A hybrid surface/subsurface flow and transport model is developed that blends distributed parameter models with simpler lumped parameter models. The hybrid model solves the channel flow and saturated groundwater flow domains in continuous time using fully distributed physically-based formulations. This system is supplemented with the overland flow and unsaturated groundwater flow that uses lumped parameter descriptions in discrete time. In the proposed model, a one-dimensional channel flow model is dynamically coupled with a two-dimensional vertically-averaged groundwater flow model along the river bed. As an alternative to the commonly applied iterative solution technique, a so-called simultaneous solution procedure is developed to provide a better understanding to the coupled flow problem. This new methodology is based on the principle of solving the two flow domains within a single matrix structure in a simultaneous manner. In addition to the flow model, a coupled contaminant transport model is also developed to simulate the migration of contaminants between surface and subsurface domains. The contaminant transport model dynamically couples a one-dimensional channel transport model with a two-dimensional vertically-averaged groundwater transport model. The coupling is performed at the river bed interface via advective and dispersive transport mechanisms. A modified extension of the proposed simultaneous solution procedure is also implemented to solve the coupled contaminant transport problem. The dynamic coupling provides the much needed understanding for the continuity of contaminants in strongly interacting surface/subsurface systems such as a river and an unconfined aquifer. The coupled flow and transport models are applied to the lower Altamaha watershed in southern Georgia. The flow model is used to perform simulations of hydrologic and hydraulic conditions along the river and in the dynamically linked surfacial aquifer. The model predicted the flood patterns including the magnitude of peaks and their arrival times with accuracy. Under the given flow conditions, the transport model is then implemented to test alternative contaminant transport patterns both in the river and within the aquifer. It has been found that the channel network would serve as a conduit for rapid transport of contaminants within the aquifer to large distances in small time frames.

A growing demand for more detailed modeling of subsurface physics as ever more challenging reservoirs - often unconventional, with significant geomechanical particularities - become production targets has moti-vated research in coupled flow and geomechanics. Reservoir rock deforms to given stress conditions, so the simplified approach of using a scalar value of the rock compressibility factor in the fluid mass balance equation to describe the geomechanical system response cannot correctly estimate multi-dimensional rock deformation. A coupled flow and geomechanics model considers flow physics and rock physics simultaneously by cou-pling different types of partial differential equations through primary variables. A number of coupled flow and geomechanics simulators have been developed and applied to describe fluid flow in deformable po-rous media but the majority of these coupled flow and geomechanics simulators have limited capabilities in modeling multiphase flow and geomechanical deformation in a heterogeneous and fractured reservoir. In addition, most simulators do not have the capability to simulate both coarse and fine scale multiphysics. In this study I developed a new, fully implicit multiphysics simulator (TAM-CFGM: Texas A&M Coupled Flow and Geomechanics simulator) that can be applied to simulate a 2D or 3D multiphase flow and rock deformation in a heterogeneous and/or fractured reservoir system. I derived a mixed finite element formu-lation that satisfies local mass conservation and provides a more accurate estimation of the velocity solu-tion in the fluid flow equations. I used a continuous Galerkin formulation to solve the geomechanics equa-tion. These formulations allowed me to use unstructured meshes, a full-tensor permeability, and elastic stiffness. I proposed a numerical upscaling of the permeability and of the elastic stiffness tensors to gener-ate a coarse-scale description of the fine-scale grid in the model, and I implemented the methodology in the simulator. I applied the code I developed to the simulation of the problem of multiphase flow in a fractured tight gas system. As a result, I observed unique phenomena (not reported before) that could not have been deter-mined without coupling. I demonstrated the importance and advantages of using unstructured meshes to effectively and realistically model a reservoir. In particular, high resolution discrete fracture models al-lowed me to obtain more detailed physics that could not be resolved with a structured grid. I performed numerical upscaling of a very heterogeneous geologic model and observed that the coarse-scale numerical solution matched the fine scale reference solution well. As a result, I believed I developed a method that can capture important physics of the fine-scale model with a reasonable computation cost.

