A Model to Estimate Sediment Yield from Field-Sized Areas: Development of Model

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

A MODEL TO ESTIMATE SEDIMENT Y I E L D FROM F I E L D - S I Z E D AREAS:

DEVELOPMENT OF MODEL

G.R. F o s t e r L . J . L a n e J.D. N o w l i n J.M. L a f l e n R.A. Y o u n g J u n e 1980 CP-80- 1 0

C o Z Z a E o r a t i v e Papers report work which has not been performed solely at the International Institute for Applied Systems Analysis and which has received only

limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organi- zations supporting the work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

G.R. FOSTER i s a H y d r a u l i c Engineer w i t h t h e U n i t e d S t a t e s Department o f A g r i c u l t u r e , Science and Education A d m i n i s t r a t i o n , A g r i c u l t u r a l Research (USDA-SEA-AR) and i s an Associate P r o f e s s o r w i t h t h e A g r i c u l t u r a l Engineer- i ng Department a t Purdue U n i v e r s i t y

,

West L a f a y e t t e

,

I n d i a n a 47907, USA.

L.J. LANE i s a H y d r o l o g i s t w i t h t h e USDA-SEA-AR, Southwest Rangeland Water- shed Research Center, 442 East Seventh, Tucson, Arizona 85705, USA.

J.D. NOWLIN i s a Computer Programmer w i t h t h e A g r i c u l t u r a l Engineering De- partment a t Purdue U n i v e r s i t y , West L a f a y e t t e , I n d i a n a 47907, USA.

J .M. LAFLEN i s an A g r i c u l t u r a l Engineer w i t h t h e USDA-SEA-AR, A g r i c u l t u r a l Engineering Department a t Iowa S t a t e U n i v e r s i t y , Ames, Iowa 50011, USA.

R.A. YOUNG i s an A g r i c u l t u r a l Engineer w i t h t h e USDA-SEA-AR, North Central S o i l Conservation Research Center, M o r r i s , Minnesota 56267, USA.

PREFACE

The e r o s i o n o f s o i l by w a t e r i s one o f t h e m a j o r u n d e s i r a b l e consequences o f a g r i c u l t u r e , as s o i l l o s s l e a d s t o a decrease i n t h e n a t u r a l p r o d u c t i v i t y o f t h e agroecosystem. Thus, t h e r e i s an u r g e n t need t o d e v e l o p methods t o p l a n f o r t h e c o n t r o l o f sediment y i e l d . The model p r e s e n t e d i n t h i s paper seems u s e f u l f o r c h e c k i n g t h e c u r r e n t s i t u a t i o n w i t h r e s p e c t t o s o i l e r o s i o n and sediment y i e l d f o r a f i e l d - s i z e d area, and f o r t r y i n g v a r i o u s management a l t e r n a t i v e s t o c o n t r o l t h e problem. W h i l e t h e model i s based on p r e v i o u s e x p e r i e n c e f r o m e x p e r i m e n t a l s t u d i e s and m o d e l l i n g o f w a t e r e r o s i o n and s e d i - m e n t a t i o n , i t goes one s t e p f u r t h e r .

T h i s paper was prepared as a c o n t r i b u t i o n t o o u r c o l l a b o r a t i v e e f f o r t s w i t h t h e U.S. Department o f A g r i c u l t u r e , Science and E d u c a t i o n A d m i n i s t r a t i o n , A g r i c u l t u r a l Research. I t f u l f i l l s t h e r e s e a r c h o b j e c t i v e s o f t h e IIASA t a s k

"Environmental Problems o f A g r i c u l t u r e .

"

Gennady N. Golubev Task Leader

Environmental Probl ems o f A g r i c u l t u r e

J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,

ACKNOWLEDGMENTS

T h i s manuscript i s a c o n t r i b u t i o n o f t h e U n i t e d S t a t e s Department o f A g r i c u l t u r e , Science and E d u c a t i o n A d m i n i s t r a t i o n , A g r i c u l t u r a l Research i n cooperat i o n w i t h t h e Purdue A g r i c u l t u r a l Experiment S t a t i o n . Purdue J o u r - n a l Paper No. 7781.

The erosion/sediment y i e l d model d e s c r i b e d here i s a component o f a comprehensive f i e l d - s c a l e model i n c l u d i n g hydrology, e r o s i o n , p e s t i c i d e , and n u t r i e n t components developed b y USDA-SEA-AR s c i e n t i s t s under t h e lead- e r s h i p o f W.G. K n i s e l , J r . A d d i t i o n a l i n f o r m a t i o n on t h e comprehensive model i s g i v e n b y K n i s e l (1978). A d d i t i o n a l support f o r t h i s p r o j e c t ,

i n c l u d i n g computer p r o g r a m i n g b y V.A. F e r r e i r a and t y p i n g b y E.S. S c h i e l d , J.J. Rocha and K.L. M e l l o r , was p r o v i d e d b y K.G. Renard o f t h e Southwest Watershed Research Center, USDA-SEA-AR, Tucson, A r i z o n a and b y K . C . Moldenhauer of t h e E r o s i o n Research U n i t , USDA-SEA-AR, L a f a y e t t e , I n d i a n a .

J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,

ABSTRACT

A t o o l f o r e v a l u a t i n g sediment y i e l d f r o m f i e l d - s i z e d areas i s needed f o r p l a n n i n g management p r a c t i c e s t o c o n t r o l sediment y i e l d . We developed a r e a s o n a b l y s i m p l e s i m u l a t i o n model which i n c o r p o r a t e s fundamental p r i n c i - p l e s o f e r o s i o n , d e p o s i t ion, and sediment t r a n s p o r t mechanics. The model summarizes t h e s t a t e - o f - t h e - a r t i n e r o s i o n and sediment y i e l d model i n g w i t h a p p r o p r i a t e s i m p l i f i c a t i o n s r e q u i r e d t o c o u p l e t h e g o v e r n i n g e q u a t i o n s .

L i m i t e d t e s t i n g showed t h a t t h e procedures developed here g i v e improv- ed e s t i m a t e s over t h e U n i v e r s a l S o i l Loss Equation. S p e c i f i c components o f t h e model were t e s t e d u s i n g e x p e r i m e n t a l d a t a from over1 and f l o w , e r o d i b l e channel, and impoundment s t u d i e s . These r e s u l t s suggest t h a t t h e model produces reasonable e s t i m a t e s o f e r o s i o n , sediment t r a n s p o r t , and deposi- t i o n under a v a r i e t y o f c i r c u m s t a n c e s common t o f i e l d - s c a l e areas.

A l t e r n a t i v e management p r a c t i c e s such as c o n s e r v a t i o n t i l l a g e , t e r r a c - i n g , and c o n t o u r i n g can be e v a l u a t e d sepa;-ately o r i n combination t o d e t e r - mine t h e i r i n f l u e n c e on sediment y i e l d . Given a p a r t i c u l a r l o c a t i o n w i t h s p e c i f i e d c h a r a c t e r i s t i c s f o r c l i m a t e , s o i l s , topography, and crops, t h e model p r o v i d e s a means o f e v a l u a t i n g a1 t e r n a t i v e management p r a c t i c e s t o

s u i t a p a r t i c u l a r f a r m i n g o p e r a t i o n .

