Computer Model for Determining Bank Storage 

At Hungry Horse Reservoir, 

Northwestern Montana 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 833 

Prepared in cooperation with the 

Bonneville Power Administration 

Computer Model for Determining Bank Storage 

At Hungry Horse Reservoir, 

Northwestern Montana 

By T. H. THOMPSON 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 833 

Prepared in cooperation with the 

Bonneville Power Administration 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1974 
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary 

GEOWGICAL SURVEY 

V. E. McKelvey, Director 

Library of Congress catalog-card No. 73-600313 

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

Washington, D.C. 20402- Price 65 cents (paper cover) 

Stock Number 2401-02485 
CONTENTS 

Page 

Abstract -----------------------------···········-·················---·············-----------·· 1 

Page 

Computation of bank storage.................................................... 4 

Introduction ................................................................................ 1 Computing bank storage from historical data................ 5 

Acknowledgments ································---·····-----------------·--· 1 Computing bank storage from assumed elevations ...... 5 

Model development ···-···---······-·-····-···--·-···---·········---·············--·-·· 1 Other model configurations ............ -------------------------------------------- 6 

Assumed conditions ·············--··------------·-··--·······--·-······-··-···· 2 

Equations developed by Rorabaugh................................ 2 

Equations used in program................................................ 3 

Model sensitivity ·······························--------------------------------··-······ 7 

Summary and conclusions ·················----------·---------------------------- 7 

Selected references ·······-------------------------------------------------·-······---- 8 

ILLUSTRATIONS 

Page 

FIGURES 1-7. Graphs showing: 

TABLE 

L Comparison of monthly accumulated volumes as computed by the digital model with those computed 

by Simons' and Rorabaugh's model and water-budget residuals ------·-·········-··-····-····-··-----------------···············- 10 

2. Daily reservoir elevations, yield from bank storage, and cumulative bank-storage volumes for the period 

December 1, 1959, through March 31, 1960 ................................................ ---------------------------------·--········-·---------- 11 

3. Cumulative reservoir draft versus cumulative yield from bank storage for the period December 1, 1959, 

through March 31, 1960 ............ ----···---······--·················--·-············------·-·································--·-·········-····--·--·---------- 12 

4. Model sensitivity to changes in aquifer width .... ---·--·-·-----·----·--·············--·····-·····-····--·-··----·-··------------------------------·· 13 

5. Model sensitivity to changes in slope or reservoir bed .......... ·-························-·-··-··---------------------------·----------------- 14 

6. Model sensitivity to changes in aquifer transmissivitY---·······------------------------------------------------------------------------------ 15 

7. Model sensitivity to changes in aquifer storage coefficient .............................................. ----------------·------------------- 16 

TABLES 

Page 

1. Cumulative monthend volumes of bank storage computed by the digital model and by Simons' and 

Rorabaugh's simplified model and the accumulated water-budget residuals ··········----········---------------------------------- 17 

2. Cumulative monthly bank-storage volumes for water years 1952 through 1972 ........... ·--·-·------------------------------------------- 17 

3. Projected monthend critical rule-curve elevations for Hungry Horse Reservoir for power years 

1971-72 through 1980-81 .......................................................... ---····--·············--·--·--·-·····-··---------------------------------------------- 17 

4. Cumulative volumes yielded from bank storage computed by using projected critical rule-curve elevations 

for power years 1971-72 through 1980-SL.................................................................................................................. 17 

m 

COMPUTER MODEL FOR DETERMINING BANK STORAGE 

AT HUNGRY HORSE RESERVOIR, NORTHWESTERN MONTANA 

By T. H. THOMPSON 

ABSTRACT 

A mathematical model to compute bank storage at Hungry 

Horse Reservoir, in northwestern Montana, was developed by 

Simons and Rorabaugh (1971). The model was programmed 

for solution on a digital computer, using daily reservoir elevations as an input parameter. Monthly accumulated bank-storage volumes for the period October 1951 through September 

1972 were calculated. 

According to computations using projected rule-curve elevations, an estimated 5.8 percent of the usable reservoir storage 

volume would be available from bank storage for at-site power 

generation and downstream benefits if the reservoir was subjected to a long-term, cyclic drawdown. 

Several model configurations, other than that recommended 

by Simons and Rorabaugh (1971), were evaluated. However, 

the accuracy of the solution using the recommended configuration is within the accuracy limits of the input parameters. 

The model sensitivity was evaluated by comparing the results 

of the model before and after a parameter was changed a fixed 

percentage while holding the other parameters constant. The 

model is most sensitive to changes in aquifer width on an annual 

basis and to changes in the storage coefficient on a seasonal 

basis. 

The computer model can be used to compute bank-storage 

volumes whenever historical or assumed elevations are available for Hungry Horse Reservoir. The parameters can be 

changed to estimate bank-storage volumes at other reservoirs 

having similar geologic, physiographic, and hydrologic conditions. 

INTRODUCTION 

The hydrology of Hungry Horse Reservoir, in northwestern Montana, was the subject of an extensive study 

by Simons and Rorabaugh ( 1971). One of the objectives of their study was to determine accurately the 

amount of water available for the generation of power 

at Hungry Horse Dam and for subsequent downstream 

uses. An apparent imbalance between inflow and outflow at Hungry Horse Reservoir led them to develop a 

water budget, which included a bank-storage component. Bank storage is defined by Lohman and others 

(1972, p. 12) as the amount of storage change in an 

aquifer resulting from a change in stage of an adjacent surface-water body such as a reservoir or river. 

Simons and Rorabaugh (1971) developed a mathematical model to compute the volume and rate of flow to 

and from bank storage that would be independent of 

the water-budget computations of bank storage. 

The computation of bank storage by hand using the 

mathematical model developed by Simons and Rorabaugh is tedious and time consuming. The solution of 

these equations is greatly facilitated by use of a digital 

computer. The purpose of this report is to describe the 

development of a computer program to calculate the 

volume of bank storage at Hungry Horse Reservoir. The 

equations and techniques of the model developed by 

Simons and Rorabaugh are briefly described. The equation and parameters used for the digital-computer solution of bank-storage volume are also described. The 

sensitivity of the mathematical model to changes in 

each of the parameters in the equations has been tested 

and the results are presented in this report. 

ACKNOWLEDGMENTS 

This study is part of the continuing program of 

hydrologic investigations in the Columbia River basin 

undertaken by the U.S. Geological Survey and the 

Bonneville Power Administration. The current study 

was requested and financed by the Bonneville Power 

Administration. 

