SOBEK 3 Technical Reference Manual

8/12/2019 SOBEK 3 Technical Reference Manual

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F TSOBEK 3Hydrodynamics

Technical Reference Manual

SOBEK in DeltaShell

Version: 3.0.1.33144

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SOBEK 3, Technical Reference Manual

Published and printed by:DeltaresBoussinesqweg 12629 HV DelftP.O. Box 1772600 MH DelftThe Netherlands

telephone: +31883358273fax: +31883358582e-mail: [email protected]: http://www.deltares.nl

For sales contact:telephone: +31883358188fax: +31883358111e-mail: [email protected]: http://www.deltaressystems.nl

For support contact:telephone: +31883358100fax: +31883358111e-mail: [email protected]: http://www.deltaressystems.nl

Copyright 2014 Deltares

All rights reserved. No part of this document may be reproduced in any form by print, photoprint, photo copy, microlm or any other means, without written permission from the publisher:Deltares.


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Contents

Contents

1 Conceptual description 11.1 Hydrodynamics 1DFLOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Hydrodynamic denitions . . . . . . . . . . . . . . . . . . . . . . . 11.1.1.1 Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1.2 Bed level . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1.3 Water depth . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1.4 Water level . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1.5 Flow area . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1.6 Storage area . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1.7 Wetted area . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1.8 Wetted perimeter . . . . . . . . . . . . . . . . . . . . . . 41.1.1.9 Flow velocity . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1.10 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1.11 Hydraulic radius . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2.1 Continuity equation (1D) . . . . . . . . . . . . . . . . . . 71.1.2.2 Momentum equation (1D) . . . . . . . . . . . . . . . . . . 71.1.2.3 Transport equation for salinity . . . . . . . . . . . . . . . . 7

1.1.3 Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.4 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Convection (1D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.6 Water level gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.7 Wind friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.8 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.9 Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.10 Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.11 Typeof Lateral Discharges andtheir Assignment toh -Calculationpoint(s) 11

1.1.11.1 Different Types of Lateral Discharges . . . . . . . . . . . . 111.1.11.2 Assignment of Lateral Discharges to h -calculation point(s) 12

1.1.12 Lateral discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.13 Bed friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.13.1 Bos-Bijkerk . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.13.2 Chzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.13.3 Manning . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.1.13.4 Nikuradse . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.1.13.5 Strickler . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.13.6 White-Colebrook . . . . . . . . . . . . . . . . . . . . . . 171.1.14 Froude number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.15 Boussinesq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.1.16 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.1.17 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.1.17.1 Advanced weir . . . . . . . . . . . . . . . . . . . . . . . . 211.1.17.2 Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.1.17.3 Compound structure . . . . . . . . . . . . . . . . . . . . 261.1.17.4 Culvert . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.1.17.5 Database structure . . . . . . . . . . . . . . . . . . . . . 29

1.1.17.6 General structure . . . . . . . . . . . . . . . . . . . . . . 301.1.17.7 Inverted siphon . . . . . . . . . . . . . . . . . . . . . . . 341.1.17.8 Orice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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1.1.17.9 Pump station and Internal Pump station . . . . . . . . . . 391.1.17.10 External Pump station . . . . . . . . . . . . . . . . . . . . 441.1.17.11 River Pump . . . . . . . . . . . . . . . . . . . . . . . . . 491.1.17.12 River Weir . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.1.17.13 Siphon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561.1.17.14 Universal Weir . . . . . . . . . . . . . . . . . . . . . . . . 591.1.17.15 Vertical obstacle friction . . . . . . . . . . . . . . . . . . . 621.1.17.16 Weir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

1.1.18 h -Calculation point . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.1.19 Methods for computing conveyance . . . . . . . . . . . . . . . . . . 65

1.1.19.1 Tabulated lumped conveyance approach . . . . . . . . . . 661.1.19.2 Vertically segmented conveyance approach . . . . . . . . 67

1.1.20 Drying/ooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731.1.21 Free board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731.1.22 Ground layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

1.1.23 Measurement station . . . . . . . . . . . . . . . . . . . . . . . . . . 741.1.24 Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741.1.24.1 Reach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751.1.24.2 Reach length . . . . . . . . . . . . . . . . . . . . . . . . 761.1.24.3 Reach segment . . . . . . . . . . . . . . . . . . . . . . . 771.1.24.4 Connection node . . . . . . . . . . . . . . . . . . . . . . 77

1.1.25 Reference level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781.1.26 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791.1.27 Sediment transport capacity . . . . . . . . . . . . . . . . . . . . . . 791.1.28 Simulation output parameters at reach segments . . . . . . . . . . . 811.1.29 Time step reductions during the simulation . . . . . . . . . . . . . . 81

1.1.30 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831.1.31 Staggered grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831.1.32 Stationary computation . . . . . . . . . . . . . . . . . . . . . . . . . 851.1.33 Summer dike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851.1.34 Super-critical ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 891.1.35 Surface level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2 Numerical concepts (1D) 912.1 Staggered grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.2 Computational grid and denitions . . . . . . . . . . . . . . . . . . . . . . . 922.3 Upwinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.4 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.4.1 Time integration of the continuity equation . . . . . . . . . . . . . . 942.4.2 Time integration of the momentum equation . . . . . . . . . . . . . 962.4.3 Time integration of the combined continuity and momentum equation 982.4.4 Numerical method for the transport equation . . . . . . . . . . . . . 992.4.5 Open boundary conditions for the transport equation . . . . . . . . . 99

2.4.5.1 Thatcher Harleman time lag . . . . . . . . . . . . . . . . 1002.4.5.2 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.4.5.3 Dispersion coefcient . . . . . . . . . . . . . . . . . . . . 1012.4.5.4 Salinity intrusion and baroclinic term . . . . . . . . . . . . 102

References 105

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List of Figures

List of Figures

1.1 Denition of model datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Denition of bed level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Denition of water depth d = h

z

b . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Denition of water level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Deniton of ow area (Af ) and storage area (As ) . . . . . . . . . . . . . . . 41.6 Wetted perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Wetted perimeter for the main channel . . . . . . . . . . . . . . . . . . . . . 51.8 Computation of hydraulic radius . . . . . . . . . . . . . . . . . . . . . . . . 61.9 Wetted perimeter for the main channel (Subdivision in main channel and ood-

plains only in SOBEK-River) . . . . . . . . . . . . . . . . . . . . . . . . . . 61.10 Wind direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 Lateral discharges, point and line source . . . . . . . . . . . . . . . . . . . . 131.12 Diffuse lateral discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.13 Denition of energy and water level . . . . . . . . . . . . . . . . . . . . . . 221.14 Pillar bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.15 A suspension bridge with abutments . . . . . . . . . . . . . . . . . . . . . . 251.16 Fixed bed bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.17 Soil bed bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.18 Side view of a culvert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.19 Good modelling practice, Culvert, Inverted Siphon and Siphon . . . . . . . . 291.20 General structure, side view . . . . . . . . . . . . . . . . . . . . . . . . . . 301.21 General structure, top view . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.22 Drowned gate ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.23 Drowned weir-ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.24 Side view of an inverted siphon . . . . . . . . . . . . . . . . . . . . . . . . . 351.25 Good modelling practice, Culvert, Inverted Siphon and Siphon . . . . . . . . 371.26 Orice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.27 Pump station with positive pump direction and two pump stages . . . . . . . 401.28 Pump station with negative pump direction and two pump stages . . . . . . . 411.29 External pump station with pump direction IN and two pump stages . . . . . 451.30 External pump station with pump direction OUT and two pump stages . . . . 451.31 River pump with Upward control and start-level above stop-level . . . . . . . 501.32 River pump with Upward control and stop-level above start-level . . . . . . . 511.33 River pump with Downward control and start-level above stop-level . . . . . . 511.34 River pump with Downward control and stop-level above start-level . . . . . . 52

1.35 Free and drowned weir ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.36 Drowned ow reduction curves . . . . . . . . . . . . . . . . . . . . . . . . . 561.37 Siphon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561.38 Good modelling practice, Culvert, Inverted Siphon and Siphon . . . . . . . . 591.39 Weir prole of a Universal weir, division in rectangular and triangular weir sec-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.40 Weir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.41 h-calculation points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651.42 Concept of the lumped conveyance approach . . . . . . . . . . . . . . . . . 661.43 Example of constructed vertical segments in a Y-Z prole . . . . . . . . . . . 671.44 Denition sketch of a vertical segment, considered in computing conveyance

for a Y-Z prole and an Asymmetrical Trapezium cross-section . . . . . . . . 691.45 Free board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731.46 Ground layer in circular cross section . . . . . . . . . . . . . . . . . . . . . 74

