Transport and Dispersion of 

Labeled Bed Material 

North Loup River, Nebraska

GEOLOGICAL SURVEY PROFESSIONAL PAPER 433-C

Prepared in cooperation <with the 

U.S. Atomic Energy Commission

-'"v
Transport and Dispersion of 

Labeled Bed Material 

North Loup River, Nebraska

By W. W. SAYRE and D. W. HUBBELL

TRANSPORT OF RADIONUCLIDES BY STREAMS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 433-C

Prepared in cooperation with the 

U.S. Atomic Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1965
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 45 cents (paper cover)
CONTENTS

Symbols._---_----___--_-__________________________

Abstract_ _.--_--__----____-___________________.__

Introduction and acknowledgments-__________________

The experiment.___________________________________

The test reach________________________________

The radioactive label and the detection system_____

Amount of radioactivity__-_------_--_-__--______

Number of tracer particles.______________________

Size of tracer particles..._____________ __________

Experimental equipment-________________________

Calibration of radiation-detection equipment- ______

Field procedures..______________________________

Radiation safety______________________________

Page 

v

Cl 

1 

3

3

4

5

6

6

6

9

10

12

Page 

Presentation and discussion of data___________________ C13

Derivation of concentration-distribution and related

functions.________-_____-_____----_--_-_-----__-_ 20

The concentration-distribution function ___________ 27

Related distribution functions._-_-__-____-_--____ 30

Application of concentration-distribution function to

experimental data_______________________________ 31

Bed-material transport._____________________________ 39

Applicability of results.______-____---____-_--_-_--__ 41

Conclusions-_______--_-_-_-_-___----_--------__---_ 43

Supplementary data point-source method for determining detection-system sensitivity _________________ 44

Literature cited____-_------------------___--------- 48

ILLUSTRATIONS

Page 

FIGURE 1. Location map of the Loup River basin.___________________________________________________________ C3

2. Sketch map of test reach._______-_____________________________________________-_---_----_-------_ 4

3. Downstream view of test reach.__________________________________________________________________ 4

4. Upstream view of test reach from about station 1000______________________________________________ 4

5. Graph showing size distributions of measured suspended sediment, labeled Ottawa sand, and bed material... _ 7 

6, 7. Photographs of the dosing apparatus.________-___________-__-_-_-___-_-__-_--_-_-__--------------- 7

8. Photograph of the sled and scintillation detector_________-__________-___--____-----------------_-_ 7

9. Diagram of the watertight casing for scintillation detector.________________________----__-__--------- 8

10. Sketch showing arrangement of boat and sled for the longitudinal traverses.___________________________ 8

11-14. Photographs showing 

11. Arrangement of instruments in the boat_____________________-___________-_-----__-__------- 9

12. Dual ultrasonic depth sounder and distance-marking system._____-_-________-_-__--__-_--_----- 9

13. Equipment for sampling and analyzing the bed material.______________________________________ 9

14. Analysis of the vertical distribution of labeled particles in the field.___-_____-----_-_-----------_ 9

15. Graph showing relations between the concentration of tracer particles in submerged sand and the adjusted

counting rate for iridium 192____-____________________________________________-________-________ 10

16. Graph showing variation in adjusted count rate with the depth to which different concentrations of tracer

particles are uniformly mixed____-________-__________-_____________--_____-_-__---_--_-------- 11

17-21. Graphs showing longitudinal distribution of labeled particles 

17. Along the right side of the channel, November 3-8, 1960__-_--------------_-------_------_-__ 14

18. Along the right side of the channel, November 10-15, 1960-_-__-----------_-----_-------_____ 15

19. Along the center of the channel, November 9 and 11, 1960 ________________________-_----_--__- 16

20. Along the left side of the channel, November 4-8, 1960__________________________________ _ ___ 17

21. Along the left side of the channel, November 10-15, 1960_____-_-_-____-________-____-______ 18

22. Photograph showing frazil ice in the stream, November 9.___________________________________________ 19

23. Graphs showing lateral distribution of labeled particles downstream from the source on November 4, I960.. 20

24. Graphs showing lateral distribution of labeled particles downstream from the source on November 5 and 8,

I960-.-_-_________________________________________________________________________ 21

25-29. Graphs showing vertical distribution of labeled particles at selected verticals on 

25. November 4, I960-________________________________________________________________________ 22

26. November 4, 5, and 9, 1960_____________________________________________________________ 23

27. November 10 and 11, I960 __________________________________________________________ 24

28. November 11 and 13, I960 ____________________________________________._________ 25

29. November 13, 1960-----__-._________________________ _____________________-..__ 26

in
IV CONTENTS

Page 

FIGURE 30. Water-and bed-surface profiles defined by the dual channel stream monitor______---__-___--__-_____- C27

31. Graph showing location of mode and mean as a function of dispersion time.___________________________ 32

32. Graph showing attenuation of observed peak relative concentration with dispersion time_________________ 33

33. Graph showing attenuation of peak according to theoretical function._________________________ .____.__ 34

34-40. Graphs showing theoretical and observed longitudinal distribution of tracer particles 

34. Runs 3R and 3L.______._____._...___.___._.__.__.___._______.__.______.-_______--_.____ _ 35

35. Runs 4R and 4L___._._.___._______________________.._____--_-____-----._--__---_----_._- 36

36. Runs 5R and 5L_______._________________________-_______-_-_-___-_-_---_-_----_--____._- 37

37. Runs 6R and 6L_________________________________________________________________________ 38

38. Run 7C..._._ ____ __.-_____.__.____.____-___ _ _. ----.-_ --- _.- --- ------ 38

39. RunsSR and 8L________________________________----____-_--_--_--------_-_-_-----_------ 39

40. Runs URand 12R_________________________________--__-____---_.-__-_---_-__------__--_-- 40

41. Schematic diagram illustrating method for estimating average depth of zone of particle movement. _______ 43

42. Schematic sketch of arrangement of equipment for point-source sensitivity calibration,__________________ 44

43. Graph showing isocounts for a 7-juc iridium 192 point source in submerged sand._______________________ 45

44. Graph showing counting rates per microcurie from point sources located in submerged sand at various radial

distances from the center of the detector crystal._________________________________________________ 46

45. Definition sketch for calculation of sensitivity from point-source calibration________---_____-_-______-_- 47

46. Graph showing relative counting rate from sources uniformly distributed throughout the volume of submerged sand within various radial distances of the center of the detector crystal._____________________ 47

TABLES

Page 

TABLE 1. Time, location, and extent of longitudinal and lateral traverses...________________________-_--_------------- C13

2. Summary of hydraulic data__________________________________________________________-__-----_---_----- 19

3. Summary of sediment data_---_--_-______________________________________--__-__---------__----------- 19

4. Some characteristics of the observed distributions._____---__-_-------_----------------------------------- 31

5. Sensitivities of three radionuclides_-_____________________________________-__-_-____----_-_--_----------- 47
SYMBOLS

Symbol Definition Dimensions Units 

a Vertical distance from surface of sand to center of the detector crystal (o=l in. L in.

for a 2X2-in. crystal).

A Area under concentration-distribution curve. FfL2 g per sq ft 

A' An event. _____--___ ________--

B Width of streambed. L ft

B' An event. __________ __________

C/\/g Resistance coefficient ratio of Chezy coefficient to square root of acceleration __________ __________

of gravity. 

Ci, C2 Constants. __________ cpm per Mc

d Average depth beneath the surface of the bed through which tracer particles are L ft

distributed; also the average depth of the zone in which particle movement

occurs. 

di Mean distance between the bed surface and the corresponding line of length lt L ft

(see fig. 41). 

E An event the movement of a tracer particle. __________ _______-_-

/( ) Probability-density function L~l or T~l __________

/(r) Counting rate per /xc from a point source located in submerged sand a distance r ______-__- cpm per /xc

from the center of the detector crystal. 

F( ) Distribution function (cumulative probability-distribution function). __________ ______----

h Horizontal distance between point source and centerline of detector. L in.

* Number of occurrences of the event E (or number of steps or waiting periods __________ ___--_----

that are associated with E). 

ib Fraction, by weight, of bed material particles within the approximate size range ________-- ________--

of the tracer particles. 

Jo( )» I\ ( ) ? /.( ) Modified Bessel functions of the first kind of orders 0, 1, and 2. __________ _____-----

ki Proportionality constant in concentration-distribution and related functions; L~l ft" 1

may also be interpreted as the mean number of steps per unit of longitudinal

distance. 

&2 Proportionality constant in concentration-distribution and related functions; T~ l hr

may also be interpreted as the mean number of waiting periods per unit of

time. 

li Length of a section (see fig. 41) defined by the lowest line that can be drawn in L ft

such a way that it is parallel to the mean bed surface and is tangent to the

bottom of a trough at its upstream end.

L Total length of reach. £ ft 

m Specific activity in the bed material. _____.---- /xC per cu in.

M Amount of radioactivity. ______---- me

n Particular number of occurrences of the event E (or number of steps or waiting _______--- ----------

periods that are associated with E). 

N Mean number of tracer particles in a given volume of bed material. __________ _____-_---

P( ) Prcbability mass function. ________-- ___-_-----

P[ ] Prc lability. _______.-- - -----

Q, Bee-material discharge. PIT tons per day 

r Radial distance from center of detector crystal to a point source of radioactivity. L m. 

n> Radial distance from the center of the detector crystal within which a radio- L in.

active source is uniformly distributed. 

Ra Adjusted counting rate. __________ cpm

#& Background counting rate. _______-_- cpm

R0 Observed counting rate. ______---- cpm

(R0 Rb) Minimum net counting rate over background that is required during an experi- __________ cpm

ment for statistical significance. 

Rr0 Counting rate of a particular detection system due to a source that is uniformly _______--- cpm

distributed throughout a volume of submerged sand within the radius r0 . 

R<a Counting rate due to a source uniformly distributed throughout an infinite __________ cpm

volume.

v
VI SYMBOLS

Symbol Definition Dimensions Units 

s Length, in the direction of flow, of a source of contaminated particles. L __________

S Sensitivity of radiation-detection system. __________ cpm per me

per cu ft 

/S(£) Function that describes the variation in strength with respect to distance from F/L __________

the origin of a longitudinally distributed source released instantaneously at

t=Q. 

S(T) Function that describes the variation in strength with respect to time of a source F/T __________

discharging continuously at the origin. 

S(£, T) Function that describes the variation in strength with respect to both distance F/LT __________

from the origin and time of a continuously discharging longitudinally distributed source. 

t Time. T min, hr, or

days 

ti Length of time during which a continuous source discharges. T __________

T Half-life of radionuclide. T days

v Vertical distance between point source and surface of sand. L in.

V Volume. L3 cu ft or cu

in.

W Weight of tracer particles released from source. F g 

x Distance downstream from source. L ft 

x' Dummy variable of integration. L __________

xm Distance that mode of concentration-distribution curve is downstream from L ft

source. 

x Distance that mean of concentration-distribution curve is downstream from L ft

source. 

xn Mean distance over which a particle would be transported during n repeated L __________

occurrences of E.

ai, «2 Constants. L~l in." 1 

y, Bulk specific weight of bed material. F/L3 Ibs per cu ft 

£ Distance from the origin (x = Q) to a point in a longitudinally distributed source. L ft 

o-f Variance of concentration-distribution curve. Lz sq ft 

T Time after t=0 during which a continuous source is discharging. T __________

0 Observed concentration of tracer particles in the bed material. F/L3 g per cu ft 

0max Observed peak concentration of tracer particles in the bed material. F/L3 g per cu ft 

tf>(x, t) Theoretical concentration of tracer particles in the bed material. F/L3 __________

$(t, x) Theoretical discharge of tracer particles through a unit cross section. F/T __________
TRANSPORT OF RADIONUCLIDES BY STREAMS

TRANSPORT AND DISPERSION OF LABELED BED MATERIAL 

NORTH LOUP RIVER, NEBRASKA

By W. W. SAYRE and D. W. HUBBELL

ABSTRACT

One significant facet of the disposal of radioactive wastes in 

alluvial streams pertains to the influence that natural sediments 

have on the transport and dispersion of radioactive wastes. 

The sorption of radioactive wastes by clay, silt, and sand (particularly clay-coated sand) is a common phenomenon; hence, the 

transport and dispersion of waste can be affected to a significant 

extent by the transport and dispersion of the individual sediment 

particles.

To study the transport and dispersion of bed-material particles 

(sand), an experiment was conducted in the North Loup River 

near Purdum, Nebr. In the experiment, sand labeled with 

radioactive iridium 192 was released from a line source on the 

bed of the river. After release, the movement of the tracer 

particles downstream was observed by monitoring the streambed 

with a sled-mounted scintillation detector. In addition, the 

vertical distribution of the tracer particles in the bed was 

observed by monitoring core samples of the bed material. The 

amount of activity and the number of tracer particles used in the 

experiment were determined from detailed calibrations in which 

field conditions were simulated. The tracer particles varied 

within a fairly narrow range about a median fall diameter of 

0.305 mm. The size of the tracer particles was selected so that 

they would be slightly coarser than the dominant size range of 

the bed material. Longitudinal traverses with the radiationdetection equipment at approximately daily intervals and 

extending over a 13-day period defined the dispersion of tracer 

particles along the channel. The traverses also indicated that 

some of the tracer particles were buried as a concentrated deposit 

for about 7 days. Presumably, these tracer particles were 

deposited in an unusually deep dune trough and were not 

reexposed until a similarly deep trough passed by. The temporary burial of the tracer particles demonstrates that deposits 

of contaminated particles in an alluvial stream can remain undisturbed for relatively long periods of time and then suddenly 

be released.

A concentration-distribution function, which is based on 

probability theory, was derived to characterize the dispersion 

process. The parameters of the function are related to the 

dispersion time and distance. Significant characteristics of the 

function are that the variance and the mean of the distribution 

are proportional to time and that for large times the maximum 

concentration varies inversely with the square root of time. 

Comparable observed and theoretical distributions agreed well 

except near the origin and at the downstream tail. The explanation for the differences is based on certain discrepancies between

physical features of the experiment and the corresponding 

assumptions on which the theoretical distribution function is 

based. The major discrepancy is that the theory applies only 

to a group of tracer particles having identical transport characteristics, whereas the tracer particles did not all have the same 

characteristics.

In addition to defining the dispersion of particles, the observed 

distribution curves provided information for computing the bedmaterial discharge. The computation of discharge was made 

with a continuity-type equation, which states simply that the 

discharge is the product of the mean velocity of the particles and 

the cross-sectional area through which they move. Although a 

direct comparison of the computed discharge and a measured 

discharge could not be made, the computed discharge compared 

favorably with the discharge computed by the modified Einstein 

procedure.

Although the concentration-distribution function was derived 

for the initial conditions of a vertical-plane source, it can be 

extended to more complex conditions by performing appropriate 

mathematical operations. In addition, the function can be 

applied to real situations if the constants in the concentrationdistribution function and the depth of the zone of particle 

movement in the bed are known. When the bed configuration 

consists of dunes or ripples, the depth of the zone of particle 

movement can be estimated from longitudinal profiles of the bed 

surface. However, until the relations between hydraulic and 

sediment parameters and the constants in the theoretical distribution function are defined, the constants can be evaluated 

only by experiment.

INTRODUCTION AND ACKNOWLEDGMENTS

The potential benefits of nuclear industries to mankind depend on the extent to which safe, reliable, and 

economically feasible means are found to dispose of 

radioactive wastes. Radioactive wastes present a unique 

disposal problem because, unlike most other industrial 

wastes, they cannot be rendered biologically harmless 

by any of the conventional treatment techniques. 

