coon (fig. 9), and Rice Creeks (fig. 10).

Drainage area, stream slope, annual mean discharge, 

and instantaneous discharge were compiled from existing 

data available from the U.S. Geological Survey, or from local 

watershed organizations that maintain stream gages on Coon 

Creek and Rice Creek. Time-of-travel estimates were determined for three flow conditions: low, median, and high for 

each of the seven tributaries using Jobson’s equations.

The equation to estimate velocity of peak concentration 

(also known as peak velocity) is:

 Vp = 0.094 + 0.0143 x (D ́a)

0.919 x (Q ́a

)

–0.469 x S 0.159 x (Q/Da

) (1)

where

Vp is the peak velocity, [m/s]

g is the gravitational constant, [9.86 m/s2

]

Da is the drainage area of the river at the point of measurement, [m2

],

Methods 3
4 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

Figure 2. Lateral mixing and longitudinal dispersion of concentration of a water-soluble tracer downstream from a single, center, slug 

injection (modified from Kilpatrick, 1993, p. 2).

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Figure 3. Definition sketch for trace-response curves (modified from Kilpatrick and Wilson, 1989, p. 3).
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Figure 4. Downstream reach of Sauk River showing selected locations of estimated time of travel points.

D ́a is the dimensionless drainage area defined as Da

1.25 x 

g0.5 /Qa,

Q is the discharge at the section of the time of measurement, [m3

/s],

Qa is the mean annual flow at the section, [m3/s],

Q ́a is the dimensionless relative discharge defined as 

Q/Qa,

S is the slope of the reach [m/m].

To help estimate a “worst case” scenario, Jobson also 

developed an equation to estimate a maximum probable velocity (Vmp). The equation for the Vmp is:

 Vmp = 0.25 + 0.02 x (D ́a)

0.919 x (Q ́a

)

–0.469 x S0.159 x (Q/Da

) (2)

In addition to knowing when the peak concentration will 

arrive at a site, it is important to understand the timing of the 

arrival of the first part of the contaminant plume. The time of 

arrival of the leading edge of the plume serves as an indication 

of when a problem first may exist. The equation for the travel 

time of the leading edge is: 

 Tl= 0.890 x Tp (3)

where 

Tl is the travel time of the leading edge(s)

Tp is the travel time of the peak concentration(s) and is 

equal to the reach distance divided by peak velocity, Vp.

Unit peak concentration is a relative term used to define 

a contaminant concentration independent of the magnitude of 

discharge. The unit peak concentration can then be used for 

simulating the concentrations expected from various contaminants for different stream discharges. The unit peak concentration is defined as 1,000,000 times the concentration produced 

in a unit discharge due to the injection of a mass of conservative soluble substance (Jobson, 1996). The unit peak concentration fits one unit of mass of tracer into one unit of flow. 

Met hods 5
6 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

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Figure 5. Downstream reach of Elk River showing selected locations of estimated time of travel points.

The presence of pools, riffles, bends and other channel 

characteristics increase the rate of longitudinal mixing and, 

therefore, affect the unit peak concentration. Jobson developed 

a regression equation that produced a reasonable estimate of 

the unit peak concentration:

Cup=1,025 x Tp

-0.887 (4)

Jobson also used other river characteristics to define the 

unit peak concentration relations including drainage area (Da), 

reach slope (S), mean annual river discharge (Qa

), and discharge at the time of the measurement (Q). Another equation 

for unit peak concentration that accounts for some of the other 

characteristics is:

Cup = 857Tp

A (5)

where 

A = -0.760(Q/Qa)

-0.079 (6)

Estimates for the travel time of the leading edge, the 

travel time of the peak concentrations, and the magnitude of 

the unit peak concentrations define two points on a tracerresponse curve. Kilpatrick and Taylor (1986) found that the 

area of a normal slug-produced tracer-response curve is nearly 

equal to the area of a scalene triangle (fig. 3), with a height 

equal to the peak concentration and the base extending from 

the leading edge to a point where the trailing edge concentration is equal to 0.1 times the unit peak concentration, with 

the area under the unit peak concentration curve equal to 

1,000,000 units. The equation for time-of-passage is:

T10d = 2,000,000/Cup (7)

Where T10d is the time of passage from the leading edge 

of a tracer response curve to a point where the concentration 

has been reduced to 10 percent of the peak concentration. 

