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Continuous Time Random Walk Model

Continuous Time Random Walk Model. Primary Sources: Berkowitz, B, G. Kosakowski, G. Margolin, and H. Scher , Application of continuuous time random walk theory to tracer test measurents in fractured and heterogeneous porous media , Ground Water 39 , 593 - 604 , 2001 .

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Continuous Time Random Walk Model

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  1. Continuous Time Random Walk Model Primary Sources: Berkowitz, B, G. Kosakowski, G. Margolin, and H. Scher, Application of continuuous time random walk theory to tracer test measurents in fractured and heterogeneous porous media, Ground Water39, 593 - 604, 2001. Berkowitz, B. and H. Scher, On characterization of anomalous dispersion in porous and fractured media, Wat. Resour. Res.31, 1461 - 1466, 1995. Mike Sukop/FIU

  2. Introduction • Continuous Time Random Walk (CTRW) models • Semiconductors [Scher and Lax, 1973] • Solute transport problems [Berkowitz and Scher, 1995]

  3. Introduction • Like FADE, CTRW solute particles move along various paths and encounter spatially varying velocities • The particle spatial transitions (direction and distance given by displacement vector s) in time t represented by a joint probability density function y(s,t) • Estimation of this function is central to application of the CTRW model

  4. Introduction • The functional form y(s,t) ~ t-1-b (b > 0) is of particular interest [Berkowitz et al, 2001] • b characterizes the nature and magnitude of the dispersive processes

  5. Ranges of b • b ≥ 2 is reported to be “…equivalent to the ADE…” • For b ≥ 2, the link between the dispersivity (a = D/v) in the ADE and CTRW dimensionless bb is bb = a/L • b between 1 and 2 reflects moderate non-Fickian behavior • 0 < b < 1 indicates strong ‘anomalous’ behavior

  6. Fitting Routines/Procedures • http://www.weizmann.ac.il/ESER/People/Brian/CTRW/ • Three parameters (b, C, and C1) are involved. For the breakthrough curves in time, the fitting routines return b, T, and r, which, for 1 < b < 2, are related to C and C1 as follows: • L is the distance from the source • Inverting these equations gives C and C1, which can then be used to compute the breakthrough curves at different locations. Thus C and C1 should be constants for a ‘stationary’ porous medium

  7. Fits

  8. Fits

  9. Conclusions • CTRW models fit breakthrough curves better

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