Invasion percolation with viscous forces

We study invasion percolation in the presence of viscous forces, as a model of the drainage of a wetting fluid from a porous medium. Using concepts from gradient percolation, we consider two different cases, depending on the magnitude of the mobility ratio M. When M is sufficiently small, the displacement can be modeled by a form of gradient percolation in a stabilizing gradient, involving a particular percolation probability profile. We develop the scaling of the front width and the saturation profile, in terms of the capillary number. In the opposite case, the displacement is described by gradient percolation in a destabilizing gradient and leads to capillary-viscous fingering. This regime is identified in the context of viscous displacements and in general differs from diffusion-limited aggregation, which also describes displacements at large M. Constraints for the validity of the two regimes are developed. Limited experimental and numerical results support the theory of stabilized displacement. The effect of heterogeneity is also discussed.