Research Project A03

In the first step, we demonstrated that the most commonly used interface conditions for coupled single-phase porous-medium and free-flow systems (conservation of mass, balance of normal forces, the Beavers-Joseph condition) are unsuitable for arbitrary flows to the fluid-porous interface. For the development of generalised interface conditions, we analysed strengths and limitations of different averaging techniques: homogenisation, volume averaging, thermodynamically constrained averaging theory (TCAT). For single-phase systems, the theory of homogenisation with two-scale asymptotic expansions and boundary layers was found to be the best choice, while for two-phase systems a combination of TCAT and vertical averaging is the most suitable averaging approach.

In the second step, we rigorously derived new, generalised interface conditions for arbitrary flow directions in coupled single-phase flow systems. In this case, the free flow is described by the Stokes equations, while single-phase Darcy's law is applied in the porous medium. With the newly derived coupling conditions we recovered the classical mass balance across the fluid-porous interface, obtained an extension to the balance of normal forces and a variation of the Beavers-Joseph interface condition. These generalised coupling conditions are derived using the theory of homogenisation and boundary layers, which allows to compute all the effective coeffcients appearing in the interface conditions numerically based on the pore geometry and independently of the flow direction. Appropriate error estimates between the effective and the pore-scale solutions for the velocity and pressure are obtained. 

In the third step, we validated the newly derived coupling conditions numerically via comparison of the pore-scale resolved models and the macroscale simulation results. For the model validation, we studied different porous media (isotropic, orthotropic, anisotropic), different shapes of solid inclusions and different flow scenarios. In all cases, the macroscale numerical simulations are in good agreement with the pore-scale simulations, that is not the case for the coupled models with the classical interface conditions. Even for parallel flows to the interface the new coupling conditions are more accurate than the classical ones.

In the fourth step, we proved the well-posedness of the Stokes-Darcy problem with new interface conditions. This is the basis for the development of the effcient domain decomposition method, which is currently work in progress.

Computable models with interface conditions for arbitrary flows in two-phase coupled systems are still an open question. A general model formulation for a complex interface in coupled two-phase flow systems is theoretically derived. This model contains several unknown coefficients still to be determined. Moreover, it needs to be coupled to the free-flow model and the porous- medium model. This general model will be studied and a hierarchy of computable models of different complexities will be produced and validated. Efficient and robust numerical methods for new coupled model formulations will be developed and analysed.