A new approximate technique for the hodograph method in groundwater flow and its application to coastal aquifers

The solution to a steady, two‐dimensional problem of flow in a coastal aquifer is derived to illustrate the application of a new approximate analytical technique. The problem involves the determination of a phreatic surface, an interface separating flowing freshwater from salt water at rest, a seepage face, and an outflow face where freshwater leaves the homogeneously permeable porous medium to flow into the sea. The solution which expresses the relation between the complex potential Ω and the complex variable z, defined in the flow domain, is obtained by application of conformal mapping techniques using the hodograph method. Three auxilliary planes are introduced: the ζ plane, a reference half plane, the plane of the specific discharge function w, which is the complex conjugate of the hodograph, and the plane of the reference function R, a function of the first derivative of z and w with respect to ζ. The boundary of the flow domain, which is not known beforehand, corresponds to a polygon composed of circular arcs and straight line segments in the w plane and to a polygon of straight lines through the origin in the R plane. The function that maps the ζ plane onto the R plane is expressed analytically. In the proposed technique, the analytic function ζ=ζ(w) is evaluated using a new boundary element approximation. It is shown that the solution to the flow problem may be derived from those two relations in terms of w.