Ghyben-Herzberg Freshwater Lens

Badon Ghyben in the Netherlands in 1888 and Alexander Herzberg in Germany in 1901 worked out the same thing independently, and it is one of the few results in groundwater you can derive on a napkin. Fresh water floats on salt water. At any depth below the interface the pressure must balance, so a column of fresh water of height $h + z$ above the interface must weigh the same as a column of sea water of height $z$. That gives $\rho_f(h+z) = \rho_s z$, and therefore $z = \frac{\rho_f}{\rho_s - \rho_f}h$. Put in 1000 kg/m³ for fresh water and 1025 for seawater and the ratio is exactly 40.

The practical meaning of that 40 is brutal and every coastal water manager should be able to recite it: drawing the water table down one metre lifts the saltwater interface forty. A modest domestic well on a small island that lowers the local water table by half a metre pulls salt water up more than twenty metres beneath it, and if the lens was thirty metres thick the well is now pumping brine. This is why coastal wells fail suddenly rather than gradually. The interface rises as a cone directly under the pumping well, a process called upconing, and the well tastes fine right up until the moment the cone reaches the screen.

Take the assumptions seriously, because there are three and all of them are wrong in interesting ways. The first is hydrostatic conditions: the derivation assumes no vertical flow, which is only approximately true and fails badly near a pumping well, exactly where you most want an answer. The second is a sharp interface. Real boundaries are diffuse mixing zones, driven by tides and dispersion, that run from a few metres thick on a quiet aquifer to tens of metres on a tidally flushed one. The 40:1 depth marks roughly the middle of that zone, not the point where clean water stops. The third is the ratio itself. Seawater density varies with salinity and temperature, and brackish or hypersaline settings can move the ratio from 40 to 20 or to 80. Enter your own densities and watch the lever change.

Two practical corollaries follow. Pump coastal wells shallow, wide, and slow rather than deep and hard: several low-rate wells spread out will draw a far shallower cone than one high-rate well, and a screen set high in the lens buys you margin that no amount of monitoring can. And monitor conductivity, not just water level, because the first sign of upconing arrives as a rise in chloride long before the water tastes of anything.