Mathematical analysis and numerical resolution of some stiff differential systems arising in biogeochemical modeling

In geothermal energy, water remediation and underground hydrogen storage, biochemical processes due to bacteria and microbes have a strong impact on the production of pathogens and must be taken into account in the evaluation of contamination risks. This is achieved through the integration of dedicated modules in porous media flow simulators. However, the development of these modules is based on empirical reaction models that are quite far from the usual chemical kinetics. At present, the underlying mathematical properties of these numerous ad hoc models, made up of differential or algebraic-differential equations, are unfortunately not well understood. Not only is their long-term stability not always ensured, but they can also be particularly stiff in certain operating ranges. Consequently, their numerical integration is delicate: even an implicit scheme often requires very small time-steps and does not always preserve the positivity of concentrations.
Under these conditions, the first objective of this thesis is to study the biogeochemical models at issue from a more mathematical point of view in order to highlight the different trajectory structures as a function of the parameters. To do so, we will use the tools of dynamical systems theory, more particularly those related to nonlinear stability analysis and bifurcations. The purpose of this thesis is not only to validate the form of the computed solutions, but also to anticipate the difficulties to be addressed later. In a second part of the project, the next objective is to improve the numerical methods in terms of accuracy and robustness with respect to the difficulties highlighted. To this end, we will try to adapt several recent techniques designed to guarantee on the one hand a better convergence of the resolution by Newton of the algebraic system  after discretization, and on the other hand a better adapted handling of the appearance or disappearance of certain species. The various methods considered will be compared first in 0-D, on pure differential systems, then in 2-D/3-D via a coupling with partial differential equations of transport in porous media.

Keywords: biogeochemistry, stiff differential systems, stability analysis, numerical integration, positivity

• Academic supervisor    Dr Guillaume DUJARDIN, Inria Lille, ORCID : 0000-0002-9085-1946
• Doctoral School    ED631 MADIS, Université Lille 
• IFPEN supervisor    Dr Benjamin BRACONNIER, benjamin.braconnier@ifpen.fr
• PhD location    IFPEN, Rueil-Malmaison, France  
• Duration and start date    3 years, starting in the fourth quarter 2024 (Novembre 4)
• Employer    IFPEN
• Academic requirements    Master’s degree in applied mathematics   
• Language requirements    English level B2 (CEFR)
• Other requirements    Numerical analysis, C++, Python....

To apply, please send your cover letter and CV to the IFPEN supervisor indicated below.