Dissolution of periodic arrays of grains: Upscaling of pore-scale simulations with fast reactions
Upscaling methods are frequently used to derive transport equations at the macroscopic scale from more fundamental equations formulated at the pore scale. These methods typically give a suitable structure for the macroscopic equations and can also provide explicit expressions for the constitutive parameters, such as permeability and dispersion coefficients. Introducing chemical reactions complicates upscaling in at least two important ways. First, the interplay between chemical reactions and transport processes introduces a new length scale, which can be much smaller than the convective or dispersive length scales. A small reactive length scale breaks one of the key assumptions in upscaling; that there is a significant separation in length between the pore-scale and macro-scale processes. The second complication is that if reactions take place at mineral surfaces (dissolution or precipitation) then the pore space itself is evolving in time. In this paper we suggest ways in which these difficulties can be approached, based on analysis of pore-scale simulation data. First, we noticed that the concentration field in successive unit cells has an almost identical spatial variation, with a single scaling factor for each unit cell that is proportional to the incoming reactant flux. Using pore-scale simulations to determine the mass transfer coefficient in a few unit cells, we can calculate the concentration field in the whole domain, even when dissolution is entirely transport limited. Second, we have noticed a time-dependent mapping of the grain shapes from different unit cells. From these observations, we can deduce constitutive relations where the only time-varying parameter is the porosity. Here, we show that a model based on these ideas can quantitatively account for the pore-scale simulation data.