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Application of the Finite Element Method to Vertically Stratified Hydrodynamic Flow and Water Quality May 1980 Approved for Public Release. Distribution Unlimited. TP-72

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4. TITLE AND SUBTITLE Application of the Finite Element Method to Vertically Stratified Hydrodynamic Flow and Water Quality

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6. AUTHOR(S) Robert C. MacArthur, William R. Norton

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7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Army Corps of Engineers Institute for Water Resources Hydrologic Engineering Center (HEC) 609 Second Street Davis, CA 95616-4687

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12. DISTRIBUTION / AVAILABILITY STATEMENT Approved for public release; distribution is unlimited. 13. SUPPLEMENTARY NOTES Presented at the third International Conference on Finite Elements in Water Resources, at the University of Mississippi, Oxford, MS, 19-23 May 1980 14. ABSTRACT Computer program RMA-7 (King, et al, 1973) has been expanded to be able to simulate density induced flows and water quality conditions typically found in deep reservoirs. The mathematical basis for the program and methods of implementing the code for various kinds of flow are discussed. Results from two different applications are presented. The first example simulates flow conditions that were measured in a physical model of a deep reservoir. Results are also included for simulations of the circulation, temperature and dissolved oxygen concentrations for Lake Taneycomo, Missouri. Comparison of measured and computed results from both applications shows that the RMA-7 model provides reasonably accurate results for both flow circulation and water quality. 15. SUBJECT TERMS finite element methods, stratified flow, two-dimensional flow, reservoir modeling, water quality, computer modeling

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Application of the Finite Element Method to Vertically Stratified Hydrodynamic Flow and Water Quality

May 1980 US Army Corps of Engineers Institute for Water Resources Hydrologic Engineering Center 609 Second Street Davis, CA 95616 (530) 756-1104 (530) 756-8250 FAX www.hec.usace.army.mil TP-72

Papers in this series have resulted from technical activities of the Hydrologic Engineering Center. Versions of some of these have been published in technical journals or in conference proceedings. The purpose of this series is to make the information available for use in the Center's training program and for distribution with the Corps of Engineers. The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. The contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products.

APPLICATION OF THE FINITE ELEMENT METHOD TO VERTICALLY STRATIFIED HYDRODYNAMIC FLOW AND WATER QUALT T Y ~

R. C. ~ a c ~ r t h u r ' and W. R. Norton 3

INTRODUCTION

Continuing i n t e r e s t i n t h e i n t e r n a l processes of r e s e r v o i r s , l a k e s and e s t u a r i e s has i n t e n s i f i e d t h e development of mathematical models f o r s imula t ion of v e r t i c a l l y s t r a t i f i e d flow. Motivated p r imar i ly by t h e long term d e s i r e t o p r e d i c t t h e water q u a l i t y response t o system modif ica t ions , c u r r e n t modeling e f f o r t s have focused on t h e need t o desc r ibe flow coupled wi th temperature and/or s a l i n i t y f i e l d s i n o rde r t:o f o r e c a s t t h e in f luence of dens i ty induced flows. Computational a lgori thms have shown s u f f i c i e n t promise t h a t e f f o r t s a r e under way t o c o l l e c t p ro to type d a t a which can be used f o r c a l i b r a t i o n and v e r i f i c a t i o n of bo th flow and water q u a l i t y models.

Among t h e s e v e r a l models which have been formulated t o s imu la t e d e n s i t y induced flows and water q u a l i t y i s one c a l l e d RMA-7. This model, which was o r i g i n a l l y developed fo r t h e Of f i ce of Water Resources Research, (King, 1973) has e x i s t e d f o r s e v e r a l y e a r s b u t has rece ived r e l a t i v e l y l i t t l e use i n pro to type a p p l i c a t i o n s . I t i s t h e i n t e n t of t h e work repor ted h e r e i n t o demonstrate t h e ope ra t ion of t h e cu r r en t v e r s i o n of RMA-7 and t o o f f e r app ropr i a t e in format ion and comments on t h e use and implementation of t h e model.

