The method of in-situ combustion is a thermal technique with great potential for use in the exploration of the offshore oil, as in the case of the pre-salt reservoirs. The modeling of combustion in porous media involves Fluid Dynamics and Chemical Kinetics. The models describing in-situ combustion are composed by reaction-convection-diffusion equations, present different scales (stiff problems) and are difficult to solve both mathematically and computationally.

This project is focused on Applied Mathematics and proposes to develop research in the intersection between areas of Partial Differential Equations and Dynamical Systems aiming at applications in Petroleum Engineering.

Chapiro, G., Mailybaev, A. A., De Souza, A. J., Marchesin, D., Bruining, J., Asymptotic approximation of long-time solution for low-temperature filtration combustion. Computational Geosciences, v. 16, p. 799-808, 2012.

Chapiro, G., Marchesin, D., Schecter, S., Combustion waves and Riemann solutions in light porous foam. Journal of Hyperbolic Differential Equations, v. 11, p. 295-328, 2014.

Chapiro, G., Furtado, L., Marchesin, D., Schecter, S., Stability of Interacting Traveling Waves in Reaction-Convection-Diffusion Systems. Discrete and Continuous Dynamical Systems, v. suppl., p. 258-266, 2015.

Chapiro, G., de Souza, A.J., Asymptotic approximation for counterflow combustion in porous media., Applicable Analysis, v. online, p. 1-15, 2015.

Chapiro, G., Marchesin, D., The effect of thermal losses on traveling waves for in-situ combustion in porous medium. Journal of Physics. Conference Series (Print), v. 633, p. 012098-12101, 2015.

Chapiro, G., Senos, L., Riemann solutions for counter flow combustion in

light porous foam. Computational and Applied Mathematics, 2017.

Pereira, W., S., Chapiro, G., Numerical Validation of Analytical Estimates for Propagation of Thermal Waves Generated by Gas-Solid Combustion, Geofluids, Volume 2017, Article ID 1806052, 2017.

Ozbag, F., Schecter, S., Chapiro, G., Traveling waves in a simplified gas-solid combustion model in porous media. Accepted in Advances in Differential Equations, 2017

The aim of this project consists in modeling complex physical phenomena in a simple way in order to make analytical techniques possible to be applied. Two interesting problems that are studied following this idea are: (1) Recovery of shale gas by using combustion and (2) Microwave stimulated water flooding.

Chapiro, G., Bruining, J., Thermal Well Stimulation in Gas Shales Through Oxygen Injection and Combustion. In: Fourth EAGE Shale Workshop, 2014

Chapiro, G., Bruining, J., Combustion enhance recovery of shale gas. Journal of Petroleum Science & Engineering, v. 127, p. 179-189, 2015.

Paz, P., Z., S., Hollmann, T., H., Kermen, E., Chapiro, G., Slob, E., Zitha, P., L., J., EM heating stimulated water flooding for medium-heavy oil recovery. Transport in Porous Media, V. 119, Issue 1, p. 57–75, 2017.

Parabolic-type problems, involving a variational complementarity formulation, arise in mathematical models of several applications in Engineering, Economy, Biology and different branches of Physics. These kinds of problems present several analytical and numerical difficulties related, for example, to time evolution and a moving boundary. We develop numerical methods that employs a global convergent nonlinear complementarity (or mixed complementarity) algorithms for solving a discretized problem at each time step. Space discretization is implemented using the finite difference implicit scheme and the finite element method.

Chapiro, G., Mazorche, S. R., Herskovits, J., Roche, J. R., Solution of the non-linear parabolic problems using nonlinear complementarity algorithm (fda-ncp). Mecánica Computacional, V. XXIX, P. 2141-2153, 2010.

Chapiro, G., Gutierrez, A., E., R., Herskovits, J., N., Mazorche, S., R., Pereira, W., S., Numerical Solution of a Class of Moving Boundary Problems with a Nonlinear Complementarity Approach. Journal of Optimization Theory and Applications, v. 168, p. 534-550, 2016.

Gutierrez, A., Mazorche, S., R., Herskovits, J., Chapiro, G. An Interior Point Algorithm for Mixed Complementarity Nonlinear Problems, Journal of Optimization Theory and Applications, online, 1-18, 2017

The global incidences of dengue and, more recently, zica virus have increased the interest in studying and understanding the mosquito population dynamics. This project is based on the study of nonlinear mathematical models dealing with the life cycle of the dengue mosquito using partial differential equations.

Yamashita, W., M., S., Takahashi, L., T., Chapiro, G., Dispersive models describing mosquitoes' population dynamics. Journal of Physics. Conference Series (Online), v. 738, p. 012065, 2016.

Yamashita, W., M., S., Takahashi, L., T., Chapiro, G., Traveling wave solutions for the dispersive models describing population dynamics of Aedes aegypti. Mathematics and Computers in Simulation, , v.146, 90-99, 2018.

Some collaboration works.

Chapiro, G., Faria, L.F.O., Maldonado, A.D., On the existence of solutions for a class of fourth order differential equations. Journal of Mathematical Analysis and Applications (Print), v. 427, p. 126-139, 2015.

A. C. Alvarez, Grigori Chapiro, G. C. Garcia, Carlos Gustavo Tamm de A. Moreira. Analysis of regularization by conjugation for bounded linear operators. Preprint. 2017