NOTEThis workbook and software are provided for free as part of the e-book: 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS, by Gary Parker.

Neither I nor any university in which I am in the employ accepts responsibility or liability for its use by third parties.

This workbook and software are provided for free as part of the e-book: 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS,

Neither I nor any university in which I am in the employ accepts

Backwater Curve Calculator: Case of Constant Cz

For a given reach of river, specifyS 0.00025 bed slopeCz 22 Chezy Friction Coefficient

5.7 water discharge/width

Parameters at Normal Flow

3.014 m

1.891 m/s

0.348

7.391Parameters at Critical Flow

1.490597 m

3.823971 m/s

Calculations with Predictor-Corrector Scheme

0.00 m starting point

30.00 m starting depth1000.00 m step length

0.00207 friction coefficient

distance depth velocity bed shear stresm m/s N/m^2

x H dH/dxpred Hpred dH/dxcorr er U0.00 30 0.00025 29.750222799 0.00025 1.634856 0.19 0.0745867769

-1000.00 29.75023 0.00025 29.500454088 0.00025 1.632072 0.191595 0.0758444528-2000.00 29.50046 0.00025 29.250691326 0.00025 1.629246 0.193217 0.0771341791-3000.00 29.25069 0.00025 29.000934719 0.00025 1.626376 0.194867 0.0784570528-4000.00 29.00094 0.00025 28.751184482 0.00025 1.623461 0.196545 0.0798142183-5000.00 28.75119 0.00025 28.501440839 0.00025 1.6205 0.198253 0.0812068695-6000.00 28.50144 0.00025 28.251704024 0.00025 1.617493 0.19999 0.0826362529-7000.00 28.25171 0.00025 28.00197428 0.00025 1.614438 0.201758 0.08410367

qw m2/s

Hn

Un

Frn

tbn Newtons/m2

Hc

Uc

x1

H1

Dx

Cf

tb

-8000.00 28.00198 0.00025 27.752251865 0.00025 1.611333 0.203557 0.0856104802-9000.00 27.75226 0.00025 27.502537044 0.00025 1.608179 0.205389 0.0871581044