J. ~ i i l s c h l e g e l i s w i t h t h e N a t i o n a l I n s t i t u t e f o r Water s u p p l y ,

CONTENTS

INTRODUCTION, 1

OVERVIEW OF THE MODEL, 2 BASIC CONCEPTS, 4

BASIC EQUATIONS, 4

Detachment-Deposition L i m i t i n g Cases, 6 Sediment C h a r a c t e r i s t i c s , 8

OVERLAND FLOW ELEMENT, 11 Detachment Equation, 11 Storm E r o s i v i t y , 11

Slope Length Exponent, 1 2

Sediment T r a n s p o r t Capacity, 12

Conversion f r o m Storm t o Rate B a s i s , 1 6 Shear S t r e s s , 16

Slope Segments, 17

S e l e c t i o n o f Parameter Values, 18 CHANNEL ELEMENT, 18

Spat i a1 l y V a r i e d Flow E q u a t i o n s

,

19 Concentrated F l ow Detachment, 21

Sediment T r a n s p o r t and P a r t i t i o n i n g o f Shear S t r e s s , 25 IMPOUNDMENT (POND) ELEMENT, 25

R u n o f f Reduction, 27

VALIDATIOf4 OF THE MODEL, 28 Cornparison w i t h Other Models, 28

Comparison o f Output f r o m Flodel w i t h Observed data, 28 SIMllLATION COSTS, 30

SUMMARY AND CONCLUSIONS, 31 References, 32

Appendix A: L i s t o f Symbol s

,

36

A MODEL TO ESTIMATE SEDIMENT YIELD FROM F IELD-SIZED AREAS: DEVELOPMENT OF MODEL G. R. F o s t e r , L. J. Lane, J. D. Nowlin, J. M. L a f l e n , and R. A. Young

INTRODUCTION

E s t i m a t e s o f e r o s i o n and sediment y i e l d on f i e l d - s i z e d areas are need- ed t o w i s e l y s e l e c t b e s t management p r a c t i c e s t o c o n t r o l e r o s i o n f o r main- tenance of s o i l p r o d u c t i v i t y and c o n t r o l of sediment y i e l d t o p r e v e n t ex- c e s s i v e d e g r a d a t i o n o f water q u a l i t y . A f i e l d i s a t y p i c a l management u n i t f o r farmers and each f i e l d has s p e c i f i c c o n d i t i o n s upon which t h e s e l e c t i o n o f a management p r a c t i c e should be based. S o i l c o n s e r v a t i o n i s t s have used t h e U n i v e r s a l S o i l Loss E q u a t i o n (USLE) (Wischrneier and Smith 1978) f o r s e v e r a l years t o s e l e c t p r a c t i c e s s p e c i f i c a l l y t a i l o r e d t o a g i v e n f a r m e r ' s s i t u a t i o n . Consequently, i f sediment y i e l d t o l e r a n c e s f o r maintenance o f water q u a l i t y are e s t a b l i s h e d f o r g i v e n l o c a l areas, b e s t management prac- t i c e s can t h e n be s e l e c t e d based on a g i v e n f a r m e r ' s needs and t h e t o l e r - a b l e water l o a d i n g f o r f i e l d s i n h i s area u s i n g a model such as t h e one d e s c r i b e d h e r e i n ( F o s t e r 1979).

Sediment y i e l d i s a f u n c t i o n o f detachment o f s o i l p a r t i c l e s and t h e subsequent t r a n s p o r t o f t h e s e p a r t i c l e s ( s e d i m e n t ) . On a g i v e n f i e l d , e i - t h e r detachment o r sediment t r a n s p o r t c a p a c i t y may l i m i t sediment y i e l d depending on topography, s o i l c h a r a c t e r i s t i c s , cover, and r a i n f a l l / r u n o f f r a t e s and amounts. C o n t r o l o f sediment y i e l d b y detachment o r t r a n s p o r t can change f r o m season t o season, from storm t o storm, and even w i t h i n a storm. The mathematical r e l a t i o n s h i p f o r detachment i s d i f f e r e n t from t h e one f o r t r a n s p o r t , so t h e y cannot be lumped i n t o a s i n g l e e q u a t i o n . Since e r o s i o n and t r a n s p o r t f o r each storm are b e s t c o n s i d e r e d s e p a r a t e l y , lurrlped e q u a t i o n s such as t h e USLE ( a n e r o s i o n e q u a t i o n ) o r W i l l i a m s ' (1975) modi- f i e d USLE ( a f l o w t r a n s p o r t sediment y i e l d e q u a t i o n ) cannot g i v e t h e b e s t r e s u l t s over a broad range o f c o n d i t i o n s on f i e l d - s i z e d areas. F u r t h e r - more, t h e i n t e r r e l a t i o n between detachment and t r a n s p o r t i s n o n l i n e a r and i n t e r a c t i v e f o r each s t o r m which p r e v e n t s u s i n g separate e q u a t i o n s t o l i n e - a r l y accumulate detachment o r sediment t r a n s p o r t c a p a c i t y over s e v e r a l

storms. T h e r e f o r e , t o s i m u l a t e e r o s i o n and sediment y i e l d on an i n d i v i d u a l s t o r m b a s i s and t o s a t i s f y t h e need f o r a c o n t i n u o u s s i m u l a t i o n model

,

a r a t h e r fundamental approach was s e l e c t e d where s e p a r a t e e q u a t i o n s a r e used f o r detachment and sediment t r a n s p o r t .

A number o f f u n d a m e n t a l l y based models (e.g., Beasley e t a1

.

1977, L i 1977) compute e r o s i o n and t r a n s p o r t a t v a r i o u s t i m e s d u r i n g t h e r u n o f f e- v e n t . A1 though t h e s e models a r e powerful

,

t h e i r e x c e s s i v e use o f computer t i m e p r a c t i c a l l y p r o h i b i t s s i m u l a t i n g 20 t o 30 y e a r s o f r e c o r d . Our model uses c h a r a c t e r i s t i c r a i n f a l l and r u n o f f f a c t o r s f o r a s t o r m t o compute ero- s i o n and sediment t r a n s p o r t f o r t h a t storm. I n terms o f computational t i m e , t h i s amounts t o a s i n g l e t i m e s t e p f o r models t h a t s i m u l a t e over t h e e n t i r e r u n o f f event.

The model i s i n t e n d e d t o be u s e f u l w i t h o u t c a l i b r a t i o n o r c o l l e c t i o n o f r e s e a r c h d a t a t o d e t e r m i ne parameter values. T h e r e f o r e , e s t a b l i shed r e - l a t i o n s h i p s such as t h e USLE were m o d i f i e d and used i n t h e model.

OVERVIEW OF THE MODEL

Every model i s a r e p r e s e n t a t i o n and a s i m p l i f i c a t i o n o f t h e p r o t o t y p e . V a r i o u s t e c h n i q u e s , i n c l u d i n g p l anes and channel s ( L i 1977)

,

square g r i d s

( B e a s l e y e t a l . 1977), c o n v e r g i n g s e c t i o n s ( s m i t h 1977), and stream tubes (Onstad and F o s t e r 1975) have been used. Most erosion-sediment y i e l d mo- d e l s have adequate degrees o f freedom t o f i t observed data. Some model s, depending on t h e i r r e p r e s e n t a t i o n scheme, d i s t o r t parameter v a l u e s more t h a n do o t h e r s . D i s t o r t i o n of parameter v a l u e s g r e a t l y reduces t h e i r t r a n s f e r a b i 1 i t y f r o m one area t o a n o t h e r (Lane e t a1

.

^1975). An o b j e c t i v e i n t h i s model development was t o r e p r e s e n t t h e f i e l d i n a way t h a t m i n i - m i z e s parameter d i s t o r t i o n .

H y d r o l o g i c i n p u t t o t h e e r o s i o n component c o n s i s t s o f r a i n f a l l volume, r a i n f a l l e r o s i v i t y , r u n o f f volume, and peak r a t e o f r u n o f f . These terms d r i v e s o i l detachment and subsequent t r a n s p o r t by o v e r l a n d and open channel f l o w .

Overl and f l o w , channel f l o w , and impoundment (pond) elements are used t o r e p r e s e n t m a j o r f e a t u r e s o f a f i e l d . The u s e r s e l e c t s t h e b e s t combina- t i o n o f elements and e n t e r s t h e a p p r o p r i a t e sequence number a c c o r d i n g t o T a b l e 1. The model (computer program) c a l l s t h e elements i n t h e p r o p e r se-

quence. T y p i c a l systems t h a t t h e model can r e p r e s e n t a r e i l l u s t r a t e d i n f i g u r e 1.

T a b l e 1. P o s s i b l e elements and t h e i r c a l l i n g sequence used t o r e p r e s e n t f i e l d - s i z e d areas.