One current disposal practice is to release wastes into 

the environment. The U.S. Geological Survey, in 

cooperation with the Division of Reactor Development 

of the Atomic Energy Commission, is engaged in 

investigations related to the management of radioactive

Cl
C2 TRANSPORT OF RADIONUCLIDES BY STREAMS

wastes that are released into natural streams. One of 

these investigations deals with aspects of the influence 

of fluvial sediment in the disposal process. The study 

reported herein is one facet of that investigation.

At the present time (1962), about 95 percent of the 

low-level, short-lived liquid waste that is released into 

the environment is disposed of in waterways such as 

streams, lakes, and reservoirs (Nace, 1959). If a 

waterway is used for the disposal of radioactive wastes, 

the possibility exists for the activity to exceed safe 

levels in and along the waterway. The concentration 

of activity in a waterway at any time and place depends 

on a variety of factors which often operate in conjunction with one another. For instance, radionuclides 

initially dispersed by turbulent diffusion to concentrations far below the maximum permissible concentration 

may be taken up and concentrated to many thousand 

times the ambient water concentration (Robeck and 

others, 1954) by aquatic biota or fluvial sediment. 

Unfortunately the various factors and processes that 

tend to concentrate and disperse activity are neither 

understood well nor easily evaluated. Additional 

complications are introduced by the exchange of 

radionuclides between the various transporting media. 

The exchange processes are usually complex and depend 

on chemical and physical factors related to the radioactive waste effluent, the transporting media, and the 

flow.

One of the major transporting media in natural 

waterways is sediment (Sayre and others, 1963). 

Abundant evidence indicates that the uptake of radioactive waste products by sediment is a common phenomenon. Fine sediment (clay and silt) tends to have a 

much higher sorption capacity per unit weight than does 

sand; however, Kennedy (1963) has shown that some 

stream sands, particularly those containing clayimpregnated rock fragments, may have relatively high 

sorption capacities. According to Kennedy, most of 

the sorption capacity of suspended sediment is due to 

the clay fraction, but most of the sorption capacity of 

the bed material may be due to sand and silt, because 

those components constitute a very high percentage 

of the bed material.

Because of the potential for high amounts of exchange, the distribution of activity in the waterway 

may be governed to a significant extent by the rates at 

which the individual contaminated sediment particles 

are transported and dispersed. If flow conditions 

induce high concentrations of contaminated sediment, 

the level of activity within the body of the flow could 

exceed a safe level. If flow conditions induce low 

transport rates or deposition of contaminated sediment, 

activity could build up in and along the channel to an 

unsafe level. Deposits might consist of sand particles,

flocculated fine materials, organic sediments, or unflocculated fine materials that have penetrated into the 

bed. Hazardous conditions can also arise as a result of 

exposure of contaminated sediment through flooding 

and fluctuating water levels.

To evaluate the various potential sediment-induced 

hazards in waterways, the characteristics of the sediment and the rates of transport and dispersion of the 

different fractions of the sediment load must be considered. Ordinarily, the total sediment load is divided 

according to the size and source of the sediment into 

the bed-material load and the fine-material load. The 

bed-material load is that part of the total sediment 

load derived from the bed material; consequently, it 

is composed of particles whose sizes are similar to those 

of bed-material particles. The bed-material load also 

includes some suspended sediment in addition to all 

the bedload. The fine-material load is that part of the 

total sediment load composed of particles whose sizes 

are finer than those of particles present in the bed 

material. Generally, particles of the fine material are 

in suspension almost continuously and are transported 

at almost the same velocities as the fluid particles; however, the average velocities of bed-material particles, 

because of a process of alternate movement and temporary 

storage in the bed, are much lower than those of fluid 

particles. Because the transport mechanisms associated 

with the two kinds of sediment loads are different, the 

ways in which they are dispersed are also different.

This report describes a field experiment, conducted 

in the North Loup River near Purdum, Nebr., in which 

the dispersion and transport of bed-material particles 

were studied by means of radioactive-tracer techniques. 

In the experiment, sand particles that were labeled 

with a radioactive nuclide were released on the riverbed. 

As the tracer particles moved and dispersed downstream 

with the natural bed-material load, their distribution 

along the channel was determined with radiationdetection equipment. Because the application of 

radio active-tracer techniques in sediment studies is 

relatively new, part of the report is devoted to the 

design, techniques, and equipment that were used in 

the experiment. The discussion of the experiment is 

followed by derivation of a theoretical concentrationdistribution function based on probability theory and 

of a bed-material transport formula. The experimental 

results are compared with predictions based on the 

derived functions, and the applicability of the combined 

results to practical situations is discussed.

Permission to conduct the experiment, which involved 

the release of radioactive materials into the North Loup 

River, was given by Dr. E. A. Rogers, Director of 

Health, State of Nebraska, and by the Licensing and 

Regulation Branch of the Atomic Energy Commission.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOTJP RIVER, NEBR. C3

Prior to the field investigations, meetings were held in 

Lincoln, Nebr., on September 14,1960, and in Thedford, 

Nebr., on September 20, 1960, to inform representatives 

of various State and local agencies of the nature of the 

experiment. In all, representatives from the Nebraska 

State Health Department, the Nebraska Water Resources Department, the Conservation and Survey 

Division of the University of Nebraska, the Water 

Resources Division of the U.S. Geological Survey, and 

a board of local ranchers were present at the meetings. 

D. M. Culbertson, district engineer of the U.S. Geological Survey, assisted with the arrangements for the 

meeting in Lincoln, Nebr., and furnished valuable 

information on the characteristics of the North Loup 

River. Dale Langford, County Extension Agent, 

arranged for the local-level meeting in Thedford, Nebr. 

Kyle Cox, a local rancher, not only allowed work to be 

carried out on his property adjacent to the test reach but 

also furnished information on the history of the reach 

and on the condition of the stream. Aspects of the 

experimental program relating to radiation safety were 

carried out in accordance with the advice of Dr. W. D. 

Carlson, Radiation Safety Officer, Colorado State 

University. W. L. Haushild of the U.S. Geological 

Survey assisted materially with many aspects of the 

field operations and participated in the data collection, 

as did A. A. El Darwish, Colorado State University 

graduate student. F. C. Ames and C. L. Scott of the 

Geological Survey also assisted in the field during the 

first few days of data collection. Dr. D. L. Bentley, 

Assistant Professor of Mathematics, and E. J. Plate,

Assistant Civil Engineer at Colorado State University, 

contributed several fundamental ideas that aided 

substantially in the development of the final form of 

the concentration-distribution function. Robert McDowell, Colorado State University graduate student, 

assisted with the analysis of data and the preparation 

of this report.

THE EXPERIMENT

The radioactive-tracer technique was selected as the 

most feasible method for studying the transport and 

dispersion of sediment particles along the channel because, with suitable instrumentation, the distribution 

of the particles at any time could be measured in place. 

With the radioactive-tracer technique, the accuracy 

of the measurements increases as the level of radioactivity increases. However, as the level of radioactivity increases, so does the possibility for harmful contamination to the stream environment. Hence, the 

experiment had to be designed to maintain satisfactory 

accuracy in the measurements and at the same time to 

limit the radioactivity to a safe level. To achieve a 

satisfactory balance, careful consideration had to be 

given to the nature and location of the test reach, the 

labeling and detecting of the tracer particles, and the 

operational procedures. These general topics are 

discussed in the following sections.

THE TEST BEACH

The study was conducted on the North Loup River 

near Purdum, Blaine County, Nebr. (see fig. 1), in a

101' 100

41° -

FIGURE 1. Location of the Loup River basin.

761-353 O 6£
C4 TRANSPORT OF RADIONUCLIDES BY STREAMS

EXPLANATION

Staff gage

PROXIMATE MEAN 

DECLINATION, 1964

FIGURE 2. Sketch of the test reach.

reach (see fig. 2) in sec. 10, T. 24 N., R. 25 W. The 

site was selected because it was particularly suitable 

for the experiment from both a hydraulic and a radiological-health standpoint. Specifically, the North Loup 

River maintains a relatively constant water discharge 

for weeks at a time; ground-water flow adjacent to the 

river is toward rather than away from the stream; the 

bed form is normally one of large dunes; the bed material is sand having a median diameter of about 0.29 

mm; the test reach is fairly straight and has stable 

banks; the channel width is narrow, and the depth is 

such that a small boat carrying radiation-detection 

equipment can be navigated; the reach is isolated from 

any community; and very few people live near the 

river.

The North Loup River flows southeastward across 

the sandhills region of north-central Nebraska. The 

topography of the region is formed by northwest-trending sand dunes that are stabilized by grass cover. The 

dune sand extends to a depth of 100 feet in some places 

and is underlain by the Ogallala Formation of Tertiary 

age. The valley floor along the North Loup River is 

composed mostly of alluvial sand. Meanders are not 

well developed, although bank erosion is prevalent and 

is the source of most of the sediment load. The flow 

is mainly derived from ground-water accretion along 

the entire channel and is about 250 cfs (cubic feet per 

second) at the test reach.

The test reach (figs. 3, 4), a manmade cutoff constructed about 1950, is bordered on the north side by 

a hay meadow and on the south side by a swamp having heavy brush growth. The alinement of the channel 

is straight except for two minor bends that are approximately 700 and 1,300 feet downstream from the beginning of the reach. The reach varies in width from about 

40 to 55 feet and is approximately 1,800 feet long.

THE RADIOACTIVE LABEL AND THE DETECTION 

SYSTEM

Although no other investigations have been made in 

which the radioactive-tracer technique has been applied 

specifically to the study of the dispersion of contaminated bed-material particles in alluvial channels, the

technique has been applied successfully in other kinds 

of sediment and hydraulic studies. Lean and Crickmore (1960) studied sand transport in a laboratory 

flume, using experimental techniques which were very 

similar to those described in this report; however, their 

research was directed primarily toward developing a 

method for measuring the bed-material discharge. 

Other, less closely related studies have been reviewed 

by Krone (1957) and by Hours and Jaffry (1959).

In nearly all the sediment studies, the radionuclide 

either has been incorporated into glass particles that 

simulated the sediment or has been plated, by an adsorption process, on the surface of natural sediment.

FIGURE 3. Downstream view of the test reach. Station 0 is indicated by the dashed

line.

FIGURE 4. Upstream view of test reach from about station 1000.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C5

Lean and Crickmore (1960), however, used natural 

sand that had been irradiated in a nuclear reactor. A 

characteristic of the glass-particle and irradiation 

methods is that the radioactivity is proportional to the 

weight of the particles rather than to their surface area. 

This characteristic, however, is not significant if the 

tracer particles are uniform in size. In this experiment, 

Ottawa sand plated with a radionuclide was used because such particles of the proper size could be purchased.

In many of the sediment investigations, the radiationdetection equipment has been adapted for underwater 

detection. However, in some studies, samples of 

sediment have been collected and subsequently analyzed for radioactivity. The use of underwater-detection apparatus virtually dictates that the tracer particles 

be labeled with a gamma-emitting radionuclide, because 

the penetration range of alpha aifd beta radiation in 

water is limited to about 1 cm or less. In this experiment, both the underwater and sample-collection 

detection methods were used, and, therefore, a gammaemitting radionuclide was most suitable. Iridium 192 

was selected because (1) the gamma rays it emits have 

energies from 0.14 to 0.90 mev (million electron volts) 

and are readily detectable under water; (2) it has a 

half-life of 74 days and, hence, would decay slowly 

enough to be readily detectable during the entire 

experiment but rapidly enough to preclude long-term 

contamination of the stream; and (3) sand particles 

already plated with iridium 192 by a high-temperature 

oxidizing process were available for purchase.

Both scintillation detectors and Geiger-Mueller 

(G-M) tubes are used for detecting radiation. G-M 

tubes generally are relatively rugged and inexpensive; 

however, scintillation detectors are more sensitive to 

gamma radiation and are usable over a much wider 

range of counting rates. In addition, scintillation 

detectors, when used in conjunction with a pulse-height 

analyzer, enable differentiation of the emission energies 

of gamma rays thus permitting the identification of 

different radionuclides. Because of its versatility and 

inherently high sensitivity, a scintillation-detection 

system was selected for this experiment. In addition 

to the scintillation detector, the system included a 

pulse-height analyzer, count-rate meter, a strip-chart 

recorder, and, for some operations, a sealer.

AMOUNT OF RADIOACTIVITY

The amount of radioactivity required for the experiment depended on (1) the level of the background 

radiation, (2) the volume of sand throughout which 

the tracer particles would be dispersed, (3) the decay 

rate of the radionuclide, (4) the radiation-absorption 

characteristics of the media between the detector and

the tracer particles, (5) the efficiency of the radiationdetection system, and (6) the geometrical orientation of 

the detector to the tracer particles. Items 4-6 were 

evaluated together by determining an overall sensitivity 

from calibration measurements with the equipment 

under conditions that simulated the experimental 

environment. A method for determining overall sensitivity with license-exempt quantities of the tracer 

radionuclide is presented on p. C44.

If the sensitivity is determined, the amount of radioactivity, M (in millicuries), required for the experiment can be computed from

(R0 -Rb -)Ve 

S

where

R 0 Rb= the minimum net counting rate over background, 

in counts per minute, that is required during the 

experiment for statistical significance

F=the estimated volume, in cubic feet, through which 

the tracer particles will be dispersed at the end 

of the experiment. If this volume is estimated 

to be the product of the length and width of the 

test reach and the depth to which the particles 

will penetrate into the bed material, the tracer 

particles can disperse within the reach to the 

maximum possible extent and counting rates 

will still be statistically significant

S=the overall sensitivity, in counts per minute 

divided by millicuries per cubic foot, for the 

conditions simulating the experimental environment (see p. C47 for S values) 

0.693^= a correction factor for radioactive decay 

Z=the duration of the experiment, in days

T the half-life of the radionuclide, in days.

In the design of this experiment, a minimum net 

counting rate equal to one-half the background rate 

was considered acceptable for the condition of a uniform distribution of tracer particles throughout the 

test reach. The volume through which the tracer 

particles would be dispersed was estimated to be 

1,800X50X1.5=135,000 cubic feet. Furthermore, it 

was estimated that the experiment would last for 1 

month. By using the sensitivity determined with 

cesium 137 (S had not been determined for iridium 192 

prior to the design of the experiment), the 74-day halflife of iridium 192, and a background counting rate of 

200 cpm (counts per minute), the required amount of 

activity was computed to be 39.4 me (millicuries). 

This amount was rounded off to 40 me, which was the 

amount used in the experiment.

After the sensitivity with iridium 192 had been determined, calculations showed that with comparable 

assumptions, only 30.6 me would have been required.
C6 TRANSPORT OF RADIONUCLIDES BY STREAMS

NUMBER OF TRACER PARTICLES

In the determination of the required amount of radioactivity, a uniformly distributed source was assumed. 

Actually, each particle is a separate source, and the 

counting rate recorded by the detection system depends not only on the amount of radioactivity within 

detection range of the scintillation crystal but also on 

the distribution of the tracer particles.

If the distribution of tracer particles throughout the 

bed is random, as the concentration of tracer particles 

increases, the relative distribution of the tracer particles 

tends to become more even and random variations in 

the number of tracer particles per unit volume of bed 

material tend to have less effect on the counting rate. 

In statistical terms, the random variations in the number of particles in a given volume can be characterized 

by the coefficient of variation, or relative standard 

deviation. Because the variation in the number of 

tracer particles in a given volume follows the Poisson 

distribution, the coefficient reduces to 100/V^V, where N 

is the mean number of tracer particles in the given 

volume of bed material.

