10 TTT TTT TT] TTT TTT
zu EXPLANATION |
8 @ Wave speed 7
3
5 - a. Discharge during 7
S L dye study 4
3 |
3 b -
2 eo
&

BOF _— ;
3 ee Prediction at
3 | — Calibration 231 m/s
1 | | | | |
10 100 1000

Flow, in cubic meters per second

Figure 7. Wave celerity in Wind/Bighorn River between Boysen and Basin, Wyoming, as a function of
discharge.

Time-of-travel data for the Wind/Bighorn River were collected on March 21-24 and June 29-30, 1971
(Lowham and Wilson, 1971), and Nordin and Sabol (1974) published the complete concentration as a
function time data for the study. Dye was injected near the Boysen gage, and concentrations were measured
at seven points downstream, the first being 9.2 km below the injection site and the last being at Greybull,
Wyoming, which is 181.4 km downstream from the injection site and about 13 km downstream from the
Basin gage. The average of the discharges measured at the injection point and seven data collection sites
during the first and second injection were 57.5 and 231.3 m’/s, respectively (Lowham and Wilson, 1971, p.
6). The travel time of the peak concentration was 53.25 and 31.65 hours, respectively, for the low- and
high-flow studies (Lowham and Wilson, 1971, p. 6). The mean velocities, during low and high flow, were
0.898 and 1.51 m/s, respectively, and the average areas were 64.0 and 153 m’, respectively. By solving
equation 16 with a flow of 57.5 m/s and a total area of 64.0 m’, the low-flow time-of-travel indicates a
value of AO = 22.4 m’,

Solving equation 16 with AO = 22.4 m? for a discharge of 231.3 m’/s yields a predicted area of 154 m*
and a velocity of 1.50 m/s. This predicted velocity is 0.7 percent lower than the value measured by the high-
flow time-of-travel study. Thus, the predicted travel time for the high-flow study would be 31.89 hours,
which is 0.24 hours (0.8 percent) longer than the observed value. An accurate prediction was obtained even
though the discharge during the predicted dye study was more than 2.5 times the highest flow where the
wave celerity was observed (see figure 7), and the wave speeds were measured more than 25 years after the
time-of-travel study was conducted.

South Fork Shenandoah River, Waynesboro to Luray, Virginia

The South Fork Shenandoah River provides an example application where flow records are more
difficult to interpret. The Shenandoah River flows generally northeast near the western boundary of Virginia
and joins the Potomac River at Harpers Ferry, West Virginia, figure 2. Travel-time data were collected from
Waynesboro, Virginia to Harpers Ferry, West Virginia (Taylor and others, 1986), but only two reaches will
be considered here. Many low-head dams are located along the South Fork Shenandoah River, although in
the most cases these dams do not store sufficient water to substantially impede the movement of dye. Only
the dam located about 8 km downstream from the Luray gage (Station 01629500), retards the flow
noticeably, particularly during low flow (Taylor and others, 1986, p. 5). The dam probably had little or no
effect on the flow upstream from the Luray gage. Table 5 lists the data for sites considered here.

13


Table A-1. Hydraulic data used to compute the inactive area of river reaches (continued)

soaogce

Site | Distance | Flow Width Slope Time to | Manning | Width | Inactive
Peak N Coeff Area
km m/s m m/1,000m hr unitless | WI m
Wind/Bighorn River Nordin and Sabol, 1974
1 9.2 54.9 55 3.30 2.75
0.035 21.5 12.1
2 32.7 54.9 66* 2.70 9.38
0.035 25.3 27.7
3 50.4 56.6 78* 1.00 16.63
0.035 24.3 3.3
4 75.3 56.6 61 1.00 25.00
0.035 18.4 8.6
5 99.9 56.6 44 1,50 32.63
0.035 18.4 43
6 141.9 56.6 61" 1.20 44.50
0.0339 21.0 0
7 181.3 68.8 62° 0.70 56.00

Estimated value based on a consistent width depth ratio or value of W1.
Computed from measured widths at 680 m*/s (Graf, 1995, p. 269), assuming W2 = 0.1.

Estimated from Wiley (1993).

Estimated from the difference in gage datum.
Estimated from the channel-forming discharge.

Slope is estimated.

34



APPENDIX. Basic Data and Predicted Times of Travel

The data used to determine the inactive flow area for each river reach used in the report is given in
table A-1, and the data used to check the predicted travel times at different flows for the river reaches
contained in table A-1 is given in table A-2. The hydraulic-geometry exponent (W2) is dimensionless, but
the hydraulic geometry coefficient (W1) has units. The value of WI in table A-1 gives the width in meters
when the discharge is given in m'/s. Some assumptions had to be made to supplement the data available in
the references. The following is a summary of assumptions made in this analysis.

Width was the most common variable to be missing. For the data in table A-1, widths were sometimes
estimated on the basis of consistent hydraulic geometry coefficients (W 1), or consistent width-depth ratios.
Reaches for which this estimation procedure was used are indicated by a symbol “

Because of the confined conditions of the Colorado River through the Grand Canyon, the variation of
width with discharge is assumed to vary less than for typical rivers. A value of W2 = 0.1 was assumed, and
the measured widths of Graf (1995, p. 269) obtained at 680 m/s were used to compute W1. The widths at
other discharges were then computed from equation |. These values are indicated by a symbol °.

Wiley (1993, p. 2) estimated the average width of the New River for each reach at a flow of 57 m’ */s.
Assuming W2 = 0.26, the value of WI was determined from equation | using the flows and estimated
widths. The widths in table A-1 are then computed from equation 1 for the indicated discharge and flagged
by use of a symbol *.

Widths and slopes were not available for the South Fork Shenandoah, but stream gages exist at
Waynesboro, Harriston, Lynnwood, and Luray, and the gage datum and drainage area are available from
the USGS web page. The slope for the reach between gages was estimated as the difference in Bage « datum
divided by the distance between gages. These numbers are flagged in table A-1 by use ofa symbol %,

The values of W1 for the South Fork Shenandoah River were estimated from the channel forming
discharge and equation 2. The average drainage area for the reach was approximated as the average of the
drainage area at each end of the reach. For example, the average drainage area between Waynesboro (site
1) and Harriston (site 4) is (329 + 549)/2 = 439 km’, or 169 mi’. Following Jennings and others (1994, p.

162), the flow with a 2-year return period (Q2) is given for the Shenandoah Valley by the equation:

Q2 = 25.2 APS $26,

in which Q2 is the peak discharge in cubic feet per second, A is the drainage area in mi, and S is slope in
feet per mile. The slope of 0.00165 m/m is equal to 8.71 ft per mile. Solving the above equation yields:

Q2 = 25.2 (169) (8.71) = 3126 cfs, or 88.5 m/s.
Solving equation 2 for the bank full width yields:
Wor = 7.5 (88.5)°° = 92.3 m.
Assuming a value of W2 = 0.26, equation | can then be solved for the hydraulic geometry coefficient:
= 92.3 / 88.5976 = 28.8,
and the width at the flow of (1.02 + 1.59)/2 = 1.30 m’/s is also computed from equation 1 as:
W = 28.8 (1.30) = 30.8 m.

Widths computed in this way are flagged using a ©
One slope on Concocheague Creek was simply estimated; it is indicated by an ‘

30


Table A-2. Computed and observed times of travel for verification runs (continued)

Site Distance Flow Time to Travel Time, in hr
Peak Observed Computed
km m/s hr
South Fork Shenandoah River Taylor and others, 1986
1 287.2 3.11 0.00
11.2 11.27
2 278.7 3.68 11.20
14.3 14.15
3 266.8 4.53 25.50
1.5 12.11
4 256.5 4.81 37.00
27.5 28.85
5 229.4 19.3 64.50
24.0 22.96
6 207.7 20.7 88.50
Tangipahoa River Nordin and Sabol, 1974
1 8.2 5.78 9.83
11.67 10.21
2 18.0 9.80 21.50
13.92 13.34
3 415 11.9 35.42
9.75 9.29
4 55.3 12.4 45.17
12.00 11.07
5 71.0 14.4 57.17
9.00 7.10
6 82.1 17.8 66.17
11.00 9.73
7 94.0 18.7 T1AT
Wind/Bighorn River Nordin and Sabol, 1974
1 9.2 233 1.44
3.44 3.72
2 32.7 235 4.88
3.79 3.72
3 50.4 235 8.67
4.83 5.37
4 75.3 224 13.50
4.58 4.68
5 99.9 215 18.08
7.92 7.62
6 141.9 218 26.00
TAT TIT
7 181.3 255 33.17

39



»
WEST
VIRGINY

EXPLANATION
Basing ----os
a USGS Gaging Station
A. Reference Point
WYOMING N

{

VIRGINIA

a
Poet
}
|
UTAH WEST VIRGINIA

NEVADA

ia
VIRGINIA

Fp6°et5s Canyon

Figure 2. Gaging stations and reference points for rivers where wave celerity and travel time are computed.

The discharge at the two gaging stations starting on November 1, 1986, is shown in figure 3. The
circular symbols define the time when a wave is assumed to have passed a site. The wave passage time is
defined as follows: The discharge before and after the rise is estimated. Then, the time when the discharge
is equal to the average of the beginning and ending flows is defined as the passage time. The wave celerity

and average discharge for each of the seven waves is listed in table 1.


2 L |
2

8

2

5 150
a

2

8 L 4
3

&

Q

S 100 Starting —
< November 11, 1995

ee

&o

=}

4

2 50

ia)

Starting a

| March 8, 1996
Lt 4

0 1 2 3 4

Time, in days

Figure 11. Low-flow discharge in the South Fork Shenandoah River at Lynnwood and Luray, Virginia.

2000

|
|
4

1500

Starting
January 18, 1996

Discharge, in cubic meters per second

Starting
——
~ October 20, 1995 =

Time, in days

Figure 12. High-flow discharge in the South Fork Shenandoah River at Lynnwood and Luray, Virginia.

