Graduate Thesis Or Dissertation
 

Multiphase Flow and Transport in Porous Media with Phase Transition at Multiple Scales: Modeling, Numerical Analysis, and Simulation

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/xs55mk93b

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  • In this dissertation we consider two application specific flow and transport models in porous media at multiple scales: 1) methane gas transport models for hydrate formation and dissociation in the subsurface under two-phase conditions, and 2) coupled flow and biomass-nutrient model for biofilm growth in complex geometries with biofilm, and its impact via upscaling from pore scale to Darcy scale on Darcy scale permeability. Both projects are motivated by the challenges from real-life applications in the subsurface. First we consider the simplified methane gas transport models at the Darcy scale under equilibrium and non-equilibrium conditions. The equilibrium model (EQ) is 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. There are two unknowns in (EQ) models which are bound together by a relationship called nonlinear complementarity constraint and represented by a multivalued graph. Our main result is the weak stability of an upwind-implicit scheme for a regularized (EQ). To our best knowledge, this is the first such result for the transport model. We also consider kinetic models which approximate (EQ) and are useful when we simulate the hydrate phase change at shorter time scales, e.g., after a seismic event. After a rigorous analysis of three kinetic models, we focus on the analysis of a particular model robust across the unsaturated and saturated conditions. We also prove the weak stability of this model and confirm the rate of convergence $O(\sqrt{h})$ for both equilibrium and kinetic models. We choose various equilibrium and non-equilibrium scenarios relevant to the applications, and we provide 1d simulation results which illustrate the theory. Next we study the coupled biomass-nutrient-flow dynamics in a complicated pore scale geometry. Our goal is to describe a new monolithic coupled flow and biomass-nutrient model and to show its robustness through various numerical experiments. The biomass-nutrient model is of variational inequality type blended with nonsingular diffusivity to ensure the volume constraint while enhancing the biofilm growth mechanism. For the flow, we consider the Brinkman flow with spatially varying permeability which accounts for the flow in (somewhat) permeable domains as well as around these. We apply the flow and biofilm growth model to the entire domain so that the model and the coupling are monolithic. Our overall scheme follows operator splitting and time lagging: we solve advection explicitly by the upwind method and diffusion-reaction together using CCFD with time-lagged diffusion coefficients. For flow, we use our version of the Marker-And-Cell method adapted to the heterogeneous Brinkman model on a time-staggered grid. We also present simulation results to show the robustness of our model. To handle the sensitivity of the biomass-nutrient model to its initial data, we introduce a new modeling construction which "promotes" the adhesion of biofilm to the surface. Then we perform the Monte Carlo simulations and construct the probability distributions of upscaled permeability which represent the randomness of complex geometry with biofilm.
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  • This research was partially funded by the National Science Foundation (NSF) project NSF DMS-1522734 "Phase transitions in porous media across multiple scales," and NSF DMS-1912938 "Modeling with Constraints and Phase Transitions in Porous Media" (PI: Malgorzata Peszynska).
  • W. Martin & Joyce B. O’Neill Endowed Fellowship.
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