Upscaling non-linear reactive transport in correlated velocity fields

Abstract:

While commonly non-local transport models have been shown to reproduce breakthrough curves resulting from transport in heterogeneous media successfully, open questions include the formal link between the upscaled governing equations and the sub-scale heterogeneity, and the ability to account for the effect of heterogeneity on effective chemical reaction rates in the presence of non-linear multi-component reactions. These approaches often rely on particle tracking approaches, which can be computationally burdensome especially when non-linear reactions are sought to be modeled. Another open question regarding upscaling techniques in heterogeneous media is how to directly relate the spatio-statistical properties of the media at small scale to the parameters of the upscaled model without conducting model calibration based on observed breakthrough curves. In this presentation, we will show that under specific conditions, the upscaled reactive transport in heterogeneous media can be represented as a Ornstein-Uhlenbeck (OU) process based on which we derive the integro-differential equations governing upscaled reactive-transport. For this purpose, we express concentration or flux of solutes as a distribution over their velocity. We then derive the integro-differential equation that governs the evolution of concentration distribution over a quantity defined as velocity-rank. In this way, the spatial evolution of breakthrough curves away from the source is predicted based on ergodic cross-sectional velocity distributions and a parameterized OU process, which expresses the spatial correlation of normal scores of velocity. We will also show how the OU process parameters controlling dispersion and molecular diffusion can be directly extracted from the spatio-statistical properties of the velocity field at small scale. We demonstrate the validity of the proposed model by comparing breakthrough curves for conservative and non-linearly reacting solutes based on realizations of hydraulic conductivity fields with the results of the upscaled model.