Predicting nonlinear flow regions in highly heterogeneous porous media for the numerical simulation of underground storage

Underground storage of CO2 and geothermal energy has become a major research area in the ongoing energy transition. In this context, it is important to model and simulate flows in highly heterogeneous porous media, characterized by irregularly distributed permeability profiles. 
It is shown experimentally that these media do not follow Darcy’s law and that Forchheimer’s quadratic law is more adequate in the fast-flow zones. Applying Forchheimer’s law globally in the domain is very costly numerically. Instead, keeping Forchheimer’s law only where necessary should improve computing cost without much accuracy loss. Two new adaptive models have been recently proposed to couple the two laws; given a threshold on the flow’s velocity, they locally select the more appropriate law as they are being solved. At the end, each mesh cell is flagged as being in the Darcy or Forchheimer subdomain. 
These new models are highly nonlinear, and so we do not expect them to be significantly faster than the global Forchheimer approach. Still, they are useful to find approximations of the subdomains; once these are known, it is fast to rerun a region-based simulation by directly applying each law in its region. In this thesis, we want to use the adaptive models to predict the subdomains as preprocessing for the region-based simulation and show whether the resulting approach is accurate and faster than the global Forchheimer simulation.
We will use the adaptive models as predictors by either running a classification algorithm over the mesh cells or by making simplifying assumptions on the shapes of the connected components of the Forchheimer subdomain. In the former case, the training will be performed by interpreting the models’ results as classifiers deciding for each cell whether it is Darcy or not. In the latter case, we will extrapolate information on the generic shapes (elliptical, tubular, graph-like, etc.) of the Forchheimer components from the known permeability profile. Neural networks will be used to predict these generic shapes as outputs with the boundary conditions as inputs. Finally, the thesis will combine the subdomain predictions with the region-based simulation and validate this combination in terms of accuracy and computing time.

Keywords: porous media, Darcy's law, Forchheimer's law, adaptive constitutive law, multiphysics coupling

• Academic supervisor    Dr HDR Quang Huy TRAN, IFPEN, ORCID: 0000-0001-7771-3154
• Doctoral School    ED580 STIC, Paris-Saclay University
• IFPEN supervisor    Dr Francesco PATACCHINI, SCOPUS: 56538679300
• PhD location    IFPEN, Rueil-Malmaison, France
• Duration and start date    3 years, starting in the first quarter 2026 (January 5)
• Employer    Paris-Saclay University (doctoral contract)
• Funding    PhD Call Stand up for Science. University audits begin in September 2025
• Academic requirements    Master’s degree in scientific computing, applied mathematics 
• Language requirements    English level B2 (CEFR), willingness to learn French 
• Other requirements    Python, C++, interest in AI. Have resided in the USA in 2025.

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