Homogenisation and model reduction for transport in porous media

We propose a new model reduction method to reduce the complexity of multiscale scalar transport problems with a dominant axial dynamic. Our approach combines the Hierarchical Model (HiMod) reduction and a two-scale asymptotic homogenisation technique. First, we extend the two-scale asymptotic expansion to any desired order. Then, we utilise high-order correctors to define the HiMod modal basis, which approximates the transverse dynamics of the flow. We use finite element discretisation to model the leading stream. We name this method HiPhom? (High-order Projection-based Homogenisation). We have successfully tested HiPhom? on both steady and unsteady advection-diffusion-reaction scenarios. The numerical results demonstrate that HiPhom? outperforms HiMod in terms of accuracy and convergence rate. Additionally, it extends the reliability of standard homogenised solutions to transient and pre-asymptotic regimes. This method has applications in various fields, such as hydraulics and haemodynamics. A reformulation of the method to derive closed high-order homogenised models for general porous media applications will also be discussed.