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Where: e i is the size of the error in NEV axis due to error source i at the current survey station (a 3x1 vector) 𝜎 𝑖 is the magnitude of the ith error source (a scalar) 𝜕𝑝 𝜕𝜀 𝑖 are the weightings functions, the effect of the ith error source on the survey measurements; md, inc and azimuth. (a 3x1 vector) 𝑑𝑟 𝑑𝑝 is the effect of the survey errors in md, inc and az on the wellbore position in the NEV axis, (i.e. a 3x3
𝒅𝑵 𝒅𝑴𝒅 𝒅𝑬 𝒅𝑴𝒅 𝒅𝑽 𝒅𝑴𝒅
𝒅𝑵 𝒅𝑰𝒏𝒄 𝒅𝑬 𝒅𝑰𝒏𝒄 𝒅𝑽 𝒅𝑰𝒏𝒄
𝒅𝑵 𝒅𝑨𝒛 𝒅𝑬 𝒅𝑨𝒛
matrix
)
𝒅𝑽 𝒅𝑨𝒛 ]
[
Wellbore positions are calculated using one of the standard methods such as minimum curvature or balanced tangential, so over an interval, the 𝒅𝒓 𝒅𝒑 matrix depends on the surveys at either end of the interval. The derivation of
𝒅𝒓 𝒅𝒑 matrix equations is given more detail in Appendix A.
the
The tool model for any particular survey instrument will usually include a number of different error sources, and we must consider all survey legs in the well and all the survey stations in each leg. So for a well we must add the error contributions over all survey legs in the well; each survey station in each leg and the contributions from each error source. Ap1.2.6 Summing Error Terms and Propagation Modes Once we have calculated the contribution to the error ellipse from each error source, at each survey station in each leg of our well we have to sum up all the contributions.
There are two basic cases: 1) The contributions are directly linked (correlated in mathematical terminology).
For example; the z-axis magnetometer bias error. If we are using the same tool and BHA, then we would expect this error source to have the same value from survey station to survey station and the effects of the error will build all the way down the wellbore.
In this case the error contributions are added in the usual arithmetic way:
𝒆 𝒕𝒐𝒕𝒂𝒍 = 𝒆 𝟏 +𝒆 𝟐
2) The contributions are not linked at all (statistically independent).
For example; if we have two independent error sources, then they could both cause a positive inclination error and add together but it is also possible that one might create a positive inclination error and the other a negative error. In which case we are taking a random value from pot 1 and a random value from pot 2 and the error contributions must be root sum squared (RSS) together:
𝟐 +𝒆
𝟐 𝟐
𝒆 𝒕𝒐𝒕𝒂𝒍 =√𝒆 𝟏
In fact it is an assumption of the model that the statistics of the various different error sources are independent so h y mus b ’d og h – for example, there is no reason why sag error would be connected to z-axis magnetometer bias or to declination error etc.
However, if we take the example from 1) above, although we can see that the z-axis magnetometer bias should remain
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