Lesson 7 – Exercise: Modeling Release from a Source Representing a Concrete Vault

Now that we have discussed the fundamentals of modeling transport of exposed mass out of a Source, in this Lesson we will work through a simple Exercise in order to gain some additional insight as to how you can model mass transport within and out of a Source.

In this Exercise, the Source represents a single concrete vault (i.e., essentially a square box).  The vault is filled with sand, and three contaminants (X, Y and Z) are disposed within the sand (but are not bound in any kind of matrix material).  All three species decay (with the same half-life of 5 years).

Rather than simply assuming that advection controls the transport of the mass out of the vault, however, we will also consider other processes that impact the rate of mass transport.  Specifically, we will consider the impacts of both solubility constraints and partitioning inside the vault. In particular, we will assume that Y has a specified solubility limit (X and Z are infinitely soluble).  In addition, we will assume Z partitions onto the sand (X and Y do not).

Water flows horizontally through the system (in a direction perpendicular to one of the walls of the vault). Prior to the failure of the concrete walls, however, the flow through the sand inside the vault (and hence the contaminants) is insignificant. The vault, therefore, can be thought of as the “package”. No mass is transferred out of the vault until it fails. The vault fails randomly according to a uniform distribution (starting at time 0).

The inputs for this model are as follows:

We will model the inside of the vault using a single Cell. We could discretize the vault into multiple sections, but due to the small size of the vault and the relatively high seepage velocity, assuming the vault is rapidly well-mixed is not an unreasonable assumption for this simple model.

To create this model, you should start with a new model and follow these steps:

1. Edit the Species element and add X, Y and Z.  Make sure each has a half-life of 5 years.
2. Create Data elements for the inputs in the table above.  Note that the Solubilities, Partition Coefficients and Initial Mass should be vectors of species.
3. Create Expression elements that represent the Flow Rate through the vault (after failure) and the total volume of the vault.
4. Edit the Water element to specify the Solubilities.
5. Create a Solid (named Sand) and specify the properties (density, porosity and partition coefficients).
6. Insert a Source element and specify the following:
   1. Specify a single barrier and define a uniform distribution with the defined failure duration.
   2. Specify the inventory.
7. Create a Cell outside of the Source.  You don’t need to specify any properties for this Cell as it will simply act as a sink so that we can compute the release rate from the Source (give it a name that starts with “Sink”).
8. Inside the Source Container, edit the single Associated Cell as follows:
   1. Specify that it contains Water and Sand and define the appropriate amounts. Because the Sand is saturated, the volume of water is the Cell volume multiplied by the porosity.
   2. Create an Outflow from the Associated Cell to the “sink” Cell outside of the Source using the flow rate you define in step 3.
9. Create a Time History Result element to plot the release rate of each species from the Associated Cell (to the sink Cell).
10. Set the Time Settings for a Duration of 5 yr and a timestep of 0.01 yr. Set the Monte Carlo Settings to run 10 realizations.

Stop now and try to build and run the model.

Once you are done with your model, save it to the “MyModels” subfolder of the “Contaminant Transport Course” folder on your desktop (call it ExerciseCT18.gsm). If, and only if, you get stuck, open and look at the worked out Exercise (ExerciseCT18_Source_Vault.gsm in the “Exercises” subfolder) to help you finish the model.

Let’s walk through the model now.

The model should looks something like this:

In this model, the various inputs have been organized into a Container.  The inside of that Container looks like this:

Note that Source_Mass, Solubilities and Partition_Coefficients are all vectors of species.

The Vault_Flow_Rate should be defined as:

The Vault_Volume is simply the product of the height and the two sides:

The Source should look like this:

Note that there is a single Inventory Cell (renamed as Interior_of_Vault in this case).

The Outer Barrier should be defined as follows:

The Inventory should be defined as follows (note that it is not assumed to be bound in a matrix):

Inside of the Source Container there is a single Associated Cell that should look like this:

It should have an Outflow to the sink Cell outside of the Source:

Make sure you save the Time Histories for this Outflow (as this is the result we are interested in).

That is, the result we are interested in is the release rate from the Source, which is simply the mass transfer rate from the Associated Cell to the Cell outside of the Source:

If you run the model and plot the result it will look similar to this:

Before we discuss this result, the following points should be noted:

• Your result will look different (since the failure time is random).  Find a realization such that the failure occurs early enough such that the release rate goes to zero before 5 years.
• Your chart labels and lines will look a bit different.  Let’s briefly review the steps we carried out here to make the chart look a bit nicer:
  • The default style for the third array item has been modified.  You can change the line style by pressing the Edit Properties button in the chart (the farthest button to the right) to view the Time History Result Properties dialog. After doing so, to the right of the Result, under the Style column, press the Edit… button and select “Edit Row Label Set ‘Species’.  The Array Label set dialog for Species will appear.  In the chart shown above, we have simply changed the line style for the third item (to a dashed rather than dotted line).
  • In the Time History Result Properties dialog we deleted the Label. By doing so, only the array item then appears in the legend (rather than repeating the result name for each output).
  • The Y-Axis label was edited. When viewing the chart itself, you can press the Chart Style button (second button from the right), to edit the Y-Axis label (it is one of the tabs). By default, it refers to a keyword (%rlabel%) that causes the entire name to be displayed:
    
     In this case, it has been simplified to simply say “Rate”:
    

Let’s now focus on the release rate result. Find a realization (like the one above) where the package fails well before 5 years so that all of the mass is released before the end of the simulation. As can be seen, the three species behave quite differently:

• X neither has a solubility limit nor is it sorbed onto the sand in the vault.  As a result, once the package fails, it is gradually flushed out of the vault.
• Z does not have a solubility limit, but it is sorbed onto the sand in the vault. As we have discussed in previous Units, this results in a lower dissolved concentration, and hence a slower flushing rate from the vault.
• Y has a solubility limit (but is not sorbed).  It displays a very different behavior.  In particular, after package failure, the concentration is held at the solubility limit, thereby limiting the release rate. Eventually enough mass is flushed out such that the concentration drops below the solubility limit.

It should also be noted that because the species decay, the time of package failure affects the release rate (the earlier the failure, the higher the peak for X and Z).

This simple Exercise again illustrates how difficult it would be for you to represent a release rate using an equation (particularly due to the processes of decay and ingrowth). To represent the various processes controlling release realistically, you need to actually model them using the Source element.