The averaging of gravity currents in porous media

We explore the problem of a moving free surface in a water-saturated porous medium that has either a homogeneous or a periodically heterogeneous permeability field. We identify scaling relations and derive similarity solutions for the homogeneous, constant coefficient case in both a Cartesian and an axisymmetric, radial coordinate system. We utilize these similarity scalings to identify half-height slumping time scales as a rough guide for field groundwater cleanup strategies involving injected brines. We derive averaged solutions using homogenization for a vertically periodic, a horizontally periodic, and a two-dimensional periodic case—the solution of which requires solving a cell problem. Using effective coefficients, we connect the first two of these homogenized solutions to the similarity scaling solution derived for the homogeneous case. By simplifying to a thin limit, retaining variations of the porous media in the horizontal direction, we derive a homogenization solution in agreement with the general horizontally layered solution and an expression for the leading-order correction. Finally, we implement two numerical solution approaches and show that self-similar scaling and agreement with leading-order averaging emerge in finite time, and demonstrate the accuracy and convergence rate of the leading order correction for both the interior and the boundary of the domain. © 2003 American Institute of Physics. @DOI: 10.1063/1.1600733#