Pore-scale study of drainage behavior in gradient porous media

Gradient porous media, characterized by continuously varying pore sizes along the flow direction, fundamentally reorganize the interplay between viscous and capillary forces. Yet, how such geometric gradients modulate pore-scale drainage dynamics and displacement efficiency remains insufficiently understood. In this study, we employ the lattice Boltzmann method to investigate immiscible drainage in a three-dimensional granular packing exhibiting a clear pore-size gradient. By imposing two opposite injection directions within the same structure-corresponding to gradually increasing or decreasing pore sizes-we systematically examine how geometric gradients influence fingering behavior, interface morphology, pressure evolution, and breakthrough efficiency across a broad range of capillary numbers and viscosity ratios. Our results show that increasing pore sizes along the flow direction promote both viscous and capillary fingering, enabling early breakthrough but reducing displacement efficiency except under stable displacement. In contrast, decreasing pore sizes suppress fingering, enhance lateral invasion, and allow high efficiency even within the capillary fingering regime. Dynamic pressure analyses further reveal that gradient-induced variations in local capillary resistance and pore-scale velocities govern the stability of the displacement front. These mechanisms collectively lead to markedly different phase diagrams for the two flow directions. Overall, the findings demonstrate that the direction of flow relative to a pore-size gradient constitutes a previously underappreciated yet powerful control parameter for optimizing immiscible displacement in heterogeneous porous formations.