Adding the time-of-passage, or duration of a solute, to the 

travel time of leading edge gives the travel time of the trailing 

edge of the contaminant plume.
Estimated Travel Times of Seven 

Mississippi River Tributaries

Watershed drainage area, channel slope, mean annual 

discharge, and instantaneous discharge were used to compute 

the time of travel variables from Jobson’s equations. Watershed drainage area and channel slope were determined from 

1:24,000 scale topographic maps. 

Mean annual discharge was determined at USGS gaging 

stations for the period of record (Mitton and others, 2002). The 

mean annual discharge was adjusted for each sampling section 

based on a simple linear relation with drainage area. 

For the Coon Creek and Rice Creek watersheds, where 

there were not sufficient long term (greater than 10 years) 

USGS discharge records, intermittent shorter term (1- 2 years) 

daily discharges were obtained from local watershed managers. The shorter term daily discharges at Coon Creek and Rice 

Creek were related to daily discharges from nearby long term 

USGS records at Elm Creek and Elk River, respectively. A 

linear regression analysis was used to determine mean annual 

discharge for the Coon and Rice Creek watersheds. About 

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Figure 6. Downstream reach of Crow River showing selected locations of estimated time of travel points.

Estimated travel times of seven Mississippi River tributaries 7
8 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

770 daily discharges for Coon Creek were related to Elm 

Creek daily discharges. The resultant mean annual discharge 

for Coon Creek was adjusted for coefficient bias and back 

transformation (Helsel and Hirsch, 1992). The adjusted mean 

annual discharge for Coon Creek was 62 ft3/s. About 2,470 

daily discharges for Rice Creek were related to Elk River 

daily discharges. The resultant mean annual discharge for Rice 

Creek was adjusted for back transformation. The adjusted 

mean annual discharge for Rice Creek was 84 ft3/s.

Coefficient bias results when one stream has greater 

variability than the other stream. Elm Creek has much greater 

variability than Coon Creek. The mean discharge of Elm 

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Figure 7. Downstream reach of Rum river showing selected locations of estimated time of travel points.
Creek does not correspond to the mean discharge of Coon 

Creek when log-transforms are used in regressions because the 

distributions of daily discharges between the two streams do 

not match. The daily discharges of Elm and Coon Creeks were 

used to estimate the coefficient bias. Back-transformation bias 

results when converting estimates in log-space to flow-space. 

The antilog of the estimate underestimates the mean of the 

actual data. The maximum likelihood estimate correction factor, defined as the antilog of 1⁄2 the residual variance, was used 

to determine the back transformation correction for Coon and 

Rice Creeks (Helsel and Hirsch, 1992).

The estimated times of travel for the leading edge, peak 

concentration, and trailing edge of the solute are listed in 

tables 1-3 for each of the seven Mississippi River tributaries 

for each of the flow conditions; low (table 1), median (table 

2), and high (table 3). Each tributary was divided into reaches 

with uniform channel slopes. Velocities of the peak concentrations were computed in meters per second and converted 

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Figure 8. Downstream reach of Elm Creek showing selected locations of estimated time of travel points.

Estimated travel times of seven Mississippi River tributaries 9
10 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

