Subsurface processes represent a class of important phenomena that occurs in many natural and anthropogenic settings. This dissertation deals with mathematical modeling of multiscale subsurface processes. Subsurface processes can be simulated at multiple scales with variable degrees of fidelity. We investigate the effect of subsurface phenomena on macroscopic dynamics in three particular cases.

In Chapter 2, we consider microscopic (pore-scale) features of reactive transport that cannot be properly resolved in macroscopic (Darcy-scale) models. While microscopic descriptors might be closer to reality, they are computationally unfeasible when deployed on a macroscale. Hybrid algorithms combine the physical fidelity of a microscopic model with the computational efficiency of its macroscopic counterpart. We develop a hybrid model of dynamic reactive fronts in an open fracture, with a chemical reaction occurring in the zone of contact between two dissolved species. Our numerical experiments demonstrate that the hybrid model outperforms its microscopic and macroscopic counterparts in terms of computational time and representational accuracy, respectively.

In Chapter 3, we consider the effect of temperature on interfacial dynamics where small fluctuations affect instabilities. Viscous fingering is a hydrodynamic instability that can be caused by a viscosity difference across a traveling thermal front. We investigate the problem of displacement stability in the filtration flow for two-phase filtration in the presence of heat transfer. Fingers instability was analyzed for isothermal and non-isothermal processes. Our synergistic numerical and theoretical investigation confirms the development of instability due to differences in viscosities. Our results demonstrate that controlling the stability of such a system requires exponentially lowering the fluid's viscosity when the temperature is increased.

Even by implementing multiple simplifying assumptions, most real-world problems are extremely complex and, therefore, quite costly to thoroughly analyze. In Chapter 4, we utilize a supporting vector regression (SVR) method to create a surrogate model, based on a limited amount of data, capable of describing the behavior of a set of wells with various input parameters. SVR is applied to analyze and to deconstruct the groundwater geochemistry observed in the regional aquifer for characterization of contaminant sources. Using synthetic and real-world data, we demonstrate that our SVR model is capable of accurately building a surrogate model using only a limited amount of experimental data and simulation results. Further, we demonstrate that global sensitivity analysis can be applied to our SVR model and it can produce accurate results at significantly lower computational cost.

Overall, this thesis investigates multiple aspects of multi-scale fluid dynamics by both developing physics-based numerical models and data-driven computational approaches.