Sequence number E l ements and t h e i r sequence

1 Over1 and

2 Over1 and-Pond

3 Over1 and-Channel

4 Over1 and-Channel -Channel

5 Over1 and-Channel -Pond

6 Over1 and-Channel -Channel -Pond

OVERLAND FLOW SLOPE REPRESENTATION

OVERLANO FLOW

I STREAM

I

-

¹

AVERAGE SLOPE \

Jx.*yC

I ^C

- - -

€,,Q S L O ~ ~ I X , .O)

( 1 ) OVERLAND FLOW

SEOUENCE AND SLOPE REPRESENTATION

OVERLAND

ImPounoucNT/

'

TERRACE

'

uNOERGROUND OUTLET

( 2 ) OVERLAND FLOW POND SEOUENCE

'ERRPCE FLOW

( 4 ) OVERLAND FLOW

CHANNEL-CHANNEL SEQUENCE

CONCENTRATE0 FLOW

( 3 ) OVERLAND FLOW

CHANNEL SEQUENCE

OVERLANO FLOW C

cnANNEL FLOW

FIELO OUTLET

( 5 ) OVERLAND FLOW CHANNEL-POND SEOUENCE

Figure 1. Schematic representation of typical field systems in the field- scale erosion/sediment yield model

.

Computations b e g i n i n t h e uppermost element, which i s always t h e o v e r - l a n d f l o w element, and proceed downstream. Sediment c o n c e n t r a t i o n f o r each p a r t i c l e t y p e i s t h e o u t p u t from each element, which becomes t h e i n p u t t o t h e n e x t element i n t h e sequence.

BASIC CONCEPTS

Sediment l o a d i s assumed t o be l i m i t e d b y e i t h e r t h e amount o f s e d i - ment made a v a i l a b l e b y detachment o r by t r a n s p o r t c a p a c i t y ( F o s t e r and Me- y e r 1975). A l s o q u a s i - s t e a d y s t a t e i s assumed so t h a t a r a i n f a l l and a r u n o f f r a t e c h a r a c t e r i s t i c o f each s t o r m can be used i n t h e computations.

BASIC EQUATIONS

The e q u a t i o n f o r c o n t i n u i t y of mass f o r sediment movement downslope i s expressed by:

where qS = sediment l o a d per u n i t w i d t h per u n i t t i m e , x = d i s t a n c e , DL

= l a t e r a l i n f l o w of sediment, and DF = detachment o r d e p o s i t i o n b y f l o w . D e l e t i o n o f t i m e t e r m s f r o m e q u a t i o n 1 i s p o s s i b l e b y t h e q u a s i - s t e a d y s t a t e assumption. The m a j o r sequence o f computat i o n s i s il l u s t r a t e d i n f i g u r e 2.

L a t e r a l sediment i n f l o w i s f r o m i n t e r r i l l e r o s i o n on o v e r l a n d f l o w elements, o r i t i s f r o m o v e r l a n d f l o w ( o r a channel i f two channel elements a r e i n t h e sequence) f o r t h e channel elements. Flow i n r i l l s on o v e r l a n d f l o w areas o r i n channels t r a n s p o r t s t h e sediment l o a d downstream. L a t e r a l sediment i n f l o w i s assumed r e g a r d l e s s o f whether t h e f l o w i s d e t a c h i n g o r d e p o s i t i n g sediment.

For a segment, e i t h e r on t h e p r o f i l e f o r t h e o v e r l a n d f l o w element o r i n a channel, t h e model computes an i n i t i a l p o t e n t i a l sediment load, which i s t h e sum o f t h e sediment l o a d f r o m t h e immediate upslope segment p l u s t h a t added b y l a t e r a l i n f l o w w i t h i n t h e segment. I f t h i s p o t e n t i a l l o a d i s l e s s t h a n t h e f l o w ' s t r a n s p o r t c a p a c i t y , detachment o c c u r s a t t h e l e s s e r of e i t h e r t h e detachment c a p a c i t y r a t e o r t h e r a t e which w i l l j u s t f i l l t r a n s - p o r t c a p a c i t y . When detachment b y f l o w occurs, i t adds p a r t i c l e s h a v i n g t h e p a r t i c l e s i z e d i s t r i b u t i o n f o r detached sediment g i v e n as i n p u t . No s o r t i n g i s a l l o w e d d u r i n g detachment.

I f t h e i n i t i a l p o t e n t i a l sediment l o a d i s g r e a t e r t h a n t h e t r a n s p o r t c a p a c i t y , d e p o s i t i o n i s assumed t o occur a t t h e r a t e o f :

START

v ,--

FROM UPSLOPE SEGMENT

ADDITION FROM L A T E R A L INFLOW

LOADS FOR AN I N I T I A L - P O T E N T I A L

SEOIMLNT LOAD

COMPUTE TRANSPORT

+

CAPACIT Y BASED ON

^I

POTENTIAL SEDIMENT

SED. LOA

COMPUTE DEPOSITION

1 - 7 2 7

LOAD LEAVING SEGMENT

SEGMENT

COMPUTE FLOW DETACHMENT

CAPACITY

COMPUTE NEW POTENTIAL SEDIMLNT LOAD AS SUM Of SEDIMENT FROM

DETACHMENT CAPACITY AND I N I T I A L - P O T E N T I A L

SLDIMENT LOAD

COMPUTE TRANSPORT CAPACITY BASED ON

NEW POTENTIAL SEDIMENT LOAD

1 I TO THAT w n r c n WILL I

JUST F I L L TRANSPORT SEDIMENT LOAD LEAVING

SEGMENT EQUALS NEW P O T E N T I A L S L D I M L N T LOAD

LEAVING SEGMENT

GO TO NEXT EOUALS TRANSPORT

SEGMENT

Figure 2. Flow chart for detachment-transport-deposition computations within a segment of overland flow or concentrated flow elements.

where D = d e p o s i t i o n r a t e ( m a s s l u n i t a r e a l u n i t t i m e ) , a = a f i r s t o r d e r r e - a c t i o n c o e f f i c i e n t (1engtI-i 1 ) , and Tc ⁼ t r a n s p o r t c a p a c i t y ( m a s s l u n i t w i d t h / u n i t t i m e ) . The c o e f f i c i e n t a i s e s t i m a t e d from:

where ^E = 0.5 f o r o v e r l a n d f l o w ( D a v i s 1978) and 1.0 f o r channel f l o w ( E i n s t e i n 1968), V S = p a r t i c l e f a l l v e l o c i t y , and qLx = qw = d i s - c h a r g e r a t e o f r u n o f f p e r u n i t w i d t h ( v o l u m e / w i d t h / t i m e ) . F a l l v e l o c i t y i s e s t i m a t e d assuming s t a n d a r d d r a g r e l a t i o n s h i p s f o r a sphere o f a g i v e n d i - ameter and d e n s i t y f a 1 1 in g i n s t i l 1 water.

Detachment-Deposi t i o n L i m i t i n g Cases

F o u r p o s s i b l e cases may e x i s t f o r a segment: ( 1 ) d e p o s i t i o n may o c c u r o v e r t h e e n t i r e segment; ( 2 ) detachment by f l o w i n t h e upper end and depo- s i t i o n i n t h e l o w e r end may o c c u r ( b u t n o t n e c e s s a r i l y ) when t r a n s p o r t ca- p a c i t y decreases w i t h i n a segment; ( 3 ) d e p o s i t i o n on t h e upper end and de- tachment by f l o w i n t h e l o w e r end may o c c u r ( b u t n o t n e c e s s a r i l y ) when t r a n s p o r t c a p a c i t y i n c r e a s e s w i t h i n t h e segment; ( 4 ) detachment by f l o w may o c c u r a l l a l o n g t h e segment.

Case 1 o c c u r s when Tc

<

qs a l l a l o n g t h e segment. Where deposi- t i o n o c c u r s over t h e e n t i r e segment l e n g t h , d e p o s i t i o n r a t e i s :

where:

where dTc/dx i s assumed c o n s t a n t o v e r t h e segment and Du ⁼ d e p o s i t i o n r a t e a t xu.

The d e p o s i t i o n r a t e Du may be e s t i m a t e d from:

where Tcu and qsu = r e s p e c t i v e l y t h e t r a n s p o r t c a p a c i t y and s e d i - ment l o a d a t xu. Sediment l o a d a t x i s :

Case 2 o c c u r s when Tcu

>

qsu, dTc/dx

<

^0, and Tc becomes l e s s t h a n qs w i t h i n t h e segment. I f dTc/dx

<

0 f o r a segment where Tcu

>

qsu, Tc may decrease below qs w i t h i n t h e segment. The p o i n t where qs = Tc i s determined as xdb which i s used f o r xu i n e q u a t i o n 5, w i t h Du

= 0. D e p o s i t i o n and sediment l o a d a r e computed f r o m e q u a t i o n s 4 , 5, and 7.

