c 2014 Sarah Berastegui-Vidalle
Contents
1. 2500 Time sec Figure 2 11 Discharge at 5000 ft downstream of the entrance of the pipe Zi 2 3 The Canal In this case the channel is prismatic with a more complex geometry see 2 13 The inflow hydrograph is Gaussian distribution and the outfall is a free overfall here are the properties of the simulation Channel Properties upstream Gaussian distribution hydrograph 700s ow O oe SS discharge m s 4000 6000 Time sec Figure 2 12 Inflow hydrograph for The Canal For routing in the SRM a time step of 36 s and a spacing of 160 m are taken The simulation is also done with HEC RAS The results for the water surface elevation and the discharge at the upstream and downstream end The results given by SRM and by HEC RAS are very close in this case The maximum error for this simulation is 4 43 ZZ 180 160 140 120 100 80 60 40 20 0O 20 40 60 80 100 120 140 160 180 cross section m Figure 2 13 Cross section geometry of The Canal Belo Monte station 0 5000 time sec Figure 2 14 Water surface elevation at the entrance of the canal 23 Belo Monte station 16000 Q 7 D oeePeee woe eco epeccccccocooocs 2000 5000 t2. station O vi a s po co x QO 2 a station 5000 Time sec Figure 3 5 results for the discharge with a spacing of 50 ft 54 First theoretical case another spacing When the spacing is changed to Ax 6250 ft the error and the oscillations become more important and the results is not relevant anymore station O station 500 station 1000 station 1500 station 2000 station 2500 vi an Z 3 oD co aie Q 2 a station 3000 station 3500 station 4000 station 4500 station 5000 Time sec Figure 3 6 results for the discharge with a spacing of 50 m 5 Other theoretical cases For all other theoretical cases the oscillations and the lack of accuracy in the results starts appearing for a spacing of Ax 6250 ft the error and the oscillations become more important and the results is not relevant anymore First slope first hydrograph First slope second hydrograph 3000 4000 5000 6000 Second slope first hydrograph 2000 3000 4000 5000 6000 Station 0 eseese station 6250 station 12500 Station 18750 Station 25000 Figure 3 7 results for the discharge with a spacing of 6250 ft 50 3 3 3 Other theoretical cases For all cases it is interesting to see the values of the ratio when the lack
3. 3 8 distribution of the ratio a for all cases at the limit spacing A l overview of the code 0 0 0 0 00 0040 A 2 set the global variable interceptor 0 0048 A 3 The junctions in SRM so cece amp be eae we aS we deed A 4 initialization steady flow calculation A 5 unsteady flow calculation 0 0 8 A 6 What to change i the main file makedecisions A T progress of the calculation 0 0 0 20 08 A 8 HEC RAS file for the Chicago River A 9 Division of the Chicago River Main stem reach A 10 Two possible directions for a reach A 11 where to find the invert elevation 0 A 12 how to set the inverts fle 0 A 13 How to classify the files 0 0 0 0 0 0048 A 14 code to create the HPG files oa aa A 15 How to check the filles 0a a a aa a vi A 16 the interceptor variable s 6 a Re oe we Se A we BAS HS A 17 the conduit field of the interceptor variable the red fields are asbent of the conduit variable created to input the UP GeV Get orien amp oe FO Ee Oe E T A 18 How to use the file containing HPGs VPGs in SRM code A 19 the boundary conditions in the code A 20 the boundary conditions in the code vil Y Oo D i in oit Qn LIST OF SYMBOLS is the cross section area in m is the channel width in m is the hydraulic d
4. 2014 Sarah Berastegui Vidalle APPLICATION AND ENHANCEMENT OF THE SRM MODEL BY SARAH BERASTEGUI VIDALLE THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Environmental Engineering in Civil Engineering in the Graduate College of the University of Illinois at Urbana Champaign 2014 Urbana Illinois Adviser Professor Marcelo H Garcia ABSTRACT The main focus of this thesis has been to apply and to develop the storage routing model The storage routing model is a method of hydraulic numer ical calculation It is based on the Saint Venant equations and works under the quasi steady dynamic wave assumption It first creates HPGs and VPGs respectively upstream water surface elevation and storage as function of the discharge and the downstream water surface elevation and uses the inter polations of these graphs at each step of the calculation This model has been developed for systems composed of circular conduits In this thesis at first the accuracy of the model has been checked for systems with circular pipes the Wallingford research station and some theoretical models from Matthew A Hoys thesis Then the model has been adapted to river sys tems that include non prismatic channels with input HPGs VPGs and has been applied to the Belo Monte canal and the Main Stem Chicago river The comparison with field data or with other models results is satisfying the error max abs
5. critical X upstream elevations Y downstream elevations Tables X upstream elevations Y downstream elevations numbers isHPGsteep length inoffset outoffset height maxtiow numbers minflow Count number of discharges usinvert dsinvert usnodedenum Dsnodenum storage name name Tables Discharges volumes TriScatteredinterp hpgMatrix TrilnterpUpstreamY functions that gives the upstream elevation from the downstream elevation the discharge and the HPG TrilnterpStorage functions that gives the storage from the downstream elevation the discharge and the HPG Figure A 17 the conduit field of the interceptor variable the red fields are asbent of the conduit variable created to input the HPGs VPGs DGGE s4BrC S B MeOf b OH BBM ID Bl stack Base fe HO 8 amp o Sete j10 ia x 8 1 function SetConduitProperties name3 roughnessi lengthi inoffseti outoffseti inletnodeil outletnodei xname 2 a he global interceptor d global g 5 6 7 8 L numc 0 ER tolerance 10 6 11 TE for i 1 size name3 2 13 in the case of input HPGs VPGs comment this part 14 upnode findnode inletnodei i 19 downnode findnode outletnode1 i 16 17 note these are just indices 18 19 for k 1 interceptor numstorage 20 r if strcmp inletnodei i interceptor storage k name 77 AI upnode interceptor storage k d
6. of accuracy Occurs It has a different value for all locations and time steps therefore the distri butions is plotted i vu E A lt a o O kon a a w gt 5 UO first slope first hydrograh first slope second hydrograh second slope first hydrograh second slope second hydrograh first theoretical case Wallingford Figure 3 8 distribution of the ratio a for all cases at the limit spacing In all cases for more than 10 of the points time step and locations the ratio a is higher than 1 It can be deduced that it is a condition to obtain a satisfying result ot Chapter 4 CONCLUSION The storage Routing Method is a method of hydraulic calculation based on Saint Venant equations that can be accurate and efficient under the following assumption It applies under the following assumptions e The flow is 1 dimensional only the variations in the flow wise direction are considered e The pressure is assumed to be hydrostatic in the entire channel the vertical acceleration are neglected e The channel can be considered as a straight line e The geometry of the channel is fixed no effect of scour and deposition e The slope is small e The fluid is incompressible Q e the Quasi steady flow assumption gt is negligible e The spacing used to create the HPGs VPGs respects the spa
7. 2 2 2 9 2 4 2 0 Wallingford Hydraulic Research Station 2 2 Theoretical Model from Matthew Hoy s Thesis TNC Canal e 4 3 oe amp Sao Gone Foe fH Rm Bee RO WKS The Chicago River Main Stem 0 Theoretical cases 2 ww a aa Chapters A SPACING CRITERIA a a4 625 k c eR a eS wa 3 1 3 2 3 3 For steady flow backwater length Another spacing criterion that applies for unsteady flow Verification of unsteady flow spacing criteria Chapter 4 CONCLUSION 0 0 0 00 0 0 0 0 00 Appendix A USER S MANUAL 0 0 0 0 Pl UG EOGUCHION ts 4 g ce a dom Boe eB ae A 2 Description of the code aoao a a a 2000048 Peo LOW touse SRM 5 amp 4 36 eS Awe ee Se Soe ew A 4 Alternatives for using SRM 0 04 1 1 L2 ZA 22 2 3 2 4 2 0 2 6 Zale 2 8 2 9 2 10 2 11 Zed 2 13 2 14 2 19 2 16 Zt 2 18 2 19 2 20 2 21 2 22 2 23 LIST OF FIGURES Example of a backwater calculation 202 Example of the division of a system Inflow hydrograph for Wallingford Hydraulic Research Station Water surface elevation at 28 4 ft downstream of the en trance of the pipe we a bm ae Gwe Bw ER ae OS ome Water surface elevation at 255 7 ft downstream of the entrance of the pipe a a a a Water surface elevation at 483 ft downstream of the en trance of the pipe 2 0 0 0 0
