THE AERATION CAPACITY 

OF STREAMS 

The Aeration Capacity 

of Streams 

By W. B. Langbein and 'W. H. Durum 

GEOLOGICAL SURVEY CIRCULAR 542 

Washington 1967 
United States Department of the Interior 

STEWART L. UDALL, Secretary 

Geological Survey 

William T. Pecora, Director 

First and second printings 1967 

Free on application to the U.S. Geological Survey, Washington, D.C. 20242 
CONTENTS 

Page 

Introduction ........................ ~ ........................................... . 1 

The coefficient of reaeration .......... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

Other influences on the reaeration of streams . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . 2 

Downstream variations .............. , ............................ , . . . . . . . . . . . . . . . 2 

Regional variations at mean flow ..... ; ........................................... 1 3 

Local variations .................... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

Distribution of assimilative-capacity ... , . . . . . . . . . . . . • . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . 5 

Summary ........................... ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

Selected references .................. , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

ILLUSTRATIONS 

Page 

FIGURES 1-4. Graph .showing1. Relation of velocity and depth to reaeration coefficient . . . . . . . . . . . . 2 

2. Decrease in the reaeration coefficient with increase in size of river, 

Kansas-Missou11i-Mississippi Rivers . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

3. Regional contrast ilil the reaeration coefficient . . . . . . . . . . . . . . . . . . . . 3 

4. Schematic variation in reaeration coefficient in pools and riffies . . . . . . 4 

TABLES 

Page 

TABLE 1. Hydraulic data for Kansas-Missouri-Mississippi Rivers ............... . 3 

2. Hydraulic factors and total assimilative capacity of streams of different 

orders ............ • ........................................... . 5 

iii 

THE AERATION CAPACITY OF STREAMS 

By W. B. Lamgbein and W. H. Durum 

I ntrodudion 

Flowing bodies of water such as streams , have 

the inherent ability to assimilate organic pollution 

from domestic and industrial waste-water discharges, from agricultural runoff, and from VB!rious 

natural sources. The sanitary engineer and the 

hydrologist are searching for ways to define the 

ability of a water body to assimilate waste, the 

proportion of capacity being used at present, and 

the indices by which the capacity can be measured. 

One of these indices is dissolved oxygen, the fuel 

required for destroying organic waste. Although 

there are other factors-such as the amount of 

dissolved solids present, temperature, suspended 

sediment, biological organisms, and the amount of 

flow-the amount of dissolved oxygen is a -q.seful 

measure of the capacity of streams to assimilate 

waste. 

When polluted water is exposed to the air, oxygen 

is absorbed to replace that consumed in the. slow 

combustion of the organic matter. This proc~ss of 

reoxygenation, or reaeration, goes on at a rate that 

is proportional to the deficit of oxygen-that is, 

the difference between the amount of oxygen the 

stream can hold at a given temperature and the 

actual content. 

The Coefficient of Reaeration 

The rate of absorption of oxygen per unit of 

time is often expressed by the simple equation, 

where cis the concentration of oxygen (milligrams 

per liter), Cs is the concentration for saturation at 

the given temperature, t is time (days), al!ld k2 

is the coefficient of reaeration. The subscript 2 

denotes this coefficient as the second coefficient in 

the Streeter-Phelps ( 1925) formulation for the deoxygenation ( k1 ) and reoxygenation ( kz) of 

streams. 

The oxygenation of a stream is a function of the 

biologic, physical, and hydraulic properties of the 

stream. Oxygen may be added by such processes as 

photosynthesis of aquatic vegetation and by mechanical aeration of the flowing water. Oxygen 

may be removed by such processes as vegetal decay 

and plant respiration, as well as by the oxidation 

of pollutants. In flowing streams mechanical aeration may be the dominant factor. 

The ·effect of hydraulic properties of rivers on 

the coefficient of reaeration is usually expressed as 

the coefficient of reaeration, k 2 • There are available 

a few measurements of the coefficient of reaeration 

that indicate a rough sort of relation with the mean 

velocity, v, and depth, H. Two sets of river data 

and two sets of laboratory results plotted in terms 

ofv/H1.ss are shown in figure 1. 

This ratio seems to accommodate both the river 

and the laboratory data referenced and graphed in 

figure 1, which suggest that k 2 = 3.3v I nus. 

The correspondence shown in figure 1 is evidence 

that velocity and stream depth are highly significant factors, although measurements of these factors alone are incomplete estimators of reaeration. 