17


Table A-2. Computed and observed times of travel for verification computations
[km, kilometer; m’/s, cubic meter per second; hr, hour]

Site Distance Flow Time to Travel Time, in hr
Peak Observed Computed
km m/s hr
Antietam Creek Nordin and Sabol, 1974
i 2.6 5.10 1.35
4.15 4.03
2 9.6 5.24 5.50
10.40 10.11
3 21.5 7.36 15.90
7.50 7.93
4 29.6 7.19 23.40
1 2.6 2.41 2.7
6.8 6.16
2 9.6 2.44 9.5
17.9 17.19
3 21.5 3.40 27.4
12.7 13.16
4 29.6 3.96 40.1
Clinch River Nordin and Sabol, 1974
1 0.69 104.8 0.175
-208 0.304
2 1.58 105.9 0.383
-259 0.271
3 2.49 104.5 0.642
274 0.364
4 3.59 110.2 0.916
374 0.424
5 4.67 104.2 1.290
380 0.549
6 5.89 101.9 1.670
1 0.69 85.8 0.160
0.273 0.330
2 1.58 79.9 0.433
0.329 0.289
3 2.49 89.2 0.762
0.378 0.388
4 3.59 86.9 1.140
0.250 0.462
5 4.67 83.8 1.390
0.720 0.597
6 5.89 85.2 2.110

35



FIGURES
1. Schematic showing uniformly progressive flow and the relation of wave speed to change in
flow area with discharge ----

2. Gaging stations and reference points for rivers where wave celerity and travel time are
computed
Discharge in the New River at Hinton and Thurmond, West Virgina -----
4. Wave celerity in the New River between Hinton and Thurmond, West Virginia, as a function
of discharge ----

»

5. Variation of cross-sectional area with discharge for the New River, West Virginia ---

6. Discharge in the Wind/Bighorn River below Boysen Reservoir and at Basin, Wyoming

7. Wave celerity in the Wind/Bighorn River between Boysen and Basin, Wyoming, as a function
of discharge ---------------------------n2-nnannnnnn anne ae

8.  Low-flow discharge in the South Fork Shenandoah River at Waynesboro and Harriston,
Virginia -----------------------------------------------~

9.  High-flow discharge in the South Fork Shenandoah River at Waynesboro and Harriston,
Virginia ----- anne nace ence enone nanennanne nanennnnnen

10. Wave celerity in the South Fork Shenandoah River between Waynesboro and Harriston,
Virginia, as a function of discharge
11. Low-flow discharge in the South Fork Shenandoah River at Lynnwood and Luray, Virginia --
12. High-flow discharge in the South Fork Shenandoah River at Lynnwood and Luray, Virginia --
13. Wave celerity in the South Fork Shenandoah River between Lynnwood and Luray, Virginia,
as a function of discharge ---------------------------------2-rrrnnrnnnn nnn
14. Discharge in the Colorado River at Lees Ferry, and above Diamond Creek, Arizona -
15. Predicted and observed travel time through each reach listed in table A-2 ---
16. Predicted and observed total travel time of each time-of-travel study listed in table A-2

TABLES
1. Celerity and flow for waves in the New River between Hinton and Thurmond, West
Virginia -

2. Time-of-travel data for the New River between Sandstone and Stone Cliff, West Virginia ---

3. Area and travel time for the New River between Sandstone and Stone Cliff, West Virginia --

4.  Celerity and flow for waves in the Wind/Bighorn River between Boysen and Basin,
WYOMING --------------~-----n--m—- nnn nnn

5.  Time-of-travel data for the South Fork Shenandoah River between Waynesboro and Luray,
Virginia ----------------------------- monncennn=

6. Effect of width and slope on the predicted travel-time of the Mississippi River
7. Effect of width and slope on the predicted travel-time of Antietam Creek, Maryland
A-1. Hydraulic data used to compute the inactive area of river reaches ----
A-2. Computed and observed times of travel for verification computations

iv

10
10

12

14
24
24
29
33


Taylor, K.R., James, R.W. Jr., and Helinsky, B.M., 1986, Traveltime and dispersion in the Shenandoah
River and its tributaries, Waynesboro, Virginia, to Harpers Ferry, West Virginia: U.S. Geological
Survey Water-Resources Investigations Report 86-4065, 60 p.

Taylor, K.R., and Solley W.B., 1971, Traveltime and Concentration Attenuation of a Soluble Dye in
Antietam and Conococheague Creeks, Maryland: Maryland Geological Survey Information Circular
12, 25 pages.

Wiley, J.B., and Appel, D. H., 1989, Hydraulic characteristics of the New River in the New River Gorge
National River, West Virginia: U.S. Geological Survey Open-File Report 89-243, 34 pages.

Wiley, Jeffrey B., 1993, Simulated flow and solute transport, and mitigation of a hypothetical soluble-
contaminant spill for the New River in the New River Gorge National River, West Virginia: U.S.
Geological Survey Water-Resources Investigations Report 93-4105, 39 p.

Wilkinson, James A., 1945, Translatory waves in natural channels: Transactions, American Society of Civil
Engineers, v. 110, p. 1203-1225.

Wilson, J.F., Jr., Cobb, E.D., and Kilpatrick, F.A., 1986, Fluorometric procedures for dye tracing: U.S.
Geological Survey Techniques of Water-Resources Investigations, book 3, chap. A12, 34 p.

Wolman, M. Gordon, and Leopold, Luna B., 1957, River flood plains: Some observations on their
formation: U.S. Geological Survey Professional Paper 282-C, p. 87-107.

28


Table 5. Time-of-travel data for the South Fork Shenandoah River between Waynesboro and Luray,
Virginia, (Taylor and others, 1986)
{km, kilometer; m/m, meter fall per meter of distance; m/s, cubic meters per second]

Site Distance Slope Low __ Flow High _ Flow
below Discharge Peak-to-peak Discharge Peak-to-peak
injection Travel time Travel time
(km) (m/m) (m/s) (Hours) (m’/s) (Hours)
Waynesboro 2.25
0.0017 1.57 71 4.04 37.0
Harriston 32.99
Lynnwood 49.99
Island Ford 59.64
.0013 9.56 108 22.26 55.7
Luray 118.23

The first reach extends from Waynesboro (Station 01626000) to Harriston (Station 01627500) where
the drainage area increases from 329 to 549 km’. Discharge hydrographs at Waynesboro and Harriston
under low-flow conditions starting November 20, 1994 are shown in figure 8. Although it is generally
recommended that the wave celerity be computed from the time lag of the midpoint of the rising hydrograph
(Wilkinson, 1945), it is seen that the hydrograph at Harriston often contains a small rise before the main
peak occurs. This is apparently the result of local runoff between the gages. It also is noted that there is
substantial inflow between the two gages, which makes it difficult to identify the passage of specific waves.
The inflow along the reach significantly decreases the apparent travel time of the midpoint of the rise and
causes unreasonably large apparent celerity, especially for the rise occurring on December 10 (figure 8).
The peaks that were judged acceptable for computing the wave celerity are indicated by circles on the
figure. The wave labeled 2, which is based on the midpoint of the rise, was considered acceptable because
the rise at Harriston had receded before the midpoint of the main wave arrived.

a

f | T 1 mI To
|
|

Wave |
Wave 2B

Harriston

Waynesboro

Discharge, in cubic meters per second

: ! f it L it
20 25 30 5 10
November 1994 December 1994

Figure 8. Low-flow discharge in the South Fork Shenandoah River at Waynesboro and Harriston, Virginia.

Discharge hydrographs at Waynesboro and Harriston under high-flow conditions starting June 22, 1995
are shown in figure 9. Again, the peaks were used to define the passage time of the waves, and the largest
wave, which occurred on June 28, was not used because of the multiple peaks at Harriston, which occurred
at 1-hour intervals.

14


Extrapolation of Transport Velocity Using a Resistance Equation

The previous section has shown that continuous discharge records can be used to predict the variation
of travel time with flow. Although the USGS maintains about 7,000 continuous stream-flow gaging stations
throughout the country, for many river reaches this information is not available or is not suitable for the
analysis described above. For example, although continuous discharge records are available on the
Mississippi River at St. Louis, Missouri and Thebes, Illinois, about 220 km downstream, no consistent
relation between wave speed and discharge could be established. Because gages often do not exist at
desired locations and because inflow or other factors can limit the ability of flow records to infer wave
speeds, it generally is necessary to predict the variation in travel time with discharge from other basic
hydrologic information.

Use of a resistance equation, such as the Manning equation, requires additional data, including channel
width, slope, and a resistance coefficient. Fortunately, the results are not very sensitive to channel width and
slope and, if the modified form of the equation (eq. 12) is used as shown below, the Manning's n varies in a
predictable manner. Channel slope can be obtained from a topographic map, and, channel width can
generally be inferred from the drainage area (see the Appendix for an example). Drainage areas are
available at many points along most streams in the United States (Seaber and others, 1984) and at other
points, the drainage area can be determined by use of a topographic map.

To demonstrate the ability of the modified Manning approach to predict the variation of travel time
with discharge, river reaches for which two or more time-of-travel studies are available are analyzed below.
Table A-1 in the appendix contains the basic data necessary to calibrate 60 river reaches on 12 rivers.
Therefore, the Manning equation can be used to estimate travel time as a function of discharge.

The following steps were used to calibrate each river reach in table A-1 to an existing time-of-travel
study:

1. The data, including slope, width, discharge, travel time, reach length, and Manning’s n were
assembled.
2. The hydraulic-geometry parameters for width were determined, so that widths could be estimated for
other flows. The hydraulic-geometry exponent (W2) was assumed to be 0.26, except for the
Colorado River through the Grand Canyon. The hydraulic-geometry coefficient (W1) was
computed using equation 1 and the known width at a specified discharge.
The Manning’s n was assumed to be equal to 0.035.
The travel-time information was converted to transport velocity.
The total flow area was computed by dividing the discharge by the transport velocity.
The active flow cross-sectional area was computed by use of equation 6.
The inactive flow area (AO) was computed as the difference between the total and active flow area
by use of equation 12.
8. If the value of AO was negative, then its value was set to zero, and a Manning’s n was computed by
use of equation 6.

The first reach of Antietam Creek and the first reach of the Mississippi River will be used to illustrate
these steps.

Referring to the data in table A-1 for Antietam Creek, it is seen that at a reach average flow of (1.16 +
1.20)/2 = 1.18 m*/s, the reach average width is (12.8 + 11.0)/2 = 11.9 m. Solving equation 1 for W1 yields:
WI = 11.9/(1.18°) = 11.4,

A time of 13.0 — 3.2 = 9.8 hours was required for the dye peak to traverse the 9.6 — 2.6 = 7.0 kilometers
from site | to 2, thus the average transport velocity is 7,000 meters/ (9.8 x 3,600 s) = 0.198 m/s. The total
flow area in the reach is subsequently computed as the discharge/velocity or:

Total area = 1.18 m’/s / 0.198 m/s = 5.95 m*.
By using the average slope (0.0019), the active flow area is computed from equation 6:

Active area = 0.035°° x 11.9% x 1.18°°/ 0.0019°? = 2.61 m’.

The difference between the total area (5.95 m?) and active area (2.61 m’) is the inactive area, AO = 3.34 m’.
In this case, the value of AO is positive and the reach is calibrated.

Applying the above analysis yielded negative values of AO for 15 percent of the river reaches in table
A-1. The first reach of the Mississippi River, which has a flat slope, was one such reach. The data in table

MANE Y

20


10 TTT TTT T4111

| _  e —

ne

Celerity, in meters per second
_

| Calibration Prediction

0.1 Tt tou pot iit | titi

1 10 100 1000

Discharge, in cubic meters per second

Figure 10. Wave celerity in South Fork Shenandoah River between Waynesboro and Harriston, Virginia, as
a function of discharge.

A major tributary, the North River, joins the South Fork Shenandoah in the reach between Harriston
and Lynnwood (Station 01628500), thus this reach was not used. The drainage area at Lynwood, which is
2,808 km’, is more than five times that at Harriston. Luray (Station 01629500) is the next stream-gaging
station below Lynnwood and has a drainage area of 3,566 km”. Dye concentrations were not available at
Lynnwood but were measured at Island Ford, which is about 10 km downstream. Discharge hydrographs for
selected storms at Lynnwood and Luray under low-flow conditions are shown in figure 1 1. The time of
passage of the selected waves are indicated by circles. Hydrographs for high-flow conditions are shown in
figure 12.

