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

2. 2 Outline zMotivation zPorous Media and Models zDispersion Processes zRepresentative Elementary Volume zConvection-Dispersion Equation z Scale Dependence z Solute Transport z Conventional and Fractional Derivatives   -Stable Probability Densities z Levy Flights z Application z Conclusions

3. 3 Motivation zScale Effects zNeed for Independent Estimation zScale Effects zNeed for Independent Estimation

4. 4 Dispersion

5. 5 Soil/Aquifer Material

6. 6 Real Soil Measurements zX-Ray Tomography

7. 7 What is Dispersion? zSpreading of dissolved constituent in space and time zThree processes operate in porous media: yDiffusion (random Brownian motion) yConvection (going with the flow) yMechanical mixing (the tough part)

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

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

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

11. 11 Solute Dispersion zWater Velocities Vary on sub-Pore Scale zMechanical Mixing in Pore Network zMixing in K Zones zWater Velocities Vary on sub-Pore Scale zMechanical Mixing in Pore Network zMixing in K Zones Modified from Serrano, 1997

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

13. 13 Representative Elementary Volume (REV) From Jacob Bear

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

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

16. 16 Aquifer (10’s m)

17. 17 Laboratory and Field Scales

18. 18 Problems with the CDE zMacroscopic, REV, Scale dependence, zBrownian Motion/Gaussian distribution

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

20. 20 Scale Dependence of Dispersivity Neuman, 1995

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

22. 22 Scale Dependence zPower law growth Deff = Dx s zPerturbation/Stochastic DEs zStatistical approaches

23. 23 Scale Dependence zSerrano, 1996

24. 24 Conventional Derivatives From Benson, 1998

25. 25 Conventional Derivatives From Benson, 1998

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

27. 27 Fractional Derivatives From Benson, 1998

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

29. 29 Another Look at Divergence From Benson, 1998

30. 30 Another Look at Divergence From Benson, 1998

31. 31 Standard Symmetric  -Stable Probability Densities

32. 32 Standard Symmetric  -Stable Probability Densities

33. 33 Standard Symmetric  -Stable Probability Densities

34. 34 Brownian Motion and Levy Flights

35. 35 Monte-Carlo Simulation of Levy Flights

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

37. 37 Ogata and Banks (1961) zSemi-infinite, initially solute-free medium zPlane source at x = 0 zStep change in concentration at t = 0

38. 38 ADE/FADE

39. 39 Error Function

40. 40  -Stable Error Function

41. 41 Scaling and Tailing  =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. 42 Scaling and Tailing

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