Kinetic Theory of Multicomponent Ostwald Ripening in Porous Media

TL;DR

This work introduces the first kinetic theory for multicomponent Ostwald ripening of bubbles in porous media, formulating the bubble population with a 3D number-density function $g(s;t)$ in the phase space $s=(R_p,S^b,y_1)$. The population balance is closed via a mean-field approach that accounts for spatial correlations in pore size and enforces mass conservation, enabling activation of interactions between distant bubbles. Validation against a pore-network model across 19 cases shows strong agreement without adjustable parameters, demonstrating the theory’s ability to predict the evolution of $g(s;t)$ and bulk bubble statistics for homogeneous, heterogeneous, correlated, and uncorrelated networks. This framework generalizes previous single-component theories, overcomes key limitations in remote-bubble interactions and mass conservation, and provides a computationally efficient tool for predicting Ostwald ripening in subsurface systems, with particular relevance to underground hydrogen storage. The theory also highlights gaps related to water storage effects and suggests pathways for methodological improvements and extension to more components and pore shapes.”

Abstract

Partially miscible bubble populations trapped in porous media are ubiquitous in subsurface applications such as underground hydrogen storage (UHS), where cyclic injections fragment gas into numerous bubbles with distributions of sizes and compositions. These bubbles exchange mass through Ostwald ripening, driven by differences in composition and interfacial curvature. While kinetic theories have been developed for single-component ripening in porous media, accounting for bubble deformation and spatial correlations in pore size, no such theory exists for multicomponent systems. We present the first kinetic theory for multicomponent Ostwald ripening of bubbles in porous media. The formulation describes the bubble population with a number-density function $g(s; t)$ in a 3D statistical space of bubble states $s = (R_p, S^b, y)$, consisting of pore size, bubble saturation, and composition. Evolution is governed by a population balance equation with closure through mean-field approximations that account for spatial correlations in pore size and ensure mass conservation. The theory generalizes previous single-component formulations, removing key limitations such as the inability to capture interactions between distant bubbles. Systematic validation against pore-network simulations across homogeneous, heterogeneous, correlated, and uncorrelated networks demonstrates good agreement without adjustable parameters. Pending challenges and limitations are discussed. Since the theory imposes no constraints on bubble count or correlation length, it enables predictions beyond the pore scale.

Figures (23)

• Figure 1: The porous microstructure is composed of a solid matrix and an interconnected void space. We conceptualize the void as a pore network, a computational graph that consists of nodes (pores) and links (throats). All pores contain a wetting phase (H$_2$O), and some are occupied by partially miscible, two-component (CO$_2$ and H$_2$) gaseous bubbles. Bubbles are confined to one pore and exchange mass via Ostwald ripening, driven by concentration gradients of dissolved species around each bubble.
• Figure 1: (Left) Schematic of the curves defined by $dS^b_s/dt\!=\!0$ and $dy_{1,s}/dt\!=\!0$ for Eq.\ref{['eq:closure_raw']} in phase space for a homogeneous network. The phase space is 2D because pore size ($R_p$) is uniform, with equivalent coordinates $(\kappa/\kappa^{mf},y_1)$ instead of $(S^b,y_1)$. The two curves form an ever narrowing corridor that bubbles must traverse to reach equilibrium (yellow dot). At some point, the corridor's width is smaller than the grid used to discretize the population balance Eq.\ref{['eq:populationBalance']}, halting further progress towards the yellow dot. (Right) The reformulation in Eq.\ref{['eq:closure_final']} removes this drawback by repositioning the black line to widen the corridor.
• Figure 2: (a) Schematic of 3D phase space used to describe bubble statistics by the number-density function $g(s;t)$. Each point in this space is a bubble state $s$ with three coordinates $R_p$, $S^b$, and $y_1$. The number of bubbles within an infinitesimal volume $d\Omega$ centered at $s$ equals $g(s;t)\,d\Omega$. The gray cloud denotes $g(s;t)$ and the contours its 2D marginals projected onto the coordinate planes. (b) Schematic of a bubble at state $s$ interacting with the rest of the population, conceptualized as a mean field.
• Figure 2: Inter-bubble spacing $l_{RR'}$ predicted from Eq.\ref{['eq:bubbleSpacing']} in the theory (solid lines) versus PNM using Eq.\ref{['eq:lrr_PNM']} (circles). For clarity, we use $R_x\!\rightarrow\!R_y$ to represent $l_{R_xR_y}$. Pore sizes are divided into $n\!=\!20$ bins, where $R_1$ is the smallest and $R_n$ the largest.
• Figure 3: (Left panel) Schematic of spatial arrangement of bubbles in a pore network. H$_2$ bubbles are colored in cyan and CO$_2$ bubbles in red. (Three rightmost panels) Translation of spatial information in the pore network into statistical information in the theory. The three panels show projections of the number-density function $g(s;t)$ onto the 2D coordinate planes of the 3D phase space (Fig.\ref{['fig:PhaseMean']}). Bubbles from the first panel are annotated on these phase-space plots to emphasize which part of $g(s;t)$ they represent. Shadows represent 2D marginals of $g(s;t)$, and pink curves along the right/bottom axes the 1D marginals.

Theorems & Definitions (3)

• Theorem 1
• Corollary 1
• Remark 1