Analysis of seepage discharge variations at the downstream end of diversion dams; Pavlovsky’s solution revisited

The construction of aprons and diversion dams on alluvial beds have important role in rivers protection. Design of such hydraulic structures on permeable foundations is required to determine hydraulic gradients and seepage discharge after the downstream end of the structure. This important issue is well done by seepage flow analysis. The seepage flow may occur in or beneath hydraulic structures and also from the bed of open channels. One of the existing methods for analyzing seepage flow and solving Laplace’s equation is the application of analytical solution which is usually based on the theory of conformal mapping. The hydraulic gradients at the end of downstream and seepage discharge passing beneath diversion dams could be controlled by cutoff walls. In the present paper, analytical closed-form equations for the variations of hydraulic gradients and seepage discharge are presented as a function of the distance from downstream end for various arrangements of cutoff walls. The porous media beneath the structure is assumed to have infinite depth. The problem is solved for four scenarios: cutoff wall at downstream end, cutoff wall at upstream end, double cutoff walls at both ends and the structure with depressed floor.

In this paper, hydraulic gradients and seepage discharge have been obtained with respect to the distance from downstream end by the use of conformal mapping and an approach based on Darcy’s equation. Indeed, this method is the extension of Pavlovsky’s solution. The Schwarz-Christoffel transformation is used in conformal mapping.

Based on the resulting equations, non-dimensional charts have been produced for the variations of hydraulic gradients and seepage discharge with respect to the distance from downstream end and the length of structure. Assuming b is the length of structure, s is the depth of cutoff wall, x is the distance from downstream end, and d is the depth of depressing floor, at constant ratios of b/s or b/d , the hydraulic gradients values are decreased with increasing the distance from the end and also at constant ratios of x/s or x/d , the hydraulic gradients values are decreased with increasing the b/s or b/d ratios. For limiting the seepage discharge in a determined value which is passed from the downstream end of the structure, the value of b/s in the case of cutoff at upstream end is greater than double cutoffs, cutoff at downstream end and, depressed floor, respectively.

Based on the obtaining charts, the results show that the cutoff wall at the downstream end is more effective than the others and the hydraulic gradients in the case of depressed floor is less than the other cases