Often along the channel/floodplain, there are locations such as a dam, bridge, or waterfall (short rapids) where the flow is rapidly varied in space rather than gradually varied. At such locations (internal boundaries), the Saint-Venant equations are not applicable since gradually varied flow is a necessary condition for their derivation. Empirical water elevation-discharge relations such as weir flow are utilized for simulating rapidly varying flow. At internal boundaries, cross sections are specified for the upstream and downstream extremities of the section where rapidly varying flow occurs. The Ax reach containing an internal boundary requires two internal boundary equations since, as with any other Ax reach, two equations equivalent to the Saint-Venant equations are required. One of the required internal boundary equations represents conservation of mass with negligible time-dependent storage, i.e.,

Dam. The second equation is usually an empirical, rapidly varied flow relation. If the internal boundary represents a dam, the following equation can be used:

in which Qs and Qb are the spillway and dam-breach flow, respectively. In this way, the flows Qj and Qi+l and the elevations /;, and hi+l are in balance with the other flows and elevations occurring simultaneously throughout the entire flow system, which may consist of additional downstream dams that are treated as additional internal boundary conditions via Eqs. (63) and (64). In fact, this approach can be used to simulate the progression of a dam-break flood through an unlimited number of reservoirs located sequentially along the valley. The downstream dams may also breach if they are sufficiently overtopped. The spillway flow (Qs) is computed from the following expression:

Qs = csLs(hi - hs)'s + CgAg(h, - hgf5 + cdLd{h, - hdf5 + Q, (65)

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