Similarity Solutions for Axisymmetric Gravity-Driven Forchheimer Flow in Porous Media

We consider an axisymmetric gravity current of a Newtonian fluid advancing through an infinite, homogeneous porous medium characterized by an effective porosity ߶ and hydraulic conductivity ܭ. The flow regime is governed by Forchheimer’s law, which establishes a quadratic relationship between the velocity and the head gradient through the coefficientܾ . An integral-type inflow boundary condition is imposed, such that the volume of the current increases as a power law in time with exponent ߙ. We derive a self-similar solution describing the water table profile, which can be obtained by numerically integrating a non-linear ordinary differential equation. In the case of a finite volume release of a fluid ( ߙൌ Ͳ), the solution is analytical. This enables us to determine the gravity current profile as a function of the problem parameters. Notably, the current decelerates for ߙ൏ ͵, propagates at a constant speed for ߙൌ ͵, and accelerates for ߙ൐ ͵. Furthermore, we derive and discuss the aspect ratio of the current, its average slope, and the slope of the current in the box- model approximation. The box-model yields a dimensionless threshold time beyond which neglecting the Darcy term in the seepage equation is justified; this threshold time is found to depend on ߙ and on the parameter ܭ ൌ ܤ ܾଶ . Examples of both dimensionless and dimensional threshold times, computed from experimental data, are reported.