**LCM**Laboratory for Computational Mathematics

## Reaction-diffusion in Porous Media

### Problem Description

The mathematical analysis and the numerical simulation of reaction-diffusion phenomena in porous media have been discussed extensively in the literature. The fundamental equation governing diffusion in porous media is the equation of mass conservation, which is of parabolic type. It is established assuming that the dispersive mass flux is given by the Fick´s law where the dispersion tensor is assumed to be independent of the concentration and its gradient. It is well-known that this equation gives rise to an infinite speed of propagation. Small-scale and large-scale heterogeneities in porous matrix and/or fluid properties are the main sources of deviations of the so-called Fickian dispersion behavior. In order to overcome this deviation, a certain memory effect should be included in the flux modeling.

### Modelling & Computational Challenges

The aim of this project is to introduce memory effects in the models for fluid flows in porous media characterized by small-scale and large-scale heterogeneities in several contexts. A parallel software package to simulate the memory models will be developed. Consequently we will be able to simulate a wide range of diffusion processes in media with different properties. The software package can be used in different kinds of applications as environmental applications or industrial applications. In what concerns industrial applications we point out that polymers are laboratory materials. Their properties can be optimized simulating the diffusion process in this media. Nowadays the pollution of soils by fertilizers or pesticides are one of the most relevant environmental problems. The simulation of the diffusion process in soils will allow us to predict the evolution of this kind of pollution. We will also be able to define bounds on the use of chemicals reducing their environment impact. Biological filtration through submersed packed bed is a technology recently looked as useful to integrate in wastewater treatment systems for carbonaceous removal, nitrification/denitrification, phosphorous and residual soluble organic and inorganic removal. The non Fickian models studied during this project and the software package that we expect to develop can also be used to study biological filters.

### Research at LCM

The diffusion in porous media has been extensively studied in several contexts. A crucial equation on the mathematical description of the physical problem is the mass conservation equation, which is of parabolic type, inducing a non physical behavior: infinite speed of propagation. As been point out, there are certain discrepancies between the experimental data and the results obtained by simulation of known models, namely for media characterized by smallscale and large-scale heterogeneities. Our purpose is to introduce a memory effect in wellknown models, such as, contaminant models with diffusion, reaction and convective transport. This memory effect is achieved by introducing a delay in the Fick´s law for the diffusion process. Our aim is to gain some insight on the mathematical analysis of systems of equations, to obtain discrete schemes with good accuracy and conservation properties. Furthermore, for the proposed schemes it is our goal to build efficient iterative-coupling algorithms which will be implemented in parallel.

