SRH-1D Users Manual 3_0

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US Department of InteriorBureau of ReclamationTechnical Service CenterSedimentation and River Hydraulics Group November 2012

Technical Report SRH-2013-11

SRH-1D 3.0 Users Manual

Sedimentation and River Hydraulics One Dimension, Version 3.0


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REPORT DOCUMENTATION PAGE Form ApprovedOMB No. 0704-0188

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gatheringand maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 J effersonDavis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing tocomply with a collection of information if it does not display a currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

9-29-2012

2. REPORT TYPE

Numerical Model Documentation and Users

Manual

3. DATES COVERED (From - To)

02-10-2000 to 9-29-2012

4. TITLE AND SUBTITLE

SRH-1D 3.0 Users Manual (Sedimentation and River Hydraulics One

Dimension, Version 3.0)

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Jianchun Victor Huang

Blair Greimann

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Sedimentation and River Hydraulics Group, Technical Service Center

Bureau of Reclamation

Denver Federal Center, Bldg. 67PO Box 25007 (86-68240)

Denver, CO 80225

8. PERFORMING ORGANIZATION REPORTNUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S)

11. SPONSOR/MONITOR'S REPORTNUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT

13. SUPPLEMENTARY NOTES

14. ABSTRACT

The sediment transport model SRH-1D 3.0 (Sedimentation and River Hydraulics One Dimension Version 3.0) isdocumented. The model is for use in alluvial rivers where the one dimension assumption is appropriate. The model can

simulate both cohesive and non-cohesive sediment transport using steady or unsteady flow solutions. Simple or complex

networks can be simulated as well as a variety of hydraulic structures. The technical background, data requirements, and data

formats are detailed.

15. SUBJECT TERMS

Sediment Transport. Alluvial Rivers. Numerical Modeling. Hydraulic Model.

16. SECURITY CLASSIFICATION OF: 17. LIMITATIONOF ABSTRACT

18. NUMBEROF PAGES

19a. NAME OF RESPONSIBLE PERSON

Blair Greimann

a. REPORT b. ABSTRACT a. THIS PAGE 19b. TELEPHONE NUMBER (Include area code)303-445-2563

Standard Form 298 (Rev. 8/98)Prescribed by ANSI Std. Z39.18


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Acknowledgments

The early stages of model development were jointly funded by the Environmental

Protection Agency under Interagency Agreement #DW14938833-01-0 and

Reclamation Science & Technology Program project number 1262 and 0092. TheTaipei Economic and Cultural Representative Office has funded the development

of bedrock scour components of the model. The following people contributedreviews, comments, and many suggestions:

UBureau of Reclamation, Technical Service Center, Denver, Colorado

Travis Bauer, Cassie Klumpp, Kent Collins, Robert Hilldale, Chris

Holmquist-Johnson, Yong Lai, Paula Makar, Timothy Randle, Chih TedYang, and Christi Young

UEnvironmental Protection Agency, Athens, Georgia

Earl Hayter

UDanauConsult Zottl & Erber Zt.-GmbH, Austria

Hannes Gabriel

UNew Zealand National Institute of Water and Atmospheric Research

Jeremy Walsh

UURS Corporation, Oakland, CA

Jeremy Bricker

Disclaimer

No warranty is expressed or implied regarding the usefulness or completeness of

the information contained in this report. References to commercial products donot imply endorsement by the Bureau of Reclamation and may not be used for

advertising or promotional purposes.

Mission Statements

The U.S. Department of the Interior protects Americas natural resources and

heritage, honors our cultures and tribal communities, and supplies the energyto power our future.

___________________________

The mission of the Bureau of Reclamation is to manage, develop, and protect

water and related resources in an environmentally and economically sound

manner in the interest of the American public.


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i

TABLE OF CONTENTS

1 INTRODUCTION.............................................................................................. 11.1BACKGROUND................................................................................................ 11.2SRH-1DCAPABILITIES .................................................................................. 11.3LIMITS OF APPLICATION................................................................................. 11.4ACQUIRING SRH-1D ...................................................................................... 21.5DISCLAIMER................................................................................................... 2

2 FLOW ROUTING THEORY ........................................................................... 32.1STEADY FLOW SOLUTION .............................................................................. 3

2.1.1 Governing Equations for a Single River ................................................ 32.1.2 Numerical Method for a Single River .................................................... 42.1.3 Governing Equations In River Networks ............................................... 52.1.4 Numerical Method for a Network .......................................................... 6

2.1.4.1 Internal sections .......................................................................... 62.1.4.2 Upstream Boundary Conditions .................................................. 72.1.4.3 Downstream Boundary Conditions ............................................. 8

2.2UNSTEADY FLOW SOLUTION.......................................................................... 82.2.1 Governing Equations ............................................................................. 82.2.2 Numerical Scheme ................................................................................. 92.2.3 Upstream Boundary Conditions .......................................................... 13

2.2.3.1 Water Discharge........................................................................ 132.2.3.2 River Stage ................................................................................ 14

2.2.4 Downstream Boundary Conditions ...................................................... 152.2.4.1 Rating Curve ............................................................................. 152.2.4.2 River Stage ................................................................................ 15

2.2.5 Network Boundary Condition .............................................................. 162.3INTERNAL BOUNDARY CONDITION .............................................................. 172.3.1 Governing Equations for Internal Boundaries .................................... 17

2.3.1.1 Time Stage Table ...................................................................... 172.3.1.2 Elevation versus discharge table ............................................... 172.3.1.3 Weir........................................................................................... 172.3.1.4 Bridge ........................................................................................ 192.3.1.5 Radial Gate................................................................................ 20

2.3.2 Implementation for Steady Flows ........................................................ 222.3.3 Implementation for Unsteady Flows .................................................... 22

3 SEDIMENT TRANSPORT ............................................................................ 233.1SEDIMENT ROUTING..................................................................................... 23

3.1.1 Exner Equation Routing ....................................................................... 233.1.1.1 Non-Cohesive Sediment Routing ............................................. 243.1.1.2 Cohesive Sediment .................................................................... 27

3.1.2 Unsteady Sediment Transport .............................................................. 283.1.2.1 Unsteady Term .......................................................................... 293.1.2.2 Convective Term ....................................................................... 30


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ii

3.1.2.3 Diffusion Term.......................................................................... 313.1.2.4 Source Term .............................................................................. 313.1.2.5 Discretized Sediment Transport Equation ................................ 32

3.1.3 Non-Cohesive Particle Fall Velocity C.................................................. 323.1.4 Non-Cohesive Sediment Transport Capacity ....................................... 33

3.1.4.1 Meyer-Peter and Mller's Formula (1948) modified by Wongand Parker (2006)C...................................................................................... 343.1.4.2 Laursen's Formula (1958) and Modified Version (Madden,1993) 353.1.4.3 Engelund and Hansen's Method (1972) .................................... 363.1.4.4 Ackers and White's Method (1973) and (HR Wallingford, 1990)

373.1.4.5 Yang's Sand (1973) and Gravel (1984) Transport Formulas .... 383.1.4.6 Yang's Sand (1979) Transport Formulas .................................. 393.1.4.7 Yang et al. 's Modified Formula for Sand Transport with HighConcentration of Wash Load (C1996C) ......................................................... 393.1.4.8 Brownlies Method ................................................................... 403.1.4.9 Parker's Method (1990) ............................................................. 413.1.4.10 Wilcock and Crowe (2003) ................................................... 423.1.4.11 Gaeuman et al (2009) ........................................................... 433.1.4.12 Wu et al. (2000) ................................................................... 433.1.4.13 Parker or Wilcock and Crowe combined with Engelund-Hansen 44

3.1.5 Cohesive Sediment Aggregation .......................................................... 443.1.6 Cohesive Sediment Deposition ............................................................. 463.1.7 Cohesive Sediment Erosion.................................................................. 47

3.2BED MATERIAL MIXING............................................................................... 493.3CONSOLIDATION .......................................................................................... 53

4 BEDROCK EROSION .................................................................................... 55

4.1HYDRAULIC EROSION OF BEDROCK............................................................. 554.2BEDROCKEROSION DUE TO SEDIMENT ABRASION....................................... 564.3ESTIMATION OF BEDROCKEROSION RATES IN SRH-1D .............................. 574.4GEOMETRICAL REPRESENTATION OF BEDROCK........................................... 58

5 BED GEOMETRY SOLUTION .................................................................... 615.1CHANNEL GEOMETRY ADJUSTMENT............................................................ 615.2CHANNEL WIDTH CHANGE USING MINIMIZATION ....................................... 625.3ANGLE OF REPOSE ADJUSTMENTS ............................................................... 62