-10000.00 27.50254 0.00025 27.252830097 0.00025 1.604972 0.207254 0.0887480278-11000.00 27.25283 0.00025 27.003131317 0.00025 1.601713 0.209153 0.0903818038-12000.00 27.00314 0.00025 26.753441009 0.00025 1.5984 0.211087 0.0920610574-13000.00 26.75345 0.00025 26.503759494 0.00025 1.595031 0.213057 0.0937874898-14000.00 26.50376 0.00025 26.254087107 0.00025 1.591606 0.215064 0.095562882-15000.00 26.25409 0.00025 26.004424201 0.00025 1.588122 0.217109 0.0973890999-16000.00 26.00443 0.00025 25.754771144 0.00025 1.584578 0.219193 0.0992680988-17000.00 25.75478 0.00025 25.505128324 0.00025 1.580973 0.221318 0.1012019289-18000.00 25.50513 0.00025 25.255496146 0.00025 1.577305 0.223484 0.1031927404-19000.00 25.2555 0.00025 25.005875038 0.00025 1.573573 0.225693 0.1052427902-20000.00 25.00588 0.00025 24.756265448 0.00025 1.569774 0.227946 0.1073544473-21000.00 24.75627 0.00025 24.506667845 0.00025 1.565907 0.230245 0.1095302003-22000.00 24.50667 0.00025 24.257082727 0.00025 1.56197 0.23259 0.1117726644-23000.00 24.25709 0.00025 24.007510613 0.00025 1.557961 0.234983 0.1140845892-24000.00 24.00752 0.00025 23.757952053 0.00025 1.553878 0.237426 0.1164688669-25000.00 23.75796 0.00025 23.508407625 0.00025 1.549719 0.23992 0.1189285417-26000.00 23.50841 0.00025 23.258877936 0.00025 1.545483 0.242466 0.1214668189-27000.00 23.25889 0.00025 23.009363629 0.00025 1.541166 0.245068 0.1240870755-28000.00 23.00937 0.00025 22.759865381 0.000249 1.536766 0.247725 0.1267928716-29000.00 22.75987 0.000249 22.510383908 0.000249 1.532282 0.250441 0.1295879615-30000.00 22.51039 0.000249 22.260919962 0.000249 1.52771 0.253216 0.1324763076-31000.00 22.26093 0.000249 22.011474344 0.000249 1.523049 0.256054 0.1354620936-32000.00 22.01148 0.000249 21.762047895 0.000249 1.518295 0.258956 0.1385497396-33000.00 21.76206 0.000249 21.51264151 0.000249 1.513445 0.261924 0.1417439184-34000.00 21.51265 0.000249 21.263256132 0.000249 1.508498 0.26496 0.145049573-35000.00 21.26327 0.000249 21.013892763 0.000249 1.503449 0.268068 0.1484719351-36000.00 21.0139 0.000249 20.764552464 0.000249 1.498295 0.271249 0.1520165459-37000.00 20.76456 0.000249 20.515236363 0.000249 1.493034 0.274506 0.155689278-38000.00 20.51525 0.000249 20.265945653 0.000249 1.487663 0.277842 0.1594963594-39000.00 20.26596 0.000249 20.016681605 0.000249 1.482176 0.28126 0.1634443996-40000.00 20.0167 0.000249 19.767445569 0.000249 1.476572 0.284762 0.1675404177-41000.00 19.76746 0.000249 19.518238982 0.000249 1.470845 0.288353 0.171791873-42000.00 19.51825 0.000249 19.269063374 0.000249 1.464993 0.292034 0.1762066986-43000.00 19.26908 0.000249 19.019920373 0.000249 1.45901 0.295811 0.1807933376-44000.00 19.01994 0.000249 18.770811719 0.000249 1.452893 0.299686 0.1855607827-45000.00 18.77083 0.000249 18.521739267 0.000249 1.446637 0.303663 0.1905186194-46000.00 18.52176 0.000249 18.272704997 0.000249 1.440237 0.307746 0.1956770734-47000.00 18.27272 0.000249 18.02371103 0.000249 1.433688 0.31194 0.2010470618-48000.00 18.02373 0.000249 17.774759632 0.000249 1.426985 0.31625 0.20664025-49000.00 17.77478 0.000249 17.525853231 0.000249 1.420123 0.320679 0.2124691135-50000.00 17.52588 0.000249 17.27699443 0.000249 1.413096 0.325233 0.2185470056-51000.00 17.27702 0.000249 17.028186024 0.000249 1.405898 0.329918 0.2248882321-52000.00 17.02821 0.000249 16.779431012 0.000249 1.398522 0.334739 0.2315081331-53000.00 16.77946 0.000249 16.53073262 0.000249 1.390963 0.339701 0.2384231737-54000.00 16.53076 0.000249 16.282094322 0.000249 1.383213 0.344812 0.2456510428-55000.00 16.28213 0.000249 16.033519859 0.000249 1.375266 0.350077 0.2532107633-56000.00 16.03355 0.000249 15.785013266 0.000248 1.367113 0.355504 0.2611228131