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1.47 Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751.48 Reach length in model network . . . . . . . . . . . . . . . . . . . . . . . . . 761.49 Reach segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771.50 Connection nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

1.51 Denition of reference level . . . . . . . . . . . . . . . . . . . . . . . . . . . 781.52 Staggered grid channel-ow . . . . . . . . . . . . . . . . . . . . . . . . . . 841.53 Staggered grid sewer-ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 841.54 Summer dike option, available in a river prole . . . . . . . . . . . . . . . . . 851.55 Flow area behind the summer dike as function of local water levels. . . . . . 871.56 Total area behind the summer dike as function of local water levels. . . . . . 88

2.1 Staggered grid for unsteady channel ow . . . . . . . . . . . . . . . . . . . 912.2 Staggered ow grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.3 Figure: Illustration of memory effect for open boundary . . . . . . . . . . . . 1012.4 Salt wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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List of Tables

List of Tables

1.3 Example of switch-on-levels and switch-off-levels at the suction-side and thedelivery-side of a pump station . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.4 Example of switch-on-levels and switch-off-levels at the suction-side and thedelivery-side of an external pump station . . . . . . . . . . . . . . . . . . . . 48

1.5 Flow-Pump station covers the options of a River pump with respect to thepump direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.6 Crest shape and coefcients for simple weir structure (default values) . . . . 55

2.1 Overview on the equation solved for the two different grid elements of a stag-gered grid and the corresponding output parameters . . . . . . . . . . . . . 91

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1 Conceptual description

In this section, we will present in detail the governing equations. Below an overview of thesymbols that are used in the equations is presented.

List of Symbols

Symbol Units Meaning

Af m2 Cross sectional ow areaAR m2 Rainfall runoff areaAs m2 Cross sectional storage areaAt m2 Cross sectional total area (i.e. Af + As )Az m2 Surface areaC m1/ 2/s Chzy coefcientC d - Drag coefcientC wind The wind friction coefcient.f d - Global design factorfor overall extreme high or low load (rain-

fall)g m/s 2 Acceleration due to gravity (9.81)h m Water leveliR mm/s Intensity of rainfallis mm/s Intensity of seepageO m Wetted perimeterq lat m2/s Lateral discharge per unit lengthQ m3/s DischargeR m Hydraulic radiust s Timewf m Cross sectional width at water levelx, y m Cartesian coordinateswind deg Angle between the wind direction and the local channel direc-

tionw kg/m 3 Water densityair kg/m 3 Density of air wind N/m 2 Wind shear stress

1.1 Hydrodynamics 1DFLOW

1.1.1 Hydrodynamic denitions

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1.1.1.1 Datum

The Model datum and reference level both refer to a horizontal plane from which elevationsare dened (positive in upward direction and negative in downward direction). By default themodel datum and reference level are equal to zero.

Figure 1.1: Denition of model datum

Note:All levels (quantities with a vertical coordinate) in SOBEK are dened with respect to themodel datum or reference level.

1.1.1.2 Bed level

The bed level is dened as the lowest point in the cross section. In the denition of the crosssection an example is given of the interpolation and extrapolation of the bed levels over areach. The used symbol in the SOBEK-Flow-module is zb.

It is given relative to a reference level, for example Mean Sea Level .

Figure 1.2: Denition of bed level

If the water level is lower than the bed level, drying occurs.

1.1.1.3 Water depth

In the SOBEK-Flow-module the water depth is the distance between the water level and thebed level. The symbol used for the water depth is d.

d = h

zb (1.1)

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Conceptual description

Figure 1.3: Denition of water depth d = h zb

1.1.1.4 Water level

The water level (h) is the level of the water surface relative to the reference level or Modeldatum (= reference level).

In the one-dimensional SOBEK-Flow-module the water level perpendicular to the ow direc-tion is assumed to be horizontal.

Figure 1.4: Denition of water level

The water level together with the discharge form the result of the SOBEK-Flow-module com-putations. Water levels are calculated at the connection nodes and the h-calculation points.

Note: In each SOBEK-Flow-model, all levels are relative to the reference level

1.1.1.5 Flow area

The ow area Af of a cross-section is the area through which water is actually owing.

In a cross-section, distinction can be made between the ow area and the storage area. Inthe latter part, water is stored only, see Figure 1.5.

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Figure 1.5: Deniton of ow area ( Af ) and storage area ( As )

The shape of the cross-section, and the distinction between the ow area and storage areaare dened for each cross-section by user input.

1.1.1.6 Storage area

The storage area As of a cross-section is the area in which only water is stored i.e. the non-conveying part of the cross-section.

In a cross-section, distinction can be made between the ow area and the storage area, seeFigure 1.5.

The shape of the cross-section, and the distinction between the ow area and storage area is

dened for each cross-section by user input.The total area At is dened as the ow area plus the storage area.

1.1.1.7 Wetted area

The wetted area (Af [m2]) is the part of the cross section that is lled with water. See Fig-ure 1.6

1.1.1.8 Wetted perimeter

The wetted perimeter (O [m]) is the interface between the soil and the owing water. SeeFigure 1.6.

In the case of a main channel and one or two oodplains in the cross-section, a wetted perime-ter is computed for each section. In that case the water interface between the sections is notincluded in the wetted perimeter. See also Figure 1.6 and Figure 1.7

The wetted perimeter is used to compute the hydraulic radius, which is important for thecomputation of the bed friction.

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Figure 1.6: Wetted perimeter

Figure 1.7: Wetted perimeter for the main channel

1.1.1.9 Flow velocity

The ow velocity (u) is dened as the average ow velocity in the ow section of the cross-section. It is by default given in [m/s].

The average ow velocity is derived by dividing the discharge [m3/s ] by the ow area [m2].Output of the overall average ow velocity, the ow velocity in the main channel, the owvelocity in oodplain 1 and the ow velocity in oodplain 2 is possible.

u = QAf

, u0 = QA0

, u1 = QA1

, u2 = QA2

. (1.2)

The indices 0, 1, 2 indicate the main channel, oodplain 1 and oodplain 2 respectively.

1.1.1.10 Velocity

The velocity (u) is dened as the average ow velocity in the wetted area of the cross section.It is given in [m/s].

u = QAf

(1.3)

Note: Discharges and velocities are dened in reach segments, whereas cross sections are

dened in h-calculation points. For the computation of the velocities according to the aboveformula. The SOBEK-Flow-module uses the upstream cross section

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1.1.1.11 Hydraulic radius

The hydraulic radius is dened as the wetted area of the cross section divided by the wettedperimeter.

R = Af O (1.4)

Figure 1.8: Computation of hydraulic radius

Figure 1.9: Wetted perimeter for the main channel (Subdivision in main channel and oodplains only in SOBEK-River)

1.1.2 Model equations

The water ow is computed by solving the complete De Saint Venant equations.

For one dimensional ow (Channel Flow and Sewer Flow modules) the following equationsare solved

continuity equation 1D momentum equation 1D

For two dimensional ow (Overland Flow module), three equations are solved:

continuity equation 2D momentum equation 2D for the x-direction momentum equation 2D for the y-direction

These equations are solved numerically using the Delft-scheme.

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Note: on the 2D equationsAs opposed to the shallow water equations, the described equations do not incorporate theturbulent stress terms, accounting for the sub grid transfer of momentum in between gridcells. These terms have been omitted because they are relatively unimportant for ood ow

computations, in order to save computational effort.The wall friction termshavebeen introduced to account for the added resistance that is causedby vertical obstacles, like houses or trees. The wall friction coefcient is based on the averagenumber and diameter of the obstacles per unit area and the average obstacle drag coefcient(C d coefcient).

