Viscous fingering in porous media at large Péclet 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.