Anomalous dynamics of Darcy flow and diffusion through heterogeneous media
Authors:   , Russian Anna 
Published: October 2023

This thesis studies diffusion phenomena in heterogeneous media, which includes Darcy flow and diffusive solute transport in geological media. Natural media are heterogeneous at different scales, which induces complexity in diffusion phenomena. The work is centered on the integration of the effects of heterogeneity on Darcy flow and solute diffusion into large scale models.
The quantification of the effects of heterogeneity in diffusion phenomena is highly important for a large number of problems such as diffusion and reaction of chemicals and radionuclides in low permeability media, which is essential in subsurface hazardous waste storage problems, CO2 sequestration performance and groundwater management.
In a stochastic framework we quantify the effects of heterogeneity in large scale models considering two interrelated strategies that can be called Ôcoefficient approachÕ, which deals with the derivation of effective coefficients to insert in equivalent homogeneous models, and Ôdynamic approachÕ, which deals with the upscaling of the local scale equations and the derivation of large scale formulations which can differ from their local counterparts. Whenever a diffusion process cannot be described in terms of effective coefficient, that behaviour is named anomalous or non-Fickian.
Anomalous diffusion behaviours observed experimentally are frequently modelled by effective theories such as fractional diffusion equations, continuous time random walks. One limitation of these models is that often they are rather phenomenological and the relation to the local scale heterogeneity and dynamics may not be clear. In the dynamic approach we derive large scale descriptions that can explain anomalous behaviour and link it with a description of the local scale medium heterogeneity. To this end, we upscale the local scale governing equations using different methods depending on the type of medium heterogeneity.
For moderately heterogeneous media we upscale flow equation by stochastic averaging. Starting from the classical flow equation at local scale determined by Darcy's law, we derive an upscaled non-local effective formulation. The non-local effective formulation is compared with its local counterpart by considering the head response for a pulse injection.
Numerically, we solve flow and diffusion in heterogeneous media using particle tracking methods. While classical random walk particle tracking is an efficient numerical tool to solve for diffusion problems in moderately heterogeneous media, strong medium contrasts, as encountered in fractured media, render this method inefficient. For highly heterogeneous media efficiency of classical random walk can be increased by the use of the time domain random walk (TDRW) method. We rigorously derive the equivalence of the TDRW algorithm and the diffusion equation and we extend the classical TDRW method to solve diffusion problem in a heterogeneous medium with multi-rate mass transfer properties. Moreover we use the TDRW method in connection with a stochastic model for the heterogeneity in order to upscale heterogeneous diffusion processes. For a certain class of heterogeneity, the upscaled dynamics obey a CTRW.
Analytically we upscale diffusion in highly heterogeneous media by using a multicontinuum representation of the media. Using volume and ensemble averaging we derive a multicontinuum model that can explain anomalous diffusion behavior and link it with a suitable local scale description of the medium heterogeneity.
Finally, we integrate the multicontinuum model derived in the context of aquifer modelling. We derive a multicontinuum catchment model that can explain anomalous behavior observed in the aquifer dynamics at basin scale. We identify the physical mechanisms that induce anomalous behaviour and we determine the time scales that control its temporal evolution.