 
Summary: The averaging of gravity currents in porous media
Daniel M. Andersona)
Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
Richard M. McLaughlinb)
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599
Cass T. Millerc)
Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill,
North Carolina 27599
Received 19 December 2002; accepted 24 June 2003; published 15 August 2003
We explore the problem of a moving free surface in a watersaturated 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 halfheight
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 twodimensional periodic casethe 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
