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Multiscaling fractional advection‐dispersion equations and their solutions
Author(s) -
Schumer Rina,
Benson David A.,
Meerschaert Mark M.,
Baeumer Boris
Publication year - 2003
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2001wr001229
Subject(s) - continuous time random walk , mathematics , fractional calculus , dispersion (optics) , statistical physics , mathematical analysis , advection , random walk , probability density function , random variable , probability distribution , physics , statistics , thermodynamics , optics
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.