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 Water 39, , 2001.
Berkowitz, B. and H. Scher, On characterization of anomalous dispersion in porous and fractured media, Wat. Resour. Res. 31, , 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
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 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

9. Conclusions
CTRW models fit breakthrough curves better