The number of tracer particles required for a particular experiment can be determined by setting limits on 

the coefficient of variation and by precisely defining 

the given volume. Intuitively, the volume can be 

defined by the practical range of the detector, and N 

can be defined as the required number of tracer particles 

within the volume at the end of the experiment when 

the tracer particles are distributed over the entire 

test reach. Inasmuch as the maximum range of the 

detector approaches infinity asymptotically and therefore is not defined, the practical range of the detector 

can be considered as the radial distance from the center 

of the detector crystal to the loci of points in the bed 

within which 50 percent of the counts originate when 

the sand bed contains a uniformly distributed source of 

infinite extent. The practical range of the detection 

system used in this experiment, as determined from 

figure 46, is 4.4 inches for iridium 192, 4.8 inches for 

gold 198, and 6.8 inches for cesium 137.

Because the counting rate varies with the source-todetector distance and because the spatial distribution 

of the tracer particles is not always precisely the same, 

the coefficient of variation, when defined in terms of the 

number of tracer particles within the practical range 

of the detector, provides only an index to the expected 

variation in counting rate attributable to random 

variations in the number of tracer particles. However, 

if a relatively long ratemeter time constant is used and 

the detector is moved along the bed, the fluctuations in 

the counting rate due to local variations in the distribution of tracer particles are damped appreciably.

As a result, the required number of tracer particles near 

the detector need not be determined precisely, and the 

measure provided by the coefficient of variation is 

adequate for the purpose of experimental design.

Because the practical range of the detector for 

iridium 192 had not been determined at the time the 

experiment was designed, the method described in the 

preceding paragraphs was not used, and the required 

number of tracer particles was estimated on the basis 

of information reported by Krone. According to Krone 

(1957), a distribution of three tracer particles per square 

centimeter on the surface of the bed would provide an 

adequate number of tracer particles near the detector. 

The number of particles required for this experiment 

was calculated on the assumption that the tracer 

particles would be evenly distributed over the entire 

test reach at the end of the experiment and that one-half 

of the particles would be either lost or buried beyond 

the range of the detector. The calculation showed 

that about 5.0 X108 particles were needed.

A post-experimental analysis showed that if the 

number of particles that were actually used in the 

experiment (approximately 4.6X108) were uniformly 

distributed throughout the test reach to a depth of 

1.5 feet, the particle population density would be 1.98 

particles per cubic inch. Because the volume within 

the practical range of the detector (4.4 in. for iridium 

192) is 116 cubic inches, about 230 particles would be 

within the practical range. Thus, the coefficient of 

variation would be ±6.6 percent.

SIZE OF TRACER PARTICLES

Another critical consideration in the design of the 

experiment was the optimum size range of tracer 

particles to use for simulating the transport and dispersion of the bed-material particles. To insure that 

the tracer particles would not be in suspension any 

significant part of the time but that the most dominant 

particle-size fraction would be represented, a narrow 

particle-size range slightly coarser than the dominant 

size range of the bed material was selected. Figure 5 

shows the size distribution of the tracer particles along 

with those of the bed material and the measured suspended sediment.

The median size of the tracer particles (0.305 mm) 

and the assumption that the tracer particles were 

spherical and had a specific gravity of 2.65 were used 

to calculate the weight of the required number of tracer 

particles (43.5 Ib). For the experiment, the quantity 

of particles was rounded off to 40 pounds.

EXPERIMENTAL EQUIPMENT

The field investigation consisted of four principal 

operations: placing the tracer particles on the stream-
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C7

99%8 

99.5 

99.0 

98

95

90

80 

70 

60 

50 

40 

30 

20 

10 

5

2

1.0 

0.5

0.2 

n i

.X

1 I 1 1 1 1 1 1

1 1 1 1 1 1 1 1 =t=.

/ </

r i/ 1 // -^ r / // / j

/ // // / 1

~y

i i 1 1 1 1 1 1

i i 1 1 1 1 1 1

r^ c^'^

EXPLANATION

 

Measured suspended 

sediment

9 

Labeled Ottawa sand 

O 

Bed materiel

i i i i i i i i

FALL DIAMETER. IN MILLIMETERS

FIGUBE 5. Size distributions of measured suspended sediment, labeled Ottawa 

sand, and bed material.

FIGURE 6. The dosing apparatus. Note the tongs used for handling the cans of 

labeled sand and the adjustable mount with parallelogram linkage, which permits 

the funnel tube to be raised and lowered vertically.

bed (dosing), tracing the particles, sampling the bed 

material, and collecting hydraulic and sediment data. 

Each of these operations, except the last, required 

special equipment that could be transported in and 

operated from a small boat.

Placement of the labeled sand on the streambed was 

accomplished by means of the apparatus shown in 

figure 6. This apparatus, which consists chiefly of an 

electric can opener and a movable funnel tube, was 

connected to the stern of the boat. In the dosing 

operation, the funnel tube was lowered until the bell 

at the bottom rested on the streambed, the storage 

cans containing the tracer particles were introduced 

into the can opener by means of tongs and opened,

and the particles were poured down the funnel and 

deposited on the streambed (see fig. 7). During the 

dosing, the boat was attached to a cable that extended 

accross the width of the stream.

Tracing of the labeled particles was accomplished 

wih a scintillation-detection system that consisted of a 

scintillation detector, a pulse-height analyzer, and a 

count-rate meter with an output to a strip-chart recorder. The scintillation detector was housed in a 

watertight aluminum casing and mounted to a sled 

that was dragged along the streambed (see fig. 8). To 

assure that the detector remained in continuous contact with the streambed, it was mounted to the sled 

by a four-bar parallelogram linkage and supported

FIGUBE 7. The dosing operation.

FIGUBE 8. Sled and scintillation detector. Note the parallelogram linkage, which 

allows the detector to move independently from the sled, and the wooden dishshaped piece that supports the detector on the bed.
C8 TRANSPORT OF RADIONUCLIDES BY STREAMS

Cables potted 

in pipe

4-in. OD aluminum cylinder 

1/4 in. wall thickness

FIGURE 9. Watertight casing for scintillation detector. Scale 1 inch =3 inches.

End plate 

1/16 in. thick

FIGURE 10. Arrangement of boat and sled for the longitudinal traverses.

on the bed by a wooden dish. The watertight casing 

for the detector (see fig. 9) was an aluminum cylinder 

sealed at the bottom with a thin aluminum plate. 

For the longitudinal traverses, the boat and detector 

sled were arranged in tandem as shown in figure 10, 

and the boat, which faced upstream and pulled the 

detector sled, was maneuvered in a downstream direction by means of an outboard motor having reverse 

controls. The pulse-height analyzer, count-rate meter, 

and recorder were mounted in the boat together with 

a 115-volt a-c power supply, which consisted of a 

1-kilovolt-ampere portable gasoline motor-generator 

unit and an electronic voltage regulator. This power 

supply proved to be sufficiently stable for satisfactory 

operation of the radiation-detection equipment. The 

arrangement of the equipment in the boat is shown in 

figure 11.

In addition, the experimental equipment included a 

dual-channel ultrasonic depth sounder (Karaki and 

others, 1961) capable of providing continuous records 

of the bed and water-surface profiles, a 400-foot 

distance-measuring cable and reel, and an automatic 

distance-marking system (see fig. 12). The distancemeasuring cable functioned both as a stay line and as a 

part of the distance-marking system. In operation, 

the cable, which was fixed at the upstream end of each 

400-foot segment of the test reach, was unwound at a 

controlled rate allowing the boat and sled to move 

downstream. When buttons that were located at 

definite intervals along the cable tripped an eventmarking switch, ticks were made on the recorder charts. 

In this way, the recorder charts were provided with a 

distance coordinate.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C9

Core samples of the bed material were collected along 

the test reach to determine the vertical distribution of 

tracer particles. The cores were sampled with the 

piston-type sampler shown in figure 13. This instrument was capable of collecting 3-foot cores, which were 

ejected with a hand-operated jack in 2-inch increments. 

Each 2-inch increment was collected in a separate 

container and analyzed for radioactivity. For this 

purpose, the sled was inverted as in figures 13 and 14, 

and a sealer was incorporated in the radiation-detection 

system so that counting rates could be determined more 

accurately.

FIGURE 11. Arrangement of instruments in the boat.

FIGURE 12. Dual ultrasonic depth sounder and distance-marking system.

FIGURE 13. Equipment for sampling and analyzing the bed material. The \yy 

inch-diameter sampler is used to collect 3-foot cores, which are ejected in 2-inchincrements with the hand jack attached to the end of the sampler plunger. Samples are counted with the detector and sled in an inverted position.

FIGURE 14. Analysis of the vertical distribution of labeled particles in the field. 

Note the sealer and the inverted detector sled on top of the canopy over the 

instruments.

CALIBRATION OF RADIATION-DETECTION 

EQUIPMENT

To establish an absolute relation between the counting rate and the concentration of tracer particles, the 

detection equipment was calibrated with tracer particles in a laboratory calibration tank (see fig. 42) 4 feet 

in diameter and about 4 feet high. In the calibration, 

known amounts of labeled sand were mixed thoroughly
CIO TRANSPORT OF RADIONUCLIDES BY STREAMS

in the tank with sand having a median diameter and size 

distribution that approximated the natural bed material 

in the North Loup River. This was done for concentrations of tracer particles that ranged approximately 

from 0.7 g per cu ft (grams per cubic foot) of sand 

mixture to 40 g per cu ft. For each concentration, an 

average counting rate was determined by placing the 

detector at different locations on the surface of the sand; 

each counting location was selected so that the geometry 

of the detector would be similar to that in the natural 

stream. To reduce the background counting rate, a 

9-inch depth of water was maintained over the sand 

bed throughout the calibration procedure. It was 

anticipated that the thickness of bed through which the 

particles would disperse would vary and that the counting rates would be affected accordingly. Therefore, 

calibrations were made for depths of uniform concentrations ranging from 2 to 18 inches. Two different 

methods were used to mix the labeled sand with the 

natural sand in the tank. Initially, a motor-driven 

post-hole auger was used as a mixer. The auger 

eventually was replaced by an ordinary concrete mixer. 

The concrete mixer yielded mixtures of more uniform 

concentration and was easier to use than the auger.

Throughout the calibration, counting rates were 

determined on both differential and integral settings 

of the pulse-height analyzer so that either setting could 

be used in the field. For integral counting, the lower 

discriminator setting was 5.0 volts and the high-voltage 

setting was 900 volts; the 900-volt setting was approximately in the center of a wide voltage range over which 

the count rates remained virtually constant (center of 

the plateau). For the differential counting, the gain 

and high-voltage settings were adjusted so that the 8.5- 

volt window width corresponded to the 0.40-0.74 Mev

o 

o

L_

1 10 

o

(£

LU

a. 

10 

S 

< 

<£ 1 C3

z.

 z. 

o

$cc. 

Z 0.1

LU 

0

z 

o 

o

0.01

1

FlGUI

= 1 1 1 1 Mil! 1 1 M 1 III

~ EXPLANATION

o 

5 Differential setting

~ o 

- Integral setting

5

^/

/

/

\ 1 1 1 1 1 1>

1 M 1 MM

/^

/

/

INI Mil

1 1 M MM

/ 

y

/

1 1 1 1 INI

i/r i 1 1 i±t

/S 

/

/ ~

~

i i Mini

0 100 1000 10,000 100,000 1,000,000 

ADJUSTED COUNTING RATE, IN COUNTS PER MINUTE

IE 15. Relations between the concentration of tracer particles in submerge'd 

sand and the adjusted counting rate for iridium 192.

energy range. To insure that instrument drift did not 

alter the range of energies from which counting rates 

were determined, the detection system was calibrated 

with a sealed cesium 137 source periodically d,uring 

each day. The periodic checks indicated that the 

system was reasonably stable.

The results of the laboratory calibrations are shown 

in figure 15. This graph is for uniform concentrations 

mixed to a depth of 8 inches. Figure 16 shows that the 

counting rate remains virtually constant for all depths 

of sand greater than about 8 inches; therefore, 8 inches 

can be considered as an infinite depth. Because the 

calibrations were made during about a week's time, 

observed counting rates were adjusted to account for 

the radioactive decay. All the calibration curves are 

based on adjusted counting rates that refer to October 

26, 1960. The adjusted counting rates were computed 

using the equation:

where

Ra =ihe adjusted counting rate 

R0=the observed counting rate at time t 

fl R = the background counting rate

f=^the number of days after October 26, 1960

T=the half-life of indium 192 (74 days).

Because the data in figure 15 exhibited some scatter, 

the curves were fitted by the method of least squares. 

Interestingly, the integral and differential counting 

curves had slopes that differed slightly from 1.0. Thus, 

the data indicated that the counting rate is not directly 

proportional to the concentration, which is contrary 

to expectation. Spot checks indicated that the calibrations were repeatable. Because the calibration conditions closely approximated the field conditions, the 

empirical calibration curves shown in figure 15 were 

used in evaluating the field data.

FIELD PROCEDURES

To insure uninterrupted operation after the tracer 

particles had been introduced into the stream, auxiliary 

equipment used during the investigation was installed 

and all experimental equipment and procedures were 

tested beforehand in practice runs. The practice runs 

also provided measurements of background radioactivity along the test reach. For the operation of the 

boat and detector sled, cables were extended across 

the stream at the upstream end of the test reach and 

at 400-foot intervals downstream. The lateral cables 

downstream were necessary because the distance- 

"measuring cable, which was only 400 feet long, had to 

be detached, rewound, and attached to the next cable
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR.

6.0 X 10 5

Cll

io 5

10 4

3.0 X IO2

o

'=4.0

0=3.12-

= 1.466

EXPLANATION

O 

Differential setting

9 

Integral setting

0=38.3

Concentration of tracer 

particles in the bed material in grams per cubic 

foot

4 6 8 10 12 14 16 

DEPTH OF UNIFORM CONCENTRATION, IN INCHES

18 20

FIGURE 16. Variation in adjusted count rate with the depth to which different concentrations of tracer particles are uniformly mixed. 

TO1-353 O 65 3
C12 TRANSPORT OF RADIONUCLIDES BY STREAMS

downstream in order for the boat to remain moored 

and for the distance-marking device to be operative 

throughout the entire reach. For the measurement of 

water-surface slope, staff gages were placed on both 

banks at the upstream end of the test reach, on the left 

bank at two different intermediate positions within 

the reach, and on both banks at the downstream end 

of the reach (see fig. 2).

The dosing operation was performed on November 3, 

1960, and consisted of depositing 2-pound lots of tracer 

particles on the streambed at 2-foot intervals across 

the entire width of the channel. Each 2-pound lot 

was labeled with 2 me of iridium 192. About 2 

hours after dosing, the first longitudinal traverse was 

made. Before this traverse, as well as all subsequent 

ones, the boat was fixed to the first lateral cable and the 

detector sled was placed by hand upstream from the 

boat along the tracing path. Prior to the release of 

the boat from the lateral cable, the background counting rate was recorded for about 2 minutes. "When the 

background counting had been completed, the boat 

was released and moved downstream under power along 

a course that was roughly a distance of one-third the 

channel width away from the bank. During the course 

of the traverse, the outboard motor was used to position 

and pull the boat as the reel operator played out the 

distance-measuring cable, which allowed the boat and 

detector sled to progress downstream at a controlled 

rate. The downstream motion was arrested intermittently to allow the scale on the ratemeter to be 

changed as required by changes in the activity level.

