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, the 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 the 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 threedimensional 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. Plain Language Summary In general, fluid flow underground is unevenly distributed. Most of the flow occurs in distinct channels called preferential flow paths, which are often associated with large-scale structures such as highly permeable layers or fractures. Preferential flow paths can also form in rock formations in which such features are absent. The only requirements are a sufficiently high level of pore size heterogeneity and/or partial disconnection of the pore space. In this study, we simulated fluid flow through heterogeneous and partially connected pore networks and used graph theory to identify and characterize the possibly existing preferential flow paths, that formed owing to the randomness of the pore space. We were thus able to establish a phase diagram showing the regions in heterogeneity/connectivity parameter space where fluid flow is evenly distributed and where it is concentrated on preferential flow paths. These two regions are separated by a broad transition zone where the flow morphology is intermediate between the two extremes. By comparing our results to those of previously published studies, we found that the phase diagram appeared consistent with real rocks data.