Other hydraulic properties, such as the occurrence 

of pools and riffles and the degree of meandering, 

also affect the rate of reaeration. 

Other relations have been prepared but none for 

t"l\e set of available data in this simple form. The 

empirical nature of the formula (fig. 1) limits its 

application to the range of data on which it is 

based; fortunately, the range happens to be fairly 

large. 

1 
.,..: 

z 

UJ 

(3 

u: 

1.1.. 

UJ 

8 

z 

0 

i= 

<( 

0:: 

UJ 

<( 

~ 0.1 

X 

EXPLANATION 

• O'Connor and Dobbins ) 

River 

o Churchill, Elmore, and Buckingham 

c,. Krenke! and Orlob ) 

Laboratory 

x Streeter, Wright, and Kehr 

FIGURE 1.-Relation of velocity and depth to reaeration coefficient. 

Other Influences on the Reaeration 

of Streams 

In general, mechanical reaeration is probably a 

dominant factor in shallow, swift streams. In deep, 

sluggish estuaries and lakes, dissolved oxygen is a 

product of photosynthesis of phytoplankton and 

benthic flora (floating and bottom plant life). 

Water temperature affects the reaeration coefficient such that it is somewhat greater in warm 

waters than in cold waters. In normal practice a 

temperature correction is applied to the basic equation. (A 1 °F [0.55°C] change modifies the reaeration coefficient by about 1 percent.) However, since 

this paper is concerned with the influence of the 

hydraulic properties, all coefficients are corrected 

to a common base of 68°F (20°C). 

The reaeration coefficient is also affected, usually 

in an adverse direction, by a pollution load, including sediment, that alters the physical and chemical 

properties of the stream. 

To sum up, the reaeration coefficient of a stream 

is a property of its velocity and depth, and many 

pertinent data are available in the records of the 

U.S. Geological Survey to. examine the extent and 

nature of the variations in this coefficient. 

2 

Downstream Variations 

As it flows downstream, a river increases in its 

discharge and, with the increase in discharge, it 

increases also in velocity and depth. However, as 

shown by Leopold and Maddock ( 1953, p. 16), the 

velocity increases only as the 0.10 power of the 

discharge, whereas the depth increases as the 0.40 

power. Inserting these relations in the formula in 

figure 1 indicates that the reaeration coefficient 

decreases in the downstream direction at the 0.43 

power of the discharge. Thus the reaeration coefficient of large rivers, despite their greater 

velocity, may be less than that of small streams. 

The downstream variation in the reaeration coefficient may be computed from the set of data for 

the Kansas-Missouri-Mississippi Rivers (Leopold 

and Maddock, 1953). As shown in table 1 and 

figure 2, the reaeration coefficient of the lower 

Mississipp' River is only a tenth or less .of that of 

the Kansas River. The data indicate that the reaeration coefficient decreases at about the square root 

of the discharge, in approximate conformance with 

that previously indicated ( 0.43 power of discharge). 
.....-

z 

LIJ 

u 

u:: 

..... 

8 

z 

0 

~ ffi 0.1 <( EXPLANATION' LIJ 

0:: • Kansas River 

o Missouri River 

• Mississippi River 

O.OlL-~L_~~LU~---L~-L~LUL---L-~~LLLU 

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

MEAN DISCHARGE, IN CUBIC FEET PER SECOND 

FIGURE 2.-Decrease in the reaeration coefficient with increase 

in size of river, Kansas-Missouri-Mississippi Rivers. 

Regional Variations at Mean Flow 

The different geomorphic character of rivers is 

reflected in their reaeration coefficients when the 

~ 10 

.....-

z 

LIJ 

(3 

Li: 

..... 

LLJ 

0 

(.) 

z 

0 

~ a:: 

LLJ 

<( 

LLJ 

a:: 

00 

• 

00 

C\:) 

• • 

• 

TABLE l.-Hyilraulic ilata for Kansas-Miss01Jiri-Mississippi 

Rivers 

[From Leopold and Maddock, 1953] 

Mean Mean Locality discharge velocity 

(cfs) (ft per sec) 

Kansas River atOgden, Kans .•.....••••• 

Wamego, Kans •.•..••••• 

Topeka, Kans ••..•.•.•. 

Bonner Springs, Kans ••.. 