16


APPENDIX

29


Predicted from equation 14

Area, in square meters

{ | \ |
| Lo | | |

0 — 1 tot

100 200 300 400 500

°

Discharge, in cubic meters per second

Figure 5. Variation of cross-sectional area with discharge for the New River, West Virginia. Symbols
represent values computed from the observed travel time, and the line is computed from
equation 14

Wind/Bighorn River, Boysen Reservoir to Basin, Wyoming

The Wind/Bighorn River flows generally north from central Wyoming to Montana where it joins the
Yellowstone River, figure 2. Stream-gaging stations are located on the Wind River just below Boysen
Reservoir (Station 0625900) and on the Bighorn River at Basin, Wyoming (Station 06274300), about 70
km south of the Montana boarder and 165 km downstream from the Boysen gage. The name of the river
changes from the Wind to Bighorn as it leaves the Wind River Canyon about 20 km downstream from
Boysen Reservoir. The USGS web site at http://waterdata.usgs.gov/nwis-w/WY lists the drainage area of
the Wind River below Boysen Reservoir as 19,945 km’ and the gage datum as 1,404.70 m. Likewise the
drainage area at Basin is 34,247 km”, and the gage datum is 1,164.73 m.

A plot of discharge at the two gaging stations for two time periods when individual waves could be.
identified is shown in figure 6. The passage time of the four waves identified on figure 6, as well as the
average discharge, and computed wave celerity for each wave are shown in table 4.

i

600


Rivers and streams form their own channel by eroding the bed and the banks until equilibrium exists
between the power of the flow to remove material and the supply and resistance of the material to be
moved. Large flows have greater power to erode material, but these flows occur infrequently. Most of the
time, a river occupies a channel that has the capacity to carry a much larger flow. The flow that is
compatible with the size of the channel is called the channel-forming discharge. The channel-forming
discharge is generally considered to be the flow that can just be carried within the banks of the river. In
other words, the channel-forming discharge is the flow that occurs just before water spills onto the flood
plane. Kilpatrick and Barnes (1964) and Wolman and Leopold (1957) suggest that the channel-forming
discharge is the flow that has a recurrence interval of about 2 years. Generally speaking, geomorphic
relations only apply for within-bank flow.

The discharge with a recurrence interval of 2 years can be estimated for any basin in the United States
from regional regression equations (Jennings and others, 1994). Areas of drainage basins, which are the
dominant variable in regression equations, have been tabulated by Seaber and others (1984) for all but the
smallest basins within the United States.

Using data from the Missouri River Basin, Osterkamp (1980) found that channel geometry is somewhat
influenced by the particle-size distribution of the streambed and bank material. He developed empirical
equations for separate particle-size distributions that relate channel width to discharge. The average of these
equations is:

0.56
Wyp =75Qp¢0°° 2)

in which Wpf is the average top width of the channel, in meters, when the river is flowing at bank-full

discharge (about the 2-year flow), Qp¢ in m’/s. Equation 2 should not be considered as a substitute for field
data, but it can provide a reasonable estimate of bank-full top width when field data are not available. For
rivers with no width information available, the value of W1 in equation 1 can be estimated by assuming a
value for W2, computing the bank-full width from equation 2, and solving equation 1 for W1 with W = Wy,
and Q = Quy.

Geomorphologists also suggest that, in an average sense, the cross-sectional area, A, of a natural
channel can be approximated by an equation of the form:

A=Al-Q”?, (3)

in which A is the cross-sectional area of the river flow, Al and A2 are constants called the hydraulic-
geometry coefficient and exponent for area, respectively, and Q is the river discharge. Theoretically, the
unitless A2 value can range from 0 to 1, but its value is usually between 0.5 and 0.8 with an average of 0.66
(Leopold and Maddock, 1953; Stall and Yang, 1970; Boning, 1974; Boyle and Spahr, 1985; Jobson, 1989).
Its value tends to be consistent for different reaches of a given river. Equations 1, 2, and 3 are not
homogeneous; thus, the units of Al and W1 depend on the units selected for A, W and Q.

Equations | and 3 are developed for within-bank flow. As the flow covers the flood plain, the width
and area increases more rapidly with discharge than for within-bank flow. The values of W2 and A2,
therefore, usually increase as the flood plain is covered.

Flow resistance

Resistance in open channels is typically quantified by using an empirical equation of either the
Manning or Chezy type. In the United States, the Manning form of the equation is most common. The
Manning equation relates the discharge, Q, to measurable quantities as:

: {ays , (4)

in which n is the Manning’s resistance coefficient, P is wetted perimeter, and S is slope. The slope of the
energy grade line should be used in the equation if non-uniform flow is being considered. The equation is
often written in terms of the bed slope, in which case it should be applied for steady uniform flow only. The
Manning’s resistance coefficient is considered to be dimensionless; therefore, when the English system of
units is used, the 1 in the numerator needs to be replaced with 1.49 to make the equation homogeneous.



Summary and Conclusions

Time-of-travel data are available for many rivers in the United States at a single flow, but few river
reaches have data available at more than a single discharge. This report provides a procedure to extrapolate
the time-of-travel information from one within-bank flow to another within-bank flow.

The report makes two points. First, that it is useful to consider the cross sectional area of a stream as
being composed of two parts - an active area, whose size varies with discharge, and an inactive area of
constant size. Second, the variation in size of the active area with discharge can be predicted either from an
analysis of wave speeds determined from streamflow-gaging records or from estimates of channel slope and
width by using a constant value of the Manning resistance coefficient.

The travel time of a dissolved constituent is directly proportional to the volume of water stored in the
river, Natural rivers usually contain a significant volume of water stored in inactive storage or pools, which
would not drain if the inflow were stopped. Dividing the inactive storage volume by the reach length yields
the average inactive area (AO), which is sometimes called the dead zone. The inactive area significantly
affects the time-of-travel of pollutants in the river but has little effect on the flow resistance or the speed of
flow waves. Physically measuring the speed of a dissolved constituent, usually dye, is the easiest way to
quantify the total volume of water stored in the river at a given flow. With the total volume known, the
active volume can be computed from either an analysis of the speed of flow waves or by use of a resistance
equation, such as the Manning equation. The inactive volume can then be computed as the difference
between the total and active volume. The variation of the active area (volume/reach length) with discharge
can be predicted using either a resistance equation or a geomorphic relation that is based on an analysis of
wave speeds. Once the active and inactive areas are known, the travel time of pollutants can be estimated by
adding the constant inactive area to the active flow area.

An analysis of the time series of discharge obtained at two or more points along the river may yield the
relation of wave speed to discharge. From this relation, the theory of uniformly progressive flow can be
used to develop the relation between the active flow area and discharge.

Geomorphic as well as Manning type resistance equations relate the size of the active flow area to
discharge. It is shown that the variable components of the Manning equation (slope, width, and flow
resistance) combine to produce the geomorphic form of the area equation. Although the slope, width, and
flow resistance are each highly variable from point to point in a river, they can be combined by use of a
resistance equation to yield a parameter that appears to have less variability and, thus, is more predictable.
By accounting for the inactive area separately, a resistance equation, such as the Manning equation, can be
applied meaningfully to long reaches of natural channels, even with limited data. When predicting the active
area for bed slopes greater than 0.0002, it is usually acceptable to assume Manning’s n is equal to 0.035.

Considering the reaches evaluated by using both the wave speeds and the modified form of the
Manning equation, the wave-speed predictions were more accurate than the predictions made by use of the
modified Manning equation. Typical prediction errors that result when using the wave speed approach were
about 3 percent, whereas typical values of the errors when using the modified Manning approach were a
little less than 10 per cent. Typical errors when using the direct form of the Manning equation were about
30 percent.

The hydraulic-geometry equations generally do not apply when the water covers the flood plain;
therefore, the procedures developed in this report should not be used for out-of-bank (flood) flows.

26


100. -— 1 a,
oO
80 }— s
| 4 |
g L_
2
=
~ 60 }— ° an
é
=
os 7
z 9 |
z 40} —
3
3 9
2 Le |
a
fe)
20 [-— °. s
“~ Conocoheague Creek
Very low flow
0 | | |

0 20 40 60 80 100
Observed travel time, in hours

Figure 16. Predicted and observed total travel time of each time-of-travel study listed in table A-2.

To illustrate the sensitivity of the results to the assumed values of width and slope, the travel times in a
large and small river were predicted for various assumed values of slope, width, and hydraulic-geometry
exponent for width. The Mississippi River at high flow is considered first, and the results are tabulated in
table 6. The minimum and maximum values represent the extremes among the reaches. Extremely small
slopes or large widths require a very small Manning's n to keep the inactive area (AQ) from being negative.
Reducing the width by 50 percent produced what is probably an unrealistic width-to-depth ratio. As can be
seen from table 6, the predicted travel time is not sensitive to the assumed values of width and slope. If the
value of the hydraulic-geometry exponent for width is reduced from 0.26 to 0.1, the width-to-depth ratio of
the Mississippi River decreases as the discharge increases, which seems unrealistic. Nevertheless, the
predicted travel times varied by only 2 or 3 percent for any of the adjustments except for the assumed value
of W2, which varied by about 5 percent.

23


CONVERSION FACTORS

Multiply By To obtain
inch 25.4 millimeter (mm)
foot(ft) 0.3048 meter (m)
foot/second (ft/s) 0.3048 meters/second (m/s)
mile (mi) 1.609 kilometer (km)
square foot (ft?) 0.09290 square meter (m’)
square mile (mi*) 2.590 square kilometer (km*)
cubic foot (f°) 0.02832 cubic meter (m’)
cubic foot per second (ft/s) 0.02832 cubic meter per second (m’/s)



LIST OF SYMBOLS

Aa
Au
AO
Al
A2

nl
n2

Qoe
Q
Q

Wor
wl
Ww2
At

cross-sectional area of flow

cross-sectional area downstream of wave
cross-sectional area upstream of wave

inactive flow area, or average cross-sectional area at zero flow
hydraulic-geometry coefficient for area
hydraulic-geometry exponent for area

celerity of moving wave

Manning’s resistance coefficient

constant in relation between Manning’s n and Q
exponent in relation between Manning’s n and Q
wetted perimeter

discharge

discharge of river flowing bank full

discharge downstream of wave

discharge upstream of wave

slope

top width of channel

average top width of a river flowing bank full
hydraulic-geometry coefficient for width
hydraulic-geometry exponent for width

time increment

distance moved by wave in time At


160

140

120

100

80

40

Discharge, in cubic meters per second

20

60 |

Boysen, March 1996

Basin, March 1996

96 144 192

Time, in hours

Figure 6. Discharge in Wind/Bighorn River below Boysen Reservoir and at Basin, Wyoming. Wave
passage indicated by arrows.