### Papers & Reports

- [1] A. Araújo, J. A. Ferreira, P. de Oliveira,
*Qualitative behavior of numerical traveling waves solutions for reaction diffusion equations with memory*, Applicable Analysis, 84 (2005) 1231-1246 - [2] A. Araújo, J. A. Ferreira, P. de Oliveira,
*The effect of memory terms in the qualitative behavior of the solution of the diffusion equations*, Journal of Computational Mathematics, (2006) 91-102 - [3] S. Barbeiro, J. A. Ferreira,
*Integro-differential models for percutaneous drug absorption*, International Journal of Computer Mathematics, 84 (2007) 451-467 - [4] J. R. Branco, J. A. Ferreira,
*A singular perturbation of the heat equation with memory*, Journal of Computational and Applied Mathematics, 218 (2008) 376-394 - [5] J.R. Branco, J. A. Ferreira, P. de Oliveira,
*Numerical methods for the generalized Fisher-Kolmogorov-Petrovskii-Piskunov equation*, Applied Numerical Mathematics, 57 (2007), 89-102 - [6] S. Barbeiro, J. A. Ferreira,
*Coupled vehicle-skin models for drug release*, Computer Methods in Applied Mechanics and Engineering, 198(27-29) (2009) 2078-2086 - [7] Sílvia Barbeiro, Mary F. Wheeler,
*A priori error estimates for the numerical solution of a coupled geomechanics and reservoir flow model with stress-dependent permeability*, Computational Geosciences, published online, 5 May 2010. - [8] J.A. Ferreira, P. de Oliveira, P.M. da Silva, A. Carreira, H. Gil, J.N. Murta,
*Sustained drug release from contact lenses*, Computer Modeling in Engineering and Science, 60, 151-180, 2010 - [9] J.R. Branco, J.A. Ferreira,
*Integro-differential IBVP versus differential IBVP: stability analysis*, Pré-Publicação do Departamento de Matemática, Universidade de Coimbra, nº09-40. - [10] J.A. Ferreira, R. Serôdio, J. Vitória,
*An iterative method to compute zeros of quaternion polynomials*, Pré-Publicação do Departamento de Matemática, Universidade de Coimbra, nº11-20. - [11] J.A. Ferreira, G. Romanazzi, L. Pinto,
*Supraconvergence and Supercloseness in Volterra equations*, Pré-Publicação do Departamento de Matemática, Universidade de Coimbra, nº11-15. - [12] J.A. Ferreira, E. Gudino and P. de Oliveira,
*Analytical and Numerical study of memory formalisms in diffusin process*, Pré-Publicação do Departamento de Matemática, Universidade de Coimbra, nº12-07 - [13] J.A. Ferreira, P. de Oliveira, P.M. da Silva,
*Reaction-diffusion in viscoelastic materials*, to appear in Journal of Computational and Applied Mathematics - [14] J.A. Ferreira, P. de Oliveira, P.M. da Silva ,
*Analitycs and numerics of drug tracking*, to appear in Journal of Computational and Applied Mathematics - [15] J.A. Ferreira, L. Pinto,
*A non Fickian single phase flow model*, CMMSE 2011, Volume II, edt. J. Vigo Aguiar, pp. 508-519, 2011 - [16] J.A. Ferreira, P. de Oliveira, P.M. da Silva ,
*A mathematical kit for simulating drug delivery through polymeric membranes*, CMMSE 2011, Volume II, edt. J. Vigo Aguiar, pp. 496-507, 2011 - [17] J.R. Branco, J.A. Ferreira, P. de Oliveira ,
*Memory in mathematical modelling oh highly diffusive tumors*, CMMSE 2011, Volume I, edt. J. Vigo Aguiar, pp. 242-253, 2011 - [18] S. Barbeiro J.A. Ferreira, S. Gholami,
*Numerical Methods for non-Fickian flows based on Laplace transforms*, (11 pages), in CD-ROM Proceedings of the Congress on Numerical Methods in Engineering (CMNE|2011), Coimbra, Portugal, June 14-17, 2011 - [19] J.A. Ferreira, P. de Oliveira, P.M. da Sil, J.N. Murta,
*Drug delivery: from contact lens to the anterior chamber*, Computer Modeling in Engineering and Science, 71, 1-14, 2011 - [20] S. Barbeiro, J.A. Ferreira, L. Pinto,
*H*, Applied Numerical Mathematics, 61, 201-215, 2011^{^1}-second order convergent estimates for non Fickian models - [21] J.A. Ferreira, P. de Oliveira, P.M. da Silva, L. Simon,
*Flux tracking in delivery polymeric systems*, Computer Applied Mathematical Modelling, 35, 4684-4696, 2011 - [22] J. A. Ferreira, S. Barbeiro, G. Pena and M. F. Wheeler,
*Modelling and Simulation in Fluid Dynamics in Porous Media (Springer Proceedings in Mathematics and Statistics)*, Springer, 2012 - [23] J. A. Ferreira and G. Pena,
*Memory effect in time and space in non Fickian diffusion phenomena*, Pré-Publicação do Departamento de Matemática, Universidade de Coimbra, nº12-31

### Project Team

- José Augusto Ferreira (LCM/CMUC, Project’s Principal Investigator)
- Adérito Araújo (LCM/CMUC)
- Ercília Sousa (LCM/CMUC)
- Giuseppe Romanazzi (LCM/CMUC)
- Gonçalo Pena (LCM/CMUC)
- Marc Baboulin (LCM/CMUC)
- Sílvia Barbeiro(LCM/CMUC)
- Cidália Fonte (DMUC)
- Luís Pinto (DMUC)
- Fernando Carapau (University of Évora)
- Gergina Pencheva (ICES, University of Texas at Austin)
- Mary Fannet Wheeler (ICES, University of Texas at Austin)
- Mojdeh Delshad (ICES, University of Texas at Austin)
- Vivette Girault (Université Pierre et Marie Curie, ParisVI, LaboratoireJacques-LouisLions)

### Project Reference

FCT Research Project UTAustin/MAT/0066/2008