B3. Approximate Solution to the Generalized Boussinesq Equation

Abstract

The Boussinesq equation describes water flows in unconfined groundwater aquifers under the Dupuit assumption that the equipotential lines are vertical, making the flow essentially horizontal. It is a nonlinear diffusion equation with diffusivity depending linearly on water head. The generalized Boussinesq equation or the porous medium equation is a diffusion equation where the diffusivity is a power law function of water head. Solutions to the generalized porous medium equation will propagate with a finite speed in case of initially dry aquifer, unlike the solutions to the linear diffusion equation that propagate with infinite speed. For certain types of initial and boundary conditions similarity reductions are possible; and the original initial-boundary value problem for the partial differential equation is reduced to a boundary value problem for a nonlinear ordinary differential equation. We construct approximate analytical solutions to the generalized Boussinesq equation that respect the scaling properties of the equations.