Understanding Diffusive Mass Flux Links

Diffusive mass flux links are used to transport mass through a stagnant or slowly moving fluid via the process of molecular diffusion. Diffusive mass transport is driven by (in fact, proportional to) a concentration difference, with mass diffusing from high concentration to low concentration. The constant of proportionality is referred to as the diffusive conductance:

In this equation, the Diffusive Mass Rate has dimensions of mass/time, the Diffusive Conductance has dimensions of volume/time, and the Concentration Difference has dimensions of mass/volume.

The diffusive conductance is a function of the properties of the species and fluids involved (multiple fluids can be involved), and the geometry of the diffusive process.

Assuming that the species are diffusing through a single fluid, the Diffusive Conductance term is computed as follows:

D is the Diffusive Conductance (L³/T);
A is the diffusive area (L²);
d is the diffusivity in the fluid (L²/T);
n is the porosity of the porous medium;
t is the tortuosity of the porous medium;
r is a reduction factor to account for the degree of saturation (1 if saturated);
L is the diffusive length (L).

Note: If the diffusion is not taking place through a porous medium, t, n and r in this equation are all effectively set to 1.

Diffusive mass flux links are appropriate for describing mass transport between pathways that are not dominated by advective processes. For example, if you were simulating the behavior of a landfill with a clay liner, given the low permeability of clay, the rate at which contaminants diffuse through the liner could be more significant than the rate of advective transport. In some cases, it may be appropriate to simultaneously specify advective and diffusive mass flux links (diffusion in slowly moving water).

Diffusive mass flux links may also be the most appropriate way to simulate mass transfer across fluid interfaces (e.g., the air-water interface in a lake or stream). Transport across such an interface is often mathematically represented as diffusion through two thin boundary layers (one on each side of the interface).

As illustrated schematically below, a diffusive mass flux link between two pathways has two "sides" (one for each of the linked pathways). Mass diffuses from one pathway, through the "diffusive layer" on one side of the link, through the "diffusive layer" on the other side of the link, and into the other pathway.

To create a diffusive mass flux link between two pathways in GoldSim, you must do the following:

• For each side of the link, a fluid must be specified. Each side of the link can contain a different fluid. This determines the diffusion coefficient that is used. The diffusion coeffient can actually be species-specific.

• For each side of the link, a diffusive length must be specified. This length represents the thickness of the "diffusive layer". It typically represents the distance from the center of the Cell to the “edge” or interface of the Cell.

• For each side of the link, you can optionally specify a porous medium contained within the "diffusive layer". If you do, the diffusive flux is multiplied by the product of the porosity and the tortuosity associated with the porous medium. The porosity can actually be species-specific, while the tortuosity is assumed constant for all species.

These inputs, along with the properties of the fluids and porous media involved, determine the value of the diffusive conductance. (For details on the manner in which diffusive mass flux links are represented mathematically, see Appendix B of the Contaminant Transport Module User’s Guide).

Typically the fluids on the two sides of the link will be identical (e.g., water), but they do not have to be (e.g., you could simulate diffusive transport across the air-water interface). The diffusive lengths on the two sides of the link do not have to be the same. In fact, in some cases, only one "diffusive layer" may be present (the diffusive length may be assumed to be 0 on one side of the link).

Mass transport in a diffusive mass flux link between two Cells can be bi-directional. That is, species can be transported in either direction along the link. The species always diffuse from high concentration to low concentration. In some cases, therefore, at a given time, different species may be diffusing in opposite directions within the same diffusive mass flux link.

The diffusive lengths and the diffusive area control the geometry of the link. Through appropriate specification of these inputs, nearly any diffusion geometry can be represented (e.g., linear, radial, spherical).

There is one important limitation on how diffusive mass flux links can be created: All diffusive mass flux links must originate in a Cell pathway. This is because, by definition, GoldSim does not allow mass to diffuse out of non-Cell (Aquifer, Pipe, External and Network) pathways. Within GoldSim, the concentration and diffusive length on the non-Cell side of a diffusive mass flux link are assumed to be zero. Hence, diffusive mass flux links involving non-Cell pathways are uni-directional. A diffusive mass flux link can only be bi-directional if it is between two Cell pathways.

Note: Although you cannot explicitly create a diffusive flux from an Aquifer to another pathway, GoldSim actually creates an (invisible) diffusive flux from the Aquifer to the downstream pathway whenever an advective flux is created. When computing the diffusion out of the Aquifer, the concentration in the downstream pathway is treated as zero (which could be a slightly conservative assumption). Note that in most cases, the impact of this diffusive flux will never be noticeable (as it will be dominated by advection).