Analytical Solutions for Confined and Unconfined Coastal Interface Flow by the Hodograph Method

We present an analytical solution describing steady, two‐dimensional groundwater flow in the vertical plane in an unconfined, coastal aquifer of finite depth. An interface separates flowing freshwater from stagnant seawater. The boundary conditions along both the phreatic surface and the interface are treated exactly by the use of the hodograph method and conformal mapping. Explicit expressions for the distance of the toe from the coast, and the distance of the tip from the coast, are presented as asymptotic expansions about small gradients valid for isotropic and vertically anisotropic aquifers. A new, and simpler form of the solution to the classical problem of confined interface flow is obtained as a limiting case of the unconfined solution. Both new solutions avoid the problem of crowding, which occurs in the existing solutions for confined interface flow, and is commonly encountered in conformal mappings of elongated domains. A comparison of the tip and toe locations is made for the unconfined and confined cases; the results compare favorably for typical values of seawater density. The exact equation for the toe length as a function of the gradient, truncated after two terms, provides a simple, accurate, and invertible expression for estimating the freshwater discharge to the coast based on field observations of the toe length.