Lecompton, Kans 

2,514 

4,114 

4,655 

5,874 

7,838 

1.9 

1.9 

2.1 

1.8 

2.3 

Missouri River atBismarck, N. Dak ...•.. 20,320 2.9 

Pierre, S. Dak •...•.•... 22,080 2.5 

St. Joseph, Mo .....••••• 35,440 3.6 

Kansas City, Mo ........ 43,710 3.4 

Hermann, Mo •..•.•.•.. 69,170 3.0 

Mississippi River atAlton, Ill • . .. .. . . .. .. • 96,670 3.0 

St. Louis, Mo ......••. 166,700 3.8 

Memphis, Tenn .•.....• 454,900 4.6 

Vicksburg, Miss ....... 554,600 5.3 

Mean 

depth 

(ft) 

3.8 

4.1 

4.6 

5.9 

4.6 

6.1 

9.1 

11.5 

11.7 

14.5 

18.6 

28.0 

51.0 

40.1 

1.07 

.96 

.92 

.56 

1.0 

.87 

.44 

.45 

.42 

.28 

.20 

.15 

.073 

.11 

data are compared in graphs similar to figure 2. 

Figure 3 shows a substantial contrast between the 

streams of the Coastal Plain and those of the basins 

of the Bighorn and Powder Rivers in the NorthEXPLANATION 

• Coastal Plain river 

o Basins of Bighorn and Powder Rivers 

0 0 

0 

0 0 

0 

0 0 0 

0 0 0 

0 

• 

• • 

• 

• • • • 

10 100 1000 10,000 • MEAN DISCHARGE, IN CUBIC FEET PER SECOND 

FIGURE 3.-Regional contrast in the reaeration coefficient. 

3 
ern Rocky Mountains region. According to inferences from river depths and velocities, the reaeration coefficient, k2, varies as 15/VQ (where Q is 

mean discharge, in cubic feet per second) in the 

Coastal Plain and as 80/VQ in the Northern Rocky 

Mountains region; other regions have values in 

between. For example, the imputed values of the 

reaeration coefficient in the Appalachian Plateaus 

region are given by 50/VQ. A look at the regional 

variations suggests that river slopes are the dominant factor and point to the obvious but perverse 

fact that the streams of the populated areas-those 

of lesser slopes-have reaeration capacities that are 

low relative to their rates of discharge. 

The values of the coefficient of reaeration for 

smaller rivers derived from this formula seem to 

exceed those customarily used in engineering practice. There may be valid reasons for the aeration 

values of polluted streams to be less than the values 

projected from experimental data. The answer is 

that more field data are needed. 

The textbook classification of rivers for estimation of the reaeration properties does not seem to 

be consistent with their geomorphic properties or 

the factors suggested by the formula. For example, 

in the guide that is commonly reported in the literature (Linsley and Franzini, 1955, p. 502), 

values are given for "sluggish streams" and "swift 

streams," and for "large streams." However, depth 

is a more significant parameter than velocity, and 

"swift" streams shown as having large values of 

the coefficient of reaeration are usually large 

streams which, in turn, are deep and should therefore have low values of the coefficient. 

Local Variations 

As a river rises in response to an increase in 

discharge, it increases its depth and velocity, a 

condition causing the reaeration coefficient to decrease. In general, rivers increase in depth and 

velocity at about the 0.4 power of the discharge. 

Hence, at any given location, the coefficient of reaeration decreases at about the 0.13 power of the 

discharge. 

These are general averages. On alluvial streams 

with shifting beds, the coefficient of reaeration 

changes in rather complex ways. For example, 

over a period of several years, the depths and 

velocities of the Kansas River, as measured at a 

4 

given section at Bonner Springs, Kans., changed 

in the following way: 

Depth Velocity ~:ffici~~~ (ft) (ft per sec) (day-1) 

Low water (riffle) .......... 1.2 1.5 4.1 

Low water ~pool) .......... 4.5 .3 .13 

Mean flow ( riffie) ··········· 2.5 2.2 2.2 

Mean flow (pool) ............ 6.5 .85 .23 

Bankfull ( riffie and pools 

"drowned" out) ........... 20 7 .43 

At low water the reaeration coefficient ranged 

between 0.13 and 4.1; the range is somewhat less 

at mean flow, whereas at bankfull the reaeration coefficient centered about 0.43. It must be 

noted that these changes in depths and velocities 

were measured at a given section. Considering 

the river as a whole, it is possible that the changes 

at one section may be compensated by changes in 

the opposite direction at another section. 