Valuable assistance and data were furnished by W. D. 

Simons and M. I. Rorabaugh of the Geological Survey 

and Mr. Douglas Hanlon of the Bonneville Power 

Administration. 

MODEL DEVELOPMENT 

The theoretical equations for flow of water to and 

from reservoir bank storage were derived beginning 

with the analogy between heat flow and ground-water 

movement with similar boundary conditions. Readers 

interested in the derivation of these equations may 

refer to Simons and Rorabaugh (1971, p. 35-38). 

1 
2 BANK STORAGE, HUNGRY HORSE RESERVOIR, MONTANA 

ASSUMED CONDffiONS 

The following assumptions about the reservoir and 

adjacent aquifer are made. The applicability and reliability of the theoretical method for solution of bankstorage volume depend upon how closely the geometric 

and hydrologic conditions of the reservoir and aquifer 

being studied match these assumptions (Simons and 

Rorabaugh, 1971, p. 34): 

1. The unconsolidated material adjacent to the reservoir is the principal aquifer in which bank 

storage takes place. 

2. The aquifer shape can be described as two identical wedges separated by the reservoir. 

3. There is a free hydraulic connection through the 

vertical planes forming the boundaries between 

the reservoir and aquifer. 

4. The vertical planes forming the landward boundaries between the aquifer and the consolidated 

rocks flanking the valley are parallel to the 

reservoir boundaries and are impervious to all 

flow. 

5. The planes forming the bottom and the downstream ends of the aquifer are impervious to 

all flow. 

6. There is no vertical recharge to the aquifer. 

7. Material is homogeneous and isotropic. Initially, 

the ground-water level is everywhere at initial 

reservoir level. 

EQUATIONS DEVELOPED BY RORABAUGH 

The equations that follow were developed by M. I. 

Rorabaugh for computing bank-storage volume (Simons and Rorabaugh, 1971, p. 38). The terms of the 

equations (dimensions in parentheses) are also from 

their report. 

Symbol Description 

a Distance from reservoir to valley wall (L) 

C Rate of change of reservoir level == dh/ dt 

(LT-1) 

e 

L 

m 

s 

T 

t 

Napierianlogbase == 2.71828 ... 

Length of reservoir pool at beginning of any 

change (L) 

Slope of reservoir bed (dimensionless) 

Coefficient of storage of the aquifer ( dimensionless) 

Transmissivity of the aquifer (L'T- 1) 

Time from beginning of change in reservoir 

level (T) 

Duration of a constant rate change in reservoir level (T) 

Symbol Description 

V Volume gain ( + ) or loss ( - ) of water in the 

aquifer (L9) 

a -rr'T/4a'S (T-1) 

For constant rate of drawdown or fill, 

{[ 

a2S 1 8 ~ e-<2n-t> 2at] 

V=2LCaSt 1- 3Tt +at ,.. 

1 (2n-1)• 

Lm 2 3Tt + 15 Tt _ _0_[_.!_ _ a2

S _!_( a 2S) 2 

-(!)" ~ %;1 c;::~·;:J}· (1) 

For Tt/a2S ~ 0.2, equation 1 can be approximated 

by 

(2) 

For small time, Tt/ a 2S < 0.2, equation 1 can be 

approximated by 

4 [Tt( 2Ct) V = 2LCaSt . ay;r~(i2S 1- 5Lm . (3) 

For drawdown or fill followed by stationary level, 

Vt~t 1 =2LCaStt{ 1- Ct1 +_!_ ._!_ 2Lm at1 '71"2 

[ 

~ e-<2n-t) 2

at ( Ct1) 

L..J (2n-1) 6 - 1- Lm 

n=1 

00 

""'"' e -<2n-t> 2 ac t-t t > Ctt 1 

n L..J = +-·- 1 ---.('=2n---::-1"'") -=-6 - Lm at1 

t e-,,.._,, •• ,_e_,,._,,., • .,_,,,]} . (4) 

n=1 (2n-1)e 

ForTt/a2S ~ 0.2 and T(t-tl)-a2S ~ 0.1, equation 

4 can be approximated by 

V1, 11 =2LCaSt,{ 1- ~ +[ 1+ f; · a~, 

_ ( 1_ Ct1 + Ct1 __ 1_)e +'lt1 JL .__!_ trtt}. (5) 

Lm Lm at1 at1 ~ 

For early times, Tt/a2S < 0.2, equation 4 can be 

approximated by 
MODEL DEVELOPMENT 3 

4 { /Tr( 2Ct) Vt~tt =2LCaS. sy.; t '\} a2S 1- 5Lm 

- ( t-t ) I T (t-tl) [ 1- 2C ( t-t1) J 1 '\}. a2S 5Lm 

+ (t-t1 ) T(t-tt) ( Ctt)}· (6) 

a2S Lm 

ForTt/a2

S > 0.2 and T(t-t1 )/a2

S < 0.1, 

· [ Ct a2S Ct a2S Vt~t 1 =2LCaSt 1- 2Lm- 8Tt+Lm ·art 

_ _!_. Ct (a2S) 2 +_!_( 1 + Ct ._!_)~e-at J 

15 Lm Tt at Lm at 7r2 

_ 2LC S __ 4_ (t-t) /T(t-t1) [ 1 2C(t-tt) 

a sy:;- 1 '\} a2S 5Lm 

_ Ct1 J. (7) Lm 

The calculation of bank-storage volume by equations 

1 and 4 was too cumbersome for hand calculation. 

Therefore, by using equations 2, 3, 5, 6, and 7 and 

with the aid of mathematical and graphical shortcuts, 

averaged reservoir elevations, and rates-of-change 

trends, bank-storage volume was solved by Simons and 

Rorabaugh ( 1971) on a monthly basis. 

