WATER RE56tfRCfiS DIV1B&0* 

Bm- 132 AT IMPORTS SECTION

Response of Gas-Purged 

Manometers to 

Oscillations in Water Level

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1869-E

Prepared in cooperation with the 

California Department of Water 

Resources

to 1m. 132 AT WATSH HESOURCES

REPORTS SECTION

Response of Gas-Purged

Manometers to LIBRARY co 

Oscillations in Water Level

By J. R. BECK and C. R. GOODWIN

RIVER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1869-E

Prepared in cooperation with the 

California Department of fFater 

Resources

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1970
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HIGKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

Washington, D.C. 20402 - Price 20 cents (paper cover)
CONTENTS

Page

Abstract. ________________-__________ ______ El

Introduction. _____--------___-__--_---__-_---__---------_-------- 2

The gas-purged manometer system__________________________--_----- 3

General description ________-____________-______------------- 3

Mathematical analysis.__________________ _________-___------ 5

Relation of subsurface pressure to oscillations of water surface._________ 7

Test program.___________________-__-__.---_-___-_------------ 10

Manometer response to cyclic change in stage _________---_----- 11

Manometer response to abrupt change in stage._______-_--------- 12

Analysis of test results_____________________________________-------- 16

Cyclic input.__-__-___-_______--_---------__--__-_------------ 16

Abrupt imput-________________-__-____-----__-_--__----------- 19

Theoretical response.________________________ _____________ 20

Prediction of error in stage recorded by a bubble gage during oscillations

in water level__-_____-_-________-__-_-___--__----___---------- 22

Summary and conclusions________________-__----___-__-_----------- 22

ILLUSTRATIONS

Figure 1. Diagrammatic sketch of U.S. Geological Survey gas- Page 

purged servocontrolled-manometer._______________ E4

2-10. Graphs showing:

2. Relation of bubble volume to bubble rate. ______ 8

3. Relation of pressure-response factor to depth of

water for various wave frequencies._________ 10

4. Error in mean stage indicated by Exactel Servomanometer and piezometer system for cyclic 

change in stage__________________________ 13

5. Error in mean stage indicated by U.S. Geological 

Survey servocontrolled-manometer and piezometer system for cyclic change in stage, delay 

circuit off_________________________________ 14

6. Error in mean stage indicated by U.S. Geological 

Survey servocontrolled-manometer and piezometer system for cyclic change in stage, delay 

circuit on_______ _________ _____ 15

7. Response rate of Exactel Servomanometer and

piezometer system for abrupt rise in stage____ 16

8. Response rate of U.S. Geological Survey servocontrolled-manometer and piezometer system 

for abrupt rise in stage__.______________ 17

m
IV CONTENTS

FIGURES 2-10. Graphs showing Continued

9. Comparison of theoretical and recorded response 

of the Exactel Servomanometer and piezometer 

system to an abrupt rise in stage___________ E20

10. Comparison of the average theoretical and 

recorded responses of the Exactel Servomanometer and piezometer system to a cyclic change 

in stage_________________________________ 21

TABLES

TABLE 1. Error in stage indicated by Exactel Servomanometer and Page 

piezometer system with cyclic change in stage.________ Ell

2. Error in stage indicated by U.S. Geological Survey servocontrolled-manometer and piezometer system with cyclic 

change in stage_______________________ ________ 12

3. Response rates of Exactel Servomanometer and piezometer system for abrupt rise in stage_ __-_-__-______ 18

4. Response rates of U.S. Geological Survey servocontrolledmanometer for abrupt rise in stage (delay circuit off)__ 18

5. Comparison of theoretical and recorded stage with mean 

surface oscillation for the Exactel Servomanometer and 

piezometer system____ __________________________ 21
RIVER HYDRAULICS

RESPONSE OF GAS-PURGED

MANOMETERS TO OSCILLATIONS

IN WATER LEVEL

J. R. BECK and C. R. GOODWIN

ABSTRACT

This report describes tests conducted to evaluate the performance of bubblegage servo controlled manometers when operating in a pool or stream whose 

water level is oscillating at amplitudes and frequencies that are likely to be found 

at medium or high stages in streams or in larger bodies of water subject to wind 

action. Two types of units were tested an Exactel Servomanometer 1 made by 

the Exactel Instrument Co. and a U.S. Geological Survey servocontrolledmanometer. Piezometer systems used in each case were those used in normal 

field applications.

These tests indicate that gas-purged manometers produce an accurate record 

of water stage under static conditions, but they record a value less than the 

average water-surface elevation when surface waves of significant amplitudes 

and frequencies are present. The test program imposed wave amplitudes of 0.1 

and 0.2 foot and frequencies ranging from 2 to 10 cycles per minute, a range of 

wave conditions that commonly occur at field installations. The magnitude of 

error is shown to be dependent upon the frequency and amplitude of wave action, 

the gas-purge rate, the depth of water over the orifice, the internal volume of the 

manometer system, and the overriding effects of the servomechanism used to 

read the basic gas-purged manometer unit.

The tendency to underregister can be reduced but not entirely eliminated 

by increasing the gas-purge rate. A mathematical analysis is given which is 

verified by the experimental work. This analysis shows how the magnitude of 

underregistration can be predicted if the controlling factors wave frequency 

and amplitude and the physical-system characteristics are known.

The value of these findings is that the presence or absence of registration 

errors can be predicted from a knowledge of the regime imposed on the bubblegage system so that the effects of such errors can be compensated for in subsequent 

analysis.

