Download presentation

Presentation is loading. Please wait.

Published byAliya Raikes Modified over 3 years ago

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

Similar presentations

OK

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 1 HW 14 More on Moderators Calculate the moderating power and ratio for pure D 2 O as well.

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh). 1 HW 14 More on Moderators Calculate the moderating power and ratio for pure D 2 O as well.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on shell scripting linux Ppt on polynomials and coordinate geometry calculator Ppt on area of parallelogram video Ppt on juvenile rheumatoid arthritis Ppt on heritage and culture of rajasthan india Download ppt on nutrition in plants and animals Ppt on first conditional activities Ppt on conservation of natural resources in india Ppt on self awareness in nursing Ppt on computer virus and antivirus