Subsurface processes represent a class of important phenomena that occurs in many nat- ural and anthropogenic settings. This dissertation deals with mathematical modeling of multi- scale subsurface processes. Subsurface processes can be simulated at multiple scales with vari- able 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 can- not 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 dis- solved 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 syn- ergistic 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 param- eters. 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 us- ing 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.

This dissertation deals with multiscale and multi-physics mathematical modeling and global sensitivity analysis. Multiscale and multi-physics problems are ubiquitous in every eld of science and engineering. For example, micro- or even nano-scale material properties often in uence their large-scale properties, and microscopic interfaces between dierent materials affect bulk transport phenomena.

We develop and deploy methods of global (variance-based) sensitivity analysis to determine how, and to what degree, nano-scale characteristics of (nano)porous materials aect and give rise to bulk (Darcy-scale) properties.

We deploy stochastic multiscale algorithms to solve several problems of relevance in materials science and biology, and conduct rigorous sensitivity analysis and uncertainty quantication. In Chapter 2, we present a novel hybrid algorithm to ameliorate high

computational costs typical of multiscale, multi-physics simulations, and apply it to solve a chemotaxis-diusion-reaction problem. In Chapter 3, we report on our pore- and multi-scale simulations and perform a global (variance based) sensitivity analysis for uncorrelated input parameters. This chapter also contains results of our uncertainty quantication analysis for this multiscale problem, which is based on a (generalized) polynomial chaos expansion (gPCE). This UQ strategy can be used to identify a set of pore-scale characteristics for robust materials design. In Chapter 4, we introduce a novel graph-theoretic approach to conduct global sensitivity analyses in the presence of correlated inputs.

Transport in networks represents a class of important phenomena that occur in many natural and anthropogenic settings. This dissertation deals with two such processes: heat transport in fracture networks in the subsurface and resiliency response of a cyber network to virus propagation. Numerical simulations of heat transfer provide insight into the performance of subsurface systems (e.g. geothermal reservoirs) and the potential for their enhancement. Standard, grid-based method of heat transfer are computationally expensive, often prohibitively so [Karra et al., 2018], due to a vast discrepancy of scales between fracture apertures (millimeter scale) and the ambient rock matrix domain (meter scale). In Chapter 2, we present a mesh-free time-domain particle method for modeling heat transfer in highly heterogeneous fractured media. Our method is orders of magnitude faster than its grid-based alternatives and is readily adaptable to different network configurations. We deploy this method to model heat extraction from geothermal reservoirs by using a fractal network representing faults and damage zones. Our analysis reveals anomalous behavior of heat transfer in fractured environments due to existence of preferential flow paths. It also demonstrates a significant impact of the networks topology on the performance of geothermal reservoirs. In Chapter 3, this method is used to investigate the impact of fracture network properties on geothermal performance. Specifically, we explore how fracture-network topology and matrix-block size distribution control, respectively, the advective and conductive mechanisms of heat transfer in fractures and ambient matrix, as well as the heat flux exchanged between these structures. To accomplish this goal, we examine two different conceptual descriptors of fracture networks over a variety of hydraulic conditions and fracture-generating parameters, and the effects of topologic properties and hydraulic conditions on computational time. In Chapter 4 we further generalize our mesh-free particle method by removing the physical assumption of one-dimensional conduction in the matrix. This is achieved by accounting for both longitudinal and transversal heat conduction in the matrix. It looks at the impact of removing set assumption and identifies the parameter set where it is of importance. Finally, in Chapter 5, we explore transport in anthropogenic networks by presenting a method to simulate the resiliency response of a cyber network to a virus propagation. We identify the need for systematic data collection and appropriate metrics to enable data driven optimization of the rule base. As well as demonstrate that the optimal number of rules necessary to regulate a cyber network efficiently is likely to be small and focused on specific critical functions that the system needs to maintain.

Nanoporous materials are used in the variety of fields, ranging from medicine and biosensors to clean energy and purification. Technological advances have enabled one to manufacture nanoporous materials with a prescribed pore structure. This raises a possibility of using controllable pore scale properties (e.g., pore size distribution, pore connectivity and tortuosity) to design materials with desired macroscopic properties (e.g., porosity, effective diffusion coefficient and effective electrical conductivity). Upscaling techniques, such as homogenization via multiple-scale expansions, provide a framework to connect these two scale. This research uses such techniques to optimize macroscopic properties of a material by using its microscopic properties as decision variables.

This research aims to obtain qualitative understanding and quantitative predictions of macroscopic properties of nanoporous materials to transmit solutes that undergo non-equilibrium adsorption and local electrochemical surface reactions at the fluid-solid interface. The first part of this work focuses on the design of hierarchical nanoporous materials with optimal permeability and sorption capacity. A class of nanoporous materials whose pore space consists of ordered nanopores interconnected by nano-channels is considered. Anisotropic effective diffusion coefficients and adsorption coefficients of such materials are expressed in terms of pore properties and connectivity Nano-channels can significantly alter diffusive properties and increase its adsorbing capacity.

The second part of this work contains a macroscopic model of ion transport in electrically charged nanoporous materials. The corresponding effective diffusion coefficients, electric conductivity and transference numbers account for dynamic changes in the electrical double layer (EDL), possible overlap of EDLs in nanopores, and electrochemical conditions (i.e., concentration of ions in the electrolyte). Our effective coefficients are derived from the first principles and vary with a range of electrochemical conditions (e.g., initial concentration of ions in the electrolyte). The resulting model predictions of the EDLC voltage response match the experimental data better than the original model does.