' Trade name of Exactel Instrument Co,

El
E2 RIVER HYDRAULICS

INTRODUCTION

Since 1958 there has been an increasing use of bubble gages, the 

term commonly used for the gas-purged servocontrolled-manometer 

water-level sensing and recording system used by the U.S. Geological 

Survey. Significant advantages of bubble gages include simplicity of 

installation resulting in lower initial cost, relative ease of changing 

orifice location in unstable channels, and high-salvage value when 

relocating or discontinuing a gaging station. Further, the recorder 

and sensing unit can be located in a gage house some distance away 

from the orifice at sites free from the threat of inundation or bank 

erosion.

Performance of bubble gages has generally been very satisfactory, 

and, because of the advantages noted above, many are now in service. 

However, in some instances anomalies in recorded stage have been 

noted, particularly in situations where short-term wave actions affect 

the hydrostatic pressure at the orifice. The pulsations under consideration here are those caused by minor wave action or oscillatory 

changes in the water surface. Field observations of this phenomenon 

show that the frequency of pulsations typically ranges from 2 to 10 

cpm (cycles per minute). Such surging makes the accurate reading 

of staff gages impractical, and gage observations made under such 

conditions are generally qualified as accurate within a specified 

tolerance perhaps ±0.1 foot.

The purpose of this study was to determine and quantify the 

effects of short-period oscillating water-surface levels on the stage 

recorded by gas-purged manometer systems. Two servocontrolledmanometer systems, an Exactel Instrument Co. Servomanometer 

and a U.S. Geological Survey bubble gage, were tested. System 

responses to both abrupt and cyclic changes in stage were tested 

using two representative lengths of tubing and normally used bubble 

rates. For testing the effects of an abrupt change in stage, step-stage 

changes of 0.2 and 0.4 foot were used; for testing the effects of cyclic 

change in stage, sinusoidal oscillations at frequencies of about 2, 5, 

and 10 cpm and amplitudes of 0.1 and 0.2 foot were used.

This study was made in cooperation with the California Department of Water Resources. All the tests were made by the U.S. Geological Survey in conjunction with the Exactel Instrument Co. of 

Mountain View, Calif., who provided the Servomanometer, shop 

facilities, and assistance.

The test procedures were established by Winchell Smith, Geological Survey hydrologist. The report was prepared under the general 

supervision of Walter Hofmann and R. Stanley Lord, successive
RESPONSE TO GAS-PURGED MANOMETERS E3

district chiefs of the Water Resources Division of the U.S. Geological 

Survey in California, and under the immediate supervision of L. E. 

Young, chief of the Menlo Park subdistrict office.

THE GAS-PURGED MANOMETER SYSTEM 

GENERAL DESCRIPTION

The bubble gage, a gas-purged servocontrolled-manometer system 

(fig. 1), is used by the Geological Survey to record water-surface levels. 

At one end of the system is a length of pressurized tubing that has one 

end submerged in the water whose level is to be recorded. The head of 

water above the tubing orifice establishes a pressure which is reflected 

through the length of tubing into a U-tube manometer at the other end 

of the system. Bottled gas is forced into the manometer system at a 

constant rate causing bubbles to rise in the water from the submerged 

orifice. The input bubble rate required depends primarily on the expected rate of change of stage of the stream and the length of tubing 

between the orifice and the manometer unit.

Mercury in a reservoir at one end of the U-tube is caused to stabilize 

at levels corresponding to the head of water above the submerged 

orifice. The mercury levels thus represent various stages of water and 

can be recorded by continuous analog or digital-chart recorders. 

Mercury levels may be detected by various means. One method, used 

in the survey system, is a float-actuated switch arrangement. As the 

mercury changes position in the reservior, a contact is made to complete a circuit through an electronic control unit to a drive motor, 

which then drives the manometer to a balanced position. The manometer movement is translated to a recorder, thus indicating a change in 

water level. A delay circuit, used in the survey system, can be manually 

switched in or out of operation. When the delay circuit is in operation, 

the manometer drive does not respond to water-level changes for 

about 35 seconds after electrical contact of the float-actuated switch. 

If the float-actuated switch opens within this time interval, no waterlevel change is recorded, thus giving the effect of damping the watersurface oscillations. Manometer response is immediate with the delay 

circuit off.

The Exactel Servomanometer uses a differential transformer to 

detect mercury levels. The transformer consists of moving coils around 

the mercury tubes. A magnetic armature floating on the mercury 

surface moves within the coils; any change in armature position 

causes an unbalanced voltage that is fed to an amplifier. The amplifier 

actuates a motor that moves the coils back to the null position and 

also drives a counter and the water-stage recorder.
Electron

servoconti

unit

P

Rese

Constant-differential

regulator 

Pressurev 

gage and ", 

regulator

FIGURE 1. Diagrammatic sketch of U.S. Geological Survey gas
RESPONSE TO GAS-PURGED MANOMETERS E5 

MATHEMATICAL ANALYSIS

The bubble gage can be visualized as a simple gas-purged manometer which is read by an electromechanical servomechanism. Mathematical analysis of the gas-purged part of this system can be made, 

and an equation relating the internal pressure of the system to elapsed 

time can be developed. Factors controlling the internal pressure are 

the gas-bubble size and injection rate to the system, the temperature 

and density of the gas, the internal volume of the manometer and 

tubing, and the rate of external pressure change imposed. An instantaneous positive step change in stage is most easily analyzed. The fallingstage response of the system is almost instantaneous as far as the 

internal pressure changes are concerned. This is so because no mechanical system regulates the gas exhaust from the orifice, as is done 

with the gas supply. The gas, therefore, escapes very rapidly in 

response to decreasing external pressure at the orifice, and a practically 

constant equilibrium is maintained. For this analysis it has, therefore, 

been assumed that no internal pressure lag occurs in the manometer 

gas system during falling stages.

