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Modeling Anomalous Dispersion with Continuous Time Random Walk Theory

This study explores the application of Continuous Time Random Walk (CTRW) models for tracer test measurements in porous media, focusing on anomalous dispersion characterized by the parameter 'b'. The research investigates the relationship between b values and dispersive processes, providing insights into solute transport problems. The fitting routines and procedures for CTRW models are discussed, emphasizing the estimation of breakthrough curves. Overall, the analysis suggests that CTRW models fit breakthrough curves more accurately in heterogeneous porous media.

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Modeling Anomalous Dispersion with Continuous Time Random Walk Theory

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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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