In many rivers the water flow is alternately 

through shallows and deeps-or riffles and pools, 

as they are often called. The contrast between 

riffles and pools is especially marked at low water; 

when the river reaches bankfull, these features 

are said to be "drowned out." Figure 4 shows the 

1 

z 

0 

~ <( 

0:: 

UJ 

<( 

UJ 

0:: 

... 

0 

1-

z 

UJ 

(3 

u:::: 

... 

UJ 

0 

() 

DISCHARGE ---~ 

FIGURE 4.-Schematic variation in reaerotion coefficient in 

pools and riffles. 

schematic variation of the coefficient of reaeration 

with discharge. The relations between velocity and 

depths in these reaches are such that the reaeration coefficient increases slowly with increasing 

stage in the pool, but it decreases rapidly in the 

riffle. 

The general trend of the coefficient of reaeration 

is downward with increasing stage, as remarked 
TABLE 2.-HydrauZic factors and total assimilative capacity of streams of different orders 

Total assimilative 

Average Coefficient Total capacity 1 

dis· Average Average of reaera- length of (Tons per day per River repre- StFeam charge deftth velocity tion streams unit deficiency sentative 

order (cfs) ( t) (ft per sec) (day-1) (miles) in dissolved oxygen) of each order 

1 0.6 . . . . . . . . . . ............ ........... 1,570,000 . .......... ..................... 2 2.8 .......... ........... ' ··········· 810,000 ........... ..................... 3 14 0.55 1.2 9.3 420,000 16,300 ..................... 4 65 .95 1.6 5.5 220,000 19,000 ..................... 5 310 1.8 1.8 2.6 116,000 20,000 Pecos 

6 1,500 2.7 2.0 1.8 61,000 30,000 Shenandoah, 

Raritan. 

7 7,000 5 2.5 1.0 30,000 31,000 Allegheny, Kansas, 

Rio Grande. 

8 33,000 12 3.0 .37 14,000 21,000 Tennessee, 

Wabash. 

9 160,000 25 4.0 .19 6,200 18,000 Columbia, Ohio 

10 700,000 45 5.0 .10 1,800 9,400 Mississippi 

1 Q L k: --, where k2 is the proportion of natural logarithmic units. However, the values of k2 given in the table are in terms of common logarithmic v 2.3 Q L k2 units, hence assimilative capacity equals . Since Q is given in cubic feet per second, L and v in miles, and k2 in days-1, the 'II 

1 QL 

formula used is-- h. The quantity Q L/v is the volume of water in the stream at mean flow. 

2700 f) 

before; but since rapid aeration at riffles is at the 

expense of lesser aeration in the pools, it is less 

evident whether the pool and riffle combination 

is more or less efficient than a relatively uniform 

channel might be. 

Distribution of Assimilative Capacity 

The quantity ~QL/v has the dimension of weight 

per unit of time and represents the assimilative 

capacity of a stream. The formula for the reaeration coefficient k2 can be used with data available 

for river discharges ( Q) , lengths ( L), and velocities ( v). Leopold ( 1962) has listed the number, 

lengths, and drainage areas of streams in the 

United States and has classified them as to their 

order. 

The drainage areas given by Leopold have been 

converted to mean rates of discharge on the assumption of a rate of 0.6 cfs per square mile, the 

national average. Average depths and velocities corresponding to rivers having these discharges are 

then determined from the data given by Leopold 

and Maddock ( 1953). The total lengths of rivers 

are those given by Leopold ( 1962). Pertinent data 

are listed in table 2. 

The coefficients of reaeration are those calculated 

by the formula given previously, and the assimilative capacity for each stream of given order has 

been computed by the formula given in the footnote of table 2. The values in the column for total 

assimilative capacity refer to the total load, in tons 

of oxygen per day, at mean flow that could be 

absorbed from the air by the river system for each 

unit (part per million) of oxygen less than the 

saturation value. This follows the original premise 

in the paper that reaeration occurs at a rate that 

is proportional to the deficit of oxygen. 

One could compute similar values for, say, average low flow for the systems (lower 25-percent 

quartile) when oxygen levels are minimized if the 

hydraulic characteristics of the channel are known. 

However, the main purpose of the computations is 

to give some order of magnitude of the capacity 

of the stream for "reconditioning" itself when 

oxygen-consuming substances are encountered. The 

computations presuppose a synoptic condition, an 

unlikely situation in any total river system which 

is subject to the vagaries of nature and whose regimen is undergoing continuous change by man. 

The results, as shown in the column for assimilative capacity, indicate that, although the total 

assimilative capacity among the several orders is 

roughly of corresponding magnitude, most of the 

assimilative capacity occurs in streams of the sixth 

and seventh orders, not the largest or the smallest. 

