Non-local models for upscaling of reaction and transport processes in porous media

In this thesis, we analyze the upscaling of reaction-transport processes in porous media in order to incorporate effects of microscale heterogeneity in the macroscale when non-local transport effects are important. The volume averaging method is used to determine effective kinetic parameters for transport-reaction systems with fast kinetics in the limit of thermodynamic equilibrium. We show that computing the effective mass transfer coefficient requires solving an eigenvalue problem, coupling the local microstructure problem with the global. The latter differs from what has been conventionally used. The theoretical predictions are illustrated using a simple advection-dissolution problem. Our approach was extended to systems with finite kinetics. For reaction-transport systems that involve a sharp reaction front, traditional upscaling fails. For such systems we proposed a hybrid algorithm coupling a pore-network with a continuum model in places where the latter fails. This is implemented on a fixed boundary, simple reaction-transport problem. Looking at the problem on a larger scale, a continuum, stochastic equation is derived to model interface growth in reaction-transport processes. This equation extends the Kardar-Parisi-Zhang (KPZ) equation to capture non-local transport effects through a Hilbert transform that can act to either stabilize or destabilize the interface. The properties of the solution of the model equation are studied in one-spatial dimension in the linear and non-linear limits, for both stable and unstable growth. We find that the early-time behavior has a power law scaling similar to the KPZ. However, in the case of stable growth, the scaling of the saturation width is logarithmic, which differs from the power law in the KPZ. This dependence reflects the stabilizing effect of non-local transport. In the unstable case, the width at late times was found to obey a power-law growth. The non-local equation in the absence of noise is illustrated through a weakly non-linear stability analysis of a reaction front in reactive infiltration in porous media. Finally, analytical expressions for the overall mass transfer coefficient in terms of the Sherwood number for dissolution-reaction transport processes in two simple geometries: a Hele-Shaw cell and a wedge-shaped corner are provided.