Table 4. Celerity and flow for waves in the Wind/Bighorn River between Boysen and Basin, Wyoming

Point Time Flow Celerity
Hours cubic meters/second meters/second

Boysen 10.76 38.14

Average 38.14 1.58
Basin 39.77 38.14

Boysen 58.52 31.08

Average 30.63 1.50
Basin 88.94 30.17

Boysen 9.50 90.63

Average 91.19 1.84
Basin 34.33 91.74

Boysen 117.50 58.28

Average 57.73 1.61
Basin 145.92 57.17

Wave celerity as a function of discharge, along with the line that represents a least-squares fit to the
data are shown in figure 7. The equation of the fitted line is:

C =0.828-9°!. (1s)

From a comparison of equations 9, 11, and 15, it is seen that the area in the Wind/Bighorn River can be

represented by:

A=

A0+1.46-Q°*". (16)

12


collected from nearly a hundred streams and rivers indicate that the unit-peak concentration tends to
attenuate in proportion to travel time to the 0.89 power (Jobson, 1996).

The cross-sectional mean river velocity that controls the travel time is, therefore, a primary variable
that needs to be predicted before any reasonable estimates can be made about pollutant concentrations.
Unfortunately, the prediction of mean water velocity in long reaches of a river has proven difficult. Even the
measurement of travel time through a river reach requires special techniques (Wilson, and others, 1986;
Kilpatrick and Wilson, 1989). These measurements generally involve a time-of-travel study where a slug of
dye is injected at one cross section and its arrival time is monitored at downstream sites. The U.S.
Geological Survey (USGS) has conducted many time-of-travel studies throughout the United States, and
Jobson (1996) has compiled references to many of these studies. Several velocity-prediction equations have
been proposed, but the accuracy of these equations is poor (Boning, 1974; Eikenberry and Davis, 1976;
Graf, 1986, Jobson, 1996).

A time-of-travel study gives an accurate measure of the reach-averaged water velocity at the river
discharge that existed during the study. Because the water velocity varies with discharge, a means of
extrapolating the velocity from one flow to another is needed. The purpose of this report is to apply
principles developed in the fields of geomorphology and uniform progressive flow to improve the
prediction of water velocity as a function of river discharge. Only the change in velocity with flow is
addressed in this report, thus the velocity at some reference flow is still required.

A brief discussion of the concepts and principles to be used will be given first, followed by the
application of these concepts, which will be illustrated using time-of-travel data collected in several rivers.
It will be shown that an analysis of discharge records at each end of the reach in question can provide
accurate estimates of the variation of water velocity with discharge. The USGS maintains discharge records
at about 7,000 locations within the United States. If these data are not available for the river reach in
question, then the extrapolation must be accomplished by use of a resistance type equation. It will be shown
that the effective resistance coefficient is much more predictable when the flow area is considered to consist
of an active and inactive part. The accuracy of the recommended procedures will be demonstrated and
contrasted to predictions made using traditional applications of the resistance equation. One of the goals of
this report is to demonstrate how this information can be used to improve estimates of the variation of travel
time with discharge for dissolved constituents.

Working Concepts

Some physical concepts that are useful in predicting the variation of average stream velocity with river
discharge are reviewed below.

Hydraulic Geometry

When rafting the Colorado River through the Grand Canyon, one is struck by the extreme variability of
the flow, which includes rapids and pools, backwater eddies and central currents. When flying over the
Grand Canyon, however, the river appears uniform and one-dimensional in nature. Geomorphology is a
field of study that specializes in viewing the extreme variability of nature and estimating general trends. A
geomorphologist tends to view a river from a distant perspective so that the general trends are apparent. An
engineer or modeler, on the other hand, tends to view a river close up, to capture all of the detail.
Sometimes a more distant perspective would be helpful, especially when attempting to predict the effect of
pollutant spills.

Much geomorphic information suggest that, in an average sense, the top-width of a natural channel, W,
can be approximated by an equation of the form:

W=wiQw?, a)

in which WI and W2 are constants called the hydraulic geometry coefficient and exponent for width,
respectively, and Q is the river discharge. The value of the unitless exponent W2 is generally found to range
from 0.1 to 0.4 with a typical value of 0.26 (Leopold and Miller, 1956; Stall and Yang, 1970; Jobson,
1989). Hydraulic geometry exponents have been found to maintain relatively consistent values both at a site
and between river reaches (Beven and Kirkby 1993).


Wave celerity as a function of discharge along with a line fitted to the data is shown in figure 13. The
equation of the fitted line is:

C= 0.990 Q*'”, (20)

As before, the hydraulic-geometry equation for the South Fork Shenandoah River between Lynnwood and
Luray becomes:

A=A0+4 1.15 Q°88 (21)

By using the low-flow data in table 5, at an average flow of 9.56 m’/s, the average velocity of the dye
between Island Ford and Luray was calculated to be 0.151 m/s, thus the average cross-sectional area was
calculated to be 63.3 m’, and the value of AO is 55.0 m’, which is 87 percent of the total area. By
extrapolating to a high flow of 22.26 m’/s, the area predicted by equation 21 is 72.5 m’, giving a predicted
travel time of 53.0 hours. This predicted value is 2.7 hours less than the measured value (table 5) and
represents an error of 4.8 percent. Although the scatter on figure 13 is large, the amount of inflow between
the gages is not large except, perhaps, for flows starting on November 11, 1995 (figure 11). The predicted
dye velocity is reasonably accurate, even though the extrapolation takes place for much lower flows than
were used to measure the wave celerity.

10 >
EF TT TTT T TT TTT T TT TT ttt

! [Pot J itil | boii

100 1000 10000
Discharge, in cubic meters per second

prediction

T
|
ibration

i
|

Celerity, in meters per second

Be
°

Figure 13. Wave celerity in South Fork Shenandoah River between Lynnwood and Luray, Virginia, as a
function of discharge.

Colorado River, Lees Ferry to Diamond Creek, Arizona

Flow in the Colorado River below Glenn Canyon Dam (see figure 2) is almost totally controlled by
dam releases, and the releases are greatly influenced by demands for power. The USGS provides real-time
data for gaging stations on the Colorado River at http://az.water.usgs.gov/. Discharges at Lees Ferry
(Station 09380000) and above Diamond Creek (Station 09404200) that were obtained from the internet on
October 19, 1999 are shown in figure 14. The times-of-occurrence as well as the magnitude of the minimum
and maximum flows were estimated for six pulses at Lees Ferry and Diamond Creek. As shown in figure
14, it takes about 2 days for a flow peak to traverse the 362.1-km reach of river. Averaging the six estimates
of peak and minimum flows indicates that the average wave speed is 2.55 m/s at 411 m*/s and 2.67 m/s at
592 m/s. Solving for Al and A2 from equation 10, yields:

A=A0+0.958-Q°5". (22)

18


\

USGS =

science for a changing world

R

Estimating the Variation of Travel Time

1g the
in Rivers by Use of Wave Speed and
Hydraulic Characteristics

Water-Resources Investigations Report 00-4187

US. Jecattiait of the interior

US. Geological Survey
U.S. Geological Survey



inverting, it is seen that the wave speed is determined as:

(i-A2)
e (10)

ALA2-

DL Uniformly progressive flow

Steady flow

Figure 1. Schematic showing uniformly progressive flow and the relation of wave speed to change in flow
area with discharge.

It can be seen from equation 9 that the speed of a flow wave is precisely controlled by the change in
storage volume, or in the average area within a river reach. As shown in equation 8, the hydraulic-geometry
parameters contain information about how the combined effects of flow resistance, slope, and width vary
with discharge. Equation 10 illustrates that the speed of flow waves passing through a river reach contains
that same information. The USGS continuously monitors the discharge at about 7,000 stream sites
throughout the United States. The arrival time of flow waves can be determined from these records. If the
amount of intervening flow is not too large, wave celerity between gages can be computed from the arrival
times.

Inactive storage areas

Based on the information given above, the following steps could be followed to estimate the travel time
of a dissolved constituent through a river reach between stream-gaging stations:

1. Measure the reach length, as well as the mean discharge and travel time of flow waves through the
reach at two different discharges.

2. Calculate the wave celerity for each wave by dividing the reach length by the travel time of the
wave. This yields two values of celerity, C, and discharge, Q.

3. By using equation 10, compute the values of Al and A2 from the two values of celerity and
discharge.

4. By using equation 3, compute the flow area for the discharge that occurs when the dissolved
constituent is to be routed through the reach.

5. Divide the discharge by the flow area to determine the transport velocity and route the dissolved
constituent using a model or the analytic solution to the convective diffusion equation.

When the above steps were applied to numerous time-of-travel studies, it was found that the predicted
travel times were always substantially smaller than the measured values, indicating that equation 3 typically
predicts areas that are smaller than observed in the field. It was finally concluded that equation 3 needed to
be modified by adding a constant term:

A=A0+Al-Q*?, ah

in which AO is the average inactive area or the cross-sectional area at zero flow. Similarly, the Manning


In most natural channels, the width is much larger than the depth: Thus, for natural channels the wetted
perimeter can be closely approximated by the top width, and the equation can be approximated as:

(5)

Equation 5 will be used to represent the Manning equation throughout this report.
By solving equation 5 for the area in terms of the discharge:

A= (a°Ss-°3 w+ pS, ©

it is seen that the Manning equation is consistent with the hydraulic-geometry equation 3. The value of A2
that is observed in the field is typically different from 0.6 because both flow resistance and width vary with
discharge. If one assumes that the Manning resistance coefficient can be approximated from discharge by
using a power form of the equation

n=nlQ”,
in which nl and n2 are constants, equations 1, 6, and 7 can be combined to yield:
A= (n19Ss-O3 wits Jylosraen2s04w2) | ®)

For typical values of W2 and A2, 0.26 and 0.66 respectively, it is seen that n2 = - 0.073. A negative value
of n2 implies that the Manning n decreases with increasing discharge. This is usually rationalized by stating
that the relative roughness decreases as the water depth increases. It is seen from equation 8 that the
hydraulic-geometry parameters for area include the combined effects of width, slope, and flow resistance.
The hydraulic-geometry parameters tend to be consistent from reach to reach for a river, whereas the width,
depth, and roughness tend to be quite variable. For this reason, the hydraulic-geometry parameters tend to
be more easily predicted and provide a reasonable means of estimating reach-averaged areas (velocities)
and widths in a modeling framework.

The Manning equation is typically applied to uniform flow. To apply it to non-uniform flow, the reach
is subdivided into small subreaches, and some means of averaging the cross-sectional properties is
incorporated. For example, the average area in a subreach is estimated as either the arithmetic or geometric
mean of the areas at each end of the subreach. The hydraulic-geometry approach, on the other hand,
considers data to be reach-averaged values.

Uniformly Progressive Flow and Moving Waves

Chow (1959) defines uniformly progressive flow as having a stable wave profile that does not change
in shape as it moves down the channel. One common type of uniformly progressive flow, which
approximates most flood waves in natural channels, is the monoclinal rising wave (Chow, 1959). Kleitz
(1877) developed the mathematical principle of uniformly progressive waves, whereas Seddon (1900) and
Wilkinson (1945) showed that the principle is applicable to natural rivers. Wilkinson found that the mid-
point of the rise or fall in stages were best suited for determining the velocity of an observed wave.

