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Don’t just settle for ‘average failure rate’. Tackle any analysis task with WellMaster’s reliability analyses and multiple failure rate distributions.

With any equipment reliability calculation run in WellMaster the end-user is presented with the results in multiple failure rate distributions. This ensures fit-for-purpose failure rates applicable for any task at hand.

The Failure rate and Reliability distribution models in WellMaster include:

**Average failure rate.** This is calculated as the number
of failures for the components in a given data set divided by the total number
of Service time years for the components in the data set. This gives an Average
Failure Rate (AFR) per year, independent of time (constant failure rate). This
is normally used as a relative indication of reliability when comparing
components for benchmarking purposes mainly. It is also the basis for the Exponential
based Mean Time To Failure (MTTF) calculation.

**Exponential based Mean time to failure (MTTF). **The
Mean Time To Failure based on the Exponential distribution model is calcualted
as: MTTF = 1 / AFR [years], when assuming a constant failure rate independent of
time. This implies that;

- A used component is stochastically as good as new, so there is no reason to replace a functioning component
- For the estimation of the reliability function, the Mean Time To Failure etc, it is sufficient to collect data on the number of hours (or years) of observed time in operational service and the number of failures in the observation period. The number of components or the age of these is of no interest when applying the Exponential distribution model.
- The Exponential distribuition model is much used and best suited to perform relative comparisons between components and run benchmarking assessments. For lifetime predictions and survival probability analyses, it is recommended to apply the Weibull distribution model instead, since this model consider the fact that the failure rate in most cases for mechanical equipment is not constant over time

**MTTF Weibull 2 parameter, with the MLE and LSE calculation
methodology. **The Weibull distribution
reliability (survivor) function is given as follows:

where ƛ is called the Scale parameter and α is called the Shape parameter. Note that when α = 1,00 the Weibull distribution is equal to the Exponential distribution (constant failure rate).

The MLE (Maximum Likelihood Estimation) and the LSE (Least Squares Estimation) methods are used for the calculations for the Weibull 2P distribution model.

The reason for that the Weibull 2P – MLE is preferred versus the Weibull 2P – LSE, is that the Weibull 2P – MLE includes the calculation of the Upper and Lower confidence limits also (90 % Confidence interval), which the LSE method does not include (there is no established accepted statistical method developed here for the LSE, reference [8]). The LSE method also often provides a slightly more conservative estimate of MTTF. See the link in reference [10] for a specific description of the MLE and LSE methods used for the Weibull 2P calculations.

The MTTF is calculated as the area under the distribution curve in the graph showing the survival probability, so the type of distribution model you select to be the basis for this, Exponential, Weibull 2P -MLE or Weibull 2P – LSE, affects the MTTF you get from running the analyses/calculations. See example in the figure.

In this example, the Kaplan-Meier plot shows that the last captured failure is around after 23,5 years, but there are still some components in service, and the data goes up to approx 26,5 years of service time. WellMaster has a function that by default sets an Assumed wear-out (failure) of the last surviving component in the data set 3 to 4 years after the longest observed service time in the Kaplan-Meier plot. Here in this case this has been (by default) set to 30 years. The MTTF of 18,95 years has thus been calculated based on the area below the curve for the Weibull 2P – MLE (Curve fit) and up to the vertical line representing the assumed wear-out set at 30 years. When there are large datasets with just one or a few failures, it is not always possible to calculate the Weibull 2P MTTF without the ‘area’ limitation under the curve as set by the Assumed wear-out function, the Weibull MTTF moves towards infinity otherwise. This function also allows for the WellMaster users to actively reset/change the Assumed wear-out to be any other values than the value set by default by the system.

If we look at a plot showing how the failure rate varies over time for a given data set as shown in the figure and try to draw a smooth curve to represent the failure rate variation over time, this curve will look like the so-called **‘Bathtub’ curve**. This curve will be a better estimate for the failure rate function.

The failure rate is often high in the initial phase, often called the ‘Burn-in’ or ‘Infant mortality’ period. Several reasons for this may be given. Infant mortality or undiscovered defects, design errors, installation errors or other failures etc show up soon the components are put into their operational service – short after when the service time started (when the well was put into operation).

When the components have survived this period, the failure rate tends to stabilize at a lower level where it may remain more or less constant for a longer period. This is often called the ‘Useful life’ time period. See figure 4 below. Then the failure rate starts to increase again, as the components tend to begin to wear-out and subsequently fails at a higher rate, and this period is called the ‘Wear-out’ period. Note that in reality for the majority of mechanical components the failure rate function will usually show a very slight increasing trend also in the ‘Useful life’ period. That is why this period of the curve is also often called the ‘Chance failure period’.