A theoretical study of combustion in porous media driven by a gravity induced gas flux is conducted. Filtration of the oxidizer carrying gas arises in response to heating of the gas due to exothermic conversion of the solid fuel. Specifically, we consider a reaction front propagating through a porous matrix consisting of reactive (fuel) and inert components. Gas, consisting of both oxidizer and inert components, filters through the matrix and reacts with the solid fuel. The hot gases in the medium rise due to buoyancy, thus drawing in fresh gas from below. We employ approximate analytical methods and numerical simulations to analyze all the basic combustion phenomena, including self ignition, external ignition, both upward and downward as well as adiabatic and nonadiabatic propagating combustion waves. Our simulations also describe the dynamics of buoyancy driven combustion waves.