1.1.2.1 Continuity equation (1D)

The ow in one dimension is described by two equations: the momentum equation and thecontinuity equation. The continuity equation reads:

A f t

+ Qx

= q lat (1.5)

1.1.2.2 Momentum equation (1D)

The ow in one dimension is described by two equations: the momentum equation and thecontinuity equation. The momentum equation reads:

Qt

+ x

Q2

Af + gAf

hx

+ gQ|Q|C 2RA f wf

windw

= 0 (1.6)

The rst term describes the inertiaThe second term describes the convectionThe third term describes the water level gradientThe fourth term describes the bed frictionThe fth term describes the wind friction

1.1.2.3 Transport equation for salinity

Salt transport in estuaries and tidal rivers can be considered as transport of conservativesubstance in water. The transport of salt is described by an advection-diffusion equation,which is called the transport equation. We remark that in SOBEK temperature cannot bemodelled yet. Otherwise, the same transport equation would have been used for temperatureas well.

Next to the additional transport equation, density differences have to be accounted for in themomentum equation of the water ow module. The water ow module is therefore coupledwith the salt intrusion module by the density and the ow eld. The salinity concentration isdenoted by C .

The transport equation for salinity is described by an advection-diffusion equation includingsource term reads:

At C t

+ x

Q C Af D C x

= S s (1.7)

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in which

C concentration of salt or chloride, averaged over the total cross-sectional area[kg/m3]

D dispersion coefcient [m2 /s]S s source term [kg/ms]Q discharge (water) [mCs/s]At total cross-sectional area [m2]Af ow area [m2]

1.1.3 Inertia

The ow in one dimension is described by two equations: the momentum equation and thecontinuity equation. The rst term in the momentum equation is the inertia term:

inertia = Q

t (1.8)Q Discharge [m3/s ]t Time [s]

1.1.4 Convection

In the SOBEK-Flow-module an convection term is used to guarantee the conservation ofmomentum in a reach (between reach segments). The same holds for a connection nodewhen two reaches are connected to this node. It that case the connection nodes could alsohave been modelled as a h-calculation point. If there are more than two reaches connected

to a connection node the convection term is set to zero.

When a reach has subsections, all subsections have an independent solution for the velocity.The velocity that is used for the conservation of momentum over the reach is the average ofthe velocities ui of the subsections in the cross section.

1.1.5 Convection (1D)

The ow in one dimension is described by two equations: the momentum equation and thecontinuity equation. One of the terms in the momentum equation is the convection term:

convection = x

Q2

Af (1.9)

Q Discharge [m3/s ]Af Wetted area [m2]x Distance [m]

1.1.6 Water level gradient

The third term in the momentum equation is the water level gradient.

water level gradient = gAf hx (1.10)

x Distance [m]

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Af Wetted area [m2]g Acceleration due to gravity [m/s 2] (9.81)h Water level [m] (with respect to the reference level)

This force tries to achieve a at water surface under inuence of the gravitational acceleration.This force together with the bed friction have the greatest effect on the water movements.

1.1.7 Wind friction

The wind friction term in the momentum equation is expressed as:

wind = air C wind u2wind cos(wind channel ) (1.11) wind Wind shear stress (positive if acting along the positive x-axis of an open chan-

nel) [N/m 2]

air Density of air, 1.205 kg/m3

C wind wind ,1 + wind ,2uwind , the wind friction coefcient. The used values are:

awind ,1 = 0 .50 103 [] (1.12)awind ,2 = 0 .06 103 [s/m] (1.13)

uwind Wind velocity [m/s]channel Clock-wise angle between the positive x-axis of the 1D open channel and the

North (Nautical convention) [degrees]wind Clock-wise angle between the wind direction and the North (Nautical conven-

tion) [degrees]

The wind direction that is specied in the SOBEK-Flow-module is the direction from which thewind is blowing with respect to the north. So, if the wind is blowing from the north, the winddirection is 0 degrees. Wind from the southeast has a wind direction of 135 degrees.

Figure 1.10: Wind direction

When a cross section has a top width smaller than 20 mm (circular, egg-shaped, or nearlyclosed tabulated cross section) than the cross section is considered closed. In that case thewind is neglected in the momentum equation

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1.1.8 Initial conditions

The initial conditions are the water levels or depths and the discharges at the beginning ofthe simulation. The initial conditions are dened over a reach (water level and depth at theh-calculation points, discharges at the reach segments). Therefore the water level that initiallyshould be taken at connection nodes is not strictly dened. This happens when, for example,reach 1 with initial water level 0.5 is connected by a node to reach 2 with initial water level 0.6.In that case the water level at the connection node is set to the lowest value of the connectedreaches (0.5). Because theh-calculation point at the end of reach 1, the connection node andthe h-calculation point at the beginning of reach 2 have the same location, the water levels ofthese three points are set to 0.5.

1.1.9 Boundary

A boundary can be applied at the locations where the model network ends with a boundary

node.In order to solve the water ow equations (continuity equation and momentum equation),information about the water ow at the model boundaries must be supplied.

At each boundary node, one condition for the water ow must be specied. The followingoptions are available:

discharge (constant, tabulated function of time, tabulated function of the water level) water level (constant, tabulated function of time)

Because in the staggered grid, used by SOBEK-Flow-module, the discharges are dened forthe reach segments, a discharge boundary is imposed on the rst reach segment next to theboundary. Therefore the h-calculation point just before this rst reach segment (rst one in areach) is undened and will not be taken into consideration during a calculation.

A water level boundary is dened in the rst calculation point next to the boundary. Thish-calculation point actually has the same coordinates as the boundary node.

Note: In channel ow, usually, the discharge will be specied at boundaries where water isowing into the model, and the water level where water is owing out of the model. In bothsewer and channel systems a dead end (or beginning) of a reach can be a connection nodeat the end of a pipe or channel. In contrast to a boundary node, this node has storage.

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1.1.10 Discharge

The discharge Q is the amount of water passing a reach segment per unit of time. It is givenin [m3/s ].

The discharges at the reach segments together with the water levels at the h-calculationpoints, form the results of a water ow simulation.

The SOBEK-Flow-module gives a positive discharge when the water ows in positive directionwith respect to the dened direction of a reach. If the water is owing from reach end to reachbeginning, the discharge is negative.

1.1.11 Type of Lateral Discharges and their Assignment to h -Calculation point(s)

All three different types of lateral discharges are assigned to a h-calculation point(s) . Moreprecisely, their lateral inow or outow volume is accounted for in the local water balance ofsuch h-calculation point (e.g. local discretization of the continuity equation, see ?? ).

Three different types of lateral discharges can be discerned:

1.1.11.1 Different Types of Lateral Discharges

Three different types of lateral discharges are discerned:

1 Lateral Discharge on a Node [m3/s ]:Examples of lateral discharges on a Node are:

Point-Lateral Discharge, that is exactly located on a h-calculation point , Connection Node with Storage and Lateral Flow , Manhole with Lateral Flow , Manhole with Lateral Disch. and Runoff Manhole with Runoff Flow - RR Connection on Flow Connection Node .

2 Point-Lateral Discharge on a Reach [m3/s ]:Examples of a point-lateral discharge on a Reach are:

Flow - Lateral Flow Node Flow - RR Connection on Channel .

3 Diffusive Lateral Discharge along a Pipe or Reach [m2/s ]:Examples of a diffusive lateral discharge along a Pipe or Reach are:

Flow - Pipe with Runoff , Flow - Dry Weather Pipe , Flow - Rain Pipe , Flow - Pipe with Inltration Flow - Channel with Lateral Discharge .

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1.1.11.2 Assignment of Lateral Discharges to h -calculation point(s)

All three different types of lateral discharges (see section 1.1.11.1) are assigned to a h- calculation point(s) . More precisely, their lateral inow or outow volume is accounted forin the local water balance of such h-calculation point (e.g. local discretization of the continuityequation, see section 1.1.12).

Two different lateral discharge assignment options are available in Settings (see 1D Flow,Simulation Settings Tab), that determine to which h-calculation (or water level) point(s) a par-ticular lateral discharge is to be assigned. The two different lateral discharge assignmentoptions are:

1 Lateral Assigned to Lowest Water Level Point,2 Lateral Assigned to Nearest Water Level Point.

Hereafter, the consequences of the two different lateral discharge assignment options arediscussed for each type of lateral discharge (see section 1.1.11.1) separately.

Lateral discharge on a Node [m3/s ]:Irrespective of the selected lateral discharge assignment option, a lateral discharge ona Node is assigned to its concerning Node (e.g. h-calculation point, Connection Node orManhole).

Point-lateral discharge on a Reach [m3/s ]:A point-lateral discharge on a Reach is either assigned to one or equally distributed overboth of the h-calculation points, that are respectively located at the left-side and the right-side of the point-lateral discharge.