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Figure 9. Downstream reach of Coon Creek showing selected locaitons of estimated time of travel points.
to feet per second for each reach using Jobson’s equations 

and the associated watershed characteristics. Computed peak 

velocities ranged from 0.48 ft/s for Elm Creek to 0.92 ft/s for 

the Elk River at low flows, and from 1.6 ft/s for Coon Creek to 

2.7 ft/s for the Crow River at high flows. Computed maximum 

probable velocities typically were about 2 times greater at 

lower flows and about 1.5 times greater at higher flows than 

the computed peak velocities. Corresponding travel times were 

estimated for each reach using the computed peak velocities 

and the reach length. The minimum travel time of peak concentration is based on the maximum probable velocity. Total 

travel times of the trailing edge to the mouth of each tributary 

are given at selected points for each of three flow conditions. The low flow values are represented by the 90 percent 

exceedance discharges (table 1), the median flow values are 

represented by the 50 percent exceedance discharges (table 

2), and the high flow values are represented by the 10 percent 

exceedance discharges (table 3) for all tributaries except Coon 

and Rice Creeks where low and high flow discharges were 

assumed due to lack of available data, and the mean annual 

discharge was used for the median flow value.

On tributaries where the main channel passes through 

storage areas of lakes, marshes, and backwater from dams, 

the computed reach velocities do not reflect or incorporate the 

inherent retention times and delays caused by these features. 

Jobson’s equations are based on stream reaches where storage 

was not a major factor. Storage effects were ignored in this 

study so that the minimum travel times could be estimated, as 

would be the case if a dam were breeched and the water stored 

behind it were totally drained. Estimation of retention times 

for these areas of storage was beyond the scope of this study. 

Users of these data must make the appropriate adjustments 

based on flow volumes and area of effective flow for storage 

areas (Mays, 2000). Jobson’s equations were based on studies 

where a soluble dye tracer was used; therefore, the time of 

travel estimates in tables 1-3 are only valid for soluble conservative contaminants.

Sauk River Time of Travel Study

A time-of-travel study, using a water-soluble-tracer dye, 

was conducted on the Sauk River, from Rockville to the confluence with the Mississippi River (fig. 4). This tracer study 

defines tracer-response curves for sampling sections in the 

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Figure 10. Downstream reach of Rice Creek showing selected locations of estimated time of travel points.

Sauk River time of travel study 11
12 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

Table 1. Estimated low-flow travel times represented by 90 percent exceedance discharge 

[ft3/s, cubic feet per second; USGS, U.S. Geological Survey; WWTP, wastewater treatment plant]