The last part of this work is devoted to derivation of macroscopic properties of three-dimensional dendritic spines. The effective diffusion coefficient estimated with this analysis is used to quantify travel time of electric signal through the spine.

Continually rising energy needs in conjunction with negative externalities of fossil fuel use demand the diversification of energy resources. However, fossil hydrocarbons are also used as raw materials for a vast web of manufacturing of industrial and consumer goods. The use of plant based raw materials as substitutes for fossil materials in fuels and manufacturing has been demonstrated successfully, and microalgae are an extremely diverse and promising resource in this category. One of the chief challenges in the implementation of a bio-economy is efficient processing and conversion of biomass to platform chemicals. In order to maximize extraction efficiency, pretreatment is employed to effect cell rupture. Many pretreatment processes have been implemented using empirical operating curves, but the fundamental energy requirements of the cell disruption process have not been thoroughly explored. To fill this void, a constitutive model of cell rupture energy is derived here and implemented for low frequency power ultrasound processing of Chlorella vulgaris. A sensitivity analysis of the constitutive model is performed, identifying cell diameter as a high sensitivity input. Measured distribution of microalgae cell diameters is then used as a fixed input to Monte Carlo simulations of cell rupture energy from the constitutive model. The influence of growth media on microalgal growth rate is then investigated, and the resultant biomass subjected to power ultrasound processing to determine the effect of media choice on processing efficiency. The theoretical kinetics of cell rupture via power ultrasound induced cavitation is developed next. A reactor model is introduced to convert reactor kinetics to reaction zone kinetics. An elementary reaction model is then developed in the context of the constitutive model, leading to the introduction of an alternative reaction mechanism employing a critical distance parameter to capture the relative energies and proximity of cells and cavitation bubbles. This reaction mechanism is extended to generate an explicit expression for the cell disruption first order rate constant in terms of cell properties and power ultrasound operating parameters. This rate constant is mapped over the parameter space, then used to back-calculate cavitation rate, and finally extended to incorporate PDF parameter inputs.

A plethora of computational techniques have been developed for computing quantities of interest in “multiscale” and “multiphysics” problems combining processes occurring on a broad spatiotemporal range. However, a dearth exists in systematic studies of the impact of random fluctuations on the predictive ability and numerical properties of these algorithms. We consider two nonintrusive approaches for multiphysics problems with random noise: domain decomposition and stochastic collocation. A mass-conserving domain decomposition achieving tight Newton- or Picard-based coupling between linear diffusion equations, one having a Gaussian white-noise source term, reveals that Newton's iteration scales linearly with noise amplitude, while Picard's iteration may scale superlinearly. For a given solution error, fully-converged (``implicit'') coupling is more efficient than single-iteration (``explicit'') coupling at low noise strength; at high noise amplitudes, this remains true provided that the time interval between two subsequent implicit coupling communications is sufficiently long. A similar strategy using Jacobian-free Newton-Krylov iteration to solve a highly nonlinear, multiscale diffusion problem forced by a truncated Gaussian boundary noise shows that ensuring path-wise continuity of the state variable and its flux, as opposed to continuity in the mean, accurately propagates random fluctuations and correctly captures system dynamics. Implicit coupling is more efficient than explicit coupling at all coefficients of variation considered, and domain decomposition with path-wise implicit coupling resolves temporally correlated boundary fluctuations when the correlation time exceeds some multiple of an appropriately defined characteristic diffusion time. Application of stochastic collocation to estimate the energy deposition into a brain tumor via X-ray irradiation with parametric uncertainty reveals that the uncertain parameters' coefficients of variation may be amplified by the problem's nonlinearity to the extent that the predictive uncertainty in the energy deposition almost equals the prediction itself. Algorithm refinement for the Ginzburg-Landau equation (GLE) demonstrates the need for adding a coarse-scale random source term to correctly propagate fine-scale Ising fluctuations throughout the computational domain. A moment-based approach with Gaussian closure enabling direct computation of the state variable's statistical moments is shown to be an accurate, and potentially more efficient, alternative to numerical time integration of the system state. A statistically learned stochastic GLE exhibits optimal predictive capacity at a complexity that may differ from that of standard models in the literature. This approach enables data-driven computation of the coarse-scale noise term's amplitude.

At their best, mathematical models of physical and biological systems strive to represent the nature faithfully while maximizing their simplicity and efficiency. Large numbers of variables involved in biophysical and geophysical processes make it necessary to develop predictive models with the minimum conceivable level of complexity. The combination of coupled phenomena, spatial and temporal scales, different regimes of behavior, and plethora of agents interacting simultaneously constitutes an unmanageable amalgamation of factors that unavoidably reduce both performance and tractability of mathematical models. Creating models with the optimal complexity requires finding the balance between the following characteristics: accuracy, computational cost, applicability to multiple scales and regimes, ability to represent realistic scenarios, and versatility. In this dissertation, we present analytical models of heat transfer in fractured porous media, in vitro and in vivo kinetics models of polymerization, and a hybrid algorithm for reaction-diffusion systems. These models provide more accurate representations of reality than their current counterparts, and they do so at the small fraction of the computational cost.