If option 1 (Lateral Assigned to lowest Water Level Point) is selected, the point-lateraldischarge is assigned to the h-calculation point having the lowest bed level. If bothcalculation points have the same bed level and the point-lateral discharge is located ata u-velocity point (e.g. centre-point of the two adjacent h-calculation points), the point-lateral discharge is equally distributed over both adjacent calculation points. Else thepoint-lateral discharge is assigned to the nearest h-calculation point.

If option 2 (Lateral Assigned to nearest Water Level Point) is selected and the point-lateral discharge is located at a u-velocity point (e.g. centre-point of the two adjacenth-calculation points), the point-lateral discharge is equally distributed over both adjacenth-calculation points. Else the point-lateral discharge is assigned to the nearest h-calculation point.

Diffusive lateral discharge along a Pipe or Reach [m2

/s ]:A diffusive lateral discharge along a Pipe or Reach is either assigned to one or equallydistributed over both of the h-calculation points, that are respectively located at the begin-ning and the end of a pipe or reachsegment, receiving a diffusive lateral discharge.

If option 1 (Lateral Assigned to lowest Water Level Point) is selected, the diffusivelateral discharge is assigned to the h-calculation point having the lowest bed level.Else the diffusive lateral discharge is equally distributed over both adjacent calculationpoints.

If option 2 (Lateral Assigned to nearest Water Level Point) is selected, the diffusivelateral discharge is equally distributed over both adjacent h-calculation points.

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1.1.12 Lateral discharge

An overview of the different type of lateral discharges is given in section 1.1.11.1. All lateraldischarges enter or leave a model at a connection node or at a h-calculation point located ona particular reach. For the assignment of lateral discharges to a particular h-calculation point,reference is made to section 1.1.11.2.

Lateral discharges are included in the continuity equation.

A f t

+ Qx

= q lat (1.14)

Af Wetted area [m2]t Time [s]Q Discharge [m3/s ]x Distance [m]q

lat Lateral discharge per unit length [m2/s ]

Figure 1.11: Lateral discharges Point source [ m 3 /s ]: q lat = Qpoint / x and Line source [ m 2 /s ]: q lat = Qdiffuse (total discharge Qdiffuse l )

The point lateral discharge can be dened as

Constant Tabulated function of time Area Based

The diffuse lateral discharge can be dened as

Constant Tabulated function of time

The two latter discharges need some more explanation:

Area Based Lateral Flow

Area based lateral ow is computed as:

Qlat = A(f dR + S ) (1.15)

Q lat Lateral discharge [m3/s ]f d Design factor for area based lateral ow (see ?? ), a model-wide parameter

dened in Settings [

].

R Rainfall [m/s ].A The catchment runoff area [m2]

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S Seepage (positive, inow of groundwater) or Inltration (negative, outow togroundwater reservoir) [m/s ].

Diffuse lateral discharge over reachA diffuse lateral discharge is dened as a lateral discharge in [m2/s ] over a whole reach.This discharge is split up into separate point source discharges into the reach segments thattogether form the reach. The value of such a point source is the diffuse lateral dischargemultiplied by the length of the reach segment into with it is discharging. Figure 1.12 shows adiffuse lateral discharge.

Figure 1.12: Diffuse lateral discharge

Note: A large withdrawal of water (negative lateral discharge) can cause problems whenthe amount of outowing water is larger than the storage available in a h-calculation point(storage; negative volumes could be computed in the continuity equation. For that reason theSOBEK-Flow-model reduces the time step temporarily when the outowing water volume islarger than half the available volume (safety factor) in the h-calculation point. The time stepis set to a value that allows a withdrawal of half the available volume in one time step. If thetime step resulting from this action is smaller than 0.01 second, the lateral withdrawal is set tozero.

1.1.13 Bed friction

The bed friction is the friction between the owing water and the channel bed. As such, itexerts a force on the owing water always in the direction opposite the water ow.

In water courses, this force together with the force caused by earth gravity usually determines

the ow conditions: the other forces are far less important.The fourth term of the momentum equation is the bed-friction term:

bed friction = gQ|Q|C 2RA f

(1.16)

where:

g Acceleration due to gravity [m/s 2] (9.81)Q Discharge [m3/s ]C Chzy coefcient [m1/ 2/s ]R Hydraulic radius [m]

Af Wetted area [m2]

Note:

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If the specied roughness parameter is equal to zero (Chzy=0, Manning=0, etc.), then thebed friction term is not taken into account in the momentum equation.

The following roughness denitions can be used: Bos-Bijkerk (1.1.13.1), Chzy (1.1.13.2),

Manning (1.1.13.3), Nikuradse (1.1.13.4), Strickler (1.1.13.5), White-Colebrook (1.1.13.6)

1.1.13.1 Bos-Bijkerk

The bed friction formulation according to de Bos-Bijkerk describes the Manning coefcient asa function of the water depth and a parameter. This parameter can be used to shape thefunction to a certain empirical curve:

km = d1/ 3 (1.17)

km Manning roughness coefcient [m1/ 3/s ]d Water depth [m] Parameter [1/s ], normally between 20 and 40

This formula can be rewritten to a formulation for the Chzy coefcient as a function of thewater depth

C = d1/ 3R1/ 6 (1.18)

C Chzy coefcient [m1/ 2/s ]R Hydraulic radius [m]

For every computational time step the new friction value is calculated according to the Bos-Bijkerk formula and sequentially applied to the hydrodynamics calculations.

1.1.13.2 Chzy

The SOBEK-Flow-module uses the Chzy bed friction value in solving the water ow equa-tions.

The following roughness formulations are possible:

Chzy coefcient Bos-Bijkerk friction shape parameter g, resulting in a Chzy value according to:

C = d1/ 3

R1/ 6

(1.19)d Water depth [m] Parameter [1/s ], normally between 20 and 40C Chzy coefcient [m1/ 2/s ]R Hydraulic radius [m]

White-Colebrook, using the Nikuradse roughness coefcient kn , results in a Chzy valueaccording to:

C = 18 10 log12Rkn

(1.20)

Manning coefcient, nm , resulting in a Chzy coefcient according to:

C = R1/ 6

nm(1.21)

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Strickler, using the Nikuradsekn roughness coefcient, results in a Chzy value accordingto:

C = 25R

kn

1/ 6(1.22)

Strickler, using the Strickler ks roughness coefcient, results in a Chzy value accordingto:

C = ks R1/ 6 (1.23)

Engelund-like roughness predictor (for main sections of SOBEK-River proles only) re-sulting in a Chzy value according to:

C = C 90 90s (1.24)For SOBEK-River, the Chzy, Nikuradse, Manning or Strickler coefcients may be

a constant spatially varying a tabulated function of the water level (h) or the total discharge (Q)

Different values or tables are possible for positive as well as negative ow. See also thechapter 1D hydraulic friction concepts for information on the different friction concepts presentin the user interface.

1.1.13.3 Manning

One of the methods to dene the bed roughness is using the Manning coefcient, symbolnm .In the SOBEK-Flow-module, the Manning coefcient is used to compute the actual value ofthe Chzy coefcient, by:

C = R1/ 6

nm(1.25)

C Chzy coefcient [m1/ 2/s ]R Hydraulic radius [m]nm Manning coefcient [s/m 1/ 3]

1.1.13.4 Nikuradse

One of the methods to dene the bed friction is by entering an equivalent roughness accordingto Nikuradse, represented by the symbol kn [m].

Values of kn for open channels with bed forms are in the same order of magnitude as theheight of the bed forms.

Using this option, the actual value of the Chzy coefcient will be computed according to theWhite-Colebrook formula.

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1.1.13.5 Strickler

One of the methods to dene the bed roughness is by using the Strickler formula. The actualvalue of the Chzy coefcient is computed using:

C = 25 Rkn

1/ 6 (1.26)

or:

C = ks R1/ 6 (1.27)

C Chzy coefcient [m1/ 2/s ]R Hydraulic radius [m]ks Strickler roughness coefcient [m1/ 3/s ]

in which kn is the Nikuradse equivalent roughness, and ks is the Strickler roughness coef-cient. You may select either ks or kn as input value.

The value of the hydraulic radius R is taken from the last iteration loop.

It is possible to enter the coefcients varying in space and depending on the local ow direc-tion.

