Joint distributions of local pore space properties quantitatively explain simulated air flow variations in paper

The gas flow through sheet-like porous materials such as paper can show marked lateral variations due to a heterogeneous, locally varying microstructure. Hence, reliable predictions of such lateral flux variations require an appropriate consideration of local variations in the microstructure. The flow through such sheet-like materials is commonly described with Darcy’s law in which permeances are formulated in terms of microstructure properties, such as porosities, tortuosities, or hydraulic radii. This work proposes an extension of existing permeance models that directly considers the variation and the cross-dependence between local microstructure properties. The extended model is applied to local air fluxes through a paper sheet to exemplarily reveal the joint impact of local porosities and local tortuosities on the air flux. The key extension is to consider a joint distribution of porosity and tortuosity. The latter is constructed from the univariate property distributions using a copula approach and yields local tortuosities including their variation for any encountered local porosity. These values jointly enter any permeance model that qualitatively captures the dependence of the air flux on the porosity. To assess the merit of the model, variations in the air flux and in the pore space properties are independently determined from the same measured microstructure of paper. Air flux variations are provided by computational fluid dynamics simulations on multiple, nonoverlapping segments taken from the microstructure. A statistical analysis of the entire microstructure provides the distribution of local porosity, tortuosity, and thicknesses. Our model quantitatively explains that porosity-dependent variations in the tortuosity, in particular the ones associated with high-volume pathways, decisively determine air flux variations.