Efficient simulation of coupled free and porous media fluid flow within channelised structures using the virtual porous medium model: application to liver interlobular blood flow

Many living tissues can be modelled as porous media containing blood vessels and numerous capillaries that act as flow channels. Although direct simulation using the Navier-Stokes equations in flow channels coupled with the Brinkman equations in porous regions offers high accuracy, it is computationally expensive. This study proposes a virtual porous medium (VPM) model that approximates capillaries as virtual porous regions with estimated porosity and permeability fields. By employing Darcy's law instead of the Navier-Stokes equations, the VPM model significantly reduces computational cost. To evaluate its accuracy and efficiency, several 2D and 3D test cases related to interlobular blood flow in the liver are presented. Each case, in fact, features blood vessels surrounding a channelised porous medium, representing liver tissue embedded with capillaries. Numerical results indicate that the VPM model generally produces acceptable predictions, with 2-norm errors for pressure and velocity fields at 3 and 2.2%, respectively. Additionally, the CPU time required is approximately 60% less compared to the direct pore-scale approach. Furthermore, the VPM model accurately captures the primary flow characteristics in channelised porous media, demonstrating its effectiveness for simulating coupled free and porous media flows.