Approximate Solution to the Generalized Boussinesq Equation

The traditional Boussinesq equation describes motion of water in groundwater flows. It models unconfined groundwater flow under the Dupuit assumption that the equipotential lines are vertical, making the flowlines horizontal. The Boussinesq equation is a nonlinear diffusion equation with diffusivity depending linearly on water head. Here we analyze a generalization of the Boussinesq equation, when the diffusivity is a power law function of water head. For example polytropic gases moving through porous media obey this equation. Solving this equation usually requires numerical approximations, but for certain classes of initial and boundary conditions an approximate analytical solution can be constructed. This work focuses on the latter approach, using the scaling properties of the equation. We consider one-dimensional semi-infinite initially empty aquifer with boundary conditions at the inlet in case of cylindrical symmetry. Such situation represents the case of an injection well. Solutions would propagate with the finite speed. We construct an approximate scaling function, and we compare the approximate solution with the direct numerical solutions obtained by using the scaling properties of the equations.