# Coupling free flow with porous media flow

#### Initial Situation

Transport phenomena in structures composed by a porous layer and adjacent free flowing fluid appear in a wide range of industrial (fuel cells, filtration and drying processes), environmental (ground water pollution, infiltration of overland flows during rainfalls) and medical (flows in biological tissues, transport of drugs and nutrients) applications.

Flow inside the porous medium is typically treated using the macroscopic Darcy law whereas the free flow region is modeled by the microscopic Navier-Stokes equations. Modeling the coupling of these flows is difficult since the governing equations are of different orders and involve both microscopic and macroscopic formulations. Two different approaches are generally proposed for treating the flow at the interface between the free fluid and the porous medium.

The "two-domain approach" is based on the Beavers and Joseph velocity jump condition, which enables coupling of the Darcy law with the Stokes equation through appropriate boundary conditions at the fluid/porous interface. The "single-domain" approach involves the solution of the Brinkman equation in the entire domain. The transition from the free fluid to the porous region is achieved through specifying the spatial variation of properties (permeability, viscosity, porosity). The single-domain approach enables us to model both the porous media and the free flow using one set of equations. Furthermore, it avoids the need to specify boundary conditions between the domains since velocity and stress continuity across the interface are readily satisfied.

Note that these approaches are developed only for single-phase flows in porous media and simple structures.

#### Current Results

We applied both approaches for coupling free flow and single-phase flow in porous medium. In the case of "single-domain" approach the SIMPLE algorithm (Semi-Implicit Method for Pressure Linked Equations) is used to solve Brinkman equation. The influence of the pressure boundary conditions on the convergence of the method is studied in the case of channel flow. For "two-domain approach", the discontinuous Galerkin method is used for Stokes problem and vertex-centered finite volume method (box-scheme) for Darcy's law. At the interface, Beavers-Joseph-Saffman contitions as well as the conditions of mass conservation and balance of normal forces are used.

 Pressure and velocity distribution.

#### Work Plan

The next step is to apply volume averaging theory for coupling free flow with two-phase flow in porous media. With this, new interface conditions will be developed, which also take multiple phases and transfer of components into account.