Reappraisal of Upscaling Descriptors for Transient Two-Phase Flows in Fibrous Media

Transient two-phase flows within fibrous media are considered at local scale. Upscaling these flows constitute a key procedure towards a tractable description in an industrial context. However, the task remains challenging as a time-dependent behaviour is observed within a geometrically complex structure with interplay of various physical phenomena (capillary effects, viscous dissipation, etc.). The usual upscaling strategies encountered in both soil sciences and composite materials communities are reviewed, compared, and finally adapted to reach a method that is relevant to describe fibrous media imbibition. Using finite element flow simulations on statistical representative volume elements, the proposed approach first considers several definitions for saturation in order to characterise the flow dynamics as well as the characteristic length associated with the transient behaviour. Next, two methods are proposed to assess a resulting capillary pressure, demonstrating the importance to properly define the capillary pressure acting on the interface. The first one considers the mean pressure jump at the interface, while the second one uses a machine-learning technique, namely Gaussian process regression, to retrieve the mean curvature of the interface. Those methods are found to be both consistent and in agreement with the results from the literature. Finally, a novel approach that stochastically describes the position of the flow front through a presence distribution is detailed. The spread of the front can be compared to the saturation length, and its value has been found to be small enough to be neglected at upper scale, justifying the use of sharp interface models for similar porous media and flow settings.