Phase diagram of sustained wave fronts opposing the flow in disordered porous media

Using lattice Boltzmann simulations, we analyze the different regimes of propagation of an autocatalytic reaction front in heterogenous porous media. The heterogeneities of the porous medium are characterized by the standard deviation of its log-normal distribution of permeability and its correlation length. We focus on the situation where chemical reaction and flow field act in opposite directions. In agreement with previous experiments we observe upstream, downstream fronts as well as static, frozen ones over a range of flow velocity which depends drastically on the heterogeneities of the flow field. The transition between the static regime and the downstream one account for large enough low-velocity zones, whereas the transition from static to upstream regime is found to be given by a kind of percolation path.