The above discussion on hydraulic factors helps 

to explain why large streams that might be sources 

of copious supply are not equally effective in disposing of wastes through self-purification. Because 

population growth tends to develop around large 

rivers or bodies of water, cities in the lower reaches 

of a basin are inherently at some disadvantage in 

view of the relatively low natural assimilative 

5 
capacity of their adjacent streams. Of course, in the 

final analysis, the extent of waste loading in relation to total oxygen available at any given time 

is the balancing factor. 

Summary 

The well-established equation for expressing the 

rate of absorption of oxygen per unit of time in 

water, de/ dt = k2 ( Cs - c), involves the concentration of oxygen, the concentration for saturation 

at the given temperature, the time in days, and the 

coefficient of reaeration, k2 • 

This study examines . in more detail the effects 

of known hydraulic properties of rivers on the 

coefficient of reaeration (k2 ). A few sets of available field and laboratory experimental data indicate that k2 = 3.3v/Ht. 33, where v is mean velocity 

of the stream, in feet per second, and H is the 

mean depth, in feet. 

It is shown that the reaeration coefficient (k2 ) 

decreases in the downstream direction at the 0.43 

power of the discharge; ~ is less for large rivers 

than for small rivers, despite the greater velocity of 

the large rivers. 

The reaeration coefficient is inferred from river 

depths and velocities and varies as 15/VQ ( Q is 

discharge in cfs) in the Coastal Plain streams and 

as 80/VQ in the Northern Rocky Mountains region. 

Thus streams of lesser slopes characteristic of populated areas have reaeration capacities that are low 

relative to their rates of discharge. For example, 

k2 of the lower Mississippi River is only a tenth, 

or less, of that of the Kansas River. 

As the river rises in response to a change in 

discharge, depth and velocity change also, and the 

reaeration coefficient decreases at about the 0.13 

power of the discharge. Where river flow is alternately through shallows and deeps, k2 increases 

slowly with increasing stage in the pool, but it 

decreases abruptly in the riffle. By use of the 

6 

formula k2QL/v to give the dimension of weight 

of oxygen per unit time, computations are made for 

the total load, in tons of oxygen per day at mean 

flow, that could be absorbed from the air by the 

river system for each unit of oxygen less than 

saturation value. Most of the assimilative capacity 

occurs in streams of the sixth and seven order, not 

in the largest or the smallest. 

The above discussion 'of hydraulic factors helps 

to explain why cities in the lower reaches of a 

basin are inherently at some disadvantage in view 

of the relatively low natural assimilative capacity 

of their adjacent streams. 

·selected References 

Churchill, M. A., Elmore, H. L., and Buckingham, R. A., 

1964, Prediction of stream reaeration rates: Am. Soc. 

Civil Engineers Trans., v. 129, p. 24-26. 

Cleary, Edward J., 1966, The reaeration of rivers: Indus. 

Water Engineering, v. 3, no. 6, p. 16-21. 

Krenke!, P. A., and Orlob, G. T., 1962, Turbulent diffusion 

and the reaeration coefficient: Jour. Sanitary Eng. Div., 

Am. Soc. Civil Engineers, v. 88, no. SA 2, p. 53-84. 

Leopold, L. B., 1962, Rivers: Am. Scientist, v. 50, p. 

511-537. 

Leopold, L. B., and Maddock, Thomas, Jr., 1953, The 

hydraulic geometry of stream channels and some physiographic implications: U.S. Geol. Survey Prof. Paper 

252, 57 p. 

Linsley, R. K., Jr., and Franzini, J. B., 1955, Elements of 

hydraulic engineering: New York, McGraw-Hill Book 

Co., 582 p. 

O'Connor, D. J., and Dobbins, W. E., 1958, The mechanism 

of reaeration in natural streams: Am. Soc. Civil Engineers Trans., v. 123, p. 641-684. 

Streeter, H. W., and Phelps, E. B., 1925, A study of the 

pollution and natural purification of the Ohio River: 

U.S. Public Health Service, Public Health Bull. 146, 

75 p. 

Streeter, H. W., ·wright, C. T., and Kehr, R. W., 1936, 

Measures of natural oxidation in polluted stre~ms, III, 

an experimental study of atmospheric reaeration under 

stream flow conditions: Sewage Works Jour., v. 8, 

no. 2, Mar., p. 282-316. 

GPO 927-511