Benchmark solutions for seawater-freshwater interface problems in aquifers of random permeability distribution

Benchmark solutions are often used for the validation of numerical codes. In the field of stochastic hydrology, few closed form solutions for the statistical moments of the solutions are available. In the present contribution, we consider the problem of a sharp interface between salt and fresh waters in an aquifer of spatially variable permeability. We assume a layered structure, with permeability, a stationary random function of the vertical coordinate, of given mean and two point covariance. The flow is shallow and it obeys Dupuit assumption. We derive an exact analytical solution of two-dimensional steady flow of fresh water in a confined aquifer, with salt water in rest. The mean interface shape is Dupuit parabola, for a constant effective permeability equal to the arithmetic mean of the variable one. The variance of the interface coordinate, and particularly of the toe, depends on the permeability variance and integral scale. The uncertainty of the interface location can be quite large. The second case is that of upconning of the interface beneath a well pumping fresh water above it. For a fixed interface depth at a given distance from the well, we determine the mean and variance of the upconned interface position.