A differential equation, using the principle of mass conservation, 

is developed below for manometer operation. Rate of mass input of 

to the system (Mf) equals rate of mass output (M0) plus rate of

change in mass storage rr of the system; therefore

Mt=M.+^f- (1)

For a rising stage, no bubbles exit from the orifice, and mass output 

is zero until bubbles appear at the orifice and the inside pressure 

stabilizes. Thus,

M,=f (2)

and

Mi=viriPi, (3) 

where

Vt volume of a gas bubble entering the system, 

rt= bubble rate into the system, and 

Pi=mass density of input gas.

The rate of change in mass storage is equal to the product of the volume 

of the system and rate of density change of the gas with time; therefore

__ 

dt ~dt

where F== total volume of system.

373-i363 tfO 2
E6 RIVER HYDRAULICS

Substituting equations 3 and 4 into equation 2 gives

F. (5)

We now digress and examine the ideal gas law, p= WFil If we gril.

assume constant temperature, we can express the term, n by

a constant of proportionality, k. Furthermore, P may be expressed as 

the sum of Pa and Ps, where: P= total absolute pressure inside the 

system, Pa= atmospheric pressure (33.9 ft of water), and Ps=inside 

pressure to balance outside static head of water. We can therefore 

rewrite the ideal gas law as follows:

(6)

The total derivative of equation 6 with respect to time gives

dpi_, dPs f-^. ~E~k ~dt' (7)

Now, substituting equations 6 and 7 into equation 5 and rearranging 

terms

Equation 8 is a first-order differential equation of the form

&, (9)

where the right-hand side of the equation for a step input is a constant 

and can be solved to give a relation between system pressure, in feet 

(Ps) , and elapsed time, in minutes (£) .

The general solution, y, of this equation is the sum of the complementary solution, yc, and the particular solution, yp, which can be solved 

by the methods for linear equations with constant coefficients and the 

inspection method of undetermined coefficients for nonhomogeneous 

equations (Rainville) .2

thus

(10)

8 Eainville, E. D., 1964, Elementary differential equations: New York, Macmillan Co., p. 11»-143.
RESPONSE TO GAS-PURGED MANOMETERS E7 

where yc is the solution of

ai -j7+a>2y=Q. (11) at

An exponential equation gives the complementary solution, thus

y.=e"'f (12) 

where

 02

m= 

The particular solution for a step input is a constant, thus yv At 

where

A=i (13)

The general solution of equation 8, with the constant of integration 

C, is now

y~Cemt-\-A, (14) 

where

for our tests; therefore, we get for a final equation

Ps=Cemt-Pa. (15)

The constants C and m in equation 15 can be determined for known 

conditions. C can be found by substituting the initial stage at zero 

tune in equation 15, and m can be found at various bubble rates 

from the following equation:

m=-^> Viri fia\ (16)

where v t can be obtained from figure 2. Equation 15 can be plotted 

to indicate pressure in the system (in terms of feet of water) versus 

elapsed tune for a rising stage-step input. Use of the equation is 

discussed on pages E20 and E22.

RELATION OF SUBSURFACE PRESSURE TO OSCILLATIONS OF

WATER SURFACE

A bubble-gage installation senses pressure at the orifice location. 

This pressure may or may not oscillate with the same amplitude as 

oscillations in water-surface level, depending on depth to the orifice
E8 RIVER HYDRAULICS

1.5

o 1.4 

X

uj S13"J

u.

O

§ 1.2 

o

z 

" 1.1

I 1.0

uj 0.9

m 

o

m 0.8 

0.7

40 80 .120 160 200 240 

BUBBLE RATE, IN BUBBLES PER MINUTE

FIGURE 2. Relation of bubble volume to bubble rate.

and the frequency of the surface oscillations. A study of surface 

wave motion and pressures indicated that surface-wave and 

subsurface-pressure profiles occur at virtually the same time (in 

phase) for both regular and slightly irregular wave oscillations. For 

highly irregular wave patterns, pressure and water depth would not 

be in phase for any appreciable length of time.

Folsom 3 discussed the subsurface pressure-response factor, K, 

considering two-dimensional irrotational theory of wave motion. K is 

defined as the ratio of pressure variation at a given point below the 

mean free surface to the equivalent pressure corresponding to a wave 

height. K is equal to a hyperbolic function which includes wave 

length or frequency, depth of water, and depth to the point of interest 

and is expressed as follows

where

s Folsom, B. G., 1947, Subsurface pressure due to oscillatory waves: Am. Geophys. Union Trans., v. 28, 

no. 5, p. 875-881.
RESPONSE TO GAS-PURGED MANOMETERS E9

and where

D=depth to bottom of water, in feet;

Z,=wave length, in feet;

Z= distance to point of interest from surface, in feet; 

AP=pressure variation at a given point; and 

AJ3== equivalent pressure corresponding to a wave height.

For our analysis, the point of interest is the orifice near the streambed. 