Location Distance 

upstream 

from 

mouth, 

in feet

Discharge,

 in ft3/s

Minimum 

travel time of 

peak 

concentration 

to mouth, 

in hours

Travel time of 

peak 

concentration 

to mouth, 

in hours

Travel time 

of leading 

edge to 

mouth, 

in hours

Duration 

of solute at 

mouth,

 in hours

Total travel 

time of 

trailing edge 

to mouth, in 

hours

Sauk River

County Road 1 328 37 0.1 0.1 0.1 0.1 0.2

Veterans Drive 21,800 37 4.0 7.6 6.8 4.0 10.8

USGS gage 05270500 27,800 37 5.2 9.9 8.8 5.1 13.9

County Road 121 51,500 37 9.9 18.9 16.8 9.1 26.0

Interstate 94 Bridge 57,000 37 11.0 21.1 18.8 10.0 28.8

County Road 139 (Rockville) 83,000 37 16.3 31.3 27.9 14.4 42.2

Rum River

Downstream from Trott Brook 48,500 111 9.1 17.4 15.4 8.0 23.4

County Road 22, USGS gage 05286000 84,200 111 15.7 29.7 26.4 12.8 39.2

Elk River

Orono Lake Dam 5,800 67 0.9 1.7 1.5 1.0 2.5

Inlet to Orono Lake 18,300 67 3.2 5.9 5.2 3.0 8.2

Railroad tracks 35,800 67 6.2 11.5 10.2 5.2 15.4

USGS gage 05275000 48,300 67 8.4 15.6 13.8 6.8 20.6

Crow River

Interstate 94 Bridge 34,900 39 8.2 17.1 15.2 10.2 25.4

St. Michael WWTP 63,700 39 14.5 29.8 26.5 17.4 43.9

USGS gage 05280000 (Rockford) 122,600 39 27.8 57.1 50.8 32.8 83.6

Elm Creek

US Hwy 169 2,400 2.0 0.6 1.3 1.2 0.8 2.0

USGS gage 05287890 26,400 2.0 7.1 16.0 14.2 9.6 23.8

Downstream from Rush Creek 33,500 2.0 9.0 20.4 18.1 12.1 30.3

93rd Avenue North 55,900 2.0 15.0 33.6 29.9 19.8 49.7

Coon Creek1

Northdale Boulevard 32,700 15 6.7 13.0 11.6 5.8 17.4

South Coon Creek Drive NW 57,100 15 11.7 23.0 20.5 9.4 29.9

Rice Creek1

Long Lake Road 34,500 20 6.6 12.6 11.2 5.7 16.9

Baldwin Lake outlet 72,700 20 14.4 27.8 24.8 11.3 36.0 1

 Assumed low-flow discharge

source-water assessment area of the Sauk River for the city of 

St. Cloud, and allows for evaluation of estimated and measured time-of-travel information in one of the seven tributaries 

described in this report. The Sauk River was chosen for the 

tracer study because of its proximity to the St. Cloud Water 

Utility intakes, and because it does not contain complicated 

hydrologic reaches involving reservoirs, lakes, or wetlands, 

thus making the comparison of estimated and measured results 

much easier. 

Tracer-response curves were determined by instantaneously injecting a measured amount of water-soluble tracer 

upstream from the sampling sections and by measuring dye 
Table 2. Estimated median-flow travel times represented by 50 percent exceedence discharge

 [ft3/s, cubic feet per second; USGS, U.S. Geological Survey; WWTP, wastewater treatment plant]

Distance 

upstream 

from mouth, 

in feet

Discharge,

 in ft3/s

Minimum 

travel time of 

peak 

concentration 

to mouth,

 in hours

Travel time of 

peak 

concentration 

to mouth,

 in hours

Travel time of 

leading edge 

to mouth, 

in hours

Duration 

of solute at 

mouth, 

in hours

Total travel 

time ot 

trailing edge 

to mouth, in 

hours

Sauk River

County Road 1 328 457 0.03 0.04 0.04 0.06 0.10

Veterans Drive 21,750 457 1.78 2.82 2.51 1.39 3.90

USGS gage 05270500 27,764 457 2.32 3.67 3.27 1.69 4.96

County Road 121 51,499 457 4.47 7.12 6.33 2.75 9.09

Interstate 94 Bridge 57,007 457 4.99 7.95 7.07 2.99 10.06

County Road 139 (Rockville) 83,055 457 7.46 11.93 10.62 4.03 14.65

Rum River

Downstream from Trott Brook 48,516 360 6.59 11.39 10.14 4.56 14.69

County Road 22, USGS gage 

05286000

84,184 360 11.28 19.42 17.28 6.99 24.27

Elk River

Orono Lake Dam 5,827 163 0.70 1.17 1.05 0.74 1.78

Inlet to Orono Lake 18,305 163 2.46 4.24 3.77 2.05 5.82

Railroad tracks 35,848 163 4.80 8.26 7.35 3.49 10.85

USGS gage 05275000 48,349 163 6.50 11.21 9.98 4.46 14.43

Crow River

Interstate 94 Bridge 34,949 270 5.31 9.42 8.39 4.19 12.58

St. Michael WWTP 63,737 270 9.12 15.96 14.20 6.50 20.70

USGS gage 05280000 (Rockford) 122,578 270 17.51 30.62 27.25 11.18 38.43

Elm Creek

US Hwy 169 2,408 11.0 0.45 0.84 0.75 0.56 1.31

USGS gage 05287890 26,366 11.0 5.47 10.79 9.60 4.90 14.50

Dwonstream from Rush Creek 33,542 11.0 6.97 13.75 12.24 6.01 18.25

93rd Avenue North 55,944 11.0 11.50 22.57 20.09 9.17 29.25

Coon Creek1

Northdale Boulevard 32,738 65.0 4.53 7.85 6.98 3.10 10.09

South Coon Creek Drive NW 57,129 65.0 8.01 13.94 12.41 4.80 17.21

Rice Creek1

Long Lake Road 34,529 93.0 4.25 7.18 6.39 2.90 9.30

Baldwin Lake outlet 72,727 93.0 9.44 16.14 14.37 5.37 19.74 1 Mean annual discharge

concentrations over time downstream at each sampling section 

(Wilson, 1986). Results were used to compare measured travel 

time to estimates of travel time from mathematical solutions at 

the same reaches on the river.

