Mobility and pore-scale fluid dynamics of rate-dependent yield-stress fluids flowing through fibrous porous media

The steady flow of viscoplastic fluids through fibrous porous media is studied numerically and theoretically. We consider fluids with a plastic yield stress and a rate-dependent viscosity that can be described by the Herschel-Bulkley model. We first investigate the pore-scale flow characteristics through numerical simulations for flow transverse to a square array of fibers with comprehensive parametric studies to independently analyze the effects of the rheological properties of the fluid and the geometrical characteristics of the fibrous medium. Our numerical simulations show that the critical Bingham number at which the flow transitions from a fully-yielded regime to locally unyielded regions depends on the medium porosity. We develop a scaling model for describing the bulk characteristics of the flow, taking into account the coupled effects of the medium porosity and the fluid rheology. This model enables us to accurately predict the pressure-drop-velocity relationship over a wide range of Bingham numbers, power-law indices, and porosities with a formulation that can be applied to a square or a hexagonal array of fibers. The ultimate result of our scaling analysis is a generalized form of Darcy's law for Herschel-Bulkley fluids with the mobility coefficient provided as a function of the system parameters. Based on this model, we construct a modified Bingham number rescaled with a suitable porosity function, which incorporates all the rheological and pore-scale parameters that are required to determine the dominant flow regime.