Linear–nonlinear flow mechanisms and pressure-drop correction model for slurry flow in rough fractures

Hydraulic fractures in unconventional reservoirs typically exhibit narrow widths and pronounced surface roughness. Under such complex geometries, slurry flow and pressure-drop behavior deviate significantly from the laminar-flow assumption. Existing models cannot consistently describe the coupled effects of fracture roughness, proppant diameter, and fracture width on the pressure-drop–velocity relationship. Based on computational fluid dynamics–discrete element method simulations of slurry flow in rough fractures, this study extends the classical Forchheimer law. A Taylor expansion is applied to the viscous term, and the inertial term is generalized using a power-law formulation. The resulting model explicitly unifies the linear viscous, nonlinear viscous, and inertial contributions and captures the coupled influence of fracture roughness, proppant diameter, and width in shaping the nonlinear pressure-drop–velocity relationship. The results indicate that rough fractures operate in a nonlinear flow regime under most conditions. Asperity-scale roughness and particle confinement reduce the effective fracture width and enhance both viscous and inertial dissipation, causing slurry flow to depart markedly from that predicted by Poiseuille's law even at relatively low Reynolds numbers. Based on these findings, this study proposes an explicit nonlinear pressure-drop correction function Φp to quantify the deviation of the actual pressure gradient from that predicted by Poiseuille's law. This correction mitigates the limited applicability of Poiseuille's law when describing flow in rough fractures. The proposed framework provides a transparent, transferable basis for interpreting nonlinear slurry flow in rough fractures and for implementing nonlinear pressure-drop corrections in field-scale slurry flow simulators.