Méthodes de réduction de modèles pour la simulation d’écoulements en milieux poreux fissurés et déformables

The study of processes such as CO2 injection in the underground or geothermal energy requires to simulate flows and geomechanics in cracked porous media in a coupled way. Cracks have a major influence on the hydraulic conductivity since they can be highly permeable. In practice uncertainties exist on the model data and the corresponding poromechanical model can be simulated several times for various input parameters. The objective of this PhD work is to develop new methodologies of model reduction to decrease the computing times. In a first time, we will design a reduced-basis algorithm to build reduced discrete spaces for the rock displacements and the pore and crack pressures. The new methodology will be based on previous works where a greedy cone-projected approach (CPG) has been introduced. The CPG algorithm was first studied for elasticity problems with contacts where it ensures the positivity of the Lagrange multipliers and the existence of an inf-sup condition for the built reduced model. We will try to obtain similar results for the considered poromechanical problem and develop tailored a posteriori error estimates. In a second time, we will consider non-linear reduction approaches for this problem. These approaches may indeed be useful to study the influence of the position of the cracks on the solutions.
 

Mots-clés : porous media flows, geomechanics, reduced basis, machine-learning, finite-volume methods, finite-element methods, virtual-element methods