A Practical Field Technique 

For Measuring Reservoir 

Evaporation Utilizing 

Mass-Transfer Theory

By G. EARL HARBECK, JR. 

STUDIES OF EVAPORATION

GEOLOGICAL SURVEY PROFESSIONAL PAPER 272-E

Evaporation from many reservoirs can be computed 

from records of reservoir stage, wind speed, 

humidity, and water-surf ace temperature

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962
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 25, D.C.
CONTENTS

Page Page

Abstract_________________________________________ 101 Evaporation-seepage measurements.__________________ 102

Introduction...____________________________________ 101 Mass-transfer coefficients.----.-.-------------------- 103

Mass-transfer theory________________________________ 101 Conclusions_________-___-____-________-_---_____--_ 104

Energy-budget studies____.__________________________ 102 References cited ___________________________________ 104

ILLUSTRATIONS

FIGTJBE 30. Relation between fall in stage, AH, of Deep Creek Reservoir No. 3 near Placid, Tex., and the product Page 

iiv(eo—ea) ------------------------------------------------------------------------------------- 103

31. Relation between mass-transfer coefficient, N, and reservoir surface area________________________________ 104

32. Representative wind profiles over grass and water surfaces___-___-___---_---_-----_-~------------------- 105

in

STUDIES OF EVAPORATION

A PRACTICAL FIELD TECHNIQUE FOR MEASURING RESERVOIR EVAPORATION

UTILIZING MASS-TRANSFER THEORY

By G. EABL HAEBECK, JR.

ABSTRACT

Studies of evaporation made in recent years nave provided 

values of the mass-transfer coefficient, N, in the equation 

E=Nut(e0—ea) for reservoirs having surface areas ranging 

from 1 to nearly 30,000 acres. The apparent correlation of N 

with reservoir surface area may in large part be associated with 

variations in the shape of the wind profile near the surface 

resulting from differences in surface roughness.

It appears that evaporation from many reservoirs can be 

determined with acceptable accuracy with a fairly simple system of instrumentation, data processing, and analysis.

INTRODUCTION

The active participation of the Geological Survey 

in studies of evaporation from water surfaces began 

with the Lake Hefner study (U.S. Geol. Survey, 1954). 

The principal objectives of that study were to test the 

validity of the energy-budget and mass-transfer theories for the measurement of evaporation from lakes and 

reservoirs. The energy-budget method is based upon 

an accounting of all incoming and outgoing energy, 

the residual being the energy available for evaporation; 

the Lake Hefner study showed that this method is 

suitable for measurement of evaporation for periods of 

10 days or longer. The mass-transfer method relates 

the exchange of water vapor between a water surface 

and the atmosphere on the basis of measurements of 

certain related parameters; the study showed that certain mass-transfer equations gave results of acceptable 

accuracy.

During the Lake Mead study (Harbeck and others, 

1958) the energy budget was the basic method used. 

A simple quasi-empirical equation previously derived 

for Lake Hefner was found to give satisfactory results on an annual basis at Lake Mead, a much larger ' C7

lake, although there was a pronounced seasonal variation in the results, owing presumably to the effects of 

atmospheric stability. The seasonal variation could be 

minimized by using wind speeds measured near the

surface of the lake. Two of the mass-transfer equations that gave acceptable results at Lake Hefne~ did 

not prove satisfactory at Lake Mead, nor did another 

previously untested equation.

Following the Lake Mead study, the energy-budget 

method has been used to determine evaporation from 

many lakes in the United States. A simpler method 

based upon mass-transfer theory, which was also successful, will be described in a section to follow. The 

results of all these determinations indicate that a practical and reasonably accurate method of determining 

reservior evaporation is now available if certain conditions can be met.

