The Dupuis paradox and mathematical simulation of unsteady filtration in a homogeneous closing dike

The aim of this study is to determine a flow rate and a shape of a depression curve in conditions of filtration through a rectangular closing dike using aperiodic solutions of the Boussinesq limit problem. We established that the formation of this curve and the seepage area (the final jump of continuity or interruption of the curve at the minimum pressure point) on the border of the downstream and porous medium, in the closing dike of finite length, occurs for a finite amount of time proportional to the square of the closing dike length. Therefore, in the short closing dike, a cut-out point does not have time to fall into the downstream during the time, it takes for the depression curve to touch the water level in the upstream. The continuous curve without seepage area always reaches the steady state in the semi-infinite closing dike for a finite amount of time.