8. 3 2 the error vs the frequency of ratio higher than 10 In the case of 10 the same observations as in the case of 1 can be done Even if the points are more spread out in the case of 100 the graph shows that the order of magnitude of the error and of the frequency are the same In the cases tested the accuracy of the result is good It should be interesting to see what happens when the oscillations start occurring and the accuracy ol O Wallingford Matthew Hoy s theoretical case O first slope first hydrograh O first slope second hydrograh O second slope first hydrograh second slope second hydrograh 0 001 Frequency gt 100 Figure 3 3 the error vs the frequency of ratio higher than 100 52 3 3 2 spacing criteria that causes oscillations Wallingford Hydraulic research Station In the case of Wallingford Hydraulic Research Station for a spacing of Ax 113 6 ft the accuracy decreases and the oscillations start occurring o D station 0 station 1136 Discharge m3 s Ww 100 Time sec Figure 3 4 results for the discharge with a spacing of 113 6 ft 59 First theoretical case In the first theoretical case described previously for a spacing of Ax 200 ft the accuracy decreases and the oscillations start occurring wilt md TTI T TTT
9. backward 94 O Organize v Include in library Share with Burn Wr EE Name j Date modified Type Size E Desktop chicagoriver f01 5 2 2014 12 05PM_ F01 File 1 K8 T Downloads _J chicagoriver g03 5 7 2014 9 58 AM G03 File 132 KB 3 Dropbox chicagoriver 001 5 2 201412 05PM 001 File 275 KB 1 Recent Places chicagoriver p01 5 2 2014 12 05 PM P01 File 4 KB L chicagoriver p01 comp_msgs 5 2 2014 12 05PM _ Text Document 1 K8 G Libraries L chicagoriver prj 5 2 201412 05PM PRJ File 1 KB LJ chicagoriver 01 5 2 201412 05PM R01 File 1 149 KB Computer amp Local Disk C ze Local Disk D GP GG cee hsl gi GP TARP cee hs amp Network Figure A 13 How to classify the files ag O _ Editor C Users sberaste Desktop livrable HECRAS createH Gel livrable gt HECRAS a sReRDVO OD AOD SS eS ee saa en aaa _ _ 28 g a ve H o i 1 0 11 x 6 Organize v A Open v Share with v Burn Ne uly clear all Name a close all x Favorites 3 fclose all E Desktop Ji backward dg a f 5 Js Downloads J forward a M 94 Dropbox HECRASEXE_batch 71 N 103 OF 8 ymax 20 Recent Places fee createHPG K e HPGexists 10 fid fopen inverts txt Libraries sents a newline fgetl fid 12 i 0 13 14 fldr cd 15 16 while ischar newline Figure A 14 code to create the HPG files How to use the created HPGs and VPGs A structure variable w
10. classified in numbered folders The numbers go from upstream to downstream 13 35 ET g a Nyi TETIT sani Serre t Main Steam File Edit Options View Tables Tools GISTools Help File Edit Options View Tables Tools GISTools Help 1467052 83 15214987 35 1467052 83 15214987 35 Figure A 10 Two possible directions for a reach When everything is set the code can be run In the main folder there should be the Matlab file createHPG once opened the the number M and N have to be changed M have to be the first number of the folders made before and N the last one The maximum water surface elevation has also to be changed see figure A 14 The code to create HPGs doesnot always work perfectly sometimes HPG files rae not created sometimes the values found for the volume of water in the VPG are all equal to 0 in these cases the HPG files have to be done 14 X uas p 2gHONE AHO iann ste HAT gt Cross Section Data mainsteam lo lo River Chicago River z Na tOn Ba M Keep PrevXS Plots _ Clear Prev 4 t ale chicagoriver Plan Plan 01 1 23 2014 SIRTE Figure A 11 where to find the invert elevation A8 A fe 13 4877 aP e c D E F G 1 12 366 2 12 5692 18 1 station 3 12 7598 17 12 4 12 9113 1
11. ee ee Water surface elevation at 710 2 ft downstream of the entrance of the pipe a u e aa a aa ap Aa Inflow hydrograph for Theoretical Model from Matthew Hoyo THO e 2 4 5 0 2 ae So 6 Ad a a a a a a SS Water surface elevation at the entrance of the pipe Water surface elevation at 2500 ft downstream of the en kance Ol WNC DPs a cs amp Ge ea He ee E Se e Water surface elevation at 5000 ft downstream of the en trance Ol UNC pipe sa pe maa eek os wt oe oo we a EA Discharge at 2500 ft downstream of the entrance of the pipe Discharge at 5000 ft downstream of the entrance of the pipe Inflow hydrograph for The Canal 2 Cross section geometry of The Canal 2 Water surface elevation at the entrance of the canal Water surface elevation at the exit of the canal Discharge at the entrance of the canal Discharge at the exit of the canal 0202 View of the Chicago River Main Stem 2 Geometry of the Chicago River Main Stem Geometry of the Chicago River Main Stem Stage hydrograph at the upstream point of Main Stem Model for the stage hydrograph at the upstream point of MAMAS UCMi 6G 8 ow oe a Sw oe bee ES eS ws Discharge in the Main Stem 0 11 2 24 Water surface elevation in the Main Stem 2 25 error between SRM and HEC RAS results 2 2 26 First Inflow hydrograph 2 e carcg 6 kw amp Be
12. stored in the global variable interceptor interceptor node i flow interceptor node i wse and interceptor conduit i storage As well as in the variables WSEM flowM and storageM Once the calculation is done some graphs with will appear You can use them as reulsts but you can also use the variables cited previously to make your own graphs or to get the numerical values It can be convenient for example to get the values in excel files int his case the commands o zlswrite flow alsx flowM o zlswrite wse rlsx WSEM o zlswrite storage xlsx storageM 71 A 4 Alternatives for using SRM By default SRM makes the calculation for circular pipes and takes the inflow hydrographs and the free outfalls as boundary conditions These options can be changed A Non circular conduits river reaches In the case on non circular conduits or river reaches it is possible to create the HPGs VPGs with another tool and use them as an input for SRM A tool to create HPGs VPGs An alternative to create the HPGs VPGs is the HPG utility for HEC RAS It uses the files from HEC RAS to create HPGS VPGS The manual HPG Utility for HEC RAS Users Manual Blake J Landry and Nils Oberg 2013 explains how to use this tool How to use this tool A matlab code has been cerated to make HPGs and VPGs for all reaches conduits In order to use it it is first necessary to split the HEC RAS river file in smaller river rea
13. straightforward and efhcient manner Matthew A Hoy 2005 The methods of calculation based on Saint Venant apply for a one dimensional flow an incompressible fluid with a hydrostatic pressure no scour or depo sition and a small slope These equations being non linear no general ana lytical solution exists In the case of unsteady flow no term is neglected the computation can be intensive On the other hand when too many terms are neglected the result might not be accurate enough In some cases the con vective acceleration can be neglected and the quasi steady dynamic wave is a good compromise between accuracy and computation velocity SRM is one method of calculation based on the quasi steady dynamic wave assumption It creates table of values of upstream water surface elevation HPGs and 60 volume of water VPGs depending on the discharge and the downstream water surface elevation for all portions of the system considered beforehand and uses interpolations of these graphs at each time step of the calculation to find the solution 61 A 2 Description of the code SRM uses SWMM files describe SWMM to read the properties of each component of the system or a river reach pipe junction weir etc It either creates or uses input HPGs VPGS for each river reach pipe With given boundary conditions inflow outflow hydrographs stage hydrographs rating curves it calculates the discharge the water surfac
