Transport of conservative solutes in simulated fracture networks: 2. Ensemble solute transport and the correspondence to operator‐stable limit distributions

Abstract

In networks where individual fracture lengths follow a fractal distribution, ensemble transport of conservative solute particles at the leading plume edge exhibit characteristics of operator‐stable densities. These densities have, as their governing equations of transport, either fractional‐order or integer‐order advection‐dispersion equations. Model selection depends on the identification of either multi‐Gaussian or operator‐stable transport regimes, which in turn depends on the power law exponent of the fracture length distribution. Low to moderately fractured networks with power law fracture length exponents less than or equal to 1.9 produce solute plumes that exhibit power law leading‐edge concentration profiles and super‐Fickian plume growth rates. For these network types, a multiscaling fractional advection‐dispersion equation (MFADE) provides a model of multidimensional solute transport where different rates of power law particle motion are defined along multiple directions. The MFADE model is parameterized by a scaling matrix to describe the super‐Fickian growth process, in which the eigenvectors correspond to primary fracture group orientations and the eigenvalues code fracture length and transmissivity. The approximation of particle clouds by a multi‐Gaussian (a subset of the operator stable) for densely fractured networks with finite variance fracture length distributions can be ascribed to the classical ADE where Fickian scaling rates pertain along orthogonal plume growth directions. Fracture networks show long‐term particle retention in low‐velocity fractures so that coupling of the equations of motion with retention models such as continuous time random walk or multirate mobile/immobile will increase accuracy near the source. Particle arrival times at exit boundaries for multi‐Gaussian plumes vary with spatial density. Generally, arrival times are faster in sparsely fractured domains where transport is governed by a few very long fractures.