6 INPUT DATA REQUIREMENTS ................................................................. 646.1MODEL PARAMETERS................................................................................... 646.2UPSTREAM FLOW BOUNDARY CONDITIONS ................................................. 656.3DOWNSTREAM FLOW BOUNDARY CONDITIONS ........................................... 666.4INTERNAL BOUNDARY CONDITIONS............................................................. 666.5LATERAL INFLOWS....................................................................................... 666.6CHANNEL GEOMETRY AND FLOW CHARACTERISTICS .................................. 676.7SEDIMENT MODEL PARAMETERS ................................................................. 69


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iii

6.8SEDIMENT BOUNDARY CONDITIONS ............................................................ 696.9LATERAL SEDIMENT DISCHARGE ................................................................. 706.10SEDIMENT BED MATERIAL......................................................................... 706.11WATERTEMPERATURE .............................................................................. 706.12EROSION AND DEPOSITION LIMITS ............................................................. 706.13SEDIMENT TRANSPORT PARAMETERS ........................................................ 716.14COHESIVE SEDIMENT TRANSPORT PARAMETERS ....................................... 716.15BEDROCKGEOMETRY AND PARAMETERS .................................................. 72

7 RUNNING SRH-1D ......................................................................................... 747.1INPUT DATA FORMAT .................................................................................. 747.2EXECUTING SRH-1D ................................................................................... 747.3OUTPUT FILES .............................................................................................. 75

8 REFERENCES ................................................................................................. 78APPENDIX A FLOW CHART OF INPUT DATA RECORDS.......................... A1

APPENDIX B ALPHABETIC LIST OF THE INPUT DATA RECORDS......... B1

APPENDIX C DESCRIPTIONS OF RECORDS ................................................ C1

APPENDIX D EXAMPLE APPLICATIONS...................................................... D1

D1TRAPEZOID CHANNEL .................................................................................. D3

D2CHANNELNETWORK.................................................................................. D17D3CALIFORNIA AQUEDUCT ............................................................................ D39


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iv

TABLE OF FIGURES

Figure 2.1 Upstream boundaries of River 2 and River 3 .....................................................7Figure 2.2 Numerical grid used for unsteady flow simulation. .........................................10Figure 2.3. Comparison between water surface elevations for unsteady flow

solutions in SRH-1D. .............................................................................................13Figure 2.4 Schematic of bridge (Source: Fread and Lewis, 1998) ....................................19Figure 2.5 Schematic of radial gate (Source: Brunner,

C

2001C

). ..........................................21Figure 3.1TTRatio between non-equilibrium concentration and carrying capacity asa function of sediment particle size (from Yang and Simes, 2002). ....................25

Figure 3.2TTEffect of the recovery parameteron the computation of non-equilibrium sediment concentrations for two sediment particle sizes. (a)

deposition and (b) erosion (from Yang and Simes, 2002). ..................................26Figure 3.3TVariation of non-equilibrium effects as a function of distance between

cross sections for deposition (a) and for erosion (b) (from Yang and

Simes, 2002). .......................................................................................................27Figure 3.4TGrid Definition for Unsteady Sediment Simulation. ........................................29Figure 3.5 Relation between particle sieve diameter and its fall velocity according

to the U.S. Interagency Committee on Water Resources Subcommittee onSedimentation (1957)C. ............................................................................................33

Figure 3.6. Comparison between Laursens (1958) function and Eq. X3.44X. ......................36Figure 3.7 The influence of sediment concentration on the settling velocity

(source: Van Rijn, 1993, figure 11.4.2) ................................................................45Figure 3.8 Input data illustration for settling velocity .......................................................46Figure 3.9 The schematic illustrates the erosional characteristics that need to be

determined from erosion tests (after: Vermeyen, C1995C) ........................................48Figure 3.10 Conceptual model of bed mixing. ..................................................................50Figure 4.1. Plot of stream power versus erodibility index taken from Annandale

(2006). ....................................................................................................................56Figure 4.2. Geometrical representation of a cross section and bed rock in SRH-1D. ..........................................................................................................................59Figure 5.1 Schematic representation of channel changes: (a) vertical adjustment

due to scour or deposition; (b) width adjustment due to scour or

deposition. ..............................................................................................................61Figure 6.1 Steady Flow Representation of a Water Discharge Hydrograph. .....................65Figure 6.2 Representation of River by Discrete Cross sections. (From Yang and

Simoes, 2002). .......................................................................................................67Figure 6.3 Representation of Cross Section by Discrete Points. .......................................69


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v

TABLE OF TABLES

Table 3.1 Sediment transport functions available in SRH-1D and its type (B = bed

load; BM = bed-material total load). .....................................................................34Table 3.2 Coefficients for the 1973 and 1990 versions of the Ackers and White

formula. ..................................................................................................................38Table 4.1. Table 4b from Sklar and Dietrich (2012) containing values ofkv. ...................57Table 6.1 Input records in Model Parameter data group. ..................................................65Table 6.2 Possible downstream boundary conditions. .......................................................66Table 6.3 Possible internal boundary conditions. ..............................................................66Table 6.4 Records used in Sediment Transport Parameters data group. ...........................71Table 6.5 Parameters necessary for cohesive sediment erosion and deposition. ...............72


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Introduction 1

1 Introduction

1.1Background

SRH-1D (Sedimentation and River Hydraulics One Dimension) is a one-dimensional hydraulic and sediment transport model for use in natural rivers and

manmade canals. It is a mobile bed model with the ability to simulate steady or

unsteady flows, internal boundary conditions, looped river networks, cohesive andnon-cohesive sediment transport, and lateral inflows. The Environmental

Protection Agency (EPA) and Bureau of Reclamation (Reclamation) were

funding partners in the original development of the SRH-1D model.

1.2SRH-1D Capabilities

SRH-1D is a hydraulic and sediment transport numerical model developed to

simulate flows in rivers and channels with or without movable boundaries. Someof the models capabilities are:

Computation of water surface profiles in a single channel or multi-channel

looped networks. Steady and unsteady flows.

Subcritical flows in a steady hydraulic simulation.

Subcritical, supercritical, and transcritical flows in an unsteady hydraulicsimulation.

Transport of cohesive and non-cohesive sediments.

Cohesive sediment aggregation, deposition, erosion, and consolidation.

Multiple non-cohesive sediment transport equations that are applicable to

a wide range of hydraulic and sediment conditions. Cross stream variation in hydraulic roughness.

Fractional sediment transport, bed sorting, and armoring. Point and non-point sources of flow and sediments.

Internal boundary conditions, such as time-stage tables, rating curves,

weirs, bridges, and radial gates. Bedrock control and erosion

1.3Limits of Application

SRH-1D is a general numerical model developed to simulate and predict cohesive

and non-cohesive sediment transport and related river morphological changes dueto natural or human influences. SRH-1D is an engineering tool for solving fluvialhydraulic problems with the following limitations:

(1) SRH-1D is a one-dimensional model for flow simulation. It should not be

applied to situations where a two-dimensional or three-dimensional model is

needed for detailed simulation of local hydraulic conditions. Phenomena such assecondary currents, lateral diffusion, superelevation, and transverse sediment

movement are ignored.


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2 SRH-1D Users Manual

(2) Many of the sediment transport modules and concepts used in SRH-1D are

simplified approximations of real phenomena. Those approximations and their

limits of validity are embedded in the model.

(3) SRH-1D is currently compiled to run on the Windows 7 64-bit operatingsystem.

(4) There are no specific system requirements, but the size of the problem may be

limited by the computer memory or operating system limitations.

1.4Acquiring SRH-1D

The latest information about SRH-1D is placed on the Web and can be found by

accessing HTUUhttp://www.usbr.gov/pmts/sedimentUUTHand following the links on the webpage. Requests may be sent directly to the Bureau of Reclamations

Sedimentation and River Hydraulics Group (Attention: SRH Support, U.S.

Bureau of Reclamation, Sedimentation and River Hydraulics Group, P.O. Box25007 (86-68540), Denver, CO 80225).

SRH-1D is under continuous development and improvement. A user is

encouraged to check the SRH-1D web page regularly for updates.

1.5Disclaimer

The program SRH-1D and information in this manual are developed for use atReclamation. Reclamation does not guarantee the performance of the program,

nor help external users solve their problems. Reclamation assumes no

responsibility for the correct use of SRH-1D and makes no warranties concerning

the accuracy, completeness, reliability, usability, or suitability for any particularpurpose of the software or the information contained in this manual. SRH-1D is a

program that requires engineering expertise to be used correctly. Like other

computer programs, SRH-1D is potentially fallible. All results obtained from theuse of the program should be carefully examined by an experienced engineer to

determine if they are reasonable and accurate. Reclamation will not be liable for

any special, collateral, incidental, or consequential damages in connection withthe use of the software.
http://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sedimenthttp://www.usbr.gov/pmts/sediment


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Flow Routing 3

2 Flow Routing Theory

This chapter describes the theoretical basis for the one-dimensional flow solutionsused in SRH-1D. SRH-1D has the capability to solve either the steady or unsteady

flow equations. The governing equations for steady flow are presented first,followed by the steady flow numerical methods for a single channel, as well as a

channel network. The governing equations of unsteady flows are given next withthe numerical solution method for simple and complex river networks. The

available boundary conditions are described last.