EQUATIONS USED IN PROGRAM 

Equation 4 was selected to be programmed for computer solution, using daily changes in reservoir elevations and solving for volume on a daily basis. Under 

this condition the term t 1 == 1. Rearranging terms, 

V==2aS(LC- c

2

) +I j LC 

2m n==1l 

[ 

1 8 (1-e(2n-1)2a)] 

2aS------- a 71"2 (2n-1) 4 

c2 1 8 [e(2n-1)2a 

+-(2aS--) 

m a 71"2 (2n-1) 4 

1 (1-e(2n1)

2

a)]} -(2n-1) 2at 

+- e • (8) a (2n-1) 6 

Solving for volume in cubic feet per second per day, 

where a is in feet, T is in square feet per day, t = 

number of days, let 

A01 ==2aS/86400==aS/43200, (9) 

2aS 1 8 A02 == ----==0.81057 A01;a, (10) 86400 a 71"2 

A (2n-1}2a (

11) 

1n==e ' 

and 

then 

Let 

A - 2aS _!_~{1-e(2n-1)2a} 

2n- 86400 a 71"2 (2n-1) 4 

=A (1-A1 ) 02 n 

(2n-1) 4 ' 

2aS 1 8 

A =----

3n 86400 a 71"2 

[

e (2n-l)'a +.!._ (l-e (2n-l)'a)] 

(2n-1) 4 a (2n-1) 6 

A A 1n 1 2n 

=Ao2 (2n-1) 4 +-:- (2n-1) 2; 

V=A01 { LC--C'} + I oo 

2m n==1 

(12) 

(13) 

(14) 

E 0 ==Elevation of the base of the reservoir (ft) 

E 1 ==Elevation of reservoir at beginning of t th day (ft) 

E 2 ==Elevation of reservoir at end of t th day (ft) 

C ==Changeinstageduringtth day=E2 -E

1 

(ft) 

(15) 

L ==Reservoir length at beginning of t th day= 

(E0 -E1)tm (ft) (16) 

c2 

X ==LC--

t 2m 

Y ==LC 

t 

z ==c2 

t m 

(17) 

(18) 

(19) 

then 

V-A X I j[y A +Z A ]A -tl (20) - 01 t + n== 1l t 2n t 3n 1n ~ · 

When t==1, 

Vt=l=AolXl + n i 11[ Y1A2n +Z1A3n }ln -lf. 
4 BANK STORAGE, HUNGRY HORSE RESERVOIR, MONTANA 

At t==2 the volume includes the cumulative effect of 

the changes for day one plus the change for day two, 

thus 

At t==3, 

at t==k, 

k k 00 

V =IAX+I I t==k t==l 01 t t=l n==l 

Let 

k k 

vt=k == ti 1Ao1xt + ti 1 

-(k-t+l) 

(YtA21 +ZtA31) All 

k -(k-t+l) 

+ ti 1(YtA22+ZtA32) Al2 

k -(k-t+l) 

.+ ti 1 (YtA23 +ZtA33) A13 

k (Y A +Z A ) A -(k-t+l) 

+ · · · + t I 1 t 29 t 39 19 

k -(k-t+l) 

+ t I 1 (YtA210 +ZtA310) AllO . 

(22) 

k 

AO == t I 1 AOlX t; (23) 

k -(k-t+l); 

Al = ti 1 (YtA21 +ZtA31) All (24) 

k -(k-t+l); 

A2 = t I 1 (YtA22 +ZtA32) Al2 (25) 

(Equations for A

3 

through A

9 (26-32) not included) 

then 

(34) 

where V is the total cumulative bank storage volume 

t 

at timet. Documentation providing a detailed descrip-

{ (Y A +Z A ) A - (k-t+l)}· 

t 2n t 3n ln 

tion of the computer program and its use is in a report 

(21) by Thompson (1972). 

Equation 21 was solved using from 5 to 30 terms of 

the infinite series for various a values. For a's greater 

than 0.007, 10 terms are sufficient for the n series for 

an accuracy well within the accuracy of the measurable 

00 10 

parameters. Assuming I ~ I , then the volume at 

n==1 n==1 

day t==k is approximated by 

COMPUTATION OF BANK STORAGE 

The following dimensional constants used in the computer program for computing bank storage at Hungry 

Horse Reservoir are the same as used by Simons and 

Rorabaugh (1971, p. 38): 
COMPUTATION OF BANK STORAGE 5 

m== - 0.00293 

3057.5- (reservoir elev at start of tth day) ft 

L==--------------- -0.00293 

a ==851 feet 

Simons and Rorabaugh (1971, p. 46-53) described 

how several sets of storage and transmissivity values 

were tested in attempting to find a satisfactory single 

solution of the model. In comparing the computed bank 

storage with the water-budget residual, they concluded 

that a single solution would not be adequate. On a 

seasonal basis, the slow response of bank-storage yield 

to changes in reservoir stage is needed, but on a short 

time basis a quick response is needed. They concluded 

that a combination of two sets of hydrologic constants, 

allowing both a quick and slow response to reservoir 

elevation changes, would best describe the aquifer configuration at Hungry Horse Reservoir. The two sets of 

aquifer constants selected to calculate volumes were 

storage (S) as 0.15 and transmissivity (T) as 326 ft2 

per day (called model 2) and S as 0.2 and T as 3,710 

ft2 per day (called model 4). The results for each set 

are then combined after multiplying each by a weighting factor as follows: 

Volume==0.8 model2+0.32 model4. (35) 

The aquifer constants and weighting factors were 

originally selected on the basis of the nature and distribution of coarse and fine sediments in the aquifer adjacent to Hungry Horse Reservoir and then adjusted on 

the basis of a comparison with the values computed by 

the water-budget method, where 

Bank storage==outflow-inflow+change 

in reservoir storage. ( 36) 

The outflow includes the measured flow below the dam, 

adjusted for local inflow, and computed evaporation 

from the reservoir surface according to the energy budget method; the inflow consists of both the gaged and 

estimated ungaged surface flow to the reservoir plus 

precipitation on the reservoir as measured at the dam. 

COMPUTING BANK STORAGE FROM mSTORICAL DATA 

The same hydrologic constants, weighting factors, 

and model configuration as used in the model recommended by Simons and Rorabaugh (1971) were used 

with daily changes in stage for the period October 1964 

through April 1967 to evaluate equations 22 through 

35 on the digital computer. The accumulated volumes 

were totaled on a monthly basis and are shown in figure 

1 and table 1, along with the monthly totals as solved 

by Simons and Rorabaugh ( 1971) with the waterbudget method ( eq 36) and the hand solution of equations 2, 3, 5, 6, and 7. The results of the computer 

solution of the model are slightly less than the computations using simplified techniques. These differences 

may be due to the use by Simons and Rorabaugh of the 

approximating equations and their assuming straightline hydrographs between breakpoints rather than using 

daily changes in reservoir stage. 

