Diffusion Through Partially Saturated Porous Media

When diffusion is taking place through a fluid (typically water or air) that only partially saturates the porous medium, this can reduce the diffusive flux. You can account for this by defining a Diffusivity Reduction Formula for each Fluid in your model.

Within GoldSim, the diffusive mass transport is computed as follows:

Diffusive Mass Rate = (Diffusive Conductance) * (Concentration Difference)

In this equation, the Diffusive Mass Rate has dimensions An output attribute for an element that defines the dimensionality (in terms of Length, Time and other fundamental dimensions) of the output. of mass/time, the Diffusive Conductance has dimensions of volume/time, and the Concentration Difference has dimensions of mass/volume.

Assuming that the species The chemical (or non-chemical, such as bacterial or viral) constituents that are stored and transported through an environmental system in a contaminant transport model. In GoldSim, the Species element defines all of the contaminant species being simulated (and their properties). are diffusing through a single fluid, the Diffusive Conductance term is computed as follows:

D = (A d t n r) / L

where:

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 the Diffusivity Reduction (the default value is 1); and
L is the diffusive length (L).

d and r are Fluid properties, and t and n are Solid properties.

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.

The Diffusivity Reduction Formula is a Fluid property (i.e., it is defined in the Reference Fluid A special type of Fluid element that provides a basis for defining partition coefficients between media for the various species in the model (i.e., the ratio of the species’ concentration in the medium to its concentration in the Reference Fluid at equilibrium). or Fluid property dialog). The Diffusivity Reduction Formula is a dimensionless value that has the impact of decreasing the effective diffusivity based on the fluid's saturation level within any Cell pathways in which it is present.

When used, this formula is intended to directly reference the fluid's saturation level using a special local available property called "Saturation". It must be referenced as "~Saturation" in the equation entered into the field:

Note that the ~Saturation local available property can only be referenced directly in this field; it cannot be used in a separate element that is then linked into this field.

When the fluid is used within a Cell pathway A transport pathway element that is mathematically equivalent to a finite difference node. Cells are commonly applied to simulate discrete compartments in an environmental system (such as ponds, lakes, shallow soil compartments, or the atmosphere)., GoldSim internally computes the saturation of the fluid in the pathway, and uses this value to compute the Diffusivity Reduction Formula (and hence the diffusive mass flux) into and out of the Cell. For each Cell, the saturation level of each fluid is computed by GoldSim as follows:

1. The combined pore volume of all (non-suspended) Solid media Materials (such as water, sand, clay, air) that constitute (are contained within) transport pathways. GoldSim provides two types of elements for defining media: Fluids and Solids. is computed as:
   
   Combined Pore Volume = Σ (Mass of Solid*Porosity/Bulk Density)
   
   where the summation is over all solids.

2. For the first Fluid in the Cell, the Saturation is computed as:
   
   Saturation = min(Fluid Volume, Combined Pore Volume) / Combined Pore Volume

3. If there are additional fluids in the Cell, the Saturation for each Fluid is computed, in turn, as:
   
   Saturation = min(Fluid Volume, Remaining Unfilled Pore Volume) / Combined Pore Volume

This method of computing saturation has two important implications:

• The first Fluid listed in the Cell is treated as the wetting fluid (it is the first fluid to fill the pores).
• If the Cell contains multiple Solids, the fluids have access to the pore space in all the Solids simultaneously (i.e., the pore space of Solids are not filled in a specific order).

Conceptually, the Diffusivity Reduction Formula should equal 1 when the saturation is equal to 1 (and 0 when the saturation is equal to 0). As a result, one convenient form for the Diffusivity Reduction Formula is:

Diffusivity Reduction Formula = ~Saturation^E

Note: The impact of saturation on the diffusion rate is complex. While the Diffusivity Reduction Formula (when defined appropriately as a function of saturation) reduces the effective diffusivity, a lower saturation changes the volume of water available. This in turn affects the concentration (and hence the concentration gradient). Moreover, decreasing the volume of water available increases the retardation due to sorption (since all of the mass of the porous medium is still assumed to be available, but only a fraction of the water is).

Note: Using the Diffusivity Reduction Formula to model diffusion through partially saturated porous media can be appropriate (if properly defined) in the case of a single porous medium and a single fluid through which the species are diffusing (typically water). However, for more complex situations involving partially saturated porous media (e.g., simultaneously considering both water and air phase diffusion and/or multiple solids), such an approach likely would be inappropriate. Fortunately, there are still ways to represent such systems in GoldSim, and this is discussed in detail here.