Bifurcating paths: The relation between preferential pathways, channel splitting, under sampled regions, and tortuosity on the Darcy scale

Abstract

Transport in porous media on the Darcy scale can be both Fickian and non-Fickian, an outcome dependent on the degree of homogeneity of the hydraulic conductivity pattern, as well as the boundary conditions and flow rate. The non-Fickian manifestation is generally associated with heterogeneous media, which promotes the formation of preferential pathways that funnel the transport. Yet this funneling occurs in both weakly and strongly heterogeneous domains, through a mechanism that is yet to be fully characterized. We model Darcy-scale transport using a particle tracking code (PT) that samples a 2D, lognormally distributed conductivity field with a variance representing a range of homogeneous to heterogeneous domains. We find that the resulting preferential pathways tend to split into more pathways (bifurcations), leaving regions into which particles do not invade, which we refer to as “under sampled regions” (USR), while forming a tortuous path. The fraction of bifurcations decreases downstream, reaching an asymptotic value, with a trend that can be fitted as a power-law of the variance. We show that the same power-law exponent relating the bifurcations to the variance holds true for the USR fraction, tortuosity, and fractal dimension with the same variance. An extension of our work is also presented for varying correlation length of the conductivity spatial distribution. We further expand our analysis to a case of impermeable fraction in a uniform conductivity field and show that the power-law fit still holds.Key PointsPreferential pathways at the Darcy scale can be characterized by their bifurcations, and USR formation formed by their tortuous path.The bifurcation fraction, USR, tortuosity, and fractal dimension scale with the conductivity heterogeneity.The same power law exponent captures the scaling of bifurcation fraction, USR, tortuosity, and fractal dimension.