MASS-TRANSFER THEORY

A complete description of mass-transfer theory is 

beyond the scope of this report. A simple example of 

the same type as the Lake Hefner quasi-empirical 

equation is

E~Nu(e0 -ea}i (1) 

in which E= evaporation, in inches per day;

#=a coefficient of proportionality, hereafter

called the mass-transfer coefficient; 

«=wind speed, in miles per hour, at some 

height above the water surface; a numerical subscript, if used, indicates the 

height in meters;

e0=saturation vapor pressure in millrTS, 

corresponding to the temperature of the 

water surface;

ea=vapor pressure of the air, in millibars; a 

numerical subscript, if used, indicates 

the height in meters.

Nearly all the mass-transfer equations to be found in 

the literature have one thing in common: evaporation 

is considered to be proportional to the product of the

61819»—-62 101
102 STUDIES OF EVAPORATION

wind speed, u, times the vapor-pressure difference,

CQ—ea. In a few equations, the wind speed, u, has an 

exponent, usually less than unity.

The mass-transfer coefficient, N, represents a combination of many variables in the published mass-transfer equations. Among these are the manner of the 

variation of wind with height, the size of the lake, the 

roughness of the water surface, atmospheric stability, 

barometric pressure, and density and kinematic 

viscosity of the air. In some of the mass-transfer 

equations these variables are combined into extremely 

complicated mathematical expressions. For some of 

them a direct solution is impossible, and indirect methods must be used. As none of the many published 

equations that were tested gave acceptable results under 

all conditions, the emphasis in the Geological Survey 

was placed upon the use of the energy-budget method 

for the measurement of evaporation, and further 

studies of mass-transfer theory were abandoned.

ENERGY-BUDGET STUDIES

After the Lake Mead study the energy-budget 

method was used to measure evaporation from selected 

reservoirs in the West, principally in Texas, Arizona, 

and California. The procedure was usually as follows: 

A group of 3 or 4 vicinal reservoirs for which evaporation data were desired was selected. The basic requirement was that net incoming radiation measured at one 

centrally located radiation station could be considered 

representative of the radiation received at all reservoirs. 

Thermal surveys to determine the changes in energy 

storage were made at each reservoir at regular intervals 

(usually about 1 month). The amount of energy 

brought into each reservoir in inflow and removed in 

outflow was computed. A raft supporting an anemometer and a water-surface temperature recorder was 

moored at the approximate center of each reservoir.

During a calibration period of a little more than 1 

year (it was considered desirable that the period include 2 summers), evaporation was determined using 

the energy-budget method for each reservoir. During 

this period conterminous records of wind speed and 

water-surface temperature were obtained at each raft 

and the vapor pressure of the air was obtained from 

meteorological data recorded at the radiation station 

or from U.S. Weather Bureau records at a nearby station. Then the mass-transfer coefficient, N9 was obtained by dividing the energy-budget evaporation, E, 

by the product u(eg— ea). The aim was to determine N 

for each reservoir so that the expensive radiation equipment could be moved to another group of reservoirs. 

Evaporation could be computed on a continuing basis 

for each of the first group of reservoirs using the experimentally determined mass-transfer coefficient, 

windspeed, water-surface temperature and humidity 

of the air.

A so-called "standard" raft was designed using 

empty steel drums to provide buoyancy. The anemometer was approximately 2 meters above the water 

surface. Because of practical considerations the standard raft design was not used at a few reservoirs. For 

example, at one large reservoir high waves were expected frequently, so that a larger-thr.n-standard raft 

was used, and because the lake was extensively used for 

boating, the raft was moored near the shore where it 

would not be as dangerous to navigation as if it were 

anchored in midlake. At another reservoir, no observer was available to visit the raft weekly and change 

the water-surface temperature chart; therefore, watersurface temperatures were measured at the edge of the 

lake and recorded on the radiation recorder. Wind 

speeds were measured on shore also. Although the 

mass-transfer coefficients thus determined were suitable 

for obtaining continuous records of evaporation at 

these reservoirs, they were later found to be in poor 

agreement with the coefficients based en data obtained 

at the standard midlake rafts.