1.1.13.6 White-Colebrook

One of the methods to dene the bed friction is by specifying an equivalent roughness ac-

cording to Nikuradse. Using this option, the value of the Chzy coefcient will be computedaccording to the White-Colebrook formula:

C = 18 10 log12Rkn

(1.28)

C Chzy coefcientR Hydraulic radiuskn Nikuradse equivalent roughness

Note:Actually, this formula is a simplication of the complete White-Colebrook formula.

1.1.14 Froude number

The Froude number is dened byu/c , withu the average ow velocity andc the wave celerity.

Herein is c as follows:

c = gAf W f (1.29)Thus

Froude number = u

gAf /W f (1.30)

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u Velocity (u = Q/A f ) [m/s]g Acceleration due to gravity [m/s 2] (9.81)Af Wetted area [m2]W f Flow width [m]

Remarks: When the Froude number is less than 1, the ow is sub-critical, when it is larger than 1,the ow is super-critical

Transitions from sub-critical to super-critical ow causes an hydraulic jump.

1.1.15 Boussinesq

The Boussinesq coefcient is a parameter in the momentum equation for the water ow. Itaccounts for the non-uniform velocity distribution in a cross-section.

The Boussinesq constant B is added to the convective term:

x B

Q2

Af (1.31)

The denition of the coefcient is:

B = Af Q2

W f

0

q (y)2

d(y) dy (1.32)

The constant of Boussinesq is computed by SOBEK. The computation is based upon theEngelund approach (Engelund and Hansen, 1967). In this approach the water level gradientand bed-friction term are assumed to be an order of magnitude larger than the other terms inthe momentum equation. The resulting equation is:

gAf hx

= gQ2

C 2RA f (1.33)

in which locally a constant water level gradient is assumed. In an arbitrary cross-section thedischarge can be expressed as:

Q =

W f

0C (y)d(y)

R(y)

h

x dy (1.34)

in which d is the water depth. Furthermore, it is assumed that the water level and water levelslope are the same for main and oodplains. Combining Equation 1.33 and Equation 1.34leads to a more accurate estimate of the Czy coefcient based upon the average of the localChzy coefcients.

C = 1

AF R W f

0C (y)d(y) R(y) dy (1.35)

Now Equation 1.32 gives combined with Equation 1.33 and Equation 1.34 the following ex-pression for the Boussinesq coefcient:

B = 1

Af RC 2 W f

0C 2(y)d(y)R(y) dy (1.36)

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Cross-sections in SOBEK may be divided in two or three parts. So the following three optionsare available.

option 1:

The cross-section is not divided into sections with different roughness (only a main channel ispresent). This gives for the Boussinesq coefcient:

B = 1 (1.37)

option 2:The cross-section is divided into two sections with different roughness for the modeling of, forexample, a cross-section with a main channel and a oodplain. During a computation thisgives two possibilities:

possibility 1:

Actual water ow is in the main channel only: B = 1 (1.38)

possibility 2:Actual water ow is in the main channel and the oodplain:

B = C 20 Af 0 R0 + C 21 Af 1 R1

C 2RA f (1.39)

The indices 0 and 1 respectively indicate the main channel and oodplain.

option 3:The cross-section is divided into three sections with different roughness (main channel, ood-plain 1 and oodplain 2). During a computation this gives three possibilities:

possibility 1:Actual water ow is in the main channel only:

B = 1 (1.40)

possibility 2:Actual water ow is in the main channel and oodplain 1:

B = C 20 Af 0 R0 + C 21 Af 1 R1

C 2RA f (h) (1.41)

The indices 0 and 1 respectively indicate the main channel and oodplain.possibility 3:Water is actually owing in all sections, main channel, oodplain 1 and oodplain 2:

B = C 20 Af 0 R0 + C 21 Af 1 R1 + C 22 Af 2 R2

C 2RA f (1.42)

The indices 0, 1 and 2 respectively indicate the main channel, oodplain 1 and oodplain2.

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1.1.16 Accuracy

Whether you are measuring in a prototype, studying water ow in a scale model or modellingwith a software system, you should always make some considerations about the accuracy ofyour activities and of the results of your study.

As for the results of a mathematical model study, the following elements play a role in estab-lishing the overall accuracy:

The reliability of the available data describing the prototype; The accuracy of the available data describing the prototype; The violation of certain assumptions underlying the mathematical modelling concept beingused;

The experience and skill of the modeller(s); The overall accuracy that is required (perhaps more accurate results could be obtainedbut the aim of the study may not require such accuracy or due to a lack of time you may

have to settle for less); The accuracy of the applied numerical modelling technique.

In general practice the rst ve items are the most important when you are using a one-dimensional mathematical model for hydrodynamic ow. The numerical accuracy is in thatcase normally of minor importance.

The SOBEK-Flow-module uses the Delft-scheme to solve the water ow equations. Thisscheme is developed with robustness as the most important design aspect. It can deal withphenomena such as drying/ooding, super-critical ow and it guarantees a solution for everytime step.

The accuracy of the solution depends on the grid size; the smaller the grid sizes, the moreaccurate the solution is. The time step can be chosen arbitrarily. It is reduced internallywhen this is necessary to guarantee stability by means of a time step estimation procedure.Obviously small grid sizes result in a network with more elements and therefore in a longersimulation time.

The Delft-scheme is designed to produce a closed water balance.

1.1.17 Structures

In the SOBEK-Flow-modules the following structure types are available:

section 1.1.17.1 Advanced weirsection 1.1.17.2 Bridgesection 1.1.17.3 Compound structuresection 1.1.17.4 Culvertsection 1.1.17.5 Database structuresection 1.1.17.6 General structuresection 1.1.17.7 Inverted siphonsection 1.1.17.8 Orice

section 1.1.17.9 Pump station and Internal Pump stationsection 1.1.17.10 External Pump stationsection 1.1.17.11 River Pump

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section 1.1.17.12 River Weirsection 1.1.17.13 Siphonsection 1.1.17.14 Universal Weirsection 1.1.17.15 Vertical obstacle friction

section 1.1.17.16 Weir

The ow through structures is computed based on:

Upstream water level or energy level (River weir, Advanced weir and General structureonly)

Downstream water level or energy level (River weir, Advanced weir and General structureonly)

Structure dimensions (some can be controlled: i.e. crest level, opening height, pump ca-pacity, opening of valve)

A number of user-dened parameters, depending on the structure type (contraction coef-cient, reduction factor, etc.).

The discharges and wetted areas that are computed in the structure formulas are imposedin the reach segment where the structure is located. The formulas use the water levels ofthe h-calculation points on either side of the reach segment. For the River weir, Advancedweir, General structure, River pump, Database structure and Compound structure yields thatSOBEK by default places a computational point 0.5 m upstream and 0.5 m downstream of thestructure location. Hence such structure is located in a reach having a default length of 1 m.

Note:

The dimensions of the structure do not contribute to the storage in the water system.

1.1.17.1 Advanced weir

Please note that the discharge through an Advanced weir is computed on basis of upstreamand downstream energy levels. Further on please note that default a computational point islocated 0.5 m in front and 0.5 m behind an Advanced weir.

NotationThe ow direction can be either positive or negative. A positive ow direction is a ow inthe direction that the branches have been specied, i.e. with increasing x-coordinates. Theupstream side, facing the beginning of the reach is denoted by the subscript 1, whereas thedownstream location facing the end of the reach is denoted with 2. The subscript s is used forlocations at the sill.

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Figure 1.13: Denition of energy and water level

The energy level is equal to the water level plus the velocity head (u2/ 2g). The water depth dis equal to the water level (h) minus the bed level (z

b).

The user must enter the following input parameters for this type of hydraulic structure:

Level of crest Z s ; Total net width W n ; Number of piers N ; Level of upstream face P ; Design head H 0 of the weir; Pier contraction coefcient K p; Abutment contraction coefcient K a .

The following parameters can be controlled by a hydraulic structure controller:

W s = Width across ow section

Z s = Crest level of weir

The discharge through the structure is computed with:

Q = C 1C 2W 2g(H 1 zs )3 (1.43)In which W is the active width and C 1 and C 2 are factors computed in the following way.

Note:if h1 < Zs then Q = 0.

