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. 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

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

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

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

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

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

Fits

Fits Length CTRW Parameters (cm) T b r C C1 11 44.1 1.68 0.11 4.01 1.92 17 68.5 1.72 0.092 4.03 2.09 23 94.2 1.73 0.084 4.10 2.23

Conclusions CTRW models fit breakthrough curves better