19.2.8 Bringing It All Together

The effect of a particular error source on the survey results (md, inc or az) is found by multiplying the error magnitude by the appropriate weighting function. It is an implicit assumption of the model that all the error
statistics are Gaussian (see section 19.3.4 on the MWD error model revisions for the one exception to this rule).

Furthermore, the error sources are assumed to be statistically independent of each other so the effects of the different error sources are summed by taking the square root of the sum of the squares (RSS). (N.B: this should not be confused with the effects of the same error source across different survey stations and survey legs.) The effects of the error sources are all assumed to be linear (the weighting functions are all calculated to first order), so doubling the size of an error magnitude will in turn double the size of the error caused by that error source at any particular survey station. That error contribution will then be RSS’d with the other error sources, so that the effect on the final survey uncertainty will not be a simple doubling.

Then uncertainty in md, inc and az is converted to a positional uncertainty in the Earth-referenced co-ordinate frame. The equations for doing this are derived in [1] and are determined from the balanced tangential method of determining the borehole trajectory.

The positional values can then be converted to an uncertainty in the borehole referenced co-ordinate frame by multiplying by a direction cosine matrix dependant on the inclination and azimuth of the borehole.

The output of the error model is a covariance matrix describing the magnitude of the survey errors in the chosen co-ordinate frame. From this covariance matrix it is then possible to generate an error ellipse showing the estimated survey uncertainty in 3-d space. This error ellipse may be displayed in 3d or it may be sliced in a plane of interest, e.g. perpendicular to the well and displayed as an error ellipse. (Note to software implementers: this step is not given in the SPE paper but the axes of the error ellipsoid are the eigenvectors of the 3×3, real symmetric covariance matrix and the axes magnitudes are the eigenvalues of the covariance matrix.)

In the SPE paper the error sources are quoted at one standard deviation and hence the final error ellipse dimensions are the 1-sigma ellipse dimensions. There is then a probability that the actual well is contained with an ellipse of this size, centred on the obtained survey results. It is common practise in anti-collision calculations to use 2-sigma results – the 2-sigma ellipse is twice as large as the 1-sigma ellipse along each of the principle axes.

It is normal for directional drilling software such as 5D or Compass to be used to determine error values either for survey results for well planning or survey design. Usually the user can select whether results are reported at 1, 2 or 3 sigma. In some instances, the magnitudes of the error sources are always quoted at 1-sigma, in others the software allows the 2 sigma values to be entered. You should check with company anti-collision policy, survey focal point or software vendor if you are uncertain how reporting is applied to the wells which you use.