Scaling of single-bubble growth in a porous medium

Mass-transfer driven growth of a single gas cluster in a porous medium under the application of a supersaturation in the far field is examined. We discuss the growth pattern and its growth rate. Contrary to compact (spherical) growth in the bulk, growth patterns in porous media are disordered and vary from percolation to diffusion-limited aggregation (DLA) as the cluster size increases. At conditions of low supersaturation, scaling laws for the boundaries that delineate these patterns and of the corresponding growth rates are derived. In three dimensions (3D), it is found that the cluster grows as ${\mathit{R}}_{\mathit{g}}$∼${\mathit{t}}_{\mathit{f}}^{1/(\mathit{D}}$-1), where ${\mathit{D}}_{\mathit{f}}$ is the pattern fractal dimension (≊2.50 for percolation or DLA). A similar result involving logarithmic corrections is found for 2D. These results generalize the classical scaling ${\mathit{R}}_{\mathit{g}}$∼${\mathit{t}}^{1/2}$ to fractal clusters.