Homogenization of a diffusion convection equation, with random source terms periodically distributed

We are interested to study $u(x,t)$ , the evolution in time of the concentration, which is transported by diffusion and convection from a "sources site" made of a large number of similar "local sources". For this we consider a "local model" based on a general diffusion convection equation: \begin{eqnarray} \label{intro_eq} \partial_t u^\eps-\mathrm {div}(a(x)\nabla u^\eps)+\mathrm {div}(b(x) u^\eps)=f^\eps;\qquad{ }\\ u^\eps\big|_{t=0}=0,\qquad \frac{\partial}{\partial n_a}u^\eps\cdot n(x)-b(x)\cdot n(x)u^\eps+ \lambda u^\eps=0 .\qquad{ } \end{eqnarray} where the sources density f^\eps comes from a set of "local sources" periodically repeated and lying on a same plan $\Sigma$; f^\eps(x,t)= \bigcup\limits_{\textbf{j}\in\mathbb Z^2}f_\textbf{j}(x,t). Assuming the release curve ( source emission vs. space and time),$f_\textbf{j}(,.)$, of each local source, being random, our aim is to give a mathematical model describing the global evolution of such a system

Two phase partially miscible flow and transport modeling in porous media: application to gas migration in a nuclear waste repository

We derive a compositional compressible two-phase, liquid and gas, flow model for numerical simulations of hydrogen migration in deep geological repository for radioactive waste. This model includes capillary effects and the gas high diffusivity. Moreover, it is written in variables (total hydrogen mass density and liquid pressure) chosen in order to be consistent with gas appearance or disappearance. We discuss the well possedness of this model and give some computational evidences of its adequacy to simulate gas generation in a water saturated repository

The COUPLEX Test Cases: Nuclear Waste Disposal Simulation

International audienceThe models appearing in the COUPLEX benchmark are a set of simplified albeit realistic test cases aimed at simulating the transport of radionuclides around a nuclear waste repository. Three different models were used: The first test case is related to simulations based on a simplified 2D far-field model close to those used for safety assessments in nuclear waste management. It leads to a classical convec-tion diffusion type problem, but with highly variable parameters in space, highly concentrated sources in space and time, very different time scales and accurate results expected even after millions of years. The second test case is a simplification of a typical 3D near-field computation, taking into account the glass dissolution of vitrified waste, and the congruent release of several radionu-clides (including daughter products), with their migration through the geological barrier. The aim of the third test case is to use the results of the near-field computation (COUPLEX 2) to drive the behavior of the nuclide source term in the Far Field computation (COUPLEX 1). The modeling of this last case was purposely left rather open, unlike the previous two, leaving the choice to participants of the way the coupling should be made

Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective

A mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards −∞ and the other towards +∞, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the infinite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator’s two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization

A two level scaling-up method for multiphase flow in porous media; numerical validation and comparison with other methods

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