EVAPORATION-SEEPAGE MEASUREMENTS

A simple technique for measuring sewage and evaporation losses from small lakes and reservoirs was first 

described by Langbein, Hains, and Culler (1951). The 

two basic assumptions of the method are (1) during 

periods of no surface inflow or outf ow, the fall in 

reservoir stage is composed of two parts, evaporation 

and seepage; and (2) when the product ufa—ea) is 

zero, evaporation is negligible.

In most instances, restriction of the- analysis to periods of no surface inflow or outflow is preferable, 

though not necessary. If inflow and outflow are 

measured, the observed change in stage can be adjusted 

accordingly, but ordinarily even srnall errors in 

measuring inflow and outflow may mrke the adjusted 

change in stage of questionable accuracy.

The second assumption is reasonable because, in the 

absence of wind and vertical convect'on, evaporation 

proceeds only by molecular diffusion, an extremely slow 

process. Even with convection resultirg from substantial differences between air and water temperature, 

evaporation in the absence of wind is quite small, as 

shown by an analysis by Marciano and Harbeck (1954) 

of an equation developed by Yamamoto (1950). Also, 

if the second term of the product u(e0—ea) is zero, the 

air is saturated and evaporation is zero.

The method is simple to use. A totalizing anemometer is mounted at a height of 2 meters above the water
RESERVOIR EVAPORATION, MASS-TRANSFER THEORY 103

surface on a standard raft anchored in midlake. A 

spring-driven or battery-powered water-temperature 

recorder designed for either 7-day or 28-day operation 

is also mounted on the raft. An auxiliary marginal 

pen records units of wind movement; the size of the 

unit may be varied to suit local conditions. A tippingbucket rain gage is operated in conjunction with the 

water-stage recorder; each increment of rainfall is recorded by a pen marking in the margin of the waterstage recorder chart. Periods of rainfall may thus be 

excluded from the analysis by scanning the chart. The 

beginning and end of each period should be chosen so 

that reservoir stages at these times are accurately denned. Times when the recorder chart indicates the 

occurrence of wind-induced surges or oscillations of the 

water surface should be avoided. Winds need not 

necessarily be light during the entire period; in fact, 

some periods having windy spells within them should 

be selected purposely so that as wide a range as possible 

of the product u(e0 —ea) is obtained. The periods may 

be of variable length as long as the same units are used. 

The change in stage, AH, is usually computed in feet 

per day, and average values of wind speed and vaporpressure difference are computed for the same period. 

Data obtained at Deep Creek Eeservoir No. 3 near 

Placid, Tex., are shown in figure 30. It is apparent 

that the Y intercept of a least-squares line would not 

differ from zero by a statistically significant amount, 

which indicates that the net seepage is zero. The slope

< 0.03

a:

LUQ.

0.02

LU 

O

0.01

25 50 75 100 125

PRODUCT „,(,,_ o »*• >« »>^ TM »°"«

2V 6. IN MILLIBARS

FIGUEH 30.—Relation between fall In stage, AH, of Deep Creek Reservoir 

No. 3 near Placid, Tex., and the product «s(eo-e«).

of the best fitting line is N, the mass-transfer coefficient. 

This method has been used to determine N for many 

small ponds in Texas. Some were flood-detention r-^servoirs for which area and capacity curves were available. 

Some, however, were small ponds in south Texas that 

were used only for evaporation-suppression studies and 

therefore were not mapped. Surface areas of these 

ponds were estimated.

MASS-TRANSFER COEFFICIENTS

The mass-transfer coefficients obtained in the course 

of the energy-budget studies and the evaporation-seepage studies are considered to be comparable. In both 

studies, the wind speeds at a height of 2 meters above 

the surface and the water-surface temperature were 

measured in midlake. Air temperature and humidity 

were measured at a site usually several miles away 

from the lake, so that the vapor pressure thus competed 

is considered to be representative of the natural humidity of the air unmodified by passage over a body 

of water.