Glass fiber based polymer composites based injection molded parts provide a light-weight high-strength alternative for use in automobile applications. These composites have enhanced mechanical properties compared to those of pure polymers, if the fibers are oriented in the right direction. One of the major challenges in processing of these composites is to control the fiber orientation in the final product. The evolution of short glass fiber orientation in a center-gated disk was experimentally determined along the radial direction at three different heights representative of the shell, transition and core layers, respectively. Orientation data along the shell and transition layers in the lubrication region show shear flow effects, which tends to align the fibers along the flow direction. In the core layer, where the extension in the "-direction dominates, fibers tend to get aligned along the "-direction. In the frontal flow region orientation in the flow direction drops in all three layers due to fountain flow effects. Fiber orientation predictions in coupled and decoupled transient simulations using the Folgar-Tucker model, and the two slow versions of the Folgar-Tucker model, namely the slip Folgar-Tucker model and the reduced strain closure (RSC) model were compared with the experimental data. Measured inlet orientation was used in all simulations and model parameters were determined by fitting model predictions to rheological data under startup of shear. Pseudo-concentration method was implemented for the modeling of the advancing front and fountain flow effects in the region near the front. Discontinuous Galerkin finite element method and a third order Runge-Kutta total variance diminishing time integration scheme were implemented for the solution of the orientation and transport equations. In the lubrication region of the shell layer, all three orientation models provided a good match with the experimental data. In the frontal region, fountain flow simulations showed characteristic features seen in r- and z-profiles of orientation, although the experimental data showed these features at a relatively larger distance behind the front while the simulations predicted these effects only up to a small distance behind the front. On the other hand, orientation predictions with the Hele-Shaw flow approximation showed significant over-predictions in the frontal region. With model parameters determined from fitting to rheological data, coupling did not show any significant improvements. However, with the use of a smaller value of the fiber interaction parameter, coupling showed significant improvement in orientation predictions in all three layers in the frontal region. The simulation scheme was extended to long fiber systems by comparing available long fiber orientation data in a center-gated disk with model predictions using the Bead-Rod model which considers fiber bending, a property exhibited by long semi-flexible fibers. The Bead-Rod model showed improvements over rigid fiber models in the lubrication region of the shell layer. However, close to the front, both models showed similar predictions. In fountain flow simulations, the flow features seen in the r- and z-profiles were much better predicted with both the models while Hele-Shaw flow approximation showed over-prediction of orientation in the flow direction, especially in the shell layer. / Ph. D.

This thesis discusses and develops one approach to solve parabolic partial differential equations with random input data. The stochastic problem is firstly transformed into a parametrized one by using finite dimensional noise assumption and the truncated Karhunen-Loeve expansion. The approach, Monte Carlo discontinuous Galerkin (MCDG) method, randomly generates $M$ realizations of uncertain coefficients and approximates the expected value of the solution by averaging M numerical solutions. This approach is applied to two numerical examples. The first example is a two-dimensional parabolic partial differential equation with random convection term and the second example is a benchmark problem coupling flow and transport equations. I first apply polynomial kernel principal component analysis of second order to generate M realizations of random permeability fields. They are used to obtain M realizations of random convection term computed from solving the flow equation. Using this approach, I solve the transport equation M times corresponding to M velocity realizations. The MCDG solution spreads toward the whole domain from the initial location and the contaminant does not leave the initial location completely as time elapses. The results show that MCDG solution is realistic, because it takes the uncertainty in velocity fields into consideration. Besides, in order to correct overshoot and undershoot solutions caused by the high level of oscillation in random velocity realizations, I solve the transport equation on meshes of finer resolution than of the permeability, and use a slope limiter as well as lower and upper bound constraints to address this difficulty. Finally, future work is proposed.

