CIRM, Luminy, Marseille
Date(s) : 31/08/2021 iCal
0 h 00 min
Journée thématique d’Analyse Appliquée: Fluide-poreux : Problèmes d’interface et d’upscaling dans les écoulements fluide-poreux
Philippe Angot (Aix-Marseille Université)
Thomas Ourmières-Bonafos (Aix-Marseille Université)contact: firstname.lastname@example.org
Generalized coupling conditions for multidimensional flows in Stokes/Darcy systems
Existing coupling concepts for the Stokes/Darcy problem are limited for one-dimensional flows (parallel or perpendicular to the interface), are not validated for multidimensional flows (arbitrary to the porous bed), or contain unknown model parameters, which need to be determined.
In this talk, we present generalized interface conditions valid for multidimensional flows in Stokes/Darcy systems. These conditions are derived rigorously using periodic homogenization and boundary layer theory, and all effective parameters can be computed based on the pore geometry. We prove the well-posedness of the coupled Stokes/Darcy problem with the generalized conditions and validate the macroscale model using pore-scale resolved simulations.
Domain decomposition methods for the Stokes-Darcy problem
The Stokes-Darcy problem has received a growing attention by the mathematical community over the last decade not only due to its many possible applications but also to its mathematical nature. Indeed, it is a good example of a multi-physics problem where two different boundary value problems are coupled into a global heterogeneous one.
The multi-physics nature of the problem makes it suitable to domain decomposition techniques which permit to recover the solution of the coupled problem by solving the fluid and the porous-medium subproblems separately in an iterative way.
This talk will provide an overview of different non-overlapping and overlapping approaches for the Stokes-Darcy problem considering the convergence and robustness of these methods as well as related modelling aspects. Some numerical examples will show the effectiveness of the methods for problems arising in practical applications.
Hybrid-dimensional Stokes-Darcy model for fractured porous media
Hybrid-dimensional models for flow and transport processes in fractured porous media are effective approximations of full-dimensional models. Traditionally applied mixed-dimensional Darcy-Darcy models fail to adequately describe flow and transport through fractures which act as conduits and separate rock matrices with significantly different permeability. We propose a hybrid-dimensional Stokes-Darcy model for flows in fractured porous media. The fractures can store and transport fluid and they are modeled as lower-dimensional entities in the surrounding porous medium. The well-posedness of the proposed coupled model is proved under a mild restriction on the fracture aperture. The developed model is validated against the full-dimensional model and compared to the hybrid-dimensional Darcy-Darcy model.
Coupling of models that should hold at different scales
Contamination of soil and groundwater is a major concern that affects all populated areas. It is all the more important that groundwater pollution can be rapid and its remediation very long. A challenge is therefore to develop models to monitor the vulnerability of aquifers in a context of very heterogeneous time scales. In this talk, we propose nonlinear moving boundary problems describing the exchanges between the overland and the underground water, saltwater intrusion in coastal areas, agricultural, industrial, or sewage pollution. Let’s use the example of the recharge of aquifers by overland water. We aim at keeping track of (at least) two types of flow: the first is essentially vertical and it is dominant on a small time scale (Richards flow) and the second, that is essentially horizontal, corresponds to a large time scale (Dupuit flow). Such a model cannot be justified by arguments of classical asymptotic analysis. Indeed, the scales allowing to justify a Richards model from Darcy’s are very different from those allowing to obtain a Dupuit model from Darcy’s. We thus propose another approach. The asymptotic analysis is not used for deriving an effective model for a given reference time; rather, we want that the new model and the exact model are associated with the same effective problem for any time scale. The process allows to define some virtual interface (that does not necessarily coincide with the water table) separating an essentially vertical flow from an essentially horizontal one. In this presentation, the asymptotic analysis approach, the definition of the interface over time, numerical illustrations and some mathematical difficulties related to the structure of the new model will be discussed…