To define the lateral distribution of the tracer 

particles, lateral traverses were made during the first 

few days. The lateral traverses were made in the 

same manner as the longitudinal traverses except that 

the detector sled was detached from the boat and pulled 

across the stream by hand.

The bed-material sampling was done initially by 

collecting cores in both the longitudinal and lateral 

directions without reference to the position of the dunes 

on the bed. However, because of the time required 

for the operation, the procedure was altered and samples 

were collected only in the troughs on the bed. In the 

operation, the position of the boat was fixed with the 

distance-measuring cable and lateral ropes. After 

positioning, the core sampler was depressed into the 

bed to its maximum extent and a 3-foot core was 

extracted. After the sample had been obtained, the 

boat was pulled to one bank and the 2-inch core increments were ejected into containers for radiation analysis. Each 2-inch increment was counted for a 1-minute 

period on the detector, which was placed in an inverted 

position in the boat.

RADIATION SAFETY

As mentioned previously, radiological health factors 

were considered in the selection of the test reach. In 

addition to considerations of the location of the test 

reach, calculations were made to insure that the activity on the tracer particles presented neither an internal 

nor an external radiation hazard. For the calculations, 

two different assumptions were considered. The first 

was that all the activity would be removed from the 

tracer particles as soon as they were introduced into the 

water. Under this assumption, computations showed 

that the activity would be diluted to the maximum permissible concentration 1 in drinking water by the quantity of water flowing past the dosing section in about 

2.3 minutes. Inasmuch as the dosing operation would 

require about 30 minutes, the maximum permissible concentration would never be exceeded. The second assumption was that all the activity would remain on the 

particles. Under this assumption, the activity on the 

tracer particles would be dispersed within the bed material to a concentration comparable to the maximum 

permissible concentration for drinking water when the 

tracer particles were uniformly distributed throughout 

a 470-foot length of the test reach. Obviously, the 

maximum permissible concentration for drinking water 

is only an approximate index when the medium is the 

bed material, because, unlike water, large quantities of 

sand would never be ingested; however, the criterion 

provides a wide safety margin and therefore is useful. 

Also, under the second assumption, the tracer particles conceivably could represent an external radiation 

hazard; however, the depth of water in the stream 

afforded more than adequate shielding at all times.

To provide quantitative information on levels of 

radiation in the stream during the experiment, eight 

pocket dosimeters encased in waterproof tubes were 

placed on the streambed near one bank at about 100- 

foot intervals along the upstream end of the test 

reach. The dosimeter readings showed no trend with 

distance from the dosing section and indicated an 

average accumulation of only 8 mr (milliroentgens) for 

the period November 2-16.

In addition, the experimental procedures were designed to reduce to a minimum the radiation exposure 

received by project personnel. Precautions were particularly necessary in the operations that required the 

handling of tracer particles. The tracer particles 

were packaged by the supplier in No. 3 cans, each of 

which contained 2 pounds of Ottawa sand labeled 

with 2 me of iridium 192. The measured and calculated exposure rates at a distance of 30 inches from

i The maximum permissible concentration of iridium 192 in drinking water for unrestricted areas, recommended by the Natl. Comm. on Radiation Protection (U.S. 

Dept. Commerce, 1959), is 4X10-' /*c per ml.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C13

one can of tracer particles were, respectively, 1.2 and 

1.1 mr per hr. The intensity of exposure varies 

approximately inversely with the square of the distance 

from the can. Thus, even though the level of radiation at 30 inches did not constitute any particular 

hazard, the cans of particles could not, with safety, be 

handled directly. To avoid direct handling of the cans, 

3-foot tongs and an electric can opener were used. 

Also, in the calibration procedure when small quantities of tracer particles were separated and weighed, a 

long-handled scoop was used.

All project personnel were equipped with pocket 

dosimeters and film badges, and records of radiation 

exposure were maintained. During the period when 

the tracer particles were being handled, the maximum 

dosage received during a single day by any individual 

was 4 mr, and the average daily dose was 1.5 mr. This 

is considerably below the average maximum permissible 

dosage 20 mr per day for workers in the atomic 

industries.

PRESENTATION AND DISCUSSION OF DATA

After the dosing and the first runs on November 3, 

the tracer particles were tracked daily by making 

longitudinal traverses with the radiation-detection 

equipment down the left and right sides of the stream. 

Exceptions to this were on November 9, when only 

one longitudinal traverse (run 7C), down the center 

of the stream, was made; on November 11, when 

additional longitudinal traverses down the center 

(run 9C) and very near the left bank (run 9La) were 

made; and on November 14, when no runs were made. 

In addition, lateral traverses were made on November 

4, 5, and 8. Table 1 is a summary of the time, lateral 

position, and length of all longitudinal traverses, 

and the time and longitudinal position of all lateral 

traverses.

Curves that indicate the variation in concentration 

of tracer particles along the test reach at various times 

are shown in figures 17-21. These longitudinal concentration-distribution curves were obtained in the following manner from the counting rates that were continuously recorded during the longitudinal traverses. On 

the recorder chart for each run, the abscissa (distance 

coordinate) was divided, on the basis of the rate of 

change of the counting rate, into increments that 

represented from 5 to 100 feet of reach length. The 

average counting rate for each distance increment was 

determined graphically, and the background counting 

rate was subtracted from the average to obtain the 

net counting rate. Net counting rates, in turn, were 

corrected for the radioactive decay that had occurred 

between the time of the run and the time for which 

the calibration curves were made (Oct. 26, 1960). By

TABLE 1. Time, location, and extent of longitudinal and lateral 

traverses

Date 

(Nov. 

1960)

3 _

3_._ 

3 _

3 

4___ 

4__ 

5 

5

6_ 

6... 

7______

7._ 

8

8 

8......

9. _

10 ___

10.. ...

11 

11. 

ll.._.__ 

!!..____ 

12_ 

12 

13 _ 

13 _. 

15 

15 -..

Run

1 L 

1 La 

1 R 

1 Ra 

2L 

2R 

2Ta* 

2Tb< 

3R 

3L 

3T* 

4R 

4L 

5R 

5L 

6R 

6L 

6T< 

7C 

8R 

8L 

9L 

9 La 

9C 

9R 

10 L 

10 R 

11 L 

11 R 

12 L 

12 R

Time of run (p.m., except as 

indicated)

Start

4:00 .....

4:30 

6:05 -----

11:00 a.m. 

12:00 a.m.

9:55 a.m. 

11:30 a.m.

12:15 

1:30 

3:45-.--

11:30 a.m. i -A<\

11:30 a.m. 

3:30 

1:45 

2:40 

5:30 

5:55 ---

11:00 a.m.

1 5O

11:50 a.m. 

3:00 _ - - 

10:15 a.m. 19-';';

End

4:40 

5:20 

6:15 

10:35 a.m. 

12:10 

1:30 

4:30 

2:50 .....

4:50 _____

11:50 a.m. 

3:00______

4:45 

6:20 

2:10. _ - .

3:55- 

5:40 

6:05. 

12:15. 

3:10 

12:40- _

4:25 _____ 

11:20 a.m. 

! 55

Mean

4:35 

5:15 

6:10 - 

11:103 

12:103. .

2:30 

2 '45

10:15 a.m. 

11:50 a.m. 

1:55- 

12:52--..

3:52 

2:10 

4:18 -----

11:40 a.m. 

2:23 -----

3:30 

12:12 .....

4:08 -

1:58 -

3:18.. -__

5:35 __ _

6:00 -

11:38 a.m. 

2:30 .....

12:15 -

3:38 --- 

10:48 a.m. 

1:25 

Time 

after 

dosing 

(hr)i

2.6

3.2

4.2

21.2 

22.2 

24.5

f)A Q

44.2 

45.8

70.9 

73.9 

96.2 

98.3 

117.7 

120.4 

121.5

142.2 

170.1

171.8 

192.0

193.3

195.6 

196.0 

213.6 

216.5 

238.2 

241.6 

284.8 

287.4

Side Of 

reach monitored

Left 

 do 

Right 

-.---doLeft .

Right 

Right -

Left. 

Right 

Left .

Right 

Loft 

Right--. 

Left 

Centerline 

Right 

Left 

___-.do 

 do« ..

Centerline 

Right. . 

Left--

Right 

Left.. 

Right 

Left 

Right--

Length 

(feet) of 

reach monitored 

(sta. to 

sta.) 2

0-200 

0-200 

0-75 

0-400 

0-400 

575 

5125 

0-800 

0-800 

« 185 

0-1200 

0-1200 

0-1800 

0-1200 

0-1800 

0-1800 

5415 

0-1800 

0-1800 

0-1800 

0-400 

0-400 

0-400 

0-400 

0-1800 

0-1800 

0-1800 

0-1800 

0-1800 

0-1800

1 Mean time of dosing, 2:00 p.m., Nov. 3,1960.

2 Dosing performed at station 0.

3 Estimated.

4 Transverse run.

5 Longitudinal station of transverse run. 

«Very near bank.

use of the calibration curves in figure 15, the corrected 

counting rates were converted to concentrations that 

express the weight of tracer particles per unit volume 

of natural sand.

The background counting rate that was used to 

compute the net counting rate for each traverse was a 

time-weighted average, which had been measured just 

upstream from the dosing section prior to the traverse. 

The individual time-weighted rates were used to compensate for any instrument drift that may have occurred 

as a result of differences in operating conditions from 

time to time and because reconnaissance measurements 

along the reach indicated that the background counting 

rate varied only with time and not location. When the 

pulse-height analyzer was on the differential setting, 

the background counting rate usually averaged about 

300 cpm; when it was on the integral setting, background 

averaged about 3,000 cpm.

The longitudinal-distribution curves indicate that the 

distributions along the left and right sides of the test 

reach were different. A significant quantity of the 

tracer particles on the left side seemingly was trapped 

in a deep trough near the dosing station and remained 

buried beyond the range of the radiation-detection 

equipment for several days. Evidently, one or more 

dunes with very deep troughs existed, at the time of the 

dosing, along the left bank of the stream at the up-
C14 TRANSPORT OF RADIONUCLIDES BY STREAMS

200 400 600 800 1000 1200 

DISTANCE FROM SOURCE, IN FEET

1400 1600 1800

FIGURE 17. Longitudinal distribution of labeled particles along the right side of the channel, November 3-8,1960.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C15

Run 9R (Nov 11)

O 

Run 10R (Nov 12)

EXPLANATION

 

Run HR(Novl3)

o 

Run 12R(Nov 15)

200 400 600 800 1000 1200 1400 

DISTANCE FROM SOURCE, IN FEET

FIGURE 18. Longitudinal distribution of labeled particles along the right side of the channel, November 10-13,15,1960.

1600 1800

stream end of the test reach. Subsequently, a train 

of small dunes passed over these deep troughs and 

trapped some of the tracer particles. It was not until 

about November 10 (run 8) that another train of dunes 

with deep troughs passed through the upstream end 

of the test reach and released the trapped particles, 

which were then free to disperse downstream in a more 

normal manner.

The dispersion of tracer particles along the right side 

of the test reach seems to have proceeded normally until 

about November 11 (run 9), when particles that had 

been trapped on the left side began to disperse laterally 

toward the right side of the stream. The distribution 

defined by run 70 on November 9, indicates that a 

relatively small quantity of tracer particles had been 

trapped near the center of the stream and began to
C16

0.5

TRANSPORT OF RADIONUCLIDES BY STREAMS

0.4

0.3

0.2

0.1

EXPLANATION

O 

Run 7C(Nov9)

 

Run9C(Nov 11)

200 400 600 800 1000 1200 1400

DISTANCE FROM SOURCE, IN FEET

FIGURE 19. Longitudinal distribution of labeled particles along the center of the channel, November 9 and 11,1960.

1600 1800

disperse downstream somewhat sooner than the particles 

that had been trapped on the left side.

As a result of the temporary storage, the distribution 

curves for the left side of the test reach are less indicative of the general distribution that applies for the 

average flow conditions than those for the right side. 

However, the fact that particles were buried demontrates that under similar conditions highly concentrated 

deposits of contaminated particles could remain buried 

for considerable periods of time and then suddenly 

be released.

The normal dispersion rate was reduced somewhat 

on the mornings of November 9 and 10, when the water 

temperature was 0°C and there was heavy frazil ice 

(fig. 22) in the stream; the ice melted by noon on both 

days. Apparently, the ice damped the turbulence in 

the flow sufficiently to reduce significantly the transport and dispersion of the bed-material load. After 

the ice melted, the transport and dispersion increased 

to its earlier rate.

The lateral distribution of the tracer particles is 

illustrated in figures 23 and 24. In the graphs for 

November 4 and 5, the differences between the amounts 

of tracer particles available for transport and dispersion 

on the right side and on the left side are evident. 

However, the graph for November 8 shows that as time 

went on, the particles became distributed reasonably

uniformly across sections that were fairly distant from 

the source. 

The vertical distribution of tracer particles in the 

streambed, as determined from the core-sampling data, 

is shown in figures 25-29. The graphs show that 

although the variation in counting rate with depth into 

the bed was considerable at most of the sampling 

locations, there was no regular vertical distribution 

pattern. Hence, a vertical concentration gradient 

cannot be evaluated with any degree of certainty. 

The variations probably resulted largely from the 

passage of dunes having different heights, mean elevations, and concentrations of tracer particles. As a 

result of the irregularities of the dunes, neither the 

distance below the water surface, distance below the bed 

surface, nor depth of activity can be used to normalize 

the vertical distribution of count rate. Hence, the 

distributions from the separate verticals cannot be 

combined to provide an average gradient. However, 

the distributions from the separate verticals can be 

combined to indicate the average depth of the zone 

through which the sediment particles moved; the 

average depth to which tracer particles extended 

below the bed surface (average depth of the zone of 

movement) was 1.45 feet.

General hydraulic and sediment data were collected 

several times during the course of the experiment.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR.

3.5 

C17

Run 21 (Nov 4)

O 

Run 3L (Nov 5)

200 400 600 800 1000 1200 

DISTANCE FROM SOURCE, IN FEET

1400 1600 1800

FIGURE 20. Longitudinal distribution of labeled particles along the left side of the channel, November 4-8,1960.
CIS TRANSPORT OF RADIONUCLIDES BY STREAMS

Peak conc-13.0 g per cu ft 

at 58 ft

Run9L(Nov 11)

o 

Run 9La(Nov 11)

Run 10L(Nov12)

9

Run HL(Nov 13)

0.2

200 400 600 800 1000 1200 

DISTANCE FROM SOURCE, IN FEET

1400 1600 1800

FIGURE 21. Longitudinal distribution of labeled particles along the left side of the channel, November 10-13,15,1960.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR.

TABLE 2. Summary of hydraulic data

C19

FIGURE 22. Frazil ice in the stream, November 9.

Summaries of these data are presented in tables 2 

and 3. The hydraulic data include information on" 

water-discharge measurements, water-surface slopes, 

roughness coefficients, water temperatures, and the

Date

(Nov. 

1960)

4 

4 

5 

5 

6 

6 

6 

7 

7 

7 

8 

8 

8 

9 

9 

9 

10 

10 

10 

11 

11 

11 

12 

12 

12 

13 

13 

15 

15 

16

Time

9:00 a.m. 

11:30 a.m. 

9: 00a.m. 

11: 00a.m. 

10: 00 a.m. 

12:00 a.m. 

5:00 p.m. 

10:00 a.m. 

2:00 p.m. 

5:00 p.m. 

11: 00a.m. 

1: 00p.m. 

2:00 p.m. 

9: 00a.m. 

5:00 p.m. 

5:00 p.m. 

9:30 a.m. 

2: 00 p.m. 

5:00 p.m. 

11: 00a.m. 

