Statistical Dependence-Structure and Process-Memory Analysis of Non-Fickian Transport Processes in Porous Media

Many applied simulations of transport processes in porous media work on a regional scale. Highly-resolved pore-scale transport simulations exist, but computational limitations constrain the volumetric size of the investigated porous media to a few cubic centimeters. This is why upscaling becomes necessary. Upscaling often means to average over a predefined (representative elementary) volume. Within this volume, all the small scale heterogeneities are averaged out. Classical transport formulations (e.g., ADE or PTRW) that do not compensate appropriately for the loss of detail due to the averaging and thus, fail in representing the early and late time arrival of contaminants. Transport processes that go beyond the classical transport descriptions are known as non-Fickian transport. Modeling frameworks for non-Fickian transport exist, but these models usually consider transport as an independent stochastic process. However, this is not always true.

In this project an empirical particle-based transport simulation at the pore scale is used, to improve our understanding of the complex dependencies of transport. A copula-based analysis tool is developed, which is able to represent the full non-linear complexity of the dependence structure. The objective is to derive an appropriate stochastic processes that represents all aspects of non-Fickian transport. This stochastic process can then be used to develop a dispersion model that accounts for the neglected heterogeneity (lost during the upscaling). Furthermore, the pore-scale transport simulation is used to analyze how non-Fickian conditions affect mixing and dilution. Therefore, the two-particle separation distance over time is evaluated and the empirical probability distribution of the squared separation distance as a function of travel time and initial separation is derived. This can be understood as the fingerprint of mixing and dilution. As the fingerprints are able to analyze all moments at once in a single framework, these represent all statistical aspects of mixing and dilution over time.

The fingerprints allow a very detailed statistical analysis of mixing and dilution processes in Fickian or non-Fickian transport regimes.