Lesson 10 - Representing Complex Barrier Failure Distributions
In previous Lessons we have discussed several simple barrier failure distributions. In this Lesson we will discuss the tools GoldSim provides to represent more complex failure distributions.
First, we should note that there is a variety of failure distributions you can specify. You can see this in the Outer Barrier Failure dialog:
We’ve seen the Immediate, Uniform and Weibull distributions in previous Lessons. In the Exponential distribution, the failure rate decreases exponentially with time. This requires an Expected Lifetime. If we assume an Expected Lifetime of 3 years, the failure distribution looks like this:
The User-Defined and Table options are advanced features that we will not discuss further here.
GoldSim provides several ways in which you can modify and combine the various distributions to represent complex failure behavior. The easiest way to demonstrate this is by exploring an Example model. Let’s begin by opening ExampleCT34_Complex_Container_Failure.gsm from the “Examples” subfolder of the “Contaminant Transport Course” folder you should have downloaded and unzipped to your Desktop.
This model has three different Sources. All three Sources have 10,000 packages with a single barrier defined. We will be examining only the Failed_Packages output, so we don’t really need to be concerned with any other inputs other than those associated with the Outer Barrier failure.
Let’s first look at the Source named “Weibull”. The Outer Barrier Failure is defined as follows:
This is simply a Weibull failure distribution with a Slope of 2 and a Mean lifetime of 3 years. If you run the model and look at the Weibull Failure Result element, you see the standard shape associated with a Weibull failure distribution:
One thing you may have noticed in all of the distributions we have looked at so far is that failures begin immediately (at the beginning of the simulation). If you think about it, however, there is no reason to believe that this must be the case. That is, packages may remain intact for a long time before they begin to fail. As a result, we need a way to delay the beginning of the failures. To see how GoldSim provides this capability, open the Source named “Weibull_Shifted”. The Outer Barrier Failure is defined as follows:
Look at the Effective Time input. You will notice that it is defined as “Etime – 2 yr”. In all previous Examples and Exercises, this has been defined as “Etime” (which is the default). So what is this?
Recall that failure distributions are frequency distributions in time. That is, they define the failure rate as a function of time. You define the shape of the distribution using one or more parameters (e.g., the uniform distribution requires a duration). It is important to note, however, that the parameters defining the container failure distributions (e.g., the Weibull slope or lifetime) cannot vary with time (i.e., a parameter that defines a distribution in time cannot vary with time). As a result, it is not possible for the failure rates to respond directly to time-varying conditions. That is, a failure distribution must be defined assuming a given set of conditions. Any temporal changes in conditions (e.g., temperature, water saturation) that may affect the failure distribution must be known prior to its specification and indirectly incorporated into the form of the distribution.
The Effective Time, however, can be used to delay and/or modify failure distributions such that they can dynamically respond to time-varying conditions. For example, the failure rate could be specified to increase with increases in temperature or aggressive chemical conditions. The Effective Time represents the failure mode's view of how far along the time axis it is relative to the start-time of the defined failure distribution.
Hence, if the Effective Time changes faster than the elapsed time, this has the effect of accelerating failures (relative to that defined by the distribution). So, for example, if the Effective Time was defined as “2*Etime”, the failure rate (as defined by the distribution) would actually be increased at every point in time by a factor of two. Similarly, if the Effective Time changes at a slower rate than the elapsed time (e.g., defined as 0.5*Etime), this has the effect of decelerating failures. Finally, if the Effective Time is negative during some portion of the simulation, it is treated as zero, and no failures at all occur during that time period.
In this Example, by defining the Effective Time as “Etime-2 yrs”, we have simply shifted (delayed) the start of the failure distribution by 2 years. That is, for the first two years of the simulation, the Effective Time is zero, and it does not start to increase until after 2 years.
We can see this by looking at the Shifted Weibull Failure Result element:
As can be seen, the failure distribution is simply shifted to the right (delayed) by 2 years.
In some cases, different mechanisms may act to fail the packages (and these mechanisms may not effect all of the packages). To model this, for a particular barrier, multiple failure modes can be defined, with each mode having a different failure distribution and a different Probability. For example, one failure mode may be assigned to represent one type of failure that perhaps only affects a small fraction of the packages (e.g., bad welds), while another may be assigned to represent another type of failure (e.g., pitting corrosion) that affects all of the packages. GoldSim combines the failure modes in an appropriate manner to obtain the total failure distribution.
To see how GoldSim provides this capability, open the Source named “Two_Modes”. If you look at how the Outer Barrier Failure is defined you will see that there are two failure modes. The first looks like this:
This failure mode represents failures due to faulty construction (that only affects a fraction of the packages). It is defined using an Exponential distribution. The key thing to focus on here, however, is the Probability. What does this represent?
This is the probability that the mode is active for a given package. For example, while all packages may be vulnerable to corrosion, only a small percentage may be vulnerable to a failure mode associated with faulty construction (e.g., bad welds). Put another way, if a barrier had only a single failure mode which had a probability of 0.5, by the end of the simulation, only 50% of the packages would have failed. The default Probability for a failure mode is 1 (all packages are susceptible). For this mode, we have defined it as 0.25 (25% of the packages are vulnerable to this failure mode).
The second mode looks like this:
This is exactly the same delayed Weibull distribution we saw in the previous Source.
We can see the resulting combined failure distribution by looking at the Two Modes Failure Result element:
As can be seen, this produces a rather complex failure distribution, with about 25% of the packages failing early, followed the remainder failing due to the second failure mode.
All three Sources are plotted together in the Comparison Result element:
One very important point to reiterate regarding the Outer Barrier is that package failure is discretized. That is, GoldSim fails packages discretely and does not fail fractional packages. We did not notice that here because we had so many packages. But if we had a smaller number of packages, it would be very noticeable. To see this, return to Edit Mode and edit the Two_Modes Source so it only has 20 packages. Then rerun the model and look at the Two Modes Failure Result element again:
As can be seen, it has the same general shape as before, but failure is clearly discretized. Moreover, if we were to run multiple realizations, we would see that each one would be slightly different.
GoldSim actually provides several options for how failure of the packages is randomly simulated in this case. To see these options, close the result, return to Edit Mode, select Model|Options from the main menu, and select the Contaminant Transport tab:
In the middle of the dialog, you will see two options in the Source Term section. The most important of these is the Barrier failure type. This has two options in the drop-list: “Random failure time” (the default), and “Predicted failure time”. If "Random failure time" is selected all the models we have run have used this default), the failure time for each package is randomly sampled from the distribution. As a result, multiple realizations of the same failure distribution will be slightly different. However, if "Predicted failure time" is selected, package failures are discretized by computing the cumulative fraction failed based on the distribution, multiplying this fraction by the total number of packages, and truncating or rounding (determined by the second option directly below) to the nearest integer number of packages. As a result, multiple realizations of the same failure distribution will be identical.
Select this option now ("Predicted failure time") and close the dialog. Then rerun the model and look at the Two Modes Failure Result element again:
You will see that the result is no longer random (although it is still discretized). Moreover, if you ran multiple realizations, they would all be the same.