14. 00 Time sec Figure 2 32 Discharge at several station for S 2 x 10 with the first hydrograph 39 m Sg station O station 12500 station 25000 ma _ p _ e z gt 2 3 vy S T 7 w pe 3 400 Time sec Figure 2 33 Water surface elevation at several station for S 2 x 107 with the first hydrograph 40 S S station O n vi O kea s J ofa Q a A station 12500 station 25000 Time sec Figure 2 34 Discharge at several station for S 2 x 10 with the second hydrograph Al station O station 12500 station 25000 a _ e Ez gt A 3 3 Ye T 7 w 3 4000 Time sec Figure 2 35 Water surface elevation at several station for S 2 x 107 with the second hydrograph 42 Chapter 3 A SPACING CRITERIA Here the question of the accuracy of SRM is raised Under the quasi steady dynamic wave assumption the convective acceleration term in oe is ne elected And the system is discretized to allow a numeric solution To make sure the result is reliable it is necessary to check if this assumption is valid in most cases it is valid and make sure that the spacing and the time step are chosen adequately It has been prov
15. 7 25 5 13 0446 18 47 6 13 2811 18 53 7 13 357 16 89 invert 13 4877 16 89 9 13 7057 17 02 10 13 9387 16 99 Figure A 12 how to set the inverts file manually To verify that all HPGs and VPGs have been created and the values are correct it is possible to run the matlab code HPGezxists If a file is missing or if it exist and the VPG comports 0 values it will notify it see figure A 15 Once all the files are created and valid the values can be stored in a Matlab structure file to be used in SRM In this file too it is necessary to set the values M and N 19 Main folder Organize v Include in library v Share with v Burn New folder Sk Favorites Name Date modified Type Size ts fil 6 16 2014 4 08 PM File folder nverts tile je forward 6 16 2014 4 13PM File folder F Drop createHPG ee oeoa a oo wen na anit a Recent Plac HECRASEXE_batch gt gt livrable gt HERAS gt backward gt _ Organize v Include in library v Share with v Burn New folder Libraries 3k Favorites Name z Date modified Type m E Desktop 6 16 2014 4 32 PM ile folder 1 Computer Downloads 6 16 2014 4 34PM ___File folder amp Local Disk C 7 Dropbox 96 6 16 2014 4 37PM File folder Local Disk D E Recent Places 97 6 16 2014 4 39 PM ile folder E GG cee hsl gt Ji 98 6 16 2014 4 42PM File folder G2 TARP cee hsl A Libraries Gu PAEA OO gt livable gt HECRAS
16. M flow upstream Hec Ras flow upstream ea LL O w lt e an Q SRM flow downstream e gI o S Hec Ras flow downstream 0 4 1 t 9 8 2008 9 9 2008 9 9 2008 9 10 2008 9 10 2008 9 11 2008 9 11 2008 9 12 2008 Figure 2 23 Discharge in the Main Stem ol SRM wse upstream SRM wse downstream Hec Ras wse upstream Hec Ras wse downstream amp e co gt 2 Lud y T 72 hes w 3 f F f i 0 5 9 8 2008 9 9 2008 9 9 2008 9 10 2008 9 10 2008 9 11 2008 9 11 2008 9 12 2008 Figure 2 24 Water surface elevation in the Main Stem 92 Figure 2 25 error between SRM and HEC RAS results 99 2 5 Theoretical cases In this case the channel is a circular 50 ft diameter pipe Four different cases are tested with two different slopes and two different inflow hydrographs The inflow hydrograph has a gaussian distribution and the outfall is a free overfall here are the properties of the simulation discharge cfs 400 Time sec Figure 2 26 First Inflow hydrograph For routing in the SRM respective time step of 2 5 and 25 s and a spacing of 250 ft are taken The results for the water surface elevation and the discharge at the upstream and d
17. Zimmer A et al 2013 1 2 1 How it works At the first time step the calculation is done assuming a steady flow for the next time steps the storage calculated at the previous time step is used to do the calculation for unsteady flow For each time step a first guess is given for the values of water surface elevation and discharge at each node and it uses Newton Raphson to make it converge to the solution To solve the equations the system is divided into nodes and links Zimmer A et al 2013 p938 939 At each node there are two unknowns the discharge and the water surface elevation The pipes or portions of pipes the river reaches the pumps the storage units and other features are considered as links Each link gives two equations one is linked to the conservation of mass the other one to the momentum equation A junction is where there are more than two links or where there are two links and an inflow At each junction ghost nodes are created correspond to every links end and every inflow in order to facilitate the numerical resolution Each junctions gives as many equations as it has nodes Therefore this becomes a system of unknowns and equations There are usually more unknowns than equations To complete the number of equations boundary conditions are necessary The equations are Zimmer A et al 2013 p947 p ta for junctions f y yy Wig f gt Vin a 2 Om f
18. ale Oe Ge ew 2 27 Second Inflow hydrograph 2 28 Discharge at several station for S 2 x 107 with the first ly MOC A es ci ae es ee Sek My Ge es Be See ed 2 29 Water surface elevation at several station for S 2 x 107 with the first hydrograph ek Gb hE wh Se eS 2 30 Discharge at several station for S 2 x 107 with the SeCOnC NV GTOSTADO s pori aop eoa OOo OSS ER Bed 2 31 Water surface elevation at several station for S 2 x 107 with the second hydrograph aL 44 kee ew AES SR poe 2 32 Discharge at several station for S 2 x 107 with the first IML OCR AO S 2 dite hat BO ee Bye PR eck ae ee Shs amp Bae Bs ies 2 33 Water surface elevation at several station for S 2 x 107 with the first hydrograph i amp oaks Be a oe a amp ee He 2 34 Discharge at several station for S 2 x 107 with the second hydrograph 24 4 2244 Gee ew ee eS Ss 2 35 Water surface elevation at several station for S 2 x 107 with the second hydrograph aw dea Mee wo Me are we ae 3 1 the error vs the frequency of ratio higher thanl 3 2 the error vs the frequency of ratio higher than 10 3 3 the error vs the frequency of ratio higher than 100 3 4 results for the discharge with a spacing of 113 6 ft 3 5 results for the discharge with a spacing of 50 ft 2 3 6 results for the discharge with a spacing of 50m 3 7 results for the discharge with a spacing of 6250 ft
19. aulic Research Station Matthew A Hoy 2005 the channel is a circular 1ft diameter pipe The inflow hydrograph is triangular and the outfall is a free overfall here are the properties of the simulation tons Fce vert Similarly to Hoy s work for routing in the SRM a time step of 2 5 s anda spacing of 28 4 ft are taken The results are compared his results obtained 10 S O Un gt P Ww amn v Q w y n Q 3 N 300 400 Time sec Figure 2 1 Inflow hydrograph for Wallingford Hydraulic Research Station with FEQ and SRM and with the data from the report by Ackers and Har rison 1964 The results for the water surface elevation at the stations 28 4 255 7 483 and 710 ft downstream of the entrance of the pipe are compared The plots show that the results found with SRM are the same as the results found previously Matthew A Hoy 2005 And they match closely the experimental results l1 Depth Comparison at Station 28 4 Experimental data from the report by Ackers and Harrison 1964 Results with FEQ Results with SRM obtained by Matthew Hoy Recent results with SRM w le _ Ez a n 3 lt w pe z w Time sec Figure 2 2 Water surface elevation at 28 4 ft downstream of the entrance
20. c Performance graph is described as Juan A Gonzales Castro and Ben Chie Yen 2000 p 16 a group of Hydraulic Performance Curves for a channel reach An hydraulic performance Curve for a given channel reach and a given discharge is a curve giving the values of the upstream wa ter surface elevation as a function of the downstream water surface elevation in the case of subcritical flow In most cases the HPG is obtained using a gradually flow calculation Some times it can be obtained through other method of calculation In the case of pipes it can be extended to include pressurized flow 1 1 4 The Volumetric Performance Graph VPG The volumetric Performance Graph is similar to the HPG Matthew A Hoy 2005 however it gives the volume of water stored in the river reach or conduit as a function of the discharge and the downstream water surface elevation for subcritical flow instead of the upstream water surface elevation It is a group of Volumetric Performance Curves for a channel reach A volumetric performance Curve for a given channel reach and a given discharge is a curve giving the values of the volume of water as a function of the downstream water surface elevation in the case of subcritical flow The VPG can be obtained with the same method as the HPG and can also be extended to include pressurized flow 1 1 5 HPG and VPG extension to pressurized flow The method of backwater calculation obtained with the Sai