The study reach was divided into areas of similar channel 

slope, and sampling sections were established near the points 

where the channel slope changed. Sample sections were established at road crossings at County Road 121, Veterans Drive, 

and County Road 1 (fig. 4).

Two liters of Rhodamine WT 20 percent concentration 

dye were injected in the Sauk River just downstream from 

County Road 139 in Rockville, Minnesota (fig. 4). The locaSauk River time of travel study 13
14 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

Table 3. Estimated high-flow travel times represented by 10 percent exceedence discharge

[ft3/s, cubic feet per second; USGS, U.S. Geological Survey; WWTP, wastewater treatment plant]

Distance 

upstream 

from mouth, 

in feet

Discharge,

 in ft3/s

Minimum 

travel time 

of peak concentration to 

mouth, 

in hours

Travel time 

of peak 

concentration to 

mouth, 

in hours

Travel time of 

leading edge 

to mouth,

 in hours

Duration 

of solute at 

mouth, 

in hours

Total travel 

time of trailing edge to 

mouth, in 

hours

Sauk River

County Road 1 328 735 0.02 0.04 0.03 0.06 0.09

Veterans Drive 21,750 735 1.47 2.26 2.01 1.16 3.17

USGS gage 05270500 27,764 735 1.91 2.95 2.63 1.40 4.02

County Road 121 51,499 735 3.68 5.73 5.10 2.24 7.33

Interstate 94 Bridge 57,007 735 4.11 6.40 5.69 2.42 8.11

County Road 139 (Rockville) 83,055 735 6.16 9.61 8.56 3.23 11.79

Rum River

Downstream from Trott Brook 48,516 1350 4.10 6.50 5.78 2.51 8.29

County Road 22, USGS gage 05286000 84,184 1350 6.98 11.05 9.84 3.68 13.51

Elk River

Orono Lake Dam 5,827 553 0.44 0.69 0.61 0.49 1.11

Inlet to Orono Lake 18,305 553 1.59 2.52 2.25 1.27 3.51

Railroad tracks 35,848 553 3.09 4.92 4.38 2.06 6.43

USGS gage 05275000 48,349 553 4.20 6.68 5.95 2.57 8.51

Crow River

Interstate 94 Bridge 34,949 2240 2.47 3.84 3.42 1.67 5.09

St. Michael WWTP 63,737 2240 4.15 6.39 5.68 2.39 8.08

USGS gage 05280000 (Rockford) 122,578 2240 7.96 12.25 10.90 3.78 14.68

Elm Creek

U.S. Hwy 169 2,408 103 0.22 0.35 0.31 0.31 0.62

USGS gage 05287890 26,366 103 2.91 4.81 4.28 1.98 6.26

Downstream from Rush Creek 33,542 103 3.71 6.13 5.46 2.36 7.82

93rd Avenue North 55,944 103 6.06 9.99 8.89 3.34 12.24

Coon Creek1

Northdale Boulevard 32,738 150 3.40 5.57 4.95 2.20 7.15

South Coon Creek Drive NW 57,129 150 6.03 9.90 8.82 3.31 12.13

Rice Creek1

Long Lake Road 34,529 200 3.22 5.18 4.61 2.10 6.72

Baldwin Lake outlet 72,727 200 7.21 11.71 10.43 3.77 14.20 1 Assumed high-flow discharge

tion was chosen because it was 6 miles upstream from the 

first sampling section, thus allowing sufficient time for the 

dye to laterally mix. It also was a good location for making an 

accurate stream-discharge measurement. The dye was premeasured before arriving at the site. A stream discharge measurement was made at the mouth of the Sauk River prior to the 

injection of the dye at Rockville. The measured discharge, 478 

ft3/s, was used in a longitudinal dispersion equation (Kilpatrick, 1989) to compute the required amount of dye necessary 

to result in an observed concentration of 2 μg/L at the mouth. 