Some of the differences between either of the models 

and the water budget shown in figure 1 during periods 

of snowmelt'and reservoir filling can be explained as a 

result of vertical flow to bank storage. Since the model 

theory assumes horizontal flow from the reservoir to 

the aquifer, the actual movement to bank storage as 

terraces are submerged is larger than calculated. This 

early filling of the aquifer leaves less room for later 

months. During periods of snowmelt on the exposed 

reservoir bed, the water enters the aquifer and is 

delayed in reaching the reservoir, whereas the water 

budget includes this volume of water as part of the 

precipitation on the reservoir surface. Also, channel 

storage between the main inflow gaging station and the 

head of the reservoir pool and bank storage in this 

channel is neglected, which accounts for some of the 

differences between the models and the water budget. 

Further discussion of errors can be found at the end of 

the section "Other Model Configurations." 

Daily changes in bank-storage volume, as well as 

cumulative volumes, may be computed by the model 

on the digital computer. By using daily changes in stage 

at Hungry Horse Reservoir for the period October 1951 

through September 1972, daily changes in bank storage 

and cumulative volumes were computed with equations 

22 through 35 as previously· described. The accumulated monthly bank-storage volumes from this computation are shown in table 2. 

The effects of step drawdown of the reservoir surface 

on changes in bank storage are illustrated in figure 2 

and 3. During the period December 1, 1959, through 

March 31, 1960, Hungry Horse Reservoir was drawn 

down in a series of steps from an elevation of 3,560.5 

feet to an elevation of 3,533.0 feet. 

Hydro graphs of daily reservoir elevations, daily cumulative bank storage volume, and daily yield from bank 

storage for the period December 1, 1959, through 

March 31, 1960, are shown in figure 2. Daily cumulative 

reservoir draft (a summation of net change in reservoir 

elevation) versus daily cumulative yield from bank 

storage for this period is shown in figure 3. The points 

generally increase chronologically from lower left to 

upper right. Successive days were connected by straight 

lines. The continuing yield from bank storage during 

periods when the reservoir elevation was held nearly 

constant is illustrated by the nearly vertical segments 

of the line in this figure. 
6 BANK STORAGE, HUNGRY HORSE RESERVOIR, MONTANA 

COMPUTING BANK STORAGE FROM 

ASSUMED ELEVATIONS 

The use of the model for determining historical values of bank storage on a daily or monthly basis has been 

demonstrated. The model may also be used for computing bank-storage changes and cumulative volumes 

for theoretical or projected reservoir-level changes. 

An example of this type of application of the model 

is the determination of the volume of water that would 

go to and from bank storage under projected or potential reservoir operating rules. The Pacific Northwest 

Utilities Conference Committee has projected monthend critical rule-curve elevations at Hungry Horse Reservoir for periods of 2 and 4 years for each of the power 

years (July through June) of 1971-72 through 1980-

81 (D. W. Hanlon, written commun, 1971). The power 

years July 1971 through June 1972 and July 1972 

through June 1973 were based on a critical period 

length of 2 years' duration, and the subsequent power 

years through 1980-81 were based on critical periods 

of 4 years' duration. These theoretical reservoir elevations are given in table 3. They were determined by 

system power loads to be met, the assumed facilities 

available for generation of power, the amount of storage 

available for flow regulation, and- a sequence of projected streamflow occurrences at each damsite. The 

streamflows used are usually based upon the minimum 

flow sequences that occurred during the 30-year historical flow period October 1928 through September 1958. 

Although the critical rule elevations are designed primarily for power purposes, several other factors are 

taken into consideration. Among these are flood control, the maintenance of maximum and minimum flows, 

the fish resource, recreational uses, and pollution abatement. Hence, projected critical rule-curve elevations 

are the elevations of a particular reservoir that would 

govern the operation of that reservoir to meet critical 

water needs during a future time period, assuming a 

particular sequence of streamflow events occurs. 

The projected critical rule-curve elevations begin 

with an assumed full pool elevation of 3,560.0 feet on 

August 15. D. W. Hanlon suggested that prior to that 

date it be assumed that the reservoir was subjected 

to a normal filling from an elevation of 3,515.0 feet 

on April 15 (written commun., 1971). Bank-storage 

volume was computed assuming a normal fill and a 

straight-line hydrograph between projected monthend 

critical rule-curve elevations (table 3) for the 2- and 

4-year periods. 

The cumulative volumes calculated from bank storage at the end of each critical period by using the projected reservoir elevations as input to the computer 

model are given in table 4. Also shown in table 4 are 

the bank-storage volumes expressed as a percentage of 

the usable reservoir volume (which is 1,503,400 cfsdays between elevations 3,560.0 and 3,336.0 ft). The 

result implies that 5.8 percent additional water would 

be available for power generation at Hungry Horse Reservoir and downstream uses if the reservoir was subjected to a cyclic drawdown over a 2-year or longer 

period to the minimum pool elevation of 3,336.0 feet. 

OTHER MODEL CONFIGURATIONS 

Several model configurations and various hydrologic 

constants were evaluated by Simons and Rorabaugh 

(1971, p. 41-54). They concluded that the bank storage computed by the combination model 2 (m2) + 0.4 

model4 (m4) gave a reasonable match to the field data, 

but it was judged high because of the high implied 

storage coefficients (S of 0.19 for fine till and S of 0.4 

for the gravel and sand). The use of the configuration 

m2 + 0.4m4 would therefore be more likely to overpredict rather than underpredlct the amount of additional 

water available for power generation. They then recalculated the water budget while assuming that the original gaging records were in error and too large. Assuming 

a reduction of 0, 1, 2, and 5 percent of the flow, the 

recalculated water-budget residuals were plotted in a 

double-mass curve versus volumes determined by m2 + 

0.4m4. The slope from the double-mass plots for the 

1-percent reduction gave a multiplier of 0.8 to be applied to the model. This configuration of 0.8 (m2 + 

0.4 m4) appeared to be the best solution when compared with solutions assuming no reduction in flow and 

the 2- and 5-percent reductions in flow. The storage 

coefficients for the configuration 0.8 (m2 + 0.4 m4) areabout 0.15 for 80 percent of the aquifer (fine till) and 

0.32 for 20 percent of the aquifer (sand and gravel). 

Because of the ease in which the hydrologic constants 

and model configurations can be changed in the computer, several other model configurations and parameter values were evaluated. The results of the computer solutions were compared with the daily and accumulated monthly water-budget residuals in an attempt 

to improve the model. However, this line of investigation was terminated because the accuracy of the solution using the model configuration 0.8 (m2 + 0.4 m4) 

was within the accuracy limits of the parameter values 

and water-budget solution. 