Case 3 occurs when Tcu< qsu, dTc/dx

>

^{0, and Tc} becomes g r e a t e r t h a n qs w i t h i n t h e segment. I n s i t u a t i o n s l i k e a grass b u f f e r s t r i p , t h e t r a n s p o r t c a p a c i t y a t t h e upper edge may d r o p a b r u p t l y below t h e sediment l o a d . W i t h i n t h e upper end o f t h e s t r i p , t h e sediment l o a d decreases due t o d e p o s i t i o n w h i l e t h e t r a n s p o r t c a p a c i t y i n c r e a s e s f r o m t h e p o i n t o f t h e a b r u p t decrease. Somewhere upslope from t h e l o w e r edge o f t h e s t r i p , t h e sediment l o a d equals t h e t r a n s p o r t c a p a c i t y . A t t h i s p o i n t , x d e , deposi- t i o n ends, i.e., Du = 0 and, Tc = qS. Downslope, detachment by f l o w occurs. The p o i n t where d e p o s i t i o n ends i s g i v e n by:

where:

and Tcu = t r a n s p o r t c a p a c i t y a f t e r t h e a b r u p t decrease a t x, i:d qSu -

-

sediment l o a d a t xu. C o n t i n u i t y o f sediment l o a d i s maintained, b u t D may be d i s c o n t i n u o u s a t segment ends.

Downslope f r o m xde, where f l o w detachment occurs, t h e sediment 1 oad i s g i v e n by:

where t h e second s u b s c r i p t u o r L i n d i c a t e s upper o r l o w e r and AX = l e n g t h o f t h e segment where detachment by f l o w i s o c c u r r i n g . I n t h i s case, Ax i s from xde t o t h e l o w e r end o f t h e segment; q s u i s a t xde, which i s Tc a t 'de ; DFu = 0 a t xde ; and DFL i s e i t h e r detachment c a p a c i t y a t x o r t h a t which w i l l j u s t f i l l t h e t r a n s p o r t c a p a c i t y .

Case 4 o c c u r s when Tc

>

qs o v e r t h e e n t i r e segment. Sediment l o a d i s computed w i t h e q u a t i o n 10.

The e q u a t i o n f o r sediment t r a n s p o r t c a p a c i t y ( d i s c u s s e d 1 a t e r ) s h i f t s t o t a l t r a n s p o r t c a p a c i t y among t h e v a r i o u s p a r t i c l e types. I f t r a n s p o r t c a p a c i t y exceeds a v a i l a b i l i t y f o r one p a r t i c l e w h i l e i t i s l e s s f o r an- o t h e r , t r a n s p o r t c a p a c i t y i s s h i f t e d f r o m t h e p a r t i c l e t y p e having t h e ex- cess t o t h e one h a v i n g t h e d e f i c i t . Furthermore, l o g i c checks w i t h i n t h e model p r e v e n t simultaneous d e p o s i t i o n and detachment o f p a r t i c l e s by f l o w .

Eroded sediment i s a m i x t u r e o f p a r t i c l e s having v a r i o u s s i z e s and d e n s i t i e s . The d i s t r i b u t i o n i s broken i n t o c l a s s e s , w i t h each c l a s s r e p r e - sented by a p a r t i c l e d i a m e t e r and d e n s i t y . E q u a t i o n s 4-10 a r e solved f o r each p a r t i c l e t y p e w i t h i n t h e g i v e n c o n s t r a i n t s .

Sediment C h a r a c t e r i s t i c s

Sediment eroded on f i e l d - s i zed areas i s a m i x t u r e o f p r i m a r y p a r t i c l e s and aggregates (conglomerates o f p r i m a r y p a r t i c l e s ) . The d i s t r i b u t i o n o f t h e s e p a r t i c l e s as t h e y a r e detached i s a f u n c t i o n o f s o i l p r o p e r t i e s , man- agement, and r a i n f a l 1 and r u n o f f c h a r a c t e r i s t i c s . I f d e p o s i t i o n occurs, u s u a l l y t h e coarse and dense p a r t i c l e s a r e d e p o s i t e d f i r s t , l e a v i n g a f i n e r sediment m i x t u r e . The i n p u t t o t h e model i s t h e d i s t r i b u t i o n o f t h e sedi- ment as i t i s detached; t h e model c a l c u l a t e s a new d i s t r i b u t i o n i f i t c a l - c u l a t e s t h a t d e p o s i t i o n occurs.

Based on o u r survey o f e x i s t i n g data, v a l u e s g i v e n i n T a b l e 2 a r e t y p - i c a l o f many Midwestern s o i l s .

I f t h e p a r t i c l e d i s t r i b u t i o n i s n o t known, t h e model assumes f i v e par- t i c l e t y p e s , and e s t i m a t e s t h e d i s t r i b u t i o n from t h e p r i m a r y p a r t i c l e s i z e d i s t r i b u t i o n .

PSA ⁼ (1.0

-

O R C L ) ~ ' ~ ~ ORSA

P S I = 0.13 O R S I (12)

PCL = 0.2 ORCL ( 1 3 )

I

^{2 ORCL} f o r ORCL

<

0.25 ( 1 4 )

0.28(ORCL

-

^0.25) + 0.5 f o r 0.25

5

ORCL

5

0.50 (15

10.57 f o r 0.5

<

ORCL ( 1 6 )

LAG = 1.0

-

PSI,

-

PSI

-

PCL

-

SAG ( 1 7 )

I f LAG

<

0.0, multiply a l l o t h e r s by t h e same r a t i o t o make

LAG = 0.0 (18)

Table 2. Sediment c h a r a c t e r i s t i c s assumed f o r detached sediment before de- p o s i t i o n . Assumed typical of many Midwestern s i l t loam s o i l s .

Fraction of t o t a l

Part i cl e S p e c i f i c amount

type Diameter g r a v i t y (mass b a s i s )

(m)

Primary c l a y 0.002 2.60 0.05

Primary s i l t 0.010 2.65 0.08

Small aggregates 0.030 1.80 0.50

Large aggregates 0.500 1.60 0 -31

Primary sand 0 .ZOO 2.65 0 -06

where O R C L , ORSI, and ORSA a r e , r e s p e c t i v e l y , f r a c t i o n s f o r primary c l a y , s i l t , and sand in t h e o r i g i n a l s o i l mass, and P C L , PSI, PSA, SAG, and LAG a r e , r e s p e c t i v e l y , f r a c t i o n s f o r primary c l a y , s i l t , sand, and small and l a r g e aggregates i n t h e detached sediment.

The diameters f o r t h e p a r t i c l e s a r e given by:

DPCL = 0.002 mm (19)

DPSI = 0.010

mm

(20)

DPSA = 0.20 mm (21)

0.03

mm

f o r O R C L

<

^{0.25 (22)}

DSAG = 0.20(0RCL

-

^{0.25) + 0.03}

rrm

f o r 0.25 ORCL 0.60 (22)

i

f o r 0.60

<

ORCL (24)

DLAG = ~ ( O R C L ) mm (25)

where D P C L , DPSI, DPSA, DSAG, and DLAG a r e , r e s p e c t i v e l y , t h e diameters of t h e primary c l a y , s i l t , and sand, and the small and l a r g e aggregates in sediment. The assumed s p e c i f i c g r a v i t i e s a r e shown in Table 2. The pri- mary p a r t i c l e composition of t h e sediment load i s estimated from:

Small aggregates:

SISAG = SAG ORSII (ORCL + ORSI) SASAC = 0.0

Large aggregates:

CLLAG = ORCL

-

^PCL

-

^CLSAG (29)

SILAG = O R S I

-

PSI

-

^SISAG ( 3 0 )

SALAG = ORSA

-

PSA (31)

where CLSAG, SISAG, and SASAG a r e f r a c t i o n s o f t h e t o t a l f o r t h e sediment o f , r e s p e c t i v e l y , p r i m a r y c l a y , s - i l t , and sand i n t h e small aggregates i n t h e scdiment l o a d , and CLLAG, SILAG, and SALAG a r e c o r r e s p o n d i n g f r a c t i o n s f o r t h e l a r g e aggregates.

