Ripening Kinetics and Equilibrium States of Bubble Populations in Porous Media

Abstract:

A population of bubbles trapped inside a porous medium are in disequilibrium at early times, because each bubble has a different capillary pressure (P_c) (Garing et al., AWR, 2017). At later times, bubbles dissolve into the surrounding fluid and collectively approach a uniform P_c, a phenomenon known as “anti-coarsening Ostwald ripening” (Xu et al., PRL, 2017). Despite valuable insights gained from recent micromodel experiments and pore-network simulations, there is still a lack of a firm theory that describes the statistical evolution dynamics of trapped bubble populations. The goal of this work is to address this knowledge gap.

We study the evolution kinetics of a random bubble population (~4x10⁴) up to equilibrium within a porous material. A fully implicit pore-network model (PNM) is developed whereby the governing equations of bubble ripening are coupled to those of molecular diffusion within the background fluid. The algorithm is designed so as to avoid numerical instability near critical values of P_c. The PNM is used to demonstrate the temporal scaling of ripening dynamics in porous media. We identify two regimes. The first is when all bubbles have an initial volume below a critical threshold, V_crit. Larger bubbles grow while absorbing smaller bubbles at a ripening rate that the harmonic average radius scales with t^2/11. The scaling is very different from that of classic Ostwald ripening (in bulk fluid), which is that the arithmetic average radius scales with t^1/3. The second regime involves bubbles with initial volumes above V_crit, which leads to smaller bubbles growing while larger ones shrink (anti-coarsening). The dynamics of arithmetic average radius, in this case, scale as t^1/3 and the total time required to reach equilibrium scales as L², where L is the domain size. The impact of various parameters such as pore-size distribution and initial bubble volumes are also investigated. The above observations are finally integrated into a coherent theory that describes the statistical evolution of bubble ripening at the Darcy scale. The work contributes to our current understanding, and engineering, of subsurface systems such as geologic CO₂ storage and petroleum migration.