Open Access

David Matthias Wiedemann

• Analytical homogenisation provides effective models for processes in multiscale media based on models at the microscale. For porous media, the pore geometry strongly affects the resulting effective models. We provide an analytical homogenisation method for complex porous media with non-periodic and evolving cavities. For this, we derive a generic framework based on coordinate transformations and homogenisation of the resulting replacement equations. We rigorously justify this approach by showing that the homogenisation of the replacement problems defined in periodically perforated domains is equivalent to the homogenisation of the original problems. A back-transformation of the homogenisation results completes the method and leads to homogenised equations taking into account the local microstructure. We apply this method for the homogenisation of quasi-stationary and instationary Stokes flow in evolving porous media. This leads to a quasi-stationary Darcy law and a Darcy law withAnalytical homogenisation provides effective models for processes in multiscale media based on models at the microscale. For porous media, the pore geometry strongly affects the resulting effective models. We provide an analytical homogenisation method for complex porous media with non-periodic and evolving cavities. For this, we derive a generic framework based on coordinate transformations and homogenisation of the resulting replacement equations. We rigorously justify this approach by showing that the homogenisation of the replacement problems defined in periodically perforated domains is equivalent to the homogenisation of the original problems. A back-transformation of the homogenisation results completes the method and leads to homogenised equations taking into account the local microstructure. We apply this method for the homogenisation of quasi-stationary and instationary Stokes flow in evolving porous media. This leads to a quasi-stationary Darcy law and a Darcy law with memory for evolving microstructure. Both translate the local microstructure into effective permeability tensors and provide an additional source term for the pressure resulting from the local change in porosity. In addition, a reaction--diffusion equation with coupled pore evolution is homogenised. The resulting homogenised reactive transport system adjusts the diffusive flux by taking into account the local microstructure and scales the growth rate for the concentration with the changing porosity. The pore evolution and hence the effective transport properties are coupled to the unknown concentration by local upscaled microscopic processes.…