Muskingum Routing: Outflow at the End of the Step

Muskingum routing moves a flood one time step down a river reach. Given the flow entering the upstream end at the start and at the end of the step, and the flow leaving the downstream end at the start, it finds what leaves at the end: a weighted sum of those three flows, $O_2 = C_1 I_1 + C_2 I_2 + C_3 O_1$. Applied step after step along an inflow hydrograph, it carries the whole flood down the reach, delaying and flattening the peak. The equation and weights follow the Natural Resources Conservation Service's National Engineering Handbook, Part 630, chapter 17: the storage relation $S = K[XI + (1 - X)O]$ with continuity gives weights built from $\Delta t$, $K$ and $X$.

$K$ is about the time the flood wave takes to travel through the reach, so $\Delta t/K$ compares the step with that travel time. $X$ sets how much the reach flattens the wave: 0 gives the most attenuation and 0.5 gives none, though Muskingum-Cunge can give a negative $X$. The weights add up to 1, so a steady flow passes through unchanged: 100 in at both ends of the step and 100 out at the start gives 100 out. A weight can even be negative: Example 17-5, with $X = -0.590$, has $C_1 = -0.187$.

The page's example is a row of the handbook's Example 17-4: a 2,600 ft reach with $X = -0.254$, $K = 605.07$ s and a time step of 0.1 h, giving weights of 0.028, 0.355 and 0.617. At 13.1 h the inflows are 672.05 and 645.30 ft³/s and the earlier outflow is 644.77, and the routed outflow is 645.72 ft³/s, the peak of the outflow. Notice where the weight sits: most on the outflow already established, a good part on the new inflow, very little on the old inflow. That weighting is how the reach damps and delays the peak.

Run backwards, the storage constant that turns the 13.1 h row into its outflow is 605.4 s against the 605.07 s the handbook computed, 0.06 percent apart, which is only the rounding of the printed values. The routing is only as good as the $K$ and $X$ you give it: with a gauge record you can calibrate them from the storage relation, and without one the Muskingum-Cunge weighting factor takes them from the channel's own hydraulics. When a reach is long, Example 17-5 cuts a floodplain reach into two 1,300 ft routing steps.