In this thesis we study mathematical and computational models for phenomena of flow and transport in porous media in the presence of changing pore scale geometries. The differential equations for the flow and transport models at Darcy scale involve the coefficients of permeability, porosity, and tortuosity which depend on the pore scale geometry. The models we propose help to understand how the presence of obstruc- tions impacts the Darcy scale models. The particular changes in pore scale geometry we consider are due to the formation of obstructions to the flow, and come from two important applications of interest, biofilm clogging and gas hydrate crystal plugging up the pores. The direct simulations or experiments of these processes at pore scale is generally unfeasible or impractical.
We propose two computationally efficient mathematical and computational models to simulate the formation of the obstructions. The first method extends the phase separation model based on the Allen-Cahn equation; in our variant we add volume constraints and additional localization functions. The second method we propose is a Markov Chain Monte Carlo method inspired by the Ising model; here we use heuristics to choose the particular coefficients which guide the formation of obstructions of a particular type.
After we generate independent realizations of the obstructed geometries, we solve flow and transport problems at pore scale. Next we use the technique called upscaling which carries the information to larger scale by averaging, and we are able to derive the ensemble of Darcy scale properties for a collection of generated pore scale geometries with obstructions. We show how these techniques can be used in synthetic geometries as well as in geometries obtained from imaging. In addition, we see that the permeability coefficient is not merely a function of porosity, but is rather highly dependent on the type of obstruction growing at the pore scale.
Funding Statement (additional comments about funding)
The research in this work was partially funded by the National Science Foundation (NSF) as part of two projects. First, project NSF DMS-1522734 “Phase transitions in porous media across multiple scales,” (Principal Investigator Malgorzata Peszynska). Second, project NSF DMS-1738014, 2017 “APG: Improve STEM graduate student pre- paredness for workforce (Internship at LNBL for Joe Umhoefer),” (Principal Investigator Malgorzata Peszynska).