Fluid flow concentration on preferential paths in heterogeneous porous media: application of graph theory

Fluid flow through geological formations is often concentrated on distinct preferential flow paths owing to the presence of fractures or large-scale permeability structures. However, existence of such structures is not a mandatory condition of preferential paths formation. Pore-scale spatial fluctuations of pore size and/or pore connectivity in statistically stationary porous media, if sufficiently large, can also lead to concentration of fluid flow on distinct pathways. In this paper, we attempted to establish the conditions of formation of preferential flow paths in heterogeneous porous media in terms of pore-size heterogeneity and pore connectivity. We simulated steady-state flow through stochastically constructed two- and three-dimensional pore networks, in which the width of the pore radius distribution and the pore coordination number (a measure of pore connectivity) were varied. We developed new techniques based on graph theory to identify potential preferential flow paths and characterize them. We observed a gradual transition from approximately uniform flow fields in low heterogeneity/high connectivity networks to flow localization on preferential paths with increasing pore-size heterogeneity and decreasing connectivity. The transition occurred at lower heterogeneity levels in three-dimensional than in two-dimensional simulations and was less influenced by pore connectivity variations. These results were summarized in a phase diagram in pore-size heterogeneity/pore connectivity parameter space, which we found consistent with relevant real rocks data.