By considering the flow into and out of the control volume ABCD of figure 1, it is shown that:

QyAt-Q,At =(Ag-A,)AX,
or

- a
cn 8X _Qs-Q 80 ©)
At Ag-A, 0A
in which C is the speed of the moving wave or celerity, AX is the distance moved by the wave in time At, Q,
and Q, are discharge upstream and downstream of the wave, respectively, and A, and Aq are the cross-
sectional areas upstream and downstream of the wave, respectively. Differentiating equation 3 and


Table A-2. Computed and observed times of travel for verification runs (continued)

Site Distance Flow Time to Travel Time, in hr
Peak Observed Computed
km m’/s hr
Copper Creek Nordin and Sabol, 1974
1 0.19 1.78 0.132
0.768 0.709
2 1.01 1.78 0.900
0.750 0.597
3 1.72 1.78 1.650
0.390 0.507
4 2.40 1.90 2.040
1.34 1.17
5 3.35 1.93 3.380
0.920 0.817
6 4.14 1.93 4.300
Mississippi River Nordin and Sabol, 1974
1 54.7 6824 10.25
7.00 7.27
2 96.5 6824 17.25
5.75 6.15
3 17.9 6824 23.00
31.25 32.88
4 294.4 6824 54.25
Monocacy River, Reach | Nordin and Sabol, 1974
1 10.3 14.3 7.10
6.50 6.06
2 18.3 15.1 13.60
6.00 6.49
3 26.8 15.9 19.60
6.20 6.26
4 34.3 18.5 25.80
Monocacy River, Reach 2 Nordin and Sabol, 1974
1 75 8.10 8.5
20.5 22.62
2 18.8 8.35 29.0
12.0 11.66
3 27.6 9.34 41.0
7.0 71S
4 33.8 9.49 48.0
1 15 20.39 5.25
10.25 12.74
2 18.8 20.39 15.50
6.50 TAL
3 27.6 22.09 22.00
5.50 4.56
4 33.8 22.09 27.50

37



At this point the reach is calibrated; however, this is a large value of Manning’s n that may indicate that the
equation is not realistic. With no other knowledge of the river, there is little basis for adjusting the
Manning’s n. With the reach calibrated, the next step is to predict the travel time at high flow. This process
is identical with what was done above except the value of AO is zero, and Manning’s n is 0.138. Thus at the
high flow of 5.17 m’/s, the width can be estimated as before as 17.5 m, and equation 6 can be used to
predict the flow area:

Flow area = 0.138°° x 17.5°4 x 5.17°°/ 0.0019"? = 16.8 m’.

Computing the velocity as the discharge divided by the area gives 5.17/16.8 = 0.308 m/s, and the travel time
through the 7-km reach becomes 7,000/0.308 = 22,700 sec or 6.3 hours. As shown in table A-2, the
observed travel time at this flow was 4.15 hours, giving an error of 52 percent.

25


Table 6. Effect of width and slope on the predicted travel time of the Mississippi River.

Condition Predicted Travel Minimum Maximum Minimum | Maximum

Travel Time Width/depth | Width/depth | Manning Manning
Time Error Ratio Ratio n n
(hour) (percent)

Published data. 46.3 5.2 60 195 0.022 0.035

Double slope 45.9 44 60 195 0.0312 0.035

Half of slope 46.9 6.5 60 195 0.0156 0.035

Double low-flow 47.1 6.9 240 780 0.0139 0.035

width

Half of low-flow 45.6 3.5 15 49 0.035 0.035

width

W2=0.1 43.8 - 0.4 50 200 0.022 0.035

W2=0.4 48.6 10.4 60 225 0.022 0.035

The effect of river size is illustrated by presenting the same analysis for the Antietam Creek in table 7.
Even though reducing the widths to half of the published values gives an extremely small width-to-depth
ratio of 3, the error in the predicted travel time is less than 5 percent.

Table 7. Effect of width and slope on the predicted travel time of the Antietam Creek, Maryland.

Condition Predicted Travel Minimum Maximum Minimum | Maximum

Travel Time Width/depth | Width/depth | Manning Manning
Time Error Ratio Ratio n n
(hour) (percent)

Published data 36.5 -15 ll 24 0.035 0.035

Double slope 35.4 -4.5 ll 24 0.035 0.035

Half of slope 37.9 2.2 ll 24 0.035 0.035

Double low-flow 38.1 2.3 28 97 0.035 0.035

width

Half of low-flow 35.4 -4.2 3 12 0.035 0.035

width

W2=0.1 35.8 3.3 ll 24 0.035 0.035

W2 = 0.4 37.1 0.2 i 30 0.035 0.035

The data in tables 6 and 7 indicate that the prediction procedure is not sensitive to errors in the width,
slope or the assumed value of W2.

When the Manning equation is applied directly for rivers with slopes greater than 0.0002 m/m, errors as
large as 50 percent were common, and typical values were about 30 percent. The procedure for calibrating a
tiver reach by applying the Manning equation directly would be as follows:

1. Assemble the data, including slope, width, discharge, travel time and reach length.
2. Determine the hydraulic-geometry parameters for width.

3. Convert the travel-time information to mean transport velocity.

4. Compute the flow area by dividing the discharge by the transport velocity.

5. Calculate Manning’s n, by use of equation 5.

Once the Manning's n is calculated for the reach, it could be used to predict flow areas at other
discharges, and the transport velocity could be computed as the discharge divided by the area.

To illustrate the direct use of the Manning equation, the travel time for the high flow in the first reach
of Antietam Creek will be estimated. The first two steps have been completed above for low flow and the
results are shown in table A-1, giving S = 0.0019, Q= 1.18m’/s, travel time = 9.8 hours, reach length =
7,000 m, and W1 = 11.4. As shown above, the transport velocity is 7,000/(9.8 x 3,600) = 0.198 m/s, and the
flow area is 1.18/0.198 = 5.95 m*. Solving equation 5 for Manning’s n:

Manning’s n= 5.95! x 0.00195 / (1.18 x 11.99) = 0.138

24



Estimating the Variation of Travel Time in Rivers
by Use of Wave Speed and Hydraulic
Characteristics

by Harvey E. Jobson

Abstract

Predicting the effect of a pollutant spill on downstream water quality is primarily dependent on the
speed of movement downstream, the rate of longitudinal mixing, and the chemical or physical reactions that
affect the total mass of pollutant in the water column. Of these three processes, the speed of movement is
probably the most important and most difficult to predict. A time-of-travel study often is conducted to
quantify the mixing properties and speed of movement of a pollutant, but these results are expensive and
applicable only for the flow conditions under which they were obtained. The purpose of this report is to
provide guidance on extrapolating travel-time information obtained at one flow to a different flow in the
same reach of river.

It will be shown that in many cases a time series of discharge data (such as provided by a U.S.
Geological Survey stream-gaging station) at two or more points along the river can provide a reasonable
basis for this extrapolation. Direct application of resistance equations, such as the Manning equation,
provides very poor estimates of the variation of travel time with discharge.

For the 60 reaches on 12 rivers studied here, the accuracy of resistance equations could be greatly
improved by assuming the total flow area is composed of two parts, an active and inactive area. An
accuracy of about 10 percent, in extrapolating travel times from one within-bank flow to another within-
bank flow, could be obtained by assuming the inactive area to be constant and predicting the active flow by
use of a constant resistance coefficient. A Manning resistance coefficient (n) of 0.035 could be used for 50
of the 51 reaches with slopes greater than 0.0002 meter/meter. The predicted travel times were not very
sensitive to the assumed values of bed slope or channel width. For slopes less than 0.0002, accurate
predictions could be made by assuming the inactive flow area to be zero and the Manning coefficient to be a
constant, but less than 0.035

Introduction

Predicting the effect of a pollutant spill on the downstream water quality is a complex problem. In
general, mathematical models are available to address the transport, dispersion, and chemical reactions that
may occur, but the model accuracy is limited by the lack of reliable basic physical data, especially flow area
and mixing coefficients.

Predicting the effect of a pollutant spill is primarily dependent on the ability to predict the speed of
movement downstream, the rate of longitudinal mixing, and the chemical or physical reactions that affect
the total mass of pollutant in the water column. It is often acceptable to assume the pollutant is conservative,
to predict the worst-case scenario. The best way to obtain speed and mixing data is by conducting a time-of-
travel study. These studies are expensive, however, and the results only apply to flow conditions that existed
during the study. This report considers procedures for extrapolating time-of-travel results from one within-
bank flow to a higher or lower within-bank flow.

A fundamental building block for predicting the effect of an accidental spill is predicting the speed of
movement and attenuation of the unit-peak concentration resulting from an instantaneous spill. The unit-
peak concentration is defined as the a million times the peak concentration of an instantaneous spill times
the river discharge divided by the mass of pollutant to pass the cross section (Jobson, 1996, p. 6). If the
mixing processes are interpreted by use of the Fickian theory of diffusion, which is the most common
approach, the unit-peak concentration attenuates in proportion to the square root of travel time. Data


150 Harriston
~ L
2
8 |
3 |
2 100 -—
2
s
af
i=}
Q
3 L
3B
&
o
2 50 |
3 Wave 6
A Wave4 Wave 5
‘Waynesboro
0 f ! | | | | |
22 25 30 1 2
June 1995 July 1995

Figure 9. High-flow discharge in the South Fork Shenandoah River at Waynesboro and Harriston, Virginia.

Wave celerity as a function of discharge for the nine waves shown in figures 8 and 9 is shown in figure 10.
A least squares fit of the points, which have considerable scatter, indicates that the wave celerity can be
approximated using the equation:

C=0.825 Q’'%. (17)

As before, the hydraulic geometry for the South Fork Shenandoah River between Waynesboro and
Harriston becomes:

A=A0+ 1.36 Q°. (18)

By use of the low-flow data in table 5, it was determined that the average velocity of the dye in this
reach was 0.120 m/s at an average flow of 1.57 m/s. This flow is much lower that the lowest discharge for
which a wave speed is known (see the calibration arrow in figure 10). The average cross-sectional area,
however, was 13.05 m’, and the value of AO in equation 18 is 11.02 m?, By extrapolating to a high flow of
4.04 m’Js, the area predicted by equation 18 is 15.75 m’, giving a predicted travel time of 33.3 hours. The
predicted value is 3.7 hours less than the measured value and represents an error of 10.0 percent. This is a
larger error than observed for either of the other examples. The large inflow between the two gages is
probably the major cause of the error. The inflow complicates the measurement of the wave speed and
probably contributes to the scatter on figure 10 and errors in equation 17. Furthermore, the value of AO
represents about 85 percent of the total area at low flow. This also may influence the accuracy, in addition
to the extrapolation that was mostly outside the range of flows where the wave celerity was observed.