21. ches files For example in the case of the Chicago River the river in the file comports 13 reaches It has to be first separated in 13 files A reach can be too long to be a precise enough spacing for SRM so it is sometimes necessary to divide it in smaller parts See figure A 9 2 0 8524 ET w 1490410 23 15160587 27 Figure A 8 HEC RAS file for the Chicago River Sometimes the river can flow in both directions as it is the case for the Chicago River Main Stem In this case it might be necessary to have all these files in both directions see figure A 10 HEC RAS specifies a direc tion for the flow A list of the invert elevation of the cross sections at each end of the parts is necessary They have to be classified in order from upstream to downstream This list has to be in a text format and called inverts This file can be made with excel the first column is the station and the second is the invert elevation see figure A 12 Once all these files are created they can be used to create HPGs and VPGs They first have to be first classified in folders The main folder has to contain the znverts file and two folders backward and forward These folders contain the HEC RAS files made previously the folder forward contains all the files were the river is flowing in the streamwise direction the older backward the same files for the opposite direction In both folders all the HEC RAS files are
22. cing cri terion linked to the backwater length e The time step is small enough to allow convergence e The spacing used to divide the conduits reaches in SRM respects the criterion linked to unsteadiness The assumption to apply Saint Venant equations and the Quasi steady wave assumption have to be verified to use SRM otherwise the method of cal culation has to be changed The model used for SRM calculations can be 08 adapted to meet the time step and spacing criteria The time step criterion is already taken into account Zimmer A et al 2013 since the code is dividing it until it finds convergence The spacing criterion linked to the backwater length has no effect on the spacing taken for SRM It has to be taken into account to create the HPGs VPGs The HPGs VPGs do not account for the change of discharge Therefore the spacing criterion linked to unsteadiness is important here The spacing for SRM has to be chosen first to meet this criterion Suggestions for future work The SRM code implemented with matlab does not account for the spac ing criterion linked to unsteadiness This can be implemented by doing HPGs VPGs for different sizes of reaches conduits and the code will choose the spacing based on the criterion calculation at each time step The second thing that can be improved is the method of convergence used by SRM Newton Raphson is efficient however there are cases for which it does not work Anoth
23. e calculation using the timeseries from SWMM as inflow hydrographs and assuming free outfalls B How to set the input The input has to be a SWMM file All the properties of the system in this file will be used for the calculation The name of the SWMM has to be reported in the main file as inputfile makedecisions of the SRM code line 6 Hea BO O b B di e d f gt H BAH IB HA Stack Base Ds S a 11 x 6 a oe close all 2 oe clear all fclose all a eis 383 your main file the outfall has been make sure that they are in the right folder and that 11 the path does not comport space ta global G a E Wi g 32 2 FR enter the value of g based on the units of your system Figure A 6 What to change i the main file makedecisions It is also necessary to specify the number of time steps numintervals and the value of the time step deltat for the calculation this can be done in the main file makedecisions of the SRM code line 7 10 Once all this is set up the simulation can be run and the progress will appear in the command window Command Window 1 ax calculation 0 444443 ang calculation 0 47222 ang calculation 0 5 ans calculation 0 527785 Figure A 7 progress of the calculation C The calculation As specified previously the values of the discharge and the water surface elevation at all nodes and of the volume of water in all links are
24. e elevation and the storage everywhere in this system The calculation is made using interpolation of HPGs VPG and the method of convergence used at each time step is Newton Raphson set_interceptor_properties readHydrograph Simplifyglobal Calls OptimizeSectionSteady probability_overflows Main file Read the main SWMM file to get the properties Read the hydrographs properties Starts the hydraulic calculation for the first step Make complete hydraulic calculation at each time step Figure A 1 overview of the code A Read the inputs The geometry The function set_interceptor_properties reads the SWMM file as a text file line by line to get the inflow hydrographs and convert all the component of the system in a global structure variable called interceptor Reading each element The outlets conduits orifices outfalls pumps weirs and storage units are saved in the global variable interceptor with their geometric properties and 62 FindRuleFromLine ReadData set_interceptor_propertie Read the hydrographs for the inflows Read all components of the system Create nodes CreateAllJunctionsInclusive N N SetNodeProperties createnode Figure A 2 set the global variable interceptor functions between at least two variables water surface elevations storage discharge all this elements are considered as links in SRM The junctions The junct
25. en that Gonzles Castro J A and Ben Chie Yen 2000 p74 76 a method based on the quasi steady dynamic wave assumption using interpolations is more accurate than the Non inertia wave method more ro bust and less sensitive with respect to discretization The chosen time step also affects the accuracy of the model A time step that is too big might not allow convergence In the methodology of SRM this is taken into account Zimmer Andrea 2013 p 42 If at a time step the conditions do not allow convergence for Newton Raphson method the time step is divided by two and the values of the inflow for this new time step Even when all the previous conditions are met the results might sometimes not be very accurate or even irrelevant Some oscillations might occur These problems may be linked to another condition The spacing is another element that must affect the accuracy of the method It has to be checked at two stages the step for the HPG VPG creation and the length of conduit reach chosen for each HPG VPG The HPG VPG ac counts for the variation in water surface elevation Even though the HPG is made for a long reach the method used to calculate might use an appro priate spacing Thus the calculation of the water surface elevation might be accurate However none of the HPGs and VPGs can account for the change in discharge in the case of unsteady flow If the variation in discharge with 43 respect to the longitudina
