Seepage to a drainage ditch and optimization of its shape

An optimal shape design problem for an empty drainage ditch is solved and it is shown that a semicircle intercepting saturated seepage from two constant potential rays is a contour of constant Darcian velocity and of maximal excavated volume at a given flow rate and ditch width. The corresponding family of isobars belongs to the class of conchoids of Nicomedes. The optimal flow is generated by a hydrodynamic dipole placed in a gravitational field.