Thus, Z=D, which simplifies the numerator of the above equation to 

cosh 0=1. The equation can be further reduced to

by use of the following equalities:

(7==1.11

where

T= period of wave oscillation, in seconds;

/= frequency of wave oscillation, in cycles per second; and

0=32.2 ft per sec.2

When pressure variation at depth is approximately 10 percent of the 

equivalent pressure corresponding to a wave height, we can consider 

the pressure variation to be negligible. Figure 3 is a plot of K values 

for various wave frequencies, in cycles per minute, as related to water 

depths. These curves indicate that pressure variations at the orifice 

will be nearly equal to surface wave heights for depths of water and 

the frequencies of wave motion that are likely to be encountered in 

field installations.

Figure 3 will be used in this report in the discussion of prediction 

of error in stage recorded by the bubble gage during cyclic oscillations 

in water level (p. E22). The tests described on the following pages are 

concerned entirely with response of the bubble gage to pressure 

changes at the gage orifice.
E10 RIVER HYDRAULICS

10 20 30 40 50 

DEPTH OF WATER ABOVE ORIFICE, IN FEET

70

FIGURE 3. Relation of pressure-response factor to depth of water for various

wave frequencies.

TEST PROGRAM

Response tests for both cyclic and abrupt changes in stage were 

made for an Exactel Servomanometer and a survey servocontrolledmanometer. For testing the effect of cyclic change in stage, the 

manometer tubing orifice was placed in a 3-inch-diameter plastic 

cylinder about 2}/£ feet long that contained water, and change of 

stage was accomplished by cycling the orifice vertically at several 

rates by means of a motor and disk crank to generate a sine-wave 

input to the orifice. For testing the effect of abrupt change in stage, 

the crank motor was disconnected and movement was accomplished 

by manually rotating the crank from a minimum to a maximum head 

on the orifice.

The Exactel Servomanometer was tested at the Exactel plant. 

Two Esterline Angus 0-1 milliampere recorders were used, one to 

record the cycling rate of the orifice and the other to record the 

Servomanometer response. Both recorders operated from a potentiometer bridge between the recorder and the Servomanometer. A 

survey float-switch servocontrolled-manometer was available for 

testing at Menlo Park, Calif. Both cycling rate and response were
RESPONSE TO GAS-PURGED MANOMETERS Ell

recorded on a Leupold Stevens A-35 continuous chart recorder driven 

by a motor through a gear train.

Orifice tubing lengths and diameters and sight-glass bubble rates 

were chosen to represent typical field installations. Tubing lengths 

of 50 and 100 feet were used for the survey unit; only the longer 

tubing (100 ft) was used for the Exactel Servomanometer because 

this unit is usually used at installations where large ranges in stage 

are expected. Tubing ID (inside diameter) was % and %6 inch for 

the Exactel Servomanometer and % inch for the survey system. 

Bubble rates of 40, 80, and 160 per minute were used for both 

manometers.

MANOMETER RESPONSE TO CYCLIC CHANGE IN STAGE

The stage oscillations were in form of a sine wave and are referred 

to in this report as a sine-wave input. Cycling rates of about 2, 5, 

and 10 cpm and amplitudes of 0.1 and 0.2 foot were selected for these 

tests. These wave amplitudes and frequencies were used for the tests 

because they typify oscillations which often occur in field installations. 

Tests conducted with the Exactel Servomanometer were limited to 

amplitudes of 0.1 foot. Both 0.1- and 0.2-foot amplitudes were used 

for the survey servocontrolled-manometer tests. Testing of the survey 

unit was done first with the delay circuit off and repeated with the 

delay circuit on. Results of these tests are summarized in tables 1 

and 2 and are plotted in figures 4-6; conclusions are discussed in the 

section on analysis.

TABLE 1. Error in stage indicated by Exactel Servomanometer and piezometer 

system with cyclic change in stage

Bubbli 

(bubbli 

mini;

Error in recorded stage (foot of water)

as per 2 cycles per minute

Minimum

Maximum

Tube length, 100 feet;

40.......

80.......

160.......

........ -0.082

........ -.059

........ -.003

-0. 097 

-.095 

-.089

Tube length, 100 fee t;

40......

80.......

160. ...

........ -0.067

_ AC7

........ -.028

-0.094 

-.091 

-.093

Mean

5 cycles per minute

Minimum

inside diameter Vi

-0.089 

-.077 

-.046

-0. 089 

-.077 

-.059

Maximum

Mean

10 cycles per minute

Minimum

Maximum

Mean

i inch; wave amplitude, 0.1 foot

-0. 097 

-.094 

-.091

inside diameter 3/16 inch;

-0.080 

-.074 

-.060

-0.075 

-.071 

-.059

-0. 091 

-.092

-.104

-0.093 

-.085 

-.075

-0.092 

-.082 

-.066

-0.097 

-.089 

-.081

-0. 095 

-.085 

-.073

wave amplitude, 0.1 foot

-0.083 

-.082 

-.082

-0. 075 

-.069 

-.058

-0.095 

-.091 

-.085

-0. 085 

-.080 

-.072
E12 RIVER HYDRAULICS

TABLE 2. Error in stage indicated by U.S. Geological Survey servocontrolledmanometer with cyclic change in stage

Error In recorded stage (foot of water)

Bubble Delay circuit off Delay circuit on rate 

(bubbles 2 cycles per minute 5 cycles per minute 10 cycles per minute 2 5 10

per cycles cycles cycles 

minute) Mini- Maxi- Mini- Maxi- Mini- Maxi- per per per 

mum mum Mean mum mum Mean mum mum Mean min- min- minute ute ute

Tube length, 100 feet; inside diameter, l/& inch; wave amplitude, 0.1 foot

40.