I f t h e c l a y i n t h e l a r g e aggregate expressed as a f r a c t i o n f o r t h a t p a r t i c l e a l o n e i s l e s s t h a n 0.5 t i m e s ORCL, t h e d i s t r i b u t i o n o f t h e p a r t i - c l e t y p e s i s recomputed so t h a t t h i s c o n s t r a i n t can be met. A sum SUMPRI i s computed whereby:

SUMPRI = PCL + PSI + PSA ( 3 2 )

The f r a c t i o n s PSA, PSI, PCL a r e n o t changed. The new SAG i s :

SAG = (0.3 + 0.5 SUMPRI)(ORCL

+

ORS1)/[1

-

^{O.~(ORCL + ORSI)] (33)}

E q u a t i o n 33 i s d e r i v e d g i v e n p r e v i o u s l y determined values f o r PCL, PSI, and PSA; t h e sum o f p r i m a r y c l a y f r a c t i o n s f o r t h e t o t a l sediment equals t h e c l a y f r a c t i o n i n t h e o r i g i n a l s o i l ; and t h e assumption t h a t t h e f r a c t i o n o f p r i m a r y c l a y i n LAG e q u a l s one h a l f o f t h e p r i m a r y c l a y i n t h e o r i g i n a l s o i l .

The model a l s o computes an enrichment r a t i o u s i n g s p e c i f i c s u r f a c e areas f o r o r g a n i c m a t t e r , c l a y , s i l t , and sand. Organic m a t t e r i s d i s t r i - b u t e d among t h e p a r t i c l e t y p e s based on t h e p r o p o r t i o n o f p r i m a r y c l a y i n each type. Enrichment r a t i o i s t h e r a t i o o f t h e t o t a l s p e c i f i c s u r f a c e a r e a f o r t h e sediment t o t h a t f o r t h e o r i g i n a l s o i l .

A1 though t h e s e r e l a t i o n s h i p s a r e a p p r o x i m a t i o n s t o t h e d a t a found i n t h e 1 i t e r a t u r e ( R . A. Young 1978 personal comnunication, USDA-SEA-AR, M o r r i s , Minnesota)

,

t h e y r e p r e s e n t t h e general t r e n d s .

OVERLAND FLOW ELEMENT Detachment Equation

Detachment on i n t e r r i l l and r i l l areas and transport and deposition by r i l l flow a r e the erosion-transport processes on the overland flow element.

Detachment i s described by a modified USLE written as:

D L i = 4 * 5 7 ( E I ) ( s + 0.014) KCP ( 0 /V )

P

u and

where D Li = i n t e r r i l l detachment r a t e (g/m I s ) , DFr 2 = r i l l detach- ment capacity r a t e (g/m2/s)

,

El = Wischmeier's r a i n f a l l e r o s i v i t y ex- pressed as t o t a l rain storm energy times maximum 30-minute i n t e n s i t y ( N l h )

, x

= distance downslope

( m ) ,

s ⁼ sine of slope angle,

m

= slope length expo- nent, K = USLE soil erodibil i t y f a c t o r [g h / ( N

m2)],

C = s o i l l o s s r a t i o of the USLE cover-management f a c t o r , P = USLE contouring f a c t o r , V u -

-

runoff volume/area

( m ) ,

and op = peak runoff r a t e expressed as volume/area/

time (m/s). The units

on

t h e USLE K (Wischmeier e t a l . 1971) must be c a r e f u l l y noted. Multiplication of K i n standard Engl ish units by 131.7 gives a metric K having u n i t s of g h / ( N

m 2 ) .

Only the contouring part of the USLE P f a c t o r i s used. Other P f a c t o r e f f e c t s such as s t r i p cropping and deposition in t e r r a c e channels a r e ac- counted f o r d i r e c t l y in the model. The model more accurately represents these f a c t o r s than do the broad averages given f o r the USLE (Wischmeier and Smith 1978)

.

Storm Erosivity

The hydrologic processes of r a i n f a l l and runoff drive tile erosion- transport processes. Storm EI, volume of runoff, and peak discharge a r e t h e variables used t o characterize hydrologic inputs. Val ues f o r these f a c t o r s a r e generated by a hydro1 ogic model

, or

observed data may be used.

Techniques are commonly available f o r estimating the runoff f a c t o r s (e.g., Schwab e t a l . 1966).

An

approximate estimate of storm EI i s (Lombardi 1979) :

EI = 0.103 VR 1.51

(36 where El = storm El ( N / h ) and V R = volume of r a i n f a l l (m). Multiplica- t i o n of EI in standard English units by 1.702 gives a metric EI having

u n i t s o f N/h. E q u a t i o n 36 was developed b y r e g r e s s i o n a n a l y s i s f r o m about 2,700 d a t a p o i n t s used i n t h e development o f t h e USLE and has a c o e f f i c i e n t o f d e t e r m i n a t i o n ( ~ 2 ) o f 0.56. T h i s r e l a t i o n s h i p should be used o n l y as a

l a s t r e s o r t .

S i n c e r a i n f a l l energy f o r h i g h e r i n t e n s i t i e s does n o t v a r y g r e a t l y w i t h i n t e n s i t y (Wischmeier and Smith 1978), t h e approximate r a i n f a l l energy per u n i t r a i n f a l l i s 27.6 J/$ /n of r a i n . An e s t i m a t e o f storm E I (N/h) i s :

where I = maximum 30 m i n u t e i n t e n s i t y (mmlh). I f t h e r a i n f a l l hyetograph i s a v a i l a b l e , storm E I can be computed from:

where e = r a i n f a l l energy per u n i t o f r a i n f a l l (J/m2/mm o f r a i n ) and i = r a i n f a1 1 i n t e n s i t y ( n / h )

.

The d i f f e r e n c e between E I computed from equa- t i o n 37 o r computed f r o m e q u a t i o n 38 where increment e n e r g i e s are computed and summed i s n e g l i g i b l e .

Slope Length Exponent

For s l o p e s l e s s t h a n 50 m, t h e s l o p e l e n g t h exponent m i s s e t t o 2, b u t f o r s l o p e s l o n g e r t h a n 50 m, m i s l i m i t e d by:

T h i s l i m i t avoids e x c e s s i v e e r o s i o n f o r v e r y l o n g slopes ( F o s t e r e t a l . 1977)

.

Sed iment T r a n s p o r t Capac i t y

The Yal i n sediment t r a n s p o r t e q u a t i o n ( Y a l i n 1963) i s used t o de- s c r i b e sediment t r a n s p o r t c a p a c i t y . It gave reasonable r e s u l t s when com- pared w i t h e x p e r i m a n t a l d a t a f o r d e p o s i t i o n o f sand and c o a l b y o v e r l a n d f l o w i n a l a b o r a t o r y s t u d y ( F o s t e r and Huggins 1977, Davis 1978) and unpu- b l is h e d f i e l d p l o t d a t a ( T a b l e 3 ) . The Y a l i n e q u a t i o n was m o d i f i e d t o d i s - t r i b u t e t r a n s p o r t c a p a c i t y among t h e v a r i o u s p a r t i c l e t y p e s . 'The d i s c u s - s i o n o f t h e method g i v e n below i s a b s t r a c t e d f r o m F o s t e r and Meyer (1972), Davis (1978), and K h a l e e l e t a l . (1980).

Tab1 e 3 . Sediment y i e l d i n over1 and f l o w f r o m concave slopes.