3: 00p.m. 

5:00 p.m. 

10: 00a.m. 

3: 00p.m. 

4: 15p.m. 

10: 00 a.m. 

5: 30 p.m. 

9: 30a.m. 

1: 00p.m. 

3: 30p.m.

Water-discharge measurement

Discharge

(cfs)

263

249

251

255

260

 3^ 

12 £~S

47.4

54

56

57

53.4

a 

cn«,

<t"§52

%

2.47

2.13

1.91

2.02

2.06

o1 

&- 

iT

117

115

1C7

115

110

£

ill

s

2.25

2.17

2.35

2.22

2.37

Water-surface 3slope X10

0.839

i.839 

.833

.844 

.805

.821

.816

.810

1.810 

.810

.883

.860

.815

.826

1.826 

.838

.816

.878

.860

1.860

c£?

Ill

«

8.7

9.0

10.5

9.6

9.9

Water tempera- 

(°0) ture 

6 

2.5

4 

3

5 

6 

3

6 

7

4

4 

4

0

1

1

0

2

3.5 

2.5

5

6

4

8 

8 

4

8 

5.5

8

depth in Mean 

(ft) reach 

2.3

2.7

2.5

2.4

2.7

1.8

2.4

2.5

2.5

2.6

i Estimated.

TABLE 3. Summary of sediment data

[Particle size distribution of all sediment determined by means of visual-accumulation tube. Bed material, collected Nov. 16, 1960, was also analyzed with sieve]

MEASURED SUSPENDED SEDIMENT

Date of collection (Nov. 1960)

4

9 _ _ _. . __ _. ___ -

12

1 £±

Time 

(p.m.)

12:30 

6:10 

5:15 

4:30

Water 

discharge 

(cfs)

263 

251 

255 

260

Water 

temperature (°C)

6 

1

8

Concentration 

(ppm)

634

648 

390 

498

Percent finer than indicated size, in millimeters

0.0625

10 

7 

15 

10

0.088

14 

12 

22 

13

0.125

28 

28 

38 

27

0.175

58 

60 

64 

61

0.250

89

88 

88 

87

0.350

100 

98 

100 

100

0.500

100

Measured 

suspendedsediment 

discharge 

(tons per day)

450 

439 

269 

350

BED MATERIAL

Depth below trough surface (in.)

(')- 

0-12_ __ _____________

1 9 94.

O A Q£

fl 19

12-24

24-36__ ______________

Percent finer than indicated size, in millimeters

0.088

0 

0 

0

0 

0

0.125

2 

1 

1 

0 

0

1 1

0.175

6 

6 

8 

3 

4 

5 

4

0.250

29 

27 

32 

18 

25 

21 

15

0.350

68 

65 

66 

55 

69 

45 

41

0.500

91

87 

91 

86 

93 

87 

70

0.700

96 

92 

97 

98 

99 

97 

84

1.000

97 

94

99

90

1.651

98 

97 

98 

100 

100 

99 

93

3.327

99

99 

100

100 

99

9.424

100 

100

100

LABELED OTTAWA SAND

Percent finer than indicated size, in millimeters

0.175

0

0.250

9

0.350

81

0.500

100

Composite of 3-ft cores collected throughout test reach. 

761-353 <
C20 TRANSPORT OF RADIONTJCLIDES BY STREAMS

15,000

LEFT BANK RIGHT BANK LEFT BANK RIGHT BANK

FIGURE 23. Lateral distribution of labeled particles downstream from the source on November 4, 1960. Left graph shows distribution 75 feet downstream (2:30 p.m.).

Right graph shows distribution 125 feet downstream (2:45 p.m.).

mean depths in the reach. The sediment data include 

information on the measured suspended sediment, bed 

material, and tracer particles. During the period 

November 4 16, the water discharge was measured 

five times and the suspended sediment was sampled 

four times. The measured water discharges ranged 

from 249 to 263 cfs and averaged 256 cfs. Suspendedsediment concentrations ranged from 390 to 648 ppm 

(parts per million) and averaged 542 ppm, and measured 

suspended-sediment discharges ranged from 269 to 

450 tons per day and averaged 377 tons per day. The 

mean depths for the entire reach, which were determined 

from the records obtained with the ultrasonic depth 

sounder, ranged from 1.8 to 2.7 feet. During the course 

of the experiment, the water temperature ranged 

from 0° to 8°C and averaged about 5°C. Usually, 

the water gradually warmed during the day. As 

previously mentioned, on the mornings of November 

9 and 10 the water temperature was 0°C and the 

stream contained heavy frazil ice.

The streambed was formed of dunes that moved 

slowly downstream. The size of the individual dunes 

varied greatly; however, an average dune was about 

1.0 1.5 feet high at the crest and about 10 15 feet 

long. Typical records of the bed-surface profile, as

defined by the ultrasonic depth sounder, are shown in 

figure 30. The roughness coefficients associated with 

the bed form were within the range of values usually

C

associated with dunes; that is, values of /=, where C \9

is the Chezy coefficient and g is the acceleration of 

gravity, ranged from 8.7 to 10.5. These values correspond, respectively, to values of the Manning n 

of 0.035 and 0.028.

DERIVATION OF CONCENTRATION-DISTRIBUTION AND 

RELATED FUNCTIONS

The part of the total sediment discharge that is composed of particles having sizes the same as those of 

particles commonly found in the bed material is called 

the bed-material discharge. The individual particles of 

the bed-material discharge move in a discontinuous 

manner involving alternate transportation and storage. 

During the storage phase, the particles are a part of 

the bed material. During the transport phase, their 

downstream progress may be by rolling or sliding in 

almost continuous contact with the streambed or by 

temporary entrainment within the body of the flow.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOTJP RIVER, NEBR. G21

Ordinarily, the sediment that moves in contact with the 

bed is called the bedload, and the sediment that is 

entrained in the flow is called the suspended load. 

Regardless of the mode of transport, all particles associated with the bed-material discharge move in a series 

of discrete steps. Einstein (1950) indicated that the 

average distance traveled between consecutive points 

of deposition may be assumed to be 100 grain diameters 

for particles of average sphericity. Whether or not 100 

diameters is a valid estimate for all sediments is of no 

particular significance in this discussion. The chief 

point is that the particles move in discrete steps. 

Presumably, the distance traveled during each step 

by the entrained particles will be greater than the 

distances traveled by bedload particles, and the lengths 

of the individual steps vary to a considerable degree. 

Thus, if a group of particles is released instantaneously 

from a point or line source, the particles will not move 

downstream in a body but will be dispersed along the 

length of the channel. In addition to being distributed 

longitudinally, the particles will be dispersed vertically 

in a layer parallel to and immediately below the bed 

surface. The lower boundary of the layer will be 

determined by the characteristics of the bed forms.

Randomness is associated with many of the various 

physical factors that combine to determine the length 

of a step and the time between the steps of a particular 

particle under a given set of conditions. Among these 

factors are turbulence, the transport characteristics of 

the particle, the location and orientation of the particle, 

and the dimensions of the bed forms. Because of the 

randomness of these physical factors, the dispersion of 

bed-material particles seems to possess the basic 

attributes of a random process. By random process is 

meant a repetitive process in which any single trial can 

have any one of a number of possible outcomes but the 

actual outcome of a trial is governed by chance and 

cannot be predicted. Thus, the use of probability 

theory is appropriate in deriving a distribution function 

for describing the dispersion of tracer particles released 

simultaneously from a source in the streambed.

Einstein (1937) considered bedload transport from 

the viewpoint of probability theory. Beginning with 

virtually the same assumptions, Einstein arrived at 

distribution functions that are identical with those 

obtained in the following derivation. However, Einstein's method of derivation was considerably different.

5000

e> 4000 

3000

= 2000 -

1000

50,000

LEFT BANK RIGHT BANK

40,000

30,000

20,000

10,000

LEFT BANK RIGHT BANK

FIGUEE 24. Lateral distribution of labeled particles downstream from the source on November 5 and 8,1960. Left graph shows distribution 185 feet downstream (Nov. 5,

1:55 p.m.). Eight graph shows distribution 415 feet downstream (Nov. 8, 3:30 p.m.).
C22

o

TRANSPORT OF RADIONUCLIDES BY STREAMS

Cfl

Station 42, 15 ft 

from left bank

Bed surface

Bottom of core

5000 10,000

x\\\\\\\\\v

x\\\\\1

^^

Station 42, 15 ft 

from right bank

Bed surface

\\X\\\\\\\\\

Bottom of core

400 800

3

Station 92, 15 ft 

from right bank

Bed surface

Bottom of core

1000

E I

4= I 

uj | t/>

Station 92, 15 ft 

from left bank

Bed surface

Bottom of core

2000 1000 

NET COUNT RATE, IN COUNTS PER MINUTE

2000

FIGURE 25. Vertical distribution of labeled particles at selected verticals on November 4,1960.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C23

Station 142, 15 ft from 

left bank, Nov 4

Bed surface

Bottom of core

I I

Station 182, 15 ft from 

left bank, Nov 5

Bed surface

Estimated

1000 2000 1000 2000

DISTANCE BELOW

m i> -I* ooM HSt

v\\\\1

ation 432, 15 ft fro 

right bank, Nov 9

Bed surface

\V\\\\V\\1

Bottom of core

m Station 492, center 

of stream, Nov 9

Bed surface

Bottom of core

2000 , 4000 1000 

NET COUNT RATE, IN COUNTS PER MINUTE

FIGURE 26. Vertical distribution of labeled particles at selected verticals on November 4, 5, and 9,1960.

2000
C24

o

TRANSPORT OF RADIONUCLIDES BY STREAMS

Station 642, center 

of stream, Nov 10

\\x\r

"Bed surface

Bottom of core

100 200

Station 492, center 

of stream, Nov 10

Bed surface

Bottom of core

200 400

CO

Station 342, 15 ft from 

left bank, Nov 10

Bed surface

Bottom of core

500 1000

Station 142, 10 ft from 

left bank, Nov 11

1000 2000

NET COUNT RATE, IN COUNTS PER MINUTE 

FIGURE 27. Vertical distribution of labeled particles at selected verticals on November 10 and 11,1960.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C25

ui 

O

u. 

o= 6

Station 152, 10 ft from 

left bank, Nov 11

s

\\\\\1

,\\\\\\\ \\' \\

tation 180,8 ft fro 

left bank, Nov 11

Bed surface

\V\\\\\\\\\ x

Bottom of core

n

r3

400 800

1000 2000

O 0

Station 77, 8 ft from 

left bank, Nov 11

Bed surface

Bottom of core

1000

S1

.\\v\\\\\\\\

X\\\\\\l

< \ \\ \\\ \\ \\\ j

\\W\\\\1

ation 362, 15 ft fro 

left bank, Nov 13

Bed surface

\\\\N

\\\\\\\\\\]

Bottom of core

m

2000 400 

NET COUNT RATE, IN COUNTS PER MINUTE

800

FIGURE 28. Vertical distribution of labeled particles at selected verticals on November 11 and 13,1960.
C26 TRANSPORT OF RADIONUCLIDES BY STREAMS

Station 362, 15 ft 

from left bank

Bed surface

Bottom of core

Station 392, 15 ft 

from left bank

Bed surface

Bottom of core

500 1000 5000 

NET COUNT RATE, IN COUNTS PER MINUTE

10,000

Station 382, 15 ft 

from left bank

800 1600 

NET COUNT RATE, IN COUNTS PER MINUTE

FIGUKE 29. Vertical distribution of labeled particles at selected verticals on November 13,1960.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C27

/

\r :C

Bed surface

Weter surface

130 250

^

11

Bed surface

Water surface

250

STATIONING, IN FEET

FIGURE 30. Water- and bed-surface profiles defined by the dual channel stream monitor. (From Hubbell and Haushild, 1962.) In top graph, traverse speed is approximately 15 feet per minute; in bottom graph, 33 feet per minute.

THE CONCENTRATION-DISTRIBUTION FUNCTION

Assume a straight, uniform alluvial channel in which 

the water and bed-material discharges per unit width 

are uniform and constant. Now, consider a group of 

tracer particles having identical transport characteristics. Let these particles be released simultaneously, 

at time t=0 and station x=0, from a uniformly distributed plane source that extends laterally across the 

channel and vertically from the surface of the bed to 

the lower limit of the zone of particle movement. 

Also, let the tracer particles move and disperse downstream in such a way that there is no tendency for a 

vertical concentration gradient of tracer particles to 

develop in the bed.

If the number of tracer particles is small enough so 

that the bed-material discharge is virtually unchanged 

by the release of the particles, yet is large enough to 

approximate an infinite population, the stepwise 

movements of the tracer particles can be considered as 

sequences of occurrences of an event in a continuum of 

space, and the longitudinal distribution of the tracer 

particles will correspond to a probability-density 

function.

Let E represent the event that a tracer particle 

begins and terminates a step in the downstream direction. Also, let the probability of E occurring while 

the particle remains within a random length increment (x, z+Az) be ki&x, where ki is a constant for
C28 TRANSPORT OF RADIONUCLIDES BY STREAMS

all tracer particles and ki&x is independent of x 

and the number of times that E has occurred previously. 

Furthermore, let Ax be sufficiently small so that the 

probability of two or more successive occurrences of E 

while the particle is in (x, x+Ax) is negligibly small. 

According to these hypotheses, a tracer particle that 

is in the increment (x, x+Ax) may, when it moves, 

either remain in the increment or be transported out 

of the increment; but, it may not be in the same 

increment at the termination of a second step. The 

corresponding probabilities are (&iAz) that E will 

occur once and (1 /^Az) that E will not occur. 

With these probabilities, a formula can be derived for 

the probability, p(i; x+Ax), that E will occur exactly 

i times while the particle is being transported over the 

distance (0, x+Ax). If this distance is divided into 

the two increments (0, x) and (x, x+Ax), E may occur 

i times in either of two mutually exclusive ways. 

These are:

1. E may occur i times in (0, x) and zero times in 

(x, x+Az).

2. E may occur i 1 times in (0, x) and one time in (z,

x+Ax).

By applying the multiplication theorem for the probabilities of independent events, the respective probabilities for these two ways are [p(i; x)][(l &iAz)], 

and [p(i 1; x')](k1 ^x}. By the addition theorem for 

the probabilities of mutually exclusive events,

or

 p(i; 

which approaches the differential equation

(1)

as a limit as Ax approaches zero. By a similar process 

of reasoning, when no events occur (? 0),

(2)

With conditions p(0; 0) = 1 and p(i; 0)=0 for i>l, 

the solution to equations 1 and 2 is

(3)

which is the Poisson probability mass function with 

parameter kiX. Equation 3 expresses the probability 

that E will occur exactly i times as the given tracer 

particle is transported the distance (0, z).

2 The notation and terminology in this section are for the most part consistent 

with those of Parzen (1960).

The distribution function corresponding to equation 

3 may be stated

(4)

where F(n l;x) is the probability that the number 

of occurrences of E as the particle is transported 

over the distance (0, x) is less than ft. Then 1 

F(n 1; x) is the probability that the number of 

occurrences of E in (0, z) is equal to or greater than 

ft, which is also the probability that at the completion 

of the ftth occurrence of E the particle will have been 

transported a distance equal to or less than (0, x). 

Hence, 1 F(n 1; x) is equivalent to the distribution 

function F(x; ft) for the location of the particle at the 

completion of the ftth event. Differentiation of 

F(x; ft) with respect to z yields the probability density 

function

n, _w-i

(5)

where T(n) is the gamma function (generalized factorial function) of n. Equation 5 is the probability 

density function for the gamma distribution with parameters n and ki. When multiplied by dx, it expresses 

the probability that at the completion of the nth step 

the given tracer particle will be located in the length 

increment (x, x+dx), which is to say that the particle 

will have been transported the exact distance (0, z). 

