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.
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 of mass/time, the Diffusive Conductance has dimensions of volume/time, and the Concentration Difference has dimensions of mass/volume.
Assuming that the species 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 or Fluid property dialog). The Diffusivity Reduction Formula is a dimensionless value that has the impact of decreasing the effective diffusivity based on the 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.
Note: Locally available
properties are discussed in detail in Chapter 10 of the GoldSim User’s
Guide.
When the fluid is used within a Cell pathway, 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 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 = ~SaturationE
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).
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