2.1Steady Flow Solution

SRH-1D uses the standard step method to solve the energy equation for steadygradually varied flows. Presently, only subcritical and critical flow profiles are

calculated when the steady flow option is used.

2.1.1 Governing Equations for a Single River

The energy equation for steady gradually varied flow between downstream cross

section 1 and upstream cross section 2 is expressed as:

cf hhg

VZ

g

VZ +=+

22

2

111

2

222

(2.1)

where: ZB1B,ZB2 B= water surface elevations at cross sections 1 and 2, respectively;

VB1B, VB2 B= average velocities at cross sections 1 and 2, respectively;

B1B, B2 B= velocity distribution coefficients at cross sections 1 and 2,respectively;

g = gravitational acceleration;

hBfB= friction loss between cross sections 1 and 2, and

hBc B= contraction or expansion losses between cross sections 1 and 2.

The equation for friction loss is calculated in two ways:

efffffa LSSh 21= (2.2)

( )efffb L

KK

Qh

2

21

2

+

= (2.3)

where:1f

S ,2f

S = friction slopes at cross sections 1 and 2, respectively;

Leff = effective distance between cross sections;

Q = total flow rate at cross section; and

KB1B, KB2B = conveyance at cross sections 1 and 2, respectively.


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The actual friction loss used is the minimum of the two:

)fbfaf hhh ,min= (2.4)

The effective distance between cross sections is:

QQsL

i

iieff =

=3

1

(2.5)

where the sum is performed over three sub-channels: left overbank, main channeland the right overbank. The other variables are:

Qi= flow in sub-channel i; and

si= distance between cross sections along flow path in sub-channel i.

For a specific discharge, the conveyance, K, is used to determine the friction slopein Eq. (X2.3 X):

2

=

K

QS

f (2.6)

where Kis computed from the Mannings equation:

2/13/22/1

fm

f SARn

CKSQ == (2.7)

where: n = Mannings coefficient;

A = cross-sectional area;

R = hydraulic radius (A/P);

P = wetted perimeter; and

CBm

B

= 1.486 for English units or 1.0 for SI units.The equation for contraction or expansion losses is expressed as:

g

V

g

VCh cc 22

2

22

2

11

= (2.8)

where: CBc B= a user defined contraction or expansion coefficient

The expansion coefficient is used when the velocity head at the downstreamsection 1 is less than that at the upstream section 2. Conversely, the contraction

coefficient is used when the velocity head at the downstream section 1 is greater

than that at the upstream section 2. This is similar to the way HEC-RAS treats

energy loss.

2.1.2 Numerical Method for a Single River

Standard step method is used to solve Eq. (X2.1X), which can be expressed as:

022

)(2

111

2

2222 =+= cf hhg

VZ

g

VZZf (2.9)


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Flow Routing 5

This nonlinear algebraic equation can be solved by the Newton-Raphson iterative

method (Jain, 2000 CC). Let*2Z be an estimate of 2Z , the Newton-Raphson method

gives a better estimate of 2Z using the following:

)(

)(*2

'

*2*

22Zf

ZfZZ = (2.10)

where:2

22

2*2

' 1)(Z

h

gR

VZf

f

= (2.11)

After the first 2 iterations, the derivative in Eq ( X2.10X) is computed by using the

previous 2 values off(ZB2B). After the updated 2Z is found, it is checked to see ifthe flow at that cross section is supercritical. If it is, then the depth is set to either

critical depth or normal depth, depending upon the input given by the user (seeData Group 1 in Chapter X5X). The iteration continues until a preset accuracy is

obtained. The model automatically switches to a bisection method if the method

described above does not reach a convergent solution.

2.1.3 Governing Equations In River Networks

SRH-1D provides solutions to both dendritic networks and looped networks. The

method used by SRH-1D for such networks is similar to that found in Chaudhry(C1993 C). However, some modifications were made to handle large numbers of

connections within a river network.

The following strategy is used to record the network connection information.

River numbering is in ascending order from upstream to downstream. Theboundary condition for each river entering a junction is the ID numbers of the

other rivers entering that junction. If the flow is into the junction, the ID number

is positive and if the flow is out of the junction the ID number is negative. In alooped network where the flow direction is unknown before the numericalsimulation, the input flow direction can be assumed by the user. A calculated

positive discharge means that the assumed flow direction is correct. A negativedischarge indicates a flow direction opposite of that initially assumed.

A numerical solution of flow in a network requires the calculation of both the

energy equation and the continuity equation. At each cross section, the flow depth

and flow discharge are initially unknown. The energy equation and the continuityequation are written for each cross section as:

04

2

1

221

11

221

111

1

=

+++

+=

+

++

+

+++

+

i

ii

i

iic

f

i

iii

i

iii

iii

A

QQ

A

QQ

g

Ch

A

QQ

A

QQ

gZZF

(2.12)

01 == + iLatiii QQQG (2.13)

whereiLat

Q = the lateral inflow at the reach between cross sections i andi+1.


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SinceA andR are functions of only water surface elevation Z, the unknowns are

water surface elevation and discharge. For a river with N+1 cross-sections, there

are 2(N+1) unknowns, but only 2Nequations forNriver reaches. Therefore, two

boundary conditions are required for a unique solution of the system and thesecan be written in a general form as:

0),( 11 == ZQfBU (2.14)

0),( 11' == ++ NN ZQfBD (2.15)

wherefandf are functions defined by the boundary conditions andBUandBD

signify the upstream and downstream boundary conditions, respectively.

2.1.4 Numerical Method for a Network

2.1.4.1 Internal sections

By expanding Eqs. ( X2.11X) to (X2.14 X) in Taylor series, the system of equationsbecome:

=

+

+

++

++

++

BD

G

F

G

F

BU

Q

Z

Q

Z

Q

Z

Q

BD

Z

BD

Q

G

Z

G

Q

G

Z

G

Q

G

Z

G

Q

G

Z

G

Q

G

Z

G

Q

G

Z

G

Q

F

Z

F

Q

F

Z

FQ

BU

Z

BU

N

N

N

N

N

NN

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

1

1

1

1

2

1

1

11

11

11

2

1

2

1

1

1

1

1

2

1

2

1

1

1

1

1

11

(2.16)

For a river network, one can add equations to the matrix in Eq. (X

2.15X

) for eachindividual cross section. However, the boundary conditions may contain the river

depth or discharge in the connected sections of adjoined rivers. For the energyequation FBiB, the four non-zero partial derivatives at the nodes joining rivers are

written as:

i

f

i

iciii

i

i

Z

h

A

B

g

CQ

Z

F

+

++=

3

2

2

21 (2.17)

i

f

i

ciii

i

i

Q

h

gA

CQ

Q

F

++

=

24

22 (2.18)

131

1121

1 221

++

+++

+ +

=

i

f

i

iciii

i

i

Zh

AB

gCQ

ZF (2.19)

12

1

1

1 4

22

++

+

+

+

=

i

f

ii

cii

i

i

Q

h

gA

CQ

Q

F(2.20)

For the continuity equation GBiB, the two non-zero partial derivatives are written as:


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Flow Routing 7

1=

i

i

Q

G(2.21)

11

=

+i

i

Q

G(2.22)

2.1.4.2 Upstream Boundary Conditions

For each individual river in a network, one upstream and one downstreamboundary condition are required. If the upstream or downstream boundary is a

junction, then the ID numbers of the other rivers comprising that junction areentered into the input file. XFigure 2.1 X illustrates a simple network where one river

splits into two.

Figure 2.1 Upstream boundaries of River 2 and River 3

River 1 enters into the junction and ie1 is the last cross section of River 1. Thefirst cross sections of rivers 2 and 3 are is2 andis3, respectively. Two equations

are necessary to define the flow rates and water surface elevations at the junction.

The equations used are the continuity equation and the energy equation, assuming

no energy loss. They can be written as:

0322 =+= inisis QQQBU (2.23)

022 22

222

223

233

33 =

+=is

isisis

is

isisis

gA

QZ

gA

QZBU (2.24)

The non-zero partial derivatives in the matrix are:

13

2

2

2 ==

isis Q

BU

Q

BU(2.25)

32

2222

2

3 1is

isisis

is gA

BQ

Z

BU +=

(2.26)

2

2

22

2

3

is

isis

is gA

Q

Q

BU =

(2.27)


18/227


33

3233

3

3 1is

isisis

is gA

BQ

Z

BU =

(2.28)

2

3

33

3

3

is

isis

is gA

Q

Q

BU =

(2.29)

If there are other rivers in the network, each river has an additional energy

equation for the upstream boundary. For a complex network where the upstreamincoming discharge is unknown, the partial derivative of the continuity equation is

also a function of the discharge of the upstream river.

