Understanding Matrix Lifetimes and Fractional Degradation Rates

The first two options for defining matrix degradation rates in the Source Inventories dialog are:

• Specify a fractional degradation rate (with units of inverse time). This represents the fraction of the existing matrix mass which degrades per unit time.

• Specify a matrix lifetime (with units of time). This represents the time period over which the matrix is uniformly degraded.

It is important to understand that a fractional degradation rate and a matrix lifetime behave quite differently in terms of the amount of contaminant mass remaining (i.e., the amount of unexposed mass). If you use a (constant) fractional degradation rate, the amount of mass remaining to be exposed decreases exponentially, while if you use a lifetime, the mass decreases linearly:

In this example, one Source is represented using a Matrix Lifetime of 10 years, while the other Source is represented using a (fractional) Matrix Degradation Rate of 1/10 yr (i.e., 10% per year). As can be seen, because the fractional degradation is exponential, it actually takes longer than 10 years to expose all of the mass.

In reality, matrix degradation rates 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 as an absolute degradation rate in the following discussion). A fractional degradation rate could then be computed as the product of the absolute degradation rate and the specific area of the matrix (defined here as the area per unit mass).

Note that 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. This is valid if degradation rates are very small, but for higher rates, 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 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.

Note: If you do choose to model time dependency of the degradation rate, note that degradation is not likely to be a function of time; rather it is likely to be a function of the time since the package failed. Although GoldSim does not start degradation until a package fails, there is no way to reference the individual package failure times when defining the degradation rate. Hence, you should only specify the degradation rate as a function of time if it actually is a function of time (rather than time since failure). This would be appropriate, for example, if all packages failed at the same time and you explicitly accounted for the failure time in the equation for the degradation rate.