Special Session 108: New Developments in porous media

Self-similar viscous gravity currents in heterogeneous porous media: Second-kind solutions

Ivan C Christov
Purdue University
Co-Author(s):    Z. Zheng, H. A. Stone
We summarize our recent combined experimental-theoretical-computational study of the effects of horizontal heterogeneities on the propagation of viscous gravity currents with applications to porous media flows. Our model geometry is a horizontal channel (specifically, a Hele-Shaw cell) with variable gap thickness in the streamwise direction in the form of a power law. We demonstrate that two types of self-similar behaviors emerge as a result of such horizontal heterogeneity: (a) a ``first-kind`` solution is found using dimensional analysis for currents that propagate away from the origin (a point of zero permeability); (b) a ``second-kind`` solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. Using the phase-plane formalism, we are able to construct the universal second-kind self-similar current shape. Additionally, still employing self-similar intermediate asymptotics and the phase-plane formalism, we identify self-similar behaviors in the post-closure regime, i.e., once the current reaches the geometric origin and begins to fill the model porous medium. The theoretical predictions show good agreement with lab-scale experiments using Hele-Shaw cells and also numerical solutions of the governing partial differential equation developed under the lubrication approximation. Z. Zheng, I.C. Christov, H.A. Stone, {\it J.\ Fluid Mech.}\ {\bf 747} (2014) 218-246, doi:10.1017/jfm.2014.148.