2.1.4.3 Downstream Boundary Conditions

For each river in the network, the energy equation is used as the downstreamboundary condition:

022 2

2

2

2

=

+=is

isisis

ie

ieieie

gA

QZ

gA

QZBD (2.30)

where ie andis denote the cross-sections of the upstream and downstream rivers,

respectively, that comprise the junction.

The non-zero partial derivatives in the matrix are:

3

2

1ie

ieieie

ie gA

BQ

ZBD = (2.31)

2

ie

ieie

ie gA

Q

QBD = (2.32)

3

2

1is

isisis

is gA

BQ

ZBD += (2.33)

2

is

isis

is gA

QQBD = (2.34)

2.2Unsteady Flow Solution

SRH-1D also has the capability to simulate unsteady flow. The theoretical basis

for the unsteady flow solution is described below.

2.2.1 Governing Equations

One-dimensional river flows are described by the conservation of mass equation,

latd q

x

Q

t

AA=

+

+ )(

(2.35)

and the momentum equation written in divergent form,

fgASxZgA

x

AQ

t

Q=

+

+

)/( 2

(2.36)

where: Q = discharge (mP3

P/s),


19/227

Flow Routing 9

A = cross section area (m P2

P),

ABdB = ineffective cross section area (m P2

P),

qBlatB = lateral inflow per unit length of channel (m P2

P/s),

t= time independent variable (s),

x = spatial independent variable (m),

g = gravity acceleration (m/sP2

P),

= velocity distribution coefficients,

Z= water surface elevation (m),

SBfB = energy slope (= 2

K

QQ), and

K= conveyance (mP3

P/s).

2.2.2 Numerical Scheme

The numerical scheme used in SRH-1D scheme uses a staggered grid, meaning

that the computational points for the flow and area are not coincident, but

alternate. The scheme also uses a non-conservative form of the momentumequation. To model transcritical flow, the conservative form of the momentum

equations should be solved (see discussion by Meselhe et al. 2005). However,

transcritical flows are not often encountered in natural channels. When they are, it

is often in cases of dam-break flows or in steep mountain streams where it isdifficult to obtain sufficiently detailed topography to resolve hydraulic jumps. In

addition, the water surface is seldom constant across the cross section in these

types of flow and the 1D flow assumptions are not valid. Therefore, this model isnot recommended to obtain detailed hydraulic information near the transition

between sub- and super-critical flows. However, the model is stable for both sub-and super-critical flow and will be accurate sufficiently far away from thetransition. The advantage to not using the non-conservative form of the

momentum equations is that mass conservation of flow can easily be ensured. In

addition, for complicated natural channels it is analytically and computationaldifficult to discretize the source terms of the conservative form of the momentum

equations to ensure that the scheme preserves a stationary solution (Sanders et al.2003). That is, it is difficult to discretization of the pressure forces of the non-

conservative form without inducing artificial flow for cases where the initial

water surface elevation is constant.

The scheme in SRH-1D uses a staggered grid, where A points are located at thecross section andQ points are located at the center of two cross sections. Figure

2.2 shows a staggered grid with A points placed at the beginning and the end of

the domain and known cross sections shown as solid lines.


20/227


Figure 2.2 Numerical grid used for unsteady flow simulation.

The discretization of the continuity equation is made with oneA-point and two Q-points giving the difference equation:

)( 111

iii

n

id

n

i

n

id

n

i QQxtAAAA

=+ +

(2.37)

where the overbar signifies a time weighted averaged value with a weighting

factor in the time dimension. The time weighted discharge, iQ , can be writtenas:

1)1( += nin

iiQQQ (2.38)

and Eq. ( X2.36X) can be written in an iteration form, with m signifying the iterationnumber;

i

m

ii

m

ii

m

i QQA ++= +1 (2.39)

where the coefficients are:

ii xt= (

X2.38 Xa)

ii x

t= (X2.38 Xb)

iii

n

id

n

i

n

id

n

ii xtQQAAAA

+++= +

)( 111

(X2.38 Xc)

The discrete form of the momentum equation is made with two A-points and three

Q-points with a weighting factor in the time dimension giving the differenceequation:

+

=+

fi

i

iiiiwe

i

n

i

n

i Ss

ZZAAgtFFstQQ

111

2)( (2.40)

where:i

iie

A

QQF

4

)( 21++=

1

2

1

4

)(

+=i

iiw

A

QQF

ABis ABi-1ABis+1 ABi ABi+1 ABie

QBis QBis+1 QBiQBi-1 QBi+1 QBie QBie+1

sBiB

xBi B


21/227

Flow Routing 11

21)(

4

+=

ii

ii

fiKK

QQS

SRH-1D provides various options for the treatment of the convective terms.

Using a weighting factor in the time dimension, Eq. ( X2.39 X) can be written initeration form as:

+++=

+

+

+

+

+

++

+++

++

fi

i

iiiiwe

i

n

i

n

i

min

i

fimin

i

fi

m

in

i

fi

i

n

i

m

i

i

n

i

m

i

ii

n

fi

i

n

i

n

i

m

i

m

i

m

in

i

wm

in

i

wm

in

i

w

m

in

i

em

in

i

em

in

i

e

i

m

i

Ss

ZZAAgtFF

xtQQ

QQ

SA

A

S

AA

S

sT

A

sT

A

AAgt

Ss

ZZAAgt

AA

FQ

Q

FQ

Q

F

AA

FQ

Q

FQ

Q

F

x

tQ

111

1

1

11

1

1

1

1

11

11

1

1

1

1

1

1

2)(

2

2 (2.41)

Substituting Eq. ( X2.38 X) into Eq. (X2.40X), results in:

i

m

ii

m

ii

m

ii dQcQbQa =++ + 11 (2.42)


++

=

++

n

i

fi

i

n

i

ii

fi

i

n

i

n

ii

in

i

w

n

i

w

i

i

A

S

sT

AAS

s

ZZgt

A

F

Q

F

s

ta

11

1

11

11

1

11

1

22

(X2.41 Xa)

+

+

+

+

+=

n

i

fi

n

i

fi

i

n

i

i

n

i

fi

i

n

i

i

ii

fi

i

n

i

n

i

ii

in

i

w

n

i

w

in

i

e

n

i

e

i

i

Q

S

A

S

sT

A

S

sT

AAgt

Ss

ZZgt

A

F

Q

F

A

F

Q

F

stb

1

1

)(2

)(2

1

11

1

1

11

1

1

(X2.41 Xb)

++

+

=

+

i

fi

ii

iifi

i

n

i

n

ii

in

i

e

n

i

e

ii

A

S

sT

AASs

ZZgt

A

F

Q

Fstc

1

2211

1 (X2.41 Xc)


22/227


( )

+

++

+++

+

+

+

=

n

i

fi

i

n

i

in

i

fi

i

n

i

iii

fi

i

n

i

n

i

iiii

n

i

win

i

eiew

i

n

i

n

ii

A

S

sTA

S

sTAA

gt

Ss

ZZAA

gt

A

F

A

FFFst

QQd

11)(2

2

11

11

1

11

1

1

1

(X2.41 Xd)

where Tis the flow top width. For a single channel withN+1 cross sections, there

are N+2 unknowns and N Eqs. (X2.41 X). One upstream and one downstreamboundary condition are therefore required.

The method as described above may become unstable for supercritical flow. SRH-

1D offers three options for the simulation of supercritical flow. Two of the

options using the Local Partial Inertia (LPI, from Fread and Lewis, 1998)technique to compute the flow. The LPI technique multiplies the convective terms

by a parameter, , as follows:

fgAS

xZgA

x

AQ

t

Q=

+

+

)/( 2

(2.43)

)( rFf= (2.44)

where FBrB is the Froude Number. SRH-1D offers two methods to compute thefunctionf(Fr):

)1,0max(5

= rF (2.45)

or

),1min(2

= rF (2.46)

The first equation is taken from FLDWAV (Fread and Lewis, 1998). The secondequation is taken from MIKE 11 (DHI Software, 2002). The function from

FLDWAV provides damping of the convective terms for Fr < 1, while thefunction from MIKE 11 does not damp the convective term until Fr > 1. The

function from FLDWAV will be generally more stable, but the function from

MIKE 11 will be generally more accurate. Neither method will accuratelysimulate the propagation of rapidly changing hydrographs that occur during dam

breaks. In addition, neither method will consistently calculate the location of

hydraulic jumps accurately.

