Lesson 8 – Modeling a Source Consisting of Multiple Packages

As we have discussed in previous Lessons, a Source consists of one or more packages, and for each individual package, barriers can be explicitly considered to exist. Mass cannot be released until the barriers, if they exist, fail. However, so far we have only considered Sources that consist of a single package (e.g., a single drum or a vault).

In many systems, however, the Source will actually consist of a large number of packages.  For example, your Source might consist of a large number (e.g., hundreds) of buried drums.  When simulating an engineered disposal facility such as one for radioactive waste, the waste will typically be placed inside a large number (perhaps thousands) of highly engineered waste packages. In this Lesson we will discuss how GoldSim represents such systems.

As we discussed in Lesson 4, barrier failure is defined in terms of failure distributions. In that Lesson we provided an example of a uniform failure distribution (over 10 years):

We pointed out that this is a probability density function in which the “height” of the curve for any given value (in this case, failure time) is not a direct measurement of the probability of failure at that corresponding time. Rather, it represents the probability density at that time. The total area under the curve integrates to 1. Therefore, integrating under the curve between any two points results in the probability of failure occurring between those two points in time. When we have just a single package, GoldSim randomly samples the distribution to select when the package would fail. Hence, the distribution represented our uncertainty in when that single package would fail. 

However, for a large number of packages, conceptually what GoldSim does is sample a failure time independently for every package (and therefore each package fails at a different time). Looked at in this way, we can think of the distribution as representing the temporal variability in the failure times of the collection of packages (i.e., some containers will fail early, and some will fail later).  That is, it is a distribution of failure times.

Here is the same failure distribution with the axis labeled slightly differently to highlight this:

Here we see that we have labeled the left Y-axis as a failure rate (the fraction of packages failing per year).  The right Y-axis is simply the integral of this, and as such represents the fraction of the total number of packages that have failed by a given time (e.g., at 5 years, 50% of the packages have failed; if we had 100 packages, this would imply that 50 have failed by 5 years).

Recall that one of the outputs of a Source is the number of failed packages. When we had just a single package (and assumed a uniform failure distribution over 10 years), we noted in Lesson 4 that this output jumped from 0 to 1 some time between 0 and 10 years (with each realization being a different time):

What would this same plot look like if we had 100 packages instead of 1?

To see this, let’s reopen the Example we were looking at in that Lesson (ExampleCT30_Simple_Container_Failure.gsm). Open the Source, change the Number of Packages to 100 and run the model. Then look at the Failed Packages result:

What we see here is that the number of failed packages looks very similar to the line we saw in the previous plot of the failure distribution showing the fraction of failed packages (in this case, we are plotting the actual number of failed packages as opposed to the fraction of failed packages). The difference is that it is not a straight line (although it is nearly one). If you were to run multiple realizations, you would find that each realization is slightly different (but all would be close to a straight line). This is simply due to the random nature of the sampling of the failure times. The greater the number of packages, the closer this plot would be to a straight line.

To better understand how to represent multiple packages in a Source, let’s close this Example and open a new one. Open ExampleCT33_Multiple_Packages.gsm from the “Examples” subfolder of the “Contaminant Transport Course” folder you should have downloaded and unzipped to your Desktop.

In this model, we are simulating 10,000 buried drums.  Each drum contains species X and Y (neither of which decay). When the wall of the drum fails, the mass inside the drum is exposed. We will look at just this exposure process: once the mass is exposed, it remains inside the drums; there is no mass transport out of the drums.

Open the Drums Source element:

You can see that the Number of Packages is defined by a Data element (that is equal to 10,000).

There is a single barrier defined.  Press Outer Barrier… to see how it is defined:

Rather than a Uniform distribution, we have specified a Weibull distribution.  A Weibull failure distribution is often used to characterize failure times in reliability models.  It has two parameters: a Slope and a Mean Lifetime. 

In this case, we have defined a Slope of 2 and a Mean Lifetime of 3 years.  This results in a failure distribution that looks like this:

As can be seen, this is a very different shape than the uniform distribution.  In particular, the failure rate is not constant over an interval.  Instead, it varies over time (starting low, reaching a peak, and then declining).

Close the Outer Barrier dialog and let’s look at the Source Inventory Settings.  What you will see is that there are two inventories. The first inventory contains only the first species (X) and it is not bound in a matrix:

The second inventory contains only the second species (Y), and this is bound in a matrix (with a matrix Lifetime of 3 years):

It is important to reiterate that when defining an inventory, the mass that you specify is the amount of mass in a single package. Hence, in this example, we have specified that there is a mass of 100g of each species in each package.  Since there are 10,000 packages, the total mass (in all the drums) is 1E06 g.

Close the Source dialog and run the model.

Let’s first look at the Failed Packages result:

As can be seen, the shape of this curve perfectly matches the shape of the fraction of failed packages displayed in the chart shown above describing the Weibull failure distribution.  (If we had a smaller number of packages, it would not match it so closely due to the random nature of the failures).

As we did in previous Examples, we have also computed the Exposure Rates for the two species.  Look at this result now:

First look at X.  Recall that X was not bound in a matrix. Therefore, the exposure rate should simply equal to the failure rate of the packages.  In fact, that is exactly what we see.  The shape of the curve is identical to the shape of the failure rate displayed in the chart shown above describing the Weibull failure distribution.  Due to the randomness of the failures and the manner in which we are computing the exposure rate (by taking a difference between two points in time) the curve is not smooth. The curve for Y is a bit different.  Recall that Y was bound in a matrix.  Therefore, the exposure rate is controlled by both the failure rate of the packages and the degradation rate of the matrix.  This delays (and disperses) the exposure rate.

The Cumulative Mass Exposed result is simply the integral of this rate:

This makes it clear that the exposure of Y is delayed and dispersed.

In this model, once the mass is exposed, it remains inside the drums; there is no mass transport out of the drums. We did not discuss the Associated Cell at all. This was intentional because in order to consider mass transport out of the drums, we need to revisit what the Associated Cell(s) actually represent. In Lesson 6, we noted that the Associated Cell(s) for a Source are intended to physically represent a single package within the Source. This worked fine when we had only a single package, but if we have multiple packages, what does this mean? That is, how can a Cell represent a single package when we have 10,000 packages?  What do the Associated Cells actually represent and how do we define their properties?  We will discuss this (and build an Exercise to illustrate this) in the next Lesson.