Upscaling chemical reactions in multicontinuum systems: When might time fractional equations work?

It may often be tempting to take a fractional diffusion equation and simply create a fractional reaction-diffusion equation by including an additional reaction term of the same form as one would include for a standard Fickian system. However, in real systems, fractional diffusion equations typically arise due to complexity and heterogeneity at some unresolved scales. Thus, while transport of a conserved scalar may upscale naturally to a fractional diffusion equation, there is no guarantee that the upscaling procedure in the presence of reactions will also result in a fractional diffusion equation with such a naive reaction term added. Here we consider a multicontinuum mobile-immobile system and demonstrate that an effective transport equation for a conserved scalar can be written that is similar to a diffusion equation but with an additional term that convolves a memory function and the time derivative term. When this memory function is a power law, this equation is a time fractional dispersion equation. Including a first-order reaction in the same system, we demonstrate that the effective equation in the presence of reaction is no longer of the same time fractional form. The presence of reaction modifies the nature of the memory function, tempering it at a rate associated with the reaction. Additionally, to arrive at a consistent effective equation the memory function must also act on the reaction term in the upscaled equation. For the case of a bimolecular mixing driven reaction, the process is more complicated and the resulting memory function is no longer stationary in time or homogenous in space. This reflects the fact that memory does not just act on the evolution of the reactant, but also depends strongly on the spatio-temporal evolution and history of the other reactant. The state of mixing in all locations and all times is needed to accurately represent the evolution of reactants. Due to this it seems impossible to write a single effective transport equation in terms of a single effective concentration without having to invoke some approximation, analogous to a closure problem. For both reactive systems simply including a naive reactive term in a fractional diffusion equation results in predictions of concentrations that can be orders of magnitude different from what they should be. We demonstrate and verify this through numerical simulations. Thus, we highlight caution is needed in proposing and developing fractional reaction-diffusion equations that are consistent with the system of interest.