Transverse dispersion from an originally sharp fresh–salt interface caused by shear flow

In this paper the influence of transversal dispersion and molecular diffusion on the distribution of salt in a plane flow through a homogeneous porous medium is studied. Since the dispersion depends on the velocity and the velocity on the distribution of salt (through the specific weight) this is a nonlinear phenomenon. In particular for the flow situation considered, this leads to a differential equation which has the character of nonlinear diffusion.

The initial situation (at t = 0) is chosen such that the fresh- and salt water are separated by an interface, and each fluid has a constant specific weight γ₁ and γ₂, respectively. For this initial situation, the solution of the nonlinear diffusion equation has the form of a similarity solution, depending only on ζ/$\sqrt t$, where ζ denotes the local coordinate normal to the original interface plane and t denotes time.

Properties of this similarity solution are discussed. In particular it is shown how to obtain this solution numerically. The interpretation of these mathematical results in terms of their hydrological significance is given for a number of worked out examples. These examples describe the distribution of salt, as a function of ζ and t, for various flow conditions at the boundaries ζ = ± ∞. Also examples are given where the molecular diffusion can be disregarded with respect to the transversal dispersion.