Geometrical approach for Porous media flows: theory and numerics – GEOPOR

The GeoPor project proposal aims to gather experienced and promising researchers from different fields of mathematics, more precisely optimal transportation for the analysis of partial differential equations (PDEs) and numerical analysis, for studying the equations governing multi-phase flows in porous media, mixing analysis, differential geometry, and scientific computing. Indeed, the project has been thought as a transversal bridge across the mathematical communities, the common factor being the problems to study, that are the equations governing multiphase flows and their wide range of applications (e.g., oil-engineering, CO2 sequestration, nuclear waste repository management, hydrogeology).

In the recent years, new and promising results have been obtained separately in the field of optimal transportation, in the design and the study of advanced numerical methods, and in the development of adaptive strategies for solving numerically some PDEs based on a posteriori error analysis. We aim to take advantage of all these new developments, and to apply this new material to the study, from a theoretical and numerical point of view, of nonlinear degenerated parabolic systems and equations.

We claim that the gradient flow approach is relevant for studying multiphase flows in porous media. Indeed, the natural and physically consistent way to write the equations governing multiphase flows in porous media let some degenerate mobilities appear leading to the study of degenerate parabolic equations and making the gradient flow writing completely natural. As a major advantage of this writing, only physically meaningful quantities appear, in contrary to the classical mathematical approaches.

Keeping the physical relevance as a constant matter, we propose to design what we call nonlinear methods for approximating the solutions of such (possibly) degenerate parabolic equations and systems. The methods we have in mind are based on schemes that are widely used in oil-engineering despite very partially studied: they are robust and allow enrichments of the physics. Taking advantage of the numerous new tools developed in the recent years we will design robust advanced numerical methods whose convergence will be proven.

Bearing in mind that we are concerned with real world applications, we will extend the recent works on a posteriori error analysis to the new methods we will design. We expect to derive fully computable, guaranteed error bounds, so that the maximal error in the calculation can be certified. Moreover, relying on mesh and time stepping adaptivity, but also on adaptive stopping criteria for linear and nonlinear solvers, we hope to propose algorithms that will save an important part of the computational time, so that the methods we will propose will be competitive, especially in the industrial context.
Last but not least, this project will create new interactions between different communities as well as with international researchers. These interactions will be consolidated thanks to the organization of several workshops and one international conference, and by the implication in the formation of young researchers in France and abroad.