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  2. FOEP as an Integral Part of Petrophysical Calculations

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- [Stein] In practice, the petrophysicist is responsible for following four point. The petrophysicist should choose the best-suited petrophysical model for porosity calculation, water saturation calculation, and so on, and define the input values to this equations with errors. The petrophysicist should also define the distribution on input-related errors. Is it symmetric or non-symmetric, which in practice mean, define the low value and the high value on each of the input-related uncertainties Finally, the petrophysicist should also define the correlation between the variables. This information can very simple be a integral part of the petrophysical graphical user interface, and here we can see an example on how it can looks like for Archie's equation Here, we see the input to Archie's equation has to be delievered with a value and uncertainty If this uncertainty is non-symmetrically distributed, then by clicking on this unsymmetrical button, this graphical user interface is open for a low value, and a high value on the input-related errors. In that case, just click this one twice, with one calculation for the low side error, and one calculation for the high side error. To define the value and the correlation between the variables, that can be done by clicking on this correlation button, then a new graphical user interface is popping up for input of both that values. Then, by feeding the computer with this information, and then pushing the calculate button, the petrophysical results is calculated with errors. Here, we can see an example on how I prefer to present the petrophysical results. I always present the petrophysical results with two tracks for each as the model. The first track illustrate the petrophysical results, with the band of error. The second track illustrate the reality contribution, the individual input-related errors had undefined calculated error, so in this case, for porosity, we clearly see this uncertainty in the rho fluid, the light blue color here, which is the dominant source to uncertainty in the porosity. For the water saturation calculation, we can clearly see it is the uncertainty in the cementation factor, the red color, together with uncertainty in the saturation exponent, the light blue color, which together, dominate the uncertainty in the water saturation calculation. This picture will, of course, be different with other value on the input parameters, and with other value on the input-related errors. This example is calculated without any correlation between the variables, and with symmetrical distribution on all input-related errors. Here are some other examples. The goal with the left hand plot there, it's only to illustrate how the correlation is included and presented, and in this case, amongst theoretically correlation between the cementation factor and saturation exponent are included, and its impact is showing up by this orange color, telling us that the correlation can have some impact on the error calculations and should be included. The goal with the middle plot there is only to show how non-symmetrical distributions is presented. In that case, the track, which showing the reality contribution, is split into two parts. The last part illustrate the reality contribution on the low side error calculation. They're marked with the dark gray color, and the right part illustrate the reality contribution on the high side error calculation, illustrated by the light gray color here. The goal with the last example is only to illustrate how the error is calculated through a shale, so shale/sand is the models, and here we see the large uncertainty in the water saturation calculation is dominated by the uncertainty in v shale calculation. Here, the green color, which again, can be traced back to the v shale calculations, this two track, v shale from the gamma ray log, and here was can see the large uncertainty in the v shale is dominated by how we defined uncertainty in the shale point, for the gamma ray log, the green color. This way to include, present, understand the error, as a integral part of our petrophysical calculations gain a lot of of benefit. First of all, and most important, are that we as petrophysicists deliver our petrophysical results with specific confidence. Addition to that, the source to error are visualized and traceable, and that is very useful when we, for instance, should arguing to provide the right data. The example I will go to later is an example on that. It is also very useful to communicate the uncertainty, for instance, into our project, into other disciplines and so on. It will also be a help and contribute to reduce the uncertainty by making better and more correct decisions regarding choice of petrophysical model, and also inputs to our models. It will also be very useful when we should perform quality check of petrophysical results, for instance, delivered by partner-related assets, When we like to playing with our parameters, it is a very powerful tool when we like to do sensitivity studies. Finally, input to geo-statistic can be produced without any extra workload for the petrophysicist. The summary and the main message for this presentation are that the petrophysical result should be delivered with quantified and traceable error estimations. The error must reflect the level of knowledge about the models and the input to the models, as well as the data quality, and very important, it should be quantified from the very beginning in the petrophysical workflow. The error propagation can easily be an integral part of the petrophysical calculations, and all method are valuable, as long as the value on the model input, value on the error, and the error propogations can be explained and traced. As an integral part of the computer state that arrived at petrophysical results, first order error propagation are the preferred method because this is analytically a more practical solution. When it came to computer coding, the method is objective and the results are reproducible. The computing time is not an issue, and this method, first order error propagation, can manage dependency between variables to the covariance in the sigma matrix. It can handle non-symmetrical distribution by running the calculation twice, with one calculation for the low side error and one calculation for the high side error. It can handle non-linear equations by including the First Order Taylor series expansion on this model, which in practice may include the partial derivative of this c vector. That was the survey that was presented at Iceland last year.