## On Error Calculation and Use of First Order Error Propagation as Integral Part of Petrophysical Calculation

### by Stein Ottar Stalheim, Statoil

was presented on Wednesday the 11^{th} of January, 2017.

#### Abstract

The presentation is based on the view that the calculation of error is an essential part of measurements and scientific calculations, and that it should be an integral part of petrophysical calculations. Presentation of errors in petrophysical results are rarely applied in formation evaluation and reservoir characterization. Calculation of petrophysical error is commonly executed for input to the geostatistics, but often only after the petrophysical calculation is completed. Sometimes the calculation of petrophysical error is excluded altogether. Petrophysical results should include traceable and quantifiable errors.

The motivation of this paper is to show that errors can easily be integrated to the petrophysical results by including the first order error propagation (FOEP) method as a part of the computer script that give the petrophysical results. Errors in the petrophysical results are related to the models used and errors in model input. Calculation of petrophysical errors involves understanding how the input errors propagate through the functions to the end product. A commonly used method is Monte Carlo, while FOEP is less used. Different views exist with respect to needs, pros and cons for the various methods, but there also are some doubts regarding limitations around the use of FOEP. The FOEP solution is the chosen method in this paper because it is an analytical and more practical solution related to implementation into the script that computes the petrophysical results.

This paper includes an introduction to the theory of FOEP in matrix form and contains examples that illustrate the application of petrophysical functions. The mathematics shows how dependencies between variables and asymmetrical distributions are included in calculation of errors. The mathematics, graphical user interfaces (GUI), and plot functionalities are scripted with the use of Python. An example of a GUI for petrophysical input, and example plots including presentation of the errors and error propagation, are also presented.

The purpose of the presentation is to increase the focus on petrophysical error calculations, and to demonstrate the advantages of error propagation as a standard part of the petrophysical results. The mathematical formulation in matrix form, which makes the computer script simpler and the computation faster, and allows the implementation of asymmetrical distributions, is not observed in the petrophysical literature.

#### Stein Ottar Stalheim, CV

Stein Ottar is a specialist in Petrophysiscs with Statoil, Norway. Stein Ottar joined Statoil in 1993. His main interests are EM physics, multivariate analysis, and scientific computing.

The main role of Stein Ottar in Statoil is related to geo-electrical modeling, dynamic petrophysics and well integrity. Stein Ottar received his MSc and PhD in physics from the university in Bergen.