Sediment T r a n s p o r t r a t e s

S p e c i f i c Shear s t r e s s Observed Cal c d l a t e d u s i ng

Diameter g r a v i t y Yal i n e q u a t i o n

(mn) ( N I I I I ~ ) ( g l m l s ) (s/m/s)

0.342 2.65 0.52 5.6 4.2

0.342 2.65 0.76 19.7 13.4

0 .I50 2.65 0.55 5.2 9 .O

0 .I50 2.65 0.70 18.8 17.6

0.342 2.65 0.40 2.2 1.4

0.342 2.65 0.60 12.8 6.6

0.342 1.60 0.30 3.5 6 .O

0.342 1.60 0.42 13.7 14.4

0 .I56 1.67 0.30 3.8 6.1

0.156 1.67 0.40 13.3 12.7

Eroded from Barnes 0.33 3.3 2.8

loam, f i e l d p l o t s

Eroded f r o m Miami s i l t 0.51 4.8 7 .O

loam, f i e l d p l o t s

Eroded from Miami s i l t 0.35 2.5 3.1

loam, f i e l d p l o t s

Eroded f r o m Miami s i 1 t

- -

^1.4

- -

loam, f i e l d p l o t s

Source: l e i b l i ng and F o s t e r ( 1 9 8 0 ) . The Yal i n e q u a t i o n i s g i v e n by:

where

o = A 6

6 = (Y/Ycr) -1 (whcn Y

<

^{Ycr, 6} = 0 )

where

Vt =

shear velocity = ( ~ / p ~ ) l ' ~ ,

r =

shear s t r e s s ,

g =

acceleration due t o gravity,

p w =

mass density of the f l u i d ,

R =

hydraulic radius,

Sf =

slope of the energy grade1 ine, Sg

=

p a r t i c l e specific gravity,

d =

parti- c l e diameter,

Ycr =

c r i t i c a l l i f t force given by the Shields' diagram ex- tended t o small p a r t i c l e Reynolds numbers (Mantz 1977), and

Ws =

trans- port capacity (massluni t timeluni t flow width). The constant

0.635

and Shields' diagram were empirically derived.

The sediment load may have fewer particles of a given type than the flow's transport capacity for that type. A t the same time, the sediment load of other p a r t i c l e types may exceed the flow's transport capacity for those types. The excess transport capacity for the d e f i c i t types i s assum- ed t o be avail able t o increase the transport capacity for the types where avai

1

able sediment exceeds transport capacity.

The Yalin equation was modified t o s h i f t excess transport capacity.

For large sediment loads (sediment loads f o r each particle type clearly in excess of the respective transport capacity for each particle type), or for small loads (sediment loads f o r each particle type clearly less than the respective transport capacity for each particle type), the flow's transport capacity i s distributed among the avail able p a r t i c l e types based on parti- c l e size

and

density and flow characteristics (Foster and Meyer 1972).

Yal in assumed that the number of particles in transport i s proportion- al t o

6 .

For a mixture, the number of particles of a given type i i s as- sumed t o be proportional t o

6 i .

Values of

6 i

f o r each p a r t i c l e type in a mixture are calculated and summed t o give a total

:

where nS

=

number of p a r t i c l e types in

the

mixture. The number of trans- ported particles of type

i

in a mixture i s given as:

where

N i =

number of particles transported in sediment of uniform type

i

f o r a

6 i .

As derived by Yal i n , the nondimensional transport, PS, of equation

40

i s proportional t o the number of particles in transport.

Then,

where ( P e ) i = t h e e f f e c t i v e P f o r p a r t i c i s t y p e i i n a m i x t u r e , and ( P S ) i i s t h e P s c a l c u l a t e d f o r u n i f o r m m a t e r i a l o f t y p e i. The t r a n s p o r t ca- p a c i t y W S i o f each p a r t i c l e t y p e i n a m i x t u r e i s t h e n expressed by:

T h i s i s t h e t r a n s p o r t c a p a c i t y assuming t h a t t h e s u p p l y o f a l l p a r t i c l e t y p e s i s e i t h e r g r e a t e r t h a n o r l e s s t h a n t h e i r r e s p e c t i v e W s i . When a v a i l a b i l i t y o f some t y p e s i s g r e a t e r t h a n t h e i r W s i and o t h e r s a r e l e s s t h a n t h e i r W S i , t r a n s p o r t c a p a c i t y s h i f t s f r o m t h o s e t y p e s where supply i s l e s s t h a n c a p a c i t y so t h a t a l l o f t h e t o t a l t r a n s p o r t c a p a c i t y i s used.

The s t e p s g i v e n below a r e f o l l o w e d t o r e d i s t r i b u t e t h e t r a n s p o r t capa- c i t y when excesses and d e f i c i t s occur.

1. For t h o s e p a r t i c l e s where W s i

2

q s i ( q s i = sediment l o a d f o r par- t i c l e t y p e i ) , compute t h e a c t u a l r e q u i r e d Pireq f r o m e q u a t i o n 40, i.e.:

and a s s i g n Tci = WSi

.

2. For t h o s e p a r t i c l e t y p e s where WSi

2

^qsi, t h e sum:

SPT = s (Pi req/Pi) ki i =l

i s computed where k i = 1 i f W s i

2

q s i and k i = 0 i f LJsi

<

q s i . The sum SPT r e p r e s e n t s t h e f r a c t i o n o f t h e t o t a l t r a n s p o r t c a p a c i t y used by t h o s e p a r t i c l e t y p e s where sediment a v a i l a b i l i t y i s l e s s than t r a n s p o r t c a p a c i t y .

3. The excess, expressed as a f r a c t i o n o f t h e t o t a l , t o be d i s t r i b u - t e d i s :

xc ⁼ 1

-

^SPT

4. F o r t h o s e p a r t i c l e t y p e s where W s i

<

q s i , sum 6 i as:

n

SDLT = zs 6i li i=l

where li = 0 i f Wsi qsi and li = 1 if WSi

<

^{q s i}

5. The excess i s d i s t r i b u t e d a c c o r d i n g t o t h e d i s t r i b u t i o n of 6 i among t h e s e p a r t i c l e t y p e s , i .e.:

F o r t h e o t h e r p a r t i c l e t y p e s :

6. Repeat s t e p s 1-5 u n t i l e i t h e r a1 1 Tci

,

( q s i o r a1 1 Tci

2

q ~ i . When t h e f o r m e r occurs, t h e p r o p e r Tci s have been found. I f t h e

l a t t e r occurs, one p a r t i c l e t y p e w i l l have a l l o f t h e excess t r a n s p o r t c a p a c i t y . The excess f o r t h i s one t y p e should be equal- l y d i s t r i b u t e d among a l l o f t h e types. T h i s i s done by:

n S

SMUS = Z (Pireq/Pi) i = l

Conversion f r o m Storm t o Rate Basis

W i t h o u t t h e

( u

/ V ) term, e q u a t i o n s 34 and 35, as o r i g i n a l l y de- veloped ( F o s t e r e t $1 .u1977) were on a storm b a s i s , whereas t h e t r a n s p o r t e q u a t i o n i s on an i n s t a n t a n e o u s r a t e b a s i s . The e q u a t i o n s a r e combined by assuming t h a t computed sediment c o n c e n t r a t i o n r e p r e s e n t s an average f o r t h e r u n o f f event, and t h a t peak d i s c h a r g e r e p r e s e n t s a c h a r a c t e r i s t i c d i s c h a r g e t h a t can be used t o compute t h e average c o n c e n t r a t i o n .

S i n c e most f i e l d - s i z e d areas a r e re1 a t i v e l y small

,

t i m e o f concentra- t i o n f o r t h e r u n o f f i s u s u a l l y small and i s assumed t o be l e s s t h a n r a i n - f a l l d u r a t i o n . Thus, f o r a g i v e n storm, d i s c h a r g e a t a l o c a t i o n i s assumed t o be d i r e c t l y p r o p o r t i o n a l t o upstream d r a i n a g e area.