S p e c i f i c a l l y , t h e paragraphs which fo l low con ta in a b r i e f d e s c r i p t i o n of t h e mathematical b a s i s of RMA-7 a s wel l a s a n example of i t s implementation on a pro to type system which approximates t h e phys ica l dimensions of t h e so-ca l led GRH flume a t t h e Corps' Waterways Experiment S t a t i o n i n Vicksburg. Also included a r e some r e s u l t s ob ta ined by t h e model from t h e s imula t ion of temperature and d isso lved oxygen f o r Lake Taneycomo i n Missouri . Severa l s t a t i s t i c a l comparisons a r e included between s imulated and measured va lues f o r Lake Taneycomo which a r e designed t o q u a n t i f y , t o some e x t e n t , t h e accuracy of t h e model.

1 Presented a t t h e 3rd I n t e r n a t i o n a l Conference on F i n i t e Elements i n Water Resources, May 1.9-23, 1980, t h e Univers i ty of M i s s i s s i p p i , Oxford, MS.

' ~ ~ d r a u l i c Engineer, t h e Hydrologic Engineering Center , Davis, Ca l i fo rn i a .

' ~ r i n c i ~ a l , Resource Management Assoc ia tes , Laf aye t f e, Cal i , forn ia .

GOVERNING EQUATIONS - DIFFERENTIAL FORMS

RMA-7 is a two dimensional mathematical model which describes the behavior of velocity, pressure, temperature and dissolved oxygen in the vertical plane with homogeneity assumed in the third (lateralj direction; the model will accommodate width gradients in both the X and Y directions. This model is formulated on the classical concepts of conservation of mass, momentum, and energy although it is somewhat unusual as it retains the complete vertical momentum equation and does not make the assumption that pressure must be hydrostatic.

Hydrodynamic Model

The equations used in RMA-7 have the followi.ng differenti.al f o m s :

Velocity equations.

Continuity equation.

Temperature equation.

aT a T a 2 ~ D - a 2 ~ - aT + Cpu - + Cpv -- - Dx - CP at ax ay Y ay2 $2 = 0 (4

where u = X direction velocity v = Y direction velocity p = pressure T = temperature, degrees C t = time p = fluid density

= f (T) g = gravitational acceleration C = specific heat

+2 = thermal heat source or sink w = width

EXX'EXY = eddy viscosity coefficients &YX, €YY D,,Dy = eddy dispersion coefficients

Water Qual i ty Model ,--, -

The equat ions used i n RMA-7 t o desc r ibe t h e behavior of d i sso lved oxygen have t h e d i f f e r e n t i a l forms:

Dissolved oxygen equation.

Carbonaceous biochemical oxygen demand (BOD) equation.

Phytoplankton equation.

where cl = dissolved oxygen concentration c2 = carbonaceous BOD concentration c3 = phytoplankton concentration (dry weight) K2 = BOD decay rate a = the mass of oxygen produced per unit mass of phyto-

plankton growth IJ = the local rate of phytoplankton growth

P I 1 - r vm'max (I + KI

where = the maximum specific phytoplankton growth rate at the local temperature

I = the intensity of light at the local depth KI = the light half saturation constant r = the phytoplankton respiration rate at the local

temperature and all other terms as previously defined.

Equations 1 through 7 r ep re sen t t h e mathematical b a s i s f o r t h e model RMA-7. It is recognized t h a t t h e s e equat ions provide a n approximate r e p r e s e n t a t i o n of t h e governing processes i n t h a t c e r t a i n second o rde r terms wi th r e s p e c t t o width have been dropped and t h a t t h e v i s c o s i t y / d i s p e r s i o n terms a r e l a r g e l y empir ica l . Notwithstanding t h e s e shortcomings, however, t h e above r e l a t i o n s h i p s have shown promise i n s imu la t ion of observed phenomena, and provide a gene ra l framework f o r t e s t i n g and improving t h e procedures necessary f o r s imu la t ion of v e r t i c a l l y s t r a t i f i e d flow.

A s can b e seen , t h e f i r s t fou r equat ions and t h e second t h r e e equat ions each form a closed s e t which must be solved s imultaneously. The coupling between t h e two sets i s mani fes t i n t h e convect ive v e l o c i t i e s u and v , and i n t h e water temperature, T , Fo r tuna te ly , t h e coupling between t h e

two s e t s is of t h e "feed forward" type i n t h a t equat ions 1 through 4 may be solved independently of equat ions 5 , 6 and 7 , wi th t h e r e s u l t s of t h e hydrodynamic s o l u t i o n a c t i n g l i k e c o e f f i c i e n t s i n t h e s o l u t i o n of t h e water q u a l i t y model.