80.

160.

-0.06 -0.10 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.08 -0.05 -0.04 -0.06

-.02 -.06 -.04 -.06 -.06 -.06 -.07 -.07 -.07 0 -.04 -.05

+.04 -.03 0 -.01 -.03 -.02 -.03 -.03 -.03 0 0 0

Tube length, 100 feet; inside diameter, % inch; wave amplitude, 0.2 foot

40- 

80- 

160.

-0.15 -0.19 -0.17 -0.15 -0.17 -0.16 -0.16 -0.16 -0.16 -0.12 -0.15 -0.13

-.11 -.18 -.14 -.13 -.16 -.14 -.15 -.15 -.15 -.15 -.10 -.14

-.06 -.12 -.09 -.10 -.12 -.11 -.14 -.14 -.14 -.05 -.08 -.09

Tube length, 50 feet; inside diameter, % inch; wave amplitude, 0.1 foot

40- 

80. 

160-

-0.04 -0.10 -0.07 -0.05 -0.07 -0.06 -0.06 -0.07 -0.06 -0.07 -0.04 -0.05

0 -.07 -.04 .05 -.05 -.05 -.05 -.05 -.05 0 -.04 -.04

+.03 -.04 0 0 -.03 -.02 -.03 -.03 -.03 0 0 -.05

Tube length, 50 feet; inside diameter, Vs inch; wave amplitude, 0.2 foot

40. 

80- 

160-

-0.12 -0.19 -0.16 -0.15 -0.16 -0.16 -0.15 -0.17 -0.16 -0.15 -0.13 -0.18

-. 10 -. 18 -. 14 -. 11 -. 13 -. 12 -. 14 -. 15 -. 14 -. 17 -. 15 -. 13

-.04 -.12 -.08 -.08 -.09 -.09 -.12 -.12 -.12 0 -.08 -.12

MANOMETER RESPONSE TO ABRUPT CHANGE IN STAGE

An abrupt change in stage (a step input) was applied to each 

system in order to determine maximum response rates. The magnitude 

of the step input for the Exactel servomanometer test was 0.2 foot, 

and for the survey servocontrolled-manometer tests, 0.2 and 0.4 

foot. These amplitudes were adequate for comparison with computed 

response rates. Tests were made only for a rising stage because the 

internal pressure of the manometers responds with negligible lag to a 

falling stage. The absence of appreciable lag on falling stage was 

verified by observation of the rapid, but short-lived, rush of bubbles 

from the orifice when the stage was abruptly lowered followed by the 

almost immediate return to the normal exit bubble rate. During such 

a rapid pressure decrease, the recorded gage height is controlled 

primarily by the response of the servomechanism. The Exactel 

servosystem reacts almost instantaneously, thus providing a close
RESPONSE TO GAS-PURGED MANOMETERS E13

replication of the actual internal-system pressure. The survey system 

is limited by the traverse rate of the drive motor and consequently 

records an abrupt pressure decrease as a relatively slow decrease 

equal to the motor traverse rate. For these tests the survey unit 

was tested with the 35-second delay circuit off rather than on, because 

the delay circuit inhibited manometer response to pressure changes in 

the system and precluded determination of its maximum response 

rate.

Results are given in tables 3 and 4 and are plotted in figures 7 and 8; 

they are discussed in the section on analysis.

-0.02

-0.04

-0.06

-0.08

uj- -0.10

100 feet of ^-inch-ID tubing 

cpm (cycles per minute)

2cpm

Q

Z -0.02

-

0.04

-0.06

-0.08

-0.10

100 feet of 3/16-inch-ID tubing

2 cpm

40 80 120 160

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE

WAVE AMPLITUDE = 0.1 FOOT

200

FIGURE 4. Error in mean stage indicated by Exactel Servomanometer and 

piezometer for cyclic change in stage.
E14 RIVER HYDRAULICS

-0.10

-0.20 

§0

-0.10

-0.20

DELAY CIRCUIT OFF

10 cpm

100 feet of i/8-inch-ID tubing 

cpm(cycles per minute)

50 feet of Vs-inch-ID tubing

10 cpm

40 80 120 160

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE

WAVE AMPLITUDE=0.1 FOOT

200

O -0.10

<

100 feet of ^-inch-ID tubing

5 cpm

_EZ n.

-0.10

-0.20

50 feet of '/g-inch-ID tubing 5 cpm

40 80 120 160

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE

WAVE AMPLITUDE=0.2 FOOT

200

FIGURE 5. Error in mean stage indicated by U.S. Geological Survey servocontrolled-manometer and piezometer system for cyclic change in stage, 

delay circuit off.
RESPONSE TO GAS-PURGED MANOMETERS E15

-o.io

-0-20

D " 

Z (- 

 UJ

o:2 O ~

-0.10

-0.20

DELAY CIRCUIT ON

100 feet of i/8-inch-ID tubing

cpm(cycles per minute)

10 cpm

50 feet of '/g-inch-ID tubing

40 80 120 160

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE

WAVE AMPLITUDE = 0.1 FOOT

200

S -o.io

-0.20

10 cpm 

100 feet of Vs-inch-ID tubing

-0.10

-0.20

10 cpm 

50 feet of Vg-inch-K) tubing

40 80 120 160

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE

WAVE AMPLITUDE=0.2 FOOT

200

FIGURE 6. Error in mean stage indicated by U.S. Geological Survey servocontrolled-manometer and piezometer system for cyclic change in stage, delay 

circuit on.
E16 RIVER HYDRAULICS

uJ 0.4

OUJ

I 2

i! s 

ssP Q.