Coupled systems are ubiquitous in modern engineering and science. Such systems can encompass fluid dynamics, structural mechanics, chemical species transport and electrostatic effects among other components, all of which can be coupled in many different ways. In addition, such models are usually multiscale, making their numerical simulation challenging, and necessitating the use of adaptive modeling techniques. The multiscale, multiphysics models of electrosomotic flow (EOF) constitute a particularly challenging coupled flow system. A special feature of such models is that the coupling between the electric physics and hydrodynamics is via the boundary. Numerical simulations of coupled systems are typically targeted towards specific Quantities of Interest (QoIs). Adjoint-based approaches offer the possibility of QoI targeted adaptive mesh refinement and efficient parameter sensitivity analysis. The formulation of appropriate adjoint problems for EOF models is particularly challenging, due to the coupling of physics via the boundary as opposed to the interior of the domain. The well-posedness of the adjoint problem for such models is also non-trivial. One contribution of this dissertation is the derivation of an appropriate adjoint problem for slip EOF models, and the development of penalty-based, adjoint-consistent variational formulations of these models. We demonstrate the use of these formulations in the simulation of EOF flows in straight and T-shaped microchannels, in conjunction with goal-oriented mesh refinement and adjoint sensitivity analysis. Complex computational models may exhibit uncertain behavior due to various reasons, ranging from uncertainty in experimentally measured model parameters to imperfections in device geometry. The last decade has seen a growing interest in the field of Uncertainty Quantification (UQ), which seeks to determine the effect of input uncertainties on the system QoIs. Monte Carlo methods remain a popular computational approach for UQ due to their ease of use and "embarassingly parallel" nature. However, a major drawback of such methods is their slow convergence rate. The second contribution of this work is the introduction of a new Monte Carlo method which utilizes local sensitivity information to build accurate surrogate models. This new method, called the Local Sensitivity Derivative Enhanced Monte Carlo (LSDEMC) method can converge at a faster rate than plain Monte Carlo, especially for problems with a low to moderate number of uncertain parameters. Adjoint-based sensitivity analysis methods enable the computation of sensitivity derivatives at virtually no extra cost after the forward solve. Thus, the LSDEMC method, in conjuction with adjoint sensitivity derivative techniques can offer a robust and efficient alternative for UQ of complex systems. The efficiency of Monte Carlo methods can be further enhanced by using stratified sampling schemes such as Latin Hypercube Sampling (LHS). However, the non-incremental nature of LHS has been identified as one of the main obstacles in its application to certain classes of complex physical systems. Current incremental LHS strategies restrict the user to at least doubling the size of an existing LHS set to retain the convergence properties of LHS. The third contribution of this research is the development of a new Hierachical LHS algorithm, that creates designs which can be used to perform LHS studies in a more flexibly incremental setting, taking a step towards adaptive LHS methods. / text

In this thesis we are concerned with the numerical simulation of the in- teraction of compressible viscous flow and an elastic structure in 2D. For the elastic deformation we use a 2D linear model and nonlinear St. Venant- Kirchhoff and neo-Hookean models. The flow is described by the compressible Navier-Stokes equations written in the arbitrary Lagrangian-Eulerian (ALE) form in order to take into account the time-dependence of the flow domain. The discretization of both the flow problem and the elasticity problem is re- alized by the discontinuous Galerkin finite element method (DGM). We focus on testing the DGM applied to the solution of the flow and elasticity prob- lems. Furthermore, we discuss the coupling algorithm and the technique, how to deal with the deformation of the computational domain for the fluid flow problem. Our work is motivated by the biomedical applications. Numerical experiments include numerical simulation of vibrations of human vocal folds induced by the compressible viscous flow.

The vast majority of oil and gas reserves are trapped in fractured carbonate reservoirs. Most carbonate reservoirs are naturally fractured, with fractures ranging from millimeter- to kilometer-scale. These fractures create complex flow behaviors which impact reservoir characterization, production performance, and, eventually, total recovery. As we know, bridging the gas from plug to near-wellbore, eventually to field scales, is a persisting challenge in modeling Naturally Fractured Reservoirs (NFRs). This dissertation will focus on assessing the fundamental flow mechanisms in fractured rocks at the plug scale, understanding the governing upscaling parameters, and ultimately, developing fit-for-purpose upscaling tools for field-scale implementation. In this dissertation, we first focus on the upscaling of rock fractures under the laminar flow regime. A novel analytical model is presented by incorporating the effects of normal aperture, roughness, and tortuosity. We then investigate the stress-dependent hydraulic behaviors of rock fractures. A new and generalized theoretical model is derived and verified by a dataset collected from public experimental resources. In addition, an efficient coupled flow-geomechanics algorithm is developed to further validate the proposed analytical model. The physics of matrix-fracture interaction and fluid leakage is modeled by a high-resolution, micro-continuum approach, called extended Darcy-Brinkman-Stokes (DBS) equations. We observe the back-flow phenomena for the first time. Machine learning is then implemented into our traditional upscaling work under complex physics (e.g., initial and Klinkenberg effects). We finally consolidate the lab-scale upscaling tools and scale them up to the field scale. We develop a fully coupled hydro-mechanical model based on the Discrete-Fracture Model (DFM) in fractured reservoirs, in which we incorporate localized effects of fracture roughness at the field-scale.