Mass transport with sorption in porous media

Abstract: Small-scale models in the form of random walks, combining Gaussian jumps, advection by mean flow field and possibly very long sorbing durations, correspond to experimental data in many porous media, in the laboratory and in the field. Within this frame-work, solutes are observed in two phases, which are mobile and immobile. For such random walks, in the hydrodynamic limit, the densities of that phases are linked by a relationship involving a fractional integral. This implies that the total density of tracer evolves according to a fractional variant of Fourier's law.

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  M. Néel,et al.  Continuous-time random-walk model of transport in variably saturated heterogeneous porous media. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  M. Joelson,et al.  Non Fickian flux for advection–dispersion with immobile periods , 2009 .

[4]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[5]  Julia,et al.  Vector-valued Laplace Transforms and Cauchy Problems , 2011 .

[6]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[7]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[8]  Enrico Scalas,et al.  Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Wojbor A. Woyczyński,et al.  Models of anomalous diffusion: the subdiffusive case , 2005 .

[10]  Marie-Christine Néel,et al.  Fractional Fick's law: the direct way , 2007 .

[11]  Rina Schumer,et al.  Fractal mobile/immobile solute transport , 2003 .

[12]  Melvin Lax,et al.  Stochastic Transport in a Disordered Solid. I. Theory , 1973 .

[13]  M. Néel,et al.  A continuous variant for Grünwald–Letnikov fractional derivatives , 2008 .

[14]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[15]  P. Levy.  Théorie de l'addition des variables aléatoires , 1955 .

[16]  David A. Benson,et al.  A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations , 2009 .

[17]  S. Samko.  A new approach to the inversion of the Riesz potential operator , 1998 .

[18]  M. V. Genuchten,et al.  Mass transfer studies in sorbing porous media. I. Analytical solutions , 1976 .

[19]  Boris Rubin,et al.  Fractional Integrals and Potentials , 1996 .

[20]  Christoph Hinz,et al.  Non‐Fickian transport in homogeneous unsaturated repacked sand , 2004 .

[21]  W. Rudin.  Real and complex analysis , 1968 .

[22]  Karina Weron,et al.  Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  M. Joelson,et al.  Mass transport subject to time-dependent flow with nonuniform sorption in porous media. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  M. V. Genuchten,et al.  Mass Transfer Studies in Sorbing Porous Media: II. Experimental Evaluation with Tritium (3H2O)1 , 1977 .

[25]  D. Benson,et al.  Moment analysis for spatiotemporal fractional dispersion , 2008 .

[26]  Francesco Mainardi,et al.  Continuous-time random walk and parametric subordination in fractional diffusion , 2007 .