0.3

Step input, 0.2 feet 

Tubing, 100 feet Vs-inch ID

.6-inch ID

40 80 120 160 

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE

200

FIGURE 7. Response rate of Exactel Servomanometer and piezometer system 

for abrupt rise in stage.

ANALYSIS OF TEST RESULTS 

CYCLIC INPUT

Minimum, maximum, and mean stages were obtained from the 

recorded response traces at the various cycling and bubble rates. 

The orifice was stopped at its extreme and midpoint positions to 

obtain the static readings. The mean recorded stage for each test was 

compared to the static midpoint, and differences were plotted in 

figures 4-6. For an amplitude of 0.1 foot, the error in recorded stage 

ranged from about 0.09 foot of water at a bubble rate of 40 per minute 

to 0.05 foot of water at a rate of 160 per minute with the Exactel 

Servomanometer, and from about 0.08 to 0 foot of water with the 

Survey system. When the amplitude was doubled to 0.2 foot in the 

Survey system test, the range of error was approximately doubled 

0.17-0.08 foot of water. Little difference was noted in the results 

between tests using 100 feet of tubing and those using 50 feet of 

tubing.

These errors were in all cases negative that is, the recorded gage 

height was less than the true mean gage height when cyclic variations 

in stage were imposed on the system.

Cycling tests were run using the survey system with the delay 

circuit both on and off, as shown in the data of table 2. The orifice 

was set at midposition, and the manometer was stabilized at this 

elevation piior to each cycling run. When the orifice was cycled, the 

manometer float needle bounced back and forth between its contacts, 

and, with the delay circuit on, did not stay on each contact long
RESPONSE TO GAS-PURGED MANOMETERS E17

0.5

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0.1

100 feet of '/4-inch ID tubing

50 feet of J/fe-inch-ID tubing

40 80 120 160 200

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE 

STEP INPUT=0.2 FOOT

50 feet of %-inch-ID tubing

100 feet of Vi-inch-ID tubing

40 80 120 160 200

SIGHT-GLASS BUBBLE RATE, IN BUBBLES PER MINUTE 

STEP !NPUT=0.4 FOOT

FIGURE 8. Response rate of U.S. Geological Survey servocontrolled-manometer and piezometer system for abrupt rise in stage.
E18 RIVER HYDRAULICS

TABLE 3. Response rates of Exactel Servomanometer and piezometer system for

abrupt rise in stage

Rate of recovery Time to stabilize (minutes)

per minute) First bubble Computed from 

rate of recovery

Tube length, 100 feet; inside diameter, Vs inch; 0.2-foot step input

40

80

160

-_-__--_--__-----_-___--______ 0.039 4.95

________-___----___-_-_.__.___- .093 1.99

_________ ____ __________ _ __ .226 .77

5.26 

2. 15

.88

Tube length, 100 feet; inside diameter, 3/16 inch; 0.2-foot step input

40

80

160

_--____-...-..-_-____________._ 0.032 5.33

____.-_-_ _____ ______________ .064 2.18

_________ _ ___________________ .173 .92

6.25 

3. 13 

1. 16

TABLE 4. Response rates of U.S. Geological Survey servocontrolkd-manometer and 

piezometer system for abrupt rise in stage (delay circuit off)

Rate of recovery Time to stabilize (minutes)

(bubbles per minute) per minute) First bubble Including Computed from 

hunting rate of recovery

Tube length, 100 feet; inside diameter, Vs inch; 0.2-foot step input

40

80

160

__.__---_ _ _ __ __ 0.056 2.8 2.8

__ ___ ___ __ _ .18 1.0 1.0

___________ ___ .__ .43 .4 2.6

3. 6 

1. 1

.5

Tube length, 100 feet; inside diameter, Vs inch; 0.4-foot step input

40

80

160

___________________ 0.037 7.4 7.4

_ ________________ .16 2.0 3.1

______ __ __________ .40 .7 1.8

10. 8 

2.5 

1.0

Tube length, 50 feet; inside diameter, Vs inch; 0.2-foot step input

40

80

160

0.11 1.5 1.5

_______-. _ ___ __ .24 .8 1.3

_____ _ - ______ - .38 .3 2.5

1. 8 

.8 

.5

Tube length, 50 feet; inside diameter, Vs inch; 0.4-foot step input

40

80

160

___________________ 0.12 2.7 3.3

._ _ _ __ ______ .40 1.0 1.0

.40 .6 3.2

3. 3 

1.0 

1. 0
RESPONSE TO GAS-PURGED MANOMETERS E19

enough to overcome the delay time (about 35 seconds). After about 

5 minutes of cycling time, the recorded trace came to reasonable 

stability and trace-position values were obtained. Bouncing of the 

float needle depends upon point-gap spacing. If the points are close 

together, there is greater oscillation of the recorder trace; if they are 

farther apart, there is less oscillation of the trace. Total gap between 

points for test purposes was one-sixteenth inch. Results obtained for 

the survey system with the delay unit on seem to be anomalous in 

that the recording error, on some of the runs, tended to increase 

when the bubble rate was raised from 40-80 bubbles per minute and 

decrease, as would be expected, when the gas feed was increased to 

160 bubbles per minute. The reason for this is not understood, but the 

significant facts are that the net recording error is of the same sign 

and general magnitude as that encountered with the delay circuit off 

and that this error can be decreased by an increase in gas-purge rate.