I n a d d i t i o n t o t h e volumetr ic terms shown above, t h e model con ta ins a number of a d d i t i o n a l f e a t u r e s t o account f o r s u r f a c e e f f e c t s and source / s i n k terms. Most important of t h e s e a r e : 1 ) bottom f r i c t i o n a t t h e soi l -water i n t e r f a c e ca l cu la t ed a s a func t ion of bottom v e l o c i t y and a Chezy c o e f f i c i e n t ; 2) s u r f a c e h e a t exchange a s a func t ion of t h e water temperature, t h e equi l ibr ium temperature and an exchange c o e f f i c i e n t ; 3) i n t e r n a l absorp t ion of s h o r t wave s o l a r r a d i a t i o n a s a func t ion of depth; and 4) s u r f a c e oxygen exchange a s a func t ion of t h e l o c a l oxygen d e f i c i t and an exchange c o e f f i c i e n t which i s a func t ion of wind speed.

SOLUTION TECHNIQUE

The governing equat ions a r e solved by t h e f i n i t e element method us ing Galerk in ' s c r i t e r i a f o r t h e method of weighted r e s i d u a l s . The formula t ion employs a mixed set of b a s i s func t ions , w i th quadra t i c func t ions used f o r a l l state v a r i a b l e s except p r e s s u r e where a l i n e a r func t ion i s used. The l i n e a r p re s su re func t ion impl ies a cons t an t element d e n s i t y , which i s c a l c u l a t e d a s a func t ion of average nodal temperatures . Green's Theorem i-s used t o lower t h e o rde r of a l l second d e r i v a t i v e s i n t h e v i s c o s i t y / d i s p e r s i o n terms, r e s u l t i n g i n s u r f a c e i n t e g r a l s which must be eva lua ted ( e i t h e r i m p l i c i t l y o r e x p l i c i t l y ) a long system e x t r e m i t i e s . Green's Theorem is a l s o used on t h e p re s su re t e r m s of equat ions 1 and 2 pe rmi t t i ng a s u r f a c e i n t e g r a l t o be used a s a d i scharge boundary cond i t i on r a t h e r than a s p e c i f i e d p re s su re va lue ; t h i s procedure permi ts r e t e n t i o n of a l l nodal con t inu i ty equat ions and s u b s t a n t i a l l y improves t h e model's performance.

RMA-7 uses an i m p l i c i t , Newton-Raphson computation scheme t o achieve a s o l u t i o n t o t h e s e t of nonl inear equat ions which de f ine t h e model. The computer program accommodates e i t h e r t r i a n g u l a r o r q u a d r i l a t e r a l i soparamet r ic elements w i th numerical i n t e g r a t i o n used t o eva lua t e a l l s u r f a c e and a r e a i n t e g r a l s . The use of t h e i sopa rame t r i c formulat ion wi th in te re lement geometric s l o p e c o n t i n u i t y al lows flow t o move p a r a l l e l t o boundaries a t a l l p o i n t s wh i l e a t t h e same t i m e p rovid ing a means f o r reasonable r e p r e s e n t a t i o n of r e a l phys i ca l systems.

EXAMPLE PROBLEMS

The model descr ibed above has been app l i ed t o s e v e r a l phys i ca l s i t u a t i o n s , two of which a r e summarized below. The f i r s t example shows r e s u l t s c a l c u l a t e d from tests conducted wi th t h e geometry of t h e GRH flume a t WES. The second shows r e s u l t s ca l cu la t ed from cond i t i ons found i n Lake Taneycomo i n Missouri . The f i r s t example was run us ing only t h e flow and temperature p o r t i o n s of t h e model, whi le t h e second inc ludes flow, temperature and d isso lved oxygen. The networks used f o r each of t h e s e example problems a r e reproduced i n f i g u r e 1.

Exam~le Problem 1 - GRH Flume

The GRH flume i s a hydrodynamic t e s t f a c i l i t y which i s 80 f e e t long and v a r i e s i n depth from approximately 1 f o o t a t i t s upper ( inf low) end t o 3 f e e t a t i t s lower (outflow) end, wi th two d i f f e r e n t c ros s s e c t i o n s along i t s l eng th . The inf low s e c t i o n , which i s 20 f e e t long, has a h o r i z o n t a l bottom and v a r i e s l i n e a r l y i n width from 1 .0 f o o t t o 2.85 f e e t . The lower s e c t i o n has a cons tan t width of 2.85 f e e t , bu t a bottom which drops 2 f e e t over i t s 60 f o o t l eng th .