The dye was injected at the center of flow at the water surface 

in about 4 feet of water. A stream discharge measurement of 
457 ft3

/s was made at the site immediately following the dye 

release.

While the Rhodamine WT dye appeared bright pink when 

it was injected at County Road 139, it was indistinguishable 

to the naked eye at all sampling sections downstream. An estimate of travel time to each sampling section was determined 

based on measured stream velocity and sampling commenced 

at 5-minute intervals at an estimated time of arrival with each 

sample being analyzed at the site using a portable fluorometer. 

Samples were collected within each sample section at left, 

center, and right portions of the river. Samples were collected 

by lowering a weighted sample bottle to just below the water 

surface. Three samples were taken every 5 minutes until after 

the measured peak concentrations and the dye trace indicated a 

recession, at which time the sampling interval was lengthened 

to 10 minutes or more. Sampling ceased when the measured 

concentrations were 10 percent of the peak concentration or 

less. This method was used at all three sampling sections.

Measurements showed that lateral mixing was fairly 

well completed at all sampling sections. There was a slight 

time difference in the tracer-response curves as is shown in 

figure 11 for the sampling section at County Road 121. The 

dye plume traveled faster at the center of the channel than at 

the sides. The times were close enough to average together in 

determining travel times at all sampling sections. The travel 

times are listed in table 4.

Peak concentrations attenuated from the first sample section to the last as a result of longitudinal dispersion. Plots of 

average concentrations for three sample sections are shown in 

figure 12. The first sample section shows a higher peak concentration and a shorter duration than the last sample section 

near the mouth.

The travel time for the leading edge of the dye plume 

from Rockville, Minnesota, to County Road 1 near the mouth 

of the Sauk River, on June 16, 2003, at a discharge of 457 ft3/s 

for the 15.7-mile reach was 13.42 hours (table 4). The duration 

of the dye plume was 7.08 hours. 

The estimated travel times were less than the measured 

travel times (table 4). The estimated times for the leading edge 

were 79 percent of the measured times at County Road 1, 85 

percent at Veterans Drive, and 90 percent at County Road 121 

for the measured flow of 457 ft3

/s at the time of the study. The 

estimated times for the trailing edge, or total travel time in 

table 4, were 71 percent of the measured times at County Road 

1, 76 percent at Veterans Drive, and 77 percent at County 

Road 121. 

Jobson’s equations give reasonable estimates of travel 

time for streams where storage is not a factor. An alternate 

system or an adjustment is needed for those streams where 

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Figure 11. Tracer-response curves at left, center, and right sampling sections on the Sauk River at County Road 121.

Sauk River time of travel study 15
16 Estimation of travel times for seven tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003

Table 4. Comparison of estimated and measured travel times for reaches in the Sauk River 

[ft3/s, cubic feet per second; USGS, U.S. Geological Survey; WWTP, wastewater treatment plant]

Sampling section (fig. 4) Distance 

upstream from 

mouth, in feet

Measured or 

planned flow,

 in ft3/s

Travel time of 

peak concentration, in hours

Travel time to 

leading edge

, in hours

Duration of 

solute, in 

hours

Total travel time 

to trailing edge, 

in hours

Sauk River - Estimated travel times from slug at Rockville

County Road 1 328 457 11.9 10.6 4.0 14.6

Veterans Drive 21,800 457 9.4 8.4 3.4 11.8

County Road 121 51,000 457 5.0 4.4 2.1 6.6

County Road 139 (Rockville) 83,000 457 0.0 0.0 0.0 0.0

Sauk River - Measured travel times from dye injection at Rockville, June 16, 2003 at 11:15 hrs

County Road 1 328 457 15.5 13.4 7.1 20.5

Veterans Drive 21,800 457 11.8 9.9 5.6 15.5

County Road 121 51,000 457 5.9 5.0 3.6 8.5

County Road 139 (Rockville) 83,000 457 0.0 0.0 0.0 0.0

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Figure 12. Tracer-response curves for three Sauk River sections.

storage is a significant factor. Time-of-travel estimates could 

be improved with additional study of storage effects on Elk 

River, due to the effects of Lake Orono; Rum River, due to the 

dam in Anoka; Elm Creek, due the effects of Hayden Lake; 

and Rice Creek, due to the effects of Baldwin and Long Lakes. 

