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A hybrid-dimensional Stokes--Brinkman--Darcy model for arbitrary flows to the fluid--porous interface

Linheng Ruan, Iryna Rybak

TL;DR

This paper introduces a hybrid-dimensional Stokes–Brinkman–Darcy model that uses a thin complex interface as a buffer between free-flow and porous regions to enable accurate coupling for arbitrary flow directions. By embedding averaged Brinkman equations within the transition zone and deriving transmission conditions, the approach captures mass and momentum exchange across the interface with higher-order corrections. Numerical validation against pore-scale simulations shows that the method achieves high accuracy and is competitive with generalized interface conditions, while outperforming classical Beavers–Joseph-based coupling in non-parallel flows. The framework provides a robust, scalable tool for simulating coupled free-flow and porous-media systems with practical applicability in engineering and biological contexts, with future work extending to inertial effects.

Abstract

Mathematical modelling of coupled flow systems containing a free-flow region in contact with a porous medium is challenging, especially for arbitrary flow directions to the fluid--porous interface. Transport processes in the free flow and porous medium are typically described by distinct equations: the Stokes equations and Darcy's law, respectively, with an appropriate set of coupling conditions at the common interface. Classical interface conditions based on the Beavers--Joseph condition are not accurate for general flows. Several generalisations are recently developed for arbitrary flows at the interface, some of them are however only theoretically formulated and still need to be validated. In this manuscript, we propose an alternative to couple free flow and porous-medium flow, namely, the hybrid-dimensional Stokes--Brinkman--Darcy model. Such formulation incorporates the averaged Brinkman equations within a complex interface between the free-flow and porous-medium regions. The complex interface acts as a buffer zone facilitating storage and transport of mass and momentum and the model is applicable for arbitrary flow directions. We validate the proposed hybrid-dimensional model against the pore-scale resolved model in multiple examples and compare numerical simulation results also with the classical and generalised coupling conditions from the literature. The proposed hybrid-dimensional model demonstrates its applicability to describe arbitrary coupled flows and shows its advantages in comparison to other generalised coupling conditions.

A hybrid-dimensional Stokes--Brinkman--Darcy model for arbitrary flows to the fluid--porous interface

TL;DR

This paper introduces a hybrid-dimensional Stokes–Brinkman–Darcy model that uses a thin complex interface as a buffer between free-flow and porous regions to enable accurate coupling for arbitrary flow directions. By embedding averaged Brinkman equations within the transition zone and deriving transmission conditions, the approach captures mass and momentum exchange across the interface with higher-order corrections. Numerical validation against pore-scale simulations shows that the method achieves high accuracy and is competitive with generalized interface conditions, while outperforming classical Beavers–Joseph-based coupling in non-parallel flows. The framework provides a robust, scalable tool for simulating coupled free-flow and porous-media systems with practical applicability in engineering and biological contexts, with future work extending to inertial effects.

Abstract

Mathematical modelling of coupled flow systems containing a free-flow region in contact with a porous medium is challenging, especially for arbitrary flow directions to the fluid--porous interface. Transport processes in the free flow and porous medium are typically described by distinct equations: the Stokes equations and Darcy's law, respectively, with an appropriate set of coupling conditions at the common interface. Classical interface conditions based on the Beavers--Joseph condition are not accurate for general flows. Several generalisations are recently developed for arbitrary flows at the interface, some of them are however only theoretically formulated and still need to be validated. In this manuscript, we propose an alternative to couple free flow and porous-medium flow, namely, the hybrid-dimensional Stokes--Brinkman--Darcy model. Such formulation incorporates the averaged Brinkman equations within a complex interface between the free-flow and porous-medium regions. The complex interface acts as a buffer zone facilitating storage and transport of mass and momentum and the model is applicable for arbitrary flow directions. We validate the proposed hybrid-dimensional model against the pore-scale resolved model in multiple examples and compare numerical simulation results also with the classical and generalised coupling conditions from the literature. The proposed hybrid-dimensional model demonstrates its applicability to describe arbitrary coupled flows and shows its advantages in comparison to other generalised coupling conditions.

Paper Structure

This paper contains 13 sections, 23 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Coupled flow models
  3. Flow domain and geometry
  4. Pore-scale model
  5. Macroscale models in two flow domains
  6. Hybrid-dimensional Stokes--Brinkman--Darcy model
  7. Classical interface conditions
  8. Generalised interface conditions
  9. Numerical results
  10. Validation against analytical solutions
  11. Lid-driven cavity over porous bed
  12. Splitting flow
  13. Discussion and conclusions

Figures (9)

  • Figure 1: Geometry of the coupled free-flow and porous-medium system
  • Figure 2: Porous-medium geometry with periodic solid inclusions (a) and unit cell (b)
  • Figure 3: Staggered grid in the coupled domain for the hybrid-dimensional model
  • Figure 4: $L_2$-error for averaged normal velocity for different $\lambda_1$, $\lambda_2$ (a) and correction terms $\xi_1$, $\xi_2$, $\xi_3$ for different thickness $d$ (b)
  • Figure 5: Lid-driven cavity: geometric setting (a) and pore-scale velocity magnitude (b)
  • ...and 4 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4