Dinâmica de bolhas no escoamento bifásico estacionário em meios porosos desordenados

Abstract

Two-phase flow through porous media leads to the formation of bubbles that eventually break and merge or can become trapped in the porous matrix. This highly complex dynamic behavior creates characteristic fluctuations in the velocity fields of the two phases and strongly influences macroscopic properties such as effective permeability. In order to better understand how the microscopic behavior affects the macroscopic properties of the flow, we simulate the velocity fields of two immiscible fluids flowing through a two-dimensional porous medium. We found that the bubble size, m, follows a power law distribution, P(m) ∝ m−ξ , where the exponent ξ depends on the capillary number, Ca, which describes the ratio between the viscous and interfacial forces. Below a characteristic capillary number, given by Ca∗ ≈ 0.046, the bubbles are large and cohesive with a constant exponent ξ ≈ 1.23 ± 0.03. Above this value, the flow is dominated by many droplets and finger-like spanning clusters. In this regime the exponent ξ increases approaching 2.05 ± 0.03 in the limit when Ca → ∞. Moreover, 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 values of Ca. The fluctuations in the velocity time series present avalanches whose sizes follow a power law distribution, P(∆t) ∝ (∆t)−η , while the velocity jumps follow a Gaussian distribution. Analyzes by the DFA method show long-range correlations in the time series. The characteristic velocity of the flow, calculated as the mean value of the time series of the mixture of the two phases, follows a generalization of Darcy’s law in the form v(m) ∝ (∇p)β , where ∇p is the applied global pressure gradient and the exponent β depends on the surface tension between the two phases. For high values of capillary number, the constant of proportionality in this relation, called mobility, increases exponentially with the saturation of the less viscous phase. This result is in agreement with previous observations for effective permeabilities in dissolved gas powered reservoirs.