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Scale-Dependent Dispersivities and The Fractional Convection - Dispersion Equation

Scale-Dependent Dispersivities and The Fractional Convection - Dispersion Equation. Primary Source: Ph.D. Dissertation David Benson University of Nevada Reno, 1998. Mike Sukop/FIU. Motivation Porous Media and Models Dispersion Processes Representative Elementary Volume

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Scale-Dependent Dispersivities and The Fractional Convection - Dispersion Equation

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  1. Scale-Dependent Dispersivities and The Fractional Convection - Dispersion Equation • Primary Source: • Ph.D. Dissertation • David Benson • University of Nevada Reno, 1998 Mike Sukop/FIU

  2. Motivation Porous Media and Models Dispersion Processes Representative Elementary Volume Convection-Dispersion Equation Scale Dependence Solute Transport Conventional and Fractional Derivatives a-Stable Probability Densities Levy Flights Application Conclusions Outline

  3. Motivation • Scale Effects • Need for Independent Estimation

  4. Dispersion

  5. Soil/Aquifer Material

  6. Real Soil Measurements • X-Ray Tomography

  7. What is Dispersion? • Spreading of dissolved constituent in space and time • Three processes operate in porous media: • Diffusion (random Brownian motion) • Convection (going with the flow) • Mechanical mixing (the tough part)

  8. Solute Dispersion Diffusion Only Time = 0 Modified from Serrano, 1997

  9. Solute Dispersion Diffusion Only Time > 0 Modified from Serrano, 1997

  10. Solute Dispersion Advection Only Average Pore Water Velocity Time > 0 x > x0 Time = 0 x = x0 Modified from Serrano, 1997

  11. Solute Dispersion • Water Velocities Vary on sub-Pore Scale • Mechanical Mixing in Pore Network • Mixing in K Zones Modified from Serrano, 1997

  12. Solute Dispersion Mechanical Dispersion, Diffusion, Advection Average Pore Water Velocity Time = 0 x = x0 Time > 0 x > x0 Modified from Serrano, 1997

  13. Representative Elementary Volume (REV) From Jacob Bear

  14. Representative Elementary Volume (REV) • General notion for all continuum mechanical problems • Size cut-offs usually arbitrary for natural media (At what scale can we afford to treat medium as deterministically variable?)

  15. Soil Blocks (0.3 m) Phillips, et al, 1992

  16. Aquifer (10’s m)

  17. Laboratory and Field Scales

  18. Problems with the CDE • Macroscopic, REV, Scale dependence, • Brownian Motion/Gaussian distribution

  19. Scale Dependence of Dispersivity Gelhar, et al, 1992

  20. Scale Dependence of Dispersivity Neuman, 1995

  21. Scale Dependence of Dispersivity Pachepsky, et al, 1999 (in review)

  22. Scale Dependence • Power law growth Deff = Dxs • Perturbation/Stochastic DEs • Statistical approaches

  23. Scale Dependence • Serrano, 1996

  24. Conventional Derivatives From Benson, 1998

  25. Conventional Derivatives From Benson, 1998

  26. Fractional Derivatives The gamma function interpolates the factorial function. For integer n, gamma(n+1) = n!

  27. Fractional Derivatives From Benson, 1998

  28. Another Look at Divergence • For integer order divergence, the ratio of surface flux to volume is forced to be a constant over different volume ranges

  29. Another Look at Divergence From Benson, 1998

  30. Another Look at Divergence From Benson, 1998

  31. Standard Symmetric a-Stable Probability Densities

  32. Standard Symmetric a-Stable Probability Densities

  33. Standard Symmetric a-Stable Probability Densities

  34. Brownian Motion and Levy Flights

  35. Monte-Carlo Simulation of Levy Flights

  36. FADE (Levy Flights) MATLAB Movie/Turbulence Analogy 500 50 100 ‘flights’, 1000 time steps each

  37. Ogata and Banks (1961) • Semi-infinite, initially solute-free medium • Plane source at x = 0 • Step change in concentration at t = 0

  38. ADE/FADE

  39. Error Function

  40. a-Stable Error Function

  41. Scaling and Tailing q=0.12 After Pachepsky Y, Benson DA, and Timlin D (2001) Transport of water and solutes in soils as in fractal porous media. In Physical and Chemical Processes of Water and Solute Transport/Retention in Soils. D. Sparks and M. Selim. Eds. Soil Sci. Soc. Am. Special Pub. 56, 51-77 with permission.

  42. Scaling and Tailing

  43. Conclusions • Fractional calculus may be more appropriate for divergence theorem application in solute transport • Levy distributions generalize the normal distribution and may more accurately reflect solute transport processes • FADE appears to provide a superior fit to solute transport data and account for scale-dependence

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