Regardless of the method of solution, SRH-1D assumes that subcritical flowoccurs at the boundaries of a river and the user must therefore supply both

upstream and downstream boundary conditions. Therefore, supercritical flow

should not occur at the upstream or downstream boundaries of any river.

A test case was simulated with each of the different method. The test caseconsisted of an 8 m wide rectangular channel with 20 m

3/s of flow and a

Mannings roughness coefficient of 0.01. There were 3 reaches. One subcritical


23/227

Flow Routing 13

reach, followed by a super critical reaches, terminating in a subcritical reach. A

analytical water surface was computed using the steady flow energy equation and

a momentum balance at the hydraulic jump. SRH-1D simulated the case using all

three options for the flow solver.

The results are shown in XFigure 2.3X. Option 2, 3, and 4 refer to the options as they

are numbered in SRH-1D. Option 2 uses the LPI technique with the damping

function of FLDWAV (X

2.45X

). Option 3 uses the LPI technique with the dampingfunction of MIKE 11 (X2.46 X). Neither option 2 and 3 captures the exact locationthat a hydraulic jump that forms.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 100 200 300 400 500 600 700 800

Distance (ft)

Elevation

(ft)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Froude

Bed Elevation (ft) Analytical Water Surface Elevation (ft)

SRH-1D (LPI-HEC-RAS) SRH-1D (LPI-MIKE11)

Froude Number

Figure 2.3. Comparison between water surface elevations for unsteady flowsolutions in SRH-1D.

2.2.3 Upstream Boundary Conditions

Two upstream boundary conditions are available: 1. known water discharge; 2.known water surface elevation

2.2.3.1 Water Discharge

The known water discharge boundary condition is summarized as:

)(tfQis = (2.47)


24/227


where isQ = the discharge at the center of the Cleft fictitious cell (XFigure 2.2X), i = 1,

outside of the model domain. The discretization of the upstream boundary

condition written in an iteration form as:

)( 1++= nn

isis tfQQ (2.48)

The above equation can be written in the following form:

is

m

isis

m

isis

m

isisdQcQbQa =++ + 11 (2.49)


0=is

a

1=isb

0=isc n

isis Qtfd = )(

These coefficients are used to replace the coefficients defined in Eqs. ( X2.41 Xa) to

(X2.41 Xd) at the upstream boundary.

2.2.3.2 River Stage

The river stage boundary condition can be written as:

)(tfHis = or )(tfAis = (2.50)

The discretized continuity equation (Eq. X2.38 X) is used to implement this boundarycondition:

m

isis

m

isis

m

isis AQQ =++ +1 (2.51)

C

where:1= mis

n

is

m

is AAA ;

ABis = the given cross section area defined in Eq. (X2.47X); and

1mis

A = the estimated cross section area of last iteration.


is

m

isis

m

isis

m

isis dQcQbQa =++ + 11 (2.52)


0=isa

isisb =

isisc =

is

m

isis Ad =


25/227

Flow Routing 15


(X2.41 Xd).

2.2.4 Downstream Boundary Condit ions

The downstream boundary conditions can also be grouped into two general types:

1. rating curve (the discharge is a function of the river stage); 2. known water

surface elevation.

2.2.4.1 Rating Curve

The rating curve boundary conditions can be expressed as:

)(1 ieie AfQ =+ (2.53)

where 1+ieQ is the discharge at the center of the Cright fictitious cellC, QBie+1B in XFigure

2.2 X, and ieA is the exit cross section area as defined in XFigure 2.2X. The

discretization of the downstream boundary condition in iteration form is:

ieie

n

ie

n

ieie AA

f

AfQQ

++= ++ )(11 (2.54)

Expression (X2.38 X) can be used to eliminate the unknown ieA in Eq. (X2.51 X),resulting in the following form:

121111 ++++++ =++ iem

ieie

m

ieie

m

ieie dQcQbQa (2.55)


ie

ie

ie A

fa

=+1

ieie

ie Afb =+ 11

01 =+iec

ieie

n

ie

n

ieie A

fQAfd

+= ++ 11 )(


(X2.41 Xd) for the downstream boundary.

2.2.4.2 River Stage

The given river stage boundary condition can be written as:

)(tfHie = or )(tfAie = (2.56)

where ie = last cross section. The discretized continuity equation (Eq. X2.38X) is

used to implement the boundary condition.

m

ieie

m

ieie

m

ieie AQQ =++ +1 (2.57)


26/227


where1= mieie

m

ie AAA andABieB is the given cross section area defined in Eq.

(X2.53 X).


121111 ++++++ =++ iem

ieie

m

ieie

m

ieie dQcQbQa (2.58)


ieiea =+1

ieieb =+1

01 =+iec

ie

m

ieieAd =+1

These coefficients are used to replace the coefficients defined in Eqs. ( X2.41 Xa) to(X2.41 Xd) for the downstream boundary.

2.2.5 Network Boundary Condition

For each river with Ncross sections, there are N+1 unknowns of discharge andN-1 equations (Eq. X2.41 X). Closing the solution system requires equations from

boundary conditions. In addition to the upstream and downstream boundary

conditions, the junction of the rivers provides the constraints required to solve thecontinuity and momentum equations.

A general case is discussed here with s rivers entering the junction and t rivers

exiting the junction. A total of s+tboundary conditions exist including one

continuity equation ands+t-1 momentum equations. No storage is allowed in thejunction. The continuity equation is written as:

01

,

1

1, = ==

+

t

l

m

isl

s

l

m

iel QQ (2.59)

wherem

ielQ 1, + is the estimated outlet discharge of river l,m

islQ , is the estimated

entrance discharge of riverl. The correction form of the discharge is written as:

01

,

1

1, = ==

+

t

l

m

isl

s

l

m

iel QQ (2.60)

A simplified momentum equation is introduced at the junction, requiring that allcross sections associated with the junction share the same water level correction.

Assuming rivertis the maximum river index of the rivers that exit the junction,the boundary condition for the riverl entering the junction is written as:

m

ist

m

iel HH ,1, = + or istm

istiel

m

iel TATA ,,1,1, // = ++ (2.61)

The area correction can be replaced by Eq. ( X2.49X), and Eq. ( X2.58 X) can be writtenas:


27/227

Flow Routing 17

istist

m

istist

m

istist

ieliel

m

ieliel

m

ieliel

TQQ

TQQ

,,1,,,,

,,1,,,,

/)(

/)(

++

=++

+

+(2.62)

The same boundary condition for the riverl exiting the junction is written as:

istist

m

istist

m

istist

islisl

m

islisl

m

islisl

TQQ

TQQ

,,1,,,,

,,1,,,,

/)(

/)(

++

=++

+

+(2.63)

2.3Internal Boundary Condition

Hydraulic structures such as dams, bridges, weirs, and gates may exist along anatural river and special treatments are required in the numerical model. For each

internal cross sectional structure, two more unknowns are introduced: the

discharge QBiB and water surface elevationZBi Bat that structure. The conservation of

mass serves as one of the equations necessary to solve for the unknowns. Theother equation depends upon the particular structure. Structures currently

supported by SRH-1D are listed in the following sections. Steady and unsteady

flow conditions use the same equations, but the interpolation in time of time seriesdata is handled differently. Internal boundary conditions are interpolated using a

step function for steady flow simulations and linearly in time for unsteady flow

simulations. Internal structures are assumed to occur between cross sections andare identified by the cross section that occurs immediately upstream.

2.3.1 Governing Equations for Internal Boundaries

2.3.1.1 Time Stage Table

For this boundary condition, the user enters a known water surface elevation

versus time at a cross section. For example, this boundary condition could

represent a pool that is controlled based upon daily operations:)(tHH = (2.64)

2.3.1.2 Elevation versus discharge table

For this boundary condition, a user inputs a table of water surface elevation versus

flow rate. The water surface elevation for each discharge is linearly interpolated

between user-entered points. No extrapolation is performed. This boundarycondition could represent many different structures that have a unique relationship

between flow rate and water surface elevation:

)(QHH = (2.65)

2.3.1.3 Weir

To simulate weirs in SRH-1D the user enters the spillway crest elevation, the weir

width, and the weir coefficient. If the downstream water surface elevation doesnot affect the upstream water surface elevation, the flow over the weir is

considered non-submerged. The submergence parameter,R, can be computed as:

SPU

SPD

ZZ

ZZR

= (2.66)


28/227


For non-submerged flow past a weir, discharge is expressed as a function of the

water surface elevation,Z, written as:

23

)( SPU ZZCBQ = if 67.0


29/227

Flow Routing 19

2.3.1.4 Bridge

Figure 2.4 Schematic of bridge (Source: Fread and Lewis, 1998)The present model uses the equations presented in FLDWAV1.0 (Fread and