RMA-7 was app l i ed t o t h e GRH flume geometry by cons t ruc t ion of a f i n i t e element network conta in ing 57 elements and 1.58 node p o i n t s as shown i n f i g u r e 1. I n cons t ruc t ing t h i s network i t was f e l t d e s i r a b l e t o al low flow t o move p a r a l l e l t o t h e flume bottom a t a l l l o c a t i o n s . For t h i s reason continuous curves were passed through t h e breakpoint on t h e f lume's bottom 20 f e e t from t h e upstream end and a t t h e t r a n s i t i o n t o t h e out- le t . This type of cons t ruc t ion permi ts continuous v e l o c i t i e s t o e x i s t a long t h e flume bottom and completely e l imina te s a r t i f i c i a l s t a g n a t i o n po in t s . The s m a l l d i scharge nozz le a t t h e o u t l e t has been included f o r each boundary cond i t i on s p e c i f i c a t i o n and does no t e x i s t on t h e phys i ca l flume.

To run RMA-7 i t is necessary t o s p e c i f y va lues f o r eddy v i s c o s i t y and eddy d i f f u s i o n c o e f f i c i e n t s . A t p r e s e n t , t h i s is done by exper ience and numerical t e s t i n g of problems which have known o r assumed v e l o c i t y and temperature d i s t r i b u t i o n s . I n t h e GRH flume case , a s e r i e s of numerical t e s t s were conducted on a s teady s t a t e problem t o determine a s e t of s a t i s f a c t o r y c o e f f i c i e n t s , and t h e network's s e n s i t i v i t y t o t h e va r ious c o e f f i c i e n t s . The va lues determined f o r t h e GRH flume had t h e r e l a t i v e va lues of cXX = 0.05, EXY = 0.0005, E~~ = 0.01, E = 0.1,

YY Dx = 0.1, and Dy = 0.0005, a f t e r accounting f o r element d i s t o r t ~ o n and s i z e .

Resu l t s from two examples a r e shown, one f o r a homogeneous flow and one f o r a nonhomogeneous flow. I n each case a flow of 10 gpm w a s introduced i n t o a s t i l l flume wi th a l i n e a r v e l o c i t y d i s t r i b u t i o n i n t h e lower element a t t h e inf low end. The homogeneous case was run i so the rma l ly a t a temperature of 10.3°C, wh i l e t h e nonhomogeneous case was s t a r t e d wi th an i n i t i a l temperature of 10.3"C and an inf low of 5OC.

Veloc i ty d i s t r i b u t i o n s produced by each cond i t i on a r e g r a p h i c a l l y compared a t t h r e e t imes i11 f i g u r e 2 . The e f f e c t s of t h e d e n s i t y s t . r a t i f i c a t i o n a r e q u i t e ev ident between t h e two cases w i th t h e co lde r , more dense water underflowing t h e l i g h t e r and warmer water near t h e su r f ace . The r e s u l t s shown a r e r e p r e s e n t a t i v e of ongoing work wi.th t h e GRH flume, a l though no measured d a t a is c u r r e n t l y a v a i l a b l e f o r model/prototype comparisons under t h e cond i t i ons s imula ted . The gene ra l shape of v e l o c i t y d i s t r i b u - t i o n s , and t h e a r r i v a l t i m e of t h e temperature f r o n t (15-18 min) a r e i n gene ra l agreement w i th measured d a t a , however, and sugges t t h e model w i l l perform w e l l when s u i t a b l e d a t a become a v a i l a b l e .

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E x a m ~ l e Problem 2 - Lake Tanevcomo

The second example problem p r e s e n t s r e s u l t s ob ta ined from t h e simulati.on of Lake Taneycomo i n southern Missouri . Lake Taneycomo is a 26-mile-long r e s e r v o i r bounded by Table Rock Dam upstream and Bull Shoals Reservoir downstream, and is used f o r power product ion and r e c r e a t i o n among o t h e r t h ings . A s low d isso lved oxygen has been observed i n t h e l ake , RMA-7 was appl ied wi th t h e goal of eva lua t ing t h e impact of changes i n r e s e r v o i r ope ra t ion on t h e ambient l e v e l s of d i sso lved oxygen.

