8 research outputs found
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Reactive Flows in Deformable, Complex Media
Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is variable, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, or biological systems. Such models include various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. Having this as the background theme, this workshop focused on novel techniques and ideas in the analysis, the numerical discretization and the upscaling of such problems, as well as on applications of major societal relevance today
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Numerical Analysis of a System of Parabolic Variational Inequalities with Application to Biofilm Growth
In this work we consider a mathematical and computational model for biofilm growth and nutrient utilization. In particular, we are interested in a model appropriate at a scale of interface. The model is a system of two coupled nonlinear diffusion--reaction partial differential equations (PDEs). One of these PDEs is subject to a constraint, which can be characterized as a parabolic variational inequality (PVI). Solutions to PVI have low regularity which limits the numerical scheme to low order. We analyze the numerical approximations of this model by implementing two numerical methods: (i) low order Galerkin finite element method, and (ii) mixed finite element method with the lowest order Raviart-Thomas elements. We show the well-posedness of the approximate problems, derive rigorous error estimates, and present numerical experiments in 1D, 2D, and 3D that confirm the predicted estimates. We also illustrate the behavior of biofilm and nutrient dynamics in simple and in complex porescale geometries
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Modeling Flow and Transport at Pore Scale with Obstructions
In this thesis we study mathematical and computational models for phenomena of flow and transport in porous media in the presence of changing pore scale geometries. The differential equations for the flow and transport models at Darcy scale involve the coefficients of permeability, porosity, and tortuosity which depend on the pore scale geometry. The models we propose help to understand how the presence of obstruc- tions impacts the Darcy scale models. The particular changes in pore scale geometry we consider are due to the formation of obstructions to the flow, and come from two important applications of interest, biofilm clogging and gas hydrate crystal plugging up the pores. The direct simulations or experiments of these processes at pore scale is generally unfeasible or impractical.
We propose two computationally efficient mathematical and computational models to simulate the formation of the obstructions. The first method extends the phase separation model based on the Allen-Cahn equation; in our variant we add volume constraints and additional localization functions. The second method we propose is a Markov Chain Monte Carlo method inspired by the Ising model; here we use heuristics to choose the particular coefficients which guide the formation of obstructions of a particular type.
After we generate independent realizations of the obstructed geometries, we solve flow and transport problems at pore scale. Next we use the technique called upscaling which carries the information to larger scale by averaging, and we are able to derive the ensemble of Darcy scale properties for a collection of generated pore scale geometries with obstructions. We show how these techniques can be used in synthetic geometries as well as in geometries obtained from imaging. In addition, we see that the permeability coefficient is not merely a function of porosity, but is rather highly dependent on the type of obstruction growing at the pore scale
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Multiphase Flow and Transport in Porous Media with Phase Transition at Multiple Scales: Modeling, Numerical Analysis, and Simulation
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 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
Tracing back the source of contamination
From the time a contaminant is detected in an observation well, the question of where and when the contaminant was introduced in the aquifer needs an answer. Many techniques have been proposed to answer this question, but virtually all of them assume that the aquifer and its dynamics are perfectly known. This work discusses a new approach for the simultaneous identification of the contaminant source location and the spatial variability of hydraulic conductivity in an aquifer which has been validated on synthetic and laboratory experiments and which is in the process of being validated on a real aquifer