Lewis, C1998 C) for highway/railway bridges and their associated earthen

embankments (as shown in XFigure 2.3X). The discharge can be expressed as:

2/32/3

2/12

1

)()(

)2/(g2

clilllcuiuuu

fiiibr

hZkLcchZkLcc

hgVZZCAQ

++

+= +(2.70)

where: 76.0if0.1 = ruu hk (2.71)

76.0if)76.0(0.1 3 >= ruruuu hhck (2.72)

96.076.0if10)78.0(133 += ruruu hhc (2.74)

)/()( 1 cuicuiru hZhZh = + (2.75)

15.00if)(02.3 015.0 += uuu hhcc (2.77)


30/227


ucuiu whZh /)( = (2.78)2)/(

ibrifKQxh = (2.79)

2/121 )2/(2 gVZZCAgQ iiibrbr += + (2.80)

ii AQV /= (2.81)

Cwhere: C= bridge coefficient,

ABbrB = cross-section flow area of the downstream end of bridgeopening which is user-specified via a tabular relation of wetted top

width versus elevation,

hBcuB = elevation of the upper embankment crest,

ZBiB = water surface elevation at section i (slightly upstream ofbridge),

ZBi+1B = water surface elevation at section i+1 (slightly downstream

of bridge),

V= velocity of flow within the bridge opening,

LBuB = length of the upper embankment crest perpendicular to the

flow direction including the length of bridge at elevation hBcuB,

kBuB = computed submergence correction factor for flow over the

upper embankment crest,

wBuB = width (parallel to flow direction) of the crest of the upper

embankment, and

di = maximum depth of flow under the bridge.

When the bridge opening is submerged, the coefficient Cin Eqs. (X2.67 X) and (X2.77 X)

is replaced by 'C for orifice flow:

CcC 0'= (2.82)

where:

=otherwise0.1

31.009.0if)09.0(0.1

0

rr

c (2.83)

and: ibri dhZr /)( = (2.84)

2.3.1.5 Radial Gate

For radial gates, SRH-1D uses equations similar to those in HEC-RAS 4.0

(USACOE, C2008 C). The schematic of radial gate is shown in XFigure 2.4X. Accordingto the upstream and downstream water surface elevations, the flow can be

categorized into three types: free flow, partially submerged flow, and fully

submerged flow.


31/227

Flow Routing 21

Figure 2.5 Schematic of radial gate (Source: Brunner, C2001C).

When the downstream tailwater elevation (ZBDB) is not high enough to cause anincrease in the upstream headwater elevation, the flow is considered to be free

flow. The discharge can be expressed as

67.0if2 = SPUSPDHEBETE

ZZZZHBWTgCQ (2.85)

where Q = flow rate (cfs); C= discharge coefficient (typically ranges from 0.6

0.8); W= width of the gate (ft); T= trunnion height (ft, from spillway crest totrunnion pivot point); TE= trunnion height exponent (typically about 0.16); B =

height of gate opening (ft);BE= gate opening exponent (typically about 0.72); H

= upstream energy head above the spillway crest (ZBUB-ZBSPB); HE= head exponent(typically about 0.62);ZBUB = elevation of the upstream energy grade line (ft);ZBDB =

elevation of the downstream water surface (ft); ZBSPB = elevation of the spillway

crest through the gate (ft).

When the downstream tailwater elevation (ZB

DB

) is high enough to cause an increasein the upstream headwater elevation, the flow is considered to be partially

submerged flow. The discharge can be expressed as

67.0if)3(2 >

=SPU

SPDHEBETE

ZZ

ZZHBWTgCQ (2.86)

where H= upstream energy head (ft) above the downstream water surface (ZBUB-ZBDB).

When the discharge is further increased, the gate is fully submerged and the

discharge can be expressed as

80.0if2 >= SPUSPD

ZZZZgHCAQ (2.87)

whereA = area of the gate opening (ft P2

P);H= upstream energy head (ft) above thedownstream water surface (ZBUB-ZBDB), andC= discharge coefficient (typically 0.8).


32/227


2.3.2 Implementation for Steady Flows

For an internal boundary, the mass conservation equation is the same as equation(X2.12 X) and the energy equation (X2.11 X) is replaced by the appropriate internal

boundary condition. The water surface elevation upstream of the internal

boundary is solved using the flow rate computed from the mass conservationequation.

The equation for an internal boundary can be written as:

0),,,( 11 =++ iiiii QYQYF (2.88)

This equation is used to replace the energy equation (Eq. X2.11X). The derivatives

i

i

Z

F

,i

i

Q

F

,1+

i

i

Z

F, and

1+

i

i

Q

Fare calculated and substituted into Eq. (X2.15 X).

2.3.3 Implementation for Unsteady Flows

All internal boundary conditions can be summarized as:

),( 1 iii AAfQ = (2.89)

where 1iA and iA are the cross section areas before and after the internal

boundary, respectively. The discretized form of the internal boundary condition

written in iteration form is:

ii

ii

n

i

n

i

n

ii AA

fA

A

fAAfQQ

+

++=

11

1 ),( (2.90)

Expression (X2.38 X) can be used to eliminate unknowns 1 iA and iA inEq. (X2.87 X), which results in the following form:

i

m

ii

m

ii

m

ii dQcQbQa =++ + 11 (2.91)


11

= ii

i A

fa

ii

ii

i A

f

A

fb

=

11

1

ii

i A

fc

=

ii

ii

ni

ni

nii A

fAfQAAfd ++=

11

1 ),(

These coefficients are used to replace the coefficients defined in Eqs. ( X2.41 Xa) to(X2.41 Xd).


33/227

Sediment Transport 23

3 Sediment Transport

This chapter describes the methods used to perform the sediment transport

calculations. SRH-1D simulates the physical processes important to both cohesiveand non-cohesive sediment transport. There are three major components of

sediment transport within SRH-1D:1. Sediment Routing

2. Bed Material Mixing

3. Cohesive Sediment Consolidation

Sediment routing is the simulation of the downstream movement of sediment inthe river flow. Bed material mixing processes include bed material sorting and

armoring. Consolidation is compaction of cohesive sediment over time. The

modeling of each of these components is described in the following sections.

3.1Sediment Routing

There are two sediment routing methods available in SRH-1D: unsteady sedimentrouting and Exner equation routing. The unsteady sediment routing computes the

changes to the suspended sediment concentration with time. The Exner equation

routing ignores changes to the suspended sediment concentration over time.Unsteady sediment routing can be used when unsteady flow is being simulated

and suspended concentrations change rapidly. In most other cases, Exner equation

routing can be used.

3.1.1 Exner Equation Routing

The Exner equation (Exner, 1920; 1925) was derived assuming that changes tothe volume of sediment in suspension are much smaller than the changes to thevolume of sediment in the bed, which is generally true for long-term simulations

where steady flow is being simulated. The mass conservation equation for

sediment reduces to,

0=

+

sds qt

A

x

Q(3.1)

where = volume of sediment in a unit bed layer volume (one minus porosity);

ABdB = volume of bed sediment per unit length; QBsB = volumetric sediment discharge;andqBsB = lateral sediment inflow per unit length. Integrating ( X3.1 X) over a control

volume centered on each cross section gives an equation for the deposition depth(ZBbB) for a single sediment size fraction at a particular cross section, i:

( ) tQQtxqZxW isisiisibiii += ,1,,, (3.2)

where W is the width of the cross section subject to erosion or deposition. Theerosion volumes for each size fraction are summed to compute the total erosion or

deposition for a particular cross section. The lateral inflows are user defined and

the erosion width is computed based upon the hydraulic calculations. The only


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unknowns remaining are the sediment transport rates. The sediment transport rate

(QBsB) can also be written as QC, where C is the computed discharge weighted

average sediment concentration. The following sections describe the numerical

solution for sediment concentration for the cases of non-cohesive sediment,floodplain routing, and cohesive sediment.

