The aim of the study, reported of in this paper, is to determine the shape and position of the interface which separates the fresh from the salt water in a coastal aquifer. In this aquifer, fresh water flows from land towards the sea because the head on the land is higher than on the seabottom. The upper boundary of the flow region, formed by the land surface and the seabottom has been approximated by a straight line. Firstly the problem is solved, by use of the method of conformal mapping and the hodograph, for the case that the head is constant along that part of the line which represents the land surface and also constant, but lower, on the part which represents the seabottom. Special attention is paid to the form and position of the interface when a drain is in operation in the fresh water region. The hodograph turns out to be multiple sheeted and contains internal branch points and poles. Therefore, a simple generalization of the Schwarz-Christoffel Integral is derived which maps such hodographs onto the upper half plane. Secondly it is shown that the solution can be generalized directly by superposition in the reference half plane. This permits the description of the flow case of an arbitrary number of drains. By superposition, also flow with drains and an arbitrary number of different levels in polders and dunes can be described. Thirdly the upconing of the interface under a drain is studied in more detail for a simple case. A test was run in a parallel plate model in order to verify some of the formulas for upcoming, derived in this paper. Test results and theory agree satisfactorily.