Finite-Depth Seepage beneath Weirs with Double Sheet Piles and a Downstream Step

Abstract

In this paper, finite-depth seepage beneath weirs equipped with double sheet piles and a downstream step is analytically investigated. The Schwarz–Christoffel transformation is employed to conformally map both the physical plane and the complex potential plane onto an auxiliary semi-infinite lower-half plane. The seepage characteristics, such as seepage discharge, exit gradient, and pressure head distribution along the weir floor, are determined by applying the similarity conditions between the original and transformed planes. The results reveal that the seepage characteristics depend on several key parameters, including the upstream sheet pile depth, downstream sheet pile depth, upstream floor length, downstream floor length, and downstream step depth. The seepage discharge diminishes as the upstream sheet pile depth, downstream sheet pile depth, upstream floor length, and downstream floor length increase. Conversely, the seepage discharge increases with an increase in the downstream step depth. Additionally, the exit gradient decreases with an increase in the upstream sheet pile depth, downstream sheet pile depth, and upstream floor length. In contrast, it attains a peak magnitude at a specific value of the downstream floor length. Moreover, the exit gradient increases with an increase in the downstream step depth. For a given sheet pile location, the pressure head increases and decreases in the upstream and downstream regions, respectively. Furthermore, an increase in the downstream step depth induces a reduction in the pressure head. The predictions of the present study are consistent with those of existing studies.