Some of the larger monthly differences between the 

computer solution and the water-budget residuals have 

been previously explained. However, the magnitude of 

the difference between the computed volume change 

and actual volume change cannot be measured. Other 

possible sources of error include inaccuracies in aquifer 

width, the possibility of bank storage in the material 

bordering the aquifer, which was assumed imperme-
SUMMARY AND CONCLUSIONS 7 

able, and errors in elements of the water budget such 

as the table of reservoir contents. 

The accumulated bank-storage volumes computed by 

the digital computer model and the water-budget residuals at the end of full-pool periods closely correlate on 

a year-to-year basis. Any improvement in the "fit" 

between the computer solution and the water-budget 

residuals would be based upon small changes in the 

model parameters and water-budget components. These 

changes would be within the accuracy limitations of 

the parameters and data, and there would be no assured 

improvement in the model results. 

MODEL SENSITIVITY 

Model (system) sensitivity refers to the change in an 

output value that can be attributed to a change in one 

of the system parameters. The sensitivity of the computer model to changes in parameter values provide 

some guidelines for determining the accuracy required 

when measuring the parameters. Sensitivity tests also 

provide some indication of the range of error in the 

bank-storage volumes computed by the computer model 

when the reliability of the input parameters are. known. 

To determine the model's sensitivity in computing 

cumulative bank-storage volume, one parameter was 

changed a fixed percentage while holding the other 

parameters constant. The cumulative monthly bankstorage volumes computed after a parameter change 

(V'), the differences between monthly volumes computed before and after a parameter change (V-V'), 

V-V' and the percentage change in volumes ( V X 100) 

are shown in figures 4 through 7. 

The period October 1964 through April 1967 was 

selected for the sensitivity test. The comparison of volumes of water from the computer model when the aquifer width (a) was changed -+20 percent is shown in 

figure 4. The comparison before and after the slope 

(m) was changed -+ 10 percent is shown in figure 5. The 

effects of changing transmissivity (T) -+25 percent are 

shown in figure 6, and the effects of changing the storage coefficient (S) -+20 percent are shown in figure 7. 

The comparison of volumes shown is self-explanatory. It should be noted, however, that although the 

percentage differences between the monthly volumes 

before and after the parameter change are large in some 

cases, the actual differences are not and that a large 

percentage difference is the result of dividing one small 

number by another nearly the same. For example, the 

70 percent difference in volumes that occurs when aquifer width is increased 20 percent for the month of 

August 1966 is a result of a difference of cumulative 

bank-storage volume of only 435 cfs-days; the cumulative bank-storage volume before the change was 

only 621 cfs-days. Thus 435/621 X 100==70 percent, a 

large percentage difference but an insignificant actual 

difference. 

It is necessary to exercise care in measuring or estimating all the parameters. The change in slope (m) 

results in a proportionate change in bank-storage volume throughout the drawdown and fill cycles. The 

changes in aquifer width (a) result in the largest percentage change on an annual basis, from the beginning 

of one drawdown to the beginning of the next. When 

the storage parameter is changed, the largest differences in minimum and maximum accumulated yield 

from bank storage occur. 

There is a shift in the date of occurrence of the maximum and minimum accumulated bank storage when 

either width, transmissivity, or storage are changed but 

no shift in date when slope is changed. 

The slope parameter can be easily and accurately 

measured. Of the remaining parameters, the accuracy 

of the storage coefficient is the most important when 

the seasonal and monthly changes in bank-storage volume are of interest. However, errors in aquifer width 

can cause differences in accumulated bank-storage volume that are nearly as much as those as caused by the 

same percentage error in the storage coefficient. 

Transmissivity appears to be the least sensitive of 

the parameters. This may be misleading, however, since 

transmissivity has the widest range of possible values 

of the parameters. Transmissivity is equal to the integration of the hydraulic conductivities across the saturated part of the aquifer perpendicular to the flow path 

(Lohman and others, 1972, p. 13). According to Davis 

and DeWiest (1966, p. 164) the average hydraulic conductivities can range from 10-2 to 10 gallons per day 

per square foot in a poor aquifer and from 104 to 106 

in a gravel, a possible range of 108 • 

SUMMARY AND CONCLUSIONS 

The theoretical equation and mathematical model 

developed by Simons and Rorabaugh ( 1971) for determining bank-storage volume at Hungry Horse Reservoir 

was programmed and tested on a digital computer. 

Daily historical reservoir elevations were used as input 

parameters for computing daily and monthly accumulated bank-storage volumes and changes in bank storage. The monthly total accumulated bank storages as 

calculated by the digital computer closely match 

those computed for the same test period by Simons and 

Rorabaugh (1971) with a desk calculator and mathematical and graphical short cuts. Further attempts at 

improving the model for computing bank storage at 

Hungry Horse Reservoir are not warranted because the 
8 BANK STORAGE, HUNGRY HORSE RESERVOIR, MONTANA 

solution developed is well within the accuracy limitations of the data. 

Accumulated bank-storage volume was computed 

using projected rule-curve elevations for the period 

1971 through 1981. The results indicate that an additional 5.8 percent of total reservoir volume is available 

for generation of power and subsequent downstream 

benefits if the reservoir is subjected to a long-term 

cyclical drawdown over a 2-year or longer period to a 

minimum pool elevation of 3,336.0 feet. 

The computer model's sensitivity to changes in each 

parameter was tested. The model is most sensitive to 

changes in the aquifer width, with the largest percentage differences occurring near the end of a full pool 

period. Changes in the storage coefficient result in the 

largest actual differences in volume that occur during 

the maximum drawdown periods. 

The digital computer model proved to be a useful 

tool in solving for bank-storage volumes at Hungry 

Horse Reservoir whenever daily or monthend reservoir 

elevations (historical or assumed) are available. The 

model parameters are easily changed. Therefore, the 

computer model can be used to estimate bank-storage 

volumes at other reservoirs where historic or assumed 

reservoir elevations are available, provided the physiographic, hydrologic, and geometric conditions of the 

reservoir being studied are the same or similar to those 

assumed for Hungry Horse Reservoir. 

SELECTED REFERENCES 

Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of heat in 

solids: London, Oxford University Press, 510 p. 

