On the selection principle for viscous fingering in porous media

Viscous fingering in porous media at large Peclet numbers is subject to an unsolved selection problem, not unlike the Saffman-Taylor problem. The mixing zone predicted by the entropy solution is found to spread much faster than is observed experimentally or from fine-scale numerical simulations. In this paper we apply a recent approach by Menon & Otto (Commun. Math. Phys., vol. 257, 2005, p. 303), to develop bounds for the mixing zone. These give growth velocities smaller than the entropy solution result (M - 1/M). In particular, for an exponential viscosity-concentration mixing rule, the mixing zone velocity is bounded by (M - 1)(2)/(M ln M), which is smaller than (M - 1/M). An extension to a porous medium with an uncorrelated random heterogeneity is also given.