Stability of a plane reaction front in a porous medium

Porous media can be dissolved by pumping acids through them. The permeability of the porous material will be raised, and this is used by the petroleum industry to increase the productivity of oil wells. The acid flows preferentially into zones of high permeability which are therefore most rapidly dissolved, leading to a further preferential enhancement of the permeability. Thus the reaction front is unstable. This paper studies a simple model which demonstrates the effect of the Damköhler number and of the Acid Capacity number on the instability of a plane reaction front. Scaling lengths by the depth of the reaction zone, it is found that perturbations with wave number k grow exponentially at a rate σ ∝ log (1 + k) rather than the σ ∝ k which occurs in the Saffman—Taylor instability. Although perturbations with the smallest wavelengths (of the order of the pore size) will grow most rapidly, fingering will be marked for all fingers with wavelengths smaller than, or similar to, the width of the reaction zone. Laboratory experiments reveal negligible fingering in sandstones, and the instability is thought to be mainly associated with the rapid reactions which occur in carbonate rocks. The results obtained here predict that the instabilities in slower reacting sandstones will only become apparent in experiments on a much larger scale.