Lesson 5 – Modeling Degradation of the Waste Matrix

In the previous Lesson we looked at a very simple Example in which a single drum (with a single barrier) contained three species.  When the wall of the drum failed, the mass inside the drum was exposed. In this Lesson, we will make one change to this model: we will assume that the mass is bound in a matrix material (e.g., grout) that must degrade in order for the mass to be exposed.

Let’s begin by opening ExampleCT31_Simple_Failure_and_Degradation.gsm from the “Examples” subfolder of the “Contaminant Transport Course” folder you should have downloaded and unzipped to your Desktop.

The only difference between this model and the Example that we looked at in the previous Lesson can be seen in the Source Inventory Settings (accessed by pressing the Edit… button):

What we see here is that we have specified a Waste Matrix, along with a property (Lifetime) that quantifies how the matrix degrades.

The Waste Matrix input field is a drop-list with four possible options. The default (None) is what we used for the previous Example, and simply indicates that the mass is not bound in any matrix. If one of the other three options is selected, it indicates that the mass is bound within some kind of matrix material (e.g., grout, glass) and that material must degrade in order for the mass to be exposed.  Regardless of which of these other three options is selected, the following two points apply:

• Species mass is exposed congruently with the degradation of the matrix. This means that the mass is assumed to be uniformly distributed through the matrix, and the fraction of the mass that is released from the matrix is equal to the fraction of the matrix that has degraded (e.g., if 50% of the matrix has degraded, then 50% of the mass has been released ftom the matrix).
• Matrix degradation in a package does not start until the barriers (Outer and, if present, Inner) for the package have failed.

The first two of the final three Waste Matrix options (after “None”) are as follows:

• Specified lifetime: The inventory is bound in a matrix, and the matrix degrades with a specified lifetime. The lifetime represents the time period over which the matrix is degraded at a uniform rate.  The value must be non-negative and have dimensions of time.
• Specified degradation rate: The inventory is bound in a matrix, and the matrix degrades with a specified rate. The rate must be a fractional degradation rate.  That is, it represents the fraction of the existing mass that degrades per unit time. The value must be non-negative and have dimensions of inverse time.

Note: There is a third Waste Matrix option referred to as “Congruent Dissolution”. In this case, the inventory is bound in a matrix, and the matrix degrades based on solubility-limited dissolution (e.g., of a uranium dioxide matrix). This is an advanced option, and we will not discuss it further here.

It is important to understand that a “Specified lifetime” and a “Specified degradation rate” behave quite differently in terms of the rate at which the matrix degrades and mass is exposed.  If you use a (constant) degradation rate, the amount of mass remaining to be exposed decreases exponentially, while if you use a lifetime, the mass decreases linearly:

In this plot, which compares the fraction of unexposed mass remaining (assuming the barrier fails immediately), one Source is represented using a Specified lifetime of 10 years, while the other Source is represented using a Specified degradation rate of 1/10 yr (i.e., 10% per year). Because the fractional degradation is exponential, it actually takes longer than 10 years to expose all of the mass.

Note: In reality, matrix degradation rates can be quite complex and are likely to change with time.  They are typically a function of available surface area, and as such, are often quantified (e.g., in experiments) in units of mass/area/time (referred to here as an absolute degradation rate).  A fractional degradation rate could then be computed as the product of the absolute degradation rate and the specific area (area per unit mass) of the matrix.  If you assume that the absolute degradation rate is constant in time, the fractional degradation rate will only be constant in time if the specific area stays constant.  In many cases, however, this is likely to be a poor assumption.  This is because for nearly all possible geometric configurations, the specific surface area of a mass will typically increase with time as the material degrades (although it could potentially stay constant or even decrease if the shape of the mass changes significantly as it degrades). Countering this to some extent is the fact that it would not be unusual for the absolute degradation rate to decrease over time (e.g., due to the build up of corrosion products, etc.).

Typically, due to the considerable uncertainty in the actual geometry of the matrix, as well as uncertainty in long-term absolute degradation rates, it is often difficult to explicitly model the time variability of the degradation rate, and specification of a constant (but uncertain) fractional degradation rate or, alternatively, an uncertain lifetime is appropriate.

