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Extension of the Feller‐Fokker‐Planck equation to two and three dimensions for modeling solute transport in fractal porous media
Author(s) -
Su Ninghu
Publication year - 1997
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 - 10.1029/97wr00387
Subject(s) - fokker–planck equation , fractal , mathematics , laplace transform , mathematical analysis , statistical physics , scaling , physics , partial differential equation , geometry
Following an earlier development of a Fokker‐Planck equation (FP) for modeling fractal‐scale‐dependent transport of solutes in one‐dimensional subsurface flow of heterogeneous porous media, this technical note extends the FP to three dimensions, and presents a two‐dimensional (2‐D) FP by reducing the 3‐D FP with the aid of the Dupuit approximation. The 2‐D FP is derived by including two fractal dispersivities in the convective‐dispersive equation leading to a generalized Feller‐Fokker‐Planck equation (GFFP) featuring both the generalized Feller equation (GF) and FP. Similarity solutions of the 2‐D GFP with two linear‐scale‐dependent dispersivities are presented which can be used as a kernel in the convolution integral to yield an output on a real timescale, and the input function can be derived by a procedure known as the inverse problem with the aid of a Laplace transform.