3.1.1.1 Non-Cohesive Sediment Routing

If the cross sections are far apart relative to the change in sediment transportcapacity, it is acceptable to assume that the bed-material load discharge equals the

sediment transport capacity of the flow; i.e., the bed-material load is transported

in an equilibrium mode (QBsB = QBcapB, where QBcapB is the transport capacity). In otherwords, the exchange of sediment between the bed and the fractions in transport is

instantaneous. However, the spatial-delay and/or time-delay effects are important

in circumstances where there are rapid hydraulic changes in short reaches. For

example, reservoir sedimentation processes are non-equilibrium processes. Inaddition, some laboratory studies have shown that it may take a significant

distance for clear water inflow to reach saturation sediment concentrations. To

take these effects into consideration, SRH-1D starts with the analytical solution to

the following equation:

( )WCVVdxdCQ

dx

dQde

s == (3.3)

where Qs = sediment transport rate, C= flow weighted concentration = Qs/Q, VBeB =

erosion velocity, and VBdB = deposition velocity. The analytical solution to the

above equation between two cross sections, say i andi-1, is:

+=

i

idi

iiii Q

xWVCCCC exp)( *1

*(3.4)

where dieii VVC =*

and is the computed sediment transport capacityconcentration; x = reach length; and i = cross-section index (increasing fromupstream to downstream). Eq. ( X3.4XX) is employed for each of the particle size

fractions in the non-cohesive range. The erosion and deposition velocity for non-

cohesive sediment are given as:

( )tottote

WLQV*= and ( )totd WLQV = (3.5)

where LtotB is the adaptation length for total load, and*tot

Q = total sediment

transport capacity. The adaptation length for total load is computed as (Greimann

et al. 2008):

( )f

sbstotWw

QfLfL

+= 1 (3.6)

where LbB is the adaptation length for bed load, and is a suspended sedimentrecovery factor and the parameterfs is the fraction of suspended load relative to

the total load.


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The concept of the adaptation length for bed load, Lb, is taken from Holly and

Rahuel (1990). It is a dimensional value that is usually on the order of large bed

features, such as bars or channel width. It is assumed proportional to the average

depth of water at the cross section; a user enters the constant of proportionality as:

hbL Lb = (3.7)

where bL is a user defined non-dimensional coefficient andh is the average depthat the cross section.

The fraction of suspended load, fs, was found to be primarily a function of thesuspension parameter, Z. Greimann et al. (2008) derived the following empirical

function forfs:

( )zs ef = 5.2,1min (3.8)

The parameters andLb control the rate at which the sediment concentration

approaches the sediment carrying capacity. Higher values of or lower values ofLBbB indicate that the concentration reaches the carrying capacity more quickly.

While some have used constants for (Han and He, 1990), Galappatti and

Vreugdenhil (1985) and Armanini and Di Silvio (1988) suggest that isdependent upon the ratio of fall velocity to shear velocity and the relative

roughness height. SRH-1D assumes that is a constant but separate values forare used for deposition versus erosion and the program automatically chooses the

correct one. Han and He (1990) CC recommend a value of 0.25 for deposition and 1.0for erosion. The asymptotic behavior of Eq. (X3.4 X) with increasing particle size is

shown in Figure 3.1 X. One can see that as d(orBfB) becomes larger*ii CC .

Figure 3.1TTRatio between non-equilibrium concentration and carrying capacityas a function of sediment particle size (from Yang and Simes, 2002).

The influence of the recovery parameter is illustrated in XFigure 3.2 XX. Thedepositional case represents a situation in which there is a sudden loss of carrying


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capacity ( 0* =iC ) from an upstream equilibrium condition (

*

11 = ii CC ). The plot

shows the actual normalized concentration for two sizes of the sediment particles.

It is clear that the non-equilibrium effect is stronger on the finer particles, and that

it diminishes as increases. The erosional case represents a sudden increase in

carrying capacity, such as when clear water enters a channel with an erodible bed.

In this case, 0*

11 == ii CC and 0* >iC . The same trend occurs as before, i.e., the

non-equilibrium effects tend to diminish with increasing particle sizes and

recovery factor.

The distance between computational cross sections, x, is another importantfactor in non-equilibrium calculations. XFigure 3.3X shows how the non-equilibrium

effects vary with distance for the same situations and particle sizes in XFigure 3.2X.

In practice, the values of vary widely. If data are available, may be acalibration parameter.

Figure 3.2TTEffect of the recovery parameteron the computation of non-equilibriumsediment concentrations for two sediment particle sizes. (a) deposition and (b) erosion

(from Yang and Simes, 2002).


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Figure 3.3TVariation of non-equilibrium effects as a function of distance between crosssections for deposition (a) and for erosion (b) (from Yang and Simes, 2002).

3.1.1.2 Cohesive Sediment

SRH-1D defines cohesive sediment as sediment with a diameter smaller than0.0625 mm. For cohesive sediment, the capacity concentration, CP

*P, is not needed

because the capacity concentration is controlled by the erosion or deposition rate

occurring in the river. In SRH-1D, the erosion of fine cohesive sediment is

prevented if the volumetric concentration in the stream exceeds a user specifiedsaturated volumetric fraction (the default value is 0.15).

The steady sediment transport equation for cohesive sediment is written as

Cdx

dQWCVV

dx

dQde

s ~)( += (3.9)

where VBeB and VBdB = cohesive sediment erosion and deposition velocities,respectively; C= cohesive sediment volumetric concentration. The computation

of the erosional and depositional velocities is given in Sections X3.1.5 X to X3.1.7 X. If

the erosion velocity is zero, and the deposition velocity is greater than zero, the

solution to (X3.18 X) is given as:

CQ

Q

Q

xWVCC dii

~exp

1

+

= (3.10)

If the deposition velocity is zero and the erosion velocity is greater than zero, the

solution to (X3.18 X) is given as:


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CQ

Q

Q

xWVCC eii

~1

+

+= (3.11)

3.1.2 Unsteady Sediment Transport

When simulating unsteady flow or sediment transport, the changes in suspendedconcentration with time may be important. To compute the changes in suspended

sediment concentration, the convection-diffusion equation with a source term for

sediment erosion/deposition is used. The 1D depth-averaged convection-diffusionequation for a particular sediment size class is:

+

=

+

x

cAfD

xx

Qc

t

Acx

(3.12)

where c = depth averaged concentration;A = cross section area, Q = the flow rate;

DB = diffusion coefficients streamwise directions, respectively; The parameter, ,

is the velocity of sediment relative to the water. Finally, = source (erosion) andsink (deposition) terms for one sediment constituent.. The source term can be

written as:( )WcVV de = (3.13)

where VBeB, VBdB = erosion and deposition velocities, respectively. The erosion and

deposition velocities have already been defined in previous sections.

SRH-1D uses the expressions developed in Greimann et al. (2008) for . Theexpression for bed load is:

[ ]

r

bedU

u

=

)5exp(11.1 17.0*(3.14)

where *u is shear velocity; Uis cross-sectional velocity; and ( )r= ,20min ; andr are Shields parameter and reference Shields parameter (assumed to be

0.035, or taken from user input), respectively. The expression for suspended loadis:

( )[ ]ZU

usus 7.2exp1

21 *

+= (3.15)

where ( )*,1min uwZ f = is the suspension parameter; fw is the sediment fall

velocity; and =Von Karman constant (= 0.4). The limit ofZ< 1.0 is imposedbecause the Rouse profile is not valid near the bed due to particle inertia effects

(Greimann et al., 1999). It should be noted there is a discontinuity in sedimentvelocity between bedload and suspended load using Eq (X3.14 X) and (X3.15 X). To

remedy this situation, the total relative velocity is computed as,

( )bedsus = ,max (3.16)

This ensures that the sediment velocity is a continuous function as it transitions

from bed load to suspended load.


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The diffusion coefficient is computed as in Fischer et al. (1979),

*

22

Hu

UWKD x= (3.17)

whereHis the average cross sectional depth, Wis channel top width, Uis cross-

sectional velocity, *u is shear velocity, andKx is a user specified value. Fischer et

al. recommends 0.011.

The differential equation is transformed to an integral equation to explicitly

conserve mass.

( ) ( ) +=+

xffxdxcfDAQccAdx

t(3.18)

where the f indicates a sum over cell boundaries, where flows or gradients out ofthe control volume are defined as positive and into the control volume are defined

as negative.

The stream is discretized into cells centered on the cross sections. The integral

equation (X3.18X) is solved for each cell by calculating the coefficients for thegeneral discrete approximation of the integral equation:

PLLPPRCACA =+ (3.19)

where the sum is over the surrounding cells. The following sections discuss the

discretization of the terms in Eq. ( X3.18 X).

Figure 3.4TGrid Definition for Unsteady Sediment Simulation.3.1.2.1 Unsteady Term

The unsteady term requires integration over the CS area. The implicit Euler

method is applied for time marching. The unsteady term is approximated by theproduct of CS area and the c value at the CS center,

PPP

xxA

t

ccAdx

t

= (3.20)

where1= nP

n

PP ccc , and Px is the length of the grid cell P. the superscripts n-1 and n denote the previous and current time steps, respectively. The unsteady

term contributions to the coefficients in Eq. ( X3.19X) and can be written as:

t

xAA PPUP

=| (3.21)

*E *EE* WW * W * P


40/227


where the subscript U indicates the contribution of the unsteady term to the ABpB

coefficient; t = time step.

3.1.2.2 Convective Term

The Lax-Wendroff TVD (Total Variation Diminishing) Method is used todiscretize the convective term. The original Lax-Wendroff Method is a second-

order accuracy scheme. However, numerical results oscillate on discontinuities.