In the Example in this Lesson, we have described the matrix degradation using a specified lifetime of 3 years.

Run the model now so we can look at the results. Like the previous Example, there are three Time History Result elements for the three Source results, and for one of the outputs of Cell1. There is also an additional Time History Result that we will discuss shortly.

The Failed Packages result is identical (so we don’t need to look at that).

Now let’s look at Unexposed Mass:

Recall that this is the mass of each species that has not yet been exposed (due to failed barriers and/or degraded matrix).  At the beginning of the simulation this is therefore equal to the total Inventory specified for the Source (in this case, 100 g for all species). Let’s first look at Z (which does not decay or ingrow). Prior to package failure, it is constant (as all of the mass remained unexposed).  Once the package fails, the unexposed mass decreases linearly over 3 years as the matrix degrades.

The behavior of X and Y are a bit more complex.  X decays into Y so the values change with time even before the package fails (at 2.7 years). After the package fails, the unexposed mass does not decrease linearly (as was the case for Z) due to the effects of decay and ingrowth.

Next, let’s look at the Cumulative Mass Exposed:

Recall that this is the cumulative mass of each species that has been exposed (due to failed barriers and/or degraded matrix) and is therefore available in the single Inventory Cell. Like in the previous Example, in this model it is zero prior to the failure of the package.  Unlike the previous Example, however, it does not instantaneously jump when the package fails.  Z is the easiest curve to understand, as after 3 years exactly 100 g of the mass has been exposed.  The situation is different for X and Y, however, due to decay and ingrowth prior to and while the matrix is degrading.

Another way to look at exposure (that we did not do in the previous Example) is to plot the exposure rate. The Source does not directly output the exposure rate (it only outputs the cumulative amount exposed).  But by using a Previous Value element to compute the cumulative exposure from the previous timestep, we can approximate the exposure rate by taking the difference between the current cumulative exposure and the previous cumulative exposure, and dividing by the timestep length (which is a Run Property provided by GoldSim).  If you look at the model, you will see that this is what we have done.

We then created a Time History Result element (Exposure Rates) to plot this:

First look at Z.  Z does not decay or ingrow, so the exposure rate is determined only by the time of package failure and the matrix degradation rate.  The matrix has a lifetime of 3 years and degrades uniformly over that time.  As a result, Z (which is released congruently as the matrix degrades) is exposed at a constant rate. The situation is a bit different for X and Y, however, due to decay and ingrowth prior to and while the matrix is degrading.

Finally, let’s look at the amount of mass in the Inventory Cell (which represents the interior of the drum).  To see this output, look at the Result element named Exposed Mass in Drum (recall that this is simply plotting the mass in Cell1):

As in the previous Example, there is no exposed mass in the drum until the package fails.  At that time (2.7 days in this case) the mass slowly starts to rise over the next three years (as the matrix degrades).   After the matrix completely degrades (5.7 years), the mass of X and Y continues to change due to decay and ingrowth.

Before we leave this Example, let’s run a Monte Carlo simulation. Return to Edit Mode, edit the Simulation Settings, and change # Realizations to 100. Then rerun the model.

Look at Exposed Mass in Drum (make sure you are looking at “Realization” for the Display). Toggle through the various realizations (here is realization #100):

You will see that they are all quite different (due to the different failure times).  Note that unlike in the previous Example, the final value (at 10 years) is NOT always the same. In particular, for those realizations in which the package failure occurs in the last three years of the simulation, not all of the mass is exposed by the end of the simulation.

This Example was a bit more complex than the first (in particular, both drum failure and matrix degradation controlled exposure of the mass).  However, the mass still never moved out of the drum after exposure. Of course, the entire purpose of a Source element is to model a source term (an input rate) to another pathway in the model (outside of the Source).  For this to happen, we need to take this Example one step further and define how mass actually leaves the Source after it is exposed. This allows us to compute a release rate from the Source (that subsequently could serve as an input rate to another pathway in the model).  We will do that in the next Lesson.