Traditional methods of upscaling, using homogenization or multiscale expansions, fail when the concentration gradients are large. However, pore-scale simulations suggest that even for transport limited kinetics, the concentration field in successive unit cells within a periodic array follow a multiplicative scaling with distance from the inlet. Thus, if the index n is used to identify unit cells successively further from the inlet (n = 0), then the concentration field in cell n can be related to a universal function C, c_n(x) = F_nC(x), where x is a point in the unit cell and F_n is the total flux of concentration entering the unit cell.

The pore-scale simulations were validated for a single disk by comparisons with microfluidic experiments (Dutka et al., Chem. Geol. 2020), other simulation methods (Molins et al., Comput. Geo. 2020), and conformal mapping Ladd et al., J. Fluid Mech., 2020). Studies of the dissolution of square and hexagonal arrays of disks are being used to infer the scaling of the concentration field described above. Constitutive models for the mass transfer in periodic arrays of disks can be inferred from these results.

A key step in using these constitutive relations is a distance-dependent rescaling of time, to allow different REV's in the array to be mapped to the dissolution of a single REV. We have noticed that the shapes of the disks in different unit cells can be mapped to each other by an appropriate rescaling of time.

In this talk I will summarize the most salient features of these studies and the key things that have been learned. Then I will describe ongoing work, which uses insights drawn from simulations of arrays of disks to suggest ways of upscaling in cases where dissolution is predominantly transport controlled.

Anthony Ladd

Appearances