ABRUPT INPUT

The rate of response to an abrupt rise in stage was obtained by 

measuring the slope of the response trace resulting from an increasingstage step input to the orifice. This was done for various bubble 

rates. Response rates are plotted in figures 7 and 8. Rates for the 

Exactel Servomanometer increase slowly with increased bubble rate 

because of the large volume of the total system; those for the survey 

system seem to vary, depending on whether the trace is increasing 

continuously or in intermittent steps and at low bubble rates, and 

are significantly greater for the shorter length of tubing. Response 

rates for the Exactel unit are more nearly equivalent to the pressure 

changes in the gas system and are not limited by the servo-operation 

because the servomanometer reacts very quickly. Rates for the 

survey system are controlled by a combination of pressure changes 

and the motor traverse rate. The maximum rate of rise of the servocontrolled-manometer trace was about 0.25 fpm (feet per minute) 

using 6-volt batteries and about 0.40 fpm using a 7%-volt power 

supply and was dependent primarily on the voltage applied and motor. 

With the fast bubble rate (160 bubbles per minute), the drive motor 

ran continuously, indicating that the motor could not keep up with 

the changing pressure in the system. In addition, a period of hunting 

was sometimes observed during stabilization. The Exactel Servomanometer was observed to follow accurately the pressure trend, 

and little or no hunting resulted.

The time required for the Exactel unit to stabilize was controlled 

by the purge rate and the time required for the relatively large internal 

volume of the manometer system (manometer tubes and piezometer
E20 RIVER HYDRAULICS

tubing) to pressurize to the new stage condition. Because the survey 

system had less volume, it did not require as much time to stabilize 

pressure. However, because of the slower drive motor, the time 

necessary for the switch to make contact (which is dependent on 

point-gap setting), and the hunting period required, the overall 

stabilization tune was increased. Operation of the survey system does 

not usually follow reproducible patterns; thus, table 4 shows internalpressure stabilization times determined by timing the first appearance 

of a bubble at the orifice, overall time including hunting periods, and 

time computed from the rate of recovery.

THEORETICAL RESPONSE

The slope of the derived step-input response equation (eq 15), 

which is plotted in figure 9 for the Exactel Servomanometer, establishes a maximum rate of pressure recovery in the Servomanometer 

system for rising stages. The curve in figure 9 approximates a straight 

line for the times shown and is used as such in the analysis. Any 

rate of rise in stage which equals or exceeds this maximum rate of 

pressure recovery will, therefore, produce the maximum response 

rate. For the sine-wave input used in the experiment, the rate of rise 

during the rising-stage section of the waves was in all cases greater 

than that required to produce the maximum response. During falling 

stages, the observed response was so rapid that it was considered to 

be immediate for the purpose of this analysis. Minor errors due to 

fluid friction in the gas-purged tube and dynamic effect of the servomechanism system itself were not included in this simplified analysis.

i.o

0.9

°

0.7

100 feet of '/g-inch-ID-tubing 

Bubble rate, 40 bubbles per minute

01234

TIME, IN MINUTES

FIGURE 9. Comparison of theoretical and recorded response of the Exactel 

Servomanometer and piezometer system to an abrupt rise in stage.
RESPONSE TO GAS-PURGED MANOMETERS E21

A theoretical response curve to a sine-wave input can be obtained 

by plotting the sine-wave input at the orifice using an amplitude 

equal to the lower excursion from static midpoint and then superimposing on the plot the theoretical maximum rising-stage response 

from figure 9. This has been done in figure 10 for a typical run. From 

the plot in figure 10, the theoretical average stage can be obtained. 

That stage can be compared with both the mean of the surface 

oscillations and the recorded mean stage. The comparison is made in 

table 5.

(£.

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100 feet of Va-inch-ID tubing

Bubble rate, 40 bubbles per minute

Orifice cycling rate, 1.6 cpm (nominally 2 cycles per minute) " r\ r\ /"

/ \ i \ . \/lnput (sine wave) y ^-amplitude X sine wt

i I

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^^\ V < ^^*^^\ j ? s^x^^O.732 theoretical average

\~l^ \-*^ ^0.728 indicates average

(obtained from recorder data)

1 1 1

TIME, IN MINUTES

FIGURE 10. Comparison of the average theoretical and recorded responses of 

the Exactel Servomanometer and piezometer system to a cyclic change in

TABLE 5, Comparison of theoretical and recorded stage with mean surface oscillation 

for the Exactel Servomanometer and piezometer system

2 cycles per minute 5 cycles per minute 10 cycles per minute

(bubbles Mean stage (foot of water) Mean stage (foot of water) Mean stage (foot of water)

Surface Theo- Recorded Surface 

retical

Theo- Recorded Surface Theoretical retical

Recorded

Tube length, 100 feet; inside diameter. % inch; wave amplitude, 0.1-foot

40_-___- 0.811 0.732 0.728 0.794 0.725 0.725

80_____ . 807 . 745 . 741 . 793 . 731 . 733

160______ .813 .775 .772 .797 .746 .743

0. 799 0. 723 0. 723

. 800 . 734 . 733

. 804 . 746 . 745
E22 RIVER HYDRAULICS

Response to a step input applied to the survey system orifice is 

limited by the relatively slow motor speed of the servosystem. The 

recorder response rate, therefore, does not always reflect the pressure 

in the gas system, but it may be indicative only of motor speed. 

