Values for the unknowns at external boundaries (the upstream and downstream extremities of the routing reach) of the channel/floodplain, must be specified to obtain solutions to the Saint-Venant equations. In fact, in most unsteady-flow applications, the unsteady disturbance is introduced at one or both of the external boundaries.
A specified discharge time series (hydrograph) of inflow to the most upstream cross section is used as the upstream boundary condition. The hydrograph should not be affected by downstream flow conditions. This hydrograph may be obtained from the following: (1) historical observations, (2) assumed design hydrograph, or (3) a runoff hydrograph from specified rainfall-runoff model using calibrated or estimated model parameters. The upstream boundary is expressed mathematically as follows:
in which Q(t) is the specified discharge time series and the subscript indicates the discharge at the first cross section, i.e., the upstream boundary. Equation (60) is used for the upstream boundary if dynamic routing (based on the discretized Saint-Venant equations) commences at this location. However, if the most upstream cross section represents the inlet to an upstream reservoir, a simple routing procedure (reservoir level-pool routing) can be used rather than the considerably more complex dynamic routing if (l) the reservoir is not excessively long and (2) the inflow hydrograph Q(t) is not rapidly changing with time. Level-pool routing errors (Fread, 1992), with a magnitude of less than about 5%, can usually be tolerated.
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