W = W k 2(NK p + K a )(H 1 zs ) (1.44)C 1 = C 0C k (1.45)

The value of C 0 is computed depending on the value of the ratio between the upstream faceand the design head:

= P H 0

(1.46)

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if < 2 then:

C 0 = (0 .1256 5 1.0178 4 + 3 3 3.94 2 + 2 .28 + 1 .66) 2g (1.47)

where is a correction factor computed with: = 0.052 3 + 0 .145 2 0.096 + 1 .01 (1.48)

if > 2 then:

C 0 = 2.1549008

2g (1.49)

The value of is computed depending on the value of the ratio between the energy levelminus the sill level and the design head:

1 = H

1 z

sH 0 (1.50)

if 1 < 1.6 then:

C k = 0 .1394 31 0.416 21 + 0 .488 1 + 0 .785 (1.51)if 1 > 1.6 then:

C k = 1 .0718224 (1.52)

The value of C t is computed depending on the value of the ratio between the energy level

minus the downstream water level and the energy level minus the sill level: 2 =

H 1 h2H 1 zs

(1.53)

if 2 < 0.7 then:

C t = (1 ( 2 0.7)20.49 ) + 27(0 .7 2)4 32 (1.54)if 2 > 0.7 then:

C t = 1 (1.55)

1.1.17.2 Bridge

A bridge is one of the structure types that can be included in the SOBEK-Flow-module. Thefollowing types of bridges can be modelled:

Pillar bridge Abutment bridge Fixed bed bridge Soil bed bridge

The general description of the discharge through a bridge is given by:Q = Af 2g(h1 h2) (1.56)

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Q Discharge through bridge [m3/s ] Coefcient derived from loss-coefcients []Af Wetted area [m2] of ow through bridge at upstream sideg Acceleration due to gravity [m/s 2] (9.81)h1 Upstream water level [m]h2 Downstream water level [m]

The different types of bridges have different denitions for the coefcient:

Pillar bridge

A pillar bridge has one or more pillars that affect the discharge through the bridge with thefollowing denition of:

= 1

y(1.57)

y Pillar loss coefcient, dened as

y = yAf

(1.58)

Parameter [] depending on shape of pillar[s] (shape factor). Normally between0.22 and 1.56Af Wetted area [m2] of ow through bridge at upstream side y Area [m2] of wetted part of pillar[s] perpendicular to the ow direction, consid-

ered at upstream side

Figure 1.14: Pillar bridge

The plate of the bridge is always so high that it does not effect the ow through the bridge. Sothe cross section is considered as open.

Note:Pillar bridges are not allowed in closed cross-sections.

Abutment bridge

For the abutment bridge the overall loss coefcient is dened as:

= 1

i + f + o(1.59)

x i Entrance loss coefcient (in). Constant

xf Friction loss coefcientxo Exit loss coefcient (out).

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The o coefcient is dened as:

o = k 1 Af Af 2

2(1.60)

k Constant exit loss coefcientAf Wetted area [m2] of ow through bridge at upstream sideAf 2 Wetted area [m2] of ow in reach at downstream side of bridge

The f coefcient is dened as:

f = 2gLC 2R

(1.61)

g Acceleration due to gravity [m/s 2] (9.81)L Length of bridge [m]C Chzy coefcient [m

1/ 2

/s ]R Hydraulic radius [m]

Figure 1.15: A suspension bridge with abutments

Here, the plate of the bridge can effect the ow through the bridge. The cross section is closed

Note:The denition of this bridge is similar to the denition of a culvert.

Fixed bed bridge

This bridge has the same formulation as the abutment bridge except it has a rectangular

prole.

Figure 1.16: Fixed bed bridge

Soil bed bridge

This bridge has the same formulation as the xed bed bridge including the rectangular prole.In addition to this it has a ground layer with a different friction formulation. This ground layercan have a zero thickness.

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Figure 1.17: Soil bed bridge

1.1.17.3 Compound structure

A compound structure consists of several hydraulic structures parallel to each other at onelocation. These hydraulic structures may be of the same type or of different types. Presentlyfollowing structure types can be placed as a member in a compound structure, viz.: Generalstructure, Database structure, Advanced weir, River weir and River pump.

Please note that a value for structure inertia damping factor can be dened for each individualmember of the compound structure.

Each member of the compound structure has its own triggers and controllers.

1.1.17.4 Culvert

A culvert is one of the structure types that can be included in the SOBEK-Flow-module. Aculvert is an underground structure that normally connects two open channels. The owthrough a culvert is affected by its upstream and downstream invert levels, the size and shapeof its closed cross section, its ground layer thickness, its entrance loss, its friction loss, itsvalve loss and its exit loss.

Figure 1.18 shows a side view of a culvert.

Figure 1.18: Side view of a culvert

Two ow conditions can occur:

Free ow when h2 < z c2 + dc2

Q = Afc 2g(h1 (zc2 + dc2)) (1.62)Submerged ow when h2 zc2 + dc2:

Q = Afc 2g(h1 h2) (1.63)Q Discharge through culvert [m3/s ]26 Deltares


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Discharge coefcient, derived from loss-coefcients []Afc Discharge culvert ow area min(Afc 1, Afcgate ) [m2]Afc 1: Flow area in the culvert at its upstream side [m2]Afcgate : Flow area under the culvert gate [m2]

g Acceleration due to gravity [m/s2

] (9.81)h1 Upstream water level [m]h2 Downstream water level [m]zc2 Downstream culvert invert level [m]dc2 Critical culvert depth at the downstream side, 3 Q2/ (gT 22 ) [m]T 2 Surface width in the culvert at its downstream side [m]

For numerical reasons the discharge coefcient () is limited to a maximum of 1.0. Thedischarge coefcient () is computed as follows:

= 1

i +

f +

v +

o

(1.64)

i Entrance loss coefcient [] f Friction loss coefcient [] v Valve loss coefcient [] o Exit loss coefcient []The entrance loss coefcient ( i) can be dened as a constant value only.

The friction loss coefcient ( f ) is computed as follows:

f = 2gL

C 21 R (1.65)

L Length of the culvert [m]C 1 Chzy coefcient in the culvert at its upstream side [m1/ 2 /s]R Hydraulic radius [m]

If h1 zc1 + GHO ; R = Rc1If h1 < z c1 + GHO ; R = RgateGHO Gate height opening [m]Rc1 Hydraulic radius in the culvert at its upstream side [m]Rgate Hydraulic radius based on actual gate height opening [m]zc1 Upstream culvert invert level [m]

The valve loss coefcient ( v) can be dened as a constant value or as a function of the ratioof the Gate height opening and the maximum inner culvert height.

Note:

In case the valve loss coefcient ( v) is not a constant, in computations the actual valveloss coefcient ( v) is derived from the user dened table, while using the ratio of theactual gate height opening and the actual maximum inner culvert height.

In case the ground layer thickness is greater than zero, both the actual gate height open-ing and the actual maximum inner culvert height will differ from the values as dened inthe user interface (see next paragraph)

The exit loss coefcient ( o) is computed as follows:

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Submerged ow (h2 = zc2 + dc2):

o = k 1 AfcAf r 2

2(1.66)

k User dened constant exit loss coefcient []Afr 2 Flow area in the reach, adjacent to the downstream culvert side [m2]Afc Culvert ow area [m2]

If h1 zc1 + GHO ; Afc = AfcgateIf h1 < z c1 + GHO ; Afc = Afc 1GHO Gate height opening [m]zc1 Upstream culvert invert level [m]Af cgate Flow area under the culvert gate [m2]Af c1 Flow area in the culvert at its upstream side [m2]

Free ow (h2 < z c2 + dc2):

o = 0 (1.67)

Culvert cross-sections, bed friction and ground layer

For a culvert all available closed cross-section types can be used. In a culvert, a groundlayer with constant thickness can be dened. Culvert friction and ground layer friction can bespecied, using any of the available bed friction formulations.

Dening a ground layer thickness > 0 implies that in culvert computations:

Dened invert levels are raised with the ground layer thickness, Gate height openings are reduced with the ground layer thickness, Maximum inner height of the culvert is reduced with the ground layer thickness, Cross-sectional parameters (such as: ow areas, hydraulic radius and so on) are com-puted based on a cross-sectional prole, that is reduced by the ground layer thickness.