Shear S t r e s s

The t r a n s p o r t e q u a t i o n r e q u i r e s an e s t i m a t e o f t h e r u n o f f ' s shear s t r e s s . The sediment t r a n s p o r t concept ( G r a f 1971), where shear i s d i v i d e d between form roughness and g r a i n roughness, i s used t o e s t i m a t e t h e shear s t r e s s a c t i n g on t h e s o i l , t h e p o r t i o n assumed r e s p o n s i b l e f o r sediment t r a n s p o r t . Mulch o r v e g e t a t i o n reduces t h i s s t r e s s . The shear s t r e s s a c t - i n g on t h e s o i l , T s o i l

,

i s e s t i m a t e d by:

where y = w e i g h t d e n s i t y o f water; y = f l o w d e p t h f o r bare, smooth s o i l ; s

= s i n e o f s l o p e angle; nbov = Manning's n f c r bare s o i 1 (0.01 f o r o v e r l a n d f l o w and 0.03 f o r channel f l o w assumed); and ncov = t o t a l Kanning's n f o r rough s u r f a c e s o r s o i l covered by mulch o r v e g e t a t i o n . F l c w d s p t h i s e s t i - mated by t h e Manning e q u a t i o n as:

where qw = d i s c h a r g e r a t e p e r u n i t w i d t h . Aithough t h e Darcy-Weisbach e q u a t i o n w i t h a v a r y i n g f r i c t i o n f a c t o r f o r l a m i n a r f l o w m i g h t be more ac- c u r a t e f o r y i n some cases, most u s e r s a r e b e t t e r a c q u a i n t e d w i t h e s t i m a t - i n g Manning's n. The e r r o r i n e s t i m a t i n g a v a l u e f o r t h e roughness f a c t o r i s p r o b a b l y g r e a t e r t h a n t h e e r r o r i n u s i n g t h e Manning e q u a t i o n f o r l a n i - n a r f l o w .

Slope Segments

Computations b e g i n a t t h e upper end o f t h e slope. Sediment i s r o u t e d downslope much t h e same as i t i s i n most e r o s i o n models. Computed o u t p u t i s t h e sediment c o n c e n t r a t i o n f o r each p a r t i c l e type. C o n c e n t r a t i o n m u l t i - p l i e d by t h e r u n o f f volume and o v e r l a n d f l o w area r e p r e s e n t e d by t h e over- l a n d f l o w p r o f i l e g i v e s t h e sediment y i e l d f o r t h e storm on t h e o v e r l a n d a r e a o f t h e f i e l d .

The o v e r l a n d f l o w area i s r e p r e s e n t e d by a t y p i c a l l a n d p r o f i l e s e l e c - t e d f r o m s e v e r a l p o s s i b l e o v e r l a n d f l o w paths. I t s shape may be u n i f o r m , convex, concave, o r a c o m b i n a t i o n o f t h e s e shapes. I n p u t s a r e t o t a l s l o p e l e n g t h , average steepness, t h e s l o p e a t t h e upper end o f t h e p r o f i l e , t h e s l o p e a t t h e l o w e r end o f t h e p r o f i l e and l o c a t i o n o f t h e end p o i n t s o f a m i d u n i f o r m s e c t i o n .

Given t h i s minimum o f i n f o r m a t i o n , t h e model e s t a b l i s h e s segments a- 1 ong t h e p r o f i l e . The procedure i s il 1 u s t r a t e d by t h e convex shape shown i n f i g u r e 3. C o o r d i n a t e s o f p o i n t s A, C, and D a r e given, as a r e s l o p e s s and sm. A q u a d r a t i c c u r v e w i l l pass t h r o u g h p o i n t C t a n g e n t t o l i n e

cb

and t h r o u g h p o i n t E t a n g e n t t o 1 in e AB. The l o c a t i o n o f p o i n t E i s t h e i n t e r s e c t i o n o f a l i n e h a v i n g a s l o p e equal t o t h e average o f sb and s, w i t h l i n e AB. I f X2 i s l e s s t h a n XI, X3 i s s h i f t e d downslope so t h a t X1 = X2.

Each u n i f o r m s e c t i o n i s one segment. I n f i g u r e 3, AE and CD a r e seg- ments. Convex s e c t i o n s l i k e EC a r e d i v i d e d i n t o o n l y t h r e e segments, be- cause detachment and t r a n s p o r t computations a r e n o t e s p e c i a l l y s e n s i t i v e t o t h e number o f segments on convex slopes. Concave segments a r e d i v i d e d i n t o 10 segments because d e p o s i t i o n computations on concave s l o p e s a r e e s p e c i a l - l y s e n s i t i v e t o t h e number o f segments. Furthermore, s e v e r a l segments a r e r e q u i r e d t o a c c u r a t e l y d e t e n i ne where d e p o s i t i o n begins.

A d i t i o n a l segment ends a r e d e s i g n a t e d where K, C, P, o r n change.

Given l o c a t i o n s where t h e s e changes o c c u r as i n p u t , t h e model computes t h e c o o r d i n a t e s f o r a1 1 t h e segments f o r t h e o v e r l a n d f l o w slope.

COORDINATES O F POINTS A, C, AND D A N 0 SLOPES S, A N D S, G I V E N A S INPUT

I 1 1 I 1 1 L 8 1

DISTANCE

F i g u r e 3. R e p r e s e n t a t i o n o f convex s l o p e p r o f i l e f o r o v e r l a n d f l o w . S e l e c t i o n o f Parameter Values

Slope l e n g t h i s , perhaps, t h e most d i f f i c u l t o f t h e o v e r l a n d f l o w par- ameters t o e s t i m a t e . N i l 1 ia ~ n s and B e r n d t ' s (1977) c o n t o u r method i s a pos- s i b l e t e c h n i q u e t o use. Another i s t o s k e t c h f l o w l i n e s f r o m t h e watershed boundary t o c o n c e n t r a t e d f l ow. Topography i n most f i e l d s converges over- l a n d f l o w i n t o c o n c e n t r a t e d f l o w w i t h i n about 100 m. C e r t a i n l y a grass wa- t e r w a y ( o r a s i m i l a r f l o w c o n c e n t r a t i o n w i t h o u t a c o n s t r u c t e d waterway where e r o s i o n may o r may n o t be a problem), a t e r r a c e channel

,

o r a d i v e r - s i o n i s t h e end o f o v e r l a n d s l o p e l e n g t h .

Values f o r t h e parameters K, C ( s o i l l o s s r a t i o ) , and P ( c o n t o u r i n g ) a r e s e l e c t e d from Wischmeier and Smith (1978) a c c o r d i n g t o c r o p stage.

Values f o r Manning's ncov may be s e l e c t e d from Lane e t a l . (1975) o r from F o s t e r e t a l . (1980).

CHANNEL ELEMENT

The channel element i s used t o r e p r e s e n t f l o w i n t e r r a c e channels, d i - v e r s i o n s , m a j o r f l o w c o n c e n t r a t i o n s where topography has caused o v e r l and f l o w t o converge, grass waterways, row m i d d l e s o r graded rows, t a i l d i t c h - es, and o t h e r s i m i l a r channels. The channel element does n o t d e s c r i b e g u l - l y o r l a r g e channel e r o s i o n .

Except t h a t shear s t r e s s and detachment by f l o w a r e e s t i m a t e d d i f f e r - e n t l y , t h e same concepts and e q u a t i o n s a r e used i n b o t h t h e channel and o v e r l and f l o w elements. Discharge a1 ong t h e channel i s assumed t o v a r y d i - r e c t l y w i t h upstream d r a i n a g e area. A d i s c h a r g e g r e a t e r t h a n zero i s per- m i t t e d a t t h e upper end t o account f o r upland c o n t r i b u t i n g areas. As w i t h

t h e o v e r l a n d f l o w element, changes i n t h e c o n t r o l l i n g v a r i a b l e s a l o n g t h e channel a r e allowed. Thus, a channel w i t h a d e c r e a s i n g s l o p e o r a change i n c o v e r can be analyzed.

S p a t i a1 l y V a r i e d F l o w E q u a t i o n s

F l o w i n most channels i n f i e l d s i s s p a t i a l l y v a r i e d , e s p e c i a l l y f o r o u t l e t s r e s t r i c t e d by r i d g e s and heavy v e g e t a t i o n , and f o r v e r y f l a t t e r - r a c e channels. A1 so, d i s c h a r g e general l y i n c r e a s e s a l o n g t h e channel

.

^The

model approximates t h e s l o p e o f t h e energy grade1 i n e ( f r i c t i o n s l o p e ) a l o n g t h e channel u s i n g a s e t o f normal i z e d curves and assuming steady f l o w a t peak discharge. As an a l t e r n a t i v e , t h e model w i l l s e t t h e f r i c t i o n s l o p e equal t o t h e channel slope.