The mean distance over which a particle would be 

transported during n repeated occurrences of E is

Thus ki=n/xn may be interpreted as the mean rate 

of occurrences of E with respect to x. Also, the mean 

step length for any single occurrence of E is 1/ki.

Considering now the movements of a tracer particle 

with respect to time, E may also be defined as the 

event that a tracer particle begins and terminates a 

rest period between successive steps in the downstream 

direction. The same considerations that apply to the 

number of occurrences of E in (0, z) apply also to the 

number of occurrences of E in. (0, /) Thus, assuming 

that the total traveltime is negligible in comparison 

to the total rest time, the probability that a tracer particle will have completed exactly i rest periods in the 

elapsed time (0,0 is expressed by the probability mass 

function

,. ,, p(»;0= *=1,2 . (6)
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C29

and the probability density function for the total 

elapsed time at the completion of the nth rest period is

(7)

where k2 is analogous to ki and may be interpreted as 

the mean rate of occurrence of rest periods with respect 

to t. Also, the mean duration of any single rest period 

is l/&2 .

Up to this point, the notation (x; ri), (i; x), (t; n), 

and so on, prefixed by p,f, or F, has been used to denote 

various probability functions. The first term in the 

parentheses denotes the variable with respect to which 

probability is distributed. The second term, following 

the semicolon, specifies a qualifying condition. For 

example, p(n; t) denotes the probability that E will 

occur exactly n times during the specified time interval 

(0, t) following release from the source. Likewise, 

f(x; n) denotes the probability density for the total 

distance, x, moved by the particle at the completion 

of the nth step. In the development that follows, it is 

convenient to use different notations to express the 

same probability functions. Usage of the concept of 

conditional probability requires that the symbol 

f(x; n) be replaced by f(x n), which denotes the conditional probability density for the total distance moved 

given that the particle has taken n steps. Also, the 

symbol p(n; t) is replaced by pt (n). In the subscript 

system of notation, the qualifying condition is denoted 

by the subscript instead of by the term following the 

semicolon.

As a tracer particle moves downstream in discrete 

steps, each rest period is followed by a step which in 

turn is followed by another rest period, and so on. 

Therefore, on the assumption that the particle is 

initially at rest and begins its first rest period at t=Q, 

during any given rest period the particle must have 

undergone the same number of complete rest periods 

as steps; hence, a particle that has taken n steps will 

have also undergone n rest periods.

Use of the concept of conditional probability enables 

derivation of the density f unction/^(x), from which the 

probability of the location of the tracer particle in any 

length increment (x-\-dx) at time t, can be obtained. 

To this end it is convenient to express equation 5 in 

the form of a conditional probability-distribution 

function as follows:

F(x\n)=\ f(x'\n)dx' = P[x'<x\i=n] (5a)

where

P denotes probability

x' = a dummy variable of integration.

If one considers the two events A' and B', the conditional probability that A' will occur given that Bf 

has occurred is defined as

where [A', B'} represents the joint occurrence of A' and 

B'. Applying this definition,

_Pt[x'<x,i=n] 

~ Pt(n)

When n is given, knowledge of t adds no information; 

hence,

F t(x\n) = F(x\n) 

and

Pits' <x, i=n] = F(x\n)pt(n). (8)

The marginal distribution function in x corresponding 

to the joint probability function in equation 8 is

[x><x>i=n] = F(x\n)p t (n)

n=0

(9)

Differentiation of equation 9 with respect to x yields 

the density function for x at time t

n=i

Introducing into this equation

n\

(5)

(6)

the density function for x at time t can be written

(10)

which transforms readily into

!rf (lOa)

where Ii(2-^k1 xk2 f) is a modified Bessel function of 

the first kind of order one. Values of this Bessel function are given in tabular form in many mathematics 

reference books, such as that of Korn and Korn (1961).
C30 TRANSPORT OF RADIONTJCLIDES BY STREAMS

Equations 10 and lOa were first derived by Einstein 

(1937).

It should be noted that the integral of the density 

function in equations 10 and lOa over the entire range 

of x is

Jor

where p t(ti) is the probability that the particle has not 

yet moved from the origin at time t. Thus the density 

function ft(x) applies only to particles that have taken 

at least one initial step.

If, as assumed initially, a very large number of tracer 

particles that are initially distributed uniformly in 

a plane extending throughout the zone of movement at 

x=0 all begin their first rest period at t=0, the density 

function, jt(%), as defined by equations 10 and lOa can 

be used to estimate the concentration <t>(x, f) of tracer 

particles in the bed at any distance, x, downstream 

from the source at time t. In fact, <f>(x, f) will be directly 

proportional to ft(x) if the following conditions are 

satisfied: (1) The average depth to which the tracer 

particles are distributed remains constant downstream 

from the source; (2) the concentration of tracer particles 

does not, on the average, vary with depth below the 

bed surface; (3) at any given instant, no significant 

fraction of the tracer particles are in suspension. Given 

these conditions, if the concentration of tracer particles 

is defined as the weight of tracer particles per unit 

volume of bed material, the concentration is

where

W <f>(x,t)=~ft(x)

W=ihe total weight of tracer particles placed in the channel 

B=the width of the channel

d=the average depth beneath the bed surface to which the 

tracer particles are distributed.

Thus, the concentration-distribution function at time 

t for tracer particles that have moved from the origin 

where they began their first rest period at t=0 is

or

Either of these forms of the concentration-distribution 

function can be used to compute the concentration of 

tracer particles in the bed at any distance downstream 

from the source at any given time after release; however, computation is considerably easier with the

0= *~ (V+V)

second form. Computation of concentrations is facilitated still further by writing equation lla in the form

(lib)

where z 2^kixk2l.

Significant properties of the concentration-distribution function with respect to x for a given t are as 

follows:

Area under the distribution curve:

Mean:

Mean rate of movement of tracer particles:

_ 

dt

Variance:

Mode:

xm =x evaluated when {* = 0> in which 

ox

case

where hC^-^kixk^i) is a modified Bessel function of the 

first kind of order two.

BELATED DISTRIBUTION FUNCTIONS

If a tracer particle is introduced into the channel at 

x=0 in such a way that it takes its first step at time 

t=0, then during any given rest period the number of 

rest periods it will have completed will be one less than 

the number of steps. Equation 9 then becomes

(12)

n=\

where the asterisk denotes the initial condition of beginning with motion. Likewise equations 10 and lOa 

become

and

f*/V> Tf. Jt (x)-ki

ft (x)

(13)

(13a)

where I0 (2^/kixk^) is a modified Bessel function of 

the first kind of order zero. As were equations 10 and 

lOa, equations 13 and 13a were first derived by Einstein (1937).
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C31

In a manner analogous to that which was used to 

obtain the concentration-distribution function, <j)(x,t}, 

which specifies the concentration of tracer particles in 

the bed at any distance x and any time t, a dischargedistribution function, \f/(t,x), which specifies the discharge of tracer particles past a cross section at any 

distance x downstream from the source at any time t, 

can be obtained from the density function Jx (t}. A 

tracer particle that begins its first rest period at the 

origin at t=0 will, as it passes a point a distance x 

downstream from the origin, have completed one less 

step than rest periods. Thus, the probability density 

fx(f), which relates to the probability that the particle 

will pass the point x during the time interval (t, t-\-dt), 

corresponds tof*(x), except that the variables x and 

t and the constants &i and k2 are interchanged so that

(n-l)I

or

(14a)

If a large number of tracer particles are placed in the 

channel, again as a uniformly distributed plane source 

at x 0, and all the particles begin their first rest period 

at Z=0, the estimated discharge \I/(t, x) of tracer particles 

in weight per unit time per unit of cross-sectional area 

is proportional tofx (t), where W/Bd is again the constant of proportionality. The discharge-distribution 

function is

Bd

or

(15a)

APPLICATION OF CONCENTRATION-DISTRIBUTION 

FUNCTION TO EXPERIMENTAL DATA

To use the theoretically derived concentrationdistribution function, the constants ki and k2 must be 

evaluated. Presumably, values for these constants are 

related to various hydraulic and sediment factors; 

however, until the nature of the relations is known, the 

constants must be determined from observed distributions. To make the determinations, certain characteristics of the observed and theoretical distributions are 

equated. These characteristics are the time rate of 

movement of the mode (peak) along the channel and 

the time rate of attenuation of the peak relative concentration, per foot of width; relative concentration is 

defined as the ratio of concentration at a point to the 

area under the concentration-distribution curve. Experimental data related to these rates are listed in 

table 4.

TABLE 4. Some characteristics of the observed distributions

Bnn

2R 

2L_........

3R 

3L._.__.___

4 R. 

4L.______._

5R. ..

5L..-.._ 

6R 

6L._-_ ..

7C...._ ....

8R 

8L_________

11 R........

12 R-. 

Time 

after 

dosing 

(hr)i

22.3 

21.2 

44.2 

45.8 

70.9 

73.9 

96.2 

98.3 

117.7 

120.4 

142.2 

170.1 

171.8 

241.6 

287.4

Distance from 

source (ft)

Mean

131

115 

244 

213 

354 

336 

458 

406 

491 

499 

547 

635 

670 

835 

930

Mode

107 

73 

182 

172 

270 

275 

347 

363 

410 

388 

475 

510 

550 

801 

869

Peak 

concentration 

(g per cu ft)

2.28 

3.26 

1.14 

1.22 

.76 

.98 

.66 

.59 

.56 

.45 

.44 

.43 

.30 

.32 

.25

Area under 

concentration-distribution 

curve 

(g per sq ft)

249 

238 

236 

204 

228 

194 

213 

2179 

217 

2170 

2197 

2199 

2168 

2166 

2162

Peak relative concentration per 

foot of 

width

0.0092 

.0137 

.0048 

.0060 

.0033 

.0051 

.0031 

.0033 

.0026 

.0026 

.0022 

.0022 

.0018 

.0019 

.0015

1 Dispersion time.

2 Based on a distribution adjusted to eliminate the influence of the delayed release 

of buried tracer particles (see figs. 34-40) .

According to the theory, the rate of movement of 

the mode is

dxm k2

dt (16)

for k2t> about 4. In figure 31, the position of the 

modes of the observed distributions are plotted as a 

function of the dispersion time. The slope of the 

straight part of the line that defines the rate of movement of the mode is

^=3.00 ft per hr«^- dt fti

According to the theoretical concentration- distribution function the peak relative concentration is given 

by

4>(X, Qma»._ <ft(£;0ma». /t(aQma».

- Bd

where /t(a;)max . is equal toft(x) evaluated for the condition

As the dispersion time increases the probability that a 

particle is still at the origin, pt(Q)=e~*2t , approaches 

zero so that for k2t > about 3

Omu. (17a)

In figure 32, the values of the observed peak relative 

concentration, 4>m^JA, are plotted on logarithmic coordinates as a function of the dispersion time. In 

figure 33 the ratio of the theoretical peak relative concentration (as given in eq 17a) to ki is plotted, also on 

logarithmic coordinates, as a function of k2t. Because 

the respective abscissas and ordinates of figures 32 and 

33 differ only by the constants of proportionality ki
C32

300

TRANSPORT OF RADIONUCLIDES BY STREAMS

200

100

per hr

EXPLANATION

Right side of stream

O 

Left side of stream

. Center of stream

 

Right side of stream

n 

Left side of stream

B

_ Center of stream

100 300 400 500 600 700 

DISTANCE FROM SOURCE, IN FEET

800 900 1000 1100

FIGURE 31. Location of mode and mean as a function of dispersion time.

and k2 , these constants may be evaluated by superimposing figure 32 over figure 33 in the position where 

(1) the experimental data best fits the theoretical 

curve, and (2) k2 /ki equals 3.00 feet per hour. When 

the data are superimposed over the theoretical curve 

in this position,

_ 36 ft

and

k2t

The theoretical attenuation curve for these values of 

ki and k2 is shown in figure 32. Because the data for 

runs 3R, 4R, 5R, 6R, and 7C are considered to be the 

most reliable, the points representing these runs were 

weighted accordingly in fitting the data. In both 

figures 31 and 32, values for the points representing 

the mode in runs 5L, 6L, 7C, 8R, 8L, HR, and 12R 

were obtained from modifications of the distribution 

curves. The curves were modified by substituting 

estimated curves for the parts that reflected the dispersion of the particles that were buried initially on the left 

side of the stream.

Theoretically derived distribution curves for times 

equal to those for the observed distributions were determined from equation 11 and the computed values 

for k\ and k2 . It is significant to note that in figure 31 

the extension of the straight line part of the curve that 

defines the rate of movement of the mean does not pass 

through the origin but intersects the x axis at x=110 

feet. The displacement of the curve from the origin 

at zero dispersion time is a major discrepancy, for the 

mean of the derived distribution function varies 

directly with the time and should be zero when the 

dispersion time is zero. To compute theoretical distribution curves that coincided well at the modes with 

the observed distributions, the origin of the theoretical 

distribution curves had to be displaced to x=110 feet. 

This procedure seemed to be justified because for dispersion times greater than about 70 hours the mean 

preceded the mode (see fig. 31) approximately by the

1 5 distance -r- (54 ft), which is in accordance with the

characteristics of the derived distribution function. 

The theoretical curves, along with the corresponding 

observed distribution curves, are shown in figures 34- 

40. The crosshatched areas under the measured
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C33

o.io

1/36

0.01

0.001

0

\

\

o

EXPLANATION

Right side of stream

O 

Left side of stream

3 

Center of stream

From equation 11

o

Normal distribution

10 12 100

DISPERSION TIME ( f ), IN HOURS 

FIGURE 32. Attenuation of observed peak relative concentration with dispersion time.

400

curves in figures 36-40 represent estimates of the 

increase in area under the distribution curve that 

resulted from the delayed release of tracer particles 

that had been buried on the left side of the stream near 

the upstream end of the test reach. Distribution

curves are not shown for run 9, because the observed 

distribution curves span only the first 400 feet of the 

test reach, and for runs 10L, 10K, 11L, and 12L, 

because the areas under the observed distribution purves 

that resulted from the delayed release of buried tracer
C34 TRANSPORT OF RADIONUCLIDES BY STREAMS

0.40

0.10

-k

0.02

-~ftM evaluated when --[

ftMn

(Normal distribution)

10

k2t

FIGURE 33. Attenuation of peak according to theoretical function.

100

particles could not be isolated with confidence from the 

residual areas. Also, no distribution curves are shown 

for runs 2R and 2L. This is because the theoretical 

concentration-distribution function is not applicable 

to the conditions of the experiment for values of k2t 

less than 2. In the range 0<&2tf<2, the peak of the 

theoretical distribution curve coincides with the origin 

where concentration is not defined.

In general, the theoretical and observed distribution 

curves agree fairly well and exhibit the same characteristics throughout almost the full range of dispersion 

times. At small dispersion times both sets of distribution curves are markedly skewed in the upstream 

direction. As the dispersion time increases, they tend 

to become symmetrical at approximately the same rate. 