For Lake Hefner, the vapor pressure measured at the 

8-meter level in midlake was used in the previously 

mentioned quasi-empirical equation (Marciano and 

Harbeck, 1954, eq 58). The authors concluded that the 

upper limit of the vapor blanket was at a heigH of 

about 8 meters above the surface in midlake. The 

vapor pressure at 8 meters could therefore be considered 

to represent the water-vapor content of the unmodified 

air at Lake Hefner. In this equation the wind speed 

used was that recorded at 8 meters. The wind s^eed 

was also recorded at 2 meters and the average ratio 

of the two, w8/^2, was 1.237 (Harbeck and others, 1958, 

table 10). In order that the mass-transfer coefficient 

given in equation 58 (Marciano and Harbeck, 1954) 

could be compared with other coefficients, the units were 

changed and the 8-meter wind speed was replaced by 

the 2-meter wind speed on the basis of the above r^.tio.

At Lake Mead, both wind speed and vapor pressure 

at the 2-meter level were used in a similar equation 

(Harbeck and others, 1958, eq 20). Because Boulder 

Basin of Lake Mead is much larger than Lake Heftier, 

the upper limit of the vapor blanket was believed to be 

probably above 8 meters, but no data were obtained 

above that level. The vapor pressure at 8 meters was 

not considered to be representative of the unmodified 

air, but at least it was closer to it than the 2-meter 

vapor pressure was. Accordingly, equation 20 was 

modified by changing units and by replacing tie 2- 

meter vapor pressure with the 8-meter vapor pressure, 

using the ratio far-eO/far-*)«=!. 105 (Harbeck and 

others, 1958, table 10).
104 STUDIES OF EVAPORATION

The mass-transfer coefficients obtained from energybudget and evaporation-seepage studies are shown in 

figure 31. The coefficient for Lake Head was considered representative of Boulder Basin (29,000 acres 

at the time of the study) rather than of the entire lake, 

as Boulder Basin, where the meteorological data were

100,000

10,000

CO

UJ

a: 

o

UJ

a:

o 

<

U_ 

CC

ZD 

CO

cr 

O 

o:

UJ 

CO 

UJ

a:

1000

100

0.001 0.002 0.005 

MASS-TRANSFER COEFFICIENT, N,

0.01

FIGCBK 31.—Kelation between mass-transfer coefficient, N, and reservoir 

surface area.

obtained, is connected to the main par*; of the reservoir 

by only a narrow channel.

Part of the scatter about the best fitting line is attributable to physiographic differences between reservoirs, particularly exposure to wind. Consider two 

nearby small ponds having the same surface area, one 

in flat terrain with no shoreline vegetation and the 

other surrounded by hills and dense shoreline vegetation. The wind profile over the 2 ponds will not be the 

same even though the windspeed at s^me height, such 

as 8 meters, may be the same. As tin standard error 

of estimate of the regression shown in figure 31 is approximately 16 percent, the coefficient N should be 

determined for each reservoir individually, but the 

basic relation illustrated in figure 31 should serve to 

prevent gross errors.

The apparent significant correlation between N and 

reservoir surface area warrants further study. It was 

stated earlier that on an annual basis the mass-transfer 

coefficients for Lake Hefner and Lake Mead agreed 

closely, which would indicate no correlation between 

N and surface area, at least for surface areas of between 2,000 and 30,000 acres. Wine1 speeds used in 

this equation (Marciano and Harbeok, 1954, eq 58) 

were measured at 8 meters, however, not at 2 meters, 

and the ratio u&/uz for Lake Hefner w*<3 quite different 

from the ratio for Lake Mead. The rHio Us/u* is proportional to the wind shear resulting from surface drag. 

Consider a very small pond in a forested area. The 

time of passage over the pond is too short to permit 

significant modification of the wind profile, and the 

surface drag is therefore representative of the upwind 

terrain, not the pond surface. Consider also a water 

surface having an infinite fetch and uniform wind 

conditions. Surface drag will approach but never reach 

zero, because some energy is expended in maintaining 

wave form and motion. At Boulder Basin of Lake 

Mead the ratio u^/uz was 1.145, and at Lake Hefner it 

was 1.23T (Harbeck and others, 1958, table 10). The 

area of Boulder Basin was 29,000 acres which is equivalent to a circular area having a radius of approximately 

4 miles. Similarly, the equivalent rach'us of Lake Hefner was approximately 1 mile.

