We develop a model to describe the spreading of a reacting liquid which is injected at a steady rate into a permeable rock. We focus on the case in which there is a density difference between the host reservoir fluid and the injected liquid. We examine reactions which lead to precipitation and a decrease in permeability or dissolution and an increase in permeability. In both cases, we assume the reaction is rapid compared to the speed of the flow. As the current spreads under gravity, we show that the interface between the injected fluid and the original fluid and also the reaction front, may be described by similarity solutions. The morphology of the two interfaces is controlled by two parameters: the permeability ratio across the reaction front, k, and the speed of the reaction front as a fraction of the interstitial speed, λ. For a precipitation reaction, the reaction front lags some distance behind the leading edge of the region occupied by the injected fluid, and tends to terminate in a sharp vertical front. In contrast, for a dissolution reaction, the reaction front migrates as a gravity-driven finger along the base of the formation. In the case of large changes in permeability, kλ > 1, this finger advances to the front of the flow, whereas for smaller increases in permeability, kλ < 1, the finger is overrun with injected fluid which has already reacted and passed through the reaction front. We illustrate how these results are affected if the density of the reacting fluid decreases across the reaction zone. In the case of precipitation, small changes in density smooth out the leading edge of the reaction front, whereas large changes in density lead to slumping of the reaction front along the base of the current, and ultimately it extends to the nose of the flow. For dissolution reactions, the decrease in density across the reaction front causes the lateral extent of the finger to increase. As a result the critical value of the permeability ratio, k, for which the reaction front reaches the nose of the current decreases.