In addition, the rates of attenuation of the peak 

concentrations with respect to time in both sets of 

curves are similar and approach the attenuation rate of 

the normal distribution (see figs. 32 and 33).3 In some 

respects, however, the theoretical and observed distribution curves do not agree. One difference is the way 

in which the concentrations decrease downstream 

from the peak; the observed concentrations decrease

3 Although for large t the distribution function ft(x) exhibits similar geometric characteristics, it does not in a' rigorous sense converge to a normal distribution.

more rapidly immediately downstream from the peak 

and less rapidly at the tail than do the theoretical 

concentrations. Another difference is the displacement of the origin of the theoretical distribution curves 

downstream from the actual source (x=0); this is 

particularly evident in the early runs. Both of these 

differences can be traced to discrepancies between 

certain physical features of the experiment and the 

corresponding assumptions and (or) hypotheses on 

which the theoretical distribution is based.

Probably the most significant discrepancy is that the 

theory applies only to a group of tracer particles that 

have identical transport characteristics, whereas the 

size (therefore the transport characteristics) of the 

tracer particles actually used in the experiment varied 

over a range. Thus, because each set of transport 

characteristics has a corresponding distribution function, the observed distribution curves are actually the 

resultant summations of many separate distribution 

functions, each of which is associated with a different 

particle size. The size distribution of the tracer 

particles indicates that at least 80 percent of the particles 

could have been transported in suspension at one time 

or another. This factor undoubtedly contributed to 

the long tails of the observed distribution curves.
0.007

0.006

0.005

0.004

0.003

0.002

~ 0.001

0.006

0.005

0.004

0.003

0.002

0.001

TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOTJP RIVER, NEBR. C35

Run 3L

Observed

Theoretical

Run 3R

Theoretical

ObservedA.

200 400 600 800 1000

DISTANCE FROM SOURCE, IN FEET 

FIGURE 34. Theoretical and observed longitudinal distribution of tracer particles, runs 3R and 3L.

1200

This conclusion is supported by the results of a similar 

experiment reported by Lean and Crickmore (1960). 

In their experiment, the tracer particles were composed 

of a representative sample of the bed material, which 

included a fairly wide range of particle sizes. For comparable dispersion times, the distribution curves resulting 

from their experiment had tails in the downstream 

direction that are much more pronounced.

The other major discrepancy between the actual 

experimental conditions and the requirements of the 

theory concerns the source. The theory requires that 

the tracer particles be released from a uniformly concentrated plane source extending from the surface of 

the bed to the lower limit of the zone of particle movement and covering the width of the channel. However, 

in the experiment the source consisted of a series of
RELATIVE CONCENTRATION, IN FEET~i
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C37

0.003

0.002

Observed Theoretical

0.001

=r 0.003

0.002

0.001

200 400 600 800 

DISTANCE FROM SOURCE, IN FEET

1000 1200

FIGURE 36. Theoretical and observed longitudinal distribution of tracer particles, runs 5R and 5L. Crosshatching denotes estimated increase in area under

distribution curve due to delayed release of buried tracer particles.
C38

0.003

0.002

0.001

TRANSPORT OF RADIONUCLIDES BY STREAMS

200 400 600 800 1000 1200 

DISTANCE FROM SOURCE, IN FEET

1400 1600 1800

FIGURE 37. Theoretical and observed longitudinal distribution of tracer particles, runs 6R and 6L. Crosshatching denotes estimated increase in area under distribution

curve due to delayed release of buried tracer particles.

0.003

0.002

0.001

200 400 600 800 1000 1200 

DISTANCE FROM SOURCE, IN FEET

1400 1600 1800

FIGURE 38. Theoretical and observed longitudinal distribution of tracer particles, run 7C. Crosshatching denotes estimated increase in area under distribution curve

due to delayed release of buried tracer particles.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C39

0.003

0.002

0.001

/Theoretical

< 0.

0.001

 T 

Run 8R

\

200 400 600 800 1000 1200 

DISTANCE FROM SOURCE, IN FEET

1400 1600 1800

FIGURE 39. Thecretical and observed longitudinal distribution of tracer particles, runs 8R and 8L. Crosshatching denotes estimated increase in area under distribution

curve due to delayed release of buried tracer particles.

point sources, on the surface of the bed, extending 

across the channel. Undoubtedly, an initial mixing 

period was required for the tracer particles to become 

dispersed uniformly throughout the zone of particle 

movement. During this mixing period the quantity of 

tracer particles available for transport and dispersion 

at any given time was probably greater than it would 

have been had the tracer particles initially been distributed uniformly with respect to depth. This is 

probably one reason why the origins of the theoretical 

and observed curves do not coincide (see fig. 31). The 

fact that the rate of movement of the mean tended to 

decrease and become constant after about 70 hours 

may also have been partly caused by a fairly rapid 

passage of the finer tracer particles through and out of 

the test reach.

A third way in which the observed distribution data 

differed from the theory was that the area under the 

observed distribution curves tended to decrease as the 

dispersion time increased. The total reduction in the 

area under the curves during the 13-day period of the 

experiment was about 35 percent of the area that was 

observed 1 day after the release of the tracer particles. 

The rate of decrease is roughly linear. In the analysis,

the decrease in area was compensated for by expressing 

the concentration of tracer particles in terms of relative 

concentration, $/A Probably, the major factor contributing to the reduction in area was the removal of 

some of the radioactive label from the tracer particles 

by abrasion. Another contributing factor may have 

been that a significant quantity of the finer tracer 

particles passed through and out of the test reach during 

the course of the experiment.

In summary, the observed distribution curves are 

approximated reasonably well by curves that are based 

on the theoretical distribution function. Probably, 

the major differences between the observed and theoretical distributions can be attributed to discrepancies 

between certain physical features of the experiment 

and the corresponding assumptions on which the theory 

is based. In addition, because the constants, ki and 

k2 , were determined by a forced match between observed 

data and a theoretical curve, their values are empirical 

and are not necessarily theoretically significant.

BED-MATERIAL TRANSPORT

Most existing measurement techniques and theories 

for determining bed-material discharge are directed
TRANSPORT OF RADIONUCLIDES BY STREAMS

0.002

0.001

Run 12R

/Theoretical

200 400 600 800 1000 1200 

DISTANCE FROM SOURCE, IN FEET

1400 1600 1800

FIGURE 40. Theoretical and observed longitudinal distribution of tracer particles, runs 11R and 12R. Crosshatching denotes estimated increase in area under distribution

curve due to delayed release of buried tracer particles.

toward evaluating the flux of sediment particles past a 

stationary cross section. With the recent advent of 

radioactive-tracer and other methods for utilizing 

labeled sediment particles, a new approach has become 

feasible. The new approach is based on the determination of how the position of a particular group of 

tagged bed-material particles varies with time. Consider, within the framework of the same restrictions 

that were assumed in the derivation of the theoretical 

concentration-distribution function, a group of tracer 

particles released from a plane source in the bed of a 

channel. Assume now, however, that the transport 

characteristics of the tracer particles simulate those of 

the bed material. As the tracer particles move downstream they disperse so that at any time, t, after release 

they are distributed about some mean distance from the 

source, x. Hence, the ratio, x/t, defines the mean 

velocity of the bed-material particles in the downstream 

direction. Correspondingly, the bed-material discharge, Qs, in weight per unit time is

Qs=y s(l-\)Bd = (18)

where

X = the fraction of volume of bed material, in place, not

occupied by particles (porosity) 

7 s = the specific weight of the bed material.

If all the bed-material particles have virtually the same 

transport characteristics, the tracer particles will be 

distributed according to equation 11 and k2/ki may be 

substituted for x/t in equation 18. If the tracer particles are selected to simulate only bed-material particles 

possessing a certain characteristic such as the same size, 

shape, or specific gravity, the discharge of such particles

(19)

where

i c=the ratio of the volume of particles possessing the characteristic to the volume of bed-material particles in the 

zone of particle movement

c= a subscript that denotes terms associated with the particles possessing the characteristic.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C41

Although both equations 18 and 19 have the form of a 

continuity equation, the concept of continuity applies 

only in a statistical sense. Actually, particles move 

only when they are exposed on the surface of the bed 

or when they are in suspension. The particles move in 

a series of discrete steps and are buried in the bed, for 

varying periods of time,_between the steps. Thus, the 

mean particle velocity, x/t, represents the average rate 

of movement during a total elapsed time. The use of 

the cross-sectional area, Bd, as the zone of movement 

is based on the assumption that each observed distribution is representative of the true distribution of the 

tracer particles at some given instant regardless of 

whether or not all the particles actually moved through 

Bd and whether or not some of the particles were in 

movement (either as bedload or as suspended load) 

during the time of the distribution observation. This 

assumption seems quite reasonable because the rate of 

change of the distribution curves is low and the time 

required to observe each distribution is short.

Equations 18 and 19 are similar to the bed-material 

transport equations derived by Lean and Crickmore 

(1960). However, in their analysis, Lean and Crickmore devoted considerable attention to establishing 

the local flux of tracer particles as a function of vertical 

position within the bed.

In the experiment, the following conditions were 

either known or determined experimentally:

i c=0.65 (fraction of the bed material within the 

approximate size range of the tracer particles, 

which is 0.225-0.420 mm) 

y s (l X) = 100 Ib per cu ft = 0.05 ton per cu ft 

W=40 lb= 18,200 g 

A = 249 g per sq ft

dx 

x/<=-^-=3 ft per hr = 72 ft per day.

The value of (Qs) c for these conditions, as computed with 

equation 19, is 171 tons per day. Unfortunately, this 

value cannot be compared with a measured discharge 

to ascertain its accuracy because, at the present time, 

available equipment is not capable of measuring the 

sediment discharge on or very near the streambed. 

However, it can be compared with a discharge computed according to some reliable method. Of available 

methods for determining bed-material discharge in the 

North Loup River, the modified Einstein procedure 

(Colby and Hembree, 1955) is probably the most reliable. The procedure was developed with data from 

Nebraska sandhill streams, and its accuracy and applicability have been verified (Colby and Hembree, 1955; 

Hubbell and Matejka, 1959).

The value of (Qs) c as computed with the modified 

Einstein procedure for the 65 percent of the bed material that is in the 0.225-0.420 mm size range is 192

tons per day. (Data from tables 2 and 3 for Nov. 9, 12, 

and 16 were used in the computation.) The agreement 

between the values of (Qs) e as computed by the two 

methods (171 tons per day vs. 192 tons per day) 

is remarkably good. Actually, the two methods should 

not give exactly the same values because the distribution of the tracer particles within the 0.225-0.420 mm 

size range differs from the distribution of the bed 

material in the same range (see fig. 5).

APPLICABILITY OF RESULTS

The theory and experiment discussed in the preceding 

sections are based on the idealized concept of an instantaneous source of contaminated particles that are 

distributed uniformly over a plane which extends 

laterally across the width of the stream and vertically 

from the surface of the bed to the lower limit of the 

zone of particle movement. Although such idealized 

source conditions exist rarely, if ever, in nature, the 

concentration-distribution function corresponding to 

an instantaneous plane source (eq 11) is of fundamental 

practical importance. This is because a concentrationdistribution function corresponding to any defined 

source distribution in space and time can be obtained 

from the instantaneous plane source function by the 

technique of mathematical superposition (that is, the 

convolution integral). The above statement applies 

to the discharge-distribution function (eq 15) also.

Some examples of concentration-distribution functions corresponding to source distributions where the 

source strength varies with longitudinal distance and 

(or) time are listed below. In these examples, uniform 

concentration distributions in the lateral and vertical 

directions are assumed.

1. Instantaneous longitudinally distributed source of 

variable strength (weight per unit length), $(£):

(x, t) = S(Qft(x-S, (20)

where

£=the distance downstream from x=0 along which 

the source is distributed over length s at time 

t=0, (0<£<s)

ft(x %, f)=the instantaneous plane source density 

function, ft(x) (eq 10), with x replaced by x £.

2. Continuous plane source at x=Q, discharging at a 

variable rate (weight per unit time), S(r):

4>(X, t)=J£ P S(r)ft(x, t-r}dr (21)

where

r = the time after t=Q during which the source is

discharging, (0<r<<i) 

ft(x, t r) = the instantaneous plane source density

function (eq 10) with t replaced by t T.
C42 TRANSPORT OF RADIONUCLIDES BY STREAMS

3. Continuous longitudinally distributed source of 

variable strength (weight per unit length), discharging at a variable rate (weight per unit time),

(22)

where

ft(x £, t r)=the instantaneous plane source density function (eq 10) with x replaced by x £, 

and t replaced by t T.

By integrations such as these, an infinite variety of 

real situations can be approximated. In addition to 

longitudinal concentration-distribution curves, the instantaneous plane source density functions can be used, 

for example, to construct concentration-time curves, 

longitudinal discharge-distribution and discharge-time 

curves, and concentration- and discharge-growth and 

decay curves. The theoretical distribution functions 

also provide a means for making estimates such as the 

time of travel and reduction of maximum concentration 

of a slug of contaminated bed material as it travels 

from one location to another location farther downstream, the duration of contamination in excess of 

some specified level at a particular site, and the time 

required for the concentration of contaminated particles released from a continuous source to reach equilibrium as well as the equilibrium concentration.

One significant assumption in the development of the 

theoretical function was that a stream can be represented as a homogeneous probability field in space and 

time. Such an assumption will not be justified in 

many cases. However, by use of available knowledge 

of stream characteristics and flow patterns, it should 

be possible to break up most streams into reaches in 

which the assumption of homogeneity is an acceptable 

approximation for limited periods of time. This would 

be analogous to the procedure commonly used in calculating backwater curves in, or routing floods through, 

nonuniform channels.

Also, in the development of the theory, the effects of 

exchange of radionuclides among various phases of the 

transporting media were not considered. It was assumed that once a particle became contaminated by 

radionuclides it would remain so. However, use of the 

mathematical device of secondary sources and sinks 

should enable allowance for the effects of the exchange 

of radionuclides between the sediment, solution, or 

biota.

If it is assumed that the mathematical model on which 

the theoretical distribution functions are based approximates reality acceptably, the validity of any predictions based thereupon depends on the accuracy with 

which the different independent variables in the theoretical distribution functions can be estimated. In 

most practical situations, the amount and initial distribution of contaminated particles, the width of the 

stream, and the bulk specific gravity of the bed material 

can be determined readily. However, three other 

variables the depth of the zone of particle movement 

(d) and the constants ki and k2 need to be known.

When the bed configuration consists of dunes or 

ripples, the depth of the zone of particle movement, d, 

can be estimated from depth-sounding data. For this 

purpose continuous longitudinal profiles of the bed 

surface are required. The method for estimating d is 

illustrated in figure 41 and corresponds to the method 

presented by Hubbell (1964). The length of the reach 

for which d is to be determined is divided into sections. 

Starting from the upstream end, each section of length 

lt extends from the dune trough at which the section 

begins to the first trough downstream that is deeper. 

After sectioning, a mean depth for each section, di} is 

determined, and the d for the total reach length, L, is 

computed as the weighted average of the rf/s for each 

section. Expressed mathematically (see fig. 41),

The reasoning behind this procedure is based upon the 

assumption that although the individual bed forms may 

change shape as they progress downstream, a statistical 

constancy of form exists over a long reach. Hence, 

quantitatively the particles subject to movement are 

those that would move if the entire profile were to 

progress downstream without changing form, and the 

depth of particle movement is defined by lines that are 

parallel to the mean bed surface and extend downstream 

from the deepest trough as illustrated.

On the basis of depth-sounding records obtained in the 

upstream half of the test reach during 10 of the runs, 

d was computed to be 1.41 feet. Because the tracer 

particles were tagged in the experiment, two additional 

independent estimates of d could be made. From the 

core-sample data in figures 25-29, the average depth 

beneath the bed surface to which the tracer particles 

were distributed was computed as 1.45 feet. From

^ w d= BA

the average depth was computed as 1.46 feet for TF=40 

pounds, 5=50 feet, and ^4=250 g per sq ft.

