Numerical simulation for solute transport in fractal porous media

A modified Fokker-Planck equation with continuous source for solute transport in fractal porous media is considered. The dispersion term of the governing equation uses a fractional-order derivative and the diffusion coefficient can be time and scale dependent. In this paper, numerical solution of the modified Fokker-Planck equation is proposed. The effects of different fractional orders and fractional power functions of time and distance are numerically investigated. The results show that motions with a heavy tailed marginal distribution can be modelled by equations that use fractional-order derivatives and/or time and scale dependent dispersivity.

[1]  H. G. E. Hentschel,et al.  Relative diffusion in turbulent media: The fractal dimension of clouds , 1984 .

[2]  I. Podlubny.  Fractional differential equations , 1998 .

[3]  A two-dimensional finite volume method for transient simulation of time- and scale-dependent transport in heterogeneous aquifer systems , 2003 .

[4]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[5]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[6]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[7]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[8]  S. Bhatia,et al.  Application of Petrov–Galerkin methods to transient boundary value problems in chemical engineering: adsorption with steep gradients in bidisperse solids , 2001 .

[9]  Fawang Liu,et al.  An unstructured mesh finite volume method for modelling saltwater intrusion into coastal aquifers , 2002 .

[10]  S. Bhatia,et al.  Numerical solution of hyperbolic models of transport in bidisperse solids , 2000 .

[11]  M. Shlesinger,et al.  Random walks with infinite spatial and temporal moments , 1982 .

[12]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[13]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[14]  S. Bhatia,et al.  Computationally efficient solution techniques for adsorption problems involving steep gradients in bidisperse particles , 1999 .

[15]  S. P. Neuman.  Universal scaling of hydraulic conductivities and dispersivities in geologic media , 1990 .

[16]  Alexander I. Saichev,et al.  Fractional kinetic equations: solutions and applications. , 1997, Chaos.