26. epth in m is a force applying to the material inside the volume V in N kg m 3 is the Froude number is an equation for the Newton Raphson method is the acceleration due to gravity in m s is the water surface elevation in m is the normal water surface elevation in m is a unit conversion coefficient Equal to 1 for I S and to 1 486 for E U is the backwater length in m length of the channel affected by unsteadiness is mannings roughness coefficient is the discharge in m s is an discharge coming in the junction in m s is an discharge going out of the junction in m s is the discharge for a normal regime on in m s viil I Yi Ylimit Yup Ydown Ax At is the lateral inflow in m s is the velocity of the flow in m s is the hydraulic radius in m is the surface delineating the volume considered form outside 2 in m is the slope of the bottom of the channel is the friction slope is the time in s 3 is the volume considered for derivations in m is the velocity of the fluid entering the volume V through dS in m s is the longitudinal dimension in m is the water surface elevation at point 2 in m is the water elevation of one arbitrary chosen node in a junction a junction is several nodes with same location in m is the water surface elevation of the upstream end of a link in m is the water surface elevation of the downstream end of a link in m is the ration of the spacing
27. er method of convergence maybe less efficient but which might work for all cases when Newton Raphson does not converge The method to create HPGs VPGs used by SRM can also be improved It only works for circular conduits in the case of a more complicated geometry it is necessary to create the HPGs VPGs with another method and use them as input to the code It may be convenient to have a code that takes many types of geometry into account other shapes and non prismatic channels 59 Appendix A USER S MANUAL A 1 Introduction Irregular precipitations as well as irregular use of water cause the flow in rivers and hydraulic infrastructure to be unsteady and non uniform Dur ing critical events heavy rainstorm for example it is useful to know some estimations of the discharge and the water surface elevation it must helps preventing overflows and floods A typical way to obtain these values is to use the Saint Venant equations and solve them numerically These equa tions are non linear and quite complex Thus is it often useful to make some assumptions to neglect some terms and simplify the calculation It may al low simpler computation and faster results without losing to much accuracy SRM storage routing method is a matlab code that applies this method under the quasi steady dynamic wave assumption It has been developed and enhanced by previous authors work in order to create a method for modeling these unsteady flows in a
28. flow the terms in a Z and 2A are equal to zero The equations of Saint Venant become one equation Oh Als g Z So 55 0 BQ 04 Which becomes a gradually varied flow equation Ven Te Chow 1959 p134 dy So Ss a i 1 3 dx cos 0 F Sie Where F is the Froude Number squared Equation 1 2b allows the calcualtion of the water elevation along the channel Terry W Sturm 2001 p200 This is the backwater calculation Typically y is a function of y and the discharge The water surface elevation is calculated knowing the downstream water surface elevation and the discharge The value of the upstream water surface elevation can be obtained from the back water calculation Yup f Osean Q Complementary calculation for the volume The cross section area is a function of the geometry of the channel and the water surface elevation Thus through the backwater calculation the cross section area is known The volume can be calculated using this formula v Adz down _ _ _ i CN an Ydown Ji A fOi 1 Q iis Figure 1 1 Example of a backwater calculation Which can be discretized and used during the backwater calculation to find the storage V Adc 1 4 Thus the volume of water in the reach is also a function of the downstream water surface elevation and the discharge V T Uema Q 1 1 3 The Hydraulic Performance Graph HPG The Hydrauli
29. function that divides the time step in this case The function AdjustTimeStep calls Op 67 timizesSection the function that applies Newton Raphson for unsteady flow It check the convergence it returns the values if it is good if not it divides the time step and calls itself twice The time step can be divided endlessly however when the time step becomes too small OptimizeSectionSteady is called to make the calculation for steady flow and return the values OptimizeSection AdjustTimeStep Newton Raphson for probability_overflows e Call OptimizeSection unsteady flow Calculation for unsteady flow e No convergence gt divide the time step by 2 and call AdjustTimeStep until convergence e the time step is too small gt call OptimiseSectionSteady OptimizeSectionSteady Newton Raphson for steady flow Figure A 5 unsteady flow calculation 68 D The output of the code The values of the discharge and the water surface elevation at all nodes and of the volume of water in all links are stored in the global variable interceptor interceptor node i flow interceptor node i wse and intercep tor conduit i storage As well as in the variables WSEM flowM and stor ageM 69 A 3 How to use SRM The Matlab code for SRM includes a function that is making HPGs and VPGs for circular pipes In this case the input of the code has to be only a SWMM file A The calculation The code is making th
30. gs gov nwis is a stage hydrograph for the upstream point This hydrograph contains a lot of steep osculations When using this hydrograph as a boundary condition the model crashes The osculations are too steep therefore it is necessary to use a smoother hydrograph 28 A c gt 7 o oO g n i w H 3 oO Q i H n a D Ac 9 8 2008 9 12 2008 9 16 2008 Figure 2 21 Stage hydrograph at the upstream point of Main Stem 2 4 3 Geometry The channel cannot be modeled as a circular conduit However the code is written to create HPGs and VPGs for circular conduits Thus the HPGs and VPGs have to be created first and use as an input to the code 2 4 4 results The same simulation is done HEC RAS to compare the results The error is calculated with the following formula QsRru QHEC RAS Ora TE nae 29 9 20 2008 field data model A c gt 7 t oO g in n w H S oO Oo i H n a D A rel 9 8 2008 9 12 2008 9 16 2008 9 20 2008 Figure 2 22 Model for the stage hydrograph at the upstream point of Main Stem The results given by SRM and by HEC RAS are very close in this case The error always stays below 10 except for the first steps The error at the first steps might be caused by the initial condition in HEC RAS 90 z S SR
31. h Based on a typical cross section geometry of a river channel Samuels P G 1989 p 575 n and is approximately equal to 1 and R is approxi mately equal to D Therefore 3 4 becomes n 10 S EN aS e E 9 9 dx 3D cos O F on The following expression is obtained from 3 5 So n noexp 3 3 zo x D 3 6 The backwater length being defined as the distance where 7 0 17 it has the following value L 0 7 0 75 3 7 45 And for larger Froude Numbers L 07 1 F2 3 8 0 To obtain accurate results it might be useful to have at least 5 spacing con tained in the backwater length L 0 15 3 9 So 46 3 2 Another spacing criterion that applies for unsteady flow In the case of unsteady flow a similar derivation can be made to find the length of the channel affected by unsteadiness It provides the following formula l a Ly 2 3 3 10 Q The spacing criteria linked to the unsteadiness has to be proportional to this derivative AT 3 11 Q Sle AET amp amp a Na 3 2 1 Derivation In the case of quasi unsteady flow the first equation conservation of mo mentum of Saint Venant is used under the assumption of steady flow The unsteadiness is given by the second equation conservation of mass 1 2b A derivation inspired by the one made by Samuels can be done introducing a small pertubation to normal and applying it to the discharge Q Q
32. ill be created and save in the matlab file conduit in the main folder This variable can now be used in the SRM code This variable contains some properties of the conduits and is ordered as the conduit field in the variable interceptor is ordered in the SRM code The created variable conduit has to be pasted in the folder Set nterceptor Properties The Matlab file SetConduitProperties needs to be changed All the instructions appear in the code as comments 16 MATLAB 7 12 0 R201 File Edi rallel Desktop Window Help et Go Tools Debug P OSG 4 BBO C i E O Current Folder c Users sberaste Desktop livrable HECRAS Figure A 15 How to check the files numjunctions node numnodes numstorages conduit numweirs Weir numsluicegates numoutlets Sluicegate numorifices numoutfalls numpumps numdwf numsiphons numrules numinflows numconduits names usnodename dsnodename Figure A 16 the interceptor variable line 7 uncomment to load the conduit variable lines 30 78 comment all these lines lines 92 142 comment all these lines line 145 uncomment to get the HPG line 155 comment this line lines 269 270 comment two ends The rest of the code is not changed Therefore it is necessary to set every thing else as usual it is still necessary to create a SWMM file but the shape of the conduits are not important in this case i conduit curves HPG Tables