Cooper, H. H., Jr., and Rorabaugh, M. 1., 1963, Ground-water 

movements and bank storage due to flood stages in surface 

streams: U.S. Geol. Survey Water-Supply Paper 1536-J, 

p. 343-366. 

Davis, S. N., and DeWiest, R. J. M., 1966, Hydrogeology: New 

York, John Wiley & Sons, Inc., 463 p. 

Hantush, Madhi S., 1961, Discussion of paper by P. P. Rowe; 

An equation for estimating transmissivity and coefficient 

of storage from river-level fluctuations: Jour. Geophys. 

Research,v.66,no.4,p. 1310-1311. 

Lohman, S. W., and others, 1972, Definitions of selected groundwater terms-Revisions and conceptual refinements: U.S. 

Geol. Survey Water-Supply Paper 1988, 21 p. 

Rorabaugh, M. I., 1964, Estimating changes in bank storage 

and ground-water contribution of streamflow, in Symposium-Surface water: Internat. Assoc. Sci. Hydrology 

Pub. 63 p, 432-441. 

Simons, W. D., and Rorabaugh, M. 1., 1971, Hydrology of 

Hungry Horse Reservoir, northwestern Montana: U.S. 

Geol. Survey Prof. Paper 682, 66 p. 

Thompson, T. H., 1972, Reservoir bank storage: Menlo Park, 

Calif., U.S. Geol. Survey Computer Contr., 29 p.; available only from U.S. Dept. Commerce, Natl. Tech. Inf. 

Service, Springfield, Va. 22151 as report PB-214 416. 
FIGURES AND TABLES 9 

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I 

I Model using digital computer with 

I daily changes in stage I 

I 

I 

Model using 

~ 

simplified techniques I 

I (from Simons and Rorabaugh, 1971) 

--+----+--

Water-budget residuals (from Simons 

50 and Rorabaugh, 1971) 

60 ~L__L __ L__L __ L__L __ L__L __ L_-L--L-~~--J_~--~~--J_~---L~~~~--~~ 

ONDJ FMAMJ J AS ONDJ FMAMJ J AS 0 N D J FMAM 

1964 1965 1966 1967 

FIGURE I.-Comparison of monthly aceumulated volumes as computed by the digital model with those computed by Simons' and 

Rorabaugh's model and water-budget residuals. 
10 

1-

w 

w 

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zw 

-> 

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144 

142 

140 

138 

BANK STORAGE, HUNGRY HORSE RESERVOIR, MONTANA 

5 10 15 20 25 

DECEMBER 

1959 

5 10 15 20 25 

JANUARY 

1960 

5 10 15 20 25 5 10 15 20 25 

FEBRUARY MARCH 

FIGURE 2.-Daily reservoir elevations, yield from bank storage, and cumulative bank-storage volumes for 

the period December 1, 1959, through March 31, 1960. 
FIGURES AND TABLES 

6 en 

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CUMULATIVE RESERVOIR DRAFT, IN FEET 

FIGURE 3.-Cum.ulative reservoir draft versus cumlative yield from bank storage for the period December 1, 

1959, through March 31, 1960. Dots represent day-end values and generally accumulate from left to right. 

11 
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(V-V') 

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v 

I I / \ UJ , I ·" \ .... .. ,,-, 

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0 N D J F M A M J J A S 0 N D J F M A M J J A S 0 N D J F M A 

1964 1965 1966 1967 

FIGURE 4.-Model sensitivity to changes in aquifer width. Volumes of bank storage computed by 

model after aquifer width (a) was changed +20 percent and -20 percent; differences between volume before (V) and after (V') changes in (a); the percent difference before and 

after (a) was changed +20 percent and -20 percent. 
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FIGURES AND TABLES 

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I I I I I I I I I I I I I I I I I I I r I T l 

/ , .... / .... ... -·.;*"""' / ·, ... ... 

' .................... , / ' ·........-m-0. 1m , ,..,.·"' ..... ' ,. ... · ' ... ,. ., .. ·--·--·--·""' '·~- ·--·--· ... 

.-·-·-·-·, .-·-·-., . ..<m+0.1m " / . ·-· . ' I '·-·' ., 

,,...... '· ....... Monthly difference .......... 

!V-V') ......... 

I I I I I I I I I I I I I I I I I I I I I I 

-

-

-

-

+12.-,-l'l--,-,l--,-l,-l'l--,-l,l--,-,-l,--,-l~l--.-.-l'l--,-1~1-.l,-,-f'T--I,-~T-.I,-T,-,--,-I,-I'I--,-,I 

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0 N 0 J F M A M J J A S 0 N 0 J F M A M J J A S 0 N 0 J F M A 

1964 1965 1966 1967 

FIGURE 5.-Model sensitivity to changes in slope of reservoir bed. Volumes of bank storage computed 

by model after slope (m} was changed +10 percent and -10 percent; differences between volumes before (V) and after (V') changes in (m); the percent difference before and after (m) was 

changed +10 percent and -10 percent. 

13 
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BANK STORAGE, HUNGRY HORSE RESERVOIR, MONTANA 

0 

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I I I I I I I I I 1,! I I I I I I I I I I I I I I 

+30 

+20 

+10 

0 

/ 

-10 

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./ \ I \ t 

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0 N D J F M A M J J A S 0 N D J F M A M J J A S 0 N D J F M A 

1964 1965 1966 1967 

FIGURE 6.-Model sensitivity to changes in aquifer transmissivity. Volumes of bank storage computed by model after transmissivity ( T) was changed + 25 percent and -25 percent; differences between volumes before ( V) and after ( V') changes in ( T) ; the percent difference before and after (T) was changed +25 percent and -25 percent. 
U) 

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FIGURES AND TABLES 

.,....... ,., 

i/ \ J• \ 

\ 

/:7::;.-·-, ;j')""';\.,_, 

':v·(S-0.2S) y,'- \ f/ . ' 

·\ \ 

\ I 

i I • 

· \ 

\. I 

i I ~\.\\\ \ I \' I \ /· \_ ....... ·;· 

I I 

\ I . 

\ : \ . \ ! / \ I 

\ I 

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I I I I I I I I I I I I I I I I I 

i--\ 

I \-S-0.2S I 

I \ 

I '\ ., ·--· ' I \ .... I ' \ , ·, ,. , ' , 

I 

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... . 

