Seawater-freshwater interface in a stratified aquifer of random permeability distribution

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 the Dupuit assumption. We derive an exact analytical solution of two-dimensional steady flow of fresh water in a c confined aquifer, with salt water at rest. The mean interface shape is a Dupuit parabola, for a constant effective permeability equal to the arithmetic mean of the random permeability. 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 large. We also investigate the uncertainty of the discharge of a coastal collector, for a given depth of the upconed interface. We also determine, by using conditional probability, the impact of measurement of the interface depth in a given cross-section on the reduction of the uncertainty of its position. The second case is the generalization of the solution by Keulegan (1954, National Bureau of Standards Report, Department of Commerce, Washington, DC) of unsteady flow that results from an abrupt removal of a thin vertical partition between salt- and freshwater bodies. The mean interface is similar to that prevailing in a homogeneous medium, i.e. a rotating line whose slope approximately corresponds to the permeability arithmetic mean. The uncertainty affecting the interface position is of a smaller extent than that in steady flow.