As previously mentioned, the constants ki and k2 

cannot be predicted until their relations to pertinent 

hydraulic and sediment parameters are established by 

further experiments. However, they can be determined
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR.

Direction of movement

C43

, /Mean bed-surface level ?

FIGURE 41. Method for estimating average depth of zone of particle movement. d=-=LI

individually by experiments as was done in this study, 

and k2/ki can be determined from

The ratio xjt can be computed from any single longitudinal concentration-distribution curve for time t 

following release from an instantaneous vertical plane 

source. In addition, the ratio can be computed from 

a known or independently computed bed-material 

discharge by using equation 19 if the stream width and 

the depth of particle movement also are known. Once 

k2 /ki and x are determined, ki can be estimated from the 

variance of the concentration-distribution curve, 0%, by 

the formula

. 2x

Unfortunately, there is no theory available at the 

present time for predicting v\ quantitatively for a 

given set of conditions.

CONCLUSIONS

1. Application of radioactive-tracer techniques to the 

study of the dispersion and transport of contaminated bed-material particles is both experimentally feasible and safe.

2. The longitudinal distribution of contaminated particles that are released instantaneously from a line 

source that extends across the channel is skewed 

in the upstream direction. The distribution is 

highly skewed initially and becomes progressively 

more uniform and symmetrical as the dispersion 

time increases.

3. When the streambed is composed of dunes and 

contaminated bed-material particles are transported by the flow, some of the contaminated 

particles will be deposited in deep troughs and 

become buried. The contaminated deposits may 

remain buried for long periods of time and then 

be reexposed when another deep trough passes by.

4. The lateral distribution of contaminated particles 

that are nonuniformly distributed in the lateral 

direction at some cross section at some initial time

tends to become uniform as the distance from the 

cross section and (or) the dispersion time increases. 

The vertical distribution of contaminated particles 

n a streambed that consists of dunes varies

considerably from place to place and follows no 

regular discernible pattern.

If contaminated particles having identical transport characteristics are released instantaneously 

from a uniformly distributed plane source that 

extends laterally across the channel and vertically 

from the surface of the bed to the lower limit 

of the zone of particle movement, and if the 

conditions within the zone of particle movement 

that govern the transport of bed-material particles 

are statistically steady and homogeneous, probability theory provides a mathematical model 

for the derivation of a concentration-distribution 

function that gives the concentration of contaminated particles in the bed at any time and 

place. The theoretical distribution function contains two constants that are related to step lengths 

and rest periods of particles.

According to the theoretical concentration -distribution function, the mean displacement from the 

source of the contaminated bed-material particles 

and the variance of the distribution are directly 

proportional to the dispersion time. The change 

in displacement of the mode is also proportional to 

the dispersion time except for short dispersion 

times. Also, at large dispersion times the rate of 

decrease of the maximum concentration varies 

inversely with the square root of the dispersion 

time, and at small or intermediate dispersion times 

the attenuation is more rapid.

The two constants in the theoretical concentrationdistribution function can be determined by 

equating the time rate of displacement of the mode 

and the rate of attenuation of the peak concentration 

for the theoretical function to the corresponding 

rates determined for the observed distributions.

The theoretical and observed concentration-distribution curves for equal dispersion times agree 

well and exhibit the same characteristics except
C44 TRANSPORT OF RADIONUCLIDES BY STREAMS

at the origin and at the downstream tails. The 

differences apparently resulted from discrepancies 

between certain physical features of the experiment and some of the corresponding assumptions 

by which the theoretical distribution function 

is restricted. For example, whereas the theoretical distribution function is restricted to contaminated particles having identical transport 

characteristics that are released from a vertical 

plane source, the tracer particles actually used 

in the experiment had a range of transport 

characteristics and were released from a line 

source at the surface of the bed.

10. Probability theory also provides a mathematical 

model for the derivation of a discharge-distribution function that expresses the discharge of 

contaminated bed-material particles past any 

cross section at any given time; the function is 

similar in form to the concentration-distribution 

function and involves the same assumptions, 

variables, and constants.

11. The discharge of any fraction of the bed material 

that is simulated by tracer particles can be 

computed by a continuity-type equation, which 

states that the discharge is proportional to the 

product of the cross-sectional area of the bed 

through which the particles move and the mean 

velocity of the particles. The cross-sectional 

area can be determined either from longitudinal 

profiles of the bed surface, the area under the 

concentration-distribution curve, or core-sampling 

measurements, which define the depth of penetration of the tracer particles. The mean velocity 

is defined by the rate at which the mean displacement of tracer particles from the source 

varies with time.

12. The discharge computed with the continuitytype equation for the part of the bed material 

represented by the tracer particles compares 

favorably with the discharge computed by the 

modified Einstein procedure (Colby and Hembree, 

1955) for approximately the same part.

13. The theoretical concentration- and dischargedistribution functions, which apply to the initial 

condition of an instantaneous vertical plane 

source, can be extended by means of the mathematical superposition principle to describe a 

variety of practical situations that involve more 

complex initial conditions. This is done by introducing functions that define the source strength as 

functions of space and (or) time and performing 

the appropriate integrations. By such devices, 

the theoretical distribution functions can be 

used, for example, to estimate concentration and

and discharge growth and decay curves, times of 

travel and reduction in maximum concentration 

of slugs of contaminated bed material, the duration 

of contamination in excess of some specified level 

at a particular site, and the level of and length 

of time required to reach an equilibrium concentration of contaminated particles released 

from a continuous source.

14. For application of the theoretical functions to real 

situations, usually all the required variables except for two constants will either be known or can 

readily be estimated. Until the relations between the two constants and the pertinent 

hydraulic and sediment parameters are defined, 

the constants can be determined only by 

experiment.

SUPPLEMENTARY DATA POINT-SOURCE METHOD 

FOR DETERMINING DETECTION-SYSTEM SENSITIVITY

During the course of the study a method was developed for estimating the overall sensitivity of a 

radiation-detection system to a source that is uniformly 

distributed throughout a volume of submerged sand. 

The method is useful for predetermining the amount of 

radioactivity required in experiments in which radioactive tracer particles become distributed throughout 

the bed of an alluvial channel, and undoubtedly it has 

other applications. In the method, the only activity 

required is a single-point source having a strength of 

from 1 to 10 MC (microcuries), which is a license-exempt 

quantity for most radionuclides.

The required experimental data are obtained by 

measuring the counting rates from the point source, 

which is buried at different locations relative to the 

detector in a tank of submerged sand (fig. 42). In the

.Waterline for flushing

-H diameter 

calibration 

tank

FIGUEE 42. Schematic sketch of arrangement of equipment for point-source 

sensitivity calibration.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C45

tank, the vertical source-detector distance, v, is varied 

by moving the source vertically in the standpipe; and 

the horizontal distance, h, is varied by moving the 

detector laterally. Readings, corrected for background, are obtained at about 80 different sourcedetector positions. Because it usually increases toward 

the wall of the tank, the level of background radiation 

is measured at several different lateral positions.

The measured counting rates from each different 

source-detector position are used to define loci of equal 

counting rate (isocounts) about the center of the 

detector crystal. Jsocounts obtained for a 7-juc iridium 

192 source are shown in figure 43; isocounts for gold 198 

and cesium 137 are similar in appearance. In figure 43, 

the isocounts are closely approximated by arcs of circles 

with centers located at the center of the detector 

crystal. Because the approximations are sufficiently 

close, the circular arcs are used to represent the isocounts. Figure 44, which is defined from the approximate isocounts in figure 43, is a plot of counting rate 

per microcurie against radial distance from the center 

of the detector crystal. The curves in figure 44 define 

attenuation functions that reflect the combined effects

of radiation absorption and scatter in the submerged 

sand and of source-detector geometry. The relative 

penetrating ability of the radiation from the different 

radionuclides is shown by the slopes of the curves.

Each attenuation function provides a basis for 

estimating the counting rate, RTo , of a particular 

detection system due to a source that is uniformly 

distributed throughout the volume of submerged 

sand within a radius, r0, of the center of the crystal. 

The estimated counting rate (see fig. 45) is

f(r)dV (23)

where

m=the activity per unit volume (specific activity) in the 

submerged sand

dV ~ 2^(7-2 -ar) dr

f(r) = the counting rate per microcurie from a point source 

of radioactivity located at a distance r from the 

center of the crystal. f(r) plotted against r (fig. 44) 

defines the attenuation function

a=the vertical distance from the surface of the sand to 

the center of the crystal. (For a 2X2-in. crystal, 

a=l in.)

HORIZONTAL DISTANCE FROM SOURCE, IN INCHES

FIGURE 43. Isocounts for a 7-juC iridium 192 point source in submerged sand. Solid line indicate observed isocounts. Dashed line indicates

circular arcs with origin at center of crystal.
C46 TRANSPORT OF RADIONTJCLIDES BY STREAMS

468 

RADIAL DISTANCE, IN INCHES

FIGURE 44. Counting rates per microcurie from point sources located in submerged sand at various radial distances from the center of the detector

crystal.
TRANSPORT, DISPERSION, LABELED BED MATERIAL, NORTH LOUP RIVER, NEBR. C47

Detector

/Water surface

FIGURE 45. Definition sketch for calculation of sensitivity from point-source 

calibration.

By substitution,

RT = (24)

If the sensitivity, S, is defined as the ratio of Rm , 

which is the counting rate due to a source uniformly 

distributed throughout an infinite volume, to the 

specific activity, m, then

(25)

For example, consider the counting rate from an 

infinitely large uniformly distributed source of iridium 

192 as measured with the equipment that was used to 

define figure 43. For iridium 192 (see fig. 44),

/j(r) = Cie-«ir, I in. <r <5.6 in. 

/2 (r) = C2e-«2 r, r >5.6 in. 

where

Cj=1.09X105 cpm per /*c, a^O.85 

C2 =2.43X 104 cpm per M c, «2 =0.58.

By substituting these attenuation functions and the 

appropriate constants into equation 25,

f

1.58X106 cpm per /*c per cu in. = 

per cu ft.

cpm per me

Sensitivities that were computed for gold 198 and 

cesium 137 from their respective attenuation functions 

are listed in table 5, together with the sensitivity for 

iridium 192. Also included in the table are the principal gamma-ray energies for these three radionuclides 

and the energy-range setting of the pulse-height

i.o

0.9

0.8

0.7

0.6

0.5 D 

O 

O

0.4

y 0.3

0.2

0.1

7/

Iridium 192

Gold 198

"Cesium 137

6 8 10 

RADIAL DISTANCE, IN INCHES

12 14 16

FIOURE 46. Relative counting rate from sources uniformly distributed throughout the volume of submerged sand within various radial distances of the center of the

detector crystal.
C48 TRANSPORT OF RADIONUCLIDES BY STREAMS

analyzer that was used for measuring the counting 

rates.

TABLE 5. Sensitivities of three radionuclides

Sensitivity

(cpm per me

per cuft)

9. 15X 10s 

3.86X106 

4.56X105

Principal gamma-ray Measured energy 

Radimudide energies 1 (Mev) range (Mev)

Iridium 192_____ 0.315, 0.470, 0.605 0. 40-0. 74

Gold 198_ ______ .411 .31-0.51

Cesium 137. __._ .662 .56-0.76 

i From Heath (1957).

Figure 46 indicates the variation of Rr /Rm , the 

relative counting rate, with r0 . The figure shows that 

if iridium 192 were uniformly distributed throughout 

an infinite volume of submerged sand, 50 percent of 

the counts would be contributed by activity within 

4.4 inches of the detector and 95 percent would be 

contributed by activity within 9.2 inches of the detector. The figure also shows that the attenuation 

of radiation from cesium 137 is considerably less than 

that from iridium 192 or gold 198.

LITERATURE CITED

Colby, B. R., and Hembree, C. H., 1955, Computations of total 

sediment discharge, Niobrara River near Cody, Nebraska: 

U.S. Geol. Survey Water-Supply Paper 1357, 187 p.

Einstein, H. A., 1937, Der Geschiebetrieb als Wahrscheinlichkeitsproblem [Bed-load transport as a probability problem]: 

Zurich, Verlag Rascher, 110 p.

 1950, The bed-load function for sediment transportation in 

open channel flows: U.S. Dept. Agriculture Tech. Bull. 

1026, 70 p. [1951].

Heath, R. L., 1957, Scintillation spectrometry gamma-ray 

spectrum catalogue: Idaho Falls, Idaho, Phillips Petroleum 

Co., Atomic Energy Div.

Hours, R., and Jaffry, P., 1959, Application des isotopes radioactifs d 1'etude des mouvements sediments et des galets 

dans les cours d'eau et en mer [The application of radioactive isotopes to the study of motion of silt and pebbles in 

the rivers and in the sea]: La Houille Blanche, v. 14, 

no. 3, p. 318-347.

Hubbell, D. W., 1964, Apparatus and techniques for measuring 

bedload: U.S. Geol. Survey Water-Supply Paper 1748, 74 p.

Hubbell, D. W., and Haushild, W. L., 1962, Discussion of 

"Dual channel stream monitor": Am. Soc. Civil Engineers 

Proc., v. 88, no. HY 4, p. 287-291.

Hubbell, D. W., and Matejka, D. Q., 1959, Investigations of 

sediment transportation, Middle Loup River at Dunning, 

Nebraska: U.S. Geol. Survey Water-Supply Paper 1476, 

123 p.

Karaki, S. S., Gray, E. E., and Collins, Jack, 1961, Dual channel 

stream monitor: Am. Soc. Civil Engineers Proc., v. 87, 

no. HY 6, p. 1-16.

Kennedy, V. C., 1963, Mineralogy and exchange capacity of 

modern fluvial sediments, in U.S. Atomic Energy Comm., 

Transport of radionuclides in fresh water systems: U.S. 

Atomic Energy Comm. Rept. TID-7664, p. 71-81.

Korn, G. A., and Korn, T. M., 1961, Mathematical handbook 

for scientists and engineers: New York, McGraw-Hill 

Book Co., 943 p.

Krone, R. B., 1957, First annual progress report on the silt 

transport studies utilizing radioisotopes: Berkeley, California Univ. Inst. Eng. Research, 118 p.

Lean, G. H., and Crickmore, M. J., I960, The laboratory 

measurement of sand transport using radioactive tracers: 

Wallingford, England, Dept. Sci. and. Indus. Research, 

Hydraulics Research Sta., 26 p.

Nace, R. L., 1959, Activities of the United States Geological 

Survey in problems of radioactive-waste disposal Hearings 

before the special subcommittee on radiation of the Joint 

Committee on Atomic Energy: U.S. 86th Cong., 1st sess. 

on industrial radioactive waste disposal, v. 4, p. 2580-2645.

Parzen, Emanuel, 1960, Modern probability theory and its 

applications: New York, John Wiley & Sons, 464 p.

Robeck, G. C., Henderson, C., and Palange, R. C., 1954, Water 

quality studies on the Columbia River: Cincinnati, U.S. 

Dept. Health, Education, Welfare, Robert A. Taft Sanitary 

Eng. Center, 99 p.

Sayre, W. W., Guy, H. P., and Chamberlain, A. R., 1963, 

Uptake and transport of radionuclides by stream sediments: 

U.S. Geol. Survey Prof. Paper 433-A, 35 p.

U.S. Department of Commerce, 1959, Maximum permissible 

body burdens and maximum permissible concentrations 

of radionuclides in air and in water for occupational exposure: Natl. Bur. Standards Handb. 69, 95 p.

U.S. GOVERNMENT PRINTING OFFICE: 1965 O 761-353