Software

The CTRW MATLAB TOOLBOX contains software to model non-Fickian (as well as Fickian) transport. The software enables both "forward" modeling and "inverse" (best-fit) modeling of experimental data. A variety of inlet and outlet boundary conditions, and types of transport, are implemented.

Version 3.1 (May 2010) contains a User's Guide and accompanying files with easy-to-use format. Several examples clearly demonstrate how to work with the Toolbox in many typical analyses. This version includes improvements to the input files and explanations in the User's Guide, as well as some small improvements to the numerics.

The CTRW Toolbox can be run in Matlab (tested on versions 6.1 and higher) on any platform.

The Toolbox (called "CTRW_v3.1.zip") can be downloaded here (300 KB).

To install the Toolbox:
Unzip all the files contained in the CTRW_v3.1.zip. The directory named CTRW_v3.1 has now been created on your disk. Run MATLAB and change the working directory to CTRW_v3.1.

A detailed, step-by-step "Practical User's Guide" appears in the CTRW_v3.0 directory, as the file "CTRW3.1.pdf".

Berkowitz, B., A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Reviews of Geophysics, 44, RG2003, doi:10.1029/2005RG000178, 2006.


Published papers that discuss in detail the specific CTRW theory and applications implemented in this Toolbox include:

Cortis, A. and B. Berkowitz, Anomalous transport in ''classical'' soil and sand columns, Soil Science Society of America Journal, 68, 1539-1548, 2004.

Cortis, A., Y. Chen, H. Scher and B. Berkowitz, Quantitative characterization of pore-scale disorder effects on transport in "homogeneous" granular media, Physical Review E, 70, 041108, doi: 10.1103/PhysRevE.70.041108, 2004.

Cortis, A., C. Gallo, H. Scher and B. Berkowitz, Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities, Water Resources Research, 40, W04209, doi:10.1029/2003WR002750, 2004.

Dentz, M., A. Cortis, H. Scher and B. Berkowitz, Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport, Advances in Water Resources, 27(2), 155-173, 2004.

Below, older versions of CTRW software remain available. This material is more limited in its application, and we strongly recommend that new users work with the CTRW MATLAB TOOLBOX given above.

Platform

The CTRW software can be compiled on either of two platforms:

CTRW functions

• FPTD (T, β, r; t) - First Passage Time Distribution. T specifies time scale, 0 < β ≤ 2 and -1 < r < 1 (for β > 1, r > 0). t is the time variable.
  Brief explanation  ;   Relevant literature (pdf files): 1 (1.15MB) 2 (529KB) 3 (83KB) 4
   
• CFPTD (T, β, r; t) - Cumulative First Passage Time Distribution. T specifies time scale, 0 < β ≤ 2 and -1 < r < 1 (for β > 1, r > 0). t is the time variable. This distribution is equivalent to the FPTD for the step flux input at the origin.
  Brief explanation  ;   Relevant literature (pdf files): 1 (1.15MB) 2 (529KB) 3 (83KB) 4
   
• SCD (R, β, κ; L) - Spatial Concentration Distribution. R specifies length scale, 0 < β ≤ 2 and -1 < κ < 1 (for β > 1, κ > 0). L is the distance from the origin.
  Brief explanation  ;   Relevant literature (pdf file): 1 (152KB)
   
• CSCD (R, β, κ; L) - Cumulative Spatial Concentration Distribution. R specifies length scale, 0 < β ≤ 2 and -1 < κ < 1 (for β > 1, κ > 0). L is the distance from the origin. This distribution is equivalent to the SCD for the step residence concentration input at the origin.
  Brief explanation  ;   Relevant literature (pdf file): 1 (152KB)
   
• MEAN (β, C, C₁; t) - the spatial mean of the SCD as a function of time t. β, C and C₁ are constants of motion.
  Brief explanation  ;   Relevant literature (pdf file): 1 (753KB)
   
• ST_D (β, C, C₁; t) - the spatial mean of the SCD as a function of time t. β, C and C₁ are constants of motion.
  Brief explanation  ;   Relevant literature (pdf file): 1 (753KB)
   
• conv_beta (T, β, r; t) - convolution of the FPTD function with the injection curve.
  Brief explanation  
  

Other functions

• ade1d (x, D, u, t) - Calculates 1D solution for the advection-dispersion equation. x specifies transport length, D is dispersion, u is velocity, t is time.
    
   
• epm1d (x, P, u, t) - Calculates 1D solution for the advection-dispersion equation. x specifies transport length, P is Peclet number, u is velocity, t is time.
    
   

Note: The files include more functions than the four functions listed below.
A short explanation appears inside the package.

CTRW functions

• FPTD ([T β r], t) - First Passage Time Distribution. T specifies time scale, 0 < β ≤ 2 and -1 < r < 1 (for β > 1, r > 0). t is the time variable.
  Brief explanation  ;   Relevant literature (pdf files): 1 (1.15MB) 2 (529KB) 3 (83KB) 4
   
• CFPTD ([T β r], t) - Cumulative First Passage Time Distribution. T specifies time scale, 0 < β ≤ 2 and -1 < r < 1 (for β > 1, r > 0). t is the time variable. This distribution is equivalent to the FPTD for the step flux input at the origin.
  Brief explanation  ;   Relevant literature (pdf files): 1 (1.15MB) 2 (529KB) 3 (83KB) 4
   
• SCD ([R β κ], L) - Spatial Concentration Distribution. R specifies length scale, 0 < β ≤ 2 and -1 < κ < 1 (for β > 1, κ > 0). L is the distance from the origin.
  Brief explanation  ;   Relevant literature (pdf file): 1 (152KB)
   
• CSCD ([R β κ], L) - Cumulative Spatial Concentration Distribution. R specifies length scale, 0 < β ≤ 2 and -1 < κ < 1 (for β > 1, κ > 0). L is the distance from the origin. This distribution is equivalent to the SCD for the step residence concentration input at the origin.
  Brief explanation  ;   Relevant literature (pdf file): 1 (152KB)