If a pond is so small that it has no appreciable effect 

on the wind profile, the ratio u%/Uz could be about 1.375, 

which corresponds to a wind profile that might be 

observed over a field of long grass or perhaps a wheatfield. These three wind ratios were used as a basis for 

the profiles shown in figure 32.

Wind speeds at 8 meters were assumed to be the same 

over all 3 surfaces, and the wind speed^ at other heights 

are shown relative to the 8-meter wine1 for the 3 different surfaces. The 2-meter wind at Lake Hefner is 12
RESERVOIR EVAPORATION, MASS-TRANSFER THEORY 105

Long grass 

— — — Lake Hefner 

Lake Mead

0.5 1.0

RATIO OF WIND SPEED AT INDICATED 

HEIGHT TO WIND SPEED AT 8 METERS

•""• 32.—Representative wind profiles over grass and water surfaces.

v:.nt less than the 2-meter wind at Lake Mead, so 

, the mass-transfer coefficient for Lake Hefner 

i i be greater than for Lake Mead. The 2-meter 

3 over a grassfield is about 24 percent less than at 

e Mead. The increase in mass-transfer coefficient 

B Lake Mead to a small pond is more than 24 per- 

;, and it is likely that other factors are involved, 

x as the decrease in evaporation with distance 

T wind. Results of investigations agree that evapoon. rates should decrease downwind. There is no 

y Tient as to the manner of variation, however, and 

^ therefore concluded that the apparent relation

-"•>en N and surface area illustrated in figure 31 is

in. part to the use of wind data at the 2-meter level

;n part to the downwind decrease in evaporation

rates. In other words, the apparent relation between 

N and surface area is due in part to the relation between 

the ratio us/u2 and the surface area.

Measurement of the wind speed at the 2-meter level 

in midlake is important, if the coefficient N is to be in 

reasonable agreement with the curve shown in figure 31. 

If wind speeds are measured at some other location, 

such as on shore, the coefficients thus determined cannot 

be expected to agree with those on figure 31. Wind 

speeds measured at heights of 2 meters on shore will 

be substantially less than those measured at the same 

height in midlake.

CONCLUSIONS

Expansion of the present program of evaporation 

measurement in the United States is practicable. The 

evaporation-seepage method can be used at a reasonable 

cost. The data required are wind speed at the 2-meter 

level and water-surface temperature in midlake, vapor 

pressure of the unmodified air, and change in lake stage. 

The change in stage is then separated into its two 

components, seepage and evaporation. A mass-transfer 

coefficient, N, can be determined for each reservoir for 

use in the equation E—Nu?. (eQ —ea ).

The energy-budget method, which requires much 

more expensive equipment and an elaborate dataprocessing and analysis procedure, can be reserved for 

measurements of evaporation from the larger lakes and 

reservoirs, or from any reservoirs for which it may be 

impracticable to use the mass-transfer method because 

inflow and outflow cannot be measured with sufficient 

accuracy.

REFERENCES CITED

Harbeck, G. E., and others, 1958, Water-loss investigations— 

Lake Mead studies: U.S. Geol. Survey Prof. Paper 298.

Langbein, W. B., Hains, 0. H., and Culler, R. 0., 1951, Hydrology of stock-water reservoirs in Arizona, progress report : U.S. Geol. Survey Circ. 110.

Marciano, J. J., and Harbeck, G. E., 1954, Mass-transfer studies, 

in Water-loss investigations—Lake Hefner studies, technical report: U.S. Geol. Survey Prof. Paper 269, p. 46-70.

U.S. Geological Survey, 1954, Water-loss investigations—Lake 

Hefner studies, technical report: U.S. Geol. Survey Prof. 

Paper 269.

Yamamoto, G., 1950, Investigations of evaporation from pans: 

Am. Geophys. Union Trans., v. 31, p. 349-356.

o