B1. Laplace - Boussinesq: 3-1.

Abstract

In 1993, Jean-Yves Parlange developed a very generic solution for the Laplace equation in the archetypal drainage geometry of an aquifer on an impermeable layer between two parallel drains with linearized boundary conditions. The solution is based on Eigenfunction expansion and has as main advantage that it can accommodate any spatial and temporal distribution of recharge into the aquifer and any shape of the initial groundwater table. A parallel solution was developed for the Boussinesq equation. As such, the solutions provide a nice “toy model” to test the validity of the Dupuit assumption that the vertical pressure distribution in an aquifer is basically hydrostatic, which underlies the Boussinesq equation. The good news is that Boussinesq is indeed valid as long as the underlying assumption is valid. There are, however, also cases in which Boussinesq fails. The presentation will go in some detail to show where trouble is to be expected, and where safe sailing reigns. Although the geometry and the linearization of the boundary conditions do not allow for a complete analysis, the analysis does have clear heuristic value. For example, at short times after sudden drawdowns, vertical water movement and associated energy losses are very relevant, limiting the validity of the Boussinesq equation. The same holds when the aquifers, or drained fields, are relatively deep in comparison to theirs widths, as is the case in the raised beds found in the wetlands of Rwanda that prompted the original research.