To conduct t h e r equ i r ed s imu la t ions a network of 92 elements and 246 node p o i n t s was cons t ruc ted a s shown i n f i g u r e 1; l a t e r a l width va r i ed from about 200 t o 1000 f e e t wi th depth a t a t y p i c a l c r o s s s e c t i o n . Di f fus ion c o e f f i c i e n t s and eddy v i s c o s i t y c o e f f i c i e n t s were chosen t o be c o n s i s t e n t wi th those used i n t h e GRH flume when sca l ed f o r element d i s t o r t i o n .

De ta i l ed water temperature and d i s so lved oxygen measurements were a v a i l a b l e f o r t h r e e s e p a r a t e week long pe r iods i n t h e f a l l of 1977 . RMA-7 was c a l i b r a t e d a g a i n s t two of t hese pe r iods and v e r i f i e d a g a i n s t t h e t h i r d . Typica l measured and s imulated v e r t i c a l p r o f i l e s f o r temperature and d i s so lved oxygen a r e presented f o r s t a t i o n s T6 and T8 i n f i g u r e 3 f o r t h e e a r l i e s t c a l i b r a t i o n per iod . It should be noted t h a t model had s imulated over f i v e days of ope ra t ion by t h e t ime shown i n t h e s e f i g u r e s , and t h a t t h e inf low hydrograph v a r i e d from 0 t o 7000 c f s i n a four t o s i x hour per iod on a regul-ar b a s i s .

I n o rde r t o b r ing a degree of q u a n t i f i c a t i o n t o t h e accuracy of t h e model, a l i n e a r l e a s t squares r e g r e s s i o n of s imulated t o observed temperature and d isso lved oxygen has been made as shown i n f i g u r e 4 , w i t h t h e s t a t i s t i c s f o r each f i t g iven i n t a b l e 1.

A s can be seen , t hese s t a t i s t i c s i n d i c a t e a f a i r l y good f i t of both temper- a t u r e and d isso lved oxygen, and seem reasonable i n l i g h t of t h e u n c e r t a i n t y i n both t h e model's upstream i n p u t s (BOD, d i sso lved oxygen, meteoro logica l d a t a , e t c . ) and t h e usual measurement e r r o r s .

Table 1.--Regression S t a t i s t i c s

Regress ion S t a t i s t i c

STATION T6

i n t e r c e p t s l o p e s.d. e r r o r c o r r e l a t i o n

STATION T8

0.58 0.45 0.65 0.49 0.44 0.30

Temperature

0.29 0.96 0.43 0.89

--

Oxygen

-- 3.30 0.39 0.30 0.81

CRLC - 0 RS x

D a t e : 10-28 sl Time: 1.3:35 ------------77 . ti 1.00 1 . z.aa

TEMP DEG C x10'

D a t e : 1 0 - 2 8 Time: 1.4: 30

YI m I---

- 5 0 1. UU 1.313 Z,UU TEMP DEG C alOJ

S t a t i o n T6 S t a t i o n T8

D a t e : 10-28 =L Time: 13:35 ," - . 0 0 1

..(a .OD I. 2 0 D. ( 3 HG/L 910'

Date: 10-28

m . 0 0 ..(a = B D I. 20 D. (3. HG/L x10'

S t a t i o n T6 S t a t i o n T8

F i g u r e 3.--Observed and Simula ted Water Qua l i t y f o r Lake Taneycomo

,O Exact correlation line + + + +

:l/---- cn 9. 00 10.00 11.00 12.00

OBS TEMP IDEG C l

0 0

'1 Exact correlation + +++ +

1 ine + -. ^ . - ." U-. *$" + + + + t S

OBS TEMP IDEG C)

Station T6 Station T8

0 0

'1 Exact correlation line

Exact correlation

4 +--------- 4. 00 5. 00 6. 00 7. 00

OBS OXY IMG/LI

. .+I-. -4- ----I--

4.00 5.00 6.00 7.00 OBS BXY IMG/LI

Station T6 Stat,ion T8

Figure 4.--Statistical Regressions for Lake Taneycomo Temperature and Dissolved Oxygen

S W Y AND CONCLUSIONS

The above information summarizes t h e a p p l i c a t i o n of a two dimensional f i n i t e element model (RMA-7) t o two pro to type s t r a t i f i e d flow s i t u a t i o n s . I n each a p p l l c a t i o a a s t r a t i f i e d flow was s imulated a s a r e s u l t of d e n s i t y d i f f e r ences a r i s i n g from temperature g r a d i e n t s , I n t h e second example d isso lved oxygen, BOD and phytoplankton were a l s o routed i n accordance wi th t h e o v e r a l l f low f i e l d s .

