Uncertainty analysis in determining the position of a drill bit

Abstract

In well-bore engineering, oil-well boreholes are made using specialized drilling rigs. The position of the drill bit needs to be indirectly determined through accelero- and magnetometer measurements. To this end, the measurement data is first converted into a survey of direction vectors by applying a series of coordinate transformations. Then, a method called Minimum Curvature Method (MCM) is applied, which outputs a close idealized approximation of the actual drill bit trajectory. However, systematic and random errors in the magnetometer measurements result in error in position vectors.

A novel solution called Multi-Station Analysis (MSA) determines the systematic (Scale and Bias) errors in the magnetometer measurement data. Using reference measurements, a non-linear least squares error function is minimized, which is equivalent to solving a non-linear system of equations. This is done numerically by the Newton-Raphson algorithm. Subsequently, the measurement data is corrected. Reapplying survey conversion and MCM results in an improved estimate for the actual position vectors.

The main objective of this thesis is to derive a method that describes the uncertainty of the MSA solution. This is primarily done through the method of Monte Carlo simulation. As part of validation, the effect of MSA on final drill bit position is studied and compared with results from an uncertainty model used by the well-bore industry. Secondary, a pessimistic quantification of the uncertainty of MSA solution is given through condition numbers of Jacobian matrices from the Newton-Raphson algorithm applied to MSA, which measure the sensitivity of the non-linear least-squares error. The question whether these condition numbers are a representative measure of MSA solution quality is answered. Finally, further potential research areas are described.