15


Table A-1. Hydraulic data used to compute the inactive area of river reaches (continued)

Site | Distance Flow Width Slope Timeto | Manning | Width | Inactive
Peak n Coeff Area
km m/s m m/1,000m, hr unitless | WI m
Copper Creek Nordin and Sabol, 1974
1 0.19 1.64 14.3 1.16 0.138
0.035 14.2 1.29
2 1.01 1.64 18.0 1.44 0.875
0.035 14.8 1.21
3 1.72 1.59 15.5 1.44 1.500
0.035 13.4 0.85
4 2.40 1.59 14.6 1.44 2.040
0.035 14.3 4.06
5 3.35 1.59 17.7 1,295 3.360
0.035 15.6 2.50
6 4.14 1.64 17.7 1.295 4.260
Mississippi River Nordin and Sabol, 1974
1 54.7 2633 468 0.118 14.10
0.0246 62.5 0
2 96.5 2633 501° 0.118 23.75
0.035 66.3 1669
3 117.9 2775 533 0.118 33.60
0.0220 87.9 0
4 294.4 2973 860 0.118 76.12
Monocacy River, Reach | Nordin and Sabol, 1974
1 10.3 5.38 29.0* 0.6 12.5
0.035 21.6 119
2 18.3 5.66 38.4* 0.6 23.0
0.035 27.3 10.1
3 26.8 6.37 48.8 0.5 33.5
0.035 26.9 13.2
4 34.3 7.65 40.5* 0.3 43.5
Monocacy River, Reach 2 Nordin and Sabol, 1974
1 75 3.06 46.6 0.3 17.5
0.035 29.8 33.0
2 18.8 3.20 33.5 0.3 64.0
0.035 31.0 17.6
3 27.6 3.45 51.2 0.6 86.0
0.035 29.9 13.7
4 33.8 3.51 31.4 0.3 99.0
New River Appel and Moles, 1987
1 85.8 263
321 1.61 7.00 0.035 74.7 48
2 64.0 280
256 1.61 6.40 0.035 58.7° 80
3 44.1 297
211 3.13 8.20 0.035 48.0° 166
4 19.8 297

32



Table A-1. Hydraulic data used to compute the inactive area of river reaches (continued)

Site | Distance Flow Width Slope Timeto | Manning | Width | Inactive
Peak n Coeff Area
km m/s m m/1,000m hr unitless wi m
Red River, Reach A Nordin and Sabol, 1974
1 8.0 140 209 0.112 3.35
0.0337 56.0 0
2 54.7 140 195° 0.151 32.00
0.0203 50.5 0
3 103.0 171 180° 0.129 52.00
0.035 44.4 44
4 159.3 185 162 0.092 87.5
Red River, Reach B Nordin and Sabol, 1974
1 12.1 108 133 0.092 7.00
0.0232 45.5 0
2 82.1 113 177° 0.089 45.00
0.0240 56.7 0
3 139.2 136 221* 0.072 79.50
0.0279 66.3 0
4 199.5 167 268 0.066 121.5
South Fork Shenandoah River Taylor and others, 1986
1 287.2 1.02 0.00
30.8° 1.65% 0.035 28.8 19
2 278.7 1.59 22.0
32.9° L654 0.035 28.8 8.1
3 266.8 1.76 48.0
33.7° 1.65% 0.035 28.8 9.4
4 256.5 1.93 71.0
61.5* 2.054 0.035 40.9 24.5
5 229.4 7.10 127.0
80.7° 1.30% 0.035 46.3 38.5
6 207.7 9.26 169.0
84.15 1.307 0.035 46.3 33.4
7 195.0 10.65 189.0
Tangipahoa River Nordin and Sabol, 1974
L 8.2 3.45 17.1 0.756 13.87
0.035 15.0 14.5
2 18.0 4.64 26.2* 071 29.87
0.035 174 2.3
3 41.5 6.94 28.7° 0.687 46.87
0.035 17.3 V7
4 55.3 8.10 29.9 0.663 58.87
0.035 20.2 8.9
5 71.0 8.61 40.2 0.569 73.40
0.035 20.4 5.7
6 82.1 9.03 31.7 0.379 82.87
0.035 19.1 16.2
7 94.0 10.85 37.8% 0.265 96.40

33



A-1 for the Mississippi River indicates that at a flow of 2,633 m’/s, the reach average width was (468 +
501)/2 = 484.5 m. Solving for W1 from equation | yields:

WI = 484.5 / 2,633°8 = 62.5.

A time of 23.75 — 14.10 = 9.65 hours was required for the dye peak to traverse the 96.5 — 54.7 = 41.8 km
from site 1 to 2. The average velocity, therefore, was 41,800 m/ (9.65 x 3,600 s) = 1.203 m/s, and the total
flow area, which was computed as the discharge/velocity, was:

Total area = 2633 m’/s / 1.203 m/s = 2188 m’,
By using the slope (0.000118), the active flow area was computed from equation 6:
Active area = 0.035% x 484.5% x 2633°° / 0.000118°° = 2699 m’.

Because the total area, 2188 m’, is less than the active area, 2699 m’, the inactive flow area (AO) is
negative, which is physically unrealistic. Therefore, its value was assumed to be zero, and the value of
Manning's n was computed from the total flow area using equation 6:

n= {2188 x 0.000118°°/(484.4"* x 2633°%) }"* = 0.0246.

Only nine reaches in table A-] have values of Manning’s n smaller than 0.035, which indicates that the
inactive flow area (AO) has been assumed to be zero. Eight of the nine small values occur where the slope
is less than 0.00015 m/m, and only two of the 12 reaches with slopes of less than 0.0002 have AO values
greater than zero.

In all cases, except for Conococheague Creek and the New River, the dye study conducted at the lowest
flow is used to calibrate the reach. The inactive flow area will be the largest portion of the total area at low
flow; thus, the calculation of AO should be most accurate. On the New River, the lowest flow with data
available in all three reaches was used for calibration. The time-of-travel data for the extremely low flow in
Concocheague Creek (the flow is exceeded 90 percent of the time [Taylor and Solley, 1971]) were
suspected of being in error; therefore, the intermediate flow (exceeded 60 percent of the time) was used for
calibration.

Once a river reach is calibrated, as shown in table A-1, the travel time at any other within-bank flow
can be computed by assuming that the inactive flow area and Manning's n do not vary with discharge. The
procedure is as follows:

1. For the discharge where the travel time is desired, compute the river width by assuming W2 = 0.26
and using the value of W1 shown in table A-1.
2. Compute the flow area by use of equation 12, which is based on the slope, width, Manning’s n, and

the inactive area shown in table A-1.

Divide the discharge by the flow area to yield the flow velocity.
4. Convert the flow velocity to travel time, based on reach length.

»

Time-of-travel data obtained at different discharges for the river reaches tabulated in table A-1 are
summarized in table A-2. To illustrate the prediction procedure, the travel time of the first reach of
Antietam Creek at high flow is computed. The average high flow shown in table A-2 for the first reach is
(5.10 + 5.24)/2 = 5.17 m/s. Assuming that W2 = 0.26 and W1 = 11.4 (from table A-1), the width at high
flow is computed from equation 1:

Width = 11.4 x 5.179% = 17.5 m.

The flow area can then be computed from equation 12:
Flow area = 3.34 + {0.035°° x 17.5°4 x 5.17°* / 0,0019°*}= 10.72 m’.
The velocity is subsequently computed as the discharge divided by the area:
Velocity = 5.17 m’/s / 10.72 m? = 0.482 m/s.

The travel time for the 7.0-km reach would be 7,000 m / 0.482 m/s = 14,523 seconds or 4.03 hours, which
is less than half of the travel time at low flow. This value compares closely with 4.15 hours, which was
observed in the field. The flow in Antietam Creek exceeds 5.17 m/s only 12 percent of the time, whereas it
exceeds the 1.18 m/s 75 percent of the time, (Taylor and Solley, 1971). The intermediate flow of Antietam
Creek is exceeded 40 percent of the time.

The above procedure was used to estimate the travel time for all reaches and flows tabulated in table A-
2, and the results are plotted on figure 15 as a function of the observed travel times.

21


CONTENTS

Abstract --
Introduction -
Working Concepts ---------------------------------------=+
Hydraulic Geometry
Flow Resistance ----------------------------------------
Uniformly Progressive Flow and Moving Waves
Inactive Storage Areas
Extrapolation of Transport Velocity Using Wave Speeds -
New River, Hinton to Thurmond, West Virginia -----
Wind/Bighorn River, Boysen Reservoir to Basin, Wyoming -
South Fork Shenandoah River, Waynesboro to Luray, Virginia
Colorado River, Lees Ferry to Diamond Creek, Arizona ---
Extrapolation of Transport Velocity Using a Resistance Equation -
Summary and Conclusions. -----------------------------------------------
References Cited -------------------------------------------
Appendix: Basic Data and Predicted Times of Travel

iii

DAWRWNNH HE


References for Appendix A

Appel, D. H. and Moles, S. B., 1987, Traveltime and dispersion in the New River, Hinton to Gauley
Bridge, West Virginia: U.S. Geological Survey Water-Resources Investigations Report 87-4012,
21p.

Graf, Julia Badal, 1995, Measured and Predicted Velocity and Longitudinal Dispersion at Steady and
Unsteady flow, Colorado River, Glen Canyon Dam to Lake Mead: Water Resources Bulletin of the
American Water Resources Association, V. 31, no. 2, p. 265-281.

Jennings, M.E., W.O. Thomas, Jr., and H.C. Riggs, 1994, Nationwide summary of U.G. Geological Survey

regional regression equations for estimating magnitude and frequency of floods for ungaged sites,
1993: U.S. Geological Survey Water-Resources Investigations Report 94-4002, 196 p.

Nordin, Jr., Carl F., and Sabol, George V., 1974, Empirical data on longitudinal dispersion in rivers: U.S.
Geological Survey Water-Resources Investigations 20-74, 340 p.

Taylor, K.R., James, Jr., R.W., and Helinsky, B.M., 1986, Traveltime and dispersion in the Shenandoah
River and its tributaries, Waynesboro, Virginia, to Harpers Ferry, West Virginia: U.S. Geological
Survey Water-Resources Investigations Report 86-4065, 60 p.

Taylor, K.R., and Solley W.B., 1971, Traveltime and Concentration Attenuation of a Soluble Dye in
Antietam and Conococheague Creeks, Maryland: Maryland Geological Survey Information
Circular 12, 25 pages.

Wiley, Jeffrey B., 1993, Simulated flow and solute transport, and mitigation of a hypothetical soluble-
contaminant spill for the New River in the New River Gorge National River, West Virginia:

U.S. Geological Survey Water-Resources Investigations Report 93-4105, 39 p.