Based on t h e d a t a contained h e r e i n i t seems reasonable t o concl.ude:

. t h e f i n i t e element method i n gene ra l and RMA-7 i n p a r t i c u l a r , i s capable of s imula t ing v e r t i c a l l y s t r a t i f i e d two dimensional flow.

. t h e proper d e f i n i t i o n of eddy v i s c o s i t y and d i s p e r s i o n c o e f f i - c i e n t s i s e s s e n t i a l t o proper model ope ra t ion , and t h a t t h e r e may be t r a n s f e r a b i l i t y of va lues from problem t o problem i f element s i z e and d i s t o r t i o n is accounted f o r .

. t h e RMA-7 model. appears t o g ive reasonable answers, bu t i t i s no t c u r r e n t l y p o s s i b l e t o eva lua t e i t s accuracy i n r e l a t i o n t o v e l o c i t y due t o a l a c k of pro to type da t a ; i n i t i a l comparisons of temperature and d isso lved oxygen d a t a a r e promising and w i l l improve a s c a l i b r a t i o n s become more r igo rous and q u a n t i f i e d .

REFERENCE

King, I. P . ; W. R. Norton; G. T. Orlob 1973. A f i n i t e element s o l u t i o n f o r two-dimensional densitiy

s t r a t i f i e d flow. Prepared for. t h e U. S . Dept. of t h e I n t e r i o r , Of f i ce of Water Resources Research. Water Resources Engineers.

Technical Paper Series TP-1 Use of Interrelated Records to Simulate Streamflow TP-2 Optimization Techniques for Hydrologic

Engineering TP-3 Methods of Determination of Safe Yield and

Compensation Water from Storage Reservoirs TP-4 Functional Evaluation of a Water Resources System TP-5 Streamflow Synthesis for Ungaged Rivers TP-6 Simulation of Daily Streamflow TP-7 Pilot Study for Storage Requirements for Low Flow

Augmentation TP-8 Worth of Streamflow Data for Project Design - A

Pilot Study TP-9 Economic Evaluation of Reservoir System

Accomplishments TP-10 Hydrologic Simulation in Water-Yield Analysis TP-11 Survey of Programs for Water Surface Profiles TP-12 Hypothetical Flood Computation for a Stream

System TP-13 Maximum Utilization of Scarce Data in Hydrologic

Design TP-14 Techniques for Evaluating Long-Tem Reservoir

Yields TP-15 Hydrostatistics - Principles of Application TP-16 A Hydrologic Water Resource System Modeling

Techniques TP-17 Hydrologic Engineering Techniques for Regional

Water Resources Planning TP-18 Estimating Monthly Streamflows Within a Region TP-19 Suspended Sediment Discharge in Streams TP-20 Computer Determination of Flow Through Bridges TP-21 An Approach to Reservoir Temperature Analysis TP-22 A Finite Difference Methods of Analyzing Liquid

Flow in Variably Saturated Porous Media TP-23 Uses of Simulation in River Basin Planning TP-24 Hydroelectric Power Analysis in Reservoir Systems TP-25 Status of Water Resource System Analysis TP-26 System Relationships for Panama Canal Water

Supply TP-27 System Analysis of the Panama Canal Water

Supply TP-28 Digital Simulation of an Existing Water Resources

System TP-29 Computer Application in Continuing Education TP-30 Drought Severity and Water Supply Dependability TP-31 Development of System Operation Rules for an

Existing System by Simulation TP-32 Alternative Approaches to Water Resources System

Simulation TP-33 System Simulation of Integrated Use of

Hydroelectric and Thermal Power Generation TP-34 Optimizing flood Control Allocation for a

Multipurpose Reservoir TP-35 Computer Models for Rainfall-Runoff and River

Hydraulic Analysis TP-36 Evaluation of Drought Effects at Lake Atitlan TP-37 Downstream Effects of the Levee Overtopping at

Wilkes-Barre, PA, During Tropical Storm Agnes TP-38 Water Quality Evaluation of Aquatic Systems

TP-39 A Method for Analyzing Effects of Dam Failures in Design Studies

TP-40 Storm Drainage and Urban Region Flood Control Planning

TP-41 HEC-5C, A Simulation Model for System Formulation and Evaluation

TP-42 Optimal Sizing of Urban Flood Control Systems TP-43 Hydrologic and Economic Simulation of Flood

