Iuliia Petrova.

O objetivo deste projeto é fornecer uma análise matematicamente rigorosa dos modelos que descrevem o fluxo de fluidos em meios porosos utilizando ferramentas das equações diferenciais parciais (EDPs) e sistemas dinâmicos. Estou focada no estudo de dois problemas que têm origem na indústria petrolífera, descrevendo a injeção de água com produtos químicos em reservatórios de petróleo. O primeiro problema está relacionado ao fluxo multifásico em espaçounidimensional. Estou interessada na existência, unicidade e estabilidade das soluções fracas do modelo de polímeros (um sistema de conservação não estritamente hiperbólico 2 2 em espaço unidimensional). O foco principal é o estudo das ondas de choque, solução de um problema de Riemann e sua estabilidade. O segundo problema está relacionado com o fluxo de uma fase em espaço multidimensional. Estou interessada em uma prova matematicamente rigorosa do crescimento linear dos dedos viscosos/gravitacionais (a deslocamento instável com formação de padrões, "viscous fingering").

This Project aims to study problems related to multiphase flow in porous media, which are modeled by non-linear partial differential equations, in particular evolution equations where the time variable is involved. The problem of multiphase flows in porous media plays an important role in a wide range of planning and operation of oil and gas wells, and also in other areas of applied sciences, such as filtration, the study of dams, etc. .In more detail, the flow of a mixture of fluids in a porous medium is described by a scalar conservation law in the class of non-regular coefficients, and the mathematical problem of proving the overall well-posedness of weak solutions is a mathematical challenge. In fact, one of the difficulties in showing the existence of weak solutions is related to the common fact that we inject (in the porous medium) a less viscous fluid (for example water) into a more viscous one (dense oil), resulting in instabilities.In this sense, several developments are also important to be studied. Namely we cite: associated parabolic equations obtained via the inclusion of diffusion processes; homogenization techniques to obtain simplified equations; and very notoriously associated compressible problems through an equation of state for the pressure as a function of the density of the mixture.This Project will then study: existence/regularity of solutions for the hyperbolic and associated parabolic problems. Homogenization of models with various porosities.Traveling wave cascade study that can be seen as an approximation to the mentioned instabilities.To carry out this Project, we will go deeper into the state of the art of the proposed problems, and we will develop pertinent, challenging and original questions. Afterwards, we will apply all the mathematical knowledge available, or to be developed, in order to solve the given problem.

Russian Science Foundation grant 21-11-00047

President grant of Russian Federation MD-1791.2019.1

Russian Science Foundation grant 19-71-30002

Projeto com a compania petrolifera "GazpromNeft"

Russian Science Foundation Grant 17-11-01003

RFBR Grant 16-01-00258a

St. Petersburg State University Grant 6.38.670.2013