Double-crossover phenomena in Laplacian growth: Effects of sticking probability and finite viscosity ratio

Effect of mobility ratio on interaction between the fingers in unstable growth processes

We investigate interactions between thin fingers formed as a result of an instability of an advancing front in growth processes. We show that the fingers can both attract and repel each other, depending on their lengths and the mobility ratio between the invading and displaced phase. To understand the origin of these interactions we introduce a simple resistor model of the fingers. The predictions of the model are then compared to the numerical simulations of two unstable growth processes: dissolution of partially cemented rock fracture and viscous fingering in a regular network of channels.

Extension of the dielectric breakdown model for simulation of viscous fingering at finite viscosity ratios

Immiscible displacement of viscous oil by water in a petroleum reservoir is often hydrodynamically unstable. Due to similarities between the physics of dielectric breakdown and immiscible flow in porous media, we extend the existing dielectric breakdown model to simulate viscous fingering patterns for a wide range of viscosity ratios (μ(r)). At low values of power-law index η, the system behaves like a stable Eden growth model and as the value of η is increased to unity, diffusion limited aggregation-like fractals appear. This model is compared with our two-dimensional (2D) experiments to develop a correlation between the viscosity ratio and the power index, i.e., η = 10(-5)μ(r)(0.8775). The 2D and three-dimensional (3D) simulation data appear scalable. The fingering pattern in 3D simulations at finite viscosity ratios appear qualitatively similar to the few experimental results published in the literature.