Research Project A03

In the first funding period, we rigorously derived and analysed generalised interface conditions for coupled single-phase free-flow and single-fluid-phase (fully saturated) porous-medium systems (Stokes-Darcy model) using the theory of homogenisation and boundary layer correctors. In this model, all the effective parameters were computed based on the pore-scale geometrical information. The coupled REV-scale model was validated numerically using the pore-scale resolved simulations and compared to other coupling conditions presented in the literature demonstrating the advantages of the developed generalised interface conditions. The well-posedness of the new REV-scale coupled model was proved.

In the current (second) funding period, we developed and analysed efficient solution strategies for coupled flow problems with the interface conditions developed in the first funding period (preconditioners and optimised Schwarz domain decomposition methods). We proposed three classes of preconditoners (block diagonal, block triangular and constraint) and conducted spectral and field-of-values analysis. We showed the efficiency and robustness of the developed solution strategies in numerical experiments. We also derived a coupling concept for single-phase free-flow and two-fluid-phase (partially saturated) porous medium using dimensional model reduction. The assumptions were: flat interface, laminar flow in the free-flow domain and non-isothermal, non-compositional system. For the resulting non-linear problem, we developed a domain decomposition method and a time-splitting scheme.

In collaboration with the research projects A02, D03 and INF, we investigated different coupling approaches (classical and generalised interface conditions and pore-network model) in a joint benchmark. Currently, we extend this work and investigate the impact of different pore geometries, surface roughness and the coupling of the Navier-Stokes to the Darcy flow using the generalised coupling conditions. In the framework of the SFB 1313 internal exchange programme with project B02, we analysed the system matrices arising from discretisation of the TPM phase-field model and studied several possible preconditioners to accelerate the convergence of the GMRES method. This lays the foundation for our joint work in the third funding period.

The former and current research directions of the project A03 are mathematical modelling of coupled free-flow and porous-medium systems on the REV scale, numerical simulation, analysis and validation of the new models. Our goal for the third funding phase is the extension of mathematical and computational models developed in the first and second funding periods taking into consideration species and energy transport. This will allow us to describe evaporation processes from the soil influenced by the free-flow dynamics and to contribute to the joint benchmark within SFB 1313 organised by project D03 and to the vision topic salt. Additionally, we adapt our models to describe fluid flow in fractured porous media extending our application areas.

The effective model parameters appearing in the new coupling conditions for our REV-scale models will be computed using the pore-scale resolved simulations in collaboration with project A02. To support the validation process for our REV-scale models, the visualisation tool "Interactive Multi-Scale Analysis" developed by project D01 will be applied. We will also provide effective model parameters for the experimental setup in project A06. Additionally, we will collaborate with projects B02 and Z02 on comparison of different models for fractured porous media and development of efficient preconditioners.

Aix-Marseille University, France
We collaborate with Prof. Philippe Angot on validation and calibration of the REV-scale model with generalised interface conditions.

Loughborough University, UK
We collaborate with Reader Dr. Marco Discacciati on domain decomposition methods for coupled flow problems with single- and two-fluid-phase porous media.