Engelund’s Two-Dimensional Drainage Equation for a Toe-Drain and the Dupuit-Forchheimer Drainage Equation for a Ditch: A Coincidental Match

Abstract

Water-table heights attributable to steady uniform surface accretion rates in drained lands overlying a horizontal impermeable bed are given analytically by Engelund’s solution for toe-drains and by the approximate Dupuit-Forchheimer analysis for ditch drains. Both give the same water tables, although for different types of drain, when the spacing of the ditches in the latter is taken to be that in which the water table is drawn down to the level of the toe-drain in Engelund’s derivation. This fortuitous match also occurs in anisotropic soils when the horizontal hydraulic conductivity component is used in the formulae with the drain spacing dependent on the anisotropy factor. For soils with an infinite anisotropy factor, the Dupuit-Forchheimer analysis is exact so that the match is no longer fortuitous.