Abstract
Piles are walls that are installed underneath hydraulic structures to control seepage flow, uplift pressure, and exit hydraulic gradient. In this study, the effect of the ratio of width of the apron to the foundation depth (B/D), the ratio of the width of the apron to the downstream pile depth (B/d(2)), and the ratio of the upstream and downstream pile depths (d(1)/d(2)) on seepage flow characteristics is numerically investigated using the finite-element method. From the numerical simulations, design charts, and explicit formulas for estimating seepage flow characteristics, such as the percentage residual uplift pressure at key points, the discharge flow factor, and the exit hydraulic gradient factor underneath aprons with end double-piles for both states of equal and unequal depths are developed. The simulations are validated with existing analytical solutions for simple cases. To assess the accuracy of the presented formulas, statistical metrics; that is, the determination coefficient, root mean square error, and percent relative error (RE%) were used. The statistical metrics are satisfactory, and their values quantified the reliability of the proposed formulas. The proposed explicit formulas, with RE% less than 5%, could be used by design engineers instead of implicit analytical equations that involve elliptic integrals.