Reaction fronts in a porous medium following injection along a temperature gradient

We consider the displacement of a 'formation fluid' by an 'injection fluid' along a temperature gradient in a reactive porous rock. We assume that the formation fluid is in chemical equilibrium initially and we allow the injection fluid to differ from the formation fluid both in temperature and in chemical composition. Two types of reaction front can form as the injection fluid propagates into the rock: (i) 'depletion fronts' caused by any undersaturation of the injection fluid and (ii) 'thermal reaction fronts' caused by any difference in temperature between the injection fluid and the trailing edge of the formation fluid. As flow proceeds the temperature gradient in the formation fluid is advected at the thermal propagation speed, but the fluid itself moves more quickly at the interstitial fluid speed. This difference in speed causes formation fluid to be carried out of chemical equilibrium and a 'gradient reaction' results. The subsequent dissolution or precipitation alters the amount of reactant present on the rock and this alters the propagation speed of any subsequent depletion front. We derive the six dimensionless parameters controlling the evolution of the system and calculate approximate solution profiles for the limit of fast kinetics and negligible diffusion. These solutions give insight into the physical processes governing the solution and are compared with numerical solutions of the full equations. The behaviour of the system is strongly influenced by three dimensionless parameters in particular. The parameters J(C) and J(T) quantify the chemical and thermal difference between the fluids at the point of injection and they control the relative positions of the depletion and thermal fronts at short times. On the other hand, the parameter a quantifies the background thermal gradient and controls the relative positions of the fronts at long times. In the cases where the depletion front moves from one side of the thermal front to the other, we derive approximate expressions for the time at which this 'crossover' occurs.