Multiscaling fractional advection‐dispersion equations and their solutions

The multiscaling fractional advection‐dispersion equation (ADE) is a multidimensional model of solute transport that encompasses linear advection, Fickian dispersion, and super‐Fickian dispersion. The super‐Fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions. The directions need not be orthogonal, so the model can be applied to irregular, noncontinuum fracture networks. The statistical model underlying multiscaling fractional dispersion is a continuous time random walk (CTRW) in which particles have arbitrary jump length distributions and finite mean waiting time distributions. The meaning of the parameters in a compound Poisson process, a subset of CTRWs, is used to develop a physical interpretation of the equation variables. The Green's function solutions are the densities of operator stable probability distributions, the limit distributions of normalized sums of independent, and identically distributed random vectors. These densities can be skewed, heavy‐tailed, and scale nonlinearly, resembling solute plumes in granular aquifers. They can also have fingers in any direction, resembling transport along discrete pathways such as fractures.

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