Matrix Diffusion Toolkit

Practical tool for understanding complex process

The Matrix Diffusion Toolkit, developed for the Department of Defense’s Environmental Security Technology Certification Program (ESTCP), is a free software tool that can assist site personnel to effectively and efficiently estimate what effects matrix diffusion will have at their site, and transfer the results to stakeholders. Matrix diffusion describes the process where compounds that have diffused into low-permeability zones such as silt and clay layers in the subsurface serve as indirect, low-level sources of contamination to transmissive zones over time. This process has the potential to sustain dissolved contaminant concentrations in groundwater after the source is removed or after remediation removes or isolates contamination from transmissive compartments. 

Quantifying how matrix diffusion processes may influence decisions is becoming a critical part of long-term management of many contaminated sites.

The Toolkit is Microsoft® Excel-based and provides the following, easy-to-use tools to calculate and evaluate matrix diffusion effects:

1) Square Root Model:

A module to provide planning-level estimates of the mass discharge (in units of grams per day) caused by release from a low-k diffusion-dominated unit (typically silt or clay) into a high permeability advection-dominated unit (typically sand or gravel). Estimates of concentration and mass remaining in the high permeability unit, after the source is removed, are also provided.

Additionally, the Square Root Model also utilizes a Monte Carlo-type approach to analyze uncertainty in the actual concentration, porosity, apparent tortuosity factor exponent, and retardation factor measurements. With this tool, groundwater practitioners can estimate the accuracy of the hydrologic measurements that are being used for the matrix diffusion calculation.

2) Dandy-Sale Model:

A module allowing users to perform: i) contaminant transport via advection and transverse diffusion in the transmissive layer, and ii) transport via transverse diffusion in the low-k zone. The module provides planning-level estimates of:

Low-k Zone:

  1.  Aqueous, sorbed, and total concentration; and
  2.  Aqueous, sorbed, and total mass.

Transmissive Zone:

  1.  Aqueous, sorbed, and total concentration;
  2.  Aqueous, sorbed, and total mass; and
  3.  Mass discharge.

3) Matrix Diffusion Related Tools

An additional feature that provides a review of theory and methods related to matrix diffusion:

  • NAPL Dissolution Calculator: a module that estimates the transverse diffusion of contaminants into the groundwater passing over the top of a Non-aqueous Phase Liquid (NAPL) pool and the transverse diffusion of contaminants into the low-k unit underlying the pool;
  • Plume Magnitude Information: a summary of the Plume Magnitude Classification System (Newell et al., 2011) and its application to site investigation and remediation;
  • Low-k Zone Remediation Alternatives: a summary of current alternatives for the remediation of low-k zones; and
  • 14-Compartment Model: a discussion on the quantitative application of the 14 Compartment Model (Sale et al., 2008a).


The Matrix Diffusion Toolkit provides a valuable tool for developing site conceptual models, supporting site characterization efforts, planning remedial designs, and determining if matrix diffusion will affect remediation goals for groundwater sites. The software can assist site personnel in updating or creating a more accurate conceptual site model, which will enable them to determine if matrix diffusion processes are significant enough to cause “rebounding” of downgradient plume concentrations above remediation goals after plume remediation or isolation is complete. Having this information available before a remedy is implemented could assist site stakeholders in selecting more appropriate remedies and effectively and efficiently addressing the potential issues of matrix diffusion with regulators. Furthermore, addressing extended remediation time frames caused by matrix diffusion would lead to savings in project costs.

Technical Support

Limited technical support is available from Shahla K. Farhat (

System Requirements

The Matrix Diffusion Toolkit requires a computer capable of running Microsoft® Excel (2007/2010) for Windows 7.

Operation requires an IBM-compatible PC equipped with a Pentium or later processor running at a minimum of 450 MHz. A minimum of 256 MB of system memory (RAM) is strongly recommended. Computers not meeting these recommendations will experience slow running times and/or problems with memory


Changes in version 1.1: Some of the apparent tortuosity factor exponents have been updated and clarified.

Changes in version 1.2: The concentration units displayed for the low-k zone aqueous and transmissive zone concentration graphs in the Dandy-Sale module have been corrected.

Changes in version 1.21: An error in the Monte Carlo analysis has been corrected (all concentrations were erroneously being multiplied by 5).

Changes in version 1.22: An error in the DSM loading concentration units has been corrected (all concentrations were erroneously being treated as mg/L).

Unzip all the files to the same folder. The zipped file contains the Matrix Diffusion Toolkit spreadsheet, associated pdf files, the Help file, and the User’s Manual.