Clustering and log-linear regression of permeability-Forchheimer relationships in porous media

Non-Darcy flow in porous media becomes significant at relatively high velocities. We observed an approximate inverse scaling relationship between the Forchheimer coefficient and permeability (k), with β∝k−0.8. This study compiled experimental data on the β–k relationship from peer-reviewed studies on non-Darcy flow in porous media. A dataset was compiled from 18 peer- reviewed papers, yielding 280 measurements covering a wide range of porous media. To investigate the potential scaling relationship between permeability and the Forchheimer coefficient, a logarithmic transformation was applied to both variables. In addition, K-means clustering was applied to identify natural groups of porous media that represent different hydraulic regimes, which helps to reduce the scatter in the Forchheimer coefficient and permeability relationship. Although the K-means method is a purely mathematical grouping method, the resulting clusters correspond to different hydraulic behaviors. Furthermore, we evaluated a range of distance metrics (Euclidean, cityblock, cosine, and correlation) for cluster numbers ranging from D = 2 to D = 5 to determine the optimal subdivision of flow regimes. Internal validation metrics, including the within-cluster sum of squares (WCSS), silhouette coefficient, and Davies-Bouldin index, indicate that D=4 provides the most meaningful clustering solution. The goal of this study is to explore the β-k relationship and identify natural clusters of porous media that may reflect different hydraulic behaviors.