Thermally-Driven Reaction Fronts in Porous Media

We present a mathematical model of the reaction fronts that develop when an undersaturated `injection fluid' displaces a saturated `formation fluid' in a chemically reactive porous medium. Flows of this kind are relevant to the geothermal and hydrocarbon industries, and are important natural geological processes. The injection and formation fluids differ both in temperature and in chemical composition. We assume that the equilibrium concentration of a reactive mineral species in the fluid is a linear function of temperature. Firstly, we consider the case where the formation fluid is isothermal at t=0. We then extend this solution to the case where the formation fluid has a constant temperature gradient at t=0. The undersaturation of the incoming fluid drives a dissolution reaction and leads to the formation of a `depletion' front. Under certain circumstances, which we describe, the temperature difference is able to drive a separate thermal reaction front. Two distinct regimes arise. If the compositional difference between the injection and formation fluids exceeds a critical value, the depletion front travels faster than the thermal front, leaving the porous medium depleted of reactant from the source to a point beyond the thermal front and no thermal reaction front develops. Conversely, if the compositional difference is smaller than the critical value, the thermal front forms downstream of the depletion front and so there is a double reaction front structure. When we allow for a temperature gradient in the formation fluid, the magnitude of this gradient dictates whether the depletion front speeds up or slows down over time. It is therefore possible for the depletion front to form downstream (upstream) of the thermal front but move upstream (downstream) of it over time. 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. We derive the six dimensionless parameters which control the evolution of the system and find some approximate solutions of the governing equations which are valid at long times. We illustrate the evolution of the thermal and compositional fields towards these asymptotic solutions with numerical simulations.