SCIENTIFIC PROGRAMME
Lectures
Alírio E. Rodrigues, University of Porto (LSRE/LCM, FEUP)
José Ferreira, University of Coimbra (CMUC-LCM)
Vítor Costa, University of Aveiro
Talks
Adérito Araújo, University of Coimbra (CMUC-LCM)
Ercília Sousa, University of Coimbra (CMUC-LCM)
Sílvia Barbeiro, University of Coimbra (CMUC-LCM)
11:00-12:00 |
Alírio Rodrigues | Diffusion in porous media: a look from chemical engineering |
12:00-12:30 |
Sílvia Barbeiro | Finite element methods in models for poroelasticity |
| Lunch | ||
14:00-15:00 |
Vítor Costa | Diffusion in porous media: physics and modeling |
15:00-15:30 |
Ercília Sousa | On the fractional diffusion equation |
| Coffee break | ||
| 16:00-16:30 | Adérito Araújo | Optimal solution of a reaction-diffusion system with a control discrete source term |
| 16:30-17:30 | José Ferreira | Supraconvergence-superconvergence in fluid flow models in porous media |
Room 17 A of the Mathematics Department of the University of Coimbra.
Abstracts:
Adérito Araújo, Maria F. Patrício, José L. Santos
We study the numerical behavior of a reaction-diffusion system with a control source point. The main goal consists on estimating the position of the source point that maximizes a given objective function. To reduce the number of variables involved in the optimization algorithm, we first consider the problem with a fixed source point and then, according to the numerical results obtained, we estimate an approximation to the objective function, adjusting, by least squares, a special class of functions that depend on a few number of parameters.
Alírio E. Rodrigues
A journey through diffusion in porous media
. Fick, Einstein, Stefan-Maxwell
. Non-Fickian (Weisz)
A Chemical Engineering view
. Multiscale modelling: Pore scale, particle scale, bed scale
. Adsorbents and catalysts:
- Particle structure: homogeneous, porous, bidisperse
- Linear driving force (LDF) models
. Adsorption columns and fixed-bed reactors
Experimental measurement of diffusivity
. Macroscopic methods
. Chromatography and ZLC (zero-length columns)
Conclusions
Ercília Sousa
The use of the conventional diffusion equation in many physical situations has been questioned in recent years and alternative diffusion models have been proposed. For instance, the basic assumption of the models for the transport of contaminants through soil is that the movements of solute particles are characterized by the Brownian motion. However, the complexity of pore space in natural porous media makes the hypothesis of the Brownian motion far too restrictive in some cases.
Fractional space derivatives are used to model anomalous diffusion. When a fractional derivative replaces the second derivative in a
diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion.
A one dimensional fractional diffusion model is considered, where the usual second-order derivative gives place to a fractional derivative of order $\alpha$, with $ 1<\alpha \leq 2$. We consider the Caputo derivative as the space derivative, which is a form of representing the fractional derivative by an integral operator. We propose some numerical methods to solve this equation and numerical tests are presented.
José Ferreira
We study numerical method for the solution of the set of coupled partial differential equations
under convenient initial and boundary conditions. Equations (1) and (2) have been considered in the literature to describe the pressure and the concentration of flow in porous media. Several approaches have been considered in the literature to increase the convergence order for numerical approximations for the fluid velocity. Without being exhaustive we mention the use of cell centered schemes, mixed finite element methods and gradient recovery techniques.
We present a finite difference method defined with nonuniform mesh for the one dimensional version of (1), (2) with Dirichlet boundary conditions. We establish second order error estimates for the pressure with respect to a discrete $H1$-norm and for the concentration with respect to a discrete $L2$-norm. As the truncation error of the method studied is of first order when the norm $\|.\|_\infty $ is considered, we establish the supraconvergence of the numerical approximations. We remark that our finite difference scheme can be seen as a fully discrete Galerkin method based on piecewise linear approximation and convenient quadrature rules. This fact enable us to conclude the superconvergence of the approximations.
Sílvia Barbeiro
The modeling of coupled mechanics and flow in porous media is of great importance in a diverse range of engineering fields. Besides the application in soil mechanics, developments in poroelasticity modeling are also contributing to important achievements in civil, petroleum and even biomedical engineering. In this work we consider the numerical solution of a coupled fluid flow and geomechanics in Biot’s consolidation model for poroelasticity. The method combines mixed finite elements for Darcy flow and continuous Galerkin finite elements for elasticity. In solving the coupled system, pressure and displacements can be solved either simultaneously in a fully coupled scheme or sequentially lagging the coupling terms in one or several steps. The degree of coupling affects the stability and accuracy of the numerical solutions. We summarize a priori convergence estimates for fully coupled schemes and for iteratively coupled schemes and provide a convergence result for a decoupled scheme. We perform numerical experiments for verifying our theory and modeling engineering applications. We give special attention to the well-known Mandel’s problem. This problem has been used as a benchmark problem for the reason that it admits an analytical solution in two dimensions on a finite domain. It is therefore very useful to verify the accuracy of discretization schemes and also provides a nice example of the dynamics involved in solid-fluid interactions.
Vítor Costa
DIFFUSION IN HOMOGENEOUS MEDIA
. Physical mechanism for diffusion
. Different ‘species’ diffusion
. Constitutive laws
DIFFUSION IN POROUS MEDIA
. Different scales approach: Micro, Meso and Macro scales
. Macro-scale approach
- Effective diffusivity for some particular situations; discussion
- One-dimensional and multidimensional approaches
- Turtuosity
- Diffusion in partially saturated porous media
- Conjugated diffusion transfer
- Knudsen diffusion
- Space and time scales: steady and unsteady situations
CONCLUSIONS
* Contacts
If you want to participate in this meeting
or you need additional information, please write an email to
silvia@mat.uc.pt
* Organizers
José Ferreira (ferreira@mat.uc.pt)
Sílvia Barbeiro (silvia@mat.uc.pt)
Ercília Sousa (ecs@mat.uc.pt)
* Support
Department of Mathematics, Faculty of Sciences and Technology of University of Coimbra
Centre for Mathematics, University of Coimbra