33. ime sec Figure 2 15 Water surface elevation at the exit of the canal 24 Belo Monte station 0 cy m eee e ag lt O 2 ge 5000 time sec Figure 2 16 Discharge at the entrance of the canal 25 Belo Monte Upstream station 16000 cy m ae id ag lt O 2 ge 5000 time sec Figure 2 17 Discharge at the exit of the canal 26 2 4 The Chicago River Main Stem In this case the calculation is done for the Main Stem part of the Chicago River as seen in the picture 2 18 J iN WO inches ES Ss s ae i Ss SSA gates j h 4 o _ a a _ i hambe Sbuth Chamber gates Figure 2 18 View of the Chicago River Main Stem the geometry is not prismatic Figure 2 19 Geometry of the Chicago River Main Stem The boundary condition at the upstream end is a stage hydrograph and at the downstream end is a rating curve 24 Y down Yo y N Q CY downs Yiake Figure 2 20 Geometry of the Chicago River Main Stem geometry prismatic duration 5 days approx ee ee length 1 3280 miles downstream rating curve 2 4 1 Rating curve The rating curve used in this case is the one for all gates opened Q 18694 Ay 2 4 2 Stage hydrograph The data given by the USGS website http waterdata us
34. incompressible fluid with a hydrostatic pressure no scour or depo sition and a small slope These equations being non linear no general ana lytical solution exists In the case of unsteady flow no term is neglected the computation can be intensive On the other hand when too many terms are neglected the result might not be accurate enough In some cases the con vective acceleration can be neglected and the quasi steady dynamic wave is a good compromise between accuracy and computation velocity SRM is one method of calculation based on the quasi steady dynamic wave assumption It creates table of values of upstream water surface elevation HPGs and volume of water VPGs depending on the discharge and the downstream water surface elevation for all portions of the system considered beforehand and uses interpolations of these graphs at each time step of the calculation to find the solution In the case of SRM an aspect that can be critical for the time of computation and the accuracy is the spacing Ax taken When it is too small the time of computation is too long and the code might take days to give some results When the spacing is large the results might lack accuracy and when it is very large the code might crash or gives meaningless results Previously a spacing criterion based on the backwater length has been es tablished for unsteady flow Samuels P G 1989 The same kind of derivation can be done to find
35. ions in SWMM become nodes or junctions in SRM A junction that joins two links only for example two conduits becomes a node A junction that joins more than two links or an inflow is saved as a junction in SRM and nodes are created for each link and inflow see Figure A 3 Each junctions is saved with the properties of the nodes it contains link inflow junction junction link link inflow Figure A 3 The junctions in SRM 63 The conduits In the case of the conduits there are two functions making links between variables the HPG the upstream water surface elevation as a function of the downstream water surface elevation and the discharge and the VPG the volume of water as a function of the downstream water surface elevation and the discharge SRM contains a function to create the HPGs VPGs given the properties of the conduits This function assumes a circular geometry In the case of another shape or a river reach it is possible to create the HPGs VPGs before running the code and use them as an input The hydrographs The function set_interceptor_properties calls subfubction to read the time series of the SWMM files This works only for inflow hydrographs If other solutions are needed it is possible to use the function readHydrograph to creates new variables for the inflow outflow stage hydrographs and use them in the next parts of the code for the calculation 64 B Initialize the calc
36. l coordinate is important the length of portion to make a HPG VPG might not be accurate enough 3 1 For steady flow backwater length For steady subcritical flow the length of the channel affected by a work downstream is called the backwater length In the case of steady flow the backwater length can be calculated with the following formula Samuels P G 1989 D Ine N l b 5 3 1 With D Or for larger Froude Numbers D Ly 0 7 1 F 7 3 2 So For a steady flow calculation in order to reduce the error the length of the backwater from 3 2 has to include at least five spatial steps The maximum spacing has to be eS Ax 0 15 1 F 3 3 0 This relationship was established assuming a trapezoidal geometry Even if the channel is not trapezoidal it gives a good idea of the backwater length and of the spacing criteria needed for hydraulic calculations 44 3 1 1 Derivation The derivation to find the backwater length is done using 1 3 introducing small perturbation to normal h h n Samuels P G 1989 it gives the following results Oe ore 5 So Sf Ox cos 0 F Oh cos 0 F On o 1 1 O ais So Ss o s r o Sy Oh 0 y y cos 0 F on p 3 4 The regime is close to normal S Sf 0 The derivative of the friction slope is Samuels P G 1989 p 575 3Ss_ g 103R 20P h T 3R h PO
37. n O Qn 6 A OA O The differential equation can be obtained 8 AA aes 3 12 and solved A Goerp F x 3 13 ro f 2 a To simplify the derivation 40 24 is assumed to be constant 3 13 be Where comes c germ 2 24 ex Bas Similarly to the backwater length the length of unsteadiness can be defined as the distance where 0 1 Therefore it me i 2 Q 48 3 9 Verification of unsteady flow spacing criteria To check if this formula is valid the simulations done previously will be done with different spacings The results will be compared and the error can be calculated assuming the results will the smaller spacing is the reference The simulation of Wallingford is done with only 20 spacings ie Ax 96 8ft The theoretical case form Hoy s Thesis is done with 50 and 25 spacings instead of 100 And all the simulations for other theoretical cases are done for spacings equal to Ax 500 1250 2500 5000m In all cases the error is calculated compared to the previously made simulations the ones with the smallest spacings a Or E R Qar Or ierminien An SRM simulation gives a solution that has the form Qij Yi j V The discharge Q and the water surface y elevation are given for each time step j and at each location 7 and the storage V is given for each time step j and for each reach subconduit 7 The ave
38. nt Venant equa tions works only for non pressurized flow In the case of pressurized flow the upstream water surface elevation and the volume of water can be calculated with another method To extend the HPG to pressurized flow the method developed by Zimmer A et al 2013 p937 was to use Darcy Weisbach equations Terry W Sturm 2001 p110 h _ f v L AR2q Where f can be expressed as Sgn f nl Ka R3 The friction loss hy gives the difference between the upstream and the down stream water surface elevation as a function of the discharge Therefore the upstream water surface elevation is given as a function of the downstream water surface elevation and the discharge and the volume can be calculated with the same method as seen previously 1 4 1 2 Storage Routing Method SRM For a given geometry of a river or a pipe network and two boundary conditions In the case of one reach two boundary conditions are necessary It can be two inflow outflow or stage hydrographs or a rating curve an unsteady flow calculation can be done SRM is one method of calculation of unsteady flow based on the Saint Venant equations under the quasi steady dynamic wave assumption It interpolates tables of values of discharges water surfaces elevations and volumes of water for each component of the system In the case of conduits reaches these tables are HPGs VPGs and are extended to pressurized flow when necessary