Control Aspects of Water Resources Systems TP-44 Sizing Flood Control Reservoir Systems by System

Analysis TP-45 Techniques for Real-Time Operation of Flood

Control Reservoirs in the Merrimack River Basin TP-46 Spatial Data Analysis of Nonstructural Measures TP-47 Comprehensive Flood Plain Studies Using Spatial

Data Management Techniques TP-48 Direct Runoff Hydrograph Parameters Versus

Urbanization TP-49 Experience of HEC in Disseminating Information

on Hydrological Models TP-50 Effects of Dam Removal: An Approach to

Sedimentation TP-51 Design of Flood Control Improvements by Systems

Analysis: A Case Study TP-52 Potential Use of Digital Computer Ground Water

Models TP-53 Development of Generalized Free Surface Flow

Models Using Finite Element Techniques TP-54 Adjustment of Peak Discharge Rates for

Urbanization TP-55 The Development and Servicing of Spatial Data

Management Techniques in the Corps of Engineers TP-56 Experiences of the Hydrologic Engineering Center

in Maintaining Widely Used Hydrologic and Water Resource Computer Models

TP-57 Flood Damage Assessments Using Spatial Data Management Techniques

TP-58 A Model for Evaluating Runoff-Quality in Metropolitan Master Planning

TP-59 Testing of Several Runoff Models on an Urban Watershed

TP-60 Operational Simulation of a Reservoir System with Pumped Storage

TP-61 Technical Factors in Small Hydropower Planning TP-62 Flood Hydrograph and Peak Flow Frequency

Analysis TP-63 HEC Contribution to Reservoir System Operation TP-64 Determining Peak-Discharge Frequencies in an

Urbanizing Watershed: A Case Study TP-65 Feasibility Analysis in Small Hydropower Planning TP-66 Reservoir Storage Determination by Computer

Simulation of Flood Control and Conservation Systems

TP-67 Hydrologic Land Use Classification Using LANDSAT

TP-68 Interactive Nonstructural Flood-Control Planning TP-69 Critical Water Surface by Minimum Specific

Energy Using the Parabolic Method

TP-70 Corps of Engineers Experience with Automatic Calibration of a Precipitation-Runoff Model

TP-71 Determination of Land Use from Satellite Imagery for Input to Hydrologic Models

TP-72 Application of the Finite Element Method to Vertically Stratified Hydrodynamic Flow and Water Quality

TP-73 Flood Mitigation Planning Using HEC-SAM TP-74 Hydrographs by Single Linear Reservoir Model TP-75 HEC Activities in Reservoir Analysis TP-76 Institutional Support of Water Resource Models TP-77 Investigation of Soil Conservation Service Urban

Hydrology Techniques TP-78 Potential for Increasing the Output of Existing

Hydroelectric Plants TP-79 Potential Energy and Capacity Gains from Flood

Control Storage Reallocation at Existing U.S. Hydropower Reservoirs

TP-80 Use of Non-Sequential Techniques in the Analysis of Power Potential at Storage Projects

TP-81 Data Management Systems of Water Resources Planning

TP-82 The New HEC-1 Flood Hydrograph Package TP-83 River and Reservoir Systems Water Quality

Modeling Capability TP-84 Generalized Real-Time Flood Control System

Model TP-85 Operation Policy Analysis: Sam Rayburn

Reservoir TP-86 Training the Practitioner: The Hydrologic

Engineering Center Program TP-87 Documentation Needs for Water Resources Models TP-88 Reservoir System Regulation for Water Quality

Control TP-89 A Software System to Aid in Making Real-Time

Water Control Decisions TP-90 Calibration, Verification and Application of a Two-

Dimensional Flow Model TP-91 HEC Software Development and Support TP-92 Hydrologic Engineering Center Planning Models TP-93 Flood Routing Through a Flat, Complex Flood

Plain Using a One-Dimensional Unsteady Flow Computer Program

TP-94 Dredged-Material Disposal Management Model TP-95 Infiltration and Soil Moisture Redistribution in

HEC-1 TP-96 The Hydrologic Engineering Center Experience in

Nonstructural Planning TP-97 Prediction of the Effects of a Flood Control Project

on a Meandering Stream TP-98 Evolution in Computer Programs Causes Evolution

in Training Needs: The Hydrologic Engineering Center Experience

TP-99 Reservoir System Analysis for Water Quality TP-100 Probable Maximum Flood Estimation - Eastern

