An Analytical solution for groundwater lens pumping in a three-dimensional rectangular island

Groundwater is an important freshwater supply for residents on islands without surface reservoirs, where groundwater floats on top of seawater, forming freshwater lenses. Previous analytical solutions of groundwater lens are developed based on two-dimensional models. In this study, we first derived an analytical solution for steady-state groundwater lens profiles, i.e., the freshwater-seawater interface, under pumping conditions for a three-dimensional rectangular island. Using this solution, critical or maximum pumping rates without extracting saltwater can be conveniently estimated for wells or drains at different locations in a three-dimensional rectangular island. Numerical models are simulated to validate the analytical solution. Results show that the critical pumping rate of a single well initially increases with the aspect ratio, i.e., the ratio between island length and width, and reaches a constant, which is limited by the short island side. In addition, the critical pumping rate of a single well is the largest in the center of the domain and decreases as the well approaches the domain boundary, as expected. When a rectangular island has an aspect ratio equal to one (i.e., a square island), the critical pumping rate is close to a circular island. For islands with a high aspect ratio, the critical pumping rate increases with the pumping drain length, which may be simplified to a two-dimensional strip island model. The derived analytical solution can be easily extended to include multiple pumping wells and drains, and can potentially be used to optimize groundwater pumping in oceanic islands.