39. of the pipe 12 Depth Comparison at Station 255 7 Experimental data from the report by Ackers and Harrison 1964 Results with FEQ Results with SRM obtained by Matthew Hoy Recent results with SRM So Ww oN n w lt p a w 3 i u 2 250 Time sec Figure 2 3 Water surface elevation at 255 7 ft downstream of the entrance of the pipe 13 Depth Comparison at Station 483 Experimental data from the report by Ackers and Harrison 1964 Results with FEQ Results with SRM obtained by Matthew Hoy Recent results with SRM So Ww oN n w lt p a w 3 i u 2 250 Time sec Figure 2 4 Water surface elevation at 483 ft downstream of the entrance of the pipe 14 Depth Comparison at Station 710 2 Experimental data from the report by Ackers and Harrison 1964 Results with FEQ Results with SRM obtained by Matthew Hoy Recent results with SRM Ww oN n a u 3 i u i 2 Time sec Figure 2 5 Water surface elevation at 710 2 ft downstream of the entrance of the pipe 15 2 2 Theoretical Model from Matthew Hoy s Thesis In this case Matthew A Hoy 2005 the channel is a circular 5ft diameter pipe The inflow hydrograph is triangular and the
40. or links f At Qin Qout Avolume conservation equation Yup CRT momentum equation e Special case of conduits reaches f At n _ our AV PG Sit Sout f Yup HPG Ucina Q e The boundary conditions Q inflow outflow inflow or outflow hydrograph f f y stage stage hydrograph f rating y Q rating curve The method used to find the roots of this system is Newton Raphson The partial derivatives of these functions with respect to the discharges and the water surface elevations at every node need to be calculated Inflow Y za Inflow ott 18 017i 16 7 Weir eo i 115 ii 4 r a re ome 13 42 11 e i jInflow I v hie Weir 3 8 7 6i rg hl l Weir a m Se a m a am o4 3 2 1 v4 ee ete o Outflow Boundary Figure 1 2 Example of the division of a system Ki 24 OX ij The iteration of Newton Raphson method is PS F R AA Until F and Fprey are close enough Chapter 2 SOME RESULTS WITH SRM Before showing any work on accuracy and spacing criterion it is interesting to see some results given by the SRM method The first two cases that will be shown were tested by Matthey Hoy and are presented in his thesis Matthew A Hoy 2005 The third case will be a simplified version a canal with a more complex cross section and the last cases will be theoretical cases of pipes 2 1 Wallingford Hydraulic Research Station In the case of Wallingford Hydr
41. outfall is a free overfall here are the properties of the simulation Theoretical Model from Matthew Hoy s Thesis ane Manning s roughness coefficient 0 015 discharge cfs 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time sec Figure 2 6 Inflow hydrograph for Theoretical Model from Matthew Hoy s Thesis For routing in the SRM a time step of 3 s and a spacing of 100 ft are taken The results are compared his results obtained with FEQ and SRM The results for the water surface elevation and the discharge at the stations 0 2500 and 5000 ft downstream of the entrance of the pipe are compared In this case also the plots show that the results found with SRM are the same as the results found previously 16 Circular Channel Theoretical Reach Comparison Station 0 Results with FEQ Results with SRM obtained by Matthew Hoy time step 3s Results with SRM obtained by Matthew Hoy time step 15s Results with SRM obtained by Matthew Hoy time step 15s Recent results with SRM Water depth ft 2500 Time sec Figure 2 7 Water surface elevation at the entrance of the pipe I7 Circular Channel Theoretical Reach Comparison Station 2500 Results with FEQ Results with SRM obtained by Matthew Hoy time step 3s Results with SRM obtained by Matthew Hoy time step 15s Results with SRM obtained by Matthew Hoy
42. over the unsteadiness criteria is the Boussinesq coefficient is the spacing in m is the time step in s is a small perturbation to normal in m is the density of the material fluid inside the volume in kg m is the angle between the channel and the horizontal slope an gle is a small perturbation of the discharge to the normal regime in m s Chapter 1 INTRODUCTION Irregular precipitations as well as irregular use of water cause the flow in rivers and hydraulic infrastructure to be unsteady and non uniform Dur ing critical events heavy rainstorm for example it is useful to know some estimations of the discharge and the water surface elevation it must helps preventing overflows and floods A typical way to obtain these values is to use the Saint Venant equations and solve them numerically These equa tions are non linear and quite complex Thus is it often useful to make some assumptions to neglect some terms and simplify the calculation It may al low simpler computation and faster results without losing to much accuracy SRM storage routing method is a matlab code that applies this method under the quasi steady dynamic wave assumption It has been developed and enhanced by previous authors work in order to create a method for modeling these unsteady flows in a straightforward and efficient manner Matthew A Hoy 2005 The methods of calculation based on Saint Venant apply for a one dimensional flow an
43. ownstream end 34 _ v J a lt Q a cs 4000 Time sec Figure 2 27 Second Inflow hydrograph station O Discharge cfs station 12500 station 25000 400 Time sec Figure 2 28 Discharge at several station for S 2 x 10 with the first hydrograph 30 station O station 12500 station 25000 _ p _ e Ez gt A c8 Ya Y T 7 w 3 400 Time sec Figure 2 29 Water surface elevation at several station for S 2 x 107 with the first hydrograph 36 station O _ vi O es n QO a A station 12500 station 25000 4000 Time sec Figure 2 30 Discharge at several station for S 2 x 107 with the second hydrograph 37 N N station O station 12500 station 25000 co z 2 pm gt 2 rT o 20 Q nv g co 4000 Time sec Figure 2 31 Water surface elevation at several station for S 2 x 10 with the second hydrograph 38 S S station O n vi O kea s J ofa Q a A station 12500 station 25000 4
44. rage discharge for each reach subconduit is 1 Qij 5 Og FO Using 3 11 and noticing that Q A x Az the ratio of the spacing over the spacing criterion a can be calculated Ax O OV Ss ae ee i Ag ae ae This equation can be discretized and applied to the results given by SRM to check if the chosen spacing is fine AQ Ax l a E Ya E Vig 7 i 3 17 Azo Qij Oi At At The ratio is calculated for each point and each time step 3 3 1 Graphs showing error vs spacing unsteadiness criteria This error is plotted versus how often the ration given by 3 17 is higher than 1 O Wallingford Matthew Hoy s theoretical case O first slope first hydrograh Ofirst slope second hydrograh Osecond slope first hydrograh second slope second hydrograh 0 1 Frequency gt 1 Figure 3 1 the error vs the frequency of ratio higher than 1 The error seems to be related to the frequency of points location and time where the ratio os higher than 1 However it stays lower than 0 01 even if 50 the frequency is close to 1 Therefore it must be interesting to see how it can be related to the frequency of points where the ratio os higher than other numbers O Wallingford Matthew Hoy s theoretical case O first slope first hydrograh O first slope second hydrograh O second slope first hydrograh second slope second hydrograh 0 01 Frequency gt 10 Figure