Other numerical solutions also could be utilized to evalu-
ate storage areas with regard to volume, slope, and effective 

conveyance areas to determine estimates of travel time (Mays, 

2000).

The study described in this report only involved the estimate of travel times within each of seven tributaries. Results 

from a tracer-response study also can provide useful information concerning flow dispersion and loss of mass of a contaminant. 

Acknowledgments

The Coon Creek Watershed District and the Rice Creek 

Watershed District provided discharge records that were used 

to determine annual mean discharges for the respective tributaries. The St. Cloud Water Utility helped collect and analyze 

water samples from its water intakes. Appreciation also goes 

to Gregory A. Payne, a U.S. Geological Survey “Volunteer for 

Science,” for his expertise and assistance in dye trace methods.

Summary

The cities of St. Cloud, Minneapolis and St. Paul, Minnesota obtain most of their drinking water from the Mississippi 

River. It is important that this source of water be protected 

from contamination caused by accidental spills or discharges. 

Water managers need a reliable estimate of the travel time of 

a contaminant from a spill to water intakes, and an estimate 

of the dispersion of the contaminant in the river. Travel times 

for seven streams tributary to the Mississippi River from St. 

Cloud to Minneapolis, Minnesota, were estimated for three 

flow conditions; low, median, and high. Travel times were estimated for Sauk, Elk, Crow, and Rum Rivers, and Elm, Coon, 

and Rice Creeks. A method, developed by the U.S. Geological 

Survey based on drainage area, slope, mean annual discharge, 

and instantaneous discharge at the time of measurement from 

more than 900 streams across the nation, was used to estimate 

travel times in a set of regression equations. Travel times were 

estimated for the leading edge, peak concentration, and trailing 

edge of a tracer-response curve.

To test the validity of these equations, a time of travel 

study, using a luminescent dye, was conducted on the Sauk 

River, from Rockville, Minnesota, to the confluence with the 

Mississippi River on June 16, 2003, at a discharge of 457 ft3

/s 

at Rockville. The time of travel study defined tracer-response 

curves for sample sections in the source-water protection area 

of the Sauk River for the city of St Cloud, and allowed for the 

comparison of estimated and measured time of travel information in one of the seven tributaries. Tracer-response curves 

were used to determine times of travel between sample sections at County Road 121, Veterans Drive, and County Road 

1 along the Sauk River. The comparison of measured travel 

times to estimated travel times of the leading edge of the dye 

plume on the Sauk River shows that the estimated travel times 

were less than measured times. The estimated travel times 

for the leading edge, peak concentration, and trailing edge at 

County Road 1 were 10.6 hrs, 11.9 hrs, and 14.6 hrs respectively. The measured travel times for the leading edge, peak 

concentration, and trailing edge were 13.4 hrs, 15.5 hrs, and 

20.5 hrs respectively for the 15.7 mile reach.

References

Helsel, D.R., and Hirsch, R.M., 1992, Statistical methods in 

water resources: New York, Elsevier, 546 p. 

Jobson, H.E., 1996, Prediction of travel time and longitudinal 

dispersion in rivers and streams: U.S. Geological Survey 

Water Resources Investigations Report 96-4013, 69 p.

Kilpatrick, F.A., 1993, Simulation of soluble waste transport 

and buildup in surface waters using tracers: U.S. Geological 

Survey Techniques of Water-Resources Investigations, book 

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Arntson and others—Estimation of Travel Times for Seven Tributaries of the Mississippi River, St. Cloud to Minneapolis, Minnesota, 2003—Scientific Investigations Report 2004-5192

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