The Lax-Wendroff TVD Method suppresses the correction term in Lax-WendroffMethod when discontinuities appear. The TVD scheme remains second-order

accurate for smooth regions but becomes a first order scheme near discontinuities

to avoid oscillations. Details of Lax-Wendroff TVD Method are discussed in

Tannehill et al. (C1997 C).

Approximations of the convective term involve values of variables at the CL

centers:

( ) ( ) ( )wef

QcQcQc = (3.22)

where e denotes the face between P andE. Using the Lax-Wendroff TVD

method, *eF can be expressed as

( ) ( ) ( ) ( )[ ]PEeEPee

ccWccQQc +=21 (3.23)

where

( ) ( ) ( )[ ]eree

QcQW += 1

where( )

PP

er xA

tQc

= and is the flux limiter computed as,

( )( )[ ]2/1,2,2min,0max rr += , 21 iiPE

cc

ccr

= (3.24)

with iB1B = P andiB2B = WifQeB is positive andiB1B =EEandiB2B =EifQBeB is negative. Eq.

(X3.29 X) is van Leers MUSCL flux limiter (van Leer, 1979). If = 1 then the 2 Pnd

P

order accurate Lax-Wendroff scheme is obtained. If = 0 then the scheme is afirst order accurate upwind scheme. The Crank-Nicolson method is applied to get

second-order accuracy in time, but it is conditionally stable. With this scheme,

( )e

Qc can be expressed as

( ) ( )( )

( )( )EnEee

p

n

Peee

ccWQ

ccWQQc

+

+++=

1

1

21

21

(3.25)

where = implicit factor ( 10


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( )

( )

( ) ( )

( ) ( )

( )11121

21

21

21

|||1|

|

|

|

++

=

++++

+++=

=

=

n

WCW

n

ECE

n

PCPC

ssnn

wweeCP

wwCW

eeCE

cAcAcAR

WQWQ

WQWQA

WQA

WQA

(3.26)

where the subscript C indicates the contribution of the convective term to the

coefficients of Eq. ( XX3.19X).

3.1.2.3 Diffusion Term

The approximation of the diffusion term in Eq. ( X3.18 X) also involves the values of

variables at the CL. First, the central differential scheme (CDS) is applied inspace, and then the Crank-Nicolson method is applied in time. The discretization

of the diffusion term is second-order accurate in both space and time.

w

W

n

WP

n

P

ww

e

P

n

PE

n

E

ee

w

WP

ww

e

PE

eel

x

CCCCaD

x

ccccaD

x

ccaD

x

ccaDdlncDh

+

+

+=

+

=

1111

)(

(3.27)

where ex , wx , sx , and nx are distances between the CS center P andneighborhood CS centers E, W, N, and S, respectively. The coefficients from the

diffusion term can now be summarized as:

( )111 |||1|

)||(|

/|

/|

++

=

+=

=

=

n

WDW

n

EDE

n

PDPD

DWDEDP

wwwDW

eeeDE

cAcAcAR

AAA

xaDA

xaDA

(3.28)

where the subscript D indicates the contribution of the diffusion term to thecoefficients of Eq. ( X3.19X).

3.1.2.4 Source Term

The only term remaining in the sediment transport is the source term from net

sediment erosion and deposition in the streamwise direction.

( )111 |||1|

)||(|

/|

/|

++

=

+=

=

=

n

WDW

n

EDE

n

PDPD

DWDEDP

wwwDW

eeeDE

cAcAcAR

AAA

xaDA

xaDA


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3.1.2.5 Discretized Sediment Transport Equation

Adding all the convective and diffusive terms gives:

RcAcALLPP =+ (3.29)

where P = CS center;L = neighborhood CS centers WorE; and:

DC

DLCLL

DPCPUPP

RRR

AAAAAAA

||

|||||

+=

+= ++= (3.30)

The matrix equation X3.29 X can be solved by a tri-diagonal matrix solver using the

Thompson Algorithm.

3.1.3 Non-Cohesive Particle Fall Veloci ty C

Computation of particle fall velocity is necessary for several non-cohesive

sediment transport capacity calculations. The fall velocity is computed using

values recommended by the U.S. Interagency Committee on Water ResourcesSubcommittee on Sedimentation ( C1957 C) ( XFigure 3.5X). SRH-1D assumes the Coreyshape factor ofSF= 0.7, which is defined as

ab

cSF = (3.31)

where a, b, andc = the length of the longest, the intermediate, and the shortestmutually perpendicular axes of the particle, respectively. For particles with

diameters above the range given in XFigure 3.5X, i.e. greater than 10 mm, thefollowing formula is used:

gdGwf )1(1.1 = (3.32)

where wf is the fall velocity, g = acceleration due to gravity, G = specific gravity

of sediments, and d = particle diameter. In some cases, it is also necessary tocompute the kinematic viscosity, for which the following formula is used:

2

6

000221.00337.00.1

10792.1

TT++

=

(3.33)

where Tis the water temperature in degrees Centigrade and in mP2

P/s.


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FallVelocity(cm/s)

Particle Size (mm)

Figure 3.5 Relation between particle sieve diameter and its fall

velocity according to the U.S. Interagency Committee on WaterResources Subcommittee on Sedimentation (1957) C.

3.1.4 Non-Cohesive Sediment Transport Capacity

The literature contains many sediment transport capacity functions. Usually, each

transport function was developed for a certain range of sediment size and flow

conditions. Computed results based on different transport functions can differsignificantly from each other and from measurements. No universal functionexists which can be applied with accuracy to all sediment and flow conditions.

SRH-1D employs a number of transport functions for non-cohesive material,

presented in XXTable 3.1XX. Yang (C1996 C) published a more detailed description ofsome of these functions including comprehensive comparisons and evaluations.

New transport formulas are often added to the code as they are developed. It is

recommended that users frequently check for the most recent version of the codeon the Reclamation web site.

Many transport capacity formula were developed assuming uniform size

gradations. In these cases, the transport capacity is modified by the fraction of the

size class in the active layer according to the following equation,

T

iii QpQ =* (3.34)

where pi is the mass fraction of that size class in the active layer andT

iQ is the

transport capacity predicted by the formula assuming uniform size.


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Table 3.1 Sediment transport functions available in SRH-1D and its type (B = bed

load; BM = bed-material total load).

Equation Type

aCMeyer-Peter and Mller (1948) and Wong and Parker (2006)C B

CLaursen (1958)C BM

ModifiedCLaursen's Formula C(Madden, 1993)CC BM

CEngelund and Hansen (1972) C BMCAckers and White (1973) C BM

CAckers and White (HR Wallingford, 1990) C BM

CYang (1973) C+ CYang (1984)C BM

CYang (1979) C+ CYang (1984)C BM

Brownlie (1981)CC BM

CYang et al. (1996)C BM

CParker (1990)C B

Wilcock and Crowe (2003) B

Wu (2004) BM

3.1.4.1 Meyer-Peter and Mller's Formula (1948) modif ied by Wong andParker (2006)C

In non-dimension form, the CMeyer-Peter and Mller's bedload formula (1948) C is:

( )

5.15.1

3047.0

)1(

)/(8

1

=

dsRSKK

dsg

q rsbi (3.35)

where and BsB = specific weights of water and sediment, respectively; R =

hydraulic radius; S = energy slope; d = mean particle diameter; s = specific

gravity of sediment; qB

bB

= volume bed load transport per unit width; and (KB

sB

/KB

rB

)S=the adjusted energy slope that is responsible for bed-load motion. The value ofKBs B

andKBrB can be computed from:

2/13/2 SRC

VKm

s = (3.36)

and

6/190

26d

Kr = (3.37)

where dB90B = the size of sediment for which 90 percent of the material is finer and

is in meters.

Wong and Parker (2006) reanalyzed the data used by CMeyer-Peter and Mller andfound that the energy slope correction in the equation is unnecessary. The

modified formula suggested by them is:

( )

5.1

30495.0

)1(97.3

1

= ds

RS

dsg

qbi (3.38)


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This is the formula used in SRH-1D.

3.1.4.2 Laursen's Formula (1958) and Modi fied Version (Madden, 1993)

CLaursen's formula (1958)C was expressed in dimensionally homogeneous forms by

an CAmerican Society of Civil Engineers Task Committee (1971) C as,

( )

=

fii

i

iit w

UfD

dpC

*6/7

101.0 (3.39)

where CBtB = sediment concentration by weight per unit volume; gDSU =*

;pBi B =

percentage of materials available in size fraction i; wfBi B = fall velocity of particles

of mean size dBiB in water;D = average water depth. The parameteri is a measureof the shear stress relative to the reference shear stress:

( )ciii = (3.40)

where BcB is the reference Shields number; and BiB = Shields parameter of the

sedime

SRH-1D Users Manual 3_0

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