The e q u a t i o n f o r s p a t i a l l y v a r i e d f l o w (Chow 1959) w i t h i n c r e a s i n g d i s c h a r g e i n a t r i a n g u l a r channel may be normal i z e d as:

where y, = y / y e , y = f l o w depth, ye = f l o w d e p t h a t t h e end o f t h e channel, S, = sLeff /ye, s = channel s l o p e , x = d i s t a n c e a1 ong channel

,

^x,

= x/Leff

,

and L e f f = e f f e c t i v e channel l e n g t h ( i e . t h e l e n g t h o f t h e channel i f i t i s extended upslope t o where d i s c h a r g e would be zero w i t h t h e g i v e n 1 a t e r a l i n f l o w r a t e )

.

Constants C1, C2, and C3 a r e g i v e n by:

where n = Manning's n, z = s i d e s l o p e o f channel, Q e = d i s c h a r g e a t end o f channel, B = energy c o e f f i c i e n t (1.56 used f r o m McCool e t a l . 1966), and g

= a c c e l e r a t i o n due t o g r a v i t y . E q u a t i o n 60 was s o l v e d f o r a range o f t y p i - c a l values o f C1, C2, and C3 f o r s u b c r i t i c a l flow. The c u r v e s g i v e n b y e q u a t i o n s 64-73 were f i t t e d by r e g r e s s i o n t o t h e s o l u t i o n s .

Range o f C3: C3

>

0.3

Where 0.0

5

S,

5

1.2 and x,

_- <

^0.9

SSF = 0.2777

-

3.3110 x

+

9.1683 x2

-

8 . 9 5 5 1 ~ 3

Where 1.2

- <

^S,

5

4.8 and x,

F

0.9

2 3

SSF = 2.6002

-

8 . 0 6 7 8 ~ ~ + 15.6502~,

-

11 - 7 9 9 8 ~ ~

Where 4.8

<

^S, 20.0 and x,

I

0.9

2 3

SSF = 3.8532

-

^{12.9501~, + 21.1788~,}

-

^12.1143~,

Where 20.0

<

^S,,

SSF = 0.0

Range o f C3: 0.3

2

^C3

L

^0.03

Where S,

>

0 and x,

5

0.8,

2 3

SSF = 2.0553

-

^6.9875~, + 11.418~,

-

^6.4588~,

Where S, = 0 and x,

5

0.9,

2 3

SSF = 0.0392

-

0.4774~, + 1.0775~,

-

1.3694~,

Range o f C3: 0.03

>

^C3

2

0.007 Where S,

>

0, and x,

5

0.8,

2 3

SSF = 1.5386

-

^5.2042x, -+ 8.4477x,

-

^4.740x,

Where S, = 0.0 and x,

1

0.9,,

2 3

SSF = 0.0014

-

^0.0162~,

-

^0.0926~,

-

^0.0377~,

Range o f C3: 0.007

>

^C3

Where S,

>

0 and x,

<

^0.7,

2 3

SSF = 1.2742

-

^4.7020~, + 8.4755~,

-

^5.3332x,

Where

S, =

0

and

x, 0.9,

SSF =

-O.O363x,

2

With these values of

SSF,

the friction slope is:

Flow depth, ye, a t the end of the channel i s estimated

by

assuming a t the user's option, either c r i t i c a l depth, depth of uniform flow in an outlet control channel, or depth from a rating curve.

A

triangular channel section was used t o develop the friction slope curves because the equations are simple. In the model , a triangular chan- nel must be used t o estimate the slope of the energy gradeline,

b u t

the user may select a triangular, rectangular, or "naturally eroded" section in other computational components of the channel element.

Concentrated Fl ow Detachment

I n

the spring after planting, concentrated flow from intense rains on a freshly prepared seedbed often erodes through the finely t i l l e d layer t o the depth of secondary t i l l a g e , or perhaps, primary ti1 lage. Once the channel erodes t o the nonerodible layer, i t widens a t a decreasing rate.

Data from observed till erosion (Lane and Foster 1980) suggests that detachment capacity

(

kg/mL/s) by f l ow over a loosely t i

11

ed seedbed may be described by:

where

Kch =

a n erodibi

1

i ty factor [ ( m 2

/ N ) ~ * O ~ (kg/m 2

/ s ) ] , r

=

average shear stress ( ~ / m ' ) of the flow in the channel, and rc,= a critical shear stress below which erosion i s negligible. Critical shear stress seems t o increase greatly over the year as the soil consolidates (Graf 1971).

Shear stress i s assumed t o be triangularly distributed in time during the runoff event in order t o estimate the time t h a t shear stress i s greater t h a n the c r i t i c a l shear stress. For the time t h a t shear stress i s greater t h a n critical shear stress, shear stress i s assumed constant

and

equal t o peak shear stress for the storm.

Until the channel reaches the nonerodible layer, an active channel i s

assumed t h a t i s rectangular with the width obtained from figures 4

and

5

and

equations 76

and 7 7 .

The solution requires finding a value of xc.

Given the discharge Q , Manning's

n ,

f r i c t i o n slope S f ,

a

value g(xc) i s cal cul ated from:

Given

a

p a r t i c u l a r value

g ( x c ) , a

value of

x

i s obtained from figure 4 . Having determined

xc, a

value f o r R* = hysraul i c radiuslwetted per-

imeter and W+ = widthlwetted perimeter i s read from figure 5. The width of the channel i s then calculated from:

0.00' ^I I 1

.OO .I0 . 2 0 . 3 0 . 4 0 . 5 0 X (DISTANCE ALONG W E T T E D PERIMETER

FROM WATER SURFACE DOWN TO POINT WHERE LOCAL SHEAR STRESS EQUALS CRITICAL SHEAR STRESS) DIVIDED B Y W E T T E D PERIMETER

Fi gure 4. Function g(xc)for equil i bri

um

eroded channel

.

CL

0 _z

-

-

I

I

k

-

GEOMETRIC PROPERTIES O F

D ERODED C H A N N E L S AT EOUlLlBRlUM

0.00 0.10 0 . 2 0 0 . 3 0 0.40 0 . 5 0

X, (DISTANCE ALONG W E T T E D P E R I M E T E R FROM WATER SURFACE DOWN TO POINT WHERE LOCAL SHEAR STRESS EQUALS

CRITICAL SHEAR S T R E S S ) DIVIDED BY W E T T E D P E R I M E T E R

F i g u r e 5. Equi l i b r i um eroded channel geometric p r o p e r t i e s .

The f u n c t i o n s shown i n f i g u r e s 4 and 5 a r e s t o r e d piecewise i n t h e model.

The channel moves downward a t t h e r a t e dch:

where D F ~ = e r o s i o n r a t e (massluni t a r e a l u n i t t i m e ) , c a l c u l a t e d u s i ng t h e maximum shear s t r e s s and p s o i l = mass d e n s i t y o f t h e s o i l i n place. Tne e r o s i o n r a t e i n t h e channel i s :

where ECh i s t h e r a t e o f s o i l l o s s per u n i t channel l e n g t h (mass/unit/

1 e n g t h l u n i t t i m e ) .

Once t h e channel h i t s t h e nonerodibl e boundary, t h e e r o s i o n r a t e be- g i n s t o decrease w i t h time. The w i d t h id o f t h e channel a t any t i m e a f t e r t h e channel has eroded t o t h e n o n e r o d i b l e l a y e r i s estimated from:

where:

where Wi = w i d t h a t t = ti, W = w i d t h a t t, Wf = f i n a l eroded w i d t h f o r t ^-t

-,

and t h e g i v e n Q, t = t i m e , and (dW/dt)i = r a t e t h a t channel widens a t t = ti. The i n i t i a l widening r a t e i s g i v e n by:

where p s o i 1 = mass d e n s i t y o f t h e s o i l i n p l a c e and T~ i s given by:

and: T = T I T - = 1.35

m max (85

where xb = f l o w d e p t h l w e t t e d p e r i m e t e r , and Tmax = maximum shear s t r e s s a t c e n t e r o f channel.

The f i n a l w i d t h Wf i s determined by f i n d i n g t h e xcf t h a t gives:

where f ( x c f ) i s t h e f u n c t i o n g i v e n by e q u a t i o n 84 and 85 and evaluated a t X c f

The f i n a l w i d t h i s :