Derivation and validation of a generalized Darcy law for Herschel-Bulkley fluids in porous media
Large resources of heavy and highly viscous crude oil exhibit non-Newtonian behavior that cannot be represented adequately by the classical Darcy law. A particularly important feature of many structured oils is the existence of a yield stress and, consequently, a threshold pressure gradient below which macroscopic filtration does not occur. This study develops an analytical generalized Darcy relation for a Herschel-Bulkley fluid flowing through a homogeneous porous medium. The derivation starts from the laminar tube-flow solution and introduces porosity, an equivalent hydraulic radius, and a pore-channel tortuosity coefficient to obtain a porous-medium velocity equation. The resulting relation is nonlinear and explicitly includes yield stress, consistency index, flow-behavior index, permeability, and porosity. Its limiting forms are examined for power-law, Newtonian, and Bingham fluids. The analytical equation is then compared with a widely used simplified semi-empirical relation for permeability values of 1.0×10⁻¹³, 2.0×10⁻¹³, and 3.0×10⁻¹³ m² and flow-behavior indices of 0.90, 1.05, and 1.20. The two formulations show similar trends over the investigated engineering range, although the deviation becomes more noticeable near the onset of flow and for stronger departures from Newtonian behavior. An explicit expression for the threshold pressure gradient is also obtained, eliminating the need to treat the corresponding coefficient as purely empirical. The proposed formulation can support screening calculations and reservoir-flow models involving yield-stress oils and other Herschel-Bulkley fluids.