Event Detail

Event Type: 
Ph.D.Defense
Date/Time: 
Monday, May 24, 2021 - 14:00 to 16:00
Location: 
Zoom - If you are interested in attending this presentation, please send an email to Nikki Sullivan - nikki.sullivan@oregonstate.edu - to request Zoom log in details.
Abstract: 

In this dissertation we consider two flow and transport models in porous media: (1) methane gas transport models for hydrate formation and dissociation at Darcy scale, and (2) coupled flow and biofilm-nutrient model at pore-scale with upscaling to Darcy-scale. Both projects are motivated by the challenges from real-life applications in the subsurface.

For (1), we consider methane gas transport in the hydrate stability zone at equilibrium and non-equilibrium conditions. The numerical analysis of the problem has the following mathematical challenges. The two unknowns in the equilibrium (EQ) model are bound together by a relationship called nonlinear complementarity constraint and represented by a multivalued graph. In the non-equilibrium (kinetic, KIN) model, this relationship is replaced by an evolution problem with a multivalued monotone graph. We define and analyze a numerical scheme which combines upwind treatment of advection with an implicit treatment of nonlinearity. To analyze, we frame the EQ model as a conservation law with a nonsmooth space-dependent flux function, similar to those that are known in other applications including the two-phase flow in a heterogeneous porous medium, traffic flow on roads, and nonlinear elasticity in mixed materials. Our main result is the weak stability of an upwind-implicit scheme for a regularized EQ model, and to our knowledge, this is the first such result for the hydrate model. For KIN model we also prove weak stability in an appropriate product space and confirm the rate of convergence $O(\sqrt{h})$ for both equilibrium and kinetic models. The KIN model is useful to simulate the hydrate phase change at shorter time scales, e.g., after a seismic event, and we focus on a particular variant which we show is robust across the unsaturated and saturated conditions. We choose various scenarios relevant to the applications to illustrate the theory.

For (2), we address the many challenges present for a coupled model in a complicated pore-scale geometry with a free boundary of biofilm phase. Our goal is to have a robust monolithic flow and nonlinear transport model with which we aim to avoid regridding when the free boundary moves. We show its robustness through various numerical experiments. The model is a variational inequality type with a nonsingular diffusivity which together ensure the volume constraint that must be satisfied. For the flow, we consider Brinkman flow with spatially varying permeability which can simulate the flow in (somewhat) permeable domains, as well as around them. We employ operator splitting and some time-lagging as well as our version of the Marker-And-Cell method adapted to the heterogeneous Brinkman model on a time-staggered grid. Additionally we introduce a new modeling construction that ``promotes'' the adhesion of biofilm to the surface. We present simulation results including the upscaling and perform Monte Carlo simulations to construct the probability distributions of upscaled permeability which represents the randomness of complex geometry with biofilm.

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