45. snodenum 23 end 24 if strcmp outletnodei i interceptor storage k name ee downnode interceptor storage k usnodenum 25 end 27 26 end Figure A 18 How to use the file containing HPGs VPGs in SRM code 18 B Other type of boundary conditions The equations for the boundary conditions can be changed in the files Opti mizeSection and OptimizeSectionSteady In both files the boundary condi tions are defined in the subfunctions FindFunctions wich defines the func tions for the Newton Raphson method and FindJacobian wich defines the derivatives of the functions the Jacobian for Newton Raphson the boundary conditions are here for i 1 NumNodes if strcmp interceptor node i type inflow eqcount eqcount 1 F eqcount 1 QO i currentflow i end if strcmp interceptor node i type outfall eqcount eqcount 1 F eqcount 1 y i ycritical i end end end Figure A 19 the boundary conditions in the code the derivatives of the boundary conditions are here for i 1i NumNodes if strcmp interceptor node i type inflow eqcount eqcount 1 aFdQ eqcount i 1 end if strcmp interceptor node i type outfall eqcount eqcount 1 dFd eqcount i NumNodes 1 adFdQ eqcount i dycrit i end end Figure A 20 the boundary conditions in the code In the case of the functions for Newton Raphson all the values are stored in the
46. the length of reach affected by unsteadiness and thus another spacing criterion The backwater length is already taken into account in the HPG the spacing criterion associated to this length does not have to be taken into account when dividing the system into parts The criterion that applies for this is the one linked to unsteadiness 1 1 Description and development of HPG and VPG 1 1 1 Saint Venant equations Saint Venant equations are a special case of the momentum 1 la and the conservation 1 1b equations Ven Te Chow et Al 1988 p272 o i ff eav mas aay SF a M eoav gp oas 1 1b It applies under the following assumptions e The flow is 1 dimensionnal only the variations in the flow wise direction are considered e The pressure is assumed to be hydrostatic in the entire channel the vertical acceleration are neglected e The channel can be considered as a straight line e The geometry of the channel is fixed no effect of scour and deposition e The slope is small e The fluid is incompressible Applying the previous assumptions the momentum and the conservation equations become respectively the first 1 2a and the second 1 2b Saint Venant equations aQ oa Oh Rtg tos Z 5 ei vom o OA bet tn ane With The Boussinesq coefficient defined by the following formula B ff aa And the friction slope 1 1 2 Gradually varied flow calculation for steady flow In the case of steady
47. time step 15s Recent results with SRM Water depth ft 2500 Time sec Figure 2 8 Water surface elevation at 2500 ft downstream of the entrance of the pipe 18 Circular Channel Theoretical Reach Comparison Station 5000 Results with FEQ Results with SRM obtained by Matthew Hoy time step 3s Results with SRM obtained by Matthew Hoy time step 15s Results with SRM obtained by Matthew Hoy time step 15s Recent results with SRM Water depth ft 2500 Time sec Figure 2 9 Water surface elevation at 5000 ft downstream of the entrance of the pipe 19 Circular Channel Theoretical Reach Comparison Station 2500 Results with FEQ Results with SRM obtained by Matthew Hoy time step 3s Results with SRM obtained by Matthew Hoy time step 15s Results with SRM obtained by Matthew Hoy time step 15s Recent results with SRM discharge cfs 2500 Time sec Figure 2 10 Discharge at 2500 ft downstream of the entrance of the pipe 20 Circular Channel Theoretical Reach Comparison Station 5000 Results with FEQ Results with SRM obtained by Matthew Hoy time step 3s Results with SRM obtained by Matthew Hoy time step 15s Results with SRM obtained by Matthew Hoy time step 15s Recent results with SRM discharge cfs
48. ulation The function Simplifyglobal calculate the discharge at each node of the system for steady flow using the first value of the given hydrograph s The values of the dicharge in all the nodes are used in the function Op timizeSectionSteady as a first guess for the Newton Raphson method for convergence for steady flow Simplifyglobal IterativeMethod Makedecisions e Initialize the discharge in all nodes Newton Raphson is aplied to the system for steady flow getPartialDerivsTriSteady suppressNAN OptimizeSectionSteady Figure A 4 initialization steady flow calculation The equations for Newton Raphson At each node there are two unknowns the discharge and the water surface elevation Each link gives two equations one is linked to the conservation of mass the other one to the momentum equation Each junctions gives as many equations as it has nodes Therefore this becomes a system of unknowns and equations There are usually more unknowns than equations To complete the number of equations boundary conditions are necessary The equations are Zimmer A et al 2013 p947 P fi e for junctions f y yy Wig a gt Om p gt Qan 69 e for links TS Omn m Con conservation equation f Yup f O A Q momentum equation e Special case of conduits reaches f Qin Qout J Yup H PG Uar Q e The boundary conditions Q inflow outflo
49. variable F The size of the vector is 1 n where n is two times the number of nodes in the system one time for the water surface elevation and one time for the discharge The values for the boundary conditions are the last values of this vector In the case of the Jacobian all the values are stored in the variable dFdQ The size of the matrix is n n The values for the boundary conditions are in the last lines of this matrix T9 C A calculation under other assumption As explained in the introduction SRM is based on the quasi steady dynamic wave assumption In the files OptimizeSection and OptimizeSectionSteady in the functions this assumption can be changed the functions for Newton Raphson and their derivatives the Jacobian can be changed S0
50. w inflow or outflow hydrograph f f y stage stage hydrograph f rating y Q rating curve The method used to find the roots of this system is Newton Raphson The partial derivatives of these functions with respect to the discharges and the water surface elevations at every node need to be calculated AF 24 Ox ij The iteration of Newton Raphson method is FeFe AT HA Until F and Fprey are close enough The equations F for Newton Raphson are set by the subfunction FindFunc tions and the derivatives AF by the subfunction FindJacobian 66 C Unsteady flow calculation The same type of calculation as in the previous part is done Under the unsteady flow assumption the equations are slightly changed Zimmer A et al 2013 p947 Fa fi for junctions f y y WiFg I Oh _ 2 O e for links f At Qin Qout Avolume conservation equation Yup Visi momentum equation Special case of conduits reaches f At i _ Ovi AVPG Sint Sout f Yup HPG Cans Q The boundary conditions f Q inflow outflow inflow or outflow hydrograph f y stage stage hydrograph f rating y Q rating curve The method of convergence used here is also Newton Raphson In the case of unsteady flow the time step has to be taken into account It influences the convergence When the time step is not precise enough there might be no convergence Therefore the code contains a
51. xmodel xdata xdata is less than 10 in all the cases In the last part a study is made on how the spacing used for the numerical calculation criteria can affect the accuracy il ACKNOWLEDGMENTS I would first like to thank Marcelo H Garcia for suggesting and guiding the research presented in this thesis and for his support throughout the duration of my education at the University of Illinois I would also like to thank Blake J Landry for his guidance his time thoughtful discussions and careful re view of the work included in this thesis Special thanks is due to Nils Oberg who developed the code used to gener ate the Hydraulic Performance Graphs and Volumetric Performance Graphs with Arturo Leon and developed the tool to generate the HPGs and VPGs using HEC RAS geometry with Andrew R Waratuke I would like to thank the numerous officemates and research assistants with whom I have had the pleasure of working with at the Ven Te Chow Hydrosys tems Laboratory at the University of Illinois Finally I would like to thank my friends and family for their support and their encouragement ill CONTENTS lS PSOE PIG URS pappa e dak e n e D eei B N aop ed DOS TCTOPFSYMBOL SD z 4444 44 44 64 oe bebo SEARS SES Chapterd INTRODUCTION weese as Gk eee eee he Se BA 1 1 E2 Description and development of HPG and VPG Storage Routing Method SRM Chapter 2 SOME RESULTS WITH SRM 0 02 ZA
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