'\. ----·-·-' .-----·, .\ .I . / ' ' ., ___ . / 

·\. __ /s+0.2S 

Monthly difference 

(V-V') 

I I I I I I I I I I I I I I I I I I I I I 

I I I I r T -r T I T 

I 

/ 

/ 

' 

I 

I l Percent difference I ' - I \ I ,v-V' x ' S-0.2S-y,-\ 100) I ' I ' v I ' - I ·' ' ' / . ,·"' .--· ' / ' , '"' ............ , ' --·--·--·--·--· ', --·-----·--·--·''·--· , __ ... I ' ,.--· I \" '·-·, /·-·-·-·--- -·-·-·-·-·--·-·\;_...·-·- ............ __ .-............. 

- S+O.~\/·/ I I I I I I I I I I I .I I I I I I I I I I I 

\ 

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/ 

/ 

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0 N D J F M A M J J A 5 0 N D J F M A M J .J A 5 0 N D J F M A 

1964 1965 1966 1967 

-

-

-

-

- . 

-

-

-

-

-

-

F'IGum!: 7.-Model sensitivity to changes in aquifer storage coefficient. Volumes of bank storage computed by model after the storage coefficient (8) was changed +20 percent and -20 percent; differences between volumes before (V) and after (V') changes in (8); the percent difference before and after (8) was changed +20 percent and -20 percent. 

15 
16 BANK STORAGE, HUNGRY HORSE RESERVOIR, MONTANA 

TABLE !.-Cumulative monthend volumes of bank storage computed by the digital model 0.8 (m:+0.4m4) and by Simons' and 

Rorabaugh's simplified model 0.8 (m:+0.4m4) and the accumulated water-budget residuals 

[In 1,000 cfs-days; plus, to reservoir; minus, from reservoir] 

Digital Simons' and Water-budget Digital Simons' and Water-budget Date model Rorabaugh's model residual Date model Rorabaugh's model residual 

1964: 1966: October ___________________ -1.2 -1.0 -2.8 January __________________ 12.9 13.6 17.4 November _______________ -1.7 -1.6 -4.0 February _________________ 17.5 18.7 20.0 December ________________ -.5 -.3 1.0 March ______________________ 21.2 22.4 25.2 

1965: April ________________________ 19.9 21.3 27.7 January __________________ 4.0 4.6 9.0 MaY-------------------·------ 13.8 15.2 12.6 

February----------------- 11.3 12.1 19.4 June _________________________ 5.3 6.5 3.0 March ______________________ 25.8 26.9 39.7 July __________________________ 2.3 3.2 2.6 

April-----------------------· 34.1 35.0 52.4 August-------------------- .6 1.6 2.6 

MaY-------------------------- 30.0 30.6 31.3 September ______________ 2.3 3.2 5.7 

June·-----------------------· 15.2 16.1 17.9 October ___________________ 8.1 9.0 12.1 July __________________________ 6.7 7.5 12.7 November _______________ 15.4 16.6 17.3 August ____________________ 3.6 4.4 8.5 December ________________ 23.3 24.6 28.9 September ______________ 1.7 2.6 3.0 1967: October ____________________ .7 1.6 2.8 JanuarY-----------------· 29.2 30.6 37.8 November _______________ 1.1 2.0 3.0 February _________________ 32.9 34.2 42.9 December ________________ 5.1 6.0 8.8 March.--------------------- 38.4 40.0 47.5 

April ... --------------------· 44.5 46.2 53.0 

TABLE 2.-Cumulative monthly bank-storage volumes for water years 1952 through 1972 

[Values X 1,000 cfs-days] 

1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 

October ------------ 4.7 57.6 116.1 139.7 142.5 141.1 135.5 142.0 143.4 145.0 142.8 November 6.0 58.2 116.2 141.2 143.2 139.3 128.6 142.6 144.2 145.4 143.3 

December -------- 4.8 54.6 116.0 139.6 142.9 132.7 120.8 142.0 144.3 145.6 140.5 January ····------ 4.6 51.2 115.2 134.0 140.5 122.8 116.8 138.6 143.4 145.7 133.2 February 5.2 51.4 111.5 129.2 134.3 115.6 115.3 133.6 140.1 143.6 125.1 

March -------------- 5.4 51.8 108.1 122.9 125.1 111.1 114.3 128.3 136.3 138.2 121.6 

April ---------------- 8.2 54.2 103.3 118.8 117.0 110.3 115.3 118.8 131.6 129.4 120.9 

May------------------ 23.2 66.4 109.5 121.9 122.8 121.8 123.0 120.9 132.4 129.5 125.6 

June------------------ 39.7 87.2 123.0 133.1 135.3 132.3 133.1 133.8 140.0 139.7 135.0 

July------------------ 47.4 100.8 131.8 138.1 139.4 137.0 137.0 139.0 142.6 142.4 139.1 August 51.9 108.0 135.4 140.2 141.3 139.2 139.1 141.0 143.7 143.6 141.0 September ------ 55.0 112.7 137.9 141.6 142.1 138.0 140.7 142.4 144.4 143.8 142.3 

1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 

October----------------------------------------- 143.3 144.0 142.8 140.9 133.5 132.6 141.8 127.9 128.3 136.0 November _____________________________________ 144.1 144.4 143.3 140.5 126.2 131.2 142.7 124.3 126.5 131.3 

December·------------------------------------· 142.3 141.2 142.1 136.5 118.3 125.8 141.0 117.4 122.5 124.3 January ________________________________________ 136.7 132.9 137.6 128.7 112.4 118.4 137.8 114.4 115.5 116.1 February ______________________________________ 134.0 126.7 130.3 124.1 108.7 115.6 132.1 113.0 114.8 109.4 

March .. -------·---------------------------------- 133.5 121.1 115.8 120.4 103.2 115.7 124.2 111.6 112.8 103.8 

April .... -----------------------------------------· 130.4 116.8 107.5 121.7 97.1 116.6 118.9 107.1 105.5 95.3 

MaY-----------------------------------------------· 131.2 119.7 111.6 127.8 100.6 121.5 123.2 111.0 113.2 98.7 

June·---------------------------------------------· 138.7 133.2 126.4 136.4 118.0 130.7 132.2 125.1 126.6 116.6 

JulY----------------------------------------------- 141.7 138.0 134.9 139.3 129.4 136.3 138.2 133.1 133.8 127.1 

August------------------------------------------ 143.0 140.2 138.0 141.0 133.9 139.1 137.4 135.2 136.9 132.9 September ____________________________________ 143.8 141.6 140.0 139.3 134.4 140.6 132.3 132.7 137.2 135.3 

