The design of hydraulic structures located near a confluence of two streams must take into consideration the flows from both of the streams. A hydraulic structure located on a small tributary that drains into a large river immediately downstream is not just affected by the flow in the tributary, but also by the backwater flow from the downstream river. Currently INDOT uses a tabular summary (Table 1.1 and Table 7.3 in the HEC-22 manual) of joint probabilities of coincident flows in designing hydraulic structures at confluences. However, the source of the table is unknown, and the tabular summary provides coincidental flows for only 1% and 10% probabilities, and thus it cannot be used directly for other probabilities. This study analyzed the interdependence of flows in mainstream and tributary and then developed a Gumbel-Hougard Copula-based procedure for estimating joint probabilities for confluences in Indiana. The study found that the mainstream and tributary streamflow are significantly correlated with Kendall’s Tau varying generally ranging from 0.5 to 0.8. Furthermore, the Kendall’s Tau, which is the key parameter for Gumbel-Hougard Copula, was found to be significantly related to drainage area ratio (DAR). Regression-based equations between DAR and τ are used as a basis to relate DAR to joint probabilities at confluences. The study also found that the currently used tabular summary (Table 1.1 and Table 7.3 in HEC-22 manual) resulted in significantly conservative design estimates and therefore overdesigned structures.

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joint probability, copula, confluence design, coincident flows, Indiana, NWM, national water model

SPR Number


Performing Organization

Joint Transportation Research Program

Publisher Place

West Lafayette, IN

Date of this Version