Drainage of Ponded Surface by an Array of Ditches

Abstract

A comprehensive analytical solution for the quantity of seepage into an array of fully penetrating ditches from a ponded surface has been obtained using hodograph and Schwarz–Christoffel transformation. The solution includes equations for the quantity of seepage from the seepage face part as well as the water depth part of the ditch. The solution also comprises expressions for the velocity potential at the stagnation point and the variation in seepage velocity. The variation in seepage quantity is like the shape of a curved channel whose boundary maps along a circle onto the hodograph plane. This shape is average of a semiellipse and a parabola. The seepage contribution from the nonseepage face is maximum for half full condition and it is half of the total seepage in an empty ditch (full seepage face). Irrespective of the spacing between ditches the quantities of seepage from the seepage face part and the nonseepage part are equal for one third full ditch. The solution also deals with special cases like single ditch, unequal spacing between ditches, and unequal depth of water in adjacent ditches. The expressions the quantity of seepage have been simplified in explicit algebraic equations through minimization of errors. The simplified expressions, which are near exact, result in answers in single step computations. Also, an example and graphs have been included to demonstrate the sensitivity of the parameters.