Continuous Time Random Walk Model

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, 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 gt 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 lt b lt 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 lt b lt
  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 CTRW Parameters CTRW Parameters CTRW Parameters 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