Stability criteria of drainage displacement in pore networks under wettability gradient and gravity contrast

Viscous fingering, a common interfacial instability in two-phase displacement, reduces displacement efficiency by creating preferential flow paths that bypass the resident fluid. To achieve active control and stabilize the displacement front, this study investigates the distinct roles of wettability gradients and gravity contrasts during drainage processes in pore networks. A generalized mathematical model is first developed, incorporating arbitrary gravity contrasts, wettability gradients, and pore-size gradients. Theoretical stability criteria for viscous fingering are established and analyzed through numerical simulations. We first compare the stabilizing effects of pore-size and wettability gradients, and then systematically examine wettability-gradient-dominated flow in a uniform pore network and gravity-dominated flow in a pore network with a pore-size gradient. Results indicate that viscous fingering is effectively suppressed when the reduction in viscous resistance during local fingering is compensated by an increased capillary force or gravity contrast. Stability phase diagrams are constructed to distinguish between viscous fingering and compact displacement under various parameter combinations. Increasing either the wettability gradient or the gravity contrast raises the critical capillary number and viscosity ratio required to trigger fingering. Furthermore, stability is found to depend not only on the gradient magnitude but also on the contact angle value. This work provides a predictive criterion and physical insight for dynamically tuning displacement stability in engineered and natural porous media.