40


Table A-2. Computed and observed times of travel for verification runs (continued)

Site Distance Flow Time to Travel Time, in hr
Peak Observed Computed
km m/s hr
New River Appel and Moles (1987)
2 64.0 91
13.00 11.76
3 44.1 91
18.50 18.66
4 19.8 91
1 85.8 518
4.80 5.44
2 64.0 527
4.00 4.89
3 44.1 538
5.60 5.80
4 19.8 538
1 85.8 62
20 13.80
2 64.0 62
3 44.1 62
24.0 26.54
4 19.8 62
1 85.8 127
11.50 9.70
2 64.0 127
3 44.1 337
7.45 7.62
4 19.8 334
1 85.8 229
7.80 7.48
2 64.0 232
3 44.1 125
14.20 14.60
4 19.8 127
Red River Reach A Nordin and Sabol, 1974
1 8.0 187 4.75
28.75 26.26
2 54.7 187 33.50
20.75 18.25
3 103.0 235 54.25
31.00 31.65
4 159.3 252 85.25
Red River Reach B Nordin and Sabol, 1974
1 5.7 235 2.50
31.00 30.12
2 75.6 245 33.50
26.50 28.15
3 132.7 249 60.00
32.50 36.21
4 193.1 249 92.50

38



2500 ->——— — - — _
Q c l | | I +
i 2000 E -
a E >\*—— Thurmond |
@ ft |
£ i 4
£1500 -
a i Hinton Wave 4
3 | Wave 2
1000 -
5
I
© | “|
g 500 [ Wave 7\ |
g Wave | -
# - |
o£ zi

1 2 3 4 5 6 7 8 9 10 Il 12 13
November, 1986

Figure 3. Discharge in the New River at Hinton and Thurmond, West Virginia. Wave passage times
indicated by circles.

The wave speeds in table | are plotted as a function of discharge on figure 4. The line on figure 4
represents a least squares fit of the points, which is defined by the equation:

C= 0.428-Q°78, (13)

in which C is wave celerity in m/s, and Q is discharge in m*/s. The arrows indicate discharges where time-
of-travel dye studies were conducted.
From equations 10, 11, and 13, it is seen that the flow area of the New River should be represented by:

A=A0+3.25-Q°7". (14)


Time-of-travel data obtained under steady-flow conditions in the New River are presented in table 2.
According to Appel and Moles (1987), the distance is 21.7 km from Sandstone to Prince and 20.0 km from
Prince to Stone Cliff. Sandstone is 5.2 km below the Hinton gage. Wiley and Appel (1989) did not specify
the exact date of the two studies they reported, thus these studies are identified by the date of the report.

Table 2. Time-of-travel data for the New River between Sandstone and Stone Cliff, West Virginia (from
Appel and Moles (1987) and Wiley and Appel (1989))
[m*/s, cubic meters per second]

Date of study Sandstone Travel Prince Travel Stone Cliff
Discharge Time Discharge Time Discharge
m/s Hours m/s Hours m/s
August 14, 1985 62.3 20.0 62.3
October 24, 1985 90.6 13.0 90.6
1989 (report date) 127.4 115 127.4
1989 (report date) 229.4 78 232.2
May 15, 1986 263.4 7.0 280.3 6.4 297.3
November 8, 1985 518.2 48 526.7 4.0 538.0

The May 15, 1986 study was selected for computing the cross-sectional area at zero flow because it had
the smallest discharge to span both subreaches. The total travel time between Sandstone and Stone Cliff is
13.4 hours; therefore, the average dye velocity is 0.864 m/s over this 41.7-km reach. Assuming an average
discharge of 280.3 m*/s, which is the average of the values measured at the three sites, the volume of water
displaced in 13.4 hours is 13.52 x10° cubic meters. This represents an average cross-sectional area along
the 41.7-km reach of 324.3 m? at this flow. Solving equation 14 yields AO = 137.3 m’.

The total area for each of the other flow conditions was then computed from the observed travel time
and distance as above, and the predicted area was computed from equation 14 with the value of AO = 137.3
m?. The results are compiled in table 3, along with the predicted travel time, as well as the percent error in
the predicted travel time. The total and predicted areas are plotted on figure 5. As shown on figure 5, the
projection of the total areas to a zero discharge would indicate a non-zero value of area. Therefore, the data
on figure 5 support the general form of equations 11 and 12. Any resistance equation with a constant
resistance coefficient and a zero value of AO would pass through the origin and likely provide only a poor
estimate of the areas on figure 5 except near the flow at which it was calibrated. As indicated by the arrows
on figure 4, travel times were extrapolated for much lower discharges than were used for the calibration or
were available for the wave-celerity measurements.

Table 3. Area and travel time for the New River, between Sandstone and Stone Cliff, West Virginia
[m’/s, cubic meters per second; m’, square meters; N/A, not applicable]

Average Effective Predicted Predicted Measured Error

Discharge Area Area Travel-time Travel-time in
m/s m m Hours Hour Percent
280.3 324.3 3 13.4 13.4 N/A
62.3 206.7 200.7 19.42 20.0 -2.9
90.6 212.0 220.3 13.51 13.0 3.9
127.4 243.1 243.4 151 15 0.1
230.8 298.7 299.9 7.83 78 0.4
527.6 400.8 431.9 9.48 8.8 +7.8

10


U.S. DEPARTMENT OF THE INTERIOR

BRUCE BABBITT, Secretary
U.S. GEOLOGICAL SURVEY
Charles G. Groat, Director

The use of trade, product, industry, or firm names is for descriptive purposes only and does not
imply endorsement by the U.S. Government.

For additional information write to: Copies of this report can be purchased from:
U.S. Geological Survey U.S. Geological Survey

Chief, Office of Surface Water Branch of Information Services

415 National Center Box 25286

Reston, VA 20192 Denver, CO 80225-0286


Table A-1. Hydraulic data used to compute the inactive area of river reaches
{km, kilometer; m/s, cubic meter per second; m, meter; m/100m, meter per 1000 meter; hr, hour;
m”, square meter]

Site | Distance Flow Width Slope Time to Manning | Width | Inactive
Peak n Coeff Area
km m/s m__| m/1,000m hr unitless_| WI mm
Antietam Creek Source of data - Nordin and Sabol, 1974
1 2.6 1.16 12.8 2.7 3.2
0.035 11.4 3.34
2 9.6 1.20 11.0 Ll 13.0
0.035 10.8 9.47
3 21.5 1.64 12.8 Ld 43.0
0.035 14.2 13.67
4 29.6 L78 19.8" 0.8 67.0
Clinch River | Nordin and Sabol, 1974
1 0.69 10.79 45.7 0.484 0.467
0.035 24.5 9.32
2 1.58 9.85 44.2 0.625 1.250
0.035 23.7 0.02
3 2.49 9.46 41.1 0.625 1.800
0.035 25.2 4.65
4 3.59 10.51 50.6 0.484 2.650
0.035 28.2 15.18
5 4.67 10.59 53.6 0.441 3.840
0.035 27.7 28.75
6 5.89 9.77 47.5 0.361 5.700
Colorado River Graf, 1995
1 57.7 425 141 18.5
0.035 | 17.6" [333
2 98.3 425 1.26 33.1
0.035 | 22.3° 71
3 123.2 430 2.74 39.3
0.035 | 17.6° 2723
4 189.3 433 1.95 58.8
0.035 12.1° 227
5 267.9 436 1.51 79.0
0.035 15.5° 196
6 343.6 436 1,34 98.4
0.035 16.3° 213
7 380.5 436 161 108.3
Conococheague Creek I Taylor and Solley, 1971
1 44 6.82 31.0° 0.7 3.0
0.035 20.0 52.8
2 8.7 6.85 35.1 0.5 6.6
0.035 24.6 21.4
3 13.4 6.88 46.0 0.7 10.3
0.035 33.7 13.8
4 19.9 6.94 65.5 0.7 15.5
0.035 32.9 49.5
5 26.0 6.94 43.3 0.6 21.3
0.035 25.4 239.1
6 33.9 7.08 41.1 0.6 33.8

31



650 | re

riety

600 J
z A
g \
5 550 ;
* ; ' 4
: 4
E500 |
3 i
£ poe
g 450)
E
s
A

|
1

400

tii

350
11 12 13 14 15 16 17 18
Day, in October 1999

Figure 14. Discharge in the Colorado River at Lees Ferry, solid curve, and above Diamond Creek, Arizona,
dashed curve.

Ata steady flow of 432 m’/s the total travel time of dye from the Nautiloid Canyon section (57.7 km
below Lees Ferry) to Gneiss Canyon (380.5 km below Lees Ferry) is 89.0 hours (Graf ,1995, p. 273). Thus
the average flow area is calculated as the discharge divided by velocity to be 428.8 m. Solving the equation
22 yields AO = 236.1 m’.

The data for a high flow time-of-travel study is available at the USGS web site at
http://az.water.usgs.gov/flood2.html. These results indicate that at a steady flow of 1,317 m?/s the travel
time from the Nautiloid Canyon section (57.8 km below Lees Ferry) to Diamond Creek (362 km below Lees
Ferry) is 46.0 hours. Thus, the high-flow average area is 717 m’. Solving the equation 22 with Q = 1,317
m/s, yields an area of 746 m? and a velocity of 1.77 m/s. The predicted travel time is 47.9 hours, which is
1.9 hours (4.1 percent) greater than the observed value. The predicted travel time is at a flow more than 2.2
times larger than the largest flow for which the wave speed is known and over three times the flow where
the low-flow dye travel time is known.

If the inflow between gages is not too large, discharge hydrographs can be used to determine
representative wave-celerity information for a river reach between the gages. The celerity information can
subsequently be used to infer the change of travel time with flow with reasonable accuracy. For the
examples given, travel time could be predicted to within 10 percent as the flow ranged from 0.22 to 4.0
times the flow rate where travel-time information was known. In general, it was found that the inactive flow
area was a significant portion of the total flow area, thus extrapolation of travel times using the Manning
equation directly will likely require the resistance coefficient to vary significantly with flow. The next
section of this report demonstrates that the modified form of the Manning equation (equation 12) can be
used with reasonable accuracy to extrapolate travel-time information to different flows while keeping the
resistance coefficient constant.

19


Estimating the Variation of Travel Time in Rivers
by Use of Wave Speed and Hydraulic
Characteristics

By Harvey E. Jobson

U.S. GEOLOGICAL SURVEY
Water-Resources Investigations Report 00-4187

Reston, Virginia
2000


References Cited

Appel. D. H. and Moles, S. B., 1987, Traveltime and dispersion in the New River, Hinton to Gauley Bridge,
West Virginia: U.S. Geological Survey Water-Resources Investigations Report 87-4012, p. 8.
Beven, Keith, and Kirkby, Michael J., 1993, Channel Network Hydrology: John Wiley & Sons, New York,

p. 91.

Boning, C.W., 1974, Generalization of stream travel rates and dispersion characteristics from time-of-travel
measurements: Journal of Research of the U.S. Geological Survey, v. 2, no. 4, July-August 1974,

p. 495-499.

Boyle, J.M., and Spahr, N.E., 1985, Traveltime, longitudinal-dispersion, reaeration, and basin
characteristics of the White River, Colorado and Utah: U.S. Geological Survey Water-Resources
Investigations Report 85-4050, 53 p.

Chow, Ven te, 1959, Open-Channel Hydraulics: McGraw-Hill Book Company, New York, p. 528.

