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

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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

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

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4 Dispersion

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5 Soil/Aquifer Material

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6 Real Soil Measurements zX-Ray Tomography

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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)

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

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

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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

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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

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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

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13 Representative Elementary Volume (REV) From Jacob Bear

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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?)

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

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16 Aquifer (10’s m)

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17 Laboratory and Field Scales

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

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19 Scale Dependence of Dispersivity Gelhar, et al, 1992

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20 Scale Dependence of Dispersivity Neuman, 1995

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

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

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23 Scale Dependence zSerrano, 1996

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24 Conventional Derivatives From Benson, 1998

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25 Conventional Derivatives From Benson, 1998

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

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27 Fractional Derivatives From Benson, 1998

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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

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29 Another Look at Divergence From Benson, 1998

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30 Another Look at Divergence From Benson, 1998

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31 Standard Symmetric -Stable Probability Densities

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32 Standard Symmetric -Stable Probability Densities

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33 Standard Symmetric -Stable Probability Densities

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34 Brownian Motion and Levy Flights

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35 Monte-Carlo Simulation of Levy Flights

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

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

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38 ADE/FADE

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39 Error Function

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40 -Stable Error Function

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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, with permission.

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42 Scaling and Tailing

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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

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