TABLE 3.-Projected monthend (except as noted in August and April) critical rule-curve elevations for Hungry Horse Reservoir 

for power years 1971-72 through 1980-81 

Water Period August August April April year length July 15 31 September October November December January February March 15 30 May June 

1971-72, 2 3,560.0 3,560.0 3,560.0 3,555.9 3,550.2 3,539.1 3,521.2 3,515.9 3,515.8 3,515.8 3,515.8 3,515.8 3,539.6 3,527.0 

1972-73. 3,499.6 3,494.2 3,488.0 3,472.2 3,454.9 3,434.4 3,379.3 3,336.0 3,336.0 3,336.0 3,336.0 3,336.0 3,421.2 3,468.8 

1973-7 4---------------- 4 3,560.0 3,560.0 3,560.0 3,555.6 3,445.1 3,526.5 3,526.5 3,526.5 3,526.5 3,526.3 3,514.6 3,514.6 3,543.9 3,560.0 

3,560.0 3,560.0 3,554.1 3,554.1 3,531.5 3,506.8 3,499.9 3,485.5 3,480.1 3,471.0 3,471.0 3,489.2 3,517.4 3,534.9 

3,538.8 3,538.6 3,535.3 3,529.3 3,509.9 3,485.6 3,467.4 3,467.4 3,462.6 3,456.9 3,456.9 3,456.9 3,599.8 3,504.6 

3,507.3 3,502.1 3,486.3 3,451.6 3,415.9 3,360.0 3,336.0 3,336.0 3,336.0 3,336.0 3,353.7 3,378.9 3,458.1 3,501.0 

197 4-75---------------- 4 3,560.0 3,560.0 3,560.0 3,555.6 3,545.1 3,525.3 3,525.3 3,525.3 3,525.3 3,525.3 3,513.6 3,513.6 3,543.0 3,560.0 

3,560.0 3,560.0 3,560.0 3,555.6 3,533.1 3,508.7 3,499.9 3,499.6 3,494.7 3,486.3 3,486.3 3,499.9 3,526.4 3,543.2 

3,546.8 3,546.8 3,544.1 3,538.4 3,519.8 3,497.2 3,479.9 3,479.9 3,475.0 3,470.4 3,470.4 3,470.4 3,510.1 3,525.0 

3,527.4 3,522.8 3,509.5 3,480.1 3,444.6 3,403.8 3,366.1 3,352.0 3,336.0 3,336.0 3,353.7 3,378.9 3,458.1 3,501.0 

1975-76, 4 3,560.0 3,560.0 3,560.0 3,557.9 3,552.6 3,539.4 3,524.0 3,524.0 3,524.0 3,524.0 3,512.1 3,512.1 3,541.7 3,560.0 

1976-77, 3,560.0 3,560.0 3,560.0 3,557.7 3,535.5 3,511.4 3,499.9 3,499.6 3,494.7 3,486.3 3,486.3 3,499.9 3,526.4 3,543.2 

1977-78. 3,546.8 3,546.8 3,544.1 3,538.4 3,519.8 3,497,2 3,483.0 3,473,0 3,468.4 3,462.8 3,462.8 3,462.8 3,504.3 3,519.7 

3,522.2 3,517.4 3,506.1 3,476.1 3,439.3 3,400.1 3,366.1 3,352.0 3,336.0 3,336.0 3,353.7 3,378.9 3,458.1 3,501.0 

1978-79, 4 3,560.0 3,560.0 3,560.0 3,557.9 3,552.6 3,539.4 3,522.3 3,522.3 3,522.3 3,522.3 3,510.4 3,510.4 3,540.2 3,560.0 

1980-81. 3,560.0 3,560.0 3,560.0 3,557.7 3,535.5 3,511.4 3,499.9 3,499.6 3,494.7 3,486.3 3,486.3 3,499.9 3,526.4 3,543.2 

3,546.8 3,546.8 3,544.1 3,538.4 3,519.8 3,497.2 3,483.0 3,473.0 3,468.4 3,462.8 3,462.8 3,462.8 3,504.3 3,504.3 

3,507.0 3,501.8 3,489.3 3,455.5 3,412.1 3,363.7 3,366.1 3,352.0 3,336.0 3,336.0 3,353.7 3,378.9 3,458.1 3,501.0 

1979-80--·------------- 4 3,560.0 3,560.0 3,560.0 3,557.9 3,552.6 3,539.4 3,522.3 3,522.3 3,522.3 3,522.3 3,510.4 3,510.4 3,540.2 3,560.0 

3,560.0 3,560.0 3,560.0 3,557.7 3,535.5 3,511.4 3,499.9 3,499.6 3,494.7 3,486.3 3,486.3 3,499.9 3,526.4 3,543.2 

3.546.8 3,546.8 3,544.1 3,538.4 3,519.8 3,497.2 3,483.0 3,473.0 3,468.4 3,462.8 3,467.0 3,467.0 3,507.5 3,507.7 

3,510.5 3,505.3 3,493.1 3,460.3 3,418.5 3,372.6 3,366.1 3,352.0 3,336.0 3,336.0 3,353.7 3,378.9 3,458.1 3,501.0 

TABLE 4.-Cumulative volumes yielded from bank storage computed by using projected critical rule-curve elevations for power 

years 1971-72 through 1980-81 

Critical Power period length year (years) 

1971-72------------------------ 2 1972-73------------------------ 2 1973-74------------------------ 4 197 4-75------------------------ 4 1975-76------------------------ 4 

[Usable reservoir volume between elevations of 3,560 ft and 3,336 is 1,503,400 cfs-days] 

Yield from bank storage 

Computed yield divided by total usable from bank storage reservoir volume 

(cfs-days) (percent) 

81,518 5.82 87,518 5.82 89,331 5.94 85,098 5.66 85,663 5.70 

Critical Computed yield Power period length from bank storage year (years) ( cfs-days ) 

1976-77------------------·----- 4 85,663 1977-78________________________ 4 85,663 

1978-79-------------------·---- 4 87,745 1979-80________________________ 4 87,327 

1980-8L. .... ----------------- 4 87,7 45 

Yield from bank storage 

divided by total usable reservoir volume 

(percent) 

Mean------------------------------------·---·---------------------------------------------···-----·-

5.70 

5.70 

5.84 

5.81 

5.84 

5.78 

tr u.s. GOVERNMENT PRINTING OFFICE: 1974-543-683/94