Bubble Dynamics in Stationary Two-phase Flow Through Disordered Porous Media

Two-phase flow through porous media leads to the formation of drops and fingers, which eventually break and merge or may be trapped behind obstacles. This complex dynamical behavior highly influences macroscopic properties such as the effective permeability and it also creates characteristic fluctuations in the velocity fields of the two phases, as well as in their relative permeability curves. In order to better understand how the microscopic behavior of the flow affects macroscopic properties of two phases, we simulate the velocity fields of two immiscible fluids flowing through a two-dimensional porous medium. By analyzing the fluctuations in the velocity fields of the two phases, we find that the system is ergodic for large volume fractions of the less viscous phase and high capillary numbers Ca. We also see that the distribution of drop sizes m follows a power-law scaling, P ( m ) ∝ m − ξ . The exponent ξ depends on the capillary number. Below a characteristic capillary number, namely Ca* ≈ 0.046, the drops are large and cohesive with a constant scaling exponent ξ ≈ 1.23 ± 0.03. Above the characteristic capillary number Ca*, the flow is dominated by many small droplets and few finger-like spanning clusters. In this regime the exponent ξ increases approaching 2.05 ± 0.03 in the limit of infinite capillary number. Our analysis also shows that the temporal mean velocity of the entire mixture can be described by a generalization of Darcy’s law of the form v ̄ ( m ) ∝ ( ∇ P ) β where the exponent β is sensitive to the surface tension between the two phases. In the limit of infinite capillary numbers the mobility term increases exponentially with the saturation of the less viscous phase. This result agrees with previous observations for effective permeabilities found in dissolved-gas-driven reservoirs.

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