Culvert, Good modelling practice aspects

It is advised to avoid that the bed level of a cross-section in front of a Culvert, Inverted Siphonor Siphon is above the ground-layer level (= invert level + ground-layer thickness), since suchsituation can result in very small computational time-steps (e.g. long required wall-clock times)or even in a termination of the simulation.This is explained as follows. Consider the situation depicted in Figure 1.38 were the bed levelin front of a Culvert (Inverted Siphon or Siphon) is 0.60 m above the ground-layer level. Thismeans that at small upstream water depths, water will be sucked into the culvert (InvertedSiphon or Siphon), resulting in large ow velocities. For the computational time-step ( t)yields that t x/U , where U is the local ow velocity and x is distance between twowater level computational points (or h-points). At very low discharges even negative waterdepths may be computed, leading to a termination of the simulation.The situation explained above can be avoided by making the bed level in front of the Culvert(Inverted Siphon or Siphon) equal to the ground-layer level. In other words by dening a bedlevel slope from h-point h2 to h3 as depicted in Figure 1.38. Providing for the parameter"Maximum Lowering of Cross-section Bed Level at Culvert" a value greater or equal to 0.60m means that before the computation starts, the bed level at h-point h3 is set equal to the

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ground-layer level. In?? it is shown how to provide a value for parameter "Maximum Loweringof Cross-section Bed Level at Culvert".

Figure 1.19: Good modelling practice, Culvert, Inverted Siphon and Siphon

1.1.17.5 Database structure

Please note that default a computational point is located 0.5 m before and 0.5 m behind aDatabase structure. In a hydraulic structure the discharge through the structure depends onupstream and downstream water levels and structure parameters that dene the dimensionof the structure etc. In other words the hydraulic behaviour of a structure can be dened inthe structure equation as a relationship between upstream and dowstream water level. In thedatabase structure however, this relationship is dened in a tabulated form which is stored ina database. The user has to dene this database.

The database consists in fact of a matrix of discharges. At every point of the matrix a dis-charge (Q) is dened as a function of two corresponding water levels: one facing the begin-ning of the branch (h1) and one facing the end (h2), are dened. There are two ways to denethe relation:

a function of both water levels Q = Q(h1 zs , h2 zs ), to be used for structures withrelatively high head differences; zs = crest level w.r.t. datum a function of the water level and the water level differenceQ = Q(h1

zs , h = h1

h2),

to be used for structures with relatively small water level differences combined with largewater level variations; zs = crest level w.r.t. datum

All discharges in a row correspond to the same water level (h1) facing the beginning of thereach segment. All discharges in a column correspond either to the same water level (h2)facing the end of the reach segment or to the water level difference ( h = h1 h2) .Thewater levels that are part of the database are dened with respect to a user dened datum.During a simulation, discharges at any point in the domain of the matrix will be obtained byinterpolation. In most cases this will be a linear interpolation in two directions. However when

the discharge is requested at a water level that lies within 10 percent of a water level denedin the matrix, cubic interpolation takes place for that water level.

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Warning: The crest level can be adjusted by a controller. This implies the assumption that thedischarge head relation is independent of the level of the crest. This is not completelycorrect. The user should be aware of this.

1.1.17.6 General structure

Please note that default a computational point is located 0.5 m in front and 0.5 m behinda general structure. In the general structure type in sobek weir and gate ow is combinedin one structure type. In addition, the general structure gives more freedom in dening thedimensions and the geometry of the hydraulic structure. The geometrical shape is given inFigure 1.20 and Figure 1.21. See ?? for the denition of input parameters. Please note thatthe discharge through a General structure is computed on basis of upstream and downstreamenergy levels. Please note as well that a structural inertia damping factor can be dened foreach individual General structure.

Figure 1.20: General structure, side view

Figure 1.21: General structure, top view

Flow across the general structure can be of the following types: drowned weir ow, free weir

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ow, drowned gate ow, and free gate ow, depending on the dimensions of the structure andthe ow conditions.

When salt intrusion is modelled, the density difference of the water over the structure is incor-

porated in the impulse balance (this is not the case with the other structure types).In the solution method for the general structure particular attention has been given to themodelling of the transition between free- and submerged ow.

For this purpose, the water level at the sill (ds ) is computed by applying an impulse balanceinstead of taking it equal to the water level further downstream.

As a result the general structure is especially attractive for those who want to simulate theshifting conditions from free- to submerged gate ow accurately and of course in case ofimportant density-differences over the structure.

Computation of downstream water level

In case of drowned gate ow or drowned weir ow the water level at the sill or downstream ofthe gate is required. This level is computed by application of the impulse balance.

drowned gate-ow

Figure 1.22: Drowned gate ow

The water depth ds can be described by a second order algebraic equation:

Awd2s + Bwds + C w = 0 (1.68)

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with:

Ag = (1 + )W sd

3 +

W 26

+ (1 )W sd

4 +

W 212

(1.69)

Bg = (1 + ) ( S 1 + S 2 d2)W

sd3 + W

26 + (2 S 1 + d2)W

sd6 +

( S 1 + 2 d2)W 26

+

+ (1 ) S 1W sd

3 + ( S 1 + d2)

W sd + W 26

+

+4c2gd

2gdd

2gW 2s

W 2d2(1 +

d2

) 4cgdgddgW s (1.70)C g = (1 + )( S 1 + S 2 d2) 2 S 1 + d2)

W sd6

+ ( S 1 + 2 d2)W 26

+

+ (1 ) ( S 1)2 W sd6 + ( S 1 + d2) W sd + W 212 +

4c2gd

2gdd

2gW 2s H s1

W 2d2(1 +

d2

) + 4 cgdgddgW s H s1 (1.71)

where is given by the expression:

= 21

(1.72)

: extra resistance of general structure can be dened for each general structure separately.

Equation 1.68 leads to:

ds = Bg + B 2g 4AgC g2Ag (1.73)

drowned weir-ow

Figure 1.23: Drowned weir-ow

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The water depth at the sill ds is described by a third order algebraic equation:

Dwd3s + Awd2s + Bwds + C w = 0 (1.74)

with:

Dw = 4c2wd W 2s

W 2d2(1 +

d2

) (1.75)

Aw = (1 + )W sd

3 +

W 26

+ (1 )W sd

4 +

W 212

+

4c2wd W

2s H s1

W 2d2(1 +

d2

) + 4 cwd W s (1.76)

Bw = (1 + ) ( S 1 + S 2 d2)W sd

3 +

W 26

+ (2 S 1 + d2)W sd

6 +

( S 1 + 2 d2)W 2W 26

+

+ (1 ) S 1W fd

3 + ( S 1 + d2)

W fd + W 26

+ 4 C wd W s H s1 (1.77)

C w = (1 + )( S 1 + S 2 d2) (2 S 1 + d2)W sd

6 + ( S 1 + 2 d2)

W 26

+

+ (1 ) ( S 1)2W sd

6 + ( S 1 + d2)

W sd + W 212

(1.78)

A direct method is applied to calculate ds from equation Equation 1.74.

Discharge equations

The following discharge equations are applied during the computations.

Free gate ow:

us = gf cgf 2g(H 1 (zs + gf dg)) (1.79)Af = W s dg (1.80)Q = us Af = gf cgf W s dg 2g(H 1 (zs + gf dg)) (1.81)

Drowned gate ow:

us = gdcgd 2g(H 1 (zs + ds )) (1.82)Af = W s dg (1.83)Q = us Af = gdcgdW s dg 2g(H 1 (zs + ds )) (1.84)(1.85)

Free weir ow:

us = cwf 23g(H 1 zs ) (1.86)Af = W s

23

(H 1

zs ) (1.87)

Q = us Af = cwf W s23 23g(H 1 zs )3/ 2 (1.88)

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Drowned weir ow:

us = cwd 2g(H 1 (zs + ds )) (1.89)Af = W s ds (1.90)Q = us Af = cwd W s ds 2g(H 1 (zs + ds )) (1.91)

where:

gf contraction coefcient for free gate owgd contraction coefcient for drowned gate ow (gd = gf )cgf correction coefcient for free gate owcgd correction coefcient for drowned gate owcwf correction coefcient for free weir owcwd correction coefcient for drowned weir ow

Note:The contraction coefcient has a maximum value = 1 .0. In case the user species a highervalue, a warning will be generated

Criteria for ow types

First it is assumed that it is weir ow. Then the water level at the sill ds and the critical depthdc = 2 / 3(H 1 zs ) are calculated.The criteria are:

ds > d c and dg > d s drowned weir ow ds < d c and dg > d s free weir ow otherwise gate ow.

In the latter case the water level at the sill ds is recalculated using the gate ow conditions.The critical depth dc is now dened as gf dg.