Thus, a response trace such as that shown by the Exactel Servomanometer and piezometer system hi figure 9 cannot be obtained 

with the survey servomanometer. A step-stage increase of 0.4 foot 

combined with a bubble rate of 40 bubbles per minute was the only 

condition under which the stage trace for the survey manometer 

closely followed the pressure in the system. The rate of recovery for 

this run was 0.037 fpm, and the average recorded stage for a sinewave input of 0.2 amplitude was 0.08 foot below water-level mean 

stage. These values compare favorably with the slope and error for 

the Exactel Servomanometer when it is used with the gas-purged 

system as shown in figures 9 and 10.

PREDICTION OF ERROR IN STAGE RECORDED BY A BUBBLE 

GAGE DURING CYCLIC OSCILLATIONS IN WATER LEVEL

Error in recorded stage can be predicted by use of figure 3 and 

equation 15, if the depth of water over the orifice and the amplitude 

and frequency of the cyclic oscillations at the water surface are known. 

Figure 9 shows that the theoretical response computed from equation 

15 closely matches the actual response for an Exactel system.

To predict error in response to water-surface oscillations of known 

amplitude and frequency, enter figure 3 with the depth of the orifice 

below water surface (abscissa) and the known frequency of the oscillation and obtain the ratio of orifice to water-surface pressures, k. 

From this ratio and the known amplitude at the surface, compute 

the effective amplitude at the orifice. The frequency of pressure oscillation at the orifice and at the surface will be approximately the same 

for all but highly irregular wave patterns. A plot can be made similar 

to figure 10 showing orifice pressure-head oscillations and the theoretical response computed from equation 15. The predicted error in 

stage is the difference between the mean of the surface oscillations and 

the mean of the theoretical response graph.

This method holds strictly true for the Exactel unit, but it must 

be modified slightly when applying it to the Survey unit. If the 

computed rate of pressure rise or the indicated rate of fall exceeds 

the motor traverse rate, the motor traverse rate must be used instead.

SUMMARY AND CONCLUSIONS

The stage indicated by a bubble gage is a reading equivalent to 

the internal pressure in the gas-purged part of the system. Under 

static conditions this internal pressure is an accurate analog of the
RESPONSE TO GAS-PURGED MANOMETERS E23

water stage, but when oscillatory pressure variations occur at the 

system orifice in response to short-period wave action, the average 

internal pressure of the system will not be equal to the average pressure 

head at the orifice. This is so because such systems adjust rapidly to 

decreasing pressures (falling stage periods) and adjust slowly at a 

rate controlled primarily by the purge rate to increasing pressures 

(rising stage periods). The net effect is that bubble gages do not 

record the mean of the surge, but they underregister by a variable 

amount when subject to wave action.

The dominant factors controlling the magnitude of registration 

errors at a given installation are the size and frequencies of the waves 

and the purge rate of gas flowing into the system. Other factors, 

which may vary with time or from system to system, are the depth 

of water over the orifice, the internal gas-filled volume of the manometer itself, and the length and size of tubing used to connect with the 

orifice.

Under adverse conditions, where the purge rate is less than 80 

bubbles per minute, where the depth of water over the orifice is 

less than 50 feet (as is usual with the survey unit), and where wave 

frequency is greater than 5 cpm, the gage height recorded may 

approach the trough of the waves. The magnitude of underregistration may be reduced by increasing the purge rate, but the laboratory 

tests herein reported still show significant errors at a purge rate of 

160 bubbles per mniute.

The significance of these registration errors must be assessed in 

relation to the characteristics of the body of water involved and the 

purpose for which the stage is being recorded. State records on lakes 

and reservoirs are generally used to document the exact elevation of 

the water surface. Registration errors hi the records may consequently 

have considerable importance. For lake gages, the errors due to wave 

action would probably occur at random times coincident with periods 

of high wind. The magnitude of errors would be less at installations 

where wave frequency was high, and the gage orifice was placed hi 

deep water. In rivers the magnitude of wave action sometimes 

referred to as surge is a frequently repetitive phenomenon closely 

associated with the stage and discharge. At low stages, streams are 

typically calm with little or no wave action, whereas at high stages 

considerable turbulence develops and a consistent characteristic 

pattern of surge is apparent. If the primary purpose of the stage record 

on such a stream is for use in the computation of discharge records, 

then registration errors will probably be compensated for in the 

development of the stage-discharge relation and they will be of no 

significance whatsoever.
E24 RIVER HYDRAULICS

The significant findings of this study are that gas-purged servocontrolled-manometer systems (bubble gages) do tend to indicate a 

stage value less than the mean of the surge when surface waves are 

present. In some places these errors are negligible, or they may be 

compensated for in the manner in which the record is used. This 

report suggests a technique whereby the magnitude of errors may be 

predicted; but, unless the wave characteristics are actually known, 

minor errors must be expected. A knowledge of this system characteristic should improve the quality of interpretation of stage records 

obtained by use of a bubble gage and permit corrective steps to be 

taken in places where real problems exist.

U.S. GOVERNMENT PRiNTING OFFICE: IS70