Publication# A Lagrangian, stochastic modeling framework for multi-phase flow in porous media

Résumé

Many of the complex physical processes relevant for compositional multi-phase flow in porous media are well understood at the pore-scale level. In order to study CO2 storage in sub-surface formations, however, it is not feasible to perform simulations at these small scales directly and effective models for multi-phase flow description at Darcy scale are needed. Unfortunately, in many cases it is not clear how the micro-scale knowledge can rigorously be translated into consistent macroscopic equations. Here, we present a new methodology, which provides a link between Lagrangian statistics of phase particle evolution and Darcy scale dynamics. Unlike in finite-volume methods, the evolution of Lagrangian particles representing small fluid phase volumes is modeled. Each particle has a state vector consisting of its position, velocity, fluid phase information and possibly other properties like phase composition. While the particles are transported through the computational domain according to their individual velocities, the properties are modeled via stochastic processes honoring specified Lagrangian statistics. Note that the conditional expectations of the particle velocities are different for different fluid phases. The goal of this paper is to present the general framework for this alternative modeling approach. Various one and two-dimensional numerical experiments demonstrate that with appropriate stochastic rules the particle solutions are consistent with a standard two-phase Darcy flow formulation. In the end, we demonstrate how to model non-equilibrium phenomena within the stochastic particle framework, which will be the main focus of the future work.

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Concepts associés (15)

Écoulement polyphasique

Un écoulement polyphasique est un écoulement d'un fluide comportant plusieurs phases. On pourra par exemple étudier le comportement d'un fluide comportant des bulles de gaz, ou encore étudier le comp

Porous medium

In materials science, a porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typica

Particule matérielle

Le terme « particule matérielle » (material particle en anglais) désigne une petite portion d'un corps, de matière solide ou fluide, constituée d'un nombre suffisamment grand de particules élémentaire

In this paper, we present finite-dimensional particle-based models for fluids which respect a number of geometric properties of the Euler equations of motion. Specifically, we use Lagrange-Poincare reduction to understand the coupling between a fluid and a set of Lagrangian particles that are supposed to simulate it. We substitute the use of principal connections in Cendra et al. (2001) [13] with vector field valued interpolations from particle velocity data. The consequence of writing evolution equations in terms of interpolation is two-fold. First, it provides estimates on the error incurred when interpolation is used to derive the evolution of the system. Second, this form of the equations of motion can inspire a family of particle and hybrid particle spectral methods, where the error analysis is "built in". We also discuss the influence of other parameters attached to the particles, such as shape, orientation, or higher-order deformations, and how they can help us achieve a particle-centric version of Kelvin's circulation theorem. (C) 2013 Elsevier B.V. All rights reserved.

Alexis Arnaudon, Vivien François Bonvin, Pascale Jablonka, Matthew Luke Nichols, Yves Revaz

Chemo-dynamical N-body simulations are an essential tool for understanding the formation and evolution of galaxies. As the number of observationally determined stellar abundances continues to climb, these simulations are able to provide new constraints on the early star formaton history and chemical evolution inside both the Milky Way and Local Group dwarf galaxies. Here, we aim to reproduce the low alpha-element scatter observed in metal-poor stars. We first demonstrate that as stellar particles inside simulations drop below a mass threshold, increases in the resolution produce an unacceptably large scatter as one particle is no longer a good approximation of an entire stellar population. This threshold occurs at around 10(3) M-circle dot, a mass limit easily reached in current (and future) simulations. By simulating the Sextans and Fornax dwarf spheroidal galaxies we show that this increase in scatter at high resolutions arises from stochastic supernovae explosions. In order to reduce this scatter down to the observed value, we show the necessity of introducing a metal mixing scheme into particle-based simulations. The impact of the method used to inject the metals into the surrounding gas is also discussed. We finally summarise the best approach for accurately reproducing the scatter in simulations of both Local Group dwarf galaxies and in the Milky Way.

In nuclear safety, most severe accident scenarios lead to the presence of fission products in aerosol form in the closed containment atmosphere. It is important to understand the particle depletion process to estimate the risk of a release of radioactivity to the environment should a containment break occur. As a model for the containment, we use the three-dimensional differentially heated cavity (DHC) problem. DHC is a cubical box with a hot wall and a cold wall on vertical opposite sides. On the other walls of the cube we have adiabatic boundary conditions. For the velocity field the no-slip boundary condition is valid. The flow of the air in the cavity is described by the Boussinesq equations. Complex flow patterns develop and the flow characteristics depend on the non-dimensional Rayleigh and Prandtl numbers. The predominant flow type in the DHC is a turbulent natural convection flow. This work aims at reaching Rayleigh numbers and turbulent levels as high as possible given the available computational resources. The method used to simulate the turbulent flow is the large eddy simulation (LES) where the dynamics of the large eddies is resolved by the computational grid and the small eddies are modelled by the introduction of subgrid scale quantities using a filter function. Numerically, the LES equations are discretized by the spectral element method. Particle trajectories are computed using the Lagrangian particle tracking method, including the relevant forces (drag, gravity, thermophoresis). Four different particle sets with each set containing one million particles and diameters of 10 μm, 15 μm, 25 μm and 35 μm are simulated. The complexity and the size of the large three-dimensional problem requires the use of massively parallel supercomputers. Spectral element methods are naturally suitable for parallelisation by distributing the elements among the processors. For the Lagrangian particle tracking we use a method where equal numbers of particles are assigned to every processor. The flow field is broadcast and every particle processor tracks the assigned particles, a procedure which leads to a perfect load balancing. Simulation results for the flow field and particle sizes from 15 μm to 35 μm at a Rayleigh number of 109 are compared to previous results from a direct numerical simulation. First order statistics of the LES flow fields are in very good agreement with the direct numerical simulation while the agreement of second order moments is fair. Also the turbulent structures associated to the maximum of turbulent kinetic energy production are correctly reproduced. Particle statistics in the LES and the direct numerical simulation were similar and the settling rates practically identical. Contrary to previous particle simulations in LES, it was found that no model was necessary for the influence of the unresolved flow scales on the particle motions. This can be explained, because the important settling mechanism is through gravity and particle deposition at the walls by turbophoresis is negligible.