Eikenberry, S.E., and Davis, L.G., 1976, A technique for estimating the time of travel of water in Indiana

streams: U.S. Geological Survey Water-Resources Investigations Report 9-76, 39 p.
Graf, J.B., 1986, Traveltime and longitudinal dispersion in Illinois streams: U.S. Geological Survey Water-
Supply Paper 2269, 65 p.

Graf, Julia Badal, 1995, Measured and Predicted Velocity and Longitudinal Dispersion at Steady and
Unsteady flow, Colorado River, Glen Canyon Dam to Lake Mead: Water Resources Bulletin of the
American Water Resources Association, Vol. 31, No. 2, p. 265-281.

Jennings, M-E., Thomas, W.O. Jr., and Riggs, H.C., 1994, Nationwide summary of U.G. Geological Survey

regional regression equations for estimating magnitude and frequency of floods for ungaged sites,
1993: U.S. Geological Survey Water-Resources Investigations Report 94-4002, 196 p.

Jobson, H.E., 1989, Users manual for an open-channel streamflow model based on the diffusion analogy:
U.S. Geological Survey Water-Resources Investigations Report 89-4133, 73 p.

Jobson, H.E., 1996, Prediction of traveltime and longitudinal dispersion in rivers and streams: U.S.
Geological Survey Water-Resources Investigations Report 96-4013, 69 p.

Kilpatrick, F.A., and Barnes, H.H. Jr., 1964, Channel geometry of Piedmont streams as related to frequency
of floods: U.S. Geological Survey Professional Paper 422-E, 10 p.

Kilpatrick, F.A., and Wilson, J.F.Jr., 1989, Measurement of time of travel and dispersion in streams by dye
tracing: U.S. Geological Survey Techniques of Water-Resources Investigations, book 3, chap. A9,
27 p.

Kleitz, M., 1877, Note sur la theorie du mouvement non permanent des liquides et sur application a la
propagation des crues des rivieres (Note on the theory of unsteady flow of liquids and on application
to flood propagation in rivers): Annales des ponts et chaussees, ser. 5, v. 16, 2e semestre, p. 133-196.

Leopold, L.B., and Maddock, Thomas, 1953, The hydraulic geometry of stream channels and some
physiographic implications: U.S. Geological Survey Professional Paper 252, 57 p.

Leopold, L.B., and Miller, J.P., 1956, Ephemeral streams--hydraulic factors and their relation to drainage
net: U.S. Geological Survey Professional Paper 282-A, 36 p.

Lowham, Hugh W., and Wilson, James F. Jr., 1971, Preliminary results of time-of-travel measurements on
Wind/Bighorn River from Boysen Dam to Greybull, Wyoming: U.S. Geological Survey Open-File
Report No. OF71-85, Reston, Virginia, 7 pages.

Nordin, Jr., Carl F., and Sabol, George V., 1974, Empirical data on longitudinal dispersion in rivers: U.S.
Geological Survey Water-Resources Investigations Report 20-74, 340 p.

Osterkamp, Waite R., 1980, Sediment-morphology relations of alluvial channels: American Society of Civil
Engineers, Proceedings of the Symposium on Watershed Management ‘80, Bosie, ID, p 188-199.

Seaber, Paul R., Kapinos, F. Paul, and Knapp, George L., 1984, State Hydrologic Unit Maps: U.S.
Geological Survey Open-File Report 84-708, 21 pages.

Seddon, James A., 1900, River hydraulics: Transactions, American Society of Civil Engineers, v. 43,

p. 179-229.
Stall, J.B., and Yang, Chih Ted, 1970, Hydraulic geometry of 12 selected stream systems of the United
States: Illinois State Water Survey Research Report No. 32, July, 73 p.

27


Table A-2. Computed and observed times of travel for verification runs (continued)

Site Distance Flow Time to Travel Time, in hr
Peak Observed Computed
km m/s hr
Colorado River Graf, 1995
1 57.8 1303 9.75
7.58 6.74
2 98.3 1291 17.33
3.92 3.25
3 123.4 1291 21.25
10.25 10.14
4 196.3 1291 31.50
10.50 8.92
5 267.9 1297 42.00
3.50 3.43
6 294.4 1354 45.50
10.25 8.69
7 362.0 1410 55.75
Conococheague Creek Taylor and Solley, 1971
1 44 29.4 1.35
1.80 1.98
2 8.7 29.4 3.15
1.85 2.24
3 13.4 29.7 5.00
2.65 3.23
4 19.9 30.0 7.65
2.75 3.28
5 26.0 30.3 10.40
4.40 5.20
6 33.9 30.6 14.80
1 44 2.58 5.4
8.6 5.77
2 8.7 2.83 14.0
10.6 §.22
3 13.4 2.89 24.6
12.9 7.10
4 19.9 2.89 37.5
Copper Creek Nordin and Sabol, 1974
1 0.19 7.05 0.070
0.275 0.396
2 1.01 7.82 0.345
0.300 0.329
3 1.72 7.93 0.645
0.250 0.295
4 2.40 7.62 0.895
0.530 0.538
5 3.35 8.10 1.425
0.305 0.409
6 4.14 8.72 1.730

36



40

30 | —-

O——new River

very low flow

Predicted travel time, in hours

Conococheague Creek
very low flow a

0 10 20 30 40
Observed travel time, in hours

Figure 15. Predicted and observed travel time through each reach listed in table A-2.

Three of the outliers shown in figure 15 represent Conococheague Creek at very low flow (2.8 m/s);
the reach was calibrated at 6.9 m°/s. The other outlier represents the upstream reach in the New River at a
flow of 62 m’/s. The New River reach was calibrated at a flow of 271 m’/s. These outliers probably
indicate that it is not advisable to extrapolate from high to low flows. It should be noted that the travel time
for low flow in the New River was predicted closely by using the wave speed, as shown in table 3.

Computing the travel time for the entire reach of each injection reduces the scatter from random errors.
The predicted and observed total travel time of each dye injection listed in table A-2 are shown in figure 16.
Although several of the errors for individual reaches were larger than 10 percent, only the low flows in
Conococheague Creek had an error in the total travel time that exceeded 10 per cent.

22


equation (Equations 6 or 8) should be modified to include the zero-flow cross-sectional area:
A = A0+(n°° $03 Wt ).Qo6, (12)

The last term in equations 11 and 12, the flow dependant term, is considered the active area, and the first
term, AO, is considered to be the inactive area, or storage zone. Stall and Yang (1970) predicted travel times
from hydraulic-geometry equations, like equation 3, and compared the results with time-of-travel studies
conducted at various flow rates. Consistent with equations 11 and 12, they concluded that the accuracy of
the approach decreased as the flow decreased (Stall and Yang, 1970, p. 38).

In natural channels the flow is seldom uniform in either width or depth. Rather, the flow tends to
consist of pools that contain rather slow moving water separated by riffles, which are relatively high points
in the bed. If the inflow to a natural channel were stopped and if water did not seep into the bed or
evaporate, then the river typically would not drain completely. Instead, there would be periodic pools of
stagnant water. These pools are easily recognized as an ephemeral stream dries up. Especially at low flow,
the water level in a stream is controlled by flow over the riffles and thus, the overall “flow resistance” is
insensitive to the depth of the pools; that is, the inactive area. Travel time of a dissolved constituent through
the reach, however, is sensitive to the depth of the pools, because it influences the volume of water in the
reach.

The equations 1, 11 and 12 are considered to be representative of area or width when averaged over a
distance of several pool/riffle sequences. They may not be representative of the area or width at a specific
cross section. Thus, the value of AO in equations 11 and 12 should be considered as the volume of water per
unit length that would not drain if the flow were shut off. The second term on the right side of equations 11
and 12 is considered to represent the active area of flow, and this is the part of the flow that is influenced by
flow resistance and that influences the wave celerity. Because the value of AO is a constant, the wave speed
is independent of AO, and equation 10 remains valid.

It will be shown below that when the values of Al and A2 are determined from wave speeds in a
channel, and when AO is determined from the travel time at a specific discharge, equation 11 provides close
predictions of the flow area as a function of discharge. Likewise, it will be shown that when the inactive
area, AO, is assumed to be zero and the Manning coefficient is determined for a reach of river from
equation 5 or 6 by using the transport velocity (area), slope and width, only poor estimates of the travel time
at another flow are possible without major adjustments to the Manning coefficient. Improved estimates of
transport velocity can be obtained by using equation 12 and a constant Manning coefficient.

Extrapolation of Transport Velocity Using Wave Speeds

Four examples are provided to illustrate how stream-gaging information can be used to extrapolate
transport velocities to different flows and to demonstrate the accuracy that might be obtained. In each
example the only hydraulic data to be used is a continuous record of river discharge at two points, a known
distance apart. These data will be used to measure the variation of wave speed with discharge, and thus, to
compute the values of Al and A2 in equation 11. All examples are for stream reaches where at least two
time-of-travel dye studies have been conducted. The low-flow dye study will then be used to "calibrate" the
reach by determining the average transport velocity, and thus, the total flow area and value of AO in
equation 11. The predictive ability of the equation will then be tested by comparing the predicted and
measured travel times obtained at other flows.

New River, Hinton to Thurmond, West Virginia

The New River flows northward from its headwaters in North Carolina, through western Virginia, and
into south-central West Virginia, where it joins the Gauley River to form the Kanawha River. The 60.51-km
reach, between Hinton and Thurmond, West Virginia, figure 2, is within the New River Gorge National
River and is widely used for white-water rafting. According to the USGS web site at
http://waterdata.usgs.gow/nwis-w/WV/, the drainage area at Hinton (Station 03184500) is 16,203 km’, and
at Thurmond (Station 03185400) it is 20,544 km’. The largest drainage into this reach of the New River
Gorge is about 350 km? and drainages generally are less than 15 km? (Wiley, 1993).


Table 1. Celerity and flow for waves in the New River between Hinton and Thurmond, West Virginia

Point Time, in Flow, in Celerity, in
Hours cubic meters per second _ Meters per second

Hinton 37.31 175.3

Average 177.2 1.75
Thurmond 46.93 179.2
Hinton 101.85 1115.9

Average 1122.4 2.82
Thurmond 107.82 1128.8
Hinton 120.79 1895.9

Average 1884.4 3.43
Thurmond 125.68 1872.8
Hinton 140.87 1390.0

Average 1383.0 3.41
Thurmond 145.80 1376.1
Hinton 207.19 702.4

Average 682.1 3.03
Thurmond 212.74 661.8
Hinton 274.01 216.5

Average 219.7 1.98
Thurmond 282.50 222.9
Hinton 297.47 182.4

Average 184.2 1.85
Thurmond 306.57 185.9

10 7 TTT T TTT TT
i
EXPLANATION

© Wave speed

Discharge during
dye study em,

Calibration dye study

eA A OM Bo

10 100 1000

Celerity, in meters per second

a i

Flow, in cubic meters per second

Figure 4, Wave celerity in the New River between Hinton and Thurmond, West Virginia, as a function of
discharge.