United States TP-101 Use of Computer Program HEC-5 for Water Supply

Analysis TP-102 Role of Calibration in the Application of HEC-6 TP-103 Engineering and Economic Considerations in

Formulating TP-104 Modeling Water Resources Systems for Water

Quality

TP-105 Use of a Two-Dimensional Flow Model to Quantify Aquatic Habitat

TP-106 Flood-Runoff Forecasting with HEC-1F TP-107 Dredged-Material Disposal System Capacity

Expansion TP-108 Role of Small Computers in Two-Dimensional

Flow Modeling TP-109 One-Dimensional Model for Mud Flows TP-110 Subdivision Froude Number TP-111 HEC-5Q: System Water Quality Modeling TP-112 New Developments in HEC Programs for Flood

Control TP-113 Modeling and Managing Water Resource Systems

for Water Quality TP-114 Accuracy of Computer Water Surface Profiles -

Executive Summary TP-115 Application of Spatial-Data Management

Techniques in Corps Planning TP-116 The HEC's Activities in Watershed Modeling TP-117 HEC-1 and HEC-2 Applications on the

Microcomputer TP-118 Real-Time Snow Simulation Model for the

Monongahela River Basin TP-119 Multi-Purpose, Multi-Reservoir Simulation on a PC TP-120 Technology Transfer of Corps' Hydrologic Models TP-121 Development, Calibration and Application of

Runoff Forecasting Models for the Allegheny River Basin

TP-122 The Estimation of Rainfall for Flood Forecasting Using Radar and Rain Gage Data

TP-123 Developing and Managing a Comprehensive Reservoir Analysis Model

TP-124 Review of U.S. Army corps of Engineering Involvement With Alluvial Fan Flooding Problems

TP-125 An Integrated Software Package for Flood Damage Analysis

TP-126 The Value and Depreciation of Existing Facilities: The Case of Reservoirs

TP-127 Floodplain-Management Plan Enumeration TP-128 Two-Dimensional Floodplain Modeling TP-129 Status and New Capabilities of Computer Program

HEC-6: "Scour and Deposition in Rivers and Reservoirs"

TP-130 Estimating Sediment Delivery and Yield on Alluvial Fans

TP-131 Hydrologic Aspects of Flood Warning - Preparedness Programs

TP-132 Twenty-five Years of Developing, Distributing, and Supporting Hydrologic Engineering Computer Programs

TP-133 Predicting Deposition Patterns in Small Basins TP-134 Annual Extreme Lake Elevations by Total

Probability Theorem TP-135 A Muskingum-Cunge Channel Flow Routing

Method for Drainage Networks TP-136 Prescriptive Reservoir System Analysis Model -

Missouri River System Application TP-137 A Generalized Simulation Model for Reservoir

System Analysis TP-138 The HEC NexGen Software Development Project TP-139 Issues for Applications Developers TP-140 HEC-2 Water Surface Profiles Program TP-141 HEC Models for Urban Hydrologic Analysis

TP-142 Systems Analysis Applications at the Hydrologic Engineering Center

TP-143 Runoff Prediction Uncertainty for Ungauged Agricultural Watersheds

TP-144 Review of GIS Applications in Hydrologic Modeling

TP-145 Application of Rainfall-Runoff Simulation for Flood Forecasting

TP-146 Application of the HEC Prescriptive Reservoir Model in the Columbia River Systems

TP-147 HEC River Analysis System (HEC-RAS) TP-148 HEC-6: Reservoir Sediment Control Applications TP-149 The Hydrologic Modeling System (HEC-HMS):

Design and Development Issues TP-150 The HEC Hydrologic Modeling System TP-151 Bridge Hydraulic Analysis with HEC-RAS TP-152 Use of Land Surface Erosion Techniques with

Stream Channel Sediment Models

TP-153 Risk-Based Analysis for Corps Flood Project Studies - A Status Report

TP-154 Modeling Water-Resource Systems for Water Quality Management

TP-155 Runoff simulation Using Radar Rainfall Data TP-156 Status of HEC Next Generation Software

Development TP-157 Unsteady Flow Model for Forecasting Missouri and

Mississippi Rivers TP-158 Corps Water Management System (CWMS) TP-159 Some History and Hydrology of the Panama Canal TP-160 Application of Risk-Based Analysis to Planning

Reservoir and Levee Flood Damage Reduction Systems

TP-161 Corps Water Management System - Capabilities and Implementation Status