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 (${\mu }_r$). 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 = ${10}^{-5}{\mu }_r{}^{0.8775}$. 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.

Figure 1

(a) Schematic showing the three possibilities of electron propagation during dielectric breakdown as described by [12] and (b) some analogous case for a meniscus movement in a porous medium.

Figure 2

A plot of growth probability distribution along a flat interface with a small perturbation at the center for different values η (or ${\mu }_r$).

Figure 3

A picture of the silica micromodel used in the study showing the location of the inlet and outlet ports and an enlarged image of the etched pore-scale pattern.

Figure 4

(a) Displacement patterns generated in a 2D silica micromodel while flooding at viscosity ratios of 0.005, 1, 200, 1000, 4000, and 10,000; (b) an expanded view of stable displacement with ${\mu }_r$ = 0.005 showing the trapped oil in the swept zone and the pore-scale perturbations that exist in even the most stable displacement; (c) saturation profile along the length of the micromodel at the time of breakthrough for ${\mu }_r$ = 200 and above.

Figure 5

(a) Water and oil distribution in 2D simulations for different values of parameter η; (b) plot of water saturation along the length of the matrix at the time of breakthrough. “Saturation XY” is the phase distribution in the XY plain and “Avg. Saturation” is the average along the length.

Figure 6

A comparison of the experimental 2D micromodel result and simulation at ${\mu }_r$ = 0.005 and η = ${10}^{-6}$, respectively. Pink color (lower half) indicates the oil phase; red (top, speckled portion) is the water phase with trapped oil.

Figure 7

A plot of recovery efficiency vs power-law factor (η) for 2D and 3D simulations.

Figure 8

Relation between η and viscosity ratios based on the 2D micromodel experimental results.

Figure 9

(a) Averaged water and oil distribution in the XY plain (Avg. Sat XY) for 3D simulation with different values of parameter η; (b) plot of water saturation along the length of the matrix at the time of breakthrough.

Figure 10

A plot showing that the recovery trends of 2D and 3D simulations are scalable.

Figure 11

Simulations of oil displacement for η = 3.16 × ${10}^{-3}$ and 1.05 × ${10}^{-3}$ to simulate the 2000 and 7000 viscosity ratio displacement experiments in Bentheimer slabs presented by Skauge et al. [21].