CE322 Basic Hydrology
Jorge A. Ramirez
Muskingum Routing - Example

The inflow and outflow hydrographs of a river reach are tabulated below.

 Time (h) Inflow (m3/s) Outflow (m3/s) 1 93 85 2 137 91 3 208 114 4 320 159 5 442 233 6 546 324 7 630 420 8 678 509 9 691 578 10 675 623 11 634 642 12 571 635 13 477 603 14 390 546 15 329 479 16 247 413 17 184 341 18 134 274 19 108 215 20 90 170
The Muskingum routing procedure is used for systems that have Storage - Discharge relationships that are hysteretic. That is, for systems for which the outflow is not a unique function of storage. The S vs. O relationship for the river reach under consideration is graphed below.

A. Parameter Estimation

Use these observations to obtain the Muskingum routing parameters k and x for this river reach. The initial storage in the system is 715,000 m3.

Graphical Procedure:

The graphical procedure consists in generating graphs of [xI + (1-x)O] vs. S for different values of x, arbitrarily selected such that 0 < x < 0.5. The optimal value of x is selected as that which produces the narrowest and straightest loop graph of [xI + (1-x)O] vs. S. The slope of the least squares linear fit to the resulting points is the estimate of k.

a) Generate accumulated storage in the system. Use continuity equation as follows:

 1 2 3 4 5 6 7 8 9 x=0.25 x=0.35 x=0.1 x=0.15 Inflow, I (m3/s) Outflow,O (m3/s) Ave. Inflow (m3/s) Ave. Outflow (m3/s) Storage (m3) Weighted Average Flux xI + (1-x)O (m3/s) 93 85 715000 87 87.8 85.8 86.21204 137 91 115 88 812200 102.5 107.1 95.6 97.96922 208 114 172.5 102.5 1064200 137.5 146.9 123.4 128.2414 320 159 264 136.5 1523200 199.25 215.35 175.1 183.3923 442 233 381 196 2189200 285.25 306.15 253.9 264.6645 546 324 494 278.5 2965000 379.5 401.7 346.2 357.634 630 420 588 372 3742600 472.5 493.5 441 451.816 678 509 654 464.5 4424800 551.25 568.15 525.9 534.6043 691 578 684.5 543.5 4932400 606.25 617.55 589.3 595.1201 675 623 683 600.5 5229400 636 641.2 628.2 630.8782 634 642 654.5 632.5 5308600 640 639.2 641.2 640.788 571 635 602.5 638.5 5179000 619 612.6 628.6 625.3037 477 603 524 619 4837000 571.5 558.9 590.4 583.9104 390 546 433.5 574.5 4329400 507 491.4 530.4 522.3653 329 479 359.5 512.5 3778600 441.5 426.5 464 456.2743 247 413 288 446 3209800 371.5 354.9 396.4 387.8502 184 341 215.5 377 2628400 301.75 286.05 325.3 317.2138 134 274 159 307.5 2093800 239 225 260 252.7894 108 215 121 244.5 1649200 188.25 177.55 204.3 198.789 90 170 99 192.5 1312600 150 142 162 157.8796
Columns 1 & 2 are given.

Columns 3 & 4 are the average inflow flux (Ii+1 + Ii)/2 and outflow flux (Oi+1 + Oi)/2, respectively.

Column 5 is the cumulative storage in the system obtained using the continuity equation below.

Columns 6 - 9 are the values of the weighted average flux [xI + (1-x)O] for different values of x. The graph of Columns 6 - 9 vs. Column 5 is shown below.

Based on these results, a value of x = 0.15 is selected. The best least squares fit to the corresponding points yields a value of k = 2.3 h.

Least Squares Procedure

 Inflow (m3/s) Outflow (m3/s) Storage O2 (m3/s)2 I2 (m3/s)2 OI (m3/s)2 SO (m6/s) SI (m6/s) 93 85 715000 7225 8649 7905 60775000 66495000 137 91 812200 8281 18769 12467 73910200 111271400 208 114 1064200 12996 43264 23712 121318800 221353600 320 159 1523200 25281 102400 50880 242188800 487424000 442 233 2189200 54289 195364 102986 510083600 967626400 546 324 2965000 104976 298116 176904 960660000 1618890000 630 420 3742600 176400 396900 264600 1571892000 2357838000 678 509 4424800 259081 459684 345102 2252223200 3000014400 691 578 4932400 334084 477481 399398 2850927200 3408288400 675 623 5229400 388129 455625 420525 3257916200 3529845000 634 642 5308600 412164 401956 407028 3408121200 3365652400 571 635 5179000 403225 326041 362585 3288665000 2957209000 477 603 4837000 363609 227529 287631 2916711000 2307249000 390 546 4329400 298116 152100 212940 2363852400 1688466000 329 479 3778600 229441 108241 157591 1809949400 1243159400 247 413 3209800 170569 61009 102011 1325647400 792820600 184 341 2628400 116281 33856 62744 896284400 483625600 134 274 2093800 75076 17956 36716 573701200 280569200 108 215 1649200 46225 11664 23220 354578000 178113600 90 170 1312600 28900 8100 15300 223142000 118134000 SO2 = 3514348 SI2 = 3804704 SIO = 3472245 SSO = 29062547000 SSI = 29184045000

Using the above equations yields:

A = 1255.626164 s

B = 7029.100513 s

k = A+B = 8284.726677 s = 2.3 h

x = A/(A + B) = 0.151559154

Observe that these results for k and x are the same as those of the graphical procedure. For comparison purposes, the observed outflow hydrograph and that predicted using the estimated values of k and x are graphed below.

B. Muskingum Routing

Use the Muskingum routing procedure to route the hydrograph tabulated below through the same river reach of Part A

Select a Dt = 1 h, as suggested by the inflow data. However, check that with the selected Dt, parameter values meet restrictions:

x < 0.5 Dt/k < 1 - x

For this case: 0.1515 < (0.5) (3600)/8284.73 < 1 - 0.1515 Thus, OK. Proceed with routing, by obtaining Co, C1, and C2.

This yields: Co = 0.061787; C1 = 0.346074; and C2 = 0.592139. Using these values in the Muskingum routing equation:

obtain the outflow hydrograph as tabulated below. The resulting hydrographs are also graphed below.
.

 Time (h) Inflow (m3/s) Co x Ii+1 (m3/s) C1 x Ii (m3/s) C2 x Oi (m3/s) Outflow (m3/s) 1 50 50 2 100 6.1787 17.3037 29.60695 53.08935 3 200 12.3574 34.6074 31.43627 78.40107 4 325 20.08078 69.2148 46.42433 135.7199 5 450 27.80415 112.4741 80.36505 220.6433 6 600 37.0722 155.7333 130.6515 323.457 7 700 43.2509 207.6444 191.5315 442.4268 8 780 48.19386 242.2518 261.9782 552.4238 9 790 48.81173 269.9377 327.1117 645.8611 10 775 47.88493 273.3985 382.4396 703.723 11 750 46.34025 268.2074 416.7018 731.2494 12 680 42.01516 259.5555 433.0013 734.572 13 590 36.45433 235.3303 434.9687 706.7534 14 500 30.8935 204.1837 418.4962 653.5734 15 420 25.95054 173.037 387.0063 585.9938 16 350 21.62545 145.3511 346.9898 513.9663 17 300 18.5361 121.1259 304.3395 444.0015 18 250 15.44675 103.8222 262.9106 382.1796 19 225 13.90208 86.5185 226.3034 326.724 20 200 12.3574 77.86665 193.466 283.6901