The criteria are.

ds > d c drowned gate ow ds < d c free gate ow

ds imaginary free gate ow (i.e. downstream water level below crest level)

Note:In case upstream water level is above gate lower edge level, there can still be drowned weirow if ds > d c and dg > d s or free weir ow if ds < d c and dg > d s .

1.1.17.7 Inverted siphon

An inverted siphon is one of the structure types, that can be included in the SOBEK-Flow-module. An inverted siphon is a structure that normally connects two open channels, that areseparated by a particular infrastructural work (e.g. dike, railroad). The inverted siphon makesan underground connection through such infrastructural work. An inverted siphon is assumedto be fully lled with water at its deepest point. In case this does not yield for your stucture, you

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o Exit loss coefcient []The entrance loss coefcient ( i ) can be dened as a constant value only.

The friction loss coefcient ( f ) is computed as follows: f =

2gLC 2R

(1.96)

L Length of the inverted siphon [m]C Chzy coefcient for fuly water lled inverted siphon [m1/ 2 /s]R Hydraulic radius [m]

If GHO = MIISH; R = RgateIf GHO > MIISH; R= R inverted siphon

GHO Gate height opening [m]MIISH Maximum inner inverted siphon height [m]R inverted siphon Hydraulic radius based on a fully water lled inverted siphon [m]Rgate Hydraulic radius based on actual gate height opening [m]

The valve loss coefcient ( v) can be dened as a constant value or as a function of the ratioof the Gate height opening and the maximum inner inverted siphon height.

Note:

In case the valve loss coefcient ( v) is not a constant, in computations the actual valveloss coefcient ( v) is derived from the user dened table, while using the ratio of theactual gate height opening and the actual maximum inner inverted siphon height.

In case the ground layer thickness is greater than zero, both the actual gate height open-

ing and the actual maximum inner inverted siphon height will differ from the values asdened in the user interface.

The exit loss coefcient ( o) is computed as follows:

Submerged ow (h2 = zc2 + dc2):

o = k 1 Af isAf r 2

2(1.97)

k User dened constant exit loss coefcient []Afr 2 Flow area in the reach, adjacent to the downstream inverted siphon side [m2]Af is Inverted siphon ow area [m2]If GHOMIISH; Af is = A sgate If GHO > MIISH; Af is = A inverted siphon GHO Gate height opening [m]MIISH Maximum inner inverted siphon height [m]A sgate Flow area under the inverted siphon gate [m2]Ainverted siphon Flow area based on a fully water lled inverted siphon [m2]

Free ow (h2 < z c2 + dc2):

o = 0 (1.98)

Inverted siphon cross-sections, bed friction and ground layer:For an inverted siphon all available closed cross-section types can be used. In an inverted

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siphon, a ground layer width constant thickness can be dened. Inverted siphon friction andground layer friction can be specied, using any of the available bed friction formulations.

Dening a ground layer thickness > 0 implies that in inverted siphon computations:

Dened invert levels are raised with the ground layer thickness, Gate height openings are reduced with the ground layer thickness, Maximum inner height of the inverted siphon is reduced with the ground layer thickness, Cross-sectional parameters (such as: ow areas, hydraulic radius and so on) are com-puted based on a cross-sectional prole, that is reduced by the ground layer thickness.

Inverted Siphon, Good modelling practice aspects

It is advised to avoid that the bed level of a cross-section in front of a Culvert, Inverted Siphonor Siphon is above the ground-layer level (= invert level + ground-layer thickness), since such

situationcan result in very small computational time-steps (e.g. long required wall-clock times)or even in a termination of the simulation.This is explained as follows. Consider the situation depicted in Figure 1.38 were the bed levelin front of a Culvert (Inverted Siphon or Siphon) is 0.60 m above the ground-layer level. Thismeans that at small upstream water depths, water will be sucked into the culvert (InvertedSiphon or Siphon), resulting in large ow velocities. For the computational time-step ( t)yields that t x/U , where U is the local ow velocity and x is distance between twowater level computational points (or h-points). At very low discharges even negative waterdepths may be computed, leading to a termination of the simulation.The situation explained above can be avoided by making the bed level in front of the Culvert(Inverted Siphon or Siphon) equal to the ground-layer level. In other words by dening a bed

level slope from h-point h2 to h3 as depicted in Figure 1.38. Providing for the parameter"Maximum Lowering of Cross-section Bed Level at Culvert" a value greater or equal to 0.60m means that before the computation starts, the bed level at h-point h3 is set equal to theground-layer level. In?? it is shown how to provide a value for parameter "Maximum Loweringof Cross-section Bed Level at Culvert".

Figure 1.25: Good modelling practice, Culvert, Inverted Siphon and Siphon

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1.1.17.8 Orice

The geometrical shape of an orice is given in Figure 1.26. Four different types of orice owcan be discerned, viz: free weir ow, submerged weir ow, free gate ow and submerged gateow. No discharge ows through the orice if its gate is closed, if the upstream water levelequals the downstream water level or if both the upstream water level and the downstreamwater level are below the crest level.

Figure 1.26: Orice

The following discharge equations are applied:

Free weir ow:

Af = W s23

(h1 zs ) (1.99)Q = cwW s

23 23g (h1 zs )3/ 2 (1.100)

Submerged weir ow:

Af = W s h1 zs us 2

2g (1.101)

Q = cecwW s h1 zs u s 2

2g 2g(h1 h2) (1.102) Free gate ow:

Af = W s dg (1.103)

Q = cw W s dg 2g(h1 (zs + dg)) (1.104) Submerged gate ow:

Af = W s dg (1.105)

Q = cw W s dg 2g(h1 h2) (1.106)Q Discharge across orice [m3/s ]Af Flow area [m2]

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W s Crest width [m]dg Opening height [m] (gate lower edge level minus crest level)g Gravity acceleration [m/s 2] (9.81)h1 Upstream water level [m]h2 Downstream water level [m]zs Crest level [m]us Velocity over crest [m/s ]

The different formulas are applied when the following conditions are met

Free weir ow:

h1 zs < 32

dg and h1 zs > 3/ 2(h2 zs ) (1.107) Submerged weir ow:

h1 zs < 32dg and h1 zs 32(h2 zs ) (1.108) Free gate ow:

h1 zs 32

dg and h2 zs + dg (1.109) Submerged gate ow:

h1 zs 32

dg and h2 > z s + dg (1.110)

1.1.17.9 Pump station and Internal Pump station

The functionality of a Pump station and an Internal Pump station is identical. The only differ-ence comprises the fact that:

Pump station: A Pump station is located on an open channel reach. The pump dischargeis determined using the water levels at the nearest h-calculation points, respectively lo-cated at the upstream-side and downstream-side of the Pump station.

Internal Pump station: An Internal Pump station is accommodated in a pipe. The pumpdischarge is determined using the water levels (or hydrostatic pressure heads) at theupstream side of the pipe and at the downstream side of the pipe.

Here after, both the Pump station and the Internal Pump station are referred to as the Pumpstation. Please note that:

When activated, water is always pumped from the suction-side towards the delivery-side, A Pump station cannot be placed in a compound structure, A Time Controller, Hydraulic controller, a PID controller or an Interval controller can over-rule the pump capacity of a Pump station.

Pump station output parameters becomes available by checking thePump Data check-boxon the 1DFLOW/Output options/Structures Tab in Settings.

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Pump direction Positive pump directionmeans that thepump discharge is owing in thepositivex-directionalong a reach (see Figure 1.27). Hence, water is pumped in downstream direction towardsthe pump-side with the highest x-coordinate.

Negative pump direction means that the pump discharge is owing in the negative x-direction along a reach (see Figure 1.28). Hence, water is pumped in upstream directiontowards the pump-side with the lowest x-coordinate.

Figure 1.27: Pump station with positive pump direction and two pump stages

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Controllers

A Time-controller, a Hydraulic-controller, a PID controller or an Interval controller can be as-signed to a pump station. A controller is only active in case the pump is triggered in ac-cordance with the Switch-on and Switch-off levels, dened for stage 1. An active controlleroverrules the pump capacity of the triggered pump stage, while the actual pump stage is setto -1. Please note that capacity reduction factors are applied to the pump capacities set by acontroller. Advice: In using a controller at a pump station, dene only one (1) pump s

SOBEK 3 Technical Reference Manual

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