-57000.00 15.78505 0.000248 15.536578901 0.000248 1.358746 0.361101 0.2694092592-58000.00 15.53662 0.000248 15.288221475 0.000248 1.350159 0.366875 0.2780939062-59000.00 15.28826 0.000248 15.039946089 0.000248 1.34134 0.372835 0.2872024613-60000.00 15.03999 0.000248 14.791758269 0.000248 1.332283 0.37899 0.2967627169-61000.00 14.7918 0.000248 14.543664012 0.000248 1.322975 0.385349 0.3068047543-62000.00 14.54371 0.000248 14.295669833 0.000248 1.313409 0.391922 0.3173611699-63000.00 14.29572 0.000248 14.047782821 0.000248 1.303572 0.398721 0.3284673285-64000.00 14.04784 0.000248 13.800010698 0.000248 1.293454 0.405756 0.3401616449-65000.00 13.80007 0.000248 13.552361889 0.000248 1.283042 0.413041 0.3524859-66000.00 13.55243 0.000248 13.3048456 0.000247 1.272324 0.420589 0.3654855939-67000.00 13.30491 0.000247 13.057471902 0.000247 1.261288 0.428413 0.3792103425-68000.00 13.05755 0.000247 12.810251834 0.000247 1.249917 0.436529 0.3937143231-69000.00 12.81033 0.000247 12.563197512 0.000247 1.238199 0.444953 0.4090567754-70000.00 12.56328 0.000247 12.316322258 0.000247 1.226117 0.453703 0.4253025658-71000.00 12.31642 0.000247 12.069640744 0.000247 1.213655 0.462797 0.4425228244-72000.00 12.06974 0.000247 11.823169156 0.000246 1.200795 0.472255 0.4607956649-73000.00 11.82328 0.000246 11.576925386 0.000246 1.187519 0.4821 0.4802069988-74000.00 11.57704 0.000246 11.330929245 0.000246 1.173807 0.492354 0.5008514577-75000.00 11.33106 0.000246 11.085202713 0.000246 1.15964 0.503042 0.5228334405-76000.00 11.08534 0.000246 10.839770223 0.000245 1.144994 0.514192 0.5462683021-77000.00 10.83992 0.000245 10.594658994 0.000245 1.129848 0.525834 0.5712837058-78000.00 10.59483 0.000245 10.349899406 0.000245 1.114176 0.537998 0.5980211622-79000.00 10.35008 0.000245 10.105525447 0.000244 1.097953 0.55072 0.6266377829-80000.00 10.10573 0.000244 9.8615752219 0.000244 1.081152 0.564037 0.6573082794-81000.00 9.861799 0.000244 9.6180915511 0.000243 1.063744 0.577988 0.6902272425-82000.00 9.618338 0.000243 9.3751226639 0.000243 1.045699 0.592618 0.7256117421-83000.00 9.375395 0.000243 9.1327230137 0.000242 1.026986 0.607974 0.7637042889-84000.00 9.133024 0.000242 8.890954231 0.000241 1.007571 0.624109 0.8047762085-85000.00 8.891289 0.000241 8.6498862428 0.000241 0.987419 0.641077 0.8491314745-86000.00 8.650258 0.000241 8.4095985896 0.00024 0.966494 0.65894 0.8971110536-87000.00 8.410013 0.00024 8.170181977 0.000239 0.944759 0.677763 0.9490978101-88000.00 8.170646 0.000239 7.931740107 0.000238 0.922175 0.697619 1.0055220092-89000.00 7.93226 0.000238 7.6943918405 0.000237 0.898702 0.718585 1.0668674433-90000.00 7.694976 0.000237 7.4582737528 0.000235 0.874299 0.740743 1.1336781706-91000.00 7.458932 0.000235 7.2235431523 0.000234 0.848927 0.764184 1.2065658068-92000.00 7.224287 0.000234 6.9903816427 0.000232 0.822545 0.789005 1.286217225-93000.00 6.991225 0.000232 6.7589993155 0.00023 0.795116 0.815308 1.3734023861-94000.00 6.759957 0.00023 6.5296396655 0.000228 0.766606 0.843201 1.4689818277-95000.00 6.530731 0.000228 6.3025853168 0.000226 0.736984 0.872797 1.5739130438-96000.00 6.303832 0.000226 6.0781646305 0.000223 0.706227 0.904212 1.6892545589-97000.00 6.079592 0.000223 5.8567592245 0.00022 0.674323 0.937563 1.8161659033-98000.00 5.858396 0.00022 5.638812355 0.000216 0.641271 0.972962 1.9559008635-99000.00 5.640694 0.000216 5.4248379726 0.000212 0.607092 1.010514 2.1097902953

-100000.00 5.427003 0.000212 5.2154300368 0.000207 0.571828 1.050303 2.2792094313-101000.00 5.217923 0.000207 5.0112713173 0.000201 0.535553 1.092389 2.4655230909-102000.00 5.01414 0.000201 4.8131403968 0.000194 0.498379 1.136785 2.6700007832-103000.00 4.816438 0.000195 4.6219148784 0.000187 0.460465 1.183447 2.8936929925-104000.00 4.625693 0.000187 4.438567923 0.000179 0.422025 1.232248 3.1372610846-105000.00 4.442877 0.000179 4.2641543164 0.000169 0.383337 1.282952 3.4007580031

-106000.00 4.269034 0.000169 4.0997815943 0.000158 0.344746 1.335197 3.683367494-107000.00 4.105254 0.000159 3.9465619168 0.000147 0.306665 1.388465 3.9831278818-108000.00 3.95262 0.000147 3.8055422028 0.000134 0.269563 1.442081 4.2966919843-109000.00 3.812139 0.000135 3.6776143312 0.00012 0.23395 1.495224 4.6192017268-110000.00 3.684654 0.000121 3.5634141588 0.000107 0.200337 1.546957 4.9443704769-111000.00 3.570748 0.000108 3.4632262312 9.26E-05 0.169205 1.596304 5.2648481125-112000.00 3.470663 9.37E-05 3.3769167936 7.91E-05 0.140947 1.642338 5.5728785801-113000.00 3.384235 8.03E-05 3.3039163985 6.64E-05 0.115833 1.68428 5.8611562415-114000.00 3.310896 6.76E-05 3.2432625879 5.47E-05 0.093985 1.721588 6.1236892029-115000.00 3.249713 5.6E-05 3.1936957788 4.45E-05 0.075366 1.754001 6.3564452091-116000.00 3.199477 4.57E-05 3.1537856702 3.56E-05 0.059805 1.781541 6.5576204962-117000.00 3.158822 3.68E-05 3.1220591472 2.82E-05 0.047026 1.80447 6.7275050223-118000.00 3.126335 2.92E-05 3.0971057699 2.21E-05 0.036692 1.823221 6.8680461287-119000.00 3.100657 2.3E-05 3.0776488177 1.72E-05 0.028447 1.83832 6.9822760621-120000.00 3.080542 1.8E-05 3.0625816483 1.33E-05 0.02194 1.850324 7.0737560816-121000.00 3.064902 1.39E-05 3.050976458 1.02E-05 0.01685 1.859766 7.1461323117-122000.00 3.052815 1.07E-05 3.0420748646 7.86E-06 0.012899 1.867129 7.2028358265-123000.00 3.043516 8.25E-06 3.0352687446 6.01E-06 0.009848 1.872834 7.2469156043

Backwater Curve Calculator: Case of Constant Cz

Chezy Friction Coefficient

Input cell

Output Cell

Plots of results given on succeeding worksheets

bed elev water surf elevm m

h x0 30

0.25 30.000230.5 30.00046

0.75 30.000691 30.00094

1.25 30.001191.5 30.00144

1.75 30.00171

2 30.001982.25 30.00226

2.5 30.002542.75 30.00283

3 30.003143.25 30.00345

3.5 30.003763.75 30.00409

4 30.004434.25 30.00478

4.5 30.005134.75 30.0055

5 30.005885.25 30.00627

5.5 30.006675.75 30.00709

6 30.007526.25 30.00796

6.5 30.008416.75 30.00889

7 30.009377.25 30.00987

7.5 30.010397.75 30.01093

8 30.011488.25 30.01206

8.5 30.012658.75 30.01327

9 30.01399.25 30.01456

9.5 30.015259.75 30.01596

10 30.016710.25 30.01746

10.5 30.0182510.75 30.01908

11 30.0199411.25 30.02083

11.5 30.0217611.75 30.02272

12 30.0237312.25 30.02478

12.5 30.0258812.75 30.02702

13 30.0282113.25 30.02946

13.5 30.0307613.75 30.03213

14 30.03355

14.25 30.0350514.5 30.03662

14.75 30.0382615 30.03999

15.25 30.041815.5 30.04371

15.75 30.0457216 30.04784

16.25 30.0500716.5 30.05243

16.75 30.0549117 30.05755

17.25 30.0603317.5 30.06328

17.75 30.0664218 30.06974

18.25 30.0732818.5 30.07704

18.75 30.0810619 30.08534

19.25 30.0899219.5 30.09483

19.75 30.1000820 30.10573

20.25 30.111820.5 30.11834

20.75 30.1253921 30.13302

21.25 30.1412921.5 30.15026

21.75 30.1600122 30.17065

22.25 30.1822622.5 30.19498

22.75 30.2089323 30.22429

23.25 30.2412223.5 30.25996

23.75 30.2807324 30.30383

24.25 30.3295924.5 30.3584

24.75 30.3906925 30.427

25.25 30.4679225.5 30.51414

25.75 30.5664426 30.62569

26.25 30.69288

26.5 30.7690326.75 30.85525

27 30.9526227.25 31.06214

27.5 31.1846527.75 31.32075

28 31.4706628.25 31.63424

28.5 31.810928.75 31.99971

29 32.1994829.25 32.40882

29.5 32.6263429.75 32.85066

30 33.0805430.25 33.3149

30.5 33.5528130.75 33.79352

-140000 -120000 -100000 -80000 -60000 -40000 -20000 00

5

10

15

20

25

30

35

HUtb

x, m

H (

m),

tb

(N

/m2

), U

(m

/s)

-140000 -120000 -100000 -80000 -60000 -40000 -20000 00

5

10

15

20

25

30

35

40

hx

x m

be

d (

h)

an

d w

ate

r s

urf

ac

e (

x)

ele

va

tio

ns

m

Formulation for mild backwater curves:Integrate upstream if starting depth is greater than critical depth.

slope, water discharge per unit width and friction coefficient, respectively.

Grid:

Note as n increases we march upstream

Finite-difference approximation:

Euler scheme

Predictor-corrector scheme

Here H1 is the starting depth and S, qw and Cf, all taken to be constant here, are the bed

dHdx

=S−S f1−Fr2

=F (H ) F (H )=S−C f

qw2

gH 3

1−qw2

gH3

H (x1 )=H 1

xn+1=xn−Δx

dHdx

≃H n−Hn+1

dx

H pred=Hn−F (Hn )ΔxHn+1=Hn−0 .5[F (Hn )+F (H pred )] Δx

Hn+1=Hn−F (Hn )Δx