Flow Routing Flow routing is a procedure to determine the time and magnitude of flow (i.e. the flow hydrograph) at a point on a watercourse from known or assumed hydrographs at one or more points upstream. We want to predict what a downstream hydrograph would be if we know the upstream hydrograph If the flow is a flood, the procedure is specifically called flood routing Lumped flow routing: flow is calculated as a function of time alone at a particular location. Also called hydrologic routing. Distributed flow routing: flow is calculated as a function of space and time throughout the space. Also called hydraulic routing.

Hydrologic Routing Continuity equation: It is not enough to know inflow hydrograph, I(t) to solve for the outflow hydrograph; Q(t) as S(t) are also unknown. A second relationship or storage function is needed: S = f (I,Q) In reservoir routing level pool routing is commonly employed where S=f(Q). Q and S are related to reservoir water level, h Runge-Kutta method is an alternative to level pool routing. It is a little bit more complicated, but does not require special computation of S=f(Q) and is more physics based In channel flow routing Muskingum Method is commonly used where S is linearly related to I and Q.

Flow Routing The relationship between outflow and storage could be either invariable or variable. Invariable storage function applies to reservoirs with a horizontal water surface. Such reservoirs are wide and deep with low flow velocities When a reservoir has a horizontal water surface there is a unique S=f(Q) relationship. For such reservoirs Qpeak occurs when I=Q, because Smax occurs when dS/dt =I-Q =0 A variable S-Q relationship applies to long, narrow reservoirs and to open channel or streams, where the water surface profile could be significantly curved due to backwater effects. The amount of storage due to backwater depends on the time rate of change of flow through the system, thus S-Q is no longer unique. Due to retarding effect of backwater, Qpeak usually occurs after I=Q

Level Pool Routing Let’s reconsider the continuity equation In discrete form, for a time interval Dt: The values Ij and Ij+1 are known. The values Qj and Sj are known at the jth time interval from calculation during previous time interval. Hence, two unknowns, Qj+1 and Sj+1. Rearranging:

In order to calculate Qj+1 a Q-S function relating (2S/Dt+Q) and Q is needed. S-h: topographic maps or field survey. Q-h: hydraulic equations For Dt use time interval of I(t) To set up the data required for the next time interval, the value of 2Sj+1/Dt +Qj+1 is calculated by

Level Pool Routing Find Ij + Ij+1, 2Sj/Dt - Qj Add (Ij + Ij+1) and (2Sj/Dt - Qj) to find (2Sj+1/Dt + Qj+1) Use the (2S/Dt+Q) vs. Q relation to find the corresponding Qj+1 Find (2Sj+1/Dt - Qj+1) by subtracting 2Qj+1 from (2Sj+1/Dt + Qj+1) Repeat calculations for next discharge value The inconvenience of this method is that (2S/Dt+Q) vs. Q relationship is dependent on Dt, i.e. for different Dt’s the relationship must be reconstructed.

Example A detention pond has vertical sites and is 1 acre in area. The Q-h and S-h relationships are given in the table. Assume S(t=0) = 0 Use level pool routing method to calculate the outflow hydrograph from the inflow hydrograph given in the table on next slide. For all elevations: A = 1 acre = 43,560 ft2 S = 43,560H * Dt = 10 min = 600 sec

S1= Q1= 0 The value of Qj+1 is found by linear interpolation. If (x,y) is between (x1,y1) and (x2,y2):

River Routing Routing in natural channels is complicated by the fact that storage is not a function of outflow alone. The storage in a stable river reach can be expected to depend primarily on the inflow to and outflow from the reach, and secondarily on the hydraulic characteristics of the channel cross section. Two types of storages can be defined in natural channels: Prism storage: Storage beneath a line parallel to the streambed drawn from the downstream end of the reach Wedge storage: Storage between this parallel line and the actual profile. Wedge storage is (+) during raising flow, and (-) during falling.

Muskingum Method Total storage = S = KQ + KX(I-Q) = K[XI + (1-X)Q] X: dimensionless constant between 0 and 0.5 with typical value of 0.2 K:travel time of a flood wave through channel Sj= K[XIj + (1-X)Qj] and Sj+1= K[XIj+1 + (1-X)Qj+1]  Sj+1 - Sj = K[XIj+1 + (1-X)Qj+1 - XIj - (1-X)Qj] Continuity: dS/dt = I – Q  Sj+1- Sj = {(Ij+Ij+1) - (Qj+Qj+1)}Dt/2  Qj+1=C1Ij+1 + C2Ij + C3Qj where

Muskingum Method Note that C1 + C2 + C3 =1. Therefore it provides a check If X = 0, then S = KQ, i.e. storage is not a function of inflow and becomes reservoir routing. If X = 0.5, then S = K(I+Q)/2, i.e. inflow and outflow have equal weights. In other words, no change in flood hydrograph. Great accuracy in determining X may not be necessary, because results are relatively insensitive to the value of this parameter. K = L/ck, where ck is wave celerity given by dQ/dA or dx/dt. Using Manning’s equation for a wide channel, it can be shown that ck = 5u/3 where u is average flow velocity. K can be approximated as

Example Qj